[{"data":1,"prerenderedAt":145009},["ShallowReactive",2],{"post-\u002Fblog\u002F2026\u002F0014-zui-you-chuan-shu-ji-he-jian-lun":3,"i-ci:tag":-1,"i-mingcute:time-line":-1},{"post":4,"nextPost":52841,"prevPost":125311},{"id":5,"title":6,"body":7,"description":13,"draft":52826,"enableCodeHighlight":52826,"enableCodeHightlight":52826,"enableComment":52827,"enableD2":52826,"enableKatex":52827,"extension":52828,"image":52812,"important":52826,"location":52829,"meta":52830,"navigation":52827,"ogImage":52829,"path":52831,"seo":52832,"stem":52833,"summary":52834,"tags":52835,"time":52839,"__hash__":52840},"blog\u002Fblog\u002F2026\u002F0014-zui-you-chuan-shu-ji-he-jian-lun.md","最优传输几何概念简论",{"type":8,"value":9,"toc":52811},"minimark",[10,14,17,21,24,27,408,752,835,902,1139,1392,1515,1864,2476,2479,2979,3098,3466,3499,3671,4046,4747,4910,5165,5197,5301,5304,5307,5370,5589,5592,6985,7250,8023,8026,9610,10133,10751,10754,10757,11112,11545,11872,11875,12248,12251,12254,12381,12606,12801,12992,13065,13588,13769,14355,14750,15158,15876,16360,16363,17297,18411,18623,19061,19064,19438,19441,19472,19716,19854,19857,20230,20233,20651,20882,21081,21563,22120,22175,22307,22595,22785,22892,22895,23113,23146,23372,23375,23378,23772,23775,23860,24600,24975,24978,25115,26061,26064,26067,26070,26273,26446,26616,27090,27530,27633,27998,28580,28583,29400,29403,29406,29467,29470,29702,30144,30525,31020,31023,32380,32503,32661,33137,33906,33909,33913,34066,34267,34472,34595,35162,35165,35226,35513,35684,36678,36681,37167,37243,37246,37709,37942,38343,40166,40242,40245,40473,40964,40967,41418,41922,41925,42064,42294,42384,42387,42992,43396,44128,44469,44811,45592,45595,45598,45601,45604,45669,45836,45986,46430,46433,46598,47053,47056,47429,47491,48118,49067,49170,49528,49729,49800,50597,50805,50808,50811,50814,50929,51417,51449,51452,52028,52031,52034,52188,52440,52443,52446,52449,52452,52484,52487,52805,52808],[11,12,13],"p",{},"我从一个很具体的困扰开始。模型对一张从没见过的图片给出的 softmax 最大值是 0.99，可我没有任何依据说它“99% 属于猫”。softmax 只做了一件事：把 logits 归一化成和为 1 的数。它回答的问题是“在已经假定的这几个类里，哪个最像”，而不是“这个样本是否属于我见过的任何一类”。",[11,15,16],{},"要判断一个样本有没有越界，我需要知道的不是分类器给它打了多少分，而是已知类别的数据在空间里占据了多少概率质量、这个样本离那片质量有多远。前者是密度，后者是距离，两者都是分布的几何性质，而不是分类器的性质。",[18,19,20],"h2",{"id":20},"搬土",[11,22,23],{},"先算一道运输题。",[11,25,26],{},"甲矿有 3 吨煤，乙矿有 1 吨；A 厂要 2 吨，B 厂要 2 吨。运价是：甲到 A 每吨 1 块，甲到 B 每吨 2 块，乙到 A 每吨 3 块，乙到 B 每吨 1 块。怎么运最省？",[11,28,29,30,77,78,137,138,190,191,330,331,407],{},"设甲运往 A 的吨数是 ",[31,32,36,58],"span",{"className":33,"translate":35},[34],"katex","no",[31,37,40],{"className":38},[39],"katex-mathml",[41,42,44],"math",{"xmlns":43},"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML",[45,46,47,54],"semantics",{},[48,49,50],"mrow",{},[51,52,53],"mi",{},"x",[55,56,53],"annotation",{"encoding":57},"application\u002Fx-tex",[31,59,63],{"className":60,"ariaHidden":62},[61],"katex-html","true",[31,64,67,72],{"className":65},[66],"base",[31,68],{"className":69,"style":71},[70],"strut","height:0.4306em;",[31,73,53],{"className":74},[75,76],"mord","mathnormal","。甲的 3 吨要分完，所以甲到 B 是 ",[31,79,81,103],{"className":80,"translate":35},[34],[31,82,84],{"className":83},[39],[41,85,86],{"xmlns":43},[45,87,88,100],{},[48,89,90,94,98],{},[91,92,93],"mn",{},"3",[95,96,97],"mo",{},"−",[51,99,53],{},[55,101,102],{"encoding":57},"3-x",[31,104,106,128],{"className":105,"ariaHidden":62},[61],[31,107,109,113,116,121,125],{"className":108},[66],[31,110],{"className":111,"style":112},[70],"height:0.7278em;vertical-align:-0.0833em;",[31,114,93],{"className":115},[75],[31,117],{"className":118,"style":120},[119],"mspace","margin-right:0.2222em;",[31,122,97],{"className":123},[124],"mbin",[31,126],{"className":127,"style":120},[119],[31,129,131,134],{"className":130},[66],[31,132],{"className":133,"style":71},[70],[31,135,53],{"className":136},[75,76],"；A 要 2 吨，所以乙到 A 是 ",[31,139,141,160],{"className":140,"translate":35},[34],[31,142,144],{"className":143},[39],[41,145,146],{"xmlns":43},[45,147,148,157],{},[48,149,150,153,155],{},[91,151,152],{},"2",[95,154,97],{},[51,156,53],{},[55,158,159],{"encoding":57},"2-x",[31,161,163,181],{"className":162,"ariaHidden":62},[61],[31,164,166,169,172,175,178],{"className":165},[66],[31,167],{"className":168,"style":112},[70],[31,170,152],{"className":171},[75],[31,173],{"className":174,"style":120},[119],[31,176,97],{"className":177},[124],[31,179],{"className":180,"style":120},[119],[31,182,184,187],{"className":183},[66],[31,185],{"className":186,"style":71},[70],[31,188,53],{"className":189},[75,76],"；乙只有 1 吨，所以乙到 B 是 ",[31,192,194,233],{"className":193,"translate":35},[34],[31,195,197],{"className":196},[39],[41,198,199],{"xmlns":43},[45,200,201,230],{},[48,202,203,206,208,212,214,216,218,221,224,226,228],{},[91,204,205],{},"1",[95,207,97],{},[95,209,211],{"stretchy":210},"false","(",[91,213,152],{},[95,215,97],{},[51,217,53],{},[95,219,220],{"stretchy":210},")",[95,222,223],{},"=",[51,225,53],{},[95,227,97],{},[91,229,205],{},[55,231,232],{"encoding":57},"1-(2-x)=x-1",[31,234,236,254,277,301,320],{"className":235,"ariaHidden":62},[61],[31,237,239,242,245,248,251],{"className":238},[66],[31,240],{"className":241,"style":112},[70],[31,243,205],{"className":244},[75],[31,246],{"className":247,"style":120},[119],[31,249,97],{"className":250},[124],[31,252],{"className":253,"style":120},[119],[31,255,257,261,265,268,271,274],{"className":256},[66],[31,258],{"className":259,"style":260},[70],"height:1em;vertical-align:-0.25em;",[31,262,211],{"className":263},[264],"mopen",[31,266,152],{"className":267},[75],[31,269],{"className":270,"style":120},[119],[31,272,97],{"className":273},[124],[31,275],{"className":276,"style":120},[119],[31,278,280,283,286,290,294,298],{"className":279},[66],[31,281],{"className":282,"style":260},[70],[31,284,53],{"className":285},[75,76],[31,287,220],{"className":288},[289],"mclose",[31,291],{"className":292,"style":293},[119],"margin-right:0.2778em;",[31,295,223],{"className":296},[297],"mrel",[31,299],{"className":300,"style":293},[119],[31,302,304,308,311,314,317],{"className":303},[66],[31,305],{"className":306,"style":307},[70],"height:0.6667em;vertical-align:-0.0833em;",[31,309,53],{"className":310},[75,76],[31,312],{"className":313,"style":120},[119],[31,315,97],{"className":316},[124],[31,318],{"className":319,"style":120},[119],[31,321,323,327],{"className":322},[66],[31,324],{"className":325,"style":326},[70],"height:0.6444em;",[31,328,205],{"className":329},[75],"。四个数都得非负，于是 ",[31,332,334,357],{"className":333,"translate":35},[34],[31,335,337],{"className":336},[39],[41,338,339],{"xmlns":43},[45,340,341,354],{},[48,342,343,345,348,350,352],{},[91,344,205],{},[95,346,347],{},"≤",[51,349,53],{},[95,351,347],{},[91,353,152],{},[55,355,356],{"encoding":57},"1\\le x\\le 2",[31,358,360,379,398],{"className":359,"ariaHidden":62},[61],[31,361,363,367,370,373,376],{"className":362},[66],[31,364],{"className":365,"style":366},[70],"height:0.7804em;vertical-align:-0.136em;",[31,368,205],{"className":369},[75],[31,371],{"className":372,"style":293},[119],[31,374,347],{"className":375},[297],[31,377],{"className":378,"style":293},[119],[31,380,382,386,389,392,395],{"className":381},[66],[31,383],{"className":384,"style":385},[70],"height:0.7719em;vertical-align:-0.136em;",[31,387,53],{"className":388},[75,76],[31,390],{"className":391,"style":293},[119],[31,393,347],{"className":394},[297],[31,396],{"className":397,"style":293},[119],[31,399,401,404],{"className":400},[66],[31,402],{"className":403,"style":326},[70],[31,405,152],{"className":406},[75],"。总运费",[31,409,412],{"className":410,"translate":35},[411],"katex-display",[31,413,415,499],{"className":414,"translate":35},[34],[31,416,418],{"className":417},[39],[41,419,421],{"xmlns":43,"display":420},"block",[45,422,423,496],{},[48,424,425,427,430,432,435,437,439,441,443,445,447,449,451,453,455,457,459,461,463,465,467,469,471,473,475,477,479,481,483,486,488,490,492],{},[51,426,53],{},[95,428,429],{},"⋅",[91,431,205],{},[95,433,434],{},"+",[95,436,211],{"stretchy":210},[91,438,93],{},[95,440,97],{},[51,442,53],{},[95,444,220],{"stretchy":210},[95,446,429],{},[91,448,152],{},[95,450,434],{},[95,452,211],{"stretchy":210},[91,454,152],{},[95,456,97],{},[51,458,53],{},[95,460,220],{"stretchy":210},[95,462,429],{},[91,464,93],{},[95,466,434],{},[95,468,211],{"stretchy":210},[51,470,53],{},[95,472,97],{},[91,474,205],{},[95,476,220],{"stretchy":210},[95,478,429],{},[91,480,205],{},[95,482,223],{},[91,484,485],{},"11",[95,487,97],{},[91,489,93],{},[51,491,53],{},[51,493,495],{"mathvariant":494},"normal",".",[55,497,498],{"encoding":57},"x\\cdot1+(3-x)\\cdot2+(2-x)\\cdot3+(x-1)\\cdot1=11-3x .",[31,500,502,521,539,560,581,599,620,641,659,680,701,719,737],{"className":501,"ariaHidden":62},[61],[31,503,505,509,512,515,518],{"className":504},[66],[31,506],{"className":507,"style":508},[70],"height:0.4445em;",[31,510,53],{"className":511},[75,76],[31,513],{"className":514,"style":120},[119],[31,516,429],{"className":517},[124],[31,519],{"className":520,"style":120},[119],[31,522,524,527,530,533,536],{"className":523},[66],[31,525],{"className":526,"style":112},[70],[31,528,205],{"className":529},[75],[31,531],{"className":532,"style":120},[119],[31,534,434],{"className":535},[124],[31,537],{"className":538,"style":120},[119],[31,540,542,545,548,551,554,557],{"className":541},[66],[31,543],{"className":544,"style":260},[70],[31,546,211],{"className":547},[264],[31,549,93],{"className":550},[75],[31,552],{"className":553,"style":120},[119],[31,555,97],{"className":556},[124],[31,558],{"className":559,"style":120},[119],[31,561,563,566,569,572,575,578],{"className":562},[66],[31,564],{"className":565,"style":260},[70],[31,567,53],{"className":568},[75,76],[31,570,220],{"className":571},[289],[31,573],{"className":574,"style":120},[119],[31,576,429],{"className":577},[124],[31,579],{"className":580,"style":120},[119],[31,582,584,587,590,593,596],{"className":583},[66],[31,585],{"className":586,"style":112},[70],[31,588,152],{"className":589},[75],[31,591],{"className":592,"style":120},[119],[31,594,434],{"className":595},[124],[31,597],{"className":598,"style":120},[119],[31,600,602,605,608,611,614,617],{"className":601},[66],[31,603],{"className":604,"style":260},[70],[31,606,211],{"className":607},[264],[31,609,152],{"className":610},[75],[31,612],{"className":613,"style":120},[119],[31,615,97],{"className":616},[124],[31,618],{"className":619,"style":120},[119],[31,621,623,626,629,632,635,638],{"className":622},[66],[31,624],{"className":625,"style":260},[70],[31,627,53],{"className":628},[75,76],[31,630,220],{"className":631},[289],[31,633],{"className":634,"style":120},[119],[31,636,429],{"className":637},[124],[31,639],{"className":640,"style":120},[119],[31,642,644,647,650,653,656],{"className":643},[66],[31,645],{"className":646,"style":112},[70],[31,648,93],{"className":649},[75],[31,651],{"className":652,"style":120},[119],[31,654,434],{"className":655},[124],[31,657],{"className":658,"style":120},[119],[31,660,662,665,668,671,674,677],{"className":661},[66],[31,663],{"className":664,"style":260},[70],[31,666,211],{"className":667},[264],[31,669,53],{"className":670},[75,76],[31,672],{"className":673,"style":120},[119],[31,675,97],{"className":676},[124],[31,678],{"className":679,"style":120},[119],[31,681,683,686,689,692,695,698],{"className":682},[66],[31,684],{"className":685,"style":260},[70],[31,687,205],{"className":688},[75],[31,690,220],{"className":691},[289],[31,693],{"className":694,"style":120},[119],[31,696,429],{"className":697},[124],[31,699],{"className":700,"style":120},[119],[31,702,704,707,710,713,716],{"className":703},[66],[31,705],{"className":706,"style":326},[70],[31,708,205],{"className":709},[75],[31,711],{"className":712,"style":293},[119],[31,714,223],{"className":715},[297],[31,717],{"className":718,"style":293},[119],[31,720,722,725,728,731,734],{"className":721},[66],[31,723],{"className":724,"style":112},[70],[31,726,485],{"className":727},[75],[31,729],{"className":730,"style":120},[119],[31,732,97],{"className":733},[124],[31,735],{"className":736,"style":120},[119],[31,738,740,743,746,749],{"className":739},[66],[31,741],{"className":742,"style":326},[70],[31,744,93],{"className":745},[75],[31,747,53],{"className":748},[75,76],[31,750,495],{"className":751},[75],[11,753,754,782,783,834],{},[31,755,757,770],{"className":756,"translate":35},[34],[31,758,760],{"className":759},[39],[41,761,762],{"xmlns":43},[45,763,764,768],{},[48,765,766],{},[51,767,53],{},[55,769,53],{"encoding":57},[31,771,773],{"className":772,"ariaHidden":62},[61],[31,774,776,779],{"className":775},[66],[31,777],{"className":778,"style":71},[70],[31,780,53],{"className":781},[75,76]," 越大越省，取 ",[31,784,786,804],{"className":785,"translate":35},[34],[31,787,789],{"className":788},[39],[41,790,791],{"xmlns":43},[45,792,793,801],{},[48,794,795,797,799],{},[51,796,53],{},[95,798,223],{},[91,800,152],{},[55,802,803],{"encoding":57},"x=2",[31,805,807,825],{"className":806,"ariaHidden":62},[61],[31,808,810,813,816,819,822],{"className":809},[66],[31,811],{"className":812,"style":71},[70],[31,814,53],{"className":815},[75,76],[31,817],{"className":818,"style":293},[119],[31,820,223],{"className":821},[297],[31,823],{"className":824,"style":293},[119],[31,826,828,831],{"className":827},[66],[31,829],{"className":830,"style":326},[70],[31,832,152],{"className":833},[75],"，总运费 5 块：甲到 A 运 2 吨，甲到 B 运 1 吨，乙到 B 运 1 吨。",[11,836,837,838,869,870,901],{},"这张“每个产地往每个销地运多少”的表，就是运输方案。Monge 在 1781 年问的是一般情形：给定一堆土 ",[31,839,841,856],{"className":840,"translate":35},[34],[31,842,844],{"className":843},[39],[41,845,846],{"xmlns":43},[45,847,848,853],{},[48,849,850],{},[51,851,852],{},"μ",[55,854,855],{"encoding":57},"\\mu",[31,857,859],{"className":858,"ariaHidden":62},[61],[31,860,862,866],{"className":861},[66],[31,863],{"className":864,"style":865},[70],"height:0.625em;vertical-align:-0.1944em;",[31,867,852],{"className":868},[75,76]," 和一个坑 ",[31,871,873,888],{"className":872,"translate":35},[34],[31,874,876],{"className":875},[39],[41,877,878],{"xmlns":43},[45,879,880,885],{},[48,881,882],{},[51,883,884],{},"ν",[55,886,887],{"encoding":57},"\\nu",[31,889,891],{"className":890,"ariaHidden":62},[61],[31,892,894,897],{"className":893},[66],[31,895],{"className":896,"style":71},[70],[31,898,884],{"className":899,"style":900},[75,76],"margin-right:0.0637em;","，怎么搬最省。",[11,903,904,905,990,991,1107,1108,1138],{},"他的写法是找一个映射 ",[31,906,908,936],{"className":907,"translate":35},[34],[31,909,911],{"className":910},[39],[41,912,913],{"xmlns":43},[45,914,915,933],{},[48,916,917,920,923,927,930],{},[51,918,919],{},"T",[95,921,922],{},":",[51,924,926],{"mathvariant":925},"script","X",[95,928,929],{},"→",[51,931,932],{"mathvariant":925},"Y",[55,934,935],{"encoding":57},"T:\\mathcal X\\to\\mathcal Y",[31,937,939,959,979],{"className":938,"ariaHidden":62},[61],[31,940,942,946,950,953,956],{"className":941},[66],[31,943],{"className":944,"style":945},[70],"height:0.6833em;",[31,947,919],{"className":948,"style":949},[75,76],"margin-right:0.1389em;",[31,951],{"className":952,"style":293},[119],[31,954,922],{"className":955},[297],[31,957],{"className":958,"style":293},[119],[31,960,962,965,970,973,976],{"className":961},[66],[31,963],{"className":964,"style":945},[70],[31,966,926],{"className":967,"style":969},[75,968],"mathcal","margin-right:0.1464em;",[31,971],{"className":972,"style":293},[119],[31,974,929],{"className":975},[297],[31,977],{"className":978,"style":293},[119],[31,980,982,986],{"className":981},[66],[31,983],{"className":984,"style":985},[70],"height:0.7805em;vertical-align:-0.0972em;",[31,987,932],{"className":988,"style":989},[75,968],"margin-right:0.0822em;","，要求“搬完之后每个地方刚好装满”，也就是推前测度 ",[31,992,994,1020],{"className":993,"translate":35},[34],[31,995,997],{"className":996},[39],[41,998,999],{"xmlns":43},[45,1000,1001,1017],{},[48,1002,1003,1011,1013,1015],{},[1004,1005,1006,1008],"msub",{},[51,1007,919],{},[51,1009,1010],{"mathvariant":494},"#",[51,1012,852],{},[95,1014,223],{},[51,1016,884],{},[55,1018,1019],{"encoding":57},"T_\\#\\mu=\\nu",[31,1021,1023,1098],{"className":1022,"ariaHidden":62},[61],[31,1024,1026,1030,1086,1089,1092,1095],{"className":1025},[66],[31,1027],{"className":1028,"style":1029},[70],"height:0.9694em;vertical-align:-0.2861em;",[31,1031,1033,1036],{"className":1032},[75],[31,1034,919],{"className":1035,"style":949},[75,76],[31,1037,1040],{"className":1038},[1039],"msupsub",[31,1041,1045,1077],{"className":1042},[1043,1044],"vlist-t","vlist-t2",[31,1046,1049,1072],{"className":1047},[1048],"vlist-r",[31,1050,1054],{"className":1051,"style":1053},[1052],"vlist","height:0.3361em;",[31,1055,1057,1062],{"style":1056},"top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;",[31,1058],{"className":1059,"style":1061},[1060],"pstrut","height:2.7em;",[31,1063,1069],{"className":1064},[1065,1066,1067,1068],"sizing","reset-size6","size3","mtight",[31,1070,1010],{"className":1071},[75,1068],[31,1073,1076],{"className":1074},[1075],"vlist-s","​",[31,1078,1080],{"className":1079},[1048],[31,1081,1084],{"className":1082,"style":1083},[1052],"height:0.2861em;",[31,1085],{},[31,1087,852],{"className":1088},[75,76],[31,1090],{"className":1091,"style":293},[119],[31,1093,223],{"className":1094},[297],[31,1096],{"className":1097,"style":293},[119],[31,1099,1101,1104],{"className":1100},[66],[31,1102],{"className":1103,"style":71},[70],[31,1105,884],{"className":1106,"style":900},[75,76],"，即对任意可测集 ",[31,1109,1111,1125],{"className":1110,"translate":35},[34],[31,1112,1114],{"className":1113},[39],[41,1115,1116],{"xmlns":43},[45,1117,1118,1123],{},[48,1119,1120],{},[51,1121,1122],{},"B",[55,1124,1122],{"encoding":57},[31,1126,1128],{"className":1127,"ariaHidden":62},[61],[31,1129,1131,1134],{"className":1130},[66],[31,1132],{"className":1133,"style":945},[70],[31,1135,1122],{"className":1136,"style":1137},[75,76],"margin-right:0.0502em;","，",[31,1140,1142],{"className":1141,"translate":35},[411],[31,1143,1145,1214],{"className":1144,"translate":35},[34],[31,1146,1148],{"className":1147},[39],[41,1149,1150],{"xmlns":43,"display":420},[45,1151,1152,1211],{},[48,1153,1154,1156,1162,1164,1166,1168,1170,1172,1174,1176,1179,1190,1192,1194,1196,1198,1200,1202,1204,1206,1208],{},[95,1155,211],{"stretchy":210},[1004,1157,1158,1160],{},[51,1159,919],{},[51,1161,1010],{"mathvariant":494},[51,1163,852],{},[95,1165,220],{"stretchy":210},[95,1167,211],{"stretchy":210},[51,1169,1122],{},[95,1171,220],{"stretchy":210},[95,1173,223],{},[51,1175,852],{},[95,1177,211],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},"1.2em",[1180,1181,1182,1184],"msup",{},[51,1183,919],{},[48,1185,1186,1188],{},[95,1187,97],{},[91,1189,205],{},[95,1191,211],{"stretchy":210},[51,1193,1122],{},[95,1195,220],{"stretchy":210},[95,1197,220],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[95,1199,223],{},[51,1201,884],{},[95,1203,211],{"stretchy":210},[51,1205,1122],{},[95,1207,220],{"stretchy":210},[95,1209,1210],{"separator":62},",",[55,1212,1213],{"encoding":57},"(T_\\#\\mu)(B)=\\mu\\bigl(T^{-1}(B)\\bigr)=\\nu(B),",[31,1215,1217,1291,1370],{"className":1216,"ariaHidden":62},[61],[31,1218,1220,1224,1227,1267,1270,1273,1276,1279,1282,1285,1288],{"className":1219},[66],[31,1221],{"className":1222,"style":1223},[70],"height:1.0361em;vertical-align:-0.2861em;",[31,1225,211],{"className":1226},[264],[31,1228,1230,1233],{"className":1229},[75],[31,1231,919],{"className":1232,"style":949},[75,76],[31,1234,1236],{"className":1235},[1039],[31,1237,1239,1259],{"className":1238},[1043,1044],[31,1240,1242,1256],{"className":1241},[1048],[31,1243,1245],{"className":1244,"style":1053},[1052],[31,1246,1247,1250],{"style":1056},[31,1248],{"className":1249,"style":1061},[1060],[31,1251,1253],{"className":1252},[1065,1066,1067,1068],[31,1254,1010],{"className":1255},[75,1068],[31,1257,1076],{"className":1258},[1075],[31,1260,1262],{"className":1261},[1048],[31,1263,1265],{"className":1264,"style":1083},[1052],[31,1266],{},[31,1268,852],{"className":1269},[75,76],[31,1271,220],{"className":1272},[289],[31,1274,211],{"className":1275},[264],[31,1277,1122],{"className":1278,"style":1137},[75,76],[31,1280,220],{"className":1281},[289],[31,1283],{"className":1284,"style":293},[119],[31,1286,223],{"className":1287},[297],[31,1289],{"className":1290,"style":293},[119],[31,1292,1294,1298,1301,1309,1346,1349,1352,1355,1361,1364,1367],{"className":1293},[66],[31,1295],{"className":1296,"style":1297},[70],"height:1.2141em;vertical-align:-0.35em;",[31,1299,852],{"className":1300},[75,76],[31,1302,1304],{"className":1303},[264],[31,1305,211],{"className":1306},[1307,1308],"delimsizing","size1",[31,1310,1312,1315],{"className":1311},[75],[31,1313,919],{"className":1314,"style":949},[75,76],[31,1316,1318],{"className":1317},[1039],[31,1319,1321],{"className":1320},[1043],[31,1322,1324],{"className":1323},[1048],[31,1325,1328],{"className":1326,"style":1327},[1052],"height:0.8641em;",[31,1329,1331,1334],{"style":1330},"top:-3.113em;margin-right:0.05em;",[31,1332],{"className":1333,"style":1061},[1060],[31,1335,1337],{"className":1336},[1065,1066,1067,1068],[31,1338,1340,1343],{"className":1339},[75,1068],[31,1341,97],{"className":1342},[75,1068],[31,1344,205],{"className":1345},[75,1068],[31,1347,211],{"className":1348},[264],[31,1350,1122],{"className":1351,"style":1137},[75,76],[31,1353,220],{"className":1354},[289],[31,1356,1358],{"className":1357},[289],[31,1359,220],{"className":1360},[1307,1308],[31,1362],{"className":1363,"style":293},[119],[31,1365,223],{"className":1366},[297],[31,1368],{"className":1369,"style":293},[119],[31,1371,1373,1376,1379,1382,1385,1388],{"className":1372},[66],[31,1374],{"className":1375,"style":260},[70],[31,1377,884],{"className":1378,"style":900},[75,76],[31,1380,211],{"className":1381},[264],[31,1383,1122],{"className":1384,"style":1137},[75,76],[31,1386,220],{"className":1387},[289],[31,1389,1210],{"className":1390},[1391],"mpunct",[11,1393,1394,1395,1514],{},"代价是 ",[31,1396,1398,1447],{"className":1397,"translate":35},[34],[31,1399,1401],{"className":1400},[39],[41,1402,1403],{"xmlns":43},[45,1404,1405,1444],{},[48,1406,1407,1410,1413,1415,1417,1419,1421,1423,1425,1427,1429,1433,1436,1438,1440,1442],{},[95,1408,1409],{},"∫",[51,1411,1412],{},"c",[95,1414,211],{"stretchy":210},[51,1416,53],{},[95,1418,1210],{"separator":62},[51,1420,919],{},[95,1422,211],{"stretchy":210},[51,1424,53],{},[95,1426,220],{"stretchy":210},[95,1428,220],{"stretchy":210},[1430,1431,1432],"mtext",{}," ",[51,1434,1435],{},"d",[51,1437,852],{},[95,1439,211],{"stretchy":210},[51,1441,53],{},[95,1443,220],{"stretchy":210},[55,1445,1446],{"encoding":57},"\\int c(x,T(x))\\,d\\mu(x)",[31,1448,1450],{"className":1449,"ariaHidden":62},[61],[31,1451,1453,1457,1464,1468,1471,1474,1477,1480,1483,1486,1489,1492,1496,1499,1502,1505,1508,1511],{"className":1452},[66],[31,1454],{"className":1455,"style":1456},[70],"height:1.1111em;vertical-align:-0.3061em;",[31,1458,1409],{"className":1459,"style":1463},[1460,1461,1462],"mop","op-symbol","small-op","margin-right:0.1945em;position:relative;top:-0.0006em;",[31,1465],{"className":1466,"style":1467},[119],"margin-right:0.1667em;",[31,1469,1412],{"className":1470},[75,76],[31,1472,211],{"className":1473},[264],[31,1475,53],{"className":1476},[75,76],[31,1478,1210],{"className":1479},[1391],[31,1481],{"className":1482,"style":1467},[119],[31,1484,919],{"className":1485,"style":949},[75,76],[31,1487,211],{"className":1488},[264],[31,1490,53],{"className":1491},[75,76],[31,1493,1495],{"className":1494},[289],"))",[31,1497],{"className":1498,"style":1467},[119],[31,1500,1435],{"className":1501},[75,76],[31,1503,852],{"className":1504},[75,76],[31,1506,211],{"className":1507},[264],[31,1509,53],{"className":1510},[75,76],[31,1512,220],{"className":1513},[289],"。于是 Monge 问题就是",[31,1516,1518],{"className":1517,"translate":35},[411],[31,1519,1521,1607],{"className":1520,"translate":35},[34],[31,1522,1524],{"className":1523},[39],[41,1525,1526],{"xmlns":43,"display":420},[45,1527,1528,1604],{},[48,1529,1530,1543,1546,1552,1554,1556,1558,1560,1562,1564,1566,1568,1570,1572,1574,1576,1578,1580,1582,1585,1588,1590,1596,1598,1600,1602],{},[1531,1532,1533,1541],"munder",{},[48,1534,1535,1538],{},[51,1536,1537],{},"inf",[95,1539,1540],{},"⁡",[51,1542,919],{},[1430,1544,1545],{}," ",[1004,1547,1548,1550],{},[95,1549,1409],{},[51,1551,926],{"mathvariant":925},[51,1553,1412],{},[95,1555,211],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[51,1557,53],{},[95,1559,1210],{"separator":62},[51,1561,919],{},[95,1563,211],{"stretchy":210},[51,1565,53],{},[95,1567,220],{"stretchy":210},[95,1569,220],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[1430,1571,1432],{},[51,1573,1435],{},[51,1575,852],{},[95,1577,211],{"stretchy":210},[51,1579,53],{},[95,1581,220],{"stretchy":210},[119,1583],{"width":1584},"1em",[1430,1586,1587],{},"s.t.",[119,1589],{"width":1584},[1004,1591,1592,1594],{},[51,1593,919],{},[51,1595,1010],{"mathvariant":494},[51,1597,852],{},[95,1599,223],{},[51,1601,884],{},[51,1603,495],{"mathvariant":494},[55,1605,1606],{"encoding":57},"\\inf_{T}\\ \\int_{\\mathcal X} c\\bigl(x,T(x)\\bigr)\\,d\\mu(x)\n\\quad\\text{s.t.}\\quad T_\\#\\mu=\\nu .",[31,1608,1610,1852],{"className":1609,"ariaHidden":62},[61],[31,1611,1613,1617,1675,1678,1681,1729,1732,1735,1741,1744,1747,1750,1753,1756,1759,1762,1768,1771,1774,1777,1780,1783,1786,1790,1797,1800,1840,1843,1846,1849],{"className":1612},[66],[31,1614],{"className":1615,"style":1616},[70],"height:2.2719em;vertical-align:-0.9119em;",[31,1618,1621],{"className":1619},[1460,1620],"op-limits",[31,1622,1624,1666],{"className":1623},[1043,1044],[31,1625,1627,1663],{"className":1626},[1048],[31,1628,1631,1647],{"className":1629,"style":1630},[1052],"height:0.6944em;",[31,1632,1634,1638],{"style":1633},"top:-2.3557em;margin-left:0em;",[31,1635],{"className":1636,"style":1637},[1060],"height:3em;",[31,1639,1641],{"className":1640},[1065,1066,1067,1068],[31,1642,1644],{"className":1643},[75,1068],[31,1645,919],{"className":1646,"style":949},[75,76,1068],[31,1648,1650,1653],{"style":1649},"top:-3em;",[31,1651],{"className":1652,"style":1637},[1060],[31,1654,1655],{},[31,1656,1658,1659],{"className":1657},[1460],"in",[31,1660,1662],{"style":1661},"margin-right:0.0778em;","f",[31,1664,1076],{"className":1665},[1075],[31,1667,1669],{"className":1668},[1048],[31,1670,1673],{"className":1671,"style":1672},[1052],"height:0.7443em;",[31,1674],{},[31,1676,1545],{"className":1677},[119],[31,1679],{"className":1680,"style":1467},[119],[31,1682,1684,1689],{"className":1683},[1460],[31,1685,1409],{"className":1686,"style":1688},[1460,1461,1687],"large-op","margin-right:0.4445em;position:relative;top:-0.0011em;",[31,1690,1692],{"className":1691},[1039],[31,1693,1695,1720],{"className":1694},[1043,1044],[31,1696,1698,1717],{"className":1697},[1048],[31,1699,1702],{"className":1700,"style":1701},[1052],"height:-0.4336em;",[31,1703,1705,1708],{"style":1704},"top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;",[31,1706],{"className":1707,"style":1061},[1060],[31,1709,1711],{"className":1710},[1065,1066,1067,1068],[31,1712,1714],{"className":1713},[75,1068],[31,1715,926],{"className":1716,"style":969},[75,968,1068],[31,1718,1076],{"className":1719},[1075],[31,1721,1723],{"className":1722},[1048],[31,1724,1727],{"className":1725,"style":1726},[1052],"height:0.9119em;",[31,1728],{},[31,1730],{"className":1731,"style":1467},[119],[31,1733,1412],{"className":1734},[75,76],[31,1736,1738],{"className":1737},[264],[31,1739,211],{"className":1740},[1307,1308],[31,1742,53],{"className":1743},[75,76],[31,1745,1210],{"className":1746},[1391],[31,1748],{"className":1749,"style":1467},[119],[31,1751,919],{"className":1752,"style":949},[75,76],[31,1754,211],{"className":1755},[264],[31,1757,53],{"className":1758},[75,76],[31,1760,220],{"className":1761},[289],[31,1763,1765],{"className":1764},[289],[31,1766,220],{"className":1767},[1307,1308],[31,1769],{"className":1770,"style":1467},[119],[31,1772,1435],{"className":1773},[75,76],[31,1775,852],{"className":1776},[75,76],[31,1778,211],{"className":1779},[264],[31,1781,53],{"className":1782},[75,76],[31,1784,220],{"className":1785},[289],[31,1787],{"className":1788,"style":1789},[119],"margin-right:1em;",[31,1791,1794],{"className":1792},[75,1793],"text",[31,1795,1587],{"className":1796},[75],[31,1798],{"className":1799,"style":1789},[119],[31,1801,1803,1806],{"className":1802},[75],[31,1804,919],{"className":1805,"style":949},[75,76],[31,1807,1809],{"className":1808},[1039],[31,1810,1812,1832],{"className":1811},[1043,1044],[31,1813,1815,1829],{"className":1814},[1048],[31,1816,1818],{"className":1817,"style":1053},[1052],[31,1819,1820,1823],{"style":1056},[31,1821],{"className":1822,"style":1061},[1060],[31,1824,1826],{"className":1825},[1065,1066,1067,1068],[31,1827,1010],{"className":1828},[75,1068],[31,1830,1076],{"className":1831},[1075],[31,1833,1835],{"className":1834},[1048],[31,1836,1838],{"className":1837,"style":1083},[1052],[31,1839],{},[31,1841,852],{"className":1842},[75,76],[31,1844],{"className":1845,"style":293},[119],[31,1847,223],{"className":1848},[297],[31,1850],{"className":1851,"style":293},[119],[31,1853,1855,1858,1861],{"className":1854},[66],[31,1856],{"className":1857,"style":71},[70],[31,1859,884],{"className":1860,"style":900},[75,76],[31,1862,495],{"className":1863},[75],[11,1865,1866,1867,1895,1896,1924,1925,1953,1954,2053,2054,2372,2373,2417,2418,2446,2447,2475],{},"这个提法有一个硬伤：",[31,1868,1870,1883],{"className":1869,"translate":35},[34],[31,1871,1873],{"className":1872},[39],[41,1874,1875],{"xmlns":43},[45,1876,1877,1881],{},[48,1878,1879],{},[51,1880,919],{},[55,1882,919],{"encoding":57},[31,1884,1886],{"className":1885,"ariaHidden":62},[61],[31,1887,1889,1892],{"className":1888},[66],[31,1890],{"className":1891,"style":945},[70],[31,1893,919],{"className":1894,"style":949},[75,76]," 必须是确定性的，一个产地只能有一条固定去向。如果 ",[31,1897,1899,1912],{"className":1898,"translate":35},[34],[31,1900,1902],{"className":1901},[39],[41,1903,1904],{"xmlns":43},[45,1905,1906,1910],{},[48,1907,1908],{},[51,1909,852],{},[55,1911,855],{"encoding":57},[31,1913,1915],{"className":1914,"ariaHidden":62},[61],[31,1916,1918,1921],{"className":1917},[66],[31,1919],{"className":1920,"style":865},[70],[31,1922,852],{"className":1923},[75,76]," 在某个点上有原子，而 ",[31,1926,1928,1941],{"className":1927,"translate":35},[34],[31,1929,1931],{"className":1930},[39],[41,1932,1933],{"xmlns":43},[45,1934,1935,1939],{},[48,1936,1937],{},[51,1938,884],{},[55,1940,887],{"encoding":57},[31,1942,1944],{"className":1943,"ariaHidden":62},[61],[31,1945,1947,1950],{"className":1946},[66],[31,1948],{"className":1949,"style":71},[70],[31,1951,884],{"className":1952,"style":900},[75,76]," 在那里要求把质量分到两个地方，就无解。举个最小的例子：全部质量集中在甲一个点上，",[31,1955,1957,1981],{"className":1956,"translate":35},[34],[31,1958,1960],{"className":1959},[39],[41,1961,1962],{"xmlns":43},[45,1963,1964,1978],{},[48,1965,1966,1968,1970],{},[51,1967,852],{},[95,1969,223],{},[1004,1971,1972,1975],{},[51,1973,1974],{},"δ",[91,1976,1977],{},"0",[55,1979,1980],{"encoding":57},"\\mu=\\delta_0",[31,1982,1984,2002],{"className":1983,"ariaHidden":62},[61],[31,1985,1987,1990,1993,1996,1999],{"className":1986},[66],[31,1988],{"className":1989,"style":865},[70],[31,1991,852],{"className":1992},[75,76],[31,1994],{"className":1995,"style":293},[119],[31,1997,223],{"className":1998},[297],[31,2000],{"className":2001,"style":293},[119],[31,2003,2005,2009],{"className":2004},[66],[31,2006],{"className":2007,"style":2008},[70],"height:0.8444em;vertical-align:-0.15em;",[31,2010,2012,2016],{"className":2011},[75],[31,2013,1974],{"className":2014,"style":2015},[75,76],"margin-right:0.0379em;",[31,2017,2019],{"className":2018},[1039],[31,2020,2022,2044],{"className":2021},[1043,1044],[31,2023,2025,2041],{"className":2024},[1048],[31,2026,2029],{"className":2027,"style":2028},[1052],"height:0.3011em;",[31,2030,2032,2035],{"style":2031},"top:-2.55em;margin-left:-0.0379em;margin-right:0.05em;",[31,2033],{"className":2034,"style":1061},[1060],[31,2036,2038],{"className":2037},[1065,1066,1067,1068],[31,2039,1977],{"className":2040},[75,1068],[31,2042,1076],{"className":2043},[1075],[31,2045,2047],{"className":2046},[1048],[31,2048,2051],{"className":2049,"style":2050},[1052],"height:0.15em;",[31,2052],{},"；而两个厂各要一半，",[31,2055,2057,2105],{"className":2056,"translate":35},[34],[31,2058,2060],{"className":2059},[39],[41,2061,2062],{"xmlns":43},[45,2063,2064,2102],{},[48,2065,2066,2068,2070,2080,2086,2088,2096],{},[51,2067,884],{},[95,2069,223],{},[2071,2072,2073],"mstyle",{"scriptlevel":1977,"displaystyle":210},[2074,2075,2076,2078],"mfrac",{},[91,2077,205],{},[91,2079,152],{},[1004,2081,2082,2084],{},[51,2083,1974],{},[91,2085,205],{},[95,2087,434],{},[2071,2089,2090],{"scriptlevel":1977,"displaystyle":210},[2074,2091,2092,2094],{},[91,2093,205],{},[91,2095,152],{},[1004,2097,2098,2100],{},[51,2099,1974],{},[91,2101,152],{},[55,2103,2104],{"encoding":57},"\\nu=\\tfrac12\\delta_1+\\tfrac12\\delta_2",[31,2106,2108,2126,2258],{"className":2107,"ariaHidden":62},[61],[31,2109,2111,2114,2117,2120,2123],{"className":2110},[66],[31,2112],{"className":2113,"style":71},[70],[31,2115,884],{"className":2116,"style":900},[75,76],[31,2118],{"className":2119,"style":293},[119],[31,2121,223],{"className":2122},[297],[31,2124],{"className":2125,"style":293},[119],[31,2127,2129,2133,2209,2249,2252,2255],{"className":2128},[66],[31,2130],{"className":2131,"style":2132},[70],"height:1.1901em;vertical-align:-0.345em;",[31,2134,2136,2140,2206],{"className":2135},[75],[31,2137],{"className":2138},[264,2139],"nulldelimiter",[31,2141,2143],{"className":2142},[2074],[31,2144,2146,2197],{"className":2145},[1043,1044],[31,2147,2149,2194],{"className":2148},[1048],[31,2150,2153,2168,2179],{"className":2151,"style":2152},[1052],"height:0.8451em;",[31,2154,2156,2159],{"style":2155},"top:-2.655em;",[31,2157],{"className":2158,"style":1637},[1060],[31,2160,2162],{"className":2161},[1065,1066,1067,1068],[31,2163,2165],{"className":2164},[75,1068],[31,2166,152],{"className":2167},[75,1068],[31,2169,2171,2174],{"style":2170},"top:-3.23em;",[31,2172],{"className":2173,"style":1637},[1060],[31,2175],{"className":2176,"style":2178},[2177],"frac-line","border-bottom-width:0.04em;",[31,2180,2182,2185],{"style":2181},"top:-3.394em;",[31,2183],{"className":2184,"style":1637},[1060],[31,2186,2188],{"className":2187},[1065,1066,1067,1068],[31,2189,2191],{"className":2190},[75,1068],[31,2192,205],{"className":2193},[75,1068],[31,2195,1076],{"className":2196},[1075],[31,2198,2200],{"className":2199},[1048],[31,2201,2204],{"className":2202,"style":2203},[1052],"height:0.345em;",[31,2205],{},[31,2207],{"className":2208},[289,2139],[31,2210,2212,2215],{"className":2211},[75],[31,2213,1974],{"className":2214,"style":2015},[75,76],[31,2216,2218],{"className":2217},[1039],[31,2219,2221,2241],{"className":2220},[1043,1044],[31,2222,2224,2238],{"className":2223},[1048],[31,2225,2227],{"className":2226,"style":2028},[1052],[31,2228,2229,2232],{"style":2031},[31,2230],{"className":2231,"style":1061},[1060],[31,2233,2235],{"className":2234},[1065,1066,1067,1068],[31,2236,205],{"className":2237},[75,1068],[31,2239,1076],{"className":2240},[1075],[31,2242,2244],{"className":2243},[1048],[31,2245,2247],{"className":2246,"style":2050},[1052],[31,2248],{},[31,2250],{"className":2251,"style":120},[119],[31,2253,434],{"className":2254},[124],[31,2256],{"className":2257,"style":120},[119],[31,2259,2261,2264,2332],{"className":2260},[66],[31,2262],{"className":2263,"style":2132},[70],[31,2265,2267,2270,2329],{"className":2266},[75],[31,2268],{"className":2269},[264,2139],[31,2271,2273],{"className":2272},[2074],[31,2274,2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必须同时等于 ",[31,2419,2421,2434],{"className":2420,"translate":35},[34],[31,2422,2424],{"className":2423},[39],[41,2425,2426],{"xmlns":43},[45,2427,2428,2432],{},[48,2429,2430],{},[91,2431,205],{},[55,2433,205],{"encoding":57},[31,2435,2437],{"className":2436,"ariaHidden":62},[61],[31,2438,2440,2443],{"className":2439},[66],[31,2441],{"className":2442,"style":326},[70],[31,2444,205],{"className":2445},[75]," 和 ",[31,2448,2450,2463],{"className":2449,"translate":35},[34],[31,2451,2453],{"className":2452},[39],[41,2454,2455],{"xmlns":43},[45,2456,2457,2461],{},[48,2458,2459],{},[91,2460,152],{},[55,2462,152],{"encoding":57},[31,2464,2466],{"className":2465,"ariaHidden":62},[61],[31,2467,2469,2472],{"className":2468},[66],[31,2470],{"className":2471,"style":326},[70],[31,2473,152],{"className":2474},[75],"，函数做不到。",[11,2477,2478],{},"可这件事在现实中天天发生：一辆车分两个厂送货而已。Kantorovich 在 1940 年代的办法是把映射换成联合分布。令",[31,2480,2482],{"className":2481,"translate":35},[411],[31,2483,2485,2610],{"className":2484,"translate":35},[34],[31,2486,2488],{"className":2487},[39],[41,2489,2490],{"xmlns":43,"display":420},[45,2491,2492,2607],{},[48,2493,2494,2497,2499,2501,2503,2505,2507,2509,2513,2516,2519,2522,2524,2526,2529,2531,2533,2535,2537,2539,2541,2558,2564,2566,2568,2570,2572,2574,2576,2590,2596,2598,2600,2602,2605],{},[51,2495,2496],{"mathvariant":494},"Π",[95,2498,211],{"stretchy":210},[51,2500,852],{},[95,2502,1210],{"separator":62},[51,2504,884],{},[95,2506,220],{"stretchy":210},[95,2508,223],{},[95,2510,2512],{"fence":62,"stretchy":62,"minsize":2511,"maxsize":2511},"1.8em","{",[51,2514,2515],{},"π",[95,2517,2518],{},"∈",[51,2520,2521],{"mathvariant":925},"P",[95,2523,211],{"stretchy":210},[51,2525,926],{"mathvariant":925},[95,2527,2528],{},"×",[51,2530,932],{"mathvariant":925},[95,2532,220],{"stretchy":210},[1430,2534,1545],{},[95,2536,922],{},[1430,2538,1545],{},[95,2540,211],{"stretchy":210},[1004,2542,2543,2556],{},[48,2544,2545,2547,2550,2553],{},[51,2546,11],{"mathvariant":494},[51,2548,2549],{"mathvariant":494},"r",[51,2551,2552],{"mathvariant":494},"o",[51,2554,2555],{"mathvariant":494},"j",[51,2557,926],{"mathvariant":925},[1004,2559,2560,2562],{},[95,2561,220],{"stretchy":210},[51,2563,1010],{"mathvariant":494},[51,2565,2515],{},[95,2567,223],{},[51,2569,852],{},[95,2571,1210],{"separator":62},[1430,2573,1545],{},[95,2575,211],{"stretchy":210},[1004,2577,2578,2588],{},[48,2579,2580,2582,2584,2586],{},[51,2581,11],{"mathvariant":494},[51,2583,2549],{"mathvariant":494},[51,2585,2552],{"mathvariant":494},[51,2587,2555],{"mathvariant":494},[51,2589,932],{"mathvariant":925},[1004,2591,2592,2594],{},[95,2593,220],{"stretchy":210},[51,2595,1010],{"mathvariant":494},[51,2597,2515],{},[95,2599,223],{},[51,2601,884],{},[95,2603,2604],{"fence":62,"stretchy":62,"minsize":2511,"maxsize":2511},"}",[95,2606,1210],{"separator":62},[55,2608,2609],{"encoding":57},"\\Pi(\\mu,\\nu)=\\Bigl\\{\\pi\\in\\mathcal P(\\mathcal X\\times\\mathcal Y)\\ :\\ (\\mathrm{proj}_{\\mathcal X})_\\#\\pi=\\mu,\\ (\\mathrm{proj}_{\\mathcal Y})_\\#\\pi=\\nu\\Bigr\\},",[31,2611,2613,2649,2676,2700,2727,2840,2961],{"className":2612,"ariaHidden":62},[61],[31,2614,2616,2619,2622,2625,2628,2631,2634,2637,2640,2643,2646],{"className":2615},[66],[31,2617],{"className":2618,"style":260},[70],[31,2620,2496],{"className":2621},[75],[31,2623,211],{"className":2624},[264],[31,2626,852],{"className":2627},[75,76],[31,2629,1210],{"className":2630},[1391],[31,2632],{"className":2633,"style":1467},[119],[31,2635,884],{"className":2636,"style":900},[75,76],[31,2638,220],{"className":2639},[289],[31,2641],{"className":2642,"style":293},[119],[31,2644,223],{"className":2645},[297],[31,2647],{"className":2648,"style":293},[119],[31,2650,2652,2656,2663,2667,2670,2673],{"className":2651},[66],[31,2653],{"className":2654,"style":2655},[70],"height:1.8em;vertical-align:-0.65em;",[31,2657,2659],{"className":2658},[264],[31,2660,2512],{"className":2661},[1307,2662],"size2",[31,2664,2515],{"className":2665,"style":2666},[75,76],"margin-right:0.0359em;",[31,2668],{"className":2669,"style":293},[119],[31,2671,2518],{"className":2672},[297],[31,2674],{"className":2675,"style":293},[119],[31,2677,2679,2682,2685,2688,2691,2694,2697],{"className":2678},[66],[31,2680],{"className":2681,"style":260},[70],[31,2683,2521],{"className":2684,"style":989},[75,968],[31,2686,211],{"className":2687},[264],[31,2689,926],{"className":2690,"style":969},[75,968],[31,2692],{"className":2693,"style":120},[119],[31,2695,2528],{"className":2696},[124],[31,2698],{"className":2699,"style":120},[119],[31,2701,2703,2706,2709,2712,2715,2718,2721,2724],{"className":2702},[66],[31,2704],{"className":2705,"style":260},[70],[31,2707,932],{"className":2708,"style":989},[75,968],[31,2710,220],{"className":2711},[289],[31,2713,1545],{"className":2714},[119],[31,2716],{"className":2717,"style":293},[119],[31,2719,922],{"className":2720},[297],[31,2722,1545],{"className":2723},[119],[31,2725],{"className":2726,"style":293},[119],[31,2728,2730,2733,2736,2787,2828,2831,2834,2837],{"className":2729},[66],[31,2731],{"className":2732,"style":1223},[70],[31,2734,211],{"className":2735},[264],[31,2737,2739,2747],{"className":2738},[75],[31,2740,2742],{"className":2741},[75],[31,2743,2746],{"className":2744},[75,2745],"mathrm","proj",[31,2748,2750],{"className":2749},[1039],[31,2751,2753,2778],{"className":2752},[1043,1044],[31,2754,2756,2775],{"className":2755},[1048],[31,2757,2760],{"className":2758,"style":2759},[1052],"height:0.2342em;",[31,2761,2763,2766],{"style":2762},"top:-2.4559em;margin-right:0.05em;",[31,2764],{"className":2765,"style":1061},[1060],[31,2767,2769],{"className":2768},[1065,1066,1067,1068],[31,2770,2772],{"className":2771},[75,1068],[31,2773,926],{"className":2774,"style":969},[75,968,1068],[31,2776,1076],{"className":2777},[1075],[31,2779,2781],{"className":2780},[1048],[31,2782,2785],{"className":2783,"style":2784},[1052],"height:0.2441em;",[31,2786],{},[31,2788,2790,2793],{"className":2789},[289],[31,2791,220],{"className":2792},[289],[31,2794,2796],{"className":2795},[1039],[31,2797,2799,2820],{"className":2798},[1043,1044],[31,2800,2802,2817],{"className":2801},[1048],[31,2803,2805],{"className":2804,"style":1053},[1052],[31,2806,2808,2811],{"style":2807},"top:-2.55em;margin-left:0em;margin-right:0.05em;",[31,2809],{"className":2810,"style":1061},[1060],[31,2812,2814],{"className":2813},[1065,1066,1067,1068],[31,2815,1010],{"className":2816},[75,1068],[31,2818,1076],{"className":2819},[1075],[31,2821,2823],{"className":2822},[1048],[31,2824,2826],{"className":2825,"style":1083},[1052],[31,2827],{},[31,2829,2515],{"className":2830,"style":2666},[75,76],[31,2832],{"className":2833,"style":293},[119],[31,2835,223],{"className":2836},[297],[31,2838],{"className":2839,"style":293},[119],[31,2841,2843,2847,2850,2853,2856,2859,2862,2909,2949,2952,2955,2958],{"className":2842},[66],[31,2844],{"className":2845,"style":2846},[70],"height:1.0622em;vertical-align:-0.3122em;",[31,2848,852],{"className":2849},[75,76],[31,2851,1210],{"className":2852},[1391],[31,2854,1545],{"className":2855},[119],[31,2857],{"className":2858,"style":1467},[119],[31,2860,211],{"className":2861},[264],[31,2863,2865,2871],{"className":2864},[75],[31,2866,2868],{"className":2867},[75],[31,2869,2746],{"className":2870},[75,2745],[31,2872,2874],{"className":2873},[1039],[31,2875,2877,2900],{"className":2876},[1043,1044],[31,2878,2880,2897],{"className":2879},[1048],[31,2881,2883],{"className":2882,"style":2759},[1052],[31,2884,2885,2888],{"style":2762},[31,2886],{"className":2887,"style":1061},[1060],[31,2889,2891],{"className":2890},[1065,1066,1067,1068],[31,2892,2894],{"className":2893},[75,1068],[31,2895,932],{"className":2896,"style":989},[75,968,1068],[31,2898,1076],{"className":2899},[1075],[31,2901,2903],{"className":2902},[1048],[31,2904,2907],{"className":2905,"style":2906},[1052],"height:0.3122em;",[31,2908],{},[31,2910,2912,2915],{"className":2911},[289],[31,2913,220],{"className":2914},[289],[31,2916,2918],{"cla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读作“从 ",[31,3041,3043,3056],{"className":3042,"translate":35},[34],[31,3044,3046],{"className":3045},[39],[41,3047,3048],{"xmlns":43},[45,3049,3050,3054],{},[48,3051,3052],{},[51,3053,53],{},[55,3055,53],{"encoding":57},[31,3057,3059],{"className":3058,"ariaHidden":62},[61],[31,3060,3062,3065],{"className":3061},[66],[31,3063],{"className":3064,"style":71},[70],[31,3066,53],{"className":3067},[75,76]," 搬到 ",[31,3070,3072,3085],{"className":3071,"translate":35},[34],[31,3073,3075],{"className":3074},[39],[41,3076,3077],{"xmlns":43},[45,3078,3079,3083],{},[48,3080,3081],{},[51,3082,3003],{},[55,3084,3003],{"encoding":57},[31,3086,3088],{"className":3087,"ariaHidden":62},[61],[31,3089,3091,3094],{"className":3090},[66],[31,3092],{"className":3093,"style":865},[70],[31,3095,3003],{"className":3096,"style":2666},[75,76]," 的那部分质量”，于是",[31,3099,3101],{"className":3100,"translate":35},[411],[31,3102,3104,3203],{"className":3103,"translate":35},[34],[31,3105,3107],{"className":3106},[39],[41,3108,3109],{"xmlns":43,"display":420},[45,3110,3111,3200],{},[48,3112,3113,3120,3122,3124,3126,3128,3130,3132,3158,3170,3172,3174,3176,3178,3180,3182,3184,3186,3188,3190,3192,3194,3196,3198],{},[1004,3114,3115,3118],{},[51,3116,3117],{},"W",[51,3119,1412],{},[95,3121,211],{"stretchy":210},[51,3123,852],{},[95,3125,1210],{"separator":62},[51,3127,884],{},[95,3129,220],{"stretchy":210},[95,3131,223],{},[1531,3133,3134,3140],{},[48,3135,3136,3138],{},[51,3137,1537],{},[95,3139,1540],{},[48,3141,3142,3144,3146,3148,3150,3152,3154,3156],{},[51,3143,2515],{},[95,3145,2518],{},[51,3147,2496],{"mathvariant":494},[95,3149,211],{"stretchy":210},[51,3151,852],{},[95,3153,1210],{"separator":62},[51,3155,884],{},[95,3157,220],{"stretchy":210},[1004,3159,3160,3162],{},[95,3161,1409],{},[48,3163,3164,3166,3168],{},[51,3165,926],{"mathvariant":925},[95,3167,2528],{},[51,3169,932],{"mathvariant":925},[51,3171,1412],{},[95,3173,211],{"stretchy":210},[51,3175,53],{},[95,3177,1210],{"separator":62},[51,3179,3003],{},[95,3181,220],{"stretchy":210},[1430,3183,1432],{},[51,3185,1435],{},[51,3187,2515],{},[95,3189,211],{"stretchy":210},[51,3191,53],{},[95,3193,1210],{"separator":62},[51,3195,3003],{},[95,3197,220],{"stretchy":210},[51,3199,495],{"mathvariant":494},[55,3201,3202],{"encoding":57},"W_c(\\mu,\\nu)=\\inf_{\\pi\\in\\Pi(\\mu,\\nu)}\\int_{\\mathcal X\\times\\mathcal 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就是那张运输表。",[11,3500,3501,3502,3641,3642,3670],{},"取 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是度量），则",[31,3672,3674],{"className":3673,"translate":35},[411],[31,3675,3677,3774],{"className":3676,"translate":35},[34],[31,3678,3680],{"className":3679},[39],[41,3681,3682],{"xmlns":43,"display":420},[45,3683,3684,3771],{},[48,3685,3686,3692,3694,3696,3698,3700,3702,3704],{},[1004,3687,3688,3690],{},[51,3689,3117],{},[51,3691,11],{},[95,3693,211],{"stretchy":210},[51,3695,852],{},[95,3697,1210],{"separator":62},[51,3699,884],{},[95,3701,220],{"stretchy":210},[95,3703,223],{},[1180,3705,3706,3762],{},[48,3707,3708,3710,3736,3738,3740,3742,3744,3746,3748,3754,3756,3758,3760],{},[95,3709,211],{"fence":62},[1531,3711,3712,3718],{},[48,3713,3714,3716],{},[51,3715,1537],{},[95,3717,1540],{},[48,3719,3720,3722,3724,3726,3728,3730,3732,3734],{},[51,3721,2515],{},[95,3723,2518],{},[51,3725,2496],{"mathvariant":494},[95,3727,211],{"stretchy":210},[51,3729,852],{},[95,3731,1210],{"separator":62},[51,3733,884],{},[95,3735,220],{"stretchy":210},[95,3737,1409],{},[51,3739,1435],{},[95,3741,211],{"stretchy":210},[51,3743,53],{},[95,3745,1210],{"separator":62},[51,3747,3003],{},[1180,3749,3750,3752],{},[95,3751,220],{"stretchy":210},[51,3753,11],{},[1430,3755,1432],{},[51,3757,1435],{},[51,3759,2515],{},[95,3761,220],{"fence":62},[48,3763,3764,3766,3769],{},[91,3765,205],{},[51,3767,3768],{"mathvariant":494},"\u002F",[51,3770,11],{},[55,3772,3773],{"encoding":57},"W_p(\\mu,\\nu)=\\left(\\inf_{\\pi\\in\\Pi(\\mu,\\nu)}\\int 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上的一个真度量。这一点比看起来重要，因为 KL 不是。取 ",[31,4137,4139,4175],{"className":4138,"translate":35},[34],[31,4140,4142],{"className":4141},[39],[41,4143,4144],{"xmlns":43},[45,4145,4146,4172],{},[48,4147,4148,4150,4152,4158,4160,4162,4164,4166],{},[51,4149,852],{},[95,4151,223],{},[1004,4153,4154,4156],{},[51,4155,1974],{},[51,4157,53],{},[95,4159,1210],{"separator":62},[1430,4161,1545],{},[51,4163,884],{},[95,4165,223],{},[1004,4167,4168,4170],{},[51,4169,1974],{},[51,4171,3003],{},[55,4173,4174],{"encoding":57},"\\mu=\\delta_x,\\ 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是个有限的数，而 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要么发散要么给出零梯度，",[31,4719,4721,4734],{"className":4720,"translate":35},[34],[31,4722,4724],{"className":4723},[39],[41,4725,4726],{"xmlns":43},[45,4727,4728,4732],{},[48,4729,4730],{},[51,4731,3117],{},[55,4733,3117],{"encoding":57},[31,4735,4737],{"className":4736,"ariaHidden":62},[61],[31,4738,4740,4743],{"className":4739},[66],[31,4741],{"className":4742,"style":945},[70],[31,4744,3117],{"className":4745,"style":949},[75,76]," 却仍然告诉你“隔了多远”。",[11,4748,4749,4750,4778,4779,4809,4810,4880,4881,4909],{},"平移的例子更直白。把 ",[31,4751,4753,4766],{"className":4752,"translate":35},[34],[31,4754,4756],{"className":4755},[39],[41,4757,4758],{"xmlns":43},[45,4759,4760,4764],{},[48,4761,4762],{},[51,4763,852],{},[55,4765,855],{"encoding":57},[31,4767,4769],{"className":4768,"ariaHidden":62},[61],[31,4770,4772,4775],{"className":4771},[66],[31,4773],{"className":4774,"style":865},[70],[31,4776,852],{"className":4777},[75,76]," 整体挪 ",[31,4780,4782,4797],{"className":4781,"translate":35},[34],[31,4783,4785],{"className":4784},[39],[41,4786,4787],{"xmlns":43},[45,4788,4789,4794],{},[48,4790,4791],{},[51,4792,4793],{},"ε",[55,4795,4796],{"encoding":57},"\\varepsilon",[31,4798,4800],{"className":4799,"ariaHidden":62},[61],[31,4801,4803,4806],{"className":4802},[66],[31,4804],{"className":4805,"style":71},[70],[31,4807,4793],{"className":4808},[75,76],"：",[31,4811,4813,4831],{"className":4812,"translate":35},[34],[31,4814,4816],{"className":4815},[39],[41,4817,4818],{"xmlns":43},[45,4819,4820,4828],{},[48,4821,4822],{},[1004,4823,4824,4826],{},[51,4825,3117],{},[51,4827,11],{},[55,4829,4830],{"encoding":57},"W_p",[31,4832,4834],{"className":4833,"ariaHidden":62},[61],[31,4835,4837,4840],{"className":4836},[66],[31,4838],{"className":4839,"style":1029},[70],[31,4841,4843,4846],{"className":4842},[75],[31,4844,3117],{"className":4845,"style":949},[75,76],[31,4847,4849],{"className":4848},[1039],[31,4850,4852,4872],{"className":4851},[1043,1044],[31,4853,4855,4869],{"className":4854},[1048],[31,4856,4858],{"className":4857,"style":3230},[1052],[31,4859,4860,4863],{"style":1056},[31,4861],{"className":4862,"style":1061},[1060],[31,4864,4866],{"className":4865},[1065,1066,1067,1068],[31,4867,11],{"className":4868},[75,76,1068],[31,4870,1076],{"className":4871},[1075],[31,4873,4875],{"className":4874},[1048],[31,4876,4878],{"className":4877,"style":1083},[1052],[31,4879],{}," 是 ",[31,4882,4884,4897],{"className":4883,"translate":35},[34],[31,4885,4887],{"className":4886},[39],[41,4888,4889],{"xmlns":43},[45,4890,4891,4895],{},[48,4892,4893],{},[51,4894,4793],{},[55,4896,4796],{"encoding":57},[31,4898,4900],{"className":4899,"ariaHidden":62},[61],[31,4901,4903,4906],{"className":4902},[66],[31,4904],{"className":4905,"style":71},[70],[31,4907,4793],{"className":4908},[75,76]," 量级，而",[31,4911,4913],{"className":4912,"translate":35},[411],[31,4914,4916,4979],{"className":4915,"translate":35},[34],[31,4917,4919],{"className":4918},[39],[41,4920,4921],{"xmlns":43,"display":420},[45,4922,4923,4976],{},[48,4924,4925,4931,4933,4936,4938,4940,4942,4944,4946,4948,4950,4952,4954,4956,4958,4960,4962,4964,4974],{},[48,4926,4927,4929],{},[51,4928,4558],{"mathvariant":494},[51,4930,4561],{"mathvariant":494},[95,4932,211],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[51,4934,4935],{"mathvariant":925},"N",[95,4937,211],{"stretchy":210},[91,4939,1977],{},[95,4941,1210],{"separator":62},[91,4943,205],{},[95,4945,220],{"stretchy":210},[51,4947,4572],{"mathvariant":494},[51,4949,4935],{"mathvariant":925},[95,4951,211],{"stretchy":210},[51,4953,4793],{},[95,4955,1210],{"separator":62},[91,4957,205],{},[95,4959,220],{"stretchy":210},[95,4961,220],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[95,4963,223],{},[2074,4965,4966,4972],{},[1180,4967,4968,4970],{},[51,4969,4793],{},[91,4971,152],{},[91,4973,152],{},[95,4975,1210],{"separator":62},[55,4977,4978],{"encoding":57},"\\mathrm{KL}\\bigl(\\mathcal N(0,1)\\|\\mathcal N(\\varepsilon,1)\\bigr)=\\frac{\\varepsilon^2}{2},",[31,4980,4982,5062],{"className":4981,"ariaHidden":62},[61],[31,4983,4985,4989,4995,5001,5005,5008,5011,5014,5017,5020,5023,5026,5029,5032,5035,5038,5041,5044,5047,5053,5056,5059],{"className":4984},[66],[31,4986],{"className":4987,"style":4988},[70],"height:1.2em;vertical-align:-0.35em;",[31,4990,4992],{"className":4991},[75],[31,4993,4606],{"className":4994},[75,2745],[31,4996,4998],{"className":4997},[264],[31,4999,211],{"className":5000},[1307,1308],[31,5002,4935],{"className":5003,"style":5004},[75,968],"margin-right:0.1474em;",[31,5006,211],{"className":5007},[264],[31,5009,1977],{"className":5010},[75],[31,5012,1210],{"className":5013},[1391],[31,5015],{"className":5016,"style":1467},[119],[31,5018,205],{"className":5019},[75],[31,5021,220],{"className":5022},[289],[31,5024,4572],{"className":5025},[75],[31,5027,4935],{"className":5028,"style":5004},[75,968],[31,5030,211],{"className":5031},[264],[31,5033,4793],{"className":5034},[75,76],[31,5036,1210],{"className":5037},[1391],[31,5039],{"className":5040,"style":1467},[119],[31,5042,205],{"className":5043},[75],[31,5045,220],{"className":5046},[289],[31,5048,5050],{"className":5049},[289],[31,5051,220],{"className":5052},[1307,1308],[31,5054],{"className":5055,"style":293},[119],[31,5057,223],{"className":5058},[297],[31,5060],{"className":5061,"style":293},[119],[31,5063,5065,5069,5162],{"className":5064},[66],[31,5066],{"className":5067,"style":5068},[70],"height:2.1771em;vertical-align:-0.686em;",[31,5070,5072,5075,5159],{"className":5071},[75],[31,5073],{"className":5074},[264,2139],[31,5076,5078],{"className":5077},[2074],[31,5079,5081,5150],{"className":5080},[1043,1044],[31,5082,5084,5147],{"className":5083},[1048],[31,5085,5088,5100,5108],{"className":5086,"style":5087},[1052],"height:1.4911em;",[31,5089,5091,5094],{"style":5090},"top:-2.314em;",[31,5092],{"className":5093,"style":1637},[1060],[31,5095,5097],{"className":5096},[75],[31,5098,152],{"className":5099},[75],[31,5101,5102,5105],{"style":2170},[31,5103],{"className":5104,"style":1637},[1060],[31,5106],{"className":5107,"style":2178},[2177],[31,5109,5111,5114],{"style":5110},"top:-3.677em;",[31,5112],{"className":5113,"style":1637},[1060],[31,5115,5117],{"className":5116},[75],[31,5118,5120,5123],{"className":5119},[75],[31,5121,4793],{"className":5122},[75,76],[31,5124,5126],{"className":5125},[1039],[31,5127,5129],{"className":5128},[1043],[31,5130,5132],{"className":5131},[1048],[31,5133,5136],{"className":5134,"style":5135},[1052],"height:0.8141em;",[31,5137,5138,5141],{"style":3631},[31,5139],{"className":5140,"style":1061},[1060],[31,5142,5144],{"className":5143},[1065,1066,1067,1068],[31,5145,152],{"className":5146},[75,1068],[31,5148,1076],{"className":5149},[1075],[31,5151,5153],{"className":5152},[1048],[31,5154,5157],{"className":5155,"style":5156},[1052],"height:0.686em;",[31,5158],{},[31,5160],{"className":5161},[289,2139],[31,5163,1210],{"className":5164},[1391],[11,5166,5167,5168,5196],{},"是二阶的。所以小位移时 KL 的梯度趋近于零，",[31,5169,5171,5184],{"className":5170,"translate":35},[34],[31,5172,5174],{"className":5173},[39],[41,5175,5176],{"xmlns":43},[45,5177,5178,5182],{},[48,5179,5180],{},[51,5181,3117],{},[55,5183,3117],{"encoding":57},[31,5185,5187],{"className":5186,"ariaHidden":62},[61],[31,5188,5190,5193],{"className":5189},[66],[31,5191],{"className":5192,"style":945},[70],[31,5194,3117],{"className":5195,"style":949},[75,76]," 的梯度是常数。生成模型里那句“Wasserstein 提供不饱和梯度”，准确的数学内容就是这一行。",[11,5198,5199,5200,5271,5272,5300],{},"顺带说一句，计算机视觉里管它叫 Earth Mover's Distance，数学里叫 Wasserstein 距离，是同一个东西；EMD 通常特指 ",[31,5201,5203,5221],{"className":5202,"translate":35},[34],[31,5204,5206],{"className":5205},[39],[41,5207,5208],{"xmlns":43},[45,5209,5210,5218],{},[48,5211,5212],{},[1004,5213,5214,5216],{},[51,5215,3117],{},[91,5217,205],{},[55,5219,5220],{"encoding":57},"W_1",[31,5222,5224],{"className":5223,"ariaHidden":62},[61],[31,5225,5227,5231],{"className":5226},[66],[31,5228],{"className":5229,"style":5230},[70],"height:0.8333em;vertical-align:-0.15em;",[31,5232,5234,5237],{"className":5233},[75],[31,5235,3117],{"className":5236,"style":949},[75,76],[31,5238,5240],{"className":5239},[1039],[31,5241,5243,5263],{"className":5242},[1043,1044],[31,5244,5246,5260],{"className":5245},[1048],[31,5247,5249],{"className":5248,"style":2028},[1052],[31,5250,5251,5254],{"style":1056},[31,5252],{"className":5253,"style":1061},[1060],[31,5255,5257],{"className":5256},[1065,1066,1067,1068],[31,5258,205],{"className":5259},[75,1068],[31,5261,1076],{"className":5262},[1075],[31,5264,5266],{"className":5265},[1048],[31,5267,5269],{"className":5268,"style":2050},[1052],[31,5270],{},"，或者指带整数质量的离散版本。把一张 28×28 的手写数字的灰度值当成质量，",[31,5273,5275,5288],{"className":5274,"translate":35},[34],[31,5276,5278],{"className":5277},[39],[41,5279,5280],{"xmlns":43},[45,5281,5282,5286],{},[48,5283,5284],{},[51,5285,3117],{},[55,5287,3117],{"encoding":57},[31,5289,5291],{"className":5290,"ariaHidden":62},[61],[31,5292,5294,5297],{"className":5293},[66],[31,5295],{"className":5296,"style":945},[70],[31,5298,3117],{"className":5299,"style":949},[75,76]," 就是“把这张画改写成那张画的最小工作量”，我最初就是靠这个画面理解它的。",[18,5302,5303],{"id":5303},"对偶",[11,5305,5306],{},"回到那道运输题。我算出最低运费是 5 块。如果有人说他能做到 4 块，我怎么证明他做不到？",[11,5308,5309,5310,5339,5340,5369],{},"不去枚举所有运输表。我给每个矿定一个“提货加价” ",[31,5311,5313,5327],{"className":5312,"translate":35},[34],[31,5314,5316],{"className":5315},[39],[41,5317,5318],{"xmlns":43},[45,5319,5320,5325],{},[48,5321,5322],{},[51,5323,5324],{},"u",[55,5326,5324],{"encoding":57},[31,5328,5330],{"className":5329,"ariaHidden":62},[61],[31,5331,5333,5336],{"className":5332},[66],[31,5334],{"className":5335,"style":71},[70],[31,5337,5324],{"className":5338},[75,76],"，每个厂定一个“到货减价” ",[31,5341,5343,5357],{"className":5342,"translate":35},[34],[31,5344,5346],{"className":5345},[39],[41,5347,5348],{"xmlns":43},[45,5349,5350,5355],{},[48,5351,5352],{},[51,5353,5354],{},"v",[55,5356,5354],{"encoding":57},[31,5358,5360],{"className":5359,"ariaHidden":62},[61],[31,5361,5363,5366],{"className":5362},[66],[31,5364],{"className":5365,"style":71},[70],[31,5367,5354],{"className":5368,"style":2666},[75,76],"，只要求一条规矩：对任意一条路线，两个价加起来不超过运费，",[31,5371,5373],{"className":5372,"translate":35},[411],[31,5374,5376,5417],{"className":5375,"translate":35},[34],[31,5377,5379],{"className":5378},[39],[41,5380,5381],{"xmlns":43,"display":420},[45,5382,5383,5414],{},[48,5384,5385,5392,5394,5400,5402,5412],{},[1004,5386,5387,5389],{},[51,5388,5324],{},[51,5390,5391],{},"i",[95,5393,434],{},[1004,5395,5396,5398],{},[51,5397,5354],{},[51,5399,2555],{},[95,5401,347],{},[1004,5403,5404,5406],{},[51,5405,1412],{},[48,5407,5408,5410],{},[51,5409,5391],{},[51,5411,2555],{},[51,5413,495],{"mathvariant":494},[55,5415,5416],{"encoding":57},"u_i+v_j\\le c_{ij}.",[31,5418,5420,5477,5535],{"className":5419,"ariaHidden":62},[61],[31,5421,5423,5427,5468,5471,5474],{"className":5422},[66],[31,5424],{"className":5425,"style":5426},[70],"height:0.7333em;vertical-align:-0.15em;",[31,5428,5430,5433],{"className":5429},[75],[31,5431,5324],{"className":5432},[75,76],[31,5434,5436],{"className":5435},[1039],[31,5437,5439,5460],{"className":5438},[1043,1044],[31,5440,5442,5457],{"className":5441},[1048],[31,5443,5446],{"className":5444,"style":5445},[1052],"height:0.3117em;",[31,5447,5448,5451],{"style":2807},[31,5449],{"className":5450,"style":1061},[1060],[31,5452,5454],{"className":5453},[1065,1066,1067,1068],[31,5455,5391],{"className":5456},[75,76,1068],[31,5458,1076],{"className":5459},[1075],[31,5461,5463],{"className":5462},[1048],[31,5464,5466],{"className":5465,"style":2050},[1052],[31,5467],{},[31,5469],{"className":5470,"style":120},[119],[31,5472,434],{"className":5473},[124],[31,5475],{"className":5476,"style":120},[119],[31,5478,5480,5484,5526,5529,5532],{"className":5479},[66],[31,5481],{"className":5482,"style":5483},[70],"height:0.9221em;vertical-align:-0.2861em;",[31,5485,5487,5490],{"className":5486},[75],[31,5488,5354],{"className":5489,"style":2666},[75,76],[31,5491,5493],{"className":5492},[1039],[31,5494,5496,5518],{"className":5495},[1043,1044],[31,5497,5499,5515],{"className":5498},[1048],[31,5500,5502],{"className":5501,"style":5445},[1052],[31,5503,5505,5508],{"style":5504},"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;",[31,5506],{"className":5507,"style":1061},[1060],[31,5509,5511],{"className":5510},[1065,1066,1067,1068],[31,5512,2555],{"className":5513,"style":5514},[75,76,1068],"margin-right:0.0572em;",[31,5516,1076],{"className":5517},[1075],[31,5519,5521],{"className":5520},[1048],[31,5522,5524],{"className":5523,"style":1083},[1052],[31,5525],{},[31,5527],{"className":5528,"style":293},[119],[31,5530,347],{"className":5531},[297],[31,5533],{"className":5534,"style":293},[119],[31,5536,5538,5542,5586],{"className":5537},[66],[31,5539],{"className":5540,"style":5541},[70],"height:0.7167em;vertical-align:-0.2861em;",[31,5543,5545,5548],{"className":5544},[75],[31,5546,1412],{"className":5547},[75,76],[31,5549,5551],{"className":5550},[1039],[31,5552,5554,5578],{"className":5553},[1043,1044],[31,5555,5557,5575],{"className":5556},[1048],[31,5558,5560],{"className":5559,"style":5445},[1052],[31,5561,5562,5565],{"style":2807},[31,5563],{"className":5564,"style":1061},[1060],[31,5566,5568],{"className":5567},[1065,1066,1067,1068],[31,5569,5571],{"className":5570},[75,1068],[31,5572,5574],{"className":5573,"style":5514},[75,76,1068],"ij",[31,5576,1076],{"className":5577},[1075],[31,5579,5581],{"className":5580},[1048],[31,5582,5584],{"className":5583,"style":1083},[1052],[31,5585],{},[31,5587,495],{"className":5588},[75],[11,5590,5591],{},"那么任何一张运单的总价都满足",[31,5593,5595],{"className":5594,"translate":35},[411],[31,5596,5598,5848],{"className":5597,"translate":35},[34],[31,5599,5601],{"className":5600},[39],[41,5602,5603],{"xmlns":43,"display":420},[45,5604,5605,5845],{},[5606,5607,5611,5701,5785],"mtable",{"rowspacing":5608,"columnalign":5609,"columnspacing":5610},"0.25em","right 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u_i\\Bigl(\\sum_j\\pi_{ij}\\Bigr)+\\sum_j v_j\\Bigl(\\sum_i\\pi_{ij}\\Bigr)\\\\\n&=\\sum_i a_iu_i+\\sum_j b_jv_j 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",[31,6989,6991,7009],{"className":6990,"translate":35},[34],[31,6992,6994],{"className":6993},[39],[41,6995,6996],{"xmlns":43},[45,6997,6998,7006],{},[48,6999,7000,7002,7004],{},[51,7001,5324],{},[95,7003,1210],{"separator":62},[51,7005,5354],{},[55,7007,7008],{"encoding":57},"u,v",[31,7010,7012],{"className":7011,"ariaHidden":62},[61],[31,7013,7015,7018,7021,7024,7027],{"className":7014},[66],[31,7016],{"className":7017,"style":865},[70],[31,7019,5324],{"className":7020},[75,76],[31,7022,1210],{"className":7023},[1391],[31,7025],{"className":7026,"style":1467},[119],[31,7028,5354],{"className":7029,"style":2666},[75,76]," 有关，跟运单无关；而中间那步把 ",[31,7032,7034,7047],{"className":7033,"translate":35},[34],[31,7035,7037],{"className":7036},[39],[41,7038,7039],{"xmlns":43},[45,7040,7041,7045],{},[48,7042,7043],{},[51,7044,2515],{},[55,7046,3485],{"encoding":57},[31,7048,7050],{"className":7049,"ariaHidden":62},[61],[31,7051,7053,7056],{"className":7052},[66],[31,7054],{"className":7055,"style":71},[70],[31,7057,2515],{"className":7058,"style":2666},[75,76]," 的边缘和换成了产量 ",[31,7061,7063,7081],{"className":7062,"translate":35},[34],[31,7064,7066],{"className":7065},[39],[41,7067,7068],{"xmlns":43},[45,7069,7070,7078],{},[48,7071,7072],{},[1004,7073,7074,7076],{},[51,7075,5813],{},[51,7077,5391],{},[55,7079,7080],{"encoding":57},"a_i",[31,7082,7084],{"className":7083,"ariaHidden":62},[61],[31,7085,7087,7091],{"className":7086},[66],[31,7088],{"className":7089,"style":7090},[70],"height:0.5806em;vertical-align:-0.15em;",[31,7092,7094,7097],{"className":7093},[75],[31,7095,5813],{"className":7096},[75,76],[31,7098,7100],{"className":7099},[1039],[31,7101,7103,7123],{"className":7102},[1043,1044],[31,7104,7106,7120],{"className":7105},[1048],[31,7107,7109],{"className":7108,"style":5445},[1052],[31,7110,7111,7114],{"style":2807},[31,7112],{"className":7113,"style":1061},[1060],[31,7115,7117],{"className":7116},[1065,1066,1067,1068],[31,7118,5391],{"className":7119},[75,76,1068],[31,7121,1076],{"className":7122},[1075],[31,7124,7126],{"className":7125},[1048],[31,7127,7129],{"className":7128,"style":2050},[1052],[31,7130],{},"、需求 ",[31,7133,7135,7153],{"className":7134,"translate":35},[34],[31,7136,7138],{"className":7137},[39],[41,7139,7140],{"xmlns":43},[45,7141,7142,7150],{},[48,7143,7144],{},[1004,7145,7146,7148],{},[51,7147,5834],{},[51,7149,2555],{},[55,7151,7152],{"encoding":57},"b_j",[31,7154,7156],{"className":7155,"ariaHidden":62},[61],[31,7157,7159,7162],{"className":7158},[66],[31,7160],{"className":7161,"style":4270},[70],[31,7163,7165,7168],{"className":7164},[75],[31,7166,5834],{"className":7167},[75,76],[31,7169,7171],{"className":7170},[1039],[31,7172,7174,7194],{"className":7173},[1043,1044],[31,7175,7177,7191],{"className":7176},[1048],[31,7178,7180],{"className":7179,"style":5445},[1052],[31,7181,7182,7185],{"style":2807},[31,7183],{"className":7184,"style":1061},[1060],[31,7186,7188],{"className":7187},[1065,1066,1067,1068],[31,7189,2555],{"className":7190,"style":5514},[75,76,1068],[31,7192,1076],{"className":7193},[1075],[31,7195,7197],{"className":7196},[1048],[31,7198,7200],{"className":7199,"style":1083},[1052],[31,7201],{},"。所以只要我找到一组 ",[31,7204,7206,7223],{"className":7205,"translate":35},[34],[31,7207,7209],{"className":7208},[39],[41,7210,7211],{"xmlns":43},[45,7212,7213,7221],{},[48,7214,7215,7217,7219],{},[51,7216,5324],{},[95,7218,1210],{"separator":62},[51,7220,5354],{},[55,7222,7008],{"encoding":57},[31,7224,7226],{"className":7225,"ariaHidden":62},[61],[31,7227,7229,7232,7235,7238,7241],{"className":7228},[66],[31,7230],{"className":7231,"style":865},[70],[31,7233,5324],{"className":7234},[75,76],[31,7236,1210],{"className":7237},[1391],[31,7239],{"className":7240,"style":1467},[119],[31,7242,5354],{"className":7243,"style":2666},[75,76],"，使这个和等于 5，我就",[7246,7247,7248],"strong",{},"证明","了运费不可能低于 5 块。",[11,7251,7252,7253,1138,7332,7406,7407,7502,7503,7598,7599,7699,7700,7800,7801,8022],{},"试一试：取 ",[31,7254,7256,7284],{"className":7255,"translate":35},[34],[31,7257,7259],{"className":7258},[39],[41,7260,7261],{"xmlns":43},[45,7262,7263,7281],{},[48,7264,7265,7267,7269,7271,7273,7275,7277,7279],{},[51,7266,5324],{},[95,7268,223],{},[95,7270,211],{"stretchy":210},[91,7272,1977],{},[95,7274,1210],{"separator":62},[95,7276,97],{},[91,7278,205],{},[95,7280,220],{"stretchy":210},[55,7282,7283],{"encoding":57},"u=(0,-1)",[31,7285,7287,7305],{"className":7286,"ariaHidden":62},[61],[31,7288,7290,7293,7296,7299,7302],{"className":7289},[66],[31,7291],{"className":7292,"style":71},[70],[31,7294,5324],{"className":7295},[75,76],[31,7297],{"className":7298,"style":293},[119],[31,7300,223],{"className":7301},[297],[31,7303],{"className":7304,"style":293},[119],[31,7306,7308,7311,7314,7317,7320,7323,7326,7329],{"className":7307},[66],[31,7309],{"className":7310,"style":260},[70],[31,7312,211],{"className":7313},[264],[31,7315,1977],{"className":7316},[75],[31,7318,1210],{"className":7319},[1391],[31,7321],{"className":7322,"style":1467},[119],[31,7324,97],{"className":7325},[75],[31,7327,205],{"className":7328},[75],[31,7330,220],{"className":7331},[289],[31,7333,7335,7361],{"className":7334,"translate":35},[34],[31,7336,7338],{"className":7337},[39],[41,7339,7340],{"xmlns":43},[45,7341,7342,7358],{},[48,7343,7344,7346,7348,7350,7352,7354,7356],{},[51,7345,5354],{},[95,7347,223],{},[95,7349,211],{"stretchy":210},[91,7351,205],{},[95,7353,1210],{"separator":62},[91,7355,152],{},[95,7357,220],{"stretchy":210},[55,7359,7360],{"encoding":57},"v=(1,2)",[31,7362,7364,7382],{"className":7363,"ariaHidden":62},[61],[31,7365,7367,7370,7373,7376,7379],{"className":7366},[66],[31,7368],{"className":7369,"style":71},[70],[31,7371,5354],{"className":7372,"style":2666},[75,76],[31,7374],{"className":7375,"style":293},[119],[31,7377,223],{"className":7378},[297],[31,7380],{"className":7381,"style":293},[119],[31,7383,7385,7388,7391,7394,7397,7400,7403],{"className":7384},[66],[31,7386],{"className":7387,"style":260},[70],[31,7389,211],{"className":7390},[264],[31,7392,205],{"className":7393},[75],[31,7395,1210],{"className":7396},[1391],[31,7398],{"className":7399,"style":1467},[119],[31,7401,152],{"className":7402},[75],[31,7404,220],{"className":7405},[289],"。四条路线逐一检查：甲到 A 是 ",[31,7408,7410,7436],{"className":7409,"translate":35},[34],[31,7411,7413],{"className":7412},[39],[41,7414,7415],{"xmlns":43},[45,7416,7417,7433],{},[48,7418,7419,7421,7423,7425,7427,7429,7431],{},[91,7420,1977],{},[95,7422,434],{},[91,7424,205],{},[95,7426,223],{},[91,7428,205],{},[95,7430,347],{},[91,7432,205],{},[55,7434,7435],{"encoding":57},"0+1=1\\le 1",[31,7437,7439,7457,7475,7493],{"className":7438,"ariaHidden":62},[61],[31,7440,7442,7445,7448,7451,7454],{"className":7441},[66],[31,7443],{"className":7444,"style":112},[70],[31,7446,1977],{"className":7447},[75],[31,7449],{"className":7450,"style":120},[119],[31,7452,434],{"className":7453},[124],[31,7455],{"className":7456,"style":120},[119],[31,7458,7460,7463,7466,7469,7472],{"className":7459},[66],[31,7461],{"className":7462,"style":326},[70],[31,7464,205],{"className":7465},[75],[31,7467],{"className":7468,"style":293},[119],[31,7470,223],{"className":7471},[297],[31,7473],{"className":7474,"style":293},[119],[31,7476,7478,7481,7484,7487,7490],{"className":7477},[66],[31,7479],{"className":7480,"style":366},[70],[31,7482,205],{"className":7483},[75],[31,7485],{"className":7486,"style":293},[119],[31,7488,347],{"className":7489},[297],[31,7491],{"className":7492,"style":293},[119],[31,7494,7496,7499],{"className":7495},[66],[31,7497],{"className":7498,"style":326},[70],[31,7500,205],{"className":7501},[75],"，甲到 B 是 ",[31,7504,7506,7532],{"className":7505,"translate":35},[34],[31,7507,7509],{"className":7508},[39],[41,7510,7511],{"xmlns":43},[45,7512,7513,7529],{},[48,7514,7515,7517,7519,7521,7523,7525,7527],{},[91,7516,1977],{},[95,7518,434],{},[91,7520,152],{},[95,7522,223],{},[91,7524,152],{},[95,7526,347],{},[91,7528,152],{},[55,7530,7531],{"encoding":57},"0+2=2\\le 2",[31,7533,7535,7553,7571,7589],{"className":7534,"ariaHidden":62},[61],[31,7536,7538,7541,7544,7547,7550],{"className":7537},[66],[31,7539],{"className":7540,"style":112},[70],[31,7542,1977],{"className":7543},[75],[31,7545],{"className":7546,"style":120},[119],[31,7548,434],{"className":7549},[124],[31,7551],{"className":7552,"style":120},[119],[31,7554,7556,7559,7562,7565,7568],{"className":7555},[66],[31,7557],{"className":7558,"style":326},[70],[31,7560,152],{"className":7561},[75],[31,7563],{"className":7564,"style":293},[119],[31,7566,223],{"className":7567},[297],[31,7569],{"className":7570,"style":293},[119],[31,7572,7574,7577,7580,7583,7586],{"className":7573},[66],[31,7575],{"className":7576,"style":366},[70],[31,7578,152],{"className":7579},[75],[31,7581],{"className":7582,"style":293},[119],[31,7584,347],{"className":7585},[297],[31,7587],{"className":7588,"style":293},[119],[31,7590,7592,7595],{"className":7591},[66],[31,7593],{"className":7594,"style":326},[70],[31,7596,152],{"className":7597},[75],"，乙到 A 是 ",[31,7600,7602,7630],{"className":7601,"translate":35},[34],[31,7603,7605],{"className":7604},[39],[41,7606,7607],{"xmlns":43},[45,7608,7609,7627],{},[48,7610,7611,7613,7615,7617,7619,7621,7623,7625],{},[95,7612,97],{},[91,7614,205],{},[95,7616,434],{},[91,7618,205],{},[95,7620,223],{},[91,7622,1977],{},[95,7624,347],{},[91,7626,93],{},[55,7628,7629],{"encoding":57},"-1+1=0\\le 3",[31,7631,7633,7654,7672,7690],{"className":7632,"ariaHidden":62},[61],[31,7634,7636,7639,7642,7645,7648,7651],{"className":7635},[66],[31,7637],{"className":7638,"style":112},[70],[31,7640,97],{"className":7641},[75],[31,7643,205],{"className":7644},[75],[31,7646],{"className":7647,"style":120},[119],[31,7649,434],{"className":7650},[124],[31,7652],{"className":7653,"style":120},[119],[31,7655,7657,7660,7663,7666,7669],{"className":7656},[66],[31,7658],{"className":7659,"style":326},[70],[31,7661,205],{"className":7662},[75],[31,7664],{"className":7665,"style":293},[119],[31,7667,223],{"className":7668},[297],[31,7670],{"className":7671,"style":293},[119],[31,7673,7675,7678,7681,7684,7687],{"className":7674},[66],[31,7676],{"className":7677,"style":366},[70],[31,7679,1977],{"className":7680},[75],[31,7682],{"className":7683,"style":293},[119],[31,7685,347],{"className":7686},[297],[31,7688],{"className":7689,"style":293},[119],[31,7691,7693,7696],{"className":7692},[66],[31,7694],{"className":7695,"style":326},[70],[31,7697,93],{"className":7698},[75],"，乙到 B 是 ",[31,7701,7703,7731],{"className":7702,"translate":35},[34],[31,7704,7706],{"className":7705},[39],[41,7707,7708],{"xmlns":43},[45,7709,7710,7728],{},[48,7711,7712,7714,7716,7718,7720,7722,7724,7726],{},[95,7713,97],{},[91,7715,205],{},[95,7717,434],{},[91,7719,152],{},[95,7721,223],{},[91,7723,205],{},[95,7725,347],{},[91,7727,205],{},[55,7729,7730],{"encoding":57},"-1+2=1\\le 1",[31,7732,7734,7755,7773,7791],{"className":7733,"ariaHidden":62},[61],[31,7735,7737,7740,7743,7746,7749,7752],{"className":7736},[66],[31,7738],{"className":7739,"style":112},[70],[31,7741,97],{"className":7742},[75],[31,7744,205],{"className":7745},[75],[31,7747],{"className":7748,"style":120},[119],[31,7750,434],{"className":7751},[124],[31,7753],{"className":7754,"style":120},[119],[31,7756,7758,7761,7764,7767,7770],{"className":7757},[66],[31,7759],{"className":7760,"style":326},[70],[31,7762,152],{"className":7763},[75],[31,7765],{"className":7766,"style":293},[119],[31,7768,223],{"className":7769},[297],[31,7771],{"className":7772,"style":293},[119],[31,7774,7776,7779,7782,7785,7788],{"className":7775},[66],[31,7777],{"className":7778,"style":366},[70],[31,7780,205],{"className":7781},[75],[31,7783],{"className":7784,"style":293},[119],[31,7786,347],{"className":7787},[297],[31,7789],{"className":7790,"style":293},[119],[31,7792,7794,7797],{"className":7793},[66],[31,7795],{"className":7796,"style":326},[70],[31,7798,205],{"className":7799},[75],"，全部成立。而 ",[31,7802,7804,7857],{"className":7803,"translate":35},[34],[31,7805,7807],{"className":7806},[39],[41,7808,7809],{"xmlns":43},[45,7810,7811,7854],{},[48,7812,7813,7815,7817,7819,7821,7823,7825,7827,7829,7831,7833,7835,7837,7839,7841,7843,7845,7847,7849,7851],{},[91,7814,93],{},[95,7816,429],{},[91,7818,1977],{},[95,7820,434],{},[91,7822,205],{},[95,7824,429],{},[95,7826,211],{"stretchy":210},[95,7828,97],{},[91,7830,205],{},[95,7832,220],{"stretchy":210},[95,7834,434],{},[91,7836,152],{},[95,7838,429],{},[91,7840,205],{},[95,7842,434],{},[91,7844,152],{},[95,7846,429],{},[91,7848,152],{},[95,7850,223],{},[91,7852,7853],{},"5",[55,7855,7856],{"encoding":57},"3\\cdot0+1\\cdot(-1)+2\\cdot1+2\\cdot2=5",[31,7858,7860,7878,7896,7914,7941,7959,7977,7995,8013],{"className":7859,"ariaHidden":62},[61],[31,7861,7863,7866,7869,7872,7875],{"className":7862},[66],[31,7864],{"className":7865,"style":326},[70],[31,7867,93],{"className":7868},[75],[31,7870],{"className":7871,"style":120},[119],[31,7873,429],{"className":7874},[124],[31,7876],{"className":7877,"style":120},[119],[31,7879,7881,7884,7887,7890,7893],{"className":7880},[66],[31,7882],{"className":7883,"style":112},[70],[31,7885,1977],{"className":7886},[75],[31,7888],{"className":7889,"style":120},[119],[31,7891,434],{"className":7892},[124],[31,7894],{"className":7895,"style":120},[119],[31,7897,7899,7902,7905,7908,7911],{"className":7898},[66],[31,7900],{"className":7901,"style":326},[70],[31,7903,205],{"className":7904},[75],[31,7906],{"className":7907,"style":120},[119],[31,7909,429],{"className":7910},[124],[31,7912],{"className":7913,"style":120},[119],[31,7915,7917,7920,7923,7926,7929,7932,7935,7938],{"className":7916},[66],[31,7918],{"className":7919,"style":260},[70],[31,7921,211],{"className":7922},[264],[31,7924,97],{"className":7925},[75],[31,7927,205],{"className":7928},[75],[31,7930,220],{"className":7931},[289],[31,7933],{"className":7934,"style":120},[119],[31,7936,434],{"className":7937},[124],[31,7939],{"className":7940,"style":120},[119],[31,7942,7944,7947,7950,7953,7956],{"className":7943},[66],[31,7945],{"className":7946,"style":326},[70],[31,7948,152],{"className":7949},[75],[31,7951],{"className":7952,"style":120},[119],[31,7954,429],{"className":7955},[124],[31,7957],{"className":7958,"style":120},[119],[31,7960,7962,7965,7968,7971,7974],{"className":7961},[66],[31,7963],{"className":7964,"style":112},[70],[31,7966,205],{"className":7967},[75],[31,7969],{"className":7970,"style":120},[119],[31,7972,434],{"className":7973},[124],[31,7975],{"className":7976,"style":120},[119],[31,7978,7980,7983,7986,7989,7992],{"className":7979},[66],[31,7981],{"className":7982,"style":326},[70],[31,7984,152],{"className":7985},[75],[31,7987],{"className":7988,"style":120},[119],[31,7990,429],{"className":7991},[124],[31,7993],{"className":7994,"style":120},[119],[31,7996,7998,8001,8004,8007,8010],{"className":7997},[66],[31,7999],{"className":8000,"style":326},[70],[31,8002,152],{"className":8003},[75],[31,8005],{"className":8006,"style":293},[119],[31,8008,223],{"className":8009},[297],[31,8011],{"className":8012,"style":293},[119],[31,8014,8016,8019],{"className":8015},[66],[31,8017],{"className":8018,"style":326},[70],[31,8020,7853],{"className":8021},[75],"。证书找到了。",[11,8024,8025],{},"这就是对偶的思想：不去找运单，而是找一张“价格表”，让价格表卡住运费的下界。最优的价格表能卡到最紧，那个值就等于最低运费。离散情形下，这件事可以写成拉格朗日形式，把两组等式约束配乘子：",[31,8027,8029],{"className":8028,"translate":35},[411],[31,8030,8032,8305],{"className":8031,"translate":35},[34],[31,8033,8035],{"className":8034},[39],[41,8036,8037],{"xmlns":43,"display":420},[45,8038,8039,8302],{},[5606,8040,8041,8192],{"rowspacing":5608,"columnalign":5609,"columnspacing":5610},[5612,8042,8043,8067],{},[5615,8044,8045],{},[2071,8046,8047],{"scriptlevel":1977,"displaystyle":62},[48,8048,8049,8051,8053,8055,8057,8060,8062,8065],{},[51,8050,4561],{},[95,8052,211],{"stretchy":210},[51,8054,2515],{},[95,8056,1210],{"separator":62},[51,8058,8059],{},"φ",[95,8061,1210],{"separator":62},[51,8063,8064],{},"ψ",[95,8066,220],{"stretchy":210},[5615,8068,8069],{},[2071,8070,8071],{"scriptlevel":1977,"displaystyle":62},[48,8072,8073,8075,8077,8087,8097,8108,8110,8116,8122,8124,8130,8132,8138,8148,8150,8152,8158,8164,8166,8172,8174,8180,8190],{},[48,8074],{},[95,8076,223],{},[1531,8078,8079,8081],{},[95,8080,5625],{},[48,8082,8083,8085],{},[51,8084,5391],{},[51,8086,2555],{},[1004,8088,8089,8091],{},[51,8090,2515],{},[48,8092,8093,8095],{},[51,8094,5391],{},[51,8096,2555],{},[1004,8098,8099,8102],{},[51,8100,8101],{},"C",[48,8103,8104,8106],{},[51,8105,5391],{},[51,8107,2555],{},[95,8109,434],{},[1531,8111,8112,8114],{},[95,8113,5625],{},[51,8115,5391],{},[1004,8117,8118,8120],{},[51,8119,8059],{},[51,8121,5391],{},[95,8123,211],{"fence":62,"stretchy":62,"minsize":2511,"maxsize":2511},[1004,8125,8126,8128],{},[51,8127,5813],{},[51,8129,5391],{},[95,8131,97],{},[1531,8133,8134,8136],{},[95,8135,5625],{},[51,8137,2555],{},[1004,8139,8140,8142],{},[51,8141,2515],{},[48,8143,8144,8146],{},[51,8145,5391],{},[51,8147,2555],{},[95,8149,220],{"fence":62,"stretchy":62,"minsize":2511,"maxsize":2511},[95,8151,434],{},[1531,8153,8154,8156],{},[95,8155,5625],{},[51,8157,2555],{},[1004,8159,8160,8162],{},[51,8161,8064],{},[51,8163,2555],{},[95,8165,211],{"fence":62,"stretchy":62,"minsize":2511,"maxsize":2511},[1004,8167,8168,8170],{},[51,8169,5834],{},[51,8171,2555],{},[95,8173,97],{},[1531,8175,8176,8178],{},[95,8177,5625],{},[51,8179,5391],{},[1004,8181,8182,8184],{},[51,8183,2515],{},[48,8185,8186,8188],{},[51,8187,5391],{},[51,8189,2555],{},[95,8191,220],{"fence":62,"stretchy":62,"minsize":2511,"maxsize":2511},[5612,8193,8194,8200],{},[5615,8195,8196],{},[2071,8197,8198],{"scriptlevel":1977,"displaystyle":62},[48,8199],{},[5615,8201,8202],{},[2071,8203,8204],{"scriptlevel":1977,"displaystyle":62},[48,8205,8206,8208,8210,8216,8222,8228,8230,8236,8242,8248,8250,8260,8270,8272,8282,8284,8290,8292,8298,8300],{},[48,8207],{},[95,8209,223],{},[1531,8211,8212,8214],{},[95,8213,5625],{},[51,8215,5391],{},[1004,8217,8218,8220],{},[51,8219,5813],{},[51,8221,5391],{},[1004,8223,8224,8226],{},[51,8225,8059],{},[51,8227,5391],{},[95,8229,434],{},[1531,8231,8232,8234],{},[95,8233,5625],{},[51,8235,2555],{},[1004,8237,8238,8240],{},[51,8239,5834],{},[51,8241,2555],{},[1004,8243,8244,8246],{},[51,8245,8064],{},[51,8247,2555],{},[95,8249,434],{},[1531,8251,8252,8254],{},[95,8253,5625],{},[48,8255,8256,8258],{},[51,8257,5391],{},[51,8259,2555],{},[1004,8261,8262,8264],{},[51,8263,2515],{},[48,8265,8266,8268],{},[51,8267,5391],{},[51,8269,2555],{},[95,8271,211],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[1004,8273,8274,8276],{},[51,8275,8101],{},[48,8277,8278,8280],{},[51,8279,5391],{},[51,8281,2555],{},[95,8283,97],{},[1004,8285,8286,8288],{},[51,8287,8059],{},[51,8289,5391],{},[95,8291,97],{},[1004,8293,8294,8296],{},[51,8295,8064],{},[51,8297,2555],{},[95,8299,220],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[51,8301,495],{"mathvariant":494},[55,8303,8304],{"encoding":57},"\\begin{aligned}\nL(\\pi,\\varphi,\\psi)\n&=\\sum_{ij}\\pi_{ij}C_{ij}+\\sum_i\\varphi_i\\Bigl(a_i-\\sum_j\\pi_{ij}\\Bigr)+\\sum_j\\psi_j\\Bigl(b_j-\\sum_i\\pi_{ij}\\Bigr)\\\\\n&=\\sum_i a_i\\varphi_i+\\sum_j b_j\\psi_j+\\sum_{ij}\\pi_{ij}\\bigl(C_{ij}-\\varphi_i-\\psi_j\\bigr).\n\\end{aligned}",[31,8306,8308],{"className":8307,"ariaHidden":62},[61],[31,8309,8311,8315],{"className":8310},[66],[31,8312],{"className":8313,"style":8314},[70],"height:5.3276em;vertical-align:-2.4138em;",[31,8316,8318],{"className":8317},[75],[31,8319,8321,8394],{"className":8320},[5606],[31,8322,8324],{"className":8323},[5867],[31,8325,8327,8385],{"className":8326},[1043,1044],[31,8328,8330,8382],{"className":8329},[1048],[31,8331,8334,8373],{"className":8332,"style":8333},[1052],"height:2.9138em;",[31,8335,8337,8340],{"style":8336},"top:-4.9138em;",[31,8338],{"className":8339,"style":5884},[1060],[31,8341,8343,8346,8349,8352,8355,8358,8361,8364,8367,8370],{"className":8342},[75],[31,8344,4561],{"className":8345},[75,76],[31,8347,211],{"className":8348},[264],[31,8350,2515],{"className":8351,"style":2666},[75,76],[31,8353,1210],{"className":8354},[1391],[31,8356],{"className":8357,"style":1467},[119],[31,8359,8059],{"className":8360},[75,76],[31,8362,1210],{"className":8363},[1391],[31,8365],{"className":8366,"style":1467},[119],[31,8368,8064],{"className":8369,"style":2666},[75,76],[31,8371,220],{"className":8372},[289],[31,8374,8376,8379],{"style":8375},"top:-2.15em;",[31,8377],{"className":8378,"style":5884},[1060],[31,8380],{"className":8381},[75],[31,8383,1076],{"className":8384},[1075],[31,8386,8388],{"className":8387},[1048],[31,8389,8392],{"className":8390,"style":8391},[1052],"height:2.4138em;",[31,8393],{},[31,8395,8397],{"className":8396},[6062],[31,8398,8400,9602],{"className":8399},[1043,1044],[31,8401,8403,9599],{"className":8402},[1048],[31,8404,8406,9058],{"className":8405,"style":8333},[1052],[31,8407,8408,8411],{"style":8336},[31,8409],{"className":8410,"style":5884},[1060],[31,8412,8414,8417,8420,8423,8426,8473,8476,8519,8564,8567,8570,8573,8617,8620,8660,8666,8706,8709,8712,8715,8759,8762,8805,8811,8814,8817,8820,8864,8867,8907,8913,8953,8956,8959,8962,9006,9009,9052],{"className":8413},[75],[31,8415],{"className":8416},[75],[31,8418],{"className":8419,"style":293},[119],[31,8421,223],{"className":8422},[297],[31,8424],{"className":8425,"style":293},[119],[31,8427,8429],{"className":8428},[1460,1620],[31,8430,8432,8465],{"className":8431},[1043,1044],[31,8433,8435,8462],{"className":8434},[1048],[31,8436,8438,8452],{"className":8437,"style":5900},[1052],[31,8439,8440,8443],{"style":5903},[31,8441],{"className":8442,"style":5907},[1060],[31,8444,8446],{"className":8445},[1065,1066,1067,1068],[31,8447,8449],{"className":8448},[75,1068],[31,8450,5574],{"className":8451,"style":5514},[75,76,1068],[31,8453,8454,8457],{"style":5919},[31,8455],{"className":8456,"style":5907},[1060],[31,8458,8459],{},[31,8460,5625],{"className":8461},[1460,1461,1687],[31,8463,1076],{"className":8464},[1075],[31,8466,8468],{"className":8467},[1048],[31,8469,8471],{"className":8470,"style":5937},[1052],[31,8472],{},[31,8474],{"className":8475,"style":1467},[119],[31,8477,8479,8482],{"className":8478},[75],[31,8480,2515],{"className":8481,"style":2666},[75,76],[31,8483,8485],{"className":8484},[1039],[31,8486,8488,8511],{"className":8487},[1043,1044],[31,8489,8491,8508],{"className":8490},[1048],[31,8492,8494],{"className":8493,"style":5445},[1052],[31,8495,8496,8499],{"style":5504},[31,8497],{"className":8498,"style":1061},[1060],[31,8500,8502],{"className":8501},[1065,1066,1067,1068],[31,8503,8505],{"className":8504},[75,1068],[31,8506,5574],{"className":8507,"style":5514},[75,76,1068],[31,8509,1076],{"className":8510},[1075],[31,8512,8514],{"className":8513},[1048],[31,8515,8517],{"className":8516,"style":1083},[1052],[31,8518],{},[31,8520,8522,8526],{"className":8521},[75],[31,8523,8101],{"className":8524,"style":8525},[75,76],"margin-right:0.0715em;",[31,8527,8529],{"className":8528},[1039],[31,8530,8532,8556],{"className":8531},[1043,1044],[31,8533,8535,8553],{"className":8534},[1048],[31,8536,8538],{"className":8537,"style":5445},[1052],[31,8539,8541,8544],{"style":8540},"top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;",[31,8542],{"className":8543,"style":1061},[1060],[31,8545,8547],{"className":8546},[1065,1066,1067,1068],[31,8548,8550],{"className":8549},[75,1068],[31,8551,5574],{"className":8552,"style":5514},[75,76,1068],[31,8554,1076],{"className":8555},[1075],[31,8557,8559],{"className":8558},[1048],[31,8560,8562],{"className":8561,"style":1083},[1052],[31,8563],{},[31,8565],{"className":8566,"style":120},[119],[31,8568,434],{"className":8569},[124],[31,8571],{"className":8572,"style":120},[119],[31,8574,8576],{"className":8575},[1460,1620],[31,8577,8579,8609],{"className":8578},[1043,1044],[31,8580,8582,8606],{"className":8581},[1048],[31,8583,8585,8596],{"className":8584,"style":5900},[1052],[31,8586,8587,8590],{"style":5903},[31,8588],{"className":8589,"style":5907},[1060],[31,8591,8593],{"className":8592},[1065,1066,1067,1068],[31,8594,5391],{"className":8595},[75,76,1068],[31,8597,8598,8601],{"style":5919},[31,8599],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",[31,9719,9721,9764],{"className":9720,"translate":35},[34],[31,9722,9724],{"className":9723},[39],[41,9725,9726],{"xmlns":43},[45,9727,9728,9761],{},[48,9729,9730,9740,9742,9748,9750,9756,9759],{},[1004,9731,9732,9734],{},[51,9733,8101],{},[48,9735,9736,9738],{},[51,9737,5391],{},[51,9739,2555],{},[95,9741,97],{},[1004,9743,9744,9746],{},[51,9745,8059],{},[51,9747,5391],{},[95,9749,97],{},[1004,9751,9752,9754],{},[51,9753,8064],{},[51,9755,2555],{},[95,9757,9758],{},"\u003C",[91,9760,1977],{},[55,9762,9763],{"encoding":57},"C_{ij}-\\varphi_i-\\psi_j\u003C0",[31,9765,9767,9825,9881,9936],{"className":9766,"ariaHidden":62},[61],[31,9768,9770,9773,9816,9819,9822],{"className":9769},[66],[31,9771],{"className":9772,"style":1029},[70],[31,9774,9776,9779],{"className":9775},[75],[31,9777,8101],{"className":9778,"style":8525},[75,76],[31,9780,9782],{"className":9781},[1039],[31,9783,9785,9808],{"className":9784},[1043,1044],[31,9786,9788,9805],{"className":9787},[1048],[31,9789,9791],{"className":9790,"style":5445},[1052],[31,9792,9793,9796],{"style":8540},[31,9794],{"className":9795,"style":1061},[1060],[31,9797,9799],{"className":9798},[1065,1066,1067,1068],[31,9800,9802],{"className":9801},[75,1068],[31,9803,5574],{"className":9804,"style":5514},[75,76,1068],[31,9806,1076],{"className":9807},[1075],[31,9809,9811],{"className":9810},[1048],[31,9812,9814],{"className":9813,"style":1083},[1052],[31,9815],{},[31,9817],{"className":9818,"style":120},[119],[31,9820,97],{"className":9821},[124],[31,9823],{"className":9824,"style":120},[119],[31,9826,9828,9832,9872,9875,9878],{"className":9827},[66],[31,9829],{"className":9830,"style":9831},[70],"height:0.7778em;vertical-align:-0.1944em;",[31,9833,9835,9838],{"className":9834},[75],[31,9836,8059],{"className":9837},[75,76],[31,9839,9841],{"className":9840},[1039],[31,9842,9844,9864],{"className":9843},[1043,1044],[31,9845,9847,9861],{"className":9846},[1048],[31,9848,9850],{"className":9849,"style":5445},[1052],[31,9851,9852,9855],{"style":2807},[31,9853],{"className":9854,"style":1061},[1060],[31,9856,9858],{"className":9857},[1065,1066,1067,1068],[31,9859,5391],{"className":9860},[75,76,1068],[31,9862,1076],{"className":9863},[1075],[31,9865,9867],{"className":9866},[1048],[31,9868,9870],{"className":9869,"style":2050},[1052],[31,9871],{},[31,9873],{"className":9874,"style":120},[119],[31,9876,97],{"className":9877},[124],[31,9879],{"className":9880,"style":120},[119],[31,9882,9884,9887,9927,9930,9933],{"className":9883},[66],[31,9885],{"className":9886,"style":4270},[70],[31,9888,9890,9893],{"className":9889},[75],[31,9891,8064],{"className":9892,"style":2666},[75,76],[31,9894,9896],{"className":9895},[1039],[31,9897,9899,9919],{"className":9898},[1043,1044],[31,9900,9902,9916],{"className":9901},[1048],[31,9903,9905],{"className":9904,"style":5445},[1052],[31,9906,9907,9910],{"style":5504},[31,9908],{"className":9909,"style":1061},[1060],[31,9911,9913],{"className":9912},[1065,1066,1067,1068],[31,9914,2555],{"className":9915,"style":5514},[75,76,1068],[31,9917,1076],{"className":9918},[1075],[31,9920,9922],{"className":9921},[1048],[31,9923,9925],{"className":9924,"style":1083},[1052],[31,9926],{},[31,9928],{"className":9929,"style":293},[119],[31,9931,9758],{"className":9932},[297],[31,9934],{"className":9935,"style":293},[119],[31,9937,9939,9942],{"className":9938},[66],[31,9940],{"className":9941,"style":326},[70],[31,9943,1977],{"className":9944},[75],"，令 ",[31,9947,9949,9975],{"className":9948,"translate":35},[34],[31,9950,9952],{"className":9951},[39],[41,9953,9954],{"xmlns":43},[45,9955,9956,9972],{},[48,9957,9958,9968,9970],{},[1004,9959,9960,9962],{},[51,9961,2515],{},[48,9963,9964,9966],{},[51,9965,5391],{},[51,9967,2555],{},[95,9969,929],{},[51,9971,4587],{"mathvariant":494},[55,9973,9974],{"encoding":57},"\\pi_{ij}\\to\\infty",[31,9976,9978,10036],{"className":9977,"ariaHidden":62},[61],[31,9979,9981,9984,10027,10030,10033],{"className":9980},[66],[31,9982],{"className":9983,"style":5541},[70],[31,9985,9987,9990],{"className":9986},[75],[31,9988,2515],{"className":9989,"style":2666},[75,76],[31,9991,9993],{"className":9992},[1039],[31,9994,9996,10019],{"className":9995},[1043,1044],[31,9997,9999,10016],{"className":9998},[1048],[31,10000,10002],{"className":10001,"style":5445},[1052],[31,10003,10004,10007],{"style":5504},[31,10005],{"className":10006,"style":1061},[1060],[31,10008,10010],{"className":10009},[1065,1066,1067,1068],[31,10011,10013],{"className":10012},[75,1068],[31,10014,5574],{"className":10015,"style":5514},[75,76,1068],[31,10017,1076],{"className":10018},[1075],[31,10020,10022],{"className":10021},[1048],[31,10023,10025],{"className":10024,"style":1083},[1052],[31,10026],{},[31,10028],{"className":10029,"style":293},[119],[31,10031,929],{"className":10032},[297],[31,10034],{"className":10035,"style":293},[119],[31,10037,10039,10042],{"className":10038},[66],[31,10040],{"className":10041,"style":71},[70],[31,10043,4587],{"className":10044},[75]," 就得到 ",[31,10047,10049,10065],{"className":10048,"translate":35},[34],[31,10050,10052],{"className":10051},[39],[41,10053,10054],{"xmlns":43},[45,10055,10056,10062],{},[48,10057,10058,10060],{},[95,10059,97],{},[51,10061,4587],{"mathvariant":494},[55,10063,10064],{"encoding":57},"-\\infty",[31,10066,10068],{"className":10067,"ariaHidden":62},[61],[31,10069,10071,10074,10077],{"className":10070},[66],[31,10072],{"className":10073,"style":307},[70],[31,10075,97],{"className":10076},[75],[31,10078,4587],{"className":10079},[75],"；否则下确界在 ",[31,10082,10084,10102],{"className":10083,"translate":35},[34],[31,10085,10087],{"className":10086},[39],[41,10088,10089],{"xmlns":43},[45,10090,10091,10099],{},[48,10092,10093,10095,10097],{},[51,10094,2515],{},[95,10096,223],{},[91,10098,1977],{},[55,10100,10101],{"encoding":57},"\\pi=0",[31,10103,10105,10123],{"className":10104,"ariaHidden":62},[61],[31,10106,10108,10111,10114,10117,10120],{"className":10107},[66],[31,10109],{"className":10110,"style":71},[70],[31,10112,2515],{"className":10113,"style":2666},[75,76],[31,10115],{"className":10116,"style":293},[119],[31,10118,223],{"className":10119},[297],[31,10121],{"className":10122,"style":293},[119],[31,10124,10126,10129],{"className":10125},[66],[31,10127],{"className":10128,"style":326},[70],[31,10130,1977],{"className":10131},[75]," 处取到。于是对偶问题就是",[31,10134,10136],{"className":10135,"translate":35},[411],[31,10137,10139,10242],{"className":10138,"translate":35},[34],[31,10140,10142],{"className":10141},[39],[41,10143,10144],{"xmlns":43,"display":420},[45,10145,10146,10239],{},[48,10147,10148,10165,10167,10173,10179,10185,10187,10193,10199,10205,10207,10209,10211,10217,10219,10225,10227,10237],{},[1531,10149,10150,10157],{},[48,10151,10152,10155],{},[51,10153,10154],{},"max",[95,10156,1540],{},[48,10158,10159,10161,10163],{},[51,10160,8059],{},[95,10162,1210],{"separator":62},[51,10164,8064],{},[1430,10166,1545],{},[1531,10168,10169,10171],{},[95,10170,5625],{},[51,10172,5391],{},[1004,10174,10175,10177],{},[51,10176,5813],{},[51,10178,5391],{},[1004,10180,10181,10183],{},[51,10182,8059],{},[51,10184,5391],{},[95,10186,434],{},[1531,10188,10189,10191],{},[95,10190,5625],{},[51,10192,2555],{},[1004,10194,10195,10197],{},[51,10196,5834],{},[51,10198,2555],{},[1004,10200,10201,10203],{},[51,10202,8064],{},[51,10204,2555],{},[119,10206],{"width":1584},[1430,10208,1587],{},[119,10210],{"width":1584},[1004,10212,10213,10215],{},[51,10214,8059],{},[51,10216,5391],{},[95,10218,434],{},[1004,10220,10221,10223],{},[51,10222,8064],{},[51,10224,2555],{},[95,10226,347],{},[1004,10228,10229,10231],{},[51,10230,8101],{},[48,10232,10233,10235],{},[51,10234,5391],{},[51,10236,2555],{},[95,10238,1210],{"separator":62},[55,10240,10241],{"encoding":57},"\\max_{\\varphi,\\psi}\\ \\sum_i a_i\\varphi_i+\\sum_j b_j\\psi_j\n\\quad\\text{s.t.}\\quad \\varphi_i+\\psi_j\\le C_{ij},",[31,10243,10245,10449,10644,10699],{"className":10244,"ariaHidden":62},[61],[31,10246,10248,10252,10307,10310,10313,10357,10360,10400,10440,10443,10446],{"className":10247},[66],[31,10249],{"className":10250,"style":10251},[70],"height:2.3277em;vertical-align:-1.2777em;",[31,10253,10255],{"className":10254},[1460,1620],[31,10256,10258,10298],{"className":10257},[1043,1044],[31,10259,10261,10295],{"className":10260},[1048],[31,10262,10264,10285],{"className":10263,"style":71},[1052],[31,10265,10267,10270],{"style":10266},"top:-2.3479em;margin-left:0em;",[31,10268],{"className":10269,"style":1637},[1060],[31,10271,10273],{"className":10272},[1065,1066,1067,1068],[31,10274,10276,10279,10282],{"className":10275},[75,1068],[31,10277,8059],{"className":10278},[75,76,1068],[31,10280,1210],{"className":10281},[1391,1068],[31,10283,8064],{"className":10284,"style":2666},[75,76,1068],[31,10286,10287,10290],{"style":1649},[31,10288],{"className":10289,"style":1637},[1060],[31,10291,10292],{},[31,10293,10154],{"className":10294},[1460],[31,10296,1076],{"className":10297},[1075],[31,10299,10301],{"className":10300},[1048],[31,10302,10305],{"className":10303,"style":10304},[1052],"height:0.8882em;",[31,10306],{},[31,10308,1545],{"className":10309},[119],[31,10311],{"className":10312,"style":1467},[119],[31,10314,10316],{"className":10315},[1460,1620],[31,10317,10319,10349],{"className":10318},[1043,1044],[31,10320,10322,10346],{"className":10321},[1048],[31,10323,10325,10336],{"className":10324,"style":5900},[1052],[31,10326,10327,10330],{"style":5903},[31,10328],{"className":10329,"style":5907},[1060],[31,10331,10333],{"className":10332},[1065,1066,1067,1068],[31,10334,5391],{"className":10335},[75,76,1068],[31,10337,10338,10341],{"style":5919},[31,10339],{"className":10340,"style":5907},[1060],[31,10342,10343],{},[31,10344,5625],{"className":10345},[1460,1461,1687],[31,10347,1076],{"className":10348},[1075],[31,10350,10352],{"className":10351},[1048],[31,10353,10355],{"className":10354,"style":6342},[1052],[31,10356],{},[31,10358],{"className":10359,"style":1467},[119],[31,10361,10363,10366],{"className":10362},[75],[31,10364,5813],{"className":10365},[75,76],[31,10367,10369],{"className":10368},[1039],[31,10370,10372,10392],{"className":10371},[1043,1044],[31,10373,10375,10389],{"className":10374},[1048],[31,10376,10378],{"className":10377,"style":5445},[1052],[31,10379,10380,10383],{"style":2807},[31,10381],{"className":10382,"style":1061},[1060],[31,10384,10386],{"className":10385},[1065,1066,1067,1068],[31,10387,5391],{"className":10388},[75,76,1068],[31,10390,1076],{"className":10391},[1075],[31,10393,10395],{"className":10394},[1048],[31,10396,10398],{"className":10397,"style":2050},[1052],[31,10399],{},[31,10401,10403,10406],{"className":10402},[75],[31,10404,8059],{"className":10405},[75,76],[31,10407,10409],{"className":10408},[1039],[31,10410,10412,10432],{"className":10411},[1043,1044],[31,10413,10415,10429],{"className":10414},[1048],[31,10416,10418],{"className":10417,"style":5445},[1052],[31,10419,10420,10423],{"style":2807},[31,10421],{"className":10422,"style":1061},[1060],[31,10424,10426],{"className":10425},[1065,1066,1067,1068],[31,10427,5391],{"className":10428},[75,76,1068],[31,10430,1076],{"className":10431},[1075],[31,10433,10435],{"className":10434},[1048],[31,10436,10438],{"className":10437,"style":2050},[1052],[31,10439],{},[31,10441],{"className":10442,"style":120},[119],[31,10444,434],{"className":10445},[124],[31,10447],{"className":10448,"style":120},[119],[31,10450,10452,10456,10500,10503,10543,10583,10586,10592,10595,10635,10638,10641],{"className":10451},[66],[31,10453],{"className":10454,"style":10455},[70],"height:2.4638em;vertical-align:-1.4138em;",[31,10457,10459],{"className":10458},[1460,1620],[31,10460,10462,10492],{"className":10461},[1043,1044],[31,10463,10465,10489],{"className":10464},[1048],[31,10466,10468,10479],{"className":10467,"style":5900},[1052],[31,10469,10470,10473],{"style":5903},[31,10471],{"className":10472,"style":5907},[1060],[31,10474,10476],{"className":10475},[1065,1066,1067,1068],[31,10477,2555],{"className":10478,"style":5514},[75,76,1068],[31,10480,10481,10484],{"style":5919},[31,10482],{"className":10483,"style":5907},[1060],[31,10485,10486],{},[31,10487,5625],{"className":10488},[1460,1461,1687],[31,10490,1076],{"className":10491},[1075],[31,10493,10495],{"className":10494},[1048],[31,10496,10498],{"className":10497,"style":5937},[1052],[31,10499],{},[31,10501],{"className":10502,"style":1467},[119],[31,10504,10506,10509],{"className":10505},[75],[31,10507,5834],{"className":10508},[75,76],[31,10510,10512],{"className":10511},[1039],[31,10513,10515,10535],{"className":10514},[1043,1044],[31,10516,10518,10532],{"className":10517},[1048],[31,10519,10521],{"className":10520,"style":5445},[1052],[31,10522,10523,10526],{"style":2807},[31,10524],{"className":10525,"style":1061},[1060],[31,10527,10529],{"className":10528},[1065,1066,1067,1068],[31,105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:\\ \\varphi(x)+\\psi(y)\\le c(x,y)\\Bigr\\}.",[31,10865,10867,10940,10983,11022,11049,11076],{"className":10866,"ariaHidden":62},[61],[31,10868,10870,10873,10913,10916,10919,10922,10925,10928,10931,10934,10937],{"className":10869},[66],[31,10871],{"className":10872,"style":260},[70],[31,10874,10876,10879],{"className":10875},[75],[31,10877,3117],{"className":10878,"style":949},[75,76],[31,10880,10882],{"className":10881},[1039],[31,10883,10885,10905],{"className":10884},[1043,1044],[31,10886,10888,10902],{"className":10887},[1048],[31,10889,10891],{"className":10890,"style":3230},[1052],[31,10892,10893,10896],{"style":1056},[31,10894],{"className":10895,"style":1061},[1060],[31,10897,10899],{"className":10898},[1065,1066,1067,1068],[31,10900,1412],{"className":10901},[75,76,1068],[31,10903,1076],{"className":10904},[1075],[31,10906,10908],{"className":10907},[1048],[31,10909,10911],{"className":10910,"style":2050},[1052],[31,10912],{},[31,10914,211],{"className":10915},[264],[31,10917,852],{"className":10918},[75,76],[31,10920,1210],{"className":10921},[1391],[31,10923],{"className":10924,"style":1467},[119],[31,10926,884],{"className":10927,"style":900},[75,76],[31,10929,220],{"className":10930},[289],[31,10932],{"className":10933,"style":293},[119],[31,10935,223],{"className":10936},[297],[31,10938],{"className":10939,"style":293},[119],[31,10941,10943,10947,10950,10956,10959,10962,10965,10968,10971,10974,10977,10980],{"className":10942},[66],[31,10944],{"className":10945,"style":10946},[70],"height:2.2222em;vertical-align:-0.8622em;",[31,10948,10792],{"className":10949},[1460],[31,10951,10953],{"className":10952},[264],[31,10954,2512],{"className":10955},[1307,2662],[31,10957,1409],{"className":10958,"style":1688},[1460,1461,1687],[31,10960],{"className":10961,"style":1467},[119],[31,10963,8059],{"className":10964},[75,76],[31,10966],{"className":10967,"style":1467},[119],[31,10969,1435],{"className":10970},[75,76],[31,10972,852],{"className":10973},[75,76],[31,10975],{"className":10976,"style":120},[119],[31,10978,434],{"className":10979},[124],[31,10981],{"className":10982,"style":120},[119],[31,10984,10986,10989,10992,10995,10998,11001,11004,11007,11010,11013,11016,11019],{"className":10985},[66],[31,10987],{"className":10988,"style":10946},[70],[31,10990,1409],{"className":10991,"style":1688},[1460,1461,1687],[31,10993],{"className":10994,"style":1467},[119],[31,10996,8064],{"className":10997,"style":2666},[75,76],[31,10999],{"className":11000,"style":1467},[119],[31,11002,1435],{"className":11003},[75,76],[31,11005,884],{"className":11006,"style":900},[75,76],[31,11008,1545],{"className":11009},[119],[31,11011],{"className":11012,"style":293},[119],[31,11014,922],{"className":11015},[297],[31,11017,1545],{"className":11018},[119],[31,11020],{"className":11021,"style":293},[119],[31,11023,11025,11028,11031,11034,11037,11040,11043,11046],{"className":11024},[66],[31,11026],{"className":11027,"style":260},[70],[31,11029,8059],{"className":11030},[75,76],[31,11032,211],{"className":11033},[264],[31,11035,53],{"className":11036},[75,76],[31,11038,220],{"className":11039},[289],[31,11041],{"className":11042,"style":120},[119],[31,11044,434],{"className":11045},[124],[31,11047],{"className":11048,"style":120},[119],[31,11050,11052,11055,11058,11061,11064,11067,11070,11073],{"className":11051},[66],[31,11053],{"className":11054,"style":260},[70],[31,11056,8064],{"className":11057,"style":2666},[75,76],[31,11059,211],{"className":11060},[264],[31,11062,3003],{"className":11063,"style":2666},[75,76],[31,11065,220],{"className":11066},[289],[31,11068],{"className":11069,"style":293},[119],[31,11071,347],{"className":11072},[297],[31,11074],{"className":11075,"style":293},[119],[31,11077,11079,11082,11085,11088,11091,11094,11097,11100,11103,11109],{"className":11078},[66],[31,11080],{"className":11081,"style":2655},[70],[31,11083,1412],{"className":11084},[75,76],[31,11086,211],{"className":11087},[264],[31,11089,53],{"className":11090},[75,76],[31,11092,1210],{"className":11093},[1391],[31,11095],{"className":11096,"style":1467},[119],[31,11098,3003],{"className":11099,"style":2666},[75,76],[31,11101,220],{"className":11102},[289],[31,11104,11106],{"className":11105},[289],[31,11107,2604],{"className":11108},[1307,2662],[31,11110,495],{"className":11111},[75],[11,11113,3501,11114,11165,11166,11297,11298,11354,11355,11486,11487,4880,11516,11544],{},[31,11115,11117,11135],{"className":11116,"translate":35},[34],[31,11118,11120],{"className":11119},[39],[41,11121,11122],{"xmlns":43},[45,11123,11124,11132],{},[48,11125,11126,11128,11130],{},[51,11127,1412],{},[95,11129,223],{},[51,11131,1435],{},[55,11133,11134],{"encoding":57},"c=d",[31,11136,11138,11156],{"className":11137,"ariaHidden":62},[61],[31,11139,11141,11144,11147,11150,11153],{"className":11140},[66],[31,11142],{"className":11143,"style":71},[70],[31,11145,1412],{"className":11146},[75,76],[31,11148],{"className":11149,"style":293},[119],[31,11151,223],{"className":11152},[297],[3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",[31,11167,11169,11213],{"className":11168,"translate":35},[34],[31,11170,11172],{"className":11171},[39],[41,11173,11174],{"xmlns":43},[45,11175,11176,11210],{},[48,11177,11178,11180,11182,11184,11186,11188,11190,11192,11194,11196,11198,11200,11202,11204,11206,11208],{},[51,11179,8059],{},[95,11181,211],{"stretchy":210},[51,11183,53],{},[95,11185,220],{"stretchy":210},[95,11187,434],{},[51,11189,8064],{},[95,11191,211],{"stretchy":210},[51,11193,3003],{},[95,11195,220],{"stretchy":210},[95,11197,347],{},[51,11199,1435],{},[95,11201,211],{"stretchy":210},[51,11203,53],{},[95,11205,1210],{"separator":62},[51,11207,3003],{},[95,11209,220],{"stretchy":210},[55,11211,11212],{"encoding":57},"\\varphi(x)+\\psi(y)\\le d(x,y)",[31,11214,11216,11243,11270],{"className":11215,"ariaHidden":62},[61],[31,11217,11219,11222,11225,11228,11231,11234,11237,11240],{"className":11218},[66],[31,11220],{"className":11221,"style":260},[70],[31,11223,8059],{"className":11224},[75,76],[31,11226,211],{"className":11227},[264],[31,11229,53],{"className":11230},[75,76],[31,11232,220],{"className":11233},[289],[31,11235],{"className":11236,"style":120},[119],[31,11238,434],{"className":11239},[124],[31,11241],{"className":11242,"style":120},[119],[31,11244,11246,11249,11252,11255,11258,11261,11264,11267],{"className":11245},[66],[31,11247],{"className":11248,"style":260},[70],[31,11250,8064],{"className":11251,"style":2666},[75,76],[31,11253,211],{"className":11254},[264],[31,11256,3003],{"className":11257,"style":2666},[75,76],[31,11259,220],{"className":11260},[289],[31,11262],{"className":11263,"style":293},[119],[31,11265,347],{"className":11266},[297],[31,11268],{"className":11269,"style":293},[119],[31,11271,11273,11276,11279,11282,11285,11288,11291,11294],{"className":11272},[66],[31,11274],{"className":11275,"style":260},[70],[31,11277,1435],{"className":11278},[75,76],[31,11280,211],{"className":11281},[264],[31,11283,53],{"className":11284},[75,76],[31,11286,1210],{"className":11287},[1391],[31,11289],{"className":11290,"style":1467},[119],[31,11292,3003],{"className":11293,"style":2666},[75,76],[31,11295,220],{"className":11296},[289]," 允许我取 ",[31,11299,11301,11321],{"className":11300,"translate":35},[34],[31,11302,11304],{"className":11303},[39],[41,11305,11306],{"xmlns":43},[45,11307,11308,11318],{},[48,11309,11310,11312,11314,11316],{},[51,11311,8064],{},[95,11313,223],{},[95,11315,97],{},[51,11317,8059],{},[55,11319,11320],{"encoding":57},"\\psi=-\\varphi",[31,11322,11324,11342],{"className":11323,"ariaHidden":62},[61],[31,11325,11327,11330,11333,11336,11339],{"className":11326},[66],[31,11328],{"className":11329,"style":4202},[70],[31,11331,8064],{"className":11332,"style":2666},[75,76],[31,11334],{"className":11335,"style":293},[119],[31,11337,223],{"className":11338},[297],[31,11340],{"className":11341,"style":293},[119],[31,11343,11345,11348,11351],{"className":11344},[66],[31,11346],{"className":11347,"style":9831},[70],[31,11349,97],{"className":11350},[75],[31,11352,8059],{"className":11353},[75,76],"，此时它变成 ",[31,11356,11358,11402],{"className":11357,"translate":35},[34],[31,11359,11361],{"className":11360},[39],[41,11362,11363],{"xmlns":43},[45,11364,11365,11399],{},[48,11366,11367,11369,11371,11373,11375,11377,11379,11381,11383,11385,11387,11389,11391,11393,11395,11397],{},[51,11368,8059],{},[95,11370,211],{"stretchy":210},[51,11372,53],{},[95,11374,220],{"stretchy":210},[95,11376,97],{},[51,11378,8059],{},[95,11380,211],{"stretchy":210},[51,11382,3003],{},[95,11384,220],{"stretchy":210},[95,11386,347],{},[51,11388,1435],{},[95,11390,211],{"stretchy":210},[51,11392,53],{},[95,11394,1210],{"separator":62},[51,11396,3003],{},[95,11398,220],{"stretchy":210},[55,11400,11401],{"encoding":57},"\\varphi(x)-\\varphi(y)\\le d(x,y)",[31,11403,11405,11432,11459],{"className":11404,"ariaHidden":62},[61],[31,11406,11408,11411,11414,11417,11420,11423,11426,11429],{"className":11407},[66],[31,11409],{"className":11410,"style":260},[70],[31,11412,8059],{"className":11413},[75,76],[31,11415,211],{"className":11416},[264],[31,11418,53],{"className":11419},[75,76],[31,11421,220],{"className":11422},[289],[31,11424],{"className":11425,"style":120},[119],[31,11427,97],{"className":11428},[124],[31,11430],{"className":11431,"style":120},[119],[31,11433,11435,11438,11441,11444,11447,11450,11453,11456],{"className":11434},[66],[31,11436],{"className":11437,"style":260},[70],[31,11439,8059],{"className":11440},[75,76],[31,11442,211],{"className":11443},[264],[31,11445,3003],{"className":11446,"style":2666},[75,76],[31,11448,220],{"className":11449},[289],[31,11451],{"className":11452,"style":293},[119],[31,11454,347],{"className":11455},[297],[31,11457],{"className":11458,"style":293},[119],[31,11460,11462,11465,11468,11471,11474,11477,11480,11483],{"className":11461},[66],[31,11463],{"className":11464,"style":260},[70],[31,11466,1435],{"className":11467},[75,76],[31,11469,211],{"className":11470},[264],[31,11472,53],{"className":11473},[75,76],[31,11475,1210],{"className":11476},[1391],[31,11478],{"className":11479,"style":1467},[119],[31,11481,3003],{"className":11482,"style":2666},[75,76],[31,11484,220],{"className":11485},[289],"，也就是 ",[31,11488,11490,11504],{"className":11489,"translate":35},[34],[31,11491,11493],{"className":11492},[39],[41,11494,11495],{"xmlns":43},[45,11496,11497,11501],{},[48,11498,11499],{},[51,11500,8059],{},[55,11502,11503],{"encoding":57},"\\varphi",[31,11505,11507],{"className":11506,"ariaHidden":62},[61],[31,11508,11510,11513],{"className":11509},[66],[31,11511],{"className":11512,"style":865},[70],[31,11514,8059],{"className":11515},[75,76],[31,11517,11519,11532],{"className":11518,"translate":35},[34],[31,11520,11522],{"className":11521},[39],[41,11523,11524],{"xmlns":43},[45,11525,11526,11530],{},[48,11527,11528],{},[91,11529,205],{},[55,11531,205],{"encoding":57},[31,11533,11535],{"className":11534,"ariaHidden":62},[61],[31,11536,11538,11541],{"className":11537},[66],[31,11539],{"className":11540,"style":326},[70],[31,11542,205],{"className":11543},[75],"-Lipschitz 的。所以",[31,11546,11548],{"className":11547,"translate":35},[411],[31,11549,11551,11631],{"className":11550,"translate":35},[34],[31,11552,11554],{"className":11553},[39],[41,11555,11556],{"xmlns":43,"display":420},[45,11557,11558,11628],{},[48,11559,11560,11566,11568,11570,11572,11574,11576,11578,11608,11610,11612,11614,11616,11618,11620,11622,11624,11626],{},[1004,11561,11562,11564],{},[51,11563,3117],{},[91,11565,205],{},[95,11567,211],{"stretchy":210},[51,11569,852],{},[95,11571,1210],{"separator":62},[51,11573,884],{},[95,11575,220],{"stretchy":210},[95,11577,223],{},[1531,11579,11580,11586],{},[48,11581,11582,11584],{},[51,11583,10792],{},[95,11585,1540],{},[48,11587,11588,11590,11592,11604,11606],{},[51,11589,4572],{"mathvariant":494},[51,11591,8059],{},[1004,11593,11594,11596],{},[51,11595,4572],{"mathvariant":494},[48,11597,11598,11600,11602],{},[51,11599,4561],{"mathvariant":494},[51,11601,5391],{"mathvariant":494},[51,11603,11],{"mathvariant":494},[95,11605,347],{},[91,11607,205],{},[95,11609,1409],{},[51,11611,8059],{},[1430,11613,1432],{},[51,11615,1435],{},[95,11617,211],{"stretchy":210},[51,11619,852],{},[95,11621,97],{},[51,11623,884],{},[95,11625,220],{"stretchy":210},[51,11627,495],{"mathvariant":494},[55,11629,11630],{"encoding":57},"W_1(\\mu,\\nu)=\\sup_{\\|\\varphi\\|_{\\mathrm{Lip}}\\le 1}\\int\\varphi\\,d(\\mu-\\nu).",[31,11632,11634,11707,11857],{"className":11633,"ariaHidden":62},[61],[31,11635,11637,11640,11680,11683,11686,11689,11692,11695,11698,11701,11704],{"className":11636},[66],[31,11638],{"className":11639,"style":260},[70],[31,11641,11643,11646],{"className":11642},[75],[31,11644,3117],{"className":11645,"style":949},[75,76],[31,11647,11649],{"className":11648},[1039],[31,11650,11652,11672],{"className":11651},[1043,1044],[31,11653,11655,11669],{"className":11654},[1048],[31,11656,11658],{"className":11657,"style":2028},[1052],[31,11659,11660,11663],{"style":1056},[31,11661],{"className":11662,"style":1061},[1060],[31,11664,11666],{"className":11665},[1065,1066,1067,1068],[31,11667,205],{"className":11668},[75,1068],[31,11670,1076],{"className":11671},[1075],[31,11673,11675],{"className":11674},[1048],[31,11676,11678],{"className":11677,"style":2050},[1052],[31,11679],{},[31,11681,211],{"className":11682},[264],[31,11684,852],{"className":11685},[75,76],[31,11687,1210],{"className":11688},[1391],[31,11690],{"className":11691,"style":1467},[119],[31,11693,884],{"className":11694,"style":900},[75,76],[31,11696,220],{"className":11697},[289],[31,11699],{"className":11700,"style":293},[119],[31,11702,223],{"className":11703},[297],[31,11705],{"className":11706,"style":293},[119],[31,11708,11710,11714,11824,11827,11830,11833,11836,11839,11842,11845,11848,11851,11854],{"className":11709},[66],[31,11711],{"className":11712,"style":11713},[70],"height:2.543em;vertical-align:-1.183em;",[31,11715,11717],{"className":11716},[1460,1620],[31,11718,11720,11815],{"className":11719},[1043,1044],[31,11721,11723,11812],{"className":11722},[1048],[31,11724,11726,11802],{"className":11725,"style":71},[1052],[31,11727,11729,11732],{"style":11728},"top:-2.1146em;margin-left:0em;",[31,11730],{"className":11731,"style":1637},[1060],[31,11733,11735],{"className":11734},[1065,1066,1067,1068],[31,11736,11738,11741,11744,11796,11799],{"className":11737},[75,1068],[31,11739,4572],{"className":11740},[75,1068],[31,11742,8059],{"className":11743},[75,76,1068],[31,11745,11747,11750],{"className":11746},[75,1068],[31,11748,4572],{"className":11749},[75,1068],[31,11751,11753],{"className":11752},[1039],[31,11754,11756,11787],{"className":11755},[1043,1044],[31,11757,11759,11784],{"className":11758},[1048],[31,11760,11763],{"className":11761,"style":11762},[1052],"height:0.3448em;",[31,11764,11766,11770],{"style":11765},"top:-2.3567em;margin-left:0em;margin-right:0.0714em;",[31,11767],{"className":11768,"style":11769},[1060],"height:2.5em;",[31,11771,11774],{"className":11772},[1065,11773,1308,1068],"reset-size3",[31,11775,11777],{"className":11776},[75,1068],[31,11778,11780],{"className":11779},[75,1068],[31,11781,11783],{"className":11782},[75,2745,1068],"Lip",[31,11785,1076],{"className":11786},[1075],[31,11788,11790],{"className":11789},[1048],[31,11791,11794],{"className":11792,"style":11793},[1052],"height:0.2822em;",[31,11795],{},[31,11797,347],{"className":11798},[297,1068],[31,11800,205],{"className":11801},[75,1068],[31,11803,11804,11807],{"style":1649},[31,11805],{"className":11806,"style":1637},[1060],[31,11808,11809],{},[31,11810,10792],{"className":11811},[1460],[31,11813,1076],{"className":11814},[1075],[31,11816,11818],{"className":11817},[1048],[31,11819,11822],{"className":11820,"style":11821},[1052],"height:1.183em;",[31,11823],{},[31,11825],{"className":11826,"style":1467},[119],[31,11828,1409],{"className":11829,"style":1688},[1460,1461,1687],[31,11831],{"className":11832,"style":1467},[119],[31,11834,8059],{"className":11835},[75,76],[31,11837],{"className":11838,"style":1467},[119],[31,11840,1435],{"className":11841},[75,76],[31,11843,211],{"className":11844},[264],[31,11846,852],{"className":11847},[75,76],[31,11849],{"className":11850,"style":120},[119],[31,11852,97],{"className":11853},[124],[31,11855],{"className":11856,"style":120},[119],[31,11858,11860,11863,11866,11869],{"className":11859},[66],[31,11861],{"className":11862,"style":260},[70],[31,11864,884],{"className":11865,"style":900},[75,76],[31,11867,220],{"className":11868},[289],[31,11870,495],{"className":11871},[75],[11,11873,11874],{},"这个式子的分量很重：左边是关于分布的优化问题，右边是关于函数的优化问题，而函数可以用神经网络参数化。WGAN 干的就是这件事，只差把 Lipschitz 约束用 weight clipping 或 gradient penalty 近似实现。",[11,11876,11877,11878,12068,12069,12097,12098,12126,12127,12184,12185,12213,12214,2372],{},"二次代价 ",[31,11879,11881,11919],{"className":11880,"translate":35},[34],[31,11882,11884],{"className":11883},[39],[41,11885,11886],{"xmlns":43},[45,11887,11888,11916],{},[48,11889,11890,11892,11894,11902,11904,11906,11908,11910],{},[51,11891,1412],{},[95,11893,223],{},[2071,11895,11896],{"scriptlevel":1977,"displaystyle":210},[2074,11897,11898,11900],{},[91,11899,205],{},[91,11901,152],{},[51,11903,4572],{"mathvariant":494},[51,11905,53],{},[95,11907,97],{},[51,11909,3003],{},[1180,11911,11912,11914],{},[51,11913,4572],{"mathvariant":494},[91,11915,152],{},[55,11917,11918],{"encoding":57},"c=\\tfrac12\\|x-y\\|^2",[31,11920,11922,11940,12029],{"className":11921,"ariaHidden":62},[61],[31,11923,11925,11928,11931,11934,11937],{"className":11924},[66],[31,11926],{"className":11927,"style":71},[70],[31,11929,1412],{"className":11930},[75,76],[31,11932],{"className":11933,"style":293},[119],[31,11935,223],{"className":11936},[297],[31,11938],{"className":11939,"style":293},[119],[31,11941,11943,11946,12014,12017,12020,12023,12026],{"className":11942},[66],[31,11944],{"className":11945,"style":2132},[70],[31,11947,11949,11952,12011],{"className":11948},[75],[31,11950],{"className":11951},[264,2139],[31,11953,11955],{"className":11954},[2074],[31,11956,11958,12003],{"className":11957},[1043,1044],[31,11959,11961,12000],{"className":11960},[1048],[31,11962,11964,11978,11986],{"className":11963,"style":2152},[1052],[31,11965,11966,11969],{"style":2155},[31,11967],{"className":11968,"style":1637},[1060],[31,11970,11972],{"className":11971},[1065,1066,1067,1068],[31,11973,11975],{"className":11974},[75,1068],[31,11976,152],{"className":11977},[75,1068],[31,11979,11980,11983],{"style":2170},[31,11981],{"className":11982,"style":1637},[1060],[31,11984],{"className":11985,"style":2178},[2177],[31,11987,11988,11991],{"style":2181},[31,11989],{"className":11990,"style":1637},[1060],[31,11992,11994],{"className":11993},[1065,1066,1067,1068],[31,11995,11997],{"className":11996},[75,1068],[31,11998,205],{"className":11999},[75,1068],[31,12001,1076],{"className":12002},[1075],[31,12004,12006],{"className":12005},[1048],[31,12007,12009],{"className":12008,"style":2203},[1052],[31,12010],{},[31,12012],{"className":12013},[289,2139],[31,12015,4572],{"className":12016},[75],[31,12018,53],{"className":12019},[75,76],[31,12021],{"className":12022,"style":120},[119],[31,12024,97],{"className":12025},[124],[31,12027],{"className":12028,"style":120},[119],[31,12030,12032,12036,12039],{"className":12031},[66],[31,12033],{"className":12034,"style":12035},[70],"height:1.0641em;vertical-align:-0.25em;",[31,12037,3003],{"className":12038,"style":2666},[75,76],[31,12040,12042,12045],{"className":12041},[75],[31,12043,4572],{"className":12044},[75],[31,12046,12048],{"className":12047},[1039],[31,12049,12051],{"className":12050},[1043],[31,12052,12054],{"className":12053},[1048],[31,12055,12057],{"className":12056,"style":5135},[1052],[31,12058,12059,12062],{"style":3631},[31,12060],{"className":12061,"style":1061},[1060],[31,12063,12065],{"className":12064},[1065,1066,1067,1068],[31,12066,152],{"className":12067},[75,1068]," 时还有一个 Brenier 定理：若 ",[31,12070,12072,12085],{"className":12071,"translate":35},[34],[31,12073,12075],{"className":12074},[39],[41,12076,12077],{"xmlns":43},[45,12078,12079,12083],{},[48,12080,12081],{},[51,12082,852],{},[55,12084,855],{"encoding":57},[31,12086,12088],{"className":12087,"ariaHidden":62},[61],[31,12089,12091,12094],{"className":12090},[66],[31,12092],{"className":12093,"style":865},[70],[31,12095,852],{"className":12096},[75,76]," 关于 Lebesgue 测度绝对连续，最优的 ",[31,12099,12101,12114],{"className":12100,"translate":35},[34],[31,12102,12104],{"className":12103},[39],[41,12105,12106],{"xmlns":43},[45,12107,12108,12112],{},[48,12109,12110],{},[51,12111,2515],{},[55,12113,3485],{"encoding":57},[31,12115,12117],{"className":12116,"ariaHidden":62},[61],[31,12118,12120,12123],{"className":12119},[66],[31,12121],{"className":12122,"style":71},[70],[31,12124,2515],{"className":12125,"style":2666},[75,76]," 由映射 ",[31,12128,12130,12151],{"className":12129,"translate":35},[34],[31,12131,12133],{"className":12132},[39],[41,12134,12135],{"xmlns":43},[45,12136,12137,12148],{},[48,12138,12139,12141,12143,12146],{},[51,12140,919],{},[95,12142,223],{},[51,12144,12145],{"mathvariant":494},"∇",[51,12147,5324],{},[55,12149,12150],{"encoding":57},"T=\\nabla u",[31,12152,12154,12172],{"className":12153,"ariaHidden":62},[61],[31,12155,12157,12160,12163,12166,12169],{"className":12156},[66],[31,12158],{"className":12159,"style":945},[70],[31,12161,919],{"className":12162,"style":949},[75,76],[31,12164],{"className":12165,"style":293},[119],[31,12167,223],{"className":12168},[297],[31,12170],{"className":12171,"style":293},[119],[31,12173,12175,12178,12181],{"className":12174},[66],[31,12176],{"className":12177,"style":945},[70],[31,12179,12145],{"className":12180},[75],[31,12182,5324],{"className":12183},[75,76]," 给出，其中 ",[31,12186,12188,12201],{"className":12187,"translate":35},[34],[31,12189,12191],{"className":12190},[39],[41,12192,12193],{"xmlns":43},[45,12194,12195,12199],{},[48,12196,12197],{},[51,12198,5324],{},[55,12200,5324],{"encoding":57},[31,12202,12204],{"className":12203,"ariaHidden":62},[61],[31,12205,12207,12210],{"className":12206},[66],[31,12208],{"className":12209,"style":71},[70],[31,12211,5324],{"className":12212},[75,76]," 是凸函数。换句话说，最优搬运方案是某个凸势函数的梯度，是一个“单调”的变换。这句话是后半段所有流模型的伏笔：如果我要把一个分布搬到另一个分布，最优的那个可逆映射长得像 ",[31,12215,12217,12233],{"className":12216,"translate":35},[34],[31,12218,12220],{"className":12219},[39],[41,12221,12222],{"xmlns":43},[45,12223,12224,12230],{},[48,12225,12226,12228],{},[51,12227,12145],{"mathvariant":494},[51,12229,5324],{},[55,12231,12232],{"encoding":57},"\\nabla u",[31,12234,12236],{"className":12235,"ariaHidden":62},[61],[31,12237,12239,12242,12245],{"className":12238},[66],[31,12240],{"className":12241,"style":945},[70],[31,12243,12145],{"className":12244},[75],[31,12246,5324],{"className":12247},[75,76],[18,12249,12250],{"id":12250},"一维与高斯",[11,12252,12253],{},"A 公司的薪资在 0 到 1 万之间均匀分布，B 公司在 0 到 2 万之间均匀分布。一个人从 A 跳到 B，平均涨多少？",[11,12255,12256,12257,12287,12288,12316,12317,12351,12352,12380],{},"“最省力”的搬法是按百分位对齐：A 的第 30 百分位是 0.3 万，B 的第 30 百分位是 0.6 万，所以 A 里处在 30 分位的那个人涨 0.3 万。一般地，分位 ",[31,12258,12260,12274],{"className":12259,"translate":35},[34],[31,12261,12263],{"className":12262},[39],[41,12264,12265],{"xmlns":43},[45,12266,12267,12272],{},[48,12268,12269],{},[51,12270,12271],{},"t",[55,12273,12271],{"encoding":57},[31,12275,12277],{"className":12276,"ariaHidden":62},[61],[31,12278,12280,12284],{"className":12279},[66],[31,12281],{"className":12282,"style":12283},[70],"height:0.6151em;",[31,12285,12271],{"className":12286},[75,76]," 的人从 ",[31,12289,12291,12304],{"className":12290,"translate":35},[34],[31,12292,12294],{"className":12293},[39],[41,12295,12296],{"xmlns":43},[45,12297,12298,12302],{},[48,12299,12300],{},[51,12301,12271],{},[55,12303,12271],{"encoding":57},[31,12305,12307],{"className":12306,"ariaHidden":62},[61],[31,12308,12310,12313],{"className":12309},[66],[31,12311],{"className":12312,"style":12283},[70],[31,12314,12271],{"className":12315},[75,76]," 万变成 ",[31,12318,12320,12336],{"className":12319,"translate":35},[34],[31,12321,12323],{"className":12322},[39],[41,12324,12325],{"xmlns":43},[45,12326,12327,12333],{},[48,12328,12329,12331],{},[91,12330,152],{},[51,12332,12271],{},[55,12334,12335],{"encoding":57},"2t",[31,12337,12339],{"className":12338,"ariaHidden":62},[61],[31,12340,12342,12345,12348],{"className":12341},[66],[31,12343],{"className":12344,"style":326},[70],[31,12346,152],{"className":12347},[75],[31,12349,12271],{"className":12350},[75,76]," 万，涨幅是 ",[31,12353,12355,12368],{"className":12354,"translate":35},[34],[31,12356,12358],{"className":12357},[39],[41,12359,12360],{"xmlns":43},[45,12361,12362,12366],{},[48,12363,12364],{},[51,12365,12271],{},[55,12367,12271],{"encoding":57},[31,12369,12371],{"className":12370,"ariaHidden":62},[61],[31,12372,12374,12377],{"className":12373},[66],[31,12375],{"className":12376,"style":12283},[70],[31,12378,12271],{"className":12379},[75,76]," 万，平均",[31,12382,12384],{"className":12383,"translate":35},[411],[31,12385,12387,12431],{"className":12386,"translate":35},[34],[31,12388,12390],{"className":12389},[39],[41,12391,12392],{"xmlns":43,"display":420},[45,12393,12394,12428],{},[48,12395,12396,12405,12407,12409,12411,12413,12415,12423,12426],{},[12397,12398,12399,12401,12403],"msubsup",{},[95,12400,1409],{},[91,12402,1977],{},[91,12404,205],{},[51,12406,12271],{},[1430,12408,1432],{},[51,12410,1435],{},[51,12412,12271],{},[95,12414,223],{},[2071,12416,12417],{"scriptlevel":1977,"displaystyle":210},[2074,12418,12419,12421],{},[91,12420,205],{},[91,12422,152],{},[1430,12424,12425],{}," 万",[51,12427,495],{"mathvariant":494},[55,12429,12430],{"encoding":57},"\\int_0^1 t\\,dt=\\tfrac12\\ \\text{万}.",[31,12432,12434,12518],{"className":12433,"ariaHidden":62},[61],[31,12435,12437,12441,12494,12497,12500,12503,12506,12509,12512,12515],{"className":12436},[66],[31,12438],{"className":12439,"style":12440},[70],"height:2.476em;vertical-align:-0.9119em;",[31,12442,12444,12447],{"className":12443},[1460],[31,12445,1409],{"className":12446,"style":1688},[1460,1461,1687],[31,12448,12450],{"className":12449},[1039],[31,12451,12453,12486],{"className":12452},[1043,1044],[31,12454,12456,12483],{"className":12455},[1048],[31,12457,12460,12471],{"className":12458,"style":12459},[1052],"height:1.564em;",[31,12461,12462,12465],{"style":1704},[31,12463],{"className":12464,"style":1061},[1060],[31,12466,12468],{"className":12467},[1065,1066,1067,1068],[31,12469,1977],{"className":12470},[75,1068],[31,12472,12474,12477],{"style":12473},"top:-3.8129em;margin-right:0.05em;",[31,12475],{"className":12476,"style":1061},[1060],[31,12478,12480],{"className":12479},[1065,1066,1067,1068],[31,12481,205],{"className":12482},[75,1068],[31,12484,1076],{"className":12485},[1075],[31,12487,12489],{"className":12488},[1048],[31,12490,12492],{"className":12491,"style":1726},[1052],[31,12493],{},[31,12495],{"className":12496,"style":1467},[119],[31,12498,12271],{"className":12499},[75,76],[31,12501],{"className":12502,"style":1467},[119],[31,12504,1435],{"className":12505},[75,76],[31,12507,12271],{"className":12508},[75,76],[31,12510],{"className":12511,"style":293},[119],[31,12513,223],{"className":12514},[297],[31,12516],{"className":12517,"style":293},[119],[31,12519,12521,12524,12592,12595,12603],{"className":12520},[66],[31,12522],{"className":12523,"style":2132},[70],[31,12525,12527,12530,12589],{"className":12526},[75],[31,12528],{"className":12529},[264,2139],[31,12531,12533],{"className":12532},[2074],[31,12534,12536,12581],{"className":12535},[1043,1044],[31,12537,12539,12578],{"className":12538},[1048],[31,12540,12542,12556,12564],{"className":12541,"style":2152},[1052],[31,12543,12544,12547],{"style":2155},[31,12545],{"className":12546,"style":1637},[1060],[31,12548,12550],{"className":12549},[1065,1066,1067,1068],[31,12551,12553],{"className":12552},[75,1068],[31,12554,152],{"className":12555},[75,1068],[31,12557,12558,12561],{"style":2170},[31,12559],{"className":12560,"style":1637},[1060],[31,12562],{"className":12563,"style":2178},[2177],[31,12565,12566,12569],{"style":2181},[31,12567],{"className":12568,"style":1637},[1060],[31,12570,12572],{"className":12571},[1065,1066,1067,1068],[31,12573,12575],{"className":12574},[75,1068],[31,12576,205],{"className":12577},[75,1068],[31,12579,1076],{"className":12580},[1075],[31,12582,12584],{"className":12583},[1048],[31,12585,12587],{"className":12586,"style":2203},[1052],[31,12588],{},[31,12590],{"className":12591},[289,2139],[31,12593,1545],{"className":12594},[119],[31,12596,12598],{"className":12597},[75,1793],[31,12599,12602],{"className":12600},[75,12601],"cjk_fallback","万",[31,12604,495],{"className":12605},[75],[11,12607,12608,12609,12665,12666,12731,12732,2372],{},"这里可以直接用 ",[31,12610,12612,12632],{"className":12611,"translate":35},[34],[31,12613,12615],{"className":12614},[39],[41,12616,12617],{"xmlns":43},[45,12618,12619,12629],{},[48,12620,12621,12623,12625,12627],{},[91,12622,152],{},[51,12624,12271],{},[95,12626,97],{},[51,12628,12271],{},[55,12630,12631],{"encoding":57},"2t-t",[31,12633,12635,12656],{"className":12634,"ariaHidden":62},[61],[31,12636,12638,12641,12644,12647,12650,12653],{"className":12637},[66],[31,12639],{"className":12640,"style":112},[70],[31,12642,152],{"className":12643},[75],[31,12645,12271],{"className":12646},[75,76],[31,12648],{"className":12649,"style":120},[119],[31,12651,97],{"className":12652},[124],[31,12654],{"className":12655,"style":120},[119],[31,12657,12659,12662],{"className":12658},[66],[31,12660],{"className":12661,"style":12283},[70],[31,12663,12271],{"className":12664},[75,76]," 而不是 ",[31,12667,12669,12694],{"className":12668,"translate":35},[34],[31,12670,12672],{"className":12671},[39],[41,12673,12674],{"xmlns":43},[45,12675,12676,12691],{},[48,12677,12678,12681,12683,12685,12687,12689],{},[51,12679,12680],{"mathvariant":494},"∣",[91,12682,152],{},[51,12684,12271],{},[95,12686,97],{},[51,12688,12271],{},[51,12690,12680],{"mathvariant":494},[55,12692,12693],{"encoding":57},"|2t-t|",[31,12695,12697,12719],{"className":12696,"ariaHidden":62},[61],[31,12698,12700,12703,12707,12710,12713,12716],{"className":12699},[66],[31,12701],{"className":12702,"style":260},[70],[31,12704,12706],{"className":12705},[75],"∣2",[31,12708,12271],{"className":12709},[75,76],[31,12711],{"className":12712,"style":120},[119],[31,12714,97],{"className":12715},[124],[31,12717],{"className":12718,"style":120},[119],[31,12720,12722,12725,12728],{"className":12721},[66],[31,12723],{"className":12724,"style":260},[70],[31,12726,12271],{"className":12727},[75,76],[31,12729,12680],{"className":12730},[75],"，因为所有人都在往同一个方向涨。而这个“平均涨幅”正是 ",[31,12733,12735,12752],{"className":12734,"translate":35},[34],[31,12736,12738],{"className":12737},[39],[41,12739,12740],{"xmlns":43},[45,12741,12742,12750],{},[48,12743,12744],{},[1004,12745,12746,12748],{},[51,12747,3117],{},[91,12749,205],{},[55,12751,5220],{"encoding":57},[31,12753,12755],{"className":12754,"ariaHidden":62},[61],[31,12756,12758,12761],{"className":12757},[66],[31,12759],{"className":12760,"style":5230},[70],[31,12762,12764,12767],{"className":12763},[75],[31,12765,3117],{"className":12766,"style":949},[75,76],[31,12768,12770],{"className":12769},[1039],[31,12771,12773,12793],{"className":12772},[1043,1044],[31,12774,12776,12790],{"className":12775},[1048],[31,12777,12779],{"className":12778,"style":2028},[1052],[31,12780,12781,12784],{"style":1056},[31,12782],{"className":12783,"style":1061},[1060],[31,12785,12787],{"className":12786},[1065,1066,1067,1068],[31,12788,205],{"className":12789},[75,1068],[31,12791,1076],{"className":12792},[1075],[31,12794,12796],{"className":12795},[1048],[31,12797,12799],{"className":12798,"style":2050},[1052],[31,12800],{},[11,12802,12803,12804,12863,12864,12921,12922,12991],{},"如果反过来，A 是 ",[31,12805,12807,12832],{"className":12806,"translate":35},[34],[31,12808,12810],{"className":12809},[39],[41,12811,12812],{"xmlns":43},[45,12813,12814,12829],{},[48,12815,12816,12819,12821,12823,12825,12827],{},[51,12817,12818],{},"U",[95,12820,211],{"stretchy":210},[91,12822,1977],{},[95,12824,1210],{"separator":62},[91,12826,152],{},[95,12828,220],{"stretchy":210},[55,12830,12831],{"encoding":57},"U(0,2)",[31,12833,12835],{"className":12834,"ariaHidden":62},[61],[31,12836,12838,12841,12845,12848,12851,12854,12857,12860],{"className":12837},[66],[31,12839],{"className":12840,"style":260},[70],[31,12842,12818],{"className":12843,"style":12844},[75,76],"margin-right:0.109em;",[31,12846,211],{"className":12847},[264],[31,12849,1977],{"className":12850},[75],[31,12852,1210],{"className":12853},[1391],[31,12855],{"className":12856,"style":1467},[119],[31,12858,152],{"className":12859},[75],[31,12861,220],{"className":12862},[289],"、B 是 ",[31,12865,12867,12891],{"className":12866,"translate":35},[34],[31,12868,12870],{"className":12869},[39],[41,12871,12872],{"xmlns":43},[45,12873,12874,12888],{},[48,12875,12876,12878,12880,12882,12884,12886],{},[51,12877,12818],{},[95,12879,211],{"stretchy":210},[91,12881,1977],{},[95,12883,1210],{"separator":62},[91,12885,205],{},[95,12887,220],{"stretchy":210},[55,12889,12890],{"encoding":57},"U(0,1)",[31,12892,12894],{"className":12893,"ariaHidden":62},[61],[31,12895,12897,12900,12903,12906,12909,12912,12915,12918],{"className":12896},[66],[31,12898],{"className":12899,"style":260},[70],[31,12901,12818],{"className":12902,"style":12844},[75,76],[31,12904,211],{"className":12905},[264],[31,12907,1977],{"className":12908},[75],[31,12910,1210],{"className":12911},[1391],[31,12913],{"className":12914,"style":1467},[119],[31,12916,205],{"className":12917},[75],[31,12919,220],{"className":12920},[289],"，那就是平均降 0.5 万，",[31,12923,12925,12942],{"className":12924,"translate":35},[34],[31,12926,12928],{"className":12927},[39],[41,12929,12930],{"xmlns":43},[45,12931,12932,12940],{},[48,12933,12934],{},[1004,12935,12936,12938],{},[51,12937,3117],{},[91,12939,205],{},[55,12941,5220],{"encoding":57},[31,12943,12945],{"className":12944,"ariaHidden":62},[61],[31,12946,12948,12951],{"className":12947},[66],[31,12949],{"className":12950,"style":5230},[70],[31,12952,12954,12957],{"className":12953},[75],[31,12955,3117],{"className":12956,"style":949},[75,76],[31,12958,12960],{"className":12959},[1039],[31,12961,12963,12983],{"className":12962},[1043,1044],[31,12964,12966,12980],{"className":12965},[1048],[31,12967,12969],{"className":12968,"style":2028},[1052],[31,12970,12971,12974],{"style":1056},[31,12972],{"className":12973,"style":1061},[1060],[31,12975,12977],{"className":12976},[1065,1066,1067,1068],[31,12978,205],{"className":12979},[75,1068],[31,12981,1076],{"className":12982},[1075],[31,12984,12986],{"className":12985},[1048],[31,12987,12989],{"className":12988,"style":2050},[1052],[31,12990],{}," 仍然是 0.5——它量的是“搬了多少”，不管涨还是跌。",[11,12993,12994,12995,13064],{},"一般地，一维的 ",[31,12996,12998,13015],{"className":12997,"translate":35},[34],[31,12999,13001],{"className":13000},[39],[41,13002,13003],{"xmlns":43},[45,13004,13005,13013],{},[48,13006,13007],{},[1004,13008,13009,13011],{},[51,13010,3117],{},[51,13012,11],{},[55,13014,4830],{"encoding":57},[31,13016,13018],{"className":13017,"ariaHidden":62},[61],[31,13019,13021,13024],{"className":13020},[66],[31,13022],{"className":13023,"style":1029},[70],[31,13025,13027,13030],{"className":13026},[75],[31,13028,3117],{"className":13029,"style":949},[75,76],[31,13031,13033],{"className":13032},[1039],[31,13034,13036,13056],{"className":13035},[1043,1044],[31,13037,13039,13053],{"className":13038},[1048],[31,13040,13042],{"className":13041,"style":3230},[1052],[31,13043,13044,13047],{"style":1056},[31,13045],{"className":13046,"style":1061},[1060],[31,13048,13050],{"className":13049},[1065,1066,1067,1068],[31,13051,11],{"className":13052},[75,76,1068],[31,13054,1076],{"className":13055},[1075],[31,13057,13059],{"className":13058},[1048],[31,13060,13062],{"className":13061,"style":1083},[1052],[31,13063],{}," 有闭式解：",[31,13066,13068],{"className":13067,"translate":35},[411],[31,13069,13071,13166],{"className":13070,"translate":35},[34],[31,13072,13074],{"className":13073},[39],[41,13075,13076],{"xmlns":43,"display":420},[45,13077,13078,13163],{},[48,13079,13080,13088,13090,13092,13094,13096,13098,13100,13108,13110,13123,13125,13127,13129,13131,13143,13145,13147,13149,13155,13157,13159,13161],{},[12397,13081,13082,13084,13086],{},[51,13083,3117],{},[51,13085,11],{},[51,13087,11],{},[95,13089,211],{"stretchy":210},[51,13091,852],{},[95,13093,1210],{"separator":62},[51,13095,884],{},[95,13097,220],{"stretchy":210},[95,13099,223],{},[12397,13101,13102,13104,13106],{},[95,13103,1409],{},[91,13105,1977],{},[91,13107,205],{},[95,13109,12680],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[12397,13111,13112,13115,13117],{},[51,13113,13114],{},"F",[51,13116,852],{},[48,13118,13119,13121],{},[95,13120,97],{},[91,13122,205],{},[95,13124,211],{"stretchy":210},[51,13126,12271],{},[95,13128,220],{"stretchy":210},[95,13130,97],{},[12397,13132,13133,13135,13137],{},[51,13134,13114],{},[51,13136,884],{},[48,13138,13139,13141],{},[95,13140,97],{},[91,13142,205],{},[95,13144,211],{"stretchy":210},[51,13146,12271],{},[95,13148,220],{"stretchy":210},[1180,13150,13151,13153],{},[95,13152,12680],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[51,13154,11],{},[1430,13156,1432],{},[51,13158,1435],{},[51,13160,12271],{},[95,13162,1210],{"separator":62},[55,13164,13165],{"encoding":57},"W_p^p(\\mu,\\nu)=\\int_0^1\\bigl|F_\\mu^{-1}(t)-F_\\nu^{-1}(t)\\bigr|^p\\,dt ,",[31,13167,13169,13256,13437],{"className":13168,"ariaHidden":62},[61],[31,13170,13172,13176,13229,13232,13235,13238,13241,13244,13247,13250,13253],{"className":13171},[66],[31,13173],{"className":13174,"style":13175},[70],"height:1.1331em;vertical-align:-0.3831em;",[31,13177,13179,13182],{"className":13178},[75],[31,13180,3117],{"className":13181,"style":949},[75,76],[31,13183,13185],{"className":13184},[1039],[31,13186,13188,13220],{"className":13187},[1043,1044],[31,13189,13191,13217],{"className":13190},[1048],[31,13192,13194,13206],{"className":13193,"style":3987},[1052],[31,13195,13197,13200],{"style":13196},"top:-2.453em;margin-left:-0.1389em;margin-right:0.05em;",[31,13198],{"className":13199,"style":1061},[1060],[31,13201,13203],{"className":13202},[1065,1066,1067,1068],[31,13204,11],{"className":13205},[75,76,1068],[31,13207,13208,13211],{"style":1330},[31,13209],{"className":13210,"style":1061},[1060],[31,13212,13214],{"className":13213},[1065,1066,1067,1068],[31,13215,11],{"className":13216},[75,76,1068],[31,13218,1076],{"className":13219},[1075],[31,13221,13223],{"className":13222},[1048],[31,13224,13227],{"className":13225,"style":13226},[1052],"height:0.3831em;",[31,13228],{},[31,13230,211],{"className":13231},[264],[31,13233,852],{"className":13234},[75,76],[31,13236,1210],{"className":13237},[1391],[31,13239],{"className":13240,"style":1467},[119],[31,13242,884],{"className":13243,"style":900},[75,76],[31,13245,220],{"className":13246},[289],[31,13248],{"className":13249,"style":293},[119],[31,13251,223],{"className":13252},[297],[31,13254],{"className":13255,"style":293},[119],[31,13257,13259,13262,13313,13362,13419,13422,13425,13428,13431,13434],{"className":13258},[66],[31,13260],{"className":13261,"style":12440},[70],[31,13263,13265,13268],{"className":13264},[1460],[31,13266,1409],{"className":13267,"style":1688},[1460,1461,1687],[31,13269,13271],{"className":13270},[1039],[31,13272,13274,13305],{"className":13273},[1043,1044],[31,13275,13277,13302],{"className":13276},[1048],[31,13278,13280,13291],{"className":13279,"style":12459},[1052],[31,13281,13282,13285],{"style":1704},[31,13283],{"className":13284,"style":1061},[1060],[31,13286,13288],{"className":13287},[1065,1066,1067,1068],[31,13289,1977],{"className":13290},[75,1068],[31,13292,13293,13296],{"style":12473},[31,13294],{"className":13295,"style":1061},[1060],[31,13297,13299],{"className":13298},[1065,1066,1067,1068],[31,13300,205],{"className":13301},[75,1068],[31,13303,1076],{"className":13304},[1075],[31,13306,13308],{"className":13307},[1048],[31,13309,13311],{"className":13310,"style":1726},[1052],[31,13312],{},[31,13314,13316],{"className":13315},[264],[31,13317,13320],{"className":13318},[1307,13319],"mult",[31,13321,13323,13353],{"className":13322},[1043,1044],[31,13324,13326,13350],{"className":13325},[1048],[31,13327,13330],{"className":13328,"style":13329},[1052],"height:0.85em;",[31,13331,13333,13337],{"style":13332},"top:-2.85em;",[31,13334],{"className":13335,"style":13336},[1060],"height:3.2em;",[31,13338,13340],{"style":13339},"width:0.333em;height:1.2em;",[13341,13342,13346],"svg",{"xmlns":13343,"width":13344,"height":1178,"viewBox":13345},"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg","0.333em","0 0 333 1200",[13347,13348],"path",{"d":13349},"M145 15 v585 v0 v585 c2.667,10,9.667,15,21,15\nc10,0,16.667,-5,20,-15 v-585 v0 v-585 c-2.667,-10,-9.667,-15,-21,-15\nc-10,0,-16.667,5,-20,15z M188 15 H145 v585 v0 v585 h43z",[31,13351,1076],{"className":13352},[1075],[31,13354,13356],{"className":13355},[1048],[31,13357,13360],{"className":13358,"style":13359},[1052],"height:0.35em;",[31,13361],{},[31,13363,13365,13368],{"className":13364},[75],[31,13366,13114],{"className":13367,"style":949},[75,76],[31,13369,13371],{"className":13370},[1039],[31,13372,13374,13411],{"className":13373},[1043,1044],[31,13375,13377,13408],{"className":13376},[1048],[31,13378,13380,13391],{"className":13379,"style":1327},[1052],[31,13381,13382,13385],{"style":13196},[31,13383],{"className":13384,"style":1061},[1060],[31,13386,13388],{"className":13387},[1065,1066,1067,1068],[31,13389,852],{"className":13390},[75,76,1068],[31,13392,13393,13396],{"style":1330},[31,13394],{"className":13395,"style":1061},[1060],[31,13397,13399],{"className":13398},[1065,1066,1067,1068],[31,13400,13402,13405],{"className":13401},[75,1068],[31,13403,97],{"className":13404},[75,1068],[31,13406,205],{"className":13407},[75,1068],[31,13409,1076],{"className":13410},[1075],[31,13412,13414],{"className":13413},[1048],[31,13415,13417],{"className":13416,"style":13226},[1052],[31,13418],{},[31,13420,211],{"className":13421},[264],[31,13423,12271],{"className":13424},[75,76],[31,13426,220],{"className":13427},[289],[31,13429],{"className":13430,"style":120},[119],[31,13432,97],{"className":13433},[124],[31,13435],{"className":13436,"style":120},[119],[31,13438,13440,13444,13502,13505,13508,13511,13576,13579,13582,13585],{"className":13439},[66],[31,13441],{"className":13442,"style":13443},[70],"height:1.2543em;vertical-align:-0.35em;",[31,13445,13447,13450],{"className":13446},[75],[31,13448,13114],{"className":13449,"style":949},[75,76],[31,13451,13453],{"className":13452},[1039],[31,13454,13456,13493],{"className":13455},[1043,1044],[31,13457,13459,13490],{"className":13458},[1048],[31,13460,13462,13473],{"className":13461,"style":1327},[1052],[31,13463,13464,13467],{"style":13196},[31,13465],{"className":13466,"style":1061},[1060],[31,13468,13470],{"className":13469},[1065,1066,1067,1068],[31,13471,884],{"className":13472,"style":900},[75,76,1068],[31,13474,13475,13478],{"style":1330},[31,13476],{"className":13477,"style":1061},[1060],[31,13479,13481],{"className":13480},[1065,1066,1067,1068],[31,13482,13484,13487],{"className":13483},[75,1068],[31,13485,97],{"className":13486},[75,1068],[31,13488,205],{"className":13489},[75,1068],[31,13491,1076],{"className":13492},[1075],[31,13494,13496],{"className":13495},[1048],[31,13497,13500],{"className":13498,"style":13499},[1052],"height:0.247em;",[31,13501],{},[31,13503,211],{"className":13504},[264],[31,13506,12271],{"className":13507},[75,76],[31,13509,220],{"className":13510},[289],[31,13512,13514,13551],{"className":13513},[289],[31,13515,13517],{"className":13516},[289],[31,13518,13520],{"className":13519},[1307,13319],[31,13521,13523,13543],{"className":13522},[1043,1044],[31,13524,13526,13540],{"className":13525},[1048],[31,13527,13529],{"className":13528,"style":13329},[1052],[31,13530,13531,13534],{"style":13332},[31,13532],{"className":13533,"style":13336},[1060],[31,13535,13536],{"style":13339},[13341,13537,13538],{"xmlns":13343,"width":13344,"height":1178,"viewBox":13345},[13347,13539],{"d":13349},[31,13541,1076],{"className":13542},[1075],[31,13544,13546],{"className":13545},[1048],[31,13547,13549],{"className":13548,"style":13359},[1052],[31,13550],{},[31,13552,13554],{"className":13553},[1039],[31,13555,13557],{"className":13556},[1043],[31,13558,13560],{"className":13559},[1048],[31,13561,13564],{"className":13562,"style":13563},[1052],"height:0.9043em;",[31,13565,13567,13570],{"style":13566},"top:-3.3029em;margin-right:0.05em;",[31,13568],{"className":13569,"style":1061},[1060],[31,13571,13573],{"className":13572},[1065,1066,1067,1068],[31,13574,11],{"className":13575},[75,76,1068],[31,13577],{"className":13578,"style":1467},[119],[31,13580,1435],{"className":13581},[75,76],[31,13583,12271],{"className":13584},[75,76],[31,13586,1210],{"className":13587},[1391],[11,13589,13590,13591,13642,13643,3641,13671,13768],{},"也就是把两条分位数曲线逐点相减再积分。",[31,13592,13594,13612],{"className":13593,"translate":35},[34],[31,13595,13597],{"className":13596},[39],[41,13598,13599],{"xmlns":43},[45,13600,13601,13609],{},[48,13602,13603,13605,13607],{},[51,13604,11],{},[95,13606,223],{},[91,13608,205],{},[55,13610,13611],{"encoding":57},"p=1",[31,13613,13615,13633],{"className":13614,"ariaHidden":62},[61],[31,13616,13618,13621,13624,13627,13630],{"className":13617},[66],[31,13619],{"className":13620,"style":865},[70],[31,13622,11],{"className":13623},[75,76],[31,13625],{"className":13626,"style":293},[119],[31,13628,223],{"className":13629},[297],[31,13631],{"className":13632,"style":293},[119],[31,13634,13636,13639],{"className":13635},[66],[31,13637],{"className":13638,"style":326},[70],[31,13640,205],{"className":13641},[75]," 时有更漂亮的形式。用上一节的对偶，对绝对连续的 ",[31,13644,13646,13659],{"className":13645,"translate":35},[34],[31,13647,13649],{"className":13648},[39],[41,13650,13651],{"xmlns":43},[45,13652,13653,13657],{},[48,13654,13655],{},[51,13656,8059],{},[55,13658,11503],{"encoding":57},[31,13660,13662],{"className":13661,"ariaHidden":62},[61],[31,13663,13665,13668],{"className":13664},[66],[31,13666],{"className":13667,"style":865},[70],[31,13669,8059],{"className":13670},[75,76],[31,13672,13674,13701],{"className":13673,"translate":35},[34],[31,13675,13677],{"className":13676},[39],[41,13678,13679],{"xmlns":43},[45,13680,13681,13698],{},[48,13682,13683,13685,13692,13694,13696],{},[51,13684,12680],{"mathvariant":494},[1180,13686,13687,13689],{},[51,13688,8059],{},[95,13690,13691],{"mathvariant":494,"lspace":5610,"rspace":5610},"′",[51,13693,12680],{"mathvariant":494},[95,13695,347],{},[91,13697,205],{},[55,13699,13700],{"encoding":57},"|\\varphi'|\\le1",[31,13702,13704,13759],{"className":13703,"ariaHidden":62},[61],[31,13705,13707,13711,13714,13747,13750,13753,13756],{"className":13706},[66],[31,13708],{"className":13709,"style":13710},[70],"height:1.0019em;vertical-align:-0.25em;",[31,13712,12680],{"className":13713},[75],[31,13715,13717,13720],{"className":13716},[75],[31,13718,8059],{"className":13719},[75,76],[31,13721,13723],{"className":13722},[1039],[31,13724,13726],{"className":13725},[1043],[31,13727,13729],{"className":13728},[1048],[31,13730,13733],{"className":13731,"style":13732},[1052],"height:0.7519em;",[31,13734,13735,13738],{"style":3631},[31,13736],{"className":13737,"style":1061},[1060],[31,13739,13741],{"className":13740},[1065,1066,1067,1068],[31,13742,13744],{"className":13743},[75,1068],[31,13745,13691],{"className":13746},[75,1068],[31,13748,12680],{"className":13749},[75],[31,13751],{"className":13752,"style":293},[119],[31,13754,347],{"className":13755},[297],[31,13757],{"className":13758,"style":293},[119],[31,13760,13762,13765],{"className":13761},[66],[31,13763],{"className":13764,"style":326},[70],[31,13766,205],{"className":13767},[75],"）分部积分，",[31,13770,13772],{"className":13771,"translate":35},[411],[31,13773,13775,13909],{"className":13774,"translate":35},[34],[31,13776,13778],{"className":13777},[39],[41,13779,13780],{"xmlns":43,"display":420},[45,13781,13782,13906],{},[48,13783,13784,13786,13788,13790,13792,13794,13796,13798,13800,13802,13804,13806,13808,13810,13812,13814,13816,13824,13826,13828,13830,13832,13840,13842,13844,13846,13848,13850,13852,13854,13856,13858,13864,13866,13868,13870,13872,13878,13880,13882,13884,13886,13892,13894,13896,13898,13900,13902,13904],{},[95,13785,1409],{},[51,13787,8059],{},[1430,13789,1432],{},[51,13791,1435],{},[95,13793,211],{"stretchy":210},[51,13795,852],{},[95,13797,97],{},[51,13799,884],{},[95,13801,220],{"stretchy":210},[95,13803,223],{},[95,13805,1409],{},[51,13807,8059],{},[95,13809,211],{"stretchy":210},[51,13811,53],{},[95,13813,220],{"stretchy":210},[95,13815,211],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[12397,13817,13818,13820,13822],{},[51,13819,13114],{},[51,13821,852],{},[95,13823,13691],{"mathvariant":494,"lspace":5610,"rspace":5610},[95,13825,211],{"stretchy":210},[51,13827,53],{},[95,13829,220],{"stretchy":210},[95,13831,97],{},[12397,13833,13834,13836,13838],{},[51,13835,13114],{},[51,13837,884],{},[95,13839,13691],{"mathvariant":494,"lspace":5610,"rspace":5610},[95,13841,211],{"stretchy":210},[51,13843,53],{},[95,13845,220],{"stretchy":210},[95,13847,220],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[51,13849,1435],{},[51,13851,53],{},[95,13853,223],{},[95,13855,97],{},[95,13857,1409],{},[1180,13859,13860,13862],{},[51,13861,8059],{},[95,13863,13691],{"mathvariant":494,"lspace":5610,"rspace":5610},[95,13865,211],{"stretchy":210},[51,13867,53],{},[95,13869,220],{"stretchy":210},[95,13871,211],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[1004,13873,13874,13876],{},[51,13875,13114],{},[51,13877,852],{},[95,13879,211],{"stretchy":210},[51,13881,53],{},[95,13883,220],{"stretchy":210},[95,13885,97],{},[1004,13887,13888,13890],{},[51,13889,13114],{},[51,13891,884],{},[95,13893,211],{"stretchy":210},[51,13895,53],{},[95,13897,220],{"stretchy":210},[95,13899,220],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[51,13901,1435],{},[51,13903,53],{},[95,13905,1210],{"separator":62},[55,13907,13908],{"encoding":57},"\\int\\varphi\\,d(\\mu-\\nu)=\\int\\varphi(x)\\bigl(F_\\mu'(x)-F_\\nu'(x)\\bigr)dx\n=-\\int\\varphi'(x)\\bigl(F_\\mu(x)-F_\\nu(x)\\bigr)dx,",[31,13910,13912,13948,13969,14072,14162,14285],{"className":13911,"ariaHidden":62},[61],[31,13913,13915,13918,13921,13924,13927,13930,13933,13936,13939,13942,13945],{"className":13914},[66],[31,13916],{"className":13917,"style":10946},[70],[31,13919,1409],{"className":13920,"style":1688},[1460,1461,1687],[31,13922],{"className":13923,"style":1467},[119],[31,13925,8059],{"className":13926},[75,76],[31,13928],{"className":13929,"style":1467},[119],[31,13931,1435],{"className":13932}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",[31,14359,14361,14399],{"className":14360,"translate":35},[34],[31,14362,14364],{"className":14363},[39],[41,14365,14366],{"xmlns":43},[45,14367,14368,14396],{},[48,14369,14370,14372,14374,14380,14382,14388,14390,14392,14394],{},[95,14371,1409],{},[51,14373,12680],{"mathvariant":494},[1004,14375,14376,14378],{},[51,14377,13114],{},[51,14379,852],{},[95,14381,97],{},[1004,14383,14384,14386],{},[51,14385,13114],{},[51,14387,884],{},[51,14389,12680],{"mathvariant":494},[1430,14391,1432],{},[51,14393,1435],{},[51,14395,53],{},[55,14397,14398],{"encoding":57},"\\int|F_\\mu-F_\\nu|\\,dx",[31,14400,14402,14466],{"className":14401,"ariaHidden":62},[61],[31,14403,14405,14408,14411,14414,14417,14457,14460,14463],{"className":14404},[66],[31,14406],{"className":14407,"style":1456},[70],[31,14409,1409],{"className":14410,"style":1463},[1460,1461,1462],[31,14412],{"className":14413,"style":1467},[119],[31,14415,12680],{"className":14416},[75],[31,14418,14420,14423],{"className":14419},[75],[31,14421,13114],{"className":14422,"style":949},[75,76],[31,14424,14426],{"className":14425},[1039],[31,14427,14429,14449],{"className":14428},[1043,1044],[31,14430,14432,14446],{"className":14431},[1048],[31,14433,14435],{"className":14434,"style":3230},[1052],[31,14436,14437,14440],{"style":1056},[31,14438],{"className":14439,"style":1061},[1060],[31,14441,14443],{"className":14442},[1065,1066,1067,1068],[31,14444,852],{"className":14445},[75,76,1068],[31,14447,1076],{"className":14448},[1075],[31,14450,14452],{"className":14451},[1048],[31,14453,14455],{"className":14454,"style":1083},[1052],[31,14456],{},[31,14458],{"className":14459,"style":120},[119],[31,14461,97],{"className":14462},[124],[31,14464],{"className":14465,"style":120},[119],[31,14467,14469,14472,14512,14515,14518,14521],{"className":14468},[66],[31,14470],{"className":14471,"style":260},[70],[31,14473,14475,14478],{"className":14474},[75],[31,14476,13114],{"className":14477,"style":949},[75,76],[31,14479,14481],{"className":14480},[1039],[31,14482,14484,14504],{"className":14483},[1043,1044],[31,14485,14487,14501],{"className":14486},[1048],[31,14488,14490],{"className":14489,"style":3230},[1052],[31,14491,14492,14495],{"style":1056},[31,14493],{"className":14494,"style":1061},[1060],[31,14496,14498],{"className":14497},[1065,1066,1067,1068],[31,14499,884],{"className":14500,"style":900},[75,76,1068],[31,14502,1076],{"className":14503},[1075],[31,14505,14507],{"className":14506},[1048],[31,14508,14510],{"className":14509,"style":2050},[1052],[31,14511],{},[31,14513,12680],{"className":14514},[75],[31,14516],{"className":14517,"style":1467},[119],[31,14519,1435],{"className":14520},[75,76],[31,14522,53],{"className":14523},[75,76],"；取 ",[31,14526,14528,14581],{"className":14527,"translate":35},[34],[31,14529,14531],{"className":14530},[39],[41,14532,14533],{"xmlns":43},[45,14534,14535,14578],{},[48,14536,14537,14543,14545,14547,14560,14562,14568,14570,14576],{},[1180,14538,14539,14541],{},[51,14540,8059],{},[95,14542,13691],{"mathvariant":494,"lspace":5610,"rspace":5610},[95,14544,223],{},[95,14546,97],{},[48,14548,14549,14552,14554,14557],{},[51,14550,14551],{"mathvariant":494},"s",[51,14553,5391],{"mathvariant":494},[51,14555,14556],{"mathvariant":494},"g",[51,14558,14559],{"mathvariant":494},"n",[95,14561,211],{"stretchy":210},[1004,14563,14564,14566],{},[51,14565,13114],{},[51,14567,852],{},[95,14569,97],{},[1004,14571,14572,14574],{},[51,14573,13114],{},[51,14575,884],{},[95,14577,220],{"stretchy":210},[55,14579,14580],{"encoding":57},"\\varphi'=-\\mathrm{sign}(F_\\mu-F_\\nu)",[31,14582,14584,14632,14700],{"className":14583,"ariaHidden":62},[61],[31,14585,14587,14591,14623,14626,14629],{"className":14586},[66],[31,14588],{"className":14589,"style":14590},[70],"height:0.9463em;vertical-align:-0.1944em;",[31,14592,14594,14597],{"className":14593},[75],[31,14595,8059],{"className":14596},[75,76],[31,14598,14600],{"className":14599},[1039],[31,14601,14603],{"className":14602},[1043],[31,14604,14606],{"className":14605},[1048],[31,14607,14609],{"className":14608,"style":13732},[1052],[31,14610,14611,14614],{"style":3631},[31,14612],{"className":14613,"style":1061},[1060],[31,14615,14617],{"className":14616},[1065,1066,1067,1068],[31,14618,14620],{"className":14619},[75,1068],[31,14621,13691],{"className":14622},[75,1068],[31,14624],{"className":14625,"style":293},[119],[31,14627,223],{"className":14628},[297],[31,14630],{"className":14631,"style":293},[119],[31,14633,14635,14638,14641,14648,14651,14691,14694,14697],{"className":14634},[66],[31,14636],{"className":14637,"style":1223},[70],[31,14639,97],{"className":14640},[75],[31,14642,14644],{"className":14643},[75],[31,14645,14647],{"className":14646},[75,2745],"sign",[31,14649,211],{"className":14650},[264],[31,14652,14654,14657],{"className":14653},[75],[31,14655,13114],{"className":14656,"style":949},[75,76],[31,14658,14660],{"className":14659},[1039],[31,14661,14663,14683],{"className":14662},[1043,1044],[31,14664,14666,14680],{"className":14665},[1048],[31,14667,14669],{"className":14668,"style":3230},[1052],[31,14670,14671,14674],{"style":1056},[31,14672],{"className":14673,"style":1061},[1060],[31,14675,14677],{"className":14676},[1065,1066,1067,1068],[31,14678,852],{"className":14679},[75,76,1068],[31,14681,1076],{"className":14682},[1075],[31,14684,14686],{"className":14685},[1048],[31,14687,14689],{"className":14688,"style":1083},[1052],[31,14690],{},[31,14692],{"className":14693,"style":120},[119],[31,14695,97],{"className":14696},[124],[31,14698],{"className":14699,"style":120},[119],[31,14701,14703,14706,14746],{"className":14702},[66],[31,14704],{"className":14705,"style":260},[70],[31,14707,14709,14712],{"className":14708},[75],[31,14710,13114],{"className":14711,"style":949},[75,76],[31,14713,14715],{"className":14714},[1039],[31,14716,14718,14738],{"className":14717},[1043,1044],[31,14719,14721,14735],{"className":14720},[1048],[31,14722,14724],{"className":14723,"style":3230},[1052],[31,14725,14726,14729],{"style":1056},[31,14727],{"className":14728,"style":1061},[1060],[31,14730,14732],{"className":14731},[1065,1066,1067,1068],[31,14733,884],{"className":14734,"style":900},[75,76,1068],[31,14736,1076],{"className":14737},[1075],[31,14739,14741],{"className":14740},[1048],[31,14742,14744],{"className":14743,"style":2050},[1052],[31,14745],{},[31,14747,220],{"className":14748},[289]," 时等号成立。于是",[31,14751,14753],{"className":14752,"translate":35},[411],[31,14754,14756,14832],{"className":14755,"translate":35},[34],[31,14757,14759],{"className":14758},[39],[41,14760,14761],{"xmlns":43,"display":420},[45,14762,14763,14829],{},[48,14764,14765,14771,14773,14775,14777,14779,14781,14783,14791,14793,14799,14801,14803,14805,14807,14813,14815,14817,14819,14821,14823,14825,14827],{},[1004,14766,14767,14769],{},[51,14768,3117],{},[91,14770,205],{},[95,14772,211],{"stretchy":210},[51,14774,852],{},[95,14776,1210],{"separator":62},[51,14778,884],{},[95,14780,220],{"stretchy":210},[95,14782,223],{},[1004,14784,14785,14787],{},[95,14786,1409],{},[51,14788,14790],{"mathvariant":14789},"double-struck","R",[95,14792,12680],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[1004,14794,14795,14797],{},[51,14796,13114],{},[51,14798,852],{},[95,14800,211],{"stretchy":210},[51,14802,53],{},[95,14804,220],{"stretchy":210},[95,14806,97],{},[1004,14808,14809,14811],{},[51,14810,13114],{},[51,14812,884],{},[95,14814,211],{"stretchy":210},[51,14816,53],{},[95,14818,220],{"stretchy":210},[95,14820,12680],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[1430,14822,1432],{},[51,14824,1435],{},[51,14826,53],{},[51,14828,495],{"mathvariant":494},[55,14830,14831],{"encoding":57},"W_1(\\mu,\\nu)=\\int_{\\mathbb R}\\bigl|F_\\mu(x)-F_\\nu(x)\\bigr|\\,dx .",[31,14833,14835,14908,15054],{"className":14834,"ariaHidden":62},[61],[31,14836,14838,14841,14881,14884,14887,14890,14893,14896,14899,14902,14905],{"className":14837},[66],[31,14839],{"className":14840,"style":260},[70],[31,14842,14844,14847],{"className":14843},[75],[31,14845,3117],{"className":14846,"style":949},[75,76],[31,14848,14850],{"className":14849},[1039],[31,14851,14853,14873],{"className":14852},[1043,1044],[31,14854,14856,14870],{"className":14855},[1048],[31,14857,14859],{"className":14858,"style":2028},[1052],[31,14860,14861,14864],{"style":1056},[31,14862],{"className":14863,"style":1061},[1060],[31,14865,14867],{"className":14866},[1065,1066,1067,1068],[31,14868,205],{"className":14869},[75,1068],[31,14871,1076],{"className":14872},[1075],[31,14874,14876],{"className":14875},[1048],[31,14877,14879],{"className":14878,"style":2050},[1052],[31,14880],{},[31,14882,211],{"className":14883},[264],[31,14885,852],{"className":14886},[75,76],[31,14888,1210],{"className":14889},[1391],[31,14891],{"className":14892,"style":1467},[119],[31,14894,884],{"className":14895,"style":900},[75,76],[31,14897,220],{"className":14898},[289],[31,14900],{"className":14901,"style":293},[119],[31,14903,223],{"className":14904},[297],[31,14906],{"className":14907,"style":293},[119],[31,14909,14911,14914,14959,14996,15036,15039,15042,15045,15048,15051],{"className":14910},[66],[31,14912],{"className":14913,"style":1616},[70],[31,14915,14917,14920],{"className":14916},[1460],[31,14918,1409],{"className":14919,"style":1688},[1460,1461,1687],[31,14921,14923],{"className":14922},[1039],[31,14924,14926,14951],{"className":14925},[1043,1044],[31,14927,14929,14948],{"className":14928},[1048],[31,14930,14933],{"className":14931,"style":14932},[1052],"height:-0.4297em;",[31,14934,14935,14938],{"style":1704},[31,14936],{"className":14937,"style":1061},[1060],[31,14939,14941],{"className":14940},[1065,1066,1067,1068],[31,14942,14944],{"className":14943},[75,1068],[31,14945,14790],{"className":14946},[75,14947,1068],"mathbb",[31,14949,1076],{"className":14950},[1075],[31,14952,14954],{"className":14953},[1048],[31,14955,14957],{"className":14956,"style":1726},[1052],[31,14958],{},[31,14960,14962],{"className":14961},[264],[31,14963,14965],{"className":14964},[1307,13319],[31,14966,14968,14988],{"className":14967},[1043,1044],[31,14969,14971,14985],{"className":14970},[1048],[31,14972,14974],{"className":14973,"style":13329},[1052],[31,14975,14976,14979],{"style":13332},[31,14977],{"className":14978,"style":13336},[1060],[31,14980,14981],{"style":13339},[13341,14982,14983],{"xmlns":13343,"width":13344,"height":1178,"viewBox":13345},[13347,14984],{"d":13349},[31,14986,1076],{"className":14987},[1075],[31,14989,14991],{"className":14990},[1048],[31,14992,14994],{"className":14993,"style":13359},[1052],[31,14995],{},[31,14997,14999,15002],{"className":14998},[75],[31,15000,13114],{"className":15001,"style":949},[75,76],[31,15003,15005],{"className":15004},[1039],[31,15006,15008,15028],{"className":15007},[1043,1044],[31,15009,15011,15025],{"className":15010},[1048],[31,15012,15014],{"className":15013,"style":3230},[1052],[31,15015,15016,15019],{"style":1056},[31,15017],{"className":15018,"style":1061},[1060],[31,15020,15022],{"className":15021},[1065,1066,1067,1068],[31,15023,852],{"className":15024},[75,76,1068],[31,15026,1076],{"className":15027},[1075],[31,15029,15031],{"className":15030},[1048],[31,15032,15034],{"className":15033,"style":1083},[1052],[31,15035],{},[31,15037,211],{"className":15038},[264],[31,15040,53],{"className":15041},[75,76],[31,15043,220],{"className":15044},[289],[31,15046],{"className":15047,"style":120},[119],[31,15049,97],{"className":15050},[124],[31,15052],{"className":15053,"style":120},[119],[31,15055,15057,15060,15100,15103,15106,15109,15146,15149,15152,15155],{"className":15056},[66],[31,15058],{"className":15059,"style":4988},[70],[31,15061,15063,15066],{"className":15062},[75],[31,15064,13114],{"className":15065,"style":949},[75,76],[31,15067,15069],{"className":15068},[1039],[31,15070,15072,15092],{"className":15071},[1043,1044],[31,15073,15075,15089],{"className":15074},[1048],[31,15076,15078],{"className":15077,"style":3230},[1052],[31,15079,15080,15083],{"style":1056},[31,15081],{"className":15082,"style":1061},[1060],[31,15084,15086],{"className":15085},[1065,1066,1067,1068],[31,15087,884],{"className":15088,"style":900},[75,76,1068],[31,15090,1076],{"className":15091},[1075],[31,15093,15095],{"className":15094},[1048],[31,15096,15098],{"className":15097,"style":2050},[1052],[31,15099],{},[31,15101,211],{"className":15102},[264],[31,15104,53],{"className":15105},[75,76],[31,15107,220],{"className":15108},[289],[31,15110,15112],{"className":15111},[289],[31,15113,15115],{"className":15114},[1307,13319],[31,15116,15118,15138],{"className":15117},[1043,1044],[31,15119,15121,15135],{"className":15120},[1048],[31,15122,15124],{"className":15123,"style":13329},[1052],[31,15125,15126,15129],{"st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CDF 曲线之间的面积，就是搬运的总量。拿刚才的薪资例验一下：在 ",[31,15162,15164,15188],{"className":15163,"translate":35},[34],[31,15165,15167],{"className":15166},[39],[41,15168,15169],{"xmlns":43},[45,15170,15171,15185],{},[48,15172,15173,15176,15178,15180,15182],{},[95,15174,15175],{"stretchy":210},"[",[91,15177,1977],{},[95,15179,1210],{"separator":62},[91,15181,205],{},[95,15183,15184],{"stretchy":210},"]",[55,15186,15187],{"encoding":57},"[0,1]",[31,15189,15191],{"className":15190,"ariaHidden":62},[61],[31,15192,15194,15197,15200,15203,15206,15209,15212],{"className":15193},[66],[31,15195],{"className":15196,"style":260},[70],[31,15198,15175],{"className":15199},[264],[31,15201,1977],{"className":15202},[75],[31,15204,1210],{"className":15205},[1391],[31,15207],{"className":15208,"style":1467},[119],[31,15210,205],{"className":15211},[75],[31,15213,15184],{"className":15214},[289]," 上 ",[31,15217,15219,15243],{"className":15218,"translate":35},[34],[31,15220,15222],{"className":15221},[39],[41,15223,15224],{"xmlns":43},[45,15225,15226,15240],{},[48,15227,15228,15230,15232,15234,15236,15238],{},[51,15229,13114],{},[95,15231,211],{"stretchy":210},[51,15233,53],{},[95,15235,220],{"stretchy":210},[95,15237,223],{},[51,15239,53],{},[55,15241,15242],{"encoding":57},"F(x)=x",[31,15244,15246,15273],{"className":15245,"ariaHidden":62},[61],[31,15247,15249,15252,15255,15258,15261,15264,15267,15270],{"className":15248},[66],[31,15250],{"className":15251,"style":260},[70],[31,15253,13114],{"className":15254,"style":949},[75,76],[31,15256,211],{"className":15257},[264],[31,15259,53],{"className":15260},[75,76],[31,15262,220],{"className":15263},[289],[31,15265],{"className":15266,"style":293},[119],[31,15268,223],{"className":15269},[297],[31,15271],{"className":15272,"style":293},[119],[31,15274,15276,15279],{"className":15275},[66],[31,15277],{"className":15278,"style":71},[70],[31,15280,53],{"className":15281},[75,76],"、",[31,15284,15286,15315],{"className":15285,"translate":35},[34],[31,15287,15289],{"className":15288},[39],[41,15290,15291],{"xmlns":43},[45,15292,15293,15312],{},[48,15294,15295,15298,15300,15302,15304,15306,15308,15310],{},[51,15296,15297],{},"G",[95,15299,211],{"stretchy":210},[51,15301,53],{},[95,15303,220],{"stretchy":210},[95,15305,223],{},[51,15307,53],{},[51,15309,3768],{"mathvariant":494},[91,15311,152],{},[55,15313,15314],{"encoding":57},"G(x)=x\u002F2",[31,15316,15318,15345],{"className":15317,"ariaHidden":62},[61],[31,15319,15321,15324,15327,15330,15333,15336,15339,15342],{"className":15320},[66],[31,15322],{"className":15323,"style":260},[70],[31,15325,15297],{"className":15326},[75,76],[31,15328,211],{"className":15329},[264],[31,15331,53],{"className":15332},[75,76],[31,15334,220],{"className":15335},[289],[31,15337],{"className":15338,"style":293},[119],[31,15340,223],{"className":15341},[297],[31,15343],{"className":15344,"style":293},[119],[31,15346,15348,15351,15354],{"className":15347},[66],[31,15349],{"className":15350,"style":260},[70],[31,15352,53],{"className":15353},[75,76],[31,15355,15357],{"className":15356},[75],"\u002F2","，差是 ",[31,15360,15362,15380],{"className":15361,"translate":35},[34],[31,15363,15365],{"className":15364},[39],[41,15366,15367],{"xmlns":43},[45,15368,15369,15377],{},[48,15370,15371,15373,15375],{},[51,15372,53],{},[51,15374,3768],{"mathvariant":494},[91,15376,152],{},[55,15378,15379],{"encoding":57},"x\u002F2",[31,15381,15383],{"className":15382,"ariaHidden":62},[61],[31,15384,15386,15389,15392],{"className":15385},[66],[31,15387],{"className":15388,"style":260},[70],[31,15390,53],{"className":15391},[75,76],[31,15393,15357],{"className":15394},[75],"，积出 ",[31,15397,15399,15420],{"className":15398,"translate":35},[34],[31,15400,15402],{"className":15401},[39],[41,15403,15404],{"xmlns":43},[45,15405,15406,15417],{},[48,15407,15408],{},[2071,15409,15410],{"scriptlevel":1977,"displaystyle":210},[2074,15411,15412,15414],{},[91,15413,205],{},[91,15415,15416],{},"4",[55,15418,15419],{"encoding":57},"\\tfrac14",[31,15421,15423],{"className":15422,"ariaHidden":62},[61],[31,15424,15426,15429],{"className":15425},[66],[31,15427],{"className":15428,"style":2132},[70],[31,15430,15432,15435,15494],{"className":15431},[75],[31,15433],{"className":15434},[264,2139],[31,15436,15438],{"className":15437},[2074],[31,15439,15441,15486],{"className":15440},[1043,1044],[31,15442,15444,15483],{"className":15443},[1048],[31,15445,15447,15461,15469],{"className":15446,"style":2152},[1052],[31,15448,15449,15452],{"style":2155},[31,15450],{"className":15451,"style":1637},[1060],[31,15453,15455],{"className":15454},[1065,1066,1067,1068],[31,15456,15458],{"className":15457},[75,1068],[31,15459,15416],{"className":15460},[75,1068],[31,15462,15463,15466],{"style":2170},[31,15464],{"className":15465,"style":1637},[1060],[31,15467],{"className":15468,"style":2178},[2177],[31,15470,15471,15474],{"style":2181},[31,15472],{"className":15473,"style":1637},[1060],[31,15475,15477],{"className":15476},[1065,1066,1067,1068],[31,15478,15480],{"className":15479},[75,1068],[31,15481,205],{"className":15482},[75,1068],[31,15484,1076],{"className":15485},[1075],[31,15487,15489],{"className":15488},[1048],[31,15490,15492],{"className":15491,"style":2203},[1052],[31,15493],{},[31,15495],{"className":15496},[289,2139],"；在 ",[31,15499,15501,15523],{"className":15500,"translate":35},[34],[31,15502,15504],{"className":15503},[39],[41,15505,15506],{"xmlns":43},[45,15507,15508,15520],{},[48,15509,15510,15512,15514,15516,15518],{},[95,15511,15175],{"stretchy":210},[91,15513,205],{},[95,15515,1210],{"separator":62},[91,15517,152],{},[95,15519,15184],{"stretchy":210},[55,15521,15522],{"encoding":57},"[1,2]",[31,15524,15526],{"className":15525,"ariaHidden":62},[61],[31,15527,15529,15532,15535,15538,15541,15544,15547],{"className":15528},[66],[31,15530],{"className":15531,"style":260},[70],[31,15533,15175],{"className":15534},[264],[31,15536,205],{"className":15537},[75],[31,15539,1210],{"className":15540},[1391],[31,15542],{"className":15543,"style":1467},[119],[31,15545,152],{"className":15546},[75],[31,15548,15184],{"className":15549},[289],[31,15551,15553,15577],{"className":15552,"translate":35},[34],[31,15554,15556],{"className":15555},[39],[41,15557,15558],{"xmlns":43},[45,15559,15560,15574],{},[48,15561,15562,15564,15566,15568,15570,15572],{},[51,15563,13114],{},[95,15565,211],{"stretchy":210},[51,15567,53],{},[95,15569,220],{"stretchy":210},[95,15571,223],{},[91,15573,205],{},[55,15575,15576],{"encoding":57},"F(x)=1",[31,15578,15580,15607],{"className":15579,"ariaHidden":62},[61],[31,15581,15583,15586,15589,15592,15595,15598,15601,15604],{"className":15582},[66],[31,15584],{"className":15585,"style":260},[70],[31,15587,13114],{"className":15588,"style":949},[75,76],[31,15590,211],{"className":15591},[264],[31,15593,53],{"className":15594},[75,76],[31,15596,220],{"className":15597},[289],[31,15599],{"className":15600,"style":293},[119],[31,15602,223],{"className":15603},[297],[31,15605],{"className":15606,"style":293},[119],[31,15608,15610,15613],{"className":15609},[66],[31,15611],{"className":15612,"style":326},[70],[31,15614,205],{"className":15615},[75],[31,15617,15619,15641],{"className":15618,"translate":35},[34],[31,15620,15622],{"className":15621},[39],[41,15623,15624],{"xmlns":43},[45,15625,15626,15638],{},[48,15627,15628,15630,15632,15634,15636],{},[91,15629,205],{},[95,15631,97],{},[51,15633,53],{},[51,15635,3768],{"mathvariant":494},[91,15637,152],{},[55,15639,15640],{"encoding":57},"1-x\u002F2",[31,15642,15644,15662],{"className":15643,"ariaHidden":62},[61],[31,15645,15647,15650,15653,15656,15659],{"className":15646},[66],[31,15648],{"className":15649,"style":112},[70],[31,15651,205],{"className":15652},[75],[31,15654],{"className":15655,"style":120},[119],[31,15657,97],{"className":15658},[124],[31,15660],{"className":15661,"style":120},[119],[31,15663,15665,15668,15671],{"className":15664},[66],[31,15666],{"className":15667,"style":260},[70],[31,15669,53],{"className":15670},[75,76],[31,15672,15357],{"className":15673},[75],"，也积出 ",[31,15676,15678,15697],{"className":15677,"translate":35},[34],[31,15679,15681],{"className":15680},[39],[41,15682,15683],{"xmlns":43},[45,15684,15685,15695],{},[48,15686,15687],{},[2071,15688,15689],{"scriptlevel":1977,"displaystyle":210},[2074,15690,15691,15693],{},[91,15692,205],{},[91,15694,15416],{},[55,15696,15419],{"encoding":57},[31,15698,15700],{"className":15699,"ariaHidden":62},[61],[31,15701,15703,15706],{"className":15702},[66],[31,15704],{"className":15705,"style":2132},[70],[31,15707,15709,15712,15771],{"className":15708},[75],[31,15710],{"className":15711},[264,2139],[31,15713,15715],{"className":15714},[2074],[31,15716,15718,15763],{"className":15717},[1043,1044],[31,15719,15721,15760],{"className":15720},[1048],[31,15722,15724,15738,15746],{"className":15723,"style":2152},[1052],[31,15725,15726,15729],{"style":2155},[31,15727],{"className":15728,"style":1637},[1060],[31,15730,15732],{"className":15731},[1065,1066,1067,1068],[31,15733,15735],{"className":15734},[75,1068],[31,15736,15416],{"className":15737},[75,1068],[31,15739,15740,15743],{"style":2170},[31,15741],{"className":15742,"style":1637},[1060],[31,15744],{"className":15745,"style":2178},[2177],[31,15747,15748,15751],{"style":2181},[31,15749],{"className":15750,"style":1637},[1060],[31,15752,15754],{"className":15753},[1065,1066,1067,1068],[31,15755,15757],{"className":15756},[75,1068],[31,15758,205],{"className":15759},[75,1068],[31,15761,1076],{"className":15762},[1075],[31,15764,15766],{"className":15765},[1048],[31,15767,15769],{"className":15768,"style":2203},[1052],[31,15770],{},[31,15772],{"className":15773},[289,2139],"。合计 ",[31,15776,15778,15798],{"className":15777,"translate":35},[34],[31,15779,15781],{"className":15780},[39],[41,15782,15783],{"xmlns":43},[45,15784,15785,15795],{},[48,15786,15787],{},[2071,15788,15789],{"scriptlevel":1977,"displaystyle":210},[2074,15790,15791,15793],{},[91,15792,205],{},[91,15794,152],{},[55,15796,15797],{"encoding":57},"\\tfrac12",[31,15799,15801],{"className":15800,"ariaHidden":62},[61],[31,15802,15804,15807],{"className":15803},[66],[31,15805],{"className":15806,"style":2132},[70],[31,15808,15810,15813,15872],{"className":15809},[75],[31,15811],{"className":15812},[264,2139],[31,15814,15816],{"className":15815},[2074],[31,15817,15819,15864],{"className":15818},[1043,1044],[31,15820,15822,15861],{"className":15821},[1048],[31,15823,15825,15839,15847],{"className":15824,"style":2152},[1052],[31,15826,15827,15830],{"style":2155},[31,15828],{"className":15829,"style":1637},[1060],[31,15831,15833],{"className":15832},[1065,1066,1067,1068],[31,15834,15836],{"className":15835},[75,1068],[31,15837,152],{"className":15838},[75,1068],[31,15840,15841,15844],{"style":2170},[31,15842],{"className":15843,"style":1637},[1060],[31,15845],{"className":15846,"style":2178},[2177],[31,15848,15849,15852],{"style":2181},[31,15850],{"className":15851,"style":1637},[1060],[31,15853,15855],{"className":15854},[1065,1066,1067,1068],[31,15856,15858],{"className":15857},[75,1068],[31,15859,205],{"className":15860},[75,1068],[31,15862,1076],{"className":15863},[1075],[31,15865,15867],{"className":15866},[1048],[31,15868,15870],{"className":15869,"style":2203},[1052],[31,15871],{},[31,15873],{"className":15874},[289,2139],"，和分位数算法的结果一致。",[11,15877,15878,15879,16013,16014,16178,16179,16359],{},"一维之所以这么简单，是因为凸代价下最优耦合是“单调”的：不会出现 ",[31,15880,15882,15908],{"className":15881,"translate":35},[34],[31,15883,15885],{"className":15884},[39],[41,15886,15887],{"xmlns":43},[45,15888,15889,15905],{},[48,15890,15891,15897,15899],{},[1004,15892,15893,15895],{},[51,15894,53],{},[91,15896,205],{},[95,15898,9758],{},[1004,15900,15901,15903],{},[51,15902,53],{},[91,15904,152],{},[55,15906,15907],{"encoding":57},"x_1\u003Cx_2",[31,15909,15911,15967],{"className":15910,"ariaHidden":62},[61],[31,15912,15914,15918,15958,15961,15964],{"className":15913},[66],[31,15915],{"className":15916,"style":15917},[70],"height:0.6891em;vertical-align:-0.15em;",[31,15919,15921,15924],{"className":15920},[75],[31,15922,53],{"className":15923},[75,76],[31,15925,15927],{"className":15926},[1039],[31,15928,15930,15950],{"className":15929},[1043,1044],[31,15931,15933,15947],{"className":15932},[1048],[31,15934,15936],{"className":15935,"style":2028},[1052],[31,15937,15938,15941],{"style":2807},[31,15939],{"className":15940,"style":1061},[1060],[31,15942,15944],{"className":15943},[1065,1066,1067,1068],[31,15945,205],{"className":15946},[75,1068],[31,15948,1076],{"className":15949},[1075],[31,15951,15953],{"className":15952},[1048],[31,15954,15956],{"className":15955,"style":2050},[1052],[31,15957],{},[31,15959],{"className":15960,"style":293},[119],[31,15962,9758],{"className":15963},[297],[31,15965],{"className":15966,"style":293},[119],[31,15968,15970,15973],{"className":15969},[66],[31,15971],{"className":15972,"style":7090},[70],[31,15974,15976,15979],{"className":15975},[75],[31,15977,53],{"className":15978},[75,76],[31,15980,15982],{"className":15981},[1039],[31,15983,15985,16005],{"className":15984},[1043,1044],[31,15986,15988,16002],{"className":15987},[1048],[31,15989,15991],{"className":15990,"style":2028},[1052],[31,15992,15993,15996],{"style":2807},[31,15994],{"className":15995,"style":1061},[1060],[31,15997,15999],{"className":15998},[1065,1066,1067,1068],[31,16000,152],{"className":16001},[75,1068],[31,16003,1076],{"className":16004},[1075],[31,16006,16008],{"className":16007},[1048],[31,16009,16011],{"className":16010,"style":2050},[1052],[31,16012],{}," 却 ",[31,16015,16017,16056],{"className":16016,"translate":35},[34],[31,16018,16020],{"className":16019},[39],[41,16021,16022],{"xmlns":43},[45,16023,16024,16053],{},[48,16025,16026,16028,16030,16036,16038,16041,16043,16045,16051],{},[51,16027,919],{},[95,16029,211],{"stretchy":210},[1004,16031,16032,16034],{},[51,16033,53],{},[91,16035,205],{},[95,16037,220],{"stretchy":210},[95,16039,16040],{},">",[51,16042,919],{},[95,16044,211],{"stretchy":210},[1004,16046,16047,16049],{},[51,16048,53],{},[91,16050,152],{},[95,16052,220],{"stretchy":210},[55,16054,16055],{"encoding":57},"T(x_1)>T(x_2)",[31,16057,16059,16123],{"className":16058,"ariaHidden":62},[61],[31,16060,16062,16065,16068,16071,16111,16114,16117,16120],{"className":16061},[66],[31,16063],{"className":16064,"style":260},[70],[31,16066,919],{"className":16067,"style":949},[75,76],[31,16069,211],{"className":16070},[264],[31,16072,16074,16077],{"className":16073},[75],[31,16075,53],{"className":16076},[75,76],[31,16078,16080],{"className":16079},[1039],[31,16081,16083,16103],{"className":16082},[1043,1044],[31,16084,16086,16100],{"className":16085},[1048],[31,16087,16089],{"className":16088,"style":2028},[1052],[31,16090,16091,16094],{"style":2807},[31,16092],{"className":16093,"style":1061},[1060],[31,16095,16097],{"className":16096},[1065,1066,1067,1068],[31,16098,205],{"className":16099},[75,1068],[31,16101,1076],{"className":16102},[1075],[31,16104,16106],{"className":16105},[1048],[31,16107,16109],{"className":16108,"style":2050},[1052],[31,16110],{},[31,16112,220],{"className":16113},[289],[31,16115],{"className":16116,"style":293},[119],[31,16118,16040],{"className":16119},[297],[31,16121],{"className":16122,"style":293},[119],[31,16124,16126,16129,16132,16135,16175],{"className":16125},[66],[31,16127],{"className":16128,"style":260},[70],[31,16130,919],{"className":16131,"style":949},[75,76],[31,16133,211],{"className":16134},[264],[31,16136,16138,16141],{"className":16137},[75],[31,16139,53],{"className":16140},[75,76],[31,16142,16144],{"className":16143},[1039],[31,16145,16147,16167],{"className":16146},[1043,1044],[31,16148,16150,16164],{"className":16149},[1048],[31,16151,16153],{"className":16152,"style":2028},[1052],[31,16154,16155,16158],{"style":2807},[31,16156],{"className":16157,"style":1061},[1060],[31,16159,16161],{"className":16160},[1065,1066,1067,1068],[31,16162,152],{"className":16163},[75,1068],[31,16165,1076],{"className":16166},[1075],[31,16168,16170],{"className":16169},[1048],[31,16171,16173],{"className":16172,"style":2050},[1052],[31,16174],{},[31,16176,220],{"className":16177},[289],"，否则交换一下就更省。所以 ",[31,16180,16182,16219],{"className":16181,"translate":35},[34],[31,16183,16185],{"className":16184},[39],[41,16186,16187],{"xmlns":43},[45,16188,16189,16216],{},[48,16190,16191,16193,16195,16207,16210],{},[51,16192,919],{},[95,16194,223],{},[12397,16196,16197,16199,16201],{},[51,16198,13114],{},[51,16200,884],{},[48,16202,16203,16205],{},[95,16204,97],{},[91,16206,205],{},[95,16208,16209],{},"∘",[1004,16211,16212,16214],{},[51,16213,13114],{},[51,16215,852],{},[55,16217,16218],{"encoding":57},"T=F_\\nu^{-1}\\circ F_\\mu",[31,16220,16222,16240,16313],{"className":16221,"ariaHidden":62},[61],[31,16223,16225,16228,16231,16234,16237],{"className":16224},[66],[31,16226],{"className":16227,"style":945},[70],[31,16229,919],{"className":16230,"style":949},[75,76],[31,16232],{"className":16233,"style":293},[119],[31,16235,223],{"className":16236},[297],[31,16238],{"className":16239,"style":293},[119],[31,16241,16243,16247,16304,16307,16310],{"className":16242},[66],[31,16244],{"className":16245,"style":16246},[70],"height:1.0611em;vertical-align:-0.247em;",[31,16248,16250,16253],{"className":16249},[75],[31,16251,13114],{"className":16252,"style":949},[75,76],[31,16254,16256],{"className":16255},[1039],[31,16257,16259,16296],{"className":16258},[1043,1044],[31,16260,16262,16293],{"className":16261},[1048],[31,16263,16265,16276],{"className":16264,"style":5135},[1052],[31,16266,16267,16270],{"style":13196},[31,16268],{"className":16269,"style":1061},[1060],[31,16271,16273],{"className":16272},[1065,1066,1067,1068],[31,16274,884],{"className":16275,"style":900},[75,76,1068],[31,16277,16278,16281],{"style":3631},[31,16279],{"className":16280,"style":1061},[1060],[31,16282,16284],{"className":16283},[1065,1066,1067,1068],[31,16285,16287,16290],{"className":16286},[75,1068],[31,16288,97],{"className":16289},[75,1068],[31,16291,205],{"className":16292},[75,1068],[31,16294,1076],{"className":16295},[1075],[31,16297,16299],{"className":16298},[1048],[31,16300,16302],{"className":16301,"style":13499},[1052],[31,16303],{},[31,16305],{"className":16306,"style":120},[119],[31,16308,16209],{"className":16309},[124],[31,16311],{"className":16312,"style":120},[119],[31,16314,16316,16319],{"className":16315},[66],[31,16317],{"className":16318,"style":1029},[70],[31,16320,16322,16325],{"className":16321},[75],[31,16323,13114],{"className":16324,"style":949},[75,76],[31,16326,16328],{"className":16327},[1039],[31,16329,16331,16351],{"className":16330},[1043,1044],[31,16332,16334,16348],{"className":16333},[1048],[31,16335,16337],{"className":16336,"style":3230},[1052],[31,16338,16339,16342],{"style":1056},[31,16340],{"className":16341,"style":1061},[1060],[31,16343,16345],{"className":16344},[1065,1066,1067,1068],[31,16346,852],{"className":16347},[75,76,1068],[31,16349,1076],{"className":16350},[1075],[31,16352,16354],{"className":16353},[1048],[31,16355,16357],{"className":16356,"style":1083},[1052],[31,16358],{}," 就是答案——按排名对齐。多维没有这种排序，所有困难都从这里开始。",[11,16361,16362],{},"高斯之间有一个著名的闭式解：",[31,16364,16366],{"className":16365,"translate":35},[411],[31,16367,16369,16541],{"className":16368,"translate":35},[34],[31,16370,16372],{"className":16371},[39],[41,16373,16374],{"xmlns":43,"display":420},[45,16375,16376,16538],{},[48,16377,16378,16386,16388,16390,16392,16399,16401,16408,16410,16412,16414,16416,16422,16424,16430,16432,16434,16436,16438,16444,16446,16452,16458,16460,16466,16468,16474,16476,16482,16484,16486,16488,16502,16508,16522,16534,16536],{},[12397,16379,16380,16382,16384],{},[51,16381,3117],{},[91,16383,152],{},[91,16385,152],{},[95,16387,211],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[51,16389,4935],{"mathvariant":925},[95,16391,211],{"stretchy":210},[1004,16393,16394,16397],{},[51,16395,16396],{},"m",[91,16398,205],{},[95,16400,1210],{"separator":62},[1004,16402,16403,16406],{},[51,16404,16405],{"mathvariant":494},"Σ",[91,16407,205],{},[95,16409,220],{"stretchy":210},[95,16411,1210],{"separator":62},[51,16413,4935],{"mathvariant":925},[95,16415,211],{"stretchy":210},[1004,16417,16418,16420],{},[51,16419,16396],{},[91,16421,152],{},[95,16423,1210],{"separator":62},[1004,16425,16426,16428],{},[51,16427,16405],{"mathvariant":494},[91,16429,152],{},[95,16431,220],{"stretchy":210},[95,16433,220],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[95,16435,223],{},[51,16437,4572],{"mathvariant":494},[1004,16439,16440,16442],{},[51,16441,16396],{},[91,16443,205],{},[95,16445,97],{},[1004,16447,16448,16450],{},[51,16449,16396],{},[91,16451,152],{},[1180,16453,16454,16456],{},[51,16455,4572],{"mathvariant":494},[91,16457,152],{},[95,16459,434],{},[48,16461,16462,16464],{},[51,16463,12271],{"mathvariant":494},[51,16465,2549],{"mathvariant":494},[95,16467,211],{"fence":62,"stretchy":62,"minsize":2511,"maxsize":2511},[1004,16469,16470,16472],{},[51,16471,16405],{"mathvariant":494},[91,16473,205],{},[95,16475,434],{},[1004,16477,16478,16480],{},[51,16479,16405],{"mathvariant":494},[91,16481,152],{},[95,16483,97],{},[91,16485,152],{},[95,16487,211],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[12397,16489,16490,16492,16494],{},[51,16491,16405],{"mathvariant":494},[91,16493,205],{},[48,16495,16496,16498,16500],{},[91,16497,205],{},[51,16499,3768],{"mathvariant":494},[91,16501,152],{},[1004,16503,16504,16506],{},[51,16505,16405],{"mathvariant":494},[91,16507,152],{},[12397,16509,16510,16512,16514],{},[51,16511,16405],{"mathvariant":494},[91,16513,205],{},[48,16515,16516,16518,16520],{},[91,16517,205],{},[51,16519,3768],{"mathvariant":494},[91,16521,152],{},[1180,16523,16524,16526],{},[95,16525,220],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[48,16527,16528,16530,16532],{},[91,16529,205],{},[51,16531,3768],{"mathvariant":494},[91,16533,152],{},[95,16535,220],{"fence":62,"stretchy":62,"minsize":2511,"maxsize":2511},[51,16537,495],{"mathvariant":494},[55,16539,16540],{"encoding":57},"W_2^2\\bigl(\\mathcal N(m_1,\\Sigma_1),\\mathcal N(m_2,\\Sigma_2)\\bigr)\n=\\|m_1-m_2\\|^2+\\mathrm{tr}\\Bigl(\\Sigma_1+\\Sigma_2-2\\bigl(\\Sigma_1^{1\u002F2}\\Sigma_2\\Sigma_1^{1\u002F2}\\bigr)^{1\u002F2}\\Bigr).",[31,16542,16544,16818,16876,16961,17029,17084],{"className":16543,"ariaHidden":62},[61],[31,16545,16547,16550,16601,16607,16610,16613,16653,16656,16659,16699,16702,16705,16708,16711,16714,16754,16757,16760,16800,16803,16809,16812,16815],{"className":16546},[66],[31,16548],{"className":16549,"style":1297},[70],[31,16551,16553,16556],{"className":16552},[75],[31,16554,3117],{"className":16555,"style":949},[75,76],[31,16557,16559],{"className":16558},[1039],[31,16560,16562,16593],{"className":16561},[1043,1044],[31,16563,16565,16590],{"className":16564},[1048],[31,16566,16568,16579],{"className":16567,"style":1327},[1052],[31,16569,16570,16573],{"style":13196},[31,16571],{"className":16572,"style":1061},[1060],[31,16574,16576],{"className":16575},[1065,1066,1067,1068],[31,16577,152],{"className":16578},[75,1068],[31,16580,16581,16584],{"style":1330},[31,16582],{"className":16583,"style":1061},[1060],[31,16585,16587],{"className":16586},[1065,1066,1067,1068],[31,16588,152],{"className":16589},[75,1068],[31,16591,1076],{"className":16592},[1075],[31,16594,16596],{"className":16595},[1048],[31,16597,16599],{"className":16598,"style":13499},[1052],[31,16600],{},[31,16602,16604],{"className":16603},[264],[31,16605,211],{"className":16606},[1307,1308],[31,16608,4935],{"className":16609,"style":5004},[75,968],[31,16611,211],{"className":16612},[264],[31,16614,16616,16619],{"className":16615},[75],[31,16617,16396],{"className":16618},[75,76],[31,16620,16622],{"className":16621},[1039],[31,16623,16625,16645],{"className":16624},[1043,1044],[31,16626,16628,16642],{"className":16627},[1048],[31,16629,16631],{"className":16630,"style":2028},[1052],[31,16632,16633,16636],{"style":2807},[31,16634],{"className":16635,"style":1061},[1060],[31,16637,16639],{"className":16638},[1065,1066,1067,1068],[31,16640,205],{"className":16641},[75,1068],[31,16643,1076],{"className":16644},[1075],[31,16646,16648],{"className":16647},[1048],[31,16649,16651],{"className":16650,"style":2050},[1052],[31,16652],{},[31,16654,1210],{"className":16655},[1391],[31,16657],{"className":16658,"style":1467},[119],[31,16660,16662,16665],{"className":16661},[75],[31,16663,16405],{"className":16664},[75],[31,16666,16668],{"className":16667},[1039],[31,16669,16671,16691],{"className":16670},[1043,1044],[31,16672,16674,16688],{"className":16673},[1048],[31,16675,16677],{"className":16676,"style":2028},[1052],[31,16678,16679,16682],{"style":2807},[31,16680],{"className":16681,"style":1061},[1060],[31,16683,16685],{"className":16684},[1065,1066,1067,1068],[31,16686,205],{"className":16687},[75,1068],[31,16689,1076],{"className":16690},[1075],[31,16692,16694],{"className":16693},[1048],[31,16695,16697],{"className":16696,"style":2050},[1052],[31,16698],{},[31,16700,220],{"className":16701},[289],[31,16703,1210],{"className":16704},[1391],[31,16706],{"className":16707,"style":1467},[119],[31,16709,4935],{"className":16710,"style":5004},[75,968],[31,16712,211],{"className":16713},[264],[31,16715,16717,16720],{"className":16716},[75],[31,16718,16396],{"className":16719},[75,76],[31,16721,16723],{"className":16722},[1039],[31,16724,16726,16746],{"className":16725},[1043,1044],[31,16727,16729,16743],{"className":16728},[1048],[31,16730,16732],{"className":16731,"style":2028},[1052],[31,16733,16734,16737],{"style":2807},[31,16735],{"className":16736,"style":1061},[1060],[31,16738,16740],{"className":16739},[1065,1066,1067,1068],[31,16741,152],{"className":16742},[75,1068],[31,16744,1076],{"className":16745},[1075],[31,16747,16749],{"className":16748},[1048],[31,16750,16752],{"className":16751,"style":2050},[1052],[31,16753],{},[31,16755,1210],{"className":16756},[1391],[31,16758],{"className":16759,"style":1467},[119],[31,16761,16763,16766],{"className":16762},[75],[31,16764,16405],{"className":16765},[75],[31,16767,16769],{"className":16768},[1039],[31,16770,16772,16792],{"className":16771},[1043,1044],[31,16773,16775,16789],{"className":16774},[1048],[31,16776,16778],{"className":16777,"style":2028},[1052],[31,16779,16780,16783],{"style":2807},[31,16781],{"className":16782,"style":1061},[1060],[31,16784,16786],{"className":16785},[1065,1066,1067,1068],[31,16787,152],{"className":16788},[75,1068],[31,16790,1076],{"className":16791},[1075],[31,16793,16795],{"className":16794},[1048],[31,16796,16798],{"className":16797,"style":2050},[1052],[31,16799],{},[31,16801,220],{"className":16802},[289],[31,16804,16806],{"className":16805},[289],[31,16807,220],{"className":16808},[1307,1308],[31,16810],{"className":16811,"style":293},[119],[31,16813,223],{"className":16814},[297],[31,16816],{"className":16817,"style":293},[119],[31,16819,16821,16824,16827,16867,16870,16873],{"className":16820},[66],[31,16822],{"className":16823,"style":260},[70],[31,16825,4572],{"className":16826},[75],[31,16828,16830,16833],{"cla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",[31,17301,17303,17362],{"className":17302,"translate":35},[34],[31,17304,17306],{"className":17305},[39],[41,17307,17308],{"xmlns":43},[45,17309,17310,17359],{},[48,17311,17312,17314,17320,17322,17328,17334,17336,17338,17345,17347,17353],{},[95,17313,211],{"stretchy":210},[1004,17315,17316,17318],{},[51,17317,16396],{},[91,17319,205],{},[95,17321,97],{},[1004,17323,17324,17326],{},[51,17325,16396],{},[91,17327,152],{},[1180,17329,17330,17332],{},[95,17331,220],{"stretchy":210},[91,17333,152],{},[95,17335,434],{},[95,17337,211],{"stretchy":210},[1004,17339,17340,17343],{},[51,17341,17342],{},"σ",[91,17344,205],{},[95,17346,97],{},[1004,17348,17349,17351],{},[51,17350,17342],{},[91,17352,152],{},[1180,17354,17355,17357],{},[95,17356,220],{"stretchy":210},[91,17358,152],{},[55,17360,17361],{"encoding":57},"(m_1-m_2)^2+(\\sigma_1-\\sigma_2)^2",[31,17363,17365,17423,17507,17565],{"className":17364,"ariaHidden":62},[61],[31,17366,17368,17371,17374,17414,17417,17420],{"className":17367},[66],[31,17369],{"className":17370,"style":260},[70],[31,17372,211],{"className":17373},[264],[31,17375,17377,17380],{"className":17376},[75],[31,17378,16396],{"className":17379},[75,76],[31,17381,17383],{"className":17382},[1039],[31,17384,17386,17406],{"className":17385},[1043,1044],[31,17387,17389,17403],{"className":17388},[1048],[31,17390,17392],{"className":17391,"style":2028},[1052],[31,17393,17394,17397],{"style":2807},[31,17395],{"className":17396,"style":1061},[1060],[31,17398,17400],{"className":17399},[1065,1066,1067,1068],[31,17401,205],{"className":17402},[75,1068],[31,17404,1076],{"className":17405},[1075],[31,17407,17409],{"className":17408},[1048],[31,17410,17412],{"className":17411,"style":2050},[1052],[31,17413],{},[31,17415],{"className":17416,"style":120},[119],[31,17418,97],{"className":17419},[124],[31,17421],{"className":17422,"style":120},[119],[31,17424,17426,17429,17469,17498,17501,17504],{"className":17425},[66],[31,17427],{"className":17428,"style":12035},[70],[31,17430,17432,17435],{"className":17431},[75],[31,17433,16396],{"className":17434},[75,76],[31,17436,17438],{"className":17437},[1039],[31,17439,17441,17461],{"className":17440},[1043,1044],[31,17442,17444,17458],{"className":17443},[1048],[31,17445,17447],{"className":17446,"style":2028},[1052],[31,17448,17449,17452],{"style":2807},[31,17450],{"className":17451,"style":1061},[1060],[31,17453,17455],{"className":17454},[1065,1066,1067,1068],[31,17456,152],{"className":17457},[75,1068],[31,17459,1076],{"className":17460},[1075],[31,17462,17464],{"className":17463},[1048],[31,17465,17467],{"className":17466,"style":2050},[1052],[31,17468],{},[31,17470,17472,17475],{"className":17471},[289],[31,17473,220],{"className":17474},[289],[31,17476,17478],{"className":17477},[1039],[31,17479,17481],{"className":17480},[1043],[31,17482,17484],{"className":17483},[1048],[31,17485,17487],{"className":17486,"style":5135},[1052],[31,17488,17489,17492],{"style":3631},[31,17490],{"className":17491,"style":1061},[1060],[31,17493,17495],{"className":17494},[1065,1066,1067,1068],[31,17496,152],{"className":17497},[75,1068],[31,17499],{"className":17500,"style":120},[119],[31,17502,434],{"className":17503},[124],[31,17505],{"className":17506,"style":120},[119],[31,17508,17510,17513,17516,17556,17559,17562],{"className":17509},[66],[31,17511],{"className":17512,"style":260},[70],[31,17514,211],{"className":17515},[264],[31,17517,17519,17522],{"className":17518},[75],[31,17520,17342],{"className":17521,"style":2666},[75,76],[31,17523,17525],{"className":17524},[1039],[31,17526,17528,17548],{"className":17527},[1043,1044],[31,17529,17531,17545],{"className":17530},[1048],[31,17532,17534],{"className":17533,"style":2028},[1052],[31,17535,17536,17539],{"style":5504},[31,17537],{"className":17538,"style":1061},[1060],[31,17540,17542],{"className":17541},[1065,1066,1067,1068],[31,17543,205],{"className":17544},[75,1068],[31,17546,1076],{"className":17547},[1075],[31,17549,17551],{"className":17550},[1048],[31,17552,17554],{"className":17553,"style":2050},[1052],[31,17555],{},[31,17557],{"className":17558,"style":120},[119],[31,17560,97],{"className":17561},[124],[31,17563],{"className":17564,"style":120},[119],[31,17566,17568,17571,17611],{"className":17567},[66],[31,17569],{"className":17570,"style":12035},[70],[31,17572,17574,17577],{"className":17573},[75],[31,17575,17342],{"className":17576,"style":2666},[75,76],[31,17578,17580],{"className":17579},[1039],[31,17581,17583,17603],{"className":17582},[1043,1044],[31,17584,17586,17600],{"className":17585},[1048],[31,17587,17589],{"className":17588,"style":2028},[1052],[31,17590,17591,17594],{"style":5504},[31,17592],{"className":17593,"style":1061},[1060],[31,17595,17597],{"className":17596},[1065,1066,1067,1068],[31,17598,152],{"className":17599},[75,1068],[31,17601,1076],{"className":17602},[1075],[31,17604,17606],{"className":17605},[1048],[31,17607,17609],{"className":17608,"style":2050},[1052],[31,17610],{},[31,17612,17614,17617],{"className":17613},[289],[31,17615,220],{"className":17616},[289],[31,17618,17620],{"className":17619},[1039],[31,17621,17623],{"className":17622},[1043],[31,17624,17626],{"className":17625},[1048],[31,17627,17629],{"className":17628,"style":5135},[1052],[31,17630,17631,17634],{"style":3631},[31,17632],{"className":17633,"style":1061},[1060],[31,17635,17637],{"className":17636},[1065,1066,1067,1068],[31,17638,152],{"className":17639},[75,1068],"，验算一下：",[31,17642,17644,17728],{"className":17643,"translate":35},[34],[31,17645,17647],{"className":17646},[39],[41,17648,17649],{"xmlns":43},[45,17650,17651,17725],{},[48,17652,17653,17659,17661,17669,17671,17679,17681,17683,17685,17691,17697,17699,17701,17703,17705,17711,17713,17719],{},[48,17654,17655,17657],{},[51,17656,12271],{"mathvariant":494},[51,17658,2549],{"mathvariant":494},[95,17660,211],{"stretchy":210},[12397,17662,17663,17665,17667],{},[51,17664,17342],{},[91,17666,205],{},[91,17668,152],{},[95,17670,434],{},[12397,17672,17673,17675,17677],{},[51,17674,17342],{},[91,17676,152],{},[91,17678,152],{},[95,17680,97],{},[91,17682,152],{},[51,17684,12680],{"mathvariant":494},[1004,17686,17687,17689],{},[51,17688,17342],{},[91,17690,205],{},[1004,17692,17693,17695],{},[51,17694,17342],{},[91,17696,152],{},[51,17698,12680],{"mathvariant":494},[95,17700,220],{"stretchy":210},[95,17702,223],{},[95,17704,211],{"stretchy":210},[1004,17706,17707,17709],{},[51,17708,17342],{},[91,17710,205],{},[95,17712,97],{},[1004,17714,17715,17717],{},[51,17716,17342],{},[91,17718,152],{},[1180,17720,17721,17723],{},[95,17722,220],{"stretchy":210},[91,17724,152],{},[55,17726,17727],{"encoding":57},"\\mathrm{tr}(\\sigma_1^2+\\sigma_2^2-2|\\sigma_1\\sigma_2|)=(\\sigma_1-\\sigma_2)^2",[31,17729,17731,17808,17875,17980,18038],{"className":17730,"ariaHidden":62},[61],[31,17732,17734,17737,17743,17746,17799,17802,17805],{"className":17733},[66],[31,17735],{"className":17736,"style":12035},[70],[31,17738,17740],{"className":17739},[75],[31,17741,16973],{"className":17742},[75,2745],[31,17744,211],{"className":17745},[264],[31,17747,17749,17752],{"className":17748},[75],[31,17750,17342],{"className":17751,"style":2666},[75,76],[31,17753,17755],{"className":17754},[1039],[31,17756,17758,17790],{"className":17757},[1043,1044],[31,17759,17761,17787],{"className":17760},[1048],[31,17762,17764,17776],{"className":17763,"style":5135},[1052],[31,17765,17767,17770],{"style":17766},"top:-2.4519em;margin-left:-0.0359em;margin-right:0.05em;",[31,17768],{"className":17769,"style":1061},[1060],[31,17771,17773],{"className":17772},[1065,1066,1067,1068],[31,17774,205],{"className":17775},[75,1068],[31,17777,17778,17781],{"style":3631},[31,17779],{"className":17780,"style":1061},[1060],[31,17782,17784],{"className":17783},[1065,1066,1067,1068],[31,17785,152],{"className":17786},[75,1068],[31,17788,1076],{"className":17789},[1075],[31,17791,17793],{"className":17792},[1048],[31,17794,17797],{"className":17795,"style":17796},[1052],"height:0.2481em;",[31,17798],{},[31,17800],{"className":17801,"style":120},[119],[31,17803,434],{"className":17804},[124],[31,17806],{"className":17807,"style":120},[119],[31,17809,17811,17815,17866,17869,17872],{"className":17810},[66],[31,17812],{"className":17813,"style":17814},[70],"height:1.0622em;vertical-align:-0.2481em;",[31,17816,17818,17821],{"className":17817},[75],[31,17819,17342],{"className":17820,"style":2666},[75,76],[31,17822,17824],{"className":17823},[1039],[31,17825,17827,17858],{"className":17826},[1043,1044],[31,17828,17830,17855],{"className":17829},[1048],[31,17831,17833,17844],{"className":17832,"style":5135},[1052],[31,17834,17835,17838],{"style":17766},[31,17836],{"className":17837,"style":1061},[1060],[31,17839,17841],{"className":17840},[1065,1066,1067,1068],[31,17842,152],{"className":17843},[75,1068],[31,17845,17846,17849],{"style":3631},[31,17847],{"className":17848,"style":1061},[1060],[31,17850,17852],{"className":17851},[1065,1066,1067,1068],[31,17853,152],{"className":17854},[75,1068],[31,17856,1076],{"className":17857},[1075],[31,17859,17861],{"className":17860},[1048],[31,17862,17864],{"className":17863,"style":17796},[1052],[31,17865],{},[31,17867],{"className":17868,"style":120},[119],[31,17870,97],{"className":17871},[124],[31,17873],{"className":17874,"style":120},[119],[31,17876,17878,17881,17885,17925,17965,17968,17971,17974,17977],{"className":17877},[66],[31,17879],{"className":17880,"style":260},[70],[31,17882,17884],{"className":17883},[75],"2∣",[31,17886,17888,17891],{"className":17887},[75],[31,17889,17342],{"className":17890,"style":2666},[75,76],[31,17892,17894],{"className":17893},[1039],[31,17895,17897,17917],{"className":17896},[1043,1044],[31,17898,17900,17914],{"className":17899},[1048],[31,17901,17903],{"className":17902,"style":2028},[1052],[31,17904,17905,17908],{"style":5504},[31,17906],{"className":17907,"style":1061},[1060],[31,17909,17911],{"className":17910},[1065,1066,1067,1068],[31,17912,205],{"className":17913},[75,1068],[31,17915,1076],{"className":17916},[1075],[31,17918,17920],{"className":17919},[1048],[31,17921,17923],{"className":17922,"style":2050},[1052],[31,17924],{},[31,17926,17928,17931],{"className":17927},[75],[31,17929,17342],{"className":17930,"style":2666},[75,76],[31,17932,17934],{"className":17933},[1039],[31,17935,17937,17957],{"className":17936},[1043,1044],[31,17938,17940,17954],{"className":17939},[1048],[31,17941,17943],{"className":17942,"style":2028},[1052],[31,17944,17945,17948],{"style":5504},[31,17946],{"className":17947,"style":1061},[1060],[31,17949,17951],{"className":17950},[1065,1066,1067,1068],[31,17952,152],{"className":17953},[75,1068],[31,17955,1076],{"className":17956},[1075],[31,17958,17960],{"className":17959},[1048],[31,17961,17963],{"className":17962,"style":2050},[1052],[31,17964],{},[31,17966,12680],{"className":17967},[75],[31,17969,220],{"className":17970},[289],[31,17972],{"className":17973,"style":293},[119],[31,17975,223],{"className":17976},[297],[31,17978],{"className":17979,"style":293},[119],[31,17981,17983,17986,17989,18029,18032,18035],{"className":17982},[66],[31,17984],{"className":17985,"style":260},[70],[31,17987,211],{"className":17988},[264],[31,17990,17992,17995],{"className":17991},[75],[31,17993,17342],{"className":17994,"style":2666},[75,76],[31,17996,17998],{"className":17997},[1039],[31,17999,18001,18021],{"className":18000},[1043,1044],[31,18002,18004,18018],{"className":18003},[1048],[31,18005,18007],{"className":18006,"style":2028},[1052],[31,18008,18009,18012],{"style":5504},[31,18010],{"className":18011,"style":1061},[1060],[31,18013,18015],{"className":18014},[1065,1066,1067,1068],[31,18016,205],{"className":18017},[75,1068],[31,18019,1076],{"className":18020},[1075],[31,18022,18024],{"className":18023},[1048],[31,18025,18027],{"className":18026,"style":2050},[1052],[31,18028],{},[31,18030],{"className":18031,"style":120},[119],[31,18033,97],{"className":18034},[124],[31,18036],{"className":18037,"style":120},[119],[31,18039,18041,18044,18084],{"className":18040},[66],[31,18042],{"className":18043,"style":12035},[70],[31,18045,18047,18050],{"className":18046},[75],[31,18048,17342],{"className":18049,"style":2666},[75,76],[31,18051,18053],{"className":18052},[1039],[31,18054,18056,18076],{"className":18055},[1043,1044],[31,18057,18059,18073],{"className":18058},[1048],[31,18060,18062],{"className":18061,"style":2028},[1052],[31,18063,18064,18067],{"style":5504},[31,18065],{"className":18066,"style":1061},[1060],[31,18068,18070],{"className":18069},[1065,1066,1067,1068],[31,18071,152],{"className":18072},[75,1068],[31,18074,1076],{"className":18075},[1075],[31,18077,18079],{"className":18078},[1048],[31,18080,18082],{"className":18081,"style":2050},[1052],[31,18083],{},[31,18085,18087,18090],{"className":18086},[289],[31,18088,220],{"className":18089},[289],[31,18091,18093],{"className":18092},[1039],[31,18094,18096],{"className":18095},[1043],[31,18097,18099],{"className":18098},[1048],[31,18100,18102],{"className":18101,"style":5135},[1052],[31,18103,18104,18107],{"style":3631},[31,18105],{"className":18106,"style":1061},[1060],[31,18108,18110],{"className":18109},[1065,1066,1067,1068],[31,18111,152],{"className":18112},[75,1068],"，对得上。取 ",[31,18115,18117,18141],{"className":18116,"translate":35},[34],[31,18118,18120],{"className":18119},[39],[41,18121,18122],{"xmlns":43},[45,18123,18124,18138],{},[48,18125,18126,18128,18130,18132,18134,18136],{},[51,18127,4935],{"mathvariant":925},[95,18129,211],{"stretchy":210},[91,18131,1977],{},[95,18133,1210],{"separator":62},[91,18135,205],{},[95,18137,220],{"stretchy":210},[55,18139,18140],{"encoding":57},"\\mathcal N(0,1)",[31,18142,18144],{"className":18143,"ariaHidden":62},[61],[31,18145,18147,18150,18153,18156,18159,18162,18165,18168],{"className":18146},[66],[31,18148],{"className":18149,"style":260},[70],[31,18151,4935],{"className":18152,"style":5004},[75,968],[31,18154,211],{"className":18155},[264],[31,18157,1977],{"className":18158},[75],[31,18160,1210],{"className":18161},[1391],[31,18163],{"className":18164,"style":1467},[119],[31,18166,205],{"className":18167},[75],[31,18169,220],{"className":18170},[289]," 与 ",[31,18173,18175,18199],{"className":18174,"translate":35},[34],[31,18176,18178],{"className":18177},[39],[41,18179,18180],{"xmlns":43},[45,18181,18182,18196],{},[48,18183,18184,18186,18188,18190,18192,18194],{},[51,18185,4935],{"mathvariant":925},[95,18187,211],{"stretchy":210},[91,18189,1977],{},[95,18191,1210],{"separator":62},[91,18193,15416],{},[95,18195,220],{"stretchy":210},[55,18197,18198],{"encoding":57},"\\mathcal N(0,4)",[31,18200,18202],{"className":18201,"ariaHidden":62},[61],[31,18203,18205,18208,18211,18214,18217,18220,18223,18226],{"className":18204},[66],[31,18206],{"className":18207,"style":260},[70],[31,18209,4935],{"className":18210,"style":5004},[75,968],[31,18212,211],{"className":18213},[264],[31,18215,1977],{"className":18216},[75],[31,18218,1210],{"className":18219},[1391],[31,18221],{"className":18222,"style":1467},[119],[31,18224,15416],{"className":18225},[75],[31,18227,220],{"className":18228},[289],"，得 ",[31,18231,18233,18255],{"className":18232,"translate":35},[34],[31,18234,18236],{"className":18235},[39],[41,18237,18238],{"xmlns":43},[45,18239,18240,18252],{},[48,18241,18242,18248,18250],{},[1004,18243,18244,18246],{},[51,18245,3117],{},[91,18247,152],{},[95,18249,223],{},[91,18251,205],{},[55,18253,18254],{"encoding":57},"W_2=1",[31,18256,18258,18313],{"className":18257,"ariaHidden":62},[61],[31,18259,18261,18264,18304,18307,18310],{"className":18260},[66],[31,18262],{"className":18263,"style":5230},[70],[31,18265,18267,18270],{"className":18266},[75],[31,18268,3117],{"className":18269,"style":949},[75,76],[31,18271,18273],{"className":18272},[1039],[31,18274,18276,18296],{"className":18275},[1043,1044],[31,18277,18279,18293],{"className":18278},[1048],[31,18280,18282],{"className":18281,"style":2028},[1052],[31,18283,18284,18287],{"style":1056},[31,18285],{"className":18286,"style":1061},[1060],[31,18288,18290],{"className":18289},[1065,1066,1067,1068],[31,18291,152],{"className":18292},[75,1068],[31,18294,1076],{"className":18295},[1075],[31,18297,18299],{"className":18298},[1048],[31,18300,18302],{"className":18301,"style":2050},[1052],[31,18303],{},[31,18305],{"className":18306,"style":293},[119],[31,18308,223],{"className":18309},[297],[31,18311],{"className":18312,"style":293},[119],[31,18314,18316,18319],{"className":18315},[66],[31,18317],{"className":18318,"style":326},[70],[31,18320,205],{"className":18321},[75],"，正好是标准差之差；而最优映射是 ",[31,18324,18326,18347],{"className":18325,"translate":35},[34],[31,18327,18329],{"className":18328},[39],[41,18330,18331],{"xmlns":43},[45,18332,18333,18344],{},[48,18334,18335,18337,18340,18342],{},[51,18336,53],{},[95,18338,18339],{},"↦",[91,18341,152],{},[51,18343,53],{},[55,18345,18346],{"encoding":57},"x\\mapsto2x",[31,18348,18350,18369],{"className":18349,"ariaHidden":62},[61],[31,18351,18353,18357,18360,18363,18366],{"className":18352},[66],[31,18354],{"className":18355,"style":18356},[70],"height:0.522em;vertical-align:-0.011em;",[31,18358,53],{"className":18359},[75,76],[31,18361],{"className":18362,"style":293},[119],[31,18364,18339],{"className":18365},[297],[31,18367],{"className":18368,"style":293},[119],[31,18370,18372,18375,18378],{"className":18371},[66],[31,18373],{"className":18374,"style":326},[70],[31,18376,152],{"className":18377},[75],[31,18379,53],{"className":18380},[75,76],"，位移的均方根也正好是 ",[31,18383,18385,18398],{"className":18384,"translate":35},[34],[31,18386,18388],{"className":18387},[39],[41,18389,18390],{"xmlns":43},[45,18391,18392,18396],{},[48,18393,18394],{},[91,18395,205],{},[55,18397,205],{"encoding":57},[31,18399,18401],{"className":18400,"ariaHidden":62},[61],[31,18402,18404,18407],{"className":18403},[66],[31,18405],{"className":18406,"style":326},[70],[31,18408,205],{"className":18409},[75],"。两个数在这里是同一个数。",[11,18412,18413,18414,18622],{},"最优映射还是仿射的：",[31,18415,18417,18464],{"className":18416,"translate":35},[34],[31,18418,18420],{"className":18419},[39],[41,18421,18422],{"xmlns":43},[45,18423,18424,18461],{},[48,18425,18426,18428,18430,18432,18434,18436,18442,18444,18447,18449,18451,18453,18459],{},[51,18427,919],{},[95,18429,211],{"stretchy":210},[51,18431,53],{},[95,18433,220],{"stretchy":210},[95,18435,223],{},[1004,18437,18438,18440],{},[51,18439,16396],{},[91,18441,152],{},[95,18443,434],{},[51,18445,18446],{},"A",[95,18448,211],{"stretchy":210},[51,18450,53],{},[95,18452,97],{},[1004,18454,18455,18457],{},[51,18456,16396],{},[91,18458,205],{},[95,18460,220],{"stretchy":210},[55,18462,18463],{"encoding":57},"T(x)=m_2+A(x-m_1)",[31,18465,18467,18494,18549,18573],{"className":18466,"ariaHidden":62},[61],[31,18468,18470,18473,18476,18479,18482,18485,18488,18491],{"className":18469},[66],[31,18471],{"className":18472,"style":260},[70],[31,18474,919],{"className":18475,"style":949},[75,76],[31,18477,211],{"className":18478},[264],[31,18480,53],{"className":18481},[75,76],[31,18483,220],{"className":18484},[289],[31,18486],{"className":18487,"style":293},[119],[31,18489,223],{"className":18490},[297],[31,18492],{"className":18493,"style":293},[119],[31,18495,18497,18500,18540,18543,18546],{"className":18496},[66],[31,18498],{"className":18499,"style":5426},[70],[31,18501,18503,18506],{"className":18502},[75],[31,18504,16396],{"className":18505},[75,76],[31,18507,18509],{"className":18508},[1039],[31,18510,18512,18532],{"className":18511},[1043,1044],[31,18513,18515,18529],{"className":18514},[1048],[31,18516,18518],{"className":18517,"style":2028},[1052],[31,18519,18520,18523],{"style":2807},[31,18521],{"className":18522,"style":1061},[1060],[31,18524,18526],{"className":18525},[1065,1066,1067,1068],[31,18527,152],{"className":18528},[75,1068],[31,18530,1076],{"className":18531},[1075],[31,18533,18535],{"className":18534},[1048],[31,18536,18538],{"className":18537,"style":2050},[1052],[31,18539],{},[31,18541],{"className":18542,"style":120},[119],[31,18544,434],{"className":18545},[124],[31,18547],{"className":18548,"style":120},[119],[31,18550,18552,18555,18558,18561,18564,18567,18570],{"className":18551},[66],[31,18553],{"className":18554,"style":260},[70],[31,18556,18446],{"className":18557},[75,76],[31,18559,211],{"className":18560},[264],[31,18562,53],{"className":18563},[75,76],[31,18565],{"className":18566,"style":120},[119],[31,18568,97],{"className":18569},[124],[31,18571],{"className":18572,"style":120},[119],[31,18574,18576,18579,18619],{"className":18575},[66],[31,18577],{"className":18578,"style":260},[70],[31,18580,18582,18585],{"className":18581},[75],[31,18583,16396],{"className":18584},[75,76],[31,18586,18588],{"className":18587},[1039],[31,18589,18591,18611],{"className":18590},[1043,1044],[31,18592,18594,18608],{"className":18593},[1048],[31,18595,18597],{"className":18596,"style":2028},[1052],[31,18598,18599,18602],{"style":2807},[31,18600],{"className":18601,"style":1061},[1060],[31,18603,18605],{"className":18604},[1065,1066,1067,1068],[31,18606,205],{"className":18607},[75,1068],[31,18609,1076],{"className":18610},[1075],[31,18612,18614],{"className":18613},[1048],[31,18615,18617],{"className":18616,"style":2050},[1052],[31,18618],{},[31,18620,220],{"className":18621},[289],"，其中",[31,18624,18626],{"className":18625,"translate":35},[411],[31,18627,18629,18727],{"className":18628,"translate":35},[34],[31,18630,18632],{"className":18631},[39],[41,18633,18634],{"xmlns":43,"display":420},[45,18635,18636,18724],{},[48,18637,18638,18640,18642,18658,18660,18674,18680,18694,18706,18722],{},[51,18639,18446],{},[95,18641,223],{},[12397,18643,18644,18646,18648],{},[51,18645,16405],{"mathvariant":494},[91,18647,205],{},[48,18649,18650,18652,18654,18656],{},[95,18651,97],{},[91,18653,205],{},[51,18655,3768],{"mathvariant":494},[91,18657,152],{},[95,18659,211],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[12397,18661,18662,18664,18666],{},[51,18663,16405],{"mathvariant":494},[91,18665,205],{},[48,18667,18668,18670,18672],{},[91,18669,205],{},[51,18671,3768],{"mathvariant":494},[91,18673,152],{},[1004,18675,18676,18678],{},[51,18677,16405],{"mathvariant":494},[91,18679,152],{},[12397,18681,18682,18684,18686],{},[51,18683,16405],{"mathvariant":494},[91,18685,205],{},[48,18687,18688,18690,18692],{},[91,18689,205],{},[51,18691,3768],{"mathvariant":494},[91,18693,152],{},[1180,18695,18696,18698],{},[95,18697,220],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[48,18699,18700,18702,18704],{},[91,18701,205],{},[51,18703,3768],{"mathvariant":494},[91,18705,152],{},[12397,18707,18708,18710,18712],{},[51,18709,16405],{"mathvariant":494},[91,18711,205],{},[48,18713,18714,18716,18718,18720],{},[95,18715,97],{},[91,18717,205],{},[51,18719,3768],{"mathvariant":494},[91,18721,152],{},[51,18723,495],{"mathvariant":494},[55,18725,18726],{"encoding":57},"A=\\Sigma_1^{-1\u002F2}\\bigl(\\Sigma_1^{1\u002F2}\\Sigma_2\\Sigma_1^{1\u002F2}\\bigr)^{1\u002F2}\\Sigma_1^{-1\u002F2}.",[31,18728,18730,18748],{"className":18729,"ariaHidden":62},[61],[31,18731,18733,18736,18739,18742,18745],{"className":18732},[66],[31,18734],{"className":18735,"style":945},[70],[31,18737,18446],{"className":18738},[75,76],[31,18740],{"className":18741,"style":293},[119],[31,18743,223],{"className":18744},[297],[31,18746],{"className":18747,"style":293},[119],[31,18749,18751,18755,18812,18818,18872,18912,18966,19001,19058],{"className":18750},[66],[31,18752],{"className":18753,"style":18754},[70],"height:1.4779em;vertical-align:-0.35em;",[31,18756,18758,18761],{"className":18757},[75],[31,18759,16405],{"className":18760},[75],[31,18762,18764],{"className":18763},[1039],[31,18765,18767,18804],{"className":18766},[1043,1044],[31,18768,18770,18801],{"className":18769},[1048],[31,18771,18773,18784],{"className":18772,"style":17117},[1052],[31,18774,18775,18778],{"style":17120},[31,18776],{"className":18777,"style":1061},[1060],[31,18779,18781],{"className":18780},[1065,1066,1067,1068],[31,18782,205],{"className":18783},[75,1068],[31,18785,18786,18789],{"style":17132},[31,18787],{"className":18788,"style":1061},[1060],[31,18790,18792],{"className":18791},[1065,1066,1067,1068],[31,18793,18795,18798],{"className":18794},[75,1068],[31,18796,97],{"className":18797},[75,1068],[31,18799,17145],{"className":18800},[75,1068],[31,18802,1076],{"className":18803},[1075],[31,18805,18807],{"className":18806},[1048],[31,18808,18810],{"className":18809,"style":17155},[1052],[31,18811],{},[31,18813,18815],{"className":18814},[264],[31,18816,211],{"className":18817},[1307,1308],[31,18819,18821,18824],{"className":18820},[75],[31,18822,16405],{"className":18823},[75],[31,18825,18827],{"className":18826},[1039],[31,18828,18830,18864],{"className":18829},[1043,1044],[31,18831,18833,18861],{"className":18832},[1048],[31,18834,18836,18847],{"className":18835,"style":17117},[1052],[31,18837,18838,18841],{"style":17120},[31,18839],{"className":18840,"style":1061},[1060],[31,18842,18844],{"className":18843},[1065,1066,1067,1068],[31,18845,205],{"className":18846},[75,1068],[31,18848,18849,18852],{"style":17132},[31,18850],{"className":18851,"style":1061},[1060],[31,18853,18855],{"className":18854},[1065,1066,1067,1068],[31,18856,18858],{"className":18857},[75,1068],[31,18859,17145],{"className":18860},[75,1068],[31,18862,1076],{"className":18863},[1075],[31,18865,18867],{"className":18866},[1048],[31,18868,18870],{"className":18869,"style":17155},[1052],[31,18871],{},[31,18873,18875,18878],{"className":18874},[75],[31,18876,16405],{"className":18877},[75],[31,18879,18881],{"className":18880},[1039],[31,18882,18884,18904],{"className":18883},[1043,1044],[31,18885,18887,18901],{"className":18886},[1048],[31,18888,18890],{"className":18889,"style":2028},[1052],[31,18891,18892,18895],{"style":2807},[31,18893],{"className":18894,"style":1061},[1060],[31,18896,18898],{"className":18897},[1065,1066,1067,1068],[31,18899,152],{"className":18900},[75,1068],[31,18902,1076],{"className":18903},[1075],[31,18905,18907],{"className":18906},[1048],[31,18908,18910],{"className":18909,"style":2050},[1052],[31,18911],{},[31,18913,18915,18918],{"className":18914},[75],[31,18916,16405],{"className":18917},[75],[31,18919,18921],{"className":18920},[1039],[31,18922,18924,18958],{"className":18923},[1043,1044],[31,18925,18927,18955],{"className":18926},[1048],[31,18928,18930,18941],{"className":18929,"style":17117},[1052],[31,18931,18932,18935],{"style":17120},[31,18933],{"className":18934,"style":1061},[1060],[31,18936,18938],{"className":18937},[1065,1066,1067,1068],[31,18939,205],{"className":18940},[75,1068],[31,18942,18943,18946],{"style":17132},[31,18944],{"className":18945,"style":1061},[1060],[31,18947,18949],{"className":18948},[1065,1066,1067,1068],[31,18950,18952],{"className":18951},[75,1068],[31,18953,17145],{"className":18954},[75,1068],[31,18956,1076],{"className":18957},[1075],[31,18959,18961],{"className":18960},[1048],[31,18962,18964],{"className":18963,"style":17155},[1052],[31,18965],{},[31,18967,18969,18975],{"className":18968},[289],[31,18970,18972],{"className":18971},[289],[31,18973,220],{"className":18974},[1307,1308],[31,18976,18978],{"className":18977},[1039],[31,18979,18981],{"className":18980},[1043],[31,18982,18984],{"className":18983},[1048],[31,18985,18987],{"className":18986,"style":17273},[1052],[31,18988,18989,18992],{"style":13566},[31,18990],{"className":18991,"style":1061},[1060],[31,18993,18995],{"className":18994},[1065,1066,1067,1068],[31,18996,18998],{"className":18997},[75,1068],[31,18999,17145],{"className":19000},[75,1068],[31,19002,19004,19007],{"className":19003},[75],[31,19005,16405],{"className":19006},[75],[31,19008,19010],{"className":19009},[1039],[31,19011,19013,19050],{"className":19012},[1043,1044],[31,19014,19016,19047],{"className":19015},[1048],[31,19017,19019,19030],{"className":19018,"style":17117},[1052],[31,19020,19021,19024],{"style":17120},[31,19022],{"className":19023,"style":1061},[1060],[31,19025,19027],{"className":19026},[1065,1066,1067,1068],[31,19028,205],{"className":19029},[75,1068],[31,19031,19032,19035],{"style":17132},[31,19033],{"className":19034,"style":1061},[1060],[31,19036,19038],{"className":19037},[1065,1066,1067,1068],[31,19039,19041,19044],{"className":19040},[75,1068],[31,19042,97],{"className":19043},[75,1068],[31,19045,17145],{"className":19046},[75,1068],[31,19048,1076],{"className":19049},[1075],[31,19051,19053],{"className":19052},[1048],[31,19054,19056],{"className":19055,"style":17155},[1052],[31,19057],{},[31,19059,495],{"className":19060},[75],[11,19062,19063],{},"写成操作就是“先白化再上色”，风格迁移里的 AdaIN 做的正是这个（对角版本）。所以“高斯之间的最优传输”这件事，工程界早就在做了，只是没有这么叫。",[11,19065,19066,19067,19137,19138,19367,19368,19437],{},"高斯集合配上 ",[31,19068,19070,19088],{"className":19069,"translate":35},[34],[31,19071,19073],{"className":19072},[39],[41,19074,19075],{"xmlns":43},[45,19076,19077,19085],{},[48,19078,19079],{},[1004,19080,19081,19083],{},[51,19082,3117],{},[91,19084,152],{},[55,19086,19087],{"encoding":57},"W_2",[31,19089,19091],{"className":19090,"ariaHidden":62},[61],[31,19092,19094,19097],{"className":19093},[66],[31,19095],{"className":19096,"style":5230},[70],[31,19098,19100,19103],{"className":19099},[75],[31,19101,3117],{"className":19102,"style":949},[75,76],[31,19104,19106],{"className":19105},[1039],[31,19107,19109,19129],{"className":19108},[1043,1044],[31,19110,19112,19126],{"className":19111},[1048],[31,19113,19115],{"className":19114,"style":2028},[1052],[31,19116,19117,19120],{"style":1056},[31,19118],{"className":19119,"style":1061},[1060],[31,19121,19123],{"className":19122},[1065,1066,1067,1068],[31,19124,152],{"className":19125},[75,1068],[31,19127,1076],{"className":19128},[1075],[31,19130,19132],{"className":19131},[1048],[31,19133,19135],{"className":19134,"style":2050},[1052],[31,19136],{}," 构成一个黎曼流形（Bures–Wasserstein 空间），两点间的测地线就是位移插值 ",[31,19139,19141,19193],{"className":19140,"translate":35},[34],[31,19142,19144],{"className":19143},[39],[41,19145,19146],{"xmlns":43},[45,19147,19148,19190],{},[48,19149,19150,19156,19158,19160,19162,19164,19166,19168,19170,19176,19178,19180,19182,19188],{},[1004,19151,19152,19154],{},[51,19153,852],{},[51,19155,12271],{},[95,19157,223],{},[95,19159,211],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[95,19161,211],{"stretchy":210},[91,19163,205],{},[95,19165,97],{},[51,19167,12271],{},[95,19169,220],{"stretchy":210},[48,19171,19172,19174],{},[51,19173,5391],{"mathvariant":494},[51,19175,1435],{"mathvariant":494},[95,19177,434],{},[51,19179,12271],{},[51,19181,919],{},[1004,19183,19184,19186],{},[95,19185,220],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[51,19187,1010],{"mathvariant":494},[51,19189,852],{},[55,19191,19192],{"encoding":57},"\\mu_t=\\bigl((1-t)\\mathrm{id}+tT\\bigr)_\\#\\mu",[31,19194,19196,19252,19279,19307],{"className":19195,"ariaHidden":62},[61],[31,19197,19199,19202,19243,19246,19249],{"className":19198},[66],[31,19200],{"className":19201,"style":865},[70],[31,19203,19205,19208],{"className":19204},[75],[31,19206,852],{"className":19207},[75,76],[31,19209,19211],{"className":19210},[1039],[31,19212,19214,19235],{"className":19213},[1043,1044],[31,19215,19217,19232],{"className":19216},[1048],[31,19218,19221],{"className":19219,"style":19220},[1052],"height:0.2806em;",[31,19222,19223,19226],{"style":2807},[31,19224],{"className":19225,"style":1061},[1060],[31,19227,19229],{"className":19228},[1065,1066,1067,1068],[31,19230,12271],{"className":19231},[75,76,1068],[31,19233,1076],{"className":19234},[1075],[31,19236,19238],{"className":19237},[1048],[31,19239,19241],{"className":19240,"style":2050},[1052],[31,19242],{},[31,19244],{"className":19245,"style":293},[119],[31,19247,223],{"className":19248},[297],[31,19250],{"className":19251,"style":293},[119],[31,19253,19255,19258,19264,19267,19270,19273,19276],{"className":19254},[66],[31,19256],{"className":19257,"style":4988},[70],[31,19259,19261],{"className":19260},[264],[31,19262,211],{"className":19263},[1307,1308],[31,19265,211],{"className":19266},[264],[31,19268,205],{"className":19269},[75],[31,19271],{"className":19272,"style":120},[119],[31,19274,97],{"className":19275},[124],[31,19277],{"className":19278,"style":120},[119],[31,19280,19282,19285,19288,19291,19298,19301,19304],{"className":19281},[66],[31,19283],{"className":19284,"style":260},[70],[31,19286,12271],{"className":19287},[75,76],[31,19289,220],{"className":19290},[289],[31,19292,19294],{"className":19293},[75],[31,19295,19297],{"className":19296},[75,2745],"id",[31,19299],{"className":19300,"style":120},[119],[31,19302,434],{"className":19303},[124],[31,19305],{"className":19306,"style":120},[119],[31,19308,19310,19314,19318,19364],{"className":19309},[66],[31,19311],{"className":19312,"style":19313},[70],"height:1.3858em;vertical-align:-0.5358em;",[31,19315,19317],{"className":19316,"style":949},[75,76],"tT",[31,19319,19321,19327],{"className":19320},[289],[31,19322,19324],{"className":19323},[289],[31,19325,220],{"className":19326},[1307,1308],[31,19328,19330],{"className":19329},[1039],[31,19331,19333,19355],{"className":19332},[1043,1044],[31,19334,19336,19352],{"className":19335},[1048],[31,19337,19340],{"className":19338,"style":19339},[1052],"height:0.0864em;",[31,19341,19343,19346],{"style":19342},"top:-2.3003em;margin-right:0.05em;",[31,19344],{"className":19345,"style":1061},[1060],[31,19347,19349],{"className":19348},[1065,1066,1067,1068],[31,19350,1010],{"className":19351},[75,1068],[31,19353,1076],{"className":19354},[1075],[31,19356,19358],{"className":19357},[1048],[31,19359,19362],{"className":19360,"style":19361},[1052],"height:0.5358em;",[31,19363],{},[31,19365,852],{"className":19366},[75,76],"。这意味着我可以在分布空间里插值和求平均。我在写 Fréchet 均值那篇时说过，在没有向量结构的空间里“平均”要重新定义；",[31,19369,19371,19388],{"className":19370,"translate":35},[34],[31,19372,19374],{"className":19373},[39],[41,19375,19376],{"xmlns":43},[45,19377,19378,19386],{},[48,19379,19380],{},[1004,19381,19382,19384],{},[51,19383,3117],{},[91,19385,152],{},[55,19387,19087],{"encoding":57},[31,19389,19391],{"className":19390,"ariaHidden":62},[61],[31,19392,19394,19397],{"className":19393},[66],[31,19395],{"className":19396,"style":5230},[70],[31,19398,19400,19403],{"className":19399},[75],[31,19401,3117],{"className":19402,"style":949},[75,76],[31,19404,19406],{"className":19405},[1039],[31,19407,19409,19429],{"className":19408},[1043,1044],[31,19410,19412,19426],{"className":19411},[1048],[31,19413,19415],{"className":19414,"style":2028},[1052],[31,19416,19417,19420],{"style":1056},[31,19418],{"className":19419,"style":1061},[1060],[31,19421,19423],{"className":19422},[1065,1066,1067,1068],[31,19424,152],{"className":19425},[75,1068],[31,19427,1076],{"className":19428},[1075],[31,19430,19432],{"className":19431},[1048],[31,19433,19435],{"className":19434,"style":2050},[1052],[31,19436],{}," 空间是最典型的例子——两个高斯的中点要沿测地线走一半，不是把参数取平均。",[18,19439,19440],{"id":19440},"计算的代价",[11,19442,19443,19471],{},[31,19444,19446,19459],{"className":19445,"translate":35},[34],[31,19447,19449],{"className":19448},[39],[41,19450,19451],{"xmlns":43},[45,19452,19453,19457],{},[48,19454,19455],{},[51,19456,3117],{},[55,19458,3117],{"encoding":57},[31,19460,19462],{"className":19461,"ariaHidden":62},[61],[31,19463,19465,19468],{"className":19464},[66],[31,19466],{"className":19467,"style":945},[70],[31,19469,3117],{"className":19470,"style":949},[75,76]," 定义了“是什么”，但没告诉我“怎么算”。",[11,19473,19474,19559,19560,19622,19623,19686,19687,19715],{},[31,19475,19477,19500],{"className":19476,"translate":35},[34],[31,19478,19480],{"className":19479},[39],[41,19481,19482],{"xmlns":43},[45,19483,19484,19497],{},[48,19485,19486,19488,19490],{},[51,19487,14559],{},[95,19489,223],{},[1180,19491,19492,19495],{},[91,19493,19494],{},"10",[91,19496,15416],{},[55,19498,19499],{"encoding":57},"n=10^4",[31,19501,19503,19521],{"className":19502,"ariaHidden":62},[61],[31,19504,19506,19509,19512,19515,19518],{"className":19505},[66],[31,19507],{"className":19508,"style":71},[70],[31,19510,14559],{"className":19511},[75,76],[31,19513],{"className":19514,"style":293},[119],[31,19516,223],{"className":19517},[297],[31,19519],{"className":19520,"style":293},[119],[31,19522,19524,19527,19530],{"className":19523},[66],[31,19525],{"className":19526,"style":5135},[70],[31,19528,205],{"className":19529},[75],[31,19531,19533,19536],{"className":19532},[75],[31,19534,1977],{"className":19535},[75],[31,19537,19539],{"className":19538},[1039],[31,19540,19542],{"className":19541},[1043],[31,19543,19545],{"className":19544},[1048],[31,19546,19548],{"className":19547,"style":5135},[1052],[31,19549,19550,19553],{"style":3631},[31,19551],{"className":19552,"style":1061},[1060],[31,19554,19556],{"className":19555},[1065,1066,1067,1068],[31,19557,15416],{"className":19558},[75,1068]," 个样本对 ",[31,19561,19563,19581],{"className":19562,"translate":35},[34],[31,19564,19566],{"className":19565},[39],[41,19567,19568],{"xmlns":43},[45,19569,19570,19578],{},[48,19571,19572],{},[1180,19573,19574,19576],{},[91,19575,19494],{},[91,19577,15416],{},[55,19579,19580],{"encoding":57},"10^4",[31,19582,19584],{"className":19583,"ariaHidden":62},[61],[31,19585,19587,19590,19593],{"className":19586},[66],[31,19588],{"className":19589,"style":5135},[70],[31,19591,205],{"className":19592},[75],[31,19594,19596,19599],{"className":19595},[75],[31,19597,1977],{"className":19598},[75],[31,19600,19602],{"className":19601},[1039],[31,19603,19605],{"className":19604},[1043],[31,19606,19608],{"className":19607},[1048],[31,19609,19611],{"className":19610,"style":5135},[1052],[31,19612,19613,19616],{"style":3631},[31,19614],{"className":19615,"style":1061},[1060],[31,19617,19619],{"className":19618},[1065,1066,1067,1068],[31,19620,15416],{"className":19621},[75,1068]," 个样本，代价矩阵就是 ",[31,19624,19626,19645],{"className":19625,"translate":35},[34],[31,19627,19629],{"className":19628},[39],[41,19630,19631],{"xmlns":43},[45,19632,19633,19642],{},[48,19634,19635],{},[1180,19636,19637,19639],{},[91,19638,19494],{},[91,19640,19641],{},"8",[55,19643,19644],{"encoding":57},"10^8",[31,19646,19648],{"className":19647,"ariaHidden":62},[61],[31,19649,19651,19654,19657],{"className":19650},[66],[31,19652],{"className":19653,"style":5135},[70],[31,19655,205],{"className":19656},[75],[31,19658,19660,19663],{"className":19659},[75],[31,19661,1977],{"className":19662},[75],[31,19664,19666],{"className":19665},[1039],[31,19667,19669],{"className":19668},[1043],[31,19670,19672],{"className":19671},[1048],[31,19673,19675],{"className":19674,"style":5135},[1052],[31,19676,19677,19680],{"style":3631},[31,19678],{"className":19679,"style":1061},[1060],[31,19681,19683],{"className":19682},[1065,1066,1067,1068],[31,19684,19641],{"className":19685},[75,1068]," 个数，float32 存下来 400 MB，还没开始迭代。我最早在自己机器上试 OT，把 ",[31,19688,19690,19703],{"className":19689,"translate":35},[34],[31,19691,19693],{"className":19692},[39],[41,19694,19695],{"xmlns":43},[45,19696,19697,19701],{},[48,19698,19699],{},[51,19700,14559],{},[55,19702,14559],{"encoding":57},[31,19704,19706],{"className":19705,"ariaHidden":62},[61],[31,19707,19709,19712],{"className":19708},[66],[31,19710],{"className":19711,"style":71},[70],[31,19713,14559],{"className":19714},[75,76]," 开到 5000 就被内存劝退了。",[11,19717,19718,19719,19795,19796,19824,19825,19853],{},"离散 OT 是线性规划，实践上是 ",[31,19720,19722,19747],{"className":19721,"translate":35},[34],[31,19723,19725],{"className":19724},[39],[41,19726,19727],{"xmlns":43},[45,19728,19729,19744],{},[48,19730,19731,19734,19736,19742],{},[51,19732,19733],{},"O",[95,19735,211],{"stretchy":210},[1180,19737,19738,19740],{},[51,19739,14559],{},[91,19741,93],{},[95,19743,220],{"stretchy":210},[55,19745,19746],{"encoding":57},"O(n^3)",[31,19748,19750],{"className":19749,"ariaHidden":62},[61],[31,19751,19753,19756,19760,19763,19792],{"className":19752},[66],[31,19754],{"className":19755,"style":12035},[70],[31,19757,19733],{"className":19758,"style":19759},[75,76],"margin-right:0.0278em;",[31,19761,211],{"className":19762},[264],[31,19764,19766,19769],{"className":19765},[75],[31,19767,14559],{"className":19768},[75,76],[31,19770,19772],{"className":19771},[1039],[31,19773,19775],{"className":19774},[1043],[31,19776,19778],{"className":19777},[1048],[31,19779,19781],{"className":19780,"style":5135},[1052],[31,19782,19783,19786],{"style":3631},[31,19784],{"className":19785,"style":1061},[1060],[31,19787,19789],{"className":19788},[1065,1066,1067,1068],[31,19790,93],{"className":19791},[75,1068],[31,19793,220],{"className":19794},[289]," 量级的算法。更麻烦的是解落在多面体的顶点上，",[31,19797,19799,19812],{"className":19798,"translate":35},[34],[31,19800,19802],{"className":19801},[39],[41,19803,19804],{"xmlns":43},[45,19805,19806,19810],{},[48,19807,19808],{},[51,19809,2515],{},[55,19811,3485],{"encoding":57},[31,19813,19815],{"className":19814,"ariaHidden":62},[61],[31,19816,19818,19821],{"className":19817},[66],[31,19819],{"className":19820,"style":71},[70],[31,19822,2515],{"className":19823,"style":2666},[75,76]," 很稀疏，而 ",[31,19826,19828,19841],{"className":19827,"translate":35},[34],[31,19829,19831],{"className":19830},[39],[41,19832,19833],{"xmlns":43},[45,19834,19835,19839],{},[48,19836,19837],{},[51,19838,3117],{},[55,19840,3117],{"encoding":57},[31,19842,19844],{"className":19843,"ariaHidden":62},[61],[31,19845,19847,19850],{"className":19846},[66],[31,19848],{"className":19849,"style":945},[70],[31,19851,3117],{"className":19852,"style":949},[75,76]," 作为代价矩阵的函数是分段线性的，梯度分段常数、在切换点跳变。这种“硬”信号在优化里很难用。",[11,19855,19856],{},"Cuturi 在 2013 年的做法是加熵正则：",[31,19858,19860],{"className":19859,"translate":35},[411],[31,19861,19863,19957],{"className":19862,"translate":35},[34],[31,19864,19866],{"className":19865},[39],[41,19867,19868],{"xmlns":43,"display":420},[45,19869,19870,19954],{},[48,19871,19872,19899,19901,19904,19906,19908,19910,19913,19915,19917,19927,19937,19940,19942,19952],{},[1531,19873,19874,19881],{},[48,19875,19876,19879],{},[51,19877,19878],{},"min",[95,19880,1540],{},[48,19882,19883,19885,19887,19889,19891,19893,19895,19897],{},[51,19884,2515],{},[95,19886,2518],{},[51,19888,2496],{"mathvariant":494},[95,19890,211],{"stretchy":210},[51,19892,852],{},[95,19894,1210],{"separator":62},[51,19896,884],{},[95,19898,220],{"stretchy":210},[1430,19900,1545],{},[95,19902,19903],{"stretchy":210},"⟨",[51,19905,8101],{},[95,19907,1210],{"separator":62},[51,19909,2515],{},[95,19911,19912],{"stretchy":210},"⟩",[95,19914,434],{},[51,19916,4793],{},[1531,19918,19919,19921],{},[95,19920,5625],{},[48,19922,19923,19925],{},[51,19924,5391],{},[51,19926,2555],{},[1004,19928,19929,19931],{},[51,19930,2515],{},[48,19932,19933,19935],{},[51,19934,5391],{},[51,19936,2555],{},[51,19938,19939],{},"log",[95,19941,1540],{},[1004,19943,19944,19946],{},[51,19945,2515],{},[48,19947,19948,19950],{},[51,19949,5391],{},[51,19951,2555],{},[51,19953,495],{"mathvariant":494},[55,19955,19956],{"encoding":57},"\\min_{\\pi\\in\\Pi(\\mu,\\nu)}\\ \\langle C,\\pi\\rangle+\\varepsilon\\sum_{ij}\\pi_{ij}\\log\\pi_{ij}.",[31,19958,19960,20066],{"className":19959,"ariaHidden":62},[61],[31,19961,19963,19967,20036,20039,20042,20045,20048,20051,20054,20057,20060,20063],{"className":19962},[66],[31,19964],{"className":19965,"style":19966},[70],"height:1.716em;vertical-align:-0.966em;",[31,19968,19970],{"className":19969},[1460,1620],[31,19971,19973,20028],{"className":19972},[1043,1044],[31,19974,19976,20025],{"className":19975},[1048],[31,19977,19980,20015],{"className":19978,"style":19979},[1052],"height:0.6679em;",[31,19981,19982,19985],{"style":3301},[31,19983],{"className":19984,"style":1637},[1060],[31,19986,19988],{"className":19987},[1065,1066,1067,1068],[31,19989,19991,19994,19997,20000,20003,20006,20009,20012],{"className":19990},[75,1068],[31,19992,2515],{"className":19993,"style":2666},[75,76,1068],[31,19995,2518],{"className":19996},[297,1068],[31,19998,2496],{"className":19999},[75,1068],[31,20001,211],{"className":20002},[264,1068],[31,20004,852],{"className":20005},[75,76,1068],[31,20007,1210],{"className":20008},[1391,1068],[31,20010,884],{"className":20011,"style":900},[75,76,1068],[31,20013,220],{"className":20014},[289,1068],[31,20016,20017,20020],{"style":1649},[31,20018],{"className":20019,"style":1637},[1060],[31,20021,20022],{},[31,20023,19878],{"className":20024},[1460],[31,20026,1076],{"className":20027},[1075],[31,20029,20031],{"className":20030},[1048],[31,20032,20034],{"className":20033,"style":3356},[1052],[31,20035],{},[31,20037,1545],{"className":20038},[119],[31,20040,19903],{"className":20041},[264],[31,20043,8101],{"className":20044,"style":8525},[75,76],[31,20046,1210],{"className":20047},[1391],[31,20049],{"className":20050,"style":1467},[119],[31,20052,2515],{"className":20053,"style":2666},[75,76],[31,20055,19912],{"className":20056},[289],[31,20058],{"className":20059,"style":120},[119],[31,20061,434],{"className":20062},[124],[31,20064],{"className":20065,"style":120},[119],[31,20067,20069,20072,20075,20078,20125,20128,20171,20174,20181,20184,20227],{"className":20068},[66],[31,20070],{"className":20071,"style":10455},[70],[31,20073,4793],{"className":20074},[75,76],[31,20076],{"className":20077,"style":1467},[119],[31,20079,20081],{"className":20080},[1460,1620],[31,20082,20084,20117],{"className":20083},[1043,1044],[31,20085,20087,20114],{"className":20086},[1048],[31,20088,20090,20104],{"className":20089,"style":5900},[1052],[31,20091,20092,20095],{"style":5903},[31,20093],{"className":20094,"style":5907},[1060],[31,20096,20098],{"className":20097},[1065,1066,1067,1068],[31,20099,20101],{"className":20100},[75,1068],[31,20102,5574],{"className":20103,"style":5514},[75,76,1068],[31,20105,20106,20109],{"style":5919},[31,20107],{"className":20108,"style":5907},[1060],[31,20110,20111],{},[31,20112,5625],{"className":20113},[1460,1461,1687],[31,20115,1076],{"className":20116},[1075],[31,20118,20120],{"className":20119},[1048],[31,20121,20123],{"className":20122,"style":5937},[1052],[31,20124],{},[31,20126],{"className":20127,"style":1467},[119],[31,20129,20131,20134],{"className":20130},[75],[31,20132,2515],{"className":20133,"style":2666},[75,76],[31,20135,20137],{"className":20136},[1039],[31,20138,20140,20163],{"className":20139},[1043,1044],[31,20141,20143,20160],{"className":20142},[1048],[31,20144,20146],{"className":20145,"style":5445},[1052],[31,20147,20148,20151],{"style":5504},[31,20149],{"className":20150,"style":1061},[1060],[31,20152,20154],{"className":20153},[1065,1066,1067,1068],[31,20155,20157],{"className":20156},[75,1068],[31,20158,5574],{"className":20159,"style":5514},[75,76,1068],[31,20161,1076],{"className":20162},[1075],[31,20164,20166],{"className":20165},[1048],[31,20167,20169],{"className":20168,"style":1083},[1052],[31,20170],{},[31,20172],{"className":20173,"style":1467},[119],[31,20175,20177,20178],{"className":20176},[1460],"lo",[31,20179,14556],{"style":20180},"margin-right:0.0139em;",[31,20182],{"className":20183,"style":1467},[119],[31,20185,20187,20190],{"className":20186},[75],[31,20188,2515],{"className":20189,"style":2666},[75,76],[31,20191,20193],{"className":20192},[1039],[31,20194,20196,20219],{"className":20195},[1043,1044],[31,20197,20199,20216],{"className":20198},[1048],[31,20200,20202],{"className":20201,"style":5445},[1052],[31,20203,20204,20207],{"style":5504},[31,20205],{"className":20206,"style":1061},[1060],[31,20208,20210],{"className":20209},[1065,1066,1067,1068],[31,20211,20213],{"className":20212},[75,1068],[31,20214,5574],{"className":20215,"style":5514},[75,76,1068],[31,20217,1076],{"className":20218},[1075],[31,20220,20222],{"className":20221},[1048],[31,20223,20225],{"className":20224,"style":1083},[1052],[31,20226],{},[31,20228,495],{"className":20229},[75],[11,20231,20232],{},"熵项把解从顶点“推”到内部，变成严格正的光滑解。用拉格朗日乘子可以解出",[31,20234,20236],{"className":20235,"translate":35},[411],[31,20237,20239,20329],{"className":20238,"translate":35},[34],[31,20240,20242],{"className":20241},[39],[41,20243,20244],{"xmlns":43,"display":420},[45,20245,20246,20326],{},[48,20247,20248,20258,20260,20266,20276,20282,20284,20287,20297,20299,20302,20304,20306,20308,20318,20320,20322,20324],{},[1004,20249,20250,20252],{},[51,20251,2515],{},[48,20253,20254,20256],{},[51,20255,5391],{},[51,20257,2555],{},[95,20259,223],{},[1004,20261,20262,20264],{},[51,20263,5324],{},[51,20265,5391],{},[1004,20267,20268,20270],{},[51,20269,4558],{},[48,20271,20272,20274],{},[51,20273,5391],{},[51,20275,2555],{},[1004,20277,20278,20280],{},[51,20279,5354],{},[51,20281,2555],{},[95,20283,1210],{"separator":62},[119,20285],{"width":20286},"2em",[1004,20288,20289,20291],{},[51,20290,4558],{},[48,20292,20293,20295],{},[51,20294,5391],{},[51,20296,2555],{},[95,20298,223],{},[51,20300,20301],{},"exp",[95,20303,1540],{},[95,20305,211],{"stretchy":210},[95,20307,97],{},[1004,20309,20310,20312],{},[51,20311,8101],{},[48,20313,20314,20316],{},[51,20315,5391],{},[51,20317,2555],{},[51,20319,3768],{"mathvariant":494},[51,20321,4793],{},[95,20323,220],{"stretchy":210},[95,20325,1210],{"separator":62},[55,20327,20328],{"encoding":57},"\\pi_{ij}=u_iK_{ij}v_j,\\qquad K_{ij}=\\exp(-C_{ij}\u002F\\varepsilon),",[31,20330,20332,20390,20581],{"className":20331,"ariaHidden":62},[61],[31,20333,20335,20338,20381,20384,20387],{"className":20334},[66],[31,20336],{"className":20337,"style":5541},[70],[31,20339,20341,20344],{"className":20340},[75],[31,20342,2515],{"className":20343,"style":2666},[75,76],[31,20345,20347],{"className":20346},[1039],[31,20348,20350,20373],{"className":20349},[1043,1044],[31,20351,20353,20370],{"className":20352},[1048],[31,20354,20356],{"className":20355,"style":5445},[1052],[31,20357,20358,20361],{"style":5504},[31,20359],{"className":20360,"style":1061},[1060],[31,20362,20364],{"className":20363},[1065,1066,1067,1068],[31,20365,20367],{"className":20366},[75,1068],[31,20368,5574],{"className":20369,"style":5514},[75,76,1068],[31,20371,1076],{"className":20372},[1075],[31,20374,20376],{"className":20375},[1048],[31,20377,20379],{"className":20378,"style":1083},[1052],[31,20380],{},[31,20382],{"className":20383,"style":293},[119],[31,20385,223],{"className":20386},[297],[31,20388],{"className":20389,"style":293},[119],[31,20391,20393,20396,20436,20479,20519,20522,20526,20529,20572,20575,20578],{"className":20392},[66],[31,20394],{"className":20395,"style":1029},[70],[31,20397,20399,20402],{"className":20398},[75],[31,20400,5324],{"className":20401},[75,76],[31,20403,20405],{"className":20404},[1039],[31,20406,20408,20428],{"className":20407},[1043,1044],[31,20409,20411,20425],{"className":20410},[1048],[31,20412,20414],{"className":20413,"style":5445},[1052],[31,20415,20416,20419],{"style":2807},[31,20417],{"className":20418,"style":1061},[1060],[31,20420,20422],{"className":20421},[1065,1066,1067,1068],[31,20423,5391],{"className":20424},[75,76,1068],[31,20426,1076],{"className":20427},[1075],[31,20429,20431],{"className":20430},[1048],[31,20432,20434],{"className":20433,"style":2050},[1052],[31,20435],{},[31,20437,20439,20442],{"className":20438},[75],[31,20440,4558],{"className":20441,"style":8525},[75,76],[31,20443,20445],{"className":20444},[1039],[31,20446,20448,20471],{"className":20447},[1043,1044],[31,20449,20451,20468],{"className":20450},[1048],[31,20452,20454],{"className":20453,"style":5445},[1052],[31,20455,20456,20459],{"style":8540},[31,20457],{"className":20458,"style":1061},[1060],[31,20460,20462],{"className":20461},[1065,1066,1067,1068],[31,20463,20465],{"className":20464},[75,1068],[31,20466,5574],{"className":20467,"style":5514},[75,76,1068],[31,20469,1076],{"className":20470},[1075],[31,20472,20474],{"className":20473},[1048],[31,20475,20477],{"className":20476,"style":1083},[1052],[31,20478],{},[31,20480,20482,20485],{"className":20481},[75],[31,20483,5354],{"className":20484,"style":2666},[75,76],[31,20486,20488],{"className":20487},[1039],[31,20489,20491,20511],{"className":20490},[1043,1044],[31,20492,20494,20508],{"className":20493},[1048],[31,20495,20497],{"className":20496,"style":5445},[1052],[31,20498,20499,20502],{"style":5504},[31,20500],{"className":20501,"style":1061},[1060],[31,20503,20505],{"className":20504},[1065,1066,1067,1068],[31,20506,2555],{"className":20507,"style":5514},[75,76,1068],[31,20509,1076],{"className":20510},[1075],[31,20512,20514],{"className":20513},[1048],[31,20515,20517],{"className":20516,"style":1083},[1052],[31,20518],{},[31,20520,1210],{"className":20521},[1391],[31,20523],{"className":20524,"style":20525},[119],"margin-right:2em;",[31,20527],{"className":20528,"style":1467},[119],[31,20530,20532,20535],{"className":20531},[75],[31,20533,4558],{"className":20534,"style":8525},[75,76],[31,20536,20538],{"className":20537},[1039],[31,20539,20541,20564],{"className":20540},[1043,1044],[31,20542,20544,20561],{"className":20543},[1048],[31,20545,20547],{"className":20546,"style":5445},[1052],[31,20548,20549,20552],{"style":8540},[31,20550],{"className":20551,"style":1061},[1060],[31,20553,20555],{"className":20554},[1065,1066,1067,1068],[31,20556,20558],{"className":20557},[75,1068],[31,20559,5574],{"className":20560,"style":5514},[75,76,1068],[31,20562,1076],{"className":20563},[1075],[31,20565,20567],{"className":20566},[1048],[31,20568,20570],{"className":20569,"style":1083},[1052],[31,20571],{},[31,20573],{"className":20574,"style":293},[119],[31,20576,223],{"className":20577},[297],[31,20579],{"className":20580,"style":293},[119],[31,20582,20584,20587,20590,20593,20596,20639,20642,20645,20648],{"className":20583},[66],[31,20585],{"className":20586,"style":1223},[70],[31,20588,20301],{"className":20589},[1460],[31,20591,211],{"className":20592},[264],[31,20594,97],{"className":20595},[75],[31,20597,20599,20602],{"className":20598},[75],[31,20600,8101],{"className":20601,"style":8525},[75,76],[31,20603,20605],{"className":20604},[1039],[31,20606,20608,20631],{"className":20607},[1043,1044],[31,20609,20611,20628],{"className":20610},[1048],[31,20612,20614],{"className":20613,"style":5445},[1052],[31,20615,20616,20619],{"style":8540},[31,20617],{"className":20618,"style":1061},[1060],[31,20620,20622],{"className":20621},[1065,1066,1067,1068],[31,20623,20625],{"className":20624},[75,1068],[31,20626,5574],{"className":20627,"style":5514},[75,76,1068],[31,20629,1076],{"className":20630},[1075],[31,20632,20634],{"className":20633},[1048],[31,20635,20637],{"className":20636,"style":1083},[1052],[31,20638],{},[31,20640,3768],{"className":20641},[75],[31,20643,4793],{"className":20644},[75,76],[31,20646,220],{"className":20647},[289],[31,20649,1210],{"className":20650},[1391],[11,20652,20653,20694,20695,15282,20772,20881],{},[31,20654,20656,20673],{"className":20655,"translate":35},[34],[31,20657,20659],{"className":20658},[39],[41,20660,20661],{"xmlns":43},[45,20662,20663,20671],{},[48,20664,20665,20667,20669],{},[51,20666,5324],{},[95,20668,1210],{"separator":62},[51,20670,5354],{},[55,20672,7008],{"encoding":57},[31,20674,20676],{"className":20675,"ariaHidden":62},[61],[31,20677,20679,20682,20685,20688,20691],{"className":20678},[66],[31,20680],{"className":20681,"style":865},[70],[31,20683,5324],{"className":20684},[75,76],[31,20686,1210],{"className":20687},[1391],[31,20689],{"className":20690,"style":1467},[119],[31,20692,5354],{"className":20693,"style":2666},[75,76]," 由边缘约束定出，交替更新 ",[31,20696,20698,20727],{"className":20697,"translate":35},[34],[31,20699,20701],{"className":20700},[39],[41,20702,20703],{"xmlns":43},[45,20704,20705,20724],{},[48,20706,20707,20709,20712,20714,20716,20718,20720,20722],{},[51,20708,5324],{},[95,20710,20711],{},"←",[51,20713,5813],{},[51,20715,3768],{"mathvariant":494},[95,20717,211],{"stretchy":210},[51,20719,4558],{},[51,20721,5354],{},[95,20723,220],{"stretchy":210},[55,20725,20726],{"encoding":57},"u\\leftarrow a\u002F(Kv)",[31,20728,20730,20748],{"className":20729,"ariaHidden":62},[61],[31,20731,20733,20736,20739,20742,20745],{"className":20732},[66],[31,20734],{"className":20735,"style":71},[70],[31,20737,5324],{"className":20738},[75,76],[31,20740],{"className":20741,"style":293},[119],[31,20743,20711],{"className":20744},[297],[31,20746],{"className":20747,"style":293},[119],[31,20749,20751,20754,20757,20760,20763,20766,20769],{"className":20750},[66],[31,20752],{"className":20753,"style":260},[70],[31,20755,5813],{"className":20756},[75,76],[31,20758,3768],{"className":20759},[75],[31,20761,211],{"className":20762},[264],[31,20764,4558],{"className":20765,"style":8525},[75,76],[31,20767,5354],{"className":20768,"style":2666},[75,76],[31,20770,220],{"className":20771},[289],[31,20773,20775,20808],{"className":20774,"translate":35},[34],[31,20776,20778],{"className":20777},[39],[41,20779,20780],{"xmlns":43},[45,20781,20782,20805],{},[48,20783,20784,20786,20788,20790,20792,20794,20801,20803],{},[51,20785,5354],{},[95,20787,20711],{},[51,20789,5834],{},[51,20791,3768],{"mathvariant":494},[95,20793,211],{"stretchy":210},[1180,20795,20796,20798],{},[51,20797,4558],{},[51,20799,20800],{"mathvariant":494},"⊤",[51,20802,5324],{},[95,20804,220],{"stretchy":210},[55,20806,20807],{"encoding":57},"v\\leftarrow b\u002F(K^\\top u)",[31,20809,20811,20829],{"className":20810,"ariaHidden":62},[61],[31,20812,20814,20817,20820,20823,20826],{"className":20813},[66],[31,20815],{"className":20816,"style":71},[70],[31,20818,5354],{"className":20819,"style":2666},[75,76],[31,20821],{"className":20822,"style":293},[119],[31,20824,20711],{"className":20825},[297],[31,20827],{"className":20828,"style":293},[119],[31,20830,20832,20836,20839,20842,20845,20875,20878],{"className":20831},[66],[31,20833],{"className":20834,"style":20835},[70],"height:1.0991em;vertical-align:-0.25em;",[31,20837,5834],{"className":20838},[75,76],[31,20840,3768],{"className":20841},[75],[31,20843,211],{"className":20844},[264],[31,20846,20848,20851],{"className":20847},[75],[31,20849,4558],{"className":20850,"style":8525},[75,76],[31,20852,20854],{"className":20853},[1039],[31,20855,20857],{"className":20856},[1043],[31,20858,20860],{"className":20859},[1048],[31,20861,20864],{"className":20862,"style":20863},[1052],"height:0.8491em;",[31,20865,20866,20869],{"style":3631},[31,20867],{"className":20868,"style":1061},[1060],[31,20870,20872],{"className":20871},[1065,1066,1067,1068],[31,20873,20800],{"className":20874},[75,1068],[31,20876,5324],{"className":20877},[75,76],[31,20879,220],{"className":20880},[289],"，每次都是矩阵—向量乘法。这就是 Sinkhorn 迭代。它的意义不只是快，而是可微、可并到 GPU 上，于是能塞进神经网络当一层用。",[11,20883,20884,20912,20913,20941,20942,20970,20971,20999,21000,21051,21052,21080],{},[31,20885,20887,20900],{"className":20886,"translate":35},[34],[31,20888,20890],{"className":20889},[39],[41,20891,20892],{"xmlns":43},[45,20893,20894,20898],{},[48,20895,20896],{},[51,20897,4793],{},[55,20899,4796],{"encoding":57},[31,20901,20903],{"className":20902,"ariaHidden":62},[61],[31,20904,20906,20909],{"className":20905},[66],[31,20907],{"className":20908,"style":71},[70],[31,20910,4793],{"className":20911},[75,76]," 可以读作一个旋钮。取三个点搬到三个点：",[31,20914,20916,20929],{"className":20915,"translate":35},[34],[31,20917,20919],{"className":20918},[39],[41,20920,20921],{"xmlns":43},[45,20922,20923,20927],{},[48,20924,20925],{},[51,20926,4793],{},[55,20928,4796],{"encoding":57},[31,20930,20932],{"className":20931,"ariaHidden":62},[61],[31,20933,20935,20938],{"className":20934},[66],[31,20936],{"className":20937,"style":71},[70],[31,20939,4793],{"className":20940},[75,76]," 很大时，",[31,20943,20945,20958],{"className":20944,"translate":35},[34],[31,20946,20948],{"className":20947},[39],[41,20949,20950],{"xmlns":43},[45,20951,20952,20956],{},[48,20953,20954],{},[51,20955,4558],{},[55,20957,4558],{"encoding":57},[31,20959,20961],{"className":20960,"ariaHidden":62},[61],[31,20962,20964,20967],{"className":20963},[66],[31,20965],{"className":20966,"style":945},[70],[31,20968,4558],{"className":20969,"style":8525},[75,76]," 接近全 ",[31,20972,20974,20987],{"className":20973,"translate":35},[34],[31,20975,20977],{"className":20976},[39],[41,20978,20979],{"xmlns":43},[45,20980,20981,20985],{},[48,20982,20983],{},[91,20984,205],{},[55,20986,205],{"encoding":57},[31,20988,20990],{"className":20989,"ariaHidden":62},[61],[31,20991,20993,20996],{"className":20992},[66],[31,20994],{"className":20995,"style":326},[70],[31,20997,205],{"className":20998},[75]," 矩阵，每个点都往每处送一点，解是稠密的、模糊的；",[31,21001,21003,21021],{"className":21002,"translate":35},[34],[31,21004,21006],{"className":21005},[39],[41,21007,21008],{"xmlns":43},[45,21009,21010,21018],{},[48,21011,21012,21014,21016],{},[51,21013,4793],{},[95,21015,929],{},[91,21017,1977],{},[55,21019,21020],{"encoding":57},"\\varepsilon\\to0",[31,21022,21024,21042],{"className":21023,"ariaHidden":62},[61],[31,21025,21027,21030,21033,21036,21039],{"className":21026},[66],[31,21028],{"className":21029,"style":71},[70],[31,21031,4793],{"className":21032},[75,76],[31,21034],{"className":21035,"style":293},[119],[31,21037,929],{"className":21038},[297],[31,21040],{"className":21041,"style":293},[119],[31,21043,21045,21048],{"className":21044},[66],[31,21046],{"className":21047,"style":326},[70],[31,21049,1977],{"className":21050},[75]," 时 ",[31,21053,21055,21068],{"className":21054,"translate":35},[34],[31,21056,21058],{"className":21057},[39],[41,21059,21060],{"xmlns":43},[45,21061,21062,21066],{},[48,21063,21064],{},[51,21065,4558],{},[55,21067,4558],{"encoding":57},[31,21069,21071],{"className":21070,"ariaHidden":62},[61],[31,21072,21074,21077],{"className":21073},[66],[31,21075],{"className":21076,"style":945},[70],[31,21078,4558],{"className":21079,"style":8525},[75,76]," 接近单位矩阵，解退化成“就近硬匹配”，也就是原始 OT。中间是连续插值。",[11,21082,1394,21083,21135,21136,21214,21215,21562],{},[31,21084,21086,21104],{"className":21085,"translate":35},[34],[31,21087,21089],{"className":21088},[39],[41,21090,21091],{"xmlns":43},[45,21092,21093,21101],{},[48,21094,21095,21097,21099],{},[51,21096,4793],{},[95,21098,16040],{},[91,21100,1977],{},[55,21102,21103],{"encoding":57},"\\varepsilon>0",[31,21105,21107,21126],{"className":21106,"ariaHidden":62},[61],[31,21108,21110,21114,21117,21120,21123],{"className":21109},[66],[31,21111],{"className":21112,"style":21113},[70],"height:0.5782em;vertical-align:-0.0391em;",[31,21115,4793],{"className":21116},[75,76],[31,21118],{"className":21119,"style":293},[119],[31,21121,16040],{"className":21122},[297],[31,21124],{"className":21125,"style":293},[119],[31,21127,21129,21132],{"className":21128},[66],[31,21130],{"className":21131,"style":326},[70],[31,21133,1977],{"className":21134},[75]," 时它不再是度量。最直接的一点：把 ",[31,21137,21139,21171],{"className":21138,"translate":35},[34],[31,21140,21142],{"className":21141},[39],[41,21143,21144],{"xmlns":43},[45,21145,21146,21168],{},[48,21147,21148,21150,21152,21162,21164,21166],{},[51,21149,2515],{},[95,21151,223],{},[48,21153,21154,21156,21158,21160],{},[51,21155,1435],{"mathvariant":494},[51,21157,5391],{"mathvariant":494},[51,21159,5813],{"mathvariant":494},[51,21161,14556],{"mathvariant":494},[95,21163,211],{"stretchy":210},[51,21165,5813],{},[95,21167,220],{"stretchy":210},[55,21169,21170],{"encoding":57},"\\pi=\\mathrm{diag}(a)",[31,21172,21174,21192],{"className":21173,"ariaHidden":62},[61],[31,21175,21177,21180,21183,21186,21189],{"className":21176},[66],[31,21178],{"className":21179,"style":71},[70],[31,21181,2515],{"className":21182,"style":2666},[75,76],[31,21184],{"className":21185,"style":293},[119],[31,21187,223],{"className":21188},[297],[31,21190],{"className":21191,"style":293},[119],[31,21193,21195,21198,21205,21208,21211],{"className":21194},[66],[31,21196],{"className":21197,"style":260},[70],[31,21199,21201],{"className":21200},[75],[31,21202,21204],{"className":21203,"style":20180},[75,2745],"diag",[31,21206,211],{"className":21207},[264],[31,21209,5813],{"className":21210},[75,76],[31,21212,220],{"className":21213},[289]," 代入，",[31,21216,21218,21277],{"className":21217,"translate":35},[34],[31,21219,21221],{"className":21220},[39],[41,21222,21223],{"xmlns":43},[45,21224,21225,21274],{},[48,21226,21227,21233,21235,21237,21239,21241,21243,21245,21247,21253,21259,21261,21263,21269,21272],{},[1004,21228,21229,21231],{},[51,21230,3117],{},[51,21232,4793],{},[95,21234,211],{"stretchy":210},[51,21236,852],{},[95,21238,1210],{"separator":62},[51,21240,852],{},[95,21242,220],{"stretchy":210},[95,21244,223],{},[51,21246,4793],{},[1004,21248,21249,21251],{},[95,21250,5625],{},[51,21252,5391],{},[1004,21254,21255,21257],{},[51,21256,5813],{},[51,21258,5391],{},[51,21260,19939],{},[95,21262,1540],{},[1004,21264,21265,21267],{},[51,21266,5813],{},[51,21268,5391],{},[95,21270,21271],{"mathvariant":494},"≠",[91,21273,1977],{},[55,21275,21276],{"encoding":57},"W_\\varepsilon(\\mu,\\mu)=\\varepsilon\\sum_i a_i\\log a_i\\ne0",[31,21278,21280,21353,21553],{"className":21279,"ariaHidden":62},[61],[31,21281,21283,21286,21326,21329,21332,21335,21338,21341,21344,21347,21350],{"className":21282},[66],[31,21284],{"className":21285,"style":260},[70],[31,21287,21289,21292],{"className":21288},[75],[31,21290,3117],{"className":21291,"style":949},[75,76],[31,21293,21295],{"className":21294},[1039],[31,21296,21298,21318],{"className":21297},[1043,1044],[31,21299,21301,21315],{"className":21300},[1048],[31,21302,21304],{"className":21303,"style":3230},[1052],[31,21305,21306,21309],{"style":1056},[31,21307],{"className":21308,"style":1061},[1060],[31,21310,21312],{"className":21311},[1065,1066,1067,1068],[31,21313,4793],{"className":21314},[75,76,1068],[31,21316,1076],{"className":21317},[1075],[31,21319,21321],{"className":21320},[1048],[31,21322,21324],{"className":21323,"style":2050},[1052],[31,21325],{},[31,21327,211],{"className":21328},[264],[31,21330,852],{"className":21331},[75,76],[31,21333,1210],{"className":21334},[1391],[31,21336],{"className":21337,"style":1467},[119],[31,21339,852],{"className":21340},[75,76],[31,21342,220],{"className":21343},[289],[31,21345],{"className":21346,"style":293},[119],[31,21348,223],{"className":21349},[297],[31,21351],{"className":21352,"style":293},[119],[31,21354,21356,21360,21363,21366,21410,21413,21453,21456,21461,21464,21504,21507,21550],{"className":21355},[66],[31,21357],{"className":21358,"style":21359},[70],"height:1.0497em;vertical-align:-0.2997em;",[31,21361,4793],{"className":21362},[75,76],[31,21364],{"className":21365,"style":1467},[119],[31,21367,21369,21373],{"className":21368},[1460],[31,21370,5625],{"className":21371,"style":21372},[1460,1461,1462],"position:relative;top:0em;",[31,21374,21376],{"className":21375},[1039],[31,21377,21379,21401],{"className":21378},[1043,1044],[31,21380,21382,21398],{"className":21381},[1048],[31,21383,21386],{"className":21384,"style":21385},[1052],"height:0.162em;",[31,21387,21389,21392],{"style":21388},"top:-2.4003em;margin-left:0em;margin-right:0.05em;",[31,21390],{"className":21391,"style":1061},[1060],[31,21393,21395],{"className":21394},[1065,1066,1067,1068],[31,21396,5391],{"className":21397},[75,76,1068],[31,21399,1076],{"className":21400},[1075],[31,21402,21404],{"className":21403},[1048],[31,21405,21408],{"className":21406,"style":21407},[1052],"height:0.2997em;",[31,21409],{},[31,21411],{"className":21412,"style":1467},[119],[31,21414,21416,21419],{"className":21415},[75],[31,21417,5813],{"className":21418},[75,76],[31,21420,21422],{"className":21421},[1039],[31,21423,21425,21445],{"className":21424},[1043,1044],[31,21426,21428,21442],{"className":21427},[1048],[31,21429,21431],{"className":21430,"style":5445},[1052],[31,21432,21433,21436],{"style":2807},[31,21434],{"className":21435,"style":1061},[1060],[31,21437,21439],{"className":21438},[1065,1066,1067,1068],[31,21440,5391],{"className":21441},[75,76,1068],[31,21443,1076],{"className":21444},[1075],[31,21446,21448],{"className":21447},[1048],[31,21449,21451],{"className":21450,"style":2050},[1052],[31,21452],{},[31,21454],{"className":21455,"style":1467},[119],[31,21457,20177,21459],{"className":21458},[1460],[31,21460,14556],{"style":20180},[31,21462],{"className":21463,"style":1467},[119],[31,21465,21467,21470],{"className":21466},[75],[31,21468,5813],{"className":21469},[75,76],[31,21471,21473],{"className":21472},[1039],[31,21474,21476,21496],{"className":21475},[1043,1044],[31,21477,21479,21493],{"className":21478},[1048],[31,21480,21482],{"className":21481,"style":5445},[1052],[31,21483,21484,21487],{"style":2807},[31,21485],{"className":21486,"style":1061},[1060],[31,21488,21490],{"className":21489},[1065,1066,1067,1068],[31,21491,5391],{"className":21492},[75,76,1068],[31,21494,1076],{"className":21495},[1075],[31,21497,21499],{"className":21498},[1048],[31,21500,21502],{"className":21501,"style":2050},[1052],[31,21503],{},[31,21505],{"className":21506,"style":293},[119],[31,21508,21510,21543,21547],{"className":21509},[297],[31,21511,21513],{"className":21512},[297],[31,21514,21517],{"className":21515},[75,21516],"vbox",[31,21518,21521],{"className":21519},[21520],"thinbox",[31,21522,21525,21528,21539],{"className":21523},[21524],"rlap",[31,21526],{"className":21527,"style":4202},[70],[31,21529,21532],{"className":21530},[21531],"inner",[31,21533,21535],{"className":21534},[75],[31,21536,21538],{"className":21537},[297],"",[31,21540],{"className":21541},[21542],"fix",[31,21544],{"className":21545},[119,21546],"nobreak",[31,21548,223],{"className":21549},[297],[31,21551],{"className":21552,"style":293},[119],[31,21554,21556,21559],{"className":21555},[66],[31,21557],{"className":21558,"style":326},[70],[31,21560,1977],{"className":21561},[75],"。所以工程上常用的 Sinkhorn 散度要先去偏：",[31,21564,21566],{"className":21565,"translate":35},[411],[31,21567,21569,21682],{"className":21568,"translate":35},[34],[31,21570,21572],{"className":21571},[39],[41,21573,21574],{"xmlns":43,"display":420},[45,21575,21576,21679],{},[48,21577,21578,21585,21587,21589,21591,21593,21595,21597,21607,21609,21611,21613,21615,21617,21619,21627,21637,21639,21641,21643,21645,21647,21649,21657,21667,21669,21671,21673,21675,21677],{},[1004,21579,21580,21583],{},[51,21581,21582],{},"S",[51,21584,4793],{},[95,21586,211],{"stretchy":210},[51,21588,852],{},[95,21590,1210],{"separator":62},[51,21592,884],{},[95,21594,220],{"stretchy":210},[95,21596,223],{},[1004,21598,21599,21605],{},[48,21600,21601,21603],{},[51,21602,19733],{"mathvariant":494},[51,21604,919],{"mathvariant":494},[51,21606,4793],{},[95,21608,211],{"stretchy":210},[51,21610,852],{},[95,21612,1210],{"separator":62},[51,21614,884],{},[95,21616,220],{"stretchy":210},[95,21618,97],{},[2071,21620,21621],{"scriptlevel":1977,"displaystyle":210},[2074,21622,21623,21625],{},[91,21624,205],{},[91,21626,152],{},[1004,21628,21629,21635],{},[48,21630,21631,21633],{},[51,21632,19733],{"mathvariant":494},[51,21634,919],{"mathvariant":494},[51,21636,4793],{},[95,21638,211],{"stretchy":210},[51,21640,852],{},[95,21642,1210],{"separator":62},[51,21644,852],{},[95,21646,220],{"stretchy":210},[95,21648,97],{},[2071,21650,21651],{"scriptlevel":1977,"displaystyle":210},[2074,21652,21653,21655],{},[91,21654,205],{},[91,21656,152],{},[1004,21658,21659,21665],{},[48,21660,21661,21663],{},[51,21662,19733],{"mathvariant":494},[51,21664,919],{"mathvariant":494},[51,21666,4793],{},[95,21668,211],{"stretchy":210},[51,21670,884],{},[95,21672,1210],{"separator":62},[51,21674,884],{},[95,21676,220],{"stretchy":210},[51,21678,495],{"mathvariant":494},[55,21680,21681],{"encoding":57},"S_\\varepsilon(\\mu,\\nu)=\\mathrm{OT}_\\varepsilon(\\mu,\\nu)-\\tfrac12\\mathrm{OT}_\\varepsilon(\\mu,\\mu)-\\tfrac12\\mathrm{OT}_\\varepsilon(\\nu,\\nu).",[31,21683,21685,21760,21838,21982],{"className":21684,"ariaHidden":62},[61],[31,21686,21688,21691,21733,21736,21739,21742,21745,21748,21751,21754,21757],{"className":21687},[66],[31,21689],{"className":21690,"style":260},[70],[31,21692,21694,21698],{"className":21693},[75],[31,21695,21582],{"className":21696,"style":21697},[75,76],"margin-right:0.0576em;",[31,21699,21701],{"className":21700},[1039],[31,21702,21704,21725],{"className":21703},[1043,1044],[31,21705,21707,21722],{"className":21706},[1048],[31,21708,21710],{"className":21709,"style":3230},[1052],[31,21711,21713,21716],{"style":21712},"top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;",[31,21714],{"className":21715,"style":1061},[1060],[31,21717,21719],{"className":21718},[1065,1066,1067,1068],[31,21720,4793],{"className":21721},[75,76,1068],[31,21723,1076],{"className":21724},[1075],[31,21726,21728],{"className":21727},[1048],[31,21729,21731],{"className":21730,"style":2050},[1052],[31,21732],{},[31,21734,211],{"className":21735},[264],[31,21737,852],{"className":21738},[75,76],[31,21740,1210],{"className":21741},[1391],[31,21743],{"className":21744,"style":1467},[119],[31,21746,884],{"className":21747,"style":900},[75,76],[31,21749,220],{"className":21750},[289],[31,21752],{"className":21753,"style":293},[119],[31,21755,223],{"className":21756},[297],[31,21758],{"className":21759,"style":293},[119],[31,21761,21763,21766,21811,21814,21817,21820,21823,21826,21829,21832,21835],{"className":21762},[66],[31,21764],{"className":21765,"style":260},[70],[31,21767,21769,21776],{"className":21768},[75],[31,21770,21772],{"className":21771},[75],[31,21773,21775],{"className":21774},[75,2745],"OT",[31,21777,21779],{"className":21778},[1039],[31,21780,21782,21803],{"className":21781},[1043,1044],[31,21783,21785,21800],{"className":21784},[1048],[31,21786,21788],{"className":21787,"style":3230},[1052],[31,21789,21791,21794],{"style":21790},"top:-2.55em;margin-right:0.05em;",[31,21792],{"className":21793,"style":1061},[1060],[31,21795,21797],{"className":21796},[1065,1066,1067,1068],[31,21798,4793],{"className":21799},[75,76,1068],[31,21801,1076],{"className":21802},[1075],[31,21804,21806],{"className":21805},[1048],[31,21807,21809],{"className":21808,"style":2050},[1052],[31,21810],{},[31,21812,211],{"className":21813},[264],[31,21815,852],{"className":21816},[75,76],[31,21818,1210],{"className":21819},[1391],[31,21821],{"className":21822,"style":1467},[119],[31,21824,884],{"className":21825,"style":900},[75,76],[31,21827,220],{"className":21828},[289],[31,21830],{"className":21831,"style":120},[119],[31,21833,97],{"className":21834},[124],[31,21836],{"className":21837,"style":120},[119],[31,21839,21841,21844,21912,21955,21958,21961,21964,21967,21970,21973,21976,21979],{"className":21840},[66],[31,21842],{"className":21843,"style":2132},[70],[31,21845,21847,21850,21909],{"className":21846},[75],[31,21848],{"className":21849},[264,2139],[31,21851,21853],{"className":21852},[2074],[31,21854,21856,21901],{"className":21855},[1043,1044],[31,21857,21859,21898],{"className":21858},[1048],[31,21860,21862,21876,21884],{"className":21861,"style":2152},[1052],[31,21863,21864,21867],{"style":2155},[31,21865],{"className":21866,"style":1637},[1060],[31,21868,21870],{"className":21869},[1065,1066,1067,1068],[31,21871,21873],{"className":21872},[75,1068],[31,21874,152],{"className":21875},[75,1068],[31,21877,21878,21881],{"style":2170},[31,21879],{"className":21880,"style":1637},[1060],[31,21882],{"className":21883,"style":2178},[2177],[31,21885,21886,21889],{"style":2181},[31,21887],{"className":21888,"style":1637},[1060],[31,21890,21892],{"className":21891},[1065,1066,1067,1068],[31,21893,21895],{"className":21894},[75,1068],[31,21896,205],{"className":21897},[75,1068],[31,21899,1076],{"className":21900},[1075],[31,21902,21904],{"className":21903},[1048],[31,21905,21907],{"className":21906,"style":2203},[1052],[31,21908],{},[31,21910],{"className":21911},[289,2139],[31,21913,21915,21921],{"className":21914},[75],[31,21916,21918],{"className":21917},[75],[31,21919,21775],{"className":21920},[75,2745],[31,21922,21924],{"className":21923},[1039],[31,21925,21927,21947],{"className":21926},[1043,1044],[31,21928,21930,21944],{"className":21929},[1048],[31,21931,21933],{"className":21932,"style":3230},[1052],[31,21934,21935,21938],{"style":21790},[31,21936],{"className":21937,"style":1061},[1060],[31,21939,21941],{"className":21940},[1065,1066,1067,1068],[31,21942,4793],{"className":21943},[75,76,1068],[31,21945,1076],{"className":21946},[1075],[31,21948,21950],{"className":21949},[1048],[31,21951,21953],{"className":21952,"style":2050},[1052],[31,21954],{},[31,21956,211],{"className":21957},[264],[31,21959,852],{"className":21960},[75,76],[31,21962,1210],{"className":21963},[1391],[31,21965],{"className":21966,"style":1467},[119],[31,21968,852],{"className":21969},[75,76],[31,21971,220],{"className":21972},[289],[31,21974],{"className":21975,"style":120},[119],[31,21977,97],{"className":21978},[124],[31,21980],{"className":21981,"style":120},[119],[31,21983,21985,21988,22056,22099,22102,22105,22108,22111,22114,22117],{"className":21984},[66],[31,21986],{"className":21987,"style":2132},[70],[31,21989,21991,21994,22053],{"className":21990},[75],[31,21992],{"className":21993},[264,2139],[31,21995,21997],{"className":21996},[2074],[31,21998,22000,22045],{"className":21999},[1043,1044],[31,22001,22003,22042],{"className":22002},[1048],[31,22004,22006,22020,22028],{"className":22005,"style":2152},[1052],[31,22007,22008,22011],{"style":2155},[31,22009],{"className":22010,"style":1637},[1060],[31,22012,22014],{"className":22013},[1065,1066,1067,1068],[31,22015,22017],{"className":22016},[75,1068],[31,22018,152],{"className":22019},[75,1068],[31,22021,22022,22025],{"style":2170},[31,22023],{"className":22024,"style":1637},[1060],[31,22026],{"className":22027,"style":2178},[2177],[31,22029,22030,22033],{"style":2181},[31,22031],{"className":22032,"style":1637},[1060],[31,22034,22036],{"className":22035},[1065,1066,1067,1068],[31,22037,22039],{"className":22038},[75,1068],[31,22040,205],{"className":22041},[75,1068],[31,22043,1076],{"className":22044},[1075],[31,22046,22048],{"className":22047},[1048],[31,22049,22051],{"className":22050,"style":2203},[1052],[31,22052],{},[31,22054],{"className":22055},[289,2139],[31,22057,22059,22065],{"className":22058},[75],[31,22060,22062],{"className":22061},[75],[31,22063,21775],{"className":22064},[75,2745],[31,22066,22068],{"className":22067},[1039],[31,22069,22071,22091],{"className":22070},[1043,1044],[31,22072,22074,22088],{"className":22073},[1048],[31,22075,22077],{"className":22076,"style":3230},[1052],[31,22078,22079,22082],{"style":21790},[31,22080],{"className":22081,"style":1061},[1060],[31,22083,22085],{"className":22084},[1065,1066,1067,1068],[31,22086,4793],{"className":22087},[75,76,1068],[31,22089,1076],{"className":22090},[1075],[31,22092,22094],{"className":22093},[1048],[31,22095,22097],{"className":22096,"style":2050},[1052],[31,22098],{},[31,22100,211],{"className":22101},[264],[31,22103,884],{"className":22104,"style":900},[75,76],[31,22106,1210],{"className":22107},[1391],[31,22109],{"className":22110,"style":1467},[119],[31,22112,884],{"className":22113,"style":900},[75,76],[31,22115,220],{"className":22116},[289],[31,22118,495],{"className":22119},[75],[11,22121,22122,22123,22174],{},"去偏之后才非负、才在 ",[31,22124,22126,22144],{"className":22125,"translate":35},[34],[31,22127,22129],{"className":22128},[39],[41,22130,22131],{"xmlns":43},[45,22132,22133,22141],{},[48,22134,22135,22137,22139],{},[51,22136,852],{},[95,22138,223],{},[51,22140,884],{},[55,22142,22143],{"encoding":57},"\\mu=\\nu",[31,22145,22147,22165],{"className":22146,"ariaHidden":62},[61],[31,22148,22150,22153,22156,22159,22162],{"className":22149},[66],[31,22151],{"className":22152,"style":865},[70],[31,22154,852],{"className":22155},[75,76],[31,22157],{"className":22158,"style":293},[119],[31,22160,223],{"className":22161},[297],[31,22163],{"className":22164,"style":293},[119],[31,22166,22168,22171],{"className":22167},[66],[31,22169],{"className":22170,"style":71},[70],[31,22172,884],{"className":22173,"style":900},[75,76]," 处为零。",[11,22176,22177,22178,22248,22249,22277,22278,22306],{},"还有一个更根本的代价，来自样本。经验分布 ",[31,22179,22181,22199],{"className":22180,"translate":35},[34],[31,22182,22184],{"className":22183},[39],[41,22185,22186],{"xmlns":43},[45,22187,22188,22196],{},[48,22189,22190],{},[1004,22191,22192,22194],{},[51,22193,852],{},[51,22195,14559],{},[55,22197,22198],{"encoding":57},"\\mu_n",[31,22200,22202],{"className":22201,"ariaHidden":62},[61],[31,22203,22205,22208],{"className":22204},[66],[31,22206],{"className":22207,"style":865},[70],[31,22209,22211,22214],{"className":22210},[75],[31,22212,852],{"className":22213},[75,76],[31,22215,22217],{"className":22216},[1039],[31,22218,22220,22240],{"className":22219},[1043,1044],[31,22221,22223,22237],{"className":22222},[1048],[31,22224,22226],{"className":22225,"style":3230},[1052],[31,22227,22228,22231],{"style":2807},[31,22229],{"className":22230,"style":1061},[1060],[31,22232,22234],{"className":22233},[1065,1066,1067,1068],[31,22235,14559],{"className":22236},[75,76,1068],[31,22238,1076],{"className":22239},[1075],[31,22241,22243],{"className":22242},[1048],[31,22244,22246],{"className":22245,"style":2050},[1052],[31,22247],{}," 到真分布 ",[31,22250,22252,22265],{"className":22251,"translate":35},[34],[31,22253,22255],{"className":22254},[39],[41,22256,22257],{"xmlns":43},[45,22258,22259,22263],{},[48,22260,22261],{},[51,22262,852],{},[55,22264,855],{"encoding":57},[31,22266,22268],{"className":22267,"ariaHidden":62},[61],[31,22269,22271,22274],{"className":22270},[66],[31,22272],{"className":22273,"style":865},[70],[31,22275,852],{"className":22276},[75,76]," 的收敛速度，在 ",[31,22279,22281,22294],{"className":22280,"translate":35},[34],[31,22282,22284],{"className":22283},[39],[41,22285,22286],{"xmlns":43},[45,22287,22288,22292],{},[48,22289,22290],{},[51,22291,1435],{},[55,22293,1435],{"encoding":57},[31,22295,22297],{"className":22296,"ariaHidden":62},[61],[31,22298,22300,22303],{"className":22299},[66],[31,22301],{"className":22302,"style":1630},[70],[31,22304,1435],{"className":22305},[75,76]," 维空间里大致是",[31,22308,22310],{"className":22309,"translate":35},[411],[31,22311,22313,22387],{"className":22312,"translate":35},[34],[31,22314,22316],{"className":22315},[39],[41,22317,22318],{"xmlns":43,"display":420},[45,22319,22320,22384],{},[48,22321,22322,22325,22327,22333,22335,22341,22343,22345,22347,22349,22352,22354,22368,22370,22372,22374,22376,22378,22380,22382],{},[51,22323,22324],{"mathvariant":14789},"E",[1430,22326,1432],{},[1004,22328,22329,22331],{},[51,22330,3117],{},[51,22332,11],{},[95,22334,211],{"stretchy":210},[1004,22336,22337,22339],{},[51,22338,852],{},[51,22340,14559],{},[95,22342,1210],{"separator":62},[51,22344,852],{},[95,22346,220],{"stretchy":210},[1430,22348,1545],{},[95,22350,22351],{},"≲",[1430,22353,1545],{},[1180,22355,22356,22358],{},[51,22357,14559],{},[48,22359,22360,22362,22364,22366],{},[95,22361,97],{},[91,22363,205],{},[51,22365,3768],{"mathvariant":494},[51,22367,1435],{},[119,22369],{"width":20286},[95,22371,211],{"stretchy":210},[51,22373,1435],{},[95,22375,16040],{},[91,22377,152],{},[51,22379,11],{},[95,22381,220],{"stretchy":210},[51,22383,495],{"mathvariant":494},[55,22385,22386],{"encoding":57},"\\mathbb E\\,W_p(\\mu_n,\\mu)\\ \\lesssim\\ n^{-1\u002Fd}\\qquad(d>2p).",[31,22388,22390,22513,22577],{"className":22389,"ariaHidden":62},[61],[31,22391,22393,22396,22399,22402,22442,22445,22485,22488,22491,22494,22497,22500,22503,22507,22510],{"className":22392},[66],[31,22394],{"className":22395,"style":1223},[70],[31,22397,22324],{"className":22398},[75,14947],[31,22400],{"className":22401,"style":1467},[119],[31,22403,22405,22408],{"className":22404},[75],[31,22406,3117],{"className":22407,"style":949},[75,76],[31,22409,22411],{"className":22410},[1039],[31,22412,22414,22434],{"className":22413},[1043,1044],[31,22415,22417,22431],{"className":22416},[1048],[31,22418,22420],{"className":22419,"style":3230},[1052],[31,22421,22422,22425],{"style":1056},[31,22423],{"className":22424,"style":1061},[1060],[31,22426,22428],{"className":22427},[1065,1066,1067,1068],[31,22429,11],{"className":22430},[75,76,1068],[31,22432,1076],{"className":22433},[1075],[31,22435,22437],{"className":22436},[1048],[31,22438,22440],{"className":22439,"style":1083},[1052],[31,22441],{},[31,22443,211],{"className":22444},[264],[31,22446,22448,22451],{"className":22447},[75],[31,22449,852],{"className":22450},[75,76],[31,22452,22454],{"className":22453},[1039],[31,22455,22457,22477],{"className":22456},[1043,1044],[31,22458,22460,22474],{"className":22459},[1048],[31,22461,22463],{"className":22462,"style":3230},[1052],[31,22464,22465,22468],{"style":2807},[31,22466],{"className":22467,"style":1061},[1060],[31,22469,22471],{"className":22470},[1065,1066,1067,1068],[31,22472,14559],{"className":22473},[75,76,1068],[31,22475,1076],{"className":22476},[1075],[31,22478,22480],{"className":22479},[1048],[31,22481,22483],{"className":22482,"style":2050},[1052],[31,22484],{},[31,22486,1210],{"className":22487},[1391],[31,22489],{"className":22490,"style":1467},[119],[31,22492,852],{"className":22493},[75,76],[31,22495,220],{"className":22496},[289],[31,22498,1545],{"className":22499},[119],[31,22501],{"className":22502,"style":293},[119],[31,22504,22351],{"className":22505},[297,22506],"amsrm",[31,22508,1545],{"className":22509},[119],[31,22511],{"className":22512,"style":293},[119],[31,22514,22516,22520,22559,22562,22565,22568,22571,22574],{"className":22515},[66],[31,22517],{"className":22518,"style":22519},[70],"height:1.188em;vertical-align:-0.25em;",[31,22521,22523,22526],{"className":22522},[75],[31,22524,14559],{"className":22525},[75,76],[31,22527,22529],{"className":22528},[1039],[31,22530,22532],{"className":22531},[1043],[31,22533,22535],{"className":22534},[1048],[31,22536,22539],{"className":22537,"style":22538},[1052],"height:0.938em;",[31,22540,22541,22544],{"style":1330},[31,22542],{"className":22543,"style":1061},[1060],[31,22545,22547],{"className":22546},[1065,1066,1067,1068],[31,22548,22550,22553,22556],{"className":22549},[75,1068],[31,22551,97],{"className":22552},[75,1068],[31,22554,4042],{"className":22555},[75,1068],[31,22557,1435],{"className":22558},[75,76,1068],[31,22560],{"className":22561,"style":20525},[119],[31,22563,211],{"className":22564},[264],[31,22566,1435],{"className":22567},[75,76],[31,22569],{"className":22570,"style":293},[119],[31,22572,16040],{"className":22573},[297],[31,22575],{"className":22576,"style":293},[119],[31,22578,22580,22583,22586,22589,22592],{"className":22579},[66],[31,22581],{"className":22582,"style":260},[70],[31,22584,152],{"className":22585},[75],[31,22587,11],{"className":22588},[75,76],[31,22590,220],{"className":22591},[289],[31,22593,495],{"className":22594},[75],[11,22596,22597,22598,22650,22651,22721,22722,22784],{},"指数太糟糕了。",[31,22599,22601,22620],{"className":22600,"translate":35},[34],[31,22602,22604],{"className":22603},[39],[41,22605,22606],{"xmlns":43},[45,22607,22608,22617],{},[48,22609,22610,22612,22614],{},[51,22611,1435],{},[95,22613,223],{},[91,22615,22616],{},"100",[55,22618,22619],{"encoding":57},"d=100",[31,22621,22623,22641],{"className":22622,"ariaHidden":62},[61],[31,22624,22626,22629,22632,22635,22638],{"className":22625},[66],[31,22627],{"className":22628,"style":1630},[70],[31,22630,1435],{"className":22631},[75,76],[31,22633],{"className":22634,"style":293},[119],[31,22636,223],{"className":22637},[297],[31,22639],{"className":22640,"style":293},[119],[31,22642,22644,22647],{"className":22643},[66],[31,22645],{"className":22646,"style":326},[70],[31,22648,22616],{"className":22649},[75]," 时误差按 ",[31,22652,22654,22677],{"className":22653,"translate":35},[34],[31,22655,22657],{"className":22656},[39],[41,22658,22659],{"xmlns":43},[45,22660,22661,22674],{},[48,22662,22663],{},[1180,22664,22665,22667],{},[51,22666,14559],{},[48,22668,22669,22671],{},[95,22670,97],{},[91,22672,22673],{},"0.01",[55,22675,22676],{"encoding":57},"n^{-0.01}",[31,22678,22680],{"className":22679,"ariaHidden":62},[61],[31,22681,22683,22686],{"className":22682},[66],[31,22684],{"className":22685,"style":5135},[70],[31,22687,22689,22692],{"className":22688},[75],[31,22690,14559],{"className":22691},[75,76],[31,22693,22695],{"className":22694},[1039],[31,22696,22698],{"className":22697},[1043],[31,22699,22701],{"className":22700},[1048],[31,22702,22704],{"className":22703,"style":5135},[1052],[31,22705,22706,22709],{"style":3631},[31,22707],{"className":22708,"style":1061},[1060],[31,22710,22712],{"className":22711},[1065,1066,1067,1068],[31,22713,22715,22718],{"className":22714},[75,1068],[31,22716,97],{"className":22717},[75,1068],[31,22719,22673],{"className":22720},[75,1068]," 衰减，想把误差减半，样本量要乘 ",[31,22723,22725,22743],{"className":22724,"translate":35},[34],[31,22726,22728],{"className":22727},[39],[41,22729,22730],{"xmlns":43},[45,22731,22732,22740],{},[48,22733,22734],{},[1180,22735,22736,22738],{},[91,22737,152],{},[91,22739,22616],{},[55,22741,22742],{"encoding":57},"2^{100}",[31,22744,22746],{"className":22745,"ariaHidden":62},[61],[31,22747,22749,22752],{"className":22748},[66],[31,22750],{"className":22751,"style":5135},[70],[31,22753,22755,22758],{"className":22754},[75],[31,22756,152],{"className":22757},[75],[31,22759,22761],{"className":22760},[1039],[31,22762,22764],{"className":22763},[1043],[31,22765,22767],{"className":22766},[1048],[31,22768,22770],{"className":22769,"style":5135},[1052],[31,22771,22772,22775],{"style":3631},[31,22773],{"className":22774,"style":1061},[1060],[31,22776,22778],{"className":22777},[1065,1066,1067,1068],[31,22779,22781],{"className":22780},[75,1068],[31,22782,22616],{"className":22783},[75,1068],"。所以“直接在高维空间算 Wasserstein”在绝大多数真实问题上不可行。",[11,22786,22787,22788,22816,22817,22891],{},"三条出路我都用过：把数据先降到低维再算；用 sliced Wasserstein，把分布投影到随机一维方向上算 ",[31,22789,22791,22804],{"className":22790,"translate":35},[34],[31,22792,22794],{"className":22793},[39],[41,22795,22796],{"xmlns":43},[45,22797,22798,22802],{},[48,22799,22800],{},[51,22801,3117],{},[55,22803,3117],{"encoding":57},[31,22805,22807],{"className":22806,"ariaHidden":62},[61],[31,22808,22810,22813],{"className":22809},[66],[31,22811],{"className":22812,"style":945},[70],[31,22814,3117],{"className":22815,"style":949},[75,76],"，再对方向平均（一维的收敛是 ",[31,22818,22820,22846],{"className":22819,"translate":35},[34],[31,22821,22823],{"className":22822},[39],[41,22824,22825],{"xmlns":43},[45,22826,22827,22843],{},[48,22828,22829],{},[1180,22830,22831,22833],{},[51,22832,14559],{},[48,22834,22835,22837,22839,22841],{},[95,22836,97],{},[91,22838,205],{},[51,22840,3768],{"mathvariant":494},[91,22842,152],{},[55,22844,22845],{"encoding":57},"n^{-1\u002F2}",[31,22847,22849],{"className":22848,"ariaHidden":62},[61],[31,22850,22852,22856],{"className":22851},[66],[31,22853],{"className":22854,"style":22855},[70],"height:0.888em;",[31,22857,22859,22862],{"className":22858},[75],[31,22860,14559],{"className":22861},[75,76],[31,22863,22865],{"className":22864},[1039],[31,22866,22868],{"className":22867},[1043],[31,22869,22871],{"className":22870},[1048],[31,22872,22874],{"className":22873,"style":22855},[1052],[31,22875,22876,22879],{"style":3631},[31,22877],{"className":22878,"style":1061},[1060],[31,22880,22882],{"className":22881},[1065,1066,1067,1068],[31,22883,22885,22888],{"className":22884},[75,1068],[31,22886,97],{"className":22887},[75,1068],[31,22889,17145],{"className":22890},[75,1068],"，一下就正常了）；或者用低秩分解把代价矩阵近似成低秩。共同点是不要在原始高维空间里做全秩的 OT。",[18,22893,22894],{"id":22894},"单纯形上的尺子",[11,22896,22897,22898,2446,22952,23005,23006,2446,23059,2372],{},"我拿两个模型对同一张图跑一遍，输出分别是 ",[31,22899,22901,22925],{"className":22900,"translate":35},[34],[31,22902,22904],{"className":22903},[39],[41,22905,22906],{"xmlns":43},[45,22907,22908,22922],{},[48,22909,22910,22912,22915,22917,22920],{},[95,22911,211],{"stretchy":210},[91,22913,22914],{},"0.90",[95,22916,1210],{"separator":62},[91,22918,22919],{},"0.10",[95,22921,220],{"stretchy":210},[55,22923,22924],{"encoding":57},"(0.90,0.10)",[31,22926,22928],{"className":22927,"ariaHidden":62},[61],[31,22929,22931,22934,22937,22940,22943,22946,22949],{"className":22930},[66],[31,22932],{"className":22933,"style":260},[70],[31,22935,211],{"className":22936},[264],[31,22938,22914],{"className":22939},[75],[31,22941,1210],{"className":22942},[1391],[31,22944],{"className":22945,"style":1467},[119],[31,22947,22919],{"className":22948},[75],[31,22950,220],{"className":22951},[289],[31,22953,22955,22978],{"className":22954,"translate":35},[34],[31,22956,22958],{"className":22957},[39],[41,22959,22960],{"xmlns":43},[45,22961,22962,22975],{},[48,22963,22964,22966,22969,22971,22973],{},[95,22965,211],{"stretchy":210},[91,22967,22968],{},"0.99",[95,22970,1210],{"separator":62},[91,22972,22673],{},[95,22974,220],{"stretchy":210},[55,22976,22977],{"encoding":57},"(0.99,0.01)",[31,22979,22981],{"className":22980,"ariaHidden":62},[61],[31,22982,22984,22987,22990,22993,22996,22999,23002],{"className":22983},[66],[31,22985],{"className":22986,"style":260},[70],[31,22988,211],{"className":22989},[264],[31,22991,22968],{"className":22992},[75],[31,22994,1210],{"className":22995},[1391],[31,22997],{"className":22998,"style":1467},[119],[31,23000,22673],{"className":23001},[75],[31,23003,220],{"className":23004},[289],"。另一张图上，两个模型的输出是 ",[31,23007,23009,23032],{"className":23008,"translate":35},[34],[31,23010,23012],{"className":23011},[39],[41,23013,23014],{"xmlns":43},[45,23015,23016,23029],{},[48,23017,23018,23020,23023,23025,23027],{},[95,23019,211],{"stretchy":210},[91,23021,23022],{},"0.50",[95,23024,1210],{"separator":62},[91,23026,23022],{},[95,23028,220],{"stretchy":210},[55,23030,23031],{"encoding":57},"(0.50,0.50)",[31,23033,23035],{"className":23034,"ariaHidden":62},[61],[31,23036,23038,23041,23044,23047,23050,23053,23056],{"className":23037},[66],[31,23039],{"className":23040,"style":260},[70],[31,23042,211],{"className":23043},[264],[31,23045,23022],{"className":23046},[75],[31,23048,1210],{"className":23049},[1391],[31,23051],{"className":23052,"style":1467},[119],[31,23054,23022],{"className":23055},[75],[31,23057,220],{"className":23058},[289],[31,23060,23062,23086],{"className":23061,"translate":35},[34],[31,23063,23065],{"className":23064},[39],[41,23066,23067],{"xmlns":43},[45,23068,23069,23083],{},[48,23070,23071,23073,23076,23078,23081],{},[95,23072,211],{"stretchy":210},[91,23074,23075],{},"0.59",[95,23077,1210],{"separator":62},[91,23079,23080],{},"0.41",[95,23082,220],{"stretchy":210},[55,23084,23085],{"encoding":57},"(0.59,0.41)",[31,23087,23089],{"className":23088,"ariaHidden":62},[61],[31,23090,23092,23095,23098,23101,23104,23107,23110],{"className":23091},[66],[31,23093],{"className":23094,"style":260},[70],[31,23096,211],{"className":23097},[264],[31,23099,23075],{"className":23100},[75],[31,23102,1210],{"className":23103},[1391],[31,23105],{"className":23106,"style":1467},[119],[31,23108,23080],{"className":23109},[75],[31,23111,220],{"className":23112},[289],[11,23114,23115,23116,23145],{},"两组变化的欧氏距离都是 ",[31,23117,23119,23133],{"className":23118,"translate":35},[34],[31,23120,23122],{"className":23121},[39],[41,23123,23124],{"xmlns":43},[45,23125,23126,23131],{},[48,23127,23128],{},[91,23129,23130],{},"0.09",[55,23132,23130],{"encoding":57},[31,23134,23136],{"className":23135,"ariaHidden":62},[61],[31,23137,23139,23142],{"className":23138},[66],[31,23140],{"className":23141,"style":326},[70],[31,23143,23130],{"className":23144},[75],"。但我的直觉说它们不是一回事：前者是从“犹豫”变成“笃定”，后者只是从不偏不倚稍微偏了一点。",[11,23147,23148,23149,4880,23202,1138,23231,4880,23284,23313,23314,23342,23343,23371],{},"这种直觉能不能写成数？能。把 Bernoulli 分布按 Fisher–Rao 距离量一下：",[31,23150,23152,23175],{"className":23151,"translate":35},[34],[31,23153,23155],{"className":23154},[39],[41,23156,23157],{"xmlns":43},[45,23158,23159,23172],{},[48,23160,23161,23163,23166,23168,23170],{},[95,23162,211],{"stretchy":210},[91,23164,23165],{},"0.5",[95,23167,1210],{"separator":62},[91,23169,23075],{},[95,23171,220],{"stretchy":210},[55,23173,23174],{"encoding":57},"(0.5,0.59)",[31,23176,23178],{"className":23177,"ariaHidden":62},[61],[31,23179,23181,23184,23187,23190,23193,23196,23199],{"className":23180},[66],[31,23182],{"className":23183,"style":260},[70],[31,23185,211],{"className":23186},[264],[31,23188,23165],{"className":23189},[75],[31,23191,1210],{"className":23192},[1391],[31,23194],{"className":23195,"style":1467},[119],[31,23197,23075],{"className":23198},[75],[31,23200,220],{"className":23201},[289],[31,23203,23205,23219],{"className":23204,"translate":35},[34],[31,23206,23208],{"className":23207},[39],[41,23209,23210],{"xmlns":43},[45,23211,23212,23217],{},[48,23213,23214],{},[91,23215,23216],{},"0.18",[55,23218,23216],{"encoding":57},[31,23220,23222],{"className":23221,"ariaHidden":62},[61],[31,23223,23225,23228],{"className":23224},[66],[31,23226],{"className":23227,"style":326},[70],[31,23229,23216],{"className":23230},[75],[31,23232,23234,23257],{"className":23233,"translate":35},[34],[31,23235,23237],{"className":23236},[39],[41,23238,23239],{"xmlns":43},[45,23240,23241,23254],{},[48,23242,23243,23245,23248,23250,23252],{},[95,23244,211],{"stretchy":210},[91,23246,23247],{},"0.9",[95,23249,1210],{"separator":62},[91,23251,22968],{},[95,23253,220],{"stretchy":210},[55,23255,23256],{"encoding":57},"(0.9,0.99)",[31,23258,23260],{"className":23259,"ariaHidden":62},[61],[31,23261,23263,23266,23269,23272,23275,23278,23281],{"className":23262},[66],[31,23264],{"className":23265,"style":260},[70],[31,23267,211],{"className":23268},[264],[31,23270,23247],{"className":23271},[75],[31,23273,1210],{"className":23274},[1391],[31,23276],{"className":23277,"style":1467},[119],[31,23279,22968],{"className":23280},[75],[31,23282,220],{"className":23283},[289],[31,23285,23287,23301],{"className":23286,"translate":35},[34],[31,23288,23290],{"className":23289},[39],[41,23291,23292],{"xmlns":43},[45,23293,23294,23299],{},[48,23295,23296],{},[91,23297,23298],{},"0.44",[55,23300,23298],{"encoding":57},[31,23302,23304],{"className":23303,"ariaHidden":62},[61],[31,23305,23307,23310],{"className":23306},[66],[31,23308],{"className":23309,"style":326},[70],[31,23311,23298],{"className":23312},[75],"，后者是前者的两倍半。同样的坐标变化，在概率接近 ",[31,23315,23317,23330],{"className":23316,"translate":35},[34],[31,23318,23320],{"className":23319},[39],[41,23321,23322],{"xmlns":43},[45,23323,23324,23328],{},[48,23325,23326],{},[91,23327,1977],{},[55,23329,1977],{"encoding":57},[31,23331,23333],{"className":23332,"ariaHidden":62},[61],[31,23334,23336,23339],{"className":23335},[66],[31,23337],{"className":23338,"style":326},[70],[31,23340,1977],{"className":23341},[75]," 或 ",[31,23344,23346,23359],{"className":23345,"translate":35},[34],[31,23347,23349],{"className":23348},[39],[41,23350,23351],{"xmlns":43},[45,23352,23353,23357],{},[48,23354,23355],{},[91,23356,205],{},[55,23358,205],{"encoding":57},[31,23360,23362],{"className":23361,"ariaHidden":62},[61],[31,23363,23365,23368],{"className":23364},[66],[31,23366],{"className":23367,"style":326},[70],[31,23369,205],{"className":23370},[75]," 的地方要走更远的路——那里是单纯形的“边缘”，流形在那边被拉长了。",[11,23373,23374],{},"那么这把尺子是怎么来的？",[11,23376,23377],{},"模型的输出住在单纯形",[31,23379,23381],{"className":23380,"translate":35},[411],[31,23382,23384,23459],{"className":23383,"translate":35},[34],[31,23385,23387],{"className":23386},[39],[41,23388,23389],{"xmlns":43,"display":420},[45,23390,23391,23456],{},[48,23392,23393,23406,23408,23410,23412,23414,23420,23422,23424,23430,23432,23434,23436,23438,23444,23450,23452,23454],{},[1180,23394,23395,23398],{},[51,23396,23397],{"mathvariant":494},"Δ",[48,23399,23400,23402,23404],{},[51,23401,4558],{},[95,23403,97],{},[91,23405,205],{},[95,23407,223],{},[95,23409,2512],{"fence":62,"stretchy":62,"minsize":2511,"maxsize":2511},[51,23411,11],{},[95,23413,2518],{},[1180,23415,23416,23418],{},[51,23417,14790],{"mathvariant":14789},[51,23419,4558],{},[95,23421,922],{},[1430,23423,1545],{},[1004,23425,23426,23428],{},[51,23427,11],{},[51,23429,5391],{},[95,23431,5662],{},[91,23433,1977],{},[95,23435,1210],{"separator":62},[1430,23437,1545],{},[1531,23439,23440,23442],{},[95,23441,5625],{},[51,23443,5391],{},[1004,23445,23446,23448],{},[51,23447,11],{},[51,23449,5391],{},[95,23451,223],{},[91,23453,205],{},[95,23455,2604],{"fence":62,"stretchy":62,"minsize":2511,"maxsize":2511},[55,23457,23458],{"encoding":57},"\\Delta^{K-1}=\\Bigl\\{p\\in\\mathbb R^K:\\ p_i\\ge0,\\ \\sum_i p_i=1\\Bigr\\}",[31,23460,23462,23516,23540,23587,23643,23757],{"className":23461,"ariaHidden":62},[61],[31,23463,23465,23469,23507,23510,23513],{"className":23464},[66],[31,23466],{"className":23467,"style":23468},[70],"height:0.8913em;",[31,23470,23472,23475],{"className":23471},[75],[31,23473,23397],{"className":23474},[75],[31,23476,23478],{"className":23477},[1039],[31,23479,23481],{"className":23480},[1043],[31,23482,23484],{"className":23483},[1048],[31,23485,23487],{"className":23486,"style":23468},[1052],[31,23488,23489,23492],{"style":1330},[31,23490],{"className":23491,"style":1061},[1060],[31,23493,23495],{"className":23494},[1065,1066,1067,1068],[31,23496,23498,23501,23504],{"className":23497},[75,1068],[31,23499,4558],{"className":23500,"style":8525},[75,76,1068],[31,23502,97],{"className":23503},[124,1068],[31,23505,205],{"className":23506},[75,1068],[31,23508],{"className":23509,"style":293},[119],[31,23511,223],{"className":23512},[297],[31,23514],{"className":23515,"style":293},[119],[31,23517,23519,23522,23528,23531,23534,23537],{"className":23518},[66],[31,23520],{"className":23521,"style":2655},[70],[31,23523,23525],{"className":23524},[264],[31,23526,2512],{"className":23527},[1307,2662],[31,23529,11],{"className":23530},[75,76],[31,23532],{"className":23533,"style":293},[119],[31,23535,2518],{"className":23536},[297],[31,23538],{"className":23539,"style":293},[119],[31,23541,23543,23546,23575,23578,23581,23584],{"className":23542},[66],[31,23544],{"className":23545,"style":23468},[70],[31,23547,23549,23552],{"className":23548},[75],[31,23550,14790],{"className":23551},[75,14947],[31,23553,23555],{"className":23554},[1039],[31,23556,23558],{"className":23557},[1043],[31,23559,23561],{"className":23560},[1048],[31,23562,23564],{"className":23563,"style":23468},[1052],[31,23565,23566,23569],{"style":1330},[31,23567],{"className":23568,"style":1061},[1060],[31,23570,23572],{"className":23571},[1065,1066,1067,1068],[31,23573,4558],{"className":23574,"style":8525},[75,76,1068],[31,23576],{"className":23577,"style":293},[119],[31,23579,922],{"className":23580},[297],[31,23582,1545],{"className":23583},[119],[31,23585],{"className":23586,"style":293},[119],[31,23588,23590,23594,23634,23637,23640],{"className":23589},[66],[31,23591],{"className":23592,"style":23593},[70],"height:0.8304em;vertical-align:-0.1944em;",[31,23595,23597,23600],{"className":23596},[75],[31,23598,11],{"className":23599},[75,76],[31,23601,23603],{"className":23602},[1039],[31,23604,23606,23626],{"className":23605},[1043,1044],[31,23607,23609,23623],{"className":23608},[1048],[31,23610,23612],{"className":23611,"style":5445},[1052],[31,23613,23614,23617],{"style":2807},[31,23615],{"className":23616,"style":1061},[1060],[31,23618,23620],{"className":23619},[1065,1066,1067,1068],[31,23621,5391],{"className":23622},[75,76,1068],[31,23624,1076],{"className":23625},[1075],[31,23627,23629],{"className":23628},[1048],[31,23630,23632],{"className":23631,"style":2050},[1052],[31,23633],{},[31,23635],{"className":23636,"style":293},[119],[31,23638,5662],{"className":23639},[297],[31,23641],{"className":23642,"style":293},[119],[31,23644,23646,23649,23652,23655,23658,23661,23705,23708,23748,23751,23754],{"className":23645},[66],[31,23647],{"className":23648,"style":10251},[70],[31,23650,1977],{"className":23651},[75],[31,23653,1210],{"className":23654},[1391],[31,23656,1545],{"className":23657},[119],[31,23659],{"className":23660,"style":1467},[119],[31,23662,23664],{"className":23663},[1460,1620],[31,23665,23667,23697],{"className":23666},[1043,1044],[31,23668,23670,23694],{"className":23669},[1048],[31,23671,23673,23684],{"className":23672,"style":5900},[1052],[31,23674,23675,23678],{"style":5903},[31,23676],{"className":23677,"style":5907},[1060],[31,23679,23681],{"className":23680},[1065,1066,1067,1068],[31,23682,5391],{"className":23683},[75,76,1068],[31,23685,23686,23689],{"style":5919},[31,23687],{"className":23688,"style":5907},[1060],[31,23690,23691],{},[31,23692,5625],{"className":23693},[1460,1461,1687],[31,23695,1076],{"className":23696},[1075],[31,23698,23700],{"className":23699},[1048],[31,23701,23703],{"className":23702,"style":6342},[1052],[31,23704],{},[31,23706],{"className":23707,"style":1467},[119],[31,23709,23711,23714],{"className":23710},[75],[31,23712,11],{"className":23713},[75,76],[31,23715,23717],{"className":23716},[1039],[31,23718,23720,23740],{"className":23719},[1043,1044],[31,23721,23723,23737],{"className":23722},[1048],[31,23724,23726],{"className":23725,"style":5445},[1052],[31,23727,23728,23731],{"style":2807},[31,23729],{"className":23730,"style":1061},[1060],[31,23732,23734],{"className":23733},[1065,1066,1067,1068],[31,23735,5391],{"className":23736},[75,76,1068],[31,23738,1076],{"className":23739},[1075],[31,23741,23743],{"className":23742},[1048],[31,23744,23746],{"className":23745,"style":2050},[1052],[31,23747],{},[31,23749],{"className":23750,"style":293},[119],[31,23752,223],{"className":23753},[297],[31,23755],{"className":23756,"style":293},[119],[31,23758,23760,23763,23766],{"className":23759},[66],[31,23761],{"className":23762,"style":2655},[70],[31,23764,205],{"className":23765},[75],[31,23767,23769],{"className":23768},[289],[31,23770,2604],{"className":23771},[1307,2662],[11,23773,23774],{},"里。问题在于这个空间有边界：越靠近顶点，同样的坐标变化对应的“实质变化”越大。欧氏距离看不见这件事。",[11,23776,23777,23778,23859],{},"Rao 在 1945 年的想法是：把参数化的分布族 ",[31,23779,23781,23804],{"className":23780,"translate":35},[34],[31,23782,23784],{"className":23783},[39],[41,23785,23786],{"xmlns":43},[45,23787,23788,23801],{},[48,23789,23790,23792,23799],{},[95,23791,2512],{"stretchy":210},[1004,23793,23794,23796],{},[51,23795,11],{},[51,23797,23798],{},"θ",[95,23800,2604],{"stretchy":210},[55,23802,23803],{"encoding":57},"\\{p_\\theta\\}",[31,23805,23807],{"className":23806,"ariaHidden":62},[61],[31,23808,23810,23813,23816,23856],{"className":23809},[66],[31,23811],{"className":23812,"style":260},[70],[31,23814,2512],{"className":23815},[264],[31,23817,23819,23822],{"className":23818},[75],[31,23820,11],{"className":23821},[75,76],[31,23823,23825],{"className":23824},[1039],[31,23826,23828,23848],{"className":23827},[1043,1044],[31,23829,23831,23845],{"className":23830},[1048],[31,23832,23834],{"className":23833,"style":1053},[1052],[31,23835,23836,23839],{"style":2807},[31,23837],{"className":23838,"style":1061},[1060],[31,23840,23842],{"className":23841},[1065,1066,1067,1068],[31,23843,23798],{"className":23844,"style":19759},[75,76,1068],[31,23846,1076],{"className":23847},[1075],[31,23849,23851],{"className":23850},[1048],[31,23852,23854],{"className":23853,"style":2050},[1052],[31,23855],{},[31,23857,2604],{"className":23858},[289]," 本身看成一个流形，在上面装一把由数据决定的尺子。这把尺子就是 Fisher 信息矩阵",[31,23861,23863],{"className":23862,"translate":35},[411],[31,23864,23866,23987],{"className":23865,"translate":35},[34],[31,23867,23869],{"className":23868},[39],[41,23870,23871],{"xmlns":43,"display":420},[45,23872,23873,23984],{},[48,23874,23875,23885,23887,23889,23891,23893,23903,23905,23912,23914,23916,23922,23924,23930,23932,23934,23940,23942,23944,23946,23956,23958,23964,23970,23972,23974,23980,23982],{},[1004,23876,23877,23879],{},[51,23878,14556],{},[48,23880,23881,23883],{},[51,23882,5391],{},[51,23884,2555],{},[95,23886,211],{"stretchy":210},[51,23888,23798],{},[95,23890,220],{"stretchy":210},[95,23892,223],{},[1004,23894,23895,23897],{},[51,23896,22324],{"mathvariant":14789},[1004,23898,23899,23901],{},[51,23900,11],{},[51,23902,23798],{},[95,23904,15175],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[1004,23906,23907,23910],{},[51,23908,23909],{"mathvariant":494},"∂",[51,23911,5391],{},[51,23913,19939],{},[95,23915,1540],{},[1004,23917,23918,23920],{},[51,23919,11],{},[51,23921,23798],{},[95,23923,429],{},[1004,23925,23926,23928],{},[51,23927,23909],{"mathvariant":494},[51,23929,2555],{},[51,23931,19939],{},[95,23933,1540],{},[1004,23935,23936,23938],{},[51,23937,11],{},[51,23939,23798],{},[95,23941,15184],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[95,23943,223],{},[95,23945,97],{},[1004,23947,23948,23950],{},[51,23949,22324],{"mathvariant":14789},[1004,23951,23952,23954],{},[51,23953,11],{},[51,23955,23798],{},[95,23957,15175],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[1004,23959,23960,23962],{},[51,23961,23909],{"mathvariant":494},[51,23963,5391],{},[1004,23965,23966,23968],{},[51,23967,23909],{"mathvariant":494},[51,23969,2555],{},[51,23971,19939],{},[95,23973,1540],{},[1004,23975,23976,23978],{},[51,23977,11],{},[51,23979,23798],{},[95,23981,15184],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[95,23983,1210],{"separator":62},[55,23985,23986],{"encoding":57},"g_{ij}(\\theta)=\\mathbb E_{p_\\theta}\\bigl[\\partial_i\\log p_\\theta\\cdot\\partial_j\\log p_\\theta\\bigr]\n=-\\mathbb E_{p_\\theta}\\bigl[\\partial_i\\partial_j\\log p_\\theta\\bigr],",[31,23988,23990,24057,24253,24365],{"className":23989,"ariaHidden":62},[61],[31,23991,23993,23996,24039,24042,24045,24048,24051,24054],{"className":23992},[66],[31,23994],{"className":23995,"style":1223},[70],[31,23997,23999,24002],{"className":23998},[75],[31,24000,14556],{"className":24001,"style":2666},[75,76],[31,24003,24005],{"className":24004},[1039],[31,24006,24008,24031],{"className":24007},[1043,1044],[31,24009,24011,24028],{"className":24010},[1048],[31,24012,24014],{"className":24013,"style":5445},[1052],[31,24015,24016,24019],{"style":5504},[31,24017],{"className":24018,"style":1061},[1060],[31,24020,24022],{"className":24021},[1065,1066,1067,1068],[31,24023,24025],{"className":24024},[75,1068],[31,24026,5574],{"className":24027,"style":5514},[75,76,1068],[31,24029,1076],{"className":24030},[1075],[31,24032,24034],{"className":24033},[1048],[31,24035,24037],{"className":24036,"style":1083},[1052],[31,24038],{},[31,24040,211],{"className":24041},[264],[31,24043,23798],{"className":24044,"style":19759},[75,76],[31,24046,220],{"className":24047},[289],[31,24049],{"className":24050,"style":293},[119],[31,24052,223],{"className":24053},[297],[31,24055],{"className":24056,"style":293},[119],[31,24058,24060,24063,24145,24151,24193,24196,24201,24204,24244,24247,24250],{"className":24059},[66],[31,24061],{"className":24062,"style":4988},[70],[31,24064,24066,24069],{"className":24065},[75],[31,24067,22324],{"className":24068},[75,14947],[31,24070,24072],{"className":24071},[1039],[31,24073,24075,24137],{"className":24074},[1043,1044],[31,24076,24078,24134],{"className":24077},[1048],[31,24079,24081],{"className":24080,"style":3230},[1052],[31,24082,24083,24086],{"style":2807},[31,24084],{"className":24085,"style":1061},[1060],[31,24087,24089],{"className":24088},[1065,1066,1067,1068],[31,24090,24092],{"className":24091},[75,1068],[31,24093,24095,24098],{"className":24094},[75,1068],[31,24096,11],{"className":24097},[75,76,1068],[31,24099,24101],{"className":24100},[1039],[31,24102,24104,24125],{"className":24103},[1043,1044],[31,24105,24107,24122],{"className":24106},[1048],[31,24108,24110],{"className":24109,"style":11762},[1052],[31,24111,24113,24116],{"style":24112},"top:-2.3488em;margin-left:0em;margin-right:0.0714em;",[31,24114],{"className":24115,"style":11769},[1060],[31,24117,24119],{"className":24118},[1065,11773,1308,1068],[31,24120,23798],{"className":24121,"style":19759},[75,76,1068],[31,24123,1076],{"className":24124},[1075],[31,24126,24128],{"className":24127},[1048],[31,24129,24132],{"className":24130,"style":24131},[1052],"height:0.1512em;",[31,24133],{},[31,24135,1076],{"className":24136},[1075],[31,24138,24140],{"className":24139},[1048],[31,24141,24143],{"className":24142,"style":1083},[1052],[31,24144],{},[31,24146,24148],{"className":24147},[264],[31,24149,15175],{"className":24150},[1307,1308],[31,24152,24154,24158],{"className":24153},[75],[31,24155,23909],{"className":24156,"style":24157},[75],"margin-right:0.0556em;",[31,24159,24161],{"className":24160},[1039],[31,24162,24164,24185],{"className":24163},[1043,1044],[31,24165,24167,24182],{"className":24166},[1048],[31,24168,24170],{"className":24169,"style":5445},[1052],[31,24171,24173,24176],{"style":24172},"top:-2.55em;margin-left:-0.0556em;margin-right:0.05em;",[31,24174],{"className":24175,"style":1061},[1060],[31,24177,24179],{"className":24178},[1065,1066,1067,1068],[31,24180,5391],{"className":24181},[75,76,1068],[31,24183,1076],{"className":24184},[1075],[31,24186,24188],{"className":24187},[1048],[31,24189,24191],{"className":24190,"style":2050},[1052],[31,24192],{},[31,24194],{"className":24195,"style":1467},[119],[31,24197,20177,24199],{"className":24198},[1460],[31,24200,14556],{"style":20180},[31,24202],{"className":24203,"style":1467},[119],[31,24205,24207,24210],{"className":24206},[75],[31,24208,11],{"className":24209},[75,76],[31,24211,24213],{"className":24212},[1039],[31,24214,24216,24236],{"className":24215},[1043,1044],[31,24217,24219,24233],{"className":24218},[1048],[31,24220,24222],{"className":24221,"style":1053},[1052],[31,24223,24224,24227],{"style":2807},[31,24225],{"className":24226,"style":1061},[1060],[31,24228,24230],{"className":24229},[1065,1066,1067,1068],[31,24231,23798],{"className":24232,"style":19759},[75,76,1068],[31,24234,1076],{"className":24235},[1075],[31,24237,24239],{"className":24238},[1048],[31,24240,24242],{"className":24241,"style":2050},[1052],[31,24243],{},[31,24245],{"className":24246,"style":120},[119],[31,24248,429],{"className":24249},[124],[31,24251],{"className":24252,"style":120},[119],[31,24254,24256,24259,24299,24302,24307,24310,24350,24356,24359,24362],{"className":24255},[66],[31,24257],{"className":24258,"style":4988},[70],[31,24260,24262,24265],{"className":24261},[75],[31,24263,23909],{"className":24264,"style":24157},[75],[31,24266,24268],{"className":24267},[1039],[31,24269,24271,24291],{"className":24270},[1043,1044],[31,24272,24274,24288],{"className":24273},[1048],[31,24275,24277],{"className":24276,"style":5445},[1052],[31,24278,24279,24282],{"style":24172},[31,24280],{"className":24281,"style":1061},[1060],[31,24283,24285],{"className":24284},[1065,1066,1067,1068],[31,24286,2555],{"className":24287,"style":5514},[75,76,1068],[31,24289,1076],{"className":24290},[1075],[31,24292,24294],{"className":24293},[1048],[31,24295,24297],{"className":24296,"style":1083},[1052],[31,24298],{},[31,24300],{"className":24301,"style":1467},[119],[31,24303,20177,24305],{"className":24304},[1460],[31,24306,14556],{"style":20180},[31,24308],{"className":24309,"style":1467},[119],[31,24311,24313,24316],{"className":24312},[75],[31,24314,11],{"className":24315},[75,76],[31,24317,24319],{"className":24318},[1039],[31,24320,24322,24342],{"className":24321},[1043,1044],[31,24323,24325,24339],{"className":24324},[1048],[31,24326,24328],{"className":24327,"style":1053},[1052],[31,24329,24330,24333],{"style":2807},[31,24331],{"className":24332,"style":1061},[1060],[31,24334,24336],{"className":24335},[1065,1066,1067,1068],[31,24337,23798],{"className":24338,"style":19759},[75,76,1068],[31,24340,1076],{"className":24341},[1075],[31,24343,24345],{"className":24344},[1048],[31,24346,24348],{"className":24347,"style":2050},[1052],[31,24349],{},[31,24351,24353],{"className":24352},[289],[31,24354,15184],{"className":24355},[1307,1308],[31,24357],{"className":24358,"style":293},[119],[31,24360,223],{"className":24361},[297],[31,24363],{"className":24364,"style":293},[119],[31,24366,24368,24371,24374,24454,24460,24500,24540,24543,24548,24551,24591,24597],{"className":24367},[66],[31,24369],{"className":24370,"style":4988},[70],[31,24372,97],{"className":24373},[75],[31,24375,24377,24380],{"className":24376},[75],[31,24378,22324],{"className":24379},[75,14947],[31,24381,24383],{"className":24382},[1039],[31,24384,24386,24446],{"className":24385},[1043,1044],[31,24387,24389,24443],{"className":24388},[1048],[31,24390,24392],{"className":24391,"style":3230},[1052],[31,24393,24394,24397],{"style":2807},[31,24395],{"className":24396,"style":1061},[1060],[31,24398,24400],{"className":24399},[1065,1066,1067,1068],[31,24401,24403],{"className":24402},[75,1068],[31,24404,24406,24409],{"className":24405},[75,1068],[31,24407,11],{"className":24408},[75,76,1068],[31,24410,24412],{"className":24411},[1039],[31,24413,24415,24435],{"className":24414},[1043,1044],[31,24416,24418,24432],{"className":24417},[1048],[31,24419,24421],{"className":24420,"style":11762},[1052],[31,24422,24423,24426],{"style":24112},[31,24424],{"className":24425,"style":11769},[1060],[31,24427,24429],{"className":24428},[1065,11773,1308,1068],[31,24430,23798],{"className":24431,"style":19759},[75,76,1068],[31,24433,1076],{"className":24434},[1075],[31,24436,24438],{"className":24437},[1048],[31,24439,24441],{"className":24440,"style":24131},[1052],[31,24442],{},[31,24444,1076],{"className":24445},[1075],[31,24447,24449],{"className":24448},[1048],[31,24450,24452],{"className":24451,"style":1083},[1052],[31,24453],{},[31,24455,24457],{"className":24456},[264],[31,24458,15175],{"className":24459},[1307,1308],[31,24461,24463,24466],{"className":24462},[75],[31,24464,23909],{"className":24465,"style":24157},[75],[31,24467,24469],{"className":24468},[1039],[31,24470,24472,24492],{"className":24471},[1043,1044],[31,24473,24475,24489],{"className":24474},[1048],[31,24476,24478],{"className":24477,"style":5445},[1052],[31,24479,24480,24483],{"style":24172},[31,24481],{"className":24482,"style":1061},[1060],[31,24484,24486],{"className":24485},[1065,1066,1067,1068],[31,24487,5391],{"className":24488},[75,76,1068],[31,24490,1076],{"className":24491},[1075],[31,24493,24495],{"className":24494},[1048],[31,24496,24498],{"className":24497,"style":2050},[1052],[31,24499],{},[31,24501,24503,24506],{"className":24502},[75],[31,24504,23909],{"className":24505,"style":24157},[75],[31,24507,24509],{"className":24508},[1039],[31,24510,24512,24532],{"className":24511},[1043,1044],[31,24513,24515,24529],{"className":24514},[1048],[31,24516,24518],{"className":24517,"style":5445},[1052],[31,24519,24520,24523],{"style":24172},[31,24521],{"className":24522,"style":1061},[1060],[31,24524,24526],{"className":24525},[1065,1066,1067,1068],[31,24527,2555],{"className":24528,"style":5514},[75,76,1068],[31,24530,1076],{"className":24531},[1075],[31,24533,24535],{"className":24534},[1048],[31,24536,24538],{"className":24537,"style":1083},[1052],[31,24539],{},[31,24541],{"className":24542,"style":1467},[119],[31,24544,20177,24546],{"className":24545},[1460],[31,24547,14556],{"style":20180},[31,24549],{"className":24550,"style":1467},[119],[31,24552,24554,24557],{"className":24553},[75],[31,24555,11],{"className":24556},[75,76],[31,24558,24560],{"className":24559},[1039],[31,24561,24563,24583],{"className":24562},[1043,1044],[31,24564,24566,24580],{"className":24565},[1048],[31,24567,24569],{"className":24568,"style":1053},[1052],[31,24570,24571,24574],{"style":2807},[31,24572],{"className":24573,"style":1061},[1060],[31,24575,24577],{"className":24576},[1065,1066,1067,1068],[31,24578,23798],{"className":24579,"style":19759},[75,76,1068],[31,24581,1076],{"className":24582},[1075],[31,24584,24586],{"className":24585},[1048],[31,24587,24589],{"className":24588,"style":2050},[1052],[31,24590],{},[31,24592,24594],{"className":24593},[289],[31,24595,15184],{"className":24596},[1307,1308],[31,24598,1210],{"className":24599},[1391],[11,24601,24602,24603,24730,24731,24760,24761,2372],{},"两个表达式的相等用到了 ",[31,24604,24606,24640],{"className":24605,"translate":35},[34],[31,24607,24609],{"className":24608},[39],[41,24610,24611],{"xmlns":43},[45,24612,24613,24637],{},[48,24614,24615,24617,24619,24625,24627,24629,24631,24633,24635],{},[51,24616,22324],{"mathvariant":14789},[95,24618,15175],{"stretchy":210},[1004,24620,24621,24623],{},[51,24622,23909],{"mathvariant":494},[51,24624,5391],{},[51,24626,19939],{},[95,24628,1540],{},[51,24630,11],{},[95,24632,15184],{"stretchy":210},[95,24634,223],{},[91,24636,1977],{},[55,24638,24639],{"encoding":57},"\\mathbb E[\\partial_i\\log p]=0",[31,24641,24643,24721],{"className":24642,"ariaHidden":62},[61],[31,24644,24646,24649,24652,24655,24695,24698,24703,24706,24709,24712,24715,24718],{"className":24645},[66],[31,24647],{"className":24648,"style":260},[70],[31,24650,22324],{"className":24651},[75,14947],[31,24653,15175],{"className":24654},[264],[31,24656,24658,24661],{"className":24657},[75],[31,24659,23909],{"className":24660,"style":24157},[75],[31,24662,24664],{"className":24663},[1039],[31,24665,24667,24687],{"className":24666},[1043,1044],[31,24668,24670,24684],{"className":24669},[1048],[31,24671,24673],{"className":24672,"style":5445},[1052],[31,24674,24675,24678],{"style":24172},[31,24676],{"className":24677,"style":1061},[1060],[31,24679,24681],{"className":24680},[1065,1066,1067,1068],[31,24682,5391],{"className":24683},[75,76,1068],[31,24685,1076],{"className":24686},[1075],[31,24688,24690],{"className":24689},[1048],[31,24691,24693],{"className":24692,"style":2050},[1052],[31,24694],{},[31,24696],{"className":24697,"style":1467},[119],[31,24699,20177,24701],{"className":24700},[1460],[31,24702,14556],{"style":20180},[31,24704],{"className":24705,"style":1467},[119],[31,24707,11],{"className":24708},[75,76],[31,24710,15184],{"className":24711},[289],[31,24713],{"className":24714,"style":293},[119],[31,24716,223],{"className":24717},[297],[31,24719],{"className":24720,"style":293},[119],[31,24722,24724,24727],{"className":24723},[66],[31,24725],{"className":24726,"style":326},[70],[31,24728,1977],{"className":24729},[75],"，因为对 ",[31,24732,24734,24748],{"className":24733,"translate":35},[34],[31,24735,24737],{"className":24736},[39],[41,24738,24739],{"xmlns":43},[45,24740,24741,24745],{},[48,24742,24743],{},[51,24744,23798],{},[55,24746,24747],{"encoding":57},"\\theta",[31,24749,24751],{"className":24750,"ariaHidden":62},[61],[31,24752,24754,24757],{"className":24753},[66],[31,24755],{"className":24756,"style":1630},[70],[31,24758,23798],{"className":24759,"style":19759},[75,76]," 求导再积分，",[31,24762,24764,24814],{"className":24763,"translate":35},[34],[31,24765,24767],{"className":24766},[39],[41,24768,24769],{"xmlns":43},[45,24770,24771,24811],{},[48,24772,24773,24775,24781,24783,24785,24787,24789,24791,24797,24799,24801,24803,24805,24807,24809],{},[95,24774,1409],{},[1004,24776,24777,24779],{},[51,24778,23909],{"mathvariant":494},[51,24780,5391],{},[51,24782,11],{},[1430,24784,1432],{},[51,24786,1435],{},[51,24788,53],{},[95,24790,223],{},[1004,24792,24793,24795],{},[51,24794,23909],{"mathvariant":494},[51,24796,5391],{},[95,24798,1409],{},[51,24800,11],{},[1430,24802,1432],{},[51,24804,1435],{},[51,24806,53],{},[95,24808,223],{},[91,24810,1977],{},[55,24812,24813],{"encoding":57},"\\int\\partial_i p\\,dx=\\partial_i\\int p\\,dx=0",[31,24815,24817,24890,24966],{"className":24816,"ariaHidden":62},[61],[31,24818,24820,24823,24826,24829,24869,24872,24875,24878,24881,24884,24887],{"className":24819},[66],[31,24821],{"className":24822,"style":1456},[70],[31,24824,1409],{"className":24825,"style":1463},[1460,1461,1462],[31,24827],{"className":24828,"style":1467},[119],[31,24830,24832,24835],{"className":24831},[75],[31,24833,23909],{"className":24834,"style":24157},[75],[31,24836,24838],{"className":24837},[1039],[31,24839,24841,24861],{"className":24840},[1043,1044],[31,24842,24844,24858],{"className":24843},[1048],[31,24845,24847],{"className":24846,"style":5445},[1052],[31,24848,24849,24852],{"style":24172},[31,24850],{"className":24851,"style":1061},[1060],[31,24853,24855],{"className":24854},[1065,1066,1067,1068],[31,24856,5391],{"className":24857},[75,76,1068],[31,24859,1076],{"className":24860},[1075],[31,24862,24864],{"className":24863},[1048],[31,24865,24867],{"className":24866,"style":2050},[1052],[31,24868],{},[31,24870,11],{"className":24871},[75,76],[31,24873],{"className":24874,"style":1467},[119],[31,24876,1435],{"className":24877},[75,76],[31,24879,53],{"className":24880},[75,76],[31,24882],{"className":24883,"style":293},[119],[31,24885,223],{"className":24886},[297],[31,24888],{"className":24889,"style":293},[119],[31,24891,24893,24896,24936,24939,24942,24945,24948,24951,24954,24957,24960,24963],{"className":24892},[66],[31,24894],{"className":24895,"style":1456},[70],[31,24897,24899,24902],{"className":24898},[75],[31,24900,23909],{"className":24901,"style":24157},[75],[31,24903,24905],{"className":24904},[1039],[31,24906,24908,24928],{"className":24907},[1043,1044],[31,24909,24911,24925],{"className":24910},[1048],[31,24912,24914],{"className":24913,"style":5445},[1052],[31,24915,24916,24919],{"style":24172},[31,24917],{"className":24918,"style":1061},[1060],[31,24920,24922],{"className":24921},[1065,1066,1067,1068],[31,24923,5391],{"className":24924},[75,76,1068],[31,24926,1076],{"className":24927},[1075],[31,24929,24931],{"className":24930},[1048],[31,24932,24934],{"className":24933,"style":2050},[1052],[31,24935],{},[31,24937],{"className":24938,"style":1467},[119],[31,24940,1409],{"className":24941,"style":1463},[1460,1461,1462],[31,24943],{"className":24944,"style":1467},[119],[31,24946,11],{"className":24947},[75,76],[31,24949],{"className":24950,"style":1467},[119],[31,24952,1435],{"className":24953},[75,76],[31,24955,53],{"className":24956},[75,76],[31,24958],{"className":24959,"style":293},[119],[31,24961,223],{"className":24962},[297],[31,24964],{"className":24965,"style":293},[119],[31,24967,24969,24972],{"className":24968},[66],[31,24970],{"className":24971,"style":326},[70],[31,24973,1977],{"className":24974},[75],[11,24976,24977],{},"为什么是它？有三个我觉得足够有说服力的理由。",[11,24979,24980,24981,25085,25086,25114],{},"第一，它是 KL 的二阶展开。把 ",[31,24982,24984,25014],{"className":24983,"translate":35},[34],[31,24985,24987],{"className":24986},[39],[41,24988,24989],{"xmlns":43},[45,24990,24991,25011],{},[48,24992,24993,24995,24997],{},[51,24994,19939],{},[95,24996,1540],{},[1004,24998,24999,25001],{},[51,25000,11],{},[48,25002,25003,25005,25007,25009],{},[51,25004,23798],{},[95,25006,434],{},[51,25008,1435],{},[51,25010,23798],{},[55,25012,25013],{"encoding":57},"\\log p_{\\theta+d\\theta}",[31,25015,25017],{"className":25016,"ariaHidden":62},[61],[31,25018,25020,25024,25029,25032],{"className":25019},[66],[31,25021],{"className":25022,"style":25023},[70],"height:0.9028em;vertical-align:-0.2083em;",[31,25025,20177,25027],{"className":25026},[1460],[31,25028,14556],{"style":20180},[31,25030],{"className":25031,"style":1467},[119],[31,25033,25035,25038],{"className":25034},[75],[31,25036,11],{"className":25037},[75,76],[31,25039,25041],{"className":25040},[1039],[31,25042,25044,25076],{"className":25043},[1043,1044],[31,25045,25047,25073],{"className":25046},[1048],[31,25048,25050],{"className":25049,"style":1053},[1052],[31,25051,25052,25055],{"style":2807},[31,25053],{"className":25054,"style":1061},[1060],[31,25056,25058],{"className":25057},[1065,1066,1067,1068],[31,25059,25061,25064,25067,25070],{"className":25060},[75,1068],[31,25062,23798],{"className":25063,"style":19759},[75,76,1068],[31,25065,434],{"className":25066},[124,1068],[31,25068,1435],{"className":25069},[75,76,1068],[31,25071,23798],{"className":25072,"style":19759},[75,76,1068],[31,25074,1076],{"className":25075},[1075],[31,25077,25079],{"className":25078},[1048],[31,25080,25083],{"className":25081,"style":25082},[1052],"height:0.2083em;",[31,25084],{}," 在 ",[31,25087,25089,25102],{"className":25088,"translate":35},[34],[31,25090,25092],{"className":25091},[39],[41,25093,25094],{"xmlns":43},[45,25095,25096,25100],{},[48,25097,25098],{},[51,25099,23798],{},[55,25101,24747],{"encoding":57},[31,25103,25105],{"className":25104,"ariaHidden":62},[61],[31,25106,25108,25111],{"className":25107},[66],[31,25109],{"className":25110,"style":1630},[70],[31,25112,23798],{"className":25113,"style":19759},[75,76]," 处展开到二阶，一阶项的期望为零，剩下",[31,25116,25118],{"className":25117,"translate":35},[411],[31,25119,25121,25311],{"className":25120,"translate":35},[34],[31,25122,25124],{"className":25123},[39],[41,25125,25126],{"xmlns":43,"display":420},[45,25127,25128,25308],{},[48,25129,25130,25136,25138,25144,25146,25160,25162,25164,25166,25174,25176,25178,25180,25186,25192,25194,25196,25202,25204,25206,25208,25214,25216,25218,25224,25226,25228,25230,25232,25234,25236,25242,25244,25246,25254,25256,25266,25268,25270,25276,25278,25280,25286,25288,25290,25292,25294,25296,25298,25304,25306],{},[48,25131,25132,25134],{},[51,25133,4558],{"mathvariant":494},[51,25135,4561],{"mathvariant":494},[95,25137,211],{"stretchy":210},[1004,25139,25140,25142],{},[51,25141,11],{},[51,25143,23798],{},[51,25145,4572],{"mathvariant":494},[1004,25147,25148,25150],{},[51,25149,11],{},[48,25151,25152,25154,25156,25158],{},[51,25153,23798],{},[95,25155,434],{},[51,25157,1435],{},[51,25159,23798],{},[95,25161,220],{"stretchy":210},[95,25163,223],{},[95,25165,97],{},[2071,25167,25168],{"scriptlevel":1977,"displaystyle":210},[2074,25169,25170,25172],{},[91,25171,205],{},[91,25173,152],{},[1430,25175,1432],{},[51,25177,22324],{"mathvariant":14789},[95,25179,15175],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[1004,25181,25182,25184],{},[51,25183,23909],{"mathvariant":494},[51,25185,5391],{},[1004,25187,25188,25190],{},[51,25189,23909],{"mathvariant":494},[51,25191,2555],{},[51,25193,19939],{},[95,25195,1540],{},[1004,25197,25198,25200],{},[51,25199,11],{},[51,25201,23798],{},[95,25203,15184],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[1430,25205,1432],{},[51,25207,1435],{},[1180,25209,25210,25212],{},[51,25211,23798],{},[51,25213,5391],{},[1430,25215,1432],{},[51,25217,1435],{},[1180,25219,25220,25222],{},[51,25221,23798],{},[51,25223,2555],{},[95,25225,434],{},[51,25227,19733],{},[95,25229,211],{"stretchy":210},[51,25231,4572],{"mathvariant":494},[51,25233,1435],{},[51,25235,23798],{},[1180,25237,25238,25240],{},[51,25239,4572],{"mathvariant":494},[91,25241,93],{},[95,25243,220],{"stretchy":210},[95,25245,223],{},[2071,25247,25248],{"scriptlevel":1977,"displaystyle":210},[2074,25249,25250,25252],{},[91,25251,205],{},[91,25253,152],{},[1430,25255,1432],{},[1004,25257,25258,25260],{},[51,25259,14556],{},[48,25261,25262,25264],{},[51,25263,5391],{},[51,25265,2555],{},[1430,25267,1432],{},[51,25269,1435],{},[1180,25271,25272,25274],{},[51,25273,23798],{},[51,25275,5391],{},[1430,25277,1432],{},[51,25279,1435],{},[1180,25281,25282,25284],{},[51,25283,23798],{},[51,25285,2555],{},[95,25287,434],{},[51,25289,19733],{},[95,25291,211],{"stretchy":210},[51,25293,4572],{"mathvariant":494},[51,25295,1435],{},[51,25297,23798],{},[1180,25299,25300,25302],{},[51,25301,4572],{"mathvariant":494},[91,25303,93],{},[95,25305,220],{"stretchy":210},[51,25307,495],{"mathvariant":494},[55,25309,25310],{"encoding":57},"\\mathrm{KL}(p_\\theta\\|p_{\\theta+d\\theta})\n=-\\tfrac12\\,\\mathbb E\\bigl[\\partial_i\\partial_j\\log p_\\theta\\bigr]\\,d\\theta^i\\,d\\theta^j+O(\\|d\\theta\\|^3)\n=\\tfrac12\\,g_{ij}\\,d\\theta^i\\,d\\theta^j+O(\\|d\\theta\\|^3).",[31,25312,25314,25436,25743,25805,26005],{"className":25313,"ariaHidden":62},[61],[31,25315,25317,25320,25326,25329,25369,25372,25424,25427,25430,25433],{"className":25316},[66],[31,25318],{"className":25319,"style":260},[70],[31,25321,25323],{"className":25322},[75],[31,25324,4606],{"className":25325},[75,2745],[31,25327,211],{"className":25328},[264],[31,25330,25332,25335],{"className":25331},[75],[31,25333,11],{"className":25334},[75,76],[31,25336,25338],{"className":25337},[1039],[31,25339,25341,25361],{"className":25340},[1043,1044],[31,25342,25344,25358],{"className":25343},[1048],[31,25345,25347],{"className":25346,"style":1053},[1052],[31,25348,25349,25352],{"style":2807},[31,25350],{"className":25351,"style":1061},[1060],[31,25353,25355],{"className":25354},[1065,1066,1067,1068],[31,25356,23798],{"className":25357,"style":19759},[75,76,1068],[31,25359,1076],{"className":25360},[1075],[31,25362,25364],{"className":25363},[1048],[31,25365,25367],{"className":25366,"style":2050},[1052],[31,25368],{},[31,25370,4572],{"className":25371},[75],[31,25373,25375,25378],{"className":25374},[75],[31,25376,11],{"className":25377},[75,76],[31,25379,25381],{"className":25380},[1039],[31,25382,25384,25416],{"className":25383},[1043,1044],[31,25385,25387,25413],{"className":25386},[1048],[31,25388,25390],{"className":25389,"style":1053},[1052],[31,25391,25392,25395],{"style":2807},[31,25393],{"className":25394,"style":1061},[1060],[31,25396,25398],{"className":25397},[1065,1066,1067,1068],[31,25399,25401,25404,25407,25410],{"className":25400},[75,1068],[31,25402,23798],{"className":25403,"style":19759},[75,76,1068],[31,25405,434],{"className":25406},[124,1068],[31,25408,1435],{"className":25409},[75,76,1068],[31,25411,23798],{"className":25412,"style":19759},[75,76,1068],[31,25414,1076],{"className":25415},[1075],[31,25417,25419],{"className":25418},[1048],[31,25420,25422],{"className":25421,"style":25082},[1052],[31,25423],{},[31,25425,220],{"className":25426},[289],[31,25428],{"className":25429,"style":293},[119],[31,25431,223],{"className":25432},[297],[31,25434],{"className":25435,"style":293},[119],[31,25437,25439,25443,25446,25514,25517,25520,25526,25566,25606,25609,25614,25617,25657,25663,25666,25669,25699,25702,25705,25734,25737,25740],{"className":25438},[66],[31,25440],{"className":25441,"style":25442},[70],"height:1.2247em;vertical-align:-0.35em;",[31,25444,97],{"className":25445},[75],[31,25447,25449,25452,25511],{"className":25448},[75],[31,25450],{"className":25451},[264,2139],[31,25453,25455],{"className":25454},[2074],[31,25456,25458,25503],{"className":25457},[1043,1044],[31,25459,25461,25500],{"className":25460},[1048],[31,25462,25464,25478,25486],{"className":25463,"style":2152},[1052],[31,25465,25466,25469],{"style":2155},[31,25467],{"className":25468,"style":1637},[1060],[31,25470,25472],{"className":25471},[1065,1066,1067,1068],[31,25473,25475],{"className":25474},[75,1068],[31,25476,152],{"className":25477},[75,1068],[31,25479,25480,25483],{"style":2170},[31,25481],{"className":25482,"style":1637},[1060],[31,25484],{"className":25485,"style":2178},[2177],[31,25487,25488,25491],{"style":2181},[31,25489],{"className":25490,"style":1637},[1060],[31,25492,25494],{"className":25493},[1065,1066,1067,1068],[31,25495,25497],{"className":25496},[75,1068],[31,25498,205],{"className":25499},[75,1068],[31,25501,1076],{"className":25502},[1075],[31,25504,25506],{"className":25505},[1048],[31,25507,25509],{"className":25508,"style":2203},[1052],[31,25510],{},[31,25512],{"className":25513},[289,2139],[31,25515],{"className":25516,"style":1467},[119],[31,25518,22324],{"className":25519},[75,14947],[31,25521,25523],{"className":25522},[264],[31,25524,15175],{"className":25525},[1307,1308],[31,25527,25529,25532],{"className":25528},[75],[31,25530,23909],{"className":25531,"style":24157},[75],[31,25533,25535],{"className":25534},[1039],[31,25536,25538,25558],{"className":25537},[1043,1044],[31,25539,25541,25555],{"className":25540},[1048],[31,25542,25544],{"className":25543,"style":5445},[1052],[31,25545,25546,25549],{"style":24172},[31,25547],{"className":25548,"style":1061},[1060],[31,25550,25552],{"className":25551},[1065,1066,1067,1068],[31,25553,5391],{"className":25554},[75,76,1068],[31,25556,1076],{"className":25557},[1075],[31,25559,25561],{"className":25560},[1048],[31,25562,25564],{"className":25563,"style":2050},[1052],[31,25565],{},[31,25567,25569,25572],{"className":25568},[75],[31,25570,23909],{"className":25571,"style":24157},[75],[31,25573,25575],{"className":25574},[1039],[31,25576,25578,25598],{"className":25577},[1043,1044],[31,25579,25581,25595],{"className":25580},[1048],[31,25582,25584],{"className":25583,"style":5445},[1052],[31,25585,25586,25589],{"style":24172},[31,25587],{"className":25588,"style":1061},[1060],[31,25590,25592],{"className":25591},[1065,106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,[75],[31,25887,14556],{"className":25888,"style":2666},[75,76],[31,25890,25892],{"className":25891},[1039],[31,25893,25895,25918],{"className":25894},[1043,1044],[31,25896,25898,25915],{"className":25897},[1048],[31,25899,25901],{"className":25900,"style":5445},[1052],[31,25902,25903,25906],{"style":5504},[31,25904],{"className":25905,"style":1061},[1060],[31,25907,25909],{"className":25908},[1065,1066,1067,1068],[31,25910,25912],{"className":25911},[75,1068],[31,25913,5574],{"className":25914,"style":5514},[75,76,1068],[31,25916,1076],{"className":25917},[1075],[31,25919,25921],{"className":25920},[1048],[31,25922,25924],{"className":25923,"style":1083},[1052],[31,25925],{},[31,25927],{"className":25928,"style":1467},[119],[31,25930,1435],{"className":25931},[75,76],[31,25933,25935,25938],{"className":25934},[75],[31,25936,23798],{"className":25937,"style":19759},[75,76],[31,25939,25941],{"className":25940},[1039],[31,25942,25944],{"className":25943},[1043],[31,25945,25947],{"className":25946},[1048],[31,25948,25950],{"className":25949,"style":25687},[1052],[31,25951,25952,25955],{"style":1330},[31,25953],{"className":25954,"style":1061},[1060],[31,25956,25958],{"className":25957},[1065,1066,1067,1068],[31,25959,5391],{"className":25960},[75,76,1068],[31,25962],{"className":25963,"style":1467},[119],[31,25965,1435],{"className":25966},[75,76],[31,25968,25970,25973],{"className":25969},[75],[31,25971,23798],{"className":25972,"style":19759},[75,76],[31,25974,25976],{"className":25975},[1039],[31,25977,25979],{"className":25978},[1043],[31,25980,25982],{"className":25981},[1048],[31,25983,25985],{"className":25984,"style":25687},[1052],[31,25986,25987,25990],{"style":1330},[31,25988],{"className":25989,"style":1061},[1060],[31,25991,25993],{"className":25992},[1065,1066,1067,1068],[31,25994,2555],{"className":25995,"style":5514},[75,76,1068],[31,25997],{"className":25998,"style":120},[119],[31,26000,434],{"className":26001},[124],[31,26003],{"className":26004,"style":120},[119],[31,26006,26008,26011,26014,26017,26020,26023,26026,26055,26058],{"className":26007},[66],[31,26009],{"className":26010,"style":16882},[70],[31,26012,19733],{"className":26013,"style":19759},[75,76],[31,26015,211],{"className":26016},[264],[31,26018,4572],{"className":26019},[75],[31,26021,1435],{"className":26022},[75,76],[31,26024,23798],{"className":26025,"style":19759},[75,76],[31,26027,26029,26032],{"className":26028},[75],[31,26030,4572],{"className":26031},[75],[31,26033,26035],{"className":26034},[1039],[31,26036,26038],{"className":26037},[1043],[31,26039,26041],{"className":26040},[1048],[31,26042,26044],{"className":26043,"style":1327},[1052],[31,26045,26046,26049],{"style":1330},[31,26047],{"className":26048,"style":1061},[1060],[31,26050,26052],{"className":26051},[1065,1066,1067,1068],[31,26053,93],{"className":26054},[75,1068],[31,26056,220],{"className":26057},[289],[31,26059,495],{"className":26060},[75],[11,26062,26063],{},"所以 KL 虽然整体上不对称，局部却退化成一个二次型，二次型的矩阵就是 Fisher。反过来说，任何“局部像 KL”的距离都会给出同一个度量。",[11,26065,26066],{},"第二，Chentsov 定理说：单纯形上的黎曼度量，若在充分统计量（更准确地说，Markov 态射）下保持不变，则差一个常数因子必为 Fisher–Rao。换句话说，这不是我随手挑的度量，而是在“只依赖分布、不依赖参数化”这个要求下唯一的选择。",[11,26068,26069],{},"第三，对指数族它恰好是势函数的 Hessian，下一节说。",[11,26071,26072,26073,1138],{},"把 Bernoulli 族算一遍。",[31,26074,26076,26128],{"className":26075,"translate":35},[34],[31,26077,26079],{"className":26078},[39],[41,26080,26081],{"xmlns":43},[45,26082,26083,26125],{},[48,26084,26085,26087,26089,26091,26093,26095,26097,26099,26105,26107,26109,26111,26113],{},[51,26086,11],{},[95,26088,211],{"stretchy":210},[51,26090,53],{},[95,26092,12680],{},[51,26094,23798],{},[95,26096,220],{"stretchy":210},[95,26098,223],{},[1180,26100,26101,26103],{},[51,26102,23798],{},[51,26104,53],{},[95,26106,211],{"stretchy":210},[91,26108,205],{},[95,26110,97],{},[51,26112,23798],{},[1180,26114,26115,26117],{},[95,26116,220],{"stretchy":210},[48,26118,26119,26121,26123],{},[91,26120,205],{},[95,26122,97],{},[51,26124,53],{},[55,26126,26127],{"encoding":57},"p(x\\mid\\theta)=\\theta^x(1-\\theta)^{1-x}",[31,26129,26131,26155,26176,26226],{"className":26130,"ariaHidden":62},[61],[31,26132,26134,26137,26140,26143,26146,26149,26152],{"className":26133},[66],[31,26135],{"className":26136,"style":260},[70],[31,26138,11],{"className":26139},[75,76],[31,26141,211],{"className":26142},[264],[31,26144,53],{"className":26145},[75,76],[31,26147],{"className":26148,"style":293},[119],[31,26150,12680],{"className":26151},[297],[31,26153],{"className":26154,"style":293},[119],[31,26156,26158,26161,26164,26167,26170,26173],{"className":26157},[66],[31,26159],{"className":26160,"style":260},[70],[31,26162,23798],{"className":26163,"style":19759},[75,76],[31,26165,220],{"className":26166},[289],[31,26168],{"className":26169,"style":293},[119],[31,26171,223],{"className":26172},[297],[31,26174],{"className":26175,"style":293},[119],[31,26177,26179,26182,26211,26214,26217,26220,26223],{"className":26178},[66],[31,26180],{"className":26181,"style":260},[70],[31,26183,26185,26188],{"className":26184},[75],[31,26186,23798],{"className":26187,"style":19759},[75,76],[31,26189,26191],{"className":26190},[1039],[31,26192,26194],{"className":26193},[1043],[31,26195,26197],{"className":26196},[1048],[31,26198,26200],{"className":26199,"style":3628},[1052],[31,26201,26202,26205],{"style":3631},[31,26203],{"className":26204,"style":1061},[1060],[31,26206,26208],{"className":26207},[1065,1066,1067,1068],[31,26209,53],{"className":26210},[75,76,1068],[31,26212,211],{"className":26213},[264],[31,26215,205],{"className":26216},[75],[31,26218],{"className":26219,"style":120},[119],[31,26221,97],{"className":26222},[124],[31,26224],{"className":26225,"style":120},[119],[31,26227,26229,26232,26235],{"className":26228},[66],[31,26230],{"className":26231,"style":12035},[70],[31,26233,23798],{"className":26234,"style":19759},[75,76],[31,26236,26238,26241],{"className":26237},[289],[31,26239,220],{"className":26240},[289],[31,26242,26244],{"className":26243},[1039],[31,26245,26247],{"className":26246},[1043],[31,26248,26250],{"className":26249},[1048],[31,26251,26253],{"className":26252,"style":5135},[1052],[31,26254,26255,26258],{"style":3631},[31,26256],{"className":26257,"style":1061},[1060],[31,26259,26261],{"className":26260},[1065,1066,1067,1068],[31,26262,26264,26267,26270],{"className":26263},[75,1068],[31,26265,205],{"className":26266},[75,1068],[31,26268,97],{"className":26269},[124,1068],[31,26271,53],{"className":26272},[75,76,1068],[31,26274,26276],{"className":26275,"translate":35},[411],[31,26277,26279,26321],{"className":26278,"translate":35},[34],[31,26280,26282],{"className":26281},[39],[41,26283,26284],{"xmlns":43,"display":420},[45,26285,26286,26318],{},[48,26287,26288,26290,26292,26294,26296,26298,26316],{},[51,26289,14556],{},[95,26291,211],{"stretchy":210},[51,26293,23798],{},[95,26295,220],{"stretchy":210},[95,26297,223],{},[2074,26299,26300,26302],{},[91,26301,205],{},[48,26303,26304,26306,26308,26310,26312,26314],{},[51,26305,23798],{},[95,26307,211],{"stretchy":210},[91,26309,205],{},[95,26311,97],{},[51,26313,23798],{},[95,26315,220],{"stretchy":210},[51,26317,495],{"mathvariant":494},[55,26319,26320],{"encoding":57},"g(\\theta)=\\frac{1}{\\theta(1-\\theta)} .",[31,26322,26324,26351],{"className":26323,"ariaHidden":62},[61],[31,26325,26327,26330,26333,26336,26339,26342,26345,26348],{"className":26326},[66],[31,26328],{"className":26329,"style":260},[70],[31,26331,14556],{"className":26332,"style":2666},[75,76],[31,26334,211],{"className":26335},[264],[31,26337,23798],{"className":26338,"style":19759},[75,76],[31,26340,220],{"className":26341},[289],[31,26343],{"className":26344,"style":293},[119],[31,26346,223],{"className":26347},[297],[31,26349],{"className":26350,"style":293},[119],[31,26352,26354,26358,26443],{"className":26353},[66],[31,26355],{"className":26356,"style":26357},[70],"height:2.2574em;vertical-align:-0.936em;",[31,26359,26361,26364,26440],{"className":26360},[75],[31,26362],{"className":26363},[264,2139],[31,26365,26367],{"className":26366},[2074],[31,26368,26370,26431],{"className":26369},[1043,1044],[31,26371,26373,26428],{"className":26372},[1048],[31,26374,26377,26409,26417],{"className":26375,"style":26376},[1052],"height:1.3214em;",[31,26378,26379,26382],{"style":5090},[31,26380],{"className":26381,"style":1637},[1060],[31,26383,26385,26388,26391,26394,26397,26400,26403,26406],{"className":26384},[75],[31,26386,23798],{"className":26387,"style":19759},[75,76],[31,26389,211],{"className":26390},[264],[31,26392,205],{"className":26393},[75],[31,26395],{"className":26396,"style":120},[119],[31,26398,97],{"className":26399},[124],[31,26401],{"className":26402,"style":120},[119],[31,26404,23798],{"className":26405,"style":19759},[75,76],[31,26407,220],{"className":26408},[289],[31,26410,26411,26414],{"style":2170},[31,26412],{"className":26413,"style":1637},[1060],[31,26415],{"className":26416,"style":2178},[2177],[31,26418,26419,26422],{"style":5110},[31,26420],{"className":26421,"style":1637},[1060],[31,26423,26425],{"className":26424},[75],[31,26426,205],{"className":26427},[75],[31,26429,1076],{"className":26430},[1075],[31,26432,26434],{"className":26433},[1048],[31,26435,26438],{"className":26436,"style":26437},[1052],"height:0.936em;",[31,26439],{},[31,26441],{"className":26442},[289,2139],[31,26444,495],{"className":26445},[75],[11,26447,26448,26449,26615],{},"弧长 ",[31,26450,26452,26493],{"className":26451,"translate":35},[34],[31,26453,26455],{"className":26454},[39],[41,26456,26457],{"xmlns":43},[45,26458,26459,26490],{},[48,26460,26461,26463,26465,26467,26469,26471,26473],{},[51,26462,1435],{},[51,26464,14551],{},[95,26466,223],{},[51,26468,1435],{},[51,26470,23798],{},[51,26472,3768],{"mathvariant":494},[26474,26475,26476],"msqrt",{},[48,26477,26478,26480,26482,26484,26486,26488],{},[51,26479,23798],{},[95,26481,211],{"stretchy":210},[91,26483,205],{},[95,26485,97],{},[51,26487,23798],{},[95,26489,220],{"stretchy":210},[55,26491,26492],{"encoding":57},"ds=d\\theta\u002F\\sqrt{\\theta(1-\\theta)}",[31,26494,26496,26517],{"className":26495,"ariaHidden":62},[61],[31,26497,26499,26502,26505,26508,26511,26514],{"className":26498},[66],[31,26500],{"className":26501,"style":1630},[70],[31,26503,1435],{"className":26504},[75,76],[31,26506,14551],{"className":26507},[75,76],[31,26509],{"className":26510,"style":293},[119],[31,26512,223],{"className":26513},[297],[31,26515],{"className":26516,"style":293},[119],[31,26518,26520,26524,26527,26530,26533],{"className":26519},[66],[31,26521],{"className":26522,"style":26523},[70],"height:1.24em;vertical-align:-0.305em;",[31,26525,1435],{"className":26526},[75,76],[31,26528,23798],{"className":26529,"style":19759},[75,76],[31,26531,3768],{"className":26532},[75],[31,26534,26537],{"className":26535},[75,26536],"sqrt",[31,26538,26540,26606],{"className":26539},[1043,1044],[31,26541,26543,26603],{"className":26542},[1048],[31,26544,26547,26583],{"className":26545,"style":26546},[1052],"height:0.935em;",[31,26548,26552,26555],{"className":26549,"style":26551},[26550],"svg-align","top:-3.2em;",[31,26553],{"className":26554,"style":13336},[1060],[31,26556,26559,26562,26565,26568,26571,26574,26577,26580],{"className":26557,"style":26558},[75],"padding-left:1em;",[31,26560,23798],{"className":26561,"style":19759},[75,76],[31,26563,211],{"className":26564},[264],[31,26566,205],{"className":26567},[75],[31,26569],{"className":26570,"style":120},[119],[31,26572,97],{"className":26573},[124],[31,26575],{"className":26576,"style":120},[119],[31,26578,23798],{"className":26579,"style":19759},[75,76],[31,26581,220],{"className":26582},[289],[31,26584,26586,26589],{"style":26585},"top:-2.895em;",[31,26587],{"className":26588,"style":13336},[1060],[31,26590,26594],{"className":26591,"style":26593},[26592],"hide-tail","min-width:1.02em;height:1.28em;",[13341,26595,26600],{"xmlns":13343,"width":26596,"height":26597,"viewBox":26598,"preserveAspectRatio":26599},"400em","1.28em","0 0 400000 1296","xMinYMin slice",[13347,26601],{"d":26602},"M263,681c0.7,0,18,39.7,52,119\nc34,79.3,68.167,158.7,102.5,238c34.3,79.3,51.8,119.3,52.5,120\nc340,-704.7,510.7,-1060.3,512,-1067\nl0 -0\nc4.7,-7.3,11,-11,19,-11\nH40000v40H1012.3\ns-271.3,567,-271.3,567c-38.7,80.7,-84,175,-136,283c-52,108,-89.167,185.3,-111.5,232\nc-22.3,46.7,-33.8,70.3,-34.5,71c-4.7,4.7,-12.3,7,-23,7s-12,-1,-12,-1\ns-109,-253,-109,-253c-72.7,-168,-109.3,-252,-110,-252c-10.7,8,-22,16.7,-34,26\nc-22,17.3,-33.3,26,-34,26s-26,-26,-26,-26s76,-59,76,-59s76,-60,76,-60z\nM1001 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的距离是 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",[31,27289,27291,27309],{"className":27290,"translate":35},[34],[31,27292,27294],{"className":27293},[39],[41,27295,27296],{"xmlns":43},[45,27297,27298,27306],{},[48,27299,27300,27302,27304],{},[51,27301,23798],{},[95,27303,929],{},[91,27305,1977],{},[55,27307,27308],{"encoding":57},"\\theta\\to0",[31,27310,27312,27330],{"className":27311,"ariaHidden":62},[61],[31,27313,27315,27318,27321,27324,27327],{"className":27314},[66],[31,27316],{"className":27317,"style":1630},[70],[31,27319,23798],{"className":27320,"style":19759},[75,76],[31,27322],{"className":27323,"style":293},[119],[31,27325,929],{"className":27326},[297],[31,27328],{"className":27329,"style":293},[119],[31,27331,27333,27336],{"className":27332},[66],[31,27334],{"className":27335,"style":326},[70],[31,27337,1977],{"className":27338},[75],[31,27340,27342,27360],{"className":27341,"translate":35},[34],[31,27343,27345],{"className":27344},[39],[41,27346,27347],{"xmlns":43},[45,27348,27349,27357],{},[48,27350,27351,27353,27355],{},[51,27352,23798],{},[95,27354,929],{},[91,27356,205],{},[55,27358,27359],{"encoding":57},"\\theta\\to1",[31,27361,27363,27381],{"className":27362,"ariaHidden":62},[61],[31,27364,27366,27369,27372,27375,27378],{"className":27365},[66],[31,27367],{"className":27368,"style":1630},[70],[31,27370,23798],{"className":27371,"style":19759},[75,76],[31,27373],{"className":27374,"style":293},[119],[31,27376,929],{"className":27377},[297],[31,27379],{"className":27380,"style":293},[119],[31,27382,27384,27387],{"className":27383},[66],[31,27385],{"className":27386,"style":326},[70],[31,27388,205],{"className":27389},[75],[31,27391,27393,27406],{"className":27392,"translate":35},[34],[31,27394,27396],{"className":27395},[39],[41,27397,27398],{"xmlns":43},[45,27399,27400,27404],{},[48,27401,27402],{},[51,27403,2515],{},[55,27405,3485],{"encoding":57},[31,27407,27409],{"className":27408,"ariaHidden":62},[61],[31,27410,27412,27415],{"className":27411},[66],[31,27413],{"className":27414,"style":71},[70],[31,27416,2515],{"className":27417,"style":2666},[75,76],"，有限。这一点我很喜欢：从“几乎确定是 0”走到“几乎确定是 1”，不管两端有多极端，路径长度都是 ",[31,27420,27422,27435],{"className":27421,"translate":35},[34],[31,27423,27425],{"className":27424},[39],[41,27426,27427],{"xmlns":43},[45,27428,27429,27433],{},[48,27430,27431],{},[51,27432,2515],{},[55,27434,3485],{"encoding":57},[31,27436,27438],{"className":27437,"ariaHidden":62},[61],[31,27439,27441,27444],{"className":27440},[66],[31,27442],{"className":27443,"style":71},[70],[31,27445,2515],{"className":27446,"style":2666},[75,76],"；而 ",[31,27449,27451,27486],{"className":27450,"translate":35},[34],[31,27452,27454],{"className":27453},[39],[41,27455,27456],{"xmlns":43},[45,27457,27458,27483],{},[48,27459,27460,27466,27468,27471,27473,27476,27478,27480],{},[48,27461,27462,27464],{},[51,27463,4558],{"mathvariant":494},[51,27465,4561],{"mathvariant":494},[95,27467,211],{"stretchy":210},[91,27469,27470],{},"0.999",[51,27472,4572],{"mathvariant":494},[91,27474,27475],{},"0.001",[95,27477,220],{"stretchy":210},[95,27479,27223],{},[91,27481,27482],{},"6.9",[55,27484,27485],{"encoding":57},"\\mathrm{KL}(0.999\\|0.001)\\approx6.9",[31,27487,27489,27520],{"className":27488,"ariaHidden":62},[61],[31,27490,27492,27495,27501,27504,27508,27511,27514,27517],{"className":27491},[66],[31,27493],{"className":27494,"style":260},[70],[31,27496,27498],{"className":27497},[75],[31,27499,4606],{"className":27500},[75,2745],[31,27502,211],{"className":27503},[264],[31,27505,27507],{"className":27506},[75],"0.999∥0.001",[31,27509,220],{"className":27510},[289],[31,27512],{"className":27513,"style":293},[119],[31,27515,27223],{"className":27516},[297],[31,27518],{"className":27519,"style":293},[119],[31,27521,27523,27526],{"className":27522},[66],[31,27524],{"className":27525,"style":326},[70],[31,27527,27482],{"className":27528},[75]," 并且随极端程度一起发散。几何距离和信息量在这里分道扬镳。",[11,27531,27532,27533,27632],{},"那个 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0 400000 1080",[13347,27618],{"d":27619},"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z",[31,27621,1076],{"className":27622},[1075],[31,27624,27626],{"className":27625},[1048],[31,27627,27630],{"className":27628,"style":27629},[1052],"height:0.1078em;",[31,27631],{}," 不是巧合。对一般的离散分布，",[31,27634,27636],{"className":27635,"translate":35},[411],[31,27637,27639,27709],{"className":27638,"translate":35},[34],[31,27640,27642],{"className":27641},[39],[41,27643,27644],{"xmlns":43,"display":420},[45,27645,27646,27706],{},[48,27647,27648,27658,27660,27662,27664,27667,27669,27671,27673,27676,27678,27680,27686,27702,27704],{},[1004,27649,27650,27652],{},[51,27651,1435],{},[48,27653,27654,27656],{},[51,27655,13114],{"mathvariant":494},[51,27657,14790],{"mathvariant":494},[95,27659,211],{"stretchy":210},[51,27661,11],{},[95,27663,1210],{"separator":62},[51,27665,27666],{},"q",[95,27668,220],{"stretchy":210},[95,27670,223],{},[91,27672,152],{},[51,27674,27675],{},"arccos",[95,27677,1540],{},[95,27679,211],{"fence":62,"stretchy":62,"minsize":2511,"maxsize":2511},[1531,27681,27682,27684],{},[95,27683,5625],{},[51,27685,5391],{},[26474,27687,27688],{},[48,27689,27690,27696],{},[1004,27691,27692,27694],{},[51,27693,11],{},[51,27695,5391],{},[1004,27697,27698,27700],{},[51,27699,27666],{},[51,27701,5391],{},[95,27703,220],{"fence":62,"stretchy":62,"minsize":2511,"maxsize":2511},[95,27705,1210],{"separator":62},[55,27707,27708],{"encoding":57},"d_{\\mathrm{FR}}(p,q)=2\\arccos\\Bigl(\\sum_i\\sqrt{p_iq_i}\\Bigr),",[31,27710,27712,27793],{"className":27711,"ariaHidden":62},[61],[31,27713,27715,27718,27766,27769,27772,27775,27778,27781,27784,27787,27790],{"className":27714},[66],[31,27716],{"className":27717,"style":260},[70],[31,27719,27721,27724],{"className":27720},[75],[31,27722,1435],{"className":27723},[75,76],[31,27725,27727],{"className":27726},[1039],[31,27728,27730,27758],{"className":27729},[1043,1044],[31,27731,27733,27755],{"className":27732},[1048],[31,27734,27737],{"className":27735,"style":27736},[1052],"height:0.3283em;",[31,27738,27739,27742],{"style":2807},[31,27740],{"className":27741,"style":1061},[1060],[31,27743,27745],{"className":27744},[1065,1066,1067,1068],[31,27746,27748],{"className":27747},[75,1068],[31,27749,27751],{"className":27750},[75,1068],[31,27752,27754],{"className":27753},[75,2745,1068],"FR",[31,27756,1076],{"className":27757},[1075],[31,27759,27761],{"className":27760},[1048],[31,27762,27764],{"className":27763,"style":2050},[1052],[31,27765],{},[31,27767,211],{"className":27768},[264],[31,27770,11],{"className":27771},[75,76],[31,27773,1210],{"className":27774},[1391],[31,27776],{"className":27777,"style":1467},[119],[31,27779,27666],{"className":27780,"style":2666},[75,76],[31,27782,220],{"className":27783},[289],[31,27785],{"className":27786,"style":293},[119],[31,27788,223],{"className":27789},[297],[31,27791],{"className":27792,"style":293},[119],[31,27794,27796,27800,27803,27806,27809,27815,27859,27862,27989,27995],{"className":27795},[66],[31,27797],{"className":27798,"style":27799},[70],"height:2.4277em;vertical-align:-1.2777em;",[31,27801,152],{"className":27802},[75],[31,27804],{"className":27805,"style":1467},[119],[31,27807,27675],{"className":27808},[1460],[31,27810,27812],{"className":27811},[264],[31,27813,211],{"className":27814},[1307,2662],[31,27816,27818],{"className":27817},[1460,1620],[31,27819,27821,27851],{"className":27820},[1043,1044],[31,27822,27824,27848],{"className":27823},[1048],[31,27825,27827,27838],{"className":27826,"style":5900},[1052],[31,27828,27829,27832],{"style":5903},[31,27830],{"className":27831,"style":5907},[1060],[31,27833,27835],{"className":27834},[1065,1066,1067,1068],[31,27836,5391],{"className":27837},[75,76,1068],[31,27839,27840,27843],{"style":5919},[31,27841],{"className":27842,"style":5907},[1060],[31,27844,27845],{},[31,27846,5625],{"className":27847},[1460,1461,1687],[31,27849,1076],{"className":27850},[1075],[31,27852,27854],{"className":27853},[1048],[31,27855,27857],{"className":27856,"style":6342},[1052],[31,27858],{},[31,27860],{"className":27861,"style":1467},[119],[31,27863,27865],{"className":27864},[75,26536],[31,27866,27868,27980],{"className":27867},[1043,1044],[31,27869,27871,27977],{"className":27870},[1048],[31,27872,27874,27964],{"className":27873,"style":13732},[1052],[31,27875,27877,27880],{"className":27876,"style":1649},[26550],[31,27878],{"className":27879,"style":1637},[1060],[31,27881,27884,27924],{"className":27882,"style":27883},[75],"padding-left:0.833em;",[31,27885,27887,27890],{"className":27886},[75],[31,27888,11],{"className":27889},[75,76],[31,27891,27893],{"className":27892},[1039],[31,27894,27896,27916],{"className":27895},[1043,1044],[31,27897,27899,27913],{"className":27898},[1048],[31,27900,27902],{"className":27901,"style":5445},[1052],[31,27903,27904,27907],{"style":2807},[31,27905],{"className":27906,"style":1061},[1060],[31,27908,27910],{"className":27909},[1065,1066,1067,1068],[31,27911,5391],{"className":27912},[75,76,1068],[31,27914,1076],{"className":27915},[1075],[31,27917,27919],{"className":27918},[1048],[31,27920,27922],{"className":27921,"style":2050},[1052],[31,27923],{},[31,27925,27927,27930],{"className":27926},[75],[31,27928,27666],{"className":27929,"style":2666},[75,76],[31,27931,27933],{"className":27932},[1039],[31,27934,27936,27956],{"className":27935},[1043,1044],[31,27937,27939,27953],{"className":27938},[1048],[31,27940,27942],{"className":27941,"style":5445},[1052],[31,27943,27944,27947],{"style":5504},[31,27945],{"className":27946,"style":1061},[1060],[31,27948,27950],{"className":27949},[1065,1066,1067,1068],[31,27951,5391],{"className":27952},[75,76,1068],[31,27954,1076],{"className":27955},[1075],[31,27957,27959],{"className":27958},[1048],[31,27960,27962],{"className":27961,"style":2050},[1052],[31,27963],{},[31,27965,27967,27970],{"style":27966},"top:-2.7119em;",[31,27968],{"className":27969,"style":1637},[1060],[31,27971,27973],{"className":27972,"style":27612},[26592],[13341,27974,27975],{"xmlns":13343,"width":26596,"height":27615,"viewBox":27616,"preserveAspectRatio":26599},[13347,27976],{"d":27619},[31,27978,1076],{"className":27979},[1075],[31,27981,27983],{"className":27982},[1048],[31,27984,27987],{"className":27985,"style":27986},[1052],"height:0.2881em;",[31,27988],{},[31,27990,27992],{"className":27991},[289],[31,27993,220],{"className":27994},[1307,2662],[31,27996,1210],{"className":27997},[1391],[11,27999,28000,28001,28127,28128,28188,28189,28217,28218,28364,28365,28579],{},"而映射 ",[31,28002,28004,28030],{"className":28003,"translate":35},[34],[31,28005,28007],{"className":28006},[39],[41,28008,28009],{"xmlns":43},[45,28010,28011,28027],{},[48,28012,28013,28015,28017,28019,28021,28023],{},[51,28014,11],{},[95,28016,18339],{},[51,28018,5324],{},[95,28020,223],{},[91,28022,152],{},[26474,28024,28025],{},[51,28026,11],{},[55,28028,28029],{"encoding":57},"p\\mapsto u=2\\sqrt p",[31,28031,28033,28052,28070],{"className":28032,"ariaHidden":62},[61],[31,28034,28036,28040,28043,28046,28049],{"className":28035},[66],[31,28037],{"className":28038,"style":28039},[70],"height:0.7054em;vertical-align:-0.1944em;",[31,28041,11],{"className":28042},[75,76],[31,28044],{"className":28045,"style":293},[119],[31,28047,18339],{"className":28048},[297],[31,28050],{"className":28051,"style":293},[119],[31,28053,28055,28058,28061,28064,28067],{"className":28054},[66],[31,28056],{"className":28057,"style":71},[70],[31,28059,5324],{"className":28060},[75,76],[31,28062],{"className":28063,"style":293},[119],[31,28065,223],{"className":28066},[297],[31,28068],{"className":28069,"style":293},[119],[31,28071,28073,28077,28080],{"className":28072},[66],[31,28074],{"className":28075,"style":28076},[70],"height:1.04em;vertical-align:-0.3369em;",[31,28078,152],{"className":28079},[75],[31,28081,28083],{"className":28082},[75,26536],[31,28084,28086,28118],{"className":28085},[1043,1044],[31,28087,28089,28115],{"className":28088},[1048],[31,28090,28093,28102],{"className":28091,"style":28092},[1052],"height:0.7031em;",[31,28094,28096,28099],{"className":28095,"style":1649},[26550],[31,28097],{"className":28098,"style":1637},[1060],[31,28100,11],{"className":28101,"style":27883},[75,76],[31,28103,28105,28108],{"style":28104},"top:-2.6631em;",[31,28106],{"className":28107,"style":1637},[1060],[31,28109,28111],{"className":28110,"style":27612},[26592],[13341,28112,28113],{"xmlns":13343,"width":26596,"height":27615,"viewBox":27616,"preserveAspectRatio":26599},[13347,28114],{"d":27619},[31,28116,1076],{"className":28117},[1075],[31,28119,28121],{"className":28120},[1048],[31,28122,28125],{"className":28123,"style":28124},[1052],"height:0.3369em;",[31,28126],{}," 把单纯形嵌入到 ",[31,28129,28131,28149],{"className":28130,"translate":35},[34],[31,28132,28134],{"className":28133},[39],[41,28135,28136],{"xmlns":43},[45,28137,28138,28146],{},[48,28139,28140],{},[1180,28141,28142,28144],{},[51,28143,14790],{"mathvariant":14789},[51,28145,4558],{},[55,28147,28148],{"encoding":57},"\\mathbb R^K",[31,28150,28152],{"className":28151,"ariaHidden":62},[61],[31,28153,28155,28159],{"className":28154},[66],[31,28156],{"className":28157,"style":28158},[70],"height:0.8413em;",[31,28160,28162,28165],{"className":28161},[75],[31,28163,14790],{"className":28164},[75,14947],[31,28166,28168],{"className":28167},[1039],[31,28169,28171],{"className":28170},[1043],[31,28172,28174],{"className":28173},[1048],[31,28175,28177],{"className":28176,"style":28158},[1052],[31,28178,28179,28182],{"style":3631},[31,28180],{"className":28181,"style":1061},[1060],[31,28183,28185],{"className":28184},[1065,1066,1067,1068],[31,28186,4558],{"className":28187,"style":8525},[75,76,1068]," 中半径为 ",[31,28190,28192,28205],{"className":28191,"translate":35},[34],[31,28193,28195],{"className":28194},[39],[41,28196,28197],{"xmlns":43},[45,28198,28199,28203],{},[48,28200,28201],{},[91,28202,152],{},[55,28204,152],{"encoding":57},[31,28206,28208],{"className":28207,"ariaHidden":62},[61],[31,28209,28211,28214],{"className":28210},[66],[31,28212],{"className":28213,"style":326},[70],[31,28215,152],{"className":28216},[75]," 的球面上（因为 ",[31,28219,28221,28251],{"className":28220,"translate":35},[34],[31,28222,28224],{"className":28223},[39],[41,28225,28226],{"xmlns":43},[45,28227,28228,28248],{},[48,28229,28230,28236,28238,28244,28246],{},[1004,28231,28232,28234],{},[95,28233,5625],{},[51,28235,5391],{},[91,28237,15416],{},[1004,28239,28240,28242],{},[51,28241,11],{},[51,28243,5391],{},[95,28245,223],{},[91,28247,15416],{},[55,28249,28250],{"encoding":57},"\\sum_i4p_i=4",[31,28252,28254,28355],{"className":28253,"ariaHidden":62},[61],[31,28255,28257,28260,28300,28303,28306,28346,28349,28352],{"className":28256},[66],[31,28258],{"className":28259,"style":21359},[70],[31,28261,28263,28266],{"className":28262},[1460],[31,28264,5625],{"className":28265,"style":21372},[1460,1461,1462],[31,28267,28269],{"className":28268},[1039],[31,28270,28272,28292],{"className":28271},[1043,1044],[31,28273,28275,28289],{"className":28274},[1048],[31,28276,28278],{"className":28277,"style":21385},[1052],[31,28279,28280,28283],{"style":21388},[31,28281],{"className":28282,"style":1061},[1060],[31,28284,28286],{"className":28285},[1065,1066,1067,1068],[31,28287,5391],{"className":28288},[75,76,1068],[31,28290,1076],{"className":28291},[1075],[31,28293,28295],{"className":28294},[1048],[31,28296,28298],{"className":28297,"style":21407},[1052],[31,28299],{},[31,28301],{"className":28302,"style":1467},[119],[31,28304,15416],{"className":28305},[75],[31,28307,28309,28312],{"className":28308},[75],[31,28310,11],{"className":28311},[75,76],[31,28313,28315],{"className":28314},[1039],[31,28316,28318,28338],{"className":28317},[1043,1044],[31,28319,28321,28335],{"className":28320},[1048],[31,28322,28324],{"className":28323,"style":5445},[1052],[31,28325,28326,28329],{"style":2807},[31,28327],{"className":28328,"style":1061},[1060],[31,28330,28332],{"className":28331},[1065,1066,1067,1068],[31,28333,5391],{"className":28334},[75,76,1068],[31,28336,1076],{"className":28337},[1075],[31,28339,28341],{"className":28340},[1048],[31,28342,28344],{"className":28343,"style":2050},[1052],[31,28345],{},[31,28347],{"className":28348,"style":293},[119],[31,28350,223],{"className":28351},[297],[31,28353],{"className":28354,"style":293},[119],[31,28356,28358,28361],{"className":28357},[66],[31,28359],{"className":28360,"style":326},[70],[31,28362,15416],{"className":28363},[75],"），Fisher–Rao 距离正好是球面上的大圆距离。这个嵌入叫 Hellinger 嵌入，",[31,28366,28368,28402],{"className":28367,"translate":35},[34],[31,28369,28371],{"className":28370},[39],[41,28372,28373],{"xmlns":43},[45,28374,28375,28399],{},[48,28376,28377,28383],{},[1004,28378,28379,28381],{},[95,28380,5625],{},[51,28382,5391],{},[26474,28384,28385],{},[48,28386,28387,28393],{},[1004,28388,28389,28391],{},[51,28390,11],{},[51,28392,5391],{},[1004,28394,28395,28397],{},[51,28396,27666],{},[51,28398,5391],{},[55,28400,28401],{"encoding":57},"\\sum_i\\sqrt{p_iq_i}",[31,28403,28405],{"className":28404,"ariaHidden":62},[61],[31,28406,28408,28412,28452,28455],{"className":28407},[66],[31,28409],{"className":28410,"style":28411},[70],"height:1.0869em;vertical-align:-0.3369em;",[31,28413,28415,28418],{"className":28414},[1460],[31,28416,5625],{"className":28417,"style":21372},[1460,1461,1462],[31,28419,28421],{"className":28420},[1039],[31,28422,28424,28444],{"className":28423},[1043,1044],[31,28425,28427,28441],{"className":28426},[1048],[31,28428,28430],{"className":28429,"style":21385},[1052],[31,28431,28432,28435],{"style":21388},[31,28433],{"className":28434,"style":1061},[1060],[31,28436,28438],{"className":28437},[1065,1066,1067,1068],[31,28439,5391],{"className":28440},[75,76,1068],[31,28442,1076],{"className":28443},[1075],[31,28445,28447],{"className":28446},[1048],[31,28448,28450],{"className":28449,"style":21407},[1052],[31,28451],{},[31,28453],{"className":28454,"style":1467},[119],[31,28456,28458],{"className":28457},[75,26536],[31,28459,28461,28571],{"className":28460},[1043,1044],[31,28462,28464,28568],{"className":28463},[1048],[31,28465,28467,28556],{"className":28466,"style":28092},[1052],[31,28468,28470,28473],{"className":28469,"style":1649},[26550],[31,28471],{"className":28472,"style":1637},[1060],[31,28474,28476,28516],{"className":28475,"style":27883},[75],[31,28477,28479,28482],{"className":28478},[75],[31,28480,11],{"className":28481},[75,76],[31,28483,28485],{"className":28484},[1039],[31,28486,28488,28508],{"className":28487},[1043,1044],[31,28489,28491,28505],{"className":28490},[1048],[31,28492,28494],{"className":28493,"style":5445},[1052],[31,28495,28496,28499],{"style":2807},[31,28497],{"className":28498,"style":1061},[1060],[31,28500,28502],{"className":28501},[1065,1066,1067,1068],[31,28503,5391],{"className":28504},[75,76,1068],[31,28506,1076],{"className":28507},[1075],[31,28509,28511],{"className":28510},[1048],[31,28512,28514],{"className":28513,"style":2050},[1052],[31,28515],{},[31,28517,28519,28522],{"className":28518},[75],[31,28520,27666],{"className":28521,"style":2666},[75,76],[31,28523,28525],{"className":28524},[1039],[31,28526,28528,28548],{"className":28527},[1043,1044],[31,28529,28531,28545],{"className":28530},[1048],[31,28532,28534],{"className":28533,"style":5445},[1052],[31,28535,28536,28539],{"style":5504},[31,28537],{"className":28538,"style":1061},[1060],[31,28540,28542],{"className":28541},[1065,1066,1067,1068],[31,28543,5391],{"className":28544},[75,76,1068],[31,28546,1076],{"className":28547},[1075],[31,28549,28551],{"className":28550},[1048],[31,28552,28554],{"className":28553,"style":2050},[1052],[31,28555],{},[31,28557,28558,28561],{"style":28104},[31,28559],{"className":28560,"style":1637},[1060],[31,28562,28564],{"className":28563,"style":27612},[26592],[13341,28565,28566],{"xmlns":13343,"width":26596,"height":27615,"viewBox":27616,"preserveAspectRatio":26599},[13347,28567],{"d":27619},[31,28569,1076],{"className":28570},[1075],[31,28572,28574],{"className":28573},[1048],[31,28575,28577],{"className":28576,"style":28124},[1052],[31,28578],{}," 叫 Bhattacharyya 系数。",[11,28581,28582],{},"有了球面，测地线就清楚了。两点之间的大圆弧写回单纯形是",[31,28584,28586],{"className":28585,"translate":35},[411],[31,28587,28589,28708],{"className":28588,"translate":35},[34],[31,28590,28592],{"className":28591},[39],[41,28593,28594],{"xmlns":43,"display":420},[45,28595,28596,28705],{},[48,28597,28598,28604,28606,28647,28649,28651,28657,28659,28665,28667,28669,28671,28673,28675,28677,28685,28687,28689,28697,28703],{},[1004,28599,28600,28602],{},[51,28601,11],{},[51,28603,12271],{},[95,28605,223],{},[2074,28607,28608,28640],{},[48,28609,28610,28612,28614,28616,28618,28620,28622,28626,28628,28630,28634],{},[95,28611,211],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[95,28613,211],{"stretchy":210},[91,28615,205],{},[95,28617,97],{},[51,28619,12271],{},[95,28621,220],{"stretchy":210},[26474,28623,28624],{},[51,28625,11],{},[95,28627,434],{},[51,28629,12271],{},[26474,28631,28632],{},[51,28633,27666],{},[1180,28635,28636,28638],{},[95,28637,220],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[91,28639,152],{},[1004,28641,28642,28645],{},[51,28643,28644],{},"Z",[51,28646,12271],{},[95,28648,1210],{"separator":62},[119,28650],{"width":20286},[1004,28652,28653,28655],{},[51,28654,28644],{},[51,28656,12271],{},[95,28658,223],{},[1531,28660,28661,28663],{},[95,28662,5625],{},[51,28664,5391],{},[95,28666,211],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[95,28668,211],{"stretchy":210},[91,28670,205],{},[95,28672,97],{},[51,28674,12271],{},[95,28676,220],{"stretchy":210},[26474,28678,28679],{},[1004,28680,28681,28683],{},[51,28682,11],{},[51,28684,5391],{},[95,28686,434],{},[51,28688,12271],{},[26474,28690,28691],{},[1004,28692,28693,28695],{},[51,28694,27666],{},[51,28696,5391],{},[1180,28698,28699,28701],{},[95,28700,220],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[91,28702,152],{},[51,28704,495],{"mathvariant":494},[55,28706,28707],{"encoding":57},"p_t=\\frac{\\bigl((1-t)\\sqrt p+t\\sqrt q\\bigr)^2}{Z_t},\\qquad\nZ_t=\\sum_i\\bigl((1-t)\\sqrt{p_i}+t\\sqrt{q_i}\\bigr)^2 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29284],{"className":29283},[75,26536],[31,29285,29287,29357],{"className":29286},[1043,1044],[31,29288,29290,29354],{"className":29289},[1048],[31,29291,29293,29342],{"className":29292,"style":13732},[1052],[31,29294,29296,29299],{"className":29295,"style":1649},[26550],[31,29297],{"className":29298,"style":1637},[1060],[31,29300,29302],{"className":29301,"style":27883},[75],[31,29303,29305,29308],{"className":29304},[75],[31,29306,27666],{"className":29307,"style":2666},[75,76],[31,29309,29311],{"className":29310},[1039],[31,29312,29314,29334],{"className":29313},[1043,1044],[31,29315,29317,29331],{"className":29316},[1048],[31,29318,29320],{"className":29319,"style":5445},[1052],[31,29321,29322,29325],{"style":5504},[31,29323],{"className":29324,"style":1061},[1060],[31,29326,29328],{"className":29327},[1065,1066,1067,1068],[31,29329,5391],{"className":29330},[75,76,1068],[31,29332,1076],{"className":29333},[1075],[31,29335,29337],{"className":29336},[1048],[31,29338,29340],{"className":29339,"style":2050},[1052],[31,29341],{},[31,29343,29344,29347],{"style":27966},[31,29345],{"className":29346,"style":1637},[1060],[31,29348,29350],{"className":29349,"style":27612},[26592],[13341,29351,29352],{"xmlns":13343,"width":26596,"height":27615,"viewBox":27616,"preserveAspectRatio":26599},[13347,29353],{"d":27619},[31,29355,1076],{"className":29356},[1075],[31,29358,29360],{"className":29359},[1048],[31,29361,29363],{"className":29362,"style":27986},[1052],[31,29364],{},[31,29366,29368,29374],{"className":29367},[289],[31,29369,29371],{"className":29370},[289],[31,29372,220],{"className":29373},[1307,1308],[31,29375,29377],{"className":29376},[1039],[31,29378,29380],{"className":29379},[1043],[31,29381,29383],{"className":29382},[1048],[31,29384,29386],{"className":29385,"style":29009},[1052],[31,29387,29388,29391],{"style":13566},[31,29389],{"className":29390,"style":1061},[1060],[31,29392,29394],{"className":29393},[1065,1066,1067,1068],[31,29395,152],{"className":29396},[75,1068],[31,29398,495],{"className":29399},[75],[11,29401,29402],{},"这不是线性插值：概率质量走的是一条弯的路。如果我在单纯形上按大圆距离做最近原型分类，两个原型之间的“垂直平分线”是一条大圆，投影回单纯形是弯的；而按线性插值它会是直的。这就是我理解的“非欧氏边界”最干净的来源。",[18,29404,29405],{"id":29405},"指数族",[11,29407,29408,29409,29437,29438,29466],{},"我抛一枚硬币 100 次，60 次正面。这句话里真正有用的信息只有两个数：100 和 60。给定正面概率 ",[31,29410,29412,29425],{"className":29411,"translate":35},[34],[31,29413,29415],{"className":29414},[39],[41,29416,29417],{"xmlns":43},[45,29418,29419,29423],{},[48,29420,29421],{},[51,29422,23798],{},[55,29424,24747],{"encoding":57},[31,29426,29428],{"className":29427,"ariaHidden":62},[61],[31,29429,29431,29434],{"className":29430},[66],[31,29432],{"className":29433,"style":1630},[70],[31,29435,23798],{"className":29436,"style":19759},[75,76],"，正面次数服从二项分布，而样本对 ",[31,29439,29441,29454],{"className":29440,"translate":35},[34],[31,29442,29444],{"className":29443},[39],[41,29445,29446],{"xmlns":43},[45,29447,29448,29452],{},[48,29449,29450],{},[51,29451,23798],{},[55,29453,24747],{"encoding":57},[31,29455,29457],{"className":29456,"ariaHidden":62},[61],[31,29458,29460,29463],{"className":29459},[66],[31,29461],{"className":29462,"style":1630},[70],[31,29464,23798],{"className":29465,"style":19759},[75,76]," 的全部影响都通过“正面次数”这个线性统计量进去——这就是充分统计量。",[11,29468,29469],{},"指数族把这个结构写成一个模板：",[31,29471,29473],{"className":29472,"translate":35},[411],[31,29474,29476,29545],{"className":29475,"translate":35},[34],[31,29477,29479],{"className":29478},[39],[41,29480,29481],{"xmlns":43,"display":420},[45,29482,29483,29542],{},[48,29484,29485,29487,29489,29491,29493,29495,29497,29499,29502,29504,29506,29508,29510,29512,29514,29520,29522,29524,29526,29528,29530,29532,29534,29536,29538,29540],{},[51,29486,11],{},[95,29488,211],{"stretchy":210},[51,29490,53],{},[95,29492,12680],{},[51,29494,23798],{},[95,29496,220],{"stretchy":210},[95,29498,223],{},[51,29500,29501],{},"h",[95,29503,211],{"stretchy":210},[51,29505,53],{},[95,29507,220],{"stretchy":210},[51,29509,20301],{},[95,29511,1540],{},[95,29513,211],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[1180,29515,29516,29518],{},[51,29517,23798],{},[51,29519,20800],{"mathvariant":494},[51,29521,919],{},[95,29523,211],{"stretchy":210},[51,29525,53],{},[95,29527,220],{"stretchy":210},[95,29529,97],{},[51,29531,8064],{},[95,29533,211],{"stretchy":210},[51,29535,23798],{},[95,29537,220],{"stretchy":210},[95,29539,220],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[51,29541,495],{"mathvariant":494},[55,29543,29544],{"encoding":57},"p(x\\mid\\theta)=h(x)\\exp\\bigl(\\theta^\\top T(x)-\\psi(\\theta)\\bigr).",[31,29546,29548,29572,29593,29675],{"className":29547,"ariaHidden":62},[61],[31,29549,29551,29554,29557,29560,29563,29566,29569],{"className":29550},[66],[31,29552],{"className":29553,"style":260},[70],[31,29555,11],{"className":29556},[75,76],[31,29558,211],{"className":29559},[264],[31,29561,53],{"className":29562},[75,76],[31,29564],{"className":29565,"style":293},[119],[31,29567,12680],{"className":29568},[297],[31,29570],{"className":29571,"style":293},[119],[31,29573,29575,29578,29581,29584,29587,29590],{"className":29574},[66],[31,29576],{"className":29577,"style":260},[70],[31,29579,23798],{"className":29580,"style":19759},[75,76],[31,29582,220],{"className":29583},[289],[31,29585],{"className":29586,"style":293},[119],[31,29588,223],{"className":29589},[297],[31,29591],{"className":29592,"style":293},[119],[31,29594,29596,29600,29603,29606,29609,29612,29615,29618,29624,29654,29657,29660,29663,29666,29669,29672],{"className":29595},[66],[31,29597],{"className":29598,"style":29599},[70],"height:1.2491em;vertical-align:-0.35em;",[31,29601,29501],{"className":29602},[75,76],[31,29604,211],{"className":29605},[264],[31,29607,53],{"className":29608},[75,76],[31,29610,220],{"className":29611},[289],[31,29613],{"className":29614,"style":1467},[119],[31,29616,20301],{"className":29617},[1460],[31,29619,29621],{"className":29620},[264],[31,29622,211],{"className":29623},[1307,1308],[31,29625,29627,29630],{"className":29626},[75],[31,29628,23798],{"className":29629,"style":19759},[75,76],[31,29631,29633],{"className":29632},[1039],[31,29634,29636],{"className":29635},[1043],[31,29637,29639],{"className":29638},[1048],[31,29640,29643],{"className":29641,"style":29642},[1052],"height:0.8991em;",[31,29644,29645,29648],{"style":1330},[31,29646],{"className":29647,"style":1061},[1060],[31,29649,29651],{"className":29650},[1065,1066,1067,1068],[31,29652,20800],{"className":29653},[75,1068],[31,29655,919],{"className":29656,"style":949},[75,76],[31,29658,211],{"className":29659},[264],[31,29661,53],{"className":29662},[75,76],[31,29664,220],{"className":29665},[289],[31,29667],{"className":29668,"style":120},[119],[31,29670,97],{"className":29671},[124],[31,29673],{"className":29674,"style":120},[119],[31,29676,29678,29681,29684,29687,29690,29693,29699],{"className":29677},[66],[31,29679],{"className":29680,"style":4988},[70],[31,29682,8064],{"className":29683,"style":2666},[75,76],[31,29685,211],{"className":29686},[264],[31,29688,23798],{"className":29689,"style":19759},[75,76],[31,29691,220],{"className":29692},[289],[31,29694,29696],{"className":29695},[289],[31,29697,220],{"className":29698},[1307,1308],[31,29700,495],{"className":29701},[75],[11,29703,29704,29732,29733,29761,29762,29791,29792,15282,29858,15282,30006,2372],{},[31,29705,29707,29720],{"className":29706,"translate":35},[34],[31,29708,29710],{"className":29709},[39],[41,29711,29712],{"xmlns":43},[45,29713,29714,29718],{},[48,29715,29716],{},[51,29717,919],{},[55,29719,919],{"encoding":57},[31,29721,29723],{"className":29722,"ariaHidden":62},[61],[31,29724,29726,29729],{"className":29725},[66],[31,29727],{"className":29728,"style":945},[70],[31,29730,919],{"className":29731,"style":949},[75,76]," 是充分统计量，",[31,29734,29736,29749],{"className":29735,"translate":35},[34],[31,29737,29739],{"className":29738},[39],[41,29740,29741],{"xmlns":43},[45,29742,29743,29747],{},[48,29744,29745],{},[51,29746,23798],{},[55,29748,24747],{"encoding":57},[31,29750,29752],{"className":29751,"ariaHidden":62},[61],[31,29753,29755,29758],{"className":29754},[66],[31,29756],{"className":29757,"style":1630},[70],[31,29759,23798],{"className":29760,"style":19759},[75,76]," 是自然参数，",[31,29763,29765,29779],{"className":29764,"translate":35},[34],[31,29766,29768],{"className":29767},[39],[41,29769,29770],{"xmlns":43},[45,29771,29772,29776],{},[48,29773,29774],{},[51,29775,8064],{},[55,29777,29778],{"encoding":57},"\\psi",[31,29780,29782],{"className":29781,"ariaHidden":62},[61],[31,29783,29785,29788],{"className":29784},[66],[31,29786],{"className":29787,"style":4202},[70],[31,29789,8064],{"className":29790,"style":2666},[75,76]," 是对数配分函数，负责把总质量归一化成 1。抛硬币就是 ",[31,29793,29795,29819],{"className":29794,"translate":35},[34],[31,29796,29798],{"className":29797},[39],[41,29799,29800],{"xmlns":43},[45,29801,29802,29816],{},[48,29803,29804,29806,29808,29810,29812,29814],{},[51,29805,919],{},[95,29807,211],{"stretchy":210},[51,29809,53],{},[95,29811,220],{"stretchy":210},[95,29813,223],{},[51,29815,53],{},[55,29817,29818],{"encoding":57},"T(x)=x",[31,29820,29822,29849],{"className":29821,"ariaHidden":62},[61],[31,29823,29825,29828,29831,29834,29837,29840,29843,29846],{"className":29824},[66],[31,29826],{"className":29827,"style":260},[70],[31,29829,919],{"className":29830,"style":949},[75,76],[31,29832,211],{"className":29833},[264],[31,29835,53],{"className":29836},[75,76],[31,29838,220],{"className":29839},[289],[31,29841],{"className":29842,"style":293},[119],[31,29844,223],{"className":29845},[297],[31,29847],{"className":29848,"style":293},[119],[31,29850,29852,29855],{"className":29851},[66],[31,29853],{"className":29854,"style":71},[70],[31,29856,53],{"className":29857},[75,76],[31,29859,29861,29893],{"className":29860,"translate":35},[34],[31,29862,29864],{"className":29863},[39],[41,29865,29866],{"xmlns":43},[45,29867,29868,29890],{},[48,29869,29870,29872,29874,29876,29878],{},[51,29871,23798],{},[95,29873,223],{},[51,29875,19939],{},[95,29877,1540],{},[2074,29879,29880,29882],{},[51,29881,11],{},[48,29883,29884,29886,29888],{},[91,29885,205],{},[95,29887,97],{},[51,29889,11],{},[55,29891,29892],{"encoding":57},"\\theta=\\log\\frac{p}{1-p}",[31,29894,29896,29914],{"className":29895,"ariaHidden":62},[61],[31,29897,29899,29902,29905,29908,29911],{"className":29898},[66],[31,29900],{"className":29901,"style":1630},[70],[31,29903,23798],{"className":29904,"style":19759},[75,76],[31,29906],{"className":29907,"style":293},[119],[31,29909,223],{"className":29910},[297],[31,29912],{"className":29913,"style":293},[119],[31,29915,29917,29921,29926,29929],{"className":29916},[66],[31,29918],{"className":29919,"style":29920},[70],"height:1.2286em;vertical-align:-0.4811em;",[31,29922,20177,29924],{"className":29923},[1460],[31,29925,14556],{"style":20180},[31,29927],{"className":29928,"style":1467},[119],[31,29930,29932,29935,30003],{"className":29931},[75],[31,29933],{"className":29934},[264,2139],[31,29936,29938],{"className":29937},[2074],[31,29939,29941,29994],{"className":29940},[1043,1044],[31,29942,29944,29991],{"className":29943},[1048],[31,29945,29948,29968,29976],{"className":29946,"style":29947},[1052],"height:0.7475em;",[31,29949,29950,29953],{"style":2155},[31,29951],{"className":29952,"style":1637},[1060],[31,29954,29956],{"className":29955},[1065,1066,1067,1068],[31,29957,29959,29962,29965],{"className":29958},[75,1068],[31,29960,205],{"className":29961},[75,1068],[31,29963,97],{"className":29964},[124,1068],[31,29966,11],{"className":29967},[75,76,1068],[31,29969,29970,29973],{"style":2170},[31,29971],{"className":29972,"style":1637},[1060],[31,29974],{"className":29975,"style":2178},[2177],[31,29977,29979,29982],{"style":29978},"top:-3.4461em;",[31,29980],{"className":29981,"style":1637},[1060],[31,29983,29985],{"className":29984},[1065,1066,1067,1068],[31,29986,29988],{"className":29987},[75,1068],[31,29989,11],{"className":29990},[75,76,1068],[31,29992,1076],{"className":29993},[1075],[31,29995,29997],{"className":29996},[1048],[31,29998,30001],{"className":29999,"style":30000},[1052],"height:0.4811em;",[31,30002],{},[31,30004],{"className":30005},[289,2139],[31,30007,30009,30050],{"className":30008,"translate":35},[34],[31,30010,30012],{"className":30011},[39],[41,30013,30014],{"xmlns":43},[45,30015,30016,30047],{},[48,30017,30018,30020,30022,30024,30026,30028,30030,30032,30034,30036,30038,30045],{},[51,30019,8064],{},[95,30021,211],{"stretchy":210},[51,30023,23798],{},[95,30025,220],{"stretchy":210},[95,30027,223],{},[51,30029,19939],{},[95,30031,1540],{},[95,30033,211],{"stretchy":210},[91,30035,205],{},[95,30037,434],{},[1180,30039,30040,30043],{},[51,30041,30042],{},"e",[51,30044,23798],{},[95,30046,220],{"stretchy":210},[55,30048,30049],{"encoding":57},"\\psi(\\theta)=\\log(1+e^\\theta)",[31,30051,30053,30080,30106],{"className":30052,"ariaHidden":62},[61],[31,30054,30056,30059,30062,30065,30068,30071,30074,30077],{"className":30055},[66],[31,30057],{"className":30058,"style":260},[70],[31,30060,8064],{"className":30061,"style":2666},[75,76],[31,30063,211],{"className":30064},[264],[31,30066,23798],{"className":30067,"style":19759},[75,76],[31,30069,220],{"className":30070},[289],[31,30072],{"className":30073,"style":293},[119],[31,30075,223],{"className":30076},[297],[31,30078],{"className":30079,"style":293},[119],[31,30081,30083,30086,30091,30094,30097,30100,30103],{"className":30082},[66],[31,30084],{"className":30085,"style":260},[70],[31,30087,20177,30089],{"className":30088},[1460],[31,30090,14556],{"style":20180},[31,30092,211],{"className":30093},[264],[31,30095,205],{"className":30096},[75],[31,30098],{"className":30099,"style":120},[119],[31,30101,434],{"className":30102},[124],[31,30104],{"className":30105,"style":120},[119],[31,30107,30109,30112,30141],{"className":30108},[66],[31,30110],{"className":30111,"style":20835},[70],[31,30113,30115,30118],{"className":30114},[75],[31,30116,30042],{"className":30117},[75,76],[31,30119,30121],{"className":30120},[1039],[31,30122,30124],{"className":30123},[1043],[31,30125,30127],{"className":30126},[1048],[31,30128,30130],{"className":30129,"style":20863},[1052],[31,30131,30132,30135],{"style":3631},[31,30133],{"className":30134,"style":1061},[1060],[31,30136,30138],{"className":30137},[1065,1066,1067,1068],[31,30139,23798],{"className":30140,"style":19759},[75,76,1068],[31,30142,220],{"className":30143},[289],[11,30145,30146,30174,30175,1138,30316,30495,30496,30524],{},[31,30147,30149,30162],{"className":30148,"translate":35},[34],[31,30150,30152],{"className":30151},[39],[41,30153,30154],{"xmlns":43},[45,30155,30156,30160],{},[48,30157,30158],{},[51,30159,8064],{},[55,30161,29778],{"encoding":57},[31,30163,30165],{"className":30164,"ariaHidden":62},[61],[31,30166,30168,30171],{"className":30167},[66],[31,30169],{"className":30170,"style":4202},[70],[31,30172,8064],{"className":30173,"style":2666},[75,76]," 的性质很好：",[31,30176,30178,30220],{"className":30177,"translate":35},[34],[31,30179,30181],{"className":30180},[39],[41,30182,30183],{"xmlns":43},[45,30184,30185,30217],{},[48,30186,30187,30189,30191,30193,30195,30197,30199,30205,30207,30209,30211,30213,30215],{},[51,30188,12145],{"mathvariant":494},[51,30190,8064],{},[95,30192,211],{"stretchy":210},[51,30194,23798],{},[95,30196,220],{"stretchy":210},[95,30198,223],{},[1004,30200,30201,30203],{},[51,30202,22324],{"mathvariant":14789},[51,30204,23798],{},[95,30206,15175],{"stretchy":210},[51,30208,919],{},[95,30210,211],{"stretchy":210},[51,30212,926],{},[95,30214,220],{"stretchy":210},[95,30216,15184],{"stretchy":210},[55,30218,30219],{"encoding":57},"\\nabla\\psi(\\theta)=\\mathbb E_\\theta[T(X)]",[31,30221,30223,30253],{"className":30222,"ariaHidden":62},[61],[31,30224,30226,30229,30232,30235,30238,30241,30244,30247,30250],{"className":30225},[66],[31,30227],{"className":30228,"style":260},[70],[31,30230,12145],{"className":30231},[75],[31,30233,8064],{"className":30234,"style":2666},[75,76],[31,30236,211],{"className":30237},[264],[31,30239,23798],{"className":30240,"style":19759},[75,76],[31,30242,220],{"className":30243},[289],[31,30245],{"className":30246,"style":293},[119],[31,30248,223],{"className":30249},[297],[31,30251],{"className":30252,"style":293},[119],[31,30254,30256,30259,30299,30302,30305,30308,30312],{"className":30255},[66],[31,30257],{"className":30258,"style":260},[70],[31,30260,30262,30265],{"className":30261},[75],[31,30263,22324],{"className":30264},[75,14947],[31,30266,30268],{"className":30267},[1039],[31,30269,30271,30291],{"className":30270},[1043,1044],[31,30272,30274,30288],{"className":30273},[1048],[31,30275,30277],{"className":30276,"style":1053},[1052],[31,30278,30279,30282],{"style":2807},[31,30280],{"className":30281,"style":1061},[1060],[31,30283,30285],{"className":30284},[1065,1066,1067,1068],[31,30286,23798],{"className":30287,"style":19759},[75,76,1068],[31,30289,1076],{"className":30290},[1075],[31,30292,30294],{"className":30293},[1048],[31,30295,30297],{"className":30296,"style":2050},[1052],[31,30298],{},[31,30300,15175],{"className":30301},[264],[31,30303,919],{"className":30304,"style":949},[75,76],[31,30306,211],{"className":30307},[264],[31,30309,926],{"className":30310,"style":30311},[75,76],"margin-right:0.0785em;",[31,30313,30315],{"className":30314},[289],")]",[31,30317,30319,30371],{"className":30318,"translate":35},[34],[31,30320,30322],{"className":30321},[39],[41,30323,30324],{"xmlns":43},[45,30325,30326,30368],{},[48,30327,30328,30334,30336,30338,30340,30342,30344,30356,30358,30360,30362,30364,30366],{},[1180,30329,30330,30332],{},[51,30331,12145],{"mathvariant":494},[91,30333,152],{},[51,30335,8064],{},[95,30337,211],{"stretchy":210},[51,30339,23798],{},[95,30341,220],{"stretchy":210},[95,30343,223],{},[1004,30345,30346,30354],{},[48,30347,30348,30350,30352],{},[51,30349,8101],{"mathvariant":494},[51,30351,2552],{"mathvariant":494},[51,30353,5354],{"mathvariant":494},[51,30355,23798],{},[95,30357,15175],{"stretchy":210},[51,30359,919],{},[95,30361,211],{"stretchy":210},[51,30363,926],{},[95,30365,220],{"stretchy":210},[95,30367,15184],{"stretchy":210},[55,30369,30370],{"encoding":57},"\\nabla^2\\psi(\\theta)=\\mathrm{Cov}_\\theta[T(X)]",[31,30372,30374,30430],{"className":30373,"ariaHidden":62},[61],[31,30375,30377,30380,30409,30412,30415,30418,30421,30424,30427],{"className":30376},[66],[31,30378],{"className":30379,"style":12035},[70],[31,30381,30383,30386],{"className":30382},[75],[31,30384,12145],{"className":30385},[75],[31,30387,30389],{"className":30388},[1039],[31,30390,30392],{"className":30391},[1043],[31,30393,30395],{"className":30394},[1048],[31,30396,30398],{"className":30397,"style":5135},[1052],[31,30399,30400,30403],{"style":3631},[31,30401],{"className":30402,"style":1061},[1060],[31,30404,30406],{"className":30405},[1065,1066,1067,1068],[31,30407,152],{"className":30408},[75,1068],[31,30410,8064],{"className":30411,"style":2666},[75,76],[31,30413,211],{"className":30414},[264],[31,30416,23798],{"className":30417,"style":19759},[75,76],[31,30419,220],{"className":30420},[289],[31,30422],{"className":30423,"style":293},[119],[31,30425,223],{"className":30426},[297],[31,30428],{"className":30429,"style":293},[119],[31,30431,30433,30436,30480,30483,30486,30489,30492],{"className":30432},[66],[31,30434],{"className":30435,"style":260},[70],[31,30437,30439,30446],{"className":30438},[75],[31,30440,30442],{"className":30441},[75],[31,30443,30445],{"className":30444,"style":20180},[75,2745],"Cov",[31,30447,30449],{"className":30448},[1039],[31,30450,30452,30472],{"className":30451},[1043,1044],[31,30453,30455,30469],{"className":30454},[1048],[31,30456,30458],{"className":30457,"style":1053},[1052],[31,30459,30460,30463],{"style":21790},[31,30461],{"className":30462,"style":1061},[1060],[31,30464,30466],{"className":30465},[1065,1066,1067,1068],[31,30467,23798],{"className":30468,"style":19759},[75,76,1068],[31,30470,1076],{"className":30471},[1075],[31,30473,30475],{"className":30474},[1048],[31,30476,30478],{"className":30477,"style":2050},[1052],[31,30479],{},[31,30481,15175],{"className":30482},[264],[31,30484,919],{"className":30485,"style":949},[75,76],[31,30487,211],{"className":30488},[264],[31,30490,926],{"className":30491,"style":30311},[75,76],[31,30493,30315],{"className":30494},[289],"。第二个等式说明 Fisher 度量就是 ",[31,30497,30499,30512],{"className":30498,"translate":35},[34],[31,30500,30502],{"className":30501},[39],[41,30503,30504],{"xmlns":43},[45,30505,30506,30510],{},[48,30507,30508],{},[51,30509,8064],{},[55,30511,29778],{"encoding":57},[31,30513,30515],{"className":30514,"ariaHidden":62},[61],[31,30516,30518,30521],{"className":30517},[66],[31,30519],{"className":30520,"style":4202},[70],[31,30522,8064],{"className":30523,"style":2666},[75,76]," 的 Hessian，上一节里“度量由凸函数给出”的说法在这里变成了具体的东西。",[11,30526,30527,30528,30600,30601,30629,30630,30858,30859,30960,30961,30989,30990,31019],{},"令 ",[31,30529,30531,30558],{"className":30530,"translate":35},[34],[31,30532,30534],{"className":30533},[39],[41,30535,30536],{"xmlns":43},[45,30537,30538,30555],{},[48,30539,30540,30543,30545,30547,30549,30551,30553],{},[51,30541,30542],{},"η",[95,30544,223],{},[51,30546,12145],{"mathvariant":494},[51,30548,8064],{},[95,30550,211],{"stretchy":210},[51,30552,23798],{},[95,30554,220],{"stretchy":210},[55,30556,30557],{"encoding":57},"\\eta=\\nabla\\psi(\\theta)",[31,30559,30561,30579],{"className":30560,"ariaHidden":62},[61],[31,30562,30564,30567,30570,30573,30576],{"className":30563},[66],[31,30565],{"className":30566,"style":865},[70],[31,30568,30542],{"className":30569,"style":2666},[75,76],[31,30571],{"className":30572,"style":293},[119],[31,30574,223],{"className":30575},[297],[31,30577],{"className":30578,"style":293},[119],[31,30580,30582,30585,30588,30591,30594,30597],{"className":30581},[66],[31,30583],{"className":30584,"style":260},[70],[31,30586,12145],{"className":30587},[75],[31,30589,8064],{"className":30590,"style":2666},[75,76],[31,30592,211],{"className":30593},[264],[31,30595,23798],{"className":30596,"style":19759},[75,76],[31,30598,220],{"className":30599},[289]," 为期望参数，它与 ",[31,30602,30604,30617],{"className":30603,"translate":35},[34],[31,30605,30607],{"className":30606},[39],[41,30608,30609],{"xmlns":43},[45,30610,30611,30615],{},[48,30612,30613],{},[51,30614,23798],{},[55,30616,24747],{"encoding":57},[31,30618,30620],{"className":30619,"ariaHidden":62},[61],[31,30621,30623,30626],{"className":30622},[66],[31,30624],{"className":30625,"style":1630},[70],[31,30627,23798],{"className":30628,"style":19759},[75,76]," 之间是 Legendre 变换的关系：",[31,30631,30633,30692],{"className":30632,"translate":35},[34],[31,30634,30636],{"className":30635},[39],[41,30637,30638],{"xmlns":43},[45,30639,30640,30689],{},[48,30641,30642,30649,30651,30653,30655,30657,30667,30669,30675,30677,30679,30681,30683,30685,30687],{},[1180,30643,30644,30646],{},[51,30645,8064],{},[95,30647,30648],{},"∗",[95,30650,211],{"stretchy":210},[51,30652,30542],{},[95,30654,220],{"stretchy":210},[95,30656,223],{},[1004,30658,30659,30665],{},[48,30660,30661,30663],{},[51,30662,10792],{},[95,30664,1540],{},[51,30666,23798],{},[95,30668,211],{"stretchy":210},[1180,30670,30671,30673],{},[51,30672,23798],{},[51,30674,20800],{"mathvariant":494},[51,30676,30542],{},[95,30678,97],{},[51,30680,8064],{},[95,30682,211],{"stretchy":210},[51,30684,23798],{},[95,30686,220],{"stretchy":210},[95,30688,220],{"stretchy":210},[55,30690,30691],{"encoding":57},"\\psi^*(\\eta)=\\sup_\\theta(\\theta^\\top\\eta-\\psi(\\theta))",[31,30693,30695,30749,30840],{"className":30694,"ariaHidden":62},[61],[31,30696,30698,30701,30731,30734,30737,30740,30743,30746],{"className":30697},[66],[31,30699],{"className":30700,"style":260},[70],[31,30702,30704,30707],{"className":30703},[75],[31,30705,8064],{"className":30706,"style":2666},[75,76],[31,30708,30710],{"className":30709},[1039],[31,30711,30713],{"className":30712},[1043],[31,30714,30716],{"className":30715},[1048],[31,30717,30720],{"className":30718,"style":30719},[1052],"height:0.6887em;",[31,30721,30722,30725],{"style":3631},[31,30723],{"className":30724,"style":1061},[1060],[31,30726,30728],{"className":30727},[1065,1066,1067,1068],[31,30729,30648],{"className":30730},[124,1068],[31,30732,211],{"className":30733},[264],[31,30735,30542],{"className":30736,"style":2666},[75,76],[31,30738,220],{"className":30739},[289],[31,30741],{"className":30742,"style":293},[119],[31,30744,223],{"className":30745},[297],[31,30747],{"className":30748,"style":293},[119],[31,30750,30752,30755,30796,30799,30828,30831,30834,30837],{"className":30751},[66],[31,30753],{"className":30754,"style":20835},[70],[31,30756,30758,30761],{"className":30757},[1460],[31,30759,10792],{"className":30760},[1460],[31,30762,30764],{"className":30763},[1039],[31,30765,30767,30788],{"className":30766},[1043,1044],[31,30768,30770,30785],{"className":30769},[1048],[31,30771,30774],{"className":30772,"style":30773},[1052],"height:0.242em;",[31,30775,30776,30779],{"style":2762},[31,30777],{"className":30778,"style":1061},[1060],[31,30780,30782],{"className":30781},[1065,1066,1067,1068],[31,30783,23798],{"className":30784,"style":19759},[75,76,1068],[31,30786,1076],{"className":30787},[1075],[31,30789,30791],{"className":30790},[1048],[31,30792,30794],{"className":30793,"style":2784},[1052],[31,30795],{},[31,30797,211],{"className":30798},[264],[31,30800,30802,30805],{"className":30801},[75],[31,30803,23798],{"className":30804,"style":19759},[75,76],[31,30806,30808],{"className":30807},[1039],[31,30809,30811],{"className":30810},[1043],[31,30812,30814],{"className":30813},[1048],[31,30815,30817],{"className":30816,"style":20863},[1052],[31,30818,30819,30822],{"style":3631},[31,30820],{"className":30821,"style":1061},[1060],[31,30823,30825],{"className":30824},[1065,1066,1067,1068],[31,30826,20800],{"className":30827},[75,1068],[31,30829,30542],{"className":30830,"style":2666},[75,76],[31,30832],{"className":30833,"style":120},[119],[31,30835,97],{"className":30836},[124],[31,30838],{"className":30839,"style":120},[119],[31,30841,30843,30846,30849,30852,30855],{"className":30842},[66],[31,30844],{"className":30845,"style":260},[70],[31,30847,8064],{"className":30848,"style":2666},[75,76],[31,30850,211],{"className":30851},[264],[31,30853,23798],{"className":30854,"style":19759},[75,76],[31,30856,1495],{"className":30857},[289],"，且 ",[31,30860,30862,30892],{"className":30861,"translate":35},[34],[31,30863,30865],{"className":30864},[39],[41,30866,30867],{"xmlns":43},[45,30868,30869,30889],{},[48,30870,30871,30873,30879,30881,30883,30885,30887],{},[51,30872,12145],{"mathvariant":494},[1180,30874,30875,30877],{},[51,30876,8064],{},[95,30878,30648],{},[95,30880,211],{"stretchy":210},[51,30882,30542],{},[95,30884,220],{"stretchy":210},[95,30886,223],{},[51,30888,23798],{},[55,30890,30891],{"encoding":57},"\\nabla\\psi^*(\\eta)=\\theta",[31,30893,30895,30951],{"className":30894,"ariaHidden":62},[61],[31,30896,30898,30901,30904,30933,30936,30939,30942,30945,30948],{"className":30897},[66],[31,30899],{"className":30900,"style":260},[70],[31,30902,12145],{"className":30903},[75],[31,30905,30907,30910],{"className":30906},[75],[31,30908,8064],{"className":30909,"style":2666},[75,76],[31,30911,30913],{"className":30912},[1039],[31,30914,30916],{"className":30915},[1043],[31,30917,30919],{"className":30918},[1048],[31,30920,30922],{"className":30921,"style":30719},[1052],[31,30923,30924,30927],{"style":3631},[31,30925],{"className":30926,"style":1061},[1060],[31,30928,30930],{"className":30929},[1065,1066,1067,1068],[31,30931,30648],{"className":30932},[124,1068],[31,30934,211],{"className":30935},[264],[31,30937,30542],{"className":30938,"style":2666},[75,76],[31,30940,220],{"className":30941},[289],[31,30943],{"className":30944,"style":293},[119],[31,30946,223],{"className":30947},[297],[31,30949],{"className":30950,"style":293},[119],[31,30952,30954,30957],{"className":30953},[66],[31,30955],{"className":30956,"style":1630},[70],[31,30958,23798],{"className":30959,"style":19759},[75,76],"。所以同一个分布有两套坐标，自然参数 ",[31,30962,30964,30977],{"className":30963,"translate":35},[34],[31,30965,30967],{"className":30966},[39],[41,30968,30969],{"xmlns":43},[45,30970,30971,30975],{},[48,30972,30973],{},[51,30974,23798],{},[55,30976,24747],{"encoding":57},[31,30978,30980],{"className":30979,"ariaHidden":62},[61],[31,30981,30983,30986],{"className":30982},[66],[31,30984],{"className":30985,"style":1630},[70],[31,30987,23798],{"className":30988,"style":19759},[75,76]," 和期望参数 ",[31,30991,30993,31007],{"className":30992,"translate":35},[34],[31,30994,30996],{"className":30995},[39],[41,30997,30998],{"xmlns":43},[45,30999,31000,31004],{},[48,31001,31002],{},[51,31003,30542],{},[55,31005,31006],{"encoding":57},"\\eta",[31,31008,31010],{"className":31009,"ariaHidden":62},[61],[31,31011,31013,31016],{"className":31012},[66],[31,31014],{"className":31015,"style":865},[70],[31,31017,30542],{"className":31018,"style":2666},[75,76],"，互为对偶。",[11,31021,31022],{},"现在算两个成员之间的 KL：",[31,31024,31026],{"className":31025,"translate":35},[411],[31,31027,31029,31341],{"className":31028,"translate":35},[34],[31,31030,31032],{"className":31031},[39],[41,31033,31034],{"xmlns":43,"display":420},[45,31035,31036,31338],{},[5606,31037,31038,31120,31196,31266],{"rowspacing":5608,"columnalign":5609,"columnspacing":5610},[5612,31039,31040,31074],{},[5615,31041,31042],{},[2071,31043,31044],{"scriptlevel":1977,"displaystyle":62},[48,31045,31046,31052,31054,31060,31062,31072],{},[48,31047,31048,31050],{},[51,31049,4558],{"mathvariant":494},[51,31051,4561],{"mathvariant":494},[95,31053,211],{"stretchy":210},[1004,31055,31056,31058],{},[51,31057,11],{},[51,31059,23798],{},[51,31061,4572],{"mathvariant":494},[1004,31063,31064,31066],{},[51,31065,11],{},[1180,31067,31068,31070],{},[51,31069,23798],{},[95,31071,13691],{"mathvariant":494,"lspace":5610,"rspace":5610},[95,31073,220],{"stretchy":210},[5615,31075,31076],{},[2071,31077,31078],{"scriptlevel":1977,"displaystyle":62},[48,31079,31080,31082,31084,31090,31092,31094,31096,31102,31104,31106,31108,31118],{},[48,31081],{},[95,31083,223],{},[1004,31085,31086,31088],{},[51,31087,22324],{"mathvariant":14789},[51,31089,23798],{},[95,31091,15175],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[51,31093,19939],{},[95,31095,1540],{},[1004,31097,31098,31100],{},[51,31099,11],{},[51,31101,23798],{},[95,31103,97],{},[51,31105,19939],{},[95,31107,1540],{},[1004,31109,31110,31112],{},[51,31111,11],{},[1180,31113,31114,31116],{},[51,31115,23798],{},[95,31117,13691],{"mathvariant":494,"lspace":5610,"rspace":5610},[95,31119,15184],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[5612,31121,31122,31128],{},[5615,31123,31124],{},[2071,31125,31126],{"scriptlevel":1977,"displaystyle":62},[48,31127],{},[5615,31129,31130],{},[2071,31131,31132],{"scriptlevel":1977,"displaystyle":62},[48,31133,31134,31136,31138,31144,31146,31148,31150,31152,31158,31164,31166,31168,31170,31172,31174,31176,31178,31180,31182,31184,31186,31192,31194],{},[48,31135],{},[95,31137,223],{},[1004,31139,31140,31142],{},[51,31141,22324],{"mathvariant":14789},[51,31143,23798],{},[95,31145,15175],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[95,31147,211],{"stretchy":210},[51,31149,23798],{},[95,31151,97],{},[1180,31153,31154,31156],{},[51,31155,23798],{},[95,31157,13691],{"mathvariant":494,"lspace":5610,"rspace":5610},[1180,31159,31160,31162],{},[95,31161,220],{"stretchy":210},[51,31163,20800],{"mathvariant":494},[51,31165,919],{},[95,31167,15184],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[95,31169,97],{},[95,31171,211],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[51,31173,8064],{},[95,31175,211],{"stretchy":210},[51,31177,23798],{},[95,31179,220],{"stretchy":210},[95,31181,97],{},[51,31183,8064],{},[95,31185,211],{"stretchy":210},[1180,31187,31188,31190],{},[51,31189,23798],{},[95,31191,13691],{"mathvariant":494,"lspace":5610,"rspace":5610},[95,31193,220],{"stretchy":210},[95,31195,220],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[5612,31197,31198,31204],{},[5615,31199,31200],{},[2071,31201,31202],{"scriptlevel":1977,"displaystyle":62},[48,31203],{},[5615,31205,31206],{},[2071,31207,31208],{"scriptlevel":1977,"displaystyle":62},[48,31209,31210,31212,31214,31216,31218,31220,31226,31232,31234,31236,31238,31240,31242,31244,31246,31248,31250,31252,31254,31256,31258,31264],{},[48,31211],{},[95,31213,223],{},[95,31215,211],{"stretchy":210},[51,31217,23798],{},[95,31219,97],{},[1180,31221,31222,31224],{},[51,31223,23798],{},[95,31225,13691],{"mathvariant":494,"lspace":5610,"rspace":5610},[1180,31227,31228,31230],{},[95,31229,220],{"stretchy":210},[51,31231,20800],{"mathvariant":494},[51,31233,12145],{"mathvariant":494},[51,31235,8064],{},[95,31237,211],{"stretchy":210},[51,31239,23798],{},[95,31241,220],{"stretchy":210},[95,31243,97],{},[51,31245,8064],{},[95,31247,211],{"stretchy":210},[51,31249,23798],{},[95,31251,220],{"stretchy":210},[95,31253,434],{},[51,31255,8064],{},[95,31257,211],{"stretchy":210},[1180,31259,31260,31262],{},[51,31261,23798],{},[95,31263,13691],{"mathvariant":494,"lspace":5610,"rspace":5610},[95,31265,220],{"stretchy":210},[5612,31267,31268,31274],{},[5615,31269,31270],{},[2071,31271,31272],{"scriptlevel":1977,"displaystyle":62},[48,31273],{},[5615,31275,31276],{},[2071,31277,31278],{"scriptlevel":1977,"displaystyle":62},[48,31279,31280,31282,31284,31286,31288,31294,31296,31298,31300,31302,31304,31306,31308,31310,31312,31314,31316,31322,31324,31330,31332,31334,31336],{},[48,31281],{},[95,31283,223],{},[51,31285,8064],{},[95,31287,211],{"stretchy":210},[1180,31289,31290,31292],{},[51,31291,23798],{},[95,31293,13691],{"mathvariant":494,"lspace":5610,"rspace":5610},[95,31295,220],{"stretchy":210},[95,31297,97],{},[51,31299,8064],{},[95,31301,211],{"stretchy":210},[51,31303,23798],{},[95,31305,220],{"stretchy":210},[95,31307,97],{},[51,31309,12145],{"mathvariant":494},[51,31311,8064],{},[95,31313,211],{"stretchy":210},[51,31315,23798],{},[1180,31317,31318,31320],{},[95,31319,220],{"stretchy":210},[51,31321,20800],{"mathvariant":494},[95,31323,211],{"stretchy":210},[1180,31325,31326,31328],{},[51,31327,23798],{},[95,31329,13691],{"mathvariant":494,"lspace":5610,"rspace":5610},[95,31331,97],{},[51,31333,23798],{},[95,31335,220],{"stretchy":210},[51,31337,495],{"mathvariant":494},[55,31339,31340],{"encoding":57},"\\begin{aligned}\n\\mathrm{KL}(p_\\theta\\|p_{\\theta'})\n&=\\mathbb E_\\theta\\bigl[\\log p_\\theta-\\log p_{\\theta'}\\bigr]\\\\\n&=\\mathbb E_\\theta\\bigl[(\\theta-\\theta')^\\top T\\bigr]-\\bigl(\\psi(\\theta)-\\psi(\\theta')\\bigr)\\\\\n&=(\\theta-\\theta')^\\top\\nabla\\psi(\\theta)-\\psi(\\theta)+\\psi(\\theta')\\\\\n&=\\psi(\\theta')-\\psi(\\theta)-\\nabla\\psi(\\theta)^\\top(\\theta'-\\theta).\n\\end{aligned}",[31,31342,31344],{"className":31343,"ariaHidden":62},[61],[31,31345,31347,31351],{"className":31346},[66],[31,31348],{"className":31349,"style":31350},[70],"height:5.8873em;vertical-align:-2.6937em;",[31,31352,31354],{"className":31353},[75],[31,31355,31357,31547],{"className":31356},[5606],[31,31358,31360],{"className":31359},[5867],[31,31361,31363,31538],{"className":31362},[1043,1044],[31,31364,31366,31535],{"className":31365},[1048],[31,31367,31370,31508,31517,31526],{"className":31368,"style":31369},[1052],"height:3.1937em;",[31,31371,31373,31376],{"style":31372},"top:-5.3437em;",[31,31374],{"className":31375,"style":1637},[1060],[31,31377,31379,31385,31388,31428,31431,31505],{"className":31378},[75],[31,31380,31382],{"className":31381},[75],[31,31383,4606],{"className":31384},[75,2745],[31,31386,211],{"className":31387},[264],[31,31389,31391,31394],{"className":31390},[75],[31,31392,11],{"className":31393},[75,76],[31,31395,31397],{"className":31396},[1039],[31,31398,31400,31420],{"className":31399},[1043,1044],[31,31401,31403,31417],{"className":31402},[1048],[31,31404,31406],{"className":31405,"style":1053},[1052],[31,31407,31408,31411],{"style":2807},[31,31409],{"className":31410,"style":1061},[1060],[31,31412,31414],{"className":31413},[1065,1066,1067,1068],[31,31415,23798],{"className":31416,"style":19759},[75,76,1068],[31,31418,1076],{"className":31419},[1075],[31,31421,31423],{"className":31422},[1048],[31,31424,31426],{"className":31425,"style":2050},[1052],[31,31427],{},[31,31429,4572],{"className":31430},[75],[31,31432,31434,31437],{"className":31433},[75],[31,31435,11],{"className":31436},[75,76],[31,31438,31440],{"className":31439},[1039],[31,31441,31443,31497],{"className":31442},[1043,1044],[31,31444,31446,31494],{"className":31445},[1048],[31,31447,31449],{"className":31448,"style":1053},[1052],[31,31450,31451,31454],{"style":2807},[31,31452],{"className":31453,"style":1061},[1060],[31,31455,31457],{"className":31456},[1065,1066,1067,1068],[31,31458,31460],{"className":31459},[75,1068],[31,31461,31463,31466],{"className":31462},[75,1068],[31,31464,23798],{"className":31465,"style":19759},[75,76,1068],[31,31467,31469],{"className":31468},[1039],[31,31470,31472],{"className":31471},[1043],[31,31473,31475],{"className":31474},[1048],[31,31476,31479],{"className":31477,"style":31478},[1052],"height:0.6828em;",[31,31480,31482,31485],{"style":31481},"top:-2.786em;margin-right:0.0714em;",[31,31483],{"className":31484,"style":11769},[1060],[31,31486,31488],{"className":31487},[1065,11773,1308,1068],[31,31489,31491],{"className":31490},[75,1068],[31,31492,13691],{"className":31493},[75,1068],[31,31495,1076],{"className":31496},[1075],[31,31498,31500],{"className":31499},[1048],[31,31501,31503],{"className":31502,"style":2050},[1052],[31,31504],{},[31,31506,220],{"className":31507},[289],[31,31509,31511,31514],{"style":31510},"top:-3.7846em;",[31,31512],{"className":31513,"style":1637},[1060],[31,31515],{"className":31516},[75],[31,31518,31520,31523],{"style":31519},"top:-2.2254em;",[31,31521],{"className":31522,"style":1637},[1060],[31,31524],{"className":31525},[75],[31,31527,31529,31532],{"style":31528},"top:-0.6663em;",[31,31530],{"className":31531,"style":1637},[1060],[31,31533],{"className":31534},[75],[31,31536,1076],{"className":31537},[1075],[31,31539,31541],{"className":31540},[1048],[31,31542,31545],{"className":31543,"style":31544},[1052],"height:2.6937em;",[31,31546],{},[31,31548,31550],{"className":31549},[6062],[31,31551,31553,32372],{"className":31552},[1043,1044],[31,31554,31556,32369],{"className":31555},[1048],[31,31557,31559,31768,32002,32184],{"className":31558,"style":31369},[1052],[31,31560,31561,31564],{"style":31372},[31,31562],{"className":31563,"style":1637},[1060],[31,31565,31567,31570,31573,31576,31579,31619,31625,31630,31633,31673,31676,31679,31682,31687,31690,31762],{"className":31566},[75],[31,31568],{"className":31569},[75],[31,31571],{"className":31572,"style":293},[119],[31,31574,223],{"className":31575},[297],[31,31577],{"className":31578,"style":293},[119],[31,31580,31582,31585],{"className":31581},[75],[31,31583,22324],{"className":31584},[75,14947],[31,31586,31588],{"className":31587},[1039],[31,31589,31591,31611],{"className":31590},[1043,1044],[31,31592,31594,31608],{"className":31593},[1048],[31,31595,31597],{"className":31596,"style":1053},[1052],[31,31598,31599,31602],{"style":2807},[31,31600],{"className":31601,"style":1061},[1060],[31,31603,31605],{"className":31604},[1065,1066,1067,1068],[31,31606,23798],{"className":31607,"style":19759},[75,76,1068],[31,31609,1076],{"className":31610},[1075],[31,31612,31614],{"className":31613},[1048],[31,31615,31617],{"className":31616,"style":2050},[1052],[31,31618],{},[31,31620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",[31,32384,32386,32399],{"className":32385,"translate":35},[34],[31,32387,32389],{"className":32388},[39],[41,32390,32391],{"xmlns":43},[45,32392,32393,32397],{},[48,32394,32395],{},[51,32396,8064],{},[55,32398,29778],{"encoding":57},[31,32400,32402],{"className":32401,"ariaHidden":62},[61],[31,32403,32405,32408],{"className":32404},[66],[31,32406],{"className":32407,"style":4202},[70],[31,32409,8064],{"className":32410,"style":2666},[75,76],[31,32412,32414,32427],{"className":32413,"translate":35},[34],[31,32415,32417],{"className":32416},[39],[41,32418,32419],{"xmlns":43},[45,32420,32421,32425],{},[48,32422,32423],{},[51,32424,23798],{},[55,32426,24747],{"encoding":57},[31,32428,32430],{"className":32429,"ariaHidden":62},[61],[31,32431,32433,32436],{"className":32432},[66],[31,32434],{"className":32435,"style":1630},[70],[31,32437,23798],{"className":32438,"style":19759},[75,76]," 处的切平面在 ",[31,32441,32443,32461],{"className":32442,"translate":35},[34],[31,32444,32446],{"className":32445},[39],[41,32447,32448],{"xmlns":43},[45,32449,32450,32458],{},[48,32451,32452],{},[1180,32453,32454,32456],{},[51,32455,23798],{},[95,32457,13691],{"mathvariant":494,"lspace":5610,"rspace":5610},[55,32459,32460],{"encoding":57},"\\theta'",[31,32462,32464],{"className":32463,"ariaHidden":62},[61],[31,32465,32467,32470],{"className":32466},[66],[31,32468],{"className":32469,"style":13732},[70],[31,32471,32473,32476],{"className":32472},[75],[31,32474,23798],{"className":32475,"style":19759},[75,76],[31,32477,32479],{"className":32478},[1039],[31,32480,32482],{"className":32481},[1043],[31,32483,32485],{"className":32484},[1048],[31,32486,32488],{"className":32487,"style":13732},[1052],[31,32489,32490,32493],{"style":3631},[31,32491],{"className":32492,"style":1061},[1060],[31,32494,32496],{"className":32495},[1065,1066,1067,1068],[31,32497,32499],{"className":32498},[75,1068],[31,32500,13691],{"className":32501},[75,1068]," 处高出的那部分。它有名字：Bregman 散度。",[11,32504,32505,32506,1138,32557,32660],{},"拿方差固定为 1 的高斯族验一下。此时 ",[31,32507,32509,32527],{"className":32508,"translate":35},[34],[31,32510,32512],{"className":32511},[39],[41,32513,32514],{"xmlns":43},[45,32515,32516,32524],{},[48,32517,32518,32520,32522],{},[51,32519,23798],{},[95,32521,223],{},[51,32523,852],{},[55,32525,32526],{"encoding":57},"\\theta=\\mu",[31,32528,32530,32548],{"className":32529,"ariaHidden":62},[61],[31,32531,32533,32536,32539,32542,32545],{"className":32532},[66],[31,32534],{"className":32535,"style":1630},[70],[31,32537,23798],{"className":32538,"style":19759},[75,76],[31,32540],{"className":32541,"style":293},[119],[31,32543,223],{"className":32544},[297],[31,32546],{"className":32547,"style":293},[119],[31,32549,32551,32554],{"className":32550},[66],[31,32552],{"className":32553,"style":865},[70],[31,32555,852],{"className":32556},[75,76],[31,32558,32560,32592],{"className":32559,"translate":35},[34],[31,32561,32563],{"className":32562},[39],[41,32564,32565],{"xmlns":43},[45,32566,32567,32589],{},[48,32568,32569,32571,32573,32575,32577,32579,32585,32587],{},[51,32570,8064],{},[95,32572,211],{"stretchy":210},[51,32574,23798],{},[95,32576,220],{"stretchy":210},[95,32578,223],{},[1180,32580,32581,32583],{},[51,32582,23798],{},[91,32584,152],{},[51,32586,3768],{"mathvariant":494},[91,32588,152],{},[55,32590,32591],{"encoding":57},"\\psi(\\theta)=\\theta^2\u002F2",[31,32593,32595,32622],{"className":32594,"ariaHidden":62},[61],[31,32596,32598,32601,32604,32607,32610,32613,32616,32619],{"className":32597},[66],[31,32599],{"className":32600,"style":260},[70],[31,32602,8064],{"className":32603,"style":2666},[75,76],[31,32605,211],{"className":32606},[264],[31,32608,23798],{"className":32609,"style":19759},[75,76],[31,32611,220],{"className":32612},[289],[31,32614],{"className":32615,"style":293},[119],[31,32617,223],{"className":32618},[297],[31,32620],{"className":32621,"style":293},[119],[31,32623,32625,32628,32657],{"className":32624},[66],[31,32626],{"className":32627,"style":12035},[70],[31,32629,32631,32634],{"className":32630},[75],[31,32632,23798],{"className":32633,"style":19759},[75,76],[31,32635,32637],{"className":32636},[1039],[31,32638,32640],{"className":32639},[1043],[31,32641,32643],{"className":32642},[1048],[31,32644,32646],{"className":32645,"style":5135},[1052],[31,32647,32648,32651],{"style":3631},[31,32649],{"className":32650,"style":1061},[1060],[31,32652,32654],{"className":32653},[1065,1066,1067,1068],[31,32655,152],{"className":32656},[75,1068],[31,32658,15357],{"className":32659},[75],"，手算",[31,32662,32664],{"className":32663,"translate":35},[411],[31,32665,32667,32755],{"className":32666,"translate":35},[34],[31,32668,32670],{"className":32669},[39],[41,32671,32672],{"xmlns":43,"display":420},[45,32673,32674,32752],{},[48,32675,32676,32682,32684,32686,32688,32694,32696,32698,32700,32702,32704,32706,32712,32714,32716,32718,32720,32722,32750],{},[48,32677,32678,32680],{},[51,32679,4558],{"mathvariant":494},[51,32681,4561],{"mathvariant":494},[95,32683,211],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[51,32685,4935],{"mathvariant":925},[95,32687,211],{"stretchy":210},[1004,32689,32690,32692],{},[51,32691,23798],{},[91,32693,205],{},[95,32695,1210],{"separator":62},[91,32697,205],{},[95,32699,220],{"stretchy":210},[95,32701,4572],{"fence":210,"stretchy":62,"minsize":1178,"maxsize":1178},[51,32703,4935],{"mathvariant":925},[95,32705,211],{"stretchy":210},[1004,32707,32708,32710],{},[51,32709,23798],{},[91,32711,152],{},[95,32713,1210],{"separator":62},[91,32715,205],{},[95,32717,220],{"stretchy":210},[95,32719,220],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[95,32721,223],{},[2074,32723,32724,32748],{},[48,32725,32726,32728,32734,32736,32742],{},[95,32727,211],{"stretchy":210},[1004,32729,32730,32732],{},[51,32731,23798],{},[91,32733,205],{},[95,32735,97],{},[1004,32737,32738,32740],{},[51,32739,23798],{},[91,32741,152],{},[1180,32743,32744,32746],{},[95,32745,220],{"stretchy":210},[91,32747,152],{},[91,32749,152],{},[95,32751,1210],{"separator":62},[55,32753,32754],{"encoding":57},"\\mathrm{KL}\\bigl(\\mathcal N(\\theta_1,1)\\big\\|\\mathcal N(\\theta_2,1)\\bigr)=\\frac{(\\theta_1-\\theta_2)^2}{2},",[31,32756,32758,32948],{"className":32757,"ariaHidden":62},[61],[31,32759,32761,32764,32770,32776,32779,32782,32822,32825,32828,32831,32834,32875,32878,32881,32921,32924,32927,32930,32933,32939,32942,32945],{"className":32760},[66],[31,32762],{"className":32763,"style":4988},[70],[31,32765,32767],{"className":32766},[75],[31,32768,4606],{"className":32769},[75,2745],[31,32771,32773],{"className":32772},[264],[31,32774,211],{"className":32775},[1307,1308],[31,32777,4935],{"className":32778,"style":5004},[75,968],[31,32780,211],{"className":32781},[264],[31,32783,32785,32788],{"className":32784},[75],[31,32786,23798],{"className":32787,"style":19759},[75,76],[31,32789,32791],{"className":32790},[1039],[31,32792,32794,32814],{"className":32793},[1043,1044],[31,32795,32797,32811],{"className":32796},[1048],[31,32798,32800],{"className":32799,"style":2028},[1052],[31,32801,32802,32805],{"style":26726},[31,32803],{"className":32804,"style":1061},[1060],[31,32806,32808],{"className":32807},[1065,1066,1067,1068],[31,32809,205],{"className":32810},[75,1068],[31,32812,1076],{"className":32813},[1075],[31,32815,32817],{"className":32816},[1048],[31,32818,32820],{"className":32819,"style":2050},[1052],[31,32821],{},[31,32823,1210],{"className":32824},[1391],[31,32826],{"className":32827,"style":1467},[119],[31,32829,205],{"className":32830},[75],[31,32832,220],{"className":32833},[289],[31,32835,32837],{"className":32836},[75],[31,32838,32840],{"className":32839},[1307,13319],[31,32841,32843,32867],{"className":32842},[1043,1044],[31,32844,32846,32864],{"className":32845},[1048],[31,32847,32849],{"className":32848,"style":13329},[1052],[31,32850,32851,32854],{"style":13332},[31,32852],{"className":32853,"style":13336},[1060],[31,32855,32857],{"style":32856},"width:0.556em;height:1.2em;",[13341,32858,32861],{"xmlns":13343,"width":32859,"height":1178,"viewBox":32860},"0.556em","0 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h43z",[31,32865,1076],{"className":32866},[1075],[31,32868,32870],{"className":32869},[1048],[31,32871,32873],{"className":32872,"style":13359},[1052],[31,32874],{},[31,32876,4935],{"className":32877,"style":5004},[75,968],[31,32879,211],{"className":32880},[264],[31,32882,32884,32887],{"className":32883},[75],[31,32885,23798],{"className":32886,"style":19759},[75,76],[31,32888,32890],{"className":32889},[1039],[31,32891,32893,32913],{"className":32892},[1043,1044],[31,32894,32896,32910],{"className":32895},[1048],[31,32897,32899],{"className":32898,"style":2028},[1052],[31,32900,32901,32904],{"style":26726},[31,32902],{"className":32903,"style":1061},[1060],[31,32905,32907],{"className":32906},[1065,1066,1067,1068],[31,32908,152],{"className":32909},[75,1068],[31,32911,1076],{"className":32912},[1075],[31,32914,32916],{"className":32915},[1048],[31,32917,32919],{"className":32918,"style":2050},[1052],[31,32920],{},[31,32922,1210],{"className":32923},[1391],[31,32925],{"className":32926,"style":1467},[119],[31,32928,205],{"className":32929},[75],[31,32931,220],{"className":32932},[289],[31,32934,32936],{"className":32935},[289],[31,32937,220],{"className":32938},[1307,1308],[31,32940],{"className":32941,"style":293},[119],[31,32943,223],{"className":32944},[297],[31,32946],{"className":32947,"style":293},[119],[31,32949,32951,32954,33134],{"className":32950},[66],[31,32952],{"className":32953,"style":5068},[70],[31,32955,32957,32960,33131],{"className":32956},[75],[31,32958],{"className":32959},[264,2139],[31,32961,32963],{"className":32962},[2074],[31,32964,32966,33123],{"className":32965},[1043,1044],[31,32967,32969,33120],{"className":32968},[1048],[31,32970,32972,32983,32991],{"className":32971,"style":5087},[1052],[31,32973,32974,32977],{"style":5090},[31,32975],{"className":32976,"style":1637},[1060],[31,32978,32980],{"className":32979},[75],[31,32981,152],{"className":32982},[75],[31,32984,32985,32988],{"style":2170},[31,32986],{"className":32987,"style":1637},[1060],[31,32989],{"className":32990,"style":2178},[2177],[31,32992,32993,32996],{"style":5110},[31,32994],{"className":32995,"style":1637},[1060],[31,32997,32999,33002,33042,33045,33048,33051,33091],{"className":32998},[75],[31,33000,211],{"className":33001},[264],[31,33003,33005,33008],{"className":33004},[75],[31,33006,23798],{"className":33007,"style":19759},[75,76],[31,33009,33011],{"className":33010},[1039],[31,33012,33014,33034],{"className":33013},[1043,1044],[31,33015,33017,33031],{"className":33016},[1048],[31,33018,33020],{"className":33019,"style":2028},[1052],[31,33021,33022,33025],{"style":26726},[31,33023],{"className":33024,"style":1061},[1060],[31,33026,33028],{"className":33027},[1065,1066,1067,1068],[31,33029,205],{"className":33030},[75,1068],[31,33032,1076],{"className":33033},[1075],[31,33035,33037],{"className":33036},[1048],[31,33038,33040],{"className":33039,"style":2050},[1052],[31,33041],{},[31,33043],{"className":33044,"style":120},[119],[31,33046,97],{"className":33047},[124],[31,33049],{"className":33050,"style":120},[119],[31,33052,33054,33057],{"className":33053},[75],[31,33055,23798],{"className":33056,"style":19759},[75,76],[31,33058,33060],{"className":33059},[1039],[31,33061,33063,33083],{"className":33062},[1043,1044],[31,33064,33066,33080],{"className":33065},[1048],[31,33067,33069],{"className":33068,"style":2028},[1052],[31,33070,33071,33074],{"style":26726},[31,33072],{"className":33073,"style":1061},[1060],[31,33075,33077],{"className":33076},[1065,1066,1067,1068],[31,33078,152],{"className":33079},[75,1068],[31,33081,1076],{"className":33082},[1075],[31,33084,33086],{"className":33085},[1048],[31,33087,33089],{"className":33088,"style":2050},[1052],[31,33090],{},[31,33092,33094,33097],{"className":33093},[289],[31,33095,220],{"className":33096},[289],[31,33098,33100],{"className":33099},[1039],[31,33101,33103],{"className":33102},[1043],[31,33104,33106],{"className":33105},[1048],[31,33107,33109],{"className":33108,"style":5135},[1052],[31,33110,33111,33114],{"style":3631},[31,33112],{"className":33113,"style":1061},[1060],[31,33115,33117],{"className":33116},[1065,1066,1067,1068],[31,33118,152],{"className":33119},[75,1068],[31,33121,1076],{"className":33122},[1075],[31,33124,33126],{"className":33125},[1048],[31,33127,33129],{"className":33128,"style":5156},[1052],[31,33130],{},[31,33132],{"className":33133},[289,2139],[31,33135,1210],{"className":33136},[1391],[11,33138,33139,33140,33168,33169,33905],{},"而 ",[31,33141,33143,33156],{"className":33142,"translate":35},[34],[31,33144,33146],{"className":33145},[39],[41,33147,33148],{"xmlns":43},[45,33149,33150,33154],{},[48,33151,33152],{},[51,33153,8064],{},[55,33155,29778],{"encoding":57},[31,33157,33159],{"className":33158,"ariaHidden":62},[61],[31,33160,33162,33165],{"className":33161},[66],[31,33163],{"className":33164,"style":4202},[70],[31,33166,8064],{"className":33167,"style":2666},[75,76]," 的 Bregman 散度是 ",[31,33170,33172,33276],{"className":33171,"translate":35},[34],[31,33173,33175],{"className":33174},[39],[41,33176,33177],{"xmlns":43},[45,33178,33179,33273],{},[48,33180,33181,33189,33197,33199,33207,33215,33217,33223,33225,33231,33233,33239,33241,33243,33251,33253,33259,33261,33267],{},[2071,33182,33183],{"scriptlevel":1977,"displaystyle":210},[2074,33184,33185,33187],{},[91,33186,205],{},[91,33188,152],{},[12397,33190,33191,33193,33195],{},[51,33192,23798],{},[91,33194,152],{},[91,33196,152],{},[95,33198,97],{},[2071,33200,33201],{"scriptlevel":1977,"displaystyle":210},[2074,33202,33203,33205],{},[91,33204,205],{},[91,33206,152],{},[12397,33208,33209,33211,33213],{},[51,33210,23798],{},[91,33212,205],{},[91,33214,152],{},[95,33216,97],{},[1004,33218,33219,33221],{},[51,33220,23798],{},[91,33222,205],{},[95,33224,211],{"stretchy":210},[1004,33226,33227,33229],{},[51,33228,23798],{},[91,33230,152],{},[95,33232,97],{},[1004,33234,33235,33237],{},[51,33236,23798],{},[91,33238,205],{},[95,33240,220],{"stretchy":210},[95,33242,223],{},[2071,33244,33245],{"scriptlevel":1977,"displaystyle":210},[2074,33246,33247,33249],{},[91,33248,205],{},[91,33250,152],{},[95,33252,211],{"stretchy":210},[1004,33254,33255,33257],{},[51,33256,23798],{},[91,33258,152],{},[95,33260,97],{},[1004,33262,33263,33265],{},[51,33264,23798],{},[91,33266,205],{},[1180,33268,33269,33271],{},[95,33270,220],{"stretchy":210},[91,33272,152],{},[55,33274,33275],{"encoding":57},"\\tfrac12\\theta_2^2-\\tfrac12\\theta_1^2-\\theta_1(\\theta_2-\\theta_1)=\\tfrac12(\\theta_2-\\theta_1)^2",[31,33277,33279,33414,33548,33646,33704,33830],{"className":33278,"ariaHidden":62},[61],[31,33280,33282,33285,33353,33405,33408,33411],{"className":33281},[66],[31,33283],{"className":33284,"style":2132},[70],[31,33286,33288,33291,33350],{"className":33287},[75],[31,33289],{"className":33290},[264,2139],[31,33292,33294],{"className":33293},[2074],[31,33295,33297,33342],{"className":33296},[1043,1044],[31,33298,33300,33339],{"className":33299},[1048],[31,33301,33303,33317,33325],{"className":33302,"style":2152},[1052],[31,33304,33305,33308],{"style":2155},[31,33306],{"className":33307,"style":1637},[1060],[31,33309,33311],{"className":33310},[1065,1066,1067,1068],[31,33312,33314],{"className":33313},[75,1068],[31,33315,152],{"className":33316},[75,1068],[31,33318,33319,33322],{"style":2170},[31,33320],{"className":33321,"style":1637},[1060],[31,33323],{"className":33324,"style":2178},[2177],[31,33326,33327,33330],{"style":2181},[31,33328],{"className":33329,"style":1637},[1060],[31,33331,33333],{"className":33332},[1065,1066,1067,1068],[31,33334,33336],{"className":33335},[75,1068],[31,33337,205],{"className":33338},[75,1068],[31,33340,1076],{"className":33341},[1075],[31,33343,33345],{"className":33344},[1048],[31,33346,33348],{"className":33347,"style":2203},[1052],[31,33349],{},[31,33351],{"className":33352},[289,2139],[31,33354,33356,33359],{"className":33355},[75],[31,33357,23798],{"className":33358,"style":19759},[75,76],[31,33360,33362],{"className":33361},[1039],[31,33363,33365,33397],{"className":33364},[1043,1044],[31,33366,33368,33394],{"className":33367},[1048],[31,33369,33371,33383],{"className":33370,"style":5135},[1052],[31,33372,33374,33377],{"style":33373},"top:-2.4519em;margin-left:-0.0278em;margin-right:0.05em;",[31,33375],{"className":33376,"style":1061},[1060],[31,33378,33380],{"className":33379},[1065,1066,1067,1068],[31,33381,152],{"className":33382},[75,1068],[31,33384,33385,33388],{"style":3631},[31,33386],{"className":33387,"style":1061},[1060],[31,33389,33391],{"className":33390},[1065,1066,1067,1068],[31,33392,152],{"className":33393},[75,1068],[31,33395,1076],{"className":33396},[1075],[31,33398,33400],{"className":33399},[1048],[31,33401,33403],{"className":33402,"style":17796},[1052],[31,33404],{},[31,33406],{"className":33407,"style":120},[119],[31,33409,97],{"className":33410},[124],[31,33412],{"className":33413,"style":120},[119],[31,33415,33417,33420,33488,33539,33542,33545],{"className":33416},[66],[31,33418],{"className":33419,"style":2132},[70],[31,33421,33423,33426,33485],{"className":33422},[75],[31,33424],{"className":33425},[264,2139],[31,33427,33429],{"className":33428},[2074],[31,33430,33432,33477],{"className":33431},[1043,1044],[31,33433,33435,33474],{"className":33434},[1048],[31,33436,33438,33452,33460],{"className":33437,"style":2152},[1052],[31,33439,33440,33443],{"style":2155},[31,33441],{"className":33442,"style":1637},[1060],[31,33444,33446],{"className":33445},[1065,1066,1067,1068],[31,33447,33449],{"className":33448},[75,1068],[31,33450,152],{"className":33451},[75,1068],[31,33453,33454,33457],{"style":2170},[31,33455],{"className":33456,"style":1637},[1060],[31,33458],{"className":33459,"style":2178},[2177],[31,33461,33462,33465],{"style":2181},[31,33463],{"className":33464,"style":1637},[1060],[31,33466,33468],{"className":33467},[1065,1066,1067,1068],[31,33469,33471],{"className":33470},[75,1068],[31,33472,20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,{"className":33772,"style":2203},[1052],[31,33774],{},[31,33776],{"className":33777},[289,2139],[31,33779,211],{"className":33780},[264],[31,33782,33784,33787],{"className":33783},[75],[31,33785,23798],{"className":33786,"style":19759},[75,76],[31,33788,33790],{"className":33789},[1039],[31,33791,33793,33813],{"className":33792},[1043,1044],[31,33794,33796,33810],{"className":33795},[1048],[31,33797,33799],{"className":33798,"style":2028},[1052],[31,33800,33801,33804],{"style":26726},[31,33802],{"className":33803,"style":1061},[1060],[31,33805,33807],{"className":33806},[1065,1066,1067,1068],[31,33808,152],{"className":33809},[75,1068],[31,33811,1076],{"className":33812},[1075],[31,33814,33816],{"className":33815},[1048],[31,33817,33819],{"className":33818,"style":2050},[1052],[31,33820],{},[31,33822],{"className":33823,"style":120},[119],[31,33825,97],{"className":33826},[124],[31,33828],{"className":33829,"style":120},[119],[31,33831,33833,33836,33876],{"className":33832},[66],[31,33834],{"className":33835,"style":12035},[70],[31,33837,33839,33842],{"className":33838},[75],[31,33840,23798],{"className":33841,"style":19759},[75,76],[31,33843,33845],{"className":33844},[1039],[31,33846,33848,33868],{"className":33847},[1043,1044],[31,33849,33851,33865],{"className":33850},[1048],[31,33852,33854],{"className":33853,"style":2028},[1052],[31,33855,33856,33859],{"style":26726},[31,33857],{"className":33858,"style":1061},[1060],[31,33860,33862],{"className":33861},[1065,1066,1067,1068],[31,33863,205],{"className":33864},[75,1068],[31,33866,1076],{"className":33867},[1075],[31,33869,33871],{"className":33870},[1048],[31,33872,33874],{"className":33873,"style":2050},[1052],[31,33875],{},[31,33877,33879,33882],{"className":33878},[289],[31,33880,220],{"className":33881},[289],[31,33883,33885],{"className":33884},[1039],[31,33886,33888],{"className":33887},[1043],[31,33889,33891],{"className":33890},[1048],[31,33892,33894],{"className":33893,"style":5135},[1052],[31,33895,33896,33899],{"style":3631},[31,33897],{"className":33898,"style":1061},[1060],[31,33900,33902],{"className":33901},[1065,1066,1067,1068],[31,33903,152],{"className":33904},[75,1068],"。同一个数。",[11,33907,33908],{},"所以在这类模型里，KL 不是某种抽象的“信息量”，它就是 Bregman 散度；而它的 Hessian 就是 Fisher 度量。散度和度量是同一个凸函数的两个侧面。这是我写这篇博客最想讲清楚的一件事。",[18,33910,33912],{"id":33911},"bregman","Bregman",[11,33914,33915,33916,33990,33991,34065],{},"先看一个会让人不舒服的例子。",[31,33917,33919,33945],{"className":33918,"translate":35},[34],[31,33920,33922],{"className":33921},[39],[41,33923,33924],{"xmlns":43},[45,33925,33926,33942],{},[48,33927,33928,33930,33932,33934,33936,33938,33940],{},[51,33929,11],{},[95,33931,223],{},[95,33933,211],{"stretchy":210},[91,33935,205],{},[95,33937,1210],{"separator":62},[91,33939,1977],{},[95,33941,220],{"stretchy":210},[55,33943,33944],{"encoding":57},"p=(1,0)",[31,33946,33948,33966],{"className":33947,"ariaHidden":62},[61],[31,33949,33951,33954,33957,33960,33963],{"className":33950},[66],[31,33952],{"className":33953,"style":865},[70],[31,33955,11],{"className":33956},[75,76],[31,33958],{"className":33959,"style":293},[119],[31,33961,223],{"className":33962},[297],[31,33964],{"className":33965,"style":293},[119],[31,33967,33969,33972,33975,33978,33981,33984,33987],{"className":33968},[66],[31,33970],{"className":33971,"style":260},[70],[31,33973,211],{"className":33974},[264],[31,33976,205],{"className":33977},[75],[31,33979,1210],{"className":33980},[1391],[31,33982],{"className":33983,"style":1467},[119],[31,33985,1977],{"className":33986},[75],[31,33988,220],{"className":33989},[289]," 是“真相”，",[31,33992,33994,34020],{"className":33993,"translate":35},[34],[31,33995,33997],{"className":33996},[39],[41,33998,33999],{"xmlns":43},[45,34000,34001,34017],{},[48,34002,34003,34005,34007,34009,34011,34013,34015],{},[51,34004,27666],{},[95,34006,223],{},[95,34008,211],{"stretchy":210},[91,34010,23165],{},[95,34012,1210],{"separator":62},[91,34014,23165],{},[95,34016,220],{"stretchy":210},[55,34018,34019],{"encoding":57},"q=(0.5,0.5)",[31,34021,34023,34041],{"className":34022,"ariaHidden":62},[61],[31,34024,34026,34029,34032,34035,34038],{"className":34025},[66],[31,34027],{"className":34028,"style":865},[70],[31,34030,27666],{"className":34031,"style":2666},[75,76],[31,34033],{"className":34034,"style":293},[119],[31,34036,223],{"className":34037},[297],[31,34039],{"className":34040,"style":293},[119],[31,34042,34044,34047,34050,34053,34056,34059,34062],{"className":34043},[66],[31,34045],{"className":34046,"style":260},[70],[31,34048,211],{"className":34049},[264],[31,34051,23165],{"className":34052},[75],[31,34054,1210],{"className":34055},[1391],[31,34057],{"className":34058,"style":1467},[119],[31,34060,23165],{"className":34061},[75],[31,34063,220],{"className":34064},[289]," 是“模型的预测”。",[31,34067,34069],{"className":34068,"translate":35},[411],[31,34070,34072,34141],{"className":34071,"translate":35},[34],[31,34073,34075],{"className":34074},[39],[41,34076,34077],{"xmlns":43,"display":420},[45,34078,34079,34138],{},[48,34080,34081,34087,34089,34091,34093,34095,34097,34099,34101,34103,34105,34107,34110,34112,34114,34120,34122,34124,34126,34128,34130,34132,34134,34136],{},[48,34082,34083,34085],{},[51,34084,4558],{"mathvariant":494},[51,34086,4561],{"mathvariant":494},[95,34088,211],{"stretchy":210},[51,34090,11],{},[51,34092,4572],{"mathvariant":494},[51,34094,27666],{},[95,34096,220],{"stretchy":210},[95,34098,223],{},[51,34100,19939],{},[95,34102,1540],{},[91,34104,152],{},[95,34106,27223],{},[91,34108,34109],{},"0.693",[95,34111,1210],{"separator":62},[119,34113],{"width":20286},[48,34115,34116,34118],{},[51,34117,4558],{"mathvariant":494},[51,34119,4561],{"mathvariant":494},[95,34121,211],{"stretchy":210},[51,34123,27666],{},[51,34125,4572],{"mathvariant":494},[51,34127,11],{},[95,34129,220],{"stretchy":210},[95,34131,223],{},[95,34133,434],{},[51,34135,4587],{"mathvariant":494},[51,34137,495],{"mathvariant":494},[55,34139,34140],{"encoding":57},"\\mathrm{KL}(p\\|q)=\\log2\\approx0.693,\n\\qquad\n\\mathrm{KL}(q\\|p)=+\\infty .",[31,34142,34144,34180,34206,34254],{"className":34143,"ariaHidden":62},[61],[31,34145,34147,34150,34156,34159,34162,34165,34168,34171,34174,34177],{"className":34146},[66],[31,34148],{"className":34149,"style":260},[70],[31,34151,34153],{"className":34152},[75],[31,34154,4606],{"className":34155},[75,2745],[31,34157,211],{"className":34158},[264],[31,34160,11],{"className":34161},[75,76],[31,34163,4572],{"className":34164},[75],[31,34166,27666],{"className":34167,"style":2666},[75,76],[31,34169,220],{"className":34170},[289],[31,34172],{"className":34173,"style":293},[119],[31,34175,223],{"className":34176},[297],[31,34178],{"className":34179,"style":293},[119],[31,34181,34183,34186,34191,34194,34197,34200,34203],{"className":34182},[66],[31,34184],{"className":34185,"style":4202},[70],[31,34187,20177,34189],{"className":34188},[1460],[31,34190,14556],{"style":20180},[31,34192],{"className":34193,"style":1467},[119],[31,34195,152],{"className":34196},[75],[31,34198],{"className":34199,"style":293},[119],[31,34201,27223],{"className":34202},[297],[31,34204],{"className":34205,"style":293},[119],[31,34207,34209,34212,34215,34218,34221,34224,34230,34233,34236,34239,34242,34245,34248,34251],{"className":34208},[66],[31,34210],{"className":34211,"style":260},[70],[31,34213,34109],{"className":34214},[75],[31,34216,1210],{"className":34217},[1391],[31,34219],{"className":34220,"style":20525},[119],[31,34222],{"className":34223,"style":1467},[119],[31,34225,34227],{"className":34226},[75],[31,34228,4606],{"className":34229},[75,2745],[31,34231,211],{"className":34232},[264],[31,34234,27666],{"className":34235,"style":2666},[75,76],[31,34237,4572],{"className":34238},[75],[31,34240,11],{"className":34241},[75,76],[31,34243,220],{"className":34244},[289],[31,34246],{"className":34247,"style":293},[119],[31,34249,223],{"className":34250},[297],[31,34252],{"className":34253,"style":293},[119],[31,34255,34257,34260,34263],{"className":34256},[66],[31,34258],{"className":34259,"style":307},[70],[31,34261,434],{"className":34262},[75],[31,34264,34266],{"className":34265},[75],"∞.",[11,34268,34269,34270,34298,34299,34327,34328,34356,34357,34385,34386,34356,34414,34442,34443,34471],{},"同一个“差”，从左往右量是 0.69，从右往左量是无穷。原因不难想：",[31,34271,34273,34286],{"className":34272,"translate":35},[34],[31,34274,34276],{"className":34275},[39],[41,34277,34278],{"xmlns":43},[45,34279,34280,34284],{},[48,34281,34282],{},[51,34283,11],{},[55,34285,11],{"encoding":57},[31,34287,34289],{"className":34288,"ariaHidden":62},[61],[31,34290,34292,34295],{"className":34291},[66],[31,34293],{"className":34294,"style":865},[70],[31,34296,11],{"className":34297},[75,76]," 在第二位上放了 0，而 ",[31,34300,34302,34315],{"className":34301,"translate":35},[34],[31,34303,34305],{"className":34304},[39],[41,34306,34307],{"xmlns":43},[45,34308,34309,34313],{},[48,34310,34311],{},[51,34312,27666],{},[55,34314,27666],{"encoding":57},[31,34316,34318],{"className":34317,"ariaHidden":62},[61],[31,34319,34321,34324],{"className":34320},[66],[31,34322],{"className":34323,"style":865},[70],[31,34325,27666],{"className":34326,"style":2666},[75,76]," 在那里放了 0.5，用 ",[31,34329,34331,34344],{"className":34330,"translate":35},[34],[31,34332,34334],{"className":34333},[39],[41,34335,34336],{"xmlns":43},[45,34337,34338,34342],{},[48,34339,34340],{},[51,34341,11],{},[55,34343,11],{"encoding":57},[31,34345,34347],{"className":34346,"ariaHidden":62},[61],[31,34348,34350,34353],{"className":34349},[66],[31,34351],{"className":34352,"style":865},[70],[31,34354,11],{"className":34355},[75,76]," 去看 ",[31,34358,34360,34373],{"className":34359,"translate":35},[34],[31,34361,34363],{"className":34362},[39],[41,34364,34365],{"xmlns":43},[45,34366,34367,34371],{},[48,34368,34369],{},[51,34370,27666],{},[55,34372,27666],{"encoding":57},[31,34374,34376],{"className":34375,"ariaHidden":62},[61],[31,34377,34379,34382],{"className":34378},[66],[31,34380],{"className":34381,"style":865},[70],[31,34383,27666],{"className":34384,"style":2666},[75,76],"，只是多花了 0.69 nat；用 ",[31,34387,34389,34402],{"className":34388,"translate":35},[34],[31,34390,34392],{"className":34391},[39],[41,34393,34394],{"xmlns":43},[45,34395,34396,34400],{},[48,34397,34398],{},[51,34399,27666],{},[55,34401,27666],{"encoding":57},[31,34403,34405],{"className":34404,"ariaHidden":62},[61],[31,34406,34408,34411],{"className":34407},[66],[31,34409],{"className":34410,"style":865},[70],[31,34412,27666],{"className":34413,"style":2666},[75,76],[31,34415,34417,34430],{"className":34416,"translate":35},[34],[31,34418,34420],{"className":34419},[39],[41,34421,34422],{"xmlns":43},[45,34423,34424,34428],{},[48,34425,34426],{},[51,34427,11],{},[55,34429,11],{"encoding":57},[31,34431,34433],{"className":34432,"ariaHidden":62},[61],[31,34434,34436,34439],{"className":34435},[66],[31,34437],{"className":34438,"style":865},[70],[31,34440,11],{"className":34441},[75,76],"，等于说“我给了一个 ",[31,34444,34446,34459],{"className":34445,"translate":35},[34],[31,34447,34449],{"className":34448},[39],[41,34450,34451],{"xmlns":43},[45,34452,34453,34457],{},[48,34454,34455],{},[51,34456,11],{},[55,34458,11],{"encoding":57},[31,34460,34462],{"className":34461,"ariaHidden":62},[61],[31,34463,34465,34468],{"className":34464},[66],[31,34466],{"className":34467,"style":865},[70],[31,34469,11],{"className":34470},[75,76]," 认为绝不可能的东西分配了概率”，这种错误在 KL 里罚到无穷。",[11,34473,34474,34475,34536,34537,34565,34566,34594],{},"这正是交叉熵写成 ",[31,34476,34478,34506],{"className":34477,"translate":35},[34],[31,34479,34481],{"className":34480},[39],[41,34482,34483],{"xmlns":43},[45,34484,34485,34503],{},[48,34486,34487,34493,34495,34497,34499,34501],{},[48,34488,34489,34491],{},[51,34490,4558],{"mathvariant":494},[51,34492,4561],{"mathvariant":494},[95,34494,211],{"stretchy":210},[51,34496,3003],{},[51,34498,4572],{"mathvariant":494},[51,34500,27666],{},[95,34502,220],{"stretchy":210},[55,34504,34505],{"encoding":57},"\\mathrm{KL}(y\\|q)",[31,34507,34509],{"className":34508,"ariaHidden":62},[61],[31,34510,34512,34515,34521,34524,34527,34530,34533],{"className":34511},[66],[31,34513],{"className":34514,"style":260},[70],[31,34516,34518],{"className":34517},[75],[31,34519,4606],{"className":34520},[75,2745],[31,34522,211],{"className":34523},[264],[31,34525,3003],{"className":34526,"style":2666},[75,76],[31,34528,4572],{"className":34529},[75],[31,34531,27666],{"className":34532,"style":2666},[75,76],[31,34534,220],{"className":34535},[289]," 而不是反过来的原因：",[31,34538,34540,34553],{"className":34539,"translate":35},[34],[31,34541,34543],{"className":34542},[39],[41,34544,34545],{"xmlns":43},[45,34546,34547,34551],{},[48,34548,34549],{},[51,34550,3003],{},[55,34552,3003],{"encoding":57},[31,34554,34556],{"className":34555,"ariaHidden":62},[61],[31,34557,34559,34562],{"className":34558},[66],[31,34560],{"className":34561,"style":865},[70],[31,34563,3003],{"className":34564,"style":2666},[75,76]," 是真相，模型 ",[31,34567,34569,34582],{"className":34568,"translate":35},[34],[31,34570,34572],{"className":34571},[39],[41,34573,34574],{"xmlns":43},[45,34575,34576,34580],{},[48,34577,34578],{},[51,34579,27666],{},[55,34581,27666],{"encoding":57},[31,34583,34585],{"className":34584,"ariaHidden":62},[61],[31,34586,34588,34591],{"className":34587},[66],[31,34589],{"className":34590,"style":865},[70],[31,34592,27666],{"className":34593,"style":2666},[75,76]," 不该在真答案上给 0 概率。",[11,34596,34597,34598,34811,34812,34993,34994,35074,35075,35103,35104,2446,35133,35161],{},"再看一个尺度上的例子。Itakura–Saito 散度 ",[31,34599,34601,34663],{"className":34600,"translate":35},[34],[31,34602,34604],{"className":34603},[39],[41,34605,34606],{"xmlns":43},[45,34607,34608,34660],{},[48,34609,34610,34622,34624,34626,34628,34630,34632,34634,34636,34638,34640,34642,34644,34646,34648,34650,34652,34654,34656,34658],{},[1004,34611,34612,34615],{},[51,34613,34614],{},"D",[48,34616,34617,34620],{},[51,34618,34619],{"mathvariant":494},"I",[51,34621,21582],{"mathvariant":494},[95,34623,211],{"stretchy":210},[51,34625,53],{},[51,34627,4572],{"mathvariant":494},[51,34629,3003],{},[95,34631,220],{"stretchy":210},[95,34633,223],{},[51,34635,53],{},[51,34637,3768],{"mathvariant":494},[51,34639,3003],{},[95,34641,97],{},[51,34643,19939],{},[95,34645,1540],{},[95,34647,211],{"stretchy":210},[51,34649,53],{},[51,34651,3768],{"mathvariant":494},[51,34653,3003],{},[95,34655,220],{"stretchy":210},[95,34657,97],{},[91,34659,205],{},[55,34661,34662],{"encoding":57},"D_{\\mathrm{IS}}(x\\|y)=x\u002Fy-\\log(x\u002Fy)-1",[31,34664,34666,34743,34767,34802],{"className":34665,"ariaHidden":62},[61],[31,34667,34669,34672,34719,34722,34725,34728,34731,34734,34737,34740],{"className":34668},[66],[31,34670],{"className":34671,"style":260},[70],[31,34673,34675,34678],{"className":34674},[75],[31,34676,34614],{"className":34677,"style":19759},[75,76],[31,34679,34681],{"className":34680},[1039],[31,34682,34684,34711],{"className":34683},[1043,1044],[31,34685,34687,34708],{"className":34686},[1048],[31,34688,34690],{"className":34689,"style":27736},[1052],[31,34691,34692,34695],{"style":26726},[31,34693],{"className":34694,"style":1061},[1060],[31,34696,34698],{"className":34697},[1065,1066,1067,1068],[31,34699,34701],{"className":34700},[75,1068],[31,34702,34704],{"className":34703},[75,1068],[31,34705,34707],{"className":34706},[75,2745,1068],"IS",[31,34709,1076],{"className":34710},[1075],[31,34712,34714],{"className":34713},[1048],[31,34715,34717],{"className":34716,"style":2050},[1052],[31,34718],{},[31,34720,211],{"className":34721},[264],[31,34723,53],{"className":34724},[75,76],[31,34726,4572],{"className":34727},[75],[31,34729,3003],{"className":34730,"style":2666},[75,76],[31,34732,220],{"className":34733},[289],[31,34735],{"className":34736,"style":293},[119],[31,34738,223],{"className":34739},[297],[31,34741],{"className":34742,"style":293},[119],[31,34744,34746,34749,34752,34755,34758,34761,34764],{"className":34745},[66],[31,34747],{"className":34748,"style":260},[70],[31,34750,53],{"className":34751},[75,76],[31,34753,3768],{"className":34754},[75],[31,34756,3003],{"className":34757,"style":2666},[75,76],[31,34759],{"className":34760,"style":120},[119],[31,34762,97],{"className":34763},[124],[31,34765],{"className":34766,"style":120},[119],[31,34768,34770,34773,34778,34781,34784,34787,34790,34793,34796,34799],{"className":34769},[66],[31,34771],{"className":34772,"style":260},[70],[31,34774,20177,34776],{"className":34775},[1460],[31,34777,14556],{"style":20180},[31,34779,211],{"className":34780},[264],[31,34782,53],{"className":34783},[75,76],[31,34785,3768],{"className":34786},[75],[31,34788,3003],{"className":34789,"style":2666},[75,76],[31,34791,220],{"className":34792},[289],[31,34794],{"className":34795,"style":120},[119],[31,34797,97],{"className":34798},[124],[31,34800],{"className":34801,"style":120},[119],[31,34803,34805,34808],{"className":34804},[66],[31,34806],{"className":34807,"style":326},[70],[31,34809,205],{"className":34810},[75],"：把音量从 1 减半到 0.5，",[31,34813,34815,34858],{"className":34814,"translate":35},[34],[31,34816,34818],{"className":34817},[39],[41,34819,34820],{"xmlns":43},[45,34821,34822,34855],{},[48,34823,34824,34834,34836,34838,34840,34842,34844,34846,34848,34850,34852],{},[1004,34825,34826,34828],{},[51,34827,34614],{},[48,34829,34830,34832],{},[51,34831,34619],{"mathvariant":494},[51,34833,21582],{"mathvariant":494},[95,34835,223],{},[91,34837,152],{},[95,34839,97],{},[51,34841,19939],{},[95,34843,1540],{},[91,34845,152],{},[95,34847,97],{},[91,34849,205],{},[95,34851,223],{},[91,34853,34854],{},"0.307",[55,34856,34857],{"encoding":57},"D_{\\mathrm{IS}}=2-\\log2-1=0.307",[31,34859,34861,34922,34940,34966,34984],{"className":34860,"ariaHidden":62},[61],[31,34862,34864,34867,34913,34916,34919],{"className":34863},[66],[31,34865],{"className":34866,"style":5230},[70],[31,34868,34870,34873],{"className":34869},[75],[31,34871,34614],{"className":34872,"style":19759},[75,76],[31,34874,34876],{"className":34875},[1039],[31,34877,34879,34905],{"className":34878},[1043,1044],[31,34880,34882,34902],{"className":34881},[1048],[31,34883,34885],{"className":34884,"style":27736},[1052],[31,34886,34887,34890],{"style":26726},[31,34888],{"className":34889,"style":1061},[1060],[31,34891,34893],{"className":34892},[1065,1066,1067,1068],[31,34894,34896],{"className":34895},[75,1068],[31,34897,34899],{"className":34898},[75,1068],[31,34900,34707],{"className":34901},[75,2745,1068],[31,34903,1076],{"className":34904},[1075],[31,34906,34908],{"className":34907},[1048],[31,34909,34911],{"className":34910,"style":2050},[1052],[31,34912],{},[31,34914],{"className":34915,"style":293},[119],[31,34917,223],{"className":34918},[297],[31,34920],{"className":34921,"style":293},[119],[31,34923,34925,34928,34931,34934,34937],{"className":34924},[66],[31,34926],{"className":34927,"style":112},[70],[31,34929,152],{"className":34930},[75],[31,34932],{"className":34933,"style":120},[119],[31,34935,97],{"className":34936},[124],[31,34938],{"className":34939,"style":120},[119],[31,34941,34943,34946,34951,34954,34957,34960,34963],{"className":34942},[66],[31,34944],{"className":34945,"style":4202},[70],[31,34947,20177,34949],{"className":34948},[1460],[31,34950,14556],{"style":20180},[31,34952],{"className":34953,"style":1467},[119],[31,34955,152],{"className":34956},[75],[31,34958],{"className":34959,"style":120},[119],[31,34961,97],{"className":34962},[124],[31,34964],{"className":34965,"style":120},[119],[31,34967,34969,34972,34975,34978,34981],{"className":34968},[66],[31,34970],{"className":34971,"style":326},[70],[31,34973,205],{"className":34974},[75],[31,34976],{"className":34977,"style":293},[119],[31,34979,223],{"className":34980},[297],[31,34982],{"className":34983,"style":293},[119],[31,34985,34987,34990],{"className":34986},[66],[31,34988],{"className":34989,"style":326},[70],[31,34991,34854],{"className":34992},[75],"；从 2 减半到 1，",[31,34995,34997,35019],{"className":34996,"translate":35},[34],[31,34998,35000],{"className":34999},[39],[41,35001,35002],{"xmlns":43},[45,35003,35004,35016],{},[48,35005,35006],{},[1004,35007,35008,35010],{},[51,35009,34614],{},[48,35011,35012,35014],{},[51,35013,34619],{"mathvariant":494},[51,35015,21582],{"mathvariant":494},[55,35017,35018],{"encoding":57},"D_{\\mathrm{IS}}",[31,35020,35022],{"className":35021,"ariaHidden":62},[61],[31,35023,35025,35028],{"className":35024},[66],[31,35026],{"className":35027,"style":5230},[70],[31,35029,35031,35034],{"className":35030},[75],[31,35032,34614],{"className":35033,"style":19759},[75,76],[31,35035,35037],{"className":35036},[1039],[31,35038,35040,35066],{"className":35039},[1043,1044],[31,35041,35043,35063],{"className":35042},[1048],[31,35044,35046],{"className":35045,"style":27736},[1052],[31,35047,35048,35051],{"style":26726},[31,35049],{"className":35050,"style":1061},[1060],[31,35052,35054],{"className":35053},[1065,1066,1067,1068],[31,35055,35057],{"className":35056},[75,1068],[31,35058,35060],{"className":35059},[75,1068],[31,35061,34707],{"className":35062},[75,2745,1068],[31,35064,1076],{"className":35065},[1075],[31,35067,35069],{"className":35068},[1048],[31,35070,35072],{"className":35071,"style":2050},[1052],[31,35073],{}," 还是 ",[31,35076,35078,35091],{"className":35077,"translate":35},[34],[31,35079,35081],{"className":35080},[39],[41,35082,35083],{"xmlns":43},[45,35084,35085,35089],{},[48,35086,35087],{},[91,35088,34854],{},[55,35090,34854],{"encoding":57},[31,35092,35094],{"className":35093,"ariaHidden":62},[61],[31,35095,35097,35100],{"className":35096},[66],[31,35098],{"className":35099,"style":326},[70],[31,35101,34854],{"className":35102},[75],"。同样的比例变化，距离一样。而欧氏距离给的是 ",[31,35105,35107,35121],{"className":35106,"translate":35},[34],[31,35108,35110],{"className":35109},[39],[41,35111,35112],{"xmlns":43},[45,35113,35114,35119],{},[48,35115,35116],{},[91,35117,35118],{},"0.25",[55,35120,35118],{"encoding":57},[31,35122,35124],{"className":35123,"ariaHidden":62},[61],[31,35125,35127,35130],{"className":35126},[66],[31,35128],{"className":35129,"style":326},[70],[31,35131,35118],{"className":35132},[75],[31,35134,35136,35149],{"className":35135,"translate":35},[34],[31,35137,35139],{"className":35138},[39],[41,35140,35141],{"xmlns":43},[45,35142,35143,35147],{},[48,35144,35145],{},[91,35146,205],{},[55,35148,205],{"encoding":57},[31,35150,35152],{"className":35151,"ariaHidden":62},[61],[31,35153,35155,35158],{"className":35154},[66],[31,35156],{"className":35157,"style":326},[70],[31,35159,205],{"className":35160},[75],"，差了四倍——因为欧氏觉得“从 2 到 1 变化更大”，可听感上这明明是同一件事。",[11,35163,35164],{},"这两个例子指向同一件事：距离可以不对称，也可以不按绝对差来量。",[11,35166,35167,35168,35197,35198,1138],{},"一般地，给定凸函数 ",[31,35169,35171,35184],{"className":35170,"translate":35},[34],[31,35172,35174],{"className":35173},[39],[41,35175,35176],{"xmlns":43},[45,35177,35178,35182],{},[48,35179,35180],{},[51,35181,1662],{},[55,35183,1662],{"encoding":57},[31,35185,35187],{"className":35186,"ariaHidden":62},[61],[31,35188,35190,35193],{"className":35189},[66],[31,35191],{"className":35192,"style":4202},[70],[31,35194,1662],{"className":35195,"style":35196},[75,76],"margin-right:0.1076em;"," 和可微点 ",[31,35199,35201,35214],{"className":35200,"translate":35},[34],[31,35202,35204],{"className":35203},[39],[41,35205,35206],{"xmlns":43},[45,35207,35208,35212],{},[48,35209,35210],{},[51,35211,3003],{},[55,35213,3003],{"encoding":57},[31,35215,35217],{"className":35216,"ariaHidden":62},[61],[31,35218,35220,35223],{"className":35219},[66],[31,35221],{"className":35222,"style":865},[70],[31,35224,3003],{"className":35225,"style":2666},[75,76],[31,35227,35229],{"className":35228,"translate":35},[411],[31,35230,35232,35308],{"className":35231,"translate":35},[34],[31,35233,35235],{"className":35234},[39],[41,35236,35237],{"xmlns":43,"display":420},[45,35238,35239,35305],{},[48,35240,35241,35247,35249,35251,35253,35255,35257,35259,35261,35263,35265,35267,35269,35271,35273,35275,35277,35279,35281,35283,35285,35287,35293,35295,35297,35299,35301,35303],{},[1004,35242,35243,35245],{},[51,35244,34614],{},[51,35246,1662],{},[95,35248,211],{"stretchy":210},[51,35250,53],{},[51,35252,4572],{"mathvariant":494},[51,35254,3003],{},[95,35256,220],{"stretchy":210},[95,35258,223],{},[51,35260,1662],{},[95,35262,211],{"stretchy":210},[51,35264,53],{},[95,35266,220],{"stretchy":210},[95,35268,97],{},[51,35270,1662],{},[95,35272,211],{"stretchy":210},[51,35274,3003],{},[95,35276,220],{"stretchy":210},[95,35278,97],{},[51,35280,12145],{"mathvariant":494},[51,35282,1662],{},[95,35284,211],{"stretchy":210},[51,35286,3003],{},[1180,35288,35289,35291],{},[95,35290,220],{"stretchy":210},[51,35292,20800],{"mathvariant":494},[95,35294,211],{"stretchy":210},[51,35296,53],{},[95,35298,97],{},[51,35300,3003],{},[95,35302,220],{"stretchy":210},[51,35304,495],{"mathvariant":494},[55,35306,35307],{"encoding":57},"D_f(x\\|y)=f(x)-f(y)-\\nabla f(y)^\\top(x-y).",[31,35309,35311,35381,35408,35435,35498],{"className":35310,"ariaHidden":62},[61],[31,35312,35314,35317,35357,35360,35363,35366,35369,35372,35375,35378],{"className":35313},[66],[31,35315],{"className":35316,"style":1223},[70],[31,35318,35320,35323],{"className":35319},[75],[31,35321,34614],{"className":35322,"style":19759},[75,76],[31,35324,35326],{"className":35325},[1039],[31,35327,35329,35349],{"className":35328},[1043,1044],[31,35330,35332,35346],{"className":35331},[1048],[31,35333,35335],{"className":35334,"style":1053},[1052],[31,35336,35337,35340],{"style":26726},[31,35338],{"className":35339,"style":1061},[1060],[31,35341,35343],{"className":35342},[1065,1066,1067,1068],[31,35344,1662],{"className":35345,"style":35196},[75,76,1068],[31,35347,1076],{"className":35348},[1075],[31,35350,35352],{"className":35351},[1048],[31,35353,35355],{"className":35354,"style":1083},[1052],[31,35356],{},[31,35358,211],{"className":35359},[264],[31,35361,53],{"className":35362},[75,76],[31,35364,4572],{"className":35365},[75],[31,35367,3003],{"className":35368,"style":2666},[75,76],[31,35370,220],{"className":35371},[289],[31,35373],{"className":35374,"style":293},[119],[31,35376,223],{"className":35377},[297],[31,35379],{"className":35380,"style":293},[119],[31,35382,35384,35387,35390,35393,35396,35399,35402,35405],{"className":35383},[66],[31,35385],{"className":35386,"style":260},[70],[31,35388,1662],{"className":35389,"style":35196},[75,76],[31,35391,211],{"className":35392},[264],[31,35394,53],{"className":35395},[75,76],[31,35397,220],{"className":35398},[289],[31,35400],{"className":35401,"style":120},[119],[31,35403,97],{"className":35404},[124],[31,35406],{"className":35407,"style":120},[119],[31,35409,35411,35414,35417,35420,35423,35426,35429,35432],{"className":35410},[66],[31,35412],{"className":35413,"style":260},[70],[31,35415,1662],{"className":35416,"style":35196},[75,76],[31,35418,211],{"className":35419},[264],[31,35421,3003],{"className":35422,"style":2666},[75,76],[31,35424,220],{"className":35425},[289],[31,35427],{"className":35428,"style":120},[119],[31,35430,97],{"className":35431},[124],[31,35433],{"className":35434,"style":120},[119],[31,35436,35438,35442,35445,35448,35451,35454,35483,35486,35489,35492,35495],{"className":35437},[66],[31,35439],{"className":35440,"style":35441},[70],"height:1.1491em;vertical-align:-0.25em;",[31,35443,12145],{"className":35444},[75],[31,35446,1662],{"className":35447,"style":35196},[75,76],[31,35449,211],{"className":35450},[264],[31,35452,3003],{"className":35453,"style":2666},[75,76],[31,35455,35457,35460],{"className":35456},[289],[31,35458,220],{"className":35459},[289],[31,35461,35463],{"className":35462},[1039],[31,35464,35466],{"className":35465},[1043],[31,35467,35469],{"className":35468},[1048],[31,35470,35472],{"className":35471,"style":29642},[1052],[31,35473,35474,35477],{"style":1330},[31,35475],{"className":35476,"style":1061},[1060],[31,35478,35480],{"className":35479},[1065,1066,1067,1068],[31,35481,20800],{"className":35482},[75,1068],[31,35484,211],{"className":35485},[264],[31,35487,53],{"className":35488},[75,76],[31,35490],{"className":35491,"style":120},[119],[31,35493,97],{"className":35494},[124],[31,35496],{"className":35497,"style":120},[119],[31,35499,35501,35504,35507,35510],{"className":35500},[66],[31,35502],{"className":35503,"style":260},[70],[31,35505,3003],{"className":35506,"style":2666},[75,76],[31,35508,220],{"className":35509},[289],[31,35511,495],{"className":35512},[75],[11,35514,35515,35516,35544,35545,35573,35574,35602,35603,35631,35632,35683],{},"几何上，这是 ",[31,35517,35519,35532],{"className":35518,"translate":35},[34],[31,35520,35522],{"className":35521},[39],[41,35523,35524],{"xmlns":43},[45,35525,35526,35530],{},[48,35527,35528],{},[51,35529,1662],{},[55,35531,1662],{"encoding":57},[31,35533,35535],{"className":35534,"ariaHidden":62},[61],[31,35536,35538,35541],{"className":35537},[66],[31,35539],{"className":35540,"style":4202},[70],[31,35542,1662],{"className":35543,"style":35196},[75,76]," 的图像在 ",[31,35546,35548,35561],{"className":35547,"translate":35},[34],[31,35549,35551],{"className":35550},[39],[41,35552,35553],{"xmlns":43},[45,35554,35555,35559],{},[48,35556,35557],{},[51,35558,53],{},[55,35560,53],{"encoding":57},[31,35562,35564],{"className":35563,"ariaHidden":62},[61],[31,35565,35567,35570],{"className":35566},[66],[31,35568],{"className":35569,"style":71},[70],[31,35571,53],{"className":35572},[75,76]," 处比 ",[31,35575,35577,35590],{"className":35576,"translate":35},[34],[31,35578,35580],{"className":35579},[39],[41,35581,35582],{"xmlns":43},[45,35583,35584,35588],{},[48,35585,35586],{},[51,35587,3003],{},[55,35589,3003],{"encoding":57},[31,35591,35593],{"className":35592,"ariaHidden":62},[61],[31,35594,35596,35599],{"className":35595},[66],[31,35597],{"className":35598,"style":865},[70],[31,35600,3003],{"className":35601,"style":2666},[75,76]," 处的切平面高出多少。",[31,35604,35606,35619],{"className":35605,"translate":35},[34],[31,35607,35609],{"className":35608},[39],[41,35610,35611],{"xmlns":43},[45,35612,35613,35617],{},[48,35614,35615],{},[51,35616,1662],{},[55,35618,1662],{"encoding":57},[31,35620,35622],{"className":35621,"ariaHidden":62},[61],[31,35623,35625,35628],{"className":35624},[66],[31,35626],{"className":35627,"style":4202},[70],[31,35629,1662],{"className":35630,"style":35196},[75,76]," 凸所以非负，严格凸所以只在 ",[31,35633,35635,35653],{"className":35634,"translate":35},[34],[31,35636,35638],{"className":35637},[39],[41,35639,35640],{"xmlns":43},[45,35641,35642,35650],{},[48,35643,35644,35646,35648],{},[51,35645,53],{},[95,35647,223],{},[51,35649,3003],{},[55,35651,35652],{"encoding":57},"x=y",[31,35654,35656,35674],{"className":35655,"ariaHidden":62},[61],[31,35657,35659,35662,35665,35668,35671],{"className":35658},[66],[31,35660],{"className":35661,"style":71},[70],[31,35663,53],{"className":35664},[75,76],[31,35666],{"className":35667,"style":293},[119],[31,35669,223],{"className":35670},[297],[31,35672],{"className":35673,"style":293},[119],[31,35675,35677,35680],{"className":35676},[66],[31,35678],{"className":35679,"style":865},[70],[31,35681,3003],{"className":35682,"style":2666},[75,76]," 时为零。它不对称，也不满足三角不等式，所以我不叫它距离，叫散度。",[11,35685,35686,35687,35869,35870,36037,36038,36255,36256,36037,36317,36493,36494,36677],{},"几个我常用的：",[31,35688,35690,35730],{"className":35689,"translate":35},[34],[31,35691,35693],{"className":35692},[39],[41,35694,35695],{"xmlns":43},[45,35696,35697,35727],{},[48,35698,35699,35701,35703,35705,35707,35709,35717,35719,35721],{},[51,35700,1662],{},[95,35702,211],{"stretchy":210},[51,35704,53],{},[95,35706,220],{"stretchy":210},[95,35708,223],{},[2071,35710,35711],{"scriptlevel":1977,"displaystyle":210},[2074,35712,35713,35715],{},[91,35714,205],{},[91,35716,152],{},[51,35718,4572],{"mathvariant":494},[51,35720,53],{},[1180,35722,35723,35725],{},[51,35724,4572],{"mathvariant":494},[91,35726,152],{},[55,35728,35729],{"encoding":57},"f(x)=\\tfrac12\\|x\\|^2",[31,35731,35733,35760],{"className":35732,"ariaHidden":62},[61],[31,35734,35736,35739,35742,35745,35748,35751,35754,35757],{"className":35735},[66],[31,35737],{"className":35738,"style":260},[70],[31,35740,1662],{"className":35741,"style":35196},[75,76],[31,35743,211],{"className":35744},[264],[31,35746,53],{"className":35747},[75,76],[31,35749,220],{"className":35750},[289],[31,35752],{"className":35753,"style":293},[119],[31,35755,223],{"className":35756},[297],[31,35758],{"className":35759,"style":293},[119],[31,35761,35763,35766,35834,35837,35840],{"className":35762},[66],[31,35764],{"className":35765,"style":2132},[70],[31,35767,35769,35772,35831],{"className":35768},[75],[31,35770],{"className":35771},[264,2139],[31,35773,35775],{"className":35774},[2074],[31,35776,35778,35823],{"className":35777},[1043,1044],[31,35779,35781,35820],{"className":35780},[1048],[31,35782,35784,35798,35806],{"className":35783,"style":2152},[1052],[31,35785,35786,35789],{"style":2155},[31,35787],{"className":35788,"style":1637},[1060],[31,35790,35792],{"className":35791},[1065,1066,1067,1068],[31,35793,35795],{"className":35794},[75,1068],[31,35796,152],{"className":35797},[75,1068],[31,35799,35800,35803],{"style":2170},[31,35801],{"className":35802,"style":1637},[1060],[31,35804],{"className":35805,"style":2178},[2177],[31,35807,35808,35811],{"style":2181},[31,35809],{"className":35810,"style":1637},[1060],[31,35812,35814],{"className":35813},[1065,1066,1067,1068],[31,35815,35817],{"className":35816},[75,1068],[31,35818,205],{"className":35819},[75,1068],[31,35821,1076],{"className":35822},[1075],[31,35824,35826],{"className":35825},[1048],[31,35827,35829],{"className":35828,"style":2203},[1052],[31,35830],{},[31,35832],{"className":35833},[289,2139],[31,35835,4572],{"className":35836},[75],[31,35838,53],{"className":35839},[75,76],[31,35841,35843,35846],{"className":35842},[75],[31,35844,4572],{"className":35845},[75],[31,35847,35849],{"className":35848},[1039],[31,35850,35852],{"className":35851},[1043],[31,35853,35855],{"className":35854},[1048],[31,35856,35858],{"className":35857,"style":5135},[1052],[31,35859,35860,35863],{"style":3631},[31,35861],{"className":35862,"style":1061},[1060],[31,35864,35866],{"className":35865},[1065,1066,1067,1068],[31,35867,152],{"className":35868},[75,1068]," 给 ",[31,35871,35873,35907],{"className":35872,"translate":35},[34],[31,35874,35876],{"className":35875},[39],[41,35877,35878],{"xmlns":43},[45,35879,35880,35904],{},[48,35881,35882,35890,35892,35894,35896,35898],{},[2071,35883,35884],{"scriptlevel":1977,"displaystyle":210},[2074,35885,35886,35888],{},[91,35887,205],{},[91,35889,152],{},[51,35891,4572],{"mathvariant":494},[51,35893,53],{},[95,35895,97],{},[51,35897,3003],{},[1180,35899,35900,35902],{},[51,35901,4572],{"mathvariant":494},[91,35903,152],{},[55,35905,35906],{"encoding":57},"\\tfrac12\\|x-y\\|^2",[31,35908,35910,35999],{"className":35909,"ariaHidden":62},[61],[31,35911,35913,35916,35984,35987,35990,35993,35996],{"className":35912},[66],[31,35914],{"className":35915,"style":2132},[70],[31,35917,35919,35922,35981],{"className":35918},[75],[31,35920],{"className":35921},[264,2139],[31,35923,35925],{"className":35924},[2074],[31,35926,35928,35973],{"className":35927},[1043,1044],[31,35929,35931,35970],{"className":35930},[1048],[31,35932,35934,35948,35956],{"className":35933,"style":2152},[1052],[31,35935,35936,35939],{"style":2155},[31,35937],{"className":35938,"style":1637},[1060],[31,35940,35942],{"className":35941},[1065,1066,1067,1068],[31,35943,35945],{"className":35944},[75,1068],[31,35946,152],{"className":35947},[75,1068],[31,35949,35950,35953],{"style":2170},[31,35951],{"className":35952,"style":1637},[1060],[31,35954],{"className":35955,"style":2178},[2177],[31,35957,35958,35961],{"style":2181},[31,35959],{"className":35960,"style":1637},[1060],[31,35962,35964],{"className":35963},[1065,1066,1067,1068],[31,35965,35967],{"className":35966},[75,1068],[31,35968,205],{"className":35969},[75,1068],[31,35971,1076],{"className":35972},[1075],[31,35974,35976],{"className":35975},[1048],[31,35977,35979],{"className":35978,"style":2203},[1052],[31,35980],{},[31,35982],{"className":35983},[289,2139],[31,35985,4572],{"className":35986},[75],[31,35988,53],{"className":35989},[75,76],[31,35991],{"className":35992,"style":120},[119],[31,35994,97],{"className":35995},[124],[31,35997],{"className":35998,"style":120},[119],[31,36000,36002,36005,36008],{"className":36001},[66],[31,36003],{"className":36004,"style":12035},[70],[31,36006,3003],{"className":36007,"style":2666},[75,76],[31,36009,36011,36014],{"className":36010},[75],[31,36012,4572],{"className":36013},[75],[31,36015,36017],{"className":36016},[1039],[31,36018,36020],{"className":36019},[1043],[31,36021,36023],{"className":36022},[1048],[31,36024,36026],{"className":36025,"style":5135},[1052],[31,36027,36028,36031],{"style":3631},[31,36029],{"className":36030,"style":1061},[1060],[31,36032,36034],{"className":36033},[1065,1066,1067,1068],[31,36035,152],{"className":36036},[75,1068],"；",[31,36039,36041,36085],{"className":36040,"translate":35},[34],[31,36042,36044],{"className":36043},[39],[41,36045,36046],{"xmlns":43},[45,36047,36048,36082],{},[48,36049,36050,36052,36054,36056,36058,36060,36066,36072,36074,36076],{},[51,36051,1662],{},[95,36053,211],{"stretchy":210},[51,36055,11],{},[95,36057,220],{"stretchy":210},[95,36059,223],{},[1004,36061,36062,36064],{},[95,36063,5625],{},[51,36065,5391],{},[1004,36067,36068,36070],{},[51,36069,11],{},[51,36071,5391],{},[51,36073,19939],{},[95,36075,1540],{},[1004,36077,36078,36080],{},[51,36079,11],{},[51,36081,5391],{},[55,36083,36084],{"encoding":57},"f(p)=\\sum_ip_i\\log p_i",[31,36086,36088,36115],{"className":36087,"ariaHidden":62},[61],[31,36089,36091,36094,36097,36100,36103,36106,36109,36112],{"className":36090},[66],[31,36092],{"className":36093,"style":260},[70],[31,36095,1662],{"className":36096,"style":35196},[75,76],[31,36098,211],{"className":36099},[264],[31,36101,11],{"className":36102},[75,76],[31,36104,220],{"className":36105},[289],[31,36107],{"className":36108,"style":293},[119],[31,36110,223],{"className":36111},[297],[31,36113],{"className":36114,"style":293},[119],[31,36116,36118,36121,36161,36164,36204,36207,36212,36215],{"className":36117},[66],[31,36119],{"className":36120,"style":21359},[70],[31,36122,36124,36127],{"className":36123},[1460],[31,36125,5625],{"className":36126,"style":21372},[1460,1461,1462],[31,36128,36130],{"className":36129},[1039],[31,36131,36133,36153],{"className":36132},[1043,1044],[31,36134,36136,36150],{"className":36135},[1048],[31,36137,36139],{"className":36138,"style":21385},[1052],[31,36140,36141,36144],{"style":21388},[31,36142],{"className":36143,"style":1061},[1060],[31,36145,36147],{"className":36146},[1065,1066,1067,1068],[31,36148,5391],{"className":36149},[75,76,1068],[31,36151,1076],{"className":36152},[1075],[31,36154,36156],{"className":36155},[1048],[31,36157,36159],{"className":36158,"style":21407},[1052],[31,36160],{},[31,36162],{"className":36163,"style":1467},[119],[31,36165,36167,36170],{"className":36166},[75],[31,36168,11],{"className":36169},[75,76],[31,36171,36173],{"className":36172},[1039],[31,36174,36176,36196],{"className":36175},[1043,1044],[31,36177,36179,36193],{"className":36178},[1048],[31,36180,36182],{"className":36181,"style":5445},[1052],[31,36183,36184,36187],{"style":2807},[31,36185],{"className":36186,"style":1061},[1060],[31,36188,36190],{"className":36189},[1065,1066,1067,1068],[31,36191,5391],{"className":36192},[75,76,1068],[31,36194,1076],{"className":36195},[1075],[31,36197,36199],{"className":36198},[1048],[31,36200,36202],{"className":36201,"style":2050},[1052],[31,36203],{},[31,36205],{"className":36206,"style":1467},[119],[31,36208,20177,36210],{"className":36209},[1460],[31,36211,14556],{"style":20180},[31,36213],{"className":36214,"style":1467},[119],[31,36216,36218,36221],{"className":36217},[75],[31,36219,11],{"className":36220},[75,76],[31,36222,36224],{"className":36223},[1039],[31,36225,36227,36247],{"className":36226},[1043,1044],[31,36228,36230,36244],{"className":36229},[1048],[31,36231,36233],{"className":36232,"style":5445},[1052],[31,36234,36235,36238],{"style":2807},[31,36236],{"className":36237,"style":1061},[1060],[31,36239,36241],{"className":36240},[1065,1066,1067,1068],[31,36242,5391],{"className":36243},[75,76,1068],[31,36245,1076],{"className":36246},[1075],[31,36248,36250],{"className":36249},[1048],[31,36251,36253],{"className":36252,"style":2050},[1052],[31,36254],{},"（负熵）给 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给 Itakura–Saito；",[31,36495,36497,36537],{"className":36496,"translate":35},[34],[31,36498,36500],{"className":36499},[39],[41,36501,36502],{"xmlns":43},[45,36503,36504,36534],{},[48,36505,36506,36508,36510,36512,36514,36516,36524,36530,36532],{},[51,36507,1662],{},[95,36509,211],{"stretchy":210},[51,36511,53],{},[95,36513,220],{"stretchy":210},[95,36515,223],{},[2071,36517,36518],{"scriptlevel":1977,"displaystyle":210},[2074,36519,36520,36522],{},[91,36521,205],{},[91,36523,152],{},[1180,36525,36526,36528],{},[51,36527,53],{},[51,36529,20800],{"mathvariant":494},[51,36531,18446],{},[51,36533,53],{},[55,36535,36536],{"encoding":57},"f(x)=\\tfrac12x^\\top 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给 Mahalanobis 距离。",[11,36679,36680],{},"它在机器学习里出现的频率比多数人以为的高。交叉熵",[31,36682,36684],{"className":36683,"translate":35},[411],[31,36685,36687,36804],{"className":36686,"translate":35},[34],[31,36688,36690],{"className":36689},[39],[41,36691,36692],{"xmlns":43,"display":420},[45,36693,36694,36801],{},[48,36695,36696,36702,36704,36706,36708,36710,36712,36714,36716,36722,36728,36730,36732,36738,36740,36746,36748,36750,36752,36754,36756,36758,36761,36763,36765,36767,36769,36779,36781,36783,36785,36787,36789,36791,36793,36795,36797,36799],{},[48,36697,36698,36700],{},[51,36699,8101],{"mathvariant":494},[51,36701,22324],{"mathvariant":494},[95,36703,211],{"stretchy":210},[51,36705,3003],{},[95,36707,1210],{"separator":62},[51,36709,27666],{},[95,36711,220],{"stretchy":210},[95,36713,223],{},[95,36715,97],{},[1531,36717,36718,36720],{},[95,36719,5625],{},[51,36721,5391],{},[1004,36723,36724,36726],{},[51,36725,3003],{},[51,36727,5391],{},[51,36729,19939],{},[95,36731,1540],{},[1004,36733,36734,36736],{},[51,36735,27666],{},[51,36737,5391],{},[95,36739,223],{},[48,36741,36742,36744],{},[51,36743,4558],{"mathvariant":494},[51,36745,4561],{"mathvariant":494},[95,36747,211],{"stretchy":210},[51,36749,3003],{},[51,36751,4572],{"mathvariant":494},[51,36753,27666],{},[95,36755,220],{"stretchy":210},[95,36757,434],{},[51,36759,36760],{},"H",[95,36762,211],{"stretchy":210},[51,36764,3003],{},[95,36766,220],{"stretchy":210},[95,36768,223],{},[1004,36770,36771,36773],{},[51,36772,34614],{},[48,36774,36775,36777],{},[95,36776,97],{},[51,36778,36760],{},[95,36780,211],{"stretchy":210},[51,36782,3003],{},[51,36784,4572],{"mathvariant":494},[51,36786,27666],{},[95,36788,220],{"stretchy":210},[95,36790,434],{},[51,36792,36760],{},[95,36794,211],{"stretchy":210},[51,36796,3003],{},[95,36798,220],{"stretchy":210},[95,36800,1210],{"separator":62},[55,36802,36803],{"encoding":57},"\\mathrm{CE}(y,q)=-\\sum_iy_i\\log q_i=\\mathrm{KL}(y\\|q)+H(y)=D_{-H}(y\\|q)+H(y),",[31,36805,36807,36847,37006,37042,37070,37146],{"className":36806,"ariaHidden":62},[61],[31,36808,36810,36813,36820,36823,36826,36829,36832,36835,36838,36841,36844],{"className":36809},[66],[31,36811],{"className":36812,"style":260},[70],[31,36814,36816],{"className":36815},[75],[31,36817,36819],{"className":36818},[75,2745],"CE",[31,36821,211],{"className":36822},[264],[31,36824,3003],{"className":36825,"style":2666},[75,76],[31,36827,1210],{"className":36828},[1391],[31,36830],{"className":36831,"style":1467},[119],[31,36833,27666],{"className":36834,"style":2666},[75,76],[31,36836,220],{"className":36837},[289],[31,36839],{"className":36840,"style":293},[119],[31,36842,223],{"className":36843},[297],[31,36845],{"className":36846,"style":293},[119],[31,36848,36850,36853,36856,36859,36903,36906,36946,36949,36954,36957,36997,37000,37003],{"className":36849},[66],[31,36851],{"className":36852,"style":10251},[70],[31,36854,97],{"className":36855},[75],[31,36857],{"className":36858,"style":1467},[119],[31,36860,36862],{"className":36861},[1460,1620],[31,36863,36865,36895],{"className":36864},[1043,1044],[31,36866,36868,36892],{"className":36867},[1048],[31,36869,36871,36882],{"className":36870,"style":5900},[1052],[31,36872,36873,36876],{"style":5903},[31,36874],{"className":36875,"style":5907},[1060],[31,36877,36879],{"className":36878},[1065,1066,1067,1068],[31,36880,5391],{"className":36881},[75,76,1068],[31,36883,36884,36887],{"style":5919},[31,36885],{"className":36886,"style":5907},[1060],[31,36888,36889],{},[31,36890,5625],{"className":36891},[1460,1461,1687],[31,36893,1076],{"className":36894},[1075],[31,36896,36898],{"className":36897},[1048],[31,36899,36901],{"className":36900,"style":6342},[1052],[31,36902],{},[31,36904],{"className":36905,"style":1467},[119],[31,36907,36909,36912],{"className":36908},[75],[31,36910,3003],{"className":36911,"style":2666},[75,76],[31,36913,36915],{"className":36914},[1039],[31,36916,36918,36938],{"className":36917},[1043,1044],[31,36919,36921,36935],{"className":36920},[1048],[31,36922,36924],{"className":36923,"style":5445},[1052],[31,36925,36926,36929],{"style":5504},[31,36927],{"className":36928,"style":1061},[1060],[31,36930,36932],{"className":36931},[1065,1066,1067,1068],[31,36933,5391],{"className":36934},[75,76,1068],[31,36936,1076],{"className":36937},[1075],[31,36939,36941],{"className":36940},[1048],[31,36942,36944],{"className":36943,"style":2050},[1052],[31,36945],{},[31,36947],{"className":36948,"style":1467},[119],[31,36950,20177,36952],{"className":36951},[1460],[31,36953,14556],{"style":20180},[31,36955],{"className":36956,"style":1467},[119],[31,36958,36960,36963],{"className":36959},[75],[31,36961,27666],{"className":36962,"style":2666},[75,76],[31,36964,36966],{"className":36965},[1039],[31,36967,36969,36989],{"className":36968},[1043,1044],[31,36970,36972,36986],{"className":36971},[1048],[31,36973,36975],{"className":36974,"style":5445},[1052],[31,36976,36977,36980],{"style":5504},[31,36978],{"className":36979,"style":1061},[1060],[31,36981,36983],{"className":36982},[1065,1066,1067,1068],[31,36984,5391],{"className":36985},[75,76,1068],[31,36987,1076],{"className":36988},[1075],[31,36990,36992],{"className":36991},[1048],[31,36993,36995],{"className":36994,"style":2050},[1052],[31,36996],{},[31,36998],{"className":36999,"style":293},[119],[31,37001,223],{"className":37002},[297],[31,37004],{"className":37005,"style":293},[119],[31,37007,37009,37012,37018,37021,37024,37027,37030,37033,37036,37039],{"className":37008},[66],[31,37010],{"className":37011,"style":260},[70],[31,37013,37015],{"className":37014},[75],[31,37016,4606],{"className":37017},[75,2745],[31,37019,211],{"className":37020},[264],[31,37022,3003],{"className":37023,"style":2666},[75,76],[31,37025,4572],{"className":37026},[75],[31,37028,27666],{"className":37029,"style":2666},[75,76],[31,37031,220],{"className":37032},[289],[31,37034],{"className":37035,"style":120},[119],[31,37037,434],{"className":37038},[124],[31,37040],{"className":37041,"style":120},[119],[31,37043,37045,37048,37052,37055,37058,37061,37064,37067],{"className":37044},[66],[31,37046],{"className":37047,"style":260},[70],[31,37049,36760],{"className":37050,"style":37051},[75,76],"margin-right:0.0813em;",[31,37053,211],{"className":37054},[264],[31,37056,3003],{"className":37057,"style":2666},[75,76],[31,37059,220],{"className":37060},[289],[31,37062],{"className":37063,"style":293},[119],[31,37065,223],{"className":37066},[297],[31,37068],{"className":37069,"style":293},[119],[31,37071,37073,37076,37122,37125,37128,37131,37134,37137,37140,37143],{"className":37072},[66],[31,37074],{"className":37075,"style":260},[70],[31,37077,37079,37082],{"className":37078},[75],[31,37080,34614],{"className":37081,"style":19759},[75,76],[31,37083,37085],{"className":37084},[1039],[31,37086,37088,37114],{"className":37087},[1043,1044],[31,37089,37091,37111],{"className":37090},[1048],[31,37092,37094],{"className":37093,"style":27736},[1052],[31,37095,37096,37099],{"style":26726},[31,37097],{"className":37098,"style":1061},[1060],[31,37100,37102],{"className":37101},[1065,1066,1067,1068],[31,37103,37105,37108],{"className":37104},[75,1068],[31,37106,97],{"className":37107},[75,1068],[31,37109,36760],{"className":37110,"style":37051},[75,76,1068],[31,37112,1076],{"className":37113},[1075],[31,37115,37117],{"className":37116},[1048],[31,37118,37120],{"className":37119,"style":25082},[1052],[31,37121],{},[31,37123,211],{"className":37124},[264],[31,37126,3003],{"className":37127,"style":2666},[75,76],[31,37129,4572],{"className":37130},[75],[31,37132,27666],{"className":37133,"style":2666},[75,76],[31,37135,220],{"className":37136},[289],[31,37138],{"className":37139,"style":120},[119],[31,37141,434],{"className":37142},[124],[31,37144],{"className":37145,"style":120},[119],[31,37147,37149,37152,37155,37158,37161,37164],{"className":37148},[66],[31,37150],{"className":37151,"style":260},[70],[31,37153,36760],{"className":37154,"style":37051},[75,76],[31,37156,211],{"className":37157},[264],[31,37159,3003],{"className":37160,"style":2666},[75,76],[31,37162,220],{"className":37163},[289],[31,37165,1210],{"className":37166},[1391],[11,37168,37169,37170,37214,37215,2372],{},"所以训练一个分类器、一个语言模型，最小化交叉熵就是在最小化一个 Bregman 散度；标签固定时 ",[31,37171,37173,37193],{"className":37172,"translate":35},[34],[31,37174,37176],{"className":37175},[39],[41,37177,37178],{"xmlns":43},[45,37179,37180,37190],{},[48,37181,37182,37184,37186,37188],{},[51,37183,36760],{},[95,37185,211],{"stretchy":210},[51,37187,3003],{},[95,37189,220],{"stretchy":210},[55,37191,37192],{"encoding":57},"H(y)",[31,37194,37196],{"className":37195,"ariaHidden":62},[61],[31,37197,37199,37202,37205,37208,37211],{"className":37198},[66],[31,37200],{"className":37201,"style":260},[70],[31,37203,36760],{"className":37204,"style":37051},[75,76],[31,37206,211],{"className":37207},[264],[31,37209,3003],{"className":37210,"style":2666},[75,76],[31,37212,220],{"className":37213},[289]," 是常数，梯度完全来自 ",[31,37216,37218,37231],{"className":37217,"translate":35},[34],[31,37219,37221],{"className":37220},[39],[41,37222,37223],{"xmlns":43},[45,37224,37225,37229],{},[48,37226,37227],{},[51,37228,34614],{},[55,37230,34614],{"encoding":57},[31,37232,37234],{"className":37233,"ariaHidden":62},[61],[31,37235,37237,37240],{"className":37236},[66],[31,37238],{"className":37239,"style":945},[70],[31,37241,34614],{"className":37242,"style":19759},[75,76],[11,37244,37245],{},"Bregman 散度还有一个我很喜欢的性质：它的质心是算术平均。对任意 Bregman 散度，",[31,37247,37249],{"className":37248,"translate":35},[411],[31,37250,37252,37329],{"className":37251,"translate":35},[34],[31,37253,37255],{"className":37254},[39],[41,37256,37257],{"xmlns":43,"display":420},[45,37258,37259,37326],{},[48,37260,37261,37264,37266,37276,37278,37284,37290,37292,37298,37300,37302,37304,37306,37312,37318,37324],{},[51,37262,37263],{},"arg",[95,37265,1540],{},[1531,37267,37268,37274],{},[48,37269,37270,37272],{},[51,37271,19878],{},[95,37273,1540],{},[51,37275,1412],{},[1430,37277,1545],{},[1531,37279,37280,37282],{},[95,37281,5625],{},[51,37283,5391],{},[1004,37285,37286,37288],{},[51,37287,34614],{},[51,37289,1662],{},[95,37291,211],{"stretchy":210},[1004,37293,37294,37296],{},[51,37295,53],{},[51,37297,5391],{},[51,37299,4572],{"mathvariant":494},[51,37301,1412],{},[95,37303,220],{"stretchy":210},[95,37305,223],{},[2074,37307,37308,37310],{},[91,37309,205],{},[51,37311,14559],{},[1531,37313,37314,37316],{},[95,37315,5625],{},[51,37317,5391],{},[1004,37319,37320,37322],{},[51,37321,53],{},[51,37323,5391],{},[51,37325,495],{"mathvariant":494},[55,37327,37328],{"encoding":57},"\\arg\\min_c\\ \\sum_iD_f(x_i\\|c)=\\frac1n\\sum_ix_i .",[31,37330,37332,37547],{"className":37331,"ariaHidden":62},[61],[31,37333,37335,37338,37344,37347,37393,37396,37399,37443,37446,37486,37489,37529,37532,37535,37538,37541,37544],{"className":37334},[66],[31,37336],{"className":37337,"style":10251},[70],[31,37339,37341,37342],{"className":37340},[1460],"ar",[31,37343,14556],{"style":20180},[31,37345],{"className":37346,"style":1467},[119],[31,37348,37350],{"className":37349},[1460,1620],[31,37351,37353,37384],{"className":37352},[1043,1044],[31,37354,37356,37381],{"className":37355},[1048],[31,37357,37359,37371],{"className":37358,"style":19979},[1052],[31,37360,37362,37365],{"style":37361},"top:-2.4em;margin-left:0em;",[31,37363],{"className":37364,"style":1637},[1060],[31,37366,37368],{"className":37367},[1065,1066,1067,1068],[31,37369,1412],{"className":37370},[75,76,1068],[31,37372,37373,37376],{"style":1649},[31,37374],{"className":37375,"style":1637},[1060],[31,37377,37378],{},[31,37379,19878],{"className":37380},[1460],[31,37382,1076],{"className":37383},[1075],[31,37385,37387],{"className":37386},[1048],[31,37388,37391],{"className":37389,"style":37390},[1052],"height:0.7em;",[31,37392],{},[31,37394,1545],{"className":37395},[119],[31,37397],{"className":37398,"style":1467},[119],[31,37400,37402],{"className":37401},[1460,1620],[31,37403,37405,37435],{"className":37404},[1043,1044],[31,37406,37408,37432],{"className":37407},[1048],[31,37409,37411,37422],{"className":37410,"style":5900},[1052],[31,37412,37413,37416],{"style":5903},[31,37414],{"className":37415,"style":5907},[1060],[31,37417,37419],{"className":37418},[1065,1066,1067,1068],[31,37420,5391],{"className":37421},[75,76,1068],[31,37423,37424,37427],{"style":5919},[31,37425],{"className":37426,"style":5907},[1060],[31,37428,37429],{},[31,37430,5625],{"className":37431},[1460,1461,1687],[31,37433,1076],{"className":37434},[1075],[31,37436,37438],{"className":37437},[1048],[31,37439,37441],{"className":37440,"style":6342},[1052],[31,37442],{},[31,37444],{"className":37445,"style":1467},[119],[31,37447,37449,37452],{"className":37448},[75],[31,37450,34614],{"className":37451,"style":19759},[75,76],[31,37453,37455],{"className":37454},[1039],[31,37456,37458,37478],{"className":37457},[1043,1044],[31,37459,37461,37475],{"className":37460},[1048],[31,37462,37464],{"className":37463,"style":1053},[1052],[31,37465,37466,37469],{"style":26726},[31,37467],{"className":37468,"style":1061},[1060],[31,37470,37472],{"className":37471},[1065,1066,1067,1068],[31,37473,1662],{"className":37474,"style":35196},[75,76,1068],[31,37476,1076],{"className":37477},[1075],[31,37479,37481],{"className":37480},[1048],[31,37482,37484],{"className":37483,"style":1083},[1052],[31,37485],{},[31,37487,211],{"className":37488},[264],[31,37490,37492,37495],{"className":37491},[75],[31,37493,53],{"className":37494},[75,76],[31,37496,37498],{"className":37497},[1039],[31,37499,37501,37521],{"className":37500},[1043,1044],[31,37502,37504,37518],{"className":37503},[1048],[31,37505,37507],{"className":37506,"style":5445},[1052],[31,37508,37509,37512],{"style":2807},[31,37510],{"className":37511,"style":1061},[1060],[31,37513,37515],{"className":37514},[1065,1066,1067,1068],[31,37516,5391],{"className":37517},[75,76,1068],[31,37519,1076],{"className":37520},[1075],[31,37522,37524],{"className":37523},[1048],[31,37525,37527],{"className":37526,"style":2050},[1052],[31,37528],{},[31,37530,4572],{"className":37531},[75],[31,37533,1412],{"className":37534},[75,76],[31,37536,220],{"className":37537},[289],[31,37539],{"className":37540,"style":293},[119],[31,37542,223],{"className":37543},[297],[31,37545],{"className":37546,"style":293},[119],[31,37548,37550,37554,37616,37619,37663,37666,37706],{"className":37549},[66],[31,37551],{"className":37552,"style":37553},[70],"height:2.5991em;vertical-align:-1.2777em;",[31,37555,37557,37560,37613],{"className":37556},[75],[31,37558],{"className":37559},[264,2139],[31,37561,37563],{"className":37562},[2074],[31,37564,37566,37605],{"className":37565},[1043,1044],[31,37567,37569,37602],{"className":37568},[1048],[31,37570,37572,37583,37591],{"className":37571,"style":26376},[1052],[31,37573,37574,37577],{"style":5090},[31,37575],{"className":37576,"style":1637},[1060],[31,37578,37580],{"className":37579},[75],[31,37581,14559],{"className":37582},[75,76],[31,37584,37585,37588],{"style":2170},[31,37586],{"className":37587,"style":1637},[1060],[31,37589],{"className":37590,"style":2178},[2177],[31,37592,37593,37596],{"style":5110},[31,37594],{"className":37595,"style":1637},[1060],[31,37597,37599],{"className":37598},[75],[31,37600,205],{"className":37601},[75],[31,37603,1076],{"className":37604},[1075],[31,37606,37608],{"className":37607},[1048],[31,37609,37611],{"className":37610,"style":5156},[1052],[31,37612],{},[31,37614],{"className":37615},[289,2139],[31,37617],{"className":37618,"style":1467},[119],[31,37620,37622],{"className":37621},[1460,1620],[31,37623,37625,37655],{"className":37624},[1043,1044],[31,37626,37628,37652],{"className":37627},[1048],[31,37629,37631,37642],{"className":37630,"style":5900},[1052],[31,37632,37633,37636],{"style":5903},[31,37634],{"className":37635,"style":5907},[1060],[31,37637,37639],{"className":37638},[1065,1066,1067,1068],[31,37640,5391],{"className":37641},[75,76,1068],[31,37643,37644,37647],{"style":5919},[31,37645],{"className":37646,"style":5907},[1060],[31,37648,37649],{},[31,37650,5625],{"className":37651},[1460,1461,1687],[31,37653,1076],{"className":37654},[1075],[31,37656,37658],{"className":37657},[1048],[31,37659,37661],{"className":37660,"style":6342},[1052],[31,37662],{},[31,37664],{"className":37665,"style":1467},[119],[31,37667,37669,37672],{"className":37668},[75],[31,37670,53],{"className":37671},[75,76],[31,37673,37675],{"className":37674},[1039],[31,37676,37678,37698],{"className":37677},[1043,1044],[31,37679,37681,37695],{"className":37680},[1048],[31,37682,37684],{"className":37683,"style":5445},[1052],[31,37685,37686,37689],{"style":2807},[31,37687],{"className":37688,"style":1061},[1060],[31,37690,37692],{"className":37691},[1065,1066,1067,1068],[31,37693,5391],{"className":37694},[75,76,1068],[31,37696,1076],{"className":37697},[1075],[31,37699,37701],{"className":37700},[1048],[31,37702,37704],{"className":37703,"style":2050},[1052],[31,37705],{},[31,37707,495],{"className":37708},[75],[11,37710,37711,37712,37941],{},"求导得到 ",[31,37713,37715,37765],{"className":37714,"translate":35},[34],[31,37716,37718],{"className":37717},[39],[41,37719,37720],{"xmlns":43},[45,37721,37722,37762],{},[48,37723,37724,37730,37736,37738,37740,37742,37744,37746,37752,37754,37756,37758,37760],{},[1004,37725,37726,37728],{},[95,37727,5625],{},[51,37729,5391],{},[1180,37731,37732,37734],{},[51,37733,12145],{"mathvariant":494},[91,37735,152],{},[51,37737,1662],{},[95,37739,211],{"stretchy":210},[51,37741,1412],{},[95,37743,220],{"stretchy":210},[95,37745,211],{"stretchy":210},[1004,37747,37748,37750],{},[51,37749,53],{},[51,37751,5391],{},[95,37753,97],{},[51,37755,1412],{},[95,37757,220],{"stretchy":210},[95,37759,223],{},[91,37761,1977],{},[55,37763,37764],{"encoding":57},"\\sum_i\\nabla^2f(c)(x_i-c)=0",[31,37766,37768,37911,37932],{"className":37767,"ariaHidden":62},[61],[31,37769,37771,37775,37815,37818,37847,37850,37853,37856,37859,37862,37902,37905,37908],{"className":37770},[66],[31,37772],{"className":37773,"style":37774},[70],"height:1.1138em;vertical-align:-0.2997em;",[31,37776,37778,37781],{"className":37777},[1460],[31,37779,5625],{"className":37780,"style":21372},[1460,1461,1462],[31,37782,37784],{"className":37783},[1039],[31,37785,37787,37807],{"className":37786},[1043,1044],[31,37788,37790,37804],{"className":37789},[1048],[31,37791,37793],{"className":37792,"style":21385},[1052],[31,37794,37795,37798],{"style":21388},[31,37796],{"className":37797,"style":1061},[1060],[31,37799,37801],{"className":37800},[1065,1066,1067,1068],[31,37802,5391],{"className":37803},[75,76,1068],[31,37805,1076],{"className":37806},[1075],[31,37808,37810],{"className":37809},[1048],[31,37811,37813],{"className":37812,"style":21407},[1052],[31,37814],{},[31,37816],{"className":37817,"style":1467},[119],[31,37819,37821,37824],{"className":37820},[75],[31,37822,12145],{"className":37823},[75],[31,37825,37827],{"className":37826},[1039],[31,37828,37830],{"className":37829},[1043],[31,37831,37833],{"className":37832},[1048],[31,37834,37836],{"className":37835,"style":5135},[1052],[31,37837,37838,37841],{"style":3631},[31,37839],{"className":37840,"style":1061},[1060],[31,37842,37844],{"className":37843},[1065,1066,1067,1068],[31,37845,152],{"className":37846},[75,1068],[31,37848,1662],{"className":37849,"style":35196},[75,76],[31,37851,211],{"className":37852},[264],[31,37854,1412],{"className":37855},[75,76],[31,37857,220],{"className":37858},[289],[31,37860,211],{"className":37861},[264],[31,37863,37865,37868],{"className":37864},[75],[31,37866,53],{"className":37867},[75,76],[31,37869,37871],{"className":37870},[1039],[31,37872,37874,37894],{"className":37873},[1043,1044],[31,37875,37877,37891],{"className":37876},[1048],[31,37878,37880],{"className":37879,"style":5445},[1052],[31,37881,37882,37885],{"style":2807},[31,37883],{"className":37884,"style":1061},[1060],[31,37886,37888],{"className":37887},[1065,1066,1067,1068],[31,37889,5391],{"className":37890},[75,76,1068],[31,37892,1076],{"className":37893},[1075],[31,37895,37897],{"className":37896},[1048],[31,37898,37900],{"className":37899,"style":2050},[1052],[31,37901],{},[31,37903],{"className":37904,"style":120},[119],[31,37906,97],{"className":37907},[124],[31,37909],{"className":37910,"style":120},[119],[31,37912,37914,37917,37920,37923,37926,37929],{"className":37913},[66],[31,37915],{"className":37916,"style":260},[70],[31,37918,1412],{"className":37919},[75,76],[31,37921,220],{"className":37922},[289],[31,37924],{"className":37925,"style":293},[119],[31,37927,223],{"className":37928},[297],[31,37930],{"className":37931,"style":293},[119],[31,37933,37935,37938],{"className":37934},[66],[31,37936],{"className":37937,"style":326},[70],[31,37939,1977],{"className":37940},[75],"，结论就是这一行。所以换散度不会改变“平均”这件事，但会改变“谁离谁近”这件事。",[11,37943,37944,37945,38069,38070,18171,38206,38342],{},"这一点值得展开，因为它关系到分类边界。设两个类原型 ",[31,37946,37948,37974],{"className":37947,"translate":35},[34],[31,37949,37951],{"className":37950},[39],[41,37952,37953],{"xmlns":43},[45,37954,37955,37971],{},[48,37956,37957,37963,37965],{},[1004,37958,37959,37961],{},[51,37960,1412],{},[91,37962,205],{},[95,37964,1210],{"separator":62},[1004,37966,37967,37969],{},[51,37968,1412],{},[91,37970,152],{},[55,37972,37973],{"encoding":57},"c_1,c_2",[31,37975,37977],{"className":37976,"ariaHidden":62},[61],[31,37978,37980,37983,38023,38026,38029],{"className":37979},[66],[31,37981],{"className":37982,"style":865},[70],[31,37984,37986,37989],{"className":37985},[75],[31,37987,1412],{"className":37988},[75,76],[31,37990,37992],{"className":37991},[1039],[31,37993,37995,38015],{"className":37994},[1043,1044],[31,37996,37998,38012],{"className":37997},[1048],[31,37999,38001],{"className":38000,"style":2028},[1052],[31,38002,38003,38006],{"style":2807},[31,38004],{"className":38005,"style":1061},[1060],[31,38007,38009],{"className":38008},[1065,1066,1067,1068],[31,38010,205],{"className":38011},[75,1068],[31,38013,1076],{"className":38014},[1075],[31,38016,38018],{"className":38017},[1048],[31,38019,38021],{"className":38020,"style":2050},[1052],[31,38022],{},[31,38024,1210],{"className":38025},[1391],[31,38027],{"className":38028,"style":1467},[119],[31,38030,38032,38035],{"className":38031},[75],[31,38033,1412],{"className":38034},[75,76],[31,38036,38038],{"className":38037},[1039],[31,38039,38041,38061],{"className":38040},[1043,1044],[31,38042,38044,38058],{"className":38043},[1048],[31,38045,38047],{"className":38046,"style":2028},[1052],[31,38048,38049,38052],{"style":2807},[31,38050],{"className":38051,"style":1061},[1060],[31,38053,38055],{"className":38054},[1065,1066,1067,1068],[31,38056,152],{"className":38057},[75,1068],[31,38059,1076],{"className":38060},[1075],[31,38062,38064],{"className":38063},[1048],[31,38065,38067],{"className":38066,"style":2050},[1052],[31,38068],{},"，最近原型判类就是比较 ",[31,38071,38073,38105],{"className":38072,"translate":35},[34],[31,38074,38076],{"className":38075},[39],[41,38077,38078],{"xmlns":43},[45,38079,38080,38102],{},[48,38081,38082,38088,38090,38092,38094,38100],{},[1004,38083,38084,38086],{},[51,38085,34614],{},[51,38087,1662],{},[95,38089,211],{"stretchy":210},[51,38091,53],{},[51,38093,4572],{"mathvariant":494},[1004,38095,38096,38098],{},[51,38097,1412],{},[91,38099,205],{},[95,38101,220],{"stretchy":210},[55,38103,38104],{"encoding":57},"D_f(x\\|c_1)",[31,38106,38108],{"className":38107,"ariaHidden":62},[61],[31,38109,38111,38114,38154,38157,38160,38163,38203],{"className":38110},[66],[31,38112],{"className":38113,"style":1223},[70],[31,38115,38117,38120],{"className":38116},[75],[31,38118,34614],{"className":38119,"style":19759},[75,76],[31,38121,38123],{"className":38122},[1039],[31,38124,38126,38146],{"className":38125},[1043,1044],[31,38127,38129,38143],{"className":38128},[1048],[31,38130,38132],{"className":38131,"style":1053},[1052],[31,38133,38134,38137],{"style":26726},[31,38135],{"className":38136,"style":1061},[1060],[31,38138,38140],{"className":38139},[1065,1066,1067,1068],[31,38141,1662],{"className":38142,"style":35196},[75,76,1068],[31,38144,1076],{"className":38145},[1075],[31,38147,38149],{"className":38148},[1048],[31,38150,38152],{"className":38151,"style":1083},[1052],[31,38153],{},[31,38155,211],{"className":38156},[264],[31,38158,53],{"className":38159},[75,76],[31,38161,4572],{"className":38162},[75],[31,38164,38166,38169],{"className":38165},[75],[31,38167,1412],{"className":38168},[75,76],[31,38170,38172],{"className":38171},[1039],[31,38173,38175,38195],{"className":38174},[1043,1044],[31,38176,38178,38192],{"className":38177},[1048],[31,38179,38181],{"className":38180,"style":2028},[1052],[31,38182,38183,38186],{"style":2807},[31,38184],{"className":38185,"style":1061},[1060],[31,38187,38189],{"className":38188},[1065,1066,1067,1068],[31,38190,205],{"className":38191},[75,1068],[31,38193,1076],{"className":38194},[1075],[31,38196,38198],{"className":38197},[1048],[31,38199,38201],{"className":38200,"style":2050},[1052],[31,38202],{},[31,38204,220],{"className":38205},[289],[31,38207,38209,38241],{"className":38208,"translate":35},[34],[31,38210,38212],{"className":38211},[39],[41,38213,38214],{"xmlns":43},[45,38215,38216,38238],{},[48,38217,38218,38224,38226,38228,38230,38236],{},[1004,38219,38220,38222],{},[51,38221,34614],{},[51,38223,1662],{},[95,38225,211],{"stretchy":210},[51,38227,53],{},[51,38229,4572],{"mathvariant":494},[1004,38231,38232,38234],{},[51,38233,1412],{},[91,38235,152],{},[95,38237,220],{"stretchy":210},[55,38239,38240],{"encoding":57},"D_f(x\\|c_2)",[31,38242,38244],{"className":38243,"ariaHidden":62},[61],[31,38245,38247,38250,38290,38293,38296,38299,38339],{"className":38246},[66],[31,38248],{"className":38249,"style":1223},[70],[31,38251,38253,38256],{"className":38252},[75],[31,38254,34614],{"className":38255,"style":19759},[75,76],[31,38257,38259],{"className":38258},[1039],[31,38260,38262,38282],{"className":38261},[1043,1044],[31,38263,38265,38279],{"className":38264},[1048],[31,38266,38268],{"className":38267,"style":1053},[1052],[31,38269,38270,38273],{"style":26726},[31,38271],{"className":38272,"style":1061},[1060],[31,38274,38276],{"className":38275},[1065,1066,1067,1068],[31,38277,1662],{"className":38278,"style":35196},[75,76,1068],[31,38280,1076],{"className":38281},[1075],[31,38283,38285],{"className":38284},[1048],[31,38286,38288],{"className":38287,"style":1083},[1052],[31,38289],{},[31,38291,211],{"className":38292},[264],[31,38294,53],{"className":38295},[75,76],[31,38297,4572],{"className":38298},[75],[31,38300,38302,38305],{"className":38301},[75],[31,38303,1412],{"className":38304},[75,76],[31,38306,38308],{"className":38307},[1039],[31,38309,38311,38331],{"className":38310},[1043,1044],[31,38312,38314,38328],{"className":38313},[1048],[31,38315,38317],{"className":38316,"style":2028},[1052],[31,38318,38319,38322],{"style":2807},[31,38320],{"className":38321,"style":1061},[1060],[31,38323,38325],{"className":38324},[1065,1066,1067,1068],[31,38326,152],{"className":38327},[75,1068],[31,38329,1076],{"className":38330},[1075],[31,38332,38334],{"className":38333},[1048],[31,38335,38337],{"className":38336,"style":2050},[1052],[31,38338],{},[31,38340,220],{"className":38341},[289],"。把定义代进去，",[31,38344,38346],{"className":38345,"translate":35},[411],[31,38347,38349,38713],{"className":38348,"translate":35},[34],[31,38350,38352],{"className":38351},[39],[41,38353,38354],{"xmlns":43,"display":420},[45,38355,38356,38710],{},[5606,38357,38358,38416,38556],{"rowspacing":5608,"columnalign":5609,"columnspacing":5610},[5612,38359,38360,38366],{},[5615,38361,38362],{},[2071,38363,38364],{"scriptlevel":1977,"displaystyle":62},[48,38365],{},[5615,38367,38368],{},[2071,38369,38370],{"scriptlevel":1977,"displaystyle":62},[48,38371,38372,38374,38380,38382,38384,38386,38392,38394,38396,38402,38404,38406,38408,38414],{},[48,38373],{},[1004,38375,38376,38378],{},[51,38377,34614],{},[51,38379,1662],{},[95,38381,211],{"stretchy":210},[51,38383,53],{},[51,38385,4572],{"mathvariant":494},[1004,38387,38388,38390],{},[51,38389,1412],{},[91,38391,205],{},[95,38393,220],{"stretchy":210},[95,38395,97],{},[1004,38397,38398,38400],{},[51,38399,34614],{},[51,38401,1662],{},[95,38403,211],{"stretchy":210},[51,38405,53],{},[51,38407,4572],{"mathvariant":494},[1004,38409,38410,38412],{},[51,38411,1412],{},[91,38413,152],{},[95,38415,220],{"stretchy":210},[5612,38417,38418,38424],{},[5615,38419,38420],{},[2071,38421,38422],{"scriptlevel":1977,"displaystyle":62},[48,38423],{},[5615,38425,38426],{},[2071,38427,38428],{"scriptlevel":1977,"displaystyle":62},[48,38429,38430,38432,38434,38436,38438,38440,38442,38444,38446,38448,38450,38456,38458,38460,38462,38464,38466,38472,38478,38480,38482,38484,38490,38492,38494,38496,38498,38500,38502,38504,38506,38508,38510,38512,38518,38520,38522,38524,38526,38528,38534,38540,38542,38544,38546,38552,38554],{},[48,38431],{},[95,38433,223],{},[95,38435,15175],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[51,38437,1662],{},[95,38439,211],{"stretchy":210},[51,38441,53],{},[95,38443,220],{"stretchy":210},[95,38445,97],{},[51,38447,1662],{},[95,38449,211],{"stretchy":210},[1004,38451,38452,38454],{},[51,38453,1412],{},[91,38455,205],{},[95,38457,220],{"stretchy":210},[95,38459,97],{},[51,38461,12145],{"mathvariant":494},[51,38463,1662],{},[95,38465,211],{"stretchy":210},[1004,38467,38468,38470],{},[51,38469,1412],{},[91,38471,205],{},[1180,38473,38474,38476],{},[95,38475,220],{"stretchy":210},[51,38477,20800],{"mathvariant":494},[95,38479,211],{"stretchy":210},[51,38481,53],{},[95,38483,97],{},[1004,38485,38486,38488],{},[51,38487,1412],{},[91,38489,205],{},[95,38491,220],{"stretchy":210},[95,38493,15184],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[95,38495,97],{},[95,38497,15175],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[51,38499,1662],{},[95,38501,211],{"stretchy":210},[51,38503,53],{},[95,38505,220],{"stretchy":210},[95,38507,97],{},[51,38509,1662],{},[95,38511,211],{"stretchy":210},[1004,38513,38514,38516],{},[51,38515,1412],{},[91,38517,152],{},[95,38519,220],{"stretchy":210},[95,38521,97],{},[51,38523,12145],{"mathvariant":494},[51,38525,1662],{},[95,38527,211],{"stretchy":210},[1004,38529,38530,38532],{},[51,38531,1412],{},[91,38533,152],{},[1180,38535,38536,38538],{},[95,38537,220],{"stretchy":210},[51,38539,20800],{"mathvariant":494},[95,38541,211],{"stretchy":210},[51,38543,53],{},[95,38545,97],{},[1004,38547,38548,38550],{},[51,38549,1412],{},[91,38551,152],{},[95,38553,220],{"stretchy":210},[95,38555,15184],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[5612,38557,38558,38564],{},[5615,38559,38560],{},[2071,38561,38562],{"scriptlevel":1977,"displaystyle":62},[48,38563],{},[5615,38565,38566],{},[2071,38567,38568],{"scriptlevel":1977,"displaystyle":62},[48,38569,38570,38572,38574,38666,38668,38670,38672,38674,38676,38682,38684,38686,38688,38690,38692,38698,38700,38706,38708],{},[48,38571],{},[95,38573,223],{},[1531,38575,38576,38663],{},[1531,38577,38578,38660],{},[48,38579,38580,38582,38584,38586,38592,38594,38596,38598,38600,38606,38608,38610,38612,38614,38616,38622,38628,38634,38636,38638,38640,38642,38648,38654],{},[95,38581,97],{},[51,38583,1662],{},[95,38585,211],{"stretchy":210},[1004,38587,38588,38590],{},[51,38589,1412],{},[91,38591,205],{},[95,38593,220],{"stretchy":210},[95,38595,434],{},[51,38597,1662],{},[95,38599,211],{"stretchy":210},[1004,38601,38602,38604],{},[51,38603,1412],{},[91,38605,152],{},[95,38607,220],{"stretchy":210},[95,38609,434],{},[51,38611,12145],{"mathvariant":494},[51,38613,1662],{},[95,38615,211],{"stretchy":210},[1004,38617,38618,38620],{},[51,38619,1412],{},[91,38621,205],{},[1180,38623,38624,38626],{},[95,38625,220],{"stretchy":210},[51,38627,20800],{"mathvariant":494},[1004,38629,38630,38632],{},[51,38631,1412],{},[91,38633,205],{},[95,38635,97],{},[51,38637,12145],{"mathvariant":494},[51,38639,1662],{},[95,38641,211],{"stretchy":210},[1004,38643,38644,38646],{},[51,38645,1412],{},[91,38647,152],{},[1180,38649,38650,38652],{},[95,38651,220],{"stretchy":210},[51,38653,20800],{"mathvariant":494},[1004,38655,38656,38658],{},[51,38657,1412],{},[91,38659,152],{},[95,38661,38662],{"stretchy":62},"⏟",[1430,38664,38665],{},"常数",[95,38667,434],{},[95,38669,211],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[51,38671,12145],{"mathvariant":494},[51,38673,1662],{},[95,38675,211],{"stretchy":210},[1004,38677,38678,38680],{},[51,38679,1412],{},[91,38681,152],{},[95,38683,220],{"stretchy":210},[95,38685,97],{},[51,38687,12145],{"mathvariant":494},[51,38689,1662],{},[95,38691,211],{"stretchy":210},[1004,38693,38694,38696],{},[51,38695,1412],{},[91,38697,205],{},[95,38699,220],{"stretchy":210},[1180,38701,38702,38704],{},[95,38703,220],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[51,38705,20800],{"mathvariant":494},[51,38707,53],{},[51,38709,495],{"mathvariant":494},[55,38711,38712],{"encoding":57},"\\begin{aligned}\n&D_f(x\\|c_1)-D_f(x\\|c_2)\\\\\n&=\\bigl[f(x)-f(c_1)-\\nabla f(c_1)^\\top(x-c_1)\\bigr]\n -\\bigl[f(x)-f(c_2)-\\nabla f(c_2)^\\top(x-c_2)\\bigr]\\\\\n&=\\underbrace{-f(c_1)+f(c_2)+\\nabla f(c_1)^\\top c_1-\\nabla f(c_2)^\\top c_2}_{\\text{常数}}\n +\\bigl(\\nabla f(c_2)-\\nabla f(c_1)\\bigr)^\\top x .\n\\end{aligned}",[31,38714,38716],{"className":38715,"ariaHidden":62},[61],[31,38717,38719,38723],{"className":38718},[66],[31,38720],{"className":38721,"style":38722},[70],"height:5.7244em;vertical-align:-2.6122em;",[31,38724,38726],{"className":38725},[75],[31,38727,38729,38782],{"className":38728},[5606],[31,38730,38732],{"className":38731},[5867],[31,38733,38735,38773],{"className":38734},[1043,1044],[31,38736,38738,38770],{"className":38737},[1048],[31,38739,38742,38752,38761],{"className":38740,"style":38741},[1052],"height:3.1122em;",[31,38743,38745,38749],{"style":38744},"top:-5.3612em;",[31,38746],{"className":38747,"style":38748},[1060],"height:3.089em;",[31,38750],{"className":38751},[75],[31,38753,38755,38758],{"style":38754},"top:-3.8021em;",[31,38756],{"className":38757,"style":38748},[1060],[31,38759],{"className":38760},[75],[31,38762,38764,38767],{"style":38763},"top:-2.0531em;",[31,38765],{"className":38766,"style":38748},[1060],[31,38768],{"className":38769},[75],[31,38771,1076],{"className":38772},[1075],[31,38774,38776],{"className":38775},[1048],[31,38777,38780],{"className":38778,"style":38779},[1052],"height:2.6122em;",[31,38781],{},[31,38783,38785],{"className":38784},[6062],[31,38786,38788,40158],{"className":38787},[1043,1044],[31,38789,38791,40155],{"className":38790},[1048],[31,38792,38794,38998,39481],{"className":38793,"style":38741},[1052],[31,38795,38796,38799],{"style":38744},[31,38797],{"className":38798,"style":38748},[1060],[31,38800,38802,38805,38845,38848,38851,38854,38894,38897,38900,38903,38906,38946,38949,38952,38955,38995],{"className":38801},[75],[31,38803],{"className":38804},[75],[31,38806,38808,38811],{"className":38807},[75],[31,38809,34614],{"className":38810,"style":19759},[75,76],[31,38812,38814],{"className":38813},[1039],[31,38815,38817,38837],{"className":38816},[1043,1044],[31,38818,38820,38834],{"className":38819},[1048],[31,38821,38823],{"className":38822,"style":1053},[1052],[31,38824,38825,38828],{"style":26726},[31,38826],{"className":38827,"style":1061},[1060],[31,38829,38831],{"className":38830},[1065,1066,1067,1068],[31,38832,1662],{"className":38833,"style":35196},[75,76,1068],[31,38835,1076],{"className":38836},[1075],[31,38838,38840],{"className":38839},[1048],[31,38841,38843],{"className":38842,"style":1083},[1052],[31,38844],{},[31,38846,211],{"className":38847},[264],[31,38849,53],{"className":38850},[75,76],[31,38852,4572],{"className":38853},[75],[31,38855,38857,38860],{"className":38856},[75],[31,38858,1412],{"className":38859},[75,76],[31,38861,38863],{"className":38862},[1039],[31,38864,38866,38886],{"className":38865},[1043,1044],[31,38867,38869,38883],{"className":38868},[1048],[31,38870,38872],{"className":38871,"style":2028},[1052],[31,38873,38874,38877],{"style":2807},[31,38875],{"className":38876,"style":1061},[1060],[31,38878,38880],{"className":38879},[1065,1066,1067,1068],[31,38881,205],{"className":38882},[75,1068],[31,38884,1076],{"className":38885},[1075],[31,38887,38889],{"className":38888},[1048],[31,38890,38892],{"className":38891,"style":2050},[1052],[31,38893],{},[31,38895,220],{"className":38896},[289],[31,38898],{"className":38899,"style":120},[119],[31,38901,97],{"className":38902},[124],[31,38904],{"className":38905,"style":120},[119],[31,38907,38909,38912],{"className":38908},[75],[31,38910,34614],{"className":38911,"style":19759},[75,76],[31,38913,38915],{"className":38914},[1039],[31,38916,38918,38938],{"className":38917},[1043,1044],[31,38919,38921,38935],{"className":38920},[1048],[31,38922,38924],{"className":38923,"style":1053},[1052],[31,38925,38926,38929],{"style":26726},[31,38927],{"className":38928,"style":1061},[1060],[31,38930,38932],{"className":38931},[1065,1066,1067,1068],[31,38933,1662],{"className":38934,"style":35196},[75,76,1068],[31,38936,1076],{"className":38937},[1075],[31,38939,38941],{"className":38940},[1048],[31,38942,38944],{"className":38943,"style":1083},[1052],[31,38945],{},[31,38947,211],{"className":38948},[264],[31,38950,53],{"className":38951},[75,76],[31,38953,4572],{"className":38954},[75],[31,38956,38958,38961],{"className":38957},[75],[31,38959,1412],{"className":38960},[75,76],[31,38962,38964],{"className":38963},[1039],[31,38965,38967,38987],{"className":38966},[1043,1044],[31,38968,38970,38984],{"className":38969},[1048],[31,38971,38973],{"className":38972,"style":2028},[1052],[31,38974,38975,38978],{"style":2807},[31,38976],{"className":38977,"style":1061},[1060],[31,38979,38981],{"className":38980},[1065,1066,1067,1068],[31,38982,152],{"className":38983},[75,1068],[31,38985,1076],{"className":38986},[1075],[31,38988,38990],{"className":38989},[1048],[31,38991,38993],{"className":38992,"style":2050},[1052],[31,38994],{},[31,38996,220],{"className":38997},[289],[31,38999,39000,39003],{"style":38754},[31,39001],{"className":39002,"style":38748},[1060],[31,39004,39006,39009,39012,39015,39018,39024,39027,39030,39033,39036,39039,39042,39045,39048,39051,39091,39094,39097,3910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20-1h17z",[31,39593,39594,39597],{"style":1649},[31,39595],{"className":39596,"style":1637},[1060],[31,39598,39600,39603,39606,39609,39649,39652,39655,39658,39661,39664,39667,39707,39710,39713,39716,39719,39722,39725,39728,39768,39797,39837,39840,39843,39846,39849,39852,39855,39895,39924],{"className":39599},[75],[31,39601,97],{"className":39602},[75],[31,39604,1662],{"className":39605,"style":35196},[75,76],[31,39607,211],{"className":39608},[264],[31,39610,39612,39615],{"className":39611},[75],[31,39613,1412],{"className":39614},[75,76],[31,39616,39618],{"className":39617},[1039],[31,39619,39621,39641],{"className":39620},[1043,1044],[31,39622,39624,39638],{"className":39623},[1048],[31,39625,39627],{"className":39626,"style":2028},[1052],[31,39628,39629,39632],{"style":2807},[31,39630],{"className":39631,"style":1061},[1060],[31,39633,39635],{"className":39634},[1065,1066,1067,1068],[31,39636,205],{"className":39637},[75,1068],[31,39639,1076],{"className":39640},[1075],[31,39642,39644],{"className":39643},[1048],[31,39645,39647],{"className":39646,"style":2050},[1052],[31,39648],{},[31,39650,220],{"className":39651},[289],[31,39653],{"className":39654,"style":120},[119],[31,39656,434],{"className":39657},[124],[31,39659],{"className":39660,"style":120},[119],[31,39662,1662],{"className":39663,"style":35196},[75,76],[31,39665,211],{"className":39666},[264],[31,39668,39670,39673],{"className":39669},[75],[31,39671,1412],{"className":39672},[75,76],[31,39674,39676],{"className":39675},[1039],[31,39677,39679,39699],{"className":39678},[1043,1044],[31,39680,39682,39696],{"className":39681},[1048],[31,39683,39685],{"className":39684,"style":2028},[1052],[31,39686,39687,39690],{"style":2807},[31,39688],{"className":39689,"style":1061},[1060],[31,39691,39693],{"className":39692},[1065,1066,1067,1068],[31,39694,152],{"className":39695},[75,1068],[31,39697,1076],{"className":39698},[1075],[31,39700,39702],{"className":39701},[1048],[31,39703,39705],{"className":39704,"style":2050},[1052],[31,39706],{},[31,39708,220],{"className":39709},[289],[31,39711],{"className":39712,"style":120},[119],[31,39714,434],{"className":39715},[124],[31,39717],{"className":39718,"style":120},[119],[31,39720,12145],{"className":39721},[75],[31,39723,1662],{"className":39724,"style":35196},[75,76],[31,39726,211],{"className":39727},[264],[31,39729,39731,39734],{"className":39730},[75],[31,39732,1412],{"className":39733},[75,76],[31,39735,39737],{"className":39736},[1039],[31,39738,39740,39760],{"className":39739},[1043,1044],[31,39741,39743,39757],{"className":39742},[1048],[31,39744,39746],{"className":39745,"style":2028},[1052],[31,39747,39748,39751],{"style":2807},[31,39749],{"className":39750,"style":1061},[1060],[31,39752,39754],{"className":39753},[1065,1066,1067,1068],[31,39755,205],{"className":39756},[75,1068],[31,39758,1076],{"className":39759},[1075],[31,39761,39763],{"className":39762},[1048],[31,39764,39766],{"className":39765,"style":2050},[1052],[31,39767],{},[31,39769,39771,39774],{"className":39770},[289],[31,39772,220],{"className":39773},[289],[31,39775,39777],{"className":39776},[1039],[31,39778,39780],{"className":39779},[1043],[31,39781,39783],{"className":39782},[1048],[31,39784,39786],{"className":39785,"style":29642},[1052],[31,39787,39788,39791],{"style":1330},[31,39789],{"className":39790,"style":1061},[1060],[31,39792,39794],{"className":39793},[1065,1066,1067,1068],[31,39795,20800],{"className":39796},[75,1068],[31,39798,39800,39803],{"className":39799},[75],[31,39801,1412],{"className":39802},[75,76],[31,39804,39806],{"className":39805},[1039],[31,39807,39809,39829],{"className":39808},[1043,1044],[31,39810,39812,39826],{"className":39811},[1048],[31,39813,39815],{"className":39814,"style":2028},[1052],[31,39816,39817,39820],{"style":2807},[31,39818],{"className":39819,"style":1061},[1060],[31,39821,39823],{"className":39822},[1065,1066,1067,1068],[31,39824,205],{"className":39825},[75,1068],[31,39827,1076],{"className":39828},[1075],[31,39830,39832],{"className":39831},[1048],[31,39833,39835],{"className":39834,"style":2050},[1052],[31,39836],{},[31,39838],{"className":39839,"style":120},[119],[31,39841,97],{"className":39842},[124],[31,39844],{"className":39845,"style":120},[119],[31,39847,12145],{"className":39848},[75],[31,39850,1662],{"className":39851,"style":35196},[75,76],[31,39853,211],{"className":39854},[264],[31,39856,39858,39861],{"className":39857},[75],[31,39859,1412],{"className":39860},[75,76],[31,39862,39864],{"className":39863},[1039],[31,39865,39867,39887],{"className":39866},[1043,1044],[31,39868,39870,39884],{"className":39869},[1048],[31,39871,39873],{"className":39872,"style":2028},[1052],[31,39874,39875,39878],{"style":2807},[31,39876],{"className":39877,"style":1061},[1060],[31,39879,39881],{"className":39880},[1065,1066,1067,1068],[31,39882,152],{"className":39883},[75,1068],[31,39885,1076],{"className":39886},[1075],[31,39888,39890],{"className":39889},[1048],[31,39891,39893],{"className":39892,"style":2050},[1052],[31,39894],{},[31,39896,39898,39901],{"className":39897},[289],[31,39899,220],{"className":39900},[289],[31,39902,39904],{"className":39903},[1039],[31,39905,39907],{"className":39906},[1043],[31,39908,39910],{"className":39909},[1048],[31,39911,39913],{"className":39912,"style":29642},[1052],[31,39914,39915,39918],{"style":1330},[31,39916],{"className":39917,"style":1061},[1060],[31,39919,39921],{"className":39920},[1065,1066,1067,1068],[31,39922,20800],{"className":39923},[75,1068],[31,39925,39927,39930],{"className":39926},[75],[31,39928,1412],{"className":39929},[75,76],[31,39931,39933],{"className":39932},[1039],[31,39934,39936,39956],{"className":39935},[1043,1044],[31,39937,39939,39953],{"className":39938},[1048],[31,39940,39942],{"className":39941,"style":2028},[1052],[31,39943,39944,39947],{"style":2807},[31,39945],{"className":39946,"style":1061},[1060],[31,39948,39950],{"className":39949},[1065,1066,1067,1068],[31,39951,152],{"className":39952},[75,1068],[31,39954,1076],{"className":39955},[1075],[31,39957,39959],{"className":39958},[1048],[31,39960,39962],{"className":39961,"style":2050},[1052],[31,39963],{},[31,39965,1076],{"className":39966},[1075],[31,39968,39970],{"className":39969},[1048],[31,39971,39974],{"className":39972,"style":39973},[1052],"height:0.898em;",[31,39975],{},[31,39977,1076],{"className":39978},[1075],[31,39980,39982],{"className":39981},[1048],[31,39983,39986],{"className":39984,"style":39985},[1052],"height:1.5763em;",[31,39987],{},[31,39989],{"className":39990,"style":120},[119],[31,39992,434],{"className":39993},[124],[31,39995],{"className":39996,"style":120},[119],[31,39998,40000],{"className":39999},[264],[31,40001,211],{"className":40002},[1307,1308],[31,40004,12145],{"className":40005},[75],[31,40007,1662],{"className":40008,"style":35196},[75,76],[31,40010,211],{"className":40011},[264],[31,40013,40015,40018],{"className":40014},[75],[31,40016,1412],{"className":40017},[75,76],[31,40019,40021],{"className":40020},[1039],[31,40022,40024,40044],{"className":40023},[1043,1044],[31,40025,40027,40041],{"className":40026},[1048],[31,40028,40030],{"className":40029,"style":2028},[1052],[31,40031,40032,40035],{"style":2807},[31,40033],{"className":40034,"style":1061},[1060],[31,40036,40038],{"className":40037},[1065,1066,1067,1068],[31,40039,152],{"className":40040},[75,1068],[31,40042,1076],{"className":40043},[1075],[31,40045,40047],{"className":40046},[1048],[31,40048,40050],{"className":40049,"style":2050},[1052],[31,40051],{},[31,40053,220],{"className":40054},[289],[31,40056],{"className":40057,"style":120},[119],[31,40059,97],{"className":40060},[124],[31,40062],{"className":40063,"style":120},[119],[31,40065,12145],{"className":40066},[75],[31,40068,1662],{"className":40069,"style":35196},[75,76],[31,40071,211],{"className":40072},[264],[31,40074,40076,40079],{"className":40075},[75],[31,40077,1412],{"className":40078},[75,76],[31,40080,40082],{"className":40081},[1039],[31,40083,40085,40105],{"className":40084},[1043,1044],[31,40086,40088,40102],{"className":40087},[1048],[31,40089,40091],{"className":40090,"style":2028},[1052],[31,40092,40093,40096],{"style":2807},[31,40094],{"className":40095,"style":1061},[1060],[31,40097,40099],{"className":40098},[1065,1066,1067,1068],[31,40100,205],{"className":40101},[75,1068],[31,40103,1076],{"className":40104},[1075],[31,40106,40108],{"className":40107},[1048],[31,40109,40111],{"className":40110,"style":2050},[1052],[31,40112],{},[31,40114,220],{"className":40115},[289],[31,40117,40119,40125],{"className":40118},[289],[31,40120,40122],{"className":40121},[289],[31,40123,220],{"className":40124},[1307,1308],[31,40126,40128],{"className":40127},[1039],[31,40129,40131],{"className":40130},[1043],[31,40132,40134],{"className":40133},[1048],[31,40135,40138],{"className":40136,"style":40137},[1052],"height:1.089em;",[31,40139,40140,40143],{"style":13566},[31,40141],{"className":40142,"style":1061},[1060],[31,40144,40146],{"className":40145},[1065,1066,1067,1068],[31,40147,20800],{"className":40148},[75,1068],[31,40150,53],{"className":40151},[75,76],[31,40153,495],{"className":40154},[75],[31,40156,1076],{"className":40157},[1075],[31,40159,40161],{"className":40160},[1048],[31,40162,40164],{"className":40163,"style":38779},[1052],[31,40165],{},[11,40167,40168,40212,40213,40241],{},[31,40169,40171,40191],{"className":40170,"translate":35},[34],[31,40172,40174],{"className":40173},[39],[41,40175,40176],{"xmlns":43},[45,40177,40178,40188],{},[48,40179,40180,40182,40184,40186],{},[51,40181,1662],{},[95,40183,211],{"stretchy":210},[51,40185,53],{},[95,40187,220],{"stretchy":210},[55,40189,40190],{"encoding":57},"f(x)",[31,40192,40194],{"className":40193,"ariaHidden":62},[61],[31,40195,40197,40200,40203,40206,40209],{"className":40196},[66],[31,40198],{"className":40199,"style":260},[70],[31,40201,1662],{"className":40202,"style":35196},[75,76],[31,40204,211],{"className":40205},[264],[31,40207,53],{"className":40208},[75,76],[31,40210,220],{"className":40211},[289]," 那一项消掉了，剩下的是 ",[31,40214,40216,40229],{"className":40215,"translate":35},[34],[31,40217,40219],{"className":40218},[39],[41,40220,40221],{"xmlns":43},[45,40222,40223,40227],{},[48,40224,40225],{},[51,40226,53],{},[55,40228,53],{"encoding":57},[31,40230,40232],{"className":40231,"ariaHidden":62},[61],[31,40233,40235,40238],{"className":40234},[66],[31,40236],{"className":40237,"style":71},[70],[31,40239,53],{"className":40240},[75,76]," 的仿射函数。所以边界仍然是一个超平面。",[11,40243,40244],{},"这个结论有点扫兴，但我觉得它比“Bregman 给出弯曲边界”更有用：Bregman 改变的不是假设空间，而是“垂直于谁”——这个超平面垂直于 Hessian 度量下的法向，而不是欧氏法向。换句话说，它换了一把尺子，没有换刀。",[11,40246,40247,40248,40299,40300,2446,40329,40358,40359,40412,40413,40442,40443,40472],{},"具体到二维单纯形（其实就是线段 ",[31,40249,40251,40272],{"className":40250,"translate":35},[34],[31,40252,40254],{"className":40253},[39],[41,40255,40256],{"xmlns":43},[45,40257,40258,40270],{},[48,40259,40260,40262,40264,40266,40268],{},[95,40261,15175],{"stretchy":210},[91,40263,1977],{},[95,40265,1210],{"separator":62},[91,40267,205],{},[95,40269,15184],{"stretchy":210},[55,40271,15187],{"encoding":57},[31,40273,40275],{"className":40274,"ariaHidden":62},[61],[31,40276,40278,40281,40284,40287,40290,40293,40296],{"className":40277},[66],[31,40279],{"className":40280,"style":260},[70],[31,40282,15175],{"className":40283},[264],[31,40285,1977],{"className":40286},[75],[31,40288,1210],{"className":40289},[1391],[31,40291],{"className":40292,"style":1467},[119],[31,40294,205],{"className":40295},[75],[31,40297,15184],{"className":40298},[289],"）上，两个类原型 ",[31,40301,40303,40317],{"className":40302,"translate":35},[34],[31,40304,40306],{"className":40305},[39],[41,40307,40308],{"xmlns":43},[45,40309,40310,40315],{},[48,40311,40312],{},[91,40313,40314],{},"0.2",[55,40316,40314],{"encoding":57},[31,40318,40320],{"className":40319,"ariaHidden":62},[61],[31,40321,40323,40326],{"className":40322},[66],[31,40324],{"className":40325,"style":326},[70],[31,40327,40314],{"className":40328},[75],[31,40330,40332,40346],{"className":40331,"translate":35},[34],[31,40333,40335],{"className":40334},[39],[41,40336,40337],{"xmlns":43},[45,40338,40339,40344],{},[48,40340,40341],{},[91,40342,40343],{},"0.6",[55,40345,40343],{"encoding":57},[31,40347,40349],{"className":40348,"ariaHidden":62},[61],[31,40350,40352,40355],{"className":40351},[66],[31,40353],{"className":40354,"style":326},[70],[31,40356,40343],{"className":40357},[75],"，用 KL 判类的分界点在 ",[31,40360,40362,40381],{"className":40361,"translate":35},[34],[31,40363,40365],{"className":40364},[39],[41,40366,40367],{"xmlns":43},[45,40368,40369,40378],{},[48,40370,40371,40373,40375],{},[51,40372,11],{},[95,40374,27223],{},[91,40376,40377],{},"0.387",[55,40379,40380],{"encoding":57},"p\\approx0.387",[31,40382,40384,40403],{"className":40383,"ariaHidden":62},[61],[31,40385,40387,40391,40394,40397,40400],{"className":40386},[66],[31,40388],{"className":40389,"style":40390},[70],"height:0.6776em;vertical-align:-0.1944em;",[31,40392,11],{"className":40393},[75,76],[31,40395],{"className":40396,"style":293},[119],[31,40398,27223],{"className":40399},[297],[31,40401],{"className":40402,"style":293},[119],[31,40404,40406,40409],{"className":40405},[66],[31,40407],{"className":40408,"style":326},[70],[31,40410,40377],{"className":40411},[75],"；欧氏中点是 ",[31,40414,40416,40430],{"className":40415,"translate":35},[34],[31,40417,40419],{"className":40418},[39],[41,40420,40421],{"xmlns":43},[45,40422,40423,40428],{},[48,40424,40425],{},[91,40426,40427],{},"0.4",[55,40429,40427],{"encoding":57},[31,40431,40433],{"className":40432,"ariaHidden":62},[61],[31,40434,40436,40439],{"className":40435},[66],[31,40437],{"className":40438,"style":326},[70],[31,40440,40427],{"className":40441},[75],"；而在 logit 坐标里取中点又得到 ",[31,40444,40446,40460],{"className":40445,"translate":35},[34],[31,40447,40449],{"className":40448},[39],[41,40450,40451],{"xmlns":43},[45,40452,40453,40458],{},[48,40454,40455],{},[91,40456,40457],{},"0.380",[55,40459,40457],{"encoding":57},[31,40461,40463],{"className":40462,"ariaHidden":62},[61],[31,40464,40466,40469],{"className":40465},[66],[31,40467],{"className":40468,"style":326},[70],[31,40470,40457],{"className":40471},[75],"。三个数，三把尺子，没有哪把是“错”的，取决于你认为什么该被同等对待。",[11,40474,40475,40476,40580,40581,25085,40906,40934,40935,40963],{},"真正弯的边界来自两个地方。一是把比较放在约束集上：单纯形加 Fisher–Rao，就是上一节的大圆。二是把它放在非线性映射之后：如果 ",[31,40477,40479,40525],{"className":40478,"translate":35},[34],[31,40480,40482],{"className":40481},[39],[41,40483,40484],{"xmlns":43},[45,40485,40486,40522],{},[48,40487,40488,40490,40492,40494,40496,40498,40514,40516,40518,40520],{},[51,40489,27666],{},[95,40491,211],{"stretchy":210},[51,40493,53],{},[95,40495,220],{"stretchy":210},[95,40497,223],{},[48,40499,40500,40502,40504,40506,40508,40510,40512],{},[51,40501,14551],{"mathvariant":494},[51,40503,2552],{"mathvariant":494},[51,40505,1662],{"mathvariant":494},[51,40507,12271],{"mathvariant":494},[51,40509,16396],{"mathvariant":494},[51,40511,5813],{"mathvariant":494},[51,40513,53],{"mathvariant":494},[95,40515,211],{"stretchy":210},[51,40517,3117],{},[51,40519,53],{},[95,40521,220],{"stretchy":210},[55,40523,40524],{"encoding":57},"q(x)=\\mathrm{softmax}(Wx)",[31,40526,40528,40555],{"className":40527,"ariaHidden":62},[61],[31,40529,40531,40534,40537,40540,40543,40546,40549,40552],{"className":40530},[66],[31,40532],{"className":40533,"style":260},[70],[31,40535,27666],{"className":40536,"style":2666},[75,76],[31,40538,211],{"className":40539},[264],[31,40541,53],{"className":40542},[75,76],[31,40544,220],{"className":40545},[289],[31,40547],{"className":40548,"style":293},[119],[31,40550,223],{"className":40551},[297],[31,40553],{"className":40554,"style":293},[119],[31,40556,40558,40561,40568,40571,40574,40577],{"className":40557},[66],[31,40559],{"className":40560,"style":260},[70],[31,40562,40564],{"className":40563},[75],[31,40565,40567],{"className":40566},[75,2745],"softmax",[31,40569,211],{"className":40570},[264],[31,40572,3117],{"className":40573,"style":949},[75,76],[31,40575,53],{"className":40576},[75,76],[31,40578,220],{"className":40579},[289],"，那么 ",[31,40582,40584,40658],{"className":40583,"translate":35},[34],[31,40585,40587],{"className":40586},[39],[41,40588,40589],{"xmlns":43},[45,40590,40591,40655],{},[48,40592,40593,40595,40597,40599,40605,40607,40609,40611,40613,40615,40617,40623,40625,40627,40633,40635,40637,40639,40641,40643,40645,40651,40653],{},[95,40594,2512],{"stretchy":210},[51,40596,53],{},[95,40598,922],{},[1004,40600,40601,40603],{},[51,40602,34614],{},[51,40604,1662],{},[95,40606,211],{"stretchy":210},[51,40608,27666],{},[95,40610,211],{"stretchy":210},[51,40612,53],{},[95,40614,220],{"stretchy":210},[51,40616,4572],{"mathvariant":494},[1004,40618,40619,40621],{},[51,40620,1412],{},[91,40622,205],{},[95,40624,220],{"stretchy":210},[95,40626,223],{},[1004,40628,40629,40631],{},[51,40630,34614],{},[51,40632,1662],{},[95,40634,211],{"stretchy":210},[51,40636,27666],{},[95,40638,211],{"stretchy":210},[51,40640,53],{},[95,40642,220],{"stretchy":210},[51,40644,4572],{"mathvariant":494},[1004,40646,40647,40649],{},[51,40648,1412],{},[91,40650,152],{},[95,40652,220],{"stretchy":210},[95,40654,2604],{"stretchy":210},[55,40656,40657],{"encoding":57},"\\{x:D_f(q(x)\\|c_1)=D_f(q(x)\\|c_2)\\}",[31,40659,40661,40682,40798],{"className":40660,"ariaHidden":62},[61],[31,40662,40664,40667,40670,40673,40676,40679],{"className":40663},[66],[31,40665],{"className":40666,"style":260},[70],[31,40668,2512],{"className":40669},[264],[31,40671,53],{"className":40672},[75,76],[31,40674],{"className":40675,"style":293},[119],[31,40677,922],{"className":40678},[297],[31,40680],{"className":40681,"style":293},[119],[31,40683,40685,40688,40728,40731,40734,40737,40740,40743,40746,40786,40789,40792,40795],{"className":40684},[66],[31,40686],{"className":40687,"style":1223},[70],[31,40689,40691,40694],{"className":40690},[75],[31,40692,34614],{"className":40693,"style":19759},[75,76],[31,40695,40697],{"className":40696},[1039],[31,40698,40700,40720],{"className":40699},[1043,1044],[31,40701,40703,40717],{"className":40702},[1048],[31,40704,40706],{"className":40705,"style":1053},[1052],[31,40707,40708,40711],{"style":26726},[31,40709],{"className":40710,"style":1061},[1060],[31,40712,40714],{"className":40713},[1065,1066,1067,1068],[31,40715,1662],{"className":40716,"style":35196},[75,76,1068],[31,40718,1076],{"className":40719},[1075],[31,40721,40723],{"className":40722},[1048],[31,40724,40726],{"className":40725,"style":1083},[1052],[31,40727],{},[31,40729,211],{"className":40730},[264],[31,40732,27666],{"className":40733,"style":2666},[75,76],[31,40735,211],{"className":40736},[264],[31,40738,53],{"className":40739},[75,76],[31,40741,220],{"className":40742},[289],[31,40744,4572],{"className":40745},[75],[31,40747,40749,40752],{"className":40748},[75],[31,40750,1412],{"className":40751},[75,76],[31,40753,40755],{"className":40754},[1039],[31,40756,40758,40778],{"className":40757},[1043,1044],[31,40759,40761,40775],{"className":40760},[1048],[31,40762,40764],{"className":40763,"style":2028},[1052],[31,40765,40766,40769],{"style":2807},[31,40767],{"className":40768,"style":1061},[1060],[31,40770,40772],{"className":40771},[1065,1066,1067,1068],[31,40773,205],{"className":40774},[75,1068],[31,40776,1076],{"className":40777},[1075],[31,40779,40781],{"className":40780},[1048],[31,40782,40784],{"className":40783,"style":2050},[1052],[31,40785],{},[31,40787,220],{"className":40788},[289],[31,40790],{"className":40791,"style":293},[119],[31,40793,223],{"className":40794},[297],[31,40796],{"className":40797,"style":293},[119],[31,40799,40801,40804,40844,40847,40850,40853,40856,40859,40862,40902],{"className":40800},[66],[31,40802],{"className":40803,"style":1223},[70],[31,40805,40807,40810],{"className":40806},[75],[31,40808,34614],{"className":40809,"style":19759},[75,76],[31,40811,40813],{"className":40812},[1039],[31,40814,40816,40836],{"className":40815},[1043,1044],[31,40817,40819,40833],{"className":40818},[1048],[31,40820,40822],{"className":40821,"style":1053},[1052],[31,40823,40824,40827],{"style":26726},[31,40825],{"className":40826,"style":1061},[1060],[31,40828,40830],{"className":40829},[1065,1066,1067,1068],[31,40831,1662],{"className":40832,"style":35196},[75,76,1068],[31,40834,1076],{"className":40835},[1075],[31,40837,40839],{"className":40838},[1048],[31,40840,40842],{"className":40841,"style":1083},[1052],[31,40843],{},[31,40845,211],{"className":40846},[264],[31,40848,27666],{"className":40849,"style":2666},[75,76],[31,40851,211],{"className":40852},[264],[31,40854,53],{"className":40855},[75,76],[31,40857,220],{"className":40858},[289],[31,40860,4572],{"className":40861},[75],[31,40863,40865,40868],{"className":40864},[75],[31,40866,1412],{"className":40867},[75,76],[31,40869,40871],{"className":40870},[1039],[31,40872,40874,40894],{"className":40873},[1043,1044],[31,40875,40877,40891],{"className":40876},[1048],[31,40878,40880],{"className":40879,"style":2028},[1052],[31,40881,40882,40885],{"style":2807},[31,40883],{"className":40884,"style":1061},[1060],[31,40886,40888],{"className":40887},[1065,1066,1067,1068],[31,40889,152],{"className":40890},[75,1068],[31,40892,1076],{"className":40893},[1075],[31,40895,40897],{"className":40896},[1048],[31,40898,40900],{"className":40899,"style":2050},[1052],[31,40901],{},[31,40903,40905],{"className":40904},[289],")}",[31,40907,40909,40922],{"className":40908,"translate":35},[34],[31,40910,40912],{"className":40911},[39],[41,40913,40914],{"xmlns":43},[45,40915,40916,40920],{},[48,40917,40918],{},[51,40919,27666],{},[55,40921,27666],{"encoding":57},[31,40923,40925],{"className":40924,"ariaHidden":62},[61],[31,40926,40928,40931],{"className":40927},[66],[31,40929],{"className":40930,"style":865},[70],[31,40932,27666],{"className":40933,"style":2666},[75,76]," 空间是超平面，在 ",[31,40936,40938,40951],{"className":40937,"translate":35},[34],[31,40939,40941],{"className":40940},[39],[41,40942,40943],{"xmlns":43},[45,40944,40945,40949],{},[48,40946,40947],{},[51,40948,53],{},[55,40950,53],{"encoding":57},[31,40952,40954],{"className":40953,"ariaHidden":62},[61],[31,40955,40957,40960],{"className":40956},[66],[31,40958],{"className":40959,"style":71},[70],[31,40961,53],{"className":40962},[75,76]," 空间是它的原像，一般是一张弯的曲面。我见到的“非欧氏边界”基本都是后者。",[11,40965,40966],{},"Bregman 还给出两个优化工具。一个是镜像下降：",[31,40968,40970],{"className":40969,"translate":35},[411],[31,40971,40973,41059],{"className":40972,"translate":35},[34],[31,40974,40976],{"className":40975},[39],[41,40977,40978],{"xmlns":43,"display":420},[45,40979,40980,41056],{},[48,40981,40982,40994,40996,40998,41000,41010,41012,41014,41020,41022,41024,41026,41028,41034,41040,41042,41044,41046,41052,41054],{},[1004,40983,40984,40986],{},[51,40985,53],{},[48,40987,40988,40990,40992],{},[51,40989,12271],{},[95,40991,434],{},[91,40993,205],{},[95,40995,223],{},[51,40997,37263],{},[95,40999,1540],{},[1531,41001,41002,41008],{},[48,41003,41004,41006],{},[51,41005,19878],{},[95,41007,1540],{},[51,41009,53],{},[1430,41011,1545],{},[95,41013,19903],{"stretchy":210},[1004,41015,41016,41018],{},[51,41017,14556],{},[51,41019,12271],{},[95,41021,1210],{"separator":62},[51,41023,53],{},[95,41025,19912],{"stretchy":210},[95,41027,434],{},[2074,41029,41030,41032],{},[91,41031,205],{},[51,41033,30542],{},[1004,41035,41036,41038],{},[51,41037,34614],{},[51,41039,1662],{},[95,41041,211],{"stretchy":210},[51,41043,53],{},[51,41045,4572],{"mathvariant":494},[1004,41047,41048,41050],{},[51,41049,53],{},[51,41051,12271],{},[95,41053,220],{"stretchy":210},[95,41055,1210],{"separator":62},[55,41057,41058],{"encoding":57},"x_{t+1}=\\arg\\min_x\\ \\langle g_t,x\\rangle+\\frac1\\eta D_f(x\\|x_t),",[31,41060,41062,41127,41253],{"className":41061,"ariaHidden":62},[61],[31,41063,41065,41069,41118,41121,41124],{"className":41064},[66],[31,41066],{"className":41067,"style":41068},[70],"height:0.6389em;vertical-align:-0.2083em;",[31,41070,41072,41075],{"className":41071},[75],[31,41073,53],{"className":41074},[75,76],[31,41076,41078],{"className":41077},[1039],[31,41079,41081,41110],{"className":41080},[1043,1044],[31,41082,41084,41107],{"className":41083},[1048],[31,41085,41087],{"className":41086,"style":2028},[1052],[31,41088,41089,41092],{"style":2807},[31,41090],{"className":41091,"style":1061},[1060],[31,41093,41095],{"className":41094},[1065,1066,1067,1068],[31,41096,41098,41101,41104],{"className":41097},[75,1068],[31,41099,12271],{"className":41100},[75,76,1068],[31,41102,434],{"className":41103},[124,1068],[31,41105,205],{"className":41106},[75,1068],[31,41108,1076],{"className":41109},[1075],[31,41111,41113],{"className":41112},[1048],[31,41114,41116],{"className":41115,"style":25082},[1052],[31,41117],{},[31,41119],{"className":41120,"style":293},[119],[31,41122,223],{"className":41123},[297],[31,41125],{"className":41126,"style":293},[119],[31,41128,41130,41134,41139,41142,41186,41189,41192,41232,41235,41238,41241,41244,41247,41250],{"className":41129},[66],[31,41131],{"className":41132,"style":41133},[70],"height:1.45em;vertical-align:-0.7em;",[31,41135,37341,41137],{"className":41136},[1460],[31,41138,14556],{"style":20180},[31,41140],{"className":41141,"style":1467},[119],[31,41143,41145],{"className":41144},[1460,1620],[31,41146,41148,41178],{"className":41147},[1043,1044],[31,41149,41151,41175],{"className":41150},[1048],[31,41152,41154,41165],{"className":41153,"style":19979},[1052],[31,41155,41156,41159],{"style":37361},[31,41157],{"className":41158,"style":1637},[1060],[31,41160,41162],{"className":41161},[1065,1066,1067,1068],[31,41163,53],{"className":41164},[75,76,1068],[31,41166,41167,41170],{"style":1649},[31,41168],{"className":41169,"style":1637},[1060],[31,41171,41172],{},[31,41173,19878],{"className":41174},[1460],[31,41176,1076],{"className":41177},[1075],[31,41179,41181],{"className":41180},[1048],[31,41182,41184],{"className":41183,"style":37390},[1052],[31,41185],{},[31,41187,1545],{"className":41188},[119],[31,41190,19903],{"className":41191},[264],[31,41193,41195,41198],{"className":41194},[75],[31,41196,14556],{"className":41197,"style":2666},[75,76],[31,41199,41201],{"className":41200},[1039],[31,41202,41204,41224],{"className":41203},[1043,1044],[31,41205,41207,41221],{"className":41206},[1048],[31,41208,41210],{"className":41209,"style":19220},[1052],[31,41211,41212,41215],{"style":5504},[31,41213],{"className":41214,"style":1061},[1060],[31,41216,41218],{"className":41217},[1065,1066,1067,1068],[31,41219,12271],{"className":41220},[75,76,1068],[31,41222,1076],{"className":41223},[1075],[31,41225,41227],{"className":41226},[1048],[31,41228,41230],{"className":41229,"style":2050},[1052],[31,41231],{},[31,41233,1210],{"className":41234},[1391],[31,41236],{"className":41237,"style":1467},[119],[31,41239,53],{"className":41240},[75,76],[31,41242,19912],{"className":41243},[289],[31,41245],{"className":41246,"style":120},[119],[31,41248,434],{"className":41249},[124],[31,41251],{"className":41252,"style":120},[119],[31,41254,41256,41260,41323,41363,41366,41369,41372,41412,41415],{"className":41255},[66],[31,41257],{"className":41258,"style":41259},[70],"height:2.2019em;vertical-align:-0.8804em;",[31,41261,41263,41266,41320],{"className":41262},[75],[31,41264],{"className":41265},[264,2139],[31,41267,41269],{"className":41268},[2074],[31,41270,41272,41311],{"className":41271},[1043,1044],[31,41273,41275,41308],{"className":41274},[1048],[31,41276,41278,41289,41297],{"className":41277,"style":26376},[1052],[31,41279,41280,41283],{"style":5090},[31,41281],{"className":41282,"style":1637},[1060],[31,41284,41286],{"className":41285},[75],[31,41287,30542],{"className":41288,"style":2666},[75,76],[31,41290,41291,41294],{"style":2170},[31,41292],{"className":41293,"style":1637},[1060],[31,41295],{"className":41296,"style":2178},[2177],[31,41298,41299,41302],{"style":5110},[31,41300],{"className":41301,"style":1637},[1060],[31,41303,41305],{"className":41304},[75],[31,41306,205],{"className":41307},[75],[31,41309,1076],{"className":41310},[1075],[31,41312,41314],{"className":41313},[1048],[31,41315,41318],{"className":41316,"style":41317},[1052],"height:0.8804em;",[31,41319],{},[31,41321],{"className":41322},[289,2139],[31,41324,41326,41329],{"className":41325},[75],[31,41327,34614],{"className":41328,"style":19759},[75,76],[31,41330,41332],{"className":41331},[1039],[31,41333,41335,41355],{"className":41334},[1043,1044],[31,41336,41338,41352],{"className":41337},[1048],[31,41339,41341],{"className":41340,"style":1053},[1052],[31,41342,41343,41346],{"style":26726},[31,41344],{"className":41345,"style":1061},[1060],[31,41347,41349],{"className":41348},[1065,1066,1067,1068],[31,41350,1662],{"className":41351,"style":35196},[75,76,1068],[31,41353,1076],{"className":41354},[1075],[31,41356,41358],{"className":41357},[1048],[31,41359,41361],{"className":41360,"style":1083},[1052],[31,41362],{},[31,41364,211],{"className":41365},[264],[31,41367,53],{"className":41368},[75,76],[31,41370,4572],{"className":41371},[75],[31,41373,41375,41378],{"className":41374},[75],[31,41376,53],{"className":41377},[75,76],[31,41379,41381],{"className":41380},[1039],[31,41382,41384,41404],{"className":41383},[1043,1044],[31,41385,41387,41401],{"className":41386},[1048],[31,41388,41390],{"className":41389,"style":19220},[1052],[31,41391,41392,41395],{"style":2807},[31,41393],{"className":41394,"style":1061},[1060],[31,41396,41398],{"className":41397},[1065,1066,1067,1068],[31,41399,12271],{"className":41400},[75,76,1068],[31,41402,1076],{"className":41403},[1075],[31,41405,41407],{"className":41406},[1048],[31,41408,41410],{"className":41409,"style":2050},[1052],[31,41411],{},[31,41413,220],{"className":41414},[289],[31,41416,1210],{"className":41417},[1391],[11,41419,41420,41421,41449,41450,41617,41618,41646,41647,41892,41893,41921],{},"直觉是“走一步，但别离原来的位置太远”，而“太远”用 ",[31,41422,41424,41437],{"className":41423,"translate":35},[34],[31,41425,41427],{"className":41426},[39],[41,41428,41429],{"xmlns":43},[45,41430,41431,41435],{},[48,41432,41433],{},[51,41434,1662],{},[55,41436,1662],{"encoding":57},[31,41438,41440],{"className":41439,"ariaHidden":62},[61],[31,41441,41443,41446],{"className":41442},[66],[31,41444],{"className":41445,"style":4202},[70],[31,41447,1662],{"className":41448,"style":35196},[75,76]," 来量。取 ",[31,41451,41453,41487],{"className":41452,"translate":35},[34],[31,41454,41456],{"className":41455},[39],[41,41457,41458],{"xmlns":43},[45,41459,41460,41484],{},[48,41461,41462,41464,41466,41474,41476,41478],{},[51,41463,1662],{},[95,41465,223],{},[2071,41467,41468],{"scriptlevel":1977,"displaystyle":210},[2074,41469,41470,41472],{},[91,41471,205],{},[91,41473,152],{},[51,41475,4572],{"mathvariant":494},[51,41477,53],{},[1180,41479,41480,41482],{},[51,41481,4572],{"mathvariant":494},[91,41483,152],{},[55,41485,41486],{"encoding":57},"f=\\tfrac12\\|x\\|^2",[31,41488,41490,41508],{"className":41489,"ariaHidden":62},[61],[31,41491,41493,41496,41499,41502,41505],{"className":41492},[66],[31,41494],{"className":41495,"style":4202},[70],[31,41497,1662],{"className":41498,"style":35196},[75,76],[31,41500],{"className":41501,"style":293},[119],[31,41503,223],{"className":41504},[297],[31,41506],{"className":41507,"style":293},[119],[31,41509,41511,41514,41582,41585,41588],{"className":41510},[66],[31,41512],{"className":41513,"style":2132},[70],[31,41515,41517,41520,41579],{"className":41516},[75],[31,41518],{"className":41519},[264,2139],[31,41521,41523],{"className":41522},[2074],[31,41524,41526,41571],{"className":41525},[1043,1044],[31,41527,41529,41568],{"className":41528},[1048],[31,41530,41532,41546,41554],{"className":41531,"style":2152},[1052],[31,41533,41534,41537],{"style":2155},[31,41535],{"className":41536,"style":1637},[1060],[31,41538,41540],{"className":41539},[1065,1066,1067,1068],[31,41541,41543],{"className":41542},[75,1068],[31,41544,152],{"className":41545},[75,1068],[31,41547,41548,41551],{"style":2170},[31,41549],{"className":41550,"style":1637},[1060],[31,41552],{"className":41553,"style":2178},[2177],[31,41555,41556,41559],{"style":2181},[31,41557],{"className":41558,"style":1637},[1060],[31,41560,41562],{"className":41561},[1065,1066,1067,1068],[31,41563,41565],{"className":41564},[75,1068],[31,41566,205],{"className":41567},[75,1068],[31,41569,1076],{"className":41570},[1075],[31,41572,41574],{"className":41573},[1048],[31,41575,41577],{"className":41576,"style":2203},[1052],[31,41578],{},[31,41580],{"className":41581},[289,2139],[31,41583,4572],{"className":41584},[75],[31,41586,53],{"className":41587},[75,76],[31,41589,41591,41594],{"className":41590},[75],[31,41592,4572],{"className":41593},[75],[31,41595,41597],{"className":41596},[1039],[31,41598,41600],{"className":41599},[1043],[31,41601,41603],{"className":41602},[1048],[31,41604,41606],{"className":41605,"style":5135},[1052],[31,41607,41608,41611],{"style":3631},[31,41609],{"className":41610,"style":1061},[1060],[31,41612,41614],{"className":41613},[1065,1066,1067,1068],[31,41615,152],{"className":41616},[75,1068]," 就是梯度下降；取 ",[31,41619,41621,41634],{"className":41620,"translate":35},[34],[31,41622,41624],{"className":41623},[39],[41,41625,41626],{"xmlns":43},[45,41627,41628,41632],{},[48,41629,41630],{},[51,41631,1662],{},[55,41633,1662],{"encoding":57},[31,41635,41637],{"className":41636,"ariaHidden":62},[61],[31,41638,41640,41643],{"className":41639},[66],[31,41641],{"className":41642,"style":4202},[70],[31,41644,1662],{"className":41645,"style":35196},[75,76]," 为负熵，在单纯形上得到指数梯度／乘法权重更新，更新式是乘性的，天然保持非负。另一个是自然梯度：",[31,41648,41650,41708],{"className":41649,"translate":35},[34],[31,41651,41653],{"className":41652},[39],[41,41654,41655],{"xmlns":43},[45,41656,41657,41705],{},[48,41658,41659,41671,41673,41679,41681,41683,41685,41687,41689,41691,41701,41703],{},[1004,41660,41661,41663],{},[51,41662,23798],{},[48,41664,41665,41667,41669],{},[51,41666,12271],{},[95,41668,434],{},[91,41670,205],{},[95,41672,223],{},[1004,41674,41675,41677],{},[51,41676,23798],{},[51,41678,12271],{},[95,41680,97],{},[51,41682,30542],{},[1430,41684,1432],{},[51,41686,15297],{},[95,41688,211],{"stretchy":210},[51,41690,23798],{},[1180,41692,41693,41695],{},[95,41694,220],{"stretchy":210},[48,41696,41697,41699],{},[95,41698,97],{},[91,41700,205],{},[51,41702,12145],{"mathvariant":494},[51,41704,4561],{},[55,41706,41707],{"encoding":57},"\\theta_{t+1}=\\theta_t-\\eta\\,G(\\theta)^{-1}\\nabla L",[31,41709,41711,41775,41830],{"className":41710,"ariaHidden":62},[61],[31,41712,41714,41717,41766,41769,41772],{"className":41713},[66],[31,41715],{"className":41716,"style":25023},[70],[31,41718,41720,41723],{"className":41719},[75],[31,41721,23798],{"className":41722,"style":19759},[75,76],[31,41724,41726],{"className":41725},[1039],[31,41727,41729,41758],{"className":41728},[1043,1044],[31,41730,41732,41755],{"className":41731},[1048],[31,41733,41735],{"className":41734,"style":2028},[1052],[31,41736,41737,41740],{"style":26726},[31,41738],{"className":41739,"style":1061},[1060],[31,41741,41743],{"className":41742},[1065,1066,1067,1068],[31,41744,41746,41749,41752],{"className":41745},[75,1068],[31,41747,12271],{"className":41748},[75,76,1068],[31,41750,434],{"className":41751},[124,1068],[31,41753,205],{"className":41754},[75,1068],[31,41756,1076],{"className":41757},[1075],[31,41759,41761],{"className":41760},[1048],[31,41762,41764],{"className":41763,"style":25082},[1052],[31,41765],{},[31,41767],{"className":41768,"style":293},[119],[31,41770,223],{"className":41771},[297],[31,41773],{"className":41774,"style":293},[119],[31,41776,41778,41781,41821,41824,41827],{"className":41777},[66],[31,41779],{"className":41780,"style":2008},[70],[31,41782,41784,41787],{"className":41783},[75],[31,41785,23798],{"className":41786,"style":19759},[75,76],[31,41788,41790],{"className":41789},[1039],[31,41791,41793,41813],{"className":41792},[1043,1044],[31,41794,41796,41810],{"className":41795},[1048],[31,41797,41799],{"className":41798,"style":19220},[1052],[31,41800,41801,41804],{"style":26726},[31,41802],{"className":41803,"style":1061},[1060],[31,41805,41807],{"className":41806},[1065,1066,1067,1068],[31,41808,12271],{"className":41809},[75,76,1068],[31,41811,1076],{"className":41812},[1075],[31,41814,41816],{"className":41815},[1048],[31,41817,41819],{"className":41818,"style":2050},[1052],[31,41820],{},[31,41822],{"className":41823,"style":120},[119],[31,41825,97],{"className":41826},[124],[31,41828],{"className":41829,"style":120},[119],[31,41831,41833,41836,41839,41842,41845,41848,41851,41886,41889],{"className":41832},[66],[31,41834],{"className":41835,"style":12035},[70],[31,41837,30542],{"className":41838,"style":2666},[75,76],[31,41840],{"className":41841,"style":1467},[119],[31,41843,15297],{"className":41844},[75,76],[31,41846,211],{"className":41847},[264],[31,41849,23798],{"className":41850,"style":19759},[75,76],[31,41852,41854,41857],{"className":41853},[289],[31,41855,220],{"className":41856},[289],[31,41858,41860],{"className":41859},[1039],[31,41861,41863],{"className":41862},[1043],[31,41864,41866],{"className":41865},[1048],[31,41867,41869],{"className":41868,"style":5135},[1052],[31,41870,41871,41874],{"style":3631},[31,41872],{"className":41873,"style":1061},[1060],[31,41875,41877],{"className":41876},[1065,1066,1067,1068],[31,41878,41880,41883],{"className":41879},[75,1068],[31,41881,97],{"className":41882},[75,1068],[31,41884,205],{"className":41885},[75,1068],[31,41887,12145],{"className":41888},[75],[31,41890,4561],{"className":41891},[75,76],"，其中 ",[31,41894,41896,41909],{"className":41895,"translate":35},[34],[31,41897,41899],{"className":41898},[39],[41,41900,41901],{"xmlns":43},[45,41902,41903,41907],{},[48,41904,41905],{},[51,41906,15297],{},[55,41908,15297],{"encoding":57},[31,41910,41912],{"className":41911,"ariaHidden":62},[61],[31,41913,41915,41918],{"className":41914},[66],[31,41916],{"className":41917,"style":945},[70],[31,41919,15297],{"className":41920},[75,76]," 是 Fisher。它是在 Fisher–Rao 度量下做最速下降，而不是在欧氏度量下。对 softmax 回归，自然梯度会消掉 logits 尺度带来的平台期——因为在 Fisher 度量下，那个尺度根本不算“距离”。",[18,41923,41924],{"id":41924},"点态密度",[11,41926,41927,41928,42034,42035,42063],{},"手上有五个一维样本：",[31,41929,41931,41973],{"className":41930,"translate":35},[34],[31,41932,41934],{"className":41933},[39],[41,41935,41936],{"xmlns":43},[45,41937,41938,41970],{},[48,41939,41940,41942,41944,41946,41949,41951,41953,41956,41958,41960,41963,41965,41967],{},[91,41941,23247],{},[95,41943,1210],{"separator":62},[1430,41945,1545],{},[91,41947,41948],{},"1.0",[95,41950,1210],{"separator":62},[1430,41952,1545],{},[91,41954,41955],{},"1.1",[95,41957,1210],{"separator":62},[1430,41959,1545],{},[91,41961,41962],{},"5.0",[95,41964,1210],{"separator":62},[1430,41966,1545],{},[91,41968,41969],{},"5.1",[55,41971,41972],{"encoding":57},"0.9,\\ 1.0,\\ 1.1,\\ 5.0,\\ 5.1",[31,41974,41976],{"className":41975,"ariaHidden":62},[61],[31,41977,41979,41983,41986,41989,41992,41995,41998,42001,42004,42007,42010,42013,42016,42019,42022,42025,42028,42031],{"className":41978},[66],[31,41980],{"className":41981,"style":41982},[70],"height:0.8389em;vertical-align:-0.1944em;",[31,41984,23247],{"className":41985},[75],[31,41987,1210],{"className":41988},[1391],[31,41990,1545],{"className":41991},[119],[31,41993],{"className":41994,"style":1467},[119],[31,41996,41948],{"className":41997},[75],[31,41999,1210],{"className":42000},[1391],[31,42002,1545],{"className":42003},[119],[31,42005],{"className":42006,"style":1467},[119],[31,42008,41955],{"className":42009},[75],[31,42011,1210],{"className":42012},[1391],[31,42014,1545],{"className":42015},[119],[31,42017],{"className":42018,"style":1467},[119],[31,42020,41962],{"className":42021},[75],[31,42023,1210],{"className":42024},[1391],[31,42026,1545],{"className":42027},[119],[31,42029],{"className":42030,"style":1467},[119],[31,42032,41969],{"className":42033},[75],"。肉眼看出两簇，中间在 ",[31,42036,42038,42051],{"className":42037,"translate":35},[34],[31,42039,42041],{"className":42040},[39],[41,42042,42043],{"xmlns":43},[45,42044,42045,42049],{},[48,42046,42047],{},[91,42048,93],{},[55,42050,93],{"encoding":57},[31,42052,42054],{"className":42053,"ariaHidden":62},[61],[31,42055,42057,42060],{"className":42056},[66],[31,42058],{"className":42059,"style":326},[70],[31,42061,93],{"className":42062},[75]," 附近是空的。",[11,42065,42066,42067,42096,42097,42125,42126,42177,42178,42207,42208,42259,42260,42289,42290,42293],{},"现在用核密度估计去拟合，只用高斯核、只改带宽。带宽取 ",[31,42068,42070,42084],{"className":42069,"translate":35},[34],[31,42071,42073],{"className":42072},[39],[41,42074,42075],{"xmlns":43},[45,42076,42077,42082],{},[48,42078,42079],{},[91,42080,42081],{},"0.1",[55,42083,42081],{"encoding":57},[31,42085,42087],{"className":42086,"ariaHidden":62},[61],[31,42088,42090,42093],{"className":42089},[66],[31,42091],{"className":42092,"style":326},[70],[31,42094,42081],{"className":42095},[75],"，每个点都是一个尖峰，曲线在两个簇内部剧烈起伏；带宽取 ",[31,42098,42100,42113],{"className":42099,"translate":35},[34],[31,42101,42103],{"className":42102},[39],[41,42104,42105],{"xmlns":43},[45,42106,42107,42111],{},[48,42108,42109],{},[91,42110,7853],{},[55,42112,7853],{"encoding":57},[31,42114,42116],{"className":42115,"ariaHidden":62},[61],[31,42117,42119,42122],{"className":42118},[66],[31,42120],{"className":42121,"style":326},[70],[31,42123,7853],{"className":42124},[75],"，曲线被抹平。抹平到什么程度？算一下会发现，",[31,42127,42129,42147],{"className":42128,"translate":35},[34],[31,42130,42132],{"className":42131},[39],[41,42133,42134],{"xmlns":43},[45,42135,42136,42144],{},[48,42137,42138,42140,42142],{},[51,42139,53],{},[95,42141,223],{},[91,42143,93],{},[55,42145,42146],{"encoding":57},"x=3",[31,42148,42150,42168],{"className":42149,"ariaHidden":62},[61],[31,42151,42153,42156,42159,42162,42165],{"className":42152},[66],[31,42154],{"className":42155,"style":71},[70],[31,42157,53],{"className":42158},[75,76],[31,42160],{"className":42161,"style":293},[119],[31,42163,223],{"className":42164},[297],[31,42166],{"className":42167,"style":293},[119],[31,42169,42171,42174],{"className":42170},[66],[31,42172],{"className":42173,"style":326},[70],[31,42175,93],{"className":42176},[75]," 那个空洞处的密度大约是 ",[31,42179,42181,42195],{"className":42180,"translate":35},[34],[31,42182,42184],{"className":42183},[39],[41,42185,42186],{"xmlns":43},[45,42187,42188,42193],{},[48,42189,42190],{},[91,42191,42192],{},"0.073",[55,42194,42192],{"encoding":57},[31,42196,42198],{"className":42197,"ariaHidden":62},[61],[31,42199,42201,42204],{"className":42200},[66],[31,42202],{"className":42203,"style":326},[70],[31,42205,42192],{"className":42206},[75],"，而簇心 ",[31,42209,42211,42229],{"className":42210,"translate":35},[34],[31,42212,42214],{"className":42213},[39],[41,42215,42216],{"xmlns":43},[45,42217,42218,42226],{},[48,42219,42220,42222,42224],{},[51,42221,53],{},[95,42223,223],{},[91,42225,205],{},[55,42227,42228],{"encoding":57},"x=1",[31,42230,42232,42250],{"className":42231,"ariaHidden":62},[61],[31,42233,42235,42238,42241,42244,42247],{"className":42234},[66],[31,42236],{"className":42237,"style":71},[70],[31,42239,53],{"className":42240},[75,76],[31,42242],{"className":42243,"style":293},[119],[31,42245,223],{"className":42246},[297],[31,42248],{"className":42249,"style":293},[119],[31,42251,42253,42256],{"className":42252},[66],[31,42254],{"className":42255,"style":326},[70],[31,42257,205],{"className":42258},[75]," 处是 ",[31,42261,42263,42277],{"className":42262,"translate":35},[34],[31,42264,42266],{"className":42265},[39],[41,42267,42268],{"xmlns":43},[45,42269,42270,42275],{},[48,42271,42272],{},[91,42273,42274],{},"0.071",[55,42276,42274],{"encoding":57},[31,42278,42280],{"className":42279,"ariaHidden":62},[61],[31,42281,42283,42286],{"className":42282},[66],[31,42284],{"className":42285,"style":326},[70],[31,42287,42274],{"className":42288},[75],"——",[7246,42291,42292],{},"空洞反而比簇心更“密”","。因为核太宽，五个点全都够到了中间，而簇心只够到自己人和远处那两个点的尾巴。",[11,42295,42296,42297,42325,42326,42354,42355,42383],{},"这个例子里，带宽就是分辨率。",[31,42298,42300,42313],{"className":42299,"translate":35},[34],[31,42301,42303],{"className":42302},[39],[41,42304,42305],{"xmlns":43},[45,42306,42307,42311],{},[48,42308,42309],{},[51,42310,29501],{},[55,42312,29501],{"encoding":57},[31,42314,42316],{"className":42315,"ariaHidden":62},[61],[31,42317,42319,42322],{"className":42318},[66],[31,42320],{"className":42321,"style":1630},[70],[31,42323,29501],{"className":42324},[75,76]," 太小，方差爆炸；",[31,42327,42329,42342],{"className":42328,"translate":35},[34],[31,42330,42332],{"className":42331},[39],[41,42333,42334],{"xmlns":43},[45,42335,42336,42340],{},[48,42337,42338],{},[51,42339,29501],{},[55,42341,29501],{"encoding":57},[31,42343,42345],{"className":42344,"ariaHidden":62},[61],[31,42346,42348,42351],{"className":42347},[66],[31,42349],{"className":42350,"style":1630},[70],[31,42352,29501],{"className":42353},[75,76]," 太大，两个类之间的空隙被填平，一个明显偏离的样本反而显得很正常。我用 KDE 做异常检测时最常踩的坑就是这个：调 ",[31,42356,42358,42371],{"className":42357,"translate":35},[34],[31,42359,42361],{"className":42360},[39],[41,42362,42363],{"xmlns":43},[45,42364,42365,42369],{},[48,42366,42367],{},[51,42368,29501],{},[55,42370,29501],{"encoding":57},[31,42372,42374],{"className":42373,"ariaHidden":62},[61],[31,42375,42377,42380],{"className":42376},[66],[31,42378],{"className":42379,"style":1630},[70],[31,42381,29501],{"className":42382},[75,76]," 其实是在调“我允许多大的类间间隙”。",[11,42385,42386],{},"形式上是",[31,42388,42390],{"className":42389,"translate":35},[411],[31,42391,42393,42508],{"className":42392,"translate":35},[34],[31,42394,42396],{"className":42395},[39],[41,42397,42398],{"xmlns":43,"display":420},[45,42399,42400,42505],{},[48,42401,42402,42414,42416,42418,42420,42422,42428,42443,42449,42451,42453,42455,42461,42463,42465,42467,42473,42475,42477,42479,42481,42491,42493,42495,42497,42499,42501,42503],{},[1004,42403,42404,42412],{},[42405,42406,42407,42409],"mover",{"accent":62},[51,42408,11],{},[95,42410,42411],{},"^",[51,42413,29501],{},[95,42415,211],{"stretchy":210},[51,42417,53],{},[95,42419,220],{"stretchy":210},[95,42421,223],{},[2074,42423,42424,42426],{},[91,42425,205],{},[51,42427,14559],{},[42429,42430,42431,42433,42441],"munderover",{},[95,42432,5625],{},[48,42434,42435,42437,42439],{},[51,42436,5391],{},[95,42438,223],{},[91,42440,205],{},[51,42442,14559],{},[1004,42444,42445,42447],{},[51,42446,4558],{},[51,42448,29501],{},[95,42450,211],{"stretchy":210},[51,42452,53],{},[95,42454,97],{},[1004,42456,42457,42459],{},[51,42458,53],{},[51,42460,5391],{},[95,42462,220],{"stretchy":210},[95,42464,1210],{"separator":62},[119,42466],{"width":20286},[1004,42468,42469,42471],{},[51,42470,4558],{},[51,42472,29501],{},[95,42474,211],{"stretchy":210},[51,42476,5324],{},[95,42478,220],{"stretchy":210},[95,42480,223],{},[1180,42482,42483,42485],{},[51,42484,29501],{},[48,42486,42487,42489],{},[95,42488,97],{},[51,42490,1435],{},[51,42492,4558],{},[95,42494,211],{"stretchy":210},[51,42496,5324],{},[51,42498,3768],{"mathvariant":494},[51,42500,29501],{},[95,42502,220],{"stretchy":210},[51,42504,495],{"mathvariant":494},[55,42506,42507],{"encoding":57},"\\hat p_h(x)=\\frac1n\\sum_{i=1}^nK_h(x-x_i),\\qquad K_h(u)=h^{-d}K(u\u002Fh).",[31,42509,42511,42618,42814,42930],{"className":42510,"ariaHidden":62},[61],[31,42512,42514,42517,42600,42603,42606,42609,42612,42615],{"className":42513},[66],[31,42515],{"className":42516,"style":260},[70],[31,42518,42520,42566],{"className":42519},[75],[31,42521,42524],{"className":42522},[75,42523],"accent",[31,42525,42527,42557],{"className":42526},[1043,1044],[31,42528,42530,42554],{"className":42529},[1048],[31,42531,42533,42541],{"className":42532,"style":1630},[1052],[31,42534,42535,42538],{"style":1649},[31,42536],{"className":42537,"style":1637},[1060],[31,42539,11],{"className":42540},[75,76],[31,42542,42543,42546],{"style":1649},[31,42544],{"className":42545,"style":1637},[1060],[31,42547,42551],{"className":42548,"style":42550},[42549],"accent-body","left:-0.1667em;",[31,42552,42411],{"className":42553},[75],[31,42555,1076],{"className":42556},[1075],[31,42558,42560],{"className":42559},[1048],[31,42561,42564],{"className":42562,"style":42563},[1052],"height:0.1944em;",[31,42565],{},[31,42567,42569],{"className":42568},[1039],[31,42570,42572,42592],{"className":42571},[1043,1044],[31,42573,42575,42589],{"className":42574},[1048],[31,42576,42578],{"className":42577,"style":1053},[1052],[31,42579,42580,42583],{"style":2807},[31,42581],{"className":42582,"style":1061},[1060],[31,42584,42586],{"className":42585},[1065,1066,1067,1068],[31,42587,29501],{"className":42588},[75,76,1068],[31,42590,1076],{"className":42591},[1075],[31,42593,42595],{"className":42594},[1048],[31,42596,42598],{"className":42597,"style":2050},[1052],[31,42599],{},[31,42601,211],{"className":42602},[264],[31,42604,53],{"className":42605},[75,76],[31,42607,220],{"className":42608},[289],[31,42610],{"className":42611,"style":293},[119],[31,42613,223],{"className":42614},[297],[31,42616],{"className":42617,"style":293},[119],[31,42619,42621,42625,42687,42690,42756,42759,42799,42802,42805,42808,42811],{"className":42620},[66],[31,42622],{"className":42623,"style":42624},[70],"height:2.9291em;vertical-align:-1.2777em;",[31,42626,42628,42631,42684],{"className":42627},[75],[31,42629],{"className":42630},[264,2139],[31,42632,42634],{"className":42633},[2074],[31,42635,42637,42676],{"className":42636},[1043,1044],[31,42638,42640,42673],{"className":42639},[1048],[31,42641,42643,42654,42662],{"className":42642,"style":26376},[1052],[31,42644,42645,42648],{"style":5090},[31,42646],{"className":42647,"style":1637},[1060],[31,42649,42651],{"className":42650},[75],[31,42652,14559],{"className":42653},[75,76],[31,42655,42656,42659],{"style":2170},[31,42657],{"className":42658,"style":1637},[1060],[31,42660],{"className":42661,"style":2178},[2177],[31,42663,42664,42667],{"style":5110},[31,42665],{"className":42666,"style":1637},[1060],[31,42668,42670],{"className":42669},[75],[31,42671,205],{"className":42672},[75],[31,42674,1076],{"className":42675},[1075],[31,42677,42679],{"className":42678},[1048],[31,42680,42682],{"className":42681,"style":5156},[1052],[31,42683],{},[31,42685],{"className":42686},[289,2139],[31,42688],{"className":42689,"style":1467},[119],[31,42691,42693],{"className":42692},[1460,1620],[31,42694,42696,42748],{"className":42695},[1043,1044],[31,42697,42699,42745],{"className":42698},[1048],[31,42700,42703,42723,42733],{"className":42701,"style":42702},[1052],"height:1.6514em;",[31,42704,42705,42708],{"style":5903},[31,42706],{"className":42707,"style":5907},[1060],[31,42709,42711],{"className":42710},[1065,1066,1067,1068],[31,42712,42714,42717,42720],{"className":42713},[75,1068],[31,42715,5391],{"className":42716},[75,76,1068],[31,42718,223],{"className":42719},[297,1068],[31,42721,205],{"className":42722},[75,1068],[31,42724,42725,42728],{"style":5919},[31,42726],{"className":42727,"style":5907},[1060],[31,42729,42730],{},[31,42731,5625],{"className":42732},[1460,1461,1687],[31,42734,42736,42739],{"style":42735},"top:-4.3em;margin-left:0em;",[31,42737],{"className":42738,"style":5907},[1060],[31,42740,42742],{"className":42741},[1065,1066,1067,1068],[31,42743,14559],{"className":42744},[75,76,1068],[31,42746,1076],{"className":42747},[1075],[31,42749,42751],{"className":42750},[1048],[31,42752,42754],{"className":42753,"style":6342},[1052],[31,42755],{},[31,42757],{"className":42758,"style":1467},[119],[31,42760,42762,42765],{"className":42761},[75],[31,42763,4558],{"className":42764,"style":8525},[75,76],[31,42766,42768],{"className":42767},[1039],[31,42769,42771,42791],{"className":42770},[1043,1044],[31,42772,42774,42788],{"className":42773},[1048],[31,42775,42777],{"className":42776,"style":1053},[1052],[31,42778,42779,42782],{"style":8540},[31,42780],{"className":42781,"style":1061},[1060],[31,42783,42785],{"className":42784},[1065,1066,1067,1068],[31,42786,29501],{"className":42787},[75,76,1068],[31,42789,1076],{"className":42790},[1075],[31,42792,42794],{"className":42793},[1048],[31,42795,42797],{"className":42796,"style":2050},[1052],[31,42798],{},[31,42800,211],{"className":42801},[264],[31,42803,53],{"className":42804},[75,76],[31,42806],{"className":42807,"style":120},[119],[31,42809,97],{"className":42810},[124],[31,42812],{"className":42813,"style":120},[119],[31,42815,42817,42820,42860,42863,42866,42869,42872,42912,42915,42918,42921,42924,42927],{"className":42816},[66],[31,42818],{"className":42819,"style":260},[70],[31,42821,42823,42826],{"className":42822},[75],[31,42824,53],{"className":42825},[75,76],[31,42827,42829],{"className":42828},[1039],[31,42830,42832,42852],{"className":42831},[1043,1044],[31,42833,42835,42849],{"className":42834},[1048],[31,42836,42838],{"className":42837,"style":5445},[1052],[31,42839,42840,42843],{"style":2807},[31,42841],{"className":42842,"style":1061},[1060],[31,42844,42846],{"className":42845},[1065,1066,1067,1068],[31,42847,5391],{"className":42848},[75,76,1068],[31,42850,1076],{"className":42851},[1075],[31,42853,42855],{"className":42854},[1048],[31,42856,42858],{"className":42857,"style":2050},[1052],[31,42859],{},[31,42861,220],{"className":42862},[289],[31,42864,1210],{"className":42865},[1391],[31,42867],{"className":42868,"style":20525},[119],[31,42870],{"className":42871,"style":1467},[119],[31,42873,42875,42878],{"className":42874},[75],[31,42876,4558],{"className":42877,"style":8525},[75,76],[31,42879,42881],{"className":42880},[1039],[31,42882,42884,42904],{"className":42883},[1043,1044],[31,42885,42887,42901],{"className":42886},[1048],[31,42888,42890],{"className":42889,"style":1053},[1052],[31,42891,42892,42895],{"style":8540},[31,42893],{"className":42894,"style":1061},[1060],[31,42896,42898],{"className":42897},[1065,1066,1067,1068],[31,42899,29501],{"className":42900},[75,76,1068],[31,42902,1076],{"className":42903},[1075],[31,42905,42907],{"className":42906},[1048],[31,42908,42910],{"className":42909,"style":2050},[1052],[31,42911],{},[31,42913,211],{"className":42914},[264],[31,42916,5324],{"className":42917},[75,76],[31,42919,220],{"className":42920},[289],[31,42922],{"className":42923,"style":293},[119],[31,42925,223],{"className":42926},[297],[31,42928],{"className":42929,"style":293},[119],[31,42931,42933,42936,42971,42974,42977,42980,42983,42986,42989],{"className":42932},[66],[31,42934],{"className":42935,"style":35441},[70],[31,42937,42939,42942],{"className":42938},[75],[31,42940,29501],{"className":42941},[75,76],[31,42943,42945],{"className":42944},[1039],[31,42946,42948],{"className":42947},[1043],[31,42949,42951],{"className":42950},[1048],[31,42952,42954],{"className":42953,"style":29642},[1052],[31,42955,42956,42959],{"style":1330},[31,42957],{"className":42958,"style":1061},[1060],[31,42960,42962],{"className":42961},[1065,1066,1067,1068],[31,42963,42965,42968],{"className":42964},[75,1068],[31,42966,97],{"className":42967},[75,1068],[31,42969,1435],{"className":42970},[75,76,1068],[31,42972,4558],{"className":42973,"style":8525},[75,76],[31,42975,211],{"className":42976},[264],[31,42978,5324],{"className":42979},[75,76],[31,42981,3768],{"className":42982},[75],[31,42984,29501],{"className":42985},[75,76],[31,42987,220],{"className":42988},[289],[31,42990,495],{"className":42991},[75],[11,42993,42994,42995,1138,43054,1138,43241,43395],{},"它的偏差和方差可以算出来（记 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n^{-1\u002F(d+4)}",[31,44236,44238,44282],{"className":44237,"ariaHidden":62},[61],[31,44239,44241,44244,44273,44276,44279],{"className":44240},[66],[31,44242],{"className":44243,"style":1630},[70],[31,44245,44247,44250],{"className":44246},[75],[31,44248,29501],{"className":44249},[75,76],[31,44251,44253],{"className":44252},[1039],[31,44254,44256],{"className":44255},[1043],[31,44257,44259],{"className":44258},[1048],[31,44260,44262],{"className":44261,"style":30719},[1052],[31,44263,44264,44267],{"style":3631},[31,44265],{"className":44266,"style":1061},[1060],[31,44268,44270],{"className":44269},[1065,1066,1067,1068],[31,44271,30648],{"className":44272},[124,1068],[31,44274],{"className":44275,"style":293},[119],[31,44277,44209],{"className":44278},[297],[31,44280],{"className":44281,"style":293},[119],[31,44283,44285,44288],{"className":44284},[66],[31,44286],{"className":44287,"style":22855},[70],[31,44289,44291,44294],{"className":44290},[75],[31,44292,14559],{"className":44293},[75,76],[31,44295,44297],{"className":44296},[1039],[31,44298,44300],{"className":44299},[1043],[31,44301,44303],{"className":44302},[1048],[31,44304,44306],{"className":44305,"style":22855},[1052],[31,44307,44308,44311],{"style":3631},[31,44309],{"className":44310,"style":1061},[1060],[31,44312,44314],{"className":44313},[1065,1066,1067,1068],[31,44315,44317,44320,44323,44326,44329,44332,44335],{"className":44316},[75,1068],[31,44318,97],{"className":44319},[75,1068],[31,44321,4042],{"className":44322},[75,1068],[31,44324,211],{"className":44325},[264,1068],[31,44327,1435],{"className":44328},[75,76,1068],[31,44330,434],{"className":44331},[124,1068],[31,44333,15416],{"className":44334},[75,1068],[31,44336,220],{"className":44337},[289,1068],"，此时 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n^{-4\u002F(d+4)}",[31,44388,44390,44412],{"className":44389,"ariaHidden":62},[61],[31,44391,44393,44396,44403,44406,44409],{"className":44392},[66],[31,44394],{"className":44395,"style":945},[70],[31,44397,44399],{"className":44398},[75],[31,44400,44402],{"className":44401},[75,2745],"MSE",[31,44404],{"className":44405,"style":293},[119],[31,44407,44209],{"className":44408},[297],[31,44410],{"className":44411,"style":293},[119],[31,44413,44415,44418],{"className":44414},[66],[31,44416],{"className":44417,"style":22855},[70],[31,44419,44421,44424],{"className":44420},[75],[31,44422,14559],{"className":44423},[75,76],[31,44425,44427],{"className":44426},[1039],[31,44428,44430],{"className":44429},[1043],[31,44431,44433],{"className":44432},[1048],[31,44434,44436],{"className":44435,"style":22855},[1052],[31,44437,44438,44441],{"style":3631},[31,44439],{"className":44440,"style":1061},[1060],[31,44442,44444],{"className":44443},[1065,1066,1067,1068],[31,44445,44447,44450,44454,44457,44460,44463,44466],{"className":44446},[75,1068],[31,44448,97],{"className":44449},[75,1068],[31,44451,44453],{"className":44452},[75,1068],"4\u002F",[31,44455,211],{"className":44456},[264,1068],[31,44458,1435],{"className":44459},[75,76,1068],[31,44461,434],{"className":44462},[124,1068],[31,44464,15416],{"className":44465},[75,1068],[31,44467,220],{"className":44468},[289,1068],[11,44470,44471,44522,44523,44597,44598,44522,44649,44724,44725,44810],{},[31,44472,44474,44492],{"className":44473,"translate":35},[34],[31,44475,44477],{"className":44476},[39],[41,44478,44479],{"xmlns":43},[45,44480,44481,44489],{},[48,44482,44483,44485,44487],{},[51,44484,1435],{},[95,44486,223],{},[91,44488,205],{},[55,44490,44491],{"encoding":57},"d=1",[31,44493,44495,44513],{"className":44494,"ariaHidden":62},[61],[31,44496,44498,44501,44504,44507,44510],{"className":44497},[66],[31,44499],{"className":44500,"style":1630},[70],[31,44502,1435],{"className":44503},[75,76],[31,44505],{"className":44506,"style":293},[119],[31,44508,223],{"className":44509},[297],[31,44511],{"className":44512,"style":293},[119],[31,44514,44516,44519],{"className":44515},[66],[31,44517],{"className":44518,"style":326},[70],[31,44520,205],{"className":44521},[75]," 时是 ",[31,44524,44526,44552],{"className":44525,"translate":35},[34],[31,44527,44529],{"className":44528},[39],[41,44530,44531],{"xmlns":43},[45,44532,44533,44549],{},[48,44534,44535],{},[1180,44536,44537,44539],{},[51,44538,14559],{},[48,44540,44541,44543,44545,44547],{},[95,44542,97],{},[91,44544,15416],{},[51,44546,3768],{"mathvariant":494},[91,44548,7853],{},[55,44550,44551],{"encoding":57},"n^{-4\u002F5}",[31,44553,44555],{"className":44554,"ariaHidden":62},[61],[31,44556,44558,44561],{"className":44557},[66],[31,44559],{"className":44560,"style":22855},[70],[31,44562,44564,44567],{"className":44563},[75],[31,44565,14559],{"className":44566},[75,76],[31,44568,44570],{"className":44569},[1039],[31,44571,44573],{"className":44572},[1043],[31,44574,44576],{"className":44575},[1048],[31,44577,44579],{"className":44578,"style":22855},[1052],[31,44580,44581,44584],{"style":3631},[31,44582],{"className":44583,"style":1061},[1060],[31,44585,44587],{"className":44586},[1065,1066,1067,1068],[31,44588,44590,44593],{"className":44589},[75,1068],[31,44591,97],{"className":44592},[75,1068],[31,44594,44596],{"className":44595},[75,1068],"4\u002F5","，很不错。",[31,44599,44601,44619],{"className":44600,"translate":35},[34],[31,44602,44604],{"className":44603},[39],[41,44605,44606],{"xmlns":43},[45,44607,44608,44616],{},[48,44609,44610,44612,44614],{},[51,44611,1435],{},[95,44613,223],{},[91,44615,19494],{},[55,44617,44618],{"encoding":57},"d=10",[31,44620,44622,44640],{"className":44621,"ariaHidden":62},[61],[31,44623,44625,44628,44631,44634,44637],{"className":44624},[66],[31,44626],{"className":44627,"style":1630},[70],[31,44629,1435],{"className":44630},[75,76],[31,44632],{"className":44633,"style":293},[119],[31,44635,223],{"className":44636},[297],[31,44638],{"className":44639,"style":293},[119],[31,44641,44643,44646],{"className":44642},[66],[31,44644],{"className":44645,"style":326},[70],[31,44647,19494],{"className":44648},[75],[31,44650,44652,44679],{"className":44651,"translate":35},[34],[31,44653,44655],{"className":44654},[39],[41,44656,44657],{"xmlns":43},[45,44658,44659,44676],{},[48,44660,44661],{},[1180,44662,44663,44665],{},[51,44664,14559],{},[48,44666,44667,44669,44671,44673],{},[95,44668,97],{},[91,44670,152],{},[51,44672,3768],{"mathvariant":494},[91,44674,44675],{},"7",[55,44677,44678],{"encoding":57},"n^{-2\u002F7}",[31,44680,44682],{"className":44681,"ariaHidden":62},[61],[31,44683,44685,44688],{"className":44684},[66],[31,44686],{"className":44687,"style":22855},[70],[31,44689,44691,44694],{"className":44690},[75],[31,44692,14559],{"className":44693},[75,76],[31,44695,44697],{"className":44696},[1039],[31,44698,44700],{"className":44699},[1043],[31,44701,44703],{"className":44702},[1048],[31,44704,44706],{"className":44705,"style":22855},[1052],[31,44707,44708,44711],{"style":3631},[31,44709],{"className":44710,"style":1061},[1060],[31,44712,44714],{"className":44713},[1065,1066,1067,1068],[31,44715,44717,44720],{"className":44716},[75,1068],[31,44718,97],{"className":44719},[75,1068],[31,44721,44723],{"className":44722},[75,1068],"2\u002F7","，想把误差减半，样本量要乘 ",[31,44726,44728,44751],{"className":44727,"translate":35},[34],[31,44729,44731],{"className":44730},[39],[41,44732,44733],{"xmlns":43},[45,44734,44735,44748],{},[48,44736,44737,44744,44746],{},[1180,44738,44739,44741],{},[91,44740,152],{},[91,44742,44743],{},"3.5",[95,44745,27223],{},[91,44747,485],{},[55,44749,44750],{"encoding":57},"2^{3.5}\\approx11",[31,44752,44754,44801],{"className":44753,"ariaHidden":62},[61],[31,44755,44757,44760,44792,44795,44798],{"className":44756},[66],[31,44758],{"className":44759,"style":5135},[70],[31,44761,44763,44766],{"className":44762},[75],[31,44764,152],{"className":44765},[75],[31,44767,44769],{"className":44768},[1039],[31,44770,44772],{"className":44771},[1043],[31,44773,44775],{"className":44774},[1048],[31,44776,44778],{"className":44777,"style":5135},[1052],[31,44779,44780,44783],{"style":3631},[31,44781],{"className":44782,"style":1061},[1060],[31,44784,44786],{"className":44785},[1065,1066,1067,1068],[31,44787,44789],{"className":44788},[75,1068],[31,44790,44743],{"className":44791},[75,1068],[31,44793],{"className":44794,"style":293},[119],[31,44796,27223],{"className":44797},[297],[31,44799],{"className":44800,"style":293},[119],[31,44802,44804,44807],{"className":44803},[66],[31,44805],{"className":44806,"style":326},[70],[31,44808,485],{"className":44809},[75],"。这个数字我建议背下来：每加几个维度，KDE 就从“能用”滑向“不能用”。",[11,44812,44813,44814,44913,44914,45006,45007,45079,45080,45181,45182,45325,45326,45397,45398,44522,45448,2372],{},"但 KDE 在高维还有一层跟带宽无关的麻烦。考虑标准正态 ",[31,44815,44817,44845],{"className":44816,"translate":35},[34],[31,44818,44820],{"className":44819},[39],[41,44821,44822],{"xmlns":43},[45,44823,44824,44842],{},[48,44825,44826,44828,44830,44832,44834,44840],{},[51,44827,4935],{"mathvariant":925},[95,44829,211],{"stretchy":210},[91,44831,1977],{},[95,44833,1210],{"separator":62},[1004,44835,44836,44838],{},[51,44837,34619],{},[51,44839,1435],{},[95,44841,220],{"stretchy":210},[55,44843,44844],{"encoding":57},"\\mathcal N(0,I_d)",[31,44846,44848],{"className":44847,"ariaHidden":62},[61],[31,44849,44851,44854,44857,44860,44863,44866,44869,44910],{"className":44850},[66],[31,44852],{"className":44853,"style":260},[70],[31,44855,4935],{"className":44856,"style":5004},[75,968],[31,44858,211],{"className":44859},[264],[31,44861,1977],{"className":44862},[75],[31,44864,1210],{"className":44865},[1391],[31,44867],{"className":44868,"style":1467},[119],[31,44870,44872,44875],{"className":44871},[75],[31,44873,34619],{"className":44874,"style":30311},[75,76],[31,44876,44878],{"className":44877},[1039],[31,44879,44881,44902],{"className":44880},[1043,1044],[31,44882,44884,44899],{"className":44883},[1048],[31,44885,44887],{"className":44886,"style":1053},[1052],[31,44888,44890,44893],{"style":44889},"top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;",[31,44891],{"className":44892,"style":1061},[1060],[31,44894,44896],{"className":44895},[1065,1066,1067,1068],[31,44897,1435],{"className":44898},[75,76,1068],[31,44900,1076],{"className":44901},[1075],[31,44903,44905],{"className":44904},[1048],[31,44906,44908],{"className":44907,"style":2050},[1052],[31,44909],{},[31,44911,220],{"className":44912},[289],"。众数在原点的密度是 ",[31,44915,44917,44949],{"className":44916,"translate":35},[34],[31,44918,44920],{"className":44919},[39],[41,44921,44922],{"xmlns":43},[45,44923,44924,44946],{},[48,44925,44926,44928,44930,44932],{},[95,44927,211],{"stretchy":210},[91,44929,152],{},[51,44931,2515],{},[1180,44933,44934,44936],{},[95,44935,220],{"stretchy":210},[48,44937,44938,44940,44942,44944],{},[95,44939,97],{},[51,44941,1435],{},[51,44943,3768],{"mathvariant":494},[91,44945,152],{},[55,44947,44948],{"encoding":57},"(2\\pi)^{-d\u002F2}",[31,44950,44952],{"className":44951,"ariaHidden":62},[61],[31,44953,44955,44959,44962,44965,44968],{"className":44954},[66],[31,44956],{"className":44957,"style":44958},[70],"height:1.138em;vertical-align:-0.25em;",[31,44960,211],{"className":44961},[264],[31,44963,152],{"className":44964},[75],[31,44966,2515],{"className":44967,"style":2666},[75,76],[31,44969,44971,44974],{"className":44970},[289],[31,44972,220],{"className":44973},[289],[31,44975,44977],{"className":44976},[1039],[31,44978,44980],{"className":44979},[1043],[31,44981,44983],{"className":44982},[1048],[31,44984,44986],{"className":44985,"style":22855},[1052],[31,44987,44988,44991],{"style":3631},[31,44989],{"className":44990,"style":1061},[1060],[31,44992,44994],{"className":44993},[1065,1066,1067,1068],[31,44995,44997,45000,45003],{"className":44996},[75,1068],[31,44998,97],{"className":44999},[75,1068],[31,45001,1435],{"className":45002},[75,76,1068],[31,45004,15357],{"className":45005},[75,1068],"；而一个“典型”样本落在半径 ",[31,45008,45010,45026],{"className":45009,"translate":35},[34],[31,45011,45013],{"className":45012},[39],[41,45014,45015],{"xmlns":43},[45,45016,45017,45023],{},[48,45018,45019],{},[26474,45020,45021],{},[51,45022,1435],{},[55,45024,45025],{"encoding":57},"\\sqrt d",[31,45027,45029],{"className":45028,"ariaHidden":62},[61],[31,45030,45032,45035],{"className":45031},[66],[31,45033],{"className":45034,"style":27567},[70],[31,45036,45038],{"className":45037},[75,26536],[31,45039,45041,45071],{"className":45040},[1043,1044],[31,45042,45044,45068],{"className":45043},[1048],[31,45045,45047,45056],{"className":45046,"style":27592},[1052],[31,45048,45050,45053],{"className":45049,"style":1649},[26550],[31,45051],{"className":45052,"style":1637},[1060],[31,45054,1435],{"className":45055,"style":27883},[75,76],[31,45057,45058,45061],{"style":27605},[31,45059],{"className":45060,"style":1637},[1060],[31,45062,45064],{"className":45063,"style":27612},[26592],[13341,45065,45066],{"xmlns":13343,"width":26596,"height":27615,"viewBox":27616,"preserveAspectRatio":26599},[13347,45067],{"d":27619},[31,45069,1076],{"className":45070},[1075],[31,45072,45074],{"className":45073},[1048],[31,45075,45077],{"className":45076,"style":27629},[1052],[31,45078],{}," 的球壳上（因为 ",[31,45081,45083,45113],{"className":45082,"translate":35},[34],[31,45084,45086],{"className":45085},[39],[41,45087,45088],{"xmlns":43},[45,45089,45090,45110],{},[48,45091,45092,45094,45096,45102,45104,45106,45108],{},[51,45093,4572],{"mathvariant":494},[51,45095,926],{},[1180,45097,45098,45100],{},[51,45099,4572],{"mathvariant":494},[91,45101,152],{},[51,45103,3768],{"mathvariant":494},[51,45105,1435],{},[95,45107,929],{},[91,45109,205],{},[55,45111,45112],{"encoding":57},"\\|X\\|^2\u002Fd\\to1",[31,45114,45116,45172],{"className":45115,"ariaHidden":62},[61],[31,45117,45119,45122,45125,45128,45157,45160,45163,45166,45169],{"className":45118},[66],[31,45120],{"className":45121,"style":12035},[70],[31,45123,4572],{"className":45124},[75],[31,45126,926],{"className":45127,"style":30311},[75,76],[31,45129,45131,45134],{"className":45130},[75],[31,45132,4572],{"className":45133},[75],[31,45135,45137],{"className":45136},[1039],[31,45138,45140],{"className":45139},[1043],[31,45141,45143],{"className":45142},[1048],[31,45144,45146],{"className":45145,"style":5135},[1052],[31,45147,45148,45151],{"style":3631},[31,45149],{"className":45150,"style":1061},[1060],[31,45152,45154],{"className":45153},[1065,1066,1067,1068],[31,45155,152],{"className":45156},[75,1068],[31,45158,3768],{"className":45159},[75],[31,45161,1435],{"className":45162},[75,76],[31,45164],{"className":45165,"style":293},[119],[31,45167,929],{"className":45168},[297],[31,45170],{"className":45171,"style":293},[119],[31,45173,45175,45178],{"className":45174},[66],[31,45176],{"className":45177,"style":326},[70],[31,45179,205],{"className":45180},[75],"），那里的密度是 ",[31,45183,45185,45231],{"className":45184,"translate":35},[34],[31,45186,45188],{"className":45187},[39],[41,45189,45190],{"xmlns":43},[45,45191,45192,45228],{},[48,45193,45194,45196,45198,45200,45214],{},[95,45195,211],{"stretchy":210},[91,45197,152],{},[51,45199,2515],{},[1180,45201,45202,45204],{},[95,45203,220],{"stretchy":210},[48,45205,45206,45208,45210,45212],{},[95,45207,97],{},[51,45209,1435],{},[51,45211,3768],{"mathvariant":494},[91,45213,152],{},[1180,45215,45216,45218],{},[51,45217,30042],{},[48,45219,45220,45222,45224,45226],{},[95,45221,97],{},[51,45223,1435],{},[51,45225,3768],{"mathvariant":494},[91,45227,152],{},[55,45229,45230],{"encoding":57},"(2\\pi)^{-d\u002F2}e^{-d\u002F2}",[31,45232,45234],{"className":45233,"ariaHidden":62},[61],[31,45235,45237,45240,45243,45246,45249,45287],{"className":45236},[66],[31,45238],{"className":45239,"style":44958},[70],[31,45241,211],{"className":45242},[264],[31,45244,152],{"className":45245},[75],[31,45247,2515],{"className":45248,"style":2666},[75,76],[31,45250,45252,45255],{"className":45251},[289],[31,45253,220],{"className":45254},[289],[31,45256,45258],{"className":45257},[1039],[31,45259,45261],{"className":45260},[1043],[31,45262,45264],{"className":45263},[1048],[31,45265,45267],{"className":45266,"style":22855},[1052],[31,45268,45269,45272],{"style":3631},[31,45270],{"className":45271,"style":1061},[1060],[31,45273,45275],{"className":45274},[1065,1066,1067,1068],[31,45276,45278,45281,45284],{"className":45277},[75,1068],[31,45279,97],{"className":45280},[75,1068],[31,45282,1435],{"className":45283},[75,76,1068],[31,45285,15357],{"className":45286},[75,1068],[31,45288,45290,45293],{"className":45289},[75],[31,45291,30042],{"className":45292},[75,76],[31,45294,45296],{"className":45295},[1039],[31,45297,45299],{"className":45298},[1043],[31,45300,45302],{"className":45301},[1048],[31,45303,45305],{"className":45304,"style":22855},[1052],[31,45306,45307,45310],{"style":3631},[31,45308],{"className":45309,"style":1061},[1060],[31,45311,45313],{"className":45312},[1065,1066,1067,1068],[31,45314,45316,45319,45322],{"className":45315},[75,1068],[31,45317,97],{"className":45318},[75,1068],[31,45320,1435],{"className":45321},[75,76,1068],[31,45323,15357],{"className":45324},[75,1068],"。两者差 ",[31,45327,45329,45353],{"className":45328,"translate":35},[34],[31,45330,45332],{"className":45331},[39],[41,45333,45334],{"xmlns":43},[45,45335,45336,45350],{},[48,45337,45338],{},[1180,45339,45340,45342],{},[51,45341,30042],{},[48,45343,45344,45346,45348],{},[51,45345,1435],{},[51,45347,3768],{"mathvariant":494},[91,45349,152],{},[55,45351,45352],{"encoding":57},"e^{d\u002F2}",[31,45354,45356],{"className":45355,"ariaHidden":62},[61],[31,45357,45359,45362],{"className":45358},[66],[31,45360],{"className":45361,"style":22855},[70],[31,45363,45365,45368],{"className":45364},[75],[31,45366,30042],{"className":45367},[75,76],[31,45369,45371],{"className":45370},[1039],[31,45372,45374],{"className":45373},[1043],[31,45375,45377],{"className":45376},[1048],[31,45378,45380],{"className":45379,"style":22855},[1052],[31,45381,45382,45385],{"style":3631},[31,45383],{"className":45384,"style":1061},[1060],[31,45386,45388],{"className":45387},[1065,1066,1067,1068],[31,45389,45391,45394],{"className":45390},[75,1068],[31,45392,1435],{"className":45393},[75,76,1068],[31,45395,15357],{"className":45396},[75,1068]," 倍，",[31,45399,45401,45418],{"className":45400,"translate":35},[34],[31,45402,45404],{"className":45403},[39],[41,45405,45406],{"xmlns":43},[45,45407,45408,45416],{},[48,45409,45410,45412,45414],{},[51,45411,1435],{},[95,45413,223],{},[91,45415,22616],{},[55,45417,22619],{"encoding":57},[31,45419,45421,45439],{"className":45420,"ariaHidden":62},[61],[31,45422,45424,45427,45430,45433,45436],{"className":45423},[66],[31,45425],{"className":45426,"style":1630},[70],[31,45428,1435],{"className":45429},[75,76],[31,45431],{"className":45432,"style":293},[119],[31,45434,223],{"className":45435},[297],[31,45437],{"className":45438,"style":293},[119],[31,45440,45442,45445],{"className":45441},[66],[31,45443],{"className":45444,"style":326},[70],[31,45446,22616],{"className":45447},[75],[31,45449,45451,45483],{"className":45450,"translate":35},[34],[31,45452,45454],{"className":45453},[39],[41,45455,45456],{"xmlns":43},[45,45457,45458,45480],{},[48,45459,45460,45467,45469,45471,45473],{},[1180,45461,45462,45464],{},[51,45463,30042],{},[91,45465,45466],{},"50",[95,45468,27223],{},[91,45470,7853],{},[95,45472,2528],{},[1180,45474,45475,45477],{},[91,45476,19494],{},[91,45478,45479],{},"21",[55,45481,45482],{"encoding":57},"e^{50}\\approx5\\times10^{21}",[31,45484,45486,45533,45551],{"className":45485,"ariaHidden":62},[61],[31,45487,45489,45492,45524,45527,45530],{"className":45488},[66],[31,45490],{"className":45491,"style":5135},[70],[31,45493,45495,45498],{"className":45494},[75],[31,45496,30042],{"className":45497},[75,76],[31,45499,45501],{"className":45500},[1039],[31,45502,45504],{"className":45503},[1043],[31,45505,45507],{"className":45506},[1048],[31,45508,45510],{"className":45509,"style":5135},[1052],[31,45511,45512,45515],{"style":3631},[31,45513],{"className":45514,"style":1061},[1060],[31,45516,45518],{"className":45517},[1065,1066,1067,1068],[31,45519,45521],{"className":45520},[75,1068],[31,45522,45466],{"className":45523},[75,1068],[31,45525],{"className":45526,"style":293},[119],[31,45528,27223],{"className":45529},[297],[31,45531],{"className":45532,"style":293},[119],[31,45534,45536,45539,45542,45545,45548],{"className":45535},[66],[31,45537],{"className":45538,"style":112},[70],[31,45540,7853],{"className":45541},[75],[31,45543],{"className":45544,"style":120},[119],[31,45546,2528],{"className":45547},[124],[31,45549],{"className":45550,"style":120},[119],[31,45552,45554,45557,45560],{"className":45553},[66],[31,45555],{"className":45556,"style":5135},[70],[31,45558,205],{"className":45559},[75],[31,45561,45563,45566],{"className":45562},[75],[31,45564,1977],{"className":45565},[75],[31,45567,45569],{"className":45568},[1039],[31,45570,45572],{"className":45571},[1043],[31,45573,45575],{"className":45574},[1048],[31,45576,45578],{"className":45577,"style":5135},[1052],[31,45579,45580,45583],{"style":3631},[31,45581],{"className":45582,"style":1061},[1060],[31,45584,45586],{"className":45585},[1065,1066,1067,1068],[31,45587,45589],{"className":45588},[75,1068],[31,45590,45479],{"className":45591},[75,1068],[11,45593,45594],{},"有个很好记的画面：一个 100 维的橙子，果皮比果肉厚得多。球的体积几乎全在表面附近那层薄壳里，维度越高这层壳占的比例越大。所以“最典型的位置”不是圆心，而是那层壳。密度的最大值在圆心，可那里几乎没有质量。",[11,45596,45597],{},"这意味着什么？如果我用“密度低于阈值”判 OOD，而阈值按众数附近校准，那几乎会把所有正常样本都判成异常。这就是典型集悖论。",[11,45599,45600],{},"所以“低密度等于分布外”这个直觉在高维是错的。密度是点态的，回答的是“这个点附近有多挤”；我真正想问的是“这个点离已知类别的概率质量有多远”。后者是分布态的问题，得用前面那些工具。",[18,45602,45603],{"id":45603},"可逆变换",[11,45605,45606,45607,45611,45612,45668],{},"写采样代码时我总会用到同一招：",[45608,45609,45610],"code",{},"random()"," 只能给 ",[31,45613,45615,45638],{"className":45614,"translate":35},[34],[31,45616,45618],{"className":45617},[39],[41,45619,45620],{"xmlns":43},[45,45621,45622,45636],{},[48,45623,45624,45626,45628,45630,45632,45634],{},[51,45625,12818],{},[95,45627,211],{"stretchy":210},[91,45629,1977],{},[95,45631,1210],{"separator":62},[91,45633,205],{},[95,45635,220],{"stretchy":210},[55,45637,12890],{"encoding":57},[31,45639,45641],{"className":45640,"ariaHidden":62},[61],[31,45642,45644,45647,45650,45653,45656,45659,45662,45665],{"className":45643},[66],[31,45645],{"className":45646,"style":260},[70],[31,45648,12818],{"className":45649,"style":12844},[75,76],[31,45651,211],{"className":45652},[264],[31,45654,1977],{"className":45655},[75],[31,45657,1210],{"className":45658},[1391],[31,45660],{"className":45661,"style":1467},[119],[31,45663,205],{"className":45664},[75],[31,45666,220],{"className":45667},[289],"，可我想要指数分布，于是取",[31,45670,45672],{"className":45671,"translate":35},[411],[31,45673,45675,45716],{"className":45674,"translate":35},[34],[31,45676,45678],{"className":45677},[39],[41,45679,45680],{"xmlns":43,"display":420},[45,45681,45682,45713],{},[48,45683,45684,45686,45688,45690,45711],{},[51,45685,53],{},[95,45687,223],{},[95,45689,97],{},[2074,45691,45692,45708],{},[48,45693,45694,45696,45698,45700,45702,45704,45706],{},[51,45695,19939],{},[95,45697,1540],{},[95,45699,211],{"stretchy":210},[91,45701,205],{},[95,45703,97],{},[51,45705,5324],{},[95,45707,220],{"stretchy":210},[51,45709,45710],{},"λ",[51,45712,495],{"mathvariant":494},[55,45714,45715],{"encoding":57},"x=-\\frac{\\log(1-u)}{\\lambda}.",[31,45717,45719,45737],{"className":45718,"ariaHidden":62},[61],[31,45720,45722,45725,45728,45731,45734],{"className":45721},[66],[31,45723],{"className":45724,"style":71},[70],[31,45726,53],{"className":45727},[75,76],[31,45729],{"className":45730,"style":293},[119],[31,45732,223],{"className":45733},[297],[31,45735],{"className":45736,"style":293},[119],[31,45738,45740,45744,45747,45833],{"className":45739},[66],[31,45741],{"className":45742,"style":45743},[70],"height:2.113em;vertical-align:-0.686em;",[31,45745,97],{"className":45746},[75],[31,45748,45750,45753,45830],{"className":45749},[75],[31,45751],{"className":45752},[264,2139],[31,45754,45756],{"className":45755},[2074],[31,45757,45759,45822],{"className":45758},[1043,1044],[31,45760,45762,45819],{"className":45761},[1048],[31,45763,45766,45777,45785],{"className":45764,"style":45765},[1052],"height:1.427em;",[31,45767,45768,45771],{"style":5090},[31,45769],{"className":45770,"style":1637},[1060],[31,45772,45774],{"className":45773},[75],[31,45775,45710],{"className":45776},[75,76],[31,45778,45779,45782],{"style":2170},[31,45780],{"className":45781,"style":1637},[1060],[31,45783],{"className":45784,"style":2178},[2177],[31,45786,45787,45790],{"style":5110},[31,45788],{"className":45789,"style":1637},[1060],[31,45791,45793,45798,45801,45804,45807,45810,45813,45816],{"className":45792},[75],[31,45794,20177,45796],{"className":45795},[1460],[31,45797,14556],{"style":20180},[31,45799,211],{"className":45800},[264],[31,45802,205],{"className":45803},[75],[31,45805],{"className":45806,"style":120},[119],[31,45808,97],{"className":45809},[124],[31,45811],{"className":45812,"style":120},[119],[31,45814,5324],{"className":45815},[75,76],[31,45817,220],{"className":45818},[289],[31,45820,1076],{"className":45821},[1075],[31,45823,45825],{"className":45824},[1048],[31,45826,45828],{"className":45827,"style":5156},[1052],[31,45829],{},[31,45831],{"className":45832},[289,2139],[31,45834,495],{"className":45835},[75],[11,45837,45838,45839,4880,45957,45985],{},"这一个式子就是一个标准化流。反过来看，",[31,45840,45842,45874],{"className":45841,"translate":35},[34],[31,45843,45845],{"className":45844},[39],[41,45846,45847],{"xmlns":43},[45,45848,45849,45871],{},[48,45850,45851,45853,45855,45857,45859],{},[51,45852,5324],{},[95,45854,223],{},[91,45856,205],{},[95,45858,97],{},[1180,45860,45861,45863],{},[51,45862,30042],{},[48,45864,45865,45867,45869],{},[95,45866,97],{},[51,45868,45710],{},[51,45870,53],{},[55,45872,45873],{"encoding":57},"u=1-e^{-\\lambda x}",[31,45875,45877,45895,45913],{"className":45876,"ariaHidden":62},[61],[31,45878,45880,45883,45886,45889,45892],{"className":45879},[66],[31,45881],{"className":45882,"style":71},[70],[31,45884,5324],{"className":45885},[75,76],[31,45887],{"className":45888,"style":293},[119],[31,45890,223],{"className":45891},[297],[31,45893],{"className":45894,"style":293},[119],[31,45896,45898,45901,45904,45907,45910],{"className":45897},[66],[31,45899],{"className":45900,"style":112},[70],[31,45902,205],{"className":45903},[75],[31,45905],{"className":45906,"style":120},[119],[31,45908,97],{"className":45909},[124],[31,45911],{"className":45912,"style":120},[119],[31,45914,45916,45919],{"className":45915},[66],[31,45917],{"className":45918,"style":20863},[70],[31,45920,45922,45925],{"className":45921},[75],[31,45923,30042],{"className":45924},[75,76],[31,45926,45928],{"className":45927},[1039],[31,45929,45931],{"className":45930},[1043],[31,45932,45934],{"className":45933},[1048],[31,45935,45937],{"className":45936,"style":20863},[1052],[31,45938,45939,45942],{"style":3631},[31,45940],{"className":45941,"style":1061},[1060],[31,45943,45945],{"className":45944},[1065,1066,1067,1068],[31,45946,45948,45951,45954],{"className":45947},[75,1068],[31,45949,97],{"className":45950},[75,1068],[31,45952,45710],{"className":45953},[75,76,1068],[31,45955,53],{"className":45956},[75,76,1068],[31,45958,45960,45973],{"className":45959,"translate":35},[34],[31,45961,45963],{"className":45962},[39],[41,45964,45965],{"xmlns":43},[45,45966,45967,45971],{},[48,45968,45969],{},[51,45970,53],{},[55,45972,53],{"encoding":57},[31,45974,45976],{"className":45975,"ariaHidden":62},[61],[31,45977,45979,45982],{"className":45978},[66],[31,45980],{"className":45981,"style":71},[70],[31,45983,53],{"className":45984},[75,76]," 的 CDF，而密度按变量替换是",[31,45987,45989],{"className":45988,"translate":35},[411],[31,45990,45992,46072],{"className":45991,"translate":35},[34],[31,45993,45995],{"className":45994},[39],[41,45996,45997],{"xmlns":43,"display":420},[45,45998,45999,46069],{},[48,46000,46001,46007,46009,46011,46013,46015,46021,46023,46025,46027,46047,46049,46051,46053,46055,46067],{},[1004,46002,46003,46005],{},[51,46004,11],{},[51,46006,53],{},[95,46008,211],{"stretchy":210},[51,46010,53],{},[95,46012,220],{"stretchy":210},[95,46014,223],{},[1004,46016,46017,46019],{},[51,46018,11],{},[51,46020,5324],{},[95,46022,211],{"stretchy":210},[51,46024,5324],{},[95,46026,220],{"stretchy":210},[48,46028,46029,46031,46045],{},[95,46030,12680],{"fence":62},[2074,46032,46033,46039],{},[48,46034,46035,46037],{},[51,46036,1435],{},[51,46038,5324],{},[48,46040,46041,46043],{},[51,46042,1435],{},[51,46044,53],{},[95,46046,12680],{"fence":62},[95,46048,223],{},[91,46050,205],{},[95,46052,429],{},[51,46054,45710],{},[1180,46056,46057,46059],{},[51,46058,30042],{},[48,46060,46061,46063,46065],{},[95,46062,97],{},[51,46064,45710],{},[51,46066,53],{},[95,46068,1210],{"separator":62},[55,46070,46071],{"encoding":57},"p_x(x)=p_u(u)\\left|\\frac{du}{dx}\\right|=1\\cdot\\lambda e^{-\\lambda 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p_x(x)=\\log p_z(z)-\\log\\bigl|\\det J_f(z)\\bigr|.",[31,47138,47140,47212,47284],{"className":47139,"ariaHidden":62},[61],[31,47141,47143,47146,47151,47154,47194,47197,47200,47203,47206,47209],{"className":47142},[66],[31,47144],{"className":47145,"style":260},[70],[31,47147,20177,47149],{"className":47148},[1460],[31,47150,14556],{"style":20180},[31,47152],{"className":47153,"style":1467},[119],[31,47155,47157,47160],{"className":47156},[75],[31,47158,11],{"className":47159},[75,76],[31,47161,47163],{"className":47162},[1039],[31,47164,47166,47186],{"className":47165},[1043,1044],[31,47167,47169,47183],{"className":47168},[1048],[31,47170,47172],{"className":47171,"style":3230},[1052],[31,47173,47174,47177],{"style":2807},[31,47175],{"className":47176,"style":1061},[1060],[31,47178,47180],{"className":47179},[1065,1066,1067,1068],[31,47181,53],{"className":47182},[75,76,1068],[31,47184,1076],{"className":47185},[1075],[31,47187,47189],{"className":47188},[1048],[31,47190,47192],{"className":47191,"style":2050},[1052],[31,47193],{},[31,47195,211],{"className":47196},[264],[31,47198,53],{"className":47199},[75,76],[31,47201,220],{"className":47202},[289],[31,47204],{"className":47205,"style":293},[119],[31,47207,223],{"className":47208},[297],[31,47210],{"className":47211,"style":293},[119],[31,47213,47215,47218,47223,47226,47266,47269,47272,47275,47278,47281],{"className":47214},[66],[31,47216],{"className":47217,"style":260},[70],[31,47219,20177,47221],{"className":47220},[1460],[31,47222,14556],{"style":20180},[31,47224],{"className":47225,"style":1467},[119],[31,47227,47229,47232],{"className":47228},[75],[31,47230,11],{"className":47231},[75,76],[31,47233,47235],{"className":47234},[1039],[31,47236,47238,47258],{"className":47237},[1043,1044],[31,47239,47241,47255],{"className":47240},[1048],[31,47242,47244],{"className":47243,"style":3230},[1052],[31,47245,47246,47249],{"style":2807},[31,47247],{"className":47248,"style":1061},[1060],[31,47250,47252],{"className":47251},[1065,1066,1067,1068],[31,47253,46458],{"className":47254,"style":46500},[75,76,1068],[31,47256,1076],{"className":47257},[1075],[31,47259,47261],{"className":47260},[1048],[31,47262,47264],{"className":47263,"style":2050},[1052],[31,47265],{},[31,47267,211],{"className":47268},[264],[31,47270,46458],{"className":47271,"style":46500},[75,76],[31,47273,220],{"className":47274},[289],[31,47276],{"className":47277,"style":120},[119],[31,47279,97],{"className":47280},[124],[31,47282],{"className":47283,"style":120},[119],[31,47285,47287,47290,47295,47332,47335,47338,47380,47383,47386,47389,47426],{"className":47286},[66],[31,47288],{"className":47289,"style":4988},[70],[31,47291,20177,47293],{"className":47292},[1460],[31,47294,14556],{"style":20180},[31,47296,47298],{"className":47297},[264],[31,47299,47301],{"className":47300},[1307,13319],[31,47302,47304,47324],{"className":47303},[1043,1044],[31,47305,47307,47321],{"className":47306},[1048],[31,47308,47310],{"className":47309,"style":13329},[1052],[31,47311,47312,47315],{"style":13332},[31,47313],{"className":47314,"style":13336},[1060],[31,47316,47317],{"style":13339},[13341,47318,47319],{"xmlns":13343,"width":13344,"height":1178,"viewBox":13345},[13347,47320],{"d":13349},[31,47322,1076],{"className":47323},[1075],[31,47325,47327],{"className":47326},[1048],[31,47328,47330],{"className":47329,"style":13359},[1052],[31,47331],{},[31,47333,46659],{"className":47334},[1460],[31,47336],{"className":47337,"style":1467},[119],[31,47339,47341,47345],{"className":47340},[75],[31,47342,47121],{"className":47343,"style":47344},[75,76],"margin-right:0.0962em;",[31,47346,47348],{"className":47347},[1039],[31,47349,47351,47372],{"className":47350},[1043,1044],[31,47352,47354,47369],{"className":47353},[1048],[31,47355,47357],{"className":47356,"style":1053},[1052],[31,47358,47360,47363],{"style":47359},"top:-2.55em;margin-left:-0.0962em;margin-right:0.05em;",[31,47361],{"className":47362,"style":1061},[1060],[31,47364,47366],{"className":47365},[1065,1066,1067,1068],[31,47367,1662],{"className":47368,"style":35196},[75,76,1068],[31,47370,1076],{"className":47371},[1075],[31,47373,47375],{"className":47374},[1048],[31,47376,47378],{"className":47377,"style":1083},[1052],[31,47379],{},[31,47381,211],{"className":47382},[264],[31,47384,46458],{"className":47385,"style":46500},[75,76],[31,47387,220],{"className":47388},[289],[31,47390,47392],{"className":47391},[289],[31,47393,47395],{"className":47394},[1307,13319],[31,47396,47398,47418],{"className":47397},[1043,1044],[31,47399,47401,47415],{"className":47400},[1048],[31,47402,47404],{"className":47403,"style":13329},[1052],[31,47405,47406,47409],{"style":13332},[31,47407],{"className":47408,"style":13336},[1060],[31,47410,47411],{"style":13339},[13341,47412,47413],{"xmlns":13343,"width":13344,"height":1178,"viewBox":13345},[13347,47414],{"d":13349},[31,47416,1076],{"className":47417},[1075],[31,47419,47421],{"className":47420},[1048],[31,47422,47424],{"className":47423,"style":13359},[1052],[31,47425],{},[31,47427,495],{"className":47428},[75],[11,47430,27532,47431,47490],{},[31,47432,47434,47458],{"className":47433,"translate":35},[34],[31,47435,47437],{"className":47436},[39],[41,47438,47439],{"xmlns":43},[45,47440,47441,47455],{},[48,47442,47443,47445,47447,47449,47451,47453],{},[51,47444,19939],{},[95,47446,1540],{},[51,47448,12680],{"mathvariant":494},[51,47450,46659],{},[95,47452,1540],{},[51,47454,12680],{"mathvariant":494},[55,47456,47457],{"encoding":57},"\\log|\\det|",[31,47459,47461],{"className":47460,"ariaHidden":62},[61],[31,47462,47464,47467,47472,47475,47478,47481,47484,47487],{"className":47463},[66],[31,47465],{"className":47466,"style":260},[70],[31,47468,20177,47470],{"className":47469},[1460],[31,47471,14556],{"style":20180},[31,47473],{"className":47474,"style":1467},[119],[31,47476,12680],{"className":47477},[75],[31,47479],{"className":47480,"style":1467},[119],[31,47482,46659],{"className":47483},[1460],[31,47485],{"className":47486,"style":1467},[119],[31,47488,12680],{"className":47489},[75]," 项就是“局部的伸缩率”：变换把一小块体积放大了多少，密度就要相应缩小多少。多层复合时 log-det 相加，所以加深网络不增加计算上的困难，只要每层的 Jacobian 行列式好算（比如三角矩阵）。",[11,47492,47493,47494,15282,47573,1138,47664,47729,47730,47758,47759,1138,47838,47952,47953,2372],{},"一维的情形可以推到极致：任意一维分布都能写成逆 CDF。取 ",[31,47495,47497,47525],{"className":47496,"translate":35},[34],[31,47498,47500],{"className":47499},[39],[41,47501,47502],{"xmlns":43},[45,47503,47504,47522],{},[48,47505,47506,47508,47510,47512,47514,47516,47518,47520],{},[51,47507,46458],{},[95,47509,46520],{},[51,47511,12818],{},[95,47513,211],{"stretchy":210},[91,47515,1977],{},[95,47517,1210],{"separator":62},[91,47519,205],{},[95,47521,220],{"stretchy":210},[55,47523,47524],{"encoding":57},"z\\sim U(0,1)",[31,47526,47528,47546],{"className":47527,"ariaHidden":62},[61],[31,47529,47531,47534,47537,47540,47543],{"className":47530},[66],[31,47532],{"className":47533,"style":71},[70],[31,47535,46458],{"className":47536,"style":46500},[75,76],[31,47538],{"className":47539,"style":293},[119],[31,47541,46520],{"className":47542},[297],[31,47544],{"className":47545,"style":293},[119],[31,47547,47549,47552,47555,47558,47561,47564,47567,47570],{"className":47548},[66],[31,47550],{"className":47551,"style":260},[70],[31,47553,12818],{"className":47554,"style":12844},[75,76],[31,47556,211],{"className":47557},[264],[31,47559,1977],{"className":47560},[75],[31,47562,1210],{"className":47563},[1391],[31,47565],{"className":47566,"style":1467},[119],[31,47568,205],{"className":47569},[75],[31,47571,220],{"className":47572},[289],[31,47574,47576,47602],{"className":47575,"translate":35},[34],[31,47577,47579],{"className":47578},[39],[41,47580,47581],{"xmlns":43},[45,47582,47583,47599],{},[48,47584,47585,47587,47589],{},[51,47586,1662],{},[95,47588,223],{},[1180,47590,47591,47593],{},[51,47592,13114],{},[48,47594,47595,47597],{},[95,47596,97],{},[91,47598,205],{},[55,47600,47601],{"encoding":57},"f=F^{-1}",[31,47603,47605,47623],{"className":47604,"ariaHidden":62},[61],[31,47606,47608,47611,47614,47617,47620],{"className":47607},[66],[31,47609],{"className":47610,"style":4202},[70],[31,47612,1662],{"className":47613,"style":35196},[75,76],[31,47615],{"className":47616,"style":293},[119],[31,47618,223],{"className":47619},[297],[31,47621],{"className":47622,"style":293},[119],[31,47624,47626,47629],{"className":47625},[66],[31,47627],{"className":47628,"style":5135},[70],[31,47630,47632,47635],{"className":47631},[75],[31,47633,13114],{"className":47634,"style":949},[75,76],[31,47636,47638],{"className":47637},[1039],[31,47639,47641],{"className":47640},[1043],[31,47642,47644],{"className":47643},[1048],[31,47645,47647],{"className":47646,"style":5135},[1052],[31,47648,47649,47652],{"style":3631},[31,47650],{"className":47651,"style":1061},[1060],[31,47653,47655],{"className":47654},[1065,1066,1067,1068],[31,47656,47658,47661],{"className":47657},[75,1068],[31,47659,97],{"className":47660},[75,1068],[31,47662,205],{"className":47663},[75,1068],[31,47665,47667,47690],{"className":47666,"translate":35},[34],[31,47668,47670],{"className":47669},[39],[41,47671,47672],{"xmlns":43},[45,47673,47674,47688],{},[48,47675,47676,47678,47680,47682,47684,47686],{},[51,47677,53],{},[95,47679,223],{},[51,47681,1662],{},[95,47683,211],{"stretchy":210},[51,47685,46458],{},[95,47687,220],{"stretchy":210},[55,47689,46463],{"encoding":57},[31,47691,47693,47711],{"className":47692,"ariaHidden":62},[61],[31,47694,47696,47699,47702,47705,47708],{"className":47695},[66],[31,47697],{"className":47698,"style":71},[70],[31,47700,53],{"className":47701},[75,76],[31,47703],{"className":47704,"style":293},[119],[31,47706,223],{"className":47707},[297],[31,47709],{"className":47710,"style":293},[119],[31,47712,47714,47717,47720,47723,47726],{"className":47713},[66],[31,47715],{"className":47716,"style":260},[70],[31,47718,1662],{"className":47719,"style":35196},[75,76],[31,47721,211],{"className":47722},[264],[31,47724,46458],{"className":47725,"style":46500},[75,76],[31,47727,220],{"className":47728},[289]," 的分布就是 ",[31,47731,47733,47746],{"className":47732,"translate":35},[34],[31,47734,47736],{"className":47735},[39],[41,47737,47738],{"xmlns":43},[45,47739,47740,47744],{},[48,47741,47742],{},[51,47743,13114],{},[55,47745,13114],{"encoding":57},[31,47747,47749],{"className":47748,"ariaHidden":62},[61],[31,47750,47752,47755],{"className":47751},[66],[31,47753],{"className":47754,"style":945},[70],[31,47756,13114],{"className":47757,"style":949},[75,76],"，这是精确的，不是近似。取 ",[31,47760,47762,47790],{"className":47761,"translate":35},[34],[31,47763,47765],{"className":47764},[39],[41,47766,47767],{"xmlns":43},[45,47768,47769,47787],{},[48,47770,47771,47773,47775,47777,47779,47781,47783,47785],{},[51,47772,46458],{},[95,47774,46520],{},[51,47776,4935],{"mathvariant":925},[95,47778,211],{"stretchy":210},[91,47780,1977],{},[95,47782,1210],{"separator":62},[91,47784,205],{},[95,47786,220],{"stretchy":210},[55,47788,47789],{"encoding":57},"z\\sim\\mathcal N(0,1)",[31,47791,47793,47811],{"className":47792,"ariaHidden":62},[61],[31,47794,47796,47799,47802,47805,47808],{"className":47795},[66],[31,47797],{"className":47798,"style":71},[70],[31,47800,46458],{"className":47801,"style":46500},[75,76],[31,47803],{"className":47804,"style":293},[119],[31,47806,46520],{"className":47807},[297],[31,47809],{"className":47810,"style":293},[119],[31,47812,47814,47817,47820,47823,47826,47829,47832,47835],{"className":47813},[66],[31,47815],{"className":47816,"style":260},[70],[31,47818,4935],{"className":47819,"style":5004},[75,968],[31,47821,211],{"className":47822},[264],[31,47824,1977],{"className":47825},[75],[31,47827,1210],{"className":47828},[1391],[31,47830],{"className":47831,"style":1467},[119],[31,47833,205],{"className":47834},[75],[31,47836,220],{"className":47837},[289],[31,47839,47841,47872],{"className":47840,"translate":35},[34],[31,47842,47844],{"className":47843},[39],[41,47845,47846],{"xmlns":43},[45,47847,47848,47869],{},[48,47849,47850,47852,47854,47864,47866],{},[51,47851,1662],{},[95,47853,223],{},[1180,47855,47856,47858],{},[51,47857,13114],{},[48,47859,47860,47862],{},[95,47861,97],{},[91,47863,205],{},[95,47865,16209],{},[51,47867,47868],{"mathvariant":494},"Φ",[55,47870,47871],{"encoding":57},"f=F^{-1}\\circ\\Phi",[31,47873,47875,47893,47943],{"className":47874,"ariaHidden":62},[61],[31,47876,47878,47881,47884,47887,47890],{"className":47877},[66],[31,47879],{"className":47880,"style":4202},[70],[31,47882,1662],{"className":47883,"style":35196},[75,76],[31,47885],{"className":47886,"style":293},[119],[31,47888,223],{"className":47889},[297],[31,47891],{"className":47892,"style":293},[119],[31,47894,47896,47899,47934,47937,47940],{"className":47895},[66],[31,47897],{"className":47898,"style":5135},[70],[31,47900,47902,47905],{"className":47901},[75],[31,47903,13114],{"className":47904,"style":949},[75,76],[31,47906,47908],{"className":47907},[1039],[31,47909,47911],{"className":47910},[1043],[31,47912,47914],{"className":47913},[1048],[31,47915,47917],{"className":47916,"style":5135},[1052],[31,47918,47919,47922],{"style":3631},[31,47920],{"className":47921,"style":1061},[1060],[31,47923,47925],{"className":47924},[1065,1066,1067,1068],[31,47926,47928,47931],{"className":47927},[75,1068],[31,47929,97],{"className":47930},[75,1068],[31,47932,205],{"className":47933},[75,1068],[31,47935],{"className":47936,"style":120},[119],[31,47938,16209],{"className":47939},[124],[31,47941],{"className":47942,"style":120},[119],[31,47944,47946,47949],{"className":47945},[66],[31,47947],{"className":47948,"style":945},[70],[31,47950,47868],{"className":47951},[75]," 就得到任意一维分布。上面那个指数分布的例子就是 ",[31,47954,47956,48006],{"className":47955,"translate":35},[34],[31,47957,47959],{"className":47958},[39],[41,47960,47961],{"xmlns":43},[45,47962,47963,48003],{},[48,47964,47965,47975,47977,47979,47981,47983,47985,47987,47989,47991,47993,47995,47997,47999,48001],{},[1180,47966,47967,47969],{},[51,47968,13114],{},[48,47970,47971,47973],{},[95,47972,97],{},[91,47974,205],{},[95,47976,211],{"stretchy":210},[51,47978,5324],{},[95,47980,220],{"stretchy":210},[95,47982,223],{},[95,47984,97],{},[51,47986,19939],{},[95,47988,1540],{},[95,47990,211],{"stretchy":210},[91,47992,205],{},[95,47994,97],{},[51,47996,5324],{},[95,47998,220],{"stretchy":210},[51,48000,3768],{"mathvariant":494},[51,48002,45710],{},[55,48004,48005],{"encoding":57},"F^{-1}(u)=-\\log(1-u)\u002F\\lambda",[31,48007,48009,48068,48100],{"className":48008,"ariaHidden":62},[61],[31,48010,48012,48015,48050,48053,48056,48059,48062,48065],{"className":48011},[66],[31,48013],{"className":48014,"style":12035},[70],[31,48016,48018,48021],{"className":48017},[75],[31,48019,13114],{"className":48020,"style":949},[75,76],[31,48022,48024],{"className":48023},[1039],[31,48025,48027],{"className":48026},[1043],[31,48028,48030],{"className":48029},[1048],[31,48031,48033],{"className":48032,"style":5135},[1052],[31,48034,48035,48038],{"style":3631},[31,48036],{"className":48037,"style":1061},[1060],[31,48039,48041],{"className":48040},[1065,1066,1067,1068],[31,48042,48044,48047],{"className":48043},[75,1068],[31,48045,97],{"className":48046},[75,1068],[31,48048,205],{"className":48049},[75,1068],[31,48051,211],{"className":48052},[264],[31,48054,5324],{"className":48055},[75,76],[31,48057,220],{"className":48058},[289],[31,48060],{"className":48061,"style":293},[119],[31,48063,223],{"className":48064},[297],[31,48066],{"className":48067,"style":293},[119],[31,48069,48071,48074,48077,48080,48085,48088,48091,48094,48097],{"className":48070},[66],[31,48072],{"className":48073,"style":260},[70],[31,48075,97],{"className":48076},[75],[31,48078],{"className":48079,"style":1467},[119],[31,48081,20177,48083],{"className":48082},[1460],[31,48084,14556],{"style":20180},[31,48086,211],{"className":48087},[264],[31,48089,205],{"className":48090},[75],[31,48092],{"className":48093,"style":120},[119],[31,48095,97],{"className":48096},[124],[31,48098],{"className":48099,"style":120},[119],[31,48101,48103,48106,48109,48112,48115],{"className":48102},[66],[31,48104],{"className":48105,"style":260},[70],[31,48107,5324],{"className":48108},[75,76],[31,48110,220],{"className":48111},[289],[31,48113,3768],{"className":48114},[75],[31,48116,45710],{"className":48117},[75,76],[11,48119,48120,48121,48413,48414,48442,48443,48735,48736,48848,48849,48901,48902,48969,48970,48998,48999,49066],{},"举个能算的例子。",[31,48122,48124,48184],{"className":48123,"translate":35},[34],[31,48125,48127],{"className":48126},[39],[41,48128,48129],{"xmlns":43},[45,48130,48131,48181],{},[48,48132,48133,48135,48137,48139,48141,48143,48151,48153,48155,48157,48159,48161,48169,48171,48173,48175,48177,48179],{},[51,48134,11],{},[95,48136,211],{"stretchy":210},[51,48138,53],{},[95,48140,220],{"stretchy":210},[95,48142,223],{},[2071,48144,48145],{"scriptlevel":1977,"displaystyle":210},[2074,48146,48147,48149],{},[91,48148,205],{},[91,48150,152],{},[51,48152,8059],{},[95,48154,211],{"stretchy":210},[51,48156,53],{},[95,48158,220],{"stretchy":210},[95,48160,434],{},[2071,48162,48163],{"scriptlevel":1977,"displaystyle":210},[2074,48164,48165,48167],{},[91,48166,205],{},[91,48168,152],{},[51,48170,8059],{},[95,48172,211],{"stretchy":210},[51,48174,53],{},[95,48176,97],{},[91,48178,93],{},[95,48180,220],{"stretchy":210},[55,48182,48183],{"encoding":57},"p(x)=\\tfrac12\\varphi(x)+\\tfrac12\\varphi(x-3)",[31,48185,48187,48214,48309,48401],{"className":48186,"ariaHidden":62},[61],[31,48188,48190,48193,48196,48199,48202,48205,48208,48211],{"className":48189},[66],[31,48191],{"className":48192,"style":260},[70],[31,48194,11],{"className":48195},[75,76],[31,48197,211],{"className":48198},[264],[31,48200,53],{"className":48201},[75,76],[31,48203,220],{"className":48204},[289],[31,48206],{"className":48207,"style":293},[119],[31,48209,223],{"className":48210},[297],[31,48212],{"className":48213,"style":293},[119],[31,48215,48217,48220,48288,48291,48294,48297,48300,48303,48306],{"className":48216},[66],[31,48218],{"className":48219,"style":2132},[70],[31,48221,48223,48226,48285],{"className":48222},[75],[31,48224],{"className":48225},[264,2139],[31,48227,48229],{"className":48228},[2074],[31,48230,48232,48277],{"className":48231},[1043,1044],[31,48233,48235,48274],{"className":48234},[1048],[31,48236,48238,48252,48260],{"className":48237,"style":2152},[1052],[31,48239,48240,48243],{"style":2155},[31,48241],{"className":48242,"style":1637},[1060],[31,48244,48246],{"className":48245},[1065,1066,1067,1068],[31,48247,48249],{"className":48248},[75,1068],[31,48250,152],{"className":48251},[75,1068],[31,48253,48254,48257],{"style":2170},[31,48255],{"className":48256,"style":1637},[1060],[31,48258],{"className":48259,"style":2178},[2177],[31,48261,48262,48265],{"style":2181},[31,48263],{"className":48264,"style":1637},[1060],[31,48266,48268],{"className":48267},[1065,1066,1067,1068],[31,48269,48271],{"className":48270},[75,1068],[31,48272,205],{"className":48273},[75,1068],[31,48275,1076],{"className":48276},[1075],[31,48278,48280],{"className":48279},[1048],[31,48281,48283],{"className":48282,"style":2203},[1052],[31,48284],{},[31,48286],{"className":48287},[289,2139],[31,48289,8059],{"className":48290},[75,76],[31,48292,211],{"className":48293},[264],[31,48295,53],{"className":48296},[75,76],[31,48298,220],{"className":48299},[289],[31,48301],{"className":48302,"style":120},[119],[31,48304,434],{"className":48305},[124],[31,48307],{"className":48308,"style":120},[119],[31,48310,48312,48315,48383,48386,48389,48392,48395,48398],{"className":48311},[66],[31,48313],{"className":48314,"style":2132},[70],[31,48316,48318,48321,48380],{"className":48317},[75],[31,48319],{"className":48320},[264,2139],[31,48322,48324],{"className":48323},[2074],[31,48325,48327,48372],{"className":48326},[1043,1044],[31,48328,48330,48369],{"className":48329},[1048],[31,48331,48333,48347,48355],{"className":48332,"style":2152},[1052],[31,48334,48335,48338],{"style":2155},[31,48336],{"className":48337,"style":1637},[1060],[31,48339,48341],{"className":48340},[1065,1066,1067,1068],[31,48342,48344],{"className":48343},[75,1068],[31,48345,152],{"className":48346},[75,1068],[31,48348,48349,48352],{"style":2170},[31,48350],{"className":48351,"style":1637},[1060],[31,48353],{"className":48354,"style":2178},[2177],[31,48356,48357,48360],{"style":2181},[31,48358],{"className":48359,"style":1637},[1060],[31,48361,48363],{"className":48362},[1065,1066,1067,1068],[31,48364,48366],{"className":48365},[75,1068],[31,48367,205],{"className":48368},[75,1068],[31,48370,1076],{"className":48371},[1075],[31,48373,48375],{"className":48374},[1048],[31,48376,48378],{"className":48377,"style":2203},[1052],[31,48379],{},[31,48381],{"className":48382},[289,2139],[31,48384,8059],{"className":48385},[75,76],[31,48387,211],{"className":48388},[264],[31,48390,53],{"className":48391},[75,76],[31,48393],{"className":48394,"style":120},[119],[31,48396,97],{"className":48397},[124],[31,48399],{"className":48400,"style":120},[119],[31,48402,48404,48407,48410],{"className":48403},[66],[31,48405],{"className":48406,"style":260},[70],[31,48408,93],{"className":48409},[75],[31,48411,220],{"className":48412},[289],"，两个单位方差高斯，中心相距 ",[31,48415,48417,48430],{"className":48416,"translate":35},[34],[31,48418,48420],{"className":48419},[39],[41,48421,48422],{"xmlns":43},[45,48423,48424,48428],{},[48,48425,48426],{},[91,48427,93],{},[55,48429,93],{"encoding":57},[31,48431,48433],{"className":48432,"ariaHidden":62},[61],[31,48434,48436,48439],{"className":48435},[66],[31,48437],{"className":48438,"style":326},[70],[31,48440,93],{"className":48441},[75],"。它的 CDF 是 ",[31,48444,48446,48506],{"className":48445,"translate":35},[34],[31,48447,48449],{"className":48448},[39],[41,48450,48451],{"xmlns":43},[45,48452,48453,48503],{},[48,48454,48455,48457,48459,48461,48463,48465,48473,48475,48477,48479,48481,48483,48491,48493,48495,48497,48499,48501],{},[51,48456,13114],{},[95,48458,211],{"stretchy":210},[51,48460,53],{},[95,48462,220],{"stretchy":210},[95,48464,223],{},[2071,48466,48467],{"scriptlevel":1977,"displaystyle":210},[2074,48468,48469,48471],{},[91,48470,205],{},[91,48472,152],{},[51,48474,47868],{"mathvariant":494},[95,48476,211],{"stretchy":210},[51,48478,53],{},[95,48480,220],{"stretchy":210},[95,48482,434],{},[2071,48484,48485],{"scriptlevel":1977,"displaystyle":210},[2074,48486,48487,48489],{},[91,48488,205],{},[91,48490,152],{},[51,48492,47868],{"mathvariant":494},[95,48494,211],{"stretchy":210},[51,48496,53],{},[95,48498,97],{},[91,48500,93],{},[95,48502,220],{"stretchy":210},[55,48504,48505],{"encoding":57},"F(x)=\\tfrac12\\Phi(x)+\\tfrac12\\Phi(x-3)",[31,48507,48509,48536,48631,48723],{"className":48508,"ariaHidden":62},[61],[31,48510,48512,48515,48518,48521,48524,48527,48530,48533],{"className":48511},[66],[31,48513],{"className":48514,"style":260},[70],[31,48516,13114],{"className":48517,"style":949},[75,76],[31,48519,211],{"className":48520},[264],[31,48522,53],{"className":48523},[75,76],[31,48525,220],{"className":48526},[289],[31,48528],{"className":48529,"style":293},[119],[31,48531,223],{"className":48532},[297],[31,48534],{"className":48535,"style":293},[119],[31,48537,48539,48542,48610,48613,48616,48619,48622,48625,48628],{"className":48538},[66],[31,48540],{"className":48541,"style":2132},[70],[31,48543,48545,48548,48607],{"className":48544},[75],[31,48546],{"className":48547},[264,2139],[31,48549,48551],{"className":48550},[2074],[31,48552,48554,48599],{"className":48553},[1043,1044],[31,48555,48557,48596],{"className":48556},[1048],[31,48558,48560,48574,48582],{"className":48559,"style":2152},[1052],[31,48561,48562,48565],{"style":2155},[31,48563],{"className":48564,"style":1637},[1060],[31,48566,48568],{"className":48567},[1065,1066,1067,1068],[31,48569,48571],{"className":48570},[75,1068],[31,48572,152],{"className":48573},[75,1068],[31,48575,48576,48579],{"style":2170},[31,48577],{"className":48578,"style":1637},[1060],[31,48580],{"className":48581,"style":2178},[2177],[31,48583,48584,48587],{"style":2181},[31,48585],{"className":48586,"style":1637},[1060],[31,48588,48590],{"className":48589},[1065,1066,1067,1068],[31,48591,48593],{"className":48592},[75,1068],[31,48594,205],{"className":48595},[75,1068],[31,48597,1076],{"className":48598},[1075],[31,48600,48602],{"className":48601},[1048],[31,48603,48605],{"className":48604,"style":2203},[1052],[31,48606],{},[31,48608],{"className":48609},[289,2139],[31,48611,47868],{"className":48612},[75],[31,48614,211],{"className":48615},[264],[31,48617,53],{"className":48618},[75,76],[31,48620,220],{"className":48621},[289],[31,48623],{"className":48624,"style":120},[119],[31,48626,434],{"className":48627},[124],[31,48629],{"className":48630,"style":120},[119],[31,48632,48634,48637,48705,48708,48711,48714,48717,48720],{"className":48633},[66],[31,48635],{"className":48636,"style":2132},[70],[31,48638,48640,48643,48702],{"className":48639},[75],[31,48641],{"className":48642},[264,2139],[31,48644,48646],{"className":48645},[2074],[31,48647,48649,48694],{"className":48648},[1043,1044],[31,48650,48652,48691],{"className":48651},[1048],[31,48653,48655,48669,48677],{"className":48654,"style":2152},[1052],[31,48656,48657,48660],{"style":2155},[31,48658],{"className":48659,"style":1637},[1060],[31,48661,48663],{"className":48662},[1065,1066,1067,1068],[31,48664,48666],{"className":48665},[75,1068],[31,48667,152],{"className":48668},[75,1068],[31,48670,48671,48674],{"style":2170},[31,48672],{"className":48673,"style":1637},[1060],[31,48675],{"className":48676,"style":2178},[2177],[31,48678,48679,48682],{"style":2181},[31,48680],{"className":48681,"style":1637},[1060],[31,48683,48685],{"className":48684},[1065,1066,1067,1068],[31,48686,48688],{"className":48687},[75,1068],[31,48689,205],{"className":48690},[75,1068],[31,48692,1076],{"className":48693},[1075],[31,48695,48697],{"className":48696},[1048],[31,48698,48700],{"className":48699,"style":2203},[1052],[31,48701],{},[31,48703],{"className":48704},[289,2139],[31,48706,47868],{"className":48707},[75],[31,48709,211],{"className":48710},[264],[31,48712,53],{"className":48713},[75,76],[31,48715],{"className":48716,"style":120},[119],[31,48718,97],{"className":48719},[124],[31,48721],{"className":48722,"style":120},[119],[31,48724,48726,48729,48732],{"className":48725},[66],[31,48727],{"className":48728,"style":260},[70],[31,48730,93],{"className":48731},[75],[31,48733,220],{"className":48734},[289],"，流就是 ",[31,48737,48739,48768],{"className":48738,"translate":35},[34],[31,48740,48742],{"className":48741},[39],[41,48743,48744],{"xmlns":43},[45,48745,48746,48766],{},[48,48747,48748,48750,48752,48762,48764],{},[51,48749,1662],{},[95,48751,223],{},[1180,48753,48754,48756],{},[51,48755,13114],{},[48,48757,48758,48760],{},[95,48759,97],{},[91,48761,205],{},[95,48763,16209],{},[51,48765,47868],{"mathvariant":494},[55,48767,47871],{"encoding":57},[31,48769,48771,48789,48839],{"className":48770,"ariaHidden":62},[61],[31,48772,48774,48777,48780,48783,48786],{"className":48773},[66],[31,48775],{"className":48776,"style":4202},[70],[31,48778,1662],{"className":48779,"style":35196},[75,76],[31,48781],{"className":48782,"style":293},[119],[31,48784,223],{"className":48785},[297],[31,48787],{"className":48788,"style":293},[119],[31,48790,48792,48795,48830,48833,48836],{"className":48791},[66],[31,48793],{"className":48794,"style":5135},[70],[31,48796,48798,48801],{"className":48797},[75],[31,48799,13114],{"className":48800,"style":949},[75,76],[31,48802,48804],{"className":48803},[1039],[31,48805,48807],{"className":48806},[1043],[31,48808,48810],{"className":48809},[1048],[31,48811,48813],{"className":48812,"style":5135},[1052],[31,48814,48815,48818],{"style":3631},[31,48816],{"className":48817,"style":1061},[1060],[31,48819,48821],{"className":48820},[1065,1066,1067,1068],[31,48822,48824,48827],{"className":48823},[75,1068],[31,48825,97],{"className":48826},[75,1068],[31,48828,205],{"className":48829},[75,1068],[31,48831],{"className":48832,"style":120},[119],[31,48834,16209],{"className":48835},[124],[31,48837],{"className":48838,"style":120},[119],[31,48840,48842,48845],{"className":48841},[66],[31,48843],{"className":48844,"style":945},[70],[31,48846,47868],{"className":48847},[75],"。两峰之间 ",[31,48850,48852,48871],{"className":48851,"translate":35},[34],[31,48853,48855],{"className":48854},[39],[41,48856,48857],{"xmlns":43},[45,48858,48859,48868],{},[48,48860,48861,48863,48865],{},[51,48862,53],{},[95,48864,223],{},[91,48866,48867],{},"1.5",[55,48869,48870],{"encoding":57},"x=1.5",[31,48872,48874,48892],{"className":48873,"ariaHidden":62},[61],[31,48875,48877,48880,48883,48886,48889],{"className":48876},[66],[31,48878],{"className":48879,"style":71},[70],[31,48881,53],{"className":48882},[75,76],[31,48884],{"className":48885,"style":293},[119],[31,48887,223],{"className":48888},[297],[31,48890],{"className":48891,"style":293},[119],[31,48893,48895,48898],{"className":48894},[66],[31,48896],{"className":48897,"style":326},[70],[31,48899,48867],{"className":48900},[75]," 处的真实密度是 ",[31,48903,48905,48930],{"className":48904,"translate":35},[34],[31,48906,48908],{"className":48907},[39],[41,48909,48910],{"xmlns":43},[45,48911,48912,48927],{},[48,48913,48914,48916,48918,48920,48922,48924],{},[51,48915,8059],{},[95,48917,211],{"stretchy":210},[91,48919,48867],{},[95,48921,220],{"stretchy":210},[95,48923,27223],{},[91,48925,48926],{},"0.1295",[55,48928,48929],{"encoding":57},"\\varphi(1.5)\\approx0.1295",[31,48931,48933,48960],{"className":48932,"ariaHidden":62},[61],[31,48934,48936,48939,48942,48945,48948,48951,48954,48957],{"className":48935},[66],[31,48937],{"className":48938,"style":260},[70],[31,48940,8059],{"className":48941},[75,76],[31,48943,211],{"className":48944},[264],[31,48946,48867],{"className":48947},[75],[31,48949,220],{"className":48950},[289],[31,48952],{"className":48953,"style":293},[119],[31,48955,27223],{"className":48956},[297],[31,48958],{"className":48959,"style":293},[119],[31,48961,48963,48966],{"className":48962},[66],[31,48964],{"className":48965,"style":326},[70],[31,48967,48926],{"className":48968},[75],"；如果错当成单个高斯（中心在 ",[31,48971,48973,48986],{"className":48972,"translate":35},[34],[31,48974,48976],{"className":48975},[39],[41,48977,48978],{"xmlns":43},[45,48979,48980,48984],{},[48,48981,48982],{},[91,48983,48867],{},[55,48985,48867],{"encoding":57},[31,48987,48989],{"className":48988,"ariaHidden":62},[61],[31,48990,48992,48995],{"className":48991},[66],[31,48993],{"className":48994,"style":326},[70],[31,48996,48867],{"className":48997},[75],"、单峰），会给出 ",[31,49000,49002,49027],{"className":49001,"translate":35},[34],[31,49003,49005],{"className":49004},[39],[41,49006,49007],{"xmlns":43},[45,49008,49009,49024],{},[48,49010,49011,49013,49015,49017,49019,49021],{},[51,49012,8059],{},[95,49014,211],{"stretchy":210},[91,49016,1977],{},[95,49018,220],{"stretchy":210},[95,49020,27223],{},[91,49022,49023],{},"0.3989",[55,49025,49026],{"encoding":57},"\\varphi(0)\\approx0.3989",[31,49028,49030,49057],{"className":49029,"ariaHidden":62},[61],[31,49031,49033,49036,49039,49042,49045,49048,49051,49054],{"className":49032},[66],[31,49034],{"className":49035,"style":260},[70],[31,49037,8059],{"className":49038},[75,76],[31,49040,211],{"className":49041},[264],[31,49043,1977],{"className":49044},[75],[31,49046,220],{"className":49047},[289],[31,49049],{"className":49050,"style":293},[119],[31,49052,27223],{"className":49053},[297],[31,49055],{"className":49056,"style":293},[119],[31,49058,49060,49063],{"className":49059},[66],[31,49061],{"className":49062,"style":326},[70],[31,49064,49023],{"className":49065},[75],"，高估三倍。流模型能把这个“鞍部”学出来，靠的正是分段可逆映射，而不是更大的方差。",[11,49068,49069,49070,49169],{},"连续情形就是 Neural ODE：",[31,49071,49073,49109],{"className":49072,"translate":35},[34],[31,49074,49076],{"className":49075},[39],[41,49077,49078],{"xmlns":43},[45,49079,49080,49106],{},[48,49081,49082,49084,49086,49088,49090,49092,49094,49096,49098,49100,49102,49104],{},[51,49083,1435],{},[51,49085,46458],{},[51,49087,3768],{"mathvariant":494},[51,49089,1435],{},[51,49091,12271],{},[95,49093,223],{},[51,49095,5354],{},[95,49097,211],{"stretchy":210},[51,49099,46458],{},[95,49101,1210],{"separator":62},[51,49103,12271],{},[95,49105,220],{"stretchy":210},[55,49107,49108],{"encoding":57},"dz\u002Fdt=v(z,t)",[31,49110,49112,49142],{"className":49111,"ariaHidden":62},[61],[31,49113,49115,49118,49121,49124,49127,49130,49133,49136,49139],{"className":49114},[66],[31,49116],{"className":49117,"style":260},[70],[31,49119,1435],{"className":49120},[75,76],[31,49122,46458],{"className":49123,"style":46500},[75,76],[31,49125,3768],{"className":49126},[75],[31,49128,1435],{"className":49129},[75,76],[31,49131,12271],{"className":49132},[75,76],[31,49134],{"className":49135,"style":293},[119],[31,49137,223],{"className":49138},[297],[31,49140],{"className":49141,"style":293},[119],[31,49143,49145,49148,49151,49154,49157,49160,49163,49166],{"className":49144},[66],[31,49146],{"className":49147,"style":260},[70],[31,49149,5354],{"className":49150,"style":2666},[75,76],[31,49152,211],{"className":49153},[264],[31,49155,46458],{"className":49156,"style":46500},[75,76],[31,49158,1210],{"className":49159},[1391],[31,49161],{"className":49162,"style":1467},[119],[31,49164,12271],{"className":49165},[75,76],[31,49167,220],{"className":49168},[289],"，密度沿轨迹演化，",[31,49171,49173],{"className":49172,"translate":35},[411],[31,49174,49176,49248],{"className":49175,"translate":35},[34],[31,49177,49179],{"className":49178},[39],[41,49180,49181],{"xmlns":43,"display":420},[45,49182,49183,49245],{},[48,49184,49185,49195,49197,49199,49205,49207,49213,49215,49217,49219,49225,49227,49241,49243],{},[2074,49186,49187,49189],{},[51,49188,1435],{},[48,49190,49191,49193],{},[51,49192,1435],{},[51,49194,12271],{},[51,49196,19939],{},[95,49198,1540],{},[1004,49200,49201,49203],{},[51,49202,11],{},[51,49204,12271],{},[95,49206,211],{"stretchy":210},[1004,49208,49209,49211],{},[51,49210,46458],{},[51,49212,12271],{},[95,49214,220],{"stretchy":210},[95,49216,223],{},[95,49218,97],{},[48,49220,49221,49223],{},[51,49222,12271],{"mathvariant":494},[51,49224,2549],{"mathvariant":494},[95,49226,211],{"fence":62,"stretchy":62,"minsize":2511,"maxsize":2511},[2074,49228,49229,49235],{},[48,49230,49231,49233],{},[51,49232,23909],{"mathvariant":494},[51,49234,5354],{},[48,49236,49237,49239],{},[51,49238,23909],{"mathvariant":494},[51,49240,46458],{},[95,49242,220],{"fence":62,"stretchy":62,"minsize":2511,"maxsize":2511},[95,49244,1210],{"separator":62},[55,49246,49247],{"encoding":57},"\\frac{d}{dt}\\log p_t(z_t)=-\\mathrm{tr}\\Bigl(\\frac{\\partial v}{\\partial z}\\Bigr),",[31,49249,49251,49430],{"className":49250,"ariaHidden":62},[61],[31,49252,49254,49258,49323,49326,49331,49334,49374,49377,49418,49421,49424,49427],{"className":49253},[66],[31,49255],{"className":49256,"style":49257},[70],"height:2.0574em;vertical-align:-0.686em;",[31,49259,49261,49264,49320],{"className":49260},[75],[31,49262],{"className":49263},[264,2139],[31,49265,49267],{"className":49266},[2074],[31,49268,49270,49312],{"className":49269},[1043,1044],[31,49271,49273,49309],{"className":49272},[1048],[31,49274,49276,49290,49298],{"className":49275,"style":46264},[1052],[31,49277,49278,49281],{"style":5090},[31,49279],{"className":49280,"style":1637},[1060],[31,49282,49284,49287],{"className":49283},[75],[31,49285,1435],{"className":49286},[75,76],[31,49288,12271],{"className":49289},[75,76],[31,49291,49292,49295],{"style":2170},[31,49293],{"className":49294,"style":1637},[1060],[31,49296],{"className":49297,"style":2178},[2177],[31,49299,49300,49303],{"style":5110},[31,49301],{"className":49302,"style":1637},[1060],[31,49304,49306],{"className":49305},[75],[31,49307,1435],{"className":49308},[75,76],[31,49310,1076],{"className":49311},[1075],[31,49313,49315],{"className":49314},[1048],[31,49316,49318],{"className":49317,"style":5156},[1052],[31,49319],{},[31,49321],{"className":49322},[289,2139],[31,49324],{"className":49325,"style":1467},[119],[31,49327,20177,49329],{"className":49328},[1460],[31,49330,14556],{"style":20180},[31,49332],{"className":49333,"style":1467},[119],[31,49335,49337,49340],{"className":49336},[75],[31,49338,11],{"className":49339},[75,76],[31,49341,49343],{"className":49342},[1039],[31,49344,49346,49366],{"className":49345},[1043,1044],[31,49347,49349,49363],{"className":49348},[1048],[31,49350,49352],{"className":49351,"style":19220},[1052],[31,49353,49354,49357],{"style":2807},[31,49355],{"className":49356,"style":1061},[1060],[31,49358,49360],{"className":49359},[1065,1066,1067,1068],[31,49361,12271],{"className":49362},[75,76,1068],[31,49364,1076],{"className":49365},[1075],[31,49367,49369],{"className":49368},[1048],[31,49370,49372],{"className":49371,"style":2050},[1052],[31,49373],{},[31,49375,211],{"className":49376},[264],[31,49378,49380,49383],{"className":49379},[75],[31,49381,46458],{"className":49382,"style":46500},[75,76],[31,49384,49386],{"className":49385},[1039],[31,49387,49389,49410],{"className":49388},[1043,1044],[31,49390,49392,49407],{"className":49391},[1048],[31,49393,49395],{"className":49394,"style":19220},[1052],[31,49396,49398,49401],{"style":49397},"top:-2.55em;margin-left:-0.044em;margin-right:0.05em;",[31,49399],{"className":49400,"style":1061},[1060],[31,49402,49404],{"className":49403},[1065,1066,1067,1068],[31,49405,12271],{"className":49406},[75,76,1068],[31,49408,1076],{"className":49409},[1075],[31,49411,49413],{"className":49412},[1048],[31,49414,49416],{"className":49415,"style":2050},[1052],[31,49417],{},[31,49419,220],{"className":49420},[289],[31,49422],{"className":49423,"style":293},[119],[31,49425,223],{"className":49426},[297],[31,49428],{"className":49429,"style":293},[119],[31,49431,49433,49436,49439,49445,49451,49519,49525],{"className":49432},[66],[31,49434],{"className":49435,"style":49257},[70],[31,49437,97],{"className":49438},[75],[31,49440,49442],{"className":49441},[75],[31,49443,16973],{"className":49444},[75,2745],[31,49446,49448],{"className":49447},[264],[31,49449,211],{"className":49450},[1307,2662],[31,49452,49454,49457,49516],{"className":49453},[75],[31,49455],{"className":49456},[264,2139],[31,49458,49460],{"className":49459},[2074],[31,49461,49463,49508],{"className":49462},[1043,1044],[31,49464,49466,49505],{"className":49465},[1048],[31,49467,49469,49483,49491],{"className":49468,"style":46264},[1052],[31,49470,49471,49474],{"style":5090},[31,49472],{"className":49473,"style":1637},[1060],[31,49475,49477,49480],{"className":49476},[75],[31,49478,23909],{"className":49479,"style":24157},[75],[31,49481,46458],{"className":49482,"style":46500},[75,76],[31,49484,49485,49488],{"style":2170},[31,49486],{"className":49487,"style":1637},[1060],[31,49489],{"className":49490,"style":2178},[2177],[31,49492,49493,49496],{"style":5110},[31,49494],{"className":49495,"style":1637},[1060],[31,49497,49499,49502],{"className":49498},[75],[31,49500,23909],{"className":49501,"style":24157},[75],[31,49503,5354],{"className":49504,"style":2666},[75,76],[31,49506,1076],{"className":49507},[1075],[31,49509,49511],{"className":49510},[1048],[31,49512,49514],{"className":49513,"style":5156},[1052],[31,49515],{},[31,49517],{"className":49518},[289,2139],[31,49520,49522],{"className":49521},[289],[31,49523,220],{"className":49524},[1307,2662],[31,49526,1210],{"className":49527},[1391],[11,49529,49530,49531,1138,49700,49728],{},"这其实是变量替换公式的无穷小版本，行列式取对数后变成迹。迹在高维不好算，实践中用 Hutchinson 估计 ",[31,49532,49534,49580],{"className":49533,"translate":35},[34],[31,49535,49537],{"className":49536},[39],[41,49538,49539],{"xmlns":43},[45,49540,49541,49577],{},[48,49542,49543,49549,49551,49553,49555,49557,49563,49565,49571,49573,49575],{},[48,49544,49545,49547],{},[51,49546,12271],{"mathvariant":494},[51,49548,2549],{"mathvariant":494},[95,49550,211],{"stretchy":210},[51,49552,18446],{},[95,49554,220],{"stretchy":210},[95,49556,27223],{},[1004,49558,49559,49561],{},[51,49560,22324],{"mathvariant":14789},[51,49562,4793],{},[95,49564,15175],{"stretchy":210},[1180,49566,49567,49569],{},[51,49568,4793],{},[51,49570,20800],{"mathvariant":494},[51,49572,18446],{},[51,49574,4793],{},[95,49576,15184],{"stretchy":210},[55,49578,49579],{"encoding":57},"\\mathrm{tr}(A)\\approx\\mathbb E_\\varepsilon[\\varepsilon^\\top A\\varepsilon]",[31,49581,49583,49613],{"className":49582,"ariaHidden":62},[61],[31,49584,49586,49589,49595,49598,49601,49604,49607,49610],{"className":49585},[66],[31,49587],{"className":49588,"style":260},[70],[31,49590,49592],{"className":49591},[75],[31,49593,16973],{"className":49594},[75,2745],[31,49596,211],{"className":49597},[264],[31,49599,18446],{"className":49600},[75,76],[31,49602,220],{"className":49603},[289],[31,49605],{"className":49606,"style":293},[119],[31,49608,27223],{"className":49609},[297],[31,49611],{"className":49612,"style":293},[119],[31,49614,49616,49619,49659,49662,49691,49694,49697],{"className":49615},[66],[31,49617],{"className":49618,"style":20835},[70],[31,49620,49622,49625],{"className":49621},[75],[31,49623,22324],{"className":49624},[75,14947],[31,49626,49628],{"className":49627},[1039],[31,49629,49631,49651],{"className":49630},[1043,1044],[31,49632,49634,49648],{"className":49633},[1048],[31,49635,49637],{"className":49636,"style":3230},[1052],[31,49638,49639,49642],{"style":2807},[31,49640],{"className":49641,"style":1061},[1060],[31,49643,49645],{"className":49644},[1065,1066,1067,1068],[31,49646,4793],{"className":49647},[75,76,1068],[31,49649,1076],{"className":49650},[1075],[31,49652,49654],{"className":49653},[1048],[31,49655,49657],{"className":49656,"style":2050},[1052],[31,49658],{},[31,49660,15175],{"className":49661},[264],[31,49663,49665,49668],{"className":49664},[75],[31,49666,4793],{"className":49667},[75,76],[31,49669,49671],{"className":49670},[1039],[31,49672,49674],{"className":49673},[1043],[31,49675,49677],{"className":49676},[1048],[31,49678,49680],{"className":49679,"style":20863},[1052],[31,49681,49682,49685],{"style":3631},[31,49683],{"className":49684,"style":1061},[1060],[31,49686,49688],{"className":49687},[1065,1066,1067,1068],[31,49689,20800],{"className":49690},[75,1068],[31,49692,18446],{"className":49693},[75,76],[31,49695,4793],{"className":49696},[75,76],[31,49698,15184],{"className":49699},[289],[31,49701,49703,49716],{"className":49702,"translate":35},[34],[31,49704,49706],{"className":49705},[39],[41,49707,49708],{"xmlns":43},[45,49709,49710,49714],{},[48,49711,49712],{},[51,49713,4793],{},[55,49715,4796],{"encoding":57},[31,49717,49719],{"className":49718,"ariaHidden":62},[61],[31,49720,49722,49725],{"className":49721},[66],[31,49723],{"className":49724,"style":71},[70],[31,49726,4793],{"className":49727},[75,76]," 取 Rademacher 向量，代价是方差。",[11,49730,49731,49732,49765,49766,49799],{},"流与最优传输有一个漂亮的接口。Brenier 说最优映射是 ",[31,49733,49735,49750],{"className":49734,"translate":35},[34],[31,49736,49738],{"className":49737},[39],[41,49739,49740],{"xmlns":43},[45,49741,49742,49748],{},[48,49743,49744,49746],{},[51,49745,12145],{"mathvariant":494},[51,49747,5324],{},[55,49749,12232],{"encoding":57},[31,49751,49753],{"className":49752,"ariaHidden":62},[61],[31,49754,49756,49759,49762],{"className":49755},[66],[31,49757],{"className":49758,"style":945},[70],[31,49760,12145],{"className":49761},[75],[31,49763,5324],{"className":49764},[75,76],"，流说数据由可逆映射得到；当映射取 ",[31,49767,49769,49784],{"className":49768,"translate":35},[34],[31,49770,49772],{"className":49771},[39],[41,49773,49774],{"xmlns":43},[45,49775,49776,49782],{},[48,49777,49778,49780],{},[51,49779,12145],{"mathvariant":494},[51,49781,5324],{},[55,49783,12232],{"encoding":57},[31,49785,49787],{"className":49786,"ariaHidden":62},[61],[31,49788,49790,49793,49796],{"className":49789},[66],[31,49791],{"className":49792,"style":945},[70],[31,49794,12145],{"className":49795},[75],[31,49797,5324],{"className":49798},[75,76]," 时两者重合，于是可以问：能不能直接学那个“最优”的流？Flow matching 就是这么做的。它的条件目标是",[31,49801,49803],{"className":49802,"translate":35},[411],[31,49804,49806,49934],{"className":49805,"translate":35},[34],[31,49807,49809],{"className":49808},[39],[41,49810,49811],{"xmlns":43,"display":420},[45,49812,49813,49931],{},[48,49814,49815,49817,49819,49843,49845,49851,49853,49859,49861,49863,49865,49867,49869,49875,49877,49883,49885,49891,49893,49895,49901,49903,49905,49907,49909,49911,49913,49919,49921,49923,49929],{},[51,49816,4561],{"mathvariant":925},[95,49818,223],{},[1004,49820,49821,49823],{},[51,49822,22324],{"mathvariant":14789},[48,49824,49825,49827,49829,49835,49837],{},[51,49826,12271],{},[95,49828,1210],{"separator":62},[1004,49830,49831,49833],{},[51,49832,53],{},[91,49834,1977],{},[95,49836,1210],{"separator":62},[1004,49838,49839,49841],{},[51,49840,53],{},[91,49842,205],{},[95,49844,4572],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[1004,49846,49847,49849],{},[51,49848,5354],{},[51,49850,23798],{},[95,49852,211],{"stretchy":210},[1004,49854,49855,49857],{},[51,49856,53],{},[51,49858,12271],{},[95,49860,1210],{"separator":62},[51,49862,12271],{},[95,49864,220],{"stretchy":210},[95,49866,97],{},[95,49868,211],{"stretchy":210},[1004,49870,49871,49873],{},[51,49872,53],{},[91,49874,205],{},[95,49876,97],{},[1004,49878,49879,49881],{},[51,49880,53],{},[91,49882,1977],{},[95,49884,220],{"stretchy":210},[1180,49886,49887,49889],{},[95,49888,4572],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[91,49890,152],{},[95,49892,1210],{"separator":62},[119,49894],{"width":20286},[1004,49896,49897,49899],{},[51,49898,53],{},[51,49900,12271],{},[95,49902,223],{},[95,49904,211],{"stretchy":210},[91,49906,205],{},[95,49908,97],{},[51,49910,12271],{},[95,49912,220],{"stretchy":210},[1004,49914,49915,49917],{},[51,49916,53],{},[91,49918,1977],{},[95,49920,434],{},[51,49922,12271],{},[1004,49924,49925,49927],{},[51,49926,53],{},[91,49928,205],{},[95,49930,1210],{"separator":62},[55,49932,49933],{"encoding":57},"\\mathcal L=\\mathbb E_{t,x_0,x_1}\\bigl\\|v_\\theta(x_t,t)-(x_1-x_0)\\bigr\\|^2,\n\\qquad x_t=(1-t)x_0+tx_1,",[31,49935,49937,49955,50234,50292,50462,50483,50544],{"className":49936,"ariaHidden":62},[61],[31,49938,49940,49943,49946,49949,49952],{"className":49939},[66],[31,49941],{"className":49942,"style":945},[70],[31,49944,4561],{"className":49945},[75,968],[31,49947],{"className":49948,"style":293},[119],[31,49950,223],{"className":49951},[297],[31,49953],{"className":49954,"style":293},[119],[31,49956,49958,49961,50093,50130,50170,50173,50213,50216,50219,50222,50225,50228,50231],{"className":49957},[66],[31,49959],{"className":49960,"style":4988},[70],[31,49962,49964,49967],{"className":49963},[75],[31,49965,22324],{"className":49966},[75,14947],[31,49968,49970],{"className":49969},[1039],[31,49971,49973,50085],{"className":49972},[1043,1044],[31,49974,49976,50082],{"className":49975},[1048],[31,49977,49979],{"className":49978,"style":19220},[1052],[31,49980,49981,49984],{"style":2807},[31,49982],{"className":49983,"style":1061},[1060],[31,49985,49987],{"className":49986},[1065,1066,1067,1068],[31,49988,49990,49993,49996,50039,50042],{"className":49989},[75,1068],[31,49991,12271],{"className":49992},[75,76,1068],[31,49994,1210],{"className":49995},[1391,1068],[31,49997,49999,50002],{"className":49998},[75,1068],[31,50000,53],{"className":50001},[75,76,1068],[31,50003,50005],{"className":50004},[1039],[31,50006,50008,50030],{"className":50007},[1043,1044],[31,50009,50011,50027],{"className":50010},[1048],[31,50012,50015],{"className":50013,"style":50014},[1052],"height:0.3173em;",[31,50016,50018,50021],{"style":50017},"top:-2.357em;margin-left:0em;margin-right:0.0714em;",[31,50019],{"className":50020,"style":11769},[1060],[31,50022,50024],{"className":50023},[1065,11773,1308,1068],[31,50025,1977],{"className":50026},[75,1068],[31,50028,1076],{"className":50029},[1075],[31,50031,50033],{"className":50032},[1048],[31,50034,50037],{"className":50035,"style":50036},[1052],"height:0.143em;",[31,50038],{},[31,50040,1210],{"className":50041},[1391,1068],[31,50043,50045,50048],{"className":50044},[75,1068],[31,50046,53],{"className":50047},[75,76,1068],[31,50049,50051],{"className":50050},[1039],[31,50052,50054,50074],{"className":50053},[1043,1044],[31,50055,50057,50071],{"className":50056},[1048],[31,50058,50060],{"className":50059,"style":50014},[1052],[31,50061,50062,50065],{"style":50017},[31,50063],{"className":50064,"style":11769},[1060],[31,50066,50068],{"className":50067},[1065,11773,1308,1068],[31,50069,205],{"className":50070},[75,1068],[31,50072,1076],{"className":50073},[1075],[31,50075,50077],{"className":50076},[1048],[31,50078,50080],{"className":50079,"style":50036},[1052],[31,50081],{},[31,50083,1076],{"className":50084},[1075],[31,50086,50088],{"className":50087},[1048],[31,50089,50091],{"className":50090,"style":1083},[1052],[31,50092],{},[31,50094,50096],{"className":50095},[264],[31,50097,50099],{"className":50098},[1307,13319],[31,50100,50102,50122],{"className":50101},[1043,1044],[31,50103,50105,50119],{"className":50104},[1048],[31,50106,50108],{"className":50107,"style":13329},[1052],[31,50109,50110,50113],{"style":13332},[31,50111],{"className":50112,"style":13336},[1060],[31,50114,50115],{"style":32856},[13341,50116,50117],{"xmlns":13343,"width":32859,"height":1178,"viewBox":32860},[13347,50118],{"d":32863},[31,50120,1076],{"className":50121},[1075],[31,50123,50125],{"className":50124},[1048],[31,50126,50128],{"className":50127,"style":13359},[1052],[31,50129],{},[31,50131,50133,50136],{"className":50132},[75],[31,50134,5354],{"className":50135,"style":2666},[75,76],[31,50137,50139],{"className":50138},[1039],[31,50140,50142,50162],{"className":50141},[1043,1044],[31,50143,50145,50159],{"className":50144},[1048],[31,50146,50148],{"className":50147,"style":1053},[1052],[31,50149,50150,50153],{"style":5504},[31,50151],{"className":50152,"style":1061},[1060],[31,50154,50156],{"className":50155},[1065,1066,1067,1068],[31,50157,23798],{"className":50158,"style":19759},[75,76,1068],[31,50160,1076],{"className":50161},[1075],[31,50163,50165],{"className":50164},[1048],[31,50166,50168],{"className":50167,"style":2050},[1052],[31,50169],{},[31,50171,211],{"className":50172},[264],[31,50174,50176,50179],{"className":50175},[75],[31,50177,53],{"className":50178},[75,76],[31,50180,50182],{"className":50181},[1039],[31,50183,50185,50205],{"className":50184},[1043,1044],[31,50186,50188,50202],{"className":50187},[1048],[31,50189,50191],{"className":50190,"style":19220},[1052],[31,50192,50193,50196],{"style":2807},[31,50194],{"className":50195,"style":1061},[1060],[31,50197,50199],{"className":50198},[1065,1066,1067,1068],[31,50200,12271],{"className":50201},[75,76,1068],[31,50203,1076],{"className":50204},[1075],[31,50206,50208],{"className":50207},[1048],[31,50209,50211],{"className":50210,"style":2050},[1052],[31,50212],{},[31,50214,1210],{"className":50215},[1391],[31,50217],{"className":50218,"style":1467},[119],[31,50220,12271],{"className":50221},[75,76],[31,50223,220],{"classNam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",[31,50601,50603,50633],{"className":50602,"translate":35},[34],[31,50604,50606],{"className":50605},[39],[41,50607,50608],{"xmlns":43},[45,50609,50610,50630],{},[48,50611,50612,50614,50620,50622,50628],{},[95,50613,211],{"stretchy":210},[1004,50615,50616,50618],{},[51,50617,53],{},[91,50619,1977],{},[95,50621,1210],{"separator":62},[1004,50623,50624,50626],{},[51,50625,53],{},[91,50627,205],{},[95,50629,220],{"stretchy":210},[55,50631,50632],{"encoding":57},"(x_0,x_1)",[31,50634,50636],{"className":50635,"ariaHidden":62},[61],[31,50637,50639,50642,50645,50685,50688,50691,50731],{"className":50638},[66],[31,50640],{"className":50641,"style":260},[70],[31,50643,211],{"className":50644},[264],[31,50646,50648,50651],{"className":50647},[75],[31,50649,53],{"className":50650},[75,76],[31,50652,50654],{"className":50653},[1039],[31,50655,50657,50677],{"className":50656},[1043,1044],[31,50658,50660,50674],{"className":50659},[1048],[31,50661,50663],{"className":50662,"style":2028},[1052],[31,50664,50665,50668],{"style":2807},[31,50666],{"className":50667,"style":1061},[1060],[31,50669,50671],{"className":50670},[1065,1066,1067,1068],[31,50672,1977],{"className":50673},[75,1068],[31,50675,1076],{"className":50676},[1075],[31,50678,50680],{"className":50679},[1048],[31,50681,50683],{"className":50682,"style":2050},[1052],[31,50684],{},[31,50686,1210],{"className":50687},[1391],[31,50689],{"className":50690,"style":1467},[119],[31,50692,50694,50697],{"className":50693},[75],[31,50695,53],{"className":50696},[75,76],[31,50698,50700],{"className":50699},[1039],[31,50701,50703,50723],{"className":50702},[1043,1044],[31,50704,50706,50720],{"className":50705},[1048],[31,50707,50709],{"className":50708,"style":2028},[1052],[31,50710,50711,50714],{"style":2807},[31,50712],{"className":50713,"style":1061},[1060],[31,50715,50717],{"className":50716},[1065,1066,1067,1068],[31,50718,205],{"className":50719},[75,1068],[31,50721,1076],{"className":50722},[1075],[31,50724,50726],{"className":50725},[1048],[31,50727,50729],{"className":50728,"style":2050},[1052],[31,50730],{},[31,50732,220],{"className":50733},[289]," 是噪声与数据的配对。如果配对来自 OT 计划，这条直线就是 ",[31,50736,50738,50755],{"className":50737,"translate":35},[34],[31,50739,50741],{"className":50740},[39],[41,50742,50743],{"xmlns":43},[45,50744,50745,50753],{},[48,50746,50747],{},[1004,50748,50749,50751],{},[51,50750,3117],{},[91,50752,152],{},[55,50754,19087],{"encoding":57},[31,50756,50758],{"className":50757,"ariaHidden":62},[61],[31,50759,50761,50764],{"className":50760},[66],[31,50762],{"className":50763,"style":5230},[70],[31,50765,50767,50770],{"className":50766},[75],[31,50768,3117],{"className":50769,"style":949},[75,76],[31,50771,50773],{"className":50772},[1039],[31,50774,50776,50796],{"className":50775},[1043,1044],[31,50777,50779,50793],{"className":50778},[1048],[31,50780,50782],{"className":50781,"style":2028},[1052],[31,50783,50784,50787],{"style":1056},[31,50785],{"className":50786,"style":1061},[1060],[31,50788,50790],{"className":50789},[1065,1066,1067,1068],[31,50791,152],{"className":50792},[75,1068],[31,50794,1076],{"className":50795},[1075],[31,50797,50799],{"className":50798},[1048],[31,50800,50802],{"className":50801,"style":2050},[1052],[31,50803],{}," 的测地线，轨迹最直，采样需要的步数最少。这是我看到的“最优传输”与“生成模型”结合得最干净的一处：OT 在这里不是损失函数，而是给流指定了走哪条路。",[11,50806,50807],{},"不过我要泼一盆冷水。流模型能给出很好的密度，但似然不等于“属于这个类”。Nalisnick 等人在 2019 年观察到，在图像上训练得很好的流模型给 OOD 样本的似然反而更高；原因还是典型集——模型把质量摊在一个高维薄壳上，OOD 样本落在壳外，却可能恰好落在密度不低的地方。所以流更适合做生成、做密度比、做精确的变量替换，而不是拿来当 OOD 分数直接用。",[18,50809,50810],{"id":50810},"落地",[11,50812,50813],{},"现在把这些工具放回机器学习里，按我实际见过、用过的顺序说。",[11,50815,50816,50817,50858,50859,50928],{},"生成模型是最直接的一站。GAN 的原始目标用 JS 散度，而 JS 在两个分布支撑不交时是个常数 ",[31,50818,50820,50838],{"className":50819,"translate":35},[34],[31,50821,50823],{"className":50822},[39],[41,50824,50825],{"xmlns":43},[45,50826,50827,50835],{},[48,50828,50829,50831,50833],{},[51,50830,19939],{},[95,50832,1540],{},[91,50834,152],{},[55,50836,50837],{"encoding":57},"\\log2",[31,50839,50841],{"className":50840,"ariaHidden":62},[61],[31,50842,50844,50847,50852,50855],{"className":50843},[66],[31,50845],{"className":50846,"style":4202},[70],[31,50848,20177,50850],{"className":50849},[1460],[31,50851,14556],{"style":20180},[31,50853],{"className":50854,"style":1467},[119],[31,50856,152],{"className":50857},[75],"，梯度为零——这是早期 GAN 训练不稳定的理论解释之一。WGAN 换成 ",[31,50860,50862,50879],{"className":50861,"translate":35},[34],[31,50863,50865],{"className":50864},[39],[41,50866,50867],{"xmlns":43},[45,50868,50869,50877],{},[48,50870,50871],{},[1004,50872,50873,50875],{},[51,50874,3117],{},[91,50876,205],{},[55,50878,5220],{"encoding":57},[31,50880,50882],{"className":50881,"ariaHidden":62},[61],[31,50883,50885,50888],{"className":50884},[66],[31,50886],{"className":50887,"style":5230},[70],[31,50889,50891,50894],{"className":50890},[75],[31,50892,3117],{"className":50893,"style":949},[75,76],[31,50895,50897],{"className":50896},[1039],[31,50898,50900,50920],{"className":50899},[1043,1044],[31,50901,50903,50917],{"className":50902},[1048],[31,50904,50906],{"className":50905,"style":2028},[1052],[31,50907,50908,50911],{"style":1056},[31,50909],{"className":50910,"style":1061},[1060],[31,50912,50914],{"className":50913},[1065,1066,1067,1068],[31,50915,205],{"className":50916},[75,1068],[31,50918,1076],{"className":50919},[1075],[31,50921,50923],{"className":50922},[1048],[31,50924,50926],{"className":50925,"style":2050},[1052],[31,50927],{},"，用对偶式",[31,50930,50932],{"className":50931,"translate":35},[411],[31,50933,50935,51037],{"className":50934,"translate":35},[34],[31,50936,50938],{"className":50937},[39],[41,50939,50940],{"xmlns":43,"display":420},[45,50941,50942,51034],{},[48,50943,50944,50954,50984,50986,50996,50998,51000,51002,51004,51006,51008,51010,51020,51022,51024,51026,51028,51030,51032],{},[1531,50945,50946,50952],{},[48,50947,50948,50950],{},[51,50949,19878],{},[95,50951,1540],{},[51,50953,15297],{},[1531,50955,50956,50962],{},[48,50957,50958,50960],{},[51,50959,10154],{},[95,50961,1540],{},[48,50963,50964,50966,50968,50980,50982],{},[51,50965,4572],{"mathvariant":494},[51,50967,1662],{},[1004,50969,50970,50972],{},[51,50971,4572],{"mathvariant":494},[48,50973,50974,50976,50978],{},[51,50975,4561],{"mathvariant":494},[51,50977,5391],{"mathvariant":494},[51,50979,11],{"mathvariant":494},[95,50981,347],{},[91,50983,205],{},[1430,50985,1545],{},[1004,50987,50988,50990],{},[51,50989,22324],{"mathvariant":14789},[1004,50991,50992,50994],{},[51,50993,2521],{},[51,50995,2549],{},[95,50997,15175],{"stretchy":210},[51,50999,1662],{},[95,51001,211],{"stretchy":210},[51,51003,53],{},[95,51005,220],{"stretchy":210},[95,51007,15184],{"stretchy":210},[95,51009,97],{},[1004,51011,51012,51014],{},[51,51013,22324],{"mathvariant":14789},[1004,51015,51016,51018],{},[51,51017,2521],{},[51,51019,14556],{},[95,51021,15175],{"stretchy":210},[51,51023,1662],{},[95,51025,211],{"stretchy":210},[51,51027,53],{},[95,51029,220],{"stretchy":210},[95,51031,15184],{"stretchy":210},[95,51033,1210],{"separator":62},[55,51035,51036],{"encoding":57},"\\min_G\\max_{\\|f\\|_{\\mathrm{Lip}}\\le1}\\ \\mathbb E_{P_r}[f(x)]-\\mathbb E_{P_g}[f(x)],",[31,51038,51040,51310],{"className":51039,"ariaHidden":62},[61],[31,51041,51043,51047,51091,51094,51197,51200,51203,51286,51289,51292,51295,51298,51301,51304,51307],{"className":51042},[66],[31,51044],{"className":51045,"style":51046},[70],"height:1.7385em;vertical-align:-0.9885em;",[31,51048,51050],{"className":51049},[1460,1620],[31,51051,51053,51083],{"className":51052},[1043,1044],[31,51054,51056,51080],{"className":51055},[1048],[31,51057,51059,51070],{"className":51058,"style":19979},[1052],[31,51060,51061,51064],{"style":1633},[31,51062],{"className":51063,"style":1637},[1060],[31,51065,51067],{"className":51066},[1065,1066,1067,1068],[31,51068,15297],{"className":51069},[75,76,1068],[31,51071,51072,51075],{"style":1649},[31,51073],{"className":51074,"style":1637},[1060],[31,51076,51077],{},[31,51078,19878],{"className":51079},[1460],[31,51081,1076],{"className":51082},[1075],[31,51084,51086],{"className":51085},[1048],[31,51087,51089],{"className":51088,"style":1672},[1052],[31,51090],{},[31,51092],{"className":51093,"style":1467},[119],[31,51095,51097],{"className":51096},[1460,1620],[31,51098,51100,51188],{"className":51099},[1043,1044],[31,51101,51103,51185],{"className":51102},[1048],[31,51104,51106,51175],{"className":51105,"style":71},[1052],[31,51107,51108,51111],{"style":3301},[31,51109],{"className":51110,"style":1637},[1060],[31,51112,51114],{"className":51113},[1065,1066,1067,1068],[31,51115,51117,51120,51123,51169,51172],{"className":51116},[75,1068],[31,51118,4572],{"className":51119},[75,1068],[31,51121,1662],{"className":51122,"style":35196},[75,76,1068],[31,51124,51126,51129],{"className":51125},[75,1068],[31,51127,4572],{"className":51128},[75,1068],[31,51130,51132],{"className":51131},[1039],[31,51133,51135,51161],{"className":51134},[1043,1044],[31,51136,51138,51158],{"className":51137},[1048],[31,51139,51141],{"className":51140,"style":11762},[1052],[31,51142,51143,51146],{"style":11765},[31,51144],{"className":51145,"style":11769},[1060],[31,51147,51149],{"className":51148},[1065,11773,1308,1068],[31,51150,51152],{"className":51151},[75,1068],[31,51153,51155],{"className":51154},[75,1068],[31,51156,11783],{"className":51157},[75,2745,1068],[31,51159,1076],{"className":51160},[1075],[31,51162,51164],{"className":51163},[1048],[31,51165,51167],{"className":51166,"style":11793},[1052],[31,51168],{},[31,51170,347],{"className":51171},[297,1068],[31,51173,205],{"className":51174},[75,1068],[31,51176,51177,51180],{"style":1649},[31,51178],{"className":51179,"style":1637},[1060],[31,51181,51182],{},[31,51183,10154],{"className":51184},[1460],[31,51186,1076],{"className":51187},[1075],[31,51189,51191],{"className":51190},[1048],[31,51192,51195],{"className":51193,"style":51194},[1052],"height:0.9885em;",[31,51196],{},[31,51198,1545],{"className":51199},[119],[31,51201],{"className":51202,"style":1467},[119],[31,51204,51206,51209],{"className":51205},[75],[31,51207,22324],{"className":51208},[75,14947],[31,51210,51212],{"className":51211},[1039],[31,51213,51215,51277],{"className":51214},[1043,1044],[31,51216,51218,51274],{"className":51217},[1048],[31,51219,51221],{"className":51220,"style":27736},[1052],[31,51222,51223,51226],{"style":2807},[31,51224],{"className":51225,"style":1061},[1060],[31,51227,51229],{"className":51228},[1065,1066,1067,1068],[31,51230,51232],{"className":51231},[75,1068],[31,51233,51235,51238],{"className":51234},[75,1068],[31,51236,2521],{"className":51237,"style":949},[75,76,1068],[31,51239,51241],{"className":51240},[1039],[31,51242,51244,51266],{"className":51243},[1043,1044],[31,51245,51247,51263],{"className":51246},[1048],[31,51248,51251],{"className":51249,"style":51250},[1052],"height:0.1645em;",[31,51252,51254,51257],{"style":51253},"top:-2.357em;margin-left:-0.1389em;margin-right:0.0714em;",[31,51255],{"className":51256,"style":11769},[1060],[31,51258,51260],{"className":51259},[1065,11773,1308,1068],[31,51261,2549],{"className":51262,"style":19759},[75,76,1068],[31,51264,1076],{"className":51265},[1075],[31,51267,51269],{"className":51268},[1048],[31,51270,51272],{"className":51271,"style":50036},[1052],[31,51273],{},[31,51275,1076],{"className":51276},[1075],[31,51278,51280],{"className":51279},[1048],[31,51281,51284],{"className":51282,"style":51283},[1052],"height:0.2501em;",[31,51285],{},[31,51287,15175],{"className":51288},[264],[31,51290,1662],{"className":51291,"style":35196},[75,76],[31,51293,211],{"className":51294},[264],[31,51296,53],{"className":51297},[75,76],[31,51299,30315],{"className":51300},[289],[31,51302],{"className":51303,"style":120},[119],[31,51305,97],{"className":51306},[124],[31,51308],{"className":51309,"style":120},[119],[31,51311,51313,51317,51399,51402,51405,51408,51411,51414],{"className":51312},[66],[31,51314],{"className":51315,"style":51316},[70],"height:1.0973em;vertical-align:-0.3473em;",[31,51318,51320,51323],{"className":51319},[75],[31,51321,22324],{"className":51322},[75,14947],[31,51324,51326],{"className":51325},[1039],[31,51327,51329,51390],{"className":51328},[1043,1044],[31,51330,51332,51387],{"className":51331},[1048],[31,51333,51335],{"className":51334,"style":27736},[1052],[31,51336,51337,51340],{"style":2807},[31,51338],{"className":51339,"style":1061},[1060],[31,51341,51343],{"className":51342},[1065,1066,1067,1068],[31,51344,51346],{"className":51345},[75,1068],[31,51347,51349,51352],{"className":51348},[75,1068],[31,51350,2521],{"className":51351,"style":949},[75,76,1068],[31,51353,51355],{"className":51354},[1039],[31,51356,51358,51378],{"className":51357},[1043,1044],[31,51359,51361,51375],{"className":51360},[1048],[31,51362,51364],{"className":51363,"style":51250},[1052],[31,51365,51366,51369],{"style":51253},[31,51367],{"className":51368,"style":11769},[1060],[31,51370,51372],{"className":51371},[1065,11773,1308,1068],[31,51373,14556],{"className":51374,"style":2666},[75,76,1068],[31,51376,1076],{"className":51377},[1075],[31,51379,51381],{"className":51380},[1048],[31,51382,51385],{"className":51383,"style":51384},[1052],"height:0.2819em;",[31,51386],{},[31,51388,1076],{"className":51389},[1075],[31,51391,51393],{"className":51392},[1048],[31,51394,51397],{"className":51395,"style":51396},[1052],"height:0.3473em;",[31,51398],{},[31,51400,15175],{"className":51401},[264],[31,51403,1662],{"className":51404,"style":35196},[75,76],[31,51406,211],{"className":51407},[264],[31,51409,53],{"className":51410},[75,76],[31,51412,30315],{"className":51413},[289],[31,51415,1210],{"className":51416},[1391],[11,51418,51419,51420,51448],{},"critic ",[31,51421,51423,51436],{"className":51422,"translate":35},[34],[31,51424,51426],{"className":51425},[39],[41,51427,51428],{"xmlns":43},[45,51429,51430,51434],{},[48,51431,51432],{},[51,51433,1662],{},[55,51435,1662],{"encoding":57},[31,51437,51439],{"className":51438,"ariaHidden":62},[61],[31,51440,51442,51445],{"className":51441},[66],[31,51443],{"className":51444,"style":4202},[70],[31,51446,1662],{"className":51447,"style":35196},[75,76]," 就是 Kantorovich 势函数，Lipschitz 约束用 weight clipping 或 gradient penalty 近似。这个改动的实质是把一个饱和的散度换成一个不饱和的度量。",[11,51450,51451],{},"扩散模型和流匹配是同一件事的两种写法，都是把噪声分布运输到数据分布。区别在路径：扩散用固定的前向过程，流匹配可以自己选，选了 OT 路径就得到更直的轨迹和更少的采样步数。",[11,51453,51454,51455,51483,51484,51608,51609,51748,51749,51777,51778,51957,51958,52027],{},"域适应与分布偏移是 ",[31,51456,51458,51471],{"className":51457,"translate":35},[34],[31,51459,51461],{"className":51460},[39],[41,51462,51463],{"xmlns":43},[45,51464,51465,51469],{},[48,51466,51467],{},[51,51468,3117],{},[55,51470,3117],{"encoding":57},[31,51472,51474],{"className":51473,"ariaHidden":62},[61],[31,51475,51477,51480],{"className":51476},[66],[31,51478],{"className":51479,"style":945},[70],[31,51481,3117],{"className":51482,"style":949},[75,76]," 最能发挥的地方。给定源域和目标域的特征分布 ",[31,51485,51487,51513],{"className":51486,"translate":35},[34],[31,51488,51490],{"className":51489},[39],[41,51491,51492],{"xmlns":43},[45,51493,51494,51510],{},[48,51495,51496,51502,51504],{},[1004,51497,51498,51500],{},[51,51499,852],{},[51,51501,21582],{},[95,51503,1210],{"separator":62},[1004,51505,51506,51508],{},[51,51507,852],{},[51,51509,919],{},[55,51511,51512],{"encoding":57},"\\mu_S,\\mu_T",[31,51514,51516],{"className":51515,"ariaHidden":62},[61],[31,51517,51519,51522,51562,51565,51568],{"className":51518},[66],[31,51520],{"className":51521,"style":865},[70],[31,51523,51525,51528],{"className":51524},[75],[31,51526,852],{"className":51527},[75,76],[31,51529,51531],{"className":51530},[1039],[31,51532,51534,51554],{"className":51533},[1043,1044],[31,51535,51537,51551],{"className":51536},[1048],[31,51538,51540],{"className":51539,"style":27736},[1052],[31,51541,51542,51545],{"style":2807},[31,51543],{"className":51544,"style":1061},[1060],[31,51546,51548],{"className":51547},[1065,1066,1067,1068],[31,51549,21582],{"className":51550,"style":21697},[75,76,1068],[31,51552,1076],{"className":51553},[1075],[31,51555,51557],{"className":51556},[1048],[31,51558,51560],{"className":51559,"style":2050},[1052],[31,51561],{},[31,51563,1210],{"className":51564},[1391],[31,51566],{"className":51567,"style":1467},[119],[31,51569,51571,51574],{"className":51570},[75],[31,51572,852],{"className":51573},[75,76],[31,51575,51577],{"className":51576},[1039],[31,51578,51580,51600],{"className":51579},[1043,1044],[31,51581,51583,51597],{"className":51582},[1048],[31,51584,51586],{"className":51585,"style":27736},[1052],[31,51587,51588,51591],{"style":2807},[31,51589],{"className":51590,"style":1061},[1060],[31,51592,51594],{"className":51593},[1065,1066,1067,1068],[31,51595,919],{"className":51596,"style":949},[75,76,1068],[31,51598,1076],{"className":51599},[1075],[31,51601,51603],{"className":51602},[1048],[31,51604,51606],{"className":51605,"style":2050},[1052],[31,51607],{},"，直接最小化 ",[31,51610,51612,51644],{"className":51611,"translate":35},[34],[31,51613,51615],{"className":51614},[39],[41,51616,51617],{"xmlns":43},[45,51618,51619,51641],{},[48,51620,51621,51623,51625,51631,51633,51639],{},[51,51622,3117],{},[95,51624,211],{"stretchy":210},[1004,51626,51627,51629],{},[51,51628,852],{},[51,51630,21582],{},[95,51632,1210],{"separator":62},[1004,51634,51635,51637],{},[51,51636,852],{},[51,51638,919],{},[95,51640,220],{"stretchy":210},[55,51642,51643],{"encoding":57},"W(\\mu_S,\\mu_T)",[31,51645,51647],{"className":51646,"ariaHidden":62},[61],[31,51648,51650,51653,51656,51659,51699,51702,51705,51745],{"className":51649},[66],[31,51651],{"className":51652,"style":260},[70],[31,51654,3117],{"className":51655,"style":949},[75,76],[31,51657,211],{"className":51658},[264],[31,51660,51662,51665],{"className":51661},[75],[31,51663,852],{"className":51664},[75,76],[31,51666,51668],{"className":51667},[1039],[31,51669,51671,51691],{"className":51670},[1043,1044],[31,51672,51674,51688],{"className":51673},[1048],[31,51675,51677],{"className":51676,"style":27736},[1052],[31,51678,51679,51682],{"style":2807},[31,51680],{"className":51681,"style":1061},[1060],[31,51683,51685],{"className":51684},[1065,1066,1067,1068],[31,51686,21582],{"className":51687,"style":21697},[75,76,1068],[31,51689,1076],{"className":51690},[1075],[31,51692,51694],{"className":51693},[1048],[31,51695,51697],{"className":51696,"style":2050},[1052],[31,51698],{},[31,51700,1210],{"className":51701},[1391],[31,51703],{"className":51704,"style":1467},[119],[31,51706,51708,51711],{"className":51707},[75],[31,51709,852],{"className":51710},[75,76],[31,51712,51714],{"className":51713},[1039],[31,51715,51717,51737],{"className":51716},[1043,1044],[31,51718,51720,51734],{"className":51719},[1048],[31,51721,51723],{"className":51722,"style":27736},[1052],[31,51724,51725,51728],{"style":2807},[31,51726],{"className":51727,"style":1061},[1060],[31,51729,51731],{"className":51730},[1065,1066,1067,1068],[31,51732,919],{"className":51733,"style":949},[75,76,1068],[31,51735,1076],{"className":51736},[1075],[31,51738,51740],{"className":51739},[1048],[31,51741,51743],{"className":51742,"style":2050},[1052],[31,51744],{},[31,51746,220],{"className":51747},[289]," 就能对齐两个域，且不需要标签。对偶形式又让它变成对抗训练：一个网络学 ",[31,51750,51752,51765],{"className":51751,"translate":35},[34],[31,51753,51755],{"className":51754},[39],[41,51756,51757],{"xmlns":43},[45,51758,51759,51763],{},[48,51760,51761],{},[51,51762,8059],{},[55,51764,11503],{"encoding":57},[31,51766,51768],{"className":51767,"ariaHidden":62},[61],[31,51769,51771,51774],{"className":51770},[66],[31,51772],{"className":51773,"style":865},[70],[31,51775,8059],{"className":51776},[75,76]," 去最大化 ",[31,51779,51781,51823],{"className":51780,"translate":35},[34],[31,51782,51784],{"className":51783},[39],[41,51785,51786],{"xmlns":43},[45,51787,51788,51820],{},[48,51789,51790,51792,51794,51796,51798,51804,51806,51808,51810,51812,51814],{},[95,51791,1409],{},[51,51793,8059],{},[1430,51795,1432],{},[51,51797,1435],{},[1004,51799,51800,51802],{},[51,51801,852],{},[51,51803,21582],{},[95,51805,97],{},[95,51807,1409],{},[51,51809,8059],{},[1430,51811,1432],{},[51,51813,1435],{},[1004,51815,51816,51818],{},[51,51817,852],{},[51,51819,919],{},[55,51821,51822],{"encoding":57},"\\int\\varphi\\,d\\mu_S-\\int\\varphi\\,d\\mu_T",[31,51824,51826,51896],{"className":51825,"ariaHidden":62},[61],[31,51827,51829,51832,51835,51838,51841,51844,51847,51887,51890,51893],{"className":51828},[66],[31,51830],{"className":51831,"style":1456},[70],[31,51833,1409],{"className":51834,"style":1463},[1460,1461,1462],[31,51836],{"className":51837,"style":1467},[119],[31,51839,8059],{"className":51840},[75,76],[31,51842],{"className":51843,"style":1467},[119],[31,51845,1435],{"className":51846},[75,76],[31,51848,51850,51853],{"className":51849},[75],[31,51851,852],{"className":51852},[75,76],[31,51854,51856],{"className":51855},[1039],[31,51857,51859,51879],{"className":51858},[1043,1044],[31,51860,51862,51876],{"className":51861},[1048],[31,51863,51865],{"className":51864,"style":27736},[1052],[31,51866,51867,51870],{"style":2807},[31,51868],{"className":51869,"style":1061},[1060],[31,51871,51873],{"className":51872},[1065,1066,1067,1068],[31,51874,21582],{"className":51875,"style":21697},[75,76,1068],[31,51877,1076],{"className":51878},[1075],[31,51880,51882],{"className":51881},[1048],[31,51883,51885],{"className":51884,"style":2050},[1052],[31,51886],{},[31,51888],{"className":51889,"style":120},[119],[31,51891,97],{"className":51892},[124],[31,51894],{"className":51895,"style":120},[119],[31,51897,51899,51902,51905,51908,51911,51914,51917],{"className":51898},[66],[31,51900],{"className":51901,"style":1456},[70],[31,51903,1409],{"className":51904,"style":1463},[1460,1461,1462],[31,51906],{"className":51907,"style":1467},[119],[31,51909,8059],{"className":51910},[75,76],[31,51912],{"className":51913,"style":1467},[119],[31,51915,1435],{"className":51916},[75,76],[31,51918,51920,51923],{"className":51919},[75],[31,51921,852],{"className":51922},[75,76],[31,51924,51926],{"className":51925},[1039],[31,51927,51929,51949],{"className":51928},[1043,1044],[31,51930,51932,51946],{"className":51931},[1048],[31,51933,51935],{"className":51934,"style":27736},[1052],[31,51936,51937,51940],{"style":2807},[31,51938],{"className":51939,"style":1061},[1060],[31,51941,51943],{"className":51942},[1065,1066,1067,1068],[31,51944,919],{"className":51945,"style":949},[75,76,1068],[31,51947,1076],{"className":51948},[1075],[31,51950,51952],{"className":51951},[1048],[31,51953,51955],{"className":51954,"style":2050},[1052],[31,51956],{},"，另一个网络调整特征表示去最小化这个差。多个源域时用 Wasserstein 重心，在 ",[31,51959,51961,51978],{"className":51960,"translate":35},[34],[31,51962,51964],{"className":51963},[39],[41,51965,51966],{"xmlns":43},[45,51967,51968,51976],{},[48,51969,51970],{},[1004,51971,51972,51974],{},[51,51973,3117],{},[91,51975,152],{},[55,51977,19087],{"encoding":57},[31,51979,51981],{"className":51980,"ariaHidden":62},[61],[31,51982,51984,51987],{"className":51983},[66],[31,51985],{"className":51986,"style":5230},[70],[31,51988,51990,51993],{"className":51989},[75],[31,51991,3117],{"className":51992,"style":949},[75,76],[31,51994,51996],{"className":51995},[1039],[31,51997,51999,52019],{"className":51998},[1043,1044],[31,52000,52002,52016],{"className":52001},[1048],[31,52003,52005],{"className":52004,"style":2028},[1052],[31,52006,52007,52010],{"style":1056},[31,52008],{"className":52009,"style":1061},[1060],[31,52011,52013],{"className":52012},[1065,1066,1067,1068],[31,52014,152],{"className":52015},[75,1068],[31,52017,1076],{"className":52018},[1075],[31,52020,52022],{"className":52021},[1048],[31,52023,52025],{"className":52024,"style":2050},[1052],[31,52026],{}," 空间里求“平均分布”，这正是 Fréchet 均值在分布空间里的实例。",[11,52029,52030],{},"OOD 检测，也就是我开头那个 0.99 的问题，我会分三层来做。",[11,52032,52033],{},"设想一条质检流水线，训练时只有“正常”“划痕”“凹陷”三类。现在来了一个从没见过的新缺陷。",[11,52035,52036,52037,52187],{},"点态一层比较样本密度：对每个类拟合 KDE 或流，取 ",[31,52038,52040,52075],{"className":52039,"translate":35},[34],[31,52041,52043],{"className":52042},[39],[41,52044,52045],{"xmlns":43},[45,52046,52047,52072],{},[48,52048,52049,52051,52053,52055,52066,52068,52070],{},[95,52050,97],{},[51,52052,19939],{},[95,52054,1540],{},[1004,52056,52057,52063],{},[42405,52058,52059,52061],{"accent":62},[51,52060,11],{},[95,52062,42411],{},[51,52064,52065],{},"k",[95,52067,211],{"stretchy":210},[51,52069,53],{},[95,52071,220],{"stretchy":210},[55,52073,52074],{"encoding":57},"-\\log\\hat p_k(x)",[31,52076,52078],{"className":52077,"ariaHidden":62},[61],[31,52079,52081,52084,52087,52090,52095,52098,52178,52181,52184],{"className":52080},[66],[31,52082],{"className":52083,"style":260},[70],[31,52085,97],{"className":52086},[75],[31,52088],{"className":52089,"style":1467},[119],[31,52091,20177,52093],{"className":52092},[1460],[31,52094,14556],{"style":20180},[31,52096],{"className":52097,"style":1467},[119],[31,52099,52101,52143],{"className":52100},[75],[31,52102,52104],{"className":52103},[75,42523],[31,52105,52107,52135],{"className":52106},[1043,1044],[31,52108,52110,52132],{"className":52109},[1048],[31,52111,52113,52121],{"className":52112,"style":1630},[1052],[31,52114,52115,52118],{"style":1649},[31,52116],{"className":52117,"style":1637},[1060],[31,52119,11],{"className":52120},[75,76],[31,52122,52123,52126],{"style":1649},[31,52124],{"className":52125,"style":1637},[1060],[31,52127,52129],{"className":52128,"style":42550},[42549],[31,52130,42411],{"className":52131},[75],[31,52133,1076],{"className":52134},[1075],[31,52136,52138],{"className":52137},[1048],[31,52139,52141],{"className":52140,"style":42563},[1052],[31,52142],{},[31,52144,52146],{"className":52145},[1039],[31,52147,52149,52170],{"className":52148},[1043,1044],[31,52150,52152,52167],{"className":52151},[1048],[31,52153,52155],{"className":52154,"style":1053},[1052],[31,52156,52157,52160],{"style":2807},[31,52158],{"className":52159,"style":1061},[1060],[31,52161,52163],{"className":52162},[1065,1066,1067,1068],[31,52164,52065],{"className":52165,"style":52166},[75,76,1068],"margin-right:0.0315em;",[31,52168,1076],{"className":52169},[1075],[31,52171,52173],{"className":52172},[1048],[31,52174,52176],{"className":52175,"style":2050},[1052],[31,52177],{},[31,52179,211],{"className":52180},[264],[31,52182,53],{"className":52183},[75,76],[31,52185,220],{"className":52186},[289]," 作为分数。它便宜、可解释，但受典型集悖论和带宽影响，高维不可靠——上一节那个“空洞比簇心更密”的例子就是反例。",[11,52189,52190,52191,52260,52261,52337,52338,52366,52367,1138,52395,52439],{},"分布一层算测试批（或测试样本的邻域）到每个类条件分布的 ",[31,52192,52194,52211],{"className":52193,"translate":35},[34],[31,52195,52197],{"className":52196},[39],[41,52198,52199],{"xmlns":43},[45,52200,52201,52209],{},[48,52202,52203],{},[1004,52204,52205,52207],{},[51,52206,3117],{},[91,52208,152],{},[55,52210,19087],{"encoding":57},[31,52212,52214],{"className":52213,"ariaHidden":62},[61],[31,52215,52217,52220],{"className":52216},[66],[31,52218],{"className":52219,"style":5230},[70],[31,52221,52223,52226],{"className":52222},[75],[31,52224,3117],{"className":52225,"style":949},[75,76],[31,52227,52229],{"className":52228},[1039],[31,52230,52232,52252],{"className":52231},[1043,1044],[31,52233,52235,52249],{"className":52234},[1048],[31,52236,52238],{"className":52237,"style":2028},[1052],[31,52239,52240,52243],{"style":1056},[31,52241],{"className":52242,"style":1061},[1060],[31,52244,52246],{"className":52245},[1065,1066,1067,1068],[31,52247,152],{"className":52248},[75,1068],[31,52250,1076],{"className":52251},[1075],[31,52253,52255],{"className":52254},[1048],[31,52256,52258],{"className":52257,"style":2050},[1052],[31,52259],{},"，取最小者作为“离最近已知类的距离”。优点是它看整片质量而不是单个点；缺点是它需要一批样本，还有 ",[31,52262,52264,52290],{"className":52263,"translate":35},[34],[31,52265,52267],{"className":52266},[39],[41,52268,52269],{"xmlns":43},[45,52270,52271,52287],{},[48,52272,52273],{},[1180,52274,52275,52277],{},[51,52276,14559],{},[48,52278,52279,52281,52283,52285],{},[95,52280,97],{},[91,52282,205],{},[51,52284,3768],{"mathvariant":494},[51,52286,1435],{},[55,52288,52289],{"encoding":57},"n^{-1\u002Fd}",[31,52291,52293],{"className":52292,"ariaHidden":62},[61],[31,52294,52296,52299],{"className":52295},[66],[31,52297],{"className":52298,"style":22855},[70],[31,52300,52302,52305],{"className":52301},[75],[31,52303,14559],{"className":52304},[75,76],[31,52306,52308],{"className":52307},[1039],[31,52309,52311],{"className":52310},[1043],[31,52312,52314],{"className":52313},[1048],[31,52315,52317],{"className":52316,"style":22855},[1052],[31,52318,52319,52322],{"style":3631},[31,52320],{"className":52321,"style":1061},[1060],[31,52323,52325],{"className":52324},[1065,1066,1067,1068],[31,52326,52328,52331,52334],{"className":52327},[75,1068],[31,52329,97],{"className":52330},[75,1068],[31,52332,4042],{"className":52333},[75,1068],[31,52335,1435],{"className":52336},[75,76,1068]," 的样本复杂度。实践中的折中是：在低维隐空间里算，或者用对偶势 ",[31,52339,52341,52354],{"className":52340,"translate":35},[34],[31,52342,52344],{"className":52343},[39],[41,52345,52346],{"xmlns":43},[45,52347,52348,52352],{},[48,52349,52350],{},[51,52351,8059],{},[55,52353,11503],{"encoding":57},[31,52355,52357],{"className":52356,"ariaHidden":62},[61],[31,52358,52360,52363],{"className":52359},[66],[31,52361],{"className":52362,"style":865},[70],[31,52364,8059],{"className":52365},[75,76],"——一旦在一批数据上解出 ",[31,52368,52370,52383],{"className":52369,"translate":35},[34],[31,52371,52373],{"className":52372},[39],[41,52374,52375],{"xmlns":43},[45,52376,52377,52381],{},[48,52378,52379],{},[51,52380,8059],{},[55,52382,11503],{"encoding":57},[31,52384,52386],{"className":52385,"ariaHidden":62},[61],[31,52387,52389,52392],{"className":52388},[66],[31,52390],{"className":52391,"style":865},[70],[31,52393,8059],{"className":52394},[75,76],[31,52396,52398,52418],{"className":52397,"translate":35},[34],[31,52399,52401],{"className":52400},[39],[41,52402,52403],{"xmlns":43},[45,52404,52405,52415],{},[48,52406,52407,52409,52411,52413],{},[51,52408,8059],{},[95,52410,211],{"stretchy":210},[51,52412,53],{},[95,52414,220],{"stretchy":210},[55,52416,52417],{"encoding":57},"\\varphi(x)",[31,52419,52421],{"className":52420,"ariaHidden":62},[61],[31,52422,52424,52427,52430,52433,52436],{"className":52423},[66],[31,52425],{"className":52426,"style":260},[70],[31,52428,8059],{"className":52429},[75,76],[31,52431,211],{"className":52432},[264],[31,52434,53],{"className":52435},[75,76],[31,52437,220],{"className":52438},[289]," 本身就是逐样本分数，可以拿去做单点判定。我很喜欢这个技巧，它把“批级别的距离”翻译成了“点级别的分数”，两个需求一次满足。",[11,52441,52442],{},"输出一层比较类别概率分布的差异：训练时记录每个类在验证集上的平均预测分布（单纯形上的一个点），测试时算测试批的平均预测分布与它的 Fisher–Rao 距离。距离大，说明模型在这批数据上的“说话方式”变了，即使每个样本的置信度都不低。这一层不需要标签也不需要密度估计，我觉得是性价比最高的一个。",[11,52444,52445],{},"分类与损失那一侧前面已经说了：交叉熵是 Bregman，自然梯度是 Fisher 预条件，镜像下降是 Bregman 投影。这三个东西实现上都很朴素，但把它们在理论上认成同一族，能省掉很多“为什么这个 trick 有效”的困惑。",[18,52447,52448],{"id":52448},"回到开头",[11,52450,52451],{},"回到那个 0.99。",[11,52453,52454,52455,52483],{},"置信度回答的是“在这几个类里最像哪个”，它是一次相对排序，分母被 softmax 强行归一化成 ",[31,52456,52458,52471],{"className":52457,"translate":35},[34],[31,52459,52461],{"className":52460},[39],[41,52462,52463],{"xmlns":43},[45,52464,52465,52469],{},[48,52466,52467],{},[91,52468,205],{},[55,52470,205],{"encoding":57},[31,52472,52474],{"className":52473,"ariaHidden":62},[61],[31,52475,52477,52480],{"className":52476},[66],[31,52478],{"className":52479,"style":326},[70],[31,52481,205],{"className":52482},[75],"。分布几何回答的是“这个点离已知类别的概率质量有多远”，它是一个绝对量，不依赖“一定属于某个类”这个假设。这两件事不该互相替代，而应该互补：分类器给候选，分布几何判越界。",[11,52485,52486],{},"如果只留一个式子，我会留 Kantorovich 对偶：",[31,52488,52490],{"className":52489,"translate":35},[411],[31,52491,52493,52573],{"className":52492,"translate":35},[34],[31,52494,52496],{"className":52495},[39],[41,52497,52498],{"xmlns":43,"display":420},[45,52499,52500,52570],{},[48,52501,52502,52508,52510,52512,52514,52516,52518,52520,52550,52552,52554,52556,52558,52560,52562,52564,52566,52568],{},[1004,52503,52504,52506],{},[51,52505,3117],{},[91,52507,205],{},[95,52509,211],{"stretchy":210},[51,52511,852],{},[95,52513,1210],{"separator":62},[51,52515,884],{},[95,52517,220],{"stretchy":210},[95,52519,223],{},[1531,52521,52522,52528],{},[48,52523,52524,52526],{},[51,52525,10792],{},[95,52527,1540],{},[48,52529,52530,52532,52534,52546,52548],{},[51,52531,4572],{"mathvariant":494},[51,52533,8059],{},[1004,52535,52536,52538],{},[51,52537,4572],{"mathvariant":494},[48,52539,52540,52542,52544],{},[51,52541,4561],{"mathvariant":494},[51,52543,5391],{"mathvariant":494},[51,52545,11],{"mathvariant":494},[95,52547,347],{},[91,52549,205],{},[95,52551,1409],{},[51,52553,8059],{},[1430,52555,1432],{},[51,52557,1435],{},[95,52559,211],{"stretchy":210},[51,52561,852],{},[95,52563,97],{},[51,52565,884],{},[95,52567,220],{"stretchy":210},[51,52569,495],{"mathvariant":494},[55,52571,52572],{"encoding":57},"W_1(\\mu,\\nu)=\\sup_{\\|\\varphi\\|_{\\mathrm{Lip}}\\le1}\\int\\varphi\\,d(\\mu-\\nu).",[31,52574,52576,52649,52790],{"className":52575,"ariaHidden":62},[61],[31,52577,52579,52582,52622,52625,52628,52631,52634,52637,52640,52643,52646],{"className":52578},[66],[31,52580],{"className":52581,"style":260},[70],[31,52583,52585,52588],{"className":52584},[75],[31,52586,3117],{"className":52587,"style":949},[75,76],[31,52589,52591],{"className":52590},[1039],[31,52592,52594,52614],{"className":52593},[1043,1044],[31,52595,52597,52611],{"className":52596},[1048],[31,52598,52600],{"className":52599,"style":2028},[1052],[31,52601,52602,52605],{"style":1056},[31,52603],{"className":52604,"style":1061},[1060],[31,52606,52608],{"className":52607},[1065,1066,1067,1068],[31,52609,205],{"className":52610},[75,1068],[31,52612,1076],{"className":52613},[1075],[31,52615,52617],{"className":52616},[1048],[31,52618,52620],{"className":52619,"style":2050},[1052],[31,52621],{},[31,52623,211],{"className":52624},[264],[31,52626,852],{"className":52627},[75,76],[31,52629,1210],{"className":52630},[1391],[31,52632],{"className":52633,"style":1467},[119],[31,52635,884],{"className":52636,"style":900},[75,76],[31,52638,220],{"className":52639},[289],[31,52641],{"className":52642,"style":293},[119],[31,52644,223],{"className":52645},[297],[31,52647],{"className":52648,"style":293},[119],[31,52650,52652,52655,52757,52760,52763,52766,52769,52772,52775,52778,52781,52784,52787],{"className":52651},[66],[31,52653],{"className":52654,"style":11713},[70],[31,52656,52658],{"className":52657},[1460,1620],[31,52659,52661,52749],{"className":52660},[1043,1044],[31,52662,52664,52746],{"className":52663},[1048],[31,52665,52667,52736],{"className":52666,"style":71},[1052],[31,52668,52669,52672],{"style":11728},[31,52670],{"className":52671,"style":1637},[1060],[31,52673,52675],{"className":52674},[1065,1066,1067,1068],[31,52676,52678,52681,52684,52730,52733],{"className":52677},[75,1068],[31,52679,4572],{"className":52680},[75,1068],[31,52682,8059],{"className":52683},[75,76,1068],[31,52685,52687,52690],{"className":52686},[75,1068],[31,52688,4572],{"className":52689},[75,1068],[31,52691,52693],{"className":52692},[1039],[31,52694,52696,52722],{"className":52695},[1043,1044],[31,52697,52699,52719],{"className":52698},[1048],[31,52700,52702],{"className":52701,"style":11762},[1052],[31,52703,52704,52707],{"style":11765},[31,52705],{"className":52706,"style":11769},[1060],[31,52708,52710],{"className":52709},[1065,11773,1308,1068],[31,52711,52713],{"className":52712},[75,1068],[31,52714,52716],{"className":52715},[75,1068],[31,52717,11783],{"className":52718},[75,2745,1068],[31,52720,1076],{"className":52721},[1075],[31,52723,52725],{"className":52724},[1048],[31,52726,52728],{"className":52727,"style":11793},[1052],[31,52729],{},[31,52731,347],{"className":52732},[297,1068],[31,52734,205],{"className":52735},[75,1068],[31,52737,52738,52741],{"style":1649},[31,52739],{"className":52740,"style":1637},[1060],[31,52742,52743],{},[31,52744,10792],{"className":52745},[1460],[31,52747,1076],{"className":52748},[1075],[31,52750,52752],{"className":52751},[1048],[31,52753,52755],{"className":52754,"style":11821},[1052],[31,52756],{},[31,52758],{"className":52759,"style":1467},[119],[31,52761,1409],{"className":52762,"style":1688},[1460,1461,1687],[31,52764],{"className":52765,"style":1467},[119],[31,52767,8059],{"className":52768},[75,76],[31,52770],{"className":52771,"style":1467},[119],[31,52773,1435],{"className":52774},[75,76],[31,52776,211],{"className":52777},[264],[31,52779,852],{"className":52780},[75,76],[31,52782],{"className":52783,"style":120},[119],[31,52785,97],{"className":52786},[124],[31,52788],{"className":52789,"style":120},[119],[31,52791,52793,52796,52799,52802],{"className":52792},[66],[31,52794],{"className":52795,"style":260},[70],[31,52797,884],{"className":52798,"style":900},[75,76],[31,52800,220],{"className":52801},[289],[31,52803,495],{"className":52804},[75],[11,52806,52807],{},"它把一个关于分布的优化问题，变成了一个关于函数的优化问题。分布是无穷维的、难参数化的；函数是可以拿神经网络拟合的。这一步转换，才是最优传输能从 1781 年的搬土问题走进今天每一篇 WGAN、每一次域适应、每一个流匹配模型的真正原因。",[11,52809,52810],{},"至于 Fisher–Rao 和 Bregman，它们给我的东西不太一样：不是算法，而是一种校准过的直觉——当我说“两个分布很近”的时候，我最好知道自己是拿哪把尺子在量，以及那把尺子在局部是不是真的像一个度量。",{"title":52812,"searchDepth":52813,"depth":52813,"links":52814},"",2,[52815,52816,52817,52818,52819,52820,52821,52822,52823,52824,52825],{"id":20,"depth":52813,"text":20},{"id":5303,"depth":52813,"text":5303},{"id":12250,"depth":52813,"text":12250},{"id":19440,"depth":52813,"text":19440},{"id":22894,"depth":52813,"text":22894},{"id":29405,"depth":52813,"text":29405},{"id":33911,"depth":52813,"text":33912},{"id":41924,"depth":52813,"text":41924},{"id":45603,"depth":52813,"text":45603},{"id":50810,"depth":52813,"text":50810},{"id":52448,"depth":52813,"text":52448},false,true,"md",null,{},"\u002Fblog\u002F2026\u002F0014-zui-you-chuan-shu-ji-he-jian-lun",{"title":6,"description":13},"blog\u002F2026\u002F0014-zui-you-chuan-shu-ji-he-jian-lun","从 Kantorovich 对偶到 Fisher–Rao 度量、Bregman 散度、核密度估计与标准化流，用同一套分布几何的语言解释为什么分布层面的判据比分类器置信度更稳健。",[52836,52837,52838],"mathematics","machine-learning","statistics","2026-10-31T20:17:23+08:00","IAT5rYeOtscnEe8Mhh1eCeIF4PJR1Ewyc86Z_G_0ncI",{"id":52842,"title":52843,"body":52844,"description":52848,"draft":52826,"enableCodeHighlight":52826,"enableCodeHightlight":52826,"enableComment":52827,"enableD2":52826,"enableKatex":52827,"extension":52828,"image":52812,"important":52826,"location":52829,"meta":125303,"navigation":52827,"ogImage":52829,"path":125304,"seo":125305,"stem":125306,"summary":125307,"tags":125308,"time":125309,"__hash__":125310},"blog\u002Fblog\u002F2026\u002F0015-the-bellman-equation-and-its-mathematical-properties.md","贝尔曼方程及其数学性质",{"type":8,"value":52845,"toc":125257},[52846,52849,52852,52855,52859,52864,54332,54791,55292,55296,56143,56838,57685,58255,58260,59022,59145,59148,59152,60072,60294,61048,61413,62569,62572,62690,62694,62698,62706,63475,63482,64333,64336,64398,64402,64468,65146,65149,65303,65577,65581,65997,66809,66992,67183,67827,68448,68752,68806,69051,69266,69815,69817,69958,69962,70040,70213,70401,70614,70617,70734,70737,71010,71013,71359,71362,71738,71741,72274,72360,72364,72367,72666,72757,73186,73447,73703,73846,73964,73968,73972,74049,74257,74419,74703,75226,75288,76087,76286,76290,77242,77395,77752,78279,78282,79047,79172,79234,79238,80051,80263,80519,80526,80807,80838,81201,81232,81519,81997,82001,83060,83184,83544,83848,84146,85157,85503,86185,86188,86671,86673,86677,86915,87113,87353,87356,88192,88358,88671,88999,89116,89120,89124,89829,89930,89934,89998,90517,90638,92145,92434,92698,92946,93725,94001,94567,95238,95729,96207,96617,96679,96801,97593,97913,97917,98087,98451,99117,99120,99668,99733,99737,100633,100979,100982,100986,101099,101402,101615,101704,101920,101980,102190,102383,102620,102852,102854,102981,103217,103519,103589,103593,103597,103708,103943,104091,105030,105035,105703,105734,105926,105930,106007,107306,107311,108453,108456,108845,108848,109424,109427,109998,110001,110427,110430,110485,110931,111140,111144,111263,111648,111651,112355,112409,112938,112992,113389,113506,113717,113986,114151,114155,114259,114429,114588,114695,114727,114933,114980,115183,115237,115241,115245,115607,116316,116562,117168,117510,118100,118395,119672,120254,120554,120558,120623,120842,120846,120849,121269,121342,121447,121450,121454,121458,121505,121567,122186,122277,122436,122440,122553,123141,123635,123740,123744,123920,124090,124621,124678,124713,124883,124886,124889,125165,125254],[11,52847,52848],{},"贝尔曼方程（Bellman Equation）是动态规划（Dynamic Programming）和强化学习（Reinforcement Learning）中的核心数学方程，用于描述一个系统在当前状态下的最优价值，如何由当前收益和未来状态的价值共同决定。",[11,52850,52851],{},"简单来说：当前状态的价值 = 当前获得的收益 + 未来可能获得的折扣收益。它的核心思想是：一个复杂的长期决策问题，可以分解为一系列相互关联的短期决策问题。",[11,52853,52854],{},"它不是某一个孤立的等式，而是一族把值函数与自身联系起来的泛函方程。它的数学内容可以概括为三个层次。第一，折扣回报在时间上的递归分解。第二，一个定义在有界函数空间上的算子。第三，该算子的不动点。动态规划与强化学习中的许多结论，本质上都是这三个层次的不同推论。",[18,52856,52858],{"id":52857},"一形式化","一、形式化",[52860,52861,52863],"h3",{"id":52862},"_11-折扣马尔可夫决策过程","1.1 折扣马尔可夫决策过程",[52865,52866,52869,53491,54212],"theorem",{"id":52867,"title":52868},"definition-mdp","定义 1.1（折扣马尔可夫决策过程）",[11,52870,52871,52872,52900,52901,52929,52930,53042,53043,53071,53072,53100,53101,53163,53164,53283,53284,53360,53361,53433,53434,53462,53463,1138],{},"设 ",[31,52873,52875,52888],{"className":52874,"translate":35},[34],[31,52876,52878],{"className":52877},[39],[41,52879,52880],{"xmlns":43},[45,52881,52882,52886],{},[48,52883,52884],{},[51,52885,21582],{},[55,52887,21582],{"encoding":57},[31,52889,52891],{"className":52890,"ariaHidden":62},[61],[31,52892,52894,52897],{"className":52893},[66],[31,52895],{"className":52896,"style":945},[70],[31,52898,21582],{"className":52899,"style":21697},[75,76]," 为状态集，",[31,52902,52904,52917],{"className":52903,"translate":35},[34],[31,52905,52907],{"className":52906},[39],[41,52908,52909],{"xmlns":43},[45,52910,52911,52915],{},[48,52912,52913],{},[51,52914,18446],{},[55,52916,18446],{"encoding":57},[31,52918,52920],{"className":52919,"ariaHidden":62},[61],[31,52921,52923,52926],{"className":52922},[66],[31,52924],{"className":52925,"style":945},[70],[31,52927,18446],{"className":52928},[75,76]," 为动作集，",[31,52931,52933,52965],{"className":52932,"translate":35},[34],[31,52934,52936],{"className":52935},[39],[41,52937,52938],{"xmlns":43},[45,52939,52940,52962],{},[48,52941,52942,52944,52946,52952,52954,52956,52958,52960],{},[51,52943,2521],{},[95,52945,211],{"stretchy":210},[1180,52947,52948,52950],{},[51,52949,14551],{},[95,52951,13691],{"mathvariant":494,"lspace":5610,"rspace":5610},[95,52953,12680],{},[51,52955,14551],{},[95,52957,1210],{"separator":62},[51,52959,5813],{},[95,52961,220],{"stretchy":210},[55,52963,52964],{"encoding":57},"P(s'\\mid s,a)",[31,52966,52968,53021],{"className":52967,"ariaHidden":62},[61],[31,52969,52971,52974,52977,52980,53012,53015,53018],{"className":52970},[66],[31,52972],{"className":52973,"style":13710},[70],[31,52975,2521],{"className":52976,"style":949},[75,76],[31,52978,211],{"className":52979},[264],[31,52981,52983,52986],{"className":52982},[75],[31,52984,14551],{"className":52985},[75,76],[31,52987,52989],{"className":52988},[1039],[31,52990,52992],{"className":52991},[1043],[31,52993,52995],{"className":52994},[1048],[31,52996,52998],{"className":52997,"style":13732},[1052],[31,52999,53000,53003],{"style":3631},[31,53001],{"className":53002,"style":1061},[1060],[31,53004,53006],{"className":53005},[1065,1066,1067,1068],[31,53007,53009],{"className":53008},[75,1068],[31,53010,13691],{"className":53011},[75,1068],[31,53013],{"className":53014,"style":293},[119],[31,53016,12680],{"className":53017},[297],[31,53019],{"className":53020,"style":293},[119],[31,53022,53024,53027,53030,53033,53036,53039],{"className":53023},[66],[31,53025],{"className":53026,"style":260},[70],[31,53028,14551],{"className":53029},[75,76],[31,53031,1210],{"className":53032},[1391],[31,53034],{"className":53035,"style":1467},[119],[31,53037,5813],{"className":53038},[75,76],[31,53040,220],{"className":53041},[289]," 为在状态 ",[31,53044,53046,53059],{"className":53045,"translate":35},[34],[31,53047,53049],{"className":53048},[39],[41,53050,53051],{"xmlns":43},[45,53052,53053,53057],{},[48,53054,53055],{},[51,53056,14551],{},[55,53058,14551],{"encoding":57},[31,53060,53062],{"className":53061,"ariaHidden":62},[61],[31,53063,53065,53068],{"className":53064},[66],[31,53066],{"className":53067,"style":71},[70],[31,53069,14551],{"className":53070},[75,76]," 执行动作 ",[31,53073,53075,53088],{"className":53074,"translate":35},[34],[31,53076,53078],{"className":53077},[39],[41,53079,53080],{"xmlns":43},[45,53081,53082,53086],{},[48,53083,53084],{},[51,53085,5813],{},[55,53087,5813],{"encoding":57},[31,53089,53091],{"className":53090,"ariaHidden":62},[61],[31,53092,53094,53097],{"className":53093},[66],[31,53095],{"className":53096,"style":71},[70],[31,53098,5813],{"className":53099},[75,76]," 后转移到状态 ",[31,53102,53104,53122],{"className":53103,"translate":35},[34],[31,53105,53107],{"className":53106},[39],[41,53108,53109],{"xmlns":43},[45,53110,53111,53119],{},[48,53112,53113],{},[1180,53114,53115,53117],{},[51,53116,14551],{},[95,53118,13691],{"mathvariant":494,"lspace":5610,"rspace":5610},[55,53120,53121],{"encoding":57},"s'",[31,53123,53125],{"className":53124,"ariaHidden":62},[61],[31,53126,53128,53131],{"className":53127},[66],[31,53129],{"className":53130,"style":13732},[70],[31,53132,53134,53137],{"className":53133},[75],[31,53135,14551],{"className":53136},[75,76],[31,53138,53140],{"className":53139},[1039],[31,53141,53143],{"className":53142},[1043],[31,53144,53146],{"className":53145},[1048],[31,53147,53149],{"className":53148,"style":13732},[1052],[31,53150,53151,53154],{"style":3631},[31,53152],{"className":53153,"style":1061},[1060],[31,53155,53157],{"className":53156},[1065,1066,1067,1068],[31,53158,53160],{"className":53159},[75,1068],[31,53161,13691],{"className":53162},[75,1068]," 的概率，",[31,53165,53167,53197],{"className":53166,"translate":35},[34],[31,53168,53170],{"className":53169},[39],[41,53171,53172],{"xmlns":43},[45,53173,53174,53194],{},[48,53175,53176,53178,53180,53182,53184,53186,53188,53190,53192],{},[51,53177,2549],{},[95,53179,922],{},[51,53181,21582],{},[95,53183,2528],{},[51,53185,18446],{},[95,53187,2528],{},[51,53189,21582],{},[95,53191,929],{},[51,53193,14790],{"mathvariant":14789},[55,53195,53196],{"encoding":57},"r:S\\times A\\times S\\to\\mathbb{R}",[31,53198,53200,53218,53237,53255,53273],{"className":53199,"ariaHidden":62},[61],[31,53201,53203,53206,53209,53212,53215],{"className":53202},[66],[31,53204],{"className":53205,"style":71},[70],[31,53207,2549],{"className":53208,"style":19759},[75,76],[31,53210],{"className":53211,"style":293},[119],[31,53213,922],{"className":53214},[297],[31,53216],{"className":53217,"style":293},[119],[31,53219,53221,53225,53228,53231,53234],{"className":53220},[66],[31,53222],{"className":53223,"style":53224},[70],"height:0.7667em;vertical-align:-0.0833em;",[31,53226,21582],{"className":53227,"style":21697},[75,76],[31,53229],{"className":53230,"style":120},[119],[31,53232,2528],{"className":53233},[124],[31,53235],{"className":53236,"style":120},[119],[31,53238,53240,53243,53246,53249,53252],{"className":53239},[66],[31,53241],{"className":53242,"style":53224},[70],[31,53244,18446],{"className":53245},[75,76],[31,53247],{"className":53248,"style":120},[119],[31,53250,2528],{"className":53251},[124],[31,53253],{"className":53254,"style":120},[119],[31,53256,53258,53261,53264,53267,53270],{"className":53257},[66],[31,53259],{"className":53260,"style":945},[70],[31,53262,21582],{"className":53263,"style":21697},[75,76],[31,53265],{"className":53266,"style":293},[119],[31,53268,929],{"className":53269},[297],[31,53271],{"className":53272,"style":293},[119],[31,53274,53276,53280],{"className":53275},[66],[31,53277],{"className":53278,"style":53279},[70],"height:0.6889em;",[31,53281,14790],{"className":53282},[75,14947]," 为奖励函数，",[31,53285,53287,53314],{"className":53286,"translate":35},[34],[31,53288,53290],{"className":53289},[39],[41,53291,53292],{"xmlns":43},[45,53293,53294,53311],{},[48,53295,53296,53299,53301,53303,53305,53307,53309],{},[51,53297,53298],{},"γ",[95,53300,2518],{},[95,53302,15175],{"stretchy":210},[91,53304,1977],{},[95,53306,1210],{"separator":62},[91,53308,205],{},[95,53310,220],{"stretchy":210},[55,53312,53313],{"encoding":57},"\\gamma\\in[0,1)",[31,53315,53317,53336],{"className":53316,"ariaHidden":62},[61],[31,53318,53320,53324,53327,53330,53333],{"className":53319},[66],[31,53321],{"className":53322,"style":53323},[70],"height:0.7335em;vertical-align:-0.1944em;",[31,53325,53298],{"className":53326,"style":24157},[75,76],[31,53328],{"className":53329,"style":293},[119],[31,53331,2518],{"className":53332},[297],[31,53334],{"className":53335,"style":293},[119],[31,53337,53339,53342,53345,53348,53351,53354,53357],{"className":53338},[66],[31,53340],{"className":53341,"style":260},[70],[31,53343,15175],{"className":53344},[264],[31,53346,1977],{"className":53347},[75],[31,53349,1210],{"className":53350},[1391],[31,53352],{"className":53353,"style":1467},[119],[31,53355,205],{"className":53356},[75],[31,53358,220],{"className":53359},[289]," 为折扣因子。记 ",[31,53362,53364,53382],{"className":53363,"translate":35},[34],[31,53365,53367],{"className":53366},[39],[41,53368,53369],{"xmlns":43},[45,53370,53371,53379],{},[48,53372,53373],{},[1004,53374,53375,53377],{},[51,53376,13114],{"mathvariant":925},[51,53378,12271],{},[55,53380,53381],{"encoding":57},"\\mathcal{F}_t",[31,53383,53385],{"className":53384,"ariaHidden":62},[61],[31,53386,53388,53391],{"className":53387},[66],[31,53389],{"className":53390,"style":5230},[70],[31,53392,53394,53398],{"className":53393},[75],[31,53395,13114],{"className":53396,"style":53397},[75,968],"margin-right:0.0993em;",[31,53399,53401],{"className":53400},[1039],[31,53402,53404,53425],{"className":53403},[1043,1044],[31,53405,53407,53422],{"className":53406},[1048],[31,53408,53410],{"className":53409,"style":19220},[1052],[31,53411,53413,53416],{"style":53412},"top:-2.55em;margin-left:-0.0993em;margin-right:0.05em;",[31,53414],{"className":53415,"style":1061},[1060],[31,53417,53419],{"className":53418},[1065,1066,1067,1068],[31,53420,12271],{"className":53421},[75,76,1068],[31,53423,1076],{"className":53424},[1075],[31,53426,53428],{"className":53427},[1048],[31,53429,53431],{"className":53430,"style":2050},[1052],[31,53432],{}," 为截至时刻 ",[31,53435,53437,53450],{"className":53436,"translate":35},[34],[31,53438,53440],{"className":53439},[39],[41,53441,53442],{"xmlns":43},[45,53443,53444,53448],{},[48,53445,53446],{},[51,53447,12271],{},[55,53449,12271],{"encoding":57},[31,53451,53453],{"className":53452,"ariaHidden":62},[61],[31,53454,53456,53459],{"className":53455},[66],[31,53457],{"className":53458,"style":12283},[70],[31,53460,12271],{"className":53461},[75,76]," 的历史信息。若对任意 ",[31,53464,53466,53479],{"className":53465,"translate":35},[34],[31,53467,53469],{"className":53468},[39],[41,53470,53471],{"xmlns":43},[45,53472,53473,53477],{},[48,53474,53475],{},[51,53476,12271],{},[55,53478,12271],{"encoding":57},[31,53480,53482],{"className":53481,"ariaHidden":62},[61],[31,53483,53485,53488],{"className":53484},[66],[31,53486],{"className":53487,"style":12283},[70],[31,53489,12271],{"className":53490},[75,76],[31,53492,53494],{"className":53493,"translate":35},[411],[31,53495,53497,53637],{"className":53496,"translate":35},[34],[31,53498,53500],{"className":53499},[39],[41,53501,53502],{"xmlns":43,"display":420},[45,53503,53504,53634],{},[5606,53505,53506],{"rowspacing":5608,"columnalign":5609,"columnspacing":5610},[5612,53507,53508,53572],{},[5615,53509,53510],{},[2071,53511,53512],{"scriptlevel":1977,"displaystyle":62},[48,53513,53514,53516,53518,53530,53532,53538,53540,53546,53548,53550,53552,53558,53560,53562,53564,53570],{},[51,53515,2521],{},[95,53517,211],{"stretchy":210},[1004,53519,53520,53522],{},[51,53521,14551],{},[48,53523,53524,53526,53528],{},[51,53525,12271],{},[95,53527,434],{},[91,53529,205],{},[95,53531,223],{},[1180,53533,53534,53536],{},[51,53535,14551],{},[95,53537,13691],{"mathvariant":494,"lspace":5610,"rspace":5610},[95,53539,12680],{},[1004,53541,53542,53544],{},[51,53543,14551],{},[51,53545,12271],{},[95,53547,223],{},[51,53549,14551],{},[95,53551,1210],{"separator":62},[1004,53553,53554,53556],{},[51,53555,5813],{},[51,53557,12271],{},[95,53559,223],{},[51,53561,5813],{},[95,53563,1210],{"separator":62},[1004,53565,53566,53568],{},[51,53567,13114],{"mathvariant":925},[51,53569,12271],{},[95,53571,220],{"stretchy":210},[5615,53573,53574],{},[2071,53575,53576],{"scriptlevel":1977,"displaystyle":62},[48,53577,53578,53580,53582,53584,53586,53598,53600,53606,53608,53614,53616,53618,53620,53626,53628,53630,53632],{},[48,53579],{},[95,53581,223],{},[51,53583,2521],{},[95,53585,211],{"stretchy":210},[1004,53587,53588,53590],{},[51,53589,14551],{},[48,53591,53592,53594,53596],{},[51,53593,12271],{},[95,53595,434],{},[91,53597,205],{},[95,53599,223],{},[1180,53601,53602,53604],{},[51,53603,14551],{},[95,53605,13691],{"mathvariant":494,"lspace":5610,"rspace":5610},[95,53607,12680],{},[1004,53609,53610,53612],{},[51,53611,14551],{},[51,53613,12271],{},[95,53615,223],{},[51,53617,14551],{},[95,53619,1210],{"separator":62},[1004,53621,53622,53624],{},[51,53623,5813],{},[51,53625,12271],{},[95,53627,223],{},[51,53629,5813],{},[95,53631,220],{"stretchy":210},[95,53633,1210],{"separator":62},[55,53635,53636],{"encoding":57},"\\begin{aligned}\nP(s_{t+1}=s'\\mid s_t=s,a_t=a,\\mathcal{F}_t)\n&=P(s_{t+1}=s'\\mid s_t=s,a_t=a),\n\\end{aligned}",[31,53638,53640],{"className":53639,"ariaHidden":62},[61],[31,53641,53643,53646],{"className":53642},[66],[31,53644],{"className":53645,"style":4988},[70],[31,53647,53649],{"className":53648},[75],[31,53650,53652,53948],{"className":53651},[5606],[31,53653,53655],{"className":53654},[5867],[31,53656,53658,53940],{"className":53657},[1043,1044],[31,53659,53661,53937],{"className":53660},[1048],[31,53662,53664],{"className":53663,"style":13329},[1052],[31,53665,53667,53670],{"style":53666},"top:-3.01em;",[31,53668],{"className":53669,"style":1637},[1060],[31,53671,53673,53676,53679,53728,53731,53734,53737,53769,53772,53775,53778,53818,53821,53824,53827,53830,53833,53836,53876,53879,53882,53885,53888,53891,53894,53934],{"className":53672},[75],[31,53674,2521],{"className":53675,"style":949},[75,76],[31,53677,211],{"className":53678},[264],[31,53680,53682,53685],{"className":53681},[75],[31,53683,14551],{"className":53684},[75,76],[31,53686,53688],{"className":53687},[1039],[31,53689,53691,53720],{"className":53690},[1043,1044],[31,53692,53694,53717],{"className":53693},[1048],[31,53695,53697],{"className":53696,"style":2028},[1052],[31,53698,53699,53702],{"style":2807},[31,53700],{"className":53701,"style":1061},[1060],[31,53703,53705],{"className":53704},[1065,1066,1067,1068],[31,53706,53708,53711,53714],{"className":53707},[75,1068],[31,53709,12271],{"className":53710},[75,76,1068],[31,53712,434],{"className":53713},[124,1068],[31,53715,205],{"className":53716},[75,1068],[31,53718,1076],{"className":53719},[1075],[31,53721,53723],{"className":53722},[1048],[31,53724,53726],{"className":53725,"style":25082},[1052],[31,53727],{},[31,53729],{"className":53730,"style":293},[119],[31,53732,223],{"className":53733},[297],[31,53735],{"className":53736,"style":293},[119],[31,53738,53740,53743],{"className":53739},[75],[31,53741,14551],{"className":53742},[75,76],[31,53744,53746],{"className":53745},[1039],[31,53747,53749],{"className":53748},[1043],[31,53750,53752],{"className":53751},[1048],[31,53753,53755],{"className":53754,"style":14017},[1052],[31,53756,53757,53760],{"style":1330},[31,53758],{"className":53759,"style":1061},[1060],[31,53761,53763],{"className":53762},[1065,1066,1067,1068],[31,53764,53766],{"className":53765},[75,1068],[31,53767,13691],{"className":53768},[75,1068],[31,53770],{"className":53771,"style":293},[119],[31,53773,12680],{"className":53774},[297],[31,53776],{"className":53777,"style":293},[119],[31,53779,53781,53784],{"className":53780},[75],[31,53782,14551],{"className":53783},[75,76],[31,53785,53787],{"className":53786},[1039],[31,53788,53790,53810],{"className":53789},[1043,1044],[31,53791,53793,53807],{"className":53792},[1048],[31,53794,53796],{"className":53795,"style":19220},[1052],[31,53797,53798,53801],{"style":2807},[31,53799],{"className":53800,"style":1061},[1060],[31,53802,53804],{"className":53803},[1065,1066,1067,1068],[31,53805,12271],{"className":53806},[75,76,1068],[31,53808,1076],{"className":53809},[1075],[31,53811,53813],{"className":53812},[1048],[31,53814,53816],{"className":53815,"style":2050},[1052],[31,53817],{},[31,53819],{"className":53820,"style":293},[119],[31,53822,223],{"className":53823},[297],[31,53825],{"className":53826,"style":293},[119],[31,53828,14551],{"className":53829},[75,76],[31,53831,1210],{"className":53832},[1391],[31,53834],{"className":53835,"style":1467},[119],[31,53837,53839,53842],{"className":53838},[75],[31,53840,5813],{"className":53841},[75,76],[31,53843,53845],{"className":53844},[1039],[31,53846,53848,53868],{"className":53847},[1043,1044],[31,53849,53851,53865],{"className":53850},[1048],[31,53852,53854],{"className":53853,"style":19220},[1052],[31,53855,53856,53859],{"style":2807},[31,53857],{"className":53858,"style":1061},[1060],[31,53860,53862],{"className":53861},[1065,1066,1067,1068],[31,53863,12271],{"className":53864},[75,76,1068],[31,53866,1076],{"className":53867},[1075],[31,53869,53871],{"className":53870},[1048],[31,53872,53874],{"className":53873,"style":2050},[1052],[31,53875],{},[31,53877],{"className":53878,"style":293},[119],[31,53880,223],{"className":53881},[297],[31,53883],{"className":53884,"style":293},[119],[31,53886,5813],{"className":53887},[75,76],[31,53889,1210],{"className":53890},[1391],[31,53892],{"className":53893,"style":1467},[119],[31,53895,53897,53900],{"className":53896},[75],[31,53898,13114],{"className":53899,"style":53397},[75,968],[31,53901,53903],{"className":53902},[1039],[31,53904,53906,53926],{"className":53905},[1043,1044],[31,53907,53909,53923],{"className":53908},[1048],[31,53910,53912],{"className":53911,"style":19220},[1052],[31,53913,53914,53917],{"style":53412},[31,53915],{"className":53916,"style":1061},[1060],[31,53918,53920],{"className":53919},[1065,1066,1067,1068],[31,53921,12271],{"className":53922},[75,76,1068],[31,53924,1076],{"className":53925},[1075],[31,53927,53929],{"className":53928},[1048],[31,53930,53932],{"className":53931,"style":2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Markov decision process，折扣 MDP），记作 ",[31,54220,54222,54260],{"className":54221,"translate":35},[34],[31,54223,54225],{"className":54224},[39],[41,54226,54227],{"xmlns":43},[45,54228,54229,54257],{},[48,54230,54231,54233,54235,54237,54239,54241,54243,54245,54247,54249,54251,54253,54255],{},[51,54232,44355],{"mathvariant":925},[95,54234,223],{},[95,54236,211],{"stretchy":210},[51,54238,21582],{},[95,54240,1210],{"separator":62},[51,54242,18446],{},[95,54244,1210],{"separator":62},[51,54246,2521],{},[95,54248,1210],{"separator":62},[51,54250,2549],{},[95,54252,1210],{"separator":62},[51,54254,53298],{},[95,54256,220],{"stretchy":210},[55,54258,54259],{"encoding":57},"\\mathcal{M}=(S,A,P,r,\\gamma)",[31,54261,54263,54281],{"className":54262,"ariaHidden":62},[61],[31,54264,54266,54269,54272,54275,54278],{"className":54265},[66],[31,54267],{"className":54268,"style":945},[70],[31,54270,44355],{"className":54271},[75,968],[31,54273],{"className":54274,"style":293},[119],[31,54276,223],{"className":54277},[297],[31,54279],{"className":54280,"style":293},[119],[31,54282,54284,54287,54290,54293,54296,54299,54302,54305,54308,54311,54314,54317,54320,54323,54326,54329],{"className":54283},[66],[31,54285],{"className":54286,"style":260},[70],[31,54288,211],{"className":54289},[264],[31,54291,21582],{"className":54292,"style":21697},[75,76],[31,54294,1210],{"className":54295},[1391],[31,54297],{"className":54298,"style":1467},[119],[31,54300,18446],{"className":54301},[75,76],[31,54303,1210],{"className":54304},[1391],[31,54306],{"className":54307,"style":1467},[119],[31,54309,2521],{"className":54310,"style":949},[75,76],[31,54312,1210],{"className":54313},[1391],[31,54315],{"className":54316,"style":1467},[119],[31,54318,2549],{"className":54319,"style":19759},[75,76],[31,54321,1210],{"className":54322},[1391],[31,54324],{"className":54325,"style":1467},[119],[31,54327,53298],{"className":54328,"style":24157},[75,76],[31,54330,220],{"className":54331},[289],[11,54333,54334,54335,18171,54363,54391,54392,54500,54501,54691,54692,54790],{},"本文默认 ",[31,54336,54338,54351],{"className":54337,"translate":35},[34],[31,54339,54341],{"className":54340},[39],[41,54342,54343],{"xmlns":43},[45,54344,54345,54349],{},[48,54346,54347],{},[51,54348,21582],{},[55,54350,21582],{"encoding":57},[31,54352,54354],{"className":54353,"ariaHidden":62},[61],[31,54355,54357,54360],{"className":54356},[66],[31,54358],{"className":54359,"style":945},[70],[31,54361,21582],{"className":54362,"style":21697},[75,76],[31,54364,54366,54379],{"className":54365,"translate":35},[34],[31,54367,54369],{"className":54368},[39],[41,54370,54371],{"xmlns":43},[45,54372,54373,54377],{},[48,54374,54375],{},[51,54376,18446],{},[55,54378,18446],{"encoding":57},[31,54380,54382],{"className":54381,"ariaHidden":62},[61],[31,54383,54385,54388],{"className":54384},[66],[31,54386],{"className":54387,"style":945},[70],[31,54389,18446],{"className":54390},[75,76]," 均为有限集，且存在常数 ",[31,54393,54395,54419],{"className":54394,"translate":35},[34],[31,54396,54398],{"className":54397},[39],[41,54399,54400],{"xmlns":43},[45,54401,54402,54416],{},[48,54403,54404,54412,54414],{},[1004,54405,54406,54408,54410],{},[51,54407,14790],{},[51,54409,10154],{},[95,54411,1540],{},[95,54413,9758],{},[51,54415,4587],{"mathvariant":494},[55,54417,54418],{"encoding":57},"R_{\\max}\u003C\\infty",[31,54420,54422,54491],{"className":54421,"ariaHidden":62},[61],[31,54423,54425,54428,54482,54485,54488],{"className":54424},[66],[31,54426],{"className":54427,"style":5230},[70],[31,54429,54431,54435],{"className":54430},[75],[31,54432,14790],{"className":54433,"style":54434},[75,76],"margin-right:0.0077em;",[31,54436,54438],{"className":54437},[1039],[31,54439,54441,54474],{"className":54440},[1043,1044],[31,54442,54444,54471],{"className":54443},[1048],[31,54445,54447],{"className":54446,"style":3230},[1052],[31,54448,54450,54453],{"style":54449},"top:-2.55em;margin-left:-0.0077em;margin-right:0.05em;",[31,54451],{"className":54452,"style":1061},[1060],[31,54454,54456],{"className":54455},[1065,1066,1067,1068],[31,54457,54459],{"className":54458},[75,1068],[31,54460,54462,54465,54468],{"className":54461},[1460,1068],[31,54463,16396],{"className":54464},[1068],[31,54466,5813],{"className":54467},[1068],[31,54469,53],{"className":54470},[1068],[31,54472,1076],{"className":54473},[1075],[31,54475,54477],{"className":54476},[1048],[31,54478,54480],{"className":54479,"style":2050},[1052],[31,54481],{},[31,54483],{"className":54484,"style":293},[119],[31,54486,9758],{"className":54487},[297],[31,54489],{"className":54490,"style":293},[119],[31,54492,54494,54497],{"className":54493},[66],[31,54495],{"className":54496,"style":71},[70],[31,54498,4587],{"className":54499},[75]," 使得 ",[31,54502,54504,54550],{"className":54503,"translate":35},[34],[31,54505,54507],{"className":54506},[39],[41,54508,54509],{"xmlns":43},[45,54510,54511,54547],{},[48,54512,54513,54515,54517,54519,54521,54523,54525,54527,54533,54535,54537,54539],{},[51,54514,12680],{"mathvariant":494},[51,54516,2549],{},[95,54518,211],{"stretchy":210},[51,54520,14551],{},[95,54522,1210],{"separator":62},[51,54524,5813],{},[95,54526,1210],{"separator":62},[1180,54528,54529,54531],{},[51,54530,14551],{},[95,54532,13691],{"mathvariant":494,"lspace":5610,"rspace":5610},[95,54534,220],{"stretchy":210},[51,54536,12680],{"mathvariant":494},[95,54538,347],{},[1004,54540,54541,54543,54545],{},[51,54542,14790],{},[51,54544,10154],{},[95,54546,1540],{},[55,54548,54549],{"encoding":57},"|r(s,a,s')|\\le R_{\\max}",[31,54551,54553,54633],{"className":54552,"ariaHidden":62},[61],[31,54554,54556,54559,54562,54565,54568,54571,54574,54577,54580,54583,54586,54618,54621,54624,54627,54630],{"className":54555},[66],[31,54557],{"className":54558,"style":13710},[70],[31,54560,12680],{"className":54561},[75],[31,54563,2549],{"className":54564,"style":19759},[75,76],[31,54566,211],{"className":54567},[264],[31,54569,14551],{"className":54570},[75,76],[31,54572,1210],{"className":54573},[1391],[31,54575],{"className":54576,"style":1467},[119],[31,54578,5813],{"className":54579},[75,76],[31,54581,1210],{"className":54582},[1391],[31,54584],{"className":54585,"style":1467},[119],[31,54587,54589,54592],{"className":54588},[75],[31,54590,14551],{"className":54591},[75,76],[31,54593,54595],{"className":54594},[1039],[31,54596,54598],{"className":54597},[1043],[31,54599,54601],{"className":54600},[1048],[31,54602,54604],{"className":54603,"style":13732},[1052],[31,54605,54606,54609],{"style":3631},[31,54607],{"className":54608,"style":1061},[1060],[31,54610,54612],{"className":54611},[1065,1066,1067,1068],[31,54613,54615],{"className":54614},[75,1068],[31,54616,13691],{"className":54617},[75,1068],[31,54619,220],{"className":54620},[289],[31,54622,12680],{"className":54623},[75],[31,54625],{"className":54626,"style":293},[119],[31,54628,347],{"className":54629},[297],[31,54631],{"className":54632,"style":293},[119],[31,54634,54636,54639],{"className":54635},[66],[31,54637],{"className":54638,"style":5230},[70],[31,54640,54642,54645],{"className":54641},[75],[31,54643,14790],{"className":54644,"style":54434},[75,76],[31,54646,54648],{"className":54647},[1039],[31,54649,54651,54683],{"className":54650},[1043,1044],[31,54652,54654,54680],{"className":54653},[1048],[31,54655,54657],{"className":54656,"style":3230},[1052],[31,54658,54659,54662],{"style":54449},[31,54660],{"className":54661,"style":1061},[1060],[31,54663,54665],{"className":54664},[1065,1066,1067,1068],[31,54666,54668],{"className":54667},[75,1068],[31,54669,54671,54674,54677],{"className":54670},[1460,1068],[31,54672,16396],{"className":54673},[1068],[31,54675,5813],{"className":54676},[1068],[31,54678,53],{"className":54679},[1068],[31,54681,1076],{"className":54682},[1075],[31,54684,54686],{"className":54685},[1048],[31,54687,54689],{"className":54688,"style":2050},[1052],[31,54690],{}," 对一切 ",[31,54693,54695,54725],{"className":54694,"translate":35},[34],[31,54696,54698],{"className":54697},[39],[41,54699,54700],{"xmlns":43},[45,54701,54702,54722],{},[48,54703,54704,54706,54708,54710,54712,54714,54720],{},[95,54705,211],{"stretchy":210},[51,54707,14551],{},[95,54709,1210],{"separator":62},[51,54711,5813],{},[95,54713,1210],{"separator":62},[1180,54715,54716,54718],{},[51,54717,14551],{},[95,54719,13691],{"mathvariant":494,"lspace":5610,"rspace":5610},[95,54721,220],{"stretchy":210},[55,54723,54724],{"encoding":57},"(s,a,s')",[31,54726,54728],{"className":54727,"ariaHidden":62},[61],[31,54729,54731,54734,54737,54740,54743,54746,54749,54752,54755,54787],{"className":54730},[66],[31,54732],{"className":54733,"style":13710},[70],[31,54735,211],{"className":54736},[264],[31,54738,14551],{"className":54739},[75,76],[31,54741,1210],{"className":54742},[1391],[31,54744],{"className":54745,"style":1467},[119],[31,54747,5813],{"className":54748},[75,76],[31,54750,1210],{"className":54751},[1391],[31,54753],{"className":54754,"style":1467},[119],[31,54756,54758,54761],{"className":54757},[75],[31,54759,14551],{"className":54760},[75,76],[31,54762,54764],{"className":54763},[1039],[31,54765,54767],{"className":54766},[1043],[31,54768,54770],{"className":54769},[1048],[31,54771,54773],{"className":54772,"style":13732},[1052],[31,54774,54775,54778],{"style":3631},[31,54776],{"className":54777,"style":1061},[1060],[31,54779,54781],{"className":54780},[1065,1066,1067,1068],[31,54782,54784],{"className":54783},[75,1068],[31,54785,13691],{"className":54786},[75,1068],[31,54788,220],{"className":54789},[289]," 成立。有限性假设保证最大值可以取到，后文会在第 7 节说明连续状态空间中的相应修改。",[11,54792,54793,54794,54797,54798,54826,54827,41892,54893,54937,54938,55004,55005,55033,55034,55062,55063,1138,55091,55156,55157,55185,55186,55214,55215,55218,55219,2372],{},"一个",[7246,54795,54796],{},"策略"," ",[31,54799,54801,54814],{"className":54800,"translate":35},[34],[31,54802,54804],{"className":54803},[39],[41,54805,54806],{"xmlns":43},[45,54807,54808,54812],{},[48,54809,54810],{},[51,54811,2515],{},[55,54813,3485],{"encoding":57},[31,54815,54817],{"className":54816,"ariaHidden":62},[61],[31,54818,54820,54823],{"className":54819},[66],[31,54821],{"className":54822,"style":71},[70],[31,54824,2515],{"className":54825,"style":2666},[75,76]," 是映射 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是动作集上的概率单纯形。",[31,54939,54941,54965],{"className":54940,"translate":35},[34],[31,54942,54944],{"className":54943},[39],[41,54945,54946],{"xmlns":43},[45,54947,54948,54962],{},[48,54949,54950,54952,54954,54956,54958,54960],{},[51,54951,2515],{},[95,54953,211],{"stretchy":210},[51,54955,5813],{},[95,54957,12680],{},[51,54959,14551],{},[95,54961,220],{"stretchy":210},[55,54963,54964],{"encoding":57},"\\pi(a\\mid s)",[31,54966,54968,54992],{"className":54967,"ariaHidden":62},[61],[31,54969,54971,54974,54977,54980,54983,54986,54989],{"className":54970},[66],[31,54972],{"className":54973,"style":260},[70],[31,54975,2515],{"className":54976,"style":2666},[75,76],[31,54978,211],{"className":54979},[264],[31,54981,5813],{"className":54982},[75,76],[31,54984],{"className":54985,"style":293},[119],[31,54987,12680],{"className":54988},[297],[31,54990],{"className":54991,"style":293},[119],[31,54993,54995,54998,55001],{"className":54994},[66],[31,54996],{"className":54997,"style":260},[70],[31,54999,14551],{"className":55000},[75,76],[31,55002,220],{"className":55003},[289]," 表示在状态 ",[31,55006,55008,55021],{"className":55007,"translate":35},[34],[31,55009,55011],{"className":55010},[39],[41,55012,55013],{"xmlns":43},[45,55014,55015,55019],{},[48,55016,55017],{},[51,55018,14551],{},[55,55020,14551],{"encoding":57},[31,55022,55024],{"className":55023,"ariaHidden":62},[61],[31,55025,55027,55030],{"className":55026},[66],[31,55028],{"className":55029,"style":71},[70],[31,55031,14551],{"className":55032},[75,76]," 选择动作 ",[31,55035,55037,55050],{"className":55036,"translate":35},[34],[31,55038,55040],{"className":55039},[39],[41,55041,55042],{"xmlns":43},[45,55043,55044,55048],{},[48,55045,55046],{},[51,55047,5813],{},[55,55049,5813],{"encoding":57},[31,55051,55053],{"className":55052,"ariaHidden":62},[61],[31,55054,55056,55059],{"className":55055},[66],[31,55057],{"className":55058,"style":71},[70],[31,55060,5813],{"className":55061},[75,76]," 的概率。若对每个 ",[31,55064,55066,55079],{"className":55065,"translate":35},[34],[31,55067,55069],{"className":55068},[39],[41,55070,55071],{"xmlns":43},[45,55072,55073,55077],{},[48,55074,55075],{},[51,55076,14551],{},[55,55078,14551],{"encoding":57},[31,55080,55082],{"className":55081,"ariaHidden":62},[61],[31,55083,55085,55088],{"className":55084},[66],[31,55086],{"className":55087,"style":71},[70],[31,55089,14551],{"className":55090},[75,76],[31,55092,55094,55117],{"className":55093,"translate":35},[34],[31,55095,55097],{"className":55096},[39],[41,55098,55099],{"xmlns":43},[45,55100,55101,55115],{},[48,55102,55103,55105,55107,55109,55111,55113],{},[51,55104,2515],{},[95,55106,211],{"stretchy":210},[51,55108,5813],{},[95,55110,12680],{},[51,55112,14551],{},[95,55114,220],{"stretchy":210},[55,55116,54964],{"encoding":57},[31,55118,55120,55144],{"className":55119,"ariaHidden":62},[61],[31,55121,55123,55126,55129,55132,55135,55138,55141],{"className":55122},[66],[31,55124],{"className":55125,"style":260},[70],[31,55127,2515],{"className":55128,"style":2666},[75,76],[31,55130,211],{"className":55131},[264],[31,55133,5813],{"className":55134},[75,76],[31,55136],{"className":55137,"style":293},[119],[31,55139,12680],{"className":55140},[297],[31,55142],{"className":55143,"style":293},[119],[31,55145,55147,55150,55153],{"className":55146},[66],[31,55148],{"className":55149,"style":260},[70],[31,55151,14551],{"className":55152},[75,76],[31,55154,220],{"className":55155},[289]," 只在某一个动作处取 ",[31,55158,55160,55173],{"className":55159,"translate":35},[34],[31,55161,55163],{"className":55162},[39],[41,55164,55165],{"xmlns":43},[45,55166,55167,55171],{},[48,55168,55169],{},[91,55170,205],{},[55,55172,205],{"encoding":57},[31,55174,55176],{"className":55175,"ariaHidden":62},[61],[31,55177,55179,55182],{"className":55178},[66],[31,55180],{"className":55181,"style":326},[70],[31,55183,205],{"className":55184},[75],"，则称 ",[31,55187,55189,55202],{"className":55188,"translate":35},[34],[31,55190,55192],{"className":55191},[39],[41,55193,55194],{"xmlns":43},[45,55195,55196,55200],{},[48,55197,55198],{},[51,55199,2515],{},[55,55201,3485],{"encoding":57},[31,55203,55205],{"className":55204,"ariaHidden":62},[61],[31,55206,55208,55211],{"className":55207},[66],[31,55209],{"className":55210,"style":71},[70],[31,55212,2515],{"className":55213,"style":2666},[75,76]," 为",[7246,55216,55217],{},"确定性策略","，并可将其视为映射 ",[31,55220,55222,55244],{"className":55221,"translate":35},[34],[31,55223,55225],{"className":55224},[39],[41,55226,55227],{"xmlns":43},[45,55228,55229,55241],{},[48,55230,55231,55233,55235,55237,55239],{},[51,55232,2515],{},[95,55234,922],{},[51,55236,21582],{},[95,55238,929],{},[51,55240,18446],{},[55,55242,55243],{"encoding":57},"\\pi:S\\to A",[31,55245,55247,55265,55283],{"className":55246,"ariaHidden":62},[61],[31,55248,55250,55253,55256,55259,55262],{"className":55249},[66],[31,55251],{"className":55252,"style":71},[70],[31,55254,2515],{"className":55255,"style":2666},[75,76],[31,55257],{"className":55258,"style":293},[119],[31,55260,922],{"className":55261},[297],[31,55263],{"className":55264,"style":293},[119],[31,55266,55268,55271,55274,55277,55280],{"className":55267},[66],[31,55269],{"className":55270,"style":945},[70],[31,55272,21582],{"className":55273,"style":21697},[75,76],[31,55275],{"className":55276,"style":293},[119],[31,55278,929],{"className":55279},[297],[31,55281],{"className":55282,"style":293},[119],[31,55284,55286,55289],{"className":55285},[66],[31,55287],{"className":55288,"style":945},[70],[31,55290,18446],{"className":55291},[75,76],[52860,55293,55295],{"id":55294},"_12-回报与值函数","1.2 回报与值函数",[52865,55297,55300,55550,55829,55865],{"id":55298,"title":55299},"definition-return","定义 1.2（折扣回报）",[11,55301,55302,55303,55549],{},"设轨迹为 ",[31,55304,55306,55353],{"className":55305,"translate":35},[34],[31,55307,55309],{"className":55308},[39],[41,55310,55311],{"xmlns":43},[45,55312,55313,55350],{},[48,55314,55315,55321,55323,55329,55331,55337,55339,55345,55347],{},[1004,55316,55317,55319],{},[51,55318,14551],{},[91,55320,1977],{},[95,55322,1210],{"separator":62},[1004,55324,55325,55327],{},[51,55326,5813],{},[91,55328,1977],{},[95,55330,1210],{"separator":62},[1004,55332,55333,55335],{},[51,55334,14551],{},[91,55336,205],{},[95,55338,1210],{"separator":62},[1004,55340,55341,55343],{},[51,55342,5813],{},[91,55344,205],{},[95,55346,1210],{"separator":62},[95,55348,55349],{},"…",[55,55351,55352],{"encoding":57},"s_0,a_0,s_1,a_1,\\dots",[31,55354,55356],{"className":55355,"ariaHidden":62},[61],[31,55357,55359,55362,55402,55405,55408,55448,55451,55454,55494,55497,55500,55540,55543,55546],{"className":55358},[66],[31,55360],{"className":55361,"style":865},[70],[31,55363,55365,55368],{"className":55364},[75],[31,55366,14551],{"className":55367},[75,76],[31,55369,55371],{"className":55370},[1039],[31,55372,55374,55394],{"className":55373},[1043,1044],[31,55375,55377,55391],{"className":55376},[1048],[31,55378,55380],{"className":55379,"style":2028},[1052],[31,55381,55382,55385],{"style":2807},[31,55383],{"className":55384,"style":1061},[1060],[31,55386,55388],{"className":55387},[1065,1066,1067,1068],[31,55389,1977],{"className":55390},[75,1068],[31,55392,1076],{"className":55393},[1075],[31,55395,55397],{"className":55396},[1048],[31,55398,55400],{"className":55399,"style":2050},[1052],[31,55401],{},[31,55403,1210],{"className":55404},[1391],[31,55406],{"className":55407,"style":1467},[119],[31,55409,55411,55414],{"className":55410},[75],[31,55412,5813],{"className":55413},[75,76],[31,55415,55417],{"className":55416},[1039],[31,55418,55420,55440],{"className":55419},[1043,1044],[31,55421,55423,55437],{"className":55422},[1048],[31,55424,55426],{"className":55425,"style":2028},[1052],[31,55427,55428,55431],{"style":2807},[31,55429],{"className":55430,"style":1061},[1060],[31,55432,55434],{"className":55433},[1065,1066,1067,1068],[31,55435,1977],{"className":55436},[75,1068],[31,55438,1076],{"className":55439},[1075],[31,55441,55443],{"className":55442},[1048],[31,55444,55446],{"className":55445,"style":2050},[1052],[31,55447],{},[31,55449,1210],{"className":55450},[1391],[31,55452],{"className":55453,"style":1467},[119],[31,55455,55457,55460],{"className":55456},[75],[31,55458,14551],{"className":55459},[75,76],[31,55461,55463],{"className":55462},[1039],[31,55464,55466,55486],{"className":55465},[1043,1044],[31,55467,55469,55483],{"className":55468},[1048],[31,55470,55472],{"className":55471,"style":2028},[1052],[31,55473,55474,55477],{"style":2807},[31,55475],{"className":55476,"style":1061},[1060],[31,55478,55480],{"className":55479},[1065,1066,1067,1068],[31,55481,205],{"className":55482},[75,1068],[31,55484,1076],{"className":55485},[1075],[31,55487,55489],{"className":55488},[1048],[31,55490,55492],{"className":55491,"style":2050},[1052],[31,55493],{},[31,55495,1210],{"className":55496},[1391],[31,55498],{"className":55499,"style":1467},[119],[31,55501,55503,55506],{"className":55502},[75],[31,55504,5813],{"className":55505},[75,76],[31,55507,55509],{"className":55508},[1039],[31,55510,55512,55532],{"className":55511},[1043,1044],[31,55513,55515,55529],{"className":55514},[1048],[31,55516,55518],{"className":55517,"style":2028},[1052],[31,55519,55520,55523],{"style":2807},[31,55521],{"className":55522,"style":1061},[1060],[31,55524,55526],{"className":55525},[1065,1066,1067,1068],[31,55527,205],{"className":55528},[75,1068],[31,55530,1076],{"className":55531},[1075],[31,55533,55535],{"className":55534},[1048],[31,55536,55538],{"className":55537,"style":2050},[1052],[31,55539],{},[31,55541,1210],{"className":55542},[1391],[31,55544],{"className":55545,"style":1467},[119],[31,55547,55349],{"className":55548},[3860],"，记",[31,55551,55553],{"className":55552,"translate":35},[411],[31,55554,55556,55612],{"className":55555,"translate":35},[34],[31,55557,55559],{"className":55558},[39],[41,55560,55561],{"xmlns":43,"display":420},[45,55562,55563,55609],{},[48,55564,55565,55571,55573,55575,55577,55583,55585,55591,55593,55605,55607],{},[1004,55566,55567,55569],{},[51,55568,2549],{},[51,55570,12271],{},[95,55572,223],{},[51,55574,2549],{},[95,55576,211],{"stretchy":210},[1004,55578,55579,55581],{},[51,55580,14551],{},[51,55582,12271],{},[95,55584,1210],{"separator":62},[1004,55586,55587,55589],{},[51,55588,5813],{},[51,55590,12271],{},[95,55592,1210],{"separator":62},[1004,55594,55595,55597],{},[51,55596,14551],{},[48,55598,55599,55601,55603],{},[51,55600,12271],{},[95,55602,434],{},[91,55604,205],{},[95,55606,220],{"stretchy":210},[51,55608,495],{"mathvariant":494},[55,55610,55611],{"encoding":57},"r_t=r(s_t,a_t,s_{t+1}).",[31,55613,55615,55670],{"className":55614,"ariaHidden":62},[61],[31,55616,55618,55621,55661,55664,55667],{"className":55617},[66],[31,55619],{"className":55620,"style":7090},[70],[31,55622,55624,55627],{"className":55623},[75],[31,55625,2549],{"className":55626,"style":19759},[75,76],[31,55628,55630],{"className":55629},[1039],[31,55631,55633,55653],{"className":55632},[1043,1044],[31,55634,55636,55650],{"className":55635},[1048],[31,55637,55639],{"className":55638,"style":19220},[1052],[31,55640,55641,55644],{"style":26726},[31,55642],{"className":55643,"style":1061},[1060],[31,55645,55647],{"className":55646},[1065,1066,1067,1068],[31,55648,12271],{"className":55649},[75,76,1068],[31,55651,1076],{"className":55652},[1075],[31,55654,55656],{"className":55655},[1048],[31,55657,55659],{"className":55658,"style":2050},[1052],[31,55660],{},[31,55662],{"className":55663,"style":293},[119],[31,55665,223],{"className":55666},[297],[31,55668],{"className":55669,"style":293},[119],[31,55671,55673,55676,55679,55682,55722,55725,55728,55768,55771,55774,55823,55826],{"className":55672},[66],[31,55674],{"className":55675,"style":260},[70],[31,55677,2549],{"className":55678,"style":19759},[75,76],[31,55680,211],{"className":55681},[264],[31,55683,55685,55688],{"className":55684},[75],[31,55686,14551],{"className":55687},[75,76],[31,55689,55691],{"className":55690},[1039],[31,55692,55694,55714],{"className":55693},[1043,1044],[31,55695,55697,55711],{"className":55696},[1048],[31,55698,55700],{"className":55699,"style":19220},[1052],[31,55701,55702,55705],{"style":2807},[31,55703],{"className":55704,"style":1061},[1060],[31,55706,55708],{"className":55707},[1065,1066,1067,1068],[31,55709,12271],{"className":55710},[75,76,1068],[31,55712,1076],{"className":55713},[1075],[31,55715,55717],{"className":55716},[1048],[31,55718,55720],{"className":55719,"style":2050},[1052],[31,55721],{},[31,55723,1210],{"className":55724},[1391],[31,55726],{"className":55727,"style":1467},[119],[31,55729,55731,55734],{"className":55730},[75],[31,55732,5813],{"className":55733},[75,76],[31,55735,55737],{"className":55736},[1039],[31,55738,55740,55760],{"className":55739},[1043,1044],[31,55741,55743,55757],{"className":55742},[1048],[31,55744,55746],{"className":55745,"style":19220},[1052],[31,55747,55748,55751],{"style":2807},[31,55749],{"className":55750,"style":1061},[1060],[31,55752,55754],{"className":55753},[1065,1066,1067,1068],[31,55755,12271],{"className":55756},[75,76,1068],[31,55758,1076],{"className":55759},[1075],[31,55761,55763],{"className":55762},[1048],[31,55764,55766],{"className":55765,"style":2050},[1052],[31,55767],{},[31,55769,1210],{"className":55770},[1391],[31,55772],{"className":55773,"style":1467},[119],[31,55775,55777,55780],{"className":55776},[75],[31,55778,14551],{"className":55779},[75,76],[31,55781,55783],{"className":55782},[1039],[31,55784,55786,55815],{"className":55785},[1043,1044],[31,55787,55789,55812],{"className":55788},[1048],[31,55790,55792],{"className":55791,"style":2028},[1052],[31,55793,55794,55797],{"style":2807},[31,55795],{"className":55796,"style":1061},[1060],[31,55798,55800],{"className":55799},[1065,1066,1067,1068],[31,55801,55803,55806,55809],{"className":55802},[75,1068],[31,55804,12271],{"className":55805},[75,76,1068],[31,55807,434],{"className":55808},[124,1068],[31,55810,205],{"className":55811},[75,1068],[31,55813,1076],{"className":55814},[1075],[31,55816,55818],{"className":55817},[1048],[31,55819,55821],{"className":55820,"style":25082},[1052],[31,55822],{},[31,55824,220],{"className":55825},[289],[31,55827,495],{"className":55828},[75],[11,55830,55831,55832,55860,55861,55864],{},"从时刻 ",[31,55833,55835,55848],{"className":55834,"translate":35},[34],[31,55836,55838],{"className":55837},[39],[41,55839,55840],{"xmlns":43},[45,55841,55842,55846],{},[48,55843,55844],{},[51,55845,12271],{},[55,55847,12271],{"encoding":57},[31,55849,55851],{"className":55850,"ariaHidden":62},[61],[31,55852,55854,55857],{"className":55853},[66],[31,55855],{"className":55856,"style":12283},[70],[31,55858,12271],{"className":55859},[75,76]," 开始的",[7246,55862,55863],{},"折扣回报","定义为",[31,55866,55868],{"className":55867,"translate":35},[411],[31,55869,55871,55925],{"className":55870,"translate":35},[34],[31,55872,55874],{"className":55873},[39],[41,55875,55876],{"xmlns":43,"display":420},[45,55877,55878,55922],{},[48,55879,55880,55886,55888,55902,55908,55920],{},[1004,55881,55882,55884],{},[51,55883,15297],{},[51,55885,12271],{},[95,55887,223],{},[42429,55889,55890,55892,55900],{},[95,55891,5625],{},[48,55893,55894,55896,55898],{},[51,55895,52065],{},[95,55897,223],{},[91,55899,1977],{},[51,55901,4587],{"mathvariant":494},[1180,55903,55904,55906],{},[51,55905,53298],{},[51,55907,52065],{},[1004,55909,55910,55912],{},[51,55911,2549],{},[48,55913,55914,55916,55918],{},[51,55915,12271],{},[95,55917,434],{},[51,55919,52065],{},[51,55921,495],{"mathvariant":494},[55,55923,55924],{"encoding":57},"G_t=\\sum_{k=0}^{\\infty}\\gamma^k r_{t+k}.",[31,55926,55928,55983],{"className":55927,"ariaHidden":62},[61],[31,55929,55931,55934,55974,55977,55980],{"className":55930},[66],[31,55932],{"className":55933,"style":5230},[70],[31,55935,55937,55940],{"className":55936},[75],[31,55938,15297],{"className":55939},[75,76],[31,55941,55943],{"className":55942},[1039],[31,55944,55946,55966],{"className":55945},[1043,1044],[31,55947,55949,55963],{"className":55948},[1048],[31,55950,55952],{"className":55951,"style":19220},[1052],[31,55953,55954,55957],{"style":2807},[31,55955],{"className":55956,"style":1061},[1060],[31,55958,55960],{"className":55959},[1065,1066,1067,1068],[31,55961,12271],{"className":55962},[75,76,1068],[31,55964,1076],{"className":55965},[1075],[31,55967,55969],{"className":55968},[1048],[31,55970,55972],{"className":55971,"style":2050},[1052],[31,55973],{},[31,55975],{"className":55976,"style":293},[119],[31,55978,223],{"className":55979},[297],[31,55981],{"className":55982,"style":293},[119],[31,55984,55986,55990,56059,56062,56091,56140],{"className":55985},[66],[31,55987],{"className":55988,"style":55989},[70],"height:2.9535em;vertical-align:-1.3021em;",[31,55991,55993],{"className":55992},[1460,1620],[31,55994,55996,56050],{"className":55995},[1043,1044],[31,55997,55999,56047],{"className":55998},[1048],[31,56000,56002,56023,56033],{"className":56001,"style":42702},[1052],[31,56003,56005,56008],{"style":56004},"top:-1.8479em;margin-left:0em;",[31,56006],{"className":56007,"style":5907},[1060],[31,56009,56011],{"className":56010},[1065,1066,1067,1068],[31,56012,56014,56017,56020],{"className":56013},[75,1068],[31,56015,52065],{"className":56016,"style":52166},[75,76,1068],[31,56018,223],{"className":56019},[297,1068],[31,56021,1977],{"className":56022},[75,1068],[31,56024,56025,56028],{"style":5919},[31,56026],{"className":56027,"style":5907},[1060],[31,56029,56030],{},[31,56031,5625],{"className":56032},[1460,1461,1687],[31,56034,56035,56038],{"style":42735},[31,56036],{"className":56037,"style":5907},[1060],[31,56039,56041],{"className":56040},[1065,1066,1067,1068],[31,56042,56044],{"className":56043},[75,1068],[31,56045,4587],{"className":56046},[75,1068],[31,56048,1076],{"className":56049},[1075],[31,56051,56053],{"className":56052},[1048],[31,56054,56057],{"className":56055,"style":56056},[1052],"height:1.3021em;",[31,56058],{},[31,56060],{"className":56061,"style":1467},[119],[31,56063,56065,56068],{"className":56064},[75],[31,56066,53298],{"className":56067,"style":24157},[75,76],[31,56069,56071],{"className":56070},[1039],[31,56072,56074],{"className":56073},[1043],[31,56075,56077],{"className":56076},[1048],[31,56078,56080],{"className":56079,"style":29642},[1052],[31,56081,56082,56085],{"style":1330},[31,56083],{"className":56084,"style":1061},[1060],[31,56086,56088],{"className":56087},[1065,1066,1067,1068],[31,56089,52065],{"className":56090,"style":52166},[75,76,1068],[31,56092,56094,56097],{"className":56093},[75],[31,56095,2549],{"className":56096,"style":19759},[75,76],[31,56098,56100],{"className":56099},[1039],[31,56101,56103,56132],{"className":56102},[1043,1044],[31,56104,56106,56129],{"className":56105},[1048],[31,56107,56109],{"className":56108,"style":1053},[1052],[31,56110,56111,56114],{"style":26726},[31,56112],{"className":56113,"style":1061},[1060],[31,56115,56117],{"className":56116},[1065,1066,1067,1068],[31,56118,56120,56123,56126],{"className":56119},[75,1068],[31,56121,12271],{"className":56122},[75,76,1068],[31,56124,434],{"className":56125},[124,1068],[31,56127,52065],{"className":56128,"style":52166},[75,76,1068],[31,56130,1076],{"className":56131},[1075],[31,56133,56135],{"className":56134},[1048],[31,56136,56138],{"className":56137,"style":25082},[1052],[31,56139],{},[31,56141,495],{"className":56142},[75],[11,56144,56145,56146,56216,56217,56287,56288,56373,56374,56508,56509,56837],{},"这里采用「转移奖励」约定，即 ",[31,56147,56149,56167],{"className":56148,"translate":35},[34],[31,56150,56152],{"className":56151},[39],[41,56153,56154],{"xmlns":43},[45,56155,56156,56164],{},[48,56157,56158],{},[1004,56159,56160,56162],{},[51,56161,2549],{},[51,56163,12271],{},[55,56165,56166],{"encoding":57},"r_t",[31,56168,56170],{"className":56169,"ariaHidden":62},[61],[31,56171,56173,56176],{"className":56172},[66],[31,56174],{"className":56175,"style":7090},[70],[31,56177,56179,56182],{"className":56178},[75],[31,56180,2549],{"className":56181,"style":19759},[75,76],[31,56183,56185],{"className":56184},[1039],[31,56186,56188,56208],{"className":56187},[1043,1044],[31,56189,56191,56205],{"className":56190},[1048],[31,56192,56194],{"className":56193,"style":19220},[1052],[31,56195,56196,56199],{"style":26726},[31,56197],{"className":56198,"style":1061},[1060],[31,56200,56202],{"className":56201},[1065,1066,1067,1068],[31,56203,12271],{"className":56204},[75,76,1068],[31,56206,1076],{"className":56207},[1075],[31,56209,56211],{"className":56210},[1048],[31,56212,56214],{"className":56213,"style":2050},[1052],[31,56215],{}," 是在从 ",[31,56218,56220,56238],{"className":56219,"translate":35},[34],[31,56221,56223],{"className":56222},[39],[41,56224,56225],{"xmlns":43},[45,56226,56227,56235],{},[48,56228,56229],{},[1004,56230,56231,56233],{},[51,56232,14551],{},[51,56234,12271],{},[55,56236,56237],{"encoding":57},"s_t",[31,56239,56241],{"className":56240,"ariaHidden":62},[61],[31,56242,56244,56247],{"className":56243},[66],[31,56245],{"className":56246,"style":7090},[70],[31,56248,56250,56253],{"className":56249},[75],[31,56251,14551],{"className":56252},[75,76],[31,56254,56256],{"className":56255},[1039],[31,56257,56259,56279],{"className":56258},[1043,1044],[31,56260,56262,56276],{"className":56261},[1048],[31,56263,56265],{"className":56264,"style":19220},[1052],[31,56266,56267,56270],{"style":2807},[31,56268],{"className":56269,"style":1061},[1060],[31,56271,56273],{"className":56272},[1065,1066,1067,1068],[31,56274,12271],{"className":56275},[75,76,1068],[31,56277,1076],{"className":56278},[1075],[31,56280,56282],{"className":56281},[1048],[31,56283,56285],{"className":56284,"style":2050},[1052],[31,56286],{}," 转移到 ",[31,56289,56291,56315],{"className":56290,"translate":35},[34],[31,56292,56294],{"className":56293},[39],[41,56295,56296],{"xmlns":43},[45,56297,56298,56312],{},[48,56299,56300],{},[1004,56301,56302,56304],{},[51,56303,14551],{},[48,56305,56306,56308,56310],{},[51,56307,12271],{},[95,56309,434],{},[91,56311,205],{},[55,56313,56314],{"encoding":57},"s_{t+1}",[31,56316,56318],{"className":56317,"ariaHidden":62},[61],[31,56319,56321,56324],{"className":56320},[66],[31,56322],{"className":56323,"style":41068},[70],[31,56325,56327,56330],{"className":56326},[75],[31,56328,14551],{"className":56329},[75,76],[31,56331,56333],{"className":56332},[1039],[31,56334,56336,56365],{"className":56335},[1043,1044],[31,56337,56339,56362],{"className":56338},[1048],[31,56340,56342],{"className":56341,"style":2028},[1052],[31,56343,56344,56347],{"style":2807},[31,56345],{"className":56346,"style":1061},[1060],[31,56348,56350],{"className":56349},[1065,1066,1067,1068],[31,56351,56353,56356,56359],{"className":56352},[75,1068],[31,56354,12271],{"className":56355},[75,76,1068],[31,56357,434],{"className":56358},[124,1068],[31,56360,205],{"className":56361},[75,1068],[31,56363,1076],{"className":56364},[1075],[31,56366,56368],{"className":56367},[1048],[31,56369,56371],{"className":56370,"style":25082},[1052],[31,56372],{}," 时获得的奖励。若奖励只依赖于 ",[31,56375,56377,56407],{"className":56376,"translate":35},[34],[31,56378,56380],{"className":56379},[39],[41,56381,56382],{"xmlns":43},[45,56383,56384,56404],{},[48,56385,56386,56388,56394,56396,56402],{},[95,56387,211],{"stretchy":210},[1004,56389,56390,56392],{},[51,56391,14551],{},[51,56393,12271],{},[95,56395,1210],{"separator":62},[1004,56397,56398,56400],{},[51,56399,5813],{},[51,56401,12271],{},[95,56403,220],{"stretchy":210},[55,56405,56406],{"encoding":57},"(s_t,a_t)",[31,56408,56410],{"className":56409,"ariaHidden":62},[61],[31,56411,56413,56416,56419,56459,56462,56465,56505],{"className":56412},[66],[31,56414],{"className":56415,"style":260},[70],[31,56417,211],{"className":56418},[264],[31,56420,56422,56425],{"className":56421},[75],[31,56423,14551],{"className":56424},[75,76],[31,56426,56428],{"className":56427},[1039],[31,56429,56431,56451],{"className":56430},[1043,1044],[31,56432,56434,56448],{"className":56433},[1048],[31,56435,56437],{"className":56436,"style":19220},[1052],[31,56438,56439,56442],{"style":2807},[31,56440],{"className":56441,"style":1061},[1060],[31,56443,56445],{"className":56444},[1065,1066,1067,1068],[31,56446,12271],{"className":56447},[75,76,1068],[31,56449,1076],{"className":56450},[1075],[31,56452,56454],{"className":56453},[1048],[31,56455,56457],{"className":56456,"style":2050},[1052],[31,56458],{},[31,56460,1210],{"className":56461},[1391],[31,56463],{"className":56464,"style":1467},[119],[31,56466,56468,56471],{"className":56467},[75],[31,56469,5813],{"className":56470},[75,76],[31,56472,56474],{"className":56473},[1039],[31,56475,56477,56497],{"className":56476},[1043,1044],[31,56478,56480,56494],{"className":56479},[1048],[31,56481,56483],{"className":56482,"style":19220},[1052],[31,56484,56485,56488],{"style":2807},[31,56486],{"className":56487,"style":1061},[1060],[31,56489,56491],{"className":56490},[1065,1066,1067,1068],[31,56492,12271],{"className":56493},[75,76,1068],[31,56495,1076],{"className":56496},[1075],[31,56498,56500],{"className":56499},[1048],[31,56501,56503],{"className":56502,"style":2050},[1052],[31,56504],{},[31,56506,220],{"className":56507},[289],"，只需令 ",[31,56510,56512,56588],{"className":56511,"translate":35},[34],[31,56513,56515],{"className":56514},[39],[41,56516,56517],{"xmlns":43},[45,56518,56519,56585],{},[48,56520,56521,56523,56525,56527,56529,56531,56533,56535,56545,56547,56549,56555,56557,56559,56561,56563,56565,56567,56569,56571,56573,56575,56577,56583],{},[51,56522,2549],{},[95,56524,211],{"stretchy":210},[51,56526,14551],{},[95,56528,1210],{"separator":62},[51,56530,5813],{},[95,56532,220],{"stretchy":210},[95,56534,223],{},[1004,56536,56537,56539],{},[95,56538,5625],{},[1180,56540,56541,56543],{},[51,56542,14551],{},[95,56544,13691],{"mathvariant":494,"lspace":5610,"rspace":5610},[51,56546,2521],{},[95,56548,211],{"stretchy":210},[1180,56550,56551,56553],{},[51,56552,14551],{},[95,56554,13691],{"mathvariant":494,"lspace":5610,"rspace":5610},[95,56556,12680],{},[51,56558,14551],{},[95,56560,1210],{"separator":62},[51,56562,5813],{},[95,56564,220],{"stretchy":210},[51,56566,2549],{},[95,56568,211],{"stretchy":210},[51,56570,14551],{},[95,56572,1210],{"separator":62},[51,56574,5813],{},[95,56576,1210],{"separator":62},[1180,56578,56579,56581],{},[51,56580,14551],{},[95,56582,13691],{"mathvariant":494,"lspace":5610,"rspace":5610},[95,56584,220],{"stretchy":210},[55,56586,56587],{"encoding":57},"r(s,a)=\\sum_{s'}P(s'\\mid s,a)r(s,a,s')",[31,56589,56591,56627,56757],{"className":56590,"ariaHidden":62},[61],[31,56592,56594,56597,56600,56603,56606,56609,56612,56615,56618,56621,56624],{"className":56593},[66],[31,56595],{"className":56596,"style":260},[70],[31,56598,2549],{"className":56599,"style":19759},[75,76],[31,56601,211],{"className":56602},[264],[31,56604,14551],{"className":56605},[75,76],[31,56607,1210],{"className":56608},[1391],[31,56610],{"className":56611,"style":1467},[119],[31,56613,5813],{"className":56614},[75,76],[31,56616,220],{"className":56617},[289],[31,56619],{"className":56620,"style":293},[119],[31,56622,223],{"className":56623},[297],[31,56625],{"className":56626,"style":293},[119],[31,56628,56630,56634,56707,56710,56713,56716,56748,56751,56754],{"className":56629},[66],[31,56631],{"className":56632,"style":56633},[70],"height:1.0516em;vertical-align:-0.2997em;",[31,56635,56637,56640],{"className":56636},[1460],[31,56638,5625],{"className":56639,"style":21372},[1460,1461,1462],[31,56641,56643],{"className":56642},[1039],[31,56644,56646,56699],{"className":56645},[1043,1044],[31,56647,56649,56696],{"className":56648},[1048],[31,56650,56653],{"className":56651,"style":56652},[1052],"height:0.1783em;",[31,56654,56655,56658],{"style":21388},[31,56656],{"className":56657,"style":1061},[1060],[31,56659,56661],{"className":56660},[1065,1066,1067,1068],[31,56662,56664],{"className":56663},[75,1068],[31,56665,56667,56670],{"className":56666},[75,1068],[31,56668,14551],{"className":56669},[75,76,1068],[31,56671,56673],{"className":56672},[1039],[31,56674,56676],{"className":56675},[1043],[31,56677,56679],{"className":56678},[1048],[31,56680,56682],{"className":56681,"style":31478},[1052],[31,56683,56684,56687],{"style":31481},[31,56685],{"className":56686,"style":11769},[1060],[31,56688,56690],{"className":56689},[1065,11773,1308,1068],[31,56691,56693],{"className":56692},[75,1068],[31,56694,13691],{"className":56695},[75,1068],[31,56697,1076],{"className":56698},[1075],[31,56700,56702],{"className":56701},[1048],[31,56703,56705],{"className":56704,"style":21407},[1052],[31,56706],{},[31,56708],{"className":56709,"style":1467},[119],[31,56711,2521],{"className":56712,"style":949},[75,76],[31,56714,211],{"className":56715},[264],[31,56717,56719,56722],{"className":56718},[75],[31,56720,14551],{"className":56721},[75,76],[31,56723,56725],{"className":56724},[1039],[31,56726,56728],{"className":56727},[1043],[31,56729,56731],{"className":56730},[1048],[31,56732,56734],{"className":56733,"style":13732},[1052],[31,56735,56736,56739],{"style":3631},[31,56737],{"className":56738,"style":1061},[1060],[31,56740,56742],{"className":56741},[1065,1066,1067,1068],[31,56743,56745],{"className":56744},[75,1068],[31,56746,13691],{"className":56747},[75,1068],[31,56749],{"className":56750,"style":293},[119],[31,56752,12680],{"className":56753},[297],[31,56755],{"className":56756,"style":293},[119],[31,56758,56760,56763,56766,56769,56772,56775,56778,56781,56784,56787,56790,56793,56796,56799,56802,56834],{"className":56759},[66],[31,56761],{"className":56762,"style":13710},[70],[31,56764,14551],{"className":56765},[75,76],[31,56767,1210],{"className":56768},[1391],[31,56770],{"className":56771,"style":1467},[119],[31,56773,5813],{"className":56774},[75,76],[31,56776,220],{"className":56777},[289],[31,56779,2549],{"className":56780,"style":19759},[75,76],[31,56782,211],{"className":56783},[264],[31,56785,14551],{"className":56786},[75,76],[31,56788,1210],{"className":56789},[1391],[31,56791],{"className":56792,"style":1467},[119],[31,56794,5813],{"className":56795},[75,76],[31,56797,1210],{"className":56798},[1391],[31,56800],{"className":56801,"style":1467},[119],[31,56803,56805,56808],{"className":56804},[75],[31,56806,14551],{"className":56807},[75,76],[31,56809,56811],{"className":56810},[1039],[31,56812,56814],{"className":56813},[1043],[31,56815,56817],{"className":56816},[1048],[31,56818,56820],{"className":56819,"style":13732},[1052],[31,56821,56822,56825],{"style":3631},[31,56823],{"className":56824,"style":1061},[1060],[31,56826,56828],{"className":56827},[1065,1066,1067,1068],[31,56829,56831],{"className":56830},[75,1068],[31,56832,13691],{"className":56833},[75,1068],[31,56835,220],{"className":56836},[289],"，以下所有结论不变。",[52865,56839,56842,56882,57624],{"id":56840,"title":56841},"definition-value","定义 1.3（状态值函数与动作值函数）",[11,56843,56844,56845,56873,56874,56877,56878,56881],{},"在策略 ",[31,56846,56848,56861],{"className":56847,"translate":35},[34],[31,56849,56851],{"className":56850},[39],[41,56852,56853],{"xmlns":43},[45,56854,56855,56859],{},[48,56856,56857],{},[51,56858,2515],{},[55,56860,3485],{"encoding":57},[31,56862,56864],{"className":56863,"ariaHidden":62},[61],[31,56865,56867,56870],{"className":56866},[66],[31,56868],{"className":56869,"style":71},[70],[31,56871,2515],{"className":56872,"style":2666},[75,76]," 下，",[7246,56875,56876],{},"状态值函数","与",[7246,56879,56880],{},"动作值函数","分别定义为",[31,56883,56885],{"className":56884,"translate":35},[411],[31,56886,56888,57041],{"className":56887,"translate":35},[34],[31,56889,56891],{"className":56890},[39],[41,56892,56893],{"xmlns":43,"display":420},[45,56894,56895,57038],{},[5606,56896,56897,56959],{"rowspacing":5608,"columnalign":5609,"columnspacing":5610},[5612,56898,56899,56917],{},[5615,56900,56901],{},[2071,56902,56903],{"scriptlevel":1977,"displaystyle":62},[48,56904,56905,56911,56913,56915],{},[1180,56906,56907,56909],{},[51,56908,43481],{},[51,56910,2515],{},[95,56912,211],{"stretchy":210},[51,56914,14551],{},[95,56916,220],{"stretchy":210},[5615,56918,56919],{},[2071,56920,56921],{"scriptlevel":1977,"displaystyle":62},[48,56922,56923,56925,56927,56933,56957],{},[48,56924],{},[95,56926,223],{},[1004,56928,56929,56931],{},[51,56930,22324],{"mathvariant":14789},[51,56932,2515],{},[48,56934,56935,56937,56943,56945,56951,56953,56955],{},[95,56936,15175],{"fence":62},[1004,56938,56939,56941],{},[51,56940,15297],{},[51,56942,12271],{},[95,56944,12680],{},[1004,56946,56947,56949],{},[51,56948,14551],{},[51,56950,12271],{},[95,56952,223],{},[51,56954,14551],{},[95,56956,15184],{"fence":62},[95,56958,1210],{"separator":62},[5612,56960,56961,56984],{},[5615,56962,56963],{},[2071,56964,56965],{"scriptlevel":1977,"displaystyle":62},[48,56966,56967,56974,56976,56978,56980,56982],{},[1180,56968,56969,56972],{},[51,56970,56971],{},"Q",[51,56973,2515],{},[95,56975,211],{"stretchy":210},[51,56977,14551],{},[95,56979,1210],{"separator":62},[51,56981,5813],{},[95,56983,220],{"stretchy":210},[5615,56985,56986],{},[2071,56987,56988],{"scriptlevel":1977,"displaystyle":62},[48,56989,56990,56992,56994,57000,57036],{},[48,56991],{},[95,56993,223],{},[1004,56995,56996,56998],{},[51,56997,22324],{"mathvariant":14789},[51,56999,2515],{},[48,57001,57002,57004,57010,57012,57018,57020,57022,57024,57030,57032,57034],{},[95,57003,15175],{"fence":62},[1004,57005,57006,57008],{},[51,57007,15297],{},[51,57009,12271],{},[95,57011,12680],{},[1004,57013,57014,57016],{},[51,57015,14551],{},[51,57017,12271],{},[95,57019,223],{},[51,57021,14551],{},[95,57023,1210],{"separator":62},[1004,57025,57026,57028],{},[51,57027,5813],{},[51,57029,12271],{},[95,57031,223],{},[51,57033,5813],{},[95,57035,15184],{"fence":62},[51,57037,495],{"mathvariant":494},[55,57039,57040],{"encoding":57},"\\begin{aligned}\nV^\\pi(s)&=\\mathbb{E}_\\pi\\left[G_t\\mid s_t=s\\right],\\\\\nQ^\\pi(s,a)&=\\mathbb{E}_\\pi\\left[G_t\\mid s_t=s,a_t=a\\right].\n\\end{aligned}",[31,57042,57044],{"className":57043,"ariaHidden":62},[61],[31,57045,57047,57051],{"className":57046},[66],[31,57048],{"className":57049,"style":57050},[70],"height:2.7em;vertical-align:-1.1em;",[31,57052,57054],{"className":57053},[75],[31,57055,57057,57185],{"className":57056},[5606],[31,57058,57060],{"className":57059},[5867],[31,57061,57063,57176],{"className":57062},[1043,1044],[31,57064,57066,57173],{"className":57065},[1048],[31,57067,57070,57117],{"className":57068,"style":57069},[1052],"height:1.6em;",[31,57071,57073,57076],{"style":57072},"top:-3.76em;",[31,57074],{"className":57075,"style":1637},[1060],[31,57077,57079,57108,57111,57114],{"className":57078},[75],[31,57080,57082,57085],{"className":57081},[75],[31,57083,43481],{"className":57084,"style":120},[75,76],[31,57086,57088],{"className":57087},[1039],[31,57089,57091],{"className":57090},[1043],[31,57092,57094],{"className":57093},[1048],[31,57095,57097],{"className":57096,"style":3987},[1052],[31,57098,57099,57102],{"style":1330},[31,57100],{"className":57101,"style":1061},[1060],[31,57103,57105],{"className":57104},[1065,1066,1067,1068],[31,57106,2515],{"className":57107,"style":2666},[75,76,1068],[31,57109,211],{"className":57110},[264],[31,57112,14551],{"className":57113},[75,76],[31,57115,220],{"className":57116},[289],[31,57118,57120,57123],{"style":57119},"top:-2.26em;",[31,57121],{"className":57122,"style":1637},[1060],[31,57124,57126,57155,57158,57161,57164,57167,57170],{"className":57125},[75],[31,57127,57129,57132],{"className":57128},[75],[31,57130,56971],{"className":57131},[75,76],[31,57133,57135],{"className":57134},[1039],[31,57136,57138],{"className":57137},[1043],[31,57139,57141],{"className":57140},[1048],[31,57142,57144],{"className":57143,"style":3987},[1052],[31,57145,57146,57149],{"style":1330},[31,57147],{"className":57148,"style":1061},[1060],[31,57150,57152],{"className":57151},[1065,1066,1067,1068],[31,57153,2515],{"className":57154,"style":2666},[75,76,1068],[31,57156,211],{"className":57157},[264],[31,57159,14551],{"className":57160},[75,76],[31,57162,1210],{"className":57163},[1391],[31,57165],{"className":57166,"style":1467},[119],[31,57168,5813],{"className":57169},[75,76],[31,57171,220],{"className":57172},[289],[31,57174,1076],{"className":57175},[1075],[31,57177,57179],{"className":57178},[1048],[31,57180,57183],{"className":57181,"style":57182},[1052],"height:1.1em;",[31,57184],{},[31,57186,57188],{"className":57187},[6062],[31,57189,57191,57616],{"className":57190},[1043,1044],[31,57192,57194,57613],{"className":57193},[1048],[31,57195,57197,57376],{"className":57196,"style":57069},[1052],[31,57198,57199,57202],{"style":57072},[31,57200],{"className":57201,"style":1637},[1060],[31,57203,57205,57208,57211,57214,57217,57257,57260,57370,57373],{"className":57204},[75],[31,57206],{"className":57207},[75],[31,57209],{"className":57210,"style":293},[119],[31,57212,223],{"className":57213},[297],[31,57215],{"className":57216,"style":293},[119],[31,57218,57220,57223],{"className":57219},[75],[31,57221,22324],{"className":57222},[75,14947],[31,57224,57226],{"className":57225},[1039],[31,57227,57229,57249],{"className":57228},[1043,1044],[31,57230,57232,57246],{"className":57231},[1048],[31,57233,57235],{"className":57234,"style":3230},[1052],[31,57236,57237,57240],{"style":2807},[31,57238],{"className":57239,"style":1061},[1060],[31,57241,57243],{"className":57242},[1065,1066,1067,1068],[31,57244,2515],{"className":57245,"style":2666},[75,76,1068],[31,57247,1076],{"className":57248},[1075],[31,57250,57252],{"className":57251},[1048],[31,57253,57255],{"className":57254,"style":2050},[1052],[31,57256],{},[31,57258],{"className":57259,"style":1467},[119],[31,57261,57263,57266,57306,57309,57312,57315,57355,57358,57361,57364,57367],{"className":57262},[3860],[31,57264,15175],{"className":57265,"style":3868},[264,3867],[31,57267,57269,57272],{"className":57268},[75],[31,57270,15297],{"className":57271},[75,76],[31,57273,57275],{"className":57274},[1039],[31,57276,57278,57298],{"className":57277},[1043,1044],[31,57279,57281,57295],{"className":57280},[1048],[31,57282,57284],{"className":57283,"style":19220},[1052],[31,57285,57286,57289],{"style":2807},[31,57287],{"className":57288,"style":1061},[1060],[31,57290,57292],{"className":57291},[1065,1066,1067,1068],[31,57293,12271],{"className":57294},[75,76,1068],[31,57296,1076],{"className":57297},[1075],[31,57299,57301],{"className":57300},[1048],[31,57302,57304],{"className":57303,"style":2050},[1052],[31,57305],{},[31,57307],{"className":57308,"style":293},[119],[31,57310,12680],{"className":57311},[297],[31,57313],{"className":57314,"style":293},[119],[31,57316,57318,57321],{"className":57317},[75],[31,57319,14551],{"className":57320},[75,76],[31,57322,57324],{"className":57323},[1039],[31,57325,57327,57347],{"className":57326},[1043,1044],[31,57328,57330,57344],{"className":57329},[1048],[31,57331,57333],{"className":57332,"style":1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由三角不等式与几何级数求和，",[31,58261,58263],{"className":58262,"translate":35},[411],[31,58264,58266,58410],{"className":58265,"translate":35},[34],[31,58267,58269],{"className":58268},[39],[41,58270,58271],{"xmlns":43,"display":420},[45,58272,58273,58407],{},[5606,58274,58275,58339],{"rowspacing":5608,"columnalign":5609,"columnspacing":5610},[5612,58276,58277,58293],{},[5615,58278,58279],{},[2071,58280,58281],{"scriptlevel":1977,"displaystyle":62},[48,58282,58283,58285,58291],{},[51,58284,12680],{"mathvariant":494},[1004,58286,58287,58289],{},[51,58288,15297],{},[51,58290,12271],{},[51,58292,12680],{"mathvariant":494},[5615,58294,58295],{},[2071,58296,58297],{"scriptlevel":1977,"displaystyle":62},[48,58298,58299,58301,58303,58317,58323,58325,58337],{},[48,58300],{},[95,58302,347],{},[42429,58304,58305,58307,58315],{},[95,58306,5625],{},[48,58308,58309,58311,58313],{},[51,58310,52065],{},[95,58312,223],{},[91,58314,1977],{},[51,58316,4587],{"mathvariant":494},[1180,58318,58319,58321],{},[51,58320,53298],{},[51,58322,52065],{},[51,58324,12680],{"mathvariant":494},[1004,58326,58327,58329],{},[51,58328,2549],{},[48,58330,58331,58333,58335],{},[51,58332,12271],{},[95,58334,434],{},[51,58336,52065],{},[51,58338,12680],{"mathvariant":494},[5612,58340,58341,58347],{},[5615,58342,58343],{},[2071,58344,58345],{"scriptlevel":1977,"displaystyle":62},[48,58346],{},[5615,58348,58349],{},[2071,58350,58351],{"scriptlevel":1977,"displaystyle":62},[48,58352,58353,58355,58357,58365,58379,58385,58387,58405],{},[48,58354],{},[95,58356,347],{},[1004,58358,58359,58361,58363],{},[51,58360,14790],{},[51,58362,10154],{},[95,58364,1540],{},[42429,58366,58367,58369,58377],{},[95,58368,5625],{},[48,58370,58371,58373,58375],{},[51,58372,52065],{},[95,58374,223],{},[91,58376,1977],{},[51,58378,4587],{"mathvariant":494},[1180,58380,58381,58383],{},[51,58382,53298],{},[51,58384,52065],{},[95,58386,223],{},[2074,58388,58389,58397],{},[1004,58390,58391,58393,58395],{},[51,58392,14790],{},[51,58394,10154],{},[95,58396,1540],{},[48,58398,58399,58401,58403],{},[91,58400,205],{},[95,58402,97],{},[51,58404,53298],{},[51,58406,495],{"mathvariant":494},[55,58408,58409],{"encoding":57},"\\begin{aligned}\n|G_t|\n&\\le\\sum_{k=0}^{\\infty}\\gamma^k|r_{t+k}|\\\\\n&\\le R_{\\max}\\sum_{k=0}^{\\infty}\\gamma^k\n=\\frac{R_{\\max}}{1-\\gamma}.\n\\end{aligned}",[31,58411,58413],{"className":58412,"ariaHidden":62},[61],[31,58414,58416,58420],{"className":58415},[66],[31,58417],{"className":58418,"style":58419},[70],"height:6.207em;vertical-align:-2.8535em;",[31,58421,58423],{"className":58422},[75],[31,58424,58426,58516],{"className":58425},[5606],[31,58427,58429],{"className":58428},[5867],[31,58430,58432,58507],{"className":58431},[1043,1044],[31,58433,58435,58504],{"className":58434},[1048],[31,58436,58439,58495],{"className":58437,"style":58438},[1052],"height:3.3535em;",[31,58440,58442,58446],{"style":58441},"top:-5.3535em;",[31,58443],{"className":58444,"style":58445},[1060],"height:3.6514em;",[31,58447,58449,58452,58492],{"className":58448},[75],[31,58450,12680],{"className":58451},[75],[31,58453,58455,58458],{"className":58454},[75],[31,58456,15297],{"className":58457},[75,76],[31,58459,58461],{"className":58460},[1039],[31,58462,58464,58484],{"className":58463},[1043,1044],[31,58465,58467,58481],{"className":58466},[1048],[31,58468,58470],{"className":58469,"style":19220},[1052],[31,58471,58472,58475],{"style":2807},[31,58473],{"className":58474,"style":1061},[1060],[31,58476,58478],{"className":58477},[1065,1066,1067,1068],[31,58479,12271],{"className":58480},[75,76,1068],[31,58482,1076],{"className":58483},[1075],[31,58485,58487],{"className":58486},[1048],[31,58488,58490],{"className":58489,"style":2050},[1052],[31,58491],{},[31,58493,12680],{"className":58494},[75],[31,58496,58498,58501],{"style":58497},"top:-2.1em;",[31,58499],{"className":58500,"style":58445},[1060],[31,58502],{"className":58503},[75],[31,58505,1076],{"className":58506},[1075],[31,58508,58510],{"className":58509},[1048],[31,58511,58514],{"className":58512,"style":58513},[1052],"height:2.8535em;",[31,58515],{},[31,58517,58519],{"className":58518},[6062],[31,58520,58522,59014],{"className":58521},[1043,1044],[31,58523,58525,59011],{"className":58524},[1048],[31,58526,58528,58702],{"className":58527,"style":58438},[1052],[31,58529,58530,58533],{"style":58441},[31,58531],{"className":58532,"style":58445},[1060],[31,58534,58536,58539,58542,58545,58548,58615,58618,58647,58650,58699],{"className":58535},[75],[31,58537],{"className":58538},[75],[31,58540],{"className":58541,"style":293},[119],[31,58543,347],{"className":58544},[297],[31,58546],{"className":58547,"style":293},[119],[31,58549,58551],{"className":58550},[1460,1620],[31,58552,58554,58607],{"className":58553},[1043,1044],[31,58555,58557,58604],{"className":58556},[1048],[31,58558,58560,58580,58590],{"className":58559,"style":42702},[1052],[31,58561,58562,58565],{"style":56004},[31,58563],{"className":58564,"style":5907},[1060],[31,58566,58568],{"className":58567},[1065,1066,1067,1068],[31,58569,58571,58574,58577],{"className":58570},[75,1068],[31,58572,52065],{"className":58573,"style":52166},[75,76,1068],[31,58575,223],{"className":58576},[297,1068],[31,58578,1977],{"className":58579},[75,1068],[31,58581,58582,58585],{"style":5919},[31,58583],{"className":58584,"style":5907},[1060],[31,58586,58587],{},[31,58588,5625],{"className":58589},[1460,1461,1687],[31,58591,58592,58595],{"style":42735},[31,58593],{"className":58594,"style":5907},[1060],[31,58596,58598],{"className":58597},[1065,1066,1067,1068],[31,58599,58601],{"className":58600},[75,1068],[31,58602,4587],{"className":58603},[75,1068],[31,58605,1076],{"className":58606},[1075],[31,58608,58610],{"className":58609},[1048],[31,58611,58613],{"className":58612,"style":56056},[1052],[31,58614],{},[31,58616],{"className":58617,"style":1467},[119],[31,58619,58621,58624],{"className":58620},[75],[31,58622,53298],{"className":58623,"style":24157},[75,76],[31,58625,58627],{"className":58626},[1039],[31,58628,58630],{"className":58629},[1043],[31,58631,58633],{"className":58632},[1048],[31,58634,58636],{"className":58635,"style":29642},[1052],[31,58637,58638,58641],{"style":1330},[31,58639],{"className":58640,"style":1061},[1060],[31,58642,58644],{"className":58643},[1065,1066,1067,1068],[31,58645,52065],{"className":58646,"style":52166},[75,76,1068],[31,58648,12680],{"className":58649},[75],[31,58651,58653,58656],{"className":58652},[75],[31,58654,2549],{"className":58655,"style":19759},[75,76],[31,58657,58659],{"className":58658},[1039],[31,58660,58662,58691],{"className":58661},[1043,1044],[31,58663,58665,58688],{"className":58664},[1048],[31,58666,58668],{"className":58667,"style":1053},[1052],[31,58669,58670,58673],{"style":26726},[31,58671],{"className":58672,"style":1061},[1060],[31,58674,58676],{"className":58675},[1065,1066,1067,1068],[31,58677,58679,58682,58685],{"className":58678},[75,1068],[31,58680,12271],{"className":58681},[75,76,1068],[31,58683,434],{"className":58684},[124,1068],[31,58686,52065],{"className":58687,"style":52166},[75,76,1068],[31,58689,1076],{"className":58690},[1075],[31,58692,58694],{"className":58693},[1048],[31,58695,58697],{"className":58696,"style":25082},[1052],[31,58698],{},[31,58700,12680],{"className":58701},[75],[31,58703,58704,58707],{"style":58497},[31,58705],{"className":58706,"style":58445},[1060],[31,58708,58710,58713,58716,58719,58722,58774,58777,58844,58847,58876,58879,58882,58885,59008],{"className":58709},[75],[31,58711],{"className":58712},[75],[31,58714],{"className":58715,"style":293},[119],[31,58717,347],{"className":58718},[297],[31,58720],{"className":58721,"style":293},[119],[31,58723,58725,58728],{"className":58724},[75],[31,58726,14790],{"className":58727,"style":54434},[75,76],[31,58729,58731],{"className":58730},[1039],[31,58732,58734,58766],{"className":58733},[1043,1044],[31,58735,58737,58763],{"className":58736},[1048],[31,58738,58740],{"className":58739,"style":3230},[1052],[31,58741,58742,58745],{"style":54449},[31,58743],{"className":58744,"style":1061},[1060],[31,58746,58748],{"className":58747},[1065,1066,1067,1068],[31,58749,58751],{"className":58750},[75,1068],[31,58752,58754,58757,58760],{"className":58753},[1460,1068],[31,58755,16396],{"className":58756},[1068],[31,58758,5813],{"className":58759},[1068],[31,58761,53],{"className":58762},[1068],[31,58764,1076],{"className":58765},[1075],[31,58767,58769],{"className":58768},[1048],[31,58770,58772],{"className":58771,"style":2050},[1052],[31,58773],{},[31,58775],{"className":58776,"style":1467},[119],[31,58778,58780],{"className":58779},[1460,1620],[31,58781,58783,58836],{"className":58782},[1043,1044],[31,58784,58786,58833],{"className":58785},[1048],[31,58787,58789,58809,58819],{"className":58788,"style":42702},[1052],[31,58790,58791,58794],{"style":56004},[31,58792],{"className":58793,"style":5907},[1060],[31,58795,58797],{"className":58796},[1065,1066,1067,1068],[31,58798,58800,58803,58806],{"className":58799},[75,1068],[31,58801,52065],{"className":58802,"style":52166},[75,76,1068],[31,58804,223],{"className":58805},[297,1068],[31,58807,1977],{"className":58808},[75,1068],[31,58810,58811,58814],{"style":5919},[31,58812],{"className":58813,"style":5907},[1060],[31,58815,58816],{},[31,58817,5625],{"className":58818},[1460,1461,1687],[31,58820,58821,58824],{"style":42735},[31,58822],{"className":58823,"style":5907},[1060],[31,58825,58827],{"className":58826},[1065,1066,1067,1068],[31,58828,58830],{"className":58829},[75,1068],[31,58831,4587],{"className":58832},[75,1068],[31,58834,1076],{"className":58835},[1075],[31,58837,58839],{"className":58838},[1048],[31,58840,58842],{"className":58841,"style":56056},[1052],[31,58843],{},[31,58845],{"className":58846,"style":1467},[119],[31,58848,58850,58853],{"className":58849},[75],[31,58851,53298],{"className":58852,"style":24157},[75,76],[31,58854,58856],{"className":58855},[1039],[31,58857,58859],{"className":58858},[1043],[31,58860,58862],{"className":58861},[1048],[31,58863,58865],{"className":58864,"style":29642},[1052],[31,58866,58867,58870],{"style":1330},[31,58868],{"className":58869,"style":1061},[1060],[31,58871,58873],{"className":58872},[1065,1066,1067,1068],[31,58874,52065],{"className":58875,"style":52166},[75,76,1068],[31,58877],{"className":58878,"style":293},[119],[31,58880,223],{"className":58881},[297],[31,58883],{"className":58884,"style":293},[119],[31,58886,58888,58891,59005],{"className":58887},[75],[31,58889],{"className":58890},[264,2139],[31,58892,58894],{"className":58893},[2074],[31,58895,58897,58997],{"className":58896},[1043,1044],[31,58898,58900,58994],{"className":58899},[1048],[31,58901,58903,58926,58934],{"className":58902,"style":58146},[1052],[31,58904,58905,58908],{"style":5090},[31,58906],{"className":58907,"style":1637},[1060],[31,58909,58911,58914,58917,58920,58923],{"className":58910},[75],[31,58912,205],{"className":58913},[75],[31,58915],{"className":58916,"style":120},[119],[31,58918,97],{"className":58919},[124],[31,58921],{"className":58922,"style":120},[119],[31,58924,53298],{"className":58925,"style":24157},[75,76],[31,58927,58928,58931],{"style":2170},[31,58929],{"className":58930,"style":1637},[1060],[31,58932],{"className":58933,"style":2178},[2177],[31,58935,58936,58939],{"style":5110},[31,58937],{"className":58938,"style":1637},[1060],[31,58940,58942],{"className":58941},[75],[31,58943,58945,58948],{"className":58944},[75],[31,58946,14790],{"className":58947,"style":54434},[75,76],[31,58949,58951],{"className":58950},[1039],[31,58952,58954,58986],{"className":58953},[1043,1044],[31,58955,58957,58983],{"className":58956},[1048],[31,58958,58960],{"className":58959,"style":3230},[1052],[31,58961,58962,58965],{"style":54449},[31,58963],{"className":58964,"style":1061},[1060],[31,58966,58968],{"className":58967},[1065,1066,1067,1068],[31,58969,58971],{"className":58970},[75,1068],[31,58972,58974,58977,58980],{"className":58973},[1460,1068],[31,58975,16396],{"className":58976},[1068],[31,58978,5813],{"className":58979},[1068],[31,58981,53],{"className":58982},[1068],[31,58984,1076],{"className":58985},[1075],[31,58987,58989],{"className":58988},[1048],[31,58990,58992],{"className":58991,"style":2050},[1052],[31,58993],{},[31,58995,1076],{"className":58996},[1075],[31,58998,59000],{"className":58999},[1048],[31,59001,59003],{"className":59002,"style":41317},[1052],[31,59004],{},[31,59006],{"className":59007},[289,2139],[31,59009,495],{"className":59010},[75],[31,59012,1076],{"className":59013},[1075],[31,59015,59017],{"className":59016},[1048],[31,59018,59020],{"className":59019,"style":58513},[1052],[31,59021],{},[11,59023,59024,59025,18171,59084,59144],{},"因此 ",[31,59026,59028,59046],{"className":59027,"translate":35},[34],[31,59029,59031],{"className":59030},[39],[41,59032,59033],{"xmlns":43},[45,59034,59035,59043],{},[48,59036,59037],{},[1180,59038,59039,59041],{},[51,59040,43481],{},[51,59042,2515],{},[55,59044,59045],{"encoding":57},"V^\\pi",[31,59047,59049],{"className":59048,"ariaHidden":62},[61],[31,59050,59052,59055],{"className":59051},[66],[31,59053],{"className":59054,"style":945},[70],[31,59056,59058,59061],{"className":59057},[75],[31,59059,43481],{"className":59060,"style":120},[75,76],[31,59062,59064],{"className":59063},[1039],[31,59065,59067],{"className":59066},[1043],[31,59068,59070],{"className":59069},[1048],[31,59071,59073],{"className":59072,"style":3628},[1052],[31,59074,59075,59078],{"style":3631},[31,59076],{"className":59077,"style":1061},[1060],[31,59079,59081],{"className":59080},[1065,1066,1067,1068],[31,59082,2515],{"className":59083,"style":2666},[75,76,1068],[31,59085,59087,59105],{"className":59086,"translate":35},[34],[31,59088,59090],{"className":59089},[39],[41,59091,59092],{"xmlns":43},[45,59093,59094,59102],{},[48,59095,59096],{},[1180,59097,59098,59100],{},[51,59099,56971],{},[51,59101,2515],{},[55,59103,59104],{"encoding":57},"Q^\\pi",[31,59106,59108],{"className":59107,"ariaHidden":62},[61],[31,59109,59111,59115],{"className":59110},[66],[31,59112],{"className":59113,"style":59114},[70],"height:0.8778em;vertical-align:-0.1944em;",[31,59116,59118,59121],{"className":59117},[75],[31,59119,56971],{"className":59120},[75,76],[31,59122,59124],{"className":59123},[1039],[31,59125,59127],{"className":59126},[1043],[31,59128,59130],{"className":59129},[1048],[31,59131,59133],{"className":59132,"style":3628},[1052],[31,59134,59135,59138],{"style":3631},[31,59136],{"className":59137,"style":1061},[1060],[31,59139,59141],{"className":59140},[1065,1066,1067,1068],[31,59142,2515],{"className":59143,"style":2666},[75,76,1068]," 均为有界函数。证毕。",[11,59146,59147],{},"有界性并不是一个无关紧要的技术条件。它保证值函数属于一个有界函数空间，而该空间在 sup 范数下是完备的。后文的压缩映射论证正是在这个空间上进行的。",[52860,59149,59151],{"id":59150},"_13-两个基本恒等式","1.3 两个基本恒等式",[52865,59153,59156,59188,59451,59454],{"id":59154,"title":59155},"proposition-basic-identities","命题 1.5（值函数的基本恒等式）",[11,59157,59158,59159,59187],{},"对任意策略 ",[31,59160,59162,59175],{"className":59161,"translate":35},[34],[31,59163,59165],{"className":59164},[39],[41,59166,59167],{"xmlns":43},[45,59168,59169,59173],{},[48,59170,59171],{},[51,59172,2515],{},[55,59174,3485],{"encoding":57},[31,59176,59178],{"className":59177,"ariaHidden":62},[61],[31,59179,59181,59184],{"className":59180},[66],[31,59182],{"className":59183,"style":71},[70],[31,59185,2515],{"className":59186,"style":2666},[75,76],"，有",[31,59189,59191],{"className":59190,"translate":35},[411],[31,59192,59194,59256],{"className":59193,"translate":35},[34],[31,59195,59197],{"className":59196},[39],[41,59198,59199],{"xmlns":43,"display":420},[45,59200,59201,59253],{},[48,59202,59203,59209,59211,59213,59215,59217,59223,59225,59227,59229,59231,59233,59235,59241,59243,59245,59247,59249,59251],{},[1180,59204,59205,59207],{},[51,59206,43481],{},[51,59208,2515],{},[95,59210,211],{"stretchy":210},[51,59212,14551],{},[95,59214,220],{"stretchy":210},[95,59216,223],{},[1531,59218,59219,59221],{},[95,59220,5625],{},[51,59222,5813],{},[51,59224,2515],{},[95,59226,211],{"stretchy":210},[51,59228,5813],{},[95,59230,12680],{},[51,59232,14551],{},[95,59234,220],{"stretchy":210},[1180,59236,59237,59239],{},[51,59238,56971],{},[51,59240,2515],{},[95,59242,211],{"stretchy":210},[51,59244,14551],{},[95,59246,1210],{"separator":62},[51,59248,5813],{},[95,59250,220],{"stretchy":210},[95,59252,1210],{"separator":62},[55,59254,59255],{"encoding":57},"V^\\pi(s)=\\sum_{a}\\pi(a\\mid s)Q^\\pi(s,a),",[31,59257,59259,59312,59389],{"className":59258,"ariaHidden":62},[61],[31,59260,59262,59265,59294,59297,59300,59303,59306,59309],{"className":59261},[66],[31,59263],{"className":59264,"style":260},[70],[31,59266,59268,59271],{"className":59267},[75],[31,59269,43481],{"className":59270,"style":120},[75,76],[31,59272,59274],{"className":59273},[1039],[31,59275,59277],{"className":59276},[1043],[31,59278,59280],{"className":59279},[1048],[31,59281,59283],{"className":59282,"style":3987},[1052],[31,59284,59285,59288],{"style":1330},[31,59286],{"className":59287,"style":1061},[1060],[31,59289,59291],{"className":59290},[1065,1066,1067,1068],[31,59292,2515],{"className":59293,"style":2666},[75,76,1068],[31,59295,211],{"className":59296},[264],[31,59298,14551],{"className":59299},[75,76],[31,59301,220],{"className":59302},[289],[31,59304],{"className":59305,"style":293},[119],[31,59307,223],{"className":59308},[297],[31,59310],{"className":59311,"style":293},[119],[31,59313,59315,59319,59368,59371,59374,59377,59380,59383,59386],{"className":59314},[66],[31,59316],{"className":59317,"style":59318},[70],"height:2.3em;vertical-align:-1.25em;",[31,59320,59322],{"className":59321},[1460,1620],[31,59323,59325,59359],{"className":59324},[1043,1044],[31,59326,59328,59356],{"className":59327},[1048],[31,59329,59331,59346],{"className":59330,"style":5900},[1052],[31,59332,59334,59337],{"style":59333},"top:-1.9em;margin-left:0em;",[31,59335],{"className":59336,"style":5907},[1060],[31,59338,59340],{"className":59339},[1065,1066,1067,1068],[31,59341,59343],{"className":59342},[75,1068],[31,59344,5813],{"className":59345},[75,76,1068],[31,59347,59348,59351],{"style":5919},[31,59349],{"className":59350,"style":5907},[1060],[31,59352,59353],{},[31,59354,5625],{"className":59355},[1460,1461,1687],[31,59357,1076],{"className":59358},[1075],[31,59360,59362],{"className":59361},[1048],[31,59363,59366],{"className":59364,"style":59365},[1052],"height:1.25em;",[31,59367],{},[31,59369],{"className":59370,"style":1467},[119],[31,59372,2515],{"className":59373,"style":2666},[75,76],[31,59375,211],{"className":59376},[264],[31,59378,5813],{"className":59379},[75,76],[31,59381],{"className":59382,"style":293},[119],[31,59384,12680],{"className":59385},[297],[31,59387],{"className":59388,"style":293},[119],[31,59390,59392,59395,59398,59401,59430,59433,59436,59439,59442,59445,59448],{"className":59391},[66],[31,59393],{"className":59394,"style":260},[70],[31,59396,14551],{"className":59397},[75,76],[31,59399,220],{"className":59400},[289],[31,59402,59404,59407],{"className":59403},[75],[31,59405,56971],{"className":59406},[75,76],[31,59408,59410],{"className":59409},[1039],[31,59411,59413],{"className":59412},[1043],[31,59414,59416],{"className":59415},[1048],[31,59417,59419],{"className":59418,"style":3987},[1052],[31,59420,59421,59424],{"style":1330},[31,59422],{"className":59423,"style":1061},[1060],[31,59425,59427],{"className":59426},[1065,1066,1067,1068],[31,59428,2515],{"className":59429,"style":2666},[75,76,1068],[31,59431,211],{"className":59432},[264],[31,59434,14551],{"className":59435},[75,76],[31,59437,1210],{"className":59438},[1391],[31,59440],{"className":59441,"style":1467},[119],[31,59443,5813],{"className":59444},[75,76],[31,59446,220],{"className":59447},[289],[31,59449,1210],{"className":59450},[1391],[11,59452,59453],{},"以及",[31,59455,59457],{"className":59456,"translate":35},[411],[31,59458,59460,59604],{"className":59459,"translate":35},[34],[31,59461,59463],{"className":59462},[39],[41,59464,59465],{"xmlns":43,"display":420},[45,59466,59467,59601],{},[5606,59468,59469,59533],{"rowspacing":5608,"columnalign":5609,"columnspacing":5610},[5612,59470,59471,59493],{},[5615,59472,59473],{},[2071,59474,59475],{"scriptlevel":1977,"displaystyle":62},[48,59476,59477,59483,59485,59487,59489,59491],{},[1180,59478,59479,59481],{},[51,59480,56971],{},[51,59482,2515],{},[95,59484,211],{"stretchy":210},[51,59486,14551],{},[95,59488,1210],{"separator":62},[51,59490,5813],{},[95,59492,220],{"stretchy":210},[5615,59494,59495],{},[2071,59496,59497],{"scriptlevel":1977,"displaystyle":62},[48,59498,59499,59501,59503,59513,59515,59517,59523,59525,59527,59529,59531],{},[48,59500],{},[95,59502,223],{},[1531,59504,59505,59507],{},[95,59506,5625],{},[1180,59508,59509,59511],{},[51,59510,14551],{},[95,59512,13691],{"mathvariant":494,"lspace":5610,"rspace":5610},[51,59514,2521],{},[95,59516,211],{"stretchy":210},[1180,59518,59519,59521],{},[51,59520,14551],{},[95,59522,13691],{"mathvariant":494,"lspace":5610,"rspace":5610},[95,59524,12680],{},[51,59526,14551],{},[95,59528,1210],{"separator":62},[51,59530,5813],{},[95,59532,220],{"stretchy":210},[5612,59534,59535,59541],{},[5615,59536,59537],{},[2071,59538,59539],{"scriptlevel":1977,"displaystyle":62},[48,59540],{},[5615,59542,59543],{},[2071,59544,59545],{"scriptlevel":1977,"displaystyle":62},[48,59546,59547,59549,59551,59553,59599],{},[48,59548],{},[119,59550],{"width":1584},[95,59552,2528],{},[48,59554,59555,59557,59559,59561,59563,59565,59567,59569,59575,59577,59579,59581,59587,59589,59595,59597],{},[95,59556,15175],{"fence":62},[51,59558,2549],{},[95,59560,211],{"stretchy":210},[51,59562,14551],{},[95,59564,1210],{"separator":62},[51,59566,5813],{},[95,59568,1210],{"separator":62},[1180,59570,59571,59573],{},[51,59572,14551],{},[95,59574,13691],{"mathvariant":494,"lspace":5610,"rspace":5610},[95,59576,220],{"stretchy":210},[95,59578,434],{},[51,59580,53298],{},[1180,59582,59583,59585],{},[51,59584,43481],{},[51,59586,2515],{},[95,59588,211],{"stretchy":210},[1180,59590,59591,59593],{},[51,59592,14551],{},[95,59594,13691],{"mathvariant":494,"lspace":5610,"rspace":5610},[95,59596,220],{"stretchy":210},[95,59598,15184],{"fence":62},[51,59600,495],{"mathvariant":494},[55,59602,59603],{"encoding":57},"\\begin{aligned}\nQ^\\pi(s,a)\n&=\\sum_{s'}P(s'\\mid s,a)\\\\\n&\\quad\\times\\left[r(s,a,s')+\\gamma V^\\pi(s')\\right].\n\\end{aligned}",[31,59605,59607],{"className":59606,"ariaHidden":62},[61],[31,59608,59610,59614],{"className":59609},[66],[31,59611],{"className":59612,"style":59613},[70],"height:3.844em;vertical-align:-1.672em;",[31,59615,59617],{"className":59616},[75],[31,59618,59620,59710],{"className":59619},[5606],[31,59621,59623],{"className":59622},[5867],[31,59624,59626,59701],{"className":59625},[1043,1044],[31,59627,59629,59698],{"className":59628},[1048],[31,59630,59633,59689],{"className":59631,"style":59632},[1052],"height:2.172em;",[31,59634,59636,59639],{"style":59635},"top:-4.172em;",[31,59637],{"className":59638,"style":5907},[1060],[31,59640,59642,59671,59674,59677,59680,59683,59686],{"className":59641},[75],[31,59643,59645,59648],{"className":59644},[75],[31,59646,56971],{"className":59647},[75,76],[31,59649,59651],{"className":59650},[1039],[31,59652,59654],{"className":59653},[1043],[31,59655,59657],{"className":59656},[1048],[31,59658,59660],{"className":59659,"style":3987},[1052],[31,59661,59662,59665],{"style":1330},[31,59663],{"className":59664,"style":1061},[1060],[31,59666,59668],{"className":59667},[1065,1066,1067,1068],[31,59669,2515],{"className":59670,"style":2666},[75,76,1068],[31,59672,211],{"className":59673},[264],[31,59675,14551],{"className":59676},[75,76],[31,59678,1210],{"className":59679},[1391],[31,59681],{"className":59682,"style":1467},[119],[31,59684,5813],{"className":59685},[75,76],[31,59687,220],{"className":59688},[289],[31,59690,59692,59695],{"style":59691},"top:-1.738em;",[31,59693],{"className":59694,"style":5907},[1060],[31,59696],{"className":59697},[75],[31,59699,1076],{"className":59700},[1075],[31,59702,59704],{"className":59703},[1048],[31,59705,59708],{"className":59706,"style":59707},[1052],"height:1.672em;",[31,59709],{},[31,59711,59713],{"className":59712},[6062],[31,59714,59716,60064],{"className":59715},[1043,1044],[31,59717,59719,60061],{"className":59718},[1048],[31,59720,59722,59885],{"className":59721,"style":59632},[1052],[31,59723,59724,59727],{"style":59635},[31,59725],{"className":59726,"style":5907},[1060],[31,59728,59730,59733,59736,59739,59742,59820,59823,59826,59829,59861,59864,59867,59870,59873,59876,59879,59882],{"className":59729},[75],[31,59731],{"className":59732},[75],[31,59734],{"className":59735,"style":293},[119],[31,59737,223],{"className":59738},[297],[31,59740],{"className":59741,"style":293},[119],[31,59743,59745],{"className":59744},[1460,1620],[31,59746,59748,59811],{"className":59747},[1043,1044],[31,59749,59751,59808],{"className":59750},[1048],[31,59752,59754,59798],{"className":59753,"style":5900},[1052],[31,59755,59757,59760],{"style":59756},"top:-1.856em;margin-left:0em;",[31,59758],{"className":59759,"style":5907},[1060],[31,59761,59763],{"className":59762},[1065,1066,1067,1068],[31,59764,59766],{"className":59765},[75,1068],[31,59767,59769,59772],{"className":59768},[75,1068],[31,59770,14551],{"className":59771},[75,76,1068],[31,59773,59775],{"className":59774},[1039],[31,59776,59778],{"className":59777},[1043],[31,59779,59781],{"className":59780},[1048],[31,59782,59784],{"className":59783,"style":31478},[1052],[31,59785,59786,59789],{"style":31481},[31,59787],{"className":59788,"style":11769},[1060],[31,59790,59792],{"className":59791},[1065,11773,1308,1068],[31,59793,59795],{"className":59794},[75,1068],[31,59796,13691],{"className":59797},[75,1068],[31,59799,59800,59803],{"style":5919},[31,59801],{"className":59802,"style":5907},[1060],[31,59804,59805],{},[31,59806,5625],{"className":59807},[1460,1461,1687],[31,59809,1076],{"className":59810},[1075],[31,59812,59814],{"className":59813},[1048],[31,59815,59818],{"className":59816,"style":59817},[1052],"height:1.294em;",[31,59819],{},[31,59821],{"className":59822,"style":1467},[119],[31,59824,2521],{"className":59825,"style":949},[75,76],[31,59827,211],{"className":59828},[264],[31,59830,59832,59835],{"className":59831},[75],[31,59833,14551],{"className":59834},[75,76],[31,59836,59838],{"className":59837},[1039],[31,59839,59841],{"className":59840},[1043],[31,59842,59844],{"className":59843},[1048],[31,59845,59847],{"className":59846,"style":14017},[1052],[31,59848,59849,59852],{"style":1330},[31,59850],{"className":59851,"style":1061},[1060],[31,59853,59855],{"className":59854},[1065,1066,1067,1068],[31,59856,59858],{"className":59857},[75,1068],[31,59859,13691],{"className":59860},[75,1068],[31,59862],{"className":59863,"style":293},[119],[31,59865,12680],{"className":59866},[297],[31,59868],{"className":59869,"style":293},[119],[31,59871,14551],{"className":59872},[75,76],[31,59874,1210],{"className":59875},[1391],[31,59877],{"className":59878,"style":1467},[119],[31,59880,5813],{"className":59881},[75,76],[31,59883,220],{"className":59884},[289],[31,59886,59887,59890],{"style":59691},[31,59888],{"className":59889,"style":5907},[1060],[31,59891,59893,59896,59899,59902,59905,59908,60055,60058],{"className":59892},[75],[31,59894],{"className":59895},[75]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第一式由全期望公式直接得到。对于第二式，利用 ",[31,60078,60080,60122],{"className":60079,"translate":35},[34],[31,60081,60083],{"className":60082},[39],[41,60084,60085],{"xmlns":43},[45,60086,60087,60119],{},[48,60088,60089,60095,60097,60103,60105,60107],{},[1004,60090,60091,60093],{},[51,60092,15297],{},[51,60094,12271],{},[95,60096,223],{},[1004,60098,60099,60101],{},[51,60100,2549],{},[51,60102,12271],{},[95,60104,434],{},[51,60106,53298],{},[1004,60108,60109,60111],{},[51,60110,15297],{},[48,60112,60113,60115,60117],{},[51,60114,12271],{},[95,60116,434],{},[91,60118,205],{},[55,60120,60121],{"encoding":57},"G_t=r_t+\\gamma 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s,a)r(s,a,s')",[31,61117,61119,61247],{"className":61118,"ariaHidden":62},[61],[31,61120,61122,61125,61197,61200,61203,61206,61238,61241,61244],{"className":61121},[66],[31,61123],{"className":61124,"style":56633},[70],[31,61126,61128,61131],{"className":61127},[1460],[31,61129,5625],{"className":61130,"style":21372},[1460,1461,1462],[31,61132,61134],{"className":61133},[1039],[31,61135,61137,61189],{"className":61136},[1043,1044],[31,61138,61140,61186],{"className":61139},[1048],[31,61141,61143],{"className":61142,"style":56652},[1052],[31,61144,61145,61148],{"style":21388},[31,61146],{"className":61147,"style":1061},[1060],[31,61149,61151],{"className":61150},[1065,1066,1067,1068],[31,61152,61154],{"className":61153},[75,1068],[31,61155,61157,61160],{"className":61156},[75,1068],[31,61158,14551],{"className":61159},[75,76,1068],[31,61161,61163],{"className":61162},[1039],[31,61164,61166],{"className":61165},[1043],[31,61167,61169],{"className":61168},[1048],[31,61170,61172],{"className":61171,"style":31478},[1052],[31,61173,61174,61177],{"style":31481},[31,61175],{"className":61176,"style":11769},[1060],[31,61178,61180],{"className":61179},[1065,11773,1308,1068],[31,61181,61183],{"className":61182},[75,1068],[31,61184,13691],{"className":61185},[75,1068],[31,61187,1076],{"className":61188},[1075],[31,61190,61192],{"className":61191},[1048],[31,61193,61195],{"className":61194,"style":21407},[1052],[31,61196],{},[31,61198],{"className":61199,"style":1467},[119],[31,61201,2521],{"className":61202,"style":949},[75,76],[31,61204,211],{"className":61205},[264],[31,61207,61209,61212],{"className":61208},[75],[31,61210,14551],{"className":61211},[75,76],[31,61213,61215],{"className":61214},[1039],[31,61216,61218],{"className":61217},[1043],[31,61219,61221],{"className":61220},[1048],[31,61222,61224],{"className":61223,"style":13732},[1052],[31,61225,61226,61229],{"style":3631},[31,61227],{"className":61228,"style":1061},[1060],[31,61230,61232],{"className":61231},[1065,1066,1067,1068],[31,61233,61235],{"className":61234},[75,1068],[31,61236,13691],{"className":61237},[75,1068],[31,61239],{"className":61240,"style":293},[119],[31,61242,12680],{"className":61243},[297],[31,61245],{"className":61246,"style":293},[119],[31,61248,61250,61253,61256,61259,61262,61265,61268,61271,61274,61277,61280,61283,61286,61289,61292,61324],{"className":61249},[66],[31,61251],{"className":61252,"style":13710},[70],[31,61254,14551],{"className":61255},[75,76],[31,61257,1210],{"className":61258},[1391],[31,61260],{"className":61261,"style":1467},[119],[31,61263,5813],{"className":61264},[75,76],[31,61266,220],{"className":61267},[289],[31,61269,2549],{"className":61270,"style":19759},[75,76],[31,61272,211],{"className":61273},[264],[31,61275,14551],{"className":61276},[75,76],[31,61278,1210],{"className":61279},[1391],[31,61281],{"className":61282,"style":1467},[119],[31,61284,5813],{"className":61285},[75,76],[31,61287,1210],{"className":61288},[1391],[31,61290],{"className":61291,"style":1467},[119],[31,61293,61295,61298],{"className":61294},[75],[31,61296,14551],{"className":61297},[75,76],[31,61299,61301],{"className":61300},[1039],[31,61302,61304],{"className":61303},[1043],[31,61305,61307],{"className":61306},[1048],[31,61308,61310],{"className":61309,"style":13732},[1052],[31,61311,61312,61315],{"style":3631},[31,61313],{"className":61314,"style":1061},[1060],[31,61316,61318],{"className":61317},[1065,1066,1067,1068],[31,61319,61321],{"className":61320},[75,1068],[31,61322,13691],{"className":61323},[75,1068],[31,61325,220],{"className":61326},[289],"。第二项先用全期望公式对 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分解，再利用马尔可夫性与策略的平稳性：",[31,61414,61416],{"className":61415,"translate":35},[411],[31,61417,61419,61655],{"className":61418,"translate":35},[34],[31,61420,61422],{"className":61421},[39],[41,61423,61424],{"xmlns":43,"display":420},[45,61425,61426,61652],{},[5606,61427,61428,61522,61586],{"rowspacing":5608,"columnalign":5609,"columnspacing":5610},[5612,61429,61430,61482],{},[5615,61431,61432],{},[2071,61433,61434],{"scriptlevel":1977,"displaystyle":62},[48,61435,61436,61442,61444,61456,61458,61464,61466,61468,61470,61476,61478,61480],{},[1004,61437,61438,61440],{},[51,61439,22324],{"mathvariant":14789},[51,61441,2515],{},[95,61443,15175],{"stretchy":210},[1004,61445,61446,61448],{},[51,61447,15297],{},[48,61449,61450,61452,61454],{},[51,61451,12271],{},[95,61453,434],{},[91,61455,205],{},[95,61457,12680],{},[1004,61459,61460,61462],{},[51,61461,14551],{},[51,61463,12271],{},[95,61465,223],{},[51,61467,14551],{},[95,61469,1210],{"separator":62},[1004,61471,61472,61474],{},[51,61473,5813],{},[51,61475,12271],{},[95,61477,223],{},[51,61479,5813],{},[95,61481,15184],{"stretchy":210},[5615,61483,61484],{},[2071,61485,61486],{"scriptlevel":1977,"displaystyle":62},[48,61487,61488,61490,61492,61502,61504,61506,61512,61514,61516,61518,61520],{},[48,61489],{},[95,61491,223],{},[1531,61493,61494,61496],{},[95,61495,5625],{},[1180,61497,61498,61500],{},[51,61499,14551],{},[95,61501,13691],{"mathvariant":494,"lspace":5610,"rspace":5610},[51,61503,2521],{},[95,61505,211],{"stretchy":210},[1180,61507,61508,61510],{},[51,61509,14551],{},[95,61511,13691],{"mathvariant":494,"lspace":5610,"rspace":5610},[95,61513,12680],{},[51,61515,14551],{},[95,61517,1210],{"separator":62},[51,61519,5813],{},[95,61521,220],{"stretchy":210},[5612,61523,61524,61530],{},[5615,61525,61526],{},[2071,61527,61528],{"scriptlevel":1977,"displaystyle":62},[48,61529],{},[5615,61531,61532],{},[2071,61533,61534],{"scriptlevel":1977,"displaystyle":62},[48,61535,61536,61538,61540,61542,61548,61550,61562,61564,61576,61578,61584],{},[48,61537],{},[119,61539],{"width":1584},[95,61541,429],{},[1004,61543,61544,61546],{},[51,61545,22324],{"mathvariant":14789},[51,61547,2515],{},[95,61549,15175],{"stretchy":210},[1004,61551,61552,61554],{},[51,61553,15297],{},[48,61555,61556,61558,61560],{},[51,61557,12271],{},[95,61559,434],{},[91,61561,205],{},[95,61563,12680],{},[1004,61565,61566,61568],{},[51,61567,14551],{},[48,61569,61570,61572,61574],{},[51,61571,12271],{},[95,61573,434],{},[91,61575,205],{},[95,61577,223],{},[1180,61579,61580,61582],{},[51,61581,14551],{},[95,61583,13691],{"mathvariant":494,"lspace":5610,"rspace":5610},[95,61585,15184],{"stretchy":210},[5612,61587,61588,61594],{},[5615,61589,61590],{},[2071,61591,61592],{"scriptlevel":1977,"displaystyle":62},[48,61593],{},[5615,61595,61596],{},[2071,61597,61598],{"scriptlevel":1977,"displaystyle":62},[48,61599,61600,61602,61604,61614,61616,61618,61624,61626,61628,61630,61632,61634,61640,61642,61648,61650],{},[48,61601],{},[95,61603,223],{},[1531,61605,61606,61608],{},[95,61607,5625],{},[1180,61609,61610,61612],{},[51,61611,14551],{},[95,61613,13691],{"mathvariant":494,"lspace":5610,"rspace":5610},[51,61615,2521],{},[95,61617,211],{"stretchy":210},[1180,61619,61620,61622],{},[51,61621,14551],{},[95,61623,13691],{"mathvariant":494,"lspace":5610,"rspace":5610},[95,61625,12680],{},[51,61627,14551],{},[95,61629,1210],{"separator":62},[51,61631,5813],{},[95,61633,220],{"stretchy":210},[1180,61635,61636,61638],{},[51,61637,43481],{},[51,61639,2515],{},[95,61641,211],{"stretchy":210},[1180,61643,61644,61646],{},[51,61645,14551],{},[95,61647,13691],{"mathvariant":494,"lspace":5610,"rspace":5610},[95,61649,220],{"stretchy":210},[51,61651,495],{"mathvariant":494},[55,61653,61654],{"encoding":57},"\\begin{aligned}\n\\mathbb{E}_\\pi[G_{t+1}\\mid s_t=s,a_t=a]\n&=\\sum_{s'}P(s'\\mid s,a)\\\\\n&\\quad\\cdot\\mathbb{E}_\\pi[G_{t+1}\\mid s_{t+1}=s']\\\\\n&=\\sum_{s'}P(s'\\mid s,a)V^\\pi(s').\n\\end{aligned}",[31,61656,61658],{"className":61657,"ariaHidden":62},[61],[31,61659,61661,61665],{"className":61660},[66],[31,61662],{"className":61663,"style":61664},[70],"height:6.488em;vertical-align:-2.994em;",[31,61666,61668],{"className":61667},[75],[31,61669,61671,61937],{"className":61670},[5606],[31,61672,61674],{"className":61673},[5867],[31,61675,61677,61928],{"className":61676},[1043,1044],[31,61678,61680,61925],{"className":61679},[1048],[31,61681,61684,61907,61916],{"className":61682,"style":61683},[1052],"height:3.494em;",[31,61685,61687,61690],{"style":61686},"top:-5.494em;",[31,61688],{"className":61689,"style":5907},[1060],[31,61691,61693,61733,61736,61785,61788,61791,61794,61834,61837,61840,61843,61846,61849,61852,61892,61895,61898,61901,61904],{"className":61692},[75],[31,61694,61696,61699],{"className":61695},[75],[31,61697,22324],{"className":61698},[75,14947],[31,61700,61702],{"className":61701},[1039],[31,61703,61705,61725],{"className":61704},[1043,1044],[31,61706,61708,61722],{"className":61707},[1048],[31,61709,61711],{"className":61710,"style":3230},[1052],[31,61712,61713,61716],{"style":2807},[31,61714],{"className":61715,"style":1061},[1060],[31,61717,61719],{"className":61718},[1065,1066,1067,1068],[31,61720,2515],{"className":61721,"style":2666},[75,76,1068],[31,61723,1076],{"className":61724},[1075],[31,61726,61728],{"className":61727},[1048],[31,61729,61731],{"className":61730,"style":2050},[1052],[31,61732],{},[31,61734,15175],{"className":61735},[264],[31,61737,61739,61742],{"className":61738},[75],[31,61740,15297],{"className":61741},[75,76],[31,61743,61745],{"className":61744},[1039],[31,61746,61748,61777],{"className":61747},[1043,1044],[31,61749,61751,61774],{"className":61750},[1048],[31,61752,61754],{"className":61753,"style":2028},[1052],[31,61755,61756,61759],{"style":2807},[31,61757],{"className":61758,"style":1061},[1060],[31,61760,61762],{"className":61761},[1065,1066,1067,1068],[31,61763,61765,61768,61771],{"className":61764},[75,1068],[31,61766,12271],{"className":61767},[75,76,1068],[31,61769,434],{"className":61770},[124,1068],[31,61772,205],{"className":61773},[75,1068],[31,61775,1076],{"className":61776},[1075],[31,61778,61780],{"className":61779},[1048],[31,61781,61783],{"className":61782,"style":25082},[1052],[31,61784],{},[31,61786],{"className":61787,"style":293},[119],[31,61789,12680],{"className":61790},[297],[31,61792],{"className":61793,"style":293},[119],[31,61795,61797,61800],{"className":61796},[75],[31,61798,14551],{"className":61799},[75,76],[31,61801,61803],{"className":61802},[1039],[31,61804,61806,61826],{"className":61805},[1043,1044],[31,61807,61809,61823],{"className":61808},[1048],[31,61810,61812],{"className":61811,"style":19220},[1052],[31,61813,61814,61817],{"style":2807},[31,61815],{"className":61816,"style":1061},[1060],[31,61818,61820],{"className":61819},[1065,1066,1067,1068],[31,61821,12271],{"className":61822},[75,76,1068],[31,61824,1076],{"className":61825},[1075],[31,61827,61829],{"className":61828},[1048],[31,61830,61832],{"className":61831,"style":2050},[1052],[31,61833],{},[31,61835],{"className":61836,"style":293},[119],[31,61838,223],{"className":61839},[297],[31,61841],{"className":61842,"style":293},[119],[31,61844,14551],{"className":61845},[75,76],[31,61847,1210],{"className":61848},[1391],[31,61850],{"className":61851,"style":1467},[119],[31,61853,61855,61858],{"className":61854},[75],[31,61856,5813],{"className":61857},[75,76],[31,61859,61861],{"className":61860},[1039],[31,61862,61864,61884],{"className":61863},[1043,1044],[31,61865,61867,61881],{"className":61866},[1048],[31,61868,61870],{"className":61869,"style":19220},[1052],[31,61871,61872,61875],{"style":2807},[31,61873],{"className":61874,"style":1061},[1060],[31,61876,61878],{"className":61877},[1065,1066,1067,1068],[31,61879,12271],{"className":61880},[75,76,1068],[31,61882,1076],{"className":61883},[1075],[31,61885,61887],{"className":61886},[1048],[31,61888,61890],{"className":61889,"style":2050},[1052],[31,61891],{},[31,61893],{"className":61894,"style":293},[119],[31,61896,223],{"className":61897},[297],[31,61899],{"className":61900,"style":293},[119],[31,61902,5813],{"className":61903},[75,76],[31,61905,15184],{"className":61906},[289],[31,61908,61910,61913],{"style":61909},"top:-3.06em;",[31,61911],{"className":61912,"style":5907},[1060],[31,61914],{"className":61915},[75],[31,61917,61919,61922],{"style":61918},"top:-1.35em;",[31,61920],{"className":61921,"style":5907},[1060],[31,61923],{"className":61924},[75],[31,61926,1076],{"className":61927},[1075],[31,61929,61931],{"className":61930},[1048],[31,61932,61935],{"className":61933,"style":61934},[1052],"height:2.994em;",[31,61936],{},[31,61938,61940],{"className":61939},[6062],[31,61941,61943,62561],{"className":61942},[1043,1044],[31,61944,61946,62558],{"className":61945},[1048],[31,61947,61949,62110,62327],{"className":61948,"style":61683},[1052],[31,61950,61951,61954],{"style":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[1065,1066,1067,1068],[31,62547,62549],{"className":62548},[75,1068],[31,62550,13691],{"className":62551},[75,1068],[31,62553,220],{"className":62554},[289],[31,62556,495],{"className":62557},[75],[31,62559,1076],{"className":62560},[1075],[31,62562,62564],{"className":62563},[1048],[31,62565,62567],{"className":62566,"style":61934},[1052],[31,62568],{},[11,62570,62571],{},"代入即得。证毕。",[11,62573,62574,62575,62660,62661,62689],{},"这里的关键是策略的平稳性。若策略依赖于时间，值函数应写成 ",[31,62576,62578,62598],{"className":62577,"translate":35},[34],[31,62579,62581],{"className":62580},[39],[41,62582,62583],{"xmlns":43},[45,62584,62585,62595],{},[48,62586,62587],{},[12397,62588,62589,62591,62593],{},[51,62590,43481],{},[51,62592,12271],{},[51,62594,2515],{},[55,62596,62597],{"encoding":57},"V_t^\\pi",[31,62599,62601],{"className":62600,"ariaHidden":62},[61],[31,62602,62604,62608],{"className":62603},[66],[31,62605],{"className":62606,"style":62607},[70],"height:0.9303em;vertical-align:-0.247em;",[31,62609,62611,62614],{"className":62610},[75],[31,62612,43481],{"className":62613,"style":120},[75,76],[31,62615,62617],{"className":62616},[1039],[31,62618,62620,62652],{"className":62619},[1043,1044],[31,62621,62623,62649],{"className":62622},[1048],[31,62624,62626,62638],{"className":62625,"style":3628},[1052],[31,62627,62629,62632],{"style":62628},"top:-2.453em;margin-left:-0.2222em;margin-right:0.05em;",[31,62630],{"className":62631,"style":1061},[1060],[31,62633,62635],{"className":62634},[1065,1066,1067,1068],[31,62636,12271],{"className":62637},[75,76,1068],[31,62639,62640,62643],{"style":3631},[31,62641],{"className":62642,"style":1061},[1060],[31,62644,62646],{"className":62645},[1065,1066,1067,1068],[31,62647,2515],{"className":62648,"style":2666},[75,76,1068],[31,62650,1076],{"className":62651},[1075],[31,62653,62655],{"className":62654},[1048],[31,62656,62658],{"className":62657,"style":13499},[1052],[31,62659],{},"，递归形式仍然成立，但函数会随 ",[31,62662,62664,62677],{"className":62663,"translate":35},[34],[31,62665,62667],{"className":62666},[39],[41,62668,62669],{"xmlns":43},[45,62670,62671,62675],{},[48,62672,62673],{},[51,62674,12271],{},[55,62676,12271],{"encoding":57},[31,62678,62680],{"className":62679,"ariaHidden":62},[61],[31,62681,62683,62686],{"className":62682},[66],[31,62684],{"className":62685,"style":12283},[70],[31,62687,12271],{"className":62688},[75,76]," 变化。本文只讨论平稳策略，因此可以省略时间下标。",[18,62691,62693],{"id":62692},"二贝尔曼期望方程","二、贝尔曼期望方程",[52860,62695,62697],{"id":62696},"_21-方程的推导","2.1 方程的推导",[11,62699,62700,62701,62705],{},"将",[5813,62702,62704],{"href":62703},"#proposition-basic-identities","命题 1.5"," 的第二式代入第一式，立即得到策略评估的核心方程。",[52865,62707,62710,62742],{"id":62708,"title":62709},"theorem-bellman-expectation","定理 2.1（贝尔曼期望方程）",[11,62711,62712,62713,62741],{},"对任意平稳策略 ",[31,62714,62716,62729],{"className":62715,"translate":35},[34],[31,62717,62719],{"className":62718},[39],[41,62720,62721],{"xmlns":43},[45,62722,62723,62727],{},[48,62724,62725],{},[51,62726,2515],{},[55,62728,3485],{"encoding":57},[31,62730,62732],{"className":62731,"ariaHidden":62},[61],[31,62733,62735,62738],{"className":62734},[66],[31,62736],{"className":62737,"style":71},[70],[31,62739,2515],{"className":62740,"style":2666},[75,76],"，状态值函数满足",[31,62743,62745],{"className":62744,"translate":35},[411],[31,62746,62748,62919],{"className":62747,"translate":35},[34],[31,62749,62751],{"className":62750},[39],[41,62752,62753],{"xmlns":43,"display":420},[45,62754,62755,62916],{},[5606,62756,62757,62835],{"rowspacing":5608,"columnalign":5609,"columnspacing":5610},[5612,62758,62759,62777],{},[5615,62760,62761],{},[2071,62762,62763],{"scriptlevel":1977,"displaystyle":62},[48,62764,62765,62771,62773,62775],{},[1180,62766,62767,62769],{},[51,62768,43481],{},[51,62770,2515],{},[95,62772,211],{"stretchy":210},[51,62774,14551],{},[95,62776,220],{"stretchy":210},[5615,62778,62779],{},[2071,62780,62781],{"scriptlevel":1977,"displaystyle":62},[48,62782,62783,62785,62787,62793,62795,62797,62799,62801,62803,62805,62815,62817,62819,62825,62827,62829,62831,62833],{},[48,62784],{},[95,62786,223],{},[1531,62788,62789,62791],{},[95,62790,5625],{},[51,62792,5813],{},[51,62794,2515],{},[95,62796,211],{"stretchy":210},[51,62798,5813],{},[95,62800,12680],{},[51,62802,14551],{},[95,62804,220],{"stretchy":210},[1531,62806,62807,62809],{},[95,62808,5625],{},[1180,62810,62811,62813],{},[51,62812,14551],{},[95,62814,13691],{"mathvariant":494,"lspace":5610,"rspace":5610},[51,62816,2521],{},[95,62818,211],{"stretchy":210},[1180,62820,62821,62823],{},[51,62822,14551],{},[95,62824,13691],{"mathvariant":494,"lspace":5610,"rspace":5610},[95,62826,12680],{},[51,62828,14551],{},[95,62830,1210],{"separator":62},[51,62832,5813],{},[95,62834,220],{"stretchy":210},[5612,62836,62837,62843],{},[5615,62838,62839],{},[2071,62840,62841],{"scriptlevel":1977,"displaystyle":62},[48,62842],{},[5615,62844,62845],{},[2071,62846,62847],{"scriptlevel":1977,"displaystyle":62},[48,62848,62849,62851,62853,62855,62901,62903,62905,62908,62910,62912,62914],{},[48,62850],{},[119,62852],{"width":1584},[95,62854,2528],{},[48,62856,62857,62859,62861,62863,62865,62867,62869,62871,62877,62879,62881,62883,62889,62891,62897,62899],{},[95,62858,15175],{"fence":62},[51,62860,2549],{},[95,62862,211],{"stretchy":210},[51,62864,14551],{},[95,62866,1210],{"separator":62},[51,62868,5813],{},[95,62870,1210],{"separator":62},[1180,62872,62873,62875],{},[51,62874,14551],{},[95,62876,13691],{"mathvariant":494,"lspace":5610,"rspace":5610},[95,62878,220],{"stretchy":210},[95,62880,434],{},[51,62882,53298],{},[1180,62884,62885,62887],{},[51,62886,43481],{},[51,62888,2515],{},[95,62890,211],{"stretchy":210},[1180,62892,62893,62895],{},[51,62894,14551],{},[95,62896,13691],{"mathvariant":494,"lspace":5610,"rspace":5610},[95,62898,220],{"stretchy":210},[95,62900,15184],{"fence":62},[95,62902,1210],{"separator":62},[119,62904],{"width":20286},[51,62906,62907],{"mathvariant":494},"∀",[51,62909,14551],{},[95,62911,2518],{},[51,62913,21582],{},[51,62915,495],{"mathvariant":494},[55,62917,62918],{"encoding":57},"\\begin{aligned}\nV^\\pi(s)\n&=\\sum_{a}\\pi(a\\mid s)\\sum_{s'}P(s'\\mid s,a)\\\\\n&\\quad\\times\\left[r(s,a,s')+\\gamma V^\\pi(s')\\right],\n\\qquad \\forall s\\in S.\n\\end{aligned}",[31,62920,62922],{"className":62921,"ariaHidden":62},[61],[31,62923,62925,62928],{"className":62924},[66],[31,62926],{"className":62927,"style":59613},[70],[31,62929,62931],{"className":62930},[75],[31,62932,62934,63011],{"className":62933},[5606],[31,62935,62937],{"className":62936},[5867],[31,62938,62940,63003],{"className":62939},[1043,1044],[31,62941,62943,63000],{"className":62942},[1048],[31,62944,62946,62992],{"className":62945,"style":59632},[1052],[31,62947,62948,62951],{"style":59635},[31,62949],{"className":62950,"style":5907},[1060],[31,62952,62954,62983,62986,62989],{"className":62953},[75],[31,62955,62957,62960],{"className":62956},[75],[31,62958,43481],{"className":62959,"style":120},[75,76],[31,62961,62963],{"className":62962},[1039],[31,62964,62966],{"className":62965},[1043],[31,62967,62969],{"className":62968},[1048],[31,62970,62972],{"className":62971,"style":3987},[1052],[31,62973,62974,62977],{"style":1330},[31,62975],{"className":62976,"style":1061},[1060],[31,62978,62980],{"className":62979},[1065,1066,1067,1068],[31,62981,2515],{"className":62982,"style":2666},[75,76,1068],[31,62984,211],{"className":62985},[264],[31,62987,14551],{"className":62988},[75,76],[31,62990,220],{"className":62991},[289],[31,62993,62994,62997],{"style":59691},[31,62995],{"className":62996,"style":5907},[1060],[31,62998],{"className":62999},[75],[31,63001,1076],{"className":63002},[1075],[31,63004,63006],{"className":63005},[1048],[31,63007,63009],{"className":63008,"style":59707},[1052],[31,63010],{},[31,63012,63014],{"className":63013},[6062],[31,63015,63017,63467],{"className":63016},[1043,1044],[31,63018,63020,63464],{"className":63019},[1048],[31,63021,63023,63261],{"className":63022,"style":59632},[1052],[31,63024,63025,63028],{"style":59635},[31,63026],{"className":63027,"style":5907},[1060],[31,63029,63031,63034,63037,63040,63043,63090,63093,63096,63099,63102,63105,63108,63111,63114,63117,63120,63196,63199,63202,63205,63237,63240,63243,63246,63249,63252,63255,63258],{"className":63030},[75],[31,63032],{"className":63033},[75],[31,63035],{"className":63036,"style":293},[119],[31,63038,223],{"className":63039},[297],[31,63041],{"className":63042,"style":293},[119],[31,63044,63046],{"className":63045},[1460,1620],[31,63047,63049,63082],{"className":63048},[1043,1044],[31,63050,63052,63079],{"className":63051},[1048],[31,63053,63055,63069],{"className":63054,"style":5900},[1052],[31,63056,63057,63060],{"style":59333},[31,63058],{"className":63059,"style":5907},[1060],[31,63061,63063],{"className":63062},[1065,1066,1067,1068],[31,63064,63066],{"className":63065},[75,1068],[31,63067,5813],{"className":63068},[75,76,1068],[31,63070,63071,63074],{"style":5919},[31,63072],{"className":63073,"style":5907},[1060],[31,63075,63076],{},[31,63077,5625],{"className":63078},[1460,1461,1687],[31,63080,1076],{"className":63081},[1075],[31,63083,63085],{"className":63084},[1048],[31,63086,63088],{"className":63087,"style":59365},[1052],[31,63089],{},[31,63091],{"className":63092,"style":1467},[119],[31,63094,2515],{"className":63095,"style":2666},[75,76],[31,63097,211],{"className":63098},[264],[31,63100,5813],{"className":63101},[75,76],[31,63103],{"className":63104,"style":293},[119],[31,63106,12680],{"className":63107},[297],[31,63109],{"className":63110,"style":293},[119],[31,63112,14551],{"className":63113},[75,76],[31,63115,220],{"className":63116},[289],[31,63118],{"className":63119,"style":1467},[119],[31,63121,63123],{"className":63122},[1460,1620],[31,63124,63126,63188],{"className":63125},[1043,1044],[31,63127,63129,63185],{"className":63128},[1048],[31,63130,63132,63175],{"className":63131,"style":5900},[1052],[31,63133,63134,63137],{"style":59756},[31,63135],{"className":63136,"style":5907},[1060],[31,63138,63140],{"className":63139},[1065,1066,1067,1068],[31,63141,63143],{"className":63142},[75,1068],[31,63144,63146,63149],{"className":63145},[75,1068],[31,63147,14551],{"className":63148},[75,76,1068],[31,63150,63152],{"className":63151},[1039],[31,63153,63155],{"className":63154},[1043],[31,63156,63158],{"className":63157},[1048],[31,63159,63161],{"className":63160,"style":31478},[1052],[31,63162,63163,63166],{"style":31481},[31,63164],{"className":63165,"style":11769},[1060],[31,63167,63169],{"className":63168},[1065,11773,1308,1068],[31,63170,63172],{"className":63171},[75,1068],[31,63173,13691],{"className":63174},[75,1068],[31,63176,63177,63180],{"style":5919},[31,63178],{"className":63179,"style":5907},[1060],[31,63181,63182],{},[31,63183,5625],{"className":63184},[1460,1461,1687],[31,63186,1076],{"className":63187},[1075],[31,63189,63191],{"className":63190},[1048],[31,63192,63194],{"className":63193,"style":59817},[1052],[31,63195],{},[31,63197],{"className":63198,"style":1467},[119],[31,63200,2521],{"className":63201,"style":949},[75,76],[31,63203,211],{"className":63204},[264],[31,63206,63208,63211],{"className":63207},[75],[31,63209,14551],{"className":63210},[75,76],[31,632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s)Q^\\pi(s,a)\\\\\n&=\\sum_{a}\\pi(a\\mid s)\\sum_{s'}P(s'\\mid s,a)\n\\left[r(s,a,s')+\\gamma V^\\pi(s')\\right].\n\\end{aligned}",[31,63679,63681],{"className":63680,"ariaHidden":62},[61],[31,63682,63684,63688],{"className":63683},[66],[31,63685],{"className":63686,"style":63687},[70],"height:4.944em;vertical-align:-2.222em;",[31,63689,63691],{"className":63690},[75],[31,63692,63694,63775],{"className":63693},[5606],[31,63695,63697],{"className":63696},[5867],[31,63698,63700,63766],{"className":63699},[1043,1044],[31,63701,63703,63763],{"className":63702},[1048],[31,63704,63707,63754],{"className":63705,"style":63706},[1052],"height:2.722em;",[31,63708,63710,63713],{"style":63709},"top:-4.722em;",[31,63711],{"className":63712,"style":5907},[1060],[31,63714,63716,63745,63748,63751],{"className":63715},[75],[31,63717,63719,63722],{"className":63718},[75],[31,63720,43481],{"className":63721,"style":120},[75,76],[31,63723,63725],{"className":63724},[1039],[31,63726,63728],{"className":63727},[1043],[31,63729,63731],{"className":63730},[1048],[31,63732,63734],{"className":63733,"style":3987},[1052],[31,63735,63736,63739],{"style":1330},[31,63737],{"className":63738,"style":1061},[1060],[31,63740,63742],{"className":63741},[1065,1066,1067,1068],[31,63743,2515],{"className":63744,"style":2666},[75,76,1068],[31,63746,211],{"className":63747},[264],[31,63749,14551],{"className":63750},[75,76],[31,63752,220],{"className":63753},[289],[31,63755,63757,63760],{"style":63756},"top:-2.122em;",[31,63758],{"className":63759,"style":5907},[1060],[31,63761],{"className":63762},[75],[31,63764,1076],{"className":63765},[1075],[31,63767,63769],{"className":63768},[1048],[31,63770,63773],{"className":63771,"style":63772},[1052],"height:2.222em;",[31,63774],{},[31,63776,63778],{"className":63777},[6062],[31,63779,63781,64325],{"className":63780},[1043,1044],[31,63782,63784,64322],{"className":63783},[1048],[31,63785,63787,63928],{"className":63786,"style":63706},[1052],[31,63788,63789,63792],{"style":63709},[31,63790],{"className":63791,"style":5907},[1060],[31,63793,63795,63798,63801,63804,63807,63854,63857,63860,63863,63866,63869,63872,63875,63878,63881,63910,63913,63916,63919,63922,63925],{"className":63794},[75],[31,63796],{"className":63797},[75],[31,63799],{"className":63800,"style":293},[119],[31,63802,223],{"className":63803},[297],[31,63805],{"className":63806,"style":293},[119],[31,63808,63810],{"className":63809},[1460,1620],[31,63811,63813,63846],{"className":63812},[1043,1044],[31,63814,63816,63843],{"className":63815},[1048],[31,63817,63819,63833],{"className":63818,"style":5900},[1052],[31,63820,63821,63824],{"style":59333},[31,63822],{"className":63823,"style":5907},[1060],[31,63825,63827],{"className":63826},[1065,1066,1067,1068],[31,63828,63830],{"className":63829},[75,1068],[31,63831,5813],{"className":63832},[75,76,1068],[31,63834,63835,63838],{"style":5919},[31,63836],{"className":63837,"style":5907},[1060],[31,63839,63840],{},[31,63841,5625],{"className":63842},[1460,1461,1687],[31,63844,1076],{"className":63845},[1075],[31,63847,63849],{"className":63848},[1048],[31,63850,63852],{"className":63851,"style":59365},[1052],[31,63853],{},[31,63855],{"className":63856,"style":1467},[119],[31,63858,2515],{"className":63859,"style":2666},[75,76],[31,63861,211],{"className":63862},[264],[31,63864,5813],{"className":63865},[75,76],[31,63867],{"className":63868,"style":293},[119],[31,63870,12680],{"className":63871},[297],[31,63873],{"className":63874,"style":293},[119],[31,63876,14551],{"className":63877},[75,76],[31,63879,220],{"className":63880},[289],[31,63882,63884,63887],{"className":63883},[75],[31,63885,56971],{"className":63886},[75,76],[31,63888,63890],{"className":63889},[1039],[31,63891,63893],{"className":63892},[1043],[31,63894,63896],{"className":63895},[1048],[31,63897,63899],{"className":63898,"style":3987},[1052],[31,63900,63901,63904],{"style":1330},[31,63902],{"className":63903,"style":1061},[1060],[31,63905,63907],{"className":63906},[1065,1066,1067,1068],[31,63908,2515],{"className":63909,"style":2666},[75,76,1068],[31,63911,211],{"className":63912},[264],[31,63914,14551],{"className":63915},[75,76],[31,63917,1210],{"className":63918},[1391],[31,63920],{"className":63921,"style":1467},[119],[31,63923,5813],{"className":63924},[75,76],[31,63926,220],{"className":63927},[289],[31,63929,63930,63933],{"style":63756},[31,63931],{"className":63932,"style":5907},[1060],[31,63934,63936,63939,63942,63945,63948,63995,63998,64001,64004,64007,64010,64013,64016,64019,64022,64025,64101,64104,64107,64110,64142,64145,64148,64151,64154,64157,64160,64163,64166,64169,64316,64319],{"className":63935},[75],[31,63937],{"className":63938},[75],[31,63940],{"className":63941,"style":293},[119],[31,63943,223],{"className":63944},[297],[31,63946],{"className":63947,"style":293},[119],[31,63949,63951],{"className":63950},[1460,1620],[31,63952,63954,63987],{"className":63953},[1043,1044],[31,63955,63957,63984],{"className":63956},[1048],[31,63958,63960,63974],{"className":63959,"style":5900},[1052],[31,63961,63962,63965],{"style":59333},[31,63963],{"className":63964,"style":5907},[1060],[31,63966,63968],{"className":63967},[1065,1066,1067,1068],[31,63969,63971],{"className":63970},[75,1068],[31,63972,5813],{"className":63973},[75,76,1068],[31,63975,63976,63979],{"style":5919},[31,63977],{"className":63978,"style":5907},[1060],[31,63980,63981],{},[31,63982,5625],{"className":63983},[1460,1461,1687],[31,63985,1076],{"className":63986},[1075],[31,63988,63990],{"className":63989},[1048],[31,63991,63993],{"className":63992,"style":59365},[1052],[31,63994],{},[31,63996],{"className":63997,"style":1467},[119],[31,63999,2515],{"className":64000,"style":2666},[75,76],[31,64002,211],{"className":64003},[264],[31,64005,5813],{"className":64006},[75,76],[31,64008],{"className":64009,"style":293},[119],[31,64011,12680],{"className":64012},[297],[31,64014],{"className":64015,"style":293},[119],[31,64017,14551],{"className":64018},[75,76],[31,64020,220],{"className":64021},[289],[31,64023],{"className":64024,"style":1467},[119],[31,64026,64028],{"className":64027},[1460,1620],[31,64029,64031,64093],{"className":64030},[1043,1044],[31,64032,64034,64090],{"className":64033},[1048],[31,64035,64037,64080],{"className":64036,"style":5900},[1052],[31,64038,64039,64042],{"style":59756},[31,64040],{"className":64041,"style":5907},[1060],[31,64043,64045],{"className":64044},[1065,1066,1067,1068],[31,64046,64048],{"className":64047},[75,1068],[31,64049,64051,64054],{"className":64050},[75,1068],[31,64052,14551],{"className":64053},[75,76,1068],[31,64055,64057],{"className":64056},[1039],[31,64058,64060],{"className":64059},[1043],[31,64061,64063],{"className":64062},[1048],[31,64064,64066],{"className":64065,"style":31478},[1052],[31,64067,64068,64071],{"style":31481},[31,64069],{"className":64070,"style":11769},[1060],[31,64072,64074],{"className":64073},[1065,11773,1308,1068],[31,64075,64077],{"className":64076},[75,1068],[31,64078,13691],{"className":64079},[75,1068],[31,64081,64082,64085],{"style":5919},[31,64083],{"className":64084,"style":5907},[1060],[31,64086,64087],{},[31,64088,5625],{"className":64089},[1460,1461,1687],[31,64091,1076],{"className":64092},[1075],[31,64094,64096],{"className":64095},[1048],[31,64097,64099],{"className":64098,"style":59817},[1052],[31,64100],{},[31,64102],{"className":64103,"style":1467},[119],[31,64105,2521],{"className":64106,"style":949},[75,76],[31,64108,211],{"className":64109},[264],[31,64111,64113,64116],{"className":64112},[75],[31,64114,14551],{"className":64115},[75,76],[31,64117,64119],{"className":64118},[1039],[31,64120,64122],{"className":64121},[1043],[31,64123,64125],{"className":64124},[1048],[31,64126,64128],{"className":64127,"style":14017},[1052],[31,64129,64130,64133],{"style":1330},[31,64131],{"className":64132,"style":1061},[1060],[31,64134,64136],{"className":64135},[1065,1066,1067,1068],[31,64137,64139],{"className":64138},[75,1068],[31,64140,13691],{"className":64141},[75,1068],[31,64143],{"className":64144,"style":293},[119],[31,64146,12680],{"className":64147},[297],[31,64149],{"className":64150,"style":293},[119],[31,64152,14551],{"className":64153},[75,76],[31,64155,1210],{"className":64156},[1391],[31,64158],{"className":64159,"style":1467},[119],[31,64161,5813],{"className":64162},[75,76],[31,64164,220],{"className":64165},[289],[31,64167],{"className":64168,"style":1467},[119],[31,64170,64172,64175,64178,64181,64184,64187,64190,64193,64196,64199,64231,64234,64237,64240,64243,64246,64275,64278,64310,64313],{"className":64171},[3860],[31,64173,15175],{"className":64174,"style":3868},[264,3867],[31,64176,2549],{"className":64177,"style":19759},[75,76],[31,64179,211],{"className":64180},[264],[31,64182,14551],{"className":64183},[75,76],[31,64185,1210],{"className":64186},[1391],[31,64188],{"className":64189,"style":1467},[119],[31,64191,5813],{"className":64192},[75,76],[31,64194,1210],{"className":64195},[1391],[31,64197],{"className":64198,"style":1467},[119],[31,64200,64202,64205],{"className":64201},[75],[31,64203,14551],{"className":64204},[75,76],[31,64206,64208],{"className":64207},[1039],[31,64209,64211],{"className":64210},[1043],[31,64212,64214],{"className":64213},[1048],[31,64215,64217],{"className":64216,"style":14017},[1052],[31,64218,64219,64222],{"style":1330},[31,64220],{"className":64221,"style":1061},[1060],[31,64223,64225],{"className":64224},[1065,1066,1067,1068],[31,64226,64228],{"className":64227},[75,1068],[31,64229,13691],{"className":64230},[75,1068],[31,64232,220],{"className":64233},[289],[31,64235],{"className":64236,"style":120},[119],[31,64238,434],{"className":64239},[124],[31,64241],{"className":64242,"style":120},[119],[31,64244,53298],{"className":64245,"style":24157},[75,76],[31,64247,64249,64252],{"className":64248},[75],[31,64250,43481],{"className":64251,"style":120},[75,76],[31,64253,64255],{"className":64254},[1039],[31,64256,64258],{"className":64257},[1043],[31,64259,64261],{"className":64260},[1048],[31,64262,64264],{"className":64263,"style":3987},[1052],[31,64265,64266,64269],{"style":1330},[31,64267],{"className":64268,"style":1061},[1060],[31,64270,64272],{"className":64271},[1065,1066,1067,1068],[31,64273,2515],{"className":64274,"style":2666},[75,76,1068],[31,64276,211],{"className":64277},[264],[31,64279,64281,64284],{"className":64280},[75],[31,64282,14551],{"className":64283},[75,76],[31,64285,64287],{"className":64286},[1039],[31,64288,64290],{"className":64289},[1043],[31,64291,64293],{"className":64292},[1048],[31,64294,64296],{"className":64295,"style":14017},[1052],[31,64297,64298,64301],{"style":1330},[31,64299],{"className":64300,"style":1061},[1060],[31,64302,64304],{"className":64303},[1065,1066,1067,1068],[31,64305,64307],{"className":64306},[75,1068],[31,64308,13691],{"className":64309},[75,1068],[31,64311,220],{"className":64312},[289],[31,64314,15184],{"className":64315,"style":3868},[289,3867],[31,64317],{"className":64318,"style":1467},[119],[31,64320,495],{"className":64321},[75],[31,64323,1076],{"className":64324},[1075],[31,64326,64328],{"className":64327},[1048],[31,64329,64331],{"className":64330,"style":63772},[1052],[31,64332],{},[11,64334,64335],{},"证毕。",[11,64337,64338,64339,64397],{},"这个方程的意义在于，它把一个涉及无限期回报的期望，转化为一个只涉及当前奖励与下一状态值函数的方程。方程右端仍然含有未知函数 ",[31,64340,64342,64359],{"className":64341,"translate":35},[34],[31,64343,64345],{"className":64344},[39],[41,64346,64347],{"xmlns":43},[45,64348,64349,64357],{},[48,64350,64351],{},[1180,64352,64353,64355],{},[51,64354,43481],{},[51,64356,2515],{},[55,64358,59045],{"encoding":57},[31,64360,64362],{"className":64361,"ariaHidden":62},[61],[31,64363,64365,64368],{"className":64364},[66],[31,64366],{"className":64367,"style":945},[70],[31,64369,64371,64374],{"className":64370},[75],[31,64372,43481],{"className":64373,"style":120},[75,76],[31,64375,64377],{"className":64376},[1039],[31,64378,64380],{"className":64379},[1043],[31,64381,64383],{"className":64382},[1048],[31,64384,64386],{"className":64385,"style":3628},[1052],[31,64387,64388,64391],{"style":3631},[31,64389],{"className":64390,"style":1061},[1060],[31,64392,64394],{"className":64393},[1065,1066,1067,1068],[31,64395,2515],{"className":64396,"style":2666},[75,76,1068],"，因此它不是普通的递推式，而是一个泛函方程。",[52860,64399,64401],{"id":64400},"_22-算子形式","2.2 算子形式",[11,64403,64404,64405,54797,64408,55214],{},"为了把上述方程写成更紧凑的形式，定义",[7246,64406,64407],{},"贝尔曼期望算子",[31,64409,64411,64429],{"className":64410,"translate":35},[34],[31,64412,64414],{"className":64413},[39],[41,64415,64416],{"xmlns":43},[45,64417,64418,64426],{},[48,64419,64420],{},[1180,64421,64422,64424],{},[51,64423,919],{"mathvariant":925},[51,64425,2515],{},[55,64427,64428],{"encoding":57},"\\mathcal{T}^\\pi",[31,64430,64432],{"className":64431,"ariaHidden":62},[61],[31,64433,64435,64438],{"className":64434},[66],[31,64436],{"className":64437,"style":945},[70],[31,64439,64441,64445],{"className":64440},[75],[31,64442,919],{"className":64443,"style":64444},[75,968],"margin-right:0.2542em;",[31,64446,64448],{"className":64447},[1039],[31,64449,64451],{"className":64450},[1043],[31,64452,64454],{"className":64453},[1048],[31,64455,64457],{"className":64456,"style":3628},[1052],[31,64458,64459,64462],{"style":3631},[31,64460],{"className":64461,"style":1061},[1060],[31,64463,64465],{"className":64464},[1065,1066,1067,1068],[31,64466,2515],{"className":64467,"style":2666},[75,76,1068],[31,64469,64471],{"className":64470,"translate":35},[411],[31,64472,64474,64634],{"className":64473,"translate":35},[34],[31,64475,64477],{"className":64476},[39],[41,64478,64479],{"xmlns":43,"display":420},[45,64480,64481,64631],{},[5606,64482,64483,64567],{"rowspacing":5608,"columnalign":5609,"columnspacing":5610},[5612,64484,64485,64509],{},[5615,64486,64487],{},[2071,64488,64489],{"scriptlevel":1977,"displaystyle":62},[48,64490,64491,64493,64499,64501,64503,64505,64507],{},[95,64492,211],{"stretchy":210},[1180,64494,64495,64497],{},[51,64496,919],{"mathvariant":925},[51,64498,2515],{},[51,64500,43481],{},[95,64502,220],{"stretchy":210},[95,64504,211],{"stretchy":210},[51,64506,14551],{},[95,64508,220],{"stretchy":210},[5615,64510,64511],{},[2071,64512,64513],{"scriptlevel":1977,"displaystyle":62},[48,64514,64515,64517,64519,64525,64527,64529,64531,64533,64535,64537,64547,64549,64551,64557,64559,64561,64563,64565],{},[48,64516],{},[95,64518,223],{},[1531,64520,64521,64523],{},[95,64522,5625],{},[51,64524,5813],{},[51,64526,2515],{},[95,64528,211],{"stretchy":210},[51,64530,5813],{},[95,64532,12680],{},[51,64534,14551],{},[95,64536,220],{"stretchy":210},[1531,64538,64539,64541],{},[95,64540,5625],{},[1180,64542,64543,64545],{},[51,64544,14551],{},[95,64546,13691],{"mathvariant":494,"lspace":5610,"rspace":5610},[51,64548,2521],{},[95,64550,211],{"stretchy":210},[1180,64552,64553,64555],{},[51,64554,14551],{},[95,64556,13691],{"mathvariant":494,"lspace":5610,"rspace":5610},[95,64558,12680],{},[51,64560,14551],{},[95,64562,1210],{"separator":62},[51,64564,5813],{},[95,64566,220],{"stretchy":210},[5612,64568,64569,64575],{},[5615,64570,64571],{},[2071,64572,64573],{"scriptlevel":1977,"displaystyle":62},[48,64574],{},[5615,64576,64577],{},[2071,64578,64579],{"scriptlevel":1977,"displaystyle":62},[48,64580,64581,64583,64585,64587,64629],{},[48,64582],{},[119,64584],{"width":1584},[95,64586,2528],{},[48,64588,64589,64591,64593,64595,64597,64599,64601,64603,64609,64611,64613,64615,64617,64619,64625,64627],{},[95,64590,15175],{"fence":62},[51,64592,2549],{},[95,64594,211],{"stretchy":210},[51,64596,14551],{},[95,64598,1210],{"separator":62},[51,64600,5813],{},[95,64602,1210],{"separator":62},[1180,64604,64605,64607],{},[51,64606,14551],{},[95,64608,13691],{"mathvariant":494,"lspace":5610,"rspace":5610},[95,64610,220],{"stretchy":210},[95,64612,434],{},[51,64614,53298],{},[51,64616,43481],{},[95,64618,211],{"stretchy":210},[1180,64620,64621,64623],{},[51,64622,14551],{},[95,64624,13691],{"mathvariant":494,"lspace":5610,"rspace":5610},[95,64626,220],{"stretchy":210},[95,64628,15184],{"fence":62},[51,64630,495],{"mathvariant":494},[55,64632,64633],{"encoding":57},"\\begin{aligned}\n(\\mathcal{T}^\\pi V)(s)\n&=\\sum_{a}\\pi(a\\mid s)\\sum_{s'}P(s'\\mid s,a)\\\\\n&\\quad\\times\\left[r(s,a,s')+\\gamma V(s')\\right].\n\\end{aligned}",[31,64635,64637],{"className":64636,"ariaHidden":62},[61],[31,64638,64640,64643],{"className":64639},[66],[31,64641],{"className":64642,"style":59613},[70],[31,64644,64646],{"className":64645},[75],[31,64647,64649,64735],{"className":64648},[5606],[31,64650,64652],{"className":64651},[5867],[31,64653,64655,64727],{"className":64654},[1043,1044],[31,64656,64658,64724],{"className":64657},[1048],[31,64659,64661,64716],{"className":64660,"style":59632},[1052],[31,64662,64663,64666],{"style":59635},[31,64664],{"className":64665,"style":5907},[1060],[31,64667,64669,64672,64701,64704,64707,64710,64713],{"className":64668},[75],[31,64670,211],{"className":64671},[264],[31,64673,64675,64678],{"className":64674},[75],[31,64676,919],{"className":64677,"style":64444},[75,968],[31,64679,64681],{"className":64680},[1039],[31,64682,64684],{"className":64683},[1043],[31,64685,64687],{"className":64686},[1048],[31,64688,64690],{"className":64689,"style":3987},[1052],[31,64691,64692,64695],{"style":1330},[31,64693],{"className":64694,"style":1061},[1060],[31,64696,64698],{"className":64697},[1065,1066,1067,1068],[31,64699,2515],{"className":64700,"style":2666},[75,76,1068],[31,64702,43481],{"className":64703,"style":120},[75,76],[31,64705,220],{"className":64706},[289],[31,64708,211],{"className":64709},[264],[31,64711,14551],{"className":64712},[75,76],[31,64714,220],{"className":64715},[289],[31,64717,64718,64721],{"style":59691},[31,64719],{"className":64720,"style":5907},[1060],[31,64722],{"className":64723},[75],[31,64725,1076],{"className":64726},[1075],[31,64728,64730],{"className":64729},[1048],[31,64731,64733],{"className":64732,"style":59707},[1052],[31,64734],{},[31,64736,64738],{"className":64737},[6062],[31,64739,64741,65138],{"className":64740},[1043,1044],[31,64742,64744,65135],{"className":64743},[1048],[31,64745,64747,64985],{"className":64746,"style":59632},[1052],[31,64748,64749,64752],{"style":59635},[31,64750],{"className":64751,"style":5907},[1060],[31,64753,64755,64758,64761,64764,64767,64814,64817,64820,64823,64826,64829,64832,64835,64838,64841,64844,64920,64923,64926,64929,64961,64964,64967,64970,64973,64976,64979,64982],{"className":64754},[75],[31,64756],{"className":64757},[75],[31,64759],{"className":64760,"style":293},[119],[31,64762,223],{"className":64763},[297],[31,64765],{"className":64766,"style":293},[119],[31,64768,64770],{"className":64769},[1460,1620],[31,64771,64773,64806],{"className":64772},[1043,1044],[31,64774,64776,64803],{"className":64775},[1048],[31,64777,64779,64793],{"className":64778,"style":5900},[1052],[31,64780,64781,64784],{"style":59333},[31,64782],{"className":64783,"style":5907},[1060],[31,64785,64787],{"className":64786},[1065,1066,1067,1068],[31,64788,64790],{"className":64789},[75,1068],[31,64791,5813],{"className":64792},[75,76,1068],[31,64794,64795,64798],{"style":5919},[31,64796],{"className":64797,"style":5907},[1060],[31,64799,64800],{},[31,64801,5625],{"className":64802},[1460,1461,1687],[31,64804,1076],{"className":64805},[1075],[31,64807,64809],{"className":64808},[1048],[31,64810,64812],{"className":64811,"style":59365},[1052],[31,64813],{},[31,64815],{"className":64816,"style":1467},[119],[31,64818,2515],{"className":64819,"style":2666},[75,76],[31,64821,211],{"className":64822},[264],[31,64824,5813],{"className":64825},[75,76],[31,64827],{"className":64828,"style":293},[119],[31,64830,12680],{"className":64831},[297],[31,64833],{"className":64834,"style":293},[119],[31,64836,14551],{"className":64837},[75,76],[31,64839,220],{"className":64840},[289],[31,64842],{"className":64843,"style":1467},[119],[31,64845,64847],{"className":64846},[1460,1620],[31,64848,64850,64912],{"className":64849},[1043,1044],[31,64851,64853,64909],{"className":64852},[1048],[31,64854,64856,64899],{"className":64855,"style":5900},[1052],[31,64857,64858,64861],{"style":59756},[31,64859],{"className":64860,"style":5907},[1060],[31,64862,64864],{"className":64863},[1065,1066,1067,1068],[31,64865,64867],{"className":64866},[75,1068],[31,64868,64870,64873],{"className":64869},[75,1068],[31,64871,14551],{"className":64872},[75,76,1068],[31,64874,64876],{"className":64875},[1039],[31,64877,64879],{"className":64878},[1043],[31,64880,64882],{"className":64881},[1048],[31,64883,64885],{"className":64884,"style":31478},[1052],[31,64886,64887,64890],{"style":31481},[31,64888],{"className":64889,"style":11769},[1060],[31,64891,64893],{"className":64892},[1065,11773,1308,1068],[31,64894,64896],{"className":64895},[75,1068],[31,64897,13691],{"className":64898},[75,1068],[31,64900,64901,64904],{"style":5919},[31,64902],{"className":64903,"style":5907},[1060],[31,64905,64906],{},[31,64907,5625],{"className":64908},[1460,1461,1687],[31,64910,1076],{"className":64911},[1075],[31,64913,64915],{"className":64914},[1048],[31,64916,64918],{"className":64917,"style":59817},[1052],[31,64919],{},[31,64921],{"className":64922,"style":1467},[119],[31,64924,2521],{"className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V^\\pi.",[31,65190,65192,65236],{"className":65191,"ariaHidden":62},[61],[31,65193,65195,65198,65227,65230,65233],{"className":65194},[66],[31,65196],{"className":65197,"style":3987},[70],[31,65199,65201,65204],{"className":65200},[75],[31,65202,43481],{"className":65203,"style":120},[75,76],[31,65205,65207],{"className":65206},[1039],[31,65208,65210],{"className":65209},[1043],[31,65211,65213],{"className":65212},[1048],[31,65214,65216],{"className":65215,"style":3987},[1052],[31,65217,65218,65221],{"style":1330},[31,65219],{"className":65220,"style":1061},[1060],[31,65222,65224],{"className":65223},[1065,1066,1067,1068],[31,65225,2515],{"className":65226,"style":2666},[75,76,1068],[31,65228],{"className":65229,"style":293},[119],[31,65231,223],{"className":65232},[297],[31,65234],{"className":65235,"style":293},[119],[31,65237,65239,65242,65271,65300],{"className":65238},[66],[31,65240],{"className":65241,"style":3987},[70],[31,65243,65245,65248],{"className":65244},[75],[31,65246,919],{"className":65247,"style":64444},[75,968],[31,65249,65251],{"className":65250},[1039],[31,65252,65254],{"className":65253},[1043],[31,65255,65257],{"className":65256},[1048],[31,65258,65260],{"className":65259,"style":3987},[1052],[31,65261,65262,65265],{"style":1330},[31,65263],{"className":65264,"style":1061},[1060],[31,65266,65268],{"className":65267},[1065,1066,1067,1068],[31,65269,2515],{"className":65270,"style":2666},[75,76,1068],[31,65272,65274,65277],{"className":65273},[75],[31,65275,43481],{"className":65276,"style":120},[75,76],[31,65278,65280],{"className":65279},[1039],[31,65281,65283],{"className":65282},[1043],[31,65284,65286],{"className":65285},[1048],[31,65287,65289],{"className":65288,"style":3987},[1052],[31,65290,65291,65294],{"style":1330},[31,65292],{"className":65293,"style":1061},[1060],[31,65295,65297],{"className":65296},[1065,1066,1067,1068],[31,65298,2515],{"className":65299,"style":2666},[75,76,1068],[31,65301,495],{"className":65302},[75],[11,65304,65305,65306,65364,65365,65423,65424,65452,65453,65517,65518,65576],{},"这里的 ",[31,65307,65309,65326],{"className":65308,"translate":35},[34],[31,65310,65312],{"className":65311},[39],[41,65313,65314],{"xmlns":43},[45,65315,65316,65324],{},[48,65317,65318],{},[1180,65319,65320,65322],{},[51,65321,43481],{},[51,65323,2515],{},[55,65325,59045],{"encoding":57},[31,65327,65329],{"className":65328,"ariaHidden":62},[61],[31,65330,65332,65335],{"className":65331},[66],[31,65333],{"className":65334,"style":945},[70],[31,65336,65338,65341],{"className":65337},[75],[31,65339,43481],{"className":65340,"style":120},[75,76],[31,65342,65344],{"className":65343},[1039],[31,65345,65347],{"className":65346},[1043],[31,65348,65350],{"className":65349},[1048],[31,65351,65353],{"className":65352,"style":3628},[1052],[31,65354,65355,65358],{"style":3631},[31,65356],{"className":65357,"style":1061},[1060],[31,65359,65361],{"className":65360},[1065,1066,1067,1068],[31,65362,2515],{"className":65363,"style":2666},[75,76,1068]," 不是某个数，而是函数空间中的一个点。算子 ",[31,65366,65368,65385],{"className":65367,"translate":35},[34],[31,65369,65371],{"className":65370},[39],[41,65372,65373],{"xmlns":43},[45,65374,65375,65383],{},[48,65376,65377],{},[1180,65378,65379,65381],{},[51,65380,919],{"mathvariant":925},[51,65382,2515],{},[55,65384,64428],{"encoding":57},[31,65386,65388],{"className":65387,"ariaHidden":62},[61],[31,65389,65391,65394],{"className":65390},[66],[31,65392],{"className":65393,"style":945},[70],[31,65395,65397,65400],{"className":65396},[75],[31,65398,919],{"className":65399,"style":64444},[75,968],[31,65401,65403],{"className":65402},[1039],[31,65404,65406],{"className":65405},[1043],[31,65407,65409],{"className":65408},[1048],[31,65410,65412],{"className":65411,"style":3628},[1052],[31,65413,65414,65417],{"style":3631},[31,65415],{"className":65416,"style":1061},[1060],[31,65418,65420],{"className":65419},[1065,1066,1067,1068],[31,65421,2515],{"className":65422,"style":2666},[75,76,1068]," 把一个有界函数 ",[31,65425,65427,65440],{"className":65426,"translate":35},[34],[31,65428,65430],{"className":65429},[39],[41,65431,65432],{"xmlns":43},[45,65433,65434,65438],{},[48,65435,65436],{},[51,65437,43481],{},[55,65439,43481],{"encoding":57},[31,65441,65443],{"className":65442,"ariaHidden":62},[61],[31,65444,65446,65449],{"className":65445},[66],[31,65447],{"className":65448,"style":945},[70],[31,65450,43481],{"className":65451,"style":120},[75,76]," 映射为另一个有界函数 ",[31,65454,65456,65476],{"className":65455,"translate":35},[34],[31,65457,65459],{"className":65458},[39],[41,65460,65461],{"xmlns":43},[45,65462,65463,65473],{},[48,65464,65465,65471],{},[1180,65466,65467,65469],{},[51,65468,919],{"mathvariant":925},[51,65470,2515],{},[51,65472,43481],{},[55,65474,65475],{"encoding":57},"\\mathcal{T}^\\pi V",[31,65477,65479],{"className":65478,"ariaHidden":62},[61],[31,65480,65482,65485,65514],{"className":65481},[66],[31,65483],{"className":65484,"style":945},[70],[31,65486,65488,65491],{"className":65487},[75],[31,65489,919],{"className":65490,"style":64444},[75,968],[31,65492,65494],{"className":65493},[1039],[31,65495,65497],{"className":65496},[1043],[31,65498,65500],{"className":65499},[1048],[31,65501,65503],{"className":65502,"style":3628},[1052],[31,65504,65505,65508],{"style":3631},[31,65506],{"className":65507,"style":1061},[1060],[31,65509,65511],{"className":65510},[1065,1066,1067,1068],[31,65512,2515],{"className":65513,"style":2666},[75,76,1068],[31,65515,43481],{"className":65516,"style":120},[75,76],"。因此，求解贝尔曼方程等价于寻找算子 ",[31,65519,65521,65538],{"className":65520,"translate":35},[34],[31,65522,65524],{"className":65523},[39],[41,65525,65526],{"xmlns":43},[45,65527,65528,65536],{},[48,65529,65530],{},[1180,65531,65532,65534],{},[51,65533,919],{"mathvariant":925},[51,65535,2515],{},[55,65537,64428],{"encoding":57},[31,65539,65541],{"className":65540,"ariaHidden":62},[61],[31,65542,65544,65547],{"className":65543},[66],[31,65545],{"className":65546,"style":945},[70],[31,65548,65550,65553],{"className":65549},[75],[31,65551,919],{"className":65552,"style":64444},[75,968],[31,65554,65556],{"className":65555},[1039],[31,65557,65559],{"className":65558},[1043],[31,65560,65562],{"className":65561},[1048],[31,65563,65565],{"className":65564,"style":3628},[1052],[31,65566,65567,65570],{"style":3631},[31,65568],{"className":65569,"style":1061},[1060],[31,65571,65573],{"className":65572},[1065,1066,1067,1068],[31,65574,2515],{"className":65575,"style":2666},[75,76,1068]," 的不动点。",[52860,65578,65580],{"id":65579},"_23-有限状态下的矩阵解","2.3 有限状态下的矩阵解",[11,65582,65583,65584,65674,65675,18171,65757,65869,65870,4809],{},"当 ",[31,65585,65587,65617],{"className":65586,"translate":35},[34],[31,65588,65590],{"className":65589},[39],[41,65591,65592],{"xmlns":43},[45,65593,65594,65614],{},[48,65595,65596,65598,65600,65602,65604,65606,65608,65610,65612],{},[51,65597,21582],{},[95,65599,223],{},[95,65601,2512],{"stretchy":210},[91,65603,205],{},[95,65605,1210],{"separator":62},[95,65607,55349],{},[95,65609,1210],{"separator":62},[51,65611,14559],{},[95,65613,2604],{"stretchy":210},[55,65615,65616],{"encoding":57},"S=\\{1,\\dots,n\\}",[31,65618,65620,65638],{"className":65619,"ariaHidden":62},[61],[31,65621,65623,65626,65629,65632,65635],{"className":65622},[66],[31,65624],{"className":65625,"style":945},[70],[31,65627,21582],{"className":65628,"style":21697},[75,76],[31,65630],{"className":65631,"style":293},[119],[31,65633,223],{"className":65634},[297],[31,65636],{"className":65637,"style":293},[119],[31,65639,65641,65644,65647,65650,65653,65656,65659,65662,65665,65668,65671],{"className":65640},[66],[31,65642],{"className":65643,"style":260},[70],[31,65645,2512],{"className":65646},[264],[31,65648,205],{"className":65649},[75],[31,65651,1210],{"className":65652},[1391],[31,65654],{"className":65655,"style":1467},[119],[31,65657,55349],{"className":65658},[3860],[31,65660],{"className":65661,"style":1467},[119],[31,65663,1210],{"className":65664},[1391],[31,65666],{"className":65667,"style":1467},[119],[31,65669,14559],{"className":65670},[75,76],[31,65672,2604],{"className":65673},[289]," 为有限集时，贝尔曼期望方程可以写成线性方程组。定义向量 ",[31,65676,65678,65700],{"className":65677,"translate":35},[34],[31,65679,65681],{"className":65680},[39],[41,65682,65683],{"xmlns":43},[45,65684,65685,65697],{},[48,65686,65687,65689,65691],{},[51,65688,43481],{},[95,65690,2518],{},[1180,65692,65693,65695],{},[51,65694,14790],{"mathvariant":14789},[51,65696,14559],{},[55,65698,65699],{"encoding":57},"V\\in\\mathbb{R}^n",[31,65701,65703,65722],{"className":65702,"ariaHidden":62},[61],[31,65704,65706,65710,65713,65716,65719],{"className":65705},[66],[31,65707],{"className":65708,"style":65709},[70],"height:0.7224em;vertical-align:-0.0391em;",[31,65711,43481],{"className":65712,"style":120},[75,76],[31,65714],{"className":65715,"style":293},[119],[31,65717,2518],{"className":65718},[297],[31,65720],{"className":65721,"style":293},[119],[31,65723,65725,65728],{"className":65724},[66],[31,65726],{"className":65727,"style":53279},[70],[31,65729,65731,65734],{"className":65730},[75],[31,65732,14790],{"className":65733},[75,14947],[31,65735,65737],{"className":65736},[1039],[31,65738,65740],{"className":65739},[1043],[31,65741,65743],{"className":65742},[1048],[31,65744,65746],{"className":65745,"style":3628},[1052],[31,65747,65748,65751],{"style":3631},[31,65749],{"className":65750,"style":1061},[1060],[31,65752,65754],{"className":65753},[1065,1066,1067,1068],[31,65755,14559],{"className":65756},[75,76,1068],[31,65758,65760,65786],{"className":65759,"translate":35},[34],[31,65761,65763],{"className":65762},[39],[41,65764,65765],{"xmlns":43},[45,65766,65767,65783],{},[48,65768,65769,65775,65777],{},[1180,65770,65771,65773],{},[51,65772,2549],{},[51,65774,2515],{},[95,65776,2518],{},[1180,65778,65779,65781],{},[51,65780,14790],{"mathvariant":14789},[51,65782,14559],{},[55,65784,65785],{"encoding":57},"r^\\pi\\in\\mathbb{R}^n",[31,65787,65789,65834],{"className":65788,"ariaHidden":62},[61],[31,65790,65792,65796,65825,65828,65831],{"className":65791},[66],[31,65793],{"className":65794,"style":65795},[70],"height:0.7035em;vertical-align:-0.0391em;",[31,65797,65799,65802],{"className":65798},[75],[31,65800,2549],{"className":65801,"style":19759},[75,76],[31,65803,65805],{"className":65804},[1039],[31,65806,65808],{"className":65807},[1043],[31,65809,65811],{"className":65810},[1048],[31,65812,65814],{"className":65813,"style":3628},[1052],[31,65815,65816,65819],{"style":3631},[31,65817],{"className":65818,"style":1061},[1060],[31,65820,65822],{"className":65821},[1065,1066,1067,1068],[31,65823,2515],{"className":65824,"style":2666},[75,76,1068],[31,65826],{"className":65827,"style":293},[119],[31,65829,2518],{"className":65830},[297],[31,65832],{"className":65833,"style":293},[119],[31,65835,65837,65840],{"className":65836},[66],[31,65838],{"className":65839,"style":53279},[70],[31,65841,65843,65846],{"className":65842},[75],[31,65844,14790],{"className":65845},[75,14947],[31,65847,65849],{"className":65848},[1039],[31,65850,65852],{"className":65851},[1043],[31,65853,65855],{"className":65854},[1048],[31,65856,65858],{"className":65857,"style":3628},[1052],[31,65859,65860,65863],{"style":3631},[31,65861],{"className":65862,"style":1061},[1060],[31,65864,65866],{"className":65865},[1065,1066,1067,1068],[31,65867,14559],{"className":65868},[75,76,1068],"，以及矩阵 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n}",[31,65906,65908,65952],{"className":65907,"ariaHidden":62},[61],[31,65909,65911,65914,65943,65946,65949],{"className":65910},[66],[31,65912],{"className":65913,"style":65709},[70],[31,65915,65917,65920],{"className":65916},[75],[31,65918,2521],{"className":65919,"style":949},[75,76],[31,65921,65923],{"className":65922},[1039],[31,65924,65926],{"className":65925},[1043],[31,65927,65929],{"className":65928},[1048],[31,65930,65932],{"className":65931,"style":3628},[1052],[31,65933,65934,65937],{"style":3631},[31,65935],{"className":65936,"style":1061},[1060],[31,65938,65940],{"className":65939},[1065,1066,1067,1068],[31,65941,2515],{"className":65942,"style":2666},[75,76,1068],[31,65944],{"className":65945,"style":293},[119],[31,65947,2518],{"className":65948},[297],[31,65950],{"className":65951,"style":293},[119],[31,65953,65955,65959],{"className":65954},[66],[31,65956],{"className":65957,"style":65958},[70],"height:0.7713em;",[31,65960,65962,65965],{"className":65961},[75],[31,65963,14790],{"className":65964},[75,14947],[31,65966,65968],{"className":65967},[1039],[31,65969,65971],{"className":65970},[1043],[31,65972,65974],{"className":65973},[1048],[31,65975,65977],{"className":65976,"style":65958},[1052],[31,65978,65979,65982],{"style":3631},[31,65980],{"className":65981,"style":1061},[1060],[31,65983,65985],{"className":65984},[1065,1066,1067,1068],[31,65986,65988,65991,65994],{"className":65987},[75,1068],[31,65989,14559],{"className":65990},[75,76,1068],[31,65992,2528],{"className":65993},[124,1068],[31,65995,14559],{"className":65996},[75,76,1068],[31,65998,66000],{"className":65999,"translate":35},[411],[31,66001,66003,66193],{"className":66002,"translate":35},[34],[31,66004,66006],{"className":66005},[39],[41,66007,66008],{"xmlns":43,"display":420},[45,66009,66010,66190],{},[5606,66011,66012,66044,66126],{"rowspacing":5608,"columnalign":5609,"columnspacing":5610},[5612,66013,66014,66024],{},[5615,66015,66016],{},[2071,66017,66018],{"scriptlevel":1977,"displaystyle":62},[1004,66019,66020,66022],{},[51,66021,43481],{},[51,66023,5391],{},[5615,66025,66026],{},[2071,66027,66028],{"scriptlevel":1977,"displaystyle":62},[48,66029,66030,66032,66034,66036,66038,66040,66042],{},[48,66031],{},[95,66033,223],{},[51,66035,43481],{},[95,66037,211],{"stretchy":210},[51,66039,5391],{},[95,66041,220],{"stretchy":210},[95,66043,1210],{"separator":62},[5612,66045,66046,66058],{},[5615,66047,66048],{},[2071,66049,66050],{"scriptlevel":1977,"displaystyle":62},[12397,66051,66052,66054,66056],{},[51,66053,2549],{},[51,66055,5391],{},[51,66057,2515],{},[5615,66059,66060],{},[2071,66061,66062],{"scriptlevel":1977,"displaystyle":62},[48,66063,66064,66066,66068,66074,66076,66078,66080,66082,66084,66086,66092,66094,66096,66098,66100,66102,66104,66106,66108,66110,66112,66114,66116,66118,66120,66122,66124],{},[48,66065],{},[95,66067,223],{},[1531,66069,66070,66072],{},[95,66071,5625],{},[51,66073,5813],{},[51,66075,2515],{},[95,66077,211],{"stretchy":210},[51,66079,5813],{},[95,66081,12680],{},[51,66083,5391],{},[95,66085,220],{"stretchy":210},[1531,66087,66088,66090],{},[95,66089,5625],{},[51,66091,2555],{},[51,66093,2521],{},[95,66095,211],{"stretchy":210},[51,66097,2555],{},[95,66099,12680],{},[51,66101,5391],{},[95,66103,1210],{"separator":62},[51,66105,5813],{},[95,66107,220],{"stretchy":210},[51,66109,2549],{},[95,66111,211],{"stretchy":210},[51,66113,5391],{},[95,66115,1210],{"separator":62},[51,66117,5813],{},[95,66119,1210],{"separator":62},[51,66121,2555],{},[95,66123,220],{"stretchy":210},[95,66125,1210],{"separator":62},[5612,66127,66128,66144],{},[5615,66129,66130],{},[2071,66131,66132],{"scriptlevel":1977,"displaystyle":62},[12397,66133,66134,66136,66142],{},[51,66135,2521],{},[48,66137,66138,66140],{},[51,66139,5391],{},[51,66141,2555],{},[51,66143,2515],{},[5615,66145,66146],{},[2071,66147,66148],{"scriptlevel":1977,"displaystyle":62},[48,66149,66150,66152,66154,66160,66162,66164,66166,66168,66170,66172,66174,66176,66178,66180,66182,66184,66186,66188],{},[48,66151],{},[95,66153,223],{},[1531,66155,66156,66158],{},[95,66157,5625],{},[51,66159,5813],{},[51,66161,2515],{},[95,66163,211],{"stretchy":210},[51,66165,5813],{},[95,66167,12680],{},[51,66169,5391],{},[95,66171,220],{"stretchy":210},[51,66173,2521],{},[95,66175,211],{"stretchy":210},[51,66177,2555],{},[95,66179,12680],{},[51,66181,5391],{},[95,66183,1210],{"separator":62},[51,66185,5813],{},[95,66187,220],{"stretchy":210},[51,66189,495],{"mathvariant":494},[55,66191,66192],{"encoding":57},"\\begin{aligned}\nV_i&=V(i),\\\\\nr^\\pi_i&=\\sum_{a}\\pi(a\\mid 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级数）",[11,67188,52871,67189,67262,67263,67349],{},[31,67190,67192,67214],{"className":67191,"translate":35},[34],[31,67193,67195],{"className":67194},[39],[41,67196,67197],{"xmlns":43},[45,67198,67199,67211],{},[48,67200,67201,67203,67205,67207,67209],{},[91,67202,1977],{},[95,67204,347],{},[51,67206,53298],{},[95,67208,9758],{},[91,67210,205],{},[55,67212,67213],{"encoding":57},"0\\le\\gamma\u003C1",[31,67215,67217,67235,67253],{"className":67216,"ariaHidden":62},[61],[31,67218,67220,67223,67226,67229,67232],{"className":67219},[66],[31,67221],{"className":67222,"style":366},[70],[31,67224,1977],{"className":67225},[75],[31,67227],{"className":67228,"style":293},[119],[31,67230,347],{"className":67231},[297],[31,67233],{"className":67234,"style":293},[119],[31,67236,67238,67241,67244,67247,67250],{"className":67237},[66],[31,67239],{"className":67240,"style":53323},[70],[31,67242,53298],{"className":67243,"style":24157},[75,76],[31,67245],{"className":67246,"style":293},[119],[31,67248,9758],{"className":67249},[297],[31,67251],{"className":67252,"style":293},[119],[31,67254,67256,67259],{"className":67255},[66],[31,67257],{"className":67258,"style":326},[70],[31,67260,205],{"className":67261},[75],"。则 ",[31,67264,67266,67290],{"className":67265,"translate":35},[34],[31,67267,67269],{"className":67268},[39],[41,67270,67271],{"xmlns":43},[45,67272,67273,67287],{},[48,67274,67275,67277,67279,67281],{},[51,67276,34619],{},[95,67278,97],{},[51,67280,53298],{},[1180,67282,67283,67285],{},[51,67284,2521],{},[51,67286,2515],{},[55,67288,67289],{"encoding":57},"I-\\gamma P^\\pi",[31,67291,67293,67311],{"className":67292,"ariaHidden":62},[61],[31,67294,67296,67299,67302,67305,67308],{"className":67295},[66],[31,67297],{"className":67298,"style":53224},[70],[31,67300,34619],{"className":67301,"style":30311},[75,76],[31,67303],{"className":67304,"style":120},[119],[31,67306,97],{"className":67307},[124],[31,67309],{"className":67310,"style":120},[119],[31,67312,67314,67317,67320],{"className":67313},[66],[31,67315],{"className":67316,"style":59114},[70],[31,67318,53298],{"className":67319,"style":24157},[75,76],[31,67321,67323,67326],{"className":67322},[75],[31,67324,2521],{"className":67325,"style":949},[75,76],[31,67327,67329],{"className":67328},[1039],[31,67330,67332],{"className":67331},[1043],[31,67333,67335],{"className":67334},[1048],[31,67336,67338],{"className":67337,"style":3628},[1052],[31,67339,67340,67343],{"style":3631},[31,67341],{"className":67342,"style":1061},[1060],[31,67344,67346],{"className":67345},[1065,1066,1067,1068],[31,67347,2515],{"className":67348,"style":2666},[75,76,1068]," 可逆，且",[31,67351,67353],{"className":67352,"translate":35},[411],[31,67354,67356,67450],{"className":67355,"translate":35},[34],[31,67357,67359],{"className":67358},[39],[41,67360,67361],{"xmlns":43,"display":420},[45,67362,67363,67447],{},[48,67364,67365,67371,67373,67375,67377,67379,67381,67387,67397,67403,67405,67419,67425,67427,67433,67439,67445],{},[1180,67366,67367,67369],{},[51,67368,43481],{},[51,67370,2515],{},[95,67372,223],{},[95,67374,211],{"stretchy":210},[51,67376,34619],{},[95,67378,97],{},[51,67380,53298],{},[1180,67382,67383,67385],{},[51,67384,2521],{},[51,67386,2515],{},[1180,67388,67389,67391],{},[95,67390,220],{"stretchy":210},[48,67392,67393,67395],{},[95,67394,97],{},[91,67396,205],{},[1180,67398,67399,67401],{},[51,67400,2549],{},[51,67402,2515],{},[95,67404,223],{},[42429,67406,67407,67409,67417],{},[95,67408,5625],{},[48,67410,67411,67413,67415],{},[51,67412,52065],{},[95,67414,223],{},[91,67416,1977],{},[51,67418,4587],{"mathvariant":494},[1180,67420,67421,67423],{},[51,67422,53298],{},[51,67424,52065],{},[95,67426,211],{"stretchy":210},[1180,67428,67429,67431],{},[51,67430,2521],{},[51,67432,2515],{},[1180,67434,67435,67437],{},[95,67436,220],{"stretchy":210},[51,67438,52065],{},[1180,67440,67441,67443],{},[51,67442,2549],{},[51,67444,2515],{},[51,67446,495],{"mathvariant":494},[55,67448,67449],{"encoding":57},"V^\\pi=(I-\\gamma P^\\pi)^{-1}r^\\pi\n=\\sum_{k=0}^{\\infty}\\gamma^k(P^\\pi)^k r^\\pi.",[31,67451,67453,67497,67518,67629],{"className":67452,"ariaHidden":62},[61],[31,67454,67456,67459,67488,67491,67494],{"className":67455},[66],[31,67457],{"className":67458,"style":3987},[70],[31,67460,67462,67465],{"className":67461},[75],[31,67463,43481],{"className":67464,"style":120},[75,76],[31,67466,67468],{"className":67467},[1039],[31,67469,67471],{"className":67470},[1043],[31,67472,67474],{"className":67473},[1048],[31,67475,67477],{"className":67476,"style":3987},[1052],[31,67478,67479,67482],{"style":1330},[31,67480],{"className":67481,"style":1061},[1060],[31,67483,67485],{"className":67484},[1065,1066,1067,1068],[31,67486,2515],{"className":67487,"style":2666},[75,76,1068],[31,67489],{"className":67490,"style":293},[119],[31,67492,223],{"className":67493},[297],[31,67495],{"className":67496,"style":293},[119],[31,67498,67500,67503,67506,67509,67512,67515],{"className":67499},[66],[31,67501],{"className":67502,"style":260},[70],[31,67504,211],{"className":67505},[264],[31,67507,34619],{"className":67508,"style":30311},[75,76],[31,67510],{"className":67511,"style":120},[119],[31,67513,97],{"className":67514},[124],[31,67516],{"className":67517,"style":120},[119],[31,67519,67521,67524,67527,67556,67591,67620,67623,67626],{"className":67520},[66],[31,67522],{"className":67523,"style":16882},[70],[31,67525,53298],{"className":67526,"style":24157},[75,76],[31,67528,67530,67533],{"className":67529},[75],[31,67531,2521],{"className":67532,"style":949},[75,76],[31,67534,67536],{"className":67535},[1039],[31,67537,67539],{"className":67538},[1043],[31,67540,67542],{"className":67541},[1048],[31,67543,67545],{"className":67544,"style":3987},[1052],[31,67546,67547,67550],{"style":1330},[31,67548],{"className":67549,"style":1061},[1060],[31,67551,67553],{"className":67552},[1065,1066,1067,1068],[31,67554,2515],{"className":67555,"style":2666},[75,76,1068],[31,67557,67559,67562],{"className":67558},[289],[31,67560,220],{"className":67561},[289],[31,67563,67565],{"className":67564},[1039],[31,67566,67568],{"className":67567},[1043],[31,67569,67571],{"className":67570},[1048],[31,67572,67574],{"className":67573,"style":1327},[1052],[31,67575,67576,67579],{"style":1330},[31,67577],{"className":67578,"style":1061},[1060],[31,67580,67582],{"className":67581},[1065,1066,1067,1068],[31,67583,67585,67588],{"className":67584},[75,1068],[31,67586,97],{"className":67587},[75,1068],[31,67589,205],{"className":67590},[75,1068],[31,67592,67594,67597],{"className":67593},[75],[31,67595,2549],{"className":67596,"style":19759},[75,76],[31,67598,67600],{"className":67599},[1039],[31,67601,67603],{"className":67602},[1043],[31,67604,67606],{"className":67605},[1048],[31,67607,67609],{"className":67608,"style":3987},[1052],[31,67610,67611,67614],{"style":1330},[31,67612],{"className":67613,"style":1061},[1060],[31,67615,67617],{"className":67616},[1065,1066,1067,1068],[31,67618,2515],{"className":67619,"style":2666},[75,76,1068],[31,67621],{"className":67622,"style":293},[119],[31,67624,223],{"className":67625},[297],[31,67627],{"className":67628,"style":293},[119],[31,67630,67632,67635,67702,67705,67734,67737,67766,67795,67824],{"className":67631},[66],[31,67633],{"className":67634,"style":55989},[70],[31,67636,67638],{"className":67637},[1460,1620],[31,67639,67641,67694],{"className":67640},[1043,1044],[31,67642,67644,67691],{"className":67643},[1048],[31,67645,67647,67667,67677],{"className":67646,"style":42702},[1052],[31,67648,67649,67652],{"style":56004},[31,67650],{"className":67651,"style":5907},[1060],[31,67653,67655],{"className":67654},[1065,1066,1067,1068],[31,67656,67658,67661,67664],{"className":67657},[75,1068],[31,67659,52065],{"className":67660,"style":52166},[75,76,1068],[31,67662,223],{"className":67663},[297,1068],[31,67665,1977],{"className":67666},[75,1068],[31,67668,67669,67672],{"style":5919},[31,67670],{"className":67671,"style":5907},[1060],[31,67673,67674],{},[31,67675,5625],{"className":67676},[1460,1461,1687],[31,67678,67679,67682],{"style":42735},[31,67680],{"className":67681,"style":5907},[1060],[31,67683,67685],{"className":67684},[1065,1066,1067,1068],[31,67686,67688],{"className":67687},[75,1068],[31,67689,4587],{"className":67690},[75,1068],[31,67692,1076],{"className":67693},[1075],[31,67695,67697],{"className":67696},[1048],[31,67698,67700],{"className":67699,"style":56056},[1052],[31,67701],{},[31,67703],{"className":67704,"style":1467},[119],[31,67706,67708,67711],{"className":67707},[75],[31,67709,53298],{"className":67710,"style":24157},[75,76],[31,67712,67714],{"className":67713},[1039],[31,67715,67717],{"className":67716},[1043],[31,67718,67720],{"className":67719},[1048],[31,67721,67723],{"className":67722,"style":29642},[1052],[31,67724,67725,67728],{"style":1330},[31,67726],{"className":67727,"style":1061},[1060],[31,67729,67731],{"className":67730},[1065,1066,1067,1068],[31,67732,52065],{"className":67733,"style":52166},[75,76,1068],[31,67735,211],{"className":67736},[264],[31,67738,67740,67743],{"className":67739},[75],[31,67741,2521],{"className":67742,"style":949},[75,76],[31,67744,67746],{"className":67745},[1039],[31,67747,67749],{"className":67748},[1043],[31,67750,67752],{"className":67751},[1048],[31,67753,67755],{"className":67754,"style":3987},[1052],[31,67756,67757,67760],{"style":1330},[31,67758],{"className":67759,"style":1061},[1060],[31,67761,67763],{"className":67762},[1065,1066,1067,1068],[31,67764,2515],{"className":67765,"style":2666},[75,76,1068],[31,67767,67769,67772],{"className":67768},[289],[31,67770,220],{"className":67771},[289],[31,67773,67775],{"className":67774},[1039],[31,67776,67778],{"className":67777},[1043],[31,67779,67781],{"className":67780},[1048],[31,67782,67784],{"className":67783,"style":29642},[1052],[31,67785,67786,67789],{"style":1330},[31,67787],{"className":67788,"style":1061},[1060],[31,67790,67792],{"className":67791},[1065,1066,1067,1068],[31,67793,52065],{"className":67794,"style":52166},[75,76,1068],[31,67796,67798,67801],{"className":67797},[75],[31,67799,2549],{"className":67800,"style":19759},[75,76],[31,67802,67804],{"className":67803},[1039],[31,67805,67807],{"className":67806},[1043],[31,67808,67810],{"className":67809},[1048],[31,67811,67813],{"className":67812,"style":3987},[1052],[31,67814,67815,67818],{"style":1330},[31,67816],{"className":67817,"style":1061},[1060],[31,67819,67821],{"className":67820},[1065,1066,1067,1068],[31,67822,2515],{"className":67823,"style":2666},[75,76,1068],[31,67825,495],{"className":67826},[75],[11,67828,67829,67831,67832,67918,67919,67978,67979,68007,68008,68140,68141,68304,68305,68447],{},[7246,67830,7248],{}," 记 ",[31,67833,67835,67859],{"className":67834,"translate":35},[34],[31,67836,67838],{"className":67837},[39],[41,67839,67840],{"xmlns":43},[45,67841,67842,67856],{},[48,67843,67844,67846,67848,67850],{},[51,67845,56971],{},[95,67847,223],{},[51,67849,53298],{},[1180,67851,67852,67854],{},[51,67853,2521],{},[51,67855,2515],{},[55,67857,67858],{"encoding":57},"Q=\\gamma P^\\pi",[31,67860,67862,67880],{"className":67861,"ariaHidden":62},[61],[31,67863,67865,67868,67871,67874,67877],{"className":67864},[66],[31,67866],{"className":67867,"style":59114},[70],[31,67869,56971],{"className":67870},[75,76],[31,67872],{"className":67873,"style":293},[119],[31,67875,223],{"className":67876},[297],[31,67878],{"className":67879,"style":293},[119],[31,67881,67883,67886,67889],{"className":67882},[66],[31,67884],{"className":67885,"style":59114},[70],[31,67887,53298],{"className":67888,"style":24157},[75,76],[31,67890,67892,67895],{"className":67891},[75],[31,67893,2521],{"className":67894,"style":949},[75,76],[31,67896,67898],{"className":67897},[1039],[31,67899,67901],{"className":67900},[1043],[31,67902,67904],{"className":67903},[1048],[31,67905,67907],{"className":67906,"style":3628},[1052],[31,67908,67909,67912],{"style":3631},[31,67910],{"className":67911,"style":1061},[1060],[31,67913,67915],{"className":67914},[1065,1066,1067,1068],[31,67916,2515],{"className":67917,"style":2666},[75,76,1068],"。由于 ",[31,67920,67922,67940],{"className":67921,"translate":35},[34],[31,67923,67925],{"className":67924},[39],[41,67926,67927],{"xmlns":43},[45,67928,67929,67937],{},[48,67930,67931],{},[1180,67932,67933,67935],{},[51,67934,2521],{},[51,67936,2515],{},[55,67938,67939],{"encoding":57},"P^\\pi",[31,67941,67943],{"className":67942,"ariaHidden":62},[61],[31,67944,67946,67949],{"className":67945},[66],[31,67947],{"className":67948,"style":945},[70],[31,67950,67952,67955],{"className":67951},[75],[31,67953,2521],{"className":67954,"style":949},[75,76],[31,67956,67958],{"className":67957},[1039],[31,67959,67961],{"className":67960},[1043],[31,67962,67964],{"className":67963},[1048],[31,67965,67967],{"className":67966,"style":3628},[1052],[31,67968,67969,67972],{"style":3631},[31,67970],{"className":67971,"style":1061},[1060],[31,67973,67975],{"className":67974},[1065,1066,1067,1068],[31,67976,2515],{"className":67977,"style":2666},[75,76,1068]," 的每一行之和为 ",[31,67980,67982,67995],{"className":67981,"translate":35},[34],[31,67983,67985],{"className":67984},[39],[41,67986,67987],{"xmlns":43},[45,67988,67989,67993],{},[48,67990,67991],{},[91,67992,205],{},[55,67994,205],{"encoding":57},[31,67996,67998],{"className":67997,"ariaHidden":62},[61],[31,67999,68001,68004],{"className":68000},[66],[31,68002],{"className":68003,"style":326},[70],[31,68005,205],{"className":68006},[75],"，有 ",[31,68009,68011,68041],{"className":68010,"translate":35},[34],[31,68012,68014],{"className":68013},[39],[41,68015,68016],{"xmlns":43},[45,68017,68018,68038],{},[48,68019,68020,68022,68028,68034,68036],{},[51,68021,4572],{"mathvariant":494},[1180,68023,68024,68026],{},[51,68025,2521],{},[51,68027,2515],{},[1004,68029,68030,68032],{},[51,68031,4572],{"mathvariant":494},[51,68033,4587],{"mathvariant":494},[95,68035,223],{},[91,68037,205],{},[55,68039,68040],{"encoding":57},"\\|P^\\pi\\|_\\infty=1",[31,68042,68044,68131],{"className":68043,"ariaHidden":62},[61],[31,68045,68047,68050,68053,68082,68122,68125,68128],{"className":68046},[66],[31,68048],{"className":68049,"style":260},[70],[31,68051,4572],{"className":68052},[75],[31,68054,68056,68059],{"className":68055},[75],[31,68057,2521],{"className":68058,"style":949},[75,76],[31,68060,68062],{"className":68061},[1039],[31,68063,68065],{"className":68064},[1043],[31,68066,68068],{"className":68067},[1048],[31,68069,68071],{"className":68070,"style":3628},[1052],[31,68072,68073,68076],{"style":3631},[31,68074],{"className":68075,"style":1061},[1060],[31,68077,68079],{"className":68078},[1065,1066,1067,1068],[31,68080,2515],{"className":68081,"style":2666},[75,76,1068],[31,68083,68085,68088],{"className":68084},[75],[31,68086,4572],{"className":68087},[75],[31,68089,68091],{"className":68090},[1039],[31,68092,68094,68114],{"className":68093},[1043,1044],[31,68095,68097,68111],{"className":68096},[1048],[31,68098,68100],{"className":68099,"style":3230},[1052],[31,68101,68102,68105],{"style":2807},[31,68103],{"className":68104,"style":1061},[1060],[31,68106,68108],{"className":68107},[1065,1066,1067,1068],[31,68109,4587],{"className":68110},[75,1068],[31,68112,1076],{"className":68113},[1075],[31,68115,68117],{"className":68116},[1048],[31,68118,68120],{"className":68119,"style":2050},[1052],[31,68121],{},[31,68123],{"className":68124,"style":293},[119],[31,68126,223],{"className":68127},[297],[31,68129],{"className":68130,"style":293},[119],[31,68132,68134,68137],{"className":68133},[66],[31,68135],{"className":68136,"style":326},[70],[31,68138,205],{"className":68139},[75],"，从而 ",[31,68142,68144,68178],{"className":68143,"translate":35},[34],[31,68145,68147],{"className":68146},[39],[41,68148,68149],{"xmlns":43},[45,68150,68151,68175],{},[48,68152,68153,68155,68161,68167,68169],{},[51,68154,4572],{"mathvariant":494},[1180,68156,68157,68159],{},[51,68158,56971],{},[51,68160,52065],{},[1004,68162,68163,68165],{},[51,68164,4572],{"mathvariant":494},[51,68166,4587],{"mathvariant":494},[95,68168,347],{},[1180,68170,68171,68173],{},[51,68172,53298],{},[51,68174,52065],{},[55,68176,68177],{"encoding":57},"\\|Q^k\\|_\\infty\\le\\gamma^k",[31,68179,68181,68268],{"className":68180,"ariaHidden":62},[61],[31,68182,68184,68187,68190,68219,68259,68262,68265],{"className":68183},[66],[31,68185],{"className":68186,"style":20835},[70],[31,68188,4572],{"className":68189},[75],[31,68191,68193,68196],{"className":68192},[75],[31,68194,56971],{"className":68195},[75,76],[31,68197,68199],{"className":68198},[1039],[31,68200,68202],{"className":68201},[1043],[31,68203,68205],{"className":68204},[1048],[31,68206,68208],{"className":68207,"style":20863},[1052],[31,68209,68210,68213],{"style":3631},[31,68211],{"className":68212,"style":1061},[1060],[31,68214,68216],{"className":68215},[1065,1066,1067,1068],[31,68217,52065],{"className":68218,"style":52166},[75,76,1068],[31,68220,68222,68225],{"className":68221},[75],[31,68223,4572],{"className":68224},[75],[31,68226,68228],{"className":68227},[1039],[31,68229,68231,68251],{"className":68230},[1043,1044],[31,68232,68234,68248],{"className":68233},[1048],[31,68235,68237],{"className":68236,"style":3230},[1052],[31,68238,68239,68242],{"style":2807},[31,68240],{"className":68241,"style":1061},[1060],[31,68243,68245],{"className":68244},[1065,1066,1067,1068],[31,68246,4587],{"className":68247},[75,1068],[31,68249,1076],{"className":68250},[1075],[31,68252,68254],{"className":68253},[1048],[31,68255,68257],{"className":68256,"style":2050},[1052],[31,68258],{},[31,68260],{"className":68261,"style":293},[119],[31,68263,347],{"className":68264},[297],[31,68266],{"className":68267,"style":293},[119],[31,68269,68271,68275],{"className":68270},[66],[31,68272],{"className":68273,"style":68274},[70],"height:1.0435em;vertical-align:-0.1944em;",[31,68276,68278,68281],{"className":68277},[75],[31,68279,53298],{"className":68280,"style":24157},[75,76],[31,68282,68284],{"className":68283},[1039],[31,68285,68287],{"className":68286},[1043],[31,68288,68290],{"className":68289},[1048],[31,68291,68293],{"className":68292,"style":20863},[1052],[31,68294,68295,68298],{"style":3631},[31,68296],{"className":68297,"style":1061},[1060],[31,68299,68301],{"className":68300},[1065,1066,1067,1068],[31,68302,52065],{"className":68303,"style":52166},[75,76,1068],"。因此 Neumann 级数 ",[31,68306,68308,68340],{"className":68307,"translate":35},[34],[31,68309,68311],{"className":68310},[39],[41,68312,68313],{"xmlns":43},[45,68314,68315,68337],{},[48,68316,68317,68331],{},[12397,68318,68319,68321,68329],{},[95,68320,5625],{},[48,68322,68323,68325,68327],{},[51,68324,52065],{},[95,68326,223],{},[91,68328,1977],{},[51,68330,4587],{"mathvariant":494},[1180,68332,68333,68335],{},[51,68334,56971],{},[51,68336,52065],{},[55,68338,68339],{"encoding":57},"\\sum_{k=0}^{\\infty}Q^k",[31,68341,68343],{"className":68342,"ariaHidden":62},[61],[31,68344,68346,68350,68415,68418],{"className":68345},[66],[31,68347],{"className":68348,"style":68349},[70],"height:1.1488em;vertical-align:-0.2997em;",[31,68351,68353,68356],{"className":68352},[1460],[31,68354,5625],{"className":68355,"style":21372},[1460,1461,1462],[31,68357,68359],{"className":68358},[1039],[31,68360,68362,68407],{"className":68361},[1043,1044],[31,68363,68365,68404],{"className":68364},[1048],[31,68366,68369,68389],{"className":68367,"style":68368},[1052],"height:0.8043em;",[31,68370,68371,68374],{"style":21388},[31,68372],{"className":68373,"style":1061},[1060],[31,68375,68377],{"className":68376},[1065,1066,1067,1068],[31,68378,68380,68383,68386],{"className":68379},[75,1068],[31,68381,52065],{"className":68382,"style":52166},[75,76,1068],[31,68384,223],{"className":68385},[297,1068],[31,68387,1977],{"className":68388},[75,1068],[31,68390,68392,68395],{"style":68391},"top:-3.2029em;margin-right:0.05em;",[31,68393],{"className":68394,"style":1061},[1060],[31,68396,68398],{"className":68397},[1065,1066,1067,1068],[31,68399,68401],{"className":68400},[75,1068],[31,68402,4587],{"className":68403},[75,1068],[31,68405,1076],{"className":68406},[1075],[31,68408,68410],{"className":68409},[1048],[31,68411,68413],{"className":68412,"style":21407},[1052],[31,68414],{},[31,68416],{"className":68417,"style":1467},[119],[31,68419,68421,68424],{"className":68420},[75],[31,68422,56971],{"className":68423},[75,76],[31,68425,68427],{"className":68426},[1039],[31,68428,68430],{"className":68429},[1043],[31,68431,68433],{"className":68432},[1048],[31,68434,68436],{"className":68435,"style":20863},[1052],[31,68437,68438,68441],{"style":3631},[31,68439],{"className":68440,"style":1061},[1060],[31,68442,68444],{"className":68443},[1065,1066,1067,1068],[31,68445,52065],{"className":68446,"style":52166},[75,76,1068]," 在算子范数下绝对收敛。并且",[31,68449,68451],{"className":68450,"translate":35},[411],[31,68452,68454,68521],{"className":68453,"translate":35},[34],[31,68455,68457],{"className":68456},[39],[41,68458,68459],{"xmlns":43,"display":420},[45,68460,68461,68518],{},[48,68462,68463,68465,68467,68469,68471,68473,68487,68493,68495,68497,68499,68511,68514,68516],{},[95,68464,211],{"stretchy":210},[51,68466,34619],{},[95,68468,97],{},[51,68470,56971],{},[95,68472,220],{"stretchy":210},[42429,68474,68475,68477,68485],{},[95,68476,5625],{},[48,68478,68479,68481,68483],{},[51,68480,52065],{},[95,68482,223],{},[91,68484,1977],{},[51,68486,14559],{},[1180,68488,68489,68491],{},[51,68490,56971],{},[51,68492,52065],{},[95,68494,223],{},[51,68496,34619],{},[95,68498,97],{},[1180,68500,68501,68503],{},[51,68502,56971],{},[48,68504,68505,68507,68509],{},[51,68506,14559],{},[95,68508,434],{},[91,68510,205],{},[95,68512,68513],{},"⟶",[51,68515,34619],{},[95,68517,1210],{"separator":62},[55,68519,68520],{"encoding":57},"(I-Q)\\sum_{k=0}^{n}Q^k=I-Q^{n+1}\\longrightarrow I,",[31,68522,68524,68545,68668,68686,68740],{"className":68523,"ariaHidden":62},[61],[31,68525,68527,68530,68533,68536,68539,68542],{"className":68526},[66],[31,68528],{"className":68529,"style":260},[70],[31,68531,211],{"className":68532},[264],[31,68534,34619],{"className":68535,"style":30311},[75,76],[31,68537],{"className":68538,"style":120},[119],[31,68540,97],{"className":68541},[124],[31,68543],{"className":68544,"style":120},[119],[31,68546,68548,68551,68554,68557,68560,68627,68630,68659,68662,68665],{"className":68547},[66],[31,68549],{"className":68550,"style":55989},[70],[31,68552,56971],{"className":68553},[75,76],[31,68555,220],{"className":68556},[289],[31,68558],{"className":68559,"style":1467},[119],[31,68561,68563],{"className":68562},[1460,1620],[31,68564,68566,68619],{"className":68565},[1043,1044],[31,68567,68569,68616],{"className":68568},[1048],[31,68570,68572,68592,68602],{"className":68571,"style":42702},[1052],[31,68573,68574,68577],{"style":56004},[31,68575],{"className":68576,"style":5907},[1060],[31,68578,68580],{"className":68579},[1065,1066,1067,1068],[31,68581,68583,68586,68589],{"className":68582},[75,1068],[31,68584,52065],{"className":68585,"style":52166},[75,76,1068],[31,68587,223],{"className":68588},[297,1068],[31,68590,1977],{"className":68591},[75,1068],[31,68593,68594,68597],{"style":5919},[31,68595],{"className":68596,"style":5907},[1060],[31,68598,68599],{},[31,68600,5625],{"className":68601},[1460,1461,1687],[31,68603,68604,68607],{"style":42735},[31,68605],{"className":68606,"style":5907},[1060],[31,68608,68610],{"className":68609},[1065,1066,1067,1068],[31,68611,68613],{"className":68612},[75,1068],[31,68614,14559],{"className":68615},[75,76,1068],[31,68617,1076],{"className":68618},[1075],[31,68620,68622],{"className":68621},[1048],[31,68623,68625],{"className":68624,"style":56056},[1052],[31,68626],{},[31,68628],{"className":68629,"style":1467},[119],[31,68631,68633,68636],{"className":68632},[75],[31,68634,56971],{"className":68635},[75,76],[31,68637,68639],{"className":68638},[1039],[31,68640,68642],{"className":68641},[1043],[31,68643,68645],{"className":68644},[1048],[31,68646,68648],{"className":68647,"style":29642},[1052],[31,68649,68650,68653],{"style":1330},[31,68651],{"className":68652,"style":1061},[1060],[31,68654,68656],{"className":68655},[1065,1066,1067,1068],[31,68657,52065],{"className":68658,"style":52166},[75,76,1068],[31,68660],{"className":68661,"style":293},[119],[31,68663,223],{"className":68664},[297],[31,68666],{"className":68667,"style":293},[119],[31,68669,68671,68674,68677,68680,68683],{"className":68670},[66],[31,68672],{"className":68673,"style":53224},[70],[31,68675,34619],{"className":68676,"style":30311},[75,76],[31,68678],{"className":68679,"style":120},[119],[31,68681,97],{"className":68682},[124],[31,68684],{"className":68685,"style":120},[119],[31,68687,68689,68693,68731,68734,68737],{"className":68688},[66],[31,68690],{"className":68691,"style":68692},[70],"height:1.0585em;vertical-align:-0.1944em;",[31,68694,68696,68699],{"className":68695},[75],[31,68697,56971],{"className":68698},[75,76],[31,68700,68702],{"className":68701},[1039],[31,68703,68705],{"className":68704},[1043],[31,68706,68708],{"className":68707},[1048],[31,68709,68711],{"className":68710,"style":1327},[1052],[31,68712,68713,68716],{"style":1330},[31,68714],{"className":68715,"style":1061},[1060],[31,68717,68719],{"className":68718},[1065,1066,1067,1068],[31,68720,68722,68725,68728],{"className":68721},[75,1068],[31,68723,14559],{"className":68724},[75,76,1068],[31,68726,434],{"className":68727},[124,1068],[31,68729,205],{"className":68730},[75,1068],[31,68732],{"className":68733,"style":293},[119],[31,68735,68513],{"className":68736},[297],[31,68738],{"className":68739,"style":293},[119],[31,68741,68743,68746,68749],{"className":68742},[66],[31,68744],{"className":68745,"style":59114},[70],[31,68747,34619],{"className":68748,"style":30311},[75,76],[31,68750,1210],{"className":68751},[1391],[11,68753,68754,68755,67349],{},"故 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2.1",[31,69059,69061,69078],{"className":69060,"translate":35},[34],[31,69062,69064],{"className":69063},[39],[41,69065,69066],{"xmlns":43},[45,69067,69068,69076],{},[48,69069,69070],{},[1180,69071,69072,69074],{},[51,69073,43481],{},[51,69075,2515],{},[55,69077,59045],{"encoding":57},[31,69079,69081],{"className":69080,"ariaHidden":62},[61],[31,69082,69084,69087],{"className":69083},[66],[31,69085],{"className":69086,"style":945},[70],[31,69088,69090,69093],{"className":69089},[75],[31,69091,43481],{"className":69092,"style":120},[75,76],[31,69094,69096],{"className":69095},[1039],[31,69097,69099],{"className":69098},[1043],[31,69100,69102],{"className":69101},[1048],[31,69103,69105],{"className":69104,"style":3628},[1052],[31,69106,69107,69110],{"style":3631},[31,69108],{"className":69109,"style":1061},[1060],[31,69111,69113],{"className":69112},[1065,1066,1067,1068],[31,69114,2515],{"className":69115,"style":2666},[75,76,1068]," 满足 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r^\\pi.\n\\end{aligned}",[31,69391,69393],{"className":69392,"ariaHidden":62},[61],[31,69394,69396,69400],{"className":69395},[66],[31,69397],{"className":69398,"style":69399},[70],"height:4.4776em;vertical-align:-1.9888em;",[31,69401,69403],{"className":69402},[75],[31,69404,69406,69478],{"className":69405},[5606],[31,69407,69409],{"className":69408},[5867],[31,69410,69412,69469],{"className":69411},[1043,1044],[31,69413,69415,69466],{"className":69414},[1048],[31,69416,69419,69457],{"className":69417,"style":69418},[1052],"height:2.4888em;",[31,69420,69422,69425],{"style":69421},"top:-5.2761em;",[31,69423],{"className":69424,"style":58445},[1060],[31,69426,69428],{"className":69427},[75],[31,69429,69431,69434],{"className":69430},[75],[31,69432,43481],{"className":69433,"style":120},[75,76],[31,69435,69437],{"className":69436},[1039],[31,69438,69440],{"className":69439},[1043],[31,69441,69443],{"className":69442},[1048],[31,69444,69446],{"className":69445,"style":3987},[1052],[31,69447,69448,69451],{"style":1330},[31,69449],{"className":69450,"style":1061},[1060],[31,69452,69454],{"className":69453},[1065,1066,1067,1068],[31,69455,2515],{"className":69456,"style":2666},[75,76,1068],[31,69458,69460,69463],{"style":69459},"top:-2.9647em;",[31,69461],{"className":69462,"style":58445},[1060],[31,69464],{"className":69465},[75],[31,69467,1076],{"className":69468},[1075],[31,69470,69472],{"className":69471},[1048],[31,69473,69476],{"className":69474,"style":69475},[1052],"height:1.9888em;",[31,69477],{},[31,69479,69481],{"className":69480},[6062],[31,69482,69484,69807],{"className":69483},[1043,1044],[31,69485,69487,69804],{"className":69486},[1048],[31,69488,69490,69592],{"className":69489,"style":69418},[1052],[31,69491,69492,69495],{"style":69421},[31,69493],{"className":69494,"style":58445},[1060],[31,69496,69498,69501,69504,69507,69510,69513,69516,69519,69522,69525,69528,69563],{"className":69497},[75],[31,69499],{"className":69500},[75],[31,69502],{"className":69503,"style":293},[119],[31,69505,223],{"className":69506},[297],[31,69508],{"className":69509,"style":293},[119],[31,69511,211],{"className":69512},[264],[31,69514,34619],{"className":69515,"style":30311},[75,76],[31,69517],{"className":69518,"style":120},[119],[31,69520,97],{"className":69521},[124],[31,69523],{"className":69524,"style":120},[119],[31,69526,56971],{"className":69527},[75,76],[31,69529,69531,69534],{"className":69530},[289],[31,69532,220],{"className":69533},[289],[31,69535,69537],{"className":69536},[1039],[31,69538,69540],{"className":69539},[1043],[31,69541,69543],{"className":69542},[1048],[31,69544,69546],{"className":69545,"style":1327},[1052],[31,69547,69548,69551],{"style":1330},[31,69549],{"className":69550,"style":1061},[1060],[31,69552,69554],{"className":69553},[1065,1066,1067,1068],[31,69555,69557,69560],{"className":69556},[75,1068],[31,69558,97],{"className":69559},[75,1068],[31,69561,205],{"className":69562},[75,1068],[31,69564,69566,69569],{"className":69565},[75],[31,69567,2549],{"className":69568,"style":19759},[75,76],[31,69570,69572],{"className":69571},[1039],[31,69573,69575],{"className":69574},[1043],[31,69576,69578],{"className":69577},[1048],[31,69579,69581],{"className":69580,"style":3987},[1052],[31,69582,69583,69586],{"style":1330},[31,69584],{"className":69585,"style":1061},[1060],[31,69587,69589],{"className":69588},[1065,1066,1067,1068],[31,69590,2515],{"className":69591,"style":2666},[75,76,1068],[31,69593,69594,69597],{"style":69459},[31,69595],{"className":69596,"style":58445},[1060],[31,69598,69600,69603,69606,69609,69612,69679,69682,69711,69714,69743,69772,69801],{"className":69599},[75],[31,69601],{"className":69602},[75],[31,69604],{"className":69605,"style":293},[119],[31,69607,223],{"className":69608},[297],[31,69610],{"className":69611,"style":293},[119],[31,69613,69615],{"className":69614},[1460,1620],[31,69616,69618,69671],{"className":69617},[1043,1044],[31,69619,69621,69668],{"className":69620},[1048],[31,69622,69624,69644,69654],{"className":69623,"style":42702},[1052],[31,69625,69626,69629],{"style":56004},[31,69627],{"className":69628,"style":5907},[1060],[31,69630,69632],{"className":69631},[1065,1066,1067,1068],[31,69633,69635,69638,69641],{"className":69634},[75,1068],[31,69636,52065],{"className":69637,"style":52166},[75,76,1068],[31,69639,223],{"className":69640},[297,1068],[31,69642,1977],{"className":69643},[75,1068],[31,69645,69646,69649],{"style":5919},[31,69647],{"className":69648,"style":5907},[1060],[31,69650,69651],{},[31,69652,5625],{"className":69653},[1460,1461,1687],[31,69655,69656,69659],{"style":42735},[31,69657],{"className":69658,"style":5907},[1060],[31,69660,69662],{"className":69661},[1065,1066,1067,1068],[31,69663,69665],{"className":69664},[75,1068],[31,69666,4587],{"className":69667},[75,1068],[31,69669,1076],{"className":69670},[1075],[31,69672,69674],{"className":69673},[1048],[31,69675,69677],{"className":69676,"style":56056},[1052],[31,69678],{},[31,69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",[31,69821,69823,69846],{"className":69822,"translate":35},[34],[31,69824,69826],{"className":69825},[39],[41,69827,69828],{"xmlns":43},[45,69829,69830,69844],{},[48,69831,69832,69834,69836,69838],{},[51,69833,34619],{},[95,69835,97],{},[51,69837,53298],{},[1180,69839,69840,69842],{},[51,69841,2521],{},[51,69843,2515],{},[55,69845,67289],{"encoding":57},[31,69847,69849,69867],{"className":69848,"ariaHidden":62},[61],[31,69850,69852,69855,69858,69861,69864],{"className":69851},[66],[31,69853],{"className":69854,"style":53224},[70],[31,69856,34619],{"className":69857,"style":30311},[75,76],[31,69859],{"className":69860,"style":120},[119],[31,69862,97],{"className":69863},[124],[31,69865],{"className":69866,"style":120},[119],[31,69868,69870,69873,69876],{"className":69869},[66],[31,69871],{"className":69872,"style":59114},[70],[31,69874,53298],{"className":69875,"style":24157},[75,76],[31,69877,69879,69882],{"className":69878},[75],[31,69880,2521],{"className":69881,"style":949},[75,76],[31,69883,69885],{"className":69884},[1039],[31,69886,69888],{"className":69887},[1043],[31,69889,69891],{"className":69890},[1048],[31,69892,69894],{"className":69893,"style":3628},[1052],[31,69895,69896,69899],{"style":3631},[31,69897],{"className":69898,"style":1061},[1060],[31,69900,69902],{"className":69901},[1065,1066,1067,1068],[31,69903,2515],{"className":69904,"style":2666},[75,76,1068],"。折扣因子 ",[31,69907,69909,69927],{"className":69908,"translate":35},[34],[31,69910,69912],{"className":69911},[39],[41,69913,69914],{"xmlns":43},[45,69915,69916,69924],{},[48,69917,69918,69920,69922],{},[51,69919,53298],{},[95,69921,9758],{},[91,69923,205],{},[55,69925,69926],{"encoding":57},"\\gamma\u003C1",[31,69928,69930,69948],{"className":69929,"ariaHidden":62},[61],[31,69931,69933,69936,69939,69942,69945],{"className":69932},[66],[31,69934],{"className":69935,"style":53323},[70],[31,69937,53298],{"className":69938,"style":24157},[75,76],[31,69940],{"className":69941,"style":293},[119],[31,69943,9758],{"className":69944},[297],[31,69946],{"className":69947,"style":293},[119],[31,69949,69951,69954],{"className":69950},[66],[31,69952],{"className":69953,"style":326},[70],[31,69955,205],{"className":69956},[75]," 保证该矩阵可逆，因此解存在且唯一。对于一般状态空间，矩阵不再有限，但同样的结论可以由第 3 节的压缩映射论证得到。",[52860,69959,69961],{"id":69960},"_24-例两状态马尔可夫链","2.4 例：两状态马尔可夫链",[11,69963,69964,69965,70039],{},"考虑两个状态 ",[31,69966,69968,69994],{"className":69967,"translate":35},[34],[31,69969,69971],{"className":69970},[39],[41,69972,69973],{"xmlns":43},[45,69974,69975,69991],{},[48,69976,69977,69979,69981,69983,69985,69987,69989],{},[51,69978,21582],{},[95,69980,223],{},[95,69982,2512],{"stretchy":210},[91,69984,205],{},[95,69986,1210],{"separator":62},[91,69988,152],{},[95,69990,2604],{"stretchy":210},[55,69992,69993],{"encoding":57},"S=\\{1,2\\}",[31,69995,69997,70015],{"className":69996,"ariaHidden":62},[61],[31,69998,70000,70003,70006,70009,70012],{"className":69999},[66],[31,70001],{"className":70002,"style":945},[70],[31,70004,21582],{"className":70005,"style":21697},[75,76],[31,70007],{"className":70008,"style":293},[119],[31,70010,223],{"className":70011},[297],[31,70013],{"className":70014,"style":293},[119],[31,70016,70018,70021,70024,70027,70030,70033,70036],{"className":70017},[66],[31,70019],{"className":70020,"style":260},[70],[31,70022,2512],{"className":70023},[264],[31,70025,205],{"className":70026},[75],[31,70028,1210],{"className":70029},[1391],[31,70031],{"className":70032,"style":1467},[119],[31,70034,152],{"className":70035},[75],[31,70037,2604],{"className":70038},[289],"，只有一个动作。设转移关系为",[31,70041,70043],{"className":70042,"translate":35},[411],[31,70044,70046,70096],{"className":70045,"translate":35},[34],[31,70047,70049],{"className":70048},[39],[41,70050,70051],{"xmlns":43,"display":420},[45,70052,70053,70093],{},[48,70054,70055,70057,70059,70061,70063,70065,70067,70069,70071,70073,70075,70077,70079,70081,70083,70085,70087,70089,70091],{},[51,70056,2521],{},[95,70058,211],{"stretchy":210},[91,70060,152],{},[95,70062,12680],{},[91,70064,205],{},[95,70066,220],{"stretchy":210},[95,70068,223],{},[91,70070,205],{},[95,70072,1210],{"separator":62},[119,70074],{"width":20286},[51,70076,2521],{},[95,70078,211],{"stretchy":210},[91,70080,205],{},[95,70082,12680],{},[91,70084,152],{},[95,70086,220],{"stretchy":210},[95,70088,223],{},[91,70090,205],{},[95,70092,1210],{"separator":62},[55,70094,70095],{"encoding":57},"P(2\\mid 1)=1,\\qquad P(1\\mid 2)=1,",[31,70097,70099,70123,70144,70180,70201],{"className":70098,"ariaHidden":62},[61],[31,70100,70102,70105,70108,70111,70114,70117,70120],{"className":70101},[66],[31,70103],{"className":70104,"style":260},[70],[31,70106,2521],{"className":70107,"style":949},[75,76],[31,70109,211],{"className":70110},[264],[31,70112,152],{"className":70113},[75],[31,70115],{"className":70116,"style":293},[119],[31,70118,12680],{"className":70119},[297],[31,70121],{"className":70122,"style":293},[119],[31,70124,70126,70129,70132,70135,70138,70141],{"className":70125},[66],[31,70127],{"className":70128,"style":260},[70],[31,70130,205],{"className":70131},[75],[31,70133,220],{"className":70134},[289],[31,70136],{"className":70137,"style":293},[119],[31,70139,223],{"className":70140},[297],[31,70142],{"className":70143,"style":293},[119],[31,70145,70147,70150,70153,70156,70159,70162,70165,70168,70171,70174,70177],{"className":70146},[66],[31,70148],{"className":70149,"style":260},[70],[31,70151,205],{"className":70152},[75],[31,70154,1210],{"className":70155},[1391],[31,70157],{"className":70158,"style":20525},[119],[31,70160],{"className":70161,"style":1467},[119],[31,70163,2521],{"className":70164,"style":949},[75,76],[31,70166,211],{"className":70167},[264],[31,70169,205],{"className":70170},[75],[31,70172],{"className":70173,"style":293},[119],[31,70175,12680],{"className":70176},[297],[31,70178],{"className":70179,"style":293},[119],[31,70181,70183,70186,70189,70192,70195,70198],{"className":70182},[66],[31,70184],{"className":70185,"style":260},[70],[31,70187,152],{"className":70188},[75],[31,70190,220],{"className":70191},[289],[31,70193],{"className":70194,"style":293},[119],[31,70196,223],{"className":70197},[297],[31,70199],{"className":70200,"style":293},[119],[31,70202,70204,70207,70210],{"className":70203},[66],[31,70205],{"className":70206,"style":41982},[70],[31,70208,205],{"className":70209},[75],[31,70211,1210],{"className":70212},[1391],[11,70214,70215,70216,1138,70282,70348,70349,70400],{},"奖励为 ",[31,70217,70219,70243],{"className":70218,"translate":35},[34],[31,70220,70222],{"className":70221},[39],[41,70223,70224],{"xmlns":43},[45,70225,70226,70240],{},[48,70227,70228,70230,70232,70234,70236,70238],{},[51,70229,2549],{},[95,70231,211],{"stretchy":210},[91,70233,205],{},[95,70235,220],{"stretchy":210},[95,70237,223],{},[91,70239,205],{},[55,70241,70242],{"encoding":57},"r(1)=1",[31,70244,70246,70273],{"className":70245,"ariaHidden":62},[61],[31,70247,70249,70252,70255,70258,70261,70264,70267,70270],{"className":70248},[66],[31,70250],{"className":70251,"style":260},[70],[31,70253,2549],{"className":70254,"style":19759},[75,76],[31,70256,211],{"className":70257},[264],[31,70259,205],{"className":70260},[75],[31,70262,220],{"className":70263},[289],[31,70265],{"className":70266,"style":293},[119],[31,70268,223],{"className":70269},[297],[31,70271],{"className":70272,"style":293},[119],[31,70274,70276,70279],{"className":70275},[66],[31,70277],{"className":70278,"style":326},[70],[31,70280,205],{"className":70281},[75],[31,70283,70285,70309],{"className":70284,"translate":35},[34],[31,70286,70288],{"className":70287},[39],[41,70289,70290],{"xmlns":43},[45,70291,70292,70306],{},[48,70293,70294,70296,70298,70300,70302,70304],{},[51,70295,2549],{},[95,70297,211],{"stretchy":210},[91,70299,152],{},[95,70301,220],{"stretchy":210},[95,70303,223],{},[91,70305,1977],{},[55,70307,70308],{"encoding":57},"r(2)=0",[31,70310,70312,70339],{"className":70311,"ariaHidden":62},[61],[31,70313,70315,70318,70321,70324,70327,70330,70333,70336],{"className":70314},[66],[31,70316],{"className":70317,"style":260},[70],[31,70319,2549],{"className":70320,"style":19759},[75,76],[31,70322,211],{"className":70323},[264],[31,70325,152],{"className":70326},[75],[31,70328,220],{"className":70329},[289],[31,70331],{"className":70332,"style":293},[119],[31,70334,223],{"className":70335},[297],[31,70337],{"className":70338,"style":293},[119],[31,70340,70342,70345],{"className":70341},[66],[31,70343],{"className":70344,"style":326},[70],[31,70346,1977],{"className":70347},[75],"，折扣因子 ",[31,70350,70352,70370],{"className":70351,"translate":35},[34],[31,70353,70355],{"className":70354},[39],[41,70356,70357],{"xmlns":43},[45,70358,70359,70367],{},[48,70360,70361,70363,70365],{},[51,70362,53298],{},[95,70364,223],{},[91,70366,23247],{},[55,70368,70369],{"encoding":57},"\\gamma=0.9",[31,70371,70373,70391],{"className":70372,"ariaHidden":62},[61],[31,70374,70376,70379,70382,70385,70388],{"className":70375},[66],[31,70377],{"className":70378,"style":865},[70],[31,70380,53298],{"className":70381,"style":24157},[75,76],[31,70383],{"className":70384,"style":293},[119],[31,70386,223],{"className":70387},[297],[31,70389],{"className":70390,"style":293},[119],[31,70392,70394,70397],{"className":70393},[66],[31,70395],{"className":70396,"style":326},[70],[31,70398,23247],{"className":70399},[75],"。贝尔曼方程为",[31,70402,70404],{"className":70403,"translate":35},[411],[31,70405,70407,70473],{"className":70406,"translate":35},[34],[31,70408,70410],{"className":70409},[39],[41,70411,70412],{"xmlns":43,"display":420},[45,70413,70414,70470],{},[48,70415,70416,70418,70420,70422,70424,70426,70428,70430,70432,70434,70436,70438,70440,70442,70444,70446,70448,70450,70452,70454,70456,70458,70460,70462,70464,70466,70468],{},[51,70417,43481],{},[95,70419,211],{"stretchy":210},[91,70421,205],{},[95,70423,220],{"stretchy":210},[95,70425,223],{},[91,70427,205],{},[95,70429,434],{},[91,70431,23247],{},[51,70433,43481],{},[95,70435,211],{"stretchy":210},[91,70437,152],{},[95,70439,220],{"stretchy":210},[95,70441,1210],{"separator":62},[119,70443],{"width":20286},[51,70445,43481],{},[95,70447,211],{"stretchy":210},[91,70449,152],{},[95,70451,220],{"stretchy":210},[95,70453,223],{},[91,70455,1977],{},[95,70457,434],{},[91,70459,23247],{},[51,70461,43481],{},[95,70463,211],{"stretchy":210},[91,70465,205],{},[95,70467,220],{"stretchy":210},[51,70469,495],{"mathvariant":494},[55,70471,70472],{"encoding":57},"V(1)=1+0.9V(2),\\qquad V(2)=0+0.9V(1).",[31,70474,70476,70503,70521,70572,70590],{"className":70475,"ariaHidden":62},[61],[31,70477,70479,70482,70485,70488,70491,70494,70497,70500],{"className":70478},[66],[31,70480],{"className":70481,"style":260},[70],[31,70483,43481],{"className":70484,"style":120},[75,76],[31,70486,211],{"className":70487},[264],[31,70489,205],{"className":70490},[75],[31,70492,220],{"className":70493},[289],[31,70495],{"className":70496,"style":293},[119],[31,70498,223],{"className":70499},[297],[31,70501],{"className":70502,"style":293},[119],[31,70504,70506,70509,70512,70515,70518],{"className":70505},[66],[31,70507],{"className":70508,"style":112},[70],[31,70510,205],{"className":70511},[75],[31,70513],{"className":70514,"style":120},[119],[31,70516,434],{"className":70517},[124],[31,70519],{"className":70520,"style":120},[119],[31,70522,70524,70527,70530,70533,70536,70539,70542,70545,70548,70551,70554,70557,70560,70563,70566,70569],{"className":70523},[66],[31,70525],{"className":70526,"style":260},[70],[31,70528,23247],{"className":70529},[75],[31,70531,43481],{"className":70532,"style":120},[75,76],[31,70534,211],{"className":70535},[264],[31,70537,152],{"className":70538},[75],[31,70540,220],{"className":70541},[289],[31,70543,1210],{"className":70544},[1391],[31,70546],{"className":70547,"style":20525},[119],[31,70549],{"className":70550,"style":1467},[119],[31,70552,43481],{"className":70553,"style":120},[75,76],[31,70555,211],{"className":70556},[264],[31,70558,152],{"className":70559},[75],[31,70561,220],{"className":70562},[289],[31,70564],{"className":70565,"style":293},[119],[31,70567,223],{"className":70568},[297],[31,70570],{"className":70571,"style":293},[119],[31,70573,70575,70578,70581,70584,70587],{"className":70574},[66],[31,70576],{"className":70577,"style":112},[70],[31,70579,1977],{"className":70580},[75],[31,70582],{"className":70583,"style":120},[119],[31,70585,434],{"className":70586},[124],[31,70588],{"className":70589,"style":120},[119],[31,70591,70593,70596,70599,70602,70605,70608,70611],{"className":70592},[66],[31,70594],{"className":70595,"style":260},[70],[31,70597,23247],{"className":70598},[75],[31,70600,43481],{"className":70601,"style":120},[75,76],[31,70603,211],{"className":70604},[264],[31,70606,205],{"className":70607},[75],[31,70609,220],{"className":70610},[289],[31,70612,495],{"className":70613},[75],[11,70615,70616],{},"将第二式代入第一式，得",[31,70618,70620],{"className":70619,"translate":35},[411],[31,70621,70623,70662],{"className":70622,"translate":35},[34],[31,70624,70626],{"className":70625},[39],[41,70627,70628],{"xmlns":43,"display":420},[45,70629,70630,70659],{},[48,70631,70632,70634,70636,70638,70640,70642,70644,70646,70649,70651,70653,70655,70657],{},[51,70633,43481],{},[95,70635,211],{"stretchy":210},[91,70637,205],{},[95,70639,220],{"stretchy":210},[95,70641,223],{},[91,70643,205],{},[95,70645,434],{},[91,70647,70648],{},"0.81",[51,70650,43481],{},[95,70652,211],{"stretchy":210},[91,70654,205],{},[95,70656,220],{"stretchy":210},[95,70658,1210],{"separator":62},[55,70660,70661],{"encoding":57},"V(1)=1+0.81V(1),",[31,70663,70665,70692,70710],{"className":70664,"ariaHidden":62},[61],[31,70666,70668,70671,70674,70677,70680,70683,70686,70689],{"className":70667},[66],[31,70669],{"className":70670,"style":260},[70],[31,70672,43481],{"className":70673,"style":120},[75,76],[31,70675,211],{"className":70676},[264],[31,70678,205],{"className":70679},[75],[31,70681,220],{"className":70682},[289],[31,70684],{"className":70685,"style":293},[119],[31,70687,223],{"className":70688},[297],[31,70690],{"className":70691,"style":293},[119],[31,70693,70695,70698,70701,70704,70707],{"className":70694},[66],[31,70696],{"className":70697,"style":112},[70],[31,70699,205],{"className":70700},[75],[31,70702],{"className":70703,"style":120},[119],[31,70705,434],{"className":70706},[124],[31,70708],{"className":70709,"style":120},[119],[31,70711,70713,70716,70719,70722,70725,70728,70731],{"className":70712},[66],[31,70714],{"className":70715,"style":260},[70],[31,70717,70648],{"className":70718},[75],[31,70720,43481],{"className":70721,"style":120},[75,76],[31,70723,211],{"className":70724},[264],[31,70726,205],{"className":70727},[75],[31,70729,220],{"className":70730},[289],[31,70732,1210],{"className":70733},[1391],[11,70735,70736],{},"因此",[31,70738,70740],{"className":70739,"translate":35},[411],[31,70741,70743,70811],{"className":70742,"translate":35},[34],[31,70744,70746],{"className":70745},[39],[41,70747,70748],{"xmlns":43,"display":420},[45,70749,70750,70808],{},[48,70751,70752,70754,70756,70758,70760,70762,70774,70776,70779,70781,70783,70785,70787,70789,70791,70793,70795,70797,70799,70801,70803,70805],{},[51,70753,43481],{},[95,70755,211],{"stretchy":210},[91,70757,205],{},[95,70759,220],{"stretchy":210},[95,70761,223],{},[2074,70763,70764,70766],{},[91,70765,205],{},[48,70767,70768,70770,70772],{},[91,70769,205],{},[95,70771,97],{},[91,70773,70648],{},[95,70775,223],{},[91,707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75],[31,72238,71869],{"className":72239},[75],[31,72241,1076],{"className":72242},[1075],[31,72244,72246],{"className":72245},[1048],[31,72247,72249],{"className":72248,"style":46243},[1052],[31,72250],{},[31,72252,72254],{"className":72253,"style":3868},[289,3867],[31,72255,220],{"className":72256},[1307,1067],[31,72258],{"className":72259,"style":1467},[119],[31,72261,495],{"className":72262},[75],[31,72264,1076],{"className":72265},[1075],[31,72267,72269],{"className":72268},[1048],[31,72270,72272],{"className":72271,"style":71937},[1052],[31,72273],{},[11,72275,72276,72277,72359],{},"这个例子说明，即使状态每两步就回到原处，只要 ",[31,72278,72280,72302],{"className":72279,"translate":35},[34],[31,72281,72283],{"className":72282},[39],[41,72284,72285],{"xmlns":43},[45,72286,72287,72299],{},[48,72288,72289,72295,72297],{},[1180,72290,72291,72293],{},[51,72292,53298],{},[91,72294,152],{},[95,72296,9758],{},[91,72298,205],{},[55,72300,72301],{"encoding":57},"\\gamma^2\u003C1",[31,72303,72305,72350],{"className":72304,"ariaHidden":62},[61],[31,72306,72308,72312,72341,72344,72347],{"className":72307},[66],[31,72309],{"className":72310,"style":72311},[70],"height:1.0085em;vertical-align:-0.1944em;",[31,72313,72315,72318],{"className":72314},[75],[31,72316,53298],{"className":72317,"style":24157},[75,76],[31,72319,72321],{"className":72320},[1039],[31,72322,72324],{"className":72323},[1043],[31,72325,72327],{"className":72326},[1048],[31,72328,72330],{"className":72329,"style":5135},[1052],[31,72331,72332,72335],{"style":3631},[31,72333],{"className":72334,"style":1061},[1060],[31,72336,72338],{"className":72337},[1065,1066,1067,1068],[31,72339,152],{"className":72340},[75,1068],[31,72342],{"className":72343,"style":293},[119],[31,72345,9758],{"className":72346},[297],[31,72348],{"className":72349,"style":293},[119],[31,72351,72353,72356],{"className":72352},[66],[31,72354],{"className":72355,"style":326},[70],[31,72357,205],{"className":72358},[75],"，折扣回报的期望仍然是有限的。值函数不是把无限多项直接相加，而是通过解线性方程组把无限和压缩成一个有限量。",[52860,72361,72363],{"id":72362},"_25-方程与回报定义的等价性","2.5 方程与回报定义的等价性",[11,72365,72366],{},"贝尔曼方程不仅是值函数的必要条件，也是充分条件。下面的命题说明，任何有界解都必然等于由折扣回报定义的值函数。",[52865,72368,72371],{"id":72369,"title":72370},"proposition-policy-verification","命题 2.3（策略评估的验证定理）",[11,72372,52871,72373,69116,72439,67262,72525,72606,72607,72665],{},[31,72374,72376,72400],{"className":72375,"translate":35},[34],[31,72377,72379],{"className":72378},[39],[41,72380,72381],{"xmlns":43},[45,72382,72383,72397],{},[48,72384,72385,72387,72389,72391,72393,72395],{},[51,72386,43481],{},[95,72388,2518],{},[51,72390,1122],{},[95,72392,211],{"stretchy":210},[51,72394,21582],{},[95,72396,220],{"stretchy":210},[55,72398,72399],{"encoding":57},"V\\in B(S)",[31,72401,72403,72421],{"className":72402,"ariaHidden":62},[61],[31,72404,72406,72409,72412,72415,72418],{"className":72405},[66],[31,72407],{"className":72408,"style":65709},[70],[31,72410,43481],{"className":72411,"style":120},[75,76],[31,72413],{"className":72414,"style":293},[119],[31,72416,2518],{"className":72417},[297],[31,72419],{"className":72420,"style":293},[119],[31,72422,72424,72427,72430,72433,72436],{"className":72423},[66],[31,72425],{"className":72426,"style":260},[70],[31,72428,1122],{"className":72429,"style":1137},[75,76],[31,72431,211],{"className":72432},[264],[31,72434,21582],{"className":72435,"style":21697},[75,76],[31,72437,220],{"className":72438},[289],[31,72440,72442,72466],{"className":72441,"translate":35},[34],[31,72443,72445],{"className":72444},[39],[41,72446,72447],{"xmlns":43},[45,72448,72449,72463],{},[48,72450,72451,72453,72455,72461],{},[51,72452,43481],{},[95,72454,223],{},[1180,72456,72457,72459],{},[51,72458,919],{"mathvariant":925},[51,72460,2515],{},[51,72462,43481],{},[55,72464,72465],{"encoding":57},"V=\\mathcal{T}^\\pi V",[31,72467,72469,72487],{"className":72468,"ariaHidden":62},[61],[31,72470,72472,72475,72478,72481,72484],{"className":72471},[66],[31,72473],{"className":72474,"style":945},[70],[31,72476,43481],{"className":72477,"style":120},[75,76],[31,72479],{"className":72480,"style":293},[119],[31,72482,223],{"className":72483},[297],[31,72485],{"className":72486,"style":293},[119],[31,72488,72490,72493,72522],{"className":72489},[66],[31,72491],{"className":72492,"style":945},[70],[31,72494,72496,72499],{"className":72495},[75],[31,72497,919],{"className":72498,"style":64444},[75,968],[31,72500,72502],{"className":72501},[1039],[31,72503,72505],{"className":72504},[1043],[31,72506,72508],{"className":72507},[1048],[31,72509,72511],{"className":72510,"style":3628},[1052],[31,72512,72513,72516],{"style":3631},[31,72514],{"className":72515,"style":1061},[1060],[31,72517,72519],{"className":72518},[1065,1066,1067,1068],[31,72520,2515],{"className":72521,"style":2666},[75,76,1068],[31,72523,43481],{"className":72524,"style":120},[75,76],[31,72526,72528,72550],{"className":72527,"translate":35},[34],[31,72529,72531],{"className":72530},[39],[41,72532,72533],{"xmlns":43},[45,72534,72535,72547],{},[48,72536,72537,72539,72541],{},[51,72538,43481],{},[95,72540,223],{},[1180,72542,72543,72545],{},[51,72544,43481],{},[51,72546,2515],{},[55,72548,72549],{"encoding":57},"V=V^\\pi",[31,72551,72553,72571],{"className":72552,"ariaHidden":62},[61],[31,72554,72556,72559,72562,72565,72568],{"className":72555},[66],[31,72557],{"className":72558,"style":945},[70],[31,72560,43481],{"className":72561,"style":120},[75,76],[31,72563],{"className":72564,"style":293},[119],[31,72566,223],{"className":72567},[297],[31,72569],{"className":72570,"style":293},[119],[31,72572,72574,72577],{"className":72573},[66],[31,72575],{"className":72576,"style":945},[70],[31,72578,72580,72583],{"className":72579},[75],[31,72581,43481],{"className":72582,"style":120},[75,76],[31,72584,72586],{"className":72585},[1039],[31,72587,72589],{"className":72588},[1043],[31,72590,72592],{"className":72591},[1048],[31,72593,72595],{"className":72594,"style":3628},[1052],[31,72596,72597,72600],{"style":3631},[31,72598],{"className":72599,"style":1061},[1060],[31,72601,72603],{"className":72602},[1065,1066,1067,1068],[31,72604,2515],{"className":72605,"style":2666},[75,76,1068],"。特别地，",[31,72608,72610,72627],{"className":72609,"translate":35},[34],[31,72611,72613],{"className":72612},[39],[41,72614,72615],{"xmlns":43},[45,72616,72617,72625],{},[48,72618,72619],{},[1180,72620,72621,72623],{},[51,72622,919],{"mathvariant":925},[51,72624,2515],{},[55,72626,64428],{"encoding":57},[31,72628,72630],{"className":72629,"ariaHidden":62},[61],[31,72631,72633,72636],{"className":72632},[66],[31,72634],{"className":72635,"style":945},[70],[31,72637,72639,72642],{"className":72638},[75],[31,72640,919],{"className":72641,"style":64444},[75,968],[31,72643,72645],{"className":72644},[1039],[31,72646,72648],{"className":72647},[1043],[31,72649,72651],{"className":72650},[1048],[31,72652,72654],{"className":72653,"style":3628},[1052],[31,72655,72656,72659],{"style":3631},[31,72657],{"className":72658,"style":1061},[1060],[31,72660,72662],{"className":72661},[1065,1066,1067,1068],[31,72663,2515],{"className":72664,"style":2666},[75,76,1068]," 在有界函数空间中至多有一个不动点。",[11,72667,72668,72670,72671,72756],{},[7246,72669,7248],{}," 设 ",[31,72672,72674,72697],{"className":72673,"translate":35},[34],[31,72675,72677],{"className":72676},[39],[41,72678,72679],{"xmlns":43},[45,72680,72681,72695],{},[48,72682,72683,72685,72687,72693],{},[51,72684,43481],{},[95,72686,223],{},[1180,72688,72689,72691],{},[51,72690,919],{"mathvariant":925},[51,72692,2515],{},[51,72694,43481],{},[55,72696,72465],{"encoding":57},[31,72698,72700,72718],{"className":72699,"ariaHidden":62},[61],[31,72701,72703,72706,72709,72712,72715],{"className":72702},[66],[31,72704],{"className":72705,"style":945},[70],[31,72707,43481],{"className":72708,"style":120},[75,76],[31,72710],{"className":72711,"style":293},[119],[31,72713,223],{"className":72714},[297],[31,72716],{"className":72717,"style":293},[119],[31,72719,72721,72724,72753],{"className":72720},[66],[31,72722],{"className":72723,"style":945},[70],[31,72725,72727,72730],{"className":72726},[75],[31,72728,919],{"className":72729,"style":64444},[75,968],[31,72731,72733],{"className":72732},[1039],[31,72734,72736],{"className":72735},[1043],[31,72737,72739],{"className":72738},[1048],[31,72740,72742],{"className":72741,"style":3628},[1052],[31,72743,72744,72747],{"style":3631},[31,72745],{"className":72746,"style":1061},[1060],[31,72748,72750],{"className":72749},[1065,1066,1067,1068],[31,72751,2515],{"className":72752,"style":2666},[75,76,1068],[31,72754,43481],{"className":72755,"style":120},[75,76],"。反复代入得",[31,72758,72760],{"className":72759,"translate":35},[411],[31,72761,72763,72851],{"className":72762,"translate":35},[34],[31,72764,72766],{"className":72765},[39],[41,72767,72768],{"xmlns":43,"display":420},[45,72769,72770,72848],{},[48,72771,72772,72774,72776,72796,72802,72804,72810,72816,72822,72824,72830,72832,72838,72844,72846],{},[51,72773,43481],{},[95,72775,223],{},[42429,72777,72778,72780,72788],{},[95,72779,5625],{},[48,72781,72782,72784,72786],{},[51,72783,52065],{},[95,72785,223],{},[91,72787,1977],{},[48,72789,72790,72792,72794],{},[51,72791,14559],{},[95,72793,97],{},[91,72795,205],{},[1180,72797,72798,72800],{},[51,72799,53298],{},[51,72801,52065],{},[95,72803,211],{"stretchy":210},[1180,72805,72806,72808],{},[51,72807,2521],{},[51,72809,2515],{},[1180,72811,72812,72814],{},[95,72813,220],{"stretchy":210},[51,72815,52065],{},[1180,72817,72818,72820],{},[51,72819,2549],{},[51,72821,2515],{},[95,72823,434],{},[1180,72825,72826,72828],{},[51,72827,53298],{},[51,72829,14559],{},[95,72831,211],{"stretchy":210},[1180,72833,72834,72836],{},[51,72835,2521],{},[51,72837,2515],{},[1180,72839,72840,72842],{},[95,72841,220],{"stretchy":210},[51,72843,14559],{},[51,72845,43481],{},[51,72847,495],{"mathvariant":494},[55,72849,72850],{"encoding":57},"V=\\sum_{k=0}^{n-1}\\gamma^k(P^\\pi)^k r^\\pi\n+\\gamma^n(P^\\pi)^n V.",[31,72852,72854,72872,73084],{"className":72853,"ariaHidden":62},[61],[31,72855,72857,72860,72863,72866,72869],{"className":72856},[66],[31,72858],{"className":72859,"style":945},[70],[31,72861,43481],{"className":72862,"style":120},[75,76],[31,72864],{"className":72865,"style":293},[119],[31,72867,223],{"className":72868},[297],[31,72870],{"className":72871,"style":293},[119],[31,72873,72875,72879,72953,72956,72985,72988,73017,73046,73075,73078,73081],{"className":72874},[66],[31,72876],{"className":72877,"style":72878},[70],"height:3.1032em;vertical-align:-1.3021em;",[31,72880,72882],{"className":72881},[1460,1620],[31,72883,72885,72945],{"className":72884},[1043,1044],[31,72886,72888,72942],{"className":72887},[1048],[31,72889,72892,72912,72922],{"className":72890,"style":72891},[1052],"height:1.8011em;",[31,72893,72894,72897],{"style":56004},[31,72895],{"className":72896,"style":5907},[1060],[31,72898,72900],{"className":72899},[1065,1066,1067,1068],[31,72901,72903,72906,72909],{"className":72902},[75,1068],[31,72904,52065],{"className":72905,"style":52166},[75,76,1068],[31,72907,223],{"className":72908},[297,1068],[31,72910,1977],{"className":72911},[75,1068],[31,72913,72914,72917],{"style":5919},[31,72915],{"className":72916,"style":5907},[1060],[31,72918,72919],{},[31,72920,5625],{"className":72921},[1460,1461,1687],[31,72923,72924,72927],{"style":42735},[31,72925],{"className":72926,"style":5907},[1060],[31,72928,72930],{"className":72929},[1065,1066,1067,1068],[31,72931,72933,72936,72939],{"className":72932},[75,1068],[31,72934,14559],{"className":72935},[75,76,1068],[31,72937,97],{"className":72938},[124,1068],[31,72940,205],{"className":72941},[75,1068],[31,72943,1076],{"className":72944},[1075],[31,72946,72948],{"className":72947},[1048],[31,72949,72951],{"className":72950,"style":56056},[1052],[31,72952],{},[31,72954],{"className":72955,"style":1467},[119],[31,72957,72959,72962],{"className":72958},[75],[31,72960,53298],{"className":72961,"style":24157},[75,76],[31,72963,72965],{"className":72964},[1039],[31,72966,72968],{"className":72967},[1043],[31,72969,72971],{"className":72970},[1048],[31,72972,72974],{"className":72973,"style":29642},[1052],[31,72975,72976,72979],{"style":1330},[31,72977],{"className":72978,"style":1061},[1060],[31,72980,72982],{"className":72981},[1065,1066,1067,1068],[31,72983,52065],{"className":72984,"style":52166},[75,76,1068],[31,72986,211],{"className":72987},[264],[31,72989,72991,72994],{"className":72990},[75],[31,72992,2521],{"className":72993,"style":949},[75,76],[31,72995,72997],{"className":72996},[1039],[31,72998,73000],{"className":72999},[1043],[31,73001,73003],{"className":73002},[1048],[31,73004,73006],{"className":73005,"style":3987},[1052],[31,73007,73008,73011],{"style":1330},[31,73009],{"className":73010,"style":1061},[1060],[31,73012,73014],{"className":73013},[1065,1066,1067,1068],[31,73015,2515],{"className":73016,"style":2666},[75,76,1068],[31,73018,73020,73023],{"className":73019},[289],[31,73021,220],{"className":73022},[289],[31,73024,73026],{"className":73025},[1039],[31,73027,73029],{"className":73028},[1043],[31,73030,73032],{"className":73031},[1048],[31,73033,73035],{"className":73034,"style":29642},[1052],[31,73036,73037,73040],{"style":1330},[31,73038],{"className":73039,"style":1061},[1060],[31,73041,73043],{"className":73042},[1065,1066,1067,1068],[31,73044,52065],{"className":73045,"style":52166},[75,76,1068],[31,73047,73049,73052],{"className":73048},[75],[31,73050,2549],{"className":73051,"style":19759},[75,76],[31,73053,73055],{"className":73054},[1039],[31,73056,73058],{"className":73057},[1043],[31,73059,73061],{"className":73060},[1048],[31,73062,73064],{"className":73063,"style":3987},[1052],[31,73065,73066,73069],{"style":1330},[31,73067],{"className":73068,"style":1061},[1060],[31,73070,73072],{"className":73071},[1065,1066,1067,1068],[31,73073,2515],{"className":73074,"style":2666},[75,76,1068],[31,73076],{"className":73077,"style":120},[119],[31,73079,434],{"className":73080},[124],[31,73082],{"className":73083,"style":120},[119],[31,73085,73087,73090,73119,73122,73151,73180,73183],{"className":73086},[66],[31,73088],{"className":73089,"style":260},[70],[31,73091,73093,73096],{"className":73092},[75],[31,73094,53298],{"className":73095,"style":24157},[75,76],[31,73097,73099],{"className":73098},[1039],[31,73100,73102],{"className":73101},[1043],[31,73103,73105],{"className":73104},[1048],[31,73106,73108],{"className":73107,"style":3987},[1052],[31,73109,73110,73113],{"style":1330},[31,73111],{"className":73112,"style":1061},[1060],[31,73114,73116],{"className":73115},[1065,1066,1067,1068],[31,73117,14559],{"className":73118},[75,76,1068],[31,73120,211],{"className":73121},[264],[31,73123,73125,73128],{"className":73124},[75],[31,73126,2521],{"className":73127,"style":949},[75,76],[31,73129,73131],{"className":73130},[1039],[31,73132,73134],{"className":73133},[1043],[31,73135,73137],{"className":73136},[1048],[31,73138,73140],{"className":73139,"style":3987},[1052],[31,73141,73142,73145],{"style":1330},[31,73143],{"className":73144,"style":1061},[1060],[31,73146,73148],{"className":73147},[1065,1066,1067,1068],[31,73149,2515],{"className":73150,"style":2666},[75,76,1068],[31,73152,73154,73157],{"className":73153},[289],[31,73155,220],{"className":73156},[289],[31,73158,73160],{"className":73159},[1039],[31,73161,73163],{"className":73162},[1043],[31,73164,73166],{"className":73165},[1048],[31,73167,73169],{"className":73168,"style":3987},[1052],[31,73170,73171,73174],{"style":1330},[31,73172],{"className":73173,"style":1061},[1060],[31,73175,73177],{"className":73176},[1065,1066,1067,1068],[31,73178,14559],{"className":73179},[75,76,1068],[31,73181,43481],{"className":73182,"style":120},[75,76],[31,73184,495],{"className":73185},[75],[11,73187,73188,73189,73417,73418,73446],{},"由于 ",[31,73190,73192,73240],{"className":73191,"translate":35},[34],[31,73193,73195],{"className":73194},[39],[41,73196,73197],{"xmlns":43},[45,73198,73199,73237],{},[48,73200,73201,73203,73205,73211,73217,73219,73225,73227,73229,73231],{},[51,73202,4572],{"mathvariant":494},[95,73204,211],{"stretchy":210},[1180,73206,73207,73209],{},[51,73208,2521],{},[51,73210,2515],{},[1180,73212,73213,73215],{},[95,73214,220],{"stretchy":210},[51,73216,14559],{},[51,73218,43481],{},[1004,73220,73221,73223],{},[51,73222,4572],{"mathvariant":494},[51,73224,4587],{"mathvariant":494},[95,73226,347],{},[51,73228,4572],{"mathvariant":494},[51,73230,43481],{},[1004,73232,73233,73235],{},[51,73234,4572],{"mathvariant":494},[51,73236,4587],{"mathvariant":494},[55,73238,73239],{"encoding":57},"\\|(P^\\pi)^n V\\|_\\infty\\le\\|V\\|_\\infty",[31,73241,73243,73365],{"className":73242,"ariaHidden":62},[61],[31,73244,73246,73249,73252,73255,73284,73313,73316,73356,73359,73362],{"className":73245},[66],[31,73247],{"className":73248,"style":260},[70],[31,73250,4572],{"className":73251},[75],[31,73253,211],{"className":73254},[264],[31,73256,73258,73261],{"className":73257},[75],[31,73259,2521],{"className":73260,"style":949},[75,76],[31,73262,73264],{"className":73263},[1039],[31,73265,73267],{"className":73266},[1043],[31,73268,73270],{"className":73269},[1048],[31,73271,73273],{"className":73272,"style":3628},[1052],[31,73274,73275,73278],{"style":3631},[31,73276],{"className":73277,"style":1061},[1060],[31,73279,73281],{"className":73280},[1065,1066,1067,1068],[31,73282,2515],{"className":73283,"style":2666},[75,76,1068],[31,73285,73287,73290],{"className":73286},[289],[31,73288,220],{"className":73289},[289],[31,73291,73293],{"className":73292},[1039],[31,73294,73296],{"className":73295},[1043],[31,73297,73299],{"className":73298},[1048],[31,73300,73302],{"className":73301,"style":3628},[1052],[31,73303,73304,73307],{"style":3631},[31,73305],{"className":73306,"style":1061},[1060],[31,73308,73310],{"className":73309},[1065,1066,1067,1068],[31,73311,14559],{"className":73312},[75,76,1068],[31,73314,43481],{"className":73315,"style":120},[75,76],[31,73317,73319,73322],{"className":73318},[75],[31,73320,4572],{"className":73321},[75],[31,73323,73325],{"className":73324},[1039],[31,73326,73328,73348],{"className":73327},[1043,1044],[31,73329,73331,73345],{"className":73330},[1048],[31,73332,73334],{"className":73333,"style":3230},[1052],[31,73335,73336,73339],{"style":2807},[31,73337],{"className":73338,"style":1061},[1060],[31,73340,73342],{"className":73341},[1065,1066,1067,1068],[31,73343,4587],{"className":73344},[75,1068],[31,73346,1076],{"className":73347},[1075],[31,73349,73351],{"className":73350},[1048],[31,73352,73354],{"className":73353,"style":2050},[1052],[31,73355],{},[31,73357],{"className":73358,"style":293},[119],[31,73360,347],{"className":73361},[297],[31,73363],{"className":73364,"style":293},[119],[31,73366,73368,73371,73374,73377],{"className":73367},[66],[31,73369],{"className":73370,"style":260},[70],[31,73372,4572],{"className":73373},[75],[31,73375,43481],{"className":73376,"style":120},[75,76],[31,73378,73380,73383],{"className":73379},[75],[31,73381,4572],{"className":73382},[75],[31,73384,73386],{"className":73385},[1039],[31,73387,73389,73409],{"className":73388},[1043,1044],[31,73390,73392,73406],{"className":73391},[1048],[31,73393,73395],{"className":73394,"style":3230},[1052],[31,73396,73397,73400],{"style":2807},[31,73398],{"className":73399,"style":1061},[1060],[31,73401,73403],{"className":73402},[1065,1066,1067,1068],[31,73404,4587],{"className":73405},[75,1068],[31,73407,1076],{"className":73408},[1075],[31,73410,73412],{"className":73411},[1048],[31,73413,73415],{"className":73414,"style":2050},[1052],[31,73416],{},"，最后一项在 sup 范数下趋于 ",[31,73419,73421,73434],{"className":73420,"translate":35},[34],[31,73422,73424],{"className":73423},[39],[41,73425,73426],{"xmlns":43},[45,73427,73428,73432],{},[48,73429,73430],{},[91,73431,1977],{},[55,73433,1977],{"encoding":57},[31,73435,73437],{"className":73436,"ariaHidden":62},[61],[31,73438,73440,73443],{"className":73439},[66],[31,73441],{"className":73442,"style":326},[70],[31,73444,1977],{"className":73445},[75],"。另一方面，",[31,73448,73450],{"className":73449,"translate":35},[411],[31,73451,73453,73505],{"className":73452,"translate":35},[34],[31,73454,73456],{"className":73455},[39],[41,73457,73458],{"xmlns":43,"display":420},[45,73459,73460,73502],{},[48,73461,73462,73476,73482,73484,73490,73496],{},[42429,73463,73464,73466,73474],{},[95,73465,5625],{},[48,73467,73468,73470,73472],{},[51,73469,52065],{},[95,73471,223],{},[91,73473,1977],{},[51,73475,4587],{"mathvariant":494},[1180,73477,73478,73480],{},[51,73479,53298],{},[51,73481,52065],{},[95,73483,211],{"stretchy":210},[1180,73485,73486,73488],{},[51,73487,2521],{},[51,73489,2515],{},[1180,73491,73492,73494],{},[95,73493,220],{"stretchy":210},[51,73495,52065],{},[1180,73497,73498,73500],{},[51,73499,2549],{},[51,73501,2515],{},[55,73503,73504],{"encoding":57},"\\sum_{k=0}^{\\infty}\\gamma^k(P^\\pi)^k r^\\pi",[31,73506,73508],{"className":73507,"ariaHidden":62},[61],[31,73509,73511,73514,73581,73584,73613,73616,73645,73674],{"className":73510},[66],[31,73512],{"className":73513,"style":55989},[70],[31,73515,73517],{"className":73516},[1460,1620],[31,73518,73520,73573],{"className":73519},[1043,1044],[31,73521,73523,73570],{"className":73522},[1048],[31,73524,73526,73546,73556],{"className":73525,"style":42702},[1052],[31,73527,73528,73531],{"style":56004},[31,73529],{"className":73530,"style":5907},[1060],[31,73532,73534],{"className":73533},[1065,1066,1067,1068],[31,73535,73537,73540,73543],{"className":73536},[75,1068],[31,73538,52065],{"className":73539,"style":52166},[75,76,1068],[31,73541,223],{"className":73542},[297,1068],[31,73544,1977],{"className":73545},[75,1068],[31,73547,73548,73551],{"style":5919},[31,73549],{"className":73550,"style":5907},[1060],[31,73552,73553],{},[31,73554,5625],{"className":73555},[1460,1461,1687],[31,73557,73558,73561],{"style":42735},[31,73559],{"className":73560,"style":5907},[1060],[31,73562,73564],{"className":73563},[1065,1066,1067,1068],[31,73565,73567],{"className":73566},[75,1068],[31,73568,4587],{"className":73569},[75,1068],[31,73571,1076],{"className":73572},[1075],[31,73574,73576],{"className":73575},[1048],[31,73577,73579],{"className":73578,"style":56056},[1052],[31,73580],{},[31,73582],{"className":73583,"style":1467},[119],[31,73585,73587,73590],{"className":73586},[75],[31,73588,53298],{"className":73589,"style":24157},[75,76],[31,73591,73593],{"className":73592},[1039],[31,73594,73596],{"className":73595},[1043],[31,73597,73599],{"className":73598},[1048],[31,73600,73602],{"className":73601,"style":29642},[1052],[31,73603,73604,73607],{"style":1330},[31,73605],{"className":73606,"style":1061},[1060],[31,73608,73610],{"className":73609},[1065,1066,1067,1068],[31,73611,52065],{"className":73612,"style":52166},[75,76,1068],[31,73614,211],{"className":73615},[264],[31,73617,73619,73622],{"className":73618},[75],[31,73620,2521],{"className":73621,"style":949},[75,76],[31,73623,73625],{"className":73624},[1039],[31,73626,73628],{"className":73627},[1043],[31,73629,73631],{"className":73630},[1048],[31,73632,73634],{"className":73633,"style":3987},[1052],[31,73635,73636,73639],{"style":1330},[31,73637],{"className":73638,"style":1061},[1060],[31,73640,73642],{"className":73641},[1065,1066,1067,1068],[31,73643,2515],{"className":73644,"style":2666},[75,76,1068],[31,73646,73648,73651],{"className":73647},[289],[31,73649,220],{"className":73650},[289],[31,73652,73654],{"className":73653},[1039],[31,73655,73657],{"className":73656},[1043],[31,73658,73660],{"className":73659},[1048],[31,73661,73663],{"className":73662,"style":29642},[1052],[31,73664,73665,73668],{"style":1330},[31,73666],{"className":73667,"style":1061},[1060],[31,73669,73671],{"className":73670},[1065,1066,1067,1068],[31,73672,52065],{"className":73673,"style":52166},[75,76,1068],[31,73675,73677,73680],{"className":73676},[75],[31,73678,2549],{"className":73679,"style":19759},[75,76],[31,73681,73683],{"className":73682},[1039],[31,73684,73686],{"className":73685},[1043],[31,73687,73689],{"className":73688},[1048],[31,73690,73692],{"className":73691,"style":3987},[1052],[31,73693,73694,73697],{"style":1330},[31,73695],{"className":73696,"style":1061},[1060],[31,73698,73700],{"className":73699},[1065,1066,1067,1068],[31,73701,2515],{"className":73702,"style":2666},[75,76,1068],[11,73704,73705,73706,73764,73765,73845],{},"正是折扣回报的期望，即 ",[31,73707,73709,73726],{"className":73708,"translate":35},[34],[31,73710,73712],{"className":73711},[39],[41,73713,73714],{"xmlns":43},[45,73715,73716,73724],{},[48,73717,73718],{},[1180,73719,73720,73722],{},[51,73721,43481],{},[51,73723,2515],{},[55,73725,59045],{"encoding":57},[31,73727,73729],{"className":73728,"ariaHidden":62},[61],[31,73730,73732,73735],{"className":73731},[66],[31,73733],{"className":73734,"style":945},[70],[31,73736,73738,73741],{"className":73737},[75],[31,73739,43481],{"className":73740,"style":120},[75,76],[31,73742,73744],{"className":73743},[1039],[31,73745,73747],{"className":73746},[1043],[31,73748,73750],{"className":73749},[1048],[31,73751,73753],{"className":73752,"style":3628},[1052],[31,73754,73755,73758],{"style":3631},[31,73756],{"className":73757,"style":1061},[1060],[31,73759,73761],{"className":73760},[1065,1066,1067,1068],[31,73762,2515],{"className":73763,"style":2666},[75,76,1068],"。故 ",[31,73766,73768,73789],{"className":73767,"translate":35},[34],[31,73769,73771],{"className":73770},[39],[41,73772,73773],{"xmlns":43},[45,73774,73775,73787],{},[48,73776,73777,73779,73781],{},[51,73778,43481],{},[95,73780,223],{},[1180,73782,73783,73785],{},[51,73784,43481],{},[51,73786,2515],{},[55,73788,72549],{"encoding":57},[31,73790,73792,73810],{"className":73791,"ariaHidden":62},[61],[31,73793,73795,73798,73801,73804,73807],{"className":73794},[66],[31,73796],{"className":73797,"style":945},[70],[31,73799,43481],{"className":73800,"style":120},[75,76],[31,73802],{"className":73803,"style":293},[119],[31,73805,223],{"className":73806},[297],[31,73808],{"className":73809,"style":293},[119],[31,73811,73813,73816],{"className":73812},[66],[31,73814],{"className":73815,"style":945},[70],[31,73817,73819,73822],{"className":73818},[75],[31,73820,43481],{"className":73821,"style":120},[75,76],[31,73823,73825],{"className":73824},[1039],[31,73826,73828],{"className":73827},[1043],[31,73829,73831],{"className":73830},[1048],[31,73832,73834],{"className":73833,"style":3628},[1052],[31,73835,73836,73839],{"style":3631},[31,73837],{"className":73838,"style":1061},[1060],[31,73840,73842],{"className":73841},[1065,1066,1067,1068],[31,73843,2515],{"className":73844,"style":2666},[75,76,1068],"。证毕。",[11,73847,73848,73849,73907,73908,73912,73913,73963],{},"这个证明不依赖矩阵的有限维结构，只用到 ",[31,73850,73852,73869],{"className":73851,"translate":35},[34],[31,73853,73855],{"className":73854},[39],[41,73856,73857],{"xmlns":43},[45,73858,73859,73867],{},[48,73860,73861],{},[1180,73862,73863,73865],{},[51,73864,2521],{},[51,73866,2515],{},[55,73868,67939],{"encoding":57},[31,73870,73872],{"className":73871,"ariaHidden":62},[61],[31,73873,73875,73878],{"className":73874},[66],[31,73876],{"className":73877,"style":945},[70],[31,73879,73881,73884],{"className":73880},[75],[31,73882,2521],{"className":73883,"style":949},[75,76],[31,73885,73887],{"className":73886},[1039],[31,73888,73890],{"className":73889},[1043],[31,73891,73893],{"className":73892},[1048],[31,73894,73896],{"className":73895,"style":3628},[1052],[31,73897,73898,73901],{"style":3631},[31,73899],{"className":73900,"style":1061},[1060],[31,73902,73904],{"className":73903},[1065,1066,1067,1068],[31,73905,2515],{"className":73906,"style":2666},[75,76,1068]," 是随机核这一事实。因此它可以直接推广到一般状态空间。",[5813,73909,73911],{"href":73910},"#proposition-policy-verification","命题 2.3"," 也说明了贝尔曼方程的一个常见误读：它并不是「用未来价值近似当前价值」，而是在精确地刻画值函数。只要 ",[31,73914,73916,73933],{"className":73915,"translate":35},[34],[31,73917,73919],{"className":73918},[39],[41,73920,73921],{"xmlns":43},[45,73922,73923,73931],{},[48,73924,73925,73927,73929],{},[51,73926,53298],{},[95,73928,9758],{},[91,73930,205],{},[55,73932,69926],{"encoding":57},[31,73934,73936,73954],{"className":73935,"ariaHidden":62},[61],[31,73937,73939,73942,73945,73948,73951],{"className":73938},[66],[31,73940],{"className":73941,"style":53323},[70],[31,73943,53298],{"className":73944,"style":24157},[75,76],[31,73946],{"className":73947,"style":293},[119],[31,73949,9758],{"className":73950},[297],[31,73952],{"className":73953,"style":293},[119],[31,73955,73957,73960],{"className":73956},[66],[31,73958],{"className":73959,"style":326},[70],[31,73961,205],{"className":73962},[75]," 且奖励有界，满足方程的函数就是值函数本身。",[18,73965,73967],{"id":73966},"三贝尔曼算子与压缩映射","三、贝尔曼算子与压缩映射",[52860,73969,73971],{"id":73970},"_31-有界函数空间","3.1 有界函数空间",[11,73973,73974,73975,74019,74020,74048],{},"记 ",[31,73976,73978,73998],{"className":73977,"translate":35},[34],[31,73979,73981],{"className":73980},[39],[41,73982,73983],{"xmlns":43},[45,73984,73985,73995],{},[48,73986,73987,73989,73991,73993],{},[51,73988,1122],{},[95,73990,211],{"stretchy":210},[51,73992,21582],{},[95,73994,220],{"stretchy":210},[55,73996,73997],{"encoding":57},"B(S)",[31,73999,74001],{"className":74000,"ariaHidden":62},[61],[31,74002,74004,74007,74010,74013,74016],{"className":74003},[66],[31,74005],{"className":74006,"style":260},[70],[31,74008,1122],{"className":74009,"style":1137},[75,76],[31,74011,211],{"className":74012},[264],[31,74014,21582],{"className":74015,"style":21697},[75,76],[31,74017,220],{"className":74018},[289]," 为 ",[31,74021,74023,74036],{"className":74022,"translate":35},[34],[31,74024,74026],{"className":74025},[39],[41,74027,74028],{"xmlns":43},[45,74029,74030,74034],{},[48,74031,74032],{},[51,74033,21582],{},[55,74035,21582],{"encoding":57},[31,74037,74039],{"className":74038,"ariaHidden":62},[61],[31,74040,74042,74045],{"className":74041},[66],[31,74043],{"className":74044,"style":945},[70],[31,74046,21582],{"className":74047,"style":21697},[75,76]," 上全体有界实值函数构成的空间，并赋予 sup 范数",[31,74050,74052],{"className":74051,"translate":35},[411],[31,74053,74055,74109],{"className":74054,"translate":35},[34],[31,74056,74058],{"className":74057},[39],[41,74059,74060],{"xmlns":43,"display":420},[45,74061,74062,74106],{},[48,74063,74064,74066,74068,74074,74076,74092,74094,74096,74098,74100,74102,74104],{},[51,74065,4572],{"mathvariant":494},[51,74067,43481],{},[1004,74069,74070,74072],{},[51,74071,4572],{"mathvariant":494},[51,74073,4587],{"mathvariant":494},[95,74075,223],{},[1531,74077,74078,74084],{},[48,74079,74080,74082],{},[51,74081,10792],{},[95,74083,1540],{},[48,74085,74086,74088,74090],{},[51,74087,14551],{},[95,74089,2518],{},[51,74091,21582],{},[51,74093,12680],{"mathvariant":494},[51,74095,43481],{},[95,74097,211],{"stretchy":210},[51,74099,14551],{},[95,74101,220],{"stretchy":210},[51,74103,12680],{"mathvariant":494},[51,74105,495],{"mathvariant":494},[55,74107,74108],{"encoding":57},"\\|V\\|_\\infty=\\sup_{s\\in S}|V(s)|.",[31,74110,74112,74173],{"className":74111,"ariaHidden":62},[61],[31,74113,74115,74118,74121,74124,74164,74167,74170],{"className":74114},[66],[31,74116],{"className":74117,"style":260},[70],[31,74119,4572],{"className":74120},[75],[31,74122,43481],{"className":74123,"style":120},[75,76],[31,74125,74127,74130],{"className":74126},[75],[31,74128,4572],{"className":74129},[75],[31,74131,74133],{"className":74132},[1039],[31,74134,74136,74156],{"className":74135},[1043,1044],[31,74137,74139,74153],{"className":74138},[1048],[31,74140,74142],{"className":74141,"style":3230},[1052],[31,74143,74144,74147],{"style":2807},[31,74145],{"className":74146,"style":1061},[1060],[31,74148,74150],{"className":74149},[1065,1066,1067,1068],[31,74151,4587],{"className":74152},[75,1068],[31,74154,1076],{"className":74155},[1075],[31,74157,74159],{"className":74158},[1048],[31,74160,74162],{"className":74161,"style":2050},[1052],[31,74163],{},[31,74165],{"className":74166,"style":293},[119],[31,74168,223],{"className":74169},[297],[31,74171],{"className":74172,"style":293},[119],[31,74174,74176,74180,74235,74238,74241,74244,74247,74250,74253],{"className":74175},[66],[31,74177],{"className":74178,"style":74179},[70],"height:1.7161em;vertical-align:-0.9661em;",[31,74181,74183],{"className":74182},[1460,1620],[31,74184,74186,74226],{"className":74185},[1043,1044],[31,74187,74189,74223],{"className":74188},[1048],[31,74190,74192,74213],{"className":74191,"style":71},[1052],[31,74193,74195,74198],{"style":74194},"top:-2.1612em;margin-left:0em;",[31,74196],{"className":74197,"style":1637},[1060],[31,74199,74201],{"className":74200},[1065,1066,1067,1068],[31,74202,74204,74207,74210],{"className":74203},[75,1068],[31,74205,14551],{"className":74206},[75,76,1068],[31,74208,2518],{"className":74209},[297,1068],[31,74211,21582],{"className":74212,"style":21697},[75,76,1068],[31,74214,74215,74218],{"style":1649},[31,74216],{"className":74217,"style":1637},[1060],[31,74219,74220],{},[31,74221,10792],{"className":74222},[1460],[31,74224,1076],{"className":74225},[1075],[31,74227,74229],{"className":74228},[1048],[31,74230,74233],{"className":74231,"style":74232},[1052],"height:0.9661em;",[31,74234],{},[31,74236],{"className":74237,"style":1467},[119],[31,74239,12680],{"className":74240},[75],[31,74242,43481],{"className":74243,"style":120},[75,76],[31,74245,211],{"className":74246},[264],[31,74248,14551],{"className":74249},[75,76],[31,74251,220],{"className":74252},[289],[31,74254,74256],{"className":74255},[75],"∣.",[11,74258,65583,74259,74287,74288,74418],{},[31,74260,74262,74275],{"className":74261,"translate":35},[34],[31,74263,74265],{"className":74264},[39],[41,74266,74267],{"xmlns":43},[45,74268,74269,74273],{},[48,74270,74271],{},[51,74272,21582],{},[55,74274,21582],{"encoding":57},[31,74276,74278],{"className":74277,"ariaHidden":62},[61],[31,74279,74281,74284],{"className":74280},[66],[31,74282],{"className":74283,"style":945},[70],[31,74285,21582],{"className":74286,"style":21697},[75,76]," 为有限集时，上确界就是最大值。",[31,74289,74291,74327],{"className":74290,"translate":35},[34],[31,74292,74294],{"className":74293},[39],[41,74295,74296],{"xmlns":43},[45,74297,74298,74324],{},[48,74299,74300,74302,74304,74306,74308,74310,74312,74314,74316,74322],{},[95,74301,211],{"stretchy":210},[51,74303,1122],{},[95,74305,211],{"stretchy":210},[51,74307,21582],{},[95,74309,220],{"stretchy":210},[95,74311,1210],{"separator":62},[51,74313,4572],{"mathvariant":494},[95,74315,429],{},[1004,74317,74318,74320],{},[51,74319,4572],{"mathvariant":494},[51,74321,4587],{"mathvariant":494},[95,74323,220],{"stretchy":210},[55,74325,74326],{"encoding":57},"(B(S),\\|\\cdot\\|_\\infty)",[31,74328,74330,74369],{"className":74329,"ariaHidden":62},[61],[31,74331,74333,74336,74339,74342,74345,74348,74351,74354,74357,74360,74363,74366],{"className":74332},[66],[31,74334],{"className":74335,"style":260},[70],[31,74337,211],{"className":74338},[264],[31,74340,1122],{"className":74341,"style":1137},[75,76],[31,74343,211],{"className":74344},[264],[31,74346,21582],{"className":74347,"style":21697},[75,76],[31,74349,220],{"className":74350},[289],[31,74352,1210],{"className":74353},[1391],[31,74355],{"className":74356,"style":1467},[119],[31,74358,4572],{"className":74359},[75],[31,74361],{"className":74362,"style":120},[119],[31,74364,429],{"className":74365},[124],[31,74367],{"className":74368,"style":120},[119],[31,74370,74372,74375,74415],{"className":74371},[66],[31,74373],{"className":74374,"style":260},[70],[31,74376,74378,74381],{"className":74377},[75],[31,74379,4572],{"className":74380},[75],[31,74382,74384],{"className":74383},[1039],[31,74385,74387,74407],{"className":74386},[1043,1044],[31,74388,74390,74404],{"className":74389},[1048],[31,74391,74393],{"className":74392,"style":3230},[1052],[31,74394,74395,74398],{"style":2807},[31,74396],{"className":74397,"style":1061},[1060],[31,74399,74401],{"className":74400},[1065,1066,1067,1068],[31,74402,4587],{"className":74403},[75,1068],[31,74405,1076],{"className":74406},[1075],[31,74408,74410],{"className":74409},[1048],[31,74411,74413],{"className":74412,"style":2050},[1052],[31,74414],{},[31,74416,220],{"className":74417},[289]," 是完备的 Banach 空间。这个完备性是 Banach 不动点定理能够使用的关键。",[11,74420,74421,74422,74480,74481,74540,74541,74584,74585,74643,74644,74702],{},"贝尔曼期望算子 ",[31,74423,74425,74442],{"className":74424,"translate":35},[34],[31,74426,74428],{"className":74427},[39],[41,74429,74430],{"xmlns":43},[45,74431,74432,74440],{},[48,74433,74434],{},[1180,74435,74436,74438],{},[51,74437,919],{"mathvariant":925},[51,74439,2515],{},[55,74441,64428],{"encoding":57},[31,74443,74445],{"className":74444,"ariaHidden":62},[61],[31,74446,74448,74451],{"className":74447},[66],[31,74449],{"className":74450,"style":945},[70],[31,74452,74454,74457],{"className":74453},[75],[31,74455,919],{"className":74456,"style":64444},[75,968],[31,74458,74460],{"className":74459},[1039],[31,74461,74463],{"className":74462},[1043],[31,74464,74466],{"className":74465},[1048],[31,74467,74469],{"className":74468,"style":3628},[1052],[31,74470,74471,74474],{"style":3631},[31,74472],{"className":74473,"style":1061},[1060],[31,74475,74477],{"className":74476},[1065,1066,1067,1068],[31,74478,2515],{"className":74479,"style":2666},[75,76,1068]," 与贝尔曼最优算子 ",[31,74482,74484,74502],{"className":74483,"translate":35},[34],[31,74485,74487],{"className":74486},[39],[41,74488,74489],{"xmlns":43},[45,74490,74491,74499],{},[48,74492,74493],{},[1180,74494,74495,74497],{},[51,74496,919],{"mathvariant":925},[95,74498,30648],{},[55,74500,74501],{"encoding":57},"\\mathcal{T}^*",[31,74503,74505],{"className":74504,"ariaHidden":62},[61],[31,74506,74508,74511],{"className":74507},[66],[31,74509],{"className":74510,"style":30719},[70],[31,74512,74514,74517],{"className":74513},[75],[31,74515,919],{"className":74516,"style":64444},[75,968],[31,74518,74520],{"className":74519},[1039],[31,74521,74523],{"className":74522},[1043],[31,74524,74526],{"className":74525},[1048],[31,74527,74529],{"className":74528,"style":30719},[1052],[31,74530,74531,74534],{"style":3631},[31,74532],{"className":74533,"style":1061},[1060],[31,74535,74537],{"className":74536},[1065,1066,1067,1068],[31,74538,30648],{"className":74539},[124,1068]," 都将 ",[31,74542,74544,74563],{"className":74543,"translate":35},[34],[31,74545,74547],{"className":74546},[39],[41,74548,74549],{"xmlns":43},[45,74550,74551,74561],{},[48,74552,74553,74555,74557,74559],{},[51,74554,1122],{},[95,74556,211],{"stretchy":210},[51,74558,21582],{},[95,74560,220],{"stretchy":210},[55,74562,73997],{"encoding":57},[31,74564,74566],{"className":74565,"ariaHidden":62},[61],[31,74567,74569,74572,74575,74578,74581],{"className":74568},[66],[31,74570],{"className":74571,"style":260},[70],[31,74573,1122],{"className":74574,"style":1137},[75,76],[31,74576,211],{"className":74577},[264],[31,74579,21582],{"className":74580,"style":21697},[75,76],[31,74582,220],{"className":74583},[289]," 映射到自身。",[31,74586,74588,74605],{"className":74587,"translate":35},[34],[31,74589,74591],{"className":74590},[39],[41,74592,74593],{"xmlns":43},[45,74594,74595,74603],{},[48,74596,74597],{},[1180,74598,74599,74601],{},[51,74600,919],{"mathvariant":925},[51,74602,2515],{},[55,74604,64428],{"encoding":57},[31,74606,74608],{"className":74607,"ariaHidden":62},[61],[31,74609,74611,74614],{"className":74610},[66],[31,74612],{"className":74613,"style":945},[70],[31,74615,74617,74620],{"className":74616},[75],[31,74618,919],{"className":74619,"style":64444},[75,968],[31,74621,74623],{"className":74622},[1039],[31,74624,74626],{"className":74625},[1043],[31,74627,74629],{"className":74628},[1048],[31,74630,74632],{"className":74631,"style":3628},[1052],[31,74633,74634,74637],{"style":3631},[31,74635],{"className":74636,"style":1061},[1060],[31,74638,74640],{"className":74639},[1065,1066,1067,1068],[31,74641,2515],{"className":74642,"style":2666},[75,76,1068]," 的定义已在 2.2 节给出，",[31,74645,74647,74664],{"className":74646,"translate":35},[34],[31,74648,74650],{"className":74649},[39],[41,74651,74652],{"xmlns":43},[45,74653,74654,74662],{},[48,74655,74656],{},[1180,74657,74658,74660],{},[51,74659,919],{"mathvariant":925},[95,74661,30648],{},[55,74663,74501],{"encoding":57},[31,74665,74667],{"className":74666,"ariaHidden":62},[61],[31,74668,74670,74673],{"className":74669},[66],[31,74671],{"className":74672,"style":30719},[70],[31,74674,74676,74679],{"className":74675},[75],[31,74677,919],{"className":74678,"style":64444},[75,968],[31,74680,74682],{"className":74681},[1039],[31,74683,74685],{"className":74684},[1043],[31,74686,74688],{"className":74687},[1048],[31,74689,74691],{"className":74690,"style":30719},[1052],[31,74692,74693,74696],{"style":3631},[31,74694],{"className":74695,"style":1061},[1060],[31,74697,74699],{"className":74698},[1065,1066,1067,1068],[31,74700,30648],{"className":74701},[124,1068]," 定义为",[31,74704,74706],{"className":74705,"translate":35},[411],[31,74707,74709,74825],{"className":74708,"translate":35},[34],[31,74710,74712],{"className":74711},[39],[41,74713,74714],{"xmlns":43,"display":420},[45,74715,74716,74822],{},[48,74717,74718,74720,74726,74728,74730,74732,74734,74736,74738,74748,74758,74760,74762,74768,74770,74772,74774,74776,74778,74820],{},[95,74719,211],{"stretchy":210},[1180,74721,74722,74724],{},[51,74723,919],{"mathvariant":925},[95,74725,30648],{},[51,74727,43481],{},[95,74729,220],{"stretchy":210},[95,74731,211],{"stretchy":210},[51,74733,14551],{},[95,74735,220],{"stretchy":210},[95,74737,223],{},[1531,74739,74740,74746],{},[48,74741,74742,74744],{},[51,74743,10154],{},[95,74745,1540],{},[51,74747,5813],{},[1531,74749,74750,74752],{},[95,74751,5625],{},[1180,74753,74754,74756],{},[51,74755,14551],{},[95,74757,13691],{"mathvariant":494,"lspace":5610,"rspace":5610},[51,74759,2521],{},[95,74761,211],{"stretchy":210},[1180,74763,74764,74766],{},[51,74765,14551],{},[95,74767,13691],{"mathvariant":494,"lspace":5610,"rspace":5610},[95,74769,12680],{},[51,74771,14551],{},[95,74773,1210],{"separator":62},[51,74775,5813],{},[95,74777,220],{"stretchy":210},[48,74779,74780,74782,74784,74786,74788,74790,74792,74794,74800,74802,74804,74806,74808,74810,74816,74818],{},[95,74781,15175],{"fence":62},[51,74783,2549],{},[95,74785,211],{"stretchy":210},[51,74787,14551],{},[95,74789,1210],{"separator":62},[51,74791,5813],{},[95,74793,1210],{"separator":62},[1180,74795,74796,74798],{},[51,74797,14551],{},[95,74799,13691],{"mathvariant":494,"lspace":5610,"rspace":5610},[95,74801,220],{"stretchy":210},[95,74803,434],{},[51,74805,53298],{},[51,74807,43481],{},[95,74809,211],{"stretchy":210},[1180,74811,74812,74814],{},[51,74813,14551],{},[95,74815,13691],{"mathvariant":494,"lspace":5610,"rspace":5610},[95,74817,220],{"stretchy":210},[95,74819,15184],{"fence":62},[51,74821,495],{"mathvariant":494},[55,74823,74824],{"encoding":57},"(\\mathcal{T}^*V)(s)\n=\\max_{a}\\sum_{s'}P(s'\\mid s,a)\n\\left[r(s,a,s')+\\gamma V(s')\\right].",[31,74826,74828,74891,75074],{"className":74827,"ariaHidden":62},[61],[31,74829,74831,74834,74837,74867,74870,74873,74876,74879,74882,74885,74888],{"className":74830},[66],[31,74832],{"className":74833,"style":260},[70],[31,74835,211],{"className":74836},[264],[31,74838,74840,74843],{"className":74839},[75],[31,74841,919],{"className":74842,"style":64444},[75,968],[31,74844,74846],{"className":74845},[1039],[31,74847,74849],{"className":74848},[1043],[31,74850,74852],{"className":74851},[1048],[31,74853,74856],{"className":74854,"style":74855},[1052],"height:0.7387em;",[31,74857,74858,74861],{"style":1330},[31,74859],{"className":74860,"style":1061},[1060],[31,74862,74864],{"className":74863},[1065,1066,1067,1068],[31,74865,30648],{"className":74866},[124,1068],[31,74868,43481],{"className":74869,"style":120},[75,76],[31,74871,220],{"className":74872},[289],[31,74874,211],{"className":74875},[264],[31,74877,14551],{"className":74878},[75,76],[31,74880,220],{"className":74881},[289],[31,74883],{"className":74884,"style":293},[119],[31,74886,223],{"className":74887},[297],[31,74889],{"className":74890,"style":293},[119],[31,74892,74894,74898,74945,74948,75024,75027,75030,75033,75065,75068,75071],{"className":74893},[66],[31,74895],{"className":74896,"style":74897},[70],"height:2.344em;vertical-align:-1.294em;",[31,74899,74901],{"className":74900},[1460,1620],[31,74902,74904,74937],{"className":74903},[1043,1044],[31,74905,74907,74934],{"className":74906},[1048],[31,74908,74910,74924],{"className":74909,"style":71},[1052],[31,74911,74912,74915],{"style":37361},[31,74913],{"className":74914,"style":1637},[1060],[31,74916,74918],{"className":74917},[1065,1066,1067,1068],[31,74919,74921],{"className":74920},[75,1068],[31,74922,5813],{"className":74923},[75,76,1068],[31,74925,74926,74929],{"style":1649},[31,74927],{"className":74928,"style":1637},[1060],[31,74930,74931],{},[31,74932,10154],{"className":74933},[1460],[31,74935,1076],{"className":74936},[1075],[31,74938,74940],{"className":74939},[1048],[31,74941,74943],{"className":74942,"style":37390},[1052],[31,74944],{},[31,74946],{"className":74947,"style":1467},[119],[31,74949,74951],{"className":74950},[1460,1620],[31,74952,74954,75016],{"className":74953},[1043,1044],[31,74955,74957,75013],{"className":74956},[1048],[31,74958,74960,75003],{"className":74959,"style":5900},[1052],[31,74961,74962,74965],{"style":59756},[31,74963],{"className":74964,"style":5907},[1060],[31,74966,74968],{"className":74967},[1065,1066,1067,1068],[31,74969,74971],{"className":74970},[75,1068],[31,74972,74974,74977],{"className":74973},[75,1068],[31,74975,14551],{"className":74976},[75,76,1068],[31,74978,74980],{"className":74979},[1039],[31,74981,74983],{"className":74982},[1043],[31,74984,74986],{"className":74985},[1048],[31,74987,74989],{"className":74988,"style":31478},[1052],[31,74990,74991,74994],{"style":31481},[31,74992],{"className":74993,"style":11769},[1060],[31,74995,74997],{"className":74996},[1065,11773,1308,1068],[31,74998,75000],{"className":74999},[75,1068],[31,75001,13691],{"className":75002},[75,1068],[31,75004,75005,75008],{"style":5919},[31,75006],{"className":75007,"style":5907},[1060],[31,75009,75010],{},[31,75011,5625],{"className":75012},[1460,1461,1687],[31,75014,1076],{"className":75015},[1075],[31,75017,75019],{"className":75018},[1048],[31,75020,75022],{"className":75021,"style":59817},[1052],[31,75023],{},[31,75025],{"className":75026,"style":1467},[119],[31,75028,2521],{"className":75029,"style":949},[75,76],[31,75031,211],{"className":75032},[264],[31,75034,75036,75039],{"className":75035},[75],[31,75037,14551],{"className":75038},[75,76],[31,75040,75042],{"className":75041},[1039],[31,75043,75045],{"className":75044},[1043],[31,75046,75048],{"className":75047},[1048],[31,75049,75051],{"className":75050,"style":14017},[1052],[31,75052,75053,75056],{"style":1330},[31,75054],{"className":75055,"style":1061},[1060],[31,75057,75059],{"className":75058},[1065,1066,1067,1068],[31,75060,75062],{"className":75061},[75,1068],[31,75063,13691],{"className":75064},[75,1068],[31,75066],{"className":75067,"style":293},[119],[31,75069,12680],{"className":75070},[297],[31,75072],{"className":75073,"style":293},[119],[31,75075,75077,75081,75084,75087,75090,75093,75096,75099,75220,75223],{"className":75076},[66],[31,75078],{"className":75079,"style":75080},[70],"height:1.0519em;vertical-align:-0.25em;",[31,75082,14551],{"className":75083},[75,76],[31,75085,1210],{"className":75086},[1391],[31,75088],{"className":75089,"style":1467},[119],[31,75091,5813],{"className":75092},[75,76],[31,75094,220],{"className":75095},[289],[31,75097],{"className":75098,"style":1467},[119],[31,75100,75102,75105,75108,75111,75114,75117,75120,75123,75126,75129,75161,75164,75167,75170,75173,75176,75179,75182,75214,75217],{"className":75101},[3860],[31,75103,15175],{"className":75104,"style":3868},[264,3867],[31,75106,2549],{"className":75107,"style":19759},[75,76],[31,75109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",[31,75230,75232,75249],{"className":75231,"translate":35},[34],[31,75233,75235],{"className":75234},[39],[41,75236,75237],{"xmlns":43},[45,75238,75239,75247],{},[48,75240,75241],{},[1180,75242,75243,75245],{},[51,75244,919],{"mathvariant":925},[51,75246,2515],{},[55,75248,64428],{"encoding":57},[31,75250,75252],{"className":75251,"ariaHidden":62},[61],[31,75253,75255,75258],{"className":75254},[66],[31,75256],{"className":75257,"style":945},[70],[31,75259,75261,75264],{"className":75260},[75],[31,75262,919],{"className":75263,"style":64444},[75,968],[31,75265,75267],{"className":75266},[1039],[31,75268,75270],{"className":75269},[1043],[31,75271,75273],{"className":75272},[1048],[31,75274,75276],{"className":75275,"style":3628},[1052],[31,75277,75278,75281],{"style":3631},[31,75279],{"className":75280,"style":1061},[1060],[31,75282,75284],{"className":75283},[1065,1066,1067,1068],[31,75285,2515],{"className":75286,"style":2666},[75,76,1068]," 为例，",[31,75289,75291],{"className":75290,"translate":35},[411],[31,75292,75294,75464],{"className":75293,"translate":35},[34],[31,75295,75297],{"className":75296},[39],[41,75298,75299],{"xmlns":43,"display":420},[45,75300,75301,75461],{},[5606,75302,75303,75419],{"rowspacing":5608,"columnalign":5609,"columnspacing":5610},[5612,75304,75305,75333],{},[5615,75306,75307],{},[2071,75308,75309],{"scriptlevel":1977,"displaystyle":62},[48,75310,75311,75313,75315,75321,75323,75325,75327,75329,75331],{},[51,75312,12680],{"mathvariant":494},[95,75314,211],{"stretchy":210},[1180,75316,75317,75319],{},[51,75318,919],{"mathvariant":925},[51,75320,2515],{},[51,75322,43481],{},[95,75324,220],{"stretchy":210},[95,75326,211],{"stretchy":210},[51,75328,14551],{},[95,75330,220],{"stretchy":210},[51,75332,12680],{"mathvariant":494},[5615,75334,75335],{},[2071,75336,75337],{"scriptlevel":1977,"displaystyle":62},[48,75338,75339,75341,75343,75349,75351,75353,75355,75357,75359,75361,75371,75373,75375,75381,75383,75385,75387,75389,75391],{},[48,75340],{},[95,75342,347],{},[1531,75344,75345,75347],{},[95,75346,5625],{},[51,75348,5813],{},[51,75350,2515],{},[95,75352,211],{"stretchy":210},[51,75354,5813],{},[95,75356,12680],{},[51,75358,14551],{},[95,75360,220],{"stretchy":210},[1531,75362,75363,75365],{},[95,75364,5625],{},[1180,75366,75367,75369],{},[51,75368,14551],{},[95,75370,13691],{"mathvariant":494,"lspace":5610,"rspace":5610},[51,75372,2521],{},[95,75374,211],{"stretchy":210},[1180,75376,75377,75379],{},[51,75378,14551],{},[95,75380,13691],{"mathvariant":494,"lspace":5610,"rspace":5610},[95,75382,12680],{},[51,75384,14551],{},[95,75386,1210],{"separator":62},[51,75388,5813],{},[95,75390,220],{"stretchy":210},[48,75392,75393,75395,75403,75405,75407,75409,75411,75417],{},[95,75394,211],{"fence":62},[1004,75396,75397,75399,75401],{},[51,75398,14790],{},[51,75400,10154],{},[95,75402,1540],{},[95,75404,434],{},[51,75406,53298],{},[51,75408,4572],{"mathvariant":494},[51,75410,43481],{},[1004,75412,75413,75415],{},[51,75414,4572],{"mathvariant":494},[51,75416,4587],{"mathvariant":494},[95,75418,220],{"fence":62},[5612,75420,75421,75427],{},[5615,75422,75423],{},[2071,75424,75425],{"scriptlevel":1977,"displaystyle":62},[48,75426],{},[5615,75428,75429],{},[2071,75430,75431],{"scriptlevel":1977,"displaystyle":62},[48,75432,75433,75435,75437,75445,75447,75449,75451,75453,75459],{},[48,75434],{},[95,75436,223],{},[1004,75438,75439,75441,75443],{},[51,75440,14790],{},[51,75442,10154],{},[95,75444,1540],{},[95,75446,434],{},[51,75448,53298],{},[51,75450,4572],{"mathvariant":494},[51,75452,43481],{},[1004,75454,75455,75457],{},[51,75456,4572],{"mathvariant":494},[51,75458,4587],{"mathvariant":494},[95,75460,1210],{"separator":62},[55,75462,75463],{"encoding":57},"\\begin{aligned}\n|(\\mathcal{T}^\\pi V)(s)|\n&\\le\\sum_{a}\\pi(a\\mid s)\\sum_{s'}P(s'\\mid s,a)\n\\left(R_{\\max}+\\gamma\\|V\\|_\\infty\\right)\\\\\n&=R_{\\max}+\\gamma\\|V\\|_\\infty,\n\\end{aligned}",[31,75465,75467],{"className":75466,"ariaHidden":62},[61],[31,75468,75470,75473],{"className":75469},[66],[31,75471],{"className":75472,"style":59613},[70],[31,75474,75476],{"className":75475},[75],[31,75477,75479,75571],{"className":75478},[5606],[31,75480,75482],{"className":75481},[5867],[31,75483,75485,75563],{"className":75484},[1043,1044],[31,75486,75488,75560],{"className":75487},[1048],[31,75489,75491,75552],{"className":75490,"style":59632},[1052],[31,75492,75493,75496],{"style":59635},[31,75494],{"className":75495,"style":5907},[1060],[31,75497,75499,75502,75505,75534,75537,75540,75543,75546,75549],{"className":75498},[75],[31,75500,12680],{"className":75501},[75],[31,75503,211],{"className":75504},[264],[31,75506,75508,75511],{"className":75507},[75],[31,75509,919],{"className":75510,"style":64444},[75,968],[31,75512,75514],{"className":75513},[1039],[31,75515,75517],{"className":75516},[1043],[31,75518,75520],{"className":75519},[1048],[31,75521,75523],{"className":75522,"style":3987},[1052],[31,75524,75525,75528],{"style":1330},[31,75526],{"className":75527,"style":1061},[1060],[31,75529,75531],{"className":75530},[1065,1066,1067,1068],[31,75532,2515],{"className":75533,"style":2666},[75,76,1068],[31,75535,43481],{"className":75536,"style":120},[75,76],[31,75538,220],{"className":75539},[289],[31,75541,211],{"className":75542},[264],[31,75544,14551],{"className":75545},[75,76],[31,75547,220],{"className":75548},[289],[31,75550,12680],{"className":75551},[75],[31,75553,75554,75557],{"style":59691},[31,75555],{"className":75556,"style":5907},[1060],[31,75558],{"className":75559},[75],[31,75561,1076],{"className":75562},[1075],[31,75564,75566],{"className":75565},[1048],[31,75567,75569],{"className":75568,"style":59707},[1052],[31,75570],{},[31,75572,75574],{"className":75573},[6062],[31,75575,75577,76079],{"className":75576},[1043,1044],[31,75578,75580,76076],{"className":75579},[1048],[31,75581,75583,75943],{"className":75582,"style":59632},[1052],[31,75584,75585,75588],{"style":59635},[31,75586],{"className":75587,"style":5907},[1060],[31,75589,75591,75594,75597,75600,75603,75650,75653,75656,75659,75662,75665,75668,75671,75674,75677,75680,75756,75759,75762,75765,75797,75800,75803,75806,75809,75812,75815,75818,75821,75824],{"className":75590},[75],[31,75592],{"className":75593},[75],[31,75595],{"className":75596,"style":293},[119],[31,75598,347],{"className":75599},[297],[31,75601],{"className":75602,"style":293},[119],[31,75604,75606],{"className":75605},[1460,1620],[31,75607,75609,75642],{"className":75608},[1043,1044],[31,75610,75612,75639],{"className":75611},[1048],[31,75613,75615,75629],{"className":75614,"style":5900},[1052],[31,75616,75617,75620],{"style":59333},[31,75618],{"className":75619,"style":5907},[1060],[31,75621,75623],{"className":75622},[1065,1066,1067,1068],[31,75624,75626],{"className":75625},[75,1068],[31,75627,5813],{"className":75628},[75,76,1068],[31,75630,75631,75634],{"style":5919},[31,75632],{"className":75633,"style":5907},[1060],[31,75635,75636],{},[31,75637,5625],{"className":75638},[1460,1461,1687],[31,75640,1076],{"className":75641},[1075],[31,75643,75645],{"className":75644},[1048],[31,75646,75648],{"className":75647,"style":59365},[1052],[31,75649],{},[31,75651],{"className":75652,"style":1467},[119],[31,75654,2515],{"className":75655,"style":2666},[75,76],[31,75657,211],{"className":75658},[264],[31,75660,5813],{"className":75661},[75,76],[31,75663],{"className":75664,"style":293},[119],[31,75666,12680],{"className":75667},[297],[31,75669],{"className":75670,"style":293},[119],[31,75672,14551],{"className":75673},[75,76],[31,75675,220],{"className":75676},[289],[31,75678],{"className":75679,"style":1467},[119],[31,75681,75683],{"className":75682},[1460,1620],[31,75684,75686,75748],{"className":75685},[1043,1044],[31,75687,75689,75745],{"className":75688},[1048],[31,75690,75692,75735],{"className":75691,"style":5900},[1052],[31,75693,75694,75697],{"style":59756},[31,75695],{"className":75696,"style":5907},[1060],[31,75698,75700],{"className":75699},[1065,1066,1067,1068],[31,75701,75703],{"className":75702},[75,1068],[31,75704,75706,75709],{"className":75705},[75,1068],[31,75707,14551],{"className":75708},[75,76,1068],[31,75710,75712],{"className":75711},[1039],[31,75713,75715],{"className":75714},[1043],[31,75716,75718],{"className":75717},[1048],[31,75719,75721],{"className":75720,"style":31478},[1052],[31,75722,75723,75726],{"style":31481},[31,75724],{"className":75725,"style":11769},[1060],[31,75727,75729],{"className":75728},[1065,11773,1308,1068],[31,75730,75732],{"className":75731},[75,1068],[31,75733,13691],{"className":75734},[75,1068],[31,75736,75737,75740],{"style":5919},[31,75738],{"className":75739,"style":5907},[1060],[31,75741,75742],{},[31,75743,5625],{"className":75744},[1460,1461,1687],[31,75746,1076],{"className":75747},[1075],[31,7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className":76050,"style":3230},[1052],[31,76052,76053,76056],{"style":2807},[31,76054],{"className":76055,"style":1061},[1060],[31,76057,76059],{"className":76058},[1065,1066,1067,1068],[31,76060,4587],{"className":76061},[75,1068],[31,76063,1076],{"className":76064},[1075],[31,76066,76068],{"className":76067},[1048],[31,76069,76071],{"className":76070,"style":2050},[1052],[31,76072],{},[31,76074,1210],{"className":76075},[1391],[31,76077,1076],{"className":76078},[1075],[31,76080,76082],{"className":76081},[1048],[31,76083,76085],{"className":76084,"style":59707},[1052],[31,76086],{},[11,76088,68754,76089,76226,76227,76285],{},[31,76090,76092,76124],{"className":76091,"translate":35},[34],[31,76093,76095],{"className":76094},[39],[41,76096,76097],{"xmlns":43},[45,76098,76099,76121],{},[48,76100,76101,76103,76109,76111,76117,76119],{},[51,76102,4572],{"mathvariant":494},[1180,76104,76105,76107],{},[51,76106,919],{"mathvariant":925},[51,76108,2515],{},[51,76110,43481],{},[1004,76112,76113,76115],{},[51,76114,4572],{"mathvariant":494},[51,76116,4587],{"mathvariant":494},[95,76118,9758],{},[51,76120,4587],{"mathvariant":494},[55,76122,76123],{"encoding":57},"\\|\\mathcal{T}^\\pi V\\|_\\infty\u003C\\infty",[31,76125,76127,76217],{"className":76126,"ariaHidden":62},[61],[31,76128,76130,76133,76136,76165,76168,76208,76211,76214],{"className":76129},[66],[31,76131],{"className":76132,"style":260},[70],[31,76134,4572],{"className":76135},[75],[31,76137,76139,76142],{"className":76138},[75],[31,76140,919],{"className":76141,"style":64444},[75,968],[31,76143,76145],{"className":76144},[1039],[31,76146,76148],{"className":76147},[1043],[31,76149,76151],{"className":76150},[1048],[31,76152,76154],{"className":76153,"style":3628},[1052],[31,76155,76156,76159],{"style":3631},[31,76157],{"className":76158,"style":1061},[1060],[31,76160,76162],{"className":76161},[1065,1066,1067,1068],[31,76163,2515],{"className":76164,"style":2666},[75,76,1068],[31,76166,43481],{"className":76167,"style":120},[75,76],[31,76169,76171,76174],{"className":76170},[75],[31,76172,4572],{"className":76173},[75],[31,76175,76177],{"className":76176},[1039],[31,76178,76180,76200],{"className":76179},[1043,1044],[31,76181,76183,76197],{"className":76182},[1048],[31,76184,76186],{"className":76185,"style":3230},[1052],[31,76187,76188,76191],{"style":2807},[31,76189],{"className":76190,"style":1061},[1060],[31,76192,76194],{"className":76193},[1065,1066,1067,1068],[31,76195,4587],{"className":76196},[75,1068],[31,76198,1076],{"className":76199},[1075],[31,76201,76203],{"className":76202},[1048],[31,76204,76206],{"className":76205,"style":2050},[1052],[31,76207],{},[31,76209],{"className":76210,"style":293},[119],[31,76212,9758],{"className":76213},[297],[31,76215],{"className":76216,"style":293},[119],[31,76218,76220,76223],{"className":76219},[66],[31,76221],{"className":76222,"style":71},[70],[31,76224,4587],{"className":76225},[75],"。对 ",[31,76228,76230,76247],{"className":76229,"translate":35},[34],[31,76231,76233],{"className":76232},[39],[41,76234,76235],{"xmlns":43},[45,76236,76237,76245],{},[48,76238,76239],{},[1180,76240,76241,76243],{},[51,76242,919],{"mathvariant":925},[95,76244,30648],{},[55,76246,74501],{"encoding":57},[31,76248,76250],{"className":76249,"ariaHidden":62},[61],[31,76251,76253,76256],{"className":76252},[66],[31,76254],{"className":76255,"style":30719},[70],[31,76257,76259,76262],{"className":76258},[75],[31,76260,919],{"className":76261,"style":64444},[75,968],[31,76263,76265],{"className":76264},[1039],[31,76266,76268],{"className":76267},[1043],[31,76269,76271],{"className":76270},[1048],[31,76272,76274],{"className":76273,"style":30719},[1052],[31,76275,76276,76279],{"style":3631},[31,76277],{"className":76278,"style":1061},[1060],[31,76280,76282],{"className":76281},[1065,1066,1067,1068],[31,76283,30648],{"className":76284},[124,1068]," 有同样的界。因此算子迭代不会离开这个空间。",[52860,76287,76289],{"id":76288},"_32-单调性与常数平移","3.2 单调性与常数平移",[52865,76291,76294,76429,76670,76724,77181],{"id":76292,"title":76293},"proposition-monotonicity","命题 3.1（单调性与常数平移）",[11,76295,52871,76296,76375,76376,76428],{},[31,76297,76299,76327],{"className":76298,"translate":35},[34],[31,76300,76302],{"className":76301},[39],[41,76303,76304],{"xmlns":43},[45,76305,76306,76324],{},[48,76307,76308,76310,76312,76314,76316,76318,76320,76322],{},[51,76309,43481],{},[95,76311,1210],{"separator":62},[51,76313,3117],{},[95,76315,2518],{},[51,76317,1122],{},[95,76319,211],{"stretchy":210},[51,76321,21582],{},[95,76323,220],{"stretchy":210},[55,76325,76326],{"encoding":57},"V,W\\in B(S)",[31,76328,76330,76357],{"className":76329,"ariaHidden":62},[61],[31,76331,76333,76336,76339,76342,76345,76348,76351,76354],{"className":76332},[66],[31,76334],{"className":76335,"style":59114},[70],[31,76337,43481],{"className":76338,"style":120},[75,76],[31,76340,1210],{"className":76341},[1391],[31,76343],{"className":76344,"style":1467},[119],[31,76346,3117],{"className":76347,"style":949},[75,76],[31,76349],{"className":76350,"style":293},[119],[31,76352,2518],{"className":76353},[297],[31,76355],{"className":76356,"style":293},[119],[31,76358,76360,76363,76366,76369,76372],{"className":76359},[66],[31,76361],{"className":76362,"style":260},[70],[31,76364,1122],{"className":76365,"style":1137},[75,76],[31,76367,211],{"className":76368},[264],[31,76370,21582],{"className":76371,"style":21697},[75,76],[31,76373,220],{"className":76374},[289]," 且 ",[31,76377,76379,76397],{"className":76378,"translate":35},[34],[31,76380,76382],{"className":76381},[39],[41,76383,76384],{"xmlns":43},[45,76385,76386,76394],{},[48,76387,76388,76390,76392],{},[51,76389,43481],{},[95,76391,347],{},[51,76393,3117],{},[55,76395,76396],{"encoding":57},"V\\le W",[31,76398,76400,76419],{"className":76399,"ariaHidden":62},[61],[31,76401,76403,76407,76410,76413,76416],{"className":76402},[66],[31,76404],{"className":76405,"style":76406},[70],"height:0.8193em;vertical-align:-0.136em;",[31,76408,43481],{"className":76409,"style":120},[75,76],[31,76411],{"className":76412,"style":293},[119],[31,76414,347],{"className":76415},[297],[31,76417],{"className":76418,"style":293},[119],[31,76420,76422,76425],{"className":76421},[66],[31,76423],{"className":76424,"style":945},[70],[31,76426,3117],{"className":76427,"style":949},[75,76]," 逐点成立。则",[31,76430,76432],{"className":76431,"translate":35},[411],[31,76433,76435,76489],{"className":76434,"translate":35},[34],[31,76436,76438],{"className":76437},[39],[41,76439,76440],{"xmlns":43,"display":420},[45,76441,76442,76486],{},[48,76443,76444,76450,76452,76454,76460,76462,76464,76466,76472,76474,76476,76482,76484],{},[1180,76445,76446,76448],{},[51,76447,919],{"mathvariant":925},[51,76449,2515],{},[51,76451,43481],{},[95,76453,347],{},[1180,76455,76456,76458],{},[51,76457,919],{"mathvariant":925},[51,76459,2515],{},[51,76461,3117],{},[95,76463,1210],{"separator":62},[119,76465],{"width":20286},[1180,76467,76468,76470],{},[51,76469,919],{"mathvariant":925},[95,76471,30648],{},[51,76473,43481],{},[95,76475,347],{},[1180,76477,76478,76480],{},[51,76479,919],{"mathvariant":925},[95,76481,30648],{},[51,76483,3117],{},[51,76485,495],{"mathvariant":494},[55,76487,76488],{"encoding":57},"\\mathcal{T}^\\pi V\\le\\mathcal{T}^\\pi W,\n\\qquad\n\\mathcal{T}^* V\\le\\mathcal{T}^* W.",[31,76490,76492,76540,76629],{"className":76491,"ariaHidden":62},[61],[31,76493,76495,76499,76528,76531,76534,76537],{"className":76494},[66],[31,76496],{"className":76497,"style":76498},[70],"height:0.8504em;vertical-align:-0.136em;",[31,76500,76502,76505],{"className":76501},[75],[31,76503,919],{"className":76504,"style":64444},[75,968],[31,76506,76508],{"className":76507},[1039],[31,76509,76511],{"className":76510},[1043],[31,76512,76514],{"className":76513},[1048],[31,76515,76517],{"className":76516,"style":3987},[1052],[31,76518,76519,76522],{"style":1330},[31,76520],{"className":76521,"style":1061},[1060],[31,76523,76525],{"className":76524},[1065,1066,1067,1068],[31,76526,2515],{"className":76527,"style":2666},[75,76,1068],[31,76529,43481],{"className":76530,"style":120},[75,76],[31,76532],{"className":76533,"style":293},[119],[31,76535,347],{"className":76536},[297],[31,76538],{"className":76539,"style":293},[119],[31,76541,76543,76547,76576,76579,76582,76585,76588,76617,76620,76623,76626],{"className":76542},[66],[31,76544],{"className":76545,"style":76546},[70],"height:0.9331em;vertical-align:-0.1944em;",[31,76548,76550,76553],{"className":76549},[75],[31,76551,919],{"className":76552,"style":64444},[75,968],[31,76554,76556],{"className":76555},[1039],[31,76557,76559],{"className":76558},[1043],[31,76560,76562],{"className":76561},[1048],[31,76563,76565],{"className":76564,"style":3987},[1052],[31,76566,76567,76570],{"style":1330},[31,76568],{"className":76569,"style":1061},[1060],[31,76571,76573],{"className":76572},[1065,1066,1067,1068],[31,76574,2515],{"className":76575,"style":2666},[75,76,1068],[31,76577,3117],{"className":76578,"style":949},[75,76],[31,76580,1210],{"className":76581},[1391],[31,76583],{"className":76584,"style":20525},[119],[31,76586],{"className":76587,"style":1467},[119],[31,76589,76591,76594],{"className":76590},[75],[31,76592,919],{"className":76593,"style":64444},[75,968],[31,76595,76597],{"className":76596},[1039],[31,76598,76600],{"className":76599},[1043],[31,76601,76603],{"className":76602},[1048],[31,76604,76606],{"className":76605,"style":74855},[1052],[31,76607,76608,76611],{"style":1330},[31,76609],{"className":76610,"style":1061},[1060],[31,76612,76614],{"className":76613},[1065,1066,1067,1068],[31,76615,30648],{"className":76616},[124,1068],[31,76618,43481],{"className":76619,"style":120},[75,76],[31,76621],{"className":76622,"style":293},[119],[31,76624,347],{"className":76625},[297],[31,76627],{"className":76628,"style":293},[119],[31,76630,76632,76635,76664,76667],{"className":76631},[66],[31,76633],{"className":76634,"style":74855},[70],[31,76636,76638,76641],{"className":76637},[75],[31,76639,919],{"className":76640,"style":64444},[75,968],[31,76642,76644],{"className":76643},[1039],[31,76645,76647],{"className":76646},[1043],[31,76648,76650],{"className":76649},[1048],[31,76651,76653],{"className":76652,"style":74855},[1052],[31,76654,76655,76658],{"style":1330},[31,76656],{"className":76657,"style":1061},[1060],[31,76659,76661],{"className":76660},[1065,1066,1067,1068],[31,76662,30648],{"className":76663},[124,1068],[31,76665,3117],{"className":76666,"style":949},[75,76],[31,76668,495],{"className":76669},[75],[11,76671,76672,76673,1138],{},"此外，对任意常数 ",[31,76674,76676,76694],{"className":76675,"translate":35},[34],[31,76677,76679],{"className":76678},[39],[41,76680,76681],{"xmlns":43},[45,76682,76683,76691],{},[48,76684,76685,76687,76689],{},[51,76686,1412],{},[95,76688,2518],{},[51,76690,14790],{"mathvariant":14789},[55,76692,76693],{"encoding":57},"c\\in\\mathbb{R}",[31,76695,76697,76715],{"className":76696,"ariaHidden":62},[61],[31,76698,76700,76703,76706,76709,76712],{"className":76699},[66],[31,76701],{"className":76702,"style":21113},[70],[31,76704,1412],{"className":76705},[75,76],[31,76707],{"className":76708,"style":293},[119],[31,76710,2518],{"className":76711},[297],[31,76713],{"className":76714,"style":293},[119],[31,76716,76718,76721],{"className":76717},[66],[31,76719],{"className":76720,"style":53279},[70],[31,76722,14790],{"className":76723},[75,14947],[31,76725,76727],{"className":76726,"translate":35},[411],[31,76728,76730,76851],{"className":76729,"translate":35},[34],[31,76731,76733],{"className":76732},[39],[41,76734,76735],{"xmlns":43,"display":420},[45,76736,76737,76848],{},[5606,76738,76739,76794],{"rowspacing":5608,"columnalign":5609,"columnspacing":5610},[5612,76740,76741,76766],{},[5615,76742,76743],{},[2071,76744,76745],{"scriptlevel":1977,"displaystyle":62},[48,76746,76747,76753,76755,76757,76759,76761,76764],{},[1180,76748,76749,76751],{},[51,76750,919],{"mathvariant":925},[51,76752,2515],{},[95,76754,211],{"stretchy":210},[51,76756,43481],{},[95,76758,434],{},[51,76760,1412],{},[91,76762,205],{"mathvariant":76763},"bold",[95,76765,220],{"stretchy":210},[5615,76767,76768],{},[2071,76769,76770],{"scriptlevel":1977,"displaystyle":62},[48,76771,76772,76774,76776,76782,76784,76786,76788,76790,76792],{},[48,76773],{},[95,76775,223],{},[1180,76777,76778,76780],{},[51,76779,919],{"mathvariant":925},[51,76781,2515],{},[51,76783,43481],{},[95,76785,434],{},[51,76787,53298],{},[51,76789,1412],{},[91,76791,205],{"mathvariant":76763},[95,76793,1210],{"separator":62},[5612,76795,76796,76820],{},[5615,76797,76798],{},[2071,76799,76800],{"scriptlevel":1977,"displaystyle":62},[48,76801,76802,76808,76810,76812,76814,76816,76818],{},[1180,76803,76804,76806],{},[51,76805,919],{"mathvariant":925},[95,76807,30648],{},[95,76809,211],{"stretchy":210},[51,76811,43481],{},[95,76813,434],{},[51,76815,1412],{},[91,76817,205],{"mathvariant":76763},[95,76819,220],{"stretchy":210},[5615,76821,76822],{},[2071,76823,76824],{"scriptlevel":1977,"displaystyle":62},[48,76825,76826,76828,76830,76836,76838,76840,76842,76844,76846],{},[48,76827],{},[95,76829,223],{},[1180,76831,76832,76834],{},[51,76833,919],{"mathvariant":925},[95,76835,30648],{},[51,76837,43481],{},[95,76839,434],{},[51,76841,53298],{},[51,76843,1412],{},[91,76845,205],{"mathvariant":76763},[95,76847,1210],{"separator":62},[55,76849,76850],{"encoding":57},"\\begin{aligned}\n\\mathcal{T}^\\pi(V+c\\mathbf{1})\n&=\\mathcal{T}^\\pi V+\\gamma c\\mathbf{1},\\\\\n\\mathcal{T}^*(V+c\\mathbf{1})\n&=\\mathcal{T}^* V+\\gamma c\\mathbf{1},\n\\end{aligned}",[31,76852,76854],{"className":76853,"ariaHidden":62},[61],[31,76855,76857,76860],{"className":76856},[66],[31,76858],{"className":76859,"style":57050},[70],[31,76861,76863],{"className":76862},[75],[31,76864,76866,77012],{"className":76865},[5606],[31,76867,76869],{"className":76868},[5867],[31,76870,76872,77004],{"className":76871},[1043,1044],[31,76873,76875,77001],{"className":76874},[1048],[31,76876,76878,76940],{"className":76877,"style":57069},[1052],[31,76879,76880,76883],{"style":57072},[31,76881],{"className":76882,"style":1637},[1060],[31,76884,76886,76915,76918,76921,76924,76927,76930,76933,76937],{"className":76885},[75],[31,76887,76889,76892],{"className":76888},[75],[31,76890,919],{"className":76891,"style":64444},[75,968],[31,76893,76895],{"className":76894},[1039],[31,76896,76898],{"className":76897},[1043],[31,76899,76901],{"className":76900},[1048],[31,76902,76904],{"className":76903,"style":3987},[1052],[31,76905,76906,76909],{"style":1330},[31,76907],{"className":76908,"style":1061},[1060],[31,76910,76912],{"className":76911},[1065,1066,1067,1068],[31,76913,2515],{"className":76914,"style":2666},[75,76,1068],[31,76916,211],{"className":76917},[264],[31,76919,43481],{"className":76920,"style":120},[75,76],[31,76922],{"className":76923,"style":120},[119],[31,76925,434],{"className":76926},[124],[31,76928],{"className":76929,"style":120},[119],[31,76931,1412],{"className":76932},[75,76],[31,76934,205],{"className":76935},[75,76936],"mathbf",[31,76938,220],{"className":76939},[289],[31,76941,76942,76945],{"style":57119},[31,76943],{"className":76944,"style":1637},[1060],[31,76946,76948,76977,76980,76983,76986,76989,76992,76995,76998],{"className":76947},[75],[31,76949,76951,76954],{"className":76950},[75],[31,76952,919],{"className":76953,"style":64444},[75,968],[31,76955,76957],{"className":76956},[1039],[31,76958,76960],{"className":76959},[1043],[31,76961,76963],{"className":76962},[1048],[31,76964,76966],{"className":76965,"style":74855},[1052],[31,76967,76968,76971],{"style":1330},[31,76969],{"className":76970,"style":1061},[1060],[31,76972,76974],{"className":76973},[1065,1066,1067,1068],[31,76975,30648],{"className":76976},[124,1068],[31,76978,211],{"className":76979},[264],[31,76981,43481],{"className":76982,"style":120},[75,76],[31,76984],{"className":76985,"style":120},[119],[31,76987,434],{"className":76988},[124],[31,76990],{"className":76991,"style":120},[119],[31,76993,1412],{"className":76994},[75,76],[31,76996,205],{"className":76997},[75,76936],[31,76999,220],{"className":77000},[289],[31,77002,1076],{"className":77003},[1075],[31,77005,77007],{"className":77006},[1048],[31,77008,77010],{"className":77009,"style":57182},[1052],[31,77011],{},[31,77013,77015],{"className":77014},[6062],[31,77016,77018,77173],{"className":77017},[1043,1044],[31,77019,77021,77170],{"className":77020},[1048],[31,77022,77024,77097],{"className":77023,"style":57069},[1052],[31,77025,77026,77029],{"style":57072},[31,77027],{"className":77028,"style":1637},[1060],[31,77030,77032,77035,77038,77041,77044,77073,77076,77079,77082,77085,77088,77091,77094],{"className":77031},[75],[31,77033],{"className":77034},[75],[31,77036],{"className":77037,"style":293},[119],[31,77039,223],{"className":77040},[297],[31,77042],{"className":77043,"style":293},[119],[31,77045,77047,77050],{"className":77046},[75],[31,77048,919],{"className":77049,"style":64444},[75,968],[31,77051,77053],{"className":77052},[1039],[31,77054,77056],{"className":77055},[1043],[31,77057,77059],{"className":77058},[1048],[31,77060,77062],{"className":77061,"style":3987},[1052],[31,77063,77064,77067],{"style":1330},[31,77065],{"className":77066,"style":1061},[1060],[31,77068,77070],{"className":77069},[1065,1066,1067,1068],[31,77071,2515],{"className":77072,"style":2666},[75,76,1068],[31,77074,43481],{"className":77075,"style":120},[75,76],[31,77077],{"className":77078,"style":120},[119],[31,77080,434],{"className":77081},[124],[31,77083],{"className":77084,"style":120},[119],[31,77086,53298],{"className":77087,"style":24157},[75,76],[31,77089,1412],{"className":77090},[75,76],[31,77092,205],{"className":77093},[75,76936],[31,77095,1210],{"className":77096},[1391],[31,77098,77099,77102],{"style":57119},[31,77100],{"className":77101,"style":1637},[1060],[31,77103,77105,77108,77111,77114,77117,77146,77149,77152,77155,77158,77161,77164,77167],{"className":77104},[75],[31,77106],{"className":77107},[75],[31,77109],{"className":77110,"style":293},[119],[31,77112,223],{"className":77113},[297],[31,77115],{"className":77116,"style":293},[119],[31,77118,77120,77123],{"className":77119},[75],[31,77121,919],{"className":77122,"style":64444},[75,968],[31,77124,77126],{"className":77125},[1039],[31,77127,77129],{"className":77128},[1043],[31,77130,77132],{"className":77131},[1048],[31,77133,77135],{"className":77134,"style":74855},[1052],[31,77136,77137,77140],{"style":1330},[31,77138],{"className":77139,"style":1061},[1060],[31,77141,77143],{"className":77142},[1065,1066,1067,1068],[31,77144,30648],{"className":77145},[124,1068],[31,77147,43481],{"className":77148,"style":120},[75,76],[31,77150],{"className":77151,"style":120},[119],[31,77153,434],{"className":77154},[124],[31,77156],{"className":77157,"style":120},[119],[31,77159,53298],{"className":77160,"style":24157},[75,76],[31,77162,1412],{"className":77163},[75,76],[31,77165,205],{"className":77166},[75,76936],[31,77168,1210],{"className":77169},[1391],[31,77171,1076],{"className":77172},[1075],[31,77174,77176],{"className":77175},[1048],[31,77177,77179],{"className":77178,"style":57182},[1052],[31,77180],{},[11,77182,50599,77183,77212,77213,77241],{},[31,77184,77186,77200],{"className":77185,"translate":35},[34],[31,77187,77189],{"className":77188},[39],[41,77190,77191],{"xmlns":43},[45,77192,77193,77197],{},[48,77194,77195],{},[91,77196,205],{"mathvariant":76763},[55,77198,77199],{"encoding":57},"\\mathbf{1}",[31,77201,77203],{"className":77202,"ariaHidden":62},[61],[31,77204,77206,77209],{"className":77205},[66],[31,77207],{"className":77208,"style":326},[70],[31,77210,205],{"className":77211},[75,76936]," 表示恒等于 ",[31,77214,77216,77229],{"className":77215,"translate":35},[34],[31,77217,77219],{"className":77218},[39],[41,77220,77221],{"xmlns":43},[45,77222,77223,77227],{},[48,77224,77225],{},[91,77226,205],{},[55,77228,205],{"encoding":57},[31,77230,77232],{"className":77231,"ariaHidden":62},[61],[31,77233,77235,77238],{"className":77234},[66],[31,77236],{"className":77237,"style":326},[70],[31,77239,205],{"className":77240},[75]," 的函数。",[11,77243,77244,77246,77247,77297,77298,1138],{},[7246,77245,7248],{}," 若 ",[31,77248,77250,77267],{"className":77249,"translate":35},[34],[31,77251,77253],{"className":77252},[39],[41,77254,77255],{"xmlns":43},[45,77256,77257,77265],{},[48,77258,77259,77261,77263],{},[51,77260,43481],{},[95,77262,347],{},[51,77264,3117],{},[55,77266,76396],{"encoding":57},[31,77268,77270,77288],{"className":77269,"ariaHidden":62},[61],[31,77271,77273,77276,77279,77282,77285],{"className":77272},[66],[31,77274],{"className":77275,"style":76406},[70],[31,77277,43481],{"className":77278,"style":120},[75,76],[31,77280],{"className":77281,"style":293},[119],[31,77283,347],{"className":77284},[297],[31,77286],{"className":77287,"style":293},[119],[31,77289,77291,77294],{"className":77290},[66],[31,77292],{"className":77293,"style":945},[70],[31,77295,3117],{"className":77296,"style":949},[75,76],"，则对每个 ",[31,77299,77301,77330],{"className":77300,"translate":35},[34],[31,77302,77304],{"className":77303},[39],[41,77305,77306],{"xmlns":43},[45,77307,77308,77328],{},[48,77309,77310,77312,77314,77316,77318,77320,77326],{},[95,77311,211],{"stretchy":210},[51,77313,14551],{},[95,77315,1210],{"separator":62},[51,77317,5813],{},[95,77319,1210],{"separator":62},[1180,77321,77322,77324],{},[51,77323,14551],{},[95,77325,13691],{"mathvariant":494,"lspace":5610,"rspace":5610},[95,77327,220],{"stretchy":210},[55,77329,54724],{"encoding":57},[31,77331,77333],{"className":77332,"ariaHidden":62},[61],[31,77334,77336,77339,77342,77345,77348,77351,77354,77357,77360,77392],{"className":77335},[66],[31,77337],{"className":77338,"style":13710},[70],[31,77340,211],{"className":77341},[264],[31,77343,14551],{"className":77344},[75,76],[31,77346,1210],{"className":77347},[1391],[31,77349],{"className":77350,"style":1467},[119],[31,77352,5813],{"className":77353},[75,76],[31,77355,1210],{"className":77356},[1391],[31,77358],{"className":77359,"style":1467},[119],[31,77361,77363,77366],{"className":77362},[75],[31,77364,14551],{"className":77365},[75,76],[31,77367,77369],{"className":77368},[1039],[31,77370,77372],{"className":77371},[1043],[31,77373,77375],{"className":77374},[1048],[31,77376,77378],{"className":77377,"style":13732},[1052],[31,77379,77380,77383],{"style":3631},[31,77381],{"className":77382,"style":1061},[1060],[31,77384,77386],{"className":77385},[1065,1066,1067,1068],[31,77387,77389],{"className":77388},[75,1068],[31,77390,13691],{"className":77391},[75,1068],[31,77393,220],{"className":77394},[289],[31,77396,77398],{"className":77397,"translate":35},[411],[31,77399,77401,77489],{"className":77400,"translate":35},[34],[31,77402,77404],{"className":77403},[39],[41,77405,77406],{"xmlns":43,"display":420},[45,77407,77408,77486],{},[48,77409,77410,77412,77414,77416,77418,77420,77422,77428,77430,77432,77434,77436,77438,77444,77446,77448,77450,77452,77454,77456,77458,77460,77466,77468,77470,77472,77474,77476,77482,77484],{},[51,77411,2549],{},[95,77413,211],{"stretchy":210},[51,77415,14551],{},[95,77417,1210],{"separator":62},[51,77419,5813],{},[95,77421,1210],{"separator":62},[1180,77423,77424,77426],{},[51,77425,14551],{},[95,77427,13691],{"mathvariant":494,"lspace":5610,"rspace":5610},[95,77429,220],{"stretchy":210},[95,77431,434],{},[51,77433,53298],{},[51,77435,43481],{},[95,77437,211],{"stretchy":210},[1180,77439,77440,77442],{},[51,77441,14551],{},[95,77443,13691],{"mathvariant":494,"lspace":5610,"rspace":5610},[95,77445,220],{"stretchy":210},[95,77447,347],{},[51,77449,2549],{},[95,77451,211],{"stretchy":210},[51,77453,14551],{},[95,77455,1210],{"separator":62},[51,77457,5813],{},[95,77459,1210],{"separator":62},[1180,77461,77462,77464],{},[51,77463,14551],{},[95,77465,13691],{"mathvariant":494,"lspace":5610,"rspace":5610},[95,77467,220],{"stretchy":210},[95,77469,434],{},[51,77471,53298],{},[51,77473,3117],{},[95,77475,211],{"stretchy":210},[1180,77477,77478,77480],{},[51,77479,14551],{},[95,77481,13691],{"mathvariant":494,"lspace":5610,"rspace":5610},[95,77483,220],{"stretchy":210},[51,77485,495],{"mathvariant":494},[55,77487,77488],{"encoding":57},"r(s,a,s')+\\gamma V(s')\\le r(s,a,s')+\\gamma W(s').",[31,77490,77492,77566,77625,77699],{"className":77491,"ariaHidden":62},[61],[31,77493,77495,77498,77501,77504,77507,77510,77513,77516,77519,77522,77554,77557,77560,77563],{"className":77494},[66],[31,77496],{"className":77497,"style":75080},[70],[31,77499,2549],{"className":77500,"style":19759},[75,76],[31,77502,211],{"className":77503},[264],[31,77505,14551],{"className":77506},[75,76],[31,77508,1210],{"className":77509},[1391],[31,77511],{"className":77512,"style":1467},[119],[31,77514,5813],{"className":77515},[75,76],[31,77517,1210],{"className":77518},[1391],[31,77520],{"className":77521,"style":1467},[119],[31,77523,77525,77528],{"className":77524},[75],[31,77526,14551],{"className":77527},[75,76],[31,77529,77531],{"className":77530},[1039],[31,77532,77534],{"className":77533},[1043],[31,77535,77537],{"className":77536},[1048],[31,77538,77540],{"className":77539,"style":14017},[1052],[31,77541,77542,77545],{"style":1330},[31,77543],{"className":77544,"style":1061},[1060],[31,77546,77548],{"className":77547},[1065,1066,1067,1068],[31,77549,77551],{"className":77550},[75,1068],[31,77552,13691],{"className":77553},[75,1068],[31,77555,220],{"className":77556},[289],[31,77558],{"className":77559,"style":120},[119],[31,77561,434],{"className":77562},[124],[31,77564],{"className":77565,"style":120},[119],[31,77567,77569,77572,77575,77578,77581,77613,77616,77619,77622],{"className":77568},[66],[31,77570],{"className":77571,"style":75080},[70],[31,77573,53298],{"className":77574,"style":24157},[75,76],[31,77576,43481],{"className":77577,"style":120},[75,76],[31,77579,211],{"className":77580},[264],[31,77582,77584,77587],{"className":77583},[75],[31,77585,14551],{"className":77586},[75,76],[31,77588,77590],{"className":77589},[1039],[31,77591,77593],{"className":77592},[1043],[31,77594,77596],{"className":77595},[1048],[31,77597,77599],{"className":77598,"style":14017},[1052],[31,77600,77601,77604],{"style":1330},[31,77602],{"className":77603,"style":1061},[1060],[31,77605,77607],{"className":77606},[1065,1066,1067,1068],[31,77608,77610],{"className":77609},[75,1068],[31,77611,13691],{"className":77612},[75,1068],[31,77614,220],{"className":77615},[289],[31,77617],{"className":77618,"style":293},[119],[31,77620,347],{"className":77621},[297],[31,77623],{"className":77624,"style":293},[119],[31,77626,77628,77631,77634,77637,77640,77643,77646,77649,77652,77655,77687,77690,77693,77696],{"className":77627},[66],[31,77629],{"className":77630,"style":75080},[70],[31,77632,2549],{"className":77633,"style":19759},[75,76],[31,77635,211],{"className":77636},[264],[31,77638,14551],{"className":77639},[75,76],[31,77641,1210],{"className":77642},[1391],[31,77644],{"className":77645,"style":1467},[119],[31,77647,5813],{"className":77648},[75,76],[31,77650,1210],{"className":77651},[1391],[31,77653],{"className":77654,"style":1467},[119],[31,77656,77658,77661],{"className":77657},[75],[31,77659,14551],{"className":77660},[75,76],[31,77662,77664],{"className":77663},[1039],[31,77665,77667],{"className":77666},[1043],[31,77668,77670],{"className":77669},[1048],[31,77671,77673],{"className":77672,"style":14017},[1052],[31,77674,77675,77678],{"style":1330},[31,77676],{"className":77677,"style":1061},[1060],[31,77679,77681],{"className":77680},[1065,1066,1067,1068],[31,77682,77684],{"className":77683},[75,1068],[31,77685,13691],{"className":77686},[75,1068],[31,77688,220],{"className":77689},[289],[31,77691],{"className":77692,"style":120},[119],[31,77694,434],{"className":77695},[124],[31,77697],{"className":77698,"style":120},[119],[31,77700,77702,77705,77708,77711,77714,77746,77749],{"className":77701},[66],[31,77703],{"className":77704,"style":75080},[70],[31,77706,53298],{"className":77707,"style":24157},[75,76],[31,77709,3117],{"className":77710,"style":949},[75,76],[31,77712,211],{"className":77713},[264],[31,77715,77717,77720],{"className":77716},[75],[31,77718,14551],{"className":77719},[75,76],[31,77721,77723],{"className":77722},[1039],[31,77724,77726],{"className":77725},[1043],[31,77727,77729],{"className":77728},[1048],[31,77730,77732],{"className":77731,"style":14017},[1052],[31,77733,77734,77737],{"style":1330},[31,77735],{"className":77736,"style":1061},[1060],[31,77738,77740],{"className":77739},[1065,1066,1067,1068],[31,77741,77743],{"className":77742},[75,1068],[31,77744,13691],{"className":77745},[75,1068],[31,77747,220],{"className":77748},[289],[31,77750,495],{"className":77751},[75],[11,77753,73188,77754,76375,77842,77976,77977,76226,78098,78156,78157,2372],{},[31,77755,77757,77785],{"className":77756,"translate":35},[34],[31,77758,77760],{"className":77759},[39],[41,77761,77762],{"xmlns":43},[45,77763,77764,77782],{},[48,77765,77766,77768,77770,77772,77774,77776,77778,77780],{},[51,77767,2515],{},[95,77769,211],{"stretchy":210},[51,77771,5813],{},[95,77773,12680],{},[51,77775,14551],{},[95,77777,220],{"stretchy":210},[95,77779,5662],{},[91,77781,1977],{},[55,77783,77784],{"encoding":57},"\\pi(a\\mid s)\\ge0",[31,77786,77788,77812,77833],{"className":77787,"ariaHidden":62},[61],[31,77789,77791,77794,77797,77800,77803,77806,77809],{"className":77790},[66],[31,77792],{"className":77793,"style":260},[70],[31,77795,2515],{"className":77796,"style":2666},[75,76],[31,77798,211],{"className":77799},[264],[31,77801,5813],{"className":77802},[75,76],[31,77804],{"className":77805,"style":293},[119],[31,77807,12680],{"className":77808},[297],[31,77810],{"className":77811,"style":293},[119],[31,77813,77815,77818,77821,77824,77827,77830],{"className":77814},[66],[31,77816],{"className":77817,"style":260},[70],[31,77819,14551],{"className":77820},[75,76],[31,77822,220],{"className":77823},[289],[31,77825],{"className":77826,"style":293},[119],[31,77828,5662],{"className":77829},[297],[31,77831],{"className":77832,"style":293},[119],[31,77834,77836,77839],{"className":77835},[66],[31,77837],{"className":77838,"style":326},[70],[31,77840,1977],{"className":77841},[75],[31,77843,77845,77881],{"className":77844,"translate":35},[34],[31,77846,77848],{"className":77847},[39],[41,77849,77850],{"xmlns":43},[45,77851,77852,77878],{},[48,77853,77854,77856,77858,77864,77866,77868,77870,77872,77874,77876],{},[51,77855,2521],{},[95,77857,211],{"stretchy":210},[1180,77859,77860,77862],{},[51,77861,14551],{},[95,77863,13691],{"mathvariant":494,"lspace":5610,"rspace":5610},[95,77865,12680],{},[51,77867,14551],{},[95,77869,1210],{"separator":62},[51,77871,5813],{},[95,77873,220],{"stretchy":210},[95,77875,5662],{},[91,77877,1977],{},[55,77879,77880],{"encoding":57},"P(s'\\mid s,a)\\ge0",[31,77882,77884,77937,77967],{"className":77883,"ariaHidden":62},[61],[31,77885,77887,77890,77893,77896,77928,77931,77934],{"className":77886},[66],[31,77888],{"className":77889,"style":13710},[70],[31,77891,2521],{"className":77892,"style":949},[75,76],[31,77894,211],{"className":77895},[264],[31,77897,77899,77902],{"className":77898},[75],[31,77900,14551],{"className":77901},[75,76],[31,77903,77905],{"className":77904},[1039],[31,77906,77908],{"className":77907},[1043],[31,77909,77911],{"className":77910},[1048],[31,77912,77914],{"className":77913,"style":13732},[1052],[31,77915,77916,77919],{"style":3631},[31,77917],{"className":77918,"style":1061},[1060],[31,77920,77922],{"className":77921},[1065,1066,1067,1068],[31,77923,77925],{"className":77924},[75,1068],[31,77926,13691],{"className":77927},[75,1068],[31,77929],{"className":77930,"style":293},[119],[31,77932,12680],{"className":77933},[297],[31,77935],{"className":77936,"style":293},[119],[31,77938,77940,77943,77946,77949,77952,77955,77958,77961,77964],{"className":77939},[66],[31,77941],{"className":77942,"style":260},[70],[31,77944,14551],{"className":77945},[75,76],[31,77947,1210],{"className":77948},[1391],[31,77950],{"className":77951,"style":1467},[119],[31,77953,5813],{"className":77954},[75,76],[31,77956,220],{"className":77957},[289],[31,77959],{"className":77960,"style":293},[119],[31,77962,5662],{"className":77963},[297],[31,77965],{"className":77966,"style":293},[119],[31,77968,77970,77973],{"className":77969},[66],[31,77971],{"className":77972,"style":326},[70],[31,77974,1977],{"className":77975},[75],"，加权求和保持不等号，故 ",[31,77978,77980,78010],{"className":77979,"translate":35},[34],[31,77981,77983],{"className":77982},[39],[41,77984,77985],{"xmlns":43},[45,77986,77987,78007],{},[48,77988,77989,77995,77997,77999,78005],{},[1180,77990,77991,77993],{},[51,77992,919],{"mathvariant":925},[51,77994,2515],{},[51,77996,43481],{},[95,77998,347],{},[1180,78000,78001,78003],{},[51,78002,919],{"mathvariant":925},[51,78004,2515],{},[51,78006,3117],{},[55,78008,78009],{"encoding":57},"\\mathcal{T}^\\pi V\\le\\mathcal{T}^\\pi W",[31,78011,78013,78060],{"className":78012,"ariaHidden":62},[61],[31,78014,78016,78019,78048,78051,78054,78057],{"className":78015},[66],[31,78017],{"className":78018,"style":76406},[70],[31,78020,78022,78025],{"className":78021},[75],[31,78023,919],{"className":78024,"style":64444},[75,968],[31,78026,78028],{"className":78027},[1039],[31,78029,78031],{"className":78030},[1043],[31,78032,78034],{"className":78033},[1048],[31,78035,78037],{"className":78036,"style":3628},[1052],[31,78038,78039,78042],{"style":3631},[31,78040],{"className":78041,"style":1061},[1060],[31,78043,78045],{"className":78044},[1065,1066,1067,1068],[31,78046,2515],{"className":78047,"style":2666},[75,76,1068],[31,78049,43481],{"className":78050,"style":120},[75,76],[31,78052],{"className":78053,"style":293},[119],[31,78055,347],{"className":78056},[297],[31,78058],{"className":78059,"style":293},[119],[31,78061,78063,78066,78095],{"className":78062},[66],[31,78064],{"className":78065,"style":945},[70],[31,78067,78069,78072],{"className":78068},[75],[31,78070,919],{"className":78071,"style":64444},[75,968],[31,78073,78075],{"className":78074},[1039],[31,78076,78078],{"className":78077},[1043],[31,78079,78081],{"className":78080},[1048],[31,78082,78084],{"className":78083,"style":3628},[1052],[31,78085,78086,78089],{"style":3631},[31,78087],{"className":78088,"style":1061},[1060],[31,78090,78092],{"className":78091},[1065,1066,1067,1068],[31,78093,2515],{"className":78094,"style":2666},[75,76,1068],[31,78096,3117],{"className":78097,"style":949},[75,76],[31,78099,78101,78118],{"className":78100,"translate":35},[34],[31,78102,78104],{"className":78103},[39],[41,78105,78106],{"xmlns":43},[45,78107,78108,78116],{},[48,78109,78110],{},[1180,78111,78112,78114],{},[51,78113,919],{"mathvariant":925},[95,78115,30648],{},[55,78117,74501],{"encoding":57},[31,78119,78121],{"className":78120,"ariaHidden":62},[61],[31,78122,78124,78127],{"className":78123},[66],[31,78125],{"className":78126,"style":30719},[70],[31,78128,78130,78133],{"className":78129},[75],[31,78131,919],{"className":78132,"style":64444},[75,968],[31,78134,78136],{"className":78135},[1039],[31,78137,78139],{"className":78138},[1043],[31,78140,78142],{"className":78141},[1048],[31,78143,78145],{"className":78144,"style":30719},[1052],[31,78146,78147,78150],{"style":3631},[31,78148],{"className":78149,"style":1061},[1060],[31,78151,78153],{"className":78152},[1065,1066,1067,1068],[31,78154,30648],{"className":78155},[124,1068],"，逐点取最大值也保持逐点序，故 ",[31,78158,78160,78190],{"className":78159,"translate":35},[34],[31,78161,78163],{"className":78162},[39],[41,78164,78165],{"xmlns":43},[45,78166,78167,78187],{},[48,78168,78169,78175,78177,78179,78185],{},[1180,78170,78171,78173],{},[51,78172,919],{"mathvariant":925},[95,78174,30648],{},[51,78176,43481],{},[95,78178,347],{},[1180,78180,78181,78183],{},[51,78182,919],{"mathvariant":925},[95,78184,30648],{},[51,78186,3117],{},[55,78188,78189],{"encoding":57},"\\mathcal{T}^*V\\le\\mathcal{T}^*W",[31,78191,78193,78241],{"className":78192,"ariaHidden":62},[61],[31,78194,78196,78200,78229,78232,78235,78238],{"className":78195},[66],[31,78197],{"className":78198,"style":78199},[70],"height:0.8247em;vertical-align:-0.136em;",[31,78201,78203,78206],{"className":78202},[75],[31,78204,919],{"className":78205,"style":64444},[75,968],[31,78207,78209],{"className":78208},[1039],[31,78210,78212],{"className":78211},[1043],[31,78213,78215],{"className":78214},[1048],[31,78216,78218],{"className":78217,"style":30719},[1052],[31,78219,78220,78223],{"style":3631},[31,78221],{"className":78222,"style":1061},[1060],[31,78224,78226],{"className":78225},[1065,1066,1067,1068],[31,78227,30648],{"className":78228},[124,1068],[31,78230,43481],{"className":78231,"style":120},[75,76],[31,78233],{"className":78234,"style":293},[119],[31,78236,347],{"className":78237},[297],[31,78239],{"className":78240,"style":293},[119],[31,78242,78244,78247,78276],{"className":78243},[66],[31,78245],{"className":78246,"style":30719},[70],[31,78248,78250,78253],{"className":78249},[75],[31,78251,919],{"className":78252,"style":64444},[75,968],[31,78254,78256],{"className":78255},[1039],[31,78257,78259],{"className":78258},[1043],[31,78260,78262],{"className":78261},[1048],[31,78263,78265],{"className":78264,"style":30719},[1052],[31,78266,78267,78270],{"style":3631},[31,78268],{"className":78269,"style":1061},[1060],[31,78271,78273],{"className":78272},[1065,1066,1067,1068],[31,78274,30648],{"className":78275},[124,1068],[31,78277,3117],{"className":78278,"style":949},[75,76],[11,78280,78281],{},"常数平移式由直接计算得到：",[31,78283,78285],{"className":78284,"translate":35},[411],[31,78286,78288,78467],{"className":78287,"translate":35},[34],[31,78289,78291],{"className":78290},[39],[41,78292,78293],{"xmlns":43,"display":420},[45,78294,78295,78464],{},[5606,78296,78297,78429],{"rowspacing":5608,"columnalign":5609,"columnspacing":5610},[5612,78298,78299,78323],{},[5615,78300,78301],{},[2071,78302,78303],{"scriptlevel":1977,"displaystyle":62},[48,78304,78305,78311,78313,78315,78317,78319,78321],{},[1180,78306,78307,78309],{},[51,78308,919],{"mathvariant":925},[51,78310,2515],{},[95,78312,211],{"stretchy":210},[51,78314,43481],{},[95,78316,434],{},[51,78318,1412],{},[91,78320,205],{"mathvariant":76763},[95,78322,220],{"stretchy":210},[5615,78324,78325],{},[2071,78326,78327],{"scriptlevel":1977,"displaystyle":62},[48,78328,78329,78331,78333,78339,78341,78343,78345,78347,78349,78351,78361,78363,78365,78371,78373,78375,78377,78379,78381],{},[48,78330],{},[95,78332,223],{},[1531,78334,78335,78337],{},[95,78336,5625],{},[51,78338,5813],{},[51,78340,2515],{},[95,78342,211],{"stretchy":210},[51,78344,5813],{},[95,78346,12680],{},[51,78348,14551],{},[95,78350,220],{"stretchy":210},[1531,78352,78353,78355],{},[95,78354,5625],{},[1180,78356,78357,78359],{},[51,78358,14551],{},[95,78360,13691],{"mathvariant":494,"lspace":5610,"rspace":5610},[51,78362,2521],{},[95,78364,211],{"stretchy":210},[1180,78366,78367,78369],{},[51,78368,14551],{},[95,78370,13691],{"mathvariant":494,"lspace":5610,"rspace":5610},[95,78372,12680],{},[51,78374,14551],{},[95,78376,1210],{"separator":62},[51,78378,5813],{},[95,78380,220],{"stretchy":210},[48,78382,78383,78385,78387,78389,78391,78393,78395,78397,78403,78405,78407,78409,78411,78413,78419,78421,78423,78425,78427],{},[95,78384,15175],{"fence":62},[51,78386,2549],{},[95,78388,211],{"stretchy":210},[51,78390,14551],{},[95,78392,1210],{"separator":62},[51,78394,5813],{},[95,78396,1210],{"separator":62},[1180,78398,78399,78401],{},[51,78400,14551],{},[95,78402,13691],{"mathvariant":494,"lspace":5610,"rspace":5610},[95,78404,220],{"stretchy":210},[95,78406,434],{},[51,78408,53298],{},[51,78410,43481],{},[95,78412,211],{"stretchy":210},[1180,78414,78415,78417],{},[51,78416,14551],{},[95,78418,13691],{"mathvariant":494,"lspace":5610,"rspace":5610},[95,78420,220],{"stretchy":210},[95,78422,434],{},[51,78424,53298],{},[51,78426,1412],{},[95,78428,15184],{"fence":62},[5612,78430,78431,78437],{},[5615,78432,78433],{},[2071,78434,78435],{"scriptlevel":1977,"displaystyle":62},[48,78436],{},[5615,78438,78439],{},[2071,78440,78441],{"scriptlevel":1977,"displaystyle":62},[48,78442,78443,78445,78447,78453,78455,78457,78459,78461],{},[48,78444],{},[95,78446,223],{},[1180,78448,78449,78451],{},[51,78450,919],{"mathvariant":925},[51,78452,2515],{},[51,78454,43481],{},[95,78456,434],{},[51,78458,53298],{},[51,78460,1412],{},[91,78462,78463],{"mathvariant":76763},"1.",[55,78465,78466],{"encoding":57},"\\begin{aligned}\n\\mathcal{T}^\\pi(V+c\\mathbf{1})\n&=\\sum_{a}\\pi(a\\mid s)\\sum_{s'}P(s'\\mid s,a)\n\\left[r(s,a,s')+\\gamma V(s')+\\gamma c\\right]\\\\\n&=\\mathcal{T}^\\pi V+\\gamma c\\mathbf{1}.\n\\end{aligned}",[31,78468,78470],{"className":78469,"ariaHidden":62},[61],[31,78471,78473,78476],{"className":78472},[66],[31,78474],{"className":78475,"style":59613},[70],[31,78477,78479],{"className":78478},[75],[31,78480,78482,78574],{"className":78481},[5606],[31,78483,78485],{"className":78484},[5867],[31,78486,78488,78566],{"className":78487},[1043,1044],[31,78489,78491,78563],{"className":78490},[1048],[31,78492,78494,78555],{"className":78493,"style":59632},[1052],[31,78495,78496,78499],{"style":59635},[31,78497],{"className":78498,"style":5907},[1060],[31,78500,78502,78531,78534,78537,78540,78543,78546,78549,78552],{"className":78501},[75],[31,78503,78505,78508],{"className":78504},[75],[31,78506,919],{"className":78507,"style":64444},[75,968],[31,78509,78511],{"className":78510},[1039],[31,78512,78514],{"className":78513},[1043],[31,78515,78517],{"className":78516},[1048],[31,78518,78520],{"className":78519,"style":3987},[1052],[31,78521,78522,78525],{"style":1330},[31,78523],{"className":78524,"style":1061},[1060],[31,78526,78528],{"className":78527},[1065,1066,1067,1068],[31,78529,2515],{"className":78530,"style":2666},[75,76,1068],[31,78532,211],{"className":78533},[264],[31,78535,43481],{"className":78536,"style":120},[75,76],[31,78538],{"className":78539,"style":120},[119],[31,78541,434],{"className":78542},[124],[31,78544],{"className":78545,"style":120},[119],[31,78547,1412],{"className":78548},[75,76],[31,78550,205],{"className":78551},[75,76936],[31,78553,220],{"className":78554},[289],[31,78556,78557,78560],{"style":59691},[31,78558],{"className":78559,"style":5907},[1060],[31,78561],{"className":78562},[75],[31,78564,1076],{"className":78565},[1075],[31,78567,78569],{"className":78568},[1048],[31,78570,78572],{"className":78571,"style":59707},[1052],[31,78573],{},[31,78575,78577],{"className":78576},[6062],[31,78578,78580,79039],{"className":78579},[1043,1044],[31,78581,78583,79036],{"className":78582},[1048],[31,78584,78586,78963],{"className":78585,"style":59632},[1052],[31,78587,78588,78591],{"style":59635},[31,78589],{"className":78590,"style":5907},[1060],[31,78592,78594,78597,78600,78603,78606,78653,78656,78659,78662,78665,78668,78671,78674,78677,78680,78683,78759,78762,78765,78768,78800,78803,78806,78809,78812,78815,78818,78821,78824,78827],{"className":78593},[75],[31,78595],{"className":78596},[75],[31,78598],{"className":78599,"style":293},[119],[31,78601,223],{"className":78602},[297],[31,78604],{"className":78605,"style":293},[119],[31,78607,78609],{"className":78608},[1460,1620],[31,78610,78612,78645],{"className":78611},[1043,1044],[31,78613,78615,78642],{"className":78614},[1048],[31,78616,78618,78632],{"className":78617,"style":5900},[1052],[31,78619,78620,78623],{"style":59333},[31,78621],{"className":78622,"style":5907},[1060],[31,78624,78626],{"className":78625},[1065,1066,1067,1068],[31,78627,78629],{"className":78628},[75,1068],[31,78630,5813],{"className":78631},[75,76,1068],[31,78633,78634,78637],{"style":5919},[31,78635],{"className":78636,"style":5907},[1060],[31,78638,78639],{},[31,78640,5625],{"className":78641},[1460,1461,1687],[31,78643,1076],{"className":78644},[1075],[31,78646,78648],{"className":78647},[1048],[31,78649,78651],{"className":78650,"style":59365},[1052],[31,78652],{},[31,78654],{"className":78655,"style":1467},[119],[31,78657,2515],{"className":78658,"style":2666},[75,76],[31,78660,211],{"className":78661},[264],[31,78663,5813],{"className":78664},[75,76],[31,78666],{"className":78667,"style":293},[119],[31,78669,12680],{"className":78670},[297],[31,78672],{"className":78673,"style":293},[119],[31,78675,14551],{"className":78676},[75,76],[31,78678,220],{"className":78679},[289],[31,78681],{"className":78682,"style":1467},[119],[31,78684,78686],{"className":78685},[1460,1620],[31,78687,78689,78751],{"className":78688},[1043,1044],[31,78690,78692,78748],{"className":78691},[1048],[31,78693,78695,78738],{"className":78694,"style":5900},[1052],[31,78696,78697,78700],{"style":59756},[31,78698],{"className":78699,"style":5907},[1060],[31,78701,78703],{"className":78702},[1065,1066,1067,1068],[31,78704,78706],{"className":78705},[75,1068],[31,78707,78709,78712],{"className":78708},[75,1068],[31,78710,14551],{"className":78711},[75,76,1068],[31,78713,78715],{"className":78714},[1039],[31,78716,78718],{"className":78717},[1043],[31,78719,78721],{"className":78720},[1048],[31,78722,78724],{"className":78723,"style":31478},[1052],[31,78725,78726,78729],{"style":31481},[31,78727],{"className":78728,"style":11769},[1060],[31,78730,78732],{"className":78731},[1065,11773,1308,1068],[31,78733,78735],{"className":78734},[75,1068],[31,78736,13691],{"className":78737},[75,1068],[31,78739,78740,78743],{"style":5919},[31,78741],{"className":78742,"style":5907},[1060],[31,78744,78745],{},[31,78746,5625],{"className":78747},[1460,1461,1687],[31,78749,1076],{"className":78750},[1075],[31,78752,78754],{"className":78753},[1048],[31,78755,78757],{"class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sName":79051,"translate":35},[34],[31,79053,79055],{"className":79054},[39],[41,79056,79057],{"xmlns":43},[45,79058,79059,79067],{},[48,79060,79061],{},[1180,79062,79063,79065],{},[51,79064,919],{"mathvariant":925},[95,79066,30648],{},[55,79068,74501],{"encoding":57},[31,79070,79072],{"className":79071,"ariaHidden":62},[61],[31,79073,79075,79078],{"className":79074},[66],[31,79076],{"className":79077,"style":30719},[70],[31,79079,79081,79084],{"className":79080},[75],[31,79082,919],{"className":79083,"style":64444},[75,968],[31,79085,79087],{"className":79086},[1039],[31,79088,79090],{"className":79089},[1043],[31,79091,79093],{"className":79092},[1048],[31,79094,79096],{"className":79095,"style":30719},[1052],[31,79097,79098,79101],{"style":3631},[31,79099],{"className":79100,"style":1061},[1060],[31,79102,79104],{"className":79103},[1065,1066,1067,1068],[31,79105,30648],{"className":79106},[124,1068],"，常数 ",[31,79109,79111,79127],{"className":79110,"translate":35},[34],[31,79112,79114],{"className":79113},[39],[41,79115,79116],{"xmlns":43},[45,79117,79118,79124],{},[48,79119,79120,79122],{},[51,79121,53298],{},[51,79123,1412],{},[55,79125,79126],{"encoding":57},"\\gamma c",[31,79128,79130],{"className":79129,"ariaHidden":62},[61],[31,79131,79133,79136,79139],{"className":79132},[66],[31,79134],{"className":79135,"style":865},[70],[31,79137,53298],{"className":79138,"style":24157},[75,76],[31,79140,1412],{"className":79141},[75,76]," 与动作 ",[31,79144,79146,79159],{"className":79145,"translate":35},[34],[31,79147,79149],{"className":79148},[39],[41,79150,79151],{"xmlns":43},[45,79152,79153,79157],{},[48,79154,79155],{},[51,79156,5813],{},[55,79158,5813],{"encoding":57},[31,79160,79162],{"className":79161,"ariaHidden":62},[61],[31,79163,79165,79168],{"className":79164},[66],[31,79166],{"className":79167,"style":71},[70],[31,79169,5813],{"className":79170},[75,76]," 无关，可以从最大值中提出。证毕。",[11,79173,79174,79175,79204,79205,79233],{},"这两个性质看起来简单，却足以推出压缩性。单调性说明算子保持函数之间的序关系，常数平移式说明算子对整体平移的响应恰好按 ",[31,79176,79178,79192],{"className":79177,"translate":35},[34],[31,79179,79181],{"className":79180},[39],[41,79182,79183],{"xmlns":43},[45,79184,79185,79189],{},[48,79186,79187],{},[51,79188,53298],{},[55,79190,79191],{"encoding":57},"\\gamma",[31,79193,79195],{"className":79194,"ariaHidden":62},[61],[31,79196,79198,79201],{"className":79197},[66],[31,79199],{"className":79200,"style":865},[70],[31,79202,53298],{"className":79203,"style":24157},[75,76]," 缩小。二者结合，就得到「两个函数经过算子作用后距离至多变为原来的 ",[31,79206,79208,79221],{"className":79207,"translate":35},[34],[31,79209,79211],{"className":79210},[39],[41,79212,79213],{"xmlns":43},[45,79214,79215,79219],{},[48,79216,79217],{},[51,79218,53298],{},[55,79220,79191],{"encoding":57},[31,79222,79224],{"className":79223,"ariaHidden":62},[61],[31,79225,79227,79230],{"className":79226},[66],[31,79228],{"className":79229,"style":865},[70],[31,79231,53298],{"className":79232,"style":24157},[75,76]," 倍」这一结论。",[52860,79235,79237],{"id":79236},"_33-压缩映射","3.3 压缩映射",[52865,79239,79242,79667],{"id":79240,"title":79241},"theorem-contraction","定理 3.2（贝尔曼算子的压缩性）",[11,79243,52871,79244,67262,79316,18171,79374,79432,79433,79476,79477,79505,79506,1138],{},[31,79245,79247,79268],{"className":79246,"translate":35},[34],[31,79248,79250],{"className":79249},[39],[41,79251,79252],{"xmlns":43},[45,79253,79254,79266],{},[48,79255,79256,79258,79260,79262,79264],{},[91,79257,1977],{},[95,79259,347],{},[51,79261,53298],{},[95,79263,9758],{},[91,79265,205],{},[55,79267,67213],{"encoding":57},[31,79269,79271,79289,79307],{"className":79270,"ariaHidden":62},[61],[31,79272,79274,79277,79280,79283,79286],{"className":79273},[66],[31,79275],{"className":79276,"style":366},[70],[31,79278,1977],{"className":79279},[75],[31,79281],{"className":79282,"style":293},[119],[31,79284,347],{"className":79285},[297],[31,79287],{"className":79288,"style":293},[119],[31,79290,79292,79295,79298,79301,79304],{"className":79291},[66],[31,79293],{"className":79294,"style":53323},[70],[31,79296,53298],{"className":79297,"style":24157},[75,76],[31,79299],{"className":79300,"style":293},[119],[31,79302,9758],{"className":79303},[297],[31,79305],{"className":79306,"style":293},[119],[31,79308,79310,79313],{"className":79309},[66],[31,79311],{"className":79312,"style":326},[70],[31,79314,205],{"className":79315},[75],[31,79317,79319,79336],{"className":79318,"translate":35},[34],[31,79320,79322],{"className":79321},[39],[41,79323,79324],{"xmlns":43},[45,79325,79326,79334],{},[48,79327,79328],{},[1180,79329,79330,79332],{},[51,79331,919],{"mathvariant":925},[51,79333,2515],{},[55,79335,64428],{"encoding":57},[31,79337,79339],{"className":79338,"ariaHidden":62},[61],[31,79340,79342,79345],{"className":79341},[66],[31,79343],{"className":79344,"style":945},[70],[31,79346,79348,79351],{"className":79347},[75],[31,79349,919],{"className":79350,"style":64444},[75,968],[31,79352,79354],{"className":79353},[1039],[31,79355,79357],{"className":79356},[1043],[31,79358,79360],{"className":79359},[1048],[31,79361,79363],{"className":79362,"style":3628},[1052],[31,79364,79365,79368],{"style":3631},[31,79366],{"className":79367,"style":1061},[1060],[31,79369,79371],{"className":79370},[1065,1066,1067,1068],[31,79372,2515],{"className":79373,"style":2666},[75,76,1068],[31,79375,79377,79394],{"className":79376,"translate":35},[34],[31,79378,79380],{"className":79379},[39],[41,79381,79382],{"xmlns":43},[45,79383,79384,79392],{},[48,79385,79386],{},[1180,79387,79388,79390],{},[51,79389,919],{"mathvariant":925},[95,79391,30648],{},[55,79393,74501],{"encoding":57},[31,79395,79397],{"className":79396,"ariaHidden":62},[61],[31,79398,79400,79403],{"className":79399},[66],[31,79401],{"className":79402,"style":30719},[70],[31,79404,79406,79409],{"className":79405},[75],[31,79407,919],{"className":79408,"style":64444},[75,968],[31,79410,79412],{"className":79411},[1039],[31,79413,79415],{"className":79414},[1043],[31,79416,79418],{"className":79417},[1048],[31,79419,79421],{"className":79420,"style":30719},[1052],[31,79422,79423,79426],{"style":3631},[31,79424],{"className":79425,"style":1061},[1060],[31,79427,79429],{"className":79428},[1065,1066,1067,1068],[31,79430,30648],{"className":79431},[124,1068]," 都是 ",[31,79434,79436,79455],{"className":79435,"translate":35},[34],[31,79437,79439],{"className":79438},[39],[41,79440,79441],{"xmlns":43},[45,79442,79443,79453],{},[48,79444,79445,79447,79449,79451],{},[51,79446,1122],{},[95,79448,211],{"stretchy":210},[51,79450,21582],{},[95,79452,220],{"stretchy":210},[55,79454,73997],{"encoding":57},[31,79456,79458],{"className":79457,"ariaHidden":62},[61],[31,79459,79461,79464,79467,79470,79473],{"className":79460},[66],[31,79462],{"className":79463,"style":260},[70],[31,79465,1122],{"className":79466,"style":1137},[75,76],[31,79468,211],{"className":79469},[264],[31,79471,21582],{"className":79472,"style":21697},[75,76],[31,79474,220],{"className":79475},[289]," 上的 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c\\mathbf{1}.",[31,80589,80591,80649,80673,80731,80789],{"className":80590,"ariaHidden":62},[61],[31,80592,80594,80597,80600,80640,80643,80646],{"className":80593},[66],[31,80595],{"className":80596,"style":5230},[70],[31,80598,919],{"className":80599,"style":64444},[75,968],[31,80601,80603,80606],{"className":80602},[75],[31,80604,43481],{"className":80605,"style":120},[75,76],[31,80607,80609],{"className":80608},[1039],[31,80610,80612,80632],{"className":80611},[1043,1044],[31,80613,80615,80629],{"className":80614},[1048],[31,80616,80618],{"className":80617,"style":2028},[1052],[31,80619,80620,80623],{"style":66251},[31,80621],{"className":80622,"style":1061},[1060],[31,80624,80626],{"className":80625},[1065,1066,1067,1068],[31,80627,152],{"className":80628},[75,1068],[31,80630,1076],{"className":80631},[1075],[31,80633,80635],{"className":80634},[1048],[31,80636,80638],{"className":80637,"style":2050},[1052],[31,80639],{},[31,80641],{"className":80642,"style":120},[119],[31,80644,97],{"className":80645},[124],[31,80647],{"className":80648,"style":120},[119],[31,80650,80652,80655,80658,80661,80664,80667,80670],{"className":80651},[66],[31,80653],{"className":80654,"style":41982},[70],[31,80656,53298],{"className":80657,"style":24157},[75,76],[31,80659,1412],{"className":80660},[75,76],[31,80662,205],{"className":80663},[75,76936],[31,80665],{"className":80666,"style":293},[119],[31,80668,347],{"className":80669},[297],[31,80671],{"className":80672,"style":293},[119],[31,80674,80676,80679,80682,80722,80725,80728],{"className":80675},[66],[31,80677],{"className":80678,"style":5230},[70],[31,80680,919],{"className":80681,"style":64444},[75,968],[31,80683,80685,80688],{"className":80684},[75],[31,80686,43481],{"className":80687,"style":120},[75,76],[31,80689,80691],{"className":80690},[1039],[31,80692,80694,80714],{"className":80693},[1043,1044],[31,80695,80697,80711],{"className":80696},[1048],[31,80698,80700],{"className":80699,"style":2028},[1052],[31,80701,80702,80705],{"style":66251},[31,80703],{"className":80704,"style":1061},[1060],[31,80706,80708],{"className":80707},[1065,1066,1067,1068],[31,80709,205],{"className":80710},[75,1068],[31,80712,1076],{"className":80713},[1075],[31,80715,80717],{"className":80716},[1048],[31,80718,80720],{"className":80719,"style":2050},[1052],[31,80721],{},[31,80723],{"className":80724,"style":293},[119],[31,80726,347],{"className":80727},[297],[31,80729],{"className":80730,"style":293},[119],[31,80732,80734,80737,80740,80780,80783,80786],{"className":80733},[66],[31,80735],{"className":80736,"style":5230},[70],[31,80738,919],{"className":80739,"style":64444},[75,968],[31,80741,80743,80746],{"className":80742},[75],[31,80744,43481],{"className":80745,"style":120},[75,76],[31,80747,80749],{"className":80748},[1039],[31,80750,80752,80772],{"className":80751},[1043,1044],[31,80753,80755,80769],{"className":80754},[1048],[31,80756,80758],{"className":80757,"style":2028},[1052],[31,80759,80760,80763],{"style":66251},[31,80761],{"className":80762,"style":1061},[1060],[31,80764,80766],{"className":80765},[1065,1066,1067,1068],[31,80767,152],{"className":80768},[75,1068],[31,80770,1076],{"className":80771},[1075],[31,80773,80775],{"className":80774},[1048],[31,80776,80778],{"className":80777,"style":2050},[1052],[31,80779],{},[31,80781],{"className":80782,"style":120},[119],[31,80784,434],{"className":80785},[124],[31,80787],{"className":80788,"style":120},[119],[31,80790,80792,80795,80798,80801,80804],{"className":80791},[66],[31,80793],{"className":80794,"style":41982},[70],[31,80796,53298],{"className":80797,"style":24157},[75,76],[31,80799,1412],{"className":80800},[75,76],[31,80802,205],{"className":80803},[75,76936],[31,80805,495],{"className":80806},[75],[11,80808,80809,80810,1138],{},"因此对一切 ",[31,80811,80813,80826],{"className":80812,"translate":35},[34],[31,80814,80816],{"className":80815},[39],[41,80817,80818],{"xmlns":43},[45,80819,80820,80824],{},[48,80821,80822],{},[51,80823,14551],{},[55,80825,14551],{"encoding":57},[31,80827,80829],{"className":80828,"ariaHidden":62},[61],[31,80830,80832,80835],{"className":80831},[66],[31,80833],{"className":80834,"style":71},[70],[31,80836,14551],{"className":80837},[75,76],[31,80839,80841],{"className":80840,"translate":35},[411],[31,80842,80844,80942],{"className":80843,"translate":35},[34],[31,80845,80847],{"className":80846},[39],[41,80848,80849],{"xmlns":43,"display":420},[45,80850,80851,80939],{},[5606,80852,80853,80913],{"rowspacing":5608,"columnalign":5609,"columnspacing":5610},[5612,80854,80855,80861],{},[5615,80856,80857],{},[2071,80858,80859],{"scriptlevel":1977,"displaystyle":62},[48,80860],{},[5615,80862,80863],{},[2071,80864,80865],{"scriptlevel":1977,"displaystyle":62},[48,80866,80867,80869],{},[48,80868],{},[48,80870,80871,80873,80875,80877,80883,80885,80887,80889,80891,80893,80895,80897,80903,80905,80907,80909,80911],{},[95,80872,12680],{"fence":62},[95,80874,211],{"stretchy":210},[51,80876,919],{"mathvariant":925},[1004,80878,80879,80881],{},[51,80880,43481],{},[91,80882,205],{},[95,80884,220],{"stretchy":210},[95,80886,211],{"stretchy":210},[51,80888,14551],{},[95,80890,220],{"stretchy":210},[95,80892,97],{},[95,80894,211],{"stretchy":210},[51,80896,919],{"mathvariant":925},[1004,80898,80899,80901],{},[51,80900,43481],{},[91,80902,152],{},[95,80904,220],{"stretchy":210},[95,80906,211],{"stretchy":210},[51,80908,14551],{},[95,80910,220],{"stretchy":210},[95,80912,12680],{"fence":62},[5612,80914,80915,80921],{},[5615,80916,80917],{},[2071,80918,80919],{"scriptlevel":1977,"displaystyle":62},[48,80920],{},[5615,80922,80923],{},[2071,80924,80925],{"scriptlevel":1977,"displaystyle":62},[48,80926,80927,80929,80931,80933,80935,80937],{},[48,80928],{},[119,80930],{"width":1584},[95,80932,347],{},[51,80934,53298],{},[51,80936,1412],{},[51,80938,495],{"mathvariant":494},[55,80940,80941],{"encoding":57},"\\begin{aligned}\n&\\left|(\\mathcal{T}V_1)(s)-(\\mathcal{T}V_2)(s)\\right|\\\\\n&\\quad\\le\\gamma c.\n\\end{aligned}",[31,80943,80945],{"className":80944,"ariaHidden":62},[61],[31,80946,80948,80951],{"className":80947},[66],[31,80949],{"className":80950,"style":57050},[70],[31,80952,80954],{"className":80953},[75],[31,80955,80957,80998],{"className":80956},[5606],[31,80958,80960],{"className":80959},[5867],[31,80961,80963,80990],{"className":80962},[1043,1044],[31,80964,80966,80987],{"className":80965},[1048],[31,80967,80969,80979],{"className":80968,"style":57069},[1052],[31,80970,80972,80976],{"style":80971},"top:-3.6em;",[31,80973],{"className":80974,"style":80975},[1060],"height:2.84em;",[31,80977],{"className":80978},[75],[31,80980,80981,80984],{"style":58497},[31,80982],{"className":80983,"style":80975},[1060],[31,80985],{"className":80986},[75],[31,80988,1076],{"className":80989},[1075],[31,80991,80993],{"className":80992},[1048],[31,80994,80996],{"className":80995,"style":57182},[1052],[31,80997],{},[31,80999,81001],{"className":81000},[6062],[31,81002,81004,81193],{"className":81003},[1043,1044],[31,81005,81007,81190],{"className":81006},[1048],[31,81008,81010,81158],{"className":81009,"style":57069},[1052],[31,81011,81012,81015],{"style":57072},[31,81013],{"className":81014,"style":1637},[1060],[31,81016,81018,81021,81024],{"className":81017},[75],[31,81019],{"className":81020},[75],[31,81022],{"className":81023,"style":1467},[119],[31,81025,81027,81030,81033,81036,81076,81079,81082,81085,81088,81091,81094,81097,81100,81103,81143,81146,81149,81152,81155],{"className":81026},[3860],[31,81028,12680],{"className":81029,"style":3868},[264,3867],[31,81031,211],{"className":81032},[264],[31,81034,919],{"className":81035,"style":64444},[75,968],[31,81037,81039,81042],{"className":81038},[75],[31,81040,43481],{"className":81041,"style":120},[75,76],[31,81043,81045],{"className":81044},[1039],[31,81046,81048,81068],{"className":81047},[1043,1044],[31,81049,81051,81065],{"className":81050},[1048],[31,81052,81054],{"className":81053,"style":2028},[1052],[31,81055,81056,81059],{"style":66251},[31,81057],{"className":81058,"style":1061},[1060],[31,81060,81062],{"className":81061},[1065,1066,1067,1068],[31,81063,205],{"className":81064},[75,1068],[31,81066,1076],{"className":81067},[1075],[31,81069,81071],{"className":81070},[1048],[31,81072,81074],{"className":81073,"style":2050},[1052],[31,81075],{},[31,81077,220],{"className":81078},[289],[31,81080,211],{"className":81081},[264],[31,81083,14551],{"className":81084},[75,76],[31,81086,220],{"className":81087},[289],[31,81089],{"className":81090,"style":120},[119],[31,81092,97],{"className":81093},[124],[31,81095],{"className":81096,"style":120},[119],[31,81098,211],{"className":81099},[264],[31,81101,919],{"className":81102,"style":64444},[75,968],[31,81104,81106,81109],{"className":81105},[75],[31,81107,43481],{"className":81108,"style":120},[75,76],[31,81110,81112],{"className":81111},[1039],[31,81113,81115,81135],{"className":81114},[1043,1044],[31,81116,81118,81132],{"className":81117},[1048],[31,81119,81121],{"className":81120,"style":2028},[1052],[31,81122,81123,81126],{"style":66251},[31,81124],{"className":81125,"style":1061},[1060],[31,81127,81129],{"className":81128},[1065,1066,1067,1068],[31,81130,152],{"className":81131},[75,1068],[31,81133,1076],{"className":81134},[1075],[31,81136,81138],{"className":81137},[1048],[31,81139,81141],{"className":81140,"style":2050},[1052],[31,81142],{},[31,81144,220],{"className":81145},[289],[31,81147,211],{"className":81148},[264],[31,81150,14551],{"className":81151},[75,76],[31,81153,220],{"className":81154},[289],[31,81156,12680],{"className":81157,"style":3868},[289,3867],[31,81159,81160,81163],{"style":57119},[31,81161],{"className":81162,"style":1637},[1060],[31,81164,81166,81169,81172,81175,81178,81181,81184,81187],{"className":81165},[75],[31,81167],{"className":81168},[75],[31,81170],{"className":81171,"style":1789},[119],[31,81173],{"className":81174,"style":293},[119],[31,81176,347],{"className":81177},[297],[31,81179],{"className":81180,"style":293},[119],[31,81182,53298],{"className":81183,"style":24157},[75,76],[31,81185,1412],{"className":81186},[75,76],[31,81188,495],{"className":81189},[75],[31,81191,1076],{"className":81192},[1075],[31,81194,81196],{"className":81195},[1048],[31,81197,81199],{"className":81198,"style":57182},[1052],[31,81200],{},[11,81202,9612,81203,81231],{},[31,81204,81206,81219],{"className":81205,"translate":35},[34],[31,81207,81209],{"className":81208},[39],[41,81210,81211],{"xmlns":43},[45,81212,81213,81217],{},[48,81214,81215],{},[51,81216,14551],{},[55,81218,14551],{"encoding":57},[31,81220,81222],{"className":81221,"ariaHidden":62},[61],[31,81223,81225,81228],{"className":81224},[66],[31,81226],{"className":81227,"style":71},[70],[31,81229,14551],{"className":81230},[75,76]," 取上确界即得结论。证毕。",[11,81233,81234,81235,81263,81264,81314,81315,81343,81344,81373,81374,1138,81403,81431,81432,81518],{},"压缩性的直观含义是：无论两个价值函数相差多大，经过一次贝尔曼更新后，它们的最大差异最多变成原来的 ",[31,81236,81238,81251],{"className":81237,"translate":35},[34],[31,81239,81241],{"className":81240},[39],[41,81242,81243],{"xmlns":43},[45,81244,81245,81249],{},[48,81246,81247],{},[51,81248,53298],{},[55,81250,79191],{"encoding":57},[31,81252,81254],{"className":81253,"ariaHidden":62},[61],[31,81255,81257,81260],{"className":81256},[66],[31,81258],{"className":81259,"style":865},[70],[31,81261,53298],{"className":81262,"style":24157},[75,76]," 倍。若 ",[31,81265,81267,81284],{"className":81266,"translate":35},[34],[31,81268,81270],{"className":81269},[39],[41,81271,81272],{"xmlns":43},[45,81273,81274,81282],{},[48,81275,81276,81278,81280],{},[51,81277,53298],{},[95,81279,223],{},[91,81281,23247],{},[55,81283,70369],{"encoding":57},[31,81285,81287,81305],{"className":81286,"ariaHidden":62},[61],[31,81288,81290,81293,81296,81299,81302],{"className":81289},[66],[31,81291],{"className":81292,"style":865},[70],[31,81294,53298],{"className":81295,"style":24157},[75,76],[31,81297],{"className":81298,"style":293},[119],[31,81300,223],{"className":81301},[297],[31,81303],{"className":81304,"style":293},[119],[31,81306,81308,81311],{"className":81307},[66],[31,81309],{"className":81310,"style":326},[70],[31,81312,23247],{"className":81313},[75],"，初始最大差异为 ",[31,81316,81318,81331],{"className":81317,"translate":35},[34],[31,81319,81321],{"className":81320},[39],[41,81322,81323],{"xmlns":43},[45,81324,81325,81329],{},[48,81326,81327],{},[91,81328,22616],{},[55,81330,22616],{"encoding":57},[31,81332,81334],{"className":81333,"ariaHidden":62},[61],[31,81335,81337,81340],{"className":81336},[66],[31,81338],{"className":81339,"style":326},[70],[31,81341,22616],{"className":81342},[75],"，则一次更新后差异不超过 ",[31,81345,81347,81361],{"className":81346,"translate":35},[34],[31,81348,81350],{"className":81349},[39],[41,81351,81352],{"xmlns":43},[45,81353,81354,81359],{},[48,81355,81356],{},[91,81357,81358],{},"90",[55,81360,81358],{"encoding":57},[31,81362,81364],{"className":81363,"ariaHidden":62},[61],[31,81365,81367,81370],{"className":81366},[66],[31,81368],{"className":81369,"style":326},[70],[31,81371,81358],{"className":81372},[75],"，两次后不超过 ",[31,81375,81377,81391],{"className":81376,"translate":35},[34],[31,81378,81380],{"className":81379},[39],[41,81381,81382],{"xmlns":43},[45,81383,81384,81389],{},[48,81385,81386],{},[91,81387,81388],{},"81",[55,81390,81388],{"encoding":57},[31,81392,81394],{"className":81393,"ariaHidden":62},[61],[31,81395,81397,81400],{"className":81396},[66],[31,81398],{"className":81399,"style":326},[70],[31,81401,81388],{"className":81402},[75],[31,81404,81406,81419],{"className":81405,"translate":35},[34],[31,81407,81409],{"className":81408},[39],[41,81410,81411],{"xmlns":43},[45,81412,81413,81417],{},[48,81414,81415],{},[51,81416,52065],{},[55,81418,52065],{"encoding":57},[31,81420,81422],{"className":81421,"ariaHidden":62},[61],[31,81423,81425,81428],{"className":81424},[66],[31,81426],{"className":81427,"style":1630},[70],[31,81429,52065],{"className":81430,"style":52166},[75,76]," 次后不超过 ",[31,81433,81435,81457],{"className":81434,"translate":35},[34],[31,81436,81438],{"className":81437},[39],[41,81439,81440],{"xmlns":43},[45,81441,81442,81454],{},[48,81443,81444,81450,81452],{},[1180,81445,81446,81448],{},[91,81447,23247],{},[51,81449,52065],{},[95,81451,429],{},[91,81453,22616],{},[55,81455,81456],{"encoding":57},"0.9^k\\cdot100",[31,81458,81460,81509],{"className":81459,"ariaHidden":62},[61],[31,81461,81463,81466,81470,81500,81503,81506],{"className":81462},[66],[31,81464],{"className":81465,"style":20863},[70],[31,81467,81469],{"className":81468},[75],"0.",[31,81471,81473,81477],{"className":81472},[75],[31,81474,81476],{"className":81475},[75],"9",[31,81478,81480],{"className":81479},[1039],[31,81481,81483],{"className":81482},[1043],[31,81484,81486],{"className":81485},[1048],[31,81487,81489],{"className":81488,"style":20863},[1052],[31,81490,81491,81494],{"style":3631},[31,81492],{"className":81493,"style":1061},[1060],[31,81495,81497],{"className":81496},[1065,1066,1067,1068],[31,81498,52065],{"className":81499,"style":52166},[75,76,1068],[31,81501],{"className":81502,"style":120},[119],[31,81504,429],{"className":81505},[124],[31,81507],{"className":81508,"style":120},[119],[31,81510,81512,81515],{"className":81511},[66],[31,81513],{"className":81514,"style":326},[70],[31,81516,22616],{"className":81517},[75],"。这正是后文几何收敛性的来源。",[11,81520,81521,81522,81572,81573,81624,81625,81996],{},"需要强调的是，压缩性依赖于 ",[31,81523,81525,81542],{"className":81524,"translate":35},[34],[31,81526,81528],{"className":81527},[39],[41,81529,81530],{"xmlns":43},[45,81531,81532,81540],{},[48,81533,81534,81536,81538],{},[51,81535,53298],{},[95,81537,9758],{},[91,81539,205],{},[55,81541,69926],{"encoding":57},[31,81543,81545,81563],{"className":81544,"ariaHidden":62},[61],[31,81546,81548,81551,81554,81557,81560],{"className":81547},[66],[31,81549],{"className":81550,"style":53323},[70],[31,81552,53298],{"className":81553,"style":24157},[75,76],[31,81555],{"className":81556,"style":293},[119],[31,81558,9758],{"className":81559},[297],[31,81561],{"className":81562,"style":293},[119],[31,81564,81566,81569],{"className":81565},[66],[31,81567],{"className":81568,"style":326},[70],[31,81570,205],{"className":81571},[75],"。当 ",[31,81574,81576,81594],{"className":81575,"translate":35},[34],[31,81577,81579],{"className":81578},[39],[41,81580,81581],{"xmlns":43},[45,81582,81583,81591],{},[48,81584,81585,81587,81589],{},[51,81586,53298],{},[95,81588,223],{},[91,81590,205],{},[55,81592,81593],{"encoding":57},"\\gamma=1",[31,81595,81597,81615],{"className":81596,"ariaHidden":62},[61],[31,81598,81600,81603,81606,81609,81612],{"className":81599},[66],[31,81601],{"className":81602,"style":865},[70],[31,81604,53298],{"className":81605,"style":24157},[75,76],[31,81607],{"className":81608,"style":293},[119],[31,81610,223],{"className":81611},[297],[31,81613],{"className":81614,"style":293},[119],[31,81616,81618,81621],{"className":81617},[66],[31,81619],{"className":81620,"style":326},[70],[31,81622,205],{"className":81623},[75]," 时，贝尔曼算子通常只是非扩张的，即只能得到 ",[31,81626,81628,81690],{"className":81627,"translate":35},[34],[31,81629,81631],{"className":81630},[39],[41,81632,81633],{"xmlns":43},[45,81634,81635,81687],{},[48,81636,81637,81639,81641,81647,81649,81651,81657,81663,81665,81667,81673,81675,81681],{},[51,81638,4572],{"mathvariant":494},[51,81640,919],{"mathvariant":925},[1004,81642,81643,81645],{},[51,81644,43481],{},[91,81646,205],{},[95,81648,97],{},[51,81650,919],{"mathvariant":925},[1004,81652,81653,81655],{},[51,81654,43481],{},[91,81656,152],{},[1004,81658,81659,81661],{},[51,81660,4572],{"mathvariant":494},[51,81662,4587],{"mathvariant":494},[95,81664,347],{},[51,81666,4572],{"mathvariant":494},[1004,81668,81669,81671],{},[51,81670,43481],{},[91,81672,205],{},[95,81674,97],{},[1004,81676,81677,81679],{},[51,81678,43481],{},[91,81680,152],{},[1004,81682,81683,81685],{},[51,81684,4572],{"mathvariant":494},[51,81686,4587],{"mathvariant":494},[55,81688,81689],{"encoding":57},"\\|\\mathcal{T}V_1-\\mathcal{T}V_2\\|_\\infty\\le\\|V_1-V_2\\|_\\infty",[31,81691,81693,81754,81852,81910],{"className":81692,"ariaHidden":62},[61],[31,81694,81696,81699,81702,81705,81745,81748,81751],{"className":81695},[66],[31,81697],{"className":81698,"style":260},[70],[31,81700,4572],{"className":81701},[75],[31,81703,919],{"className":81704,"style":64444},[75,968],[31,81706,81708,81711],{"className":81707},[75],[31,81709,43481],{"className":81710,"style":120},[75,76],[31,81712,81714],{"className":81713},[1039],[31,81715,81717,81737],{"className":81716},[1043,1044],[31,81718,81720,81734],{"className":81719},[1048],[31,81721,81723],{"className":81722,"style":2028},[1052],[31,81724,81725,81728],{"style":66251},[31,81726],{"className":81727,"style":1061},[1060],[31,81729,81731],{"className":81730},[1065,1066,1067,1068],[31,81732,205],{"className":81733},[75,1068],[31,81735,1076],{"className":81736},[1075],[31,81738,81740],{"className":81739},[1048],[31,81741,81743],{"className":81742,"style":2050},[1052],[31,81744],{},[31,81746],{"className":81747,"style":120},[119],[31,81749,97],{"className":81750},[124],[31,81752],{"className":81753,"style":120},[119],[31,81755,81757,81760,81763,81803,81843,81846,81849],{"className":81756},[66],[31,81758],{"className":81759,"style":260},[70],[31,81761,919],{"className":81762,"style":64444},[75,968],[31,81764,81766,81769],{"className":81765},[75],[31,81767,43481],{"className":81768,"style":120},[75,76],[31,81770,81772],{"className":81771},[1039],[31,81773,81775,81795],{"className":81774},[1043,1044],[31,81776,81778,81792],{"className":81777},[1048],[31,81779,81781],{"className":81780,"style":2028},[1052],[31,81782,81783,81786],{"style":66251},[31,81784],{"className":81785,"style":1061},[1060],[31,81787,81789],{"className":81788},[1065,1066,1067,1068],[31,81790,152],{"className":81791},[75,1068],[31,81793,1076],{"className":81794},[1075],[31,81796,81798],{"className":81797},[1048],[31,81799,81801],{"className":81800,"style":2050},[1052],[31,81802],{},[31,81804,81806,81809],{"className":81805},[75],[31,81807,4572],{"className":81808},[75],[31,81810,81812],{"className":81811},[1039],[31,81813,81815,81835],{"className":81814},[1043,1044],[31,81816,81818,81832],{"className":81817},[1048],[31,81819,81821],{"className":81820,"style":3230},[1052],[31,81822,81823,81826],{"style":2807},[31,81824],{"className":81825,"style":1061},[1060],[31,81827,81829],{"className":81828},[1065,1066,1067,1068],[31,81830,4587],{"className":81831},[75,1068],[31,81833,1076],{"className":81834},[1075],[31,81836,81838],{"className":81837},[1048],[31,81839,81841],{"className":81840,"style":2050},[1052],[31,81842],{},[31,81844],{"className":81845,"style":293},[119],[31,81847,347],{"className":81848},[297],[31,81850],{"className":81851,"style":293},[119],[31,81853,81855,81858,81861,81901,81904,81907],{"className":81854},[66],[31,81856],{"className":81857,"style":260},[70],[31,81859,4572],{"className":81860},[75],[31,81862,81864,81867],{"className":81863},[75],[31,81865,43481],{"className":81866,"style":120},[75,76],[31,81868,81870],{"className":81869},[1039],[31,81871,81873,81893],{"className":81872},[1043,1044],[31,81874,81876,81890],{"className":81875},[1048],[31,81877,81879],{"className":81878,"style":2028},[1052],[31,81880,81881,81884],{"style":66251},[31,81882],{"className":81883,"style":1061},[1060],[31,81885,81887],{"className":81886},[1065,1066,1067,1068],[31,81888,205],{"className":81889},[75,1068],[31,81891,1076],{"className":81892},[1075],[31,81894,81896],{"className":81895},[1048],[31,81897,81899],{"className":81898,"style":2050},[1052],[31,81900],{},[31,81902],{"className":81903,"style":120},[119],[31,81905,97],{"className":81906},[124],[31,81908],{"className":81909,"style":120},[119],[31,81911,81913,81916,81956],{"className":81912},[66],[31,81914],{"className":81915,"style":260},[70],[31,81917,81919,81922],{"className":81918},[75],[31,81920,43481],{"className":81921,"style":120},[75,76],[31,81923,81925],{"className":81924},[1039],[31,81926,81928,81948],{"className":81927},[1043,1044],[31,81929,81931,81945],{"className":81930},[1048],[31,81932,81934],{"className":81933,"style":2028},[1052],[31,81935,81936,81939],{"style":66251},[31,81937],{"className":81938,"style":1061},[1060],[31,81940,81942],{"className":81941},[1065,1066,1067,1068],[31,81943,152],{"className":81944},[75,1068],[31,81946,1076],{"className":81947},[1075],[31,81949,81951],{"className":81950},[1048],[31,81952,81954],{"className":81953,"style":2050},[1052],[31,81955],{},[31,81957,81959,81962],{"className":81958},[75],[31,81960,4572],{"className":81961},[75],[31,81963,81965],{"className":81964},[1039],[31,81966,81968,81988],{"className":81967},[1043,1044],[31,81969,81971,81985],{"className":81970},[1048],[31,81972,81974],{"className":81973,"style":3230},[1052],[31,81975,81976,81979],{"style":2807},[31,81977],{"className":81978,"style":1061},[1060],[31,81980,81982],{"className":81981},[1065,1066,1067,1068],[31,81983,4587],{"className":81984},[75,1068],[31,81986,1076],{"className":81987},[1075],[31,81989,81991],{"className":81990},[1048],[31,81992,81994],{"className":81993,"style":2050},[1052],[31,81995],{},"，不能保证距离严格缩小。第 5.4 节会给出反例。",[52860,81998,82000],{"id":81999},"_34-banach-不动点定理","3.4 Banach 不动点定理",[52865,82002,82005,82272,82402,82768],{"id":82003,"title":82004},"theorem-banach","定理 3.3（Banach 不动点定理）",[11,82006,52871,82007,82059,82060,82133,82134,82207,82208,1138],{},[31,82008,82010,82032],{"className":82009,"translate":35},[34],[31,82011,82013],{"className":82012},[39],[41,82014,82015],{"xmlns":43},[45,82016,82017,82029],{},[48,82018,82019,82021,82023,82025,82027],{},[95,82020,211],{"stretchy":210},[51,82022,926],{},[95,82024,1210],{"separator":62},[51,82026,1435],{},[95,82028,220],{"stretchy":210},[55,82030,82031],{"encoding":57},"(X,d)",[31,82033,82035],{"className":82034,"ariaHidden":62},[61],[31,82036,82038,82041,82044,82047,82050,82053,82056],{"className":82037},[66],[31,82039],{"className":82040,"style":260},[70],[31,82042,211],{"className":82043},[264],[31,82045,926],{"className":82046,"style":30311},[75,76],[31,82048,1210],{"className":82049},[1391],[31,82051],{"className":82052,"style":1467},[119],[31,82054,1435],{"className":82055},[75,76],[31,82057,220],{"className":82058},[289]," 是完备度量空间，",[31,82061,82063,82085],{"className":82062,"translate":35},[34],[31,82064,82066],{"className":82065},[39],[41,82067,82068],{"xmlns":43},[45,82069,82070,82082],{},[48,82071,82072,82074,82076,82078,82080],{},[51,82073,919],{},[95,82075,922],{},[51,82077,926],{},[95,82079,929],{},[51,82081,926],{},[55,82083,82084],{"encoding":57},"T:X\\to X",[31,82086,82088,82106,82124],{"className":82087,"ariaHidden":62},[61],[31,82089,82091,82094,82097,82100,82103],{"className":82090},[66],[31,82092],{"className":82093,"style":945},[70],[31,82095,919],{"className":82096,"style":949},[75,76],[31,82098],{"className":82099,"style":293},[119],[31,82101,922],{"className":82102},[297],[31,82104],{"className":82105,"style":293},[119],[31,82107,82109,82112,82115,82118,82121],{"className":82108},[66],[31,82110],{"className":82111,"style":945},[70],[31,82113,926],{"className":82114,"style":30311},[75,76],[31,82116],{"className":82117,"style":293},[119],[31,82119,929],{"className":82120},[297],[31,82122],{"className":82123,"style":293},[119],[31,82125,82127,82130],{"className":82126},[66],[31,82128],{"className":82129,"style":945},[70],[31,82131,926],{"className":82132,"style":30311},[75,76]," 满足：存在 ",[31,82135,82137,82162],{"className":82136,"translate":35},[34],[31,82138,82140],{"className":82139},[39],[41,82141,82142],{"xmlns":43},[45,82143,82144,82160],{},[48,82145,82146,82148,82150,82152,82154,82156,82158],{},[51,82147,53298],{},[95,82149,2518],{},[95,82151,15175],{"stretchy":210},[91,82153,1977],{},[95,82155,1210],{"separator":62},[91,82157,205],{},[95,82159,220],{"stretchy":210},[55,82161,53313],{"encoding":57},[31,82163,82165,82183],{"className":82164,"ariaHidden":62},[61],[31,82166,82168,82171,82174,82177,82180],{"className":82167},[66],[31,82169],{"className":82170,"style":53323},[70],[31,82172,53298],{"className":82173,"style":24157},[75,76],[31,82175],{"className":82176,"style":293},[119],[31,82178,2518],{"className":82179},[297],[31,82181],{"className":82182,"style":293},[119],[31,82184,82186,82189,82192,82195,82198,82201,82204],{"className":82185},[66],[31,82187],{"className":82188,"style":260},[70],[31,82190,15175],{"className":82191},[264],[31,82193,1977],{"className":82194},[75],[31,82196,1210],{"className":82197},[1391],[31,82199],{"className":82200,"style":1467},[119],[31,82202,205],{"className":82203},[75],[31,82205,220],{"className":82206},[289],"，使得对任意 ",[31,82209,82211,82233],{"className":82210,"translate":35},[34],[31,82212,82214],{"className":82213},[39],[41,82215,82216],{"xmlns":43},[45,82217,82218,82230],{},[48,82219,82220,82222,82224,82226,82228],{},[51,82221,53],{},[95,82223,1210],{"separator":62},[51,82225,3003],{},[95,82227,2518],{},[51,82229,926],{},[55,82231,82232],{"encoding":57},"x,y\\in X",[31,82234,82236,82263],{"className":82235,"ariaHidden":62},[61],[31,82237,82239,82242,82245,82248,82251,82254,82257,82260],{"className":82238},[66],[31,82240],{"className":82241,"style":53323},[70],[31,82243,53],{"className":82244},[75,76],[31,82246,1210],{"className":82247},[1391],[31,82249],{"className":82250,"style":1467},[119],[31,82252,3003],{"className":82253,"style":2666},[75,76],[31,82255],{"className":82256,"style":293},[119],[31,82258,2518],{"className":82259},[297],[31,82261],{"className":82262,"style":293},[119],[31,82264,82266,82269],{"className":82265},[66],[31,82267],{"className":82268,"style":945},[70],[31,82270,926],{"className":82271,"style":30311},[75,76],[31,82273,82275],{"className":82274,"translate":35},[411],[31,82276,82278,82324],{"className":82277,"translate":35},[34],[31,82279,82281],{"className":82280},[39],[41,82282,82283],{"xmlns":43,"display":420},[45,82284,82285,82321],{},[48,82286,82287,82289,82291,82293,82295,82297,82299,82301,82303,82305,82307,82309,82311,82313,82315,82317,82319],{},[51,82288,1435],{},[95,82290,211],{"stretchy":210},[51,82292,919],{},[51,82294,53],{},[95,82296,1210],{"separator":62},[51,82298,919],{},[51,82300,3003],{},[95,82302,220],{"stretchy":210},[95,82304,347],{},[51,82306,53298],{},[51,82308,1435],{},[95,82310,211],{"stretchy":210},[51,82312,53],{},[95,82314,1210],{"separator":62},[51,82316,3003],{},[95,82318,220],{"stretchy":210},[51,82320,495],{"mathvariant":494},[55,82322,82323],{"encoding":57},"d(Tx,Ty)\\le\\gamma d(x,y).",[31,82325,82327,82369],{"className":82326,"ariaHidden":62},[61],[31,82328,82330,82333,82336,82339,82342,82345,82348,82351,82354,82357,82360,82363,82366],{"className":82329},[66],[31,82331],{"className":82332,"style":260},[70],[31,82334,1435],{"className":82335},[75,76],[31,82337,211],{"className":82338},[264],[31,82340,919],{"className":82341,"style":949},[75,76],[31,82343,53],{"className":82344},[75,76],[31,82346,1210],{"className":82347},[1391],[31,82349],{"className":82350,"style":1467},[119],[31,82352,919],{"className":82353,"style":949},[75,76],[31,82355,3003],{"className":82356,"style":2666},[75,76],[31,82358,220],{"className":82359},[289],[31,82361],{"className":82362,"style":293},[119],[31,82364,347],{"className":82365},[297],[31,82367],{"className":82368,"style":293},[119],[31,82370,82372,82375,82378,82381,82384,82387,82390,82393,82396,82399],{"className":82371},[66],[31,82373],{"className":82374,"style":260},[70],[31,82376,53298],{"className":82377,"style":24157},[75,76],[31,82379,1435],{"className":82380},[75,76],[31,82382,211],{"className":82383},[264],[31,82385,53],{"className":82386},[75,76],[31,82388,1210],{"className":82389},[1391],[31,82391],{"className":82392,"style":1467},[119],[31,82394,3003],{"className":82395,"style":2666},[75,76],[31,82397,220],{"className":82398},[289],[31,82400,495],{"className":82401},[75],[11,82403,66811,82404,25085,82432,82460,82461,82520,82521,82613,82614,82767],{},[31,82405,82407,82420],{"className":82406,"translate":35},[34],[31,82408,82410],{"className":82409},[39],[41,82411,82412],{"xmlns":43},[45,82413,82414,82418],{},[48,82415,82416],{},[51,82417,919],{},[55,82419,919],{"encoding":57},[31,82421,82423],{"className":82422,"ariaHidden":62},[61],[31,82424,82426,82429],{"className":82425},[66],[31,82427],{"className":82428,"style":945},[70],[31,82430,919],{"className":82431,"style":949},[75,76],[31,82433,82435,82448],{"className":82434,"translate":35},[34],[31,82436,82438],{"className":82437},[39],[41,82439,82440],{"xmlns":43},[45,82441,82442,82446],{},[48,82443,82444],{},[51,82445,926],{},[55,82447,926],{"encoding":57},[31,82449,82451],{"className":82450,"ariaHidden":62},[61],[31,82452,82454,82457],{"className":82453},[66],[31,82455],{"className":82456,"style":945},[70],[31,82458,926],{"className":82459,"style":30311},[75,76]," 中存在唯一不动点 ",[31,82462,82464,82482],{"className":82463,"translate":35},[34],[31,82465,82467],{"className":82466},[39],[41,82468,82469],{"xmlns":43},[45,82470,82471,82479],{},[48,82472,82473],{},[1180,82474,82475,82477],{},[51,82476,53],{},[95,82478,30648],{},[55,82480,82481],{"encoding":57},"x^*",[31,82483,82485],{"className":82484,"ariaHidden":62},[61],[31,82486,82488,82491],{"className":82487},[66],[31,82489],{"className":82490,"style":30719},[70],[31,82492,82494,82497],{"className":82493},[75],[31,82495,53],{"className":82496},[75,76],[31,82498,82500],{"className":82499},[1039],[31,82501,82503],{"className":82502},[1043],[31,82504,82506],{"className":82505},[1048],[31,82507,82509],{"className":82508,"style":30719},[1052],[31,82510,82511,82514],{"style":3631},[31,82512],{"className":82513,"style":1061},[1060],[31,82515,82517],{"className":82516},[1065,1066,1067,1068],[31,82518,30648],{"className":82519},[124,1068],"。此外，对任意初始点 ",[31,82522,82524,82546],{"className":82523,"translate":35},[34],[31,82525,82527],{"className":82526},[39],[41,82528,82529],{"xmlns":43},[45,82530,82531,82543],{},[48,82532,82533,82539,82541],{},[1004,82534,82535,82537],{},[51,82536,53],{},[91,82538,1977],{},[95,82540,2518],{},[51,82542,926],{},[55,82544,82545],{"encoding":57},"x_0\\in X",[31,82547,82549,82604],{"className":82548,"ariaHidden":62},[61],[31,82550,82552,82555,82595,82598,82601],{"className":82551},[66],[31,82553],{"className":82554,"style":15917},[70],[31,82556,82558,82561],{"className":82557},[75],[31,82559,53],{"className":82560},[75,76],[31,82562,82564],{"className":82563},[1039],[31,82565,82567,82587],{"className":82566},[1043,1044],[31,82568,82570,82584],{"className":82569},[1048],[31,82571,82573],{"className":82572,"style":2028},[1052],[31,82574,82575,82578],{"style":2807},[31,82576],{"className":82577,"style":1061},[1060],[31,82579,82581],{"className":82580},[1065,1066,1067,1068],[31,82582,1977],{"className":82583},[75,1068],[31,82585,1076],{"className":82586},[1075],[31,82588,82590],{"className":82589},[1048],[31,82591,82593],{"className":82592,"style":2050},[1052],[31,82594],{},[31,82596],{"className":82597,"style":293},[119],[31,82599,2518],{"className":82600},[297],[31,82602],{"className":82603,"style":293},[119],[31,82605,82607,82610],{"className":82606},[66],[31,82608],{"className":82609,"style":945},[70],[31,82611,926],{"className":82612,"style":30311},[75,76],"，迭代 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满足",[31,82769,82771],{"className":82770,"translate":35},[411],[31,82772,82774,82836],{"className":82773,"translate":35},[34],[31,82775,82777],{"className":82776},[39],[41,82778,82779],{"xmlns":43,"display":420},[45,82780,82781,82833],{},[48,82782,82783,82785,82787,82793,82795,82801,82803,82805,82811,82813,82815,82821,82823,82829,82831],{},[51,82784,1435],{},[95,82786,211],{"stretchy":210},[1004,82788,82789,82791],{},[51,82790,53],{},[51,82792,52065],{},[95,82794,1210],{"separator":62},[1180,82796,82797,82799],{},[51,82798,53],{},[95,82800,30648],{},[95,82802,220],{"stretchy":210},[95,82804,347],{},[1180,82806,82807,82809],{},[51,82808,53298],{},[51,82810,52065],{},[51,82812,1435],{},[95,82814,211],{"stretchy":210},[1004,82816,82817,82819],{},[51,82818,53],{},[91,82820,1977],{},[95,82822,1210],{"separator":62},[1180,82824,82825,82827],{},[51,82826,53],{},[95,82828,30648],{},[95,82830,220],{"stretchy":210},[51,82832,495],{"mathvariant":494},[55,82834,82835],{"encoding":57},"d(x_k,x^*)\\le\\gamma^k 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是 Cauchy 列。由 ",[31,85241,85243,85256],{"className":85242,"translate":35},[34],[31,85244,85246],{"className":85245},[39],[41,85247,85248],{"xmlns":43},[45,85249,85250,85254],{},[48,85251,85252],{},[51,85253,926],{},[55,85255,926],{"encoding":57},[31,85257,85259],{"className":85258,"ariaHidden":62},[61],[31,85260,85262,85265],{"className":85261},[66],[31,85263],{"className":85264,"style":945},[70],[31,85266,926],{"className":85267,"style":30311},[75,76]," 完备，存在 ",[31,85270,85272,85294],{"className":85271,"translate":35},[34],[31,85273,85275],{"className":85274},[39],[41,85276,85277],{"xmlns":43},[45,85278,85279,85291],{},[48,85280,85281,85287,85289],{},[1180,85282,85283,85285],{},[51,85284,53],{},[95,85286,30648],{},[95,85288,2518],{},[51,85290,926],{},[55,85292,85293],{"encoding":57},"x^*\\in X",[31,85295,85297,85342],{"className":85296,"ariaHidden":62},[61],[31,85298,85300,85304,85333,85336,85339],{"className":85299},[66],[31,85301],{"className":85302,"style":85303},[70],"height:0.7278em;vertical-align:-0.0391em;",[31,85305,85307,85310],{"className":85306},[75],[31,85308,53],{"className":85309},[75,76],[31,85311,85313],{"className":85312},[1039],[31,85314,85316],{"className":85315},[1043],[31,85317,85319],{"className":85318},[1048],[31,85320,85322],{"className":85321,"style":30719},[1052],[31,85323,85324,85327],{"style":3631},[31,85325],{"className":85326,"style":1061},[1060],[31,85328,85330],{"className":85329},[1065,1066,1067,1068],[31,85331,30648],{"className":85332},[124,1068],[31,85334],{"className":85335,"style":293},[119],[31,85337,2518],{"className":85338},[297],[31,85340],{"className":85341,"style":293},[119],[31,85343,85345,85348],{"className":85344},[66],[31,85346],{"className":85347,"style":945},[70],[31,85349,926],{"className":85350,"style":30311},[75,76],[31,85352,85354,85380],{"className":85353,"translate":35},[34],[31,85355,85357],{"className":85356},[39],[41,85358,85359],{"xmlns":43},[45,85360,85361,85377],{},[48,85362,85363,85369,85371],{},[1004,85364,85365,85367],{},[51,85366,53],{},[51,85368,14559],{},[95,85370,929],{},[1180,85372,85373,85375],{},[51,85374,53],{},[95,85376,30648],{},[55,85378,85379],{"encoding":57},"x_n\\to x^*",[31,85381,85383,85438],{"className":85382,"ariaHidden":62},[61],[31,85384,85386,85389,85429,85432,85435],{"className":85385},[66],[31,85387],{"className":85388,"style":7090},[70],[31,85390,85392,85395],{"className":85391},[75],[31,85393,53],{"className":85394},[75,76],[31,85396,85398],{"className":85397},[1039],[31,85399,85401,85421],{"className":85400},[1043,1044],[31,85402,85404,85418],{"className":85403},[1048],[31,85405,85407],{"className":85406,"style":3230},[1052],[31,85408,85409,85412],{"style":2807},[31,85410],{"className":85411,"style":1061},[1060],[31,85413,85415],{"className":85414},[1065,1066,1067,1068],[31,85416,14559],{"className":85417},[75,76,1068],[31,85419,1076],{"className":85420},[1075],[31,85422,85424],{"className":85423},[1048],[31,85425,85427],{"className":85426,"style":2050},[1052],[31,85428],{},[31,85430],{"className":85431,"style":293},[119],[31,85433,929],{"className":85434},[297],[31,85436],{"className":85437,"style":293},[119],[31,85439,85441,85444],{"className":85440},[66],[31,85442],{"className":85443,"style":30719},[70],[31,85445,85447,85450],{"className":85446},[75],[31,85448,53],{"className":85449},[75,76],[31,85451,85453],{"className":85452},[1039],[31,85454,85456],{"className":85455},[1043],[31,85457,85459],{"className":85458},[1048],[31,85460,85462],{"className":85461,"style":30719},[1052],[31,85463,85464,85467],{"style":3631},[31,85465],{"className":85466,"style":1061},[1060],[31,85468,85470],{"className":85469},[1065,1066,1067,1068],[31,85471,30648],{"className":85472},[124,1068],"。算子 ",[31,85475,85477,85490],{"className":85476,"translate":35},[34],[31,85478,85480],{"className":85479},[39],[41,85481,85482],{"xmlns":43},[45,85483,85484,85488],{},[48,85485,85486],{},[51,85487,919],{},[55,85489,919],{"encoding":57},[31,85491,85493],{"className":85492,"ariaHidden":62},[61],[31,85494,85496,85499],{"className":85495},[66],[31,85497],{"className":85498,"style":945},[70],[31,85500,919],{"className":85501,"style":949},[75,76]," 是 Lipschitz 连续的，故",[31,85504,85506],{"className":85505,"translate":35},[411],[31,85507,85509,85660],{"className":85508,"translate":35},[34],[31,85510,85512],{"className":85511},[39],[41,85513,85514],{"xmlns":43,"display":420},[45,85515,85516,85657],{},[5606,85517,85518,85601],{"rowspacing":5608,"columnalign":5609,"columnspacing":5610},[5612,85519,85520,85534],{},[5615,85521,85522],{},[2071,85523,85524],{"scriptlevel":1977,"displaystyle":62},[48,85525,85526,85528],{},[51,85527,919],{},[1180,85529,85530,85532],{},[51,85531,53],{},[95,85533,30648],{},[5615,85535,85536],{},[2071,85537,85538],{"scriptlevel":1977,"displaystyle":62},[48,85539,85540,85542,85544,85546,85575,85577,85593,85595],{},[48,85541],{},[95,85543,223],{},[51,85545,919],{},[48,85547,85548,85550,85567,85573],{},[95,85549,211],{"fence":62},[1531,85551,85552,85559],{},[48,85553,85554,85557],{},[51,85555,85556],{},"lim",[95,85558,1540],{},[48,85560,85561,85563,85565],{},[51,85562,14559],{},[95,85564,929],{},[51,85566,4587],{"mathvariant":494},[1004,85568,85569,85571],{},[51,85570,53],{},[51,85572,14559],{},[95,85574,220],{"fence":62},[95,85576,223],{},[1531,85578,85579,85585],{},[48,85580,85581,85583],{},[51,85582,85556],{},[95,85584,1540],{},[48,85586,85587,85589,85591],{},[51,85588,14559],{},[95,85590,929],{},[51,85592,4587],{"mathvariant":494},[51,85594,919],{},[1004,85596,85597,85599],{},[51,85598,53],{},[51,85600,14559],{},[5612,85602,85603,85609],{},[5615,85604,85605],{},[2071,85606,85607],{"scriptlevel":1977,"displaystyle":62},[48,85608],{},[5615,85610,85611],{},[2071,85612,85613],{"scriptlevel":1977,"displaystyle":62},[48,85614,85615,85617,85619,85635,85647,85649,85655],{},[48,85616],{},[95,85618,223],{},[1531,85620,85621,85627],{},[48,85622,85623,85625],{},[51,85624,85556],{},[95,85626,1540],{},[48,85628,85629,85631,85633],{},[51,85630,14559],{},[95,85632,929],{},[51,85634,4587],{"mathvariant":494},[1004,85636,85637,85639],{},[51,85638,53],{},[48,85640,85641,85643,85645],{},[51,85642,14559],{},[95,85644,434],{},[91,85646,205],{},[95,85648,223],{},[1180,85650,85651,85653],{},[51,85652,53],{},[95,85654,30648],{},[51,85656,495],{"mathvariant":494},[55,85658,85659],{"encoding":57},"\\begin{aligned}\nTx^*&=T\\left(\\lim_{n\\to\\infty}x_n\\right)\n=\\lim_{n\\to\\infty}Tx_n\\\\\n&=\\lim_{n\\to\\infty}x_{n+1}=x^*.\n\\end{aligned}",[31,85661,85663],{"className":85662,"ariaHidden":62},[61],[31,85664,85666,85670],{"className":85665},[66],[31,85667],{"className":85668,"style":85669},[70],"height:3.69em;vertical-align:-1.595em;",[31,85671,85673],{"className":85672},[75],[31,85674,85676,85751],{"className":85675},[5606],[31,85677,85679],{"className":85678},[5867],[31,85680,85682,85742],{"className":85681},[1043,1044],[31,85683,85685,85739],{"className":85684},[1048],[31,85686,85689,85730],{"className":85687,"style":85688},[1052],"height:2.095em;",[31,85690,85692,85695],{"style":85691},"top:-4.095em;",[31,85693],{"className":85694,"style":5884},[1060],[31,85696,85698,85701],{"className":85697},[75],[31,85699,919],{"className":85700,"style":949},[75,76],[31,85702,85704,85707],{"className":85703},[75],[31,85705,53],{"className":85706},[75,76],[31,85708,85710],{"className":85709},[1039],[31,85711,85713],{"className":85712},[1043],[31,85714,85716],{"className":85715},[1048],[31,85717,85719],{"className":85718,"style":74855},[1052],[31,85720,85721,85724],{"style":1330},[31,85722],{"className":85723,"style":1061},[1060],[31,85725,85727],{"className":85726},[1065,1066,1067,1068],[31,85728,30648],{"className":85729},[124,1068],[31,85731,85733,85736],{"style":85732},"top:-2.255em;",[31,85734],{"className":85735,"style":5884},[1060],[31,85737],{"className":85738},[75],[31,85740,1076],{"className":85741},[1075],[31,85743,85745],{"className":85744},[1048],[31,85746,85749],{"className":85747,"style":85748},[1052],"height:1.595em;",[31,85750],{},[31,85752,85754],{"className":85753},[6062],[31,85755,85757,86177],{"className":85756},[1043,1044],[31,85758,85760,86174],{"className":85759},[1048],[31,85761,85763,86008],{"className":85762,"style":85688},[1052],[31,85764,85765,85768],{"style":85691},[31,85766],{"className":85767,"style":5884},[1060],[31,85769,85771,85774,85777,85780,85783,85786,85789,85900,85903,85906,85909,85962,85965,85968],{"className":85770},[75],[31,85772],{"className":85773},[75],[31,85775],{"className":85776,"style":293},[119],[31,85778,223],{"className":85779},[297],[31,85781],{"className":85782,"style":293},[119],[31,85784,919],{"className":85785,"style":949},[75,76],[31,85787],{"className":85788,"style":1467},[119],[31,85790,85792,85798,85851,85854,85894],{"className":85791},[3860],[31,85793,85795],{"className":85794,"style":3868},[264,3867],[31,85796,211],{"className":85797},[1307,2662],[31,85799,85801],{"className":85800},[1460,1620],[31,85802,85804,85843],{"className":85803},[1043,1044],[31,85805,85807,85840],{"className":85806},[1048],[31,85808,85810,85830],{"className":85809,"style":1630},[1052],[31,85811,85812,85815],{"style":37361},[31,85813],{"className":85814,"style":1637},[1060],[31,85816,85818],{"className":85817},[1065,1066,1067,1068],[31,85819,85821,85824,85827],{"className":85820},[75,1068],[31,85822,14559],{"className":85823},[75,76,1068],[31,85825,929],{"className":85826},[297,1068],[31,85828,4587],{"className":85829},[75,1068],[31,85831,85832,85835],{"style":1649},[31,85833],{"className":85834,"style":1637},[1060],[31,85836,85837],{},[31,85838,85556],{"className":85839},[1460],[31,85841,1076],{"className":85842},[1075],[31,85844,85846],{"className":85845},[1048],[31,85847,85849],{"className":85848,"style":37390},[1052],[31,85850],{},[31,85852],{"className":85853,"style":1467},[119],[31,85855,85857,85860],{"className":85856},[75],[31,85858,53],{"className":85859},[75,76],[31,85861,85863],{"className":85862},[1039],[31,85864,85866,85886],{"className":85865},[1043,1044],[31,85867,85869,85883],{"className":85868},[1048],[31,85870,85872],{"className":85871,"style":3230},[1052],[31,85873,85874,85877],{"style":2807},[31,85875],{"className":85876,"style":1061},[1060],[31,85878,85880],{"className":85879},[1065,1066,1067,1068],[31,85881,14559],{"className":85882},[75,76,1068],[31,85884,1076],{"className":85885},[1075],[31,85887,85889],{"className":85888},[1048],[31,85890,85892],{"className":85891,"style":2050},[1052],[31,85893],{},[31,85895,85897],{"className":85896,"style":3868},[289,3867],[31,85898,220],{"className":85899},[1307,2662],[31,85901],{"className":85902,"style":293},[119],[31,85904,223],{"className":85905},[297],[31,85907],{"className":85908,"style":293},[119],[31,85910,85912],{"className":85911},[1460,1620],[31,85913,85915,85954],{"className":85914},[1043,1044],[31,85916,85918,85951],{"className":85917},[1048],[31,85919,85921,85941],{"className":85920,"style":1630},[1052],[31,85922,85923,85926],{"style":37361},[31,85924],{"className":85925,"style":1637},[1060],[31,85927,85929],{"className":85928},[1065,1066,1067,1068],[31,85930,85932,85935,85938],{"className":85931},[75,1068],[31,85933,14559],{"className":85934},[75,76,1068],[31,85936,929],{"className":85937},[297,1068],[31,85939,4587],{"className":85940},[75,1068],[31,85942,85943,85946],{"style":1649},[31,85944],{"className":85945,"style":1637},[1060],[31,85947,85948],{},[31,85949,85556],{"className":85950},[1460],[31,85952,1076],{"className":85953},[1075],[31,85955,85957],{"className":85956},[1048],[31,85958,85960],{"className":85959,"style":37390},[1052],[31,85961],{},[31,85963],{"className":85964,"style":1467},[119],[31,85966,919],{"className":85967,"style":949},[75,76],[31,85969,85971,85974],{"className":85970},[75],[31,85972,53],{"className":85973},[75,76],[31,85975,85977],{"className":85976},[1039],[31,85978,85980,86000],{"className":85979},[1043,1044],[31,85981,85983,85997],{"className":85982},[1048],[31,85984,85986],{"className":85985,"style":3230},[1052],[31,85987,85988,85991],{"style":2807},[31,85989],{"className":85990,"style":1061},[1060],[31,85992,85994],{"className":85993},[1065,1066,1067,1068],[31,85995,14559],{"className":85996},[75,76,1068],[31,85998,1076],{"className":85999},[1075],[31,86001,86003],{"className":86002},[1048],[31,86004,86006],{"className":86005,"style":2050},[1052],[31,86007],{},[31,86009,86010,86013],{"style":85732},[31,86011],{"className":86012,"style":5884},[1060],[31,86014,86016,86019,86022,86025,86028,86081,86084,86133,86136,86139,86142,86171],{"className":86015},[75],[31,86017],{"className":86018},[75],[31,86020],{"className":86021,"style":293},[119],[31,86023,223],{"className":86024},[297],[31,86026],{"className":86027,"style":293},[119],[31,86029,86031],{"className":86030},[1460,1620],[31,86032,86034,86073],{"className":86033},[1043,1044],[31,86035,86037,86070],{"className":86036},[1048],[31,86038,86040,86060],{"className":86039,"style":1630},[1052],[31,86041,86042,86045],{"style":37361},[31,86043],{"className":86044,"style":1637},[1060],[31,86046,86048],{"className":86047},[1065,1066,1067,1068],[31,86049,86051,86054,86057],{"className":86050},[75,1068],[31,86052,14559],{"className":86053},[75,76,1068],[31,86055,929],{"className":86056},[297,1068],[31,86058,4587],{"className":86059},[75,1068],[31,86061,86062,86065],{"style":1649},[31,86063],{"className":86064,"style":1637},[1060],[31,86066,86067],{},[31,86068,85556],{"className":86069},[1460],[31,86071,1076],{"className":86072},[1075],[31,86074,86076],{"className":86075},[1048],[31,86077,86079],{"className":86078,"style":37390},[1052],[31,86080],{},[31,86082],{"className":86083,"style":1467},[119],[31,86085,86087,86090],{"className":86086},[75],[31,86088,53],{"className":86089},[75,76],[31,86091,86093],{"className":86092},[1039],[31,86094,86096,86125],{"className":86095},[1043,1044],[31,86097,86099,86122],{"className":86098},[1048],[31,86100,86102],{"className":86101,"style":2028},[1052],[31,86103,86104,86107],{"style":2807},[31,86105],{"className":86106,"style":1061},[1060],[31,86108,86110],{"className":86109},[1065,1066,1067,1068],[31,86111,86113,86116,86119],{"className":86112},[75,1068],[31,86114,14559],{"className":86115},[75,76,1068],[31,86117,434],{"className":86118},[124,1068],[31,86120,205],{"className":86121},[75,1068],[31,86123,1076],{"className":86124},[1075],[31,86126,86128],{"className":86127},[1048],[31,86129,86131],{"className":86130,"style":25082},[1052],[31,86132],{},[31,86134],{"className":86135,"style":293},[119],[31,86137,223],{"className":86138},[297],[31,86140],{"className":86141,"style":293},[119],[31,86143,86145,86148],{"className":86144},[75],[31,86146,53],{"className":86147},[75,76],[31,86149,86151],{"className":86150},[1039],[31,86152,86154],{"className":86153},[1043],[31,86155,86157],{"className":86156},[1048],[31,86158,86160],{"className":86159,"style":74855},[1052],[31,86161,86162,86165],{"style":1330},[31,86163],{"className":86164,"style":1061},[1060],[31,86166,86168],{"className":86167},[1065,1066,1067,1068],[31,86169,30648],{"className":86170},[124,1068],[31,86172,495],{"className":86173},[75],[31,86175,1076],{"className":86176},[1075],[31,86178,86180],{"className":86179},[1048],[31,86181,86183],{"className":86182,"style":85748},[1052],[31,86184],{},[11,86186,86187],{},"最后，由压缩性直接得到",[31,86189,86191],{"className":86190,"translate":35},[411],[31,86192,86194,86290],{"className":86193,"translate":35},[34],[31,86195,86197],{"className":86196},[39],[41,86198,86199],{"xmlns":43,"display":420},[45,86200,86201,86287],{},[48,86202,86203,86205,86207,86213,86215,86221,86223,86225,86227,86229,86235,86241,86243,86249,86255,86257,86259,86265,86267,86269,86275,86277,86283,86285],{},[51,86204,1435],{},[95,86206,211],{"stretchy":210},[1004,86208,86209,86211],{},[51,86210,53],{},[51,86212,52065],{},[95,86214,1210],{"separator":62},[1180,86216,86217,86219],{},[51,86218,53],{},[95,86220,30648],{},[95,86222,220],{"stretchy":210},[95,86224,223],{},[51,86226,1435],{},[95,86228,211],{"stretchy":210},[1180,86230,86231,86233],{},[51,86232,919],{},[51,86234,52065],{},[1004,86236,86237,86239],{},[51,86238,53],{},[91,86240,1977],{},[95,86242,1210],{"separator":62},[1180,86244,86245,86247],{},[51,86246,919],{},[51,86248,52065],{},[1180,86250,86251,86253],{},[51,86252,53],{},[95,86254,30648],{},[95,86256,220],{"stretchy":210},[95,86258,347],{},[1180,86260,86261,86263],{},[51,86262,53298],{},[51,86264,52065],{},[51,86266,1435],{},[95,86268,211],{"stretchy":210},[1004,86270,86271,86273],{},[51,86272,53],{},[91,86274,1977],{},[95,86276,1210],{"separator":62},[1180,86278,86279,86281],{},[51,86280,53],{},[95,86282,30648],{},[95,86284,220],{"stretchy":210},[51,86286,495],{"mathvariant":494},[55,86288,86289],{"encoding":57},"d(x_k,x^*)=d(T^kx_0,T^kx^*)\\le\\gamma^k d(x_0,x^*).",[31,86291,86293,86392,86549],{"className":86292,"ariaHidden":62},[61],[31,86294,86296,86299,86302,86305,86345,86348,86351,86380,86383,86386,86389],{"className":86295},[66],[31,86297],{"className":86298,"style":260},[70],[31,86300,1435],{"className":86301},[75,76],[31,86303,211],{"className":86304},[264],[31,86306,86308,86311],{"className":86307},[75],[31,86309,53],{"className":86310},[75,76],[31,86312,86314],{"className":86313},[1039],[31,86315,86317,86337],{"className":86316},[1043,1044],[31,86318,86320,86334],{"className":86319},[1048],[31,86321,86323],{"className":86322,"style":1053},[1052],[31,86324,86325,86328],{"style":2807},[31,86326],{"className":86327,"style":1061},[1060],[31,86329,86331],{"className":86330},[1065,1066,1067,1068],[31,86332,52065],{"className":86333,"style":52166},[75,76,1068],[31,86335,1076],{"className":86336},[1075],[31,86338,86340],{"className":86339},[1048],[31,86341,86343],{"className":86342,"style":2050},[1052],[31,86344],{},[31,86346,1210],{"className":86347},[1391],[31,86349],{"className":86350,"style":1467},[119],[31,86352,86354,86357],{"className":86353},[75],[31,86355,53],{"className":86356},[75,76],[31,86358,86360],{"className":86359},[1039],[31,86361,86363],{"className":86362},[1043],[31,86364,86366],{"className":86365},[1048],[31,86367,86369],{"className":86368,"style":74855},[1052],[31,86370,86371,86374],{"style":1330},[31,86372],{"className":86373,"style":1061},[1060],[31,86375,86377],{"className":86376},[1065,1066,1067,1068],[31,86378,30648],{"className":86379},[124,1068],[31,86381,220],{"className":86382},[289],[31,86384],{"className":86385,"style":293},[119],[31,86387,223],{"className":86388},[297],[31,86390],{"className":86391,"style":293},[119],[31,86393,86395,86398,86401,86404,86433,86473,86476,86479,86508,86537,86540,86543,86546],{"className":86394},[66],[31,86396],{"className":86397,"style":35441},[70],[31,86399,1435],{"className":86400},[75,76],[31,86402,211],{"className":86403},[264],[31,86405,86407,86410],{"className":86406},[75],[31,86408,919],{"className":86409,"style":949},[75,76],[31,86411,86413],{"className":86412},[1039],[31,86414,86416],{"className":86415},[1043],[31,86417,86419],{"className":86418},[1048],[31,86420,86422],{"className":86421,"style":29642},[1052],[31,86423,86424,86427],{"style":1330},[31,86425],{"className":86426,"style":1061},[1060],[31,86428,86430],{"className":86429},[1065,1066,1067,1068],[31,86431,52065],{"className":86432,"style":52166},[75,76,1068],[31,86434,86436,86439],{"className":86435},[75],[31,86437,53],{"className":86438},[75,76],[31,86440,86442],{"className":86441},[1039],[31,86443,86445,86465],{"className":86444},[1043,1044],[31,86446,86448,86462],{"className":86447},[1048],[31,86449,86451],{"className":86450,"style":2028},[1052],[31,86452,86453,86456],{"style":2807},[31,86454],{"className":86455,"style":1061},[1060],[31,86457,86459],{"className":86458},[1065,1066,1067,1068],[31,86460,1977],{"className":86461},[75,1068],[31,86463,1076],{"className":86464},[1075],[31,86466,86468],{"className":86467},[1048],[31,86469,86471],{"className":86470,"style":2050},[1052],[31,86472],{},[31,86474,1210],{"className":86475},[1391],[31,86477],{"className":86478,"style":1467},[119],[31,86480,86482,86485],{"className":86481},[75],[31,86483,919],{"className":86484,"style":949},[75,76],[31,86486,86488],{"className":86487},[1039],[31,86489,86491],{"className":86490},[1043],[31,86492,86494],{"className":86493},[1048],[31,86495,86497],{"className":86496,"style":29642},[1052],[31,86498,86499,86502],{"style":1330},[31,86500],{"className":86501,"style":1061},[1060],[31,86503,86505],{"className":86504},[1065,1066,1067,1068],[31,86506,52065],{"className":86507,"style":52166},[75,76,1068],[31,86509,86511,86514],{"className":86510},[75],[31,86512,53],{"className":86513},[75,76],[31,86515,86517],{"className":86516},[1039],[31,86518,86520],{"className":86519},[1043],[31,86521,86523],{"className":86522},[1048],[31,86524,86526],{"className":86525,"style":74855},[1052],[31,86527,86528,86531],{"style":1330},[31,86529],{"className":86530,"style":1061},[1060],[31,86532,86534],{"className":86533},[1065,1066,1067,1068],[31,86535,30648],{"className":86536},[124,1068],[31,86538,220],{"className":86539},[289],[31,86541],{"className":86542,"style":293},[119],[31,86544,347],{"className":86545},[297],[31,86547],{"className":86548,"style":293},[119],[31,86550,86552,86555,86584,86587,86590,86630,86633,86636,86665,86668],{"className":86551},[66],[31,86553],{"className":86554,"style":35441},[70],[31,86556,86558,86561],{"className":86557},[75],[31,86559,53298],{"className":86560,"style":24157},[75,76],[31,86562,86564],{"className":86563},[1039],[31,86565,86567],{"className":86566},[1043],[31,86568,86570],{"className":86569},[1048],[31,86571,86573],{"className":86572,"style":29642},[1052],[31,86574,86575,86578],{"style":1330},[31,86576],{"className":86577,"style":1061},[1060],[31,86579,86581],{"className":86580},[1065,1066,1067,1068],[31,86582,52065],{"c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唯一性与上下界",[52865,86678,86681],{"id":86679,"title":86680},"corollary-unique-fixed-point","推论 3.4（不动点的存在唯一性）",[11,86682,52871,86683,67262,86755,18171,86813,25085,86871,86914],{},[31,86684,86686,86707],{"className":86685,"translate":35},[34],[31,86687,86689],{"className":86688},[39],[41,86690,86691],{"xmlns":43},[45,86692,86693,86705],{},[48,86694,86695,86697,86699,86701,86703],{},[91,86696,1977],{},[95,86698,347],{},[51,86700,53298],{},[95,86702,9758],{},[91,86704,205],{},[55,86706,67213],{"encoding":57},[31,86708,86710,86728,86746],{"className":86709,"ariaHidden":62},[61],[31,86711,86713,86716,86719,86722,86725],{"className":86712},[66],[31,86714],{"className":86715,"style":366},[70],[31,86717,1977],{"className":86718},[75],[31,86720],{"className":86721,"style":293},[119],[31,86723,347],{"className":86724},[297],[31,86726],{"className":86727,"style":293},[119],[31,86729,86731,86734,86737,86740,86743],{"className":86730},[66],[31,86732],{"className":86733,"style":53323},[70],[31,86735,53298],{"className":86736,"style":24157},[75,76],[31,86738],{"className":86739,"style":293},[119],[31,86741,9758],{"className":86742},[297],[31,86744],{"className":86745,"style":293},[119],[31,86747,86749,86752],{"className":86748},[66],[31,86750],{"className":86751,"style":326},[70],[31,86753,205],{"className":86754},[75],[31,86756,86758,86775],{"className":86757,"translate":35},[34],[31,86759,86761],{"className":86760},[39],[41,86762,86763],{"xmlns":43},[45,86764,86765,86773],{},[48,86766,86767],{},[1180,86768,86769,86771],{},[51,86770,919],{"mathvariant":925},[51,86772,2515],{},[55,86774,64428],{"encoding":57},[31,86776,86778],{"className":86777,"ariaHidden":62},[61],[31,86779,86781,86784],{"className":86780},[66],[31,86782],{"className":86783,"style":945},[70],[31,86785,86787,86790],{"className":86786},[75],[31,86788,919],{"className":86789,"style":64444},[75,968],[31,86791,86793],{"className":86792},[1039],[31,86794,86796],{"className":86795},[1043],[31,86797,86799],{"className":86798},[1048],[31,86800,86802],{"className":86801,"style":3628},[1052],[31,86803,86804,86807],{"style":3631},[31,86805],{"className":86806,"style":1061},[1060],[31,86808,86810],{"className":86809},[1065,1066,1067,1068],[31,86811,2515],{"className":86812,"style":2666},[75,76,1068],[31,86814,86816,86833],{"className":86815,"translate":35},[34],[31,86817,86819],{"className":86818},[39],[41,86820,86821],{"xmlns":43},[45,86822,86823,86831],{},[48,86824,86825],{},[1180,86826,86827,86829],{},[51,86828,919],{"mathvariant":925},[95,86830,30648],{},[55,86832,74501],{"encoding":57},[31,86834,86836],{"className":86835,"ariaHidden":62},[61],[31,86837,86839,86842],{"className":86838},[66],[31,86840],{"className":86841,"style":30719},[70],[31,86843,86845,86848],{"className":86844},[75],[31,86846,919],{"className":86847,"style":64444},[75,968],[31,86849,86851],{"className":86850},[1039],[31,86852,86854],{"className":86853},[1043],[31,86855,86857],{"className":86856},[1048],[31,86858,86860],{"className":86859,"style":30719},[1052],[31,86861,86862,86865],{"style":3631},[31,86863],{"className":86864,"style":1061},[1060],[31,86866,86868],{"className":86867},[1065,1066,1067,1068],[31,86869,30648],{"className":86870},[124,1068],[31,86872,86874,86893],{"className":86873,"translate":35},[34],[31,86875,86877],{"className":86876},[39],[41,86878,86879],{"xmlns":43},[45,86880,86881,86891],{},[48,86882,86883,86885,86887,86889],{},[51,86884,1122],{},[95,86886,211],{"stretchy":210},[51,86888,21582],{},[95,86890,220],{"stretchy":210},[55,86892,73997],{"encoding":57},[31,86894,86896],{"className":86895,"ariaHidden":62},[61],[31,86897,86899,86902,86905,86908,86911],{"className":86898},[66],[31,86900],{"className":86901,"style":260},[70],[31,86903,1122],{"className":86904,"style":1137},[75,76],[31,86906,211],{"className":86907},[264],[31,86909,21582],{"className":86910,"style":21697},[75,76],[31,86912,220],{"className":86913},[289]," 中各自存在唯一不动点。",[11,86916,86917,54797,86919,86962,86963,18171,87021,79432,87079,87107,87108,87112],{},[7246,86918,7248],{},[31,86920,86922,86941],{"className":86921,"translate":35},[34],[31,86923,86925],{"className":86924},[39],[41,86926,86927],{"xmlns":43},[45,86928,86929,86939],{},[48,86930,86931,86933,86935,86937],{},[51,86932,1122],{},[95,86934,211],{"stretchy":210},[51,86936,21582],{},[95,86938,220],{"stretchy":210},[55,86940,73997],{"encoding":57},[31,86942,86944],{"className":86943,"ariaHidden":62},[61],[31,86945,86947,86950,86953,86956,86959],{"className":86946},[66],[31,86948],{"className":86949,"style":260},[70],[31,86951,1122],{"className":86952,"style":1137},[75,76],[31,86954,211],{"className":86955},[264],[31,86957,21582],{"className":86958,"style":21697},[75,76],[31,86960,220],{"className":86961},[289]," 在 sup 范数下完备，",[31,86964,86966,86983],{"className":86965,"translate":35},[34],[31,86967,86969],{"className":86968},[39],[41,86970,86971],{"xmlns":43},[45,86972,86973,86981],{},[48,86974,86975],{},[1180,86976,86977,86979],{},[51,86978,919],{"mathvariant":925},[51,86980,2515],{},[55,86982,64428],{"encoding":57},[31,86984,86986],{"className":86985,"ariaHidden":62},[61],[31,86987,86989,86992],{"className":86988},[66],[31,86990],{"className":86991,"style":945},[70],[31,86993,86995,86998],{"className":86994},[75],[31,86996,919],{"className":86997,"style":64444},[75,968],[31,86999,87001],{"className":87000},[1039],[31,87002,87004],{"className":87003},[1043],[31,87005,87007],{"className":87006},[1048],[31,87008,87010],{"className":87009,"style":3628},[1052],[31,87011,87012,87015],{"style":3631},[31,87013],{"className":87014,"style":1061},[1060],[31,87016,87018],{"className":87017},[1065,1066,1067,1068],[31,87019,2515],{"className":87020,"style":2666},[75,76,1068],[31,87022,87024,87041],{"className":87023,"translate":35},[34],[31,87025,87027],{"className":87026},[39],[41,87028,87029],{"xmlns":43},[45,87030,87031,87039],{},[48,87032,87033],{},[1180,87034,87035,87037],{},[51,87036,919],{"mathvariant":925},[95,87038,30648],{},[55,87040,74501],{"encoding":57},[31,87042,87044],{"className":87043,"ariaHidden":62},[61],[31,87045,87047,87050],{"className":87046},[66],[31,87048],{"className":87049,"style":30719},[70],[31,87051,87053,87056],{"className":87052},[75],[31,87054,919],{"className":87055,"style":64444},[75,968],[31,87057,87059],{"className":87058},[1039],[31,87060,87062],{"className":87061},[1043],[31,87063,87065],{"className":87064},[1048],[31,87066,87068],{"className":87067,"style":30719},[1052],[31,87069,87070,87073],{"style":3631},[31,87071],{"className":87072,"style":1061},[1060],[31,87074,87076],{"className":87075},[1065,1066,1067,1068],[31,87077,30648],{"className":87078},[124,1068],[31,87080,87082,87095],{"className":87081,"translate":35},[34],[31,87083,87085],{"className":87084},[39],[41,87086,87087],{"xmlns":43},[45,87088,87089,87093],{},[48,87090,87091],{},[51,87092,53298],{},[55,87094,79191],{"encoding":57},[31,87096,87098],{"className":87097,"ariaHidden":62},[61],[31,87099,87101,87104],{"className":87100},[66],[31,87102],{"className":87103,"style":865},[70],[31,87105,53298],{"className":87106,"style":24157},[75,76],"-压缩映射。由 ",[5813,87109,87111],{"href":87110},"#theorem-banach","定理 3.3"," 即得。证毕。",[11,87114,9612,87115,1138,87173,87175,87176,76226,87234,87292,87293,87352],{},[31,87116,87118,87135],{"className":87117,"translate":35},[34],[31,87119,87121],{"className":87120},[39],[41,87122,87123],{"xmlns":43},[45,87124,87125,87133],{},[48,87126,87127],{},[1180,87128,87129,87131],{},[51,87130,919],{"mathvariant":925},[51,87132,2515],{},[55,87134,64428],{"encoding":57},[31,87136,87138],{"className":87137,"ariaHidden":62},[61],[31,87139,87141,87144],{"className":87140},[66],[31,87142],{"className":87143,"style":945},[70],[31,87145,87147,87150],{"className":87146},[75],[31,87148,919],{"className":87149,"style":64444},[75,968],[31,87151,87153],{"className":87152},[1039],[31,87154,87156],{"className":87155},[1043],[31,87157,87159],{"className":87158},[1048],[31,87160,87162],{"className":87161,"style":3628},[1052],[31,87163,87164,87167],{"style":3631},[31,87165],{"className":87166,"style":1061},[1060],[31,87168,87170],{"className":87169},[1065,1066,1067,1068],[31,87171,2515],{"className":87172,"style":2666},[75,76,1068],[5813,87174,73911],{"href":73910}," 已经指出这个唯一不动点就是 ",[31,87177,87179,87196],{"className":87178,"translate":35},[34],[31,87180,87182],{"className":87181},[39],[41,87183,87184],{"xmlns":43},[45,87185,87186,87194],{},[48,87187,87188],{},[1180,87189,87190,87192],{},[51,87191,43481],{},[51,87193,2515],{},[55,87195,59045],{"encoding":57},[31,87197,87199],{"className":87198,"ariaHidden":62},[61],[31,87200,87202,87205],{"className":87201},[66],[31,87203],{"className":87204,"style":945},[70],[31,87206,87208,87211],{"className":87207},[75],[31,87209,43481],{"className":87210,"style":120},[75,76],[31,87212,87214],{"className":87213},[1039],[31,87215,87217],{"className":87216},[1043],[31,87218,87220],{"className":87219},[1048],[31,87221,87223],{"className":87222,"style":3628},[1052],[31,87224,87225,87228],{"style":3631},[31,87226],{"className":87227,"style":1061},[1060],[31,87229,87231],{"className":87230},[1065,1066,1067,1068],[31,87232,2515],{"className":87233,"style":2666},[75,76,1068],[31,87235,87237,87254],{"className":87236,"translate":35},[34],[31,87238,87240],{"className":87239},[39],[41,87241,87242],{"xmlns":43},[45,87243,87244,87252],{},[48,87245,87246],{},[1180,87247,87248,87250],{},[51,87249,919],{"mathvariant":925},[95,87251,30648],{},[55,87253,74501],{"encoding":57},[31,87255,87257],{"className":87256,"ariaHidden":62},[61],[31,87258,87260,87263],{"className":87259},[66],[31,87261],{"className":87262,"style":30719},[70],[31,87264,87266,87269],{"className":87265},[75],[31,87267,919],{"className":87268,"style":64444},[75,968],[31,87270,87272],{"className":87271},[1039],[31,87273,87275],{"className":87274},[1043],[31,87276,87278],{"className":87277},[1048],[31,87279,87281],{"className":87280,"style":30719},[1052],[31,87282,87283,87286],{"style":3631},[31,87284],{"className":87285,"style":1061},[1060],[31,87287,87289],{"className":87288},[1065,1066,1067,1068],[31,87290,30648],{"className":87291},[124,1068],"，第 4 节将证明它的唯一不动点就是最优值函数 ",[31,87294,87296,87314],{"className":87295,"translate":35},[34],[31,87297,87299],{"className":87298},[39],[41,87300,87301],{"xmlns":43},[45,87302,87303,87311],{},[48,87304,87305],{},[1180,87306,87307,87309],{},[51,87308,43481],{},[95,87310,30648],{},[55,87312,87313],{"encoding":57},"V^*",[31,87315,87317],{"className":87316,"ariaHidden":62},[61],[31,87318,87320,87323],{"className":87319},[66],[31,87321],{"className":87322,"style":30719},[70],[31,87324,87326,87329],{"className":87325},[75],[31,87327,43481],{"className":87328,"style":120},[75,76],[31,87330,87332],{"className":87331},[1039],[31,87333,87335],{"className":87334},[1043],[31,87336,87338],{"className":87337},[1048],[31,87339,87341],{"className":87340,"style":30719},[1052],[31,87342,87343,87346],{"style":3631},[31,87344],{"className":87345,"style":1061},[1060],[31,87347,87349],{"className":87348},[1065,1066,1067,1068],[31,87350,30648],{"className":87351},[124,1068],"。因此，「贝尔曼方程有唯一解」并不是一个额外的假设，而是压缩性的直接推论。",[11,87354,87355],{},"单调性还给出了一个在理论上很有用的上下界判据。",[52865,87357,87360],{"id":87358,"title":87359},"corollary-order-bounds","推论 3.5（上下界判据）",[11,87361,52871,87362,74019,87391,23342,87449,87507,87508,87578,87579,87744,87745,87578,87890,87744,88049,2372],{},[31,87363,87365,87379],{"className":87364,"translate":35},[34],[31,87366,87368],{"className":87367},[39],[41,87369,87370],{"xmlns":43},[45,87371,87372,87376],{},[48,87373,87374],{},[51,87375,919],{"mathvariant":925},[55,87377,87378],{"encoding":57},"\\mathcal{T}",[31,87380,87382],{"className":87381,"ariaHidden":62},[61],[31,87383,87385,87388],{"className":87384},[66],[31,87386],{"className":87387,"style":945},[70],[31,87389,919],{"className":87390,"style":64444},[75,968],[31,87392,87394,87411],{"className":87393,"translate":35},[34],[31,87395,87397],{"className":87396},[39],[41,87398,87399],{"xmlns":43},[45,87400,87401,87409],{},[48,87402,87403],{},[1180,87404,87405,87407],{},[51,87406,919],{"mathvariant":925},[51,87408,2515],{},[55,87410,64428],{"encoding":57},[31,87412,87414],{"className":87413,"ariaHidden":62},[61],[31,87415,87417,87420],{"className":87416},[66],[31,87418],{"className":87419,"style":945},[70],[31,87421,87423,87426],{"className":87422},[75],[31,87424,919],{"className":87425,"style":64444},[75,968],[31,87427,87429],{"className":87428},[1039],[31,87430,87432],{"className":87431},[1043],[31,87433,87435],{"className":87434},[1048],[31,87436,87438],{"className":87437,"style":3628},[1052],[31,87439,87440,87443],{"style":3631},[31,87441],{"className":87442,"style":1061},[1060],[31,87444,87446],{"className":87445},[1065,1066,1067,1068],[31,87447,2515],{"className":87448,"style":2666},[75,76,1068],[31,87450,87452,87469],{"className":87451,"translate":35},[34],[31,87453,87455],{"className":87454},[39],[41,87456,87457],{"xmlns":43},[45,87458,87459,87467],{},[48,87460,87461],{},[1180,87462,87463,87465],{},[51,87464,919],{"mathvariant":925},[95,87466,30648],{},[55,87468,74501],{"encoding":57},[31,87470,87472],{"className":87471,"ariaHidden":62},[61],[31,87473,87475,87478],{"className":87474},[66],[31,87476],{"className":87477,"style":30719},[70],[31,87479,87481,87484],{"className":87480},[75],[31,87482,919],{"className":87483,"style":64444},[75,968],[31,87485,87487],{"className":87486},[1039],[31,87488,87490],{"className":87489},[1043],[31,87491,87493],{"className":87492},[1048],[31,87494,87496],{"className":87495,"style":30719},[1052],[31,87497,87498,87501],{"style":3631},[31,87499],{"className":87500,"style":1061},[1060],[31,87502,87504],{"className":87503},[1065,1066,1067,1068],[31,87505,30648],{"className":87506},[124,1068],"，其唯一不动点为 ",[31,87509,87511,87529],{"className":87510,"translate":35},[34],[31,87512,87514],{"className":87513},[39],[41,87515,87516],{"xmlns":43},[45,87517,87518,87526],{},[48,87519,87520],{},[1004,87521,87522,87524],{},[51,87523,43481],{},[51,87525,4587],{"mathvariant":494},[55,87527,87528],{"encoding":57},"V_\\infty",[31,87530,87532],{"className":87531,"ariaHidden":62},[61],[31,87533,87535,87538],{"className":87534},[66],[31,87536],{"className":87537,"style":5230},[70],[31,87539,87541,87544],{"className":87540},[75],[31,87542,43481],{"className":87543,"style":120},[75,76],[31,87545,87547],{"className":87546},[1039],[31,87548,87550,87570],{"className":87549},[1043,1044],[31,87551,87553,87567],{"className":87552},[1048],[31,87554,87556],{"className":87555,"style":3230},[1052],[31,87557,87558,87561],{"style":66251},[31,87559],{"className":87560,"style":1061},[1060],[31,87562,87564],{"className":87563},[1065,1066,1067,1068],[31,87565,4587],{"className":87566},[75,1068],[31,87568,1076],{"className":87569},[1075],[31,87571,87573],{"className":87572},[1048],[31,87574,87576],{"className":87575,"style":2050},[1052],[31,87577],{},"。若 ",[31,87580,87582,87624],{"className":87581,"translate":35},[34],[31,87583,87585],{"className":87584},[39],[41,87586,87587],{"xmlns":43},[45,87588,87589,87621],{},[48,87590,87591,87605,87607,87609],{},[1004,87592,87593,87595],{},[51,87594,43481],{},[48,87596,87597,87600,87602],{},[51,87598,87599],{"mathvariant":494},"l",[51,87601,2552],{"mathvariant":494},[51,87603,87604],{"mathvariant":494},"w",[95,87606,347],{},[51,87608,919],{"mathvariant":925},[1004,87610,87611,87613],{},[51,87612,43481],{},[48,87614,87615,87617,87619],{},[51,87616,87599],{"mathvariant":494},[51,87618,2552],{"mathvariant":494},[51,87620,87604],{"mathvariant":494},[55,87622,87623],{"encoding":57},"V_{\\mathrm{low}}\\le\\mathcal{T}V_{\\mathrm{low}}",[31,87625,87627,87689],{"className":87626,"ariaHidden":62},[61],[31,87628,87630,87633,87680,87683,87686],{"className":87629},[66],[31,87631],{"className":87632,"style":5230},[70],[31,87634,87636,87639],{"className":87635},[75],[31,87637,43481],{"className":87638,"style":120},[75,76],[31,87640,87642],{"className":87641},[1039],[31,87643,87645,87672],{"className":87644},[1043,1044],[31,87646,87648,87669],{"className":87647},[1048],[31,87649,87651],{"className":87650,"style":1053},[1052],[31,87652,87653,87656],{"style":66251},[31,87654],{"className":87655,"style":1061},[1060],[31,87657,87659],{"className":87658},[1065,1066,1067,1068],[31,87660,87662],{"className":87661},[75,1068],[31,87663,87665],{"className":87664},[75,1068],[31,87666,87668],{"className":87667,"style":20180},[75,2745,1068],"low",[31,87670,1076],{"className":87671},[1075],[31,87673,87675],{"className":87674},[1048],[31,87676,87678],{"className":87677,"style":2050},[1052],[31,87679],{},[31,87681],{"className":87682,"style":293},[119],[31,87684,347],{"className":87685},[297],[31,87687],{"className":87688,"style":293},[119],[31,87690,87692,87695,87698],{"className":87691},[66],[31,87693],{"className":87694,"style":5230},[70],[31,87696,919],{"className":87697,"style":64444},[75,968],[31,87699,87701,87704],{"className":87700},[75],[31,87702,43481],{"className":87703,"style":120},[75,76],[31,87705,87707],{"className":87706},[1039],[31,87708,87710,87736],{"className":87709},[1043,1044],[31,87711,87713,87733],{"className":87712},[1048],[31,87714,87716],{"className":87715,"style":1053},[1052],[31,87717,87718,87721],{"style":66251},[31,87719],{"className":87720,"style":1061},[1060],[31,87722,87724],{"className":87723},[1065,1066,1067,1068],[31,87725,87727],{"className":87726},[75,1068],[31,87728,87730],{"className":87729},[75,1068],[31,87731,87668],{"className":87732,"style":20180},[75,2745,1068],[31,87734,1076],{"className":87735},[1075],[31,87737,87739],{"className":87738},[1048],[31,87740,87742],{"className":87741,"style":2050},[1052],[31,87743],{},"，则 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V_\\infty",[31,88715,88717,88807],{"className":88716,"ariaHidden":62},[61],[31,88718,88720,88723,88752,88798,88801,88804],{"className":88719},[66],[31,88721],{"className":88722,"style":5230},[70],[31,88724,88726,88729],{"className":88725},[75],[31,88727,919],{"className":88728,"style":64444},[75,968],[31,88730,88732],{"className":88731},[1039],[31,88733,88735],{"className":88734},[1043],[31,88736,88738],{"className":88737},[1048],[31,88739,88741],{"className":88740,"style":3628},[1052],[31,88742,88743,88746],{"style":3631},[31,88744],{"className":88745,"style":1061},[1060],[31,88747,88749],{"className":88748},[1065,1066,1067,1068],[31,88750,14559],{"className":88751},[75,76,1068],[31,88753,88755,88758],{"className":88754},[75],[31,88756,43481],{"className":88757,"style":120},[75,76],[31,88759,88761],{"className":88760},[1039],[31,88762,88764,88790],{"className":88763},[1043,1044],[31,88765,88767,88787],{"className":88766},[1048],[31,88768,88770],{"className":88769,"style":1053},[1052],[31,88771,88772,88775],{"style":66251},[31,88773],{"className":88774,"style":1061},[1060],[31,88776,88778],{"className":88777},[1065,1066,1067,1068],[31,88779,88781],{"className":88780},[75,1068],[31,88782,88784],{"className":88783},[75,1068],[31,88785,87668],{"className":88786,"style":20180},[75,2745,1068],[31,88788,1076],{"className":88789},[1075],[31,88791,88793],{"className":88792},[1048],[31,88794,88796],{"className":88795,"style":2050},[1052],[31,88797],{},[31,88799],{"className":88800,"style":293},[119],[31,88802,929],{"className":88803},[297],[31,88805],{"className":88806,"style":293},[119],[31,88808,88810,88813],{"className":88809},[66],[31,88811],{"className":88812,"style":5230},[70],[31,88814,88816,88819],{"className":88815},[75],[31,88817,43481],{"className":88818,"style":120},[75,76],[31,88820,88822],{"className":88821},[1039],[31,88823,88825,88845],{"className":88824},[1043,1044],[31,88826,88828,88842],{"className":88827},[1048],[31,88829,88831],{"className":88830,"style":3230},[1052],[31,88832,88833,88836],{"style":66251},[31,88834],{"className":88835,"style":1061},[1060],[31,88837,88839],{"className":88838},[1065,1066,1067,1068],[31,88840,4587],{"className":88841},[75,1068],[31,88843,1076],{"className":88844},[1075],[31,88846,88848],{"className":88847},[1048],[31,88849,88851],{"className":88850,"style":2050},[1052],[31,88852],{},"，故 ",[31,88855,88857,88888],{"className":88856,"translate":35},[34],[31,88858,88860],{"className":88859},[39],[41,88861,88862],{"xmlns":43},[45,88863,88864,88886],{},[48,88865,88866,88878,88880],{},[1004,88867,88868,88870],{},[51,88869,43481],{},[48,88871,88872,88874,88876],{},[51,88873,87599],{"mathvariant":494},[51,88875,2552],{"mathvariant":494},[51,88877,87604],{"mathvariant":494},[95,88879,347],{},[1004,88881,88882,88884],{},[51,88883,43481],{},[51,88885,4587],{"mathvariant":494},[55,88887,87779],{"encoding":57},[31,88889,88891,88952],{"className":88890,"ariaHidden":62},[61],[31,88892,88894,88897,88943,88946,88949],{"className":88893},[66],[31,88895],{"className":88896,"style":5230},[70],[31,88898,88900,88903],{"className":88899},[75],[31,88901,43481],{"className":88902,"style":120},[75,76],[31,88904,88906],{"className":88905},[1039],[31,88907,88909,88935],{"className":88908},[1043,1044],[31,88910,88912,88932],{"className":88911},[1048],[31,88913,88915],{"className":88914,"style":1053},[1052],[31,88916,88917,88920],{"style":66251},[31,88918],{"className":88919,"style":1061},[1060],[31,88921,88923],{"className":88922},[1065,1066,1067,1068],[31,88924,88926],{"className":88925},[75,1068],[31,88927,88929],{"className":88928},[75,1068],[31,88930,87668],{"className":88931,"style":20180},[75,2745,1068],[31,88933,1076],{"className":88934},[1075],[31,88936,88938],{"className":88937},[1048],[31,88939,88941],{"className":88940,"style":2050},[1052],[31,88942],{},[31,88944],{"className":88945,"style":293},[119],[31,88947,347],{"className":88948},[297],[31,88950],{"className":88951,"style":293},[119],[31,88953,88955,88958],{"className":88954},[66],[31,88956],{"className":88957,"style":5230},[70],[31,88959,88961,88964],{"className":88960},[75],[31,88962,43481],{"className":88963,"style":120},[75,76],[31,88965,88967],{"className":88966},[1039],[31,88968,88970,88990],{"className":88969},[1043,1044],[31,88971,88973,88987],{"className":88972},[1048],[31,88974,88976],{"className":88975,"style":3230},[1052],[31,88977,88978,88981],{"style":66251},[31,88979],{"className":88980,"style":1061},[1060],[31,88982,88984],{"className":88983},[1065,1066,1067,1068],[31,88985,4587],{"className":88986},[75,1068],[31,88988,1076],{"className":88989},[1075],[31,88991,88993],{"className":88992},[1048],[31,88994,88996],{"className":88995,"style":2050},[1052],[31,88997],{},"。上界情形同理，只是迭代序列单调下降。证毕。",[11,89000,89001,89002,89058,89059,89115],{},"这个判据说明，只要能够构造一个满足 ",[31,89003,89005,89025],{"className":89004,"translate":35},[34],[31,89006,89008],{"className":89007},[39],[41,89009,89010],{"xmlns":43},[45,89011,89012,89022],{},[48,89013,89014,89016,89018,89020],{},[51,89015,43481],{},[95,89017,347],{},[51,89019,919],{"mathvariant":925},[51,89021,43481],{},[55,89023,89024],{"encoding":57},"V\\le\\mathcal{T}V",[31,89026,89028,89046],{"className":89027,"ariaHidden":62},[61],[31,89029,89031,89034,89037,89040,89043],{"className":89030},[66],[31,89032],{"className":89033,"style":76406},[70],[31,89035,43481],{"className":89036,"style":120},[75,76],[31,89038],{"className":89039,"style":293},[119],[31,89041,347],{"className":89042},[297],[31,89044],{"className":89045,"style":293},[119],[31,89047,89049,89052,89055],{"className":89048},[66],[31,89050],{"className":89051,"style":945},[70],[31,89053,919],{"className":89054,"style":64444},[75,968],[31,89056,43481],{"className":89057,"style":120},[75,76]," 的下界函数，它就一直位于真实值函数之下。满足 ",[31,89060,89062,89082],{"className":89061,"translate":35},[34],[31,89063,89065],{"className":89064},[39],[41,89066,89067],{"xmlns":43},[45,89068,89069,89079],{},[48,89070,89071,89073,89075,89077],{},[51,89072,43481],{},[95,89074,5662],{},[51,89076,919],{"mathvariant":925},[51,89078,43481],{},[55,89080,89081],{"encoding":57},"V\\ge\\mathcal{T}V",[31,89083,89085,89103],{"className":89084,"ariaHidden":62},[61],[31,89086,89088,89091,89094,89097,89100],{"className":89087},[66],[31,89089],{"className":89090,"style":76406},[70],[31,89092,43481],{"className":89093,"style":120},[75,76],[31,89095],{"className":89096,"style":293},[119],[31,89098,5662],{"className":89099},[297],[31,89101],{"className":89102,"style":293},[119],[31,89104,89106,89109,89112],{"className":89105},[66],[31,89107],{"className":89108,"style":945},[70],[31,89110,919],{"className":89111,"style":64444},[75,968],[31,89113,43481],{"className":89114,"style":120},[75,76]," 的上界函数则一直位于真实值函数之上。这为值函数的区间估计提供了严格的数学基础。",[18,89117,89119],{"id":89118},"四贝尔曼最优方程","四、贝尔曼最优方程",[52860,89121,89123],{"id":89122},"_41-最优值函数","4.1 最优值函数",[52865,89125,89128,89131,89559],{"id":89126,"title":89127},"definition-optimal-value","定义 4.1（最优值函数）",[11,89129,89130],{},"最优状态值函数与最优动作值函数分别定义为",[31,89132,89134],{"className":89133,"translate":35},[411],[31,89135,89137,89235],{"className":89136,"translate":35},[34],[31,89138,89140],{"className":89139},[39],[41,89141,89142],{"xmlns":43,"display":420},[45,89143,89144,89232],{},[48,89145,89146,89152,89154,89156,89158,89160,89170,89176,89178,89180,89182,89184,89186,89192,89194,89196,89198,89200,89202,89204,89214,89220,89222,89224,89226,89228,89230],{},[1180,89147,89148,89150],{},[51,89149,43481],{},[95,89151,30648],{},[95,89153,211],{"stretchy":210},[51,89155,14551],{},[95,89157,220],{"stretchy":210},[95,89159,223],{},[1531,89161,89162,89168],{},[48,89163,89164,89166],{},[51,89165,10792],{},[95,89167,1540],{},[51,89169,2515],{},[1180,89171,89172,89174],{},[51,89173,43481],{},[51,89175,2515],{},[95,89177,211],{"stretchy":210},[51,89179,14551],{},[95,89181,220],{"stretchy":210},[95,89183,1210],{"separator":62},[119,89185],{"width":20286},[1180,89187,89188,89190],{},[51,89189,56971],{},[95,89191,30648],{},[95,89193,211],{"stretchy":210},[51,89195,14551],{},[95,89197,1210],{"separator":62},[51,89199,5813],{},[95,89201,220],{"stretchy":210},[95,89203,223],{},[1531,89205,89206,89212],{},[48,89207,89208,89210],{},[51,89209,10792],{},[95,89211,1540],{},[51,89213,2515],{},[1180,89215,89216,89218],{},[51,89217,56971],{},[51,89219,2515],{},[95,89221,211],{"stretchy":210},[51,89223,14551],{},[95,89225,1210],{"separator":62},[51,89227,5813],{},[95,89229,220],{"stretchy":210},[51,89231,495],{"mathvariant":494},[55,89233,89234],{"encoding":57},"V^*(s)=\\sup_{\\pi}V^\\pi(s),\n\\qquad\nQ^*(s,a)=\\sup_{\\pi}Q^\\pi(s,a).",[31,89236,89238,89291,89453],{"className":89237,"ariaHidden":62},[61],[31,89239,89241,89244,89273,89276,89279,89282,89285,89288],{"className":89240},[66],[31,89242],{"className":89243,"style":260},[70],[31,89245,89247,89250],{"className":89246},[75],[31,89248,43481],{"className":89249,"style":120},[75,76],[31,89251,89253],{"className":89252},[1039],[31,89254,89256],{"className":89255},[1043],[31,89257,89259],{"className":89258},[1048],[31,89260,89262],{"className":89261,"style":74855},[1052],[31,89263,89264,89267],{"style":1330},[31,89265],{"className":89266,"style":1061},[1060],[31,89268,89270],{"className":89269},[1065,1066,1067,1068],[31,89271,30648],{"className":89272},[124,1068],[31,89274,211],{"className":89275},[264],[31,89277,14551],{"className":89278},[75,76],[31,89280,220],{"className":89281},[289],[31,89283],{"className":89284,"style":293},[119],[31,89286,223],{"className":89287},[297],[31,89289],{"className":89290,"style":293},[119],[31,89292,89294,89298,89347,89350,89379,89382,89385,89388,89391,89394,89397,89426,89429,89432,89435,89438,89441,89444,89447,89450],{"className":89293},[66],[31,89295],{"className":89296,"style":89297},[70],"height:1.6444em;vertical-align:-0.8944em;",[31,89299,89301],{"className":89300},[1460,1620],[31,89302,89304,89338],{"className":89303},[1043,1044],[31,89305,89307,89335],{"className":89306},[1048],[31,89308,89310,89325],{"className":89309,"style":71},[1052],[31,89311,89313,89316],{"style":89312},"top:-2.2056em;margin-left:0em;",[31,89314],{"className":89315,"style":1637},[1060],[31,89317,89319],{"className":89318},[1065,1066,1067,1068],[31,89320,89322],{"className":89321},[75,1068],[31,89323,2515],{"className":89324,"style":2666},[75,76,1068],[31,89326,89327,89330],{"style":1649},[31,89328],{"className":89329,"style":1637},[1060],[31,89331,89332],{},[31,89333,10792],{"className":89334},[1460],[31,89336,1076],{"className":89337},[1075],[31,89339,89341],{"className":89340},[1048],[31,89342,89345],{"className":89343,"style":89344},[1052],"height:0.8944em;",[31,89346],{},[31,89348],{"className":89349,"style":1467},[119],[31,89351,89353,89356],{"className":89352},[75],[31,89354,43481],{"className":89355,"style":120},[75,76],[31,89357,89359],{"className":89358},[1039],[31,89360,89362],{"className":89361},[1043],[31,89363,89365],{"className":89364},[1048],[31,89366,89368],{"className":89367,"style":3987},[1052],[31,89369,89370,89373],{"style":1330},[31,89371],{"className":89372,"style":1061},[1060],[31,89374,89376],{"className":89375},[1065,1066,1067,1068],[31,89377,2515],{"className":89378,"style":2666},[75,76,1068],[31,89380,211],{"className":89381},[264],[31,89383,14551],{"className":89384},[75,76],[31,89386,220],{"className":89387},[289],[31,89389,1210],{"className":89390},[1391],[31,89392],{"className":89393,"style":20525},[119],[31,89395],{"className":89396,"style":1467},[119],[31,89398,89400,89403],{"className":89399},[75],[31,89401,56971],{"className":89402},[75,76],[31,89404,89406],{"className":89405},[1039],[31,89407,89409],{"className":89408},[1043],[31,89410,89412],{"className":89411},[1048],[31,89413,89415],{"className":89414,"style":74855},[1052],[31,89416,89417,89420],{"style":1330},[31,89418],{"className":89419,"style":1061},[1060],[31,89421,89423],{"className":89422},[1065,1066,1067,1068],[31,89424,30648],{"className":89425},[124,1068],[31,89427,2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",[31,89563,89565,89583],{"className":89564,"translate":35},[34],[31,89566,89568],{"className":89567},[39],[41,89569,89570],{"xmlns":43},[45,89571,89572,89580],{},[48,89573,89574],{},[1180,89575,89576,89578],{},[51,89577,2515],{},[95,89579,30648],{},[55,89581,89582],{"encoding":57},"\\pi^*",[31,89584,89586],{"className":89585,"ariaHidden":62},[61],[31,89587,89589,89592],{"className":89588},[66],[31,89590],{"className":89591,"style":30719},[70],[31,89593,89595,89598],{"className":89594},[75],[31,89596,2515],{"className":89597,"style":2666},[75,76],[31,89599,89601],{"className":89600},[1039],[31,89602,89604],{"className":89603},[1043],[31,89605,89607],{"className":89606},[1048],[31,89608,89610],{"className":89609,"style":30719},[1052],[31,89611,89612,89615],{"style":3631},[31,89613],{"className":89614,"style":1061},[1060],[31,89616,89618],{"className":89617},[1065,1066,1067,1068],[31,89619,30648],{"className":89620},[124,1068],[31,89622,89624,89654],{"className":89623,"translate":35},[34],[31,89625,89627],{"className":89626},[39],[41,89628,89629],{"xmlns":43},[45,89630,89631,89651],{},[48,89632,89633,89643,89645],{},[1180,89634,89635,89637],{},[51,89636,43481],{},[1180,89638,89639,89641],{},[51,89640,2515],{},[95,89642,30648],{},[95,89644,223],{},[1180,89646,89647,89649],{},[51,89648,43481],{},[95,89650,30648],{},[55,89652,89653],{"encoding":57},"V^{\\pi^*}=V^*",[31,89655,89657,89733],{"className":89656,"ariaHidden":62},[61],[31,89658,89660,89664,89724,89727,89730],{"className":89659},[66],[31,89661],{"className":89662,"style":89663},[70],"height:0.8973em;",[31,89665,89667,89670],{"className":89666},[75],[31,89668,43481],{"className":89669,"style":120},[75,76],[31,89671,89673],{"className":89672},[1039],[31,89674,89676],{"className":89675},[1043],[31,89677,89679],{"className":89678},[1048],[31,89680,89682],{"className":89681,"style":89663},[1052],[31,89683,89684,89687],{"style":3631},[31,89685],{"className":89686,"style":1061},[1060],[31,89688,89690],{"className":89689},[1065,1066,1067,1068],[31,89691,89693],{"className":89692},[75,1068],[31,89694,89696,89699],{"className":89695},[75,1068],[31,89697,2515],{"className":89698,"style":2666},[75,76,1068],[31,89700,89702],{"className":89701},[1039],[31,89703,89705],{"className":89704},[1043],[31,89706,89708],{"className":89707},[1048],[31,89709,89712],{"className":89710,"style":89711},[1052],"height:0.7633em;",[31,89713,89715,89718],{"style":89714},"top:-2.931em;margin-right:0.0714em;",[31,89716],{"className":89717,"style":11769},[1060],[31,89719,89721],{"className":89720},[1065,11773,1308,1068],[31,89722,30648],{"className":89723},[124,1068],[31,89725],{"className":89726,"style":293},[119],[31,89728,223],{"className":89729},[297],[31,89731],{"className":89732,"style":293},[119],[31,89734,89736,89739],{"className":89735},[66],[31,89737],{"className":89738,"style":30719},[70],[31,89740,89742,89745],{"className":89741},[75],[31,89743,43481],{"className":89744,"style":120},[75,76],[31,89746,89748],{"className":89747},[1039],[31,89749,89751],{"className":89750},[1043],[31,89752,89754],{"className":89753},[1048],[31,89755,89757],{"className":89756,"style":30719},[1052],[31,89758,89759,89762],{"style":3631},[31,89760],{"className":89761,"style":1061},[1060],[31,89763,89765],{"className":89764},[1065,1066,1067,1068],[31,89766,30648],{"className":89767},[124,1068],[31,89769,89771,89788],{"className":89770,"translate":35},[34],[31,89772,89774],{"className":89773},[39],[41,89775,89776],{"xmlns":43},[45,89777,89778,89786],{},[48,89779,89780],{},[1180,89781,89782,89784],{},[51,89783,2515],{},[95,89785,30648],{},[55,89787,89582],{"encoding":57},[31,89789,89791],{"className":89790,"ariaHidden":62},[61],[31,89792,89794,89797],{"className":89793},[66],[31,89795],{"className":89796,"style":30719},[70],[31,89798,89800,89803],{"className":89799},[75],[31,89801,2515],{"className":89802,"style":2666},[75,76],[31,89804,89806],{"className":89805},[1039],[31,89807,89809],{"className":89808},[1043],[31,89810,89812],{"className":89811},[1048],[31,89813,89815],{"className":89814,"style":30719},[1052],[31,89816,89817,89820],{"style":3631},[31,89818],{"className":89819,"style":1061},[1060],[31,89821,89823],{"className":89822},[1065,1066,1067,1068],[31,89824,30648],{"className":89825},[124,1068],[7246,89827,89828],{},"最优策略",[11,89830,89831,89832,89836,89837,89868,89869,89900,89901,89929],{},"在有限状态与有限动作的折扣 MDP 中，上确界实际上可以取到，而且存在一个平稳确定性最优策略。这一点将由",[5813,89833,89835],{"href":89834},"#theorem-bellman-optimality","定理 4.2"," 的证明一并给出。因此，本节的 ",[31,89838,89840,89856],{"className":89839,"translate":35},[34],[31,89841,89843],{"className":89842},[39],[41,89844,89845],{"xmlns":43},[45,89846,89847,89853],{},[48,89848,89849,89851],{},[51,89850,10792],{},[95,89852,1540],{},[55,89854,89855],{"encoding":57},"\\sup",[31,89857,89859],{"className":89858,"ariaHidden":62},[61],[31,89860,89862,89865],{"className":89861},[66],[31,89863],{"className":89864,"style":865},[70],[31,89866,10792],{"className":89867},[1460]," 可以理解为 ",[31,89870,89872,89888],{"className":89871,"translate":35},[34],[31,89873,89875],{"className":89874},[39],[41,89876,89877],{"xmlns":43},[45,89878,89879,89885],{},[48,89880,89881,89883],{},[51,89882,10154],{},[95,89884,1540],{},[55,89886,89887],{"encoding":57},"\\max",[31,89889,89891],{"className":89890,"ariaHidden":62},[61],[31,89892,89894,89897],{"className":89893},[66],[31,89895],{"className":89896,"style":71},[70],[31,89898,10154],{"className":89899},[1460],"。本文的 ",[31,89902,89904,89917],{"className":89903,"translate":35},[34],[31,89905,89907],{"className":89906},[39],[41,89908,89909],{"xmlns":43},[45,89910,89911,89915],{},[48,89912,89913],{},[51,89914,2515],{},[55,89916,3485],{"encoding":57},[31,89918,89920],{"className":89919,"ariaHidden":62},[61],[31,89921,89923,89926],{"className":89922},[66],[31,89924],{"className":89925,"style":71},[70],[31,89927,2515],{"className":89928,"style":2666},[75,76]," 均指平稳策略。对于折扣有限 MDP，平稳策略类已经包含最优策略，因此这一限制不失一般性。",[52860,89931,89933],{"id":89932},"_42-最优方程的证明","4.2 最优方程的证明",[11,89935,89936,89937,54797,89940,4809],{},"第 3.1 节已经定义了",[7246,89938,89939],{},"贝尔曼最优算子",[31,89941,89943,89960],{"className":89942,"translate":35},[34],[31,89944,89946],{"className":89945},[39],[41,89947,89948],{"xmlns":43},[45,89949,89950,89958],{},[48,89951,89952],{},[1180,89953,89954,89956],{},[51,89955,919],{"mathvariant":925},[95,89957,30648],{},[55,89959,74501],{"encoding":57},[31,89961,89963],{"className":89962,"ariaHidden":62},[61],[31,89964,89966,89969],{"className":89965},[66],[31,89967],{"className":89968,"style":30719},[70],[31,89970,89972,89975],{"className":89971},[75],[31,89973,919],{"className":89974,"style":64444},[75,968],[31,89976,89978],{"className":89977},[1039],[31,89979,89981],{"className":89980},[1043],[31,89982,89984],{"className":89983},[1048],[31,89985,89987],{"className":89986,"style":30719},[1052],[31,89988,89989,89992],{"style":3631},[31,89990],{"className":89991,"style":1061},[1060],[31,89993,89995],{"className":89994},[1065,1066,1067,1068],[31,89996,30648],{"className":89997},[124,1068],[31,89999,90001],{"className":90000,"translate":35},[411],[31,90002,90004,90119],{"className":90003,"translate":35},[34],[31,90005,90007],{"className":90006},[39],[41,90008,90009],{"xmlns":43,"display":420},[45,90010,90011,90117],{},[48,90012,90013,90015,90021,90023,90025,90027,90029,90031,90033,90043,90053,90055,90057,90063,90065,90067,90069,90071,90073,90115],{},[95,90014,211],{"stretchy":210},[1180,90016,90017,90019],{},[51,90018,919],{"mathvariant":925},[95,90020,30648],{},[51,90022,43481],{},[95,90024,220],{"stretchy":210},[95,90026,211],{"stretchy":210},[51,90028,14551],{},[95,90030,220],{"stretchy":210},[95,90032,223],{},[1531,90034,90035,90041],{},[48,90036,90037,90039],{},[51,90038,10154],{},[95,90040,1540],{},[51,90042,5813],{},[1531,90044,90045,90047],{},[95,90046,5625],{},[1180,90048,90049,90051],{},[51,90050,14551],{},[95,90052,13691],{"mathvariant":494,"lspace":5610,"rspace":5610},[51,90054,2521],{},[95,90056,211],{"stretchy":210},[1180,90058,90059,90061],{},[51,90060,14551],{},[95,90062,13691],{"mathvariant":494,"lspace":5610,"rspace":5610},[95,90064,12680],{},[51,90066,14551],{},[95,90068,1210],{"separator":62},[51,90070,5813],{},[95,90072,220],{"stretchy":210},[48,90074,90075,90077,90079,90081,90083,90085,90087,90089,90095,90097,90099,90101,90103,90105,90111,90113],{},[95,90076,15175],{"fence":62},[51,90078,2549],{},[95,90080,211],{"stretchy":210},[51,90082,14551],{},[95,90084,1210],{"separator":62},[51,90086,5813],{},[95,90088,1210],{"separator":62},[1180,90090,90091,90093],{},[51,90092,14551],{},[95,90094,13691],{"mathvariant":494,"lspace":5610,"rspace":5610},[95,90096,220],{"stretchy":210},[95,90098,434],{},[51,90100,53298],{},[51,90102,43481],{},[95,90104,211],{"stretchy":210},[1180,90106,90107,90109],{},[51,90108,14551],{},[95,90110,13691],{"mathvariant":494,"lspace":5610,"rspace":5610},[95,90112,220],{"stretchy":210},[95,90114,15184],{"fence":62},[51,90116,495],{"mathvariant":494},[55,90118,74824],{"encoding":57},[31,90120,90122,90184,90366],{"className":90121,"ariaHidden":62},[61],[31,90123,90125,90128,90131,90160,90163,90166,90169,90172,90175,90178,90181],{"className":90124},[66],[31,90126],{"className":90127,"style":260},[70],[31,90129,211],{"className":90130},[264],[31,90132,90134,90137],{"className":90133},[75],[31,90135,919],{"className":90136,"style":64444},[75,968],[31,90138,90140],{"className":90139},[1039],[31,90141,90143],{"className":90142},[1043],[31,90144,90146],{"className":90145},[1048],[31,90147,90149],{"className":90148,"style":74855},[1052],[31,90150,90151,90154],{"style":1330},[31,90152],{"className":90153,"style":1061},[1060],[31,90155,90157],{"className":90156},[1065,1066,1067,1068],[31,90158,30648],{"className":90159},[124,1068],[31,90161,43481],{"className":90162,"style":120},[75,76],[31,90164,220],{"className":90165},[289],[31,90167,211],{"className":90168},[264],[31,90170,14551],{"className":90171},[75,76],[31,90173,220],{"className":90174},[289],[31,90176],{"className":90177,"style":293},[119],[31,90179,223],{"className":90180},[297],[31,90182],{"className":90183,"style":293},[119],[31,90185,90187,90190,90237,90240,90316,90319,90322,90325,90357,90360,90363],{"className":90186},[66],[31,90188],{"className":90189,"style":74897},[70],[31,90191,90193],{"className":90192},[1460,1620],[31,90194,90196,90229],{"className":90195},[1043,1044],[31,90197,90199,90226],{"className":90198},[1048],[31,90200,90202,90216],{"className":90201,"style":71},[1052],[31,90203,90204,90207],{"style":37361},[31,90205],{"className":90206,"style":1637},[1060],[31,90208,90210],{"className":90209},[1065,1066,1067,1068],[31,90211,90213],{"className":90212},[75,1068],[31,90214,5813],{"className":90215},[75,76,1068],[31,90217,90218,90221],{"style":1649},[31,90219],{"className":90220,"style":1637},[1060],[31,90222,90223],{},[31,90224,10154],{"className":90225},[1460],[31,90227,1076],{"className":90228},[1075],[31,90230,90232],{"className":90231},[1048],[31,90233,90235],{"className":90234,"style":37390},[1052],[31,90236],{},[31,90238],{"className":90239,"style":1467},[119],[31,90241,90243],{"className":90242},[1460,1620],[31,90244,90246,90308],{"className":90245},[1043,1044],[31,90247,90249,90305],{"className":90248},[1048],[31,90250,90252,90295],{"className":90251,"style":5900},[1052],[31,90253,90254,90257],{"style":59756},[31,90255],{"className":90256,"style":5907},[1060],[31,90258,90260],{"className":90259},[1065,1066,1067,1068],[31,90261,90263],{"className":90262},[75,1068],[31,90264,90266,90269],{"className":90265},[75,1068],[31,90267,14551],{"className":90268},[75,76,1068],[31,90270,90272],{"className":90271},[1039],[31,90273,90275],{"className":90274},[1043],[31,90276,90278],{"className":90277},[1048],[31,90279,90281],{"className":90280,"style":31478},[1052],[31,90282,90283,90286],{"style":31481},[31,90284],{"className":90285,"style":11769},[1060],[31,90287,90289],{"className":90288},[1065,11773,1308,1068],[31,90290,90292],{"className":90291},[75,1068],[31,90293,13691],{"className":90294},[75,1068],[31,90296,90297,90300],{"style":5919},[31,90298],{"className":90299,"style":5907},[1060],[31,90301,90302],{},[31,90303,5625],{"className":90304},[1460,1461,1687],[31,90306,1076],{"className":90307},[1075],[31,90309,90311],{"className":90310},[1048],[31,90312,90314],{"className":90313,"style":59817},[1052],[31,90315],{},[31,90317],{"className":90318,"style":1467},[119],[31,90320,2521],{"className":90321,"style":949},[75,76],[31,90323,211],{"className":90324},[264],[31,90326,90328,90331],{"className":90327},[75],[31,90329,14551],{"className":90330},[75,76],[31,90332,90334],{"className":90333},[1039],[31,90335,90337],{"className":90336},[1043],[31,90338,90340],{"className":90339},[1048],[31,90341,90343],{"className":90342,"style":14017},[1052],[31,90344,90345,90348],{"style":1330},[31,90346],{"className":90347,"style":1061},[1060],[31,90349,90351],{"className":90350},[1065,1066,1067,1068],[31,90352,90354],{"className":90353},[75,1068],[31,90355,13691],{"className":90356},[75,1068],[31,90358],{"className":90359,"style":293},[119],[31,90361,12680],{"className":90362},[297],[31,90364],{"className":90365,"style":293},[119],[31,90367,90369,90372,90375,90378,90381,90384,90387,90390,90511,90514],{"className":90368},[66],[31,90370],{"className":90371,"style":75080},[70],[31,90373,14551],{"className":90374},[75,76],[31,90376,1210],{"className":90377},[1391],[31,90379],{"className":90380,"style":1467},[119],[31,90382,5813],{"className":90383},[75,76],[31,90385,220],{"className":90386},[289],[31,90388],{"className":90389,"style":1467},[119],[31,90391,90393,90396,90399,90402,90405,90408,90411,90414,90417,90420,90452,90455,90458,90461,90464,90467,90470,90473,90505,90508],{"className":90392},[3860],[31,90394,15175],{"className":90395,"style":3868},[264,3867],[31,90397,2549],{"className":90398,"style":19759},[75,76],[31,90400,211],{"className":90401},[264],[31,90403,14551],{"className":90404},[75,76],[31,90406,1210],{"className":90407},[1391],[31,90409],{"className":90410,"style":1467},[119],[31,90412,5813],{"className":90413},[75,76],[31,90415,1210],{"className":90416},[1391],[31,90418],{"className":90419,"style":1467},[119],[31,90421,90423,90426],{"className":90422},[75],[31,90424,14551],{"className":90425},[75,76],[31,90427,90429],{"className":90428},[1039],[31,90430,90432],{"className":90431},[1043],[31,90433,90435],{"className":90434},[1048],[31,90436,90438],{"className":90437,"style":14017},[1052],[31,90439,90440,90443],{"style":1330},[31,90441],{"className":90442,"style":1061},[1060],[31,90444,90446],{"className":90445},[1065,1066,1067,1068],[31,90447,90449],{"className":90448},[75,1068],[31,90450,13691],{"className":90451},[75,1068],[31,90453,220],{"className":90454},[289],[31,90456],{"className":90457,"style":120},[119],[31,90459,434],{"className":90460},[124],[31,90462],{"className":90463,"style":120},[119],[31,90465,53298],{"className":90466,"style":24157},[75,76],[31,90468,43481],{"className":90469,"style":120},[75,76],[31,90471,211],{"className":90472},[264],[31,90474,90476,90479],{"className":90475},[75],[31,90477,14551],{"className":90478},[75,76],[31,90480,90482],{"className":90481},[1039],[31,90483,90485],{"className":90484},[1043],[31,90486,90488],{"className":90487},[1048],[31,90489,90491],{"className":90490,"style":14017},[1052],[31,90492,90493,90496],{"style":1330},[31,90494],{"className":90495,"style":1061},[1060],[31,90497,90499],{"className":90498},[1065,1066,1067,1068],[31,90500,90502],{"className":90501},[75,1068],[31,90503,13691],{"className":90504},[75,1068],[31,90506,220],{"className":90507},[289],[31,90509,15184],{"className":90510,"style":3868},[289,3867],[31,90512],{"className":90513,"style":1467},[119],[31,90515,495],{"className":90516},[75],[11,90518,90519,90520,90578,90579,90637],{},"与 ",[31,90521,90523,90540],{"className":90522,"translate":35},[34],[31,90524,90526],{"className":90525},[39],[41,90527,90528],{"xmlns":43},[45,90529,90530,90538],{},[48,90531,90532],{},[1180,90533,90534,90536],{},[51,90535,919],{"mathvariant":925},[51,90537,2515],{},[55,90539,64428],{"encoding":57},[31,90541,90543],{"className":90542,"ariaHidden":62},[61],[31,90544,90546,90549],{"className":90545},[66],[31,90547],{"className":90548,"style":945},[70],[31,90550,90552,90555],{"className":90551},[75],[31,90553,919],{"className":90554,"style":64444},[75,968],[31,90556,90558],{"className":90557},[1039],[31,90559,90561],{"className":90560},[1043],[31,90562,90564],{"className":90563},[1048],[31,90565,90567],{"className":90566,"style":3628},[1052],[31,90568,90569,90572],{"style":3631},[31,90570],{"className":90571,"style":1061},[1060],[31,90573,90575],{"className":90574},[1065,1066,1067,1068],[31,90576,2515],{"className":90577,"style":2666},[75,76,1068]," 不同，",[31,90580,90582,90599],{"className":90581,"translate":35},[34],[31,90583,90585],{"className":90584},[39],[41,90586,90587],{"xmlns":43},[45,90588,90589,90597],{},[48,90590,90591],{},[1180,90592,90593,90595],{},[51,90594,919],{"mathvariant":925},[95,90596,30648],{},[55,90598,74501],{"encoding":57},[31,90600,90602],{"className":90601,"ariaHidden":62},[61],[31,90603,90605,90608],{"className":90604},[66],[31,90606],{"className":90607,"style":30719},[70],[31,90609,90611,90614],{"className":90610},[75],[31,90612,919],{"className":90613,"style":64444},[75,968],[31,90615,90617],{"className":90616},[1039],[31,90618,90620],{"className":90619},[1043],[31,90621,90623],{"className":90622},[1048],[31,90624,90626],{"className":90625,"style":30719},[1052],[31,90627,90628,90631],{"style":3631},[31,90629],{"className":90630,"style":1061},[1060],[31,90632,90634],{"className":90633},[1065,1066,1067,1068],[31,90635,30648],{"className":90636},[124,1068]," 先对每个动作计算期望，再在动作上取最大值。这个顺序不能交换，原因见 4.3 节。",[52865,90639,90642,90876,91508,91598],{"id":90640,"title":90641},"theorem-bellman-optimality","定理 4.2（贝尔曼最优方程）",[11,90643,52871,90644,1138,90717,18171,90745,90773,90774,4880,90832,90875],{},[31,90645,90647,90672],{"className":90646,"translate":35},[34],[31,90648,90650],{"className":90649},[39],[41,90651,90652],{"xmlns":43},[45,90653,90654,90670],{},[48,90655,90656,90658,90660,90662,90664,90666,90668],{},[51,90657,53298],{},[95,90659,2518],{},[95,90661,15175],{"stretchy":210},[91,90663,1977],{},[95,90665,1210],{"separator":62},[91,90667,205],{},[95,90669,220],{"stretchy":210},[55,90671,53313],{"encoding":57},[31,90673,90675,90693],{"className":90674,"ariaHidden":62},[61],[31,90676,90678,90681,90684,90687,90690],{"className":90677},[66],[31,90679],{"className":90680,"style":53323},[70],[31,90682,53298],{"className":90683,"style":24157},[75,76],[31,90685],{"className":90686,"style":293},[119],[31,90688,2518],{"className":90689},[297],[31,90691],{"className":90692,"style":293},[119],[31,90694,90696,90699,90702,90705,90708,90711,90714],{"className":90695},[66],[31,90697],{"className":90698,"style":260},[70],[31,90700,15175],{"className":90701},[264],[31,90703,1977],{"className":90704},[75],[31,90706,1210],{"className":90707},[1391],[31,90709],{"className":90710,"style":1467},[119],[31,90712,205],{"className":90713},[75],[31,90715,220],{"className":90716},[289],[31,90718,90720,90733],{"className":90719,"translate":35},[34],[31,90721,90723],{"className":90722},[39],[41,90724,90725],{"xmlns":43},[45,90726,90727,90731],{},[48,90728,90729],{},[51,90730,21582],{},[55,90732,21582],{"encoding":57},[31,90734,90736],{"className":90735,"ariaHidden":62},[61],[31,90737,90739,90742],{"className":90738},[66],[31,90740],{"className":90741,"style":945},[70],[31,90743,21582],{"className":90744,"style":21697},[75,76],[31,90746,90748,90761],{"className":90747,"translate":35},[34],[31,90749,90751],{"className":90750},[39],[41,90752,90753],{"xmlns":43},[45,90754,90755,90759],{},[48,90756,90757],{},[51,90758,18446],{},[55,90760,18446],{"encoding":57},[31,90762,90764],{"className":90763,"ariaHidden":62},[61],[31,90765,90767,90770],{"className":90766},[66],[31,90768],{"className":90769,"style":945},[70],[31,90771,18446],{"className":90772},[75,76]," 有限。则 ",[31,90775,90777,90794],{"className":90776,"translate":35},[34],[31,90778,90780],{"className":90779},[39],[41,90781,90782],{"xmlns":43},[45,90783,90784,90792],{},[48,90785,90786],{},[1180,90787,90788,90790],{},[51,90789,43481],{},[95,90791,30648],{},[55,90793,87313],{"encoding":57},[31,90795,90797],{"className":90796,"ariaHidden":62},[61],[31,90798,90800,90803],{"className":90799},[66],[31,90801],{"className":90802,"style":30719},[70],[31,90804,90806,90809],{"className":90805},[75],[31,90807,43481],{"className":90808,"style":120},[75,76],[31,90810,90812],{"className":90811},[1039],[31,90813,90815],{"className":90814},[1043],[31,90816,90818],{"className":90817},[1048],[31,90819,90821],{"className":90820,"style":30719},[1052],[31,90822,90823,90826],{"style":3631},[31,90824],{"className":90825,"style":1061},[1060],[31,90827,90829],{"className":90828},[1065,1066,1067,1068],[31,90830,30648],{"className":90831},[124,1068],[31,90833,90835,90854],{"className":90834,"translate":35},[34],[31,90836,90838],{"className":90837},[39],[41,90839,90840],{"xmlns":43},[45,90841,90842,90852],{},[48,90843,90844,90846,90848,90850],{},[51,90845,1122],{},[95,90847,211],{"stretchy":210},[51,90849,21582],{},[95,90851,220],{"stretchy":210},[55,90853,73997],{"encoding":57},[31,90855,90857],{"className":90856,"ariaHidden":62},[61],[31,90858,90860,90863,90866,90869,90872],{"className":90859},[66],[31,90861],{"className":90862,"style":260},[70],[31,90864,1122],{"className":90865,"style":1137},[75,76],[31,90867,211],{"className":90868},[264],[31,90870,21582],{"className":90871,"style":21697},[75,76],[31,90873,220],{"className":90874},[289]," 中满足",[31,90877,90879],{"className":90878,"translate":35},[411],[31,90880,90882,91034],{"className":90881,"translate":35},[34],[31,90883,90885],{"className":90884},[39],[41,90886,90887],{"xmlns":43,"display":420},[45,90888,90889,91031],{},[5606,90890,90891,90957],{"rowspacing":5608,"columnalign":5609,"columnspacing":5610},[5612,90892,90893,90907],{},[5615,90894,90895],{},[2071,90896,90897],{"scriptlevel":1977,"displaystyle":62},[48,90898,90899,90901,90903,90905],{},[51,90900,43481],{},[95,90902,211],{"stretchy":210},[51,90904,14551],{},[95,90906,220],{"stretchy":210},[5615,90908,90909],{},[2071,90910,90911],{"scriptlevel":1977,"displaystyle":62},[48,90912,90913,90915,90917,90927,90937,90939,90941,90947,90949,90951,90953,90955],{},[48,90914],{},[95,90916,223],{},[1531,90918,90919,90925],{},[48,90920,90921,90923],{},[51,90922,10154],{},[95,90924,1540],{},[51,90926,5813],{},[1531,90928,90929,90931],{},[95,90930,5625],{},[1180,90932,90933,90935],{},[51,90934,14551],{},[95,90936,13691],{"mathvariant":494,"lspace":5610,"rspace":5610},[51,90938,2521],{},[95,90940,211],{"stretchy":210},[1180,90942,90943,90945],{},[51,90944,14551],{},[95,90946,13691],{"mathvariant":494,"lspace":5610,"rspace":5610},[95,90948,12680],{},[51,90950,14551],{},[95,90952,1210],{"separator":62},[51,90954,5813],{},[95,90956,220],{"stretchy":210},[5612,90958,90959,90965],{},[5615,90960,90961],{},[2071,90962,90963],{"scriptlevel":1977,"displaystyle":62},[48,90964],{},[5615,90966,90967],{},[2071,90968,90969],{"scriptlevel":1977,"displaystyle":62},[48,90970,90971,90973,90975,90977,91019,91021,91023,91025,91027,91029],{},[48,90972],{},[119,90974],{"width":1584},[95,90976,2528],{},[48,90978,90979,90981,90983,90985,90987,90989,90991,90993,90999,91001,91003,91005,91007,91009,91015,91017],{},[95,90980,15175],{"fence":62},[51,90982,2549],{},[95,90984,211],{"stretchy":210},[51,90986,14551],{},[95,90988,1210],{"separator":62},[51,90990,5813],{},[95,90992,1210],{"separator":62},[1180,90994,90995,90997],{},[51,90996,14551],{},[95,90998,13691],{"mathvariant":494,"lspace":5610,"rspace":5610},[95,91000,220],{"stretchy":210},[95,91002,434],{},[51,91004,53298],{},[51,91006,43481],{},[95,91008,211],{"stretchy":210},[1180,91010,91011,91013],{},[51,91012,14551],{},[95,91014,13691],{"mathvariant":494,"lspace":5610,"rspace":5610},[95,91016,220],{"stretchy":210},[95,91018,15184],{"fence":62},[95,91020,1210],{"separator":62},[119,91022],{"width":20286},[51,91024,62907],{"mathvariant":494},[51,91026,14551],{},[95,91028,2518],{},[51,91030,21582],{},[55,91032,91033],{"encoding":57},"\\begin{aligned}\nV(s)&=\\max_{a}\\sum_{s'}P(s'\\mid s,a)\\\\\n&\\quad\\times\\left[r(s,a,s')+\\gamma V(s')\\right],\n\\qquad \\forall s\\in S\n\\end{aligned}",[31,91035,91037],{"className":91036,"ariaHidden":62},[61],[31,91038,91040,91043],{"className":91039},[66],[31,91041],{"className":91042,"style":59613},[70],[31,91044,91046],{"className":91045},[75],[31,91047,91049,91100],{"className":91048},[5606],[31,91050,91052],{"className":91051},[5867],[31,91053,91055,91092],{"className":91054},[1043,1044],[31,91056,91058,91089],{"className":91057},[1048],[31,91059,91061,91081],{"className":91060,"style":59632},[1052],[31,91062,91063,91066],{"style":59635},[31,91064],{"className":91065,"style":5907},[1060],[31,91067,91069,91072,91075,91078],{"className":91068},[75],[31,91070,43481],{"className":91071,"style":120},[75,76],[31,91073,211],{"className":91074},[264],[31,91076,14551],{"className":91077},[75,76],[31,91079,220],{"className":91080},[289],[31,91082,91083,91086],{"style":59691},[31,91084],{"className":91085,"style":5907},[1060],[31,91087],{"className":91088},[75],[31,91090,1076],{"className":91091},[1075],[31,91093,91095],{"className":91094},[1048],[31,91096,91098],{"className":91097,"style":59707},[1052],[31,91099],{},[31,91101,91103],{"className":91102},[6062],[31,91104,91106,91500],{"className":91105},[1043,1044],[31,91107,91109,91497],{"className":91108},[1048],[31,91110,91112,91323],{"className":91111,"style":59632},[1052],[31,91113,91114,91117],{"style":59635},[31,91115],{"className":91116,"style":5907},[1060],[31,91118,91120,91123,91126,91129,91132,91179,91182,91258,91261,91264,91267,91299,91302,91305,91308,91311,91314,91317,91320],{"className":91119},[75],[31,91121],{"className":91122},[75],[31,91124],{"className":91125,"style":293},[119],[31,91127,223],{"className":91128},[297],[31,91130],{"className":91131,"style":293},[119],[31,91133,91135],{"className":91134},[1460,1620],[31,91136,91138,91171],{"className":91137},[1043,1044],[31,91139,91141,91168],{"className":91140},[1048],[31,91142,91144,91158],{"className":91143,"style":71},[1052],[31,91145,91146,91149],{"style":37361},[31,91147],{"className":91148,"style":1637},[1060],[31,91150,91152],{"className":91151},[1065,1066,1067,1068],[31,91153,91155],{"className":91154},[75,1068],[31,91156,5813],{"className":91157},[75,76,1068],[31,91159,91160,91163],{"style":1649},[31,91161],{"className":91162,"style":1637},[1060],[31,91164,91165],{},[31,91166,10154],{"className":91167},[1460],[31,91169,1076],{"className":91170},[1075],[31,91172,91174],{"className":91173},[1048],[31,91175,91177],{"className":91176,"style":37390},[1052],[31,91178],{},[31,91180],{"className":91181,"style":1467},[119],[31,91183,91185],{"className":91184},[1460,1620],[31,91186,91188,91250],{"className":91187},[1043,1044],[31,91189,91191,91247],{"className":91190},[1048],[31,91192,91194,91237],{"className":91193,"style":5900},[1052],[31,91195,91196,91199],{"style":59756},[31,91197],{"className":91198,"style":5907},[1060],[31,91200,91202],{"className":91201},[1065,1066,1067,1068],[31,91203,91205],{"className":91204},[75,1068],[31,91206,91208,91211],{"className":91207},[75,1068],[31,91209,14551],{"className":91210},[75,76,1068],[31,91212,91214],{"className":91213},[1039],[31,91215,91217],{"className":91216},[1043],[31,91218,91220],{"className":91219},[1048],[31,91221,91223],{"className":91222,"style":31478},[1052],[31,91224,91225,91228],{"style":31481},[31,91226],{"className":91227,"style":11769},[1060],[31,91229,91231],{"className":91230},[1065,11773,1308,1068],[31,91232,91234],{"className":91233},[75,1068],[31,91235,13691],{"className":91236},[75,1068],[31,91238,91239,91242],{"style":5919},[31,91240],{"className":91241,"style":5907},[1060],[31,91243,91244],{},[31,91245,5625],{"className":91246},[1460,1461,1687],[31,91248,1076],{"className":91249},[1075],[31,91251,91253],{"className":91252},[1048],[31,91254,91256],{"className":91255,"style":59817},[1052],[31,91257],{},[31,91259],{"className":91260,"style":1467},[119],[31,91262,2521],{"className":91263,"style":949},[75,76],[31,91265,211],{"className":91266},[264],[31,91268,91270,91273],{"className":91269},[75],[31,91271,14551],{"className":91272},[75,76],[31,91274,91276],{"className":91275},[1039],[31,91277,91279],{"className":91278},[1043],[31,91280,91282],{"className":91281},[1048],[31,91283,91285],{"className":91284,"style":14017},[1052],[31,91286,91287,91290],{"style":1330},[31,91288],{"className":91289,"style":1061},[1060],[31,91291,91293],{"className":91292},[1065,1066,1067,1068],[31,91294,91296],{"className":91295},[75,1068],[31,91297,13691],{"className":91298},[75,1068],[31,91300],{"className":91301,"style":293},[119],[31,91303,12680],{"className":91304},[297],[31,91306],{"className":91307,"style":293},[119],[31,91309,14551],{"className":91310},[75,76],[31,91312,1210],{"className":91313},[1391],[31,91315],{"className":91316,"style":1467},[119],[31,91318,5813],{"className":91319},[75,76],[31,91321,220],{"className":91322},[289],[31,91324,91325,91328],{"style":59691},[31,91326],{"className":91327,"style":5907},[1060],[31,91329,91331,91334,91337,91340,91343,91346,91467,91470,91473,91476,91479,91482,91485,91488,91491,91494],{"className":91330},[75],[31,91332],{"className":91333},[75],[31,91335],{"className":91336,"style":1789},[119],[31,91338],{"className":91339,"style":120},[119],[31,91341,2528],{"className":91342},[124],[31,91344],{"className":91345,"style":120},[119],[31,91347,91349,91352,91355,91358,91361,91364,91367,91370,91373,91376,91408,91411,91414,91417,91420,91423,91426,91429,91461,91464],{"className":91348},[3860],[31,91350,15175],{"className":91351,"style":3868},[264,3867],[31,91353,2549],{"className":91354,"style":19759},[75,76],[31,91356,211],{"className":91357},[264],[31,91359,14551],{"className":91360},[75,76],[31,91362,1210],{"className":91363},[1391],[31,91365],{"className":91366,"style":1467},[119],[31,91368,5813],{"className":91369},[75,76],[31,91371,1210],{"className":91372},[1391],[31,91374],{"className":91375,"style":1467},[119],[31,91377,91379,91382],{"className":91378},[75],[31,91380,14551],{"className":91381},[75,76],[31,91383,91385],{"className":91384},[1039],[31,91386,91388],{"className":91387},[1043],[31,91389,91391],{"className":91390},[1048],[31,91392,91394],{"className":91393,"style":14017},[1052],[31,91395,91396,91399],{"style":1330},[31,91397],{"className":91398,"style":1061},[1060],[31,91400,91402],{"className":91401},[1065,1066,1067,1068],[31,91403,91405],{"className":91404},[75,1068],[31,91406,13691],{"className":91407},[75,1068],[31,91409,220],{"className":91410},[289],[31,91412],{"className":91413,"style":120},[119],[31,91415,434],{"className":91416},[124],[31,91418],{"className":91419,"style":120},[119],[31,91421,53298],{"className":91422,"style":24157},[75,76],[31,91424,43481],{"className":91425,"style":120},[75,76],[31,91427,211],{"className":91428},[264],[31,91430,91432,91435],{"className":91431},[75],[31,91433,14551],{"className":91434},[75,76],[31,91436,91438],{"className":91437},[1039],[31,91439,91441],{"className":91440},[1043],[31,91442,91444],{"className":91443},[1048],[31,91445,91447],{"className":91446,"style":14017},[1052],[31,91448,91449,91452],{"style":1330},[31,91450],{"className":91451,"style":1061},[1060],[31,91453,91455],{"className":91454},[1065,1066,1067,1068],[31,91456,91458],{"className":91457},[75,1068],[31,91459,13691],{"className":91460},[75,1068],[31,91462,220],{"className":91463},[289],[31,91465,15184],{"className":91466,"style":3868},[289,3867],[31,91468],{"className":91469,"style":1467},[119],[31,91471,1210],{"className":91472},[1391],[31,91474],{"className":91475,"style":20525},[119],[31,91477],{"className":91478,"style":1467},[119],[31,91480,62907],{"className":91481},[75],[31,91483,14551],{"className":91484},[75,76],[31,91486],{"className":91487,"style":293},[119],[31,91489,2518],{"className":91490},[297],[31,91492],{"className":91493,"style":293},[119],[31,91495,21582],{"className":91496,"style":21697},[75,76],[31,91498,1076],{"className":91499},[1075],[31,91501,91503],{"className":91502},[1048],[31,91504,91506],{"className":91505,"style":59707},[1052],[31,91507],{},[11,91509,91510,91511,91569,91570,1138],{},"的唯一有界解。此外，确定性策略 ",[31,91512,91514,91531],{"className":91513,"translate":35},[34],[31,91515,91517],{"className":91516},[39],[41,91518,91519],{"xmlns":43},[45,91520,91521,91529],{},[48,91522,91523],{},[1180,91524,91525,91527],{},[51,91526,2515],{},[95,91528,30648],{},[55,91530,89582],{"encoding":57},[31,91532,91534],{"className":91533,"ariaHidden":62},[61],[31,91535,91537,91540],{"className":91536},[66],[31,91538],{"className":91539,"style":30719},[70],[31,91541,91543,91546],{"className":91542},[75],[31,91544,2515],{"className":91545,"style":2666},[75,76],[31,91547,91549],{"className":91548},[1039],[31,91550,91552],{"className":91551},[1043],[31,91553,91555],{"className":91554},[1048],[31,91556,91558],{"className":91557,"style":30719},[1052],[31,91559,91560,91563],{"style":3631},[31,91561],{"className":91562,"style":1061},[1060],[31,91564,91566],{"className":91565},[1065,1066,1067,1068],[31,91567,30648],{"className":91568},[124,1068]," 最优当且仅当对每个 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s,a)\n\\left[r(s,a,s')+\\gamma V^*(s')\\right].",[31,91723,91725,91778,91968],{"className":91724,"ariaHidden":62},[61],[31,91726,91728,91731,91760,91763,91766,91769,91772,91775],{"className":91727},[66],[31,91729],{"className":91730,"style":260},[70],[31,91732,91734,91737],{"className":91733},[75],[31,91735,2515],{"className":91736,"style":2666},[75,76],[31,91738,91740],{"className":91739},[1039],[31,91741,91743],{"className":91742},[1043],[31,91744,91746],{"className":91745},[1048],[31,91747,91749],{"className":91748,"style":74855},[1052],[31,91750,91751,91754],{"style":1330},[31,91752],{"className":91753,"style":1061},[1060],[31,91755,91757],{"className":91756},[1065,1066,1067,1068],[31,91758,30648],{"className":91759},[124,1068],[31,91761,211],{"className":91762},[264],[31,91764,14551],{"className":91765},[75,76],[31,91767,220],{"className":91768},[289],[31,91770],{"className":91771,"style":293},[119],[31,91773,2518],{"className":91774},[297],[31,91776],{"className":91777,"style":293},[119],[31,91779,91781,91784,91789,91792,91839,91842,91918,91921,91924,91927,91959,91962,91965],{"className":91780},[66],[31,91782],{"className":91783,"style":74897},[70],[31,91785,37341,91787],{"className":91786},[1460],[31,91788,14556],{"style":20180},[31,91790],{"className":91791,"style":1467},[119],[31,91793,91795],{"className":91794},[1460,1620],[31,91796,91798,91831],{"className":91797},[1043,1044],[31,91799,91801,91828],{"className":91800},[1048],[31,91802,91804,91818],{"className":91803,"style":71},[1052],[31,91805,91806,91809],{"style":37361},[31,91807],{"className":91808,"style":1637},[1060],[31,91810,91812],{"className":91811},[1065,1066,1067,1068],[31,91813,91815],{"className":91814},[75,1068],[31,91816,5813],{"className":91817},[75,76,1068],[31,91819,91820,91823],{"style":1649},[31,91821],{"className":91822,"style":1637},[1060],[31,91824,91825],{},[31,91826,10154],{"className":91827},[1460],[31,91829,1076],{"className":91830},[1075],[31,91832,91834],{"className":91833},[1048],[31,91835,91837],{"className":91836,"style":37390},[1052],[31,91838],{},[31,91840],{"className":91841,"style":1467},[119],[31,91843,91845],{"className":91844},[1460,1620],[31,91846,91848,91910],{"className":91847},[1043,1044],[31,91849,91851,91907],{"className":91850},[1048],[31,91852,91854,91897],{"className":91853,"style":5900},[1052],[31,91855,91856,91859],{"style":59756},[31,91857],{"className":91858,"style":5907},[1060],[31,91860,91862],{"className":91861},[1065,1066,1067,1068],[31,91863,91865],{"className":91864},[75,1068],[31,91866,91868,91871],{"className":91867},[75,1068],[31,91869,14551],{"className":91870},[75,76,1068],[31,91872,91874],{"className":91873},[1039],[31,91875,91877],{"className":91876},[1043],[31,91878,91880],{"className":91879},[1048],[31,91881,91883],{"className":91882,"style":31478},[1052],[31,91884,91885,91888],{"style":31481},[31,91886],{"className":91887,"style":11769},[1060],[31,91889,91891],{"className":91890},[1065,11773,1308,1068],[31,91892,91894],{"className":91893},[75,1068],[31,91895,13691],{"className":91896},[75,1068],[31,91898,91899,91902],{"style":5919},[31,91900],{"className":91901,"style":5907},[1060],[31,91903,91904],{},[31,91905,5625],{"className":91906},[1460,1461,1687],[31,91908,1076],{"className":91909},[1075],[31,91911,91913],{"className":91912},[1048],[31,91914,91916],{"className":91915,"style":59817},[1052],[31,91917],{},[31,91919],{"className":91920,"style":1467},[119],[31,91922,2521],{"className":91923,"style":949},[75,76],[31,91925,211],{"className":91926},[264],[31,91928,91930,91933],{"className":91929},[75],[31,91931,14551],{"className":91932},[75,76],[31,91934,91936],{"className":91935},[1039],[31,91937,91939],{"className":91938},[1043],[31,91940,91942],{"className":91941},[1048],[31,91943,91945],{"className":91944,"style":14017},[1052],[31,91946,91947,91950],{"style":1330},[31,91948],{"className":91949,"style":1061},[1060],[31,91951,91953],{"className":91952},[1065,1066,1067,1068],[31,91954,91956],{"className":91955},[75,1068],[31,91957,13691],{"className":91958},[75,1068],[31,91960],{"className":91961,"style":293},[119],[31,91963,12680],{"className":91964},[297],[31,91966],{"className":91967,"style":293},[119],[31,91969,91971,91974,91977,91980,91983,91986,91989,91992,92139,92142],{"className":91970},[66],[31,91972],{"className":91973,"style":75080},[70],[31,91975,14551],{"className":91976},[75,76],[31,91978,1210],{"className":91979},[1391],[31,91981],{"className":91982,"style":1467},[119],[31,91984,5813],{"className":91985},[75,76],[31,91987,220],{"className":91988},[289],[31,91990],{"className":91991,"style":1467},[119],[31,91993,91995,91998,92001,92004,92007,92010,92013,92016,92019,92022,92054,92057,92060,92063,92066,92069,92098,92101,92133,92136],{"className":91994},[3860],[31,91996,15175],{"className":91997,"style":3868},[264,3867],[31,91999,2549],{"className":92000,"style":19759},[75,76],[31,92002,211],{"className":92003},[264],[31,92005,14551],{"className":92006},[75,76],[31,92008,1210],{"className":92009},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3.4",[31,92154,92156,92173],{"className":92155,"translate":35},[34],[31,92157,92159],{"className":92158},[39],[41,92160,92161],{"xmlns":43},[45,92162,92163,92171],{},[48,92164,92165],{},[1180,92166,92167,92169],{},[51,92168,919],{"mathvariant":925},[95,92170,30648],{},[55,92172,74501],{"encoding":57},[31,92174,92176],{"className":92175,"ariaHidden":62},[61],[31,92177,92179,92182],{"className":92178},[66],[31,92180],{"className":92181,"style":30719},[70],[31,92183,92185,92188],{"className":92184},[75],[31,92186,919],{"className":92187,"style":64444},[75,968],[31,92189,92191],{"className":92190},[1039],[31,92192,92194],{"className":92193},[1043],[31,92195,92197],{"className":92196},[1048],[31,92198,92200],{"className":92199,"style":30719},[1052],[31,92201,92202,92205],{"style":3631},[31,92203],{"className":92204,"style":1061},[1060],[31,92206,92208],{"className":92207},[1065,1066,1067,1068],[31,92209,30648],{"className":92210},[124,1068],[31,92212,92214,92233],{"className":92213,"translate":35},[34],[31,92215,92217],{"className":92216},[39],[41,92218,92219],{"xmlns":43},[45,92220,92221,92231],{},[48,92222,92223,92225,92227,92229],{},[51,92224,1122],{},[95,92226,211],{"stretchy":210},[51,92228,21582],{},[95,92230,220],{"stretchy":210},[55,92232,73997],{"encoding":57},[31,92234,92236],{"className":92235,"ariaHidden":62},[61],[31,92237,92239,92242,92245,92248,92251],{"className":92238},[66],[31,92240],{"className":92241,"style":260},[70],[31,92243,1122],{"className":92244,"style":1137},[75,76],[31,92246,211],{"className":92247},[264],[31,92249,21582],{"className":92250,"style":21697},[75,76],[31,92252,220],{"className":92253},[289]," 中存在唯一不动点，记作 ",[31,92256,92258,92277],{"className":92257,"translate":35},[34],[31,92259,92261],{"className":92260},[39],[41,92262,92263],{"xmlns":43},[45,92264,92265,92274],{},[48,92266,92267],{},[42405,92268,92269,92271],{"accent":62},[51,92270,43481],{},[95,92272,92273],{},"ˉ",[55,92275,92276],{"encoding":57},"\\bar V",[31,92278,92280],{"className":92279,"ariaHidden":62},[61],[31,92281,92283,92287],{"className":92282},[66],[31,92284],{"className":92285,"style":92286},[70],"height:0.8201em;",[31,92288,92290],{"className":92289},[75,42523],[31,92291,92293],{"className":92292},[1043],[31,92294,92296],{"className":92295},[1048],[31,92297,92299,92307],{"className":92298,"style":92286},[1052],[31,92300,92301,92304],{"style":1649},[31,92302],{"className":92303,"style":1637},[1060],[31,92305,43481],{"className":92306,"style":120},[75,76],[31,92308,92310,92313],{"style":92309},"top:-3.2523em;",[31,92311],{"className":92312,"style":1637},[1060],[31,92314,92317],{"className":92315,"style":92316},[42549],"left:-0.25em;",[31,92318,92273],{"className":92319},[75],"。下面证明 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V=V^*",[31,92351,92353,92399],{"className":92352,"ariaHidden":62},[61],[31,92354,92356,92359,92390,92393,92396],{"className":92355},[66],[31,92357],{"className":92358,"style":92286},[70],[31,92360,92362],{"className":92361},[75,42523],[31,92363,92365],{"className":92364},[1043],[31,92366,92368],{"className":92367},[1048],[31,92369,92371,92379],{"className":92370,"style":92286},[1052],[31,92372,92373,92376],{"style":1649},[31,92374],{"className":92375,"style":1637},[1060],[31,92377,43481],{"className":92378,"style":120},[75,76],[31,92380,92381,92384],{"style":92309},[31,92382],{"className":92383,"style":1637},[1060],[31,92385,92387],{"className":92386,"style":92316},[42549],[31,92388,92273],{"className":92389},[75],[31,92391],{"className":92392,"style":293},[119],[31,92394,223],{"className":92395},[297],[31,92397],{"className":92398,"style":293},[119],[31,92400,92402,92405],{"className":92401},[66],[31,92403],{"className":92404,"style":30719},[70],[31,92406,92408,92411],{"className":92407},[75],[31,92409,43481],{"className":92410,"style":120},[75,76],[31,92412,92414],{"className":92413},[1039],[31,92415,92417],{"className":92416},[1043],[31,92418,92420],{"className":92419},[1048],[31,92421,92423],{"className":92422,"style":30719},[1052],[31,92424,92425,92428],{"style":3631},[31,92426],{"className":92427,"style":1061},[1060],[31,92429,92431],{"className":92430},[1065,1066,1067,1068],[31,92432,30648],{"className":92433},[124,1068],[11,92435,92436,92437,92550,92551,92579,92580,92638,92639,92697],{},"先证 ",[31,92438,92440,92466],{"className":92439,"translate":35},[34],[31,92441,92443],{"className":92442},[39],[41,92444,92445],{"xmlns":43},[45,92446,92447,92463],{},[48,92448,92449,92455,92457],{},[1180,92450,92451,92453],{},[51,92452,43481],{},[95,92454,30648],{},[95,92456,347],{},[42405,92458,92459,92461],{"accent":62},[51,92460,43481],{},[95,92462,92273],{},[55,92464,92465],{"encoding":57},"V^*\\le\\bar V",[31,92467,92469,92513],{"className":92468,"ariaHidden":62},[61],[31,92470,92472,92475,92504,92507,92510],{"className":92471},[66],[31,92473],{"className":92474,"style":78199},[70],[31,92476,92478,92481],{"className":92477},[75],[31,92479,43481],{"className":92480,"style":120},[75,76],[31,92482,92484],{"className":92483},[1039],[31,92485,92487],{"className":92486},[1043],[31,92488,92490],{"className":92489},[1048],[31,92491,92493],{"className":92492,"style":30719},[1052],[31,92494,92495,92498],{"style":3631},[31,92496],{"className":92497,"style":1061},[1060],[31,92499,92501],{"className":92500},[1065,1066,1067,1068],[31,92502,30648],{"className":92503},[124,1068],[31,92505],{"className":92506,"style":293},[119],[31,92508,347],{"className":92509},[297],[31,92511],{"className":92512,"style":293},[119],[31,92514,92516,92519],{"className":92515},[66],[31,92517],{"className":92518,"style":92286},[70],[31,92520,92522],{"className":92521},[75,42523],[31,92523,92525],{"className":92524},[1043],[31,92526,92528],{"className":92527},[1048],[31,92529,92531,92539],{"className":92530,"style":92286},[1052],[31,92532,92533,92536],{"style":1649},[31,92534],{"className":92535,"style":1637},[1060],[31,92537,43481],{"className":92538,"style":120},[75,76],[31,92540,92541,92544],{"style":92309},[31,92542],{"className":92543,"style":1637},[1060],[31,92545,92547],{"className":92546,"style":92316},[42549],[31,92548,92273],{"className":92549},[75],"。对任意策略 ",[31,92552,92554,92567],{"className":92553,"translate":35},[34],[31,92555,92557],{"className":92556},[39],[41,92558,92559],{"xmlns":43},[45,92560,92561,92565],{},[48,92562,92563],{},[51,92564,2515],{},[55,92566,3485],{"encoding":57},[31,92568,92570],{"className":92569,"ariaHidden":62},[61],[31,92571,92573,92576],{"className":92572},[66],[31,92574],{"className":92575,"style":71},[70],[31,92577,2515],{"className":92578,"style":2666},[75,76],"，由于 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在动作上取最大值，故逐点有",[31,92699,92701],{"className":92700,"translate":35},[411],[31,92702,92704,92752],{"className":92703,"translate":35},[34],[31,92705,92707],{"className":92706},[39],[41,92708,92709],{"xmlns":43,"display":420},[45,92710,92711,92749],{},[48,92712,92713,92719,92725,92727,92733,92739,92741,92747],{},[1180,92714,92715,92717],{},[51,92716,919],{"mathvariant":925},[51,92718,2515],{},[42405,92720,92721,92723],{"accent":62},[51,92722,43481],{},[95,92724,92273],{},[95,92726,347],{},[1180,92728,92729,92731],{},[51,92730,919],{"mathvariant":925},[95,92732,30648],{},[42405,92734,92735,92737],{"accent":62},[51,92736,43481],{},[95,92738,92273],{},[95,92740,223],{},[42405,92742,92743,92745],{"accent":62},[51,92744,43481],{},[95,92746,92273],{},[51,92748,495],{"mathvariant":494},[55,92750,92751],{"encoding":57},"\\mathcal{T}^\\pi\\bar V\\le\\mathcal{T}^*\\bar V=\\bar V.",[31,92753,92755,92831,92906],{"className":92754,"ariaHidden":62},[61],[31,92756,92758,92762,92791,92822,92825,92828],{"className":92757},[66],[31,92759],{"className":92760,"style":92761},[70],"height:0.9561em;vertical-align:-0.136em;",[31,92763,92765,92768],{"className":92764},[75],[31,92766,919],{"className":92767,"style":64444},[75,968],[31,92769,92771],{"className":92770},[1039],[31,92772,92774],{"className":92773},[1043],[31,92775,92777],{"className":92776},[1048],[31,92778,92780],{"className":92779,"style":3987},[1052],[31,92781,92782,92785],{"style":1330},[31,92783],{"className":92784,"style":1061},[1060],[31,92786,92788],{"className":92787},[1065,1066,1067,1068],[31,92789,2515],{"className":92790,"style":2666},[75,76,1068],[31,92792,92794],{"className":92793},[75,42523],[31,92795,92797],{"className":92796},[1043],[31,92798,92800],{"className":92799},[1048],[31,92801,92803,92811],{"className":92802,"style":92286},[1052],[31,92804,92805,92808],{"style":1649},[31,92806],{"className":92807,"style":1637},[1060],[31,92809,43481],{"className":92810,"style":120},[75,76],[31,92812,92813,92816],{"style":92309},[31,92814],{"className":92815,"style":1637},[1060],[31,92817,92819],{"className":92818,"style":92316},[42549],[31,92820,92273],{"className":92821},[75],[31,92823],{"className":92824,"style":293},[119],[31,92826,347],{"className":92827},[297],[31,92829],{"className":92830,"style":293},[119],[31,92832,92834,92837,92866,92897,92900,92903],{"className":92833},[66],[31,92835],{"className":92836,"style":92286},[70],[31,92838,92840,92843],{"className":92839},[75],[31,92841,919],{"className":92842,"style":64444},[75,968],[31,92844,92846],{"className":92845},[1039],[31,92847,92849],{"className":92848},[1043],[31,92850,92852],{"className":92851},[1048],[31,92853,92855],{"className":92854,"style":74855},[1052],[31,92856,92857,92860],{"style":1330},[31,92858],{"className":92859,"style":1061},[1060],[31,92861,92863],{"className":92862},[1065,1066,1067,1068],[31,92864,30648],{"className":92865},[124,1068],[31,92867,92869],{"className":92868},[75,42523],[31,92870,92872],{"className":92871},[1043],[31,92873,92875],{"className":92874},[1048],[31,92876,92878,92886],{"className":92877,"style":92286},[1052],[31,92879,92880,92883],{"style":1649},[31,92881],{"className":92882,"style":1637},[1060],[31,92884,43481],{"className":92885,"style":120},[75,76],[31,92887,92888,92891],{"style":92309},[31,92889],{"className":92890,"style":1637},[1060],[31,92892,92894],{"className":92893,"style":92316},[42549],[31,92895,92273],{"className":92896},[75],[31,92898],{"className":92899,"style":293},[119],[31,92901,223],{"className":92902},[297],[31,92904],{"className":92905,"style":293},[119],[31,92907,92909,92912,92943],{"className":92908},[66],[31,92910],{"className":92911,"style":92286},[70],[31,92913,92915],{"className":92914},[75,42523],[31,92916,92918],{"className":92917},[1043],[31,92919,92921],{"className":92920},[1048],[31,92922,92924,92932],{"className":92923,"style":92286},[1052],[31,92925,92926,92929],{"style":1649},[31,92927],{"className":92928,"style":1637},[1060],[31,92930,43481],{"className":92931,"style":120},[75,76],[31,92933,92934,92937],{"style":92309},[31,92935],{"className":92936,"style":1637},[1060],[31,92938,92940],{"className":92939,"style":92316},[42549],[31,92941,92273],{"className":92942},[75],[31,92944,495],{"className":92945},[75],[11,92947,92948,92949,92977,92978,93169,93170,93221,93222,93280,93281,1138,93283,88853,93471,76226,93584,93612,93613,2372],{},"由单调性，对一切 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V",[31,93022,93024,93132],{"className":93023,"ariaHidden":62},[61],[31,93025,93027,93031,93034,93063,93092,93123,93126,93129],{"className":93026},[66],[31,93028],{"className":93029,"style":93030},[70],"height:1.0701em;vertical-align:-0.25em;",[31,93032,211],{"className":93033},[264],[31,93035,93037,93040],{"className":93036},[75],[31,93038,919],{"className":93039,"style":64444},[75,968],[31,93041,93043],{"className":93042},[1039],[31,93044,93046],{"className":93045},[1043],[31,93047,93049],{"className":93048},[1048],[31,93050,93052],{"className":93051,"style":3628},[1052],[31,93053,93054,93057],{"style":3631},[31,93055],{"className":93056,"style":1061},[1060],[31,93058,93060],{"className":93059},[1065,1066,1067,1068],[31,93061,2515],{"className":93062,"style":2666},[75,76,1068],[31,93064,93066,93069],{"className":93065},[289],[31,93067,220],{"className":93068},[289],[31,93070,93072],{"className":93071},[1039],[31,93073,93075],{"className":93074},[1043],[31,93076,93078],{"className":93077},[1048],[31,93079,93081],{"className":93080,"style":3628},[1052],[31,93082,93083,93086],{"style":3631},[31,93084],{"className":93085,"style":1061},[1060],[31,93087,93089],{"className":93088},[1065,1066,1067,1068],[31,93090,14559],{"className":93091},[75,76,1068],[31,93093,93095],{"className":93094},[75,42523],[31,93096,93098],{"className":93097},[1043],[31,93099,93101],{"className":93100},[1048],[31,93102,93104,93112],{"className":93103,"style":92286},[1052],[31,93105,93106,93109],{"style":1649},[31,93107],{"className":93108,"style":1637},[1060],[31,93110,43481],{"className":93111,"style":120},[75,76],[31,93113,93114,93117],{"style":92309},[31,93115],{"className":93116,"style":1637},[1060],[31,93118,93120],{"className":93119,"style":92316},[42549],[31,93121,92273],{"className":93122},[75],[31,93124],{"className":93125,"style":293},[119],[31,93127,347],{"className":93128},[297],[31,93130],{"className":93131,"style":293},[119],[31,93133,93135,93138],{"className":93134},[66],[31,93136],{"className":93137,"style":92286},[70],[31,93139,93141],{"className":93140},[75,42523],[31,93142,93144],{"className":93143},[1043],[31,93145,93147],{"className":93146},[1048],[31,93148,93150,93158],{"className":93149,"style":92286},[1052],[31,93151,93152,93155],{"style":1649},[31,93153],{"className":93154,"style":1637},[1060],[31,93156,43481],{"className":93157,"style":120},[75,76],[31,93159,93160,93163],{"style":92309},[31,93161],{"className":93162,"style":1637},[1060],[31,93164,93166],{"className":93165,"style":92316},[42549],[31,93167,92273],{"className":93168},[75],"。令 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V=V^{\\pi_g}",[31,94890,94892,94938],{"className":94891,"ariaHidden":62},[61],[31,94893,94895,94898,94929,94932,94935],{"className":94894},[66],[31,94896],{"className":94897,"style":92286},[70],[31,94899,94901],{"className":94900},[75,42523],[31,94902,94904],{"className":94903},[1043],[31,94905,94907],{"className":94906},[1048],[31,94908,94910,94918],{"className":94909,"style":92286},[1052],[31,94911,94912,94915],{"style":1649},[31,94913],{"className":94914,"style":1637},[1060],[31,94916,43481],{"className":94917,"style":120},[75,76],[31,94919,94920,94923],{"style":92309},[31,94921],{"className":94922,"style":1637},[1060],[31,94924,94926],{"className":94925,"style":92316},[42549],[31,94927,92273],{"className":94928},[75],[31,94930],{"className":94931,"style":293},[119],[31,94933,223],{"className":94934},[297],[31,94936],{"className":94937,"style":293},[119],[31,94939,94941,94944],{"className":94940},[66],[31,94942],{"className":94943,"style":945},[70],[31,94945,94947,94950],{"className":94946},[75],[31,94948,43481],{"className":94949,"style":120},[75,76],[31,94951,94953],{"className":94952},[1039],[31,94954,94956],{"className":94955},[1043],[31,94957,94959],{"className":94958},[1048],[31,94960,94962],{"className":94961,"style":3628},[1052],[31,94963,94964,94967],{"style":3631},[31,94965],{"className":94966,"style":1061},[1060],[31,94968,94970],{"className":94969},[1065,1066,1067,1068],[31,94971,94973],{"className":94972},[75,1068],[31,94974,94976,94979],{"className":94975},[75,1068],[31,94977,2515],{"className":94978,"style":2666},[75,76,1068],[31,94980,94982],{"className":94981},[1039],[31,94983,94985,95005],{"className":94984},[1043,1044],[31,94986,94988,95002],{"className":94987},[1048],[31,94989,94991],{"className":94990,"style":51250},[1052],[31,94992,94993,94996],{"style":94680},[31,94994],{"className":94995,"style":11769},[1060],[31,94997,94999],{"className":94998},[1065,11773,1308,1068],[31,95000,14556],{"className":95001,"style":2666},[75,76,1068],[31,95003,1076],{"className":95004},[1075],[31,95006,95008],{"className":95007},[1048],[31,95009,95011],{"className":95010,"style":51384},[1052],[31,95012],{},[31,95014,95016,95041],{"className":95015,"translate":35},[34],[31,95017,95019],{"className":95018},[39],[41,95020,95021],{"xmlns":43},[45,95022,95023,95039],{},[48,95024,95025,95031,95033],{},[42405,95026,95027,95029],{"accent":62},[51,95028,43481],{},[95,95030,92273],{},[95,95032,347],{},[1180,95034,95035,95037],{},[51,95036,43481],{},[95,95038,30648],{},[55,95040,93756],{"encoding":57},[31,95042,95044,95090],{"className":95043,"ariaHidden":62},[61],[31,95045,95047,95050,95081,95084,95087],{"className":95046},[66],[31,95048],{"className":95049,"style":92761},[70],[31,95051,95053],{"className":95052},[75,42523],[31,95054,95056],{"className":95055},[1043],[31,95057,95059],{"className":95058},[1048],[31,95060,95062,95070],{"className":95061,"style":92286},[1052],[31,95063,95064,95067],{"style":1649},[31,95065],{"className":95066,"style":1637},[1060],[31,95068,43481],{"className":95069,"style":120},[75,76],[31,95071,95072,95075],{"style":92309},[31,95073],{"className":95074,"style":1637},[1060],[31,95076,95078],{"className":95077,"style":92316},[42549],[31,95079,92273],{"className":95080},[75],[31,95082],{"className":95083,"style":293},[119],[31,95085,347],{"className":95086},[297],[31,95088],{"className":95089,"style":293},[119],[31,95091,95093,95096],{"className":95092},[66],[31,95094],{"className":95095,"style":30719},[70],[31,95097,95099,95102],{"className":95098},[75],[31,95100,43481],{"className":95101,"style":120},[75,76],[31,95103,95105],{"className":95104},[1039],[31,95106,95108],{"className":95107},[1043],[31,95109,95111],{"className":95110},[1048],[31,95112,95114],{"className":95113,"style":30719},[1052],[31,95115,95116,95119],{"style":3631},[31,95117],{"className":95118,"style":1061},[1060],[31,95120,95122],{"className":95121},[1065,1066,1067,1068],[31,95123,30648],{"className":95124},[124,1068],"。结合两个不等式得 ",[31,95127,95129,95154],{"className":95128,"translate":35},[34],[31,95130,95132],{"className":95131},[39],[41,95133,95134],{"xmlns":43},[45,95135,95136,95152],{},[48,95137,95138,95144,95146],{},[42405,95139,95140,95142],{"accent":62},[51,95141,43481],{},[95,95143,92273],{},[95,95145,223],{},[1180,95147,95148,95150],{},[51,95149,43481],{},[95,95151,30648],{},[55,95153,92349],{"encoding":57},[31,95155,95157,95203],{"className":95156,"ariaHidden":62},[61],[31,95158,95160,95163,95194,95197,95200],{"className":95159},[66],[31,95161],{"className":95162,"style":92286},[70],[31,95164,95166],{"className":95165},[75,42523],[31,95167,95169],{"className":95168},[1043],[31,95170,95172],{"className":95171},[1048],[31,95173,95175,95183],{"className":95174,"style":92286},[1052],[31,95176,95177,95180],{"style":1649},[31,95178],{"className":95179,"style":1637},[1060],[31,95181,43481],{"className":95182,"style":120},[75,76],[31,95184,95185,95188],{"style":92309},[31,95186],{"className":95187,"style":1637},[1060],[31,95189,95191],{"className":95190,"style":92316},[42549],[31,95192,92273],{"className":95193},[75],[31,95195],{"className":95196,"style":293},[119],[31,95198,223],{"className":95199},[297],[31,95201],{"className":95202,"style":293},[119],[31,95204,95206,95209],{"className":95205},[66],[31,95207],{"className":95208,"style":30719},[70],[31,95210,95212,95215],{"className":95211},[75],[31,95213,43481],{"className":95214,"style":120},[75,76],[31,95216,95218],{"className":95217},[1039],[31,95219,95221],{"className":95220},[1043],[31,95222,95224],{"className":95223},[1048],[31,95225,95227],{"className":95226,"style":30719},[1052],[31,95228,95229,95232],{"style":3631},[31,95230],{"className":95231,"style":1061},[1060],[31,95233,95235],{"className":95234},[1065,1066,1067,1068],[31,95236,30648],{"className":95237},[124,1068],[11,95239,95240,95241,95269,95270,95503,95504,95506,95507,95618,95619,46597],{},"最后证明策略的最优性判据。若确定性策略 ",[31,95242,95244,95257],{"className":95243,"translate":35},[34],[31,95245,95247],{"className":95246},[39],[41,95248,95249],{"xmlns":43},[45,95250,95251,95255],{},[48,95252,95253],{},[51,95254,2515],{},[55,95256,3485],{"encoding":57},[31,95258,95260],{"className":95259,"ariaHidden":62},[61],[31,95261,95263,95266],{"className":95262},[66],[31,95264],{"className":95265,"style":71},[70],[31,95267,2515],{"className":95268,"style":2666},[75,76]," 在每个状态都取到最大值，则 ",[31,95271,95273,95319],{"className":95272,"translate":35},[34],[31,95274,95276],{"className":95275},[39],[41,95277,95278],{"xmlns":43},[45,95279,95280,95316],{},[48,95281,95282,95288,95294,95296,95302,95308,95310],{},[1180,95283,95284,95286],{},[51,95285,919],{"mathvariant":925},[51,95287,2515],{},[1180,95289,95290,95292],{},[51,95291,43481],{},[95,95293,30648],{},[95,95295,223],{},[1180,95297,95298,95300],{},[51,95299,919],{"mathvariant":925},[95,95301,30648],{},[1180,95303,95304,95306],{},[51,95305,43481],{},[95,95307,30648],{},[95,95309,223],{},[1180,95311,95312,95314],{},[51,95313,43481],{},[95,95315,30648],{},[55,95317,95318],{"encoding":57},"\\mathcal{T}^\\pi V^*=\\mathcal{T}^*V^*=V^*",[31,95320,95322,95395,95468],{"className":95321,"ariaHidden":62},[61],[31,95323,95325,95328,95357,95386,95389,95392],{"className":95324},[66],[31,95326],{"className":95327,"style":30719},[70],[31,95329,95331,95334],{"className":95330},[75],[31,95332,919],{"className":95333,"style":64444},[75,968],[31,95335,95337],{"className":95336},[1039],[31,95338,95340],{"className":95339},[1043],[31,95341,95343],{"className":95342},[1048],[31,95344,95346],{"className":95345,"style":3628},[1052],[31,95347,95348,95351],{"style":3631},[31,95349],{"className":95350,"style":1061},[1060],[31,95352,95354],{"className":95353},[1065,1066,1067,1068],[31,95355,2515],{"className":95356,"style":2666},[75,76,1068],[31,95358,95360,95363],{"className":95359},[75],[31,95361,43481],{"className":95362,"style":120},[75,76],[31,95364,95366],{"className":95365},[1039],[31,95367,95369],{"className":95368},[1043],[31,95370,95372],{"className":95371},[1048],[31,95373,95375],{"className":95374,"style":30719},[1052],[31,95376,95377,95380],{"style":3631},[31,95378],{"className":95379,"style":1061},[1060],[31,95381,95383],{"className":95382},[1065,1066,1067,1068],[31,95384,30648],{"className":95385},[124,1068],[31,95387],{"className":95388,"style":293},[119],[31,95390,223],{"className":95391},[297],[31,95393],{"className":95394,"style":293},[119],[31,95396,95398,95401,95430,95459,95462,95465],{"className":95397},[66],[31,95399],{"className":95400,"style":30719},[70],[31,95402,95404,95407],{"className":95403},[75],[31,95405,919],{"className":95406,"style":64444},[75,968],[31,95408,95410],{"className":95409},[1039],[31,95411,95413],{"className":95412},[1043],[31,95414,95416],{"className":95415},[1048],[31,95417,95419],{"className":95418,"style":30719},[1052],[31,95420,95421,95424],{"style":3631},[31,95422],{"className":95423,"style":1061},[1060],[31,95425,95427],{"className":95426},[1065,1066,1067,1068],[31,95428,30648],{"className":95429},[124,1068],[31,95431,95433,95436],{"className":95432},[75],[31,95434,43481],{"className":95435,"style":120},[75,76],[31,95437,95439],{"className":95438},[1039],[31,95440,95442],{"className":95441},[1043],[31,95443,95445],{"className":95444},[1048],[31,95446,95448],{"className":95447,"style":30719},[1052],[31,95449,95450,95453],{"style":3631},[31,95451],{"className":95452,"style":1061},[1060],[31,95454,95456],{"className":95455},[1065,1066,1067,1068],[31,95457,30648],{"className":95458},[124,1068],[31,95460],{"className":95461,"style":293},[119],[31,95463,223],{"className":95464},[297],[31,95466],{"className":95467,"style":293},[119],[31,95469,95471,95474],{"className":95470},[66],[31,95472],{"className":95473,"style":30719},[70],[31,95475,95477,95480],{"className":95476},[75],[31,95478,43481],{"className":95479,"style":120},[75,76],[31,95481,95483],{"className":95482},[1039],[31,95484,95486],{"className":95485},[1043],[31,95487,95489],{"className":95488},[1048],[31,95490,95492],{"className":95491,"style":30719},[1052],[31,95493,95494,95497],{"style":3631},[31,95495],{"className":95496,"style":1061},[1060],[31,95498,95500],{"className":95499},[1065,1066,1067,1068],[31,95501,30648],{"className":95502},[124,1068],"，由",[5813,95505,73911],{"href":73910}," 得 ",[31,95508,95510,95536],{"className":95509,"translate":35},[34],[31,95511,95513],{"className":95512},[39],[41,95514,95515],{"xmlns":43},[45,95516,95517,95533],{},[48,95518,95519,95525,95527],{},[1180,95520,95521,95523],{},[51,95522,43481],{},[51,95524,2515],{},[95,95526,223],{},[1180,95528,95529,95531],{},[51,95530,43481],{},[95,95532,30648],{},[55,95534,95535],{"encoding":57},"V^\\pi=V^*",[31,95537,95539,95583],{"className":95538,"ariaHidden":62},[61],[31,95540,95542,95545,95574,95577,95580],{"className":95541},[66],[31,95543],{"className":95544,"style":945},[70],[31,95546,95548,95551],{"className":95547},[75],[31,95549,43481],{"className":95550,"style":120},[75,76],[31,95552,95554],{"className":95553},[1039],[31,95555,95557],{"className":95556},[1043],[31,95558,95560],{"className":95559},[1048],[31,95561,95563],{"className":95562,"style":3628},[1052],[31,95564,95565,95568],{"style":3631},[31,95566],{"className":95567,"style":1061},[1060],[31,95569,95571],{"className":95570},[1065,1066,1067,1068],[31,95572,2515],{"className":95573,"style":2666},[75,76,1068],[31,95575],{"className":95576,"style":293},[119],[31,95578,223],{"className":95579},[297],[31,95581],{"className":95582,"style":293},[119],[31,95584,95586,95589],{"className":95585},[66],[31,95587],{"className":95588,"style":30719},[70],[31,95590,95592,95595],{"className":95591},[75],[31,95593,43481],{"className":95594,"style":120},[75,76],[31,95596,95598],{"className":95597},[1039],[31,95599,95601],{"className":95600},[1043],[31,95602,95604],{"className":95603},[1048],[31,95605,95607],{"className":95606,"style":30719},[1052],[31,95608,95609,95612],{"style":3631},[31,95610],{"className":95611,"style":1061},[1060],[31,95613,95615],{"className":95614},[1065,1066,1067,1068],[31,95616,30648],{"className":95617},[124,1068],"。反之，若 ",[31,95620,95622,95647],{"className":95621,"translate":35},[34],[31,95623,95625],{"className":95624},[39],[41,95626,95627],{"xmlns":43},[45,95628,95629,95645],{},[48,95630,95631,95637,95639],{},[1180,95632,95633,95635],{},[51,95634,43481],{},[51,95636,2515],{},[95,95638,223],{},[1180,95640,95641,95643],{},[51,95642,43481],{},[95,95644,30648],{},[55,95646,95535],{"encoding":57},[31,95648,95650,95694],{"className":95649,"ariaHidden":62},[61],[31,95651,95653,95656,95685,95688,95691],{"className":95652},[66],[31,95654],{"className":95655,"style":945},[70],[31,95657,95659,95662],{"className":95658},[75],[31,95660,43481],{"className":95661,"style":120},[75,76],[31,95663,95665],{"className":95664},[1039],[31,95666,95668],{"className":95667},[1043],[31,95669,95671],{"className":95670},[1048],[31,95672,95674],{"className":95673,"style":3628},[1052],[31,95675,95676,95679],{"style":3631},[31,95677],{"className":95678,"style":1061},[1060],[31,95680,95682],{"className":95681},[1065,1066,1067,1068],[31,95683,2515],{"className":95684,"style":2666},[75,76,1068],[31,95686],{"className":95687,"style":293},[119],[31,95689,223],{"className":95690},[297],[31,95692],{"className":95693,"style":293},[119],[31,95695,95697,95700],{"className":95696},[66],[31,95698],{"className":95699,"style":30719},[70],[31,95701,95703,95706],{"className":95702},[75],[31,95704,43481],{"className":95705,"style":120},[75,76],[31,95707,95709],{"className":95708},[1039],[31,95710,95712],{"className":95711},[1043],[31,95713,95715],{"className":95714},[1048],[31,95716,95718],{"className":95717,"style":30719},[1052],[31,95719,95720,95723],{"style":3631},[31,95721],{"className":95722,"style":1061},[1060],[31,95724,95726],{"className":95725},[1065,1066,1067,1068],[31,95727,30648],{"className":95728},[124,1068],[31,95730,95732],{"className":95731,"translate":35},[411],[31,95733,95735,95837],{"className":95734,"translate":35},[34],[31,95736,95738],{"className":95737},[39],[41,95739,95740],{"xmlns":43,"display":420},[45,95741,95742,95834],{},[5606,95743,95744,95794],{"rowspacing":5608,"columnalign":5609,"columnspacing":5610},[5612,95745,95746,95764],{},[5615,95747,95748],{},[2071,95749,95750],{"scriptlevel":1977,"displaystyle":62},[48,95751,95752,95758],{},[1180,95753,95754,95756],{},[51,95755,919],{"mathvariant":925},[51,95757,2515],{},[1180,95759,95760,95762],{},[51,95761,43481],{},[95,95763,30648],{},[5615,95765,95766],{},[2071,95767,95768],{"scriptlevel":1977,"displaystyle":62},[48,95769,95770,95772,95774,95780,95786,95788],{},[48,95771],{},[95,95773,223],{},[1180,95775,95776,95778],{},[51,95777,919],{"mathvariant":925},[51,95779,2515],{},[1180,95781,95782,95784],{},[51,95783,43481],{},[51,95785,2515],{},[95,95787,223],{},[1180,95789,95790,95792],{},[51,95791,43481],{},[51,95793,2515],{},[5612,95795,95796,95802],{},[5615,95797,95798],{},[2071,95799,95800],{"scriptlevel":1977,"displaystyle":62},[48,95801],{},[5615,95803,95804],{},[2071,95805,95806],{"scriptlevel":1977,"displaystyle":62},[48,95807,95808,95810,95812,95818,95820,95826,95832],{},[48,95809],{},[95,95811,223],{},[1180,95813,95814,95816],{},[51,95815,43481],{},[95,95817,30648],{},[95,95819,223],{},[1180,95821,95822,95824],{},[51,95823,919],{"mathvariant":925},[95,95825,30648],{},[1180,95827,95828,95830],{},[51,95829,43481],{},[95,95831,30648],{},[51,95833,495],{"mathvariant":494},[55,95835,95836],{"encoding":57},"\\begin{aligned}\n\\mathcal{T}^\\pi V^*\n&=\\mathcal{T}^\\pi V^\\pi=V^\\pi\\\\\n&=V^*=\\mathcal{T}^*V^*.\n\\end{aligned}",[31,95838,95840],{"className":95839,"ariaHidden":62},[61],[31,95841,95843,95846],{"className":95842},[66],[31,95844],{"className":95845,"style":57050},[70],[31,95847,95849],{"className":95848},[75],[31,95850,95852,95949],{"className":95851},[5606],[31,95853,95855],{"className":95854},[5867],[31,95856,95858,95941],{"className":95857},[1043,1044],[31,95859,95861,95938],{"className":95860},[1048],[31,95862,95864,95930],{"className":95863,"style":57069},[1052],[31,95865,95866,95869],{"style":57072},[31,95867],{"className":95868,"style":1637},[1060],[31,95870,95872,95901],{"className":95871},[75],[31,95873,95875,95878],{"className":95874},[75],[31,95876,919],{"className":95877,"style":64444},[75,968],[31,95879,95881],{"className":95880},[1039],[31,95882,95884],{"className":95883},[1043],[31,95885,95887],{"className":95886},[1048],[31,95888,95890],{"className":95889,"style":3987},[1052],[31,95891,95892,95895],{"style":1330},[31,95893],{"className":95894,"style":1061},[1060],[31,95896,95898],{"className":95897},[1065,1066,1067,1068],[31,95899,2515],{"className":95900,"style":2666},[75,76,1068],[31,95902,95904,95907],{"className":95903},[75],[31,95905,43481],{"className":95906,"style":120},[75,76],[31,95908,95910],{"className":95909},[1039],[31,95911,95913],{"className":95912},[1043],[31,95914,95916],{"className":95915},[1048],[31,95917,95919],{"className":95918,"style":74855},[1052],[31,95920,95921,95924],{"style":1330},[31,95922],{"className":95923,"style":1061},[1060],[31,95925,95927],{"className":95926},[1065,1066,1067,1068],[31,95928,30648],{"className":95929},[124,1068],[31,95931,95932,95935],{"style":57119},[31,95933],{"className":95934,"style":1637},[1060],[31,95936],{"className":95937},[75],[31,95939,1076],{"className":95940},[1075],[31,95942,95944],{"className":95943},[1048],[31,95945,95947],{"className":95946,"style":57182},[1052],[31,95948],{},[31,95950,95952],{"className":95951},[6062],[31,95953,95955,96199],{"className":95954},[1043,1044],[31,95956,95958,96196],{"className":95957},[1048],[31,95959,95961,96077],{"className":95960,"style":57069},[1052],[31,95962,95963,95966],{"style":57072},[31,95964],{"className":95965,"style":1637},[1060],[31,95967,95969,95972,95975,95978,95981,96010,96039,96042,96045,96048],{"className":95968},[75],[31,95970],{"className":95971},[75],[31,95973],{"className":95974,"style":293},[119],[31,95976,223],{"className":95977},[297],[31,95979],{"className":95980,"style":293},[119],[31,95982,95984,95987],{"className":95983},[75],[31,95985,919],{"className":95986,"style":64444},[75,968],[31,95988,95990],{"className":95989},[1039],[31,95991,95993],{"className":95992},[1043],[31,95994,95996],{"className":95995},[1048],[31,95997,95999],{"className":95998,"style":3987},[1052],[31,96000,96001,96004],{"style":1330},[31,96002],{"className":96003,"style":1061},[1060],[31,96005,96007],{"className":96006},[1065,1066,1067,1068],[31,96008,2515],{"className":96009,"style":2666},[75,76,1068],[31,96011,96013,96016],{"className":96012},[75],[31,96014,43481],{"className":96015,"style":120},[75,76],[31,96017,96019],{"className":96018},[1039],[31,96020,96022],{"className":96021},[1043],[31,96023,96025],{"className":96024},[1048],[31,96026,96028],{"className":96027,"style":3987},[1052],[31,96029,96030,96033],{"style":1330},[31,96031],{"className":96032,"style":1061},[1060],[31,96034,96036],{"className":96035},[1065,1066,1067,1068],[31,96037,2515],{"className":96038,"style":2666},[75,76,1068],[31,96040],{"className":96041,"style":293},[119],[31,96043,223],{"className":96044},[297],[31,96046],{"className":96047,"style":293},[119],[31,96049,96051,96054],{"className":96050},[75],[31,96052,43481],{"className":96053,"style":120},[75,76],[31,96055,96057],{"className":96056},[1039],[31,96058,96060],{"className":96059},[1043],[31,96061,96063],{"className":96062},[1048],[31,96064,96066],{"className":96065,"style":3987},[1052],[31,96067,96068,96071],{"style":1330},[31,96069],{"className":96070,"style":1061},[1060],[31,96072,96074],{"className":96073},[1065,1066,1067,1068],[31,96075,2515],{"className":96076,"style":2666},[75,76,1068],[31,96078,96079,96082],{"style":57119},[31,96080],{"className":96081,"style":1637},[1060],[31,96083,96085,96088,96091,96094,96097,96126,96129,96132,96135,96164,96193],{"className":96084},[75],[31,96086],{"className":96087},[75],[31,96089],{"className":96090,"style":293},[119],[31,96092,223],{"className":96093},[297],[31,96095],{"className":96096,"style":293},[119],[31,96098,96100,96103],{"className":96099},[75],[31,96101,43481],{"className":96102,"style":120},[75,76],[31,96104,96106],{"className":96105},[1039],[31,96107,96109],{"className":96108},[1043],[31,96110,96112],{"className":96111},[1048],[31,96113,96115],{"className":96114,"style":74855},[1052],[31,96116,96117,96120],{"style":1330},[31,96118],{"className":96119,"style":1061},[1060],[31,96121,96123],{"className":96122},[1065,1066,1067,1068],[31,96124,30648],{"clas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V^*(s)",[31,96243,96245],{"className":96244,"ariaHidden":62},[61],[31,96246,96248,96251,96280,96309,96312,96315],{"className":96247},[66],[31,96249],{"className":96250,"style":260},[70],[31,96252,96254,96257],{"className":96253},[75],[31,96255,919],{"className":96256,"style":64444},[75,968],[31,96258,96260],{"className":96259},[1039],[31,96261,96263],{"className":96262},[1043],[31,96264,96266],{"className":96265},[1048],[31,96267,96269],{"className":96268,"style":3628},[1052],[31,96270,96271,96274],{"style":3631},[31,96272],{"className":96273,"style":1061},[1060],[31,96275,96277],{"className":96276},[1065,1066,1067,1068],[31,96278,2515],{"className":96279,"style":2666},[75,76,1068],[31,96281,96283,96286],{"className":96282},[75],[31,96284,43481],{"className":96285,"style":120},[75,76],[31,96287,96289],{"className":96288},[1039],[31,96290,96292],{"className":96291},[1043],[31,96293,96295],{"className":96294},[1048],[31,96296,96298],{"className":96297,"style":30719},[1052],[31,96299,96300,96303],{"style":3631},[31,96301],{"className":96302,"style":1061},[1060],[31,96304,96306],{"className":96305},[1065,1066,1067,1068],[31,96307,30648],{"className":96308},[124,1068],[31,96310,211],{"className":96311},[264],[31,96313,14551],{"className":96314},[75,76],[31,96316,220],{"className":96317},[289],[31,96319,96321,96351],{"className":96320,"translate":35},[34],[31,96322,96324],{"className":96323},[39],[41,96325,96326],{"xmlns":43},[45,96327,96328,96348],{},[48,96329,96330,96336,96342,96344,96346],{},[1180,96331,96332,96334],{},[51,96333,919],{"mathvariant":925},[95,96335,30648],{},[1180,96337,96338,96340],{},[51,96339,43481],{},[95,96341,30648],{},[95,96343,211],{"stretchy":210},[51,96345,14551],{},[95,96347,220],{"stretchy":210},[55,96349,96350],{"encoding":57},"\\mathcal{T}^*V^*(s)",[31,96352,96354],{"className":96353,"ariaHidden":62},[61],[31,96355,96357,96360,96389,96418,96421,96424],{"className":96356},[66],[31,96358],{"className":96359,"style":260},[70],[31,96361,96363,96366],{"className":96362},[75],[31,96364,919],{"className":96365,"style":64444},[75,968],[31,96367,96369],{"className":96368},[1039],[31,96370,96372],{"className":96371},[1043],[31,96373,96375],{"className":96374},[1048],[31,96376,96378],{"className":96377,"style":30719},[1052],[31,96379,96380,96383],{"style":3631},[31,96381],{"className":96382,"style":1061},[1060],[31,96384,96386],{"className":96385},[1065,1066,1067,1068],[31,96387,30648],{"className":96388},[124,1068],[31,96390,96392,96395],{"className":96391},[75],[31,96393,43481],{"className":96394,"style":120},[75,76],[31,96396,96398],{"className":96397},[1039],[31,96399,96401],{"className":96400},[1043],[31,96402,96404],{"className":96403},[1048],[31,96405,96407],{"className":96406,"style":30719},[1052],[31,96408,96409,96412],{"style":3631},[31,96410],{"className":96411,"style":1061},[1060],[31,96413,96415],{"className":96414},[1065,1066,1067,1068],[31,96416,30648],{"className":96417},[124,1068],[31,96419,211],{"className":96420},[264],[31,96422,14551],{"className":96423},[75,76],[31,96425,220],{"className":96426},[289]," 是这些动作值的最大值，等号成立当且仅当 ",[31,96429,96431,96444],{"className":96430,"translate":35},[34],[31,96432,96434],{"className":96433},[39],[41,96435,96436],{"xmlns":43},[45,96437,96438,96442],{},[48,96439,96440],{},[51,96441,2515],{},[55,96443,3485],{"encoding":57},[31,96445,96447],{"className":96446,"ariaHidden":62},[61],[31,96448,96450,96453],{"className":96449},[66],[31,96451],{"className":96452,"style":71},[70],[31,96454,2515],{"className":96455,"style":2666},[75,76]," 在该状态以概率 ",[31,96458,96460,96473],{"className":96459,"translate":35},[34],[31,96461,96463],{"className":96462},[39],[41,96464,96465],{"xmlns":43},[45,96466,96467,96471],{},[48,96468,96469],{},[91,96470,205],{},[55,96472,205],{"encoding":57},[31,96474,96476],{"className":96475,"ariaHidden":62},[61],[31,96477,96479,96482],{"className":96478},[66],[31,96480],{"className":96481,"style":326},[70],[31,96483,205],{"className":96484},[75]," 选择取得最大值的动作。对确定性策略，这正是 ",[31,96487,96489,96527],{"className":96488,"translate":35},[34],[31,96490,96492],{"className":96491},[39],[41,96493,96494],{"xmlns":43},[45,96495,96496,96524],{},[48,96497,96498,96500,96502,96504,96506,96508,96510,96512,96522],{},[51,96499,2515],{},[95,96501,211],{"stretchy":210},[51,96503,14551],{},[95,96505,220],{"stretchy":210},[95,96507,2518],{},[51,96509,37263],{},[95,96511,1540],{},[1004,96513,96514,96520],{},[48,96515,96516,96518],{},[51,96517,10154],{},[95,96519,1540],{},[51,96521,5813],{},[95,96523,88425],{},[55,96525,96526],{"encoding":57},"\\pi(s)\\in\\arg\\max_a\\cdots",[31,96528,96530,96557],{"className":96529,"ariaHidden":62},[61],[31,96531,96533,96536,96539,96542,96545,96548,96551,96554],{"className":96532},[66],[31,96534],{"className":96535,"style":260},[70],[31,96537,2515],{"className":96538,"style":2666},[75,76],[31,96540,211],{"className":96541},[264],[31,96543,14551],{"className":96544},[75,76],[31,96546,220],{"className":96547},[289],[31,96549],{"className":96550,"style":293},[119],[31,96552,2518],{"className":96553},[297],[31,96555],{"className":96556,"style":293},[119],[31,96558,96560,96563,96568,96571,96611,96614],{"className":96559},[66],[31,96561],{"className":96562,"style":865},[70],[31,96564,37341,96566],{"className":96565},[1460],[31,96567,14556],{"style":20180},[31,96569],{"className":96570,"style":1467},[119],[31,96572,96574,96577],{"className":96573},[1460],[31,96575,10154],{"className":96576},[1460],[31,96578,96580],{"className":96579},[1039],[31,96581,96583,96603],{"className":96582},[1043,1044],[31,96584,96586,96600],{"className":96585},[1048],[31,96587,96589],{"className":96588,"style":3230},[1052],[31,96590,96591,96594],{"style":21790},[31,96592],{"className":96593,"style":1061},[1060],[31,96595,96597],{"className":96596},[1065,1066,1067,1068],[31,96598,5813],{"className":96599},[75,76,1068],[31,96601,1076],{"className":96602},[1075],[31,96604,96606],{"className":96605},[1048],[31,96607,96609],{"className":96608,"style":2050},[1052],[31,96610],{},[31,96612],{"className":96613,"style":1467},[119],[31,96615,88425],{"className":96616},[3860],[11,96618,96619,96620,96678],{},"这个证明的结构值得注意。它没有直接对无穷期问题做「先选动作再假设后续最优」的循环论证，而是把最优性分解为两个方向。一方面，任何策略都不如 ",[31,96621,96623,96640],{"className":96622,"translate":35},[34],[31,96624,96626],{"className":96625},[39],[41,96627,96628],{"xmlns":43},[45,96629,96630,96638],{},[48,96631,96632],{},[1180,96633,96634,96636],{},[51,96635,919],{"mathvariant":925},[95,96637,30648],{},[55,96639,74501],{"encoding":57},[31,96641,96643],{"className":96642,"ariaHidden":62},[61],[31,96644,96646,96649],{"className":96645},[66],[31,96647],{"className":96648,"style":30719},[70],[31,96650,96652,96655],{"className":96651},[75],[31,96653,919],{"className":96654,"style":64444},[75,968],[31,96656,96658],{"className":96657},[1039],[31,96659,96661],{"className":96660},[1043],[31,96662,96664],{"className":96663},[1048],[31,96665,96667],{"className":96666,"style":30719},[1052],[31,96668,96669,96672],{"style":3631},[31,96670],{"className":96671,"style":1061},[1060],[31,96673,96675],{"className":96674},[1065,1066,1067,1068],[31,96676,30648],{"className":96677},[124,1068]," 的不动点。另一方面，关于该不动点的贪婪策略恰好达到它。两个方向合起来，既证明了最优方程，也证明了最优策略的存在性。",[11,96680,69053,96681,18171,96683,96742,96743,96771,96772,96800],{},[5813,96682,89835],{"href":89834},[31,96684,96686,96704],{"className":96685,"translate":35},[34],[31,96687,96689],{"className":96688},[39],[41,96690,96691],{"xmlns":43},[45,96692,96693,96701],{},[48,96694,96695],{},[1180,96696,96697,96699],{},[51,96698,56971],{},[95,96700,30648],{},[55,96702,96703],{"encoding":57},"Q^*",[31,96705,96707],{"className":96706,"ariaHidden":62},[61],[31,96708,96710,96713],{"className":96709},[66],[31,96711],{"className":96712,"style":83153},[70],[31,96714,96716,96719],{"className":96715},[75],[31,96717,56971],{"className":96718},[75,76],[31,96720,96722],{"className":96721},[1039],[31,96723,96725],{"className":96724},[1043],[31,96726,96728],{"className":96727},[1048],[31,96729,96731],{"className":96730,"style":30719},[1052],[31,96732,96733,96736],{"style":3631},[31,96734],{"className":96735,"style":1061},[1060],[31,96737,96739],{"className":96738},[1065,1066,1067,1068],[31,96740,30648],{"className":96741},[124,1068]," 的定义，还可以得到最优动作值函数的表达式。若在状态 ",[31,96744,96746,96759],{"className":96745,"translate":35},[34],[31,96747,96749],{"className":96748},[39],[41,96750,96751],{"xmlns":43},[45,96752,96753,96757],{},[48,96754,96755],{},[51,96756,14551],{},[55,96758,14551],{"encoding":57},[31,96760,96762],{"className":96761,"ariaHidden":62},[61],[31,96763,96765,96768],{"className":96764},[66],[31,96766],{"className":96767,"style":71},[70],[31,96769,14551],{"className":96770},[75,76]," 先固定动作 ",[31,96773,96775,96788],{"className":96774,"translate":35},[34],[31,96776,96778],{"className":96777},[39],[41,96779,96780],{"xmlns":43},[45,96781,96782,96786],{},[48,96783,96784],{},[51,96785,5813],{},[55,96787,5813],{"encoding":57},[31,96789,96791],{"className":96790,"ariaHidden":62},[61],[31,96792,96794,96797],{"className":96793},[66],[31,96795],{"className":96796,"style":71},[70],[31,96798,5813],{"className":96799},[75,76],"，此后采用最优策略，则",[31,96802,96804],{"className":96803,"translate":35},[411],[31,96805,96807,96989],{"className":96806,"translate":35},[34],[31,96808,96810],{"className":96809},[39],[41,96811,96812],{"xmlns":43,"display":420},[45,96813,96814,96986],{},[5606,96815,96816,96928],{"rowspacing":5608,"columnalign":5609,"columnspacing":5610},[5612,96817,96818,96840],{},[5615,96819,96820],{},[2071,96821,96822],{"scriptlevel":1977,"displaystyle":62},[48,96823,96824,96830,96832,96834,96836,96838],{},[1180,96825,96826,96828],{},[51,96827,56971],{},[95,96829,30648],{},[95,96831,211],{"stretchy":210},[51,96833,14551],{},[95,96835,1210],{"separator":62},[51,96837,5813],{},[95,96839,220],{"stretchy":210},[5615,96841,96842],{},[2071,96843,96844],{"scriptlevel":1977,"displaystyle":62},[48,96845,96846,96848,96850,96860,96862,96864,96870,96872,96874,96876,96878,96880,96926],{},[48,96847],{},[95,96849,223],{},[1531,96851,96852,96854],{},[95,96853,5625],{},[1180,96855,96856,96858],{},[51,96857,14551],{},[95,96859,13691],{"mathvariant":494,"lspace":5610,"rspace":5610},[51,96861,2521],{},[95,96863,211],{"stretchy":210},[1180,96865,96866,96868],{},[51,96867,14551],{},[95,96869,13691],{"mathvariant":494,"lspace":5610,"rspace":5610},[95,96871,12680],{},[51,96873,14551],{},[95,96875,1210],{"separator":62},[51,96877,5813],{},[95,96879,220],{"stretchy":210},[48,96881,96882,96884,96886,96888,96890,96892,96894,96896,96902,96904,96906,96908,96914,96916,96922,96924],{},[95,96883,15175],{"fence":62},[51,96885,2549],{},[95,96887,211],{"stretchy":210},[51,96889,14551],{},[95,96891,1210],{"separator":62},[51,96893,5813],{},[95,96895,1210],{"separator":62},[1180,96897,96898,96900],{},[51,96899,14551],{},[95,96901,13691],{"mathvariant":494,"lspace":5610,"rspace":5610},[95,96903,220],{"stretchy":210},[95,96905,434],{},[51,96907,53298],{},[1180,96909,96910,96912],{},[51,96911,43481],{},[95,96913,30648],{},[95,96915,211],{"stretchy":210},[1180,96917,96918,96920],{},[51,96919,14551],{},[95,96921,13691],{"mathvariant":494,"lspace":5610,"rspace":5610},[95,96923,220],{"stretchy":210},[95,96925,15184],{"fence":62},[95,96927,1210],{"separator":62},[5612,96929,96930,96948],{},[5615,96931,96932],{},[2071,96933,96934],{"scriptlevel":1977,"displaystyle":62},[48,96935,96936,96942,96944,96946],{},[1180,96937,96938,96940],{},[51,96939,43481],{},[95,96941,30648],{},[95,96943,211],{"stretchy":210},[51,96945,14551],{},[95,96947,220],{"stretchy":210},[5615,96949,96950],{},[2071,96951,96952],{"scriptlevel":1977,"displaystyle":62},[48,96953,96954,96956,96958,96968,96974,96976,96978,96980,96982,96984],{},[48,96955],{},[95,96957,223],{},[1531,96959,96960,96966],{},[48,96961,96962,96964],{},[51,96963,10154],{},[95,96965,1540],{},[51,96967,5813],{},[1180,96969,96970,96972],{},[51,96971,56971],{},[95,96973,30648],{},[95,96975,211],{"stretchy":210},[51,96977,14551],{},[95,96979,1210],{"separator":62},[51,96981,5813],{},[95,96983,220],{"stretchy":210},[51,96985,495],{"mathvariant":494},[55,96987,96988],{"encoding":57},"\\begin{aligned}\nQ^*(s,a)&=\\sum_{s'}P(s'\\mid s,a)\n\\left[r(s,a,s')+\\gamma V^*(s')\\right],\\\\\nV^*(s)&=\\max_{a}Q^*(s,a).\n\\end{aligned}",[31,96990,96992],{"className":96991,"ariaHidden":62},[61],[31,96993,96995,96999],{"className":96994},[66],[31,96996],{"className":96997,"style":96998},[70],"height:4.184em;vertical-align:-1.842em;",[31,97000,97002],{"className":97001},[75],[31,97003,97005,97133],{"className":97004},[5606],[31,97006,97008],{"className":97007},[5867],[31,97009,97011,97124],{"className":97010},[1043,1044],[31,97012,97014,97121],{"className":97013},[1048],[31,97015,97018,97074],{"className":97016,"style":97017},[1052],"height:2.342em;",[31,97019,97021,97024],{"style":97020},"top:-4.342em;",[31,97022],{"className":97023,"style":5907},[1060],[31,97025,97027,97056,97059,97062,97065,97068,97071],{"className":97026},[75],[31,97028,97030,97033],{"className":97029},[75],[31,97031,56971],{"className":97032},[75,76],[31,97034,97036],{"className":97035},[1039],[31,97037,97039],{"className":97038},[1043],[31,97040,97042],{"className":97041},[1048],[31,97043,97045],{"className":97044,"style":74855},[1052],[31,97046,97047,97050],{"style":1330},[31,97048],{"className":97049,"style":1061},[1060],[31,97051,97053],{"className":97052},[1065,1066,1067,1068],[31,97054,30648],{"className":97055},[124,1068],[31,97057,211],{"className":97058},[264],[31,97060,14551],{"className":97061},[75,76],[31,97063,1210],{"className":97064},[1391],[31,97066],{"className":97067,"style":1467},[119],[31,97069,5813],{"className":97070},[75,76],[31,97072,220],{"className":97073},[289],[31,97075,97077,97080],{"style":97076},"top:-1.908em;",[31,97078],{"className":97079,"style":5907},[1060],[31,97081,97083,97112,97115,97118],{"className":97082},[75],[31,97084,97086,97089],{"className":97085},[75],[31,97087,43481],{"className":97088,"style":120},[75,76],[31,97090,97092],{"className":97091},[1039],[31,97093,97095],{"className":97094},[1043],[31,97096,97098],{"className":97097},[1048],[31,97099,97101],{"className":97100,"style":74855},[1052],[31,97102,97103,97106],{"style":1330},[31,97104],{"className":97105,"style":1061},[1060],[31,97107,97109],{"className":97108},[1065,1066,1067,1068],[31,97110,30648],{"className":97111},[124,1068],[31,97113,211],{"className":97114},[264],[31,97116,14551],{"className":97117},[75,76],[31,97119,220],{"className":97120},[289],[31,97122,1076],{"className":97123},[1075],[31,97125,97127],{"className":97126},[1048],[31,97128,97131],{"className":97129,"style":97130},[1052],"height:1.842em;",[31,97132],{},[31,97134,97136],{"className":97135},[6062],[31,97137,97139,97585],{"className":97138},[1043,1044],[31,97140,97142,97582],{"className":97141},[1048],[31,97143,97145,97462],{"className":97144,"style":97017},[1052],[31,97146,97147,97150],{"style":97020},[31,97148],{"className":97149,"style":5907},[1060],[31,97151,97153,97156,97159,97162,97165,97241,97244,97247,97250,97282,97285,97288,97291,97294,97297,97300,97303,97306,97309,97456,97459],{"className":97152},[75],[31,97154],{"className":97155},[75],[31,97157],{"className":97158,"style":293},[119],[31,97160,223],{"className":97161},[297],[31,97163],{"className":97164,"style":293},[119],[31,97166,97168],{"className":97167},[1460,1620],[31,97169,97171,97233],{"className":97170},[1043,1044],[31,97172,97174,97230],{"className":97173},[1048],[31,97175,97177,97220],{"className":97176,"style":5900},[1052],[31,97178,97179,97182],{"style":59756},[31,97180],{"className":97181,"style":5907},[1060],[31,97183,97185],{"className":97184},[1065,1066,1067,1068],[31,97186,97188],{"className":97187},[75,1068],[31,97189,97191,97194],{"className":97190},[75,1068],[31,97192,14551],{"className":97193},[75,76,1068],[31,97195,97197],{"className":97196},[1039],[31,97198,97200],{"className":97199},[1043],[31,97201,97203],{"className":97202},[1048],[31,97204,97206],{"className":97205,"style":31478},[1052],[31,97207,97208,97211],{"style":31481},[31,97209],{"className":97210,"style":11769},[1060],[31,97212,97214],{"className":97213},[1065,11773,1308,1068],[31,97215,97217],{"className":97216},[75,1068],[31,97218,13691],{"className":97219},[75,1068],[31,97221,97222,97225],{"style":5919},[31,97223],{"className":97224,"style":5907},[1060],[31,97226,97227],{},[31,97228,5625],{"className":97229},[1460,1461,1687],[31,97231,1076],{"className":97232},[1075],[31,97234,97236],{"className":97235},[1048],[31,97237,97239],{"className":97238,"style":59817},[1052],[31,97240],{},[31,97242],{"className":97243,"style":1467},[119],[31,97245,2521],{"className":97246,"style":949},[75,76],[31,97248,211],{"className":97249},[264],[31,97251,97253,97256],{"className":97252},[75],[31,97254,14551],{"className":97255},[75,76],[31,97257,97259],{"className":97258},[1039],[31,97260,97262],{"className":97261},[1043],[31,97263,97265],{"className":97264},[1048],[31,97266,97268],{"className":97267,"style":14017},[1052],[31,97269,97270,97273],{"style":1330},[31,97271],{"className":97272,"style":1061},[1060],[31,97274,97276],{"className":97275},[1065,1066,1067,1068],[31,97277,97279],{"className":97278},[75,1068],[31,97280,13691],{"className":97281},[75,1068],[31,97283],{"className":97284,"style":293},[119],[31,97286,12680],{"className":97287},[297],[31,97289],{"className":97290,"style":293},[119],[31,97292,14551],{"className":97293},[75,76],[31,97295,1210],{"className":97296},[1391],[31,97298],{"className":97299,"style":1467},[119],[31,97301,5813],{"className":97302},[75,76],[31,97304,220],{"className":97305},[289],[31,97307],{"className":97308,"style":1467},[119],[31,97310,97312,97315,97318,97321,97324,97327,97330,97333,97336,97339,97371,97374,97377,97380,97383,97386,97415,97418,97450,97453],{"className":97311},[3860],[31,97313,15175],{"className":97314,"style":3868},[264,3867],[31,97316,2549],{"className":97317,"style":19759},[75,76],[31,97319,211],{"className":97320},[264],[31,97322,14551],{"className":97323},[75,76],[31,97325,1210],{"className":97326},[1391],[31,97328],{"className":97329,"style":1467},[119],[31,97331,5813],{"className":97332},[75,76],[31,97334,1210],{"className":97335},[1391],[31,97337],{"className":97338,"style":1467},[119],[31,97340,97342,97345],{"className":97341},[75],[31,97343,14551],{"className":97344},[75,76],[31,97346,97348],{"className":97347},[1039],[31,97349,97351],{"className":97350},[1043],[31,97352,97354],{"className":97353},[1048],[31,97355,97357],{"className":97356,"style":14017},[1052],[31,97358,97359,97362],{"style":1330},[31,97360],{"className":97361,"style":1061},[1060],[31,97363,97365],{"className":97364},[1065,1066,1067,1068],[31,97366,97368],{"className":97367},[75,1068],[31,97369,13691],{"className":97370},[75,1068],[31,97372,220],{"className":97373},[289],[31,97375],{"className":97376,"style":120},[119],[31,97378,434],{"className":97379},[124],[31,97381],{"className":97382,"style":120},[119],[31,97384,53298],{"className":97385,"style":24157},[75,76],[31,97387,97389,97392],{"className":97388},[75],[31,97390,43481],{"className":97391,"style":120},[75,76],[31,97393,97395],{"className":97394},[1039],[31,97396,97398],{"className":97397},[1043],[31,97399,97401],{"className":97400},[1048],[31,97402,97404],{"className":97403,"style":74855},[1052],[31,97405,97406,97409],{"style":1330},[31,97407],{"className":97408,"style":1061},[1060],[31,97410,97412],{"className":97411},[1065,1066,1067,1068],[31,97413,30648],{"className":97414},[124,1068],[31,97416,211],{"className":97417},[264],[31,97419,97421,97424],{"className":97420},[75],[31,97422,14551],{"className":97423},[75,76],[31,97425,97427],{"className":97426},[1039],[31,97428,97430],{"className":97429},[1043],[31,97431,97433],{"className":97432},[1048],[31,97434,97436],{"className":97435,"style":14017},[1052],[31,97437,97438,97441],{"style":1330},[31,97439],{"className":97440,"style":1061},[1060],[31,97442,97444],{"className":97443},[1065,1066,1067,1068],[31,97445,97447],{"className":97446},[75,1068],[31,97448,13691],{"className":97449},[75,1068],[31,97451,220],{"className":97452},[289],[31,97454,15184],{"className":97455,"style":3868},[289,3867],[31,97457],{"className":97458,"style":1467},[119],[31,97460,1210],{"className":97461},[1391],[31,97463,97464,97467],{"style":97076},[31,97465],{"className":97466,"style":5907},[1060],[31,97468,97470,97473,97476,97479,97482,97529,97532,97561,97564,97567,97570,97573,97576,97579],{"className":97469},[75],[31,97471],{"className":97472},[75],[31,97474],{"className":97475,"style":293},[119],[31,97477,223],{"className":97478},[297],[31,97480],{"className":97481,"style":293},[119],[31,97483,97485],{"className":97484},[1460,1620],[31,97486,97488,97521],{"className":97487},[1043,1044],[31,97489,97491,97518],{"className":97490},[1048],[31,97492,97494,97508],{"className":97493,"style":71},[1052],[31,97495,97496,97499],{"style":37361},[31,97497],{"className":97498,"style":1637},[1060],[31,97500,97502],{"className":97501},[1065,1066,1067,1068],[31,97503,97505],{"className":97504},[75,1068],[31,97506,5813],{"className":97507},[75,76,1068],[31,97509,97510,97513],{"style":1649},[31,97511],{"className":97512,"style":1637},[1060],[31,97514,97515],{},[31,97516,10154],{"className":97517},[1460],[31,97519,1076],{"className":97520},[1075],[31,97522,97524],{"className":97523},[1048],[31,97525,97527],{"className":97526,"style":37390},[1052],[31,97528],{},[31,97530],{"className":97531,"style":1467},[119],[31,97533,97535,97538],{"className":97534},[75],[31,97536,56971],{"className":97537},[75,76],[31,97539,97541],{"className":97540},[1039],[31,97542,97544],{"className":97543},[1043],[31,97545,97547],{"className":97546},[1048],[31,97548,97550],{"className":97549,"style":74855},[1052],[31,97551,97552,97555],{"style":1330},[31,97553],{"className":97554,"style":1061},[1060],[31,97556,97558],{"className":97557},[1065,1066,1067,1068],[31,97559,30648],{"className":97560},[124,1068],[31,97562,211],{"className":97563},[264],[31,97565,14551],{"className":97566},[75,76],[31,97568,1210],{"className":97569},[1391],[31,97571],{"className":97572,"style":1467},[119],[31,97574,5813],{"className":97575},[75,76],[31,97577,220],{"className":97578},[289],[31,97580,495],{"className":97581},[75],[31,97583,1076],{"className":97584},[1075],[31,97586,97588],{"className":97587},[1048],[31,97589,97591],{"className":97590,"style":97130},[1052],[31,97592],{},[11,97594,97595,97596,97707,97708,97795,97796,18171,97854,97912],{},"第一式的上界由 ",[31,97597,97599,97625],{"className":97598,"translate":35},[34],[31,97600,97602],{"className":97601},[39],[41,97603,97604],{"xmlns":43},[45,97605,97606,97622],{},[48,97607,97608,97614,97616],{},[1180,97609,97610,97612],{},[51,97611,43481],{},[51,97613,2515],{},[95,97615,347],{},[1180,97617,97618,97620],{},[51,97619,43481],{},[95,97621,30648],{},[55,97623,97624],{"encoding":57},"V^\\pi\\le V^*",[31,97626,97628,97672],{"className":97627,"ariaHidden":62},[61],[31,97629,97631,97634,97663,97666,97669],{"className":97630},[66],[31,97632],{"className":97633,"style":76406},[70],[31,97635,97637,97640],{"className":97636},[75],[31,97638,43481],{"className":97639,"style":120},[75,76],[31,97641,97643],{"className":97642},[1039],[31,97644,97646],{"className":97645},[1043],[31,97647,97649],{"className":97648},[1048],[31,97650,97652],{"className":97651,"style":3628},[1052],[31,97653,97654,97657],{"style":3631},[31,97655],{"className":97656,"style":1061},[1060],[31,97658,97660],{"className":97659},[1065,1066,1067,1068],[31,97661,2515],{"className":97662,"style":2666},[75,76,1068],[31,97664],{"className":97665,"style":293},[119],[31,97667,347],{"className":97668},[297],[31,97670],{"className":97671,"style":293},[119],[31,97673,97675,97678],{"className":97674},[66],[31,97676],{"className":97677,"style":30719},[70],[31,97679,97681,97684],{"className":97680},[75],[31,97682,43481],{"className":97683,"style":120},[75,76],[31,97685,97687],{"className":97686},[1039],[31,97688,97690],{"className":97689},[1043],[31,97691,97693],{"className":97692},[1048],[31,97694,97696],{"className":97695,"style":30719},[1052],[31,97697,97698,97701],{"style":3631},[31,97699],{"className":97700,"style":1061},[1060],[31,97702,97704],{"className":97703},[1065,1066,1067,1068],[31,97705,30648],{"className":97706},[124,1068]," 得到，下界由在后续状态选取最优续策略达到。第二式说明，最优动作正是使 ",[31,97709,97711,97739],{"className":97710,"translate":35},[34],[31,97712,97714],{"className":97713},[39],[41,97715,97716],{"xmlns":43},[45,97717,97718,97736],{},[48,97719,97720,97726,97728,97730,97732,97734],{},[1180,97721,97722,97724],{},[51,97723,56971],{},[95,97725,30648],{},[95,97727,211],{"stretchy":210},[51,97729,14551],{},[95,97731,1210],{"separator":62},[51,97733,5813],{},[95,97735,220],{"stretchy":210},[55,97737,97738],{"encoding":57},"Q^*(s,a)",[31,97740,97742],{"className":97741,"ariaHidden":62},[61],[31,97743,97745,97748,97777,97780,97783,97786,97789,97792],{"className":97744},[66],[31,97746],{"className":97747,"style":260},[70],[31,97749,97751,97754],{"className":97750},[75],[31,97752,56971],{"className":97753},[75,76],[31,97755,97757],{"className":97756},[1039],[31,97758,97760],{"className":97759},[1043],[31,97761,97763],{"className":97762},[1048],[31,97764,97766],{"className":97765,"style":30719},[1052],[31,97767,97768,97771],{"style":3631},[31,97769],{"className":97770,"style":1061},[1060],[31,97772,97774],{"className":97773},[1065,1066,1067,1068],[31,97775,30648],{"className":97776},[124,1068],[31,97778,211],{"className":97779},[264],[31,97781,14551],{"className":97782},[75,76],[31,97784,1210],{"className":97785},[1391],[31,97787],{"className":97788,"style":1467},[119],[31,97790,5813],{"className":97791},[75,76],[31,97793,220],{"className":97794},[289]," 最大的动作。因此，",[31,97797,97799,97816],{"className":97798,"translate":35},[34],[31,97800,97802],{"className":97801},[39],[41,97803,97804],{"xmlns":43},[45,97805,97806,97814],{},[48,97807,97808],{},[1180,97809,97810,97812],{},[51,97811,56971],{},[95,97813,30648],{},[55,97815,96703],{"encoding":57},[31,97817,97819],{"className":97818,"ariaHidden":62},[61],[31,97820,97822,97825],{"className":97821},[66],[31,97823],{"className":97824,"style":83153},[70],[31,97826,97828,97831],{"className":97827},[75],[31,97829,56971],{"className":97830},[75,76],[31,97832,97834],{"className":97833},[1039],[31,97835,97837],{"className":97836},[1043],[31,97838,97840],{"className":97839},[1048],[31,97841,97843],{"className":97842,"style":30719},[1052],[31,97844,97845,97848],{"style":3631},[31,97846],{"className":97847,"style":1061},[1060],[31,97849,97851],{"className":97850},[1065,1066,1067,1068],[31,97852,30648],{"className":97853},[124,1068],[31,97855,97857,97874],{"className":97856,"translate":35},[34],[31,97858,97860],{"className":97859},[39],[41,97861,97862],{"xmlns":43},[45,97863,97864,97872],{},[48,97865,97866],{},[1180,97867,97868,97870],{},[51,97869,43481],{},[95,97871,30648],{},[55,97873,87313],{"encoding":57},[31,97875,97877],{"className":97876,"ariaHidden":62},[61],[31,97878,97880,97883],{"className":97879},[66],[31,97881],{"className":97882,"style":30719},[70],[31,97884,97886,97889],{"className":97885},[75],[31,97887,43481],{"className":97888,"style":120},[75,76],[31,97890,97892],{"className":97891},[1039],[31,97893,97895],{"className":97894},[1043],[31,97896,97898],{"className":97897},[1048],[31,97899,97901],{"className":97900,"style":30719},[1052],[31,97902,97903,97906],{"style":3631},[31,97904],{"className":97905,"style":1061},[1060],[31,97907,97909],{"className":97908},[1065,1066,1067,1068],[31,97910,30648],{"className":97911},[124,1068]," 的差别只在于第一步是否已经固定动作。",[52860,97914,97916],{"id":97915},"_43-max-与期望不可交换","4.3 max 与期望不可交换",[11,97918,97919,97920,98003,98004,98086],{},"最优贝尔曼方程中是 ",[31,97921,97923,97947],{"className":97922,"translate":35},[34],[31,97924,97926],{"className":97925},[39],[41,97927,97928],{"xmlns":43},[45,97929,97930,97944],{},[48,97931,97932,97942],{},[1004,97933,97934,97940],{},[48,97935,97936,97938],{},[51,97937,10154],{},[95,97939,1540],{},[51,97941,5813],{},[51,97943,22324],{"mathvariant":14789},[55,97945,97946],{"encoding":57},"\\max_a\\mathbb{E}",[31,97948,97950],{"className":97949,"ariaHidden":62},[61],[31,97951,97953,97957,97997,98000],{"className":97952},[66],[31,97954],{"className":97955,"style":97956},[70],"height:0.8389em;vertical-align:-0.15em;",[31,97958,97960,97963],{"className":97959},[1460],[31,97961,10154],{"className":97962},[1460],[31,97964,97966],{"className":97965},[1039],[31,97967,97969,97989],{"className":97968},[1043,1044],[31,97970,97972,97986],{"className":97971},[1048],[31,97973,97975],{"className":97974,"style":3230},[1052],[31,97976,97977,97980],{"style":21790},[31,97978],{"className":97979,"style":1061},[1060],[31,97981,97983],{"className":97982},[1065,1066,1067,1068],[31,97984,5813],{"className":97985},[75,76,1068],[31,97987,1076],{"className":97988},[1075],[31,97990,97992],{"className":97991},[1048],[31,97993,97995],{"className":97994,"style":2050},[1052],[31,97996],{},[31,97998],{"className":97999,"style":1467},[119],[31,98001,22324],{"className":98002},[75,14947],"，而不是 ",[31,98005,98007,98031],{"className":98006,"translate":35},[34],[31,98008,98010],{"className":98009},[39],[41,98011,98012],{"xmlns":43},[45,98013,98014,98028],{},[48,98015,98016,98018],{},[51,98017,22324],{"mathvariant":14789},[1004,98019,98020,98026],{},[48,98021,98022,98024],{},[51,98023,10154],{},[95,98025,1540],{},[51,98027,5813],{},[55,98029,98030],{"encoding":57},"\\mathbb{E}\\max_a",[31,98032,98034],{"className":98033,"ariaHidden":62},[61],[31,98035,98037,98040,98043,98046],{"className":98036},[66],[31,98038],{"className":98039,"style":97956},[70],[31,98041,22324],{"className":98042},[75,14947],[31,98044],{"className":98045,"style":1467},[119],[31,98047,98049,98052],{"className":98048},[1460],[31,98050,10154],{"className":98051},[1460],[31,98053,98055],{"className":98054},[1039],[31,98056,98058,98078],{"className":98057},[1043,1044],[31,98059,98061,98075],{"className":98060},[1048],[31,98062,98064],{"className":98063,"style":3230},[1052],[31,98065,98066,98069],{"style":21790},[31,98067],{"className":98068,"style":1061},[1060],[31,98070,98072],{"className":98071},[1065,1066,1067,1068],[31,98073,5813],{"className":98074},[75,76,1068],[31,98076,1076],{"className":98077},[1075],[31,98079,98081],{"className":98080},[1048],[31,98082,98084],{"className":98083,"style":2050},[1052],[31,98085],{},"。这两个量一般不相等，原因在于动作必须在观察到下一状态之前选定。",[11,98088,98089,98090,98118,98119,98243,98244,98276,98277,98347,98348,98276,98380,98450],{},"考虑一个状态 ",[31,98091,98093,98106],{"className":98092,"translate":35},[34],[31,98094,98096],{"className":98095},[39],[41,98097,98098],{"xmlns":43},[45,98099,98100,98104],{},[48,98101,98102],{},[51,98103,14551],{},[55,98105,14551],{"encoding":57},[31,98107,98109],{"className":98108,"ariaHidden":62},[61],[31,98110,98112,98115],{"className":98111},[66],[31,98113],{"className":98114,"style":71},[70],[31,98116,14551],{"className":98117},[75,76],"，两个动作 ",[31,98120,98122,98148],{"className":98121,"translate":35},[34],[31,98123,98125],{"className":98124},[39],[41,98126,98127],{"xmlns":43},[45,98128,98129,98145],{},[48,98130,98131,98137,98139],{},[1004,98132,98133,98135],{},[51,98134,5813],{},[91,98136,205],{},[95,98138,1210],{"separator":62},[1004,98140,98141,98143],{},[51,98142,5813],{},[91,98144,152],{},[55,98146,98147],{"encoding":57},"a_1,a_2",[31,98149,98151],{"className":98150,"ariaHidden":62},[61],[31,98152,98154,98157,98197,98200,98203],{"className":98153},[66],[31,98155],{"className":98156,"style":865},[70],[31,98158,98160,98163],{"className":98159},[75],[31,98161,5813],{"className":98162},[75,76],[31,98164,98166],{"className":98165},[1039],[31,98167,98169,98189],{"className":98168},[1043,1044],[31,98170,98172,98186],{"className":98171},[1048],[31,98173,98175],{"className":98174,"style":2028},[1052],[31,98176,98177,98180],{"style":2807},[31,98178],{"className":98179,"style":1061},[1060],[31,98181,98183],{"className":98182},[1065,1066,1067,1068],[31,98184,205],{"className":98185},[75,1068],[31,98187,1076],{"className":98188},[1075],[31,98190,98192],{"className":98191},[1048],[31,98193,98195],{"className":98194,"style":2050},[1052],[31,98196],{},[31,98198,1210],{"className":98199},[1391],[31,98201],{"className":98202,"style":1467},[119],[31,98204,98206,98209],{"className":98205},[75],[31,98207,5813],{"className":98208},[75,76],[31,98210,98212],{"className":98211},[1039],[31,98213,98215,98235],{"className":98214},[1043,1044],[31,98216,98218,98232],{"className":98217},[1048],[31,98219,98221],{"className":98220,"style":2028},[1052],[31,98222,98223,98226],{"style":2807},[31,98224],{"className":98225,"style":1061},[1060],[31,98227,98229],{"className":98228},[1065,1066,1067,1068],[31,98230,152],{"className":98231},[75,1068],[31,98233,1076],{"className":98234},[1075],[31,98236,98238],{"className":98237},[1048],[31,98239,98241],{"className":98240,"style":2050},[1052],[31,98242],{},"。下一状态以 ",[31,98245,98247,98264],{"className":98246,"translate":35},[34],[31,98248,98250],{"className":98249},[39],[41,98251,98252],{"xmlns":43},[45,98253,98254,98262],{},[48,98255,98256,98258,98260],{},[91,98257,205],{},[51,98259,3768],{"mathvariant":494},[91,98261,152],{},[55,98263,17145],{"encoding":57},[31,98265,98267],{"className":98266,"ariaHidden":62},[61],[31,98268,98270,98273],{"className":98269},[66],[31,98271],{"className":98272,"style":260},[70],[31,98274,17145],{"className":98275},[75]," 的概率为 ",[31,98278,98280,98298],{"className":98279,"translate":35},[34],[31,98281,98283],{"className":98282},[39],[41,98284,98285],{"xmlns":43},[45,98286,98287,98295],{},[48,98288,98289],{},[1004,98290,98291,98293],{},[51,98292,14551],{},[91,98294,205],{},[55,98296,98297],{"encoding":57},"s_1",[31,98299,98301],{"className":98300,"ariaHidden":62},[61],[31,98302,98304,98307],{"className":98303},[66],[31,98305],{"className":98306,"style":7090},[70],[31,98308,98310,98313],{"className":98309},[75],[31,98311,14551],{"className":98312},[75,76],[31,98314,98316],{"className":98315},[1039],[31,98317,98319,98339],{"className":98318},[1043,1044],[31,98320,98322,98336],{"className":98321},[1048],[31,98323,98325],{"className":98324,"style":2028},[1052],[31,98326,98327,98330],{"style":2807},[31,98328],{"className":98329,"style":1061},[1060],[31,98331,98333],{"className":98332},[1065,1066,1067,1068],[31,98334,205],{"className":98335},[75,1068],[31,98337,1076],{"className":98338},[1075],[31,98340,98342],{"className":98341},[1048],[31,98343,98345],{"className":98344,"style":2050},[1052],[31,98346],{},"，以 ",[31,98349,98351,98368],{"className":98350,"translate":35},[34],[31,98352,98354],{"className":98353},[39],[41,98355,98356],{"xmlns":43},[45,98357,98358,98366],{},[48,98359,98360,98362,98364],{},[91,98361,205],{},[51,98363,3768],{"mathvariant":494},[91,98365,152],{},[55,98367,17145],{"encoding":57},[31,98369,98371],{"className":98370,"ariaHidden":62},[61],[31,98372,98374,98377],{"className":98373},[66],[31,98375],{"className":98376,"style":260},[70],[31,98378,17145],{"className":98379},[75],[31,98381,98383,98401],{"className":98382,"translate":35},[34],[31,98384,98386],{"className":98385},[39],[41,98387,98388],{"xmlns":43},[45,98389,98390,98398],{},[48,98391,98392],{},[1004,98393,98394,98396],{},[51,98395,14551],{},[91,98397,152],{},[55,98399,98400],{"encoding":57},"s_2",[31,98402,98404],{"className":98403,"ariaHidden":62},[61],[31,98405,98407,98410],{"className":98406},[66],[31,98408],{"className":98409,"style":7090},[70],[31,98411,98413,98416],{"className":98412},[75],[31,98414,14551],{"className":98415},[75,76],[31,98417,98419],{"className":98418},[1039],[31,98420,98422,98442],{"className":98421},[1043,1044],[31,98423,98425,98439],{"className":98424},[1048],[31,98426,98428],{"className":98427,"style":2028},[1052],[31,98429,98430,98433],{"style":2807},[31,98431],{"className":98432,"style":1061},[1060],[31,98434,98436],{"className":98435},[1065,1066,1067,1068],[31,98437,152],{"className":98438},[75,1068],[31,98440,1076],{"className":98441},[1075],[31,98443,98445],{"className":98444},[1048],[31,98446,98448],{"className":98447,"style":2050},[1052],[31,98449],{},"，并设未来价值为零。奖励为",[31,98452,98454],{"className":98453,"translate":35},[411],[31,98455,98457,98593],{"className":98456,"translate":35},[34],[31,98458,98460],{"className":98459},[39],[41,98461,98462],{"xmlns":43,"display":420},[45,98463,98464,98590],{},[48,98465,98466,98468,98470,98472,98474,98480,98482,98488,98490,98492,98494,98496,98498,98500,98502,98504,98506,98512,98514,98520,98522,98524,98526,98528,98530,98532,98534,98536,98538,98544,98546,98552,98554,98556,98558,98560,98562,98564,98566,98568,98570,98576,98578,98584,98586,98588],{},[51,98467,2549],{},[95,98469,211],{"stretchy":210},[51,98471,14551],{},[95,98473,1210],{"separator":62},[1004,98475,98476,98478],{},[51,98477,5813],{},[91,98479,205],{},[95,98481,1210],{"separator":62},[1004,98483,98484,98486],{},[51,98485,14551],{},[91,98487,205],{},[95,98489,220],{"stretchy":210},[95,98491,223],{},[91,98493,205],{},[95,98495,1210],{"separator":62},[119,98497],{"width":1584},[51,98499,2549],{},[95,98501,211],{"stretchy":210},[51,98503,14551],{},[95,98505,1210],{"separator":62},[1004,98507,98508,98510],{},[51,98509,5813],{},[91,98511,205],{},[95,98513,1210],{"separator":62},[1004,98515,98516,98518],{},[51,98517,14551],{},[91,98519,152],{},[95,98521,220],{"stretchy":210},[95,98523,223],{},[91,98525,1977],{},[95,98527,1210],{"separator":62},[119,98529],{"width":20286},[51,98531,2549],{},[95,98533,211],{"stretchy":210},[51,98535,14551],{},[95,98537,1210],{"separator":62},[1004,98539,98540,98542],{},[51,98541,5813],{},[91,98543,152],{},[95,98545,1210],{"separator":62},[1004,98547,98548,98550],{},[51,98549,14551],{},[91,98551,205],{},[95,98553,220],{"stretchy":210},[95,98555,223],{},[91,98557,1977],{},[95,98559,1210],{"separator":62},[119,98561],{"width":1584},[51,98563,2549],{},[95,98565,211],{"stretchy":210},[51,98567,14551],{},[95,98569,1210],{"separator":62},[1004,98571,98572,98574],{},[51,98573,5813],{},[91,98575,152],{},[95,98577,1210],{"separator":62},[1004,98579,98580,98582],{},[51,98581,14551],{},[91,98583,152],{},[95,98585,220],{"stretchy":210},[95,98587,223],{},[91,98589,78463],{},[55,98591,98592],{"encoding":57},"r(s,a_1,s_1)=1,\\quad r(s,a_1,s_2)=0,\n\\qquad\nr(s,a_2,s_1)=0,\\quad r(s,a_2,s_2)=1.",[31,98594,98596,98715,98846,98977,99108],{"className":98595,"ariaHidden":62},[61],[31,98597,98599,98602,98605,98608,98611,98614,98617,98657,98660,98663,98703,98706,98709,98712],{"className":98598},[66],[31,98600],{"className":98601,"style":260},[70],[31,98603,2549],{"className":98604,"style":19759},[75,76],[31,98606,211],{"className":98607},[264],[31,98609,14551],{"className":98610},[75,76],[31,98612,1210],{"className":98613},[1391],[31,98615],{"className":98616,"style":1467},[119],[31,98618,98620,98623],{"className":98619},[75],[31,98621,5813],{"className":98622},[75,76],[31,98624,98626],{"className":98625},[1039],[31,98627,98629,98649],{"className":98628},[1043,1044],[31,98630,98632,98646],{"className":98631},[1048],[31,98633,98635],{"className":98634,"style":2028},[1052],[31,98636,98637,98640],{"style":2807},[31,98638],{"className":98639,"style":1061},[1060],[31,98641,98643],{"className":98642},[1065,1066,1067,1068],[31,98644,205],{"className":98645},[75,1068],[31,98647,1076],{"className":98648},[1075],[31,98650,98652],{"className":98651},[1048],[31,98653,98655],{"className":98654,"style":2050},[1052],[31,98656],{},[31,98658,1210],{"className":98659},[1391],[31,98661],{"className":98662,"style":1467},[119],[31,98664,98666,98669],{"className":98665},[75],[31,98667,14551],{"className":98668},[75,76],[31,98670,98672],{"className":98671},[1039],[31,98673,98675,98695],{"className":98674},[1043,1044],[31,98676,98678,98692],{"className":98677},[1048],[31,98679,98681],{"className":98680,"style":2028},[1052],[31,98682,98683,98686],{"style":2807},[31,98684],{"className":98685,"style":1061},[1060],[31,98687,98689],{"className":98688},[1065,1066,1067,1068],[31,98690,205],{"className":98691},[75,1068],[31,98693,1076],{"className":98694},[1075],[31,98696,98698],{"className":98697},[1048],[31,98699,98701],{"className":98700,"style":2050},[1052],[31,98702],{},[31,98704,220],{"className":98705},[289],[31,98707],{"className":98708,"style":293},[119],[31,98710,223],{"className":98711},[297],[31,98713],{"className":98714,"style":293},[119],[31,98716,98718,98721,98724,98727,98730,98733,98736,98739,98742,98745,98748,98788,98791,98794,98834,98837,98840,98843],{"className":98717},[66],[31,98719],{"className":98720,"style":260},[70],[31,98722,205],{"className":98723},[75],[31,98725,1210],{"className":98726},[1391],[31,98728],{"className":98729,"style":1789},[119],[31,98731],{"className":98732,"style":1467},[119],[31,98734,2549],{"className":98735,"style":19759},[75,76],[31,98737,211],{"className":98738},[264],[31,98740,14551],{"className":98741},[75,76],[31,98743,1210],{"className":98744},[1391],[31,98746],{"className":98747,"style":1467},[119],[31,98749,98751,98754],{"className":98750},[75],[31,98752,5813],{"className":98753},[75,76],[31,98755,98757],{"className":98756},[1039],[31,98758,98760,98780],{"className":98759},[1043,1044],[31,98761,98763,98777],{"className":98762},[1048],[31,98764,98766],{"className":98765,"style":2028},[1052],[31,98767,98768,98771],{"style":2807},[31,98769],{"className":98770,"style":1061},[1060],[31,98772,98774],{"className":98773},[1065,1066,1067,1068],[31,98775,205],{"className":98776},[75,1068],[31,98778,1076],{"className":98779},[1075],[31,98781,98783],{"className":98782},[1048],[31,98784,98786],{"className":98785,"style":2050},[1052],[31,98787],{},[31,98789,1210],{"className":98790},[1391],[31,98792],{"className":98793,"style":1467},[119],[31,98795,98797,98800],{"className":98796},[75],[31,98798,14551],{"className":98799},[75,76],[31,98801,98803],{"className":98802},[1039],[31,98804,98806,98826],{"className":98805},[1043,1044],[31,98807,98809,98823],{"className":98808},[1048],[31,98810,98812],{"className":98811,"style":2028},[1052],[31,98813,98814,98817],{"style":2807},[31,98815],{"className":98816,"style":1061},[1060],[31,98818,98820],{"className":98819},[1065,1066,1067,1068],[31,98821,152],{"className":98822},[75,1068],[31,98824,1076],{"className":98825},[1075],[31,98827,98829],{"className":98828},[1048],[31,98830,98832],{"className":98831,"style":2050},[1052],[31,98833],{},[31,98835,220],{"className":98836},[289],[31,98838],{"className":98839,"style":293},[119],[31,98841,223],{"className":98842},[297],[31,98844],{"className":98845,"style":293},[119],[31,98847,98849,98852,98855,98858,98861,98864,98867,98870,98873,98876,98879,98919,98922,98925,98965,98968,98971,98974],{"className":98848},[66],[31,98850],{"className":98851,"style":260},[70],[31,98853,1977],{"className":98854},[75],[31,98856,1210],{"className":98857},[1391],[31,98859],{"className":98860,"style":20525},[119],[31,98862],{"className":98863,"style":1467},[119],[31,98865,2549],{"className":98866,"style":19759},[75,76],[31,98868,211],{"className":98869},[264],[31,98871,14551],{"className":98872},[75,76],[31,98874,1210],{"className":98875},[1391],[31,98877],{"className":98878,"style":1467},[119],[31,98880,98882,98885],{"className":98881},[75],[31,98883,5813],{"className":98884},[75,76],[31,98886,98888],{"className":9888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之后才选动作，因此获得了更大的灵活性，但它不是原问题的可行决策顺序。贝尔曼最优方程采用前者，是对决策时序的正确编码。",[52860,99734,99736],{"id":99735},"_44-验证定理与最优策略","4.4 验证定理与最优策略",[52865,99738,99741,100034,100521],{"id":99739,"title":99740},"corollary-optimal-verification","推论 4.3（最优性的验证定理）",[11,99742,52871,99743,69116,99808,67262,99894,99975,99976,100004,100005,100033],{},[31,99744,99746,99769],{"className":99745,"translate":35},[34],[31,99747,99749],{"className":99748},[39],[41,99750,99751],{"xmlns":43},[45,99752,99753,99767],{},[48,99754,99755,99757,99759,99761,99763,99765],{},[51,99756,43481],{},[95,99758,2518],{},[51,99760,1122],{},[95,99762,211],{"stretchy":210},[51,99764,21582],{},[95,99766,220],{"stretchy":210},[55,99768,72399],{"encoding":57},[31,99770,99772,99790],{"className":99771,"ariaHidden":62},[61],[31,99773,99775,99778,99781,99784,99787],{"className":99774},[66],[31,99776],{"className":99777,"style":65709},[70],[31,99779,43481],{"className":99780,"style":120},[75,76],[31,99782],{"className":99783,"style":293},[119],[31,99785,2518],{"className":99786},[297],[31,99788],{"className":99789,"style":293},[119],[31,99791,99793,99796,99799,99802,99805],{"className":99792},[66],[31,99794],{"className":99795,"style":260},[70],[31,99797,1122],{"className":99798,"style":1137},[75,76],[31,99800,211],{"className":99801},[264],[31,99803,21582],{"className":99804,"style":21697},[75,76],[31,99806,220],{"className":99807},[289],[31,99809,99811,99835],{"className":99810,"translate":35},[34],[31,99812,99814],{"className":99813},[39],[41,99815,99816],{"xmlns":43},[45,99817,99818,99832],{},[48,99819,99820,99822,99824,99830],{},[51,99821,43481],{},[95,99823,223],{},[1180,99825,99826,99828],{},[51,99827,919],{"mathvariant":925},[95,99829,30648],{},[51,99831,43481],{},[55,99833,99834],{"encoding":57},"V=\\mathcal{T}^*V",[31,99836,99838,99856],{"className":99837,"ariaHidden":62},[61],[31,99839,99841,99844,99847,99850,99853],{"className":99840},[66],[31,99842],{"className":99843,"style":945},[70],[31,99845,43481],{"className":99846,"style":120},[75,76],[31,99848],{"className":99849,"style":293},[119],[31,99851,223],{"className":99852},[297],[31,99854],{"className":99855,"style":293},[119],[31,99857,99859,99862,99891],{"className":99858},[66],[31,99860],{"className":99861,"style":30719},[70],[31,99863,99865,99868],{"className":99864},[75],[31,99866,919],{"className":99867,"style":64444},[75,968],[31,99869,99871],{"className":99870},[1039],[31,99872,99874],{"className":99873},[1043],[31,99875,99877],{"className":99876},[1048],[31,99878,99880],{"className":99879,"style":30719},[1052],[31,99881,99882,99885],{"style":3631},[31,99883],{"className":99884,"style":1061},[1060],[31,99886,99888],{"className":99887},[1065,1066,1067,1068],[31,99889,30648],{"className":99890},[124,1068],[31,99892,43481],{"className":99893,"style":120},[75,76],[31,99895,99897,99919],{"className":99896,"translate":35},[34],[31,99898,99900],{"className":99899},[39],[41,99901,99902],{"xmlns":43},[45,99903,99904,99916],{},[48,99905,99906,99908,99910],{},[51,99907,43481],{},[95,99909,223],{},[1180,99911,99912,99914],{},[51,99913,43481],{},[95,99915,30648],{},[55,99917,99918],{"encoding":57},"V=V^*",[31,99920,99922,99940],{"className":99921,"ariaHidden":62},[61],[31,99923,99925,99928,99931,99934,99937],{"className":99924},[66],[31,99926],{"className":99927,"style":945},[70],[31,99929,43481],{"className":99930,"style":120},[75,76],[31,99932],{"className":99933,"style":293},[119],[31,99935,223],{"className":99936},[297],[31,99938],{"className":99939,"style":293},[119],[31,99941,99943,99946],{"className":99942},[66],[31,99944],{"className":99945,"style":30719},[70],[31,99947,99949,99952],{"className":99948},[75],[31,99950,43481],{"className":99951,"style":120},[75,76],[31,99953,99955],{"className":99954},[1039],[31,99956,99958],{"className":99957},[1043],[31,99959,99961],{"className":99960},[1048],[31,99962,99964],{"className":99963,"style":30719},[1052],[31,99965,99966,99969],{"style":3631},[31,99967],{"className":99968,"style":1061},[1060],[31,99970,99972],{"className":99971},[1065,1066,1067,1068],[31,99973,30648],{"className":99974},[124,1068],"。若策略 ",[31,99977,99979,99992],{"className":99978,"translate":35},[34],[31,99980,99982],{"className":99981},[39],[41,99983,99984],{"xmlns":43},[45,99985,99986,99990],{},[48,99987,99988],{},[51,99989,2515],{},[55,99991,3485],{"encoding":57},[31,99993,99995],{"className":99994,"ariaHidden":62},[61],[31,99996,99998,100001],{"className":99997},[66],[31,99999],{"className":100000,"style":71},[70],[31,100002,2515],{"className":100003,"style":2666},[75,76]," 关于 ",[31,100006,100008,100021],{"className":100007,"translate":35},[34],[31,100009,100011],{"className":100010},[39],[41,100012,100013],{"xmlns":43},[45,100014,100015,100019],{},[48,100016,100017],{},[51,100018,43481],{},[55,100020,43481],{"encoding":57},[31,100022,100024],{"className":100023,"ariaHidden":62},[61],[31,100025,100027,100030],{"className":100026},[66],[31,100028],{"className":100029,"style":945},[70],[31,100031,43481],{"className":100032,"style":120},[75,76]," 贪婪，即",[31,100035,100037],{"className":100036,"translate":35},[411],[31,100038,100040,100150],{"className":100039,"translate":35},[34],[31,100041,100043],{"className":100042},[39],[41,100044,100045],{"xmlns":43,"display":420},[45,100046,100047,100147],{},[48,100048,100049,100051,100053,100055,100057,100059,100061,100063,100073,100083,100085,100087,100093,100095,100097,100099,100101,100103,100145],{},[51,100050,2515],{},[95,100052,211],{"stretchy":210},[51,100054,14551],{},[95,100056,220],{"stretchy":210},[95,100058,2518],{},[51,100060,37263],{},[95,100062,1540],{},[1531,100064,100065,100071],{},[48,100066,100067,100069],{},[51,100068,10154],{},[95,100070,1540],{},[51,100072,5813],{},[1531,100074,100075,100077],{},[95,100076,5625],{},[1180,100078,100079,100081],{},[51,100080,14551],{},[95,100082,13691],{"mathvariant":494,"lspace":5610,"rspace":5610},[51,100084,2521],{},[95,100086,211],{"stretchy":210},[1180,100088,100089,100091],{},[51,100090,14551],{},[95,100092,13691],{"mathvariant":494,"lspace":5610,"rspace":5610},[95,100094,12680],{},[51,100096,14551],{},[95,100098,1210],{"separator":62},[51,100100,5813],{},[95,100102,220],{"stretchy":210},[48,100104,100105,100107,100109,100111,100113,100115,100117,100119,100125,100127,100129,100131,100133,100135,100141,100143],{},[95,100106,15175],{"fence":62},[51,100108,2549],{},[95,100110,211],{"stretchy":210},[51,100112,14551],{},[95,100114,1210],{"separator":62},[51,100116,5813],{},[95,100118,1210],{"separator":62},[1180,100120,100121,100123],{},[51,100122,14551],{},[95,100124,13691],{"mathvariant":494,"lspace":5610,"rspace":5610},[95,100126,220],{"stretchy":210},[95,100128,434],{},[51,100130,53298],{},[51,100132,43481],{},[95,100134,211],{"stretchy":210},[1180,100136,100137,100139],{},[51,100138,14551],{},[95,100140,13691],{"mathvariant":494,"lspace":5610,"rspace":5610},[95,100142,220],{"stretchy":210},[95,100144,15184],{"fence":62},[95,100146,1210],{"separator":62},[55,100148,100149],{"encoding":57},"\\pi(s)\\in\\arg\\max_{a}\\sum_{s'}P(s'\\mid s,a)\n\\left[r(s,a,s')+\\gamma V(s')\\right],",[31,100151,100153,100180,100370],{"className":100152,"ariaHidden":62},[61],[31,100154,100156,100159,100162,100165,100168,100171,100174,100177],{"className":100155},[66],[31,100157],{"className":100158,"style":260},[70],[31,100160,2515],{"className":100161,"style":2666},[75,76],[31,100163,211],{"className":100164},[264],[31,100166,14551],{"className":100167},[75,76],[31,100169,220],{"className":100170},[289],[31,100172],{"className":100173,"style":293},[119],[31,100175,2518],{"className":100176},[297],[31,100178],{"className":100179,"style":293},[119],[31,100181,100183,100186,100191,100194,100241,100244,100320,100323,100326,100329,100361,100364,100367],{"className":100182},[66],[31,100184],{"className":100185,"style":74897},[70],[31,100187,37341,100189],{"className":100188},[1460],[31,100190,14556],{"style":20180},[31,100192],{"className":100193,"style":1467},[119],[31,100195,100197],{"className":100196},[1460,1620],[31,100198,100200,100233],{"className":100199},[1043,1044],[31,100201,100203,100230],{"className":100202},[1048],[31,100204,100206,100220],{"className":100205,"style":71},[1052],[31,100207,100208,100211],{"style":37361},[31,100209],{"className":100210,"style":1637},[1060],[31,100212,100214],{"className":100213},[1065,1066,1067,1068],[31,100215,100217],{"className":100216},[75,1068],[31,100218,5813],{"className":100219},[75,76,1068],[31,100221,100222,100225],{"style":1649},[31,100223],{"className":100224,"style":1637},[1060],[31,100226,100227],{},[31,100228,10154],{"className":100229},[1460],[31,100231,1076],{"className":100232},[1075],[31,100234,100236],{"className":100235},[1048],[31,100237,100239],{"className":100238,"style":37390},[1052],[31,100240],{},[31,100242],{"className":100243,"style":1467},[119],[31,100245,100247],{"className":100246},[1460,1620],[31,100248,100250,100312],{"className":100249},[1043,1044],[31,100251,100253,100309],{"className":100252},[1048],[31,100254,100256,100299],{"className":100255,"style":5900},[1052],[31,100257,100258,100261],{"style":59756},[31,100259],{"className":100260,"style":5907},[1060],[31,100262,100264],{"className":100263},[1065,1066,1067,1068],[31,100265,100267],{"className":100266},[75,1068],[31,100268,100270,100273],{"className":100269},[75,1068],[31,100271,14551],{"className":100272},[75,76,1068],[31,100274,100276],{"className":100275},[1039],[31,100277,100279],{"className":100278},[1043],[31,100280,100282],{"className":100281},[1048],[31,100283,100285],{"className":100284,"style":31478},[1052],[31,100286,100287,100290],{"style":31481},[31,100288],{"className":100289,"style":11769},[1060],[31,100291,100293],{"className":100292},[1065,11773,1308,1068],[31,100294,100296],{"className":100295},[75,1068],[31,100297,13691],{"className":100298},[75,1068],[31,100300,100301,100304],{"style":5919},[31,100302],{"className":100303,"style":5907},[1060],[31,100305,100306],{},[31,100307,5625],{"className":100308},[1460,1461,1687],[31,100310,1076],{"className":100311},[1075],[31,100313,100315],{"className":100314},[1048],[31,100316,100318],{"className":100317,"style":59817},[1052],[31,100319],{},[31,100321],{"className":100322,"style":1467},[119],[31,100324,2521],{"className":100325,"style":949},[75,76],[31,100327,211],{"className":100328},[264],[31,100330,100332,100335],{"className":100331},[75],[31,100333,14551],{"className":100334},[75,76],[31,100336,100338],{"className":100337},[1039],[31,100339,100341],{"className":100340},[1043],[31,100342,100344],{"className":100343},[1048],[31,100345,100347],{"className":100346,"style":14017},[1052],[31,100348,100349,100352],{"style":1330},[31,100350],{"className":100351,"style":1061},[1060],[31,100353,100355],{"className":100354},[1065,1066,1067,1068],[31,100356,100358],{"className":100357},[75,1068],[31,100359,13691],{"className":100360},[75,1068],[31,100362],{"className":100363,"style":293},[119],[31,100365,12680],{"className":100366},[297],[31,100368],{"className":100369,"style":293},[119],[31,100371,100373,100376,100379,100382,100385,100388,100391,100394,100515,100518],{"className":100372},[66],[31,100374],{"className":100375,"style":75080},[70],[31,100377,14551],{"className":100378},[75,76],[31,100380,1210],{"className":100381},[1391],[31,100383],{"className":100384,"style":1467},[119],[31,100386,5813],{"className":100387},[75,76],[31,100389,220],{"className":100390},[289],[31,100392],{"className":100393,"style":1467},[119],[31,100395,100397,100400,100403,100406,100409,100412,100415,100418,100421,100424,100456,100459,100462,100465,100468,100471,100474,100477,100509,100512],{"className":100396},[3860],[31,100398,15175],{"className":100399,"style":3868},[264,3867],[31,100401,2549],{"className":100402,"style":19759},[75,76],[31,100404,211],{"className":100405},[264],[31,100407,14551],{"className":100408},[75,76],[31,100410,1210],{"className":100411},[1391],[31,100413],{"className":100414,"style":1467},[119],[31,100416,5813],{"className":100417},[75,76],[31,100419,1210],{"className":100420},[1391],[31,1004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是最优策略。",[11,100634,100635,100637,100638,100696,100697,100004,100725,100753,100754,94853,100897,1138,100899,73845],{},[7246,100636,7248],{}," 第一式由 ",[31,100639,100641,100658],{"className":100640,"translate":35},[34],[31,100642,100644],{"className":100643},[39],[41,100645,100646],{"xmlns":43},[45,100647,100648,100656],{},[48,100649,100650],{},[1180,100651,100652,100654],{},[51,100653,919],{"mathvariant":925},[95,100655,30648],{},[55,100657,74501],{"encoding":57},[31,100659,100661],{"className":100660,"ariaHidden":62},[61],[31,100662,100664,100667],{"className":100663},[66],[31,100665],{"className":100666,"style":30719},[70],[31,100668,100670,100673],{"className":100669},[75],[31,100671,919],{"className":100672,"style":64444},[75,968],[31,100674,100676],{"className":100675},[1039],[31,100677,100679],{"className":100678},[1043],[31,100680,100682],{"className":100681},[1048],[31,100683,100685],{"className":100684,"style":30719},[1052],[31,100686,100687,100690],{"style":3631},[31,100688],{"className":100689,"style":1061},[1060],[31,100691,100693],{"className":100692},[1065,1066,1067,1068],[31,100694,30648],{"className":100695},[124,1068]," 的不动点唯一性立即得到。若 ",[31,100698,100700,100713],{"className":100699,"translate":35},[34],[31,100701,100703],{"className":100702},[39],[41,100704,100705],{"xmlns":43},[45,100706,100707,100711],{},[48,100708,100709],{},[51,100710,2515],{},[55,100712,3485],{"encoding":57},[31,100714,100716],{"className":100715,"ariaHidden":62},[61],[31,100717,100719,100722],{"className":100718},[66],[31,100720],{"className":100721,"style":71},[70],[31,100723,2515],{"className":100724,"style":2666},[75,76],[31,100726,100728,100741],{"className":100727,"translate":35},[34],[31,100729,100731],{"className":100730},[39],[41,100732,100733],{"xmlns":43},[45,100734,100735,100739],{},[48,100736,100737],{},[51,100738,43481],{},[55,100740,43481],{"encoding":57},[31,100742,100744],{"className":100743,"ariaHidden":62},[61],[31,100745,100747,100750],{"className":100746},[66],[31,100748],{"className":100749,"style":945},[70],[31,100751,43481],{"className":100752,"style":120},[75,76]," 贪婪，则 ",[31,100755,100757,100791],{"className":100756,"translate":35},[34],[31,100758,100760],{"className":100759},[39],[41,100761,100762],{"xmlns":43},[45,100763,100764,100788],{},[48,100765,100766,100772,100774,100776,100782,100784,100786],{},[1180,100767,100768,100770],{},[51,100769,919],{"mathvariant":925},[51,100771,2515],{},[51,100773,43481],{},[95,100775,223],{},[1180,100777,100778,100780],{},[51,100779,919],{"mathvariant":925},[95,100781,30648],{},[51,100783,43481],{},[95,100785,223],{},[51,100787,43481],{},[55,100789,100790],{"encoding":57},"\\mathcal{T}^\\pi V=\\mathcal{T}^*V=V",[31,100792,100794,100841,100888],{"className":100793,"ariaHidden":62},[61],[31,100795,100797,100800,100829,100832,100835,100838],{"className":100796},[66],[31,100798],{"className":100799,"style":945},[70],[31,100801,100803,100806],{"className":100802},[75],[31,100804,919],{"className":100805,"style":64444},[75,968],[31,100807,100809],{"className":100808},[1039],[31,100810,100812],{"className":100811},[1043],[31,100813,100815],{"className":100814},[1048],[31,100816,100818],{"className":100817,"style":3628},[1052],[31,100819,100820,100823],{"style":3631},[31,100821],{"className":100822,"style":1061},[1060],[31,100824,100826],{"className":100825},[1065,1066,1067,1068],[31,100827,2515],{"className":100828,"style":2666},[75,76,1068],[31,100830,43481],{"className":100831,"style":120},[75,76],[31,100833],{"className":100834,"style":293},[119],[31,100836,223],{"className":100837},[297],[31,100839],{"className":100840,"style":293},[119],[31,100842,100844,100847,100876,100879,100882,100885],{"className":100843},[66],[31,100845],{"className":100846,"style":30719},[70],[31,100848,100850,100853],{"className":100849},[75],[31,100851,919],{"className":100852,"style":64444},[75,968],[31,100854,100856],{"className":100855},[1039],[31,100857,100859],{"className":100858},[1043],[31,100860,100862],{"className":100861},[1048],[31,100863,100865],{"className":100864,"style":30719},[1052],[31,100866,100867,100870],{"style":3631},[31,100868],{"className":100869,"style":1061},[1060],[31,100871,100873],{"className":100872},[1065,1066,1067,1068],[31,100874,30648],{"className":100875},[124,1068],[31,100877,43481],{"className":100878,"style":120},[75,76],[31,100880],{"className":100881,"style":293},[119],[31,100883,223],{"className":100884},[297],[31,100886],{"className":100887,"style":293},[119],[31,100889,100891,100894],{"className":100890},[66],[31,100892],{"className":100893,"style":945},[70],[31,100895,43481],{"className":100896,"style":120},[75,76],[5813,100898,73911],{"href":73910},[31,100900,100902,100923],{"className":100901,"translate":35},[34],[31,100903,100905],{"className":100904},[39],[41,100906,100907],{"xmlns":43},[45,100908,100909,100921],{},[48,100910,100911,100917,100919],{},[1180,100912,100913,100915],{},[51,100914,43481],{},[51,100916,2515],{},[95,100918,223],{},[51,100920,43481],{},[55,100922,100547],{"encoding":57},[31,100924,100926,100970],{"className":100925,"ariaHidden":62},[61],[31,100927,100929,100932,100961,100964,100967],{"className":100928},[66],[31,100930],{"className":100931,"style":945},[70],[31,100933,100935,100938],{"className":100934},[75],[31,100936,43481],{"className":100937,"style":120},[75,76],[31,100939,100941],{"className":100940},[1039],[31,100942,100944],{"className":100943},[1043],[31,100945,100947],{"className":100946},[1048],[31,100948,100950],{"className":100949,"style":3628},[1052],[31,100951,100952,100955],{"style":3631},[31,100953],{"className":100954,"style":1061},[1060],[31,100956,100958],{"className":100957},[1065,1066,1067,1068],[31,100959,2515],{"className":100960,"style":2666},[75,76,1068],[31,100962],{"className":100963,"style":293},[119],[31,100965,223],{"className":100966},[297],[31,100968],{"className":100969,"style":293},[119],[31,100971,100973,100976],{"className":100972},[66],[31,100974],{"className":100975,"style":945},[70],[31,100977,43481],{"className":100978,"style":120},[75,76],[11,100980,100981],{},"这个推论是贝尔曼方程「充分性」的精确表述。它说明，只要能够找到一个有界函数满足最优方程，就不需要再与其他策略比较，因为它必然等于最优值函数。反过来，最优值函数也必然满足该方程。因此，在折扣有界 MDP 中，贝尔曼最优方程既必要又充分。",[52860,100983,100985],{"id":100984},"_45-例带决策的路径","4.5 例：带决策的路径",[11,100987,100988,100989,41892,101070,101098],{},"考虑如下路径选择问题。状态为 ",[31,100990,100992,101022],{"className":100991,"translate":35},[34],[31,100993,100995],{"className":100994},[39],[41,100996,100997],{"xmlns":43},[45,100998,100999,101019],{},[48,101000,101001,101003,101005,101007,101009,101011,101013,101015,101017],{},[51,101002,18446],{},[95,101004,1210],{"separator":62},[51,101006,1122],{},[95,101008,1210],{"separator":62},[51,101010,8101],{},[95,101012,1210],{"separator":62},[51,101014,34614],{},[95,101016,1210],{"separator":62},[51,101018,22324],{},[55,101020,101021],{"encoding":57},"A,B,C,D,E",[31,101023,101025],{"className":101024,"ariaHidden":62},[61],[31,101026,101028,101031,101034,101037,101040,101043,101046,101049,101052,101055,101058,101061,101064,101067],{"className":101027},[66],[31,101029],{"className":101030,"style":59114},[70],[31,101032,18446],{"className":101033},[75,76],[31,101035,1210],{"className":101036},[1391],[31,101038],{"className":101039,"style":1467},[119],[31,101041,1122],{"className":101042,"style":1137},[75,76],[31,101044,1210],{"className":101045},[1391],[31,101047],{"className":101048,"style":1467},[119],[31,101050,8101],{"className":101051,"style":8525},[75,76],[31,101053,1210],{"className":101054},[1391],[31,101056],{"className":101057,"style":1467},[119],[31,101059,34614],{"className":101060,"style":19759},[75,76],[31,101062,1210],{"className":101063},[1391],[31,101065],{"className":101066,"style":1467},[119],[31,101068,22324],{"className":101069,"style":21697},[75,76],[31,101071,101073,101086],{"className":101072,"translate":35},[34],[31,101074,101076],{"className":101075},[39],[41,101077,101078],{"xmlns":43},[45,101079,101080,101084],{},[48,101081,101082],{},[51,101083,8101],{},[55,101085,8101],{"encoding":57},[31,101087,101089],{"className":101088,"ariaHidden":62},[61],[31,101090,101092,101095],{"className":101091},[66],[31,101093],{"className":101094,"style":945},[70],[31,101096,8101],{"className":101097,"style":8525},[75,76]," 为终止状态。转移关系为",[31,101100,101102],{"className":101101,"translate":35},[411],[31,101103,101105,101195],{"className":101104,"translate":35},[34],[31,101106,101108],{"className":101107},[39],[41,101109,101110],{"xmlns":43,"display":420},[45,101111,101112,101192],{},[5606,101113,101114,101158],{"rowspacing":5608,"columnalign":5609,"columnspacing":5610},[5612,101115,101116,101122],{},[5615,101117,101118],{},[2071,101119,101120],{"scriptlevel":1977,"displaystyle":62},[48,101121],{},[5615,101123,101124],{},[2071,101125,101126],{"scriptlevel":1977,"displaystyle":62},[48,101127,101128,101130,101132,101134,101136,101138,101140,101142,101144,101146,101148,101150,101152,101154,101156],{},[48,101129],{},[51,101131,18446],{},[95,101133,929],{},[51,101135,1122],{},[95,101137,1210],{"separator":62},[119,101139],{"width":1584},[51,101141,1122],{},[95,101143,929],{},[51,101145,8101],{},[95,101147,1210],{"separator":62},[119,101149],{"width":1584},[51,101151,18446],{},[95,101153,929],{},[51,101155,34614],{},[95,101157,1210],{"separator":62},[5612,101159,101160,101166],{},[5615,101161,101162],{},[2071,101163,101164],{"scriptlevel":1977,"displaystyle":62},[48,101165],{},[5615,101167,101168],{},[2071,101169,101170],{"scriptlevel":1977,"displaystyle":62},[48,101171,101172,101174,101176,101178,101180,101182,101184,101186,101188,101190],{},[48,101173],{},[51,101175,34614],{},[95,101177,929],{},[51,101179,22324],{},[95,101181,1210],{"separator":62},[119,101183],{"width":1584},[51,101185,22324],{},[95,101187,929],{},[51,101189,8101],{},[51,101191,495],{"mathvariant":494},[55,101193,101194],{"encoding":57},"\\begin{aligned}\n&A\\to B,\\quad B\\to C,\\quad A\\to D,\\\\\n&D\\to E,\\quad E\\to C.\n\\end{aligned}",[31,101196,101198],{"className":101197,"ariaHidden":62},[61],[31,101199,101201,101204],{"className":101200},[66],[31,101202],{"className":101203,"style":57050},[70],[31,101205,101207],{"className":101206},[75],[31,101208,101210,101249],{"className":101209},[5606],[31,101211,101213],{"className":101212},[5867],[31,101214,101216,101241],{"className":101215},[1043,1044],[31,101217,101219,101238],{"className":101218},[1048],[31,101220,101222,101230],{"className":101221,"style":57069},[1052],[31,101223,101224,101227],{"style":80971},[31,101225],{"className":101226,"style":80975},[1060],[31,101228],{"className":101229},[75],[31,101231,101232,101235],{"style":58497},[31,101233],{"className":101234,"style":80975},[1060],[31,101236],{"className":101237},[75],[31,101239,1076],{"className":101240},[1075],[31,101242,101244],{"className":101243},[1048],[31,101245,101247],{"className":101246,"style":57182},[1052],[31,101248],{},[31,101250,101252],{"className":101251},[6062],[31,101253,101255,101394],{"className":101254},[1043,1044],[31,101256,101258,101391],{"className":101257},[1048],[31,101259,101261,101338],{"className":101260,"style":57069},[1052],[31,101262,101263,101266],{"style":57072},[31,101264],{"className":101265,"style":1637},[1060],[31,101267,101269,101272,101275,101278,101281,101284,101287,101290,101293,101296,101299,101302,101305,101308,101311,101314,101317,101320,101323,101326,101329,101332,101335],{"className":101268},[75],[31,101270],{"className":101271},[75],[31,101273,18446],{"className":101274},[75,76],[31,101276],{"className":101277,"style":293},[119],[31,101279,929],{"className":101280},[297],[31,101282],{"className":101283,"style":293},[119],[31,101285,1122],{"className":101286,"style":1137},[75,76],[31,101288,1210],{"className":101289},[1391],[31,101291],{"className":101292,"style":1789},[119],[31,101294],{"className":101295,"style":1467},[119],[31,101297,1122],{"className":101298,"style":1137},[75,76],[31,101300],{"className":101301,"style":293},[119],[31,101303,929],{"className":101304},[297],[31,101306],{"className":101307,"style":293},[119],[31,101309,8101],{"className":101310,"style":8525},[75,76],[31,101312,1210],{"className":101313},[1391],[31,101315],{"className":101316,"style":1789},[119],[31,101318],{"className":101319,"style":1467},[119],[31,101321,18446],{"className":101322},[75,76],[31,101324],{"className":101325,"style":293},[119],[31,101327,929],{"className":101328},[297],[31,101330],{"className":101331,"style":293},[119],[31,101333,34614],{"className":101334,"style":19759},[75,76],[31,101336,1210],{"className":101337},[1391],[31,101339,101340,101343],{"style":57119},[31,101341],{"className":101342,"style":1637},[1060],[31,101344,101346,101349,101352,101355,101358,101361,101364,101367,101370,101373,101376,101379,101382,101385,101388],{"className":101345},[75],[31,101347],{"className":101348},[75],[31,101350,34614],{"className":101351,"style":19759},[75,76],[31,101353],{"className":101354,"style":293},[119],[31,101356,929],{"className":101357},[297],[31,101359],{"className":101360,"style":293},[119],[31,101362,22324],{"className":101363,"style":21697},[75,76],[31,101365,1210],{"className":101366},[1391],[31,101368],{"className":101369,"style":1789},[119],[31,101371],{"className":101372,"style":1467},[119],[31,101374,22324],{"className":101375,"style":21697},[75,76],[31,101377],{"className":101378,"style":293},[119],[31,101380,929],{"className":101381},[297],[31,101383],{"className":101384,"style":293},[119],[31,101386,8101],{"className":101387,"style":8525},[75,76],[31,101389,495],{"className":101390},[75],[31,101392,1076],{"className":101393},[1075],[31,101395,101397],{"className":101396},[1048],[31,101398,101400],{"className":101399,"style":57182},[1052],[31,101401],{},[11,101403,101404,101405,101439,101440,101468,101469,101497,101498,101548,101549,2372],{},"每次转移获得奖励 ",[31,101406,101408,101424],{"className":101407,"translate":35},[34],[31,101409,101411],{"className":101410},[39],[41,101412,101413],{"xmlns":43},[45,101414,101415,101421],{},[48,101416,101417,101419],{},[95,101418,97],{},[91,101420,205],{},[55,101422,101423],{"encoding":57},"-1",[31,101425,101427],{"className":101426,"ariaHidden":62},[61],[31,101428,101430,101433,101436],{"className":101429},[66],[31,101431],{"className":101432,"style":112},[70],[31,101434,97],{"className":101435},[75],[31,101437,205],{"className":101438},[75],"，进入终止状态 ",[31,101441,101443,101456],{"className":101442,"translate":35},[34],[31,101444,101446],{"className":101445},[39],[41,101447,101448],{"xmlns":43},[45,101449,101450,101454],{},[48,101451,101452],{},[51,101453,8101],{},[55,101455,8101],{"encoding":57},[31,101457,101459],{"className":101458,"ariaHidden":62},[61],[31,101460,101462,101465],{"className":101461},[66],[31,101463],{"className":101464,"style":945},[70],[31,101466,8101],{"className":101467,"style":8525},[75,76]," 时额外获得奖励 ",[31,101470,101472,101485],{"className":101471,"translate":35},[34],[31,101473,101475],{"className":101474},[39],[41,101476,101477],{"xmlns":43},[45,101478,101479,101483],{},[48,101480,101481],{},[91,101482,19494],{},[55,101484,19494],{"encoding":57},[31,101486,101488],{"className":101487,"ariaHidden":62},[61],[31,101489,101491,101494],{"className":101490},[66],[31,101492],{"className":101493,"style":326},[70],[31,101495,19494],{"className":101496},[75],"。设 ",[31,101499,101501,101518],{"className":101500,"translate":35},[34],[31,101502,101504],{"className":101503},[39],[41,101505,101506],{"xmlns":43},[45,101507,101508,101516],{},[48,101509,101510,101512,101514],{},[51,101511,53298],{},[95,101513,223],{},[91,101515,23247],{},[55,101517,70369],{"encoding":57},[31,101519,101521,101539],{"className":101520,"ariaHidden":62},[61],[31,101522,101524,101527,101530,101533,101536],{"className":101523},[66],[31,101525],{"className":101526,"style":865},[70],[31,101528,53298],{"className":101529,"style":24157},[75,76],[31,101531],{"className":101532,"style":293},[119],[31,101534,223],{"className":101535},[297],[31,101537],{"className":101538,"style":293},[119],[31,101540,101542,101545],{"className":101541},[66],[31,101543],{"className":101544,"style":326},[70],[31,101546,23247],{"className":101547},[75],"，并令 ",[31,101550,101552,101576],{"className":101551,"translate":35},[34],[31,101553,101555],{"className":101554},[39],[41,101556,101557],{"xmlns":43},[45,101558,101559,101573],{},[48,101560,101561,101563,101565,101567,101569,101571],{},[51,101562,43481],{},[95,101564,211],{"stretchy":210},[51,101566,8101],{},[95,101568,220],{"stretchy":210},[95,101570,223],{},[91,101572,1977],{},[55,101574,101575],{"encoding":57},"V(C)=0",[31,101577,101579,101606],{"className":101578,"ariaHidden":62},[61],[31,101580,101582,101585,101588,101591,101594,101597,101600,101603],{"className":101581},[66],[31,101583],{"className":101584,"style":260},[70],[31,101586,43481],{"className":101587,"style":120},[75,76],[31,101589,211],{"className":101590},[264],[31,101592,8101],{"className":101593,"style":8525},[75,76],[31,101595,220],{"className":101596},[289],[31,101598],{"className":101599,"style":293},[119],[31,101601,223],{"className":101602},[297],[31,101604],{"className":101605,"style":293},[119],[31,101607,101609,101612],{"className":101608},[66],[31,101610],{"className":101611,"style":326},[70],[31,101613,1977],{"className":101614},[75],[11,101616,101617,101618,18171,101646,101674,101675,101703],{},"在 ",[31,101619,101621,101634],{"className":101620,"translate":35},[34],[31,101622,101624],{"className":101623},[39],[41,101625,101626],{"xmlns":43},[45,101627,101628,101632],{},[48,101629,101630],{},[51,101631,1122],{},[55,101633,1122],{"encoding":57},[31,101635,101637],{"className":101636,"ariaHidden":62},[61],[31,101638,101640,101643],{"className":101639},[66],[31,101641],{"className":101642,"style":945},[70],[31,101644,1122],{"className":101645,"style":1137},[75,76],[31,101647,101649,101662],{"className":101648,"translate":35},[34],[31,101650,101652],{"className":101651},[39],[41,101653,101654],{"xmlns":43},[45,101655,101656,101660],{},[48,101657,101658],{},[51,101659,22324],{},[55,101661,22324],{"encoding":57},[31,101663,101665],{"className":101664,"ariaHidden":62},[61],[31,101666,101668,101671],{"className":101667},[66],[31,101669],{"className":101670,"style":945},[70],[31,101672,22324],{"className":101673,"style":21697},[75,76],"，只有一条通往 ",[31,101676,101678,101691],{"className":101677,"translate":35},[34],[31,101679,101681],{"className":101680},[39],[41,101682,101683],{"xmlns":43},[45,101684,101685,101689],{},[48,101686,101687],{},[51,101688,8101],{},[55,101690,8101],{"encoding":57},[31,101692,101694],{"className":101693,"ariaHidden":62},[61],[31,101695,101697,101700],{"className":101696},[66],[31,101698],{"className":101699,"style":945},[70],[31,101701,8101],{"className":101702,"style":8525},[75,76]," 的动作，因此",[31,101705,101707],{"className":101706,"translate":35},[411],[31,101708,101710,101773],{"className":101709,"translate":35},[34],[31,101711,101713],{"className":101712},[39],[41,101714,101715],{"xmlns":43,"display":420},[45,101716,101717,101770],{},[48,101718,101719,101721,101723,101725,101727,101729,101731,101733,101735,101737,101739,101741,101743,101745,101747,101749,101751,101753,101755,101757,101759,101761,101763,101765,101767],{},[51,101720,43481],{},[95,101722,211],{"stretchy":210},[51,101724,1122],{},[95,101726,220],{"stretchy":210},[95,101728,223],{},[95,101730,97],{},[91,101732,205],{},[95,101734,434],{},[91,101736,19494],{},[95,101738,434],{},[91,101740,23247],{},[51,101742,43481],{},[95,101744,211],{"stretchy":210},[51,101746,8101],{},[95,101748,220],{"stretchy":210},[95,101750,223],{},[91,101752,81476],{},[95,101754,1210],{"separator":62},[119,101756],{"width":20286},[51,101758,43481],{},[95,101760,211],{"stretchy":210},[51,101762,22324],{},[95,101764,220],{"stretchy":210},[95,101766,223],{},[91,101768,101769],{},"9.",[55,101771,101772],{"encoding":57},"V(B)=-1+10+0.9V(C)=9,\n\\qquad\nV(E)=9.",[31,101774,101776,101803,101824,101842,101872,101911],{"className":101775,"ariaHidden":62},[61],[31,101777,101779,101782,101785,101788,101791,101794,101797,101800],{"className":101778},[66],[31,101780],{"className":101781,"style":260},[70],[31,101783,43481],{"className":101784,"style":120},[75,76],[31,101786,211],{"className":101787},[264],[31,101789,1122],{"className":101790,"style":1137},[75,76],[31,101792,220],{"className":101793},[289],[31,101795],{"className":101796,"style":293},[119],[31,101798,223],{"className":101799},[297],[31,101801],{"className":101802,"style":293},[119],[31,101804,101806,101809,101812,101815,101818,101821],{"className":101805},[66],[31,101807],{"className":101808,"style":112},[70],[31,101810,97],{"className":101811},[75],[31,101813,205],{"className":101814},[75],[31,101816],{"className":101817,"style":120},[119],[31,101819,434],{"className":101820},[124],[31,101822],{"className":101823,"style":120},[119],[31,101825,101827,101830,101833,101836,101839],{"className":101826},[66],[31,101828],{"className":101829,"style":112},[70],[31,101831,19494],{"className":101832},[75],[31,101834],{"className":101835,"style":120},[119],[31,101837,434],{"className":101838},[124],[31,101840],{"className":101841,"style":120},[119],[31,101843,101845,101848,101851,101854,101857,101860,101863,101866,101869],{"className":101844},[66],[31,101846],{"className":101847,"style":260},[70],[31,101849,23247],{"className":101850},[75],[31,101852,43481],{"className":101853,"style":120},[75,76],[31,101855,211],{"className":101856},[264],[31,101858,8101],{"className":101859,"style":8525},[75,76],[31,101861,220],{"className":101862},[289],[31,101864],{"className":101865,"style":293},[119],[31,101867,223],{"className":101868},[297],[31,101870],{"className":101871,"style":293},[119],[31,101873,101875,101878,101881,101884,101887,101890,101893,101896,101899,101902,101905,101908],{"className":101874},[66],[31,101876],{"className":101877,"style":260},[70],[31,101879,81476],{"className":101880},[75],[31,101882,1210],{"className":101883},[1391],[31,101885],{"className":101886,"style":20525},[119],[31,101888],{"className":101889,"style":1467},[119],[31,101891,43481],{"className":101892,"style":120},[75,76],[31,101894,211],{"className":101895},[264],[31,101897,22324],{"className":101898,"style":21697},[75,76],[31,101900,220],{"className":101901},[289],[31,101903],{"className":101904,"style":293},[119],[31,101906,223],{"className":101907},[297],[31,101909],{"className":101910,"style":293},[119],[31,101912,101914,101917],{"className":101913},[66],[31,101915],{"className":101916,"style":326},[70],[31,101918,101769],{"className":101919},[75],[11,101921,101617,101922,101950,101951,101979],{},[31,101923,101925,101938],{"className":101924,"translate":35},[34],[31,101926,101928],{"className":101927},[39],[41,101929,101930],{"xmlns":43},[45,101931,101932,101936],{},[48,101933,101934],{},[51,101935,34614],{},[55,101937,34614],{"encoding":57},[31,101939,101941],{"className":101940,"ariaHidden":62},[61],[31,101942,101944,101947],{"className":101943},[66],[31,101945],{"className":101946,"style":945},[70],[31,101948,34614],{"className":101949,"style":19759},[75,76],"，唯一动作通向 ",[31,101952,101954,101967],{"className":101953,"translate":35},[34],[31,101955,101957],{"className":101956},[39],[41,101958,101959],{"xmlns":43},[45,101960,101961,101965],{},[48,101962,101963],{},[51,101964,22324],{},[55,101966,22324],{"encoding":57},[31,101968,101970],{"className":101969,"ariaHidden":62},[61],[31,101971,101973,101976],{"className":101972},[66],[31,101974],{"className":101975,"style":945},[70],[31,101977,22324],{"className":101978,"style":21697},[75,76],"，故",[31,101981,101983],{"className":101982,"translate":35},[411],[31,101984,101986,102043],{"className":101985,"translate":35},[34],[31,101987,101989],{"className":101988},[39],[41,101990,101991],{"xmlns":43,"display":420},[45,101992,101993,102040],{},[48,101994,101995,101997,101999,102001,102003,102005,102007,102009,102011,102013,102015,102017,102019,102021,102023,102025,102027,102029,102031,102033,102035,102037],{},[51,101996,43481],{},[95,101998,211],{"stretchy":210},[51,102000,34614],{},[95,102002,220],{"stretchy":210},[95,102004,223],{},[95,102006,97],{},[91,102008,205],{},[95,102010,434],{},[91,102012,23247],{},[51,102014,43481],{},[95,102016,211],{"stretchy":210},[51,102018,22324],{},[95,102020,220],{"stretchy":210},[95,102022,223],{},[95,102024,97],{},[91,102026,205],{},[95,102028,434],{},[91,102030,23247],{},[95,102032,2528],{},[91,102034,81476],{},[95,102036,223],{},[91,102038,102039],{},"7.1.",[55,102041,102042],{"encoding":57},"V(D)=-1+0.9V(E)=-1+0.9\\times9=7.1.",[31,102044,102046,102073,102094,102124,102145,102163,102181],{"className":102045,"ariaHidden":62},[61],[31,102047,102049,102052,102055,102058,102061,102064,102067,102070],{"className":102048},[66],[31,102050],{"className":102051,"style":260},[70],[31,102053,43481],{"className":102054,"style":120},[75,76],[31,102056,211],{"className":102057},[264],[31,102059,34614],{"className":102060,"style":19759},[75,76],[31,102062,220],{"className":102063},[289],[31,102065],{"className":102066,"style":293},[119],[31,102068,223],{"className":102069},[297],[31,102071],{"className":102072,"style":293},[119],[31,102074,102076,102079,102082,102085,102088,102091],{"className":102075},[66],[31,102077],{"className":102078,"style":112},[70],[31,102080,97],{"className":102081},[75],[31,102083,205],{"className":102084},[75],[31,102086],{"className":102087,"style":120},[119],[31,102089,434],{"className":102090},[124],[31,102092],{"className":102093,"style":120},[119],[31,102095,102097,102100,102103,102106,102109,102112,102115,102118,102121],{"className":102096},[66],[31,102098],{"className":102099,"style":260},[70],[31,102101,23247],{"className":102102},[75],[31,102104,43481],{"className":102105,"style":120},[75,76],[31,102107,211],{"className":102108},[264],[31,102110,22324],{"className":102111,"style":21697},[75,76],[31,102113,220],{"className":102114},[289],[31,102116],{"className":102117,"style":293},[119],[31,102119,223],{"className":102120},[297],[31,102122],{"className":102123,"style":293},[119],[31,102125,102127,102130,102133,102136,102139,102142],{"className":102126},[66],[31,102128],{"className":102129,"style":112},[70],[31,102131,97],{"className":102132},[75],[31,102134,205],{"className":102135},[75],[31,102137],{"className":102138,"style":120},[119],[31,102140,434],{"className":102141},[124],[31,102143],{"className":102144,"style":120},[119],[31,102146,102148,102151,102154,102157,102160],{"className":102147},[66],[31,102149],{"className":102150,"style":112},[70],[31,102152,23247],{"className":102153},[75],[31,102155],{"className":102156,"style":120},[119],[31,102158,2528],{"className":102159},[124],[31,102161],{"className":102162,"style":120},[119],[31,102164,102166,102169,102172,102175,102178],{"className":102165},[66],[31,102167],{"className":102168,"style":326},[70],[31,102170,81476],{"className":102171},[75],[31,102173],{"className":102174,"style":293},[119],[31,102176,223],{"className":102177},[297],[31,102179],{"className":102180,"style":293},[119],[31,102182,102184,102187],{"className":102183},[66],[31,102185],{"className":102186,"style":326},[70],[31,102188,102039],{"className":102189},[75],[11,102191,101617,102192,102220,102221,18171,102287,102353,102354,102382],{},[31,102193,102195,102208],{"className":102194,"translate":35},[34],[31,102196,102198],{"className":102197},[39],[41,102199,102200],{"xmlns":43},[45,102201,102202,102206],{},[48,102203,102204],{},[51,102205,18446],{},[55,102207,18446],{"encoding":57},[31,102209,102211],{"className":102210,"ariaHidden":62},[61],[31,102212,102214,102217],{"className":102213},[66],[31,102215],{"className":102216,"style":945},[70],[31,102218,18446],{"className":102219},[75,76],"，有两条动作。记 ",[31,102222,102224,102248],{"className":102223,"translate":35},[34],[31,102225,102227],{"className":102226},[39],[41,102228,102229],{"xmlns":43},[45,102230,102231,102245],{},[48,102232,102233,102235,102237,102239,102241,102243],{},[51,102234,56971],{},[95,102236,211],{"stretchy":210},[51,102238,18446],{},[95,102240,929],{},[51,102242,1122],{},[95,102244,220],{"stretchy":210},[55,102246,102247],{"encoding":57},"Q(A\\to B)",[31,102249,102251,102275],{"className":102250,"ariaHidden":62},[61],[31,102252,102254,102257,102260,102263,102266,102269,102272],{"className":102253},[66],[31,102255],{"className":102256,"style":260},[70],[31,102258,56971],{"className":102259},[75,76],[31,102261,211],{"className":102262},[264],[31,102264,18446],{"className":102265},[75,76],[31,102267],{"className":102268,"style":293},[119],[31,102270,929],{"className":102271},[297],[31,102273],{"className":102274,"style":293},[119],[31,102276,102278,102281,102284],{"className":102277},[66],[31,102279],{"className":102280,"style":260},[70],[31,102282,1122],{"className":102283,"style":1137},[75,76],[31,102285,220],{"className":102286},[289],[31,102288,102290,102314],{"className":102289,"translate":35},[34],[31,102291,102293],{"className":102292},[39],[41,102294,102295],{"xmlns":43},[45,102296,102297,102311],{},[48,102298,102299,102301,102303,102305,102307,102309],{},[51,102300,56971],{},[95,102302,211],{"stretchy":210},[51,102304,18446],{},[95,102306,929],{},[51,102308,34614],{},[95,102310,220],{"stretchy":210},[55,102312,102313],{"encoding":57},"Q(A\\to D)",[31,102315,102317,102341],{"className":102316,"ariaHidden":62},[61],[31,102318,102320,102323,102326,102329,102332,102335,102338],{"className":102319},[66],[31,102321],{"className":102322,"style":260},[70],[31,102324,56971],{"className":102325},[75,76],[31,102327,211],{"className":102328},[264],[31,102330,18446],{"className":102331},[75,76],[31,102333],{"className":102334,"style":293},[119],[31,102336,929],{"className":102337},[297],[31,102339],{"className":102340,"style":293},[119],[31,102342,102344,102347,102350],{"className":102343},[66],[31,102345],{"className":102346,"style":260},[70],[31,102348,34614],{"className":102349,"style":19759},[75,76],[31,102351,220],{"className":102352},[289]," 分别为在 ",[31,102355,102357,102370],{"className":102356,"translate":35},[34],[31,102358,102360],{"className":102359},[39],[41,102361,102362],{"xmlns":43},[45,102363,102364,102368],{},[48,102365,102366],{},[51,102367,18446],{},[55,102369,18446],{"encoding":57},[31,102371,102373],{"className":102372,"ariaHidden":62},[61],[31,102374,102376,102379],{"className":102375},[66],[31,102377],{"className":102378,"style":945},[70],[31,102380,18446],{"className":102381},[75,76]," 选择这两条动作的动作值，则",[31,102384,102386],{"className":102385,"translate":35},[411],[31,102387,102389,102452],{"className":102388,"translate":35},[34],[31,102390,102392],{"className":102391},[39],[41,102393,102394],{"xmlns":43,"display":420},[45,102395,102396,102449],{},[48,102397,102398,102400,102402,102404,102406,102408,102410,102412,102414,102416,102418,102420,102422,102424,102426,102428,102430,102432,102434,102436,102438,102440,102442,102444,102447],{},[51,102399,56971],{},[95,102401,211],{"stretchy":210},[51,102403,18446],{},[95,102405,929],{},[51,102407,1122],{},[95,102409,220],{"stretchy":210},[95,102411,223],{},[95,102413,97],{},[91,102415,205],{},[95,102417,434],{},[91,102419,23247],{},[51,102421,43481],{},[95,102423,211],{"stretchy":210},[51,102425,1122],{},[95,102427,220],{"stretchy":210},[95,102429,223],{},[95,102431,97],{},[91,102433,205],{},[95,102435,434],{},[91,102437,23247],{},[95,102439,2528],{},[91,102441,81476],{},[95,102443,223],{},[91,102445,102446],{},"7.1",[95,102448,1210],{"separator":62},[55,102450,102451],{"encoding":57},"Q(A\\to B)=-1+0.9V(B)=-1+0.9\\times9=7.1,",[31,102453,102455,102479,102500,102521,102551,102572,102590,102608],{"className":102454,"ariaHidden":62},[61],[31,102456,102458,102461,102464,102467,102470,102473,102476],{"className":102457},[66],[31,102459],{"className":102460,"style":260},[70],[31,102462,56971],{"className":102463},[75,76],[31,102465,211],{"className":102466},[264],[31,102468,18446],{"className":102469},[75,76],[31,102471],{"className":102472,"style":293},[119],[31,102474,929],{"className":102475},[297],[31,102477],{"className":102478,"style":293},[119],[31,102480,102482,102485,102488,102491,102494,102497],{"className":102481},[66],[31,102483],{"className":102484,"style":260},[70],[31,102486,1122],{"className":102487,"style":1137},[75,76],[31,102489,220],{"className":102490},[289],[31,102492],{"className":102493,"style":293},[119],[31,102495,223],{"className":102496},[297],[31,102498],{"className":102499,"style":293},[119],[31,102501,102503,102506,102509,102512,102515,102518],{"className":102502},[66],[31,102504],{"className":102505,"style":112},[70],[31,102507,97],{"className":102508},[75],[31,102510,205],{"className":102511},[75],[31,102513],{"className":102514,"style":120},[119],[31,102516,434],{"className":102517},[124],[31,102519],{"className":102520,"style":120},[119],[31,102522,102524,102527,102530,102533,102536,102539,102542,102545,102548],{"className":102523},[66],[31,102525],{"className":102526,"style":260},[70],[31,102528,23247],{"className":102529},[75],[31,102531,43481],{"className":102532,"style":120},[75,76],[31,102534,211],{"className":102535},[264],[31,102537,1122],{"className":102538,"style":1137},[75,76],[31,102540,220],{"className":102541},[289],[31,102543],{"className":102544,"style":293},[119],[31,102546,223],{"className":102547},[297],[31,102549],{"className":102550,"style":293},[119],[31,102552,102554,102557,102560,102563,102566,102569],{"className":102553},[66],[31,102555],{"className":102556,"style":112},[70],[31,102558,97],{"className":102559},[75],[31,102561,205],{"className":102562},[75],[31,102564],{"className":102565,"style":120},[119],[31,102567,434],{"className":102568},[124],[31,102570],{"className":102571,"style":120},[119],[31,102573,102575,102578,102581,102584,102587],{"className":102574},[66],[31,102576],{"className":102577,"style":112},[70],[31,102579,23247],{"className":102580},[75],[31,102582],{"className":102583,"style":120},[119],[31,102585,2528],{"className":102586},[124],[31,102588],{"className":102589,"style":120},[119],[31,102591,102593,102596,102599,102602,102605],{"className":102592},[66],[31,102594],{"className":102595,"style":326},[70],[31,102597,81476],{"className":102598},[75],[31,102600],{"className":102601,"style":293},[119],[31,102603,223],{"className":102604},[297],[31,102606],{"className":102607,"style":293},[119],[31,102609,102611,102614,102617],{"className":102610},[66],[31,102612],{"className":102613,"style":41982},[70],[31,102615,102446],{"className":102616},[75],[31,102618,1210],{"className":102619},[1391],[31,102621,102623],{"className":102622,"translate":35},[411],[31,102624,102626,102687],{"className":102625,"translate":35},[34],[31,102627,102629],{"className":102628},[39],[41,102630,102631],{"xmlns":43,"display":420},[45,102632,102633,102684],{},[48,102634,102635,102637,102639,102641,102643,102645,102647,102649,102651,102653,102655,102657,102659,102661,102663,102665,102667,102669,102671,102673,102675,102677,102679,102681],{},[51,102636,56971],{},[95,102638,211],{"stretchy":210},[51,102640,18446],{},[95,102642,929],{},[51,102644,34614],{},[95,102646,220],{"stretchy":210},[95,102648,223],{},[95,102650,97],{},[91,102652,205],{},[95,102654,434],{},[91,102656,23247],{},[51,102658,43481],{},[95,102660,211],{"stretchy":210},[51,102662,34614],{},[95,102664,220],{"stretchy":210},[95,102666,223],{},[95,102668,97],{},[91,102670,205],{},[95,102672,434],{},[91,102674,23247],{},[95,102676,2528],{},[91,102678,102446],{},[95,102680,223],{},[91,102682,102683],{},"5.39.",[55,102685,102686],{"encoding":57},"Q(A\\to D)=-1+0.9V(D)=-1+0.9\\times7.1=5.39.",[31,102688,102690,102714,102735,102756,102786,102807,102825,102843],{"className":102689,"ariaHidden":62},[61],[31,102691,102693,102696,102699,102702,102705,102708,102711],{"className":102692},[66],[31,102694],{"className":102695,"style":260},[70],[31,102697,56971],{"className":102698},[75,76],[31,102700,211],{"className":102701},[264],[31,102703,18446],{"className":102704},[75,76],[31,102706],{"className":102707,"style":293},[119],[31,102709,929],{"className":102710},[297],[31,102712],{"className":102713,"style":293},[119],[31,102715,102717,102720,102723,102726,102729,102732],{"className":102716},[66],[31,102718],{"className":102719,"style":260},[70],[31,102721,34614],{"className":102722,"style":19759},[75,76],[31,102724,220],{"className":102725},[289],[31,102727],{"className":102728,"style":293},[119],[31,102730,223],{"className":102731},[297],[31,102733],{"className":102734,"style":293},[119],[31,102736,102738,102741,102744,102747,102750,102753],{"className":102737},[66],[31,102739],{"className":102740,"style":112},[70],[31,102742,97],{"className":102743},[75],[31,102745,205],{"className":102746},[75],[31,102748],{"className":102749,"style":120},[119],[31,102751,434],{"className":102752},[124],[31,102754],{"className":102755,"style":120},[119],[31,102757,102759,102762,102765,102768,102771,102774,102777,102780,102783],{"className":102758},[66],[31,102760],{"className":102761,"style":260},[70],[31,102763,23247],{"className":102764},[75],[31,102766,43481],{"className":102767,"style":120},[75,76],[31,102769,211],{"className":102770},[264],[31,102772,34614],{"className":102773,"style":19759},[75,76],[31,102775,220],{"className":102776},[289],[31,102778],{"className":102779,"style":293},[119],[31,102781,223],{"className":102782},[297],[31,102784],{"className":102785,"style":293},[119],[31,102787,102789,102792,102795,102798,102801,102804],{"className":102788},[66],[31,102790],{"className":102791,"style":112},[70],[31,102793,97],{"className":102794},[75],[31,102796,205],{"className":102797},[75],[31,102799],{"className":102800,"style":120},[119],[31,102802,434],{"className":102803},[124],[31,102805],{"className":102806,"style":120},[119],[31,102808,102810,102813,102816,102819,102822],{"className":102809},[66],[31,102811],{"className":102812,"style":112},[70],[31,102814,23247],{"className":102815},[75],[31,102817],{"className":102818,"style":120},[119],[31,102820,2528],{"className":102821},[124],[31,102823],{"className":102824,"style":120},[119],[31,102826,102828,102831,102834,102837,102840],{"className":102827},[66],[31,102829],{"className":102830,"style":326},[70],[31,102832,102446],{"className":102833},[75],[31,102835],{"className":102836,"style":293},[119],[31,102838,223],{"className":102839},[297],[31,102841],{"className":102842,"style":293},[119],[31,102844,102846,102849],{"className":102845},[66],[31,102847],{"className":102848,"style":326},[70],[31,102850,102683],{"className":102851},[75],[11,102853,70736],{},[31,102855,102857],{"className":102856,"translate":35},[411],[31,102858,102860,102903],{"className":102859,"translate":35},[34],[31,102861,102863],{"className":102862},[39],[41,102864,102865],{"xmlns":43,"display":420},[45,102866,102867,102900],{},[48,102868,102869,102871,102873,102875,102877,102879,102881,102883,102885,102887,102889,102892,102894,102896,102898],{},[51,102870,43481],{},[95,102872,211],{"stretchy":210},[51,102874,18446],{},[95,102876,220],{"stretchy":210},[95,102878,223],{},[51,102880,10154],{},[95,102882,1540],{},[95,102884,2512],{"stretchy":210},[91,102886,102446],{},[95,102888,1210],{"separator":62},[91,102890,102891],{},"5.39",[95,102893,2604],{"stretchy":210},[95,102895,223],{},[91,102897,102446],{},[95,102899,1210],{"separator":62},[55,102901,102902],{"encoding":57},"V(A)=\\max\\{7.1,5.39\\}=7.1,",[31,102904,102906,102933,102969],{"className":102905,"ariaHidden":62},[61],[31,102907,102909,102912,102915,102918,102921,102924,102927,102930],{"className":102908},[66],[31,102910],{"className":102911,"style":260},[70],[31,102913,43481],{"className":102914,"style":120},[75,76],[31,102916,211],{"className":102917},[264],[31,102919,18446],{"className":102920},[75,76],[31,102922,220],{"className":102923},[289],[31,102925],{"className":102926,"style":293},[119],[31,102928,223],{"className":102929},[297],[31,102931],{"className":102932,"style":293},[119],[31,102934,102936,102939,102942,102945,102948,102951,102954,102957,102960,102963,102966],{"className":102935},[66],[31,102937],{"className":102938,"style":260},[70],[31,102940,10154],{"className":102941},[1460],[31,102943,2512],{"className":102944},[264],[31,102946,102446],{"className":102947},[75],[31,102949,1210],{"className":102950},[1391],[31,102952],{"className":102953,"style":1467},[119],[31,102955,102891],{"className":102956},[75],[31,102958,2604],{"className":102959},[289],[31,102961],{"className":102962,"style":293},[119],[31,102964,223],{"className":102965},[297],[31,102967],{"className":102968,"style":293},[119],[31,102970,102972,102975,102978],{"className":102971},[66],[31,102973],{"className":102974,"style":41982},[70],[31,102976,102446],{"className":102977},[75],[31,102979,1210],{"className":102980},[1391],[11,102982,102983,102984,103012,103013,103086,103087,103182,103183,103216],{},"最优动作是走向 ",[31,102985,102987,103000],{"className":102986,"translate":35},[34],[31,102988,102990],{"className":102989},[39],[41,102991,102992],{"xmlns":43},[45,102993,102994,102998],{},[48,102995,102996],{},[51,102997,1122],{},[55,102999,1122],{"encoding":57},[31,103001,103003],{"className":103002,"ariaHidden":62},[61],[31,103004,103006,103009],{"className":103005},[66],[31,103007],{"className":103008,"style":945},[70],[31,103010,1122],{"className":103011,"style":1137},[75,76],"。路线 ",[31,103014,103016,103038],{"className":103015,"translate":35},[34],[31,103017,103019],{"className":103018},[39],[41,103020,103021],{"xmlns":43},[45,103022,103023,103035],{},[48,103024,103025,103027,103029,103031,103033],{},[51,103026,18446],{},[95,103028,929],{},[51,103030,1122],{},[95,103032,929],{},[51,103034,8101],{},[55,103036,103037],{"encoding":57},"A\\to B\\to C",[31,103039,103041,103059,103077],{"className":103040,"ariaHidden":62},[61],[31,103042,103044,103047,103050,103053,103056],{"className":103043},[66],[31,103045],{"className":103046,"style":945},[70],[31,103048,18446],{"className":103049},[75,76],[31,103051],{"className":103052,"style":293},[119],[31,103054,929],{"className":103055},[297],[31,103057],{"className":103058,"style":293},[119],[31,103060,103062,103065,103068,103071,103074],{"className":103061},[66],[31,103063],{"className":103064,"style":945},[70],[31,103066,1122],{"className":103067,"style":1137},[75,76],[31,103069],{"className":103070,"style":293},[119],[31,103072,929],{"className":103073},[297],[31,103075],{"className":103076,"style":293},[119],[31,103078,103080,103083],{"className":103079},[66],[31,103081],{"className":103082,"style":945},[70],[31,103084,8101],{"className":103085,"style":8525},[75,76]," 虽然与 ",[31,103088,103090,103116],{"className":103089,"translate":35},[34],[31,103091,103093],{"className":103092},[39],[41,103094,103095],{"xmlns":43},[45,103096,103097,103113],{},[48,103098,103099,103101,103103,103105,103107,103109,103111],{},[51,103100,18446],{},[95,103102,929],{},[51,103104,34614],{},[95,103106,929],{},[51,103108,22324],{},[95,103110,929],{},[51,103112,8101],{},[55,103114,103115],{"encoding":57},"A\\to D\\to E\\to C",[31,103117,103119,103137,103155,103173],{"className":103118,"ariaHidden":62},[61],[31,103120,103122,103125,103128,103131,103134],{"className":103121},[66],[31,103123],{"className":103124,"style":945},[70],[31,103126,18446],{"className":103127},[75,76],[31,103129],{"className":103130,"style":293},[119],[31,103132,929],{"className":103133},[297],[31,103135],{"className":103136,"style":293},[119],[31,103138,103140,103143,103146,103149,103152],{"className":103139},[66],[31,103141],{"className":103142,"style":945},[70],[31,103144,34614],{"className":103145,"style":19759},[75,76],[31,103147],{"className":103148,"style":293},[119],[31,103150,929],{"className":103151},[297],[31,103153],{"className":103154,"style":293},[119],[31,103156,103158,103161,103164,103167,103170],{"className":103157},[66],[31,103159],{"className":103160,"style":945},[70],[31,103162,22324],{"className":103163,"style":21697},[75,76],[31,103165],{"className":103166,"style":293},[119],[31,103168,929],{"className":103169},[297],[31,103171],{"className":103172,"style":293},[119],[31,103174,103176,103179],{"className":103175},[66],[31,103177],{"className":103178,"style":945},[70],[31,103180,8101],{"className":103181,"style":8525},[75,76]," 具有相同的终止奖励，但后者多走一步，多承担一次 ",[31,103184,103186,103201],{"className":103185,"translate":35},[34],[31,103187,103189],{"className":103188},[39],[41,103190,103191],{"xmlns":43},[45,103192,103193,103199],{},[48,103194,103195,103197],{},[95,103196,97],{},[91,103198,205],{},[55,103200,101423],{"encoding":57},[31,103202,103204],{"className":103203,"ariaHidden":62},[61],[31,103205,103207,103210,103213],{"className":103206},[66],[31,103208],{"className":103209,"style":112},[70],[31,103211,97],{"className":103212},[75],[31,103214,205],{"className":103215},[75]," 的即时奖励，并通过折扣因子进一步缩小了终止奖励的现值，因此其价值更低。",[11,103218,103219,103220,103248,103249,103277,103278,103306,103307,103358,103359,103387,103388,103306,103416,103489,103490,103518],{},"这个例子同时展示了最优子结构的两层含义。第一，从 ",[31,103221,103223,103236],{"className":103222,"translate":35},[34],[31,103224,103226],{"className":103225},[39],[41,103227,103228],{"xmlns":43},[45,103229,103230,103234],{},[48,103231,103232],{},[51,103233,18446],{},[55,103235,18446],{"encoding":57},[31,103237,103239],{"className":103238,"ariaHidden":62},[61],[31,103240,103242,103245],{"className":103241},[66],[31,103243],{"className":103244,"style":945},[70],[31,103246,18446],{"className":103247},[75,76]," 出发的最优路线选择了 ",[31,103250,103252,103265],{"className":103251,"translate":35},[34],[31,103253,103255],{"className":103254},[39],[41,103256,103257],{"xmlns":43},[45,103258,103259,103263],{},[48,103260,103261],{},[51,103262,1122],{},[55,103264,1122],{"encoding":57},[31,103266,103268],{"className":103267,"ariaHidden":62},[61],[31,103269,103271,103274],{"className":103270},[66],[31,103272],{"className":103273,"style":945},[70],[31,103275,1122],{"className":103276,"style":1137},[75,76],"，而从 ",[31,103279,103281,103294],{"className":103280,"translate":35},[34],[31,103282,103284],{"className":103283},[39],[41,103285,103286],{"xmlns":43},[45,103287,103288,103292],{},[48,103289,103290],{},[51,103291,1122],{},[55,103293,1122],{"encoding":57},[31,103295,103297],{"className":103296,"ariaHidden":62},[61],[31,103298,103300,103303],{"className":103299},[66],[31,103301],{"className":103302,"style":945},[70],[31,103304,1122],{"className":103305,"style":1137},[75,76]," 出发的剩余路线 ",[31,103308,103310,103328],{"className":103309,"translate":35},[34],[31,103311,103313],{"className":103312},[39],[41,103314,103315],{"xmlns":43},[45,103316,103317,103325],{},[48,103318,103319,103321,103323],{},[51,103320,1122],{},[95,103322,929],{},[51,103324,8101],{},[55,103326,103327],{"encoding":57},"B\\to C",[31,103329,103331,103349],{"className":103330,"ariaHidden":62},[61],[31,103332,103334,103337,103340,103343,103346],{"className":103333},[66],[31,103335],{"className":103336,"style":945},[70],[31,103338,1122],{"className":103339,"style":1137},[75,76],[31,103341],{"className":103342,"style":293},[119],[31,103344,929],{"className":103345},[297],[31,103347],{"className":103348,"style":293},[119],[31,103350,103352,103355],{"className":103351},[66],[31,103353],{"className":103354,"style":945},[70],[31,103356,8101],{"className":103357,"style":8525},[75,76]," 本身是 ",[31,103360,103362,103375],{"className":103361,"translate":35},[34],[31,103363,103365],{"className":103364},[39],[41,103366,103367],{"xmlns":43},[45,103368,103369,103373],{},[48,103370,103371],{},[51,103372,1122],{},[55,103374,1122],{"encoding":57},[31,103376,103378],{"className":103377,"ariaHidden":62},[61],[31,103379,103381,103384],{"className":103380},[66],[31,103382],{"className":103383,"style":945},[70],[31,103385,1122],{"className":103386,"style":1137},[75,76]," 处子问题的最优解。第二，从 ",[31,103389,103391,103404],{"className":103390,"translate":35},[34],[31,103392,103394],{"className":103393},[39],[41,103395,103396],{"xmlns":43},[45,103397,103398,103402],{},[48,103399,103400],{},[51,103401,34614],{},[55,103403,34614],{"encoding":57},[31,103405,103407],{"className":103406,"ariaHidden":62},[61],[31,103408,103410,103413],{"className":103409},[66],[31,103411],{"className":103412,"style":945},[70],[31,103414,34614],{"className":103415,"style":19759},[75,76],[31,103417,103419,103441],{"className":103418,"translate":35},[34],[31,103420,103422],{"className":103421},[39],[41,103423,103424],{"xmlns":43},[45,103425,103426,103438],{},[48,103427,103428,103430,103432,103434,103436],{},[51,103429,34614],{},[95,103431,929],{},[51,103433,22324],{},[95,103435,929],{},[51,103437,8101],{},[55,103439,103440],{"encoding":57},"D\\to E\\to C",[31,103442,103444,103462,103480],{"className":103443,"ariaHidden":62},[61],[31,103445,103447,103450,103453,103456,103459],{"className":103446},[66],[31,103448],{"className":103449,"style":945},[70],[31,103451,34614],{"className":103452,"style":19759},[75,76],[31,103454],{"className":103455,"style":293},[119],[31,103457,929],{"className":103458},[297],[31,103460],{"className":103461,"style":293},[119],[31,103463,103465,103468,103471,103474,103477],{"className":103464},[66],[31,103466],{"className":103467,"style":945},[70],[31,103469,22324],{"className":103470,"style":21697},[75,76],[31,103472],{"className":103473,"style":293},[119],[31,103475,929],{"className":103476},[297],[31,103478],{"className":103479,"style":293},[119],[31,103481,103483,103486],{"className":103482},[66],[31,103484],{"className":103485,"style":945},[70],[31,103487,8101],{"className":103488,"style":8525},[75,76]," 也是 ",[31,103491,103493,103506],{"className":103492,"translate":35},[34],[31,103494,103496],{"className":103495},[39],[41,103497,103498],{"xmlns":43},[45,103499,103500,103504],{},[48,103501,103502],{},[51,103503,34614],{},[55,103505,34614],{"encoding":57},[31,103507,103509],{"className":103508,"ariaHidden":62},[61],[31,103510,103512,103515],{"className":103511},[66],[31,103513],{"className":103514,"style":945},[70],[31,103516,34614],{"className":103517,"style":19759},[75,76]," 处子问题的最优解。整体最优并不是把各段的局部最优任意拼接，而是沿着实际到达的后继状态，取该后继状态的最优续策略。",[11,103520,103521,103522,103588],{},"若改用 ",[31,103523,103525,103549],{"className":103524,"translate":35},[34],[31,103526,103528],{"className":103527},[39],[41,103529,103530],{"xmlns":43},[45,103531,103532,103546],{},[48,103533,103534,103536,103538,103540,103542,103544],{},[51,103535,43481],{},[95,103537,211],{"stretchy":210},[51,103539,8101],{},[95,103541,220],{"stretchy":210},[95,103543,223],{},[91,103545,19494],{},[55,103547,103548],{"encoding":57},"V(C)=10",[31,103550,103552,103579],{"className":103551,"ariaHidden":62},[61],[31,103553,103555,103558,103561,103564,103567,103570,103573,103576],{"className":103554},[66],[31,103556],{"className":103557,"style":260},[70],[31,103559,43481],{"className":103560,"style":120},[75,76],[31,103562,211],{"className":103563},[264],[31,103565,8101],{"className":103566,"style":8525},[75,76],[31,103568,220],{"className":103569},[289],[31,103571],{"className":103572,"style":293},[119],[31,103574,223],{"className":103575},[297],[31,103577],{"className":103578,"style":293},[119],[31,103580,103582,103585],{"className":103581},[66],[31,103583],{"className":103584,"style":326},[70],[31,103586,19494],{"className":103587},[75]," 的约定，即把终止奖励直接记在终止状态上，而不是记在进入终止状态的转移上，则上述数值会改变，但最优动作不变。两种约定的差别只在于奖励被记在转移上还是被记在状态上，贝尔曼方程的形式必须与所采用的约定保持一致。",[18,103590,103592],{"id":103591},"五收敛性与误差界","五、收敛性与误差界",[52860,103594,103596],{"id":103595},"_51-不动点迭代","5.1 不动点迭代",[11,103598,103599,103600,103707],{},"压缩映射不仅证明了解的存在唯一性，还给出了构造解的方法。任取初始函数 ",[31,103601,103603,103631],{"className":103602,"translate":35},[34],[31,103604,103606],{"className":103605},[39],[41,103607,103608],{"xmlns":43},[45,103609,103610,103628],{},[48,103611,103612,103618,103620,103622,103624,103626],{},[1004,103613,103614,103616],{},[51,103615,43481],{},[91,103617,1977],{},[95,103619,2518],{},[51,103621,1122],{},[95,103623,211],{"stretchy":210},[51,103625,21582],{},[95,103627,220],{"stretchy":210},[55,103629,103630],{"encoding":57},"V_0\\in B(S)",[31,103632,103634,103689],{"className":103633,"ariaHidden":62},[61],[31,103635,103637,103640,103680,103683,103686],{"className":103636},[66],[31,103638],{"className":103639,"style":5230},[70],[31,103641,103643,103646],{"className":103642},[75],[31,103644,43481],{"className":103645,"style":120},[75,76],[31,103647,103649],{"className":103648},[1039],[31,103650,103652,103672],{"className":103651},[1043,1044],[31,103653,103655,103669],{"className":103654},[1048],[31,103656,103658],{"className":103657,"style":2028},[1052],[31,103659,103660,103663],{"style":66251},[31,103661],{"className":103662,"style":1061},[1060],[31,103664,103666],{"className":103665},[1065,1066,1067,1068],[31,103667,1977],{"className":103668},[75,1068],[31,103670,1076],{"className":103671},[1075],[31,103673,103675],{"className":103674},[1048],[31,103676,103678],{"className":103677,"style":2050},[1052],[31,103679],{},[31,103681],{"className":103682,"style":293},[119],[31,103684,2518],{"className":103685},[297],[31,103687],{"className":103688,"style":293},[119],[31,103690,103692,103695,103698,103701,103704],{"className":103691},[66],[31,103693],{"className":103694,"style":260},[70],[31,103696,1122],{"className":103697,"style":1137},[75,76],[31,103699,211],{"className":103700},[264],[31,103702,21582],{"className":103703,"style":21697},[75,76],[31,103705,220],{"className":103706},[289],"，定义迭代",[31,103709,103711],{"className":103710,"translate":35},[411],[31,103712,103714,103770],{"className":103713,"translate":35},[34],[31,103715,103717],{"className":103716},[39],[41,103718,103719],{"xmlns":43,"display":420},[45,103720,103721,103767],{},[48,103722,103723,103735,103737,103739,103745,103747,103749,103751,103753,103755,103757,103759,103761,103763,103765],{},[1004,103724,103725,103727],{},[51,103726,43481],{},[48,103728,103729,103731,103733],{},[51,103730,52065],{},[95,103732,434],{},[91,103734,205],{},[95,103736,223],{},[51,103738,919],{"mathvariant":925},[1004,103740,103741,103743],{},[51,103742,43481],{},[51,103744,52065],{},[95,103746,1210],{"separator":62},[119,103748],{"width":20286},[51,103750,52065],{},[95,103752,223],{},[91,103754,1977],{},[95,103756,1210],{"separator":62},[91,103758,205],{},[95,103760,1210],{"separator":62},[91,103762,152],{},[95,103764,1210],{"separator":62},[95,103766,55349],{},[55,103768,103769],{"encoding":57},"V_{k+1}=\\mathcal{T}V_k,\n\\qquad k=0,1,2,\\dots",[31,103771,103773,103837,103907],{"className":103772,"ariaHidden":62},[61],[31,103774,103776,103779,103828,103831,103834],{"className":103775},[66],[31,103777],{"className":103778,"style":60241},[70],[31,103780,103782,103785],{"className":103781},[75],[31,103783,43481],{"className":103784,"style":120},[75,76],[31,103786,103788],{"className":103787},[1039],[31,103789,103791,103820],{"className":103790},[1043,1044],[31,103792,103794,103817],{"className":103793},[1048],[31,103795,103797],{"className":103796,"style":1053},[1052],[31,103798,103799,103802],{"style":66251},[31,103800],{"className":103801,"style":1061},[1060],[31,103803,103805],{"className":103804},[1065,1066,1067,1068],[31,103806,103808,103811,103814],{"className":103807},[75,1068],[31,103809,52065],{"className":103810,"style":52166},[75,76,1068],[31,103812,434],{"className":103813},[124,1068],[31,103815,205],{"className":103816},[75,1068],[31,103818,1076],{"className":103819},[1075],[31,103821,103823],{"className":103822},[1048],[31,103824,103826],{"className":103825,"style":25082},[1052],[31,103827],{},[31,103829],{"className":103830,"style":293},[119],[31,103832,223],{"className":103833},[297],[31,103835],{"className":103836,"style":293},[119],[31,103838,103840,103843,103846,103886,103889,103892,103895,103898,103901,103904],{"className":103839},[66],[31,103841],{"className":103842,"style":4202},[70],[31,103844,919],{"className":103845,"style":64444},[75,968],[31,103847,103849,103852],{"className":103848},[75],[31,103850,43481],{"className":103851,"style":120},[75,76],[31,103853,103855],{"className":103854},[1039],[31,103856,103858,103878],{"className":103857},[1043,1044],[31,103859,103861,103875],{"className":103860},[1048],[31,103862,103864],{"className":103863,"style":1053},[1052],[31,103865,103866,103869],{"style":66251},[31,103867],{"className":103868,"style":1061},[1060],[31,103870,103872],{"className":103871},[1065,1066,1067,1068],[31,103873,52065],{"className":103874,"style":52166},[75,76,1068],[31,103876,1076],{"className":103877},[1075],[31,103879,103881],{"className":103880},[1048],[31,103882,103884],{"className":103883,"style":2050},[1052],[31,103885],{},[31,103887,1210],{"className":103888},[1391],[31,103890],{"className":103891,"style":20525},[119],[31,103893],{"className":103894,"style":1467},[119],[31,103896,52065],{"className":103897,"style":52166},[75,76],[31,103899],{"className":103900,"style":293},[119],[31,103902,223],{"className":103903},[297],[31,103905],{"className":103906,"style":293},[119],[31,103908,103910,103913,103916,103919,103922,103925,103928,103931,103934,103937,103940],{"className":103909},[66],[31,103911],{"className":103912,"style":41982},[70],[31,103914,1977],{"className":103915},[75],[31,103917,1210],{"className":103918},[1391],[31,103920],{"className":103921,"style":1467},[119],[31,103923,205],{"className":103924},[75],[31,103926,1210],{"className":103927},[1391],[31,103929],{"className":103930,"style":1467},[119],[31,103932,152],{"className":103933},[75],[31,103935,1210],{"className":103936},[1391],[31,103938],{"className":103939,"style":1467},[119],[31,103941,55349],{"className":103942},[3860],[11,103944,50599,103945,103973,103974,23342,104032,104090],{},[31,103946,103948,103961],{"className":103947,"translate":35},[34],[31,103949,103951],{"className":103950},[39],[41,103952,103953],{"xmlns":43},[45,103954,103955,103959],{},[48,103956,103957],{},[51,103958,919],{"mathvariant":925},[55,103960,87378],{"encoding":57},[31,103962,103964],{"className":103963,"ariaHidden":62},[61],[31,103965,103967,103970],{"className":103966},[66],[31,103968],{"className":103969,"style":945},[70],[31,103971,919],{"className":103972,"style":64444},[75,968]," 可以是 ",[31,103975,103977,103994],{"className":103976,"translate":35},[34],[31,103978,103980],{"className":103979},[39],[41,103981,103982],{"xmlns":43},[45,103983,103984,103992],{},[48,103985,103986],{},[1180,103987,103988,103990],{},[51,103989,919],{"mathvariant":925},[51,103991,2515],{},[55,103993,64428],{"encoding":57},[31,103995,103997],{"className":103996,"ariaHidden":62},[61],[31,103998,104000,104003],{"className":103999},[66],[31,104001],{"className":104002,"style":945},[70],[31,104004,104006,104009],{"className":104005},[75],[31,104007,919],{"className":104008,"style":64444},[75,968],[31,104010,104012],{"className":104011},[1039],[31,104013,104015],{"className":104014},[1043],[31,104016,104018],{"className":104017},[1048],[31,104019,104021],{"className":104020,"style":3628},[1052],[31,104022,104023,104026],{"style":3631},[31,104024],{"className":104025,"style":1061},[1060],[31,104027,104029],{"className":104028},[1065,1066,1067,1068],[31,104030,2515],{"className":104031,"style":2666},[75,76,1068],[31,104033,104035,104052],{"className":104034,"translate":35},[34],[31,104036,104038],{"className":104037},[39],[41,104039,104040],{"xmlns":43},[45,104041,104042,104050],{},[48,104043,104044],{},[1180,104045,104046,104048],{},[51,104047,919],{"mathvariant":925},[95,104049,30648],{},[55,104051,74501],{"encoding":57},[31,104053,104055],{"className":104054,"ariaHidden":62},[61],[31,104056,104058,104061],{"className":104057},[66],[31,104059],{"className":104060,"style":30719},[70],[31,104062,104064,104067],{"className":104063},[75],[31,104065,919],{"className":104066,"style":64444},[75,968],[31,104068,104070],{"className":104069},[1039],[31,104071,104073],{"className":104072},[1043],[31,104074,104076],{"className":104075},[1048],[31,104077,104079],{"className":104078,"style":30719},[1052],[31,104080,104081,104084],{"style":3631},[31,104082],{"className":104083,"style":1061},[1060],[31,104085,104087],{"className":104086},[1065,1066,1067,1068],[31,104088,30648],{"className":104089},[124,1068],"。在强化学习中，这一迭代称为值迭代。由 Banach 不动点定理，迭代序列必然收敛到相应的不动点。",[52865,104092,104095,104489,104893],{"id":104093,"title":104094},"theorem-convergence","定理 5.1（几何收敛）",[11,104096,52871,104097,1138,104169,74019,104197,23342,104255,87507,104313,104382,104383,1138],{},[31,104098,104100,104121],{"className":104099,"translate":35},[34],[31,104101,104103],{"className":104102},[39],[41,104104,104105],{"xmlns":43},[45,104106,104107,104119],{},[48,104108,104109,104111,104113,104115,104117],{},[91,104110,1977],{},[95,104112,347],{},[51,104114,53298],{},[95,104116,9758],{},[91,104118,205],{},[55,104120,67213],{"encoding":57},[31,104122,104124,104142,104160],{"className":104123,"ariaHidden":62},[61],[31,104125,104127,104130,104133,104136,104139],{"className":104126},[66],[31,104128],{"className":104129,"style":366},[70],[31,104131,1977],{"className":104132},[75],[31,104134],{"className":104135,"style":293},[119],[31,104137,347],{"className":104138},[297],[31,104140],{"className":104141,"style":293},[119],[31,104143,104145,104148,104151,104154,104157],{"className":104144},[66],[31,104146],{"className":104147,"style":53323},[70],[31,104149,53298],{"className":104150,"style":24157},[75,76],[31,104152],{"className":104153,"style":293},[119],[31,104155,9758],{"className":104156},[297],[31,104158],{"className":104159,"style":293},[119],[31,104161,104163,104166],{"className":104162},[66],[31,104164],{"className":104165,"style":326},[70],[31,104167,205],{"className":104168},[75],[31,104170,104172,104185],{"className":104171,"translate":35},[34],[31,104173,104175],{"className":104174},[39],[41,104176,104177],{"xmlns":43},[45,104178,104179,104183],{},[48,104180,104181],{},[51,104182,919],{"mathvariant":925},[55,104184,87378],{"encoding":57},[31,104186,104188],{"className":104187,"ariaHidden":62},[61],[31,104189,104191,104194],{"className":104190},[66],[31,104192],{"className":104193,"style":945},[70],[31,104195,919],{"className":104196,"style":64444},[75,968],[31,104198,104200,104217],{"className":104199,"translate":35},[34],[31,104201,104203],{"className":104202},[39],[41,104204,104205],{"xmlns":43},[45,104206,104207,104215],{},[48,104208,104209],{},[1180,104210,104211,104213],{},[51,104212,919],{"mathvariant":925},[51,104214,2515],{},[55,104216,64428],{"encoding":57},[31,104218,104220],{"className":104219,"ariaHidden":62},[61],[31,104221,104223,104226],{"className":104222},[66],[31,104224],{"className":104225,"style":945},[70],[31,104227,104229,104232],{"className":104228},[75],[31,104230,919],{"className":104231,"style":64444},[75,968],[31,104233,104235],{"className":104234},[1039],[31,104236,104238],{"className":104237},[1043],[31,104239,104241],{"className":104240},[1048],[31,104242,104244],{"className":104243,"style":3628},[1052],[31,104245,104246,104249],{"style":3631},[31,104247],{"className":104248,"style":1061},[1060],[31,104250,104252],{"className":104251},[1065,1066,1067,1068],[31,104253,2515],{"className":104254,"style":2666},[75,76,1068],[31,104256,104258,104275],{"className":104257,"translate":35},[34],[31,104259,104261],{"className":104260},[39],[41,104262,104263],{"xmlns":43},[45,104264,104265,104273],{},[48,104266,104267],{},[1180,104268,104269,104271],{},[51,104270,919],{"mathvariant":925},[95,104272,30648],{},[55,104274,74501],{"encoding":57},[31,104276,104278],{"className":104277,"ariaHidden":62},[61],[31,104279,104281,104284],{"className":104280},[66],[31,104282],{"className":104283,"style":30719},[70],[31,104285,104287,104290],{"className":104286},[75],[31,104288,919],{"className":104289,"style":64444},[75,968],[31,104291,104293],{"className":104292},[1039],[31,104294,104296],{"className":104295},[1043],[31,104297,104299],{"className":104298},[1048],[31,104300,104302],{"className":104301,"style":30719},[1052],[31,104303,104304,104307],{"style":3631},[31,104305],{"className":104306,"style":1061},[1060],[31,104308,104310],{"className":104309},[1065,1066,1067,1068],[31,104311,30648],{"className":104312},[124,1068],[31,104314,104316,104333],{"className":104315,"translate":35},[34],[31,104317,104319],{"className":104318},[39],[41,104320,104321],{"xmlns":43},[45,104322,104323,104331],{},[48,104324,104325],{},[1004,104326,104327,104329],{},[51,104328,43481],{},[51,104330,4587],{"mathvariant":494},[55,104332,87528],{"encoding":57},[31,104334,104336],{"className":104335,"ariaHidden":62},[61],[31,104337,104339,104342],{"className":104338},[66],[31,104340],{"className":104341,"style":5230},[70],[31,104343,104345,104348],{"className":104344},[75],[31,104346,43481],{"className":104347,"style":120},[75,76],[31,104349,104351],{"className":104350},[1039],[31,104352,104354,104374],{"className":104353},[1043,1044],[31,104355,104357,104371],{"className":104356},[1048],[31,104358,104360],{"className":104359,"style":3230},[1052],[31,104361,104362,104365],{"style":66251},[31,104363],{"className":104364,"style":1061},[1060],[31,104366,104368],{"className":104367},[1065,1066,1067,1068],[31,104369,4587],{"className":104370},[75,1068],[31,104372,1076],{"className":104373},[1075],[31,104375,104377],{"className":104376},[1048],[31,104378,104380],{"className":104379,"style":2050},[1052],[31,104381],{},"。则对任意 ",[31,104384,104386,104413],{"className":104385,"translate":35},[34],[31,104387,104389],{"className":104388},[39],[41,104390,104391],{"xmlns":43},[45,104392,104393,104411],{},[48,104394,104395,104401,104403,104405,104407,104409],{},[1004,104396,104397,104399],{},[51,104398,43481],{},[91,104400,1977],{},[95,104402,2518],{},[51,104404,1122],{},[95,104406,211],{"stretchy":210},[51,104408,21582],{},[95,104410,220],{"stretchy":210},[55,104412,103630],{"encoding":57},[31,104414,104416,104471],{"className":104415,"ariaHidden":62},[61],[31,104417,104419,104422,104462,104465,104468],{"className":104418},[66],[31,104420],{"className":104421,"style":5230},[70],[31,104423,104425,104428],{"className":104424},[75],[31,104426,43481],{"className":104427,"style":120},[75,76],[31,104429,104431],{"className":104430},[1039],[31,104432,104434,104454],{"className":104433},[1043,1044],[31,104435,104437,104451],{"className":104436},[1048],[31,104438,104440],{"className":104439,"style":2028},[1052],[31,104441,104442,104445],{"style":66251},[31,104443],{"className":104444,"style":1061},[1060],[31,104446,104448],{"className":104447},[1065,1066,1067,1068],[31,104449,1977],{"className":104450},[75,1068],[31,104452,1076],{"className":104453},[1075],[31,104455,104457],{"className":104456},[1048],[31,104458,104460],{"className":104459,"style":2050},[1052],[31,104461],{},[31,104463],{"className":104464,"style":293},[119],[31,104466,2518],{"className":104467},[297],[31,104469],{"className":104470,"style":293},[119],[31,104472,104474,104477,104480,104483,104486],{"className":104473},[66],[31,104475],{"className":104476,"style":260},[70],[31,104478,1122],{"className":104479,"style":1137},[75,76],[31,104481,211],{"className":104482},[264],[31,104484,21582],{"className":104485,"style":21697},[75,76],[31,104487,220],{"className":104488},[289],[31,104490,104492],{"className":104491,"translate":35},[411],[31,104493,104495,104561],{"className":104494,"translate":35},[34],[31,104496,104498],{"className":104497},[39],[41,104499,104500],{"xmlns":43,"display":420},[45,104501,104502,104558],{},[48,104503,104504,104506,104512,104514,104520,104526,104528,104534,104536,104542,104544,104550,104556],{},[51,104505,4572],{"mathvariant":494},[1004,104507,104508,104510],{},[51,104509,43481],{},[51,104511,52065],{},[95,104513,97],{},[1004,104515,104516,104518],{},[51,104517,43481],{},[51,104519,4587],{"mathvariant":494},[1004,104521,104522,104524],{},[51,104523,4572],{"mathvariant":494},[51,104525,4587],{"mathvariant":494},[95,104527,347],{},[1180,104529,104530,104532],{},[51,104531,53298],{},[51,104533,52065],{},[51,104535,4572],{"mathvariant":494},[1004,104537,104538,104540],{},[51,104539,43481],{},[91,104541,1977],{},[95,104543,97],{},[1004,104545,104546,104548],{},[51,104547,43481],{},[51,104549,4587],{"mathvariant":494},[1004,104551,104552,104554],{},[51,104553,4572],{"mathvariant":494},[51,104555,4587],{"mathvariant":494},[51,104557,495],{"mathvariant":494},[55,104559,104560],{"encoding":57},"\\|V_k-V_\\infty\\|_\\infty\n\\le\\gamma^k\\|V_0-V_\\infty\\|_\\infty.",[31,104562,104564,104622,104717,104804],{"className":104563,"ariaHidden":62},[61],[31,104565,104567,104570,104573,104613,104616,104619],{"className":104566},[66],[31,104568],{"className":104569,"style":260},[70],[31,104571,4572],{"className":104572},[75],[31,104574,104576,104579],{"className":104575},[75],[31,104577,43481],{"className":104578,"style":120},[75,76],[31,104580,104582],{"className":104581},[1039],[31,104583,104585,104605],{"className":104584},[1043,1044],[31,104586,104588,104602],{"className":104587},[1048],[31,104589,104591],{"className":104590,"style":1053},[1052],[31,104592,104593,104596],{"style":66251},[31,104594],{"className":104595,"style":1061},[1060],[31,104597,104599],{"className":104598},[1065,1066,1067,1068],[31,104600,52065],{"className":104601,"style":52166},[75,76,1068],[31,104603,1076],{"className":104604},[1075],[31,104606,104608],{"className":104607},[1048],[31,104609,104611],{"className":104610,"style":2050},[1052],[31,104612],{},[31,104614],{"className":104615,"style":120},[119],[31,104617,97],{"className":104618},[124],[31,104620],{"className":104621,"style":120},[119],[31,104623,104625,104628,104668,104708,104711,104714],{"className":104624},[66],[31,104626],{"className":104627,"style":260},[70],[31,104629,104631,104634],{"className":104630},[75],[31,104632,43481],{"className":104633,"style":120},[75,76],[31,104635,104637],{"className":104636},[1039],[31,104638,104640,104660],{"className":104639},[1043,1044],[31,104641,104643,104657],{"className":104642},[1048],[31,104644,104646],{"className":104645,"style":3230},[1052],[31,104647,104648,104651],{"style":66251},[31,104649],{"className":104650,"style":1061},[1060],[31,104652,104654],{"className":104653},[1065,1066,1067,1068],[31,104655,4587],{"className":104656},[75,1068],[31,104658,1076],{"className":104659},[1075],[31,104661,104663],{"className":104662},[1048],[31,104664,104666],{"className":104665,"style":2050},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V_\\infty",[31,104926,104928,104983],{"className":104927,"ariaHidden":62},[61],[31,104929,104931,104934,104974,104977,104980],{"className":104930},[66],[31,104932],{"className":104933,"style":5230},[70],[31,104935,104937,104940],{"className":104936},[75],[31,104938,43481],{"className":104939,"style":120},[75,76],[31,104941,104943],{"className":104942},[1039],[31,104944,104946,104966],{"className":104945},[1043,1044],[31,104947,104949,104963],{"className":104948},[1048],[31,104950,104952],{"className":104951,"style":1053},[1052],[31,104953,104954,104957],{"style":66251},[31,104955],{"className":104956,"style":1061},[1060],[31,104958,104960],{"className":104959},[1065,1066,1067,1068],[31,104961,52065],{"className":104962,"style":52166},[75,76,1068],[31,104964,1076],{"className":104965},[1075],[31,104967,104969],{"className":104968},[1048],[31,104970,104972],{"className":104971,"style":2050},[1052],[31,104973],{},[31,104975],{"className":104976,"style":293},[119],[31,104978,929],{"className":104979},[297],[31,104981],{"className":104982,"style":293},[119],[31,104984,104986,104989],{"className":104985},[66],[31,104987],{"className":104988,"style":5230},[70],[31,104990,104992,104995],{"className":104991},[75],[31,104993,43481],{"className":104994,"style":120},[75,76],[31,104996,104998],{"className":104997},[1039],[31,104999,105001,105021],{"className":105000},[1043,1044],[31,105002,105004,105018],{"className":105003},[1048],[31,105005,105007],{"className":105006,"style":3230},[1052],[31,105008,105009,105012],{"style":66251},[31,105010],{"className":105011,"style":1061},[1060],[31,105013,105015],{"className":105014},[1065,1066,1067,1068],[31,105016,4587],{"className":105017},[75,1068],[31,105019,1076],{"className":105020},[1075],[31,105022,105024],{"className":105023},[1048],[31,105025,105027],{"className":105026,"style":2050},[1052],[31,105028],{}," 在 sup 范数下成立。",[11,105031,105032,105034],{},[7246,105033,7248],{}," 由压缩性，",[31,105036,105038],{"className":105037,"translate":35},[411],[31,105039,105041,105169],{"className":105040,"translate":35},[34],[31,105042,105044],{"className":105043},[39],[41,105045,105046],{"xmlns":43,"display":420},[45,105047,105048,105166],{},[5606,105049,105050,105122],{"rowspacing":5608,"columnalign":5609,"columnspacing":5610},[5612,105051,105052,105086],{},[5615,105053,105054],{},[2071,105055,105056],{"scriptlevel":1977,"displaystyle":62},[48,105057,105058,105060,105072,105074,105080],{},[51,105059,4572],{"mathvariant":494},[1004,105061,105062,105064],{},[51,105063,43481],{},[48,105065,105066,105068,105070],{},[51,105067,52065],{},[95,105069,434],{},[91,105071,205],{},[95,105073,97],{},[1004,105075,105076,105078],{},[51,105077,43481],{},[51,105079,4587],{"mathvariant":494},[1004,105081,105082,105084],{},[51,105083,4572],{"mathvariant":494},[51,105085,4587],{"mathvariant":494},[5615,105087,105088],{},[2071,105089,105090],{"scriptlevel":1977,"displaystyle":62},[48,105091,105092,105094,105096,105098,105100,105106,105108,105110,105116],{},[48,105093],{},[95,105095,223],{},[51,105097,4572],{"mathvariant":494},[51,105099,919],{"mathvariant":925},[1004,105101,105102,105104],{},[51,105103,43481],{},[51,105105,52065],{},[95,105107,97],{},[51,105109,919],{"mathvariant":925},[1004,105111,105112,105114],{},[51,105113,43481],{},[51,105115,4587],{"mathvariant":494},[1004,105117,105118,105120],{},[51,105119,4572],{"mathvariant":494},[51,105121,4587],{"mathvariant":494},[5612,105123,105124,105130],{},[5615,105125,105126],{},[2071,105127,105128],{"scriptlevel":1977,"displaystyle":62},[48,105129],{},[5615,105131,105132],{},[2071,105133,105134],{"scriptlevel":1977,"displaystyle":62},[48,105135,105136,105138,105140,105142,105144,105150,105152,105158,105164],{},[48,105137],{},[95,105139,347],{},[51,105141,53298],{},[51,105143,4572],{"mathvariant":494},[1004,105145,105146,105148],{},[51,105147,43481],{},[51,105149,52065],{},[95,105151,97],{},[1004,105153,105154,105156],{},[51,105155,43481],{},[51,105157,4587],{"mathvariant":494},[1004,105159,105160,105162],{},[51,105161,4572],{"mathvariant":494},[51,105163,4587],{"mathvariant":494},[51,105165,495],{"mathvariant":494},[55,105167,105168],{"encoding":57},"\\begin{aligned}\n\\|V_{k+1}-V_\\infty\\|_\\infty\n&=\\|\\mathcal{T}V_k-\\mathcal{T}V_\\infty\\|_\\infty\\\\\n&\\le\\gamma\\|V_k-V_\\infty\\|_\\infty.\n\\end{aligned}",[31,105170,105172],{"className":105171,"ariaHidden":62},[61],[31,105173,105175,105178],{"className":105174},[66],[31,105176],{"className":105177,"style":57050},[70],[31,105179,105181],{"className":105180},[75],[31,105182,105184,105364],{"className":105183},[5606],[31,105185,105187],{"className":105186},[5867],[31,105188,105190,105356],{"className":105189},[1043,1044],[31,105191,105193,105353],{"className":105192},[1048],[31,105194,105196,105345],{"className":105195,"style":57069},[1052],[31,105197,105198,105201],{"style":57072},[31,105199],{"className":105200,"style":1637},[1060],[31,105202,105204,105207,105256,105259,105262,105265,105305],{"className":105203},[75],[31,105205,4572],{"className":105206},[75],[31,105208,105210,105213],{"className":105209},[75],[31,105211,43481],{"className":105212,"style":120},[75,76],[31,105214,105216],{"className":105215},[1039],[31,105217,105219,105248],{"className":105218},[1043,1044],[31,105220,105222,105245],{"className":105221},[1048],[31,105223,105225],{"className":105224,"style":1053},[1052],[31,105226,105227,105230],{"style":66251},[31,105228],{"className":105229,"style":1061},[1060],[31,105231,105233],{"className":105232},[1065,1066,1067,1068],[31,105234,105236,105239,105242],{"className":105235},[75,1068],[31,105237,52065],{"className":105238,"style":52166},[75,76,1068],[31,105240,434],{"className":105241},[124,1068],[31,105243,205],{"className":105244},[75,1068],[31,105246,1076],{"className":105247},[1075],[31,105249,105251],{"className":105250},[1048],[31,105252,105254],{"className":105253,"style":25082},[1052],[31,105255],{},[31,105257],{"className":105258,"style":120},[119],[31,105260,97],{"className":105261},[124],[31,105263],{"className":105264,"style":120},[119],[31,105266,105268,105271],{"className":105267},[75],[31,105269,43481],{"className":105270,"style":120},[75,76],[31,105272,105274],{"className":105273},[1039],[31,105275,105277,105297],{"className":105276},[1043,1044],[31,105278,105280,105294],{"className":105279},[1048],[31,105281,105283],{"className":105282,"style":3230},[1052],[31,105284,105285,105288],{"style":66251},[31,105286],{"className":105287,"style":1061},[1060],[31,105289,105291],{"className":105290},[1065,1066,1067,1068],[31,105292,4587],{"className":105293},[75,1068],[31,105295,1076],{"className":105296},[1075],[31,105298,105300],{"className":105299},[1048],[31,105301,105303],{"className":105302,"style":2050},[1052],[31,105304],{},[31,105306,105308,105311],{"className":105307},[75],[31,105309,4572],{"className":105310},[75],[31,105312,105314],{"className":105313},[1039],[31,105315,105317,105337],{"className":105316},[1043,1044],[31,105318,105320,105334],{"className":105319},[1048],[31,105321,105323],{"className":105322,"style":3230},[1052],[31,105324,105325,105328],{"style":2807},[31,105326],{"className":105327,"style":1061},[1060],[31,105329,105331],{"className":105330},[1065,1066,1067,1068],[31,105332,4587],{"className":105333},[75,1068],[31,105335,1076],{"className":105336},[1075],[31,105338,105340],{"className":105339},[1048],[31,105341,105343],{"className":105342,"style":2050},[1052],[31,105344],{},[31,105346,105347,105350],{"style":57119},[31,105348],{"className":105349,"style":1637},[1060],[31,105351],{"className":105352},[75],[31,105354,1076],{"className":105355},[1075],[31,105357,105359],{"className":105358},[1048],[31,105360,105362],{"className":105361,"style":57182},[1052],[31,105363],{},[31,105365,105367],{"className":105366},[6062],[31,105368,105370,105695],{"className":105369},[1043,1044],[31,105371,105373,105692],{"className":105372},[1048],[31,105374,105376,105534],{"className":105375,"style":57069},[1052],[31,105377,105378,105381],{"style":57072},[31,105379],{"className":105380,"style":1637},[1060],[31,105382,105384,105387,105390,105393,105396,105399,105402,105442,105445,105448,105451,105454,105494],{"className":105383},[75],[31,105385],{"className":105386},[75],[31,105388],{"className":105389,"style":293},[119],[31,105391,223],{"className":105392},[297],[31,105394],{"className":105395,"style":293},[119],[31,105397,4572],{"className":105398},[75],[31,105400,919],{"className":105401,"style":64444},[75,968],[31,105403,105405,105408],{"className":105404},[75],[31,105406,43481],{"className":105407,"style":120},[75,76],[31,105409,105411],{"className":105410},[1039],[31,105412,105414,105434],{"className":105413},[1043,1044],[31,105415,105417,105431],{"className":105416},[1048],[31,105418,105420],{"className":105419,"style":1053},[1052],[31,105421,105422,105425],{"style":66251},[31,105423],{"className":105424,"style":1061},[1060],[31,105426,105428],{"className":105427},[1065,1066,1067,1068],[31,105429,52065],{"className":105430,"style":52166},[75,76,1068],[31,105432,1076],{"className":105433},[1075],[31,105435,105437],{"className":105436},[1048],[31,105438,105440],{"className":105439,"style":2050},[1052],[31,105441],{},[31,105443],{"className":105444,"style":120},[119],[31,105446,97],{"className":105447},[124],[31,105449],{"className":105450,"style":120},[119],[31,105452,919],{"className":105453,"style":64444},[75,968],[31,105455,105457,105460],{"className":105456},[75],[31,105458,43481],{"className":105459,"style":120},[75,76],[31,105461,105463],{"className":105462},[1039],[31,105464,105466,105486],{"className":105465},[1043,1044],[31,105467,105469,105483],{"className":105468},[1048],[31,105470,105472],{"className":105471,"style":3230},[1052],[31,105473,105474,105477],{"style":66251},[31,105475],{"className":105476,"style":1061},[1060],[31,105478,105480],{"className":105479},[1065,1066,1067,1068],[31,105481,4587],{"className":105482},[75,1068],[31,105484,1076],{"className":105485},[1075],[31,105487,105489],{"className":105488},[1048],[31,105490,105492],{"className":105491,"style":2050},[1052],[31,105493],{},[31,105495,105497,105500],{"className":105496},[75],[31,105498,4572],{"className":105499},[75],[31,105501,105503],{"className":105502},[1039],[31,105504,105506,105526],{"className":105505},[1043,1044],[31,105507,105509,105523],{"className":105508},[1048],[31,105510,105512],{"className":105511,"style":3230},[1052],[31,105513,105514,105517],{"style":2807},[31,105515],{"className":105516,"style":1061},[1060],[31,105518,105520],{"className":105519},[1065,1066,1067,1068],[31,105521,4587],{"className":105522},[75,1068],[31,105524,1076],{"className":105525},[1075],[31,105527,105529],{"className":105528},[1048],[31,105530,105532],{"className":105531,"style":2050},[1052],[31,105533],{},[31,105535,105536,105539],{"style":57119},[31,105537],{"className":105538,"style":1637},[1060],[31,105540,105542,105545,105548,105551,105554,105557,105560,105600,105603,105606,105609,105649,105689],{"className":105541},[75],[31,105543],{"className":105544},[75],[31,105546],{"className":105547,"style":293},[119],[31,105549,347],{"className":105550},[297],[31,105552],{"className":105553,"style":293},[119],[31,105555,53298],{"className":105556,"style":24157},[75,76],[31,105558,4572],{"className":105559},[75],[31,105561,105563,105566],{"className":105562},[75],[31,105564,43481],{"className":105565,"style":120},[75,76],[31,105567,105569],{"className":105568},[1039],[31,105570,105572,105592],{"className":105571},[1043,1044],[31,105573,105575,105589],{"className":105574},[1048],[31,105576,105578],{"className":105577,"style":1053},[1052],[31,105579,105580,105583],{"style":66251},[31,105581],{"className":105582,"style":1061},[1060],[31,105584,105586],{"className":105585},[1065,1066,1067,1068],[31,105587,52065],{"className":105588,"style":52166},[75,76,1068],[31,105590,1076],{"className":105591},[1075],[31,105593,105595],{"className":105594},[1048],[31,105596,105598],{"className":105597,"style":2050},[1052],[31,105599],{},[31,105601],{"className":105602,"style":120},[119],[31,105604,97],{"className":105605},[124],[31,105607],{"className":105608,"style":120},[119],[31,105610,105612,105615],{"className":105611},[75],[31,105613,43481],{"className":105614,"style":120},[75,76],[31,105616,105618],{"className":105617},[1039],[31,105619,105621,105641],{"className":105620},[1043,1044],[31,105622,105624,105638],{"className":105623},[1048],[31,105625,105627],{"className":105626,"style":3230},[1052],[31,105628,105629,105632],{"style":66251},[31,105630],{"className":105631,"style":1061},[1060],[31,105633,105635],{"className":105634},[1065,1066,1067,1068],[31,105636,4587],{"className":105637},[75,1068],[31,105639,1076],{"className":105640},[1075],[31,105642,105644],{"className":105643},[1048],[31,105645,105647],{"className":105646,"style":2050},[1052],[31,105648],{},[31,105650,105652,105655],{"className":105651},[75],[31,105653,4572],{"className":105654},[75],[31,105656,105658],{"className":105657},[1039],[31,105659,105661,105681],{"className":105660},[1043,1044],[31,105662,105664,105678],{"className":105663},[1048],[31,105665,105667],{"className":105666,"style":3230},[1052],[31,105668,105669,105672],{"style":2807},[31,105670],{"className":105671,"style":1061},[1060],[31,105673,105675],{"className":105674},[1065,1066,1067,1068],[31,105676,4587],{"className":105677},[75,1068],[31,105679,1076],{"className":105680},[1075],[31,105682,105684],{"className":105683},[1048],[31,105685,105687],{"className":105686,"style":2050},[1052],[31,105688],{},[31,105690,495],{"className":105691},[75],[31,105693,1076],{"className":105694},[1075],[31,105696,105698],{"className":105697},[1048],[31,105699,105701],{"className":105700,"style":57182},[1052],[31,105702],{},[11,105704,9612,105705,105733],{},[31,105706,105708,105721],{"className":105707,"translate":35},[34],[31,105709,105711],{"className":105710},[39],[41,105712,105713],{"xmlns":43},[45,105714,105715,105719],{},[48,105716,105717],{},[51,105718,52065],{},[55,105720,52065],{"encoding":57},[31,105722,105724],{"className":105723,"ariaHidden":62},[61],[31,105725,105727,105730],{"className":105726},[66],[31,105728],{"className":105729,"style":1630},[70],[31,105731,52065],{"className":105732,"style":52166},[75,76]," 反复迭代即得。证毕。",[11,105735,105736,105737,105807,105808,105867,105868,105896,105897,105925],{},"这个定理说明，初始估计 ",[31,105738,105740,105758],{"className":105739,"translate":35},[34],[31,105741,105743],{"className":105742},[39],[41,105744,105745],{"xmlns":43},[45,105746,105747,105755],{},[48,105748,105749],{},[1004,105750,105751,105753],{},[51,105752,43481],{},[91,105754,1977],{},[55,105756,105757],{"encoding":57},"V_0",[31,105759,105761],{"className":105760,"ariaHidden":62},[61],[31,105762,105764,105767],{"className":105763},[66],[31,105765],{"className":105766,"style":5230},[70],[31,105768,105770,105773],{"className":105769},[75],[31,105771,43481],{"className":105772,"style":120},[75,76],[31,105774,105776],{"className":105775},[1039],[31,105777,105779,105799],{"className":105778},[1043,1044],[31,105780,105782,105796],{"className":105781},[1048],[31,105783,105785],{"className":105784,"style":2028},[1052],[31,105786,105787,105790],{"style":66251},[31,105788],{"className":105789,"style":1061},[1060],[31,105791,105793],{"className":105792},[1065,1066,1067,1068],[31,105794,1977],{"className":105795},[75,1068],[31,105797,1076],{"className":105798},[1075],[31,105800,105802],{"className":105801},[1048],[31,105803,105805],{"className":105804,"style":2050},[1052],[31,105806],{}," 不需要很准确。无论从哪个有界函数出发，误差都会按 ",[31,105809,105811,105829],{"className":105810,"translate":35},[34],[31,105812,105814],{"className":105813},[39],[41,105815,105816],{"xmlns":43},[45,105817,105818,105826],{},[48,105819,105820],{},[1180,105821,105822,105824],{},[51,105823,53298],{},[51,105825,52065],{},[55,105827,105828],{"encoding":57},"\\gamma^k",[31,105830,105832],{"className":105831,"ariaHidden":62},[61],[31,105833,105835,105838],{"className":105834},[66],[31,105836],{"className":105837,"style":68274},[70],[31,105839,105841,105844],{"className":105840},[75],[31,105842,53298],{"className":105843,"style":24157},[75,76],[31,105845,105847],{"className":105846},[1039],[31,105848,105850],{"className":105849},[1043],[31,105851,105853],{"className":105852},[1048],[31,105854,105856],{"className":105855,"style":20863},[1052],[31,105857,105858,105861],{"style":3631},[31,105859],{"className":105860,"style":1061},[1060],[31,105862,105864],{"className":105863},[1065,1066,1067,1068],[31,105865,52065],{"className":105866,"style":52166},[75,76,1068]," 的速度衰减。",[31,105869,105871,105884],{"className":105870,"translate":35},[34],[31,105872,105874],{"className":105873},[39],[41,105875,105876],{"xmlns":43},[45,105877,105878,105882],{},[48,105879,105880],{},[51,105881,53298],{},[55,105883,79191],{"encoding":57},[31,105885,105887],{"className":105886,"ariaHidden":62},[61],[31,105888,105890,105893],{"className":105889},[66],[31,105891],{"className":105892,"style":865},[70],[31,105894,53298],{"className":105895,"style":24157},[75,76]," 越接近 ",[31,105898,105900,105913],{"className":105899,"translate":35},[34],[31,105901,105903],{"className":105902},[39],[41,105904,105905],{"xmlns":43},[45,105906,105907,105911],{},[48,105908,105909],{},[91,105910,205],{},[55,105912,205],{"encoding":57},[31,105914,105916],{"className":105915,"ariaHidden":62},[61],[31,105917,105919,105922],{"className":105918},[66],[31,105920],{"className":105921,"style":326},[70],[31,105923,205],{"className":105924},[75],"，衰减越慢，所需的迭代次数越多。这也是折扣因子在数值上既影响问题定义、又影响求解难度的体现。",[52860,105927,105929],{"id":105928},"_52-误差上界","5.2 误差上界",[11,105931,105932,105936,105937,106006],{},[5813,105933,105935],{"href":105934},"#theorem-convergence","定理 5.1"," 给出的误差界含有未知的 ",[31,105938,105940,105957],{"className":105939,"translate":35},[34],[31,105941,105943],{"className":105942},[39],[41,105944,105945],{"xmlns":43},[45,105946,105947,105955],{},[48,105948,105949],{},[1004,105950,105951,105953],{},[51,105952,43481],{},[51,105954,4587],{"mathvariant":494},[55,105956,87528],{"encoding":57},[31,105958,105960],{"className":105959,"ariaHidden":62},[61],[31,105961,105963,105966],{"className":105962},[66],[31,105964],{"className":105965,"style":5230},[70],[31,105967,105969,105972],{"className":105968},[75],[31,105970,43481],{"className":105971,"style":120},[75,76],[31,105973,105975],{"className":105974},[1039],[31,105976,105978,105998],{"className":105977},[1043,1044],[31,105979,105981,105995],{"className":105980},[1048],[31,105982,105984],{"className":105983,"style":3230},[1052],[31,105985,105986,105989],{"style":66251},[31,105987],{"className":105988,"style":1061},[1060],[31,105990,105992],{"className":105991},[1065,1066,1067,1068],[31,105993,4587],{"className":105994},[75,1068],[31,105996,1076],{"className":105997},[1075],[31,105999,106001],{"className":106000},[1048],[31,106002,106004],{"className":106003,"style":2050},[1052],[31,106005],{},"，在实际计算中不能直接使用。下面的命题给出只依赖迭代差分与初始差分的界。",[52865,106008,106011,106345,106832,106834],{"id":106009,"title":106010},"proposition-error-bounds","命题 5.2（先验与后验误差界）",[11,106012,52871,106013,74019,106041,106069,106070,1138,106223,106292,106293,1138],{},[31,106014,106016,106029],{"className":106015,"translate":35},[34],[31,106017,106019],{"className":106018},[39],[41,106020,106021],{"xmlns":43},[45,106022,106023,106027],{},[48,106024,106025],{},[51,106026,919],{"mathvariant":925},[55,106028,87378],{"encoding":57},[31,106030,106032],{"className":106031,"ariaHidden":62},[61],[31,106033,106035,106038],{"className":106034},[66],[31,106036],{"className":106037,"style":945},[70],[31,106039,919],{"className":106040,"style":64444},[75,968],[31,106042,106044,106057],{"className":106043,"translate":35},[34],[31,106045,106047],{"className":106046},[39],[41,106048,106049],{"xmlns":43},[45,106050,106051,106055],{},[48,106052,106053],{},[51,106054,53298],{},[55,106056,79191],{"encoding":57},[31,106058,106060],{"className":106059,"ariaHidden":62},[61],[31,106061,106063,106066],{"className":106062},[66],[31,106064],{"className":106065,"style":865},[70],[31,106067,53298],{"className":106068,"style":24157},[75,76],"-压缩映射，",[31,106071,106073,106107],{"className":106072,"translate":35},[34],[31,106074,106076],{"className":106075},[39],[41,106077,106078],{"xmlns":43},[45,106079,106080,106104],{},[48,106081,106082,106094,106096,106098],{},[1004,106083,106084,106086],{},[51,106085,43481],{},[48,106087,106088,106090,106092],{},[51,106089,52065],{},[95,106091,434],{},[91,106093,205],{},[95,106095,223],{},[51,106097,919],{"mathvariant":925},[1004,106099,106100,106102],{},[51,106101,43481],{},[51,106103,52065],{},[55,106105,106106],{"encoding":57},"V_{k+1}=\\mathcal{T}V_k",[31,106108,106110,106174],{"className":106109,"ariaHidden":62},[61],[31,106111,106113,106116,106165,106168,106171],{"className":106112},[66],[31,106114],{"className":106115,"style":60241},[70],[31,106117,106119,106122],{"className":106118},[75],[31,106120,43481],{"className":106121,"style":120},[75,76],[31,106123,106125],{"className":106124},[1039],[31,106126,106128,106157],{"className":106127},[1043,1044],[31,106129,106131,106154],{"className":106130},[1048],[31,106132,106134],{"className":106133,"style":1053},[1052],[31,106135,106136,106139],{"style":66251},[31,106137],{"className":106138,"style":1061},[1060],[31,106140,106142],{"className":106141},[1065,1066,1067,1068],[31,106143,106145,106148,106151],{"className":106144},[75,1068],[31,106146,52065],{"className":106147,"style":52166},[75,76,1068],[31,106149,434],{"className":106150},[124,1068],[31,106152,205],{"className":106153},[75,1068],[31,106155,1076],{"className":106156},[1075],[31,106158,106160],{"className":106159},[1048],[31,106161,106163],{"className":106162,"style":25082},[1052],[31,106164],{},[31,106166],{"className":106167,"style":293},[119],[31,106169,223],{"className":106170},[297],[31,106172],{"className":106173,"style":293},[119],[31,106175,106177,106180,106183],{"className":106176},[66],[31,106178],{"className":106179,"style":5230},[70],[31,106181,919],{"className":106182,"style":64444},[75,968],[31,106184,106186,106189],{"className":106185},[75],[31,106187,43481],{"className":106188,"style":120},[75,76],[31,106190,106192],{"className":106191},[1039],[31,106193,106195,106215],{"className":106194},[1043,1044],[31,106196,106198,106212],{"className":106197},[1048],[31,106199,106201],{"className":106200,"style":1053},[1052],[31,106202,106203,106206],{"style":66251},[31,106204],{"className":106205,"style":1061},[1060],[31,106207,106209],{"className":106208},[1065,1066,1067,1068],[31,106210,52065],{"className":106211,"style":52166},[75,76,1068],[31,106213,1076],{"className":106214},[1075],[31,106216,106218],{"className":106217},[1048],[31,106219,106221],{"className":106220,"style":2050},[1052],[31,106222],{},[31,106224,106226,106243],{"className":106225,"translate":35},[34],[31,106227,106229],{"className":106228},[39],[41,106230,106231],{"xmlns":43},[45,106232,106233,106241],{},[48,106234,106235],{},[1004,106236,106237,106239],{},[51,106238,43481],{},[51,106240,4587],{"mathvariant":494},[55,106242,87528],{"encoding":57},[31,106244,106246],{"className":106245,"ariaHidden":62},[61],[31,106247,106249,106252],{"className":106248},[66],[31,106250],{"className":106251,"style":5230},[70],[31,106253,106255,106258],{"className":106254},[75],[31,106256,43481],{"className":106257,"style":120},[75,76],[31,106259,106261],{"className":106260},[1039],[31,106262,106264,106284],{"className":106263},[1043,1044],[31,106265,106267,106281],{"className":106266},[1048],[31,106268,106270],{"className":106269,"style":3230},[1052],[31,106271,106272,106275],{"style":66251},[31,106273],{"className":106274,"style":1061},[1060],[31,106276,106278],{"className":106277},[1065,1066,1067,1068],[31,106279,4587],{"className":106280},[75,1068],[31,106282,1076],{"className":106283},[1075],[31,106285,106287],{"className":106286},[1048],[31,106288,106290],{"className":106289,"style":2050},[1052],[31,106291],{}," 为不动点。则对任意 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点。由第二式，当 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时，若",[31,110486,110488],{"className":110487,"translate":35},[411],[31,110489,110491,110587],{"className":110490,"translate":35},[34],[31,110492,110494],{"className":110493},[39],[41,110495,110496],{"xmlns":43,"display":420},[45,110497,110498,110584],{},[5606,110499,110500,110544],{"rowspacing":5608,"columnalign":5609,"columnspacing":5610},[5612,110501,110502,110508],{},[5615,110503,110504],{},[2071,110505,110506],{"scriptlevel":1977,"displaystyle":62},[48,110507],{},[5615,110509,110510],{},[2071,110511,110512],{"scriptlevel":1977,"displaystyle":62},[48,110513,110514,110516,110518,110524,110526,110538],{},[48,110515],{},[51,110517,4572],{"mathvariant":494},[1004,110519,110520,110522],{},[51,110521,43481],{},[51,110523,52065],{},[95,110525,97],{},[1004,110527,110528,110530],{},[51,110529,43481],{},[48,110531,110532,110534,110536],{},[51,110533,52065],{},[95,110535,97],{},[91,110537,205],{},[1004,110539,110540,110542],{},[51,110541,4572],{"mathvariant":494},[51,110543,4587],{"mathvariant":494},[5612,110545,110546,110552],{},[5615,110547,110548],{},[2071,110549,110550],{"scriptlevel":1977,"displaystyle":62},[48,110551],{},[5615,110553,110554],{},[2071,110555,110556],{"scriptlevel":1977,"displaystyle":62},[48,110557,110558,110560,110562,110564,110582],{},[48,110559],{},[119,110561],{"width":1584},[95,110563,347],{},[2074,110565,110566,110580],{},[48,110567,110568,110570,110572,110574,110576,110578],{},[51,110569,4793],{},[95,110571,211],{"stretchy":210},[91,110573,205],{},[95,110575,97],{},[51,110577,53298],{},[95,110579,220],{"stretchy":210},[51,110581,53298],{},[95,110583,1210],{"separator":62},[55,110585,110586],{"encoding":57},"\\begin{aligned}\n&\\|V_k-V_{k-1}\\|_\\infty\\\\\n&\\quad\\le\\frac{\\varepsilon(1-\\gamma)}{\\gamma},\n\\end{aligned}",[31,110588,110590],{"className":110589,"ariaHidden":62},[61],[31,110591,110593,110597],{"className":110592},[66],[31,110594],{"className":110595,"style":110596},[70],"height:3.8074em;vertical-align:-1.6537em;",[31,110598,110600],{"className":110599},[75],[31,110601,110603,110647],{"className":110602},[5606],[31,110604,110606],{"className":110605},[5867],[31,110607,110609,110638],{"className":110608},[1043,1044],[31,110610,110612,110635],{"className":110611},[1048],[31,110613,110616,110626],{"className":110614,"style":110615},[1052],"height:2.1537em;",[31,110617,110619,110623],{"style":110618},"top:-4.7407em;",[31,110620],{"className":110621,"style":110622},[1060],"height:3.427em;",[31,110624],{"className":110625},[75],[31,110627,110629,110632],{"style":110628},"top:-2.6537em;",[31,110630],{"className":110631,"style":110622},[1060],[31,110633],{"className":110634},[75],[31,110636,1076],{"className":110637},[1075],[31,110639,110641],{"className":110640},[1048],[31,110642,110645],{"className":110643,"style":110644},[1052],"height:1.6537em;",[31,110646],{},[31,110648,110650],{"className":110649},[6062],[31,110651,110653,110923],{"className":110652},[1043,1044],[31,110654,110656,110920],{"className":110655},[1048],[31,110657,110659,110811],{"className":110658,"style":110615},[1052],[31,110660,110661,110664],{"style":110618},[31,110662],{"className":110663,"style":110622},[1060],[31,110665,110667,110670,110673,110713,110716,110719,110722,110771],{"className":110666},[75],[31,110668],{"className":110669},[75],[31,110671,4572],{"className":110672},[75],[31,110674,110676,110679],{"className":110675},[75],[31,110677,43481],{"className":110678,"style":120},[75,76],[31,110680,110682],{"className":110681},[1039],[31,110683,110685,110705],{"className":110684},[1043,1044],[31,110686,110688,110702],{"className":110687},[1048],[31,110689,110691],{"className":110690,"style":1053},[1052],[31,110692,110693,110696],{"style":66251},[31,110694],{"className":110695,"style":1061},[1060],[31,110697,110699],{"className":110698},[1065,1066,1067,1068],[31,110700,52065],{"className":110701,"style":52166},[75,76,1068],[31,110703,1076],{"className":110704},[1075],[31,110706,110708],{"className":110707},[1048],[31,110709,110711],{"className":110710,"style":2050},[1052],[31,110712],{},[31,110714],{"className":110715,"style":120},[119],[31,110717,97],{"className":110718},[124],[31,110720],{"className":110721,"style":120},[119],[31,110723,110725,110728],{"className":110724},[75],[31,110726,43481],{"className":110727,"style":120},[75,76],[31,110729,110731],{"className":110730},[1039],[31,110732,110734,110763],{"className":110733},[1043,1044],[31,110735,110737,110760],{"className":110736},[1048],[31,110738,110740],{"className":110739,"style":1053},[1052],[31,110741,110742,110745],{"style":66251},[31,110743],{"className":110744,"style":1061},[1060],[31,110746,110748],{"className":110747},[1065,1066,1067,1068],[31,110749,110751,110754,110757],{"className":110750},[75,1068],[31,110752,52065],{"className":110753,"style":52166},[75,76,1068],[31,110755,97],{"className":110756},[124,1068],[31,110758,205],{"className":110759},[75,1068],[31,110761,1076],{"className":110762},[1075],[31,110764,110766],{"className":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",[31,110935,110937,110975],{"className":110936,"translate":35},[34],[31,110938,110940],{"className":110939},[39],[41,110941,110942],{"xmlns":43},[45,110943,110944,110972],{},[48,110945,110946,110948,110954,110956,110962,110968,110970],{},[51,110947,4572],{"mathvariant":494},[1004,110949,110950,110952],{},[51,110951,43481],{},[51,110953,52065],{},[95,110955,97],{},[1004,110957,110958,110960],{},[51,110959,43481],{},[51,110961,4587],{"mathvariant":494},[1004,110963,110964,110966],{},[51,110965,4572],{"mathvariant":494},[51,110967,4587],{"mathvariant":494},[95,110969,347],{},[51,110971,4793],{},[55,110973,110974],{"encoding":57},"\\|V_k-V_\\infty\\|_\\infty\\le\\varepsilon",[31,110976,110978,111036,111131],{"className":110977,"ariaHidden":62},[61],[31,110979,110981,110984,110987,111027,111030,111033],{"className":110980},[66],[31,110982],{"className":110983,"style":260},[70],[31,110985,4572],{"className":110986},[75],[31,110988,110990,110993],{"className":110989},[75],[31,110991,43481],{"className":110992,"style":120},[75,76],[31,110994,110996],{"className":110995},[1039],[31,110997,110999,111019],{"className":110998},[1043,1044],[31,111000,111002,111016],{"className":111001},[1048],[31,111003,111005],{"className":111004,"style":1053},[1052],[31,111006,111007,111010],{"style":66251},[31,111008],{"className":111009,"style":1061},[1060],[31,111011,111013],{"className":111012},[1065,1066,1067,1068],[31,111014,52065],{"className":111015,"style":52166},[75,76,1068],[31,111017,1076],{"className":111018},[1075],[31,111020,111022],{"className":111021},[1048],[31,111023,111025],{"className":111024,"style":2050},[1052],[31,111026],{},[31,111028],{"className":111029,"style":120},[119],[31,111031,97],{"className":111032},[124],[31,111034],{"className":111035,"style":120},[119],[31,111037,111039,111042,111082,111122,111125,111128],{"className":111038},[66],[31,111040],{"className":111041,"style":260},[70],[31,111043,111045,111048],{"className":111044},[75],[31,111046,43481],{"className":111047,"style":120},[75,76],[31,111049,111051],{"className":111050},[1039],[31,111052,111054,111074],{"className":111053},[1043,1044],[31,111055,111057,111071],{"className":111056},[1048],[31,111058,111060],{"className":111059,"style":3230},[1052],[31,111061,111062,111065],{"style":66251},[31,111063],{"className":111064,"style":1061},[1060],[31,111066,111068],{"className":111067},[1065,1066,1067,1068],[31,111069,4587],{"className":111070},[75,1068],[31,111072,1076],{"className":111073},[1075],[31,111075,111077],{"className":111076},[1048],[31,111078,111080],{"className":111079,"style":2050},[1052],[31,111081],{},[31,111083,111085,111088],{"className":111084},[75],[31,111086,4572],{"className":111087},[75],[31,111089,111091],{"className":111090},[1039],[31,111092,111094,111114],{"className":111093},[1043,1044],[31,111095,111097,111111],{"className":111096},[1048],[31,111098,111100],{"className":111099,"style":3230},[1052],[31,111101,111102,111105],{"style":2807},[31,111103],{"className":111104,"style":1061},[1060],[31,111106,111108],{"className":111107},[1065,1066,1067,1068],[31,111109,4587],{"className":111110},[75,1068],[31,111112,1076],{"className":111113},[1075],[31,111115,111117],{"className":111116},[1048],[31,111118,111120],{"className":111119,"style":2050},[1052],[31,111121],{},[31,111123],{"className":111124,"style":293},[119],[31,111126,347],{"className":111127},[297],[31,111129],{"className":111130,"style":293},[119],[31,111132,111134,111137],{"className":111133},[66],[31,111135],{"className":111136,"style":71},[70],[31,111138,4793],{"className":111139},[75,76],[52860,111141,111143],{"id":111142},"_53-例显式迭代","5.3 例：显式迭代",[11,111145,111146,111147,111262],{},"回到 2.4 节的两状态马尔可夫链。令 ",[31,111148,111150,111180],{"className":111149,"translate":35},[34],[31,111151,111153],{"className":111152},[39],[41,111154,111155],{"xmlns":43},[45,111156,111157,111177],{},[48,111158,111159,111165,111167,111169,111171,111173,111175],{},[1004,111160,111161,111163],{},[51,111162,43481],{},[91,111164,1977],{},[95,111166,223],{},[95,111168,211],{"stretchy":210},[91,111170,1977],{},[95,111172,1210],{"separator":62},[91,111174,1977],{},[95,111176,220],{"stretchy":210},[55,111178,111179],{"encoding":57},"V_0=(0,0)",[31,111181,111183,111238],{"className":111182,"ariaHidden":62},[61],[31,111184,111186,111189,111229,111232,111235],{"className":111185},[66],[31,111187],{"className":111188,"style":5230},[70],[31,111190,111192,111195],{"className":111191},[75],[31,111193,43481],{"className":111194,"style":120},[75,76],[31,111196,111198],{"className":111197},[1039],[31,111199,111201,111221],{"className":111200},[1043,1044],[31,111202,111204,111218],{"className":111203},[1048],[31,111205,111207],{"className":111206,"style":2028},[1052],[31,111208,111209,111212],{"style":66251},[31,111210],{"className":111211,"style":1061},[1060],[31,111213,111215],{"className":111214},[1065,1066,1067,1068],[31,111216,1977],{"className":111217},[75,1068],[31,111219,1076],{"className":111220},[1075],[31,111222,111224],{"className":111223},[1048],[31,111225,111227],{"className":111226,"style":2050},[1052],[31,111228],{},[31,111230],{"className":111231,"style":293},[119],[31,111233,223],{"className":111234},[297],[31,111236],{"className":111237,"style":293},[119],[31,111239,111241,111244,111247,111250,111253,111256,111259],{"className":111240},[66],[31,111242],{"className":111243,"style":260},[70],[31,111245,211],{"className":111246},[264],[31,111248,1977],{"className":111249},[75],[31,111251,1210],{"className":111252},[1391],[31,111254],{"className":111255,"style":1467},[119],[31,111257,1977],{"className":111258},[75],[31,111260,220],{"className":111261},[289],"，迭代为",[31,111264,111266],{"className":111265,"translate":35},[411],[31,111267,111269,111359],{"className":111268,"translate":35},[34],[31,111270,111272],{"className":111271},[39],[41,111273,111274],{"xmlns":43,"display":420},[45,111275,111276,111356],{},[48,111277,111278,111290,111292,111294,111296,111298,111300,111302,111304,111310,111312,111314,111316,111318,111320,111332,111334,111336,111338,111340,111342,111348,111350,111352,111354],{},[1004,111279,111280,111282],{},[51,111281,43481],{},[48,111283,111284,111286,111288],{},[51,111285,52065],{},[95,111287,434],{},[91,111289,205],{},[95,111291,211],{"stretchy":210},[91,111293,205],{},[95,111295,220],{"stretchy":210},[95,111297,223],{},[91,111299,205],{},[95,111301,434],{},[91,111303,23247],{},[1004,111305,111306,111308],{},[51,111307,43481],{},[51,111309,52065],{},[95,111311,211],{"stretchy":210},[91,111313,152],{},[95,111315,220],{"stretchy":210},[95,111317,1210],{"separator":62},[119,111319],{"width":20286},[1004,111321,111322,111324],{},[51,111323,43481],{},[48,111325,111326,111328,111330],{},[51,111327,52065],{},[95,111329,434],{},[91,111331,205],{},[95,111333,211],{"stretchy":210},[91,111335,152],{},[95,111337,220],{"stretchy":210},[95,111339,223],{},[91,111341,23247],{},[1004,111343,111344,111346],{},[51,111345,43481],{},[51,111347,52065],{},[95,111349,211],{"stretchy":210},[91,111351,205],{},[95,111353,220],{"stretchy":210},[51,111355,495],{"mathvariant":494},[55,111357,111358],{"encoding":57},"V_{k+1}(1)=1+0.9V_k(2),\n\\qquad\nV_{k+1}(2)=0.9V_k(1).",[31,111360,111362,111435,111453,111587],{"className":111361,"ariaHidden":62},[61],[31,111363,111365,111368,111417,111420,111423,111426,111429,111432],{"className":111364},[66],[31,111366],{"className":111367,"style":260},[70],[31,111369,111371,111374],{"className":111370},[75],[31,111372,43481],{"className":111373,"style":120},[75,76],[31,111375,111377],{"className":111376},[1039],[31,111378,111380,111409],{"className":111379},[1043,1044],[31,111381,111383,111406],{"className":111382},[1048],[31,111384,111386],{"className":111385,"style":1053},[1052],[31,111387,111388,111391],{"style":66251},[31,111389],{"className":111390,"style":1061},[1060],[31,111392,111394],{"className":111393},[1065,1066,1067,1068],[31,111395,111397,111400,111403],{"className":111396},[75,1068],[31,111398,52065],{"className":111399,"style":52166},[75,76,1068],[31,111401,434],{"className":111402},[124,1068],[31,111404,205],{"className":111405},[75,1068],[31,111407,1076],{"className":111408},[1075],[31,111410,111412],{"className":111411},[1048],[31,111413,111415],{"className":111414,"style":25082},[1052],[31,111416],{},[31,111418,211],{"className":111419},[264],[31,111421,205],{"className":111422},[75],[31,111424,220],{"className":111425},[289],[31,111427],{"className":111428,"style":293},[119],[31,111430,223],{"className":111431},[297],[31,111433],{"className":111434,"style":293},[119],[31,111436,111438,111441,111444,111447,111450],{"className":111437},[66],[31,111439],{"className":111440,"style":112},[70],[31,111442,205],{"className":111443},[75],[31,111445],{"className":111446,"style":120},[119],[31,111448,434],{"className":111449},[124],[31,111451],{"className":111452,"style":120},[119],[31,111454,111456,111459,111462,111502,111505,111508,111511,111514,111517,111520,111569,111572,111575,111578,111581,111584],{"className":111455},[66],[31,111457],{"className":111458,"style":260},[70],[31,111460,23247],{"className":111461},[75],[31,111463,111465,111468],{"className":111464},[75],[31,111466,43481],{"className":111467,"style":120},[75,76],[31,111469,111471],{"className":111470},[1039],[31,111472,111474,111494],{"className":111473},[1043,1044],[31,111475,111477,111491],{"className":111476},[1048],[31,111478,111480],{"className":111479,"style":1053},[1052],[31,111481,111482,111485],{"style":66251},[31,111483],{"className":111484,"style":1061},[1060],[31,111486,111488],{"className":111487},[1065,1066,1067,1068],[31,111489,52065],{"className":111490,"style":52166},[75,76,1068],[31,111492,1076],{"className":111493},[1075],[31,111495,111497],{"className":111496},[1048],[31,111498,111500],{"className":111499,"style":2050},[1052],[31,111501],{},[31,111503,211],{"className":111504},[264],[31,111506,152],{"className":111507},[75],[31,111509,220],{"className":111510},[289],[31,111512,1210],{"className":111513},[1391],[31,111515],{"className":111516,"style":20525},[119],[31,111518],{"className":111519,"style":1467},[119],[31,111521,111523,111526],{"className":111522},[75],[31,111524,43481],{"className":111525,"style":120},[75,76],[31,111527,111529],{"className":111528},[1039],[31,111530,111532,111561],{"className":111531},[1043,1044],[31,111533,111535,111558],{"className":111534},[1048],[31,111536,111538],{"className":111537,"style":1053},[1052],[31,111539,111540,111543],{"style":66251},[31,111541],{"className":111542,"style":1061},[1060],[31,111544,111546],{"className":111545},[1065,1066,1067,1068],[31,111547,111549,111552,111555],{"className":111548},[75,1068],[31,111550,52065],{"className":111551,"style":52166},[75,76,1068],[31,111553,434],{"className":111554},[124,1068],[31,111556,205],{"className":111557},[75,1068],[31,111559,1076],{"className":111560},[1075],[31,111562,111564],{"className":111563},[1048],[31,111565,111567],{"className":111566,"style":25082},[1052],[31,111568],{},[31,111570,211],{"className":111571},[264],[31,111573,152],{"className":111574},[75],[31,111576,220],{"className":111577},[289],[31,111579],{"className":111580,"style":293},[119],[31,111582,223],{"className":111583},[297],[31,111585],{"className":111586,"style":293},[119],[31,111588,111590,111593,111596,111636,111639,111642,111645],{"className":111589},[66],[31,111591],{"className":111592,"style":260},[70],[31,111594,23247],{"className":111595},[75],[31,111597,111599,111602],{"className":111598},[75],[31,111600,43481],{"className":111601,"style":120},[75,76],[31,111603,111605],{"className":111604},[1039],[31,111606,111608,111628],{"className":111607},[1043,1044],[31,111609,111611,111625],{"className":111610},[1048],[31,111612,111614],{"className":111613,"style":1053},[1052],[31,111615,111616,111619],{"style":66251},[31,111617],{"className":111618,"style":1061},[1060],[31,111620,111622],{"className":111621},[1065,1066,1067,1068],[31,111623,52065],{"className":111624,"style":52166},[75,76,1068],[31,111626,1076],{"className":111627},[1075],[31,111629,111631],{"className":111630},[1048],[31,111632,111634],{"className":111633,"style":2050},[1052],[31,111635],{},[31,111637,211],{"className":111638},[264],[31,111640,205],{"className":111641},[75],[31,111643,220],{"className":111644},[289],[31,111646,495],{"className":111647},[75],[11,111649,111650],{},"前几步为",[31,111652,111654],{"className":111653,"translate":35},[411],[31,111655,111657,111842],{"className":111656,"translate":35},[34],[31,111658,111660],{"className":111659},[39],[41,111661,111662],{"xmlns":43,"display":420},[45,111663,111664,111839],{},[5606,111665,111666,111700,111734,111769,111804],{"rowspacing":5608,"columnalign":5609,"columnspacing":5610},[5612,111667,111668,111678],{},[5615,111669,111670],{},[2071,111671,111672],{"scriptlevel":1977,"displaystyle":62},[1004,111673,111674,111676],{},[51,111675,43481],{},[91,111677,205],{},[5615,111679,111680],{},[2071,111681,111682],{"scriptlevel":1977,"displaystyle":62},[48,111683,111684,111686,111688,111690,111692,111694,111696,111698],{},[48,111685],{},[95,111687,223],{},[95,111689,211],{"stretchy":210},[91,111691,205],{},[95,111693,1210],{"separator":62},[91,111695,1977],{},[95,111697,220],{"stretchy":210},[95,111699,1210],{"separator":62},[5612,111701,111702,111712],{},[5615,111703,111704],{},[2071,111705,111706],{"scriptlevel":1977,"displaystyle":62},[1004,111707,111708,111710],{},[51,111709,43481],{},[91,111711,152],{},[5615,111713,111714],{},[2071,111715,111716],{"scriptlevel":1977,"displaystyle":62},[48,111717,111718,111720,111722,111724,111726,111728,111730,111732],{},[48,111719],{},[95,111721,223],{},[95,111723,211],{"stretchy":210},[91,111725,205],{},[95,111727,1210],{"separator":62},[91,111729,23247],{},[95,111731,220],{"stretchy":210},[95,111733,1210],{"separator":62},[5612,111735,111736,111746],{},[5615,111737,111738],{},[2071,111739,111740],{"scriptlevel":1977,"displaystyle":62},[1004,111741,111742,111744],{},[51,111743,43481],{},[91,111745,93],{},[5615,111747,111748],{},[2071,111749,111750],{"scriptlevel":1977,"displaystyle":62},[48,111751,111752,111754,111756,111758,111761,111763,111765,111767],{},[48,111753],{},[95,111755,223],{},[95,111757,211],{"stretchy":210},[91,111759,111760],{},"1.81",[95,111762,1210],{"separator":62},[91,111764,23247],{},[95,111766,220],{"stretchy":210},[95,111768,1210],{"separator":62},[5612,111770,111771,111781],{},[5615,111772,111773],{},[2071,111774,111775],{"scriptlevel":1977,"displaystyle":62},[1004,111776,111777,111779],{},[51,111778,43481],{},[91,111780,15416],{},[5615,111782,111783],{},[2071,111784,111785],{"scriptlevel":1977,"displaystyle":62},[48,111786,111787,111789,111791,111793,111795,111797,111800,111802],{},[48,111788],{},[95,111790,223],{},[95,111792,211],{"stretchy":210},[91,111794,111760],{},[95,111796,1210],{"separator":62},[91,111798,111799],{},"1.629",[95,111801,220],{"stretchy":210},[95,111803,1210],{"separator":62},[5612,111805,111806,111816],{},[5615,111807,111808],{},[2071,111809,111810],{"scriptlevel":1977,"displaystyle":62},[1004,111811,111812,111814],{},[51,111813,43481],{},[91,111815,7853],{},[5615,111817,111818],{},[2071,111819,111820],{"scriptlevel":1977,"displaystyle":62},[48,111821,111822,111824,111826,111828,111831,111833,111835,111837],{},[48,111823],{},[95,111825,223],{},[95,111827,211],{"stretchy":210},[91,111829,111830],{},"2.4661",[95,111832,1210],{"separator":62},[91,111834,111799],{},[95,111836,220],{"stretchy":210},[51,111838,495],{"mathvariant":494},[55,111840,111841],{"encoding":57},"\\begin{aligned}\nV_1&=(1,0),\\\\\nV_2&=(1,0.9),\\\\\nV_3&=(1.81,0.9),\\\\\nV_4&=(1.81,1.629),\\\\\nV_5&=(2.4661,1.629).\n\\end{aligned}",[31,111843,111845],{"className":111844,"ariaHidden":62},[61],[31,111846,111848,111852],{"className":111847},[66],[31,111849],{"className":111850,"style":111851},[70],"height:7.2em;vertical-align:-3.35em;",[31,111853,111855],{"className":111854},[75],[31,111856,111858,112127],{"className":111857},[5606],[31,111859,111861],{"className":111860},[5867],[31,111862,111864,112118],{"className":111863},[1043,1044],[31,111865,111867,112115],{"className":111866},[1048],[31,111868,111871,111920,111969,112017,112066],{"className":111869,"style":111870},[1052],"height:3.85em;",[31,111872,111874,111877],{"style":111873},"top:-6.01em;",[31,111875],{"className":111876,"style":1637},[1060],[31,111878,111880],{"className":111879},[75],[31,111881,111883,111886],{"className":111882},[75],[31,111884,43481],{"className":111885,"style":120},[75,76],[31,111887,111889],{"className":111888},[1039],[31,111890,111892,111912],{"className":111891},[1043,1044],[31,111893,111895,111909],{"className":111894},[1048],[31,111896,111898],{"className":111897,"style":2028},[1052],[31,111899,111900,111903],{"style":66251},[31,111901],{"className":111902,"style":1061},[1060],[31,111904,111906],{"className":111905},[1065,1066,1067,1068],[31,111907,205],{"className":111908},[75,1068],[31,111910,1076],{"className":111911},[1075],[31,111913,111915],{"className":111914},[1048],[31,111916,111918],{"className":111917,"style":2050},[1052],[31,111919],{},[31,111921,111923,111926],{"style":111922},"top:-4.51em;",[31,111924],{"className":111925,"style":1637},[1060],[31,111927,111929],{"className":111928},[75],[31,111930,111932,111935],{"className":111931},[75],[31,111933,43481],{"className":111934,"style":120},[75,76],[31,111936,111938],{"className":111937},[1039],[31,111939,111941,111961],{"className":111940},[1043,1044],[31,111942,111944,111958],{"className":111943},[1048],[31,111945,111947],{"className":111946,"style":2028},[1052],[31,111948,111949,111952],{"style":66251},[31,111950],{"className":111951,"style":1061},[1060],[31,111953,111955],{"className":111954},[1065,1066,1067,1068],[31,111956,152],{"className":111957},[75,1068],[31,111959,1076],{"className":111960},[1075],[31,111962,111964],{"className":111963},[1048],[31,111965,111967],{"className":111966,"style":2050},[1052],[31,111968],{},[31,111970,111971,111974],{"style":53666},[31,111972],{"className":111973,"style":1637},[1060],[31,111975,111977],{"className":111976},[75],[31,111978,111980,111983],{"className":111979},[75],[31,111981,43481],{"className":111982,"style":120},[75,76],[31,111984,111986],{"className":111985},[1039],[31,111987,111989,112009],{"className":111988},[1043,1044],[31,111990,111992,112006],{"className":111991},[1048],[31,111993,111995],{"className":111994,"style":2028},[1052],[31,111996,111997,112000],{"style":66251},[31,111998],{"className":111999,"style":1061},[1060],[31,112001,112003],{"className":112002},[1065,1066,1067,1068],[31,112004,93],{"className":112005},[75,1068],[31,112007,1076],{"className":112008},[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lassName":112937},[75],[11,112939,30527,112940,112991],{},[31,112941,112943,112961],{"className":112942,"translate":35},[34],[31,112944,112946],{"className":112945},[39],[41,112947,112948],{"xmlns":43},[45,112949,112950,112958],{},[48,112951,112952,112954,112956],{},[51,112953,16396],{},[95,112955,929],{},[51,112957,4587],{"mathvariant":494},[55,112959,112960],{"encoding":57},"m\\to\\infty",[31,112962,112964,112982],{"className":112963,"ariaHidden":62},[61],[31,112965,112967,112970,112973,112976,112979],{"className":112966},[66],[31,112968],{"className":112969,"style":71},[70],[31,112971,16396],{"className":112972},[75,76],[31,112974],{"className":112975,"style":293},[119],[31,112977,929],{"className":112978},[297],[31,112980],{"className":112981,"style":293},[119],[31,112983,112985,112988],{"className":112984},[66],[31,112986],{"className":112987,"style":71},[70],[31,112989,4587],{"className":112990},[75],"，得到",[31,112993,112995],{"className":112994,"translate":35},[411],[31,112996,112998,113074],{"className":112997,"translate":35},[34],[31,112999,113001],{"className":113000},[39],[41,113002,113003],{"xmlns":43,"display":420},[45,113004,113005,113071],{},[48,113006,113007,113009,113011,113013,113015,113017,113033,113035,113037,113039,113041,113043,113045,113047,113049,113051,113067,113069],{},[51,113008,43481],{},[95,113010,211],{"stretchy":210},[91,113012,205],{},[95,113014,220],{"stretchy":210},[95,113016,223],{},[2074,113018,113019,113021],{},[91,113020,205],{},[48,113022,113023,113025,113027],{},[91,113024,205],{},[95,113026,97],{},[1180,113028,113029,113031],{},[91,113030,23247],{},[91,113032,152],{},[95,113034,223],{},[91,113036,70778],{},[95,113038,1210],{"separator":62},[119,113040],{"width":20286},[51,113042,43481],{},[95,113044,211],{"stretchy":210},[91,113046,152],{},[95,113048,220],{"stretchy":210},[95,113050,223],{},[2074,113052,113053,113055],{},[91,113054,23247],{},[48,113056,113057,113059,113061],{},[91,113058,205],{},[95,113060,97],{},[1180,113062,113063,113065],{},[91,113064,23247],{},[91,113066,152],{},[95,113068,223],{},[91,113070,70807],{},[55,113072,113073],{"encoding":57},"V(1)=\\frac{1}{1-0.9^2}=5.2632,\n\\qquad\nV(2)=\\frac{0.9}{1-0.9^2}=4.7368.",[31,113075,113077,113104,113223,113262,113380],{"className":113076,"ariaHidden":62},[61],[31,113078,113080,113083,113086,113089,113092,113095,113098,113101],{"className":113079},[66],[31,113081],{"className":113082,"style":260},[70],[31,113084,43481],{"className":113085,"style":120},[75,76],[31,113087,211],{"className":113088},[264],[31,113090,205],{"className":113091},[75],[31,113093,220],{"className":113094},[289],[31,113096],{"className":113097,"style":293},[119],[31,113099,223],{"className":113100},[297],[31,113102],{"className":113103,"style":293},[119],[31,113105,113107,113110,113214,113217,113220],{"className":113106},[66],[31,113108],{"className":113109,"style":70847},[70],[31,113111,113113,113116,113211],{"className":113112},[75],[31,113114],{"className":113115},[264,2139],[31,113117,113119],{"className":113118},[2074],[31,113120,113122,113203],{"className":113121},[1043,1044],[31,113123,113125,113200],{"className":113124},[1048],[31,113126,113128,113181,113189],{"className":113127,"style":26376},[1052],[31,113129,113130,113133],{"style":5090},[31,113131],{"className":113132,"style":1637},[1060],[31,113134,113136,113139,113142,113145,113148,113151],{"className":113135},[75],[31,113137,205],{"className":113138},[75],[31,113140],{"className":113141,"style":120},[119],[31,113143,97],{"className":113144},[124],[31,113146],{"className":113147,"style":120},[119],[31,113149,81469],{"className":113150},[75],[31,113152,113154,113157],{"className":113153},[75],[31,113155,81476],{"className":113156},[75],[31,113158,113160],{"className":113159},[1039],[31,113161,113163],{"className":113162},[1043],[31,113164,113166],{"className":113165},[1048],[31,113167,113170],{"className":113168,"style":113169},[1052],"height:0.7401em;",[31,113171,113172,113175],{"style":44016},[31,113173],{"className":113174,"style":1061},[1060],[31,113176,113178],{"className":113177},[1065,1066,1067,1068],[31,113179,152],{"className":113180},[75,1068],[31,113182,113183,113186],{"style":2170},[31,113184],{"className":113185,"style":1637},[1060],[31,113187],{"className":113188,"style":2178},[2177],[31,113190,113191,113194],{"style":5110},[31,113192],{"className":113193,"style":1637},[1060],[31,113195,113197],{"className":113196},[75],[31,113198,205],{"className":113199},[75],[31,113201,1076],{"className":113202},[1075],[31,113204,113206],{"className":113205},[1048],[31,113207,113209],{"className":113208,"style":70917},[1052],[31,113210],{},[31,113212],{"className":113213},[289,2139],[31,113215],{"className":113216,"style":293},[119],[31,113218,223],{"className":113219},[297],[31,113221],{"className":113222,"style":293},[119],[31,113224,113226,113229,113232,113235,113238,113241,113244,113247,113250,113253,113256,113259],{"className":113225},[66],[31,113227],{"className":113228,"style":260},[70],[31,113230,70778],{"className":113231},[75],[31,113233,1210],{"className":113234},[1391],[31,113236],{"className":113237,"style":20525},[119],[31,113239],{"className":113240,"style":1467},[119],[31,113242,43481],{"className":113243,"style":120},[75,76],[31,113245,211],{"className":113246},[264],[31,113248,152],{"className":113249},[75],[31,113251,220],{"className":113252},[289],[31,113254],{"className":113255,"style":293},[119],[31,113257,223],{"className":113258},[297],[31,113260],{"className":113261,"style":293},[119],[31,113263,113265,113268,113371,113374,113377],{"className":113264},[66],[31,113266],{"className":113267,"style":70847},[70],[31,113269,113271,113274,113368],{"className":113270},[75],[31,113272],{"className":113273},[264,2139],[31,113275,113277],{"className":113276},[2074],[31,113278,113280,113360],{"className":113279},[1043,1044],[31,113281,113283,113357],{"className":113282},[1048],[31,113284,113286,113338,113346],{"className":113285,"style":26376},[1052],[31,113287,113288,113291],{"style":5090},[31,113289],{"className":113290,"style":1637},[1060],[31,113292,113294,113297,113300,113303,113306,113309],{"className":113293},[75],[31,113295,205],{"className":113296},[75],[31,113298],{"className":113299,"style":120},[119],[31,113301,97],{"className":113302},[124],[31,113304],{"className":113305,"style":120},[119],[31,113307,81469],{"className":113308},[75],[31,113310,113312,113315],{"className":113311},[75],[31,113313,81476],{"className":113314},[75],[31,113316,113318],{"className":113317},[1039],[31,113319,113321],{"className":113320},[1043],[31,113322,113324],{"className":113323},[1048],[31,113325,113327],{"className":113326,"style":113169},[1052],[31,113328,113329,113332],{"style":44016},[31,113330],{"className":113331,"style":1061},[1060],[31,113333,113335],{"className":113334},[1065,1066,1067,1068],[31,113336,152],{"className":113337},[75,1068],[31,113339,113340,113343],{"style":2170},[31,113341],{"className":113342,"style":1637},[1060],[31,113344],{"className":113345,"style":2178},[2177],[31,113347,113348,113351],{"style":5110},[31,113349],{"className":113350,"style":1637},[1060],[31,113352,113354],{"className":113353},[75],[31,113355,23247],{"className":113356},[75],[31,113358,1076],{"className":113359},[1075],[31,113361,113363],{"className":113362},[1048],[31,113364,113366],{"className":113365,"style":70917},[1052],[31,113367],{},[31,113369],{"className":113370},[289,2139],[31,113372],{"className":113373,"style":293},[119],[31,113375,223],{"className":113376},[297],[31,113378],{"className":113379,"style":293},[119],[31,113381,113383,113386],{"className":113382},[66],[31,113384],{"className":113385,"style":326},[70],[31,113387,70807],{"className":113388},[75],[11,113390,113391,113392,113476,113477,113505],{},"这个例子的结构值得注意。若只看奇数步或偶数步的子序列，误差按 ",[31,113393,113395,113417],{"className":113394,"translate":35},[34],[31,113396,113398],{"className":113397},[39],[41,113399,113400],{"xmlns":43},[45,113401,113402,113414],{},[48,113403,113404,113410,113412],{},[1180,113405,113406,113408],{},[91,113407,23247],{},[91,113409,152],{},[95,113411,223],{},[91,113413,70648],{},[55,113415,113416],{"encoding":57},"0.9^2=0.81",[31,113418,113420,113467],{"className":113419,"ariaHidden":62},[61],[31,113421,113423,113426,113429,113458,113461,113464],{"className":113422},[66],[31,113424],{"className":113425,"style":5135},[70],[31,113427,81469],{"className":113428},[75],[31,113430,113432,113435],{"className":113431},[75],[31,113433,81476],{"className":113434},[75],[31,113436,113438],{"className":113437},[1039],[31,113439,113441],{"className":113440},[1043],[31,113442,113444],{"className":113443},[1048],[31,113445,113447],{"className":113446,"style":5135},[1052],[31,113448,113449,113452],{"style":3631},[31,113450],{"className":113451,"style":1061},[1060],[31,113453,113455],{"className":113454},[1065,1066,1067,1068],[31,113456,152],{"className":113457},[75,1068],[31,113459],{"className":113460,"style":293},[119],[31,113462,223],{"className":113463},[297],[31,113465],{"className":113466,"style":293},[119],[31,113468,113470,113473],{"className":113469},[66],[31,113471],{"className":113472,"style":326},[70],[31,113474,70648],{"className":113475},[75]," 的比例衰减，因为状态每两步回到自身。但逐步的 sup 范数误差仍然按 ",[31,113478,113480,113493],{"className":113479,"translate":35},[34],[31,113481,113483],{"className":113482},[39],[41,113484,113485],{"xmlns":43},[45,113486,113487,113491],{},[48,113488,113489],{},[91,113490,23247],{},[55,113492,23247],{"encoding":57},[31,113494,113496],{"className":113495,"ariaHidden":62},[61],[31,113497,113499,113502],{"className":113498},[66],[31,113500],{"className":113501,"style":326},[70],[31,113503,23247],{"className":113504},[75]," 衰减，只是最大误差所在的分量在两步之间交替。一般理论给出的是逐步最坏情形下的统一上界，具体问题的子序列可能具有更快的收敛速度。",[11,113507,69053,113508,113510,113511,101979],{},[5813,113509,105935],{"href":105934},"，初始误差为 ",[31,113512,113514,113552],{"className":113513,"translate":35},[34],[31,113515,113517],{"className":113516},[39],[41,113518,113519],{"xmlns":43},[45,113520,113521,113549],{},[48,113522,113523,113525,113531,113533,113539,113545,113547],{},[51,113524,4572],{"mathvariant":494},[1004,113526,113527,113529],{},[51,113528,43481],{},[91,113530,1977],{},[95,113532,97],{},[1004,113534,113535,113537],{},[51,113536,43481],{},[51,113538,4587],{"mathvariant":494},[1004,113540,113541,113543],{},[51,113542,4572],{"mathvariant":494},[51,113544,4587],{"mathvariant":494},[95,113546,223],{},[91,113548,70778],{},[55,113550,113551],{"encoding":57},"\\|V_0-V_\\infty\\|_\\infty=5.2632",[31,113553,113555,113613,113708],{"className":113554,"ariaHidden":62},[61],[31,113556,113558,113561,113564,113604,113607,113610],{"className":113557},[66],[31,113559],{"className":113560,"style":260},[70],[31,113562,4572],{"className":113563},[75],[31,113565,113567,113570],{"className":113566},[75],[31,113568,43481],{"className":113569,"style":120},[75,76],[31,113571,113573],{"className":113572},[1039],[31,113574,113576,113596],{"className":113575},[1043,1044],[31,113577,113579,113593],{"className":113578},[1048],[31,113580,113582],{"className":113581,"style":2028},[1052],[31,113583,113584,113587],{"style":66251},[31,113585],{"className":113586,"style":1061},[1060],[31,113588,113590],{"className":113589},[1065,1066,1067,1068],[31,113591,1977],{"className":113592},[75,1068],[31,113594,1076],{"className":113595},[1075],[31,113597,113599],{"className":113598},[1048],[31,113600,113602],{"className":113601,"style":2050},[1052],[31,113603],{},[31,113605],{"className":113606,"style":120},[119],[31,113608,97],{"className":113609},[124],[31,113611],{"className":113612,"style":120},[119],[31,113614,113616,113619,113659,113699,113702,113705],{"className":113615},[66],[31,113617],{"className":113618,"style":260},[70],[31,113620,113622,113625],{"className":113621},[75],[31,113623,43481],{"className":113624,"style":120},[75,76],[31,113626,113628],{"className":113627},[1039],[31,113629,113631,113651],{"className":113630},[1043,1044],[31,113632,113634,113648],{"className":113633},[1048],[31,113635,113637],{"className":113636,"style":3230},[1052],[31,113638,113639,113642],{"style":66251},[31,113640],{"className":113641,"style":1061},[1060],[31,113643,113645],{"className":113644},[1065,1066,1067,1068],[31,113646,4587],{"className":113647},[75,1068],[31,113649,1076],{"className":113650},[1075],[31,113652,113654],{"className":113653},[1048],[31,113655,113657],{"className":113656,"style":2050},[1052],[31,113658],{},[31,113660,113662,113665],{"className":113661},[75],[31,113663,4572],{"className":113664},[75],[31,113666,113668],{"className":113667},[1039],[31,113669,113671,113691],{"className":113670},[1043,1044],[31,113672,113674,113688],{"className":113673},[1048],[31,113675,113677],{"className":113676,"style":3230},[1052],[31,113678,113679,113682],{"style":2807},[31,113680],{"className":113681,"style":1061},[1060],[31,113683,113685],{"className":113684},[1065,1066,1067,1068],[31,113686,4587],{"className":113687},[75,1068],[31,113689,1076],{"className":113690},[1075],[31,113692,113694],{"className":113693},[1048],[31,113695,113697],{"className":113696,"style":2050},[1052],[31,113698],{},[31,113700],{"className":113701,"style":293},[119],[31,113703,223],{"className":113704},[297],[31,113706],{"className":113707,"style":293},[119],[31,113709,113711,113714],{"className":113710},[66],[31,113712],{"className":113713,"style":326},[70],[31,113715,70778],{"className":113716},[75],[31,113718,113720],{"className":113719,"translate":35},[411],[31,113721,113723,113771],{"className":113722,"translate":35},[34],[31,113724,113726],{"className":113725},[39],[41,113727,113728],{"xmlns":43,"display":420},[45,113729,113730,113768],{},[48,113731,113732,113734,113740,113742,113748,113754,113756,113758,113760,113766],{},[51,113733,4572],{"mathvariant":494},[1004,113735,113736,113738],{},[51,113737,43481],{},[51,113739,52065],{},[95,113741,97],{},[1004,113743,113744,113746],{},[51,113745,43481],{},[51,113747,4587],{"mathvariant":494},[1004,113749,113750,113752],{},[51,113751,4572],{"mathvariant":494},[51,113753,4587],{"mathvariant":494},[95,113755,347],{},[91,113757,70778],{},[95,113759,2528],{},[1180,113761,113762,113764],{},[91,113763,23247],{},[51,113765,52065],{},[51,113767,495],{"mathvariant":494},[55,113769,113770],{"encoding":57},"\\|V_k-V_\\infty\\|_\\infty\\le5.2632\\times0.9^k.",[31,113772,113774,113832,113927,113945],{"className":113773,"ariaHidden":62},[61],[31,113775,113777,113780,113783,113823,113826,113829],{"className":113776},[66],[31,113778],{"className":113779,"style":260},[70],[31,113781,4572],{"className":113782},[75],[31,113784,113786,113789],{"className":113785},[75],[31,113787,43481],{"className":113788,"style":120},[75,76],[31,113790,113792],{"className":113791},[1039],[31,113793,113795,113815],{"className":113794},[1043,1044],[31,113796,113798,113812],{"className":113797},[1048],[31,113799,113801],{"className":113800,"style":1053},[1052],[31,113802,113803,113806],{"style":66251},[31,113804],{"className":113805,"style":1061},[1060],[31,113807,113809],{"className":113808},[1065,1066,1067,1068],[31,113810,52065],{"className":113811,"style":52166},[75,76,1068],[31,113813,1076],{"className":113814},[1075],[31,113816,113818],{"className":113817},[1048],[31,113819,113821],{"className":113820,"style":2050},[1052],[31,113822],{},[31,113824],{"className":113825,"style":120},[119],[31,113827,97],{"className":113828},[124],[31,113830],{"className":113831,"style":120},[119],[31,113833,113835,113838,113878,113918,113921,113924],{"className":113834},[66],[31,113836],{"className":113837,"style":260},[70],[31,113839,113841,113844],{"className":113840},[75],[31,113842,43481],{"className":113843,"style":120},[75,76],[31,113845,113847],{"className":113846},[1039],[31,113848,113850,113870],{"className":113849},[1043,1044],[31,113851,113853,113867],{"className":113852},[1048],[31,113854,113856],{"className":113855,"style":3230},[1052],[31,113857,113858,113861],{"style":66251},[31,113859],{"className":113860,"style":1061},[1060],[31,113862,113864],{"className":113863},[1065,1066,1067,1068],[31,113865,4587],{"className":113866},[75,1068],[31,113868,1076],{"className":113869},[1075],[31,113871,113873],{"className":113872},[1048],[31,113874,113876],{"className":113875,"style":2050},[1052],[31,113877],{},[31,113879,113881,113884],{"className":113880},[75],[31,113882,4572],{"className":113883},[75],[31,113885,113887],{"className":113886},[1039],[31,113888,113890,113910],{"className":113889},[1043,1044],[31,113891,113893,113907],{"className":113892},[1048],[31,113894,113896],{"className":113895,"style":3230},[1052],[31,113897,113898,113901],{"style":2807},[31,113899],{"className":113900,"style":1061},[1060],[31,113902,113904],{"className":113903},[1065,1066,1067,1068],[31,113905,4587],{"className":113906},[75,1068],[31,113908,1076],{"className":113909},[1075],[31,113911,113913],{"className":113912},[1048],[31,113914,113916],{"className":113915,"style":2050},[1052],[31,113917],{},[31,113919],{"className":113920,"style":293},[119],[31,113922,347],{"className":113923},[297],[31,113925],{"className":113926,"style":293},[119],[31,113928,113930,113933,113936,113939,113942],{"className":113929},[66],[31,113931],{"className":113932,"style":112},[70],[31,113934,70778],{"className":113935},[75],[31,113937],{"className":113938,"style":120},[119],[31,113940,2528],{"className":113941},[124],[31,113943],{"className":113944,"style":120},[119],[31,113946,113948,113951,113954,113983],{"className":113947},[66],[31,113949],{"className":113950,"style":29642},[70],[31,113952,81469],{"className":113953},[75],[31,113955,113957,113960],{"className":113956},[75],[31,113958,81476],{"className":113959},[75],[31,113961,113963],{"className":113962},[1039],[31,113964,113966],{"className":113965},[1043],[31,113967,113969],{"className":113968},[1048],[31,113970,113972],{"className":113971,"style":29642},[1052],[31,113973,113974,113977],{"style":1330},[31,113975],{"className":113976,"style":1061},[1060],[31,113978,113980],{"className":113979},[1065,1066,1067,1068],[31,113981,52065],{"className":113982,"style":52166},[75,76,1068],[31,113984,495],{"className":113985},[75],[11,113987,65583,113988,114040,114041,81572,114070,114040,114121,114150],{},[31,113989,113991,114010],{"className":113990,"translate":35},[34],[31,113992,113994],{"className":113993},[39],[41,113995,113996],{"xmlns":43},[45,113997,113998,114007],{},[48,113999,114000,114002,114004],{},[51,114001,52065],{},[95,114003,223],{},[91,114005,114006],{},"20",[55,114008,114009],{"encoding":57},"k=20",[31,114011,114013,114031],{"className":114012,"ariaHidden":62},[61],[31,114014,114016,114019,114022,114025,114028],{"className":114015},[66],[31,114017],{"className":114018,"style":1630},[70],[31,114020,52065],{"className":114021,"style":52166},[75,76],[31,114023],{"className":114024,"style":293},[119],[31,114026,223],{"className":114027},[297],[31,114029],{"className":114030,"style":293},[119],[31,114032,114034,114037],{"className":114033},[66],[31,114035],{"className":114036,"style":326},[70],[31,114038,114006],{"className":114039},[75]," 时，右端约为 ",[31,114042,114044,114058],{"className":114043,"translate":35},[34],[31,114045,114047],{"className":114046},[39],[41,114048,114049],{"xmlns":43},[45,114050,114051,114056],{},[48,114052,114053],{},[91,114054,114055],{},"0.64",[55,114057,114055],{"encoding":57},[31,114059,114061],{"className":114060,"ariaHidden":62},[61],[31,114062,114064,114067],{"className":114063},[66],[31,114065],{"className":114066,"style":326},[70],[31,114068,114055],{"className":114069},[75],[31,114071,114073,114091],{"className":114072,"translate":35},[34],[31,114074,114076],{"className":114075},[39],[41,114077,114078],{"xmlns":43},[45,114079,114080,114088],{},[48,114081,114082,114084,114086],{},[51,114083,52065],{},[95,114085,223],{},[91,114087,45466],{},[55,114089,114090],{"encoding":57},"k=50",[31,114092,114094,114112],{"className":114093,"ariaHidden":62},[61],[31,114095,114097,114100,114103,114106,114109],{"className":114096},[66],[31,114098],{"className":114099,"style":1630},[70],[31,114101,52065],{"className":114102,"style":52166},[75,76],[31,114104],{"className":114105,"style":293},[119],[31,114107,223],{"className":114108},[297],[31,114110],{"className":114111,"style":293},[119],[31,114113,114115,114118],{"className":114114},[66],[31,114116],{"className":114117,"style":326},[70],[31,114119,45466],{"className":114120},[75],[31,114122,114124,114138],{"className":114123,"translate":35},[34],[31,114125,114127],{"className":114126},[39],[41,114128,114129],{"xmlns":43},[45,114130,114131,114136],{},[48,114132,114133],{},[91,114134,114135],{},"0.027",[55,114137,114135],{"encoding":57},[31,114139,114141],{"className":114140,"ariaHidden":62},[61],[31,114142,114144,114147],{"className":114143},[66],[31,114145],{"className":114146,"style":326},[70],[31,114148,114135],{"className":114149},[75],"。这与几何收敛的预期一致。",[52860,114152,114154],{"id":114153},"_54-折扣因子的必要性","5.4 折扣因子的必要性",[11,114156,114157,114158,87578,114208,114258],{},"压缩性依赖 ",[31,114159,114161,114178],{"className":114160,"translate":35},[34],[31,114162,114164],{"className":114163},[39],[41,114165,114166],{"xmlns":43},[45,114167,114168,114176],{},[48,114169,114170,114172,114174],{},[51,114171,53298],{},[95,114173,9758],{},[91,114175,205],{},[55,114177,69926],{"encoding":57},[31,114179,114181,114199],{"className":114180,"ariaHidden":62},[61],[31,114182,114184,114187,114190,114193,114196],{"className":114183},[66],[31,114185],{"className":114186,"style":53323},[70],[31,114188,53298],{"className":114189,"style":24157},[75,76],[31,114191],{"className":114192,"style":293},[119],[31,114194,9758],{"className":114195},[297],[31,114197],{"className":114198,"style":293},[119],[31,114200,114202,114205],{"className":114201},[66],[31,114203],{"className":114204,"style":326},[70],[31,114206,205],{"className":114207},[75],[31,114209,114211,114228],{"className":114210,"translate":35},[34],[31,114212,114214],{"className":114213},[39],[41,114215,114216],{"xmlns":43},[45,114217,114218,114226],{},[48,114219,114220,114222,114224],{},[51,114221,53298],{},[95,114223,223],{},[91,114225,205],{},[55,114227,81593],{"encoding":57},[31,114229,114231,114249],{"className":114230,"ariaHidden":62},[61],[31,114232,114234,114237,114240,114243,114246],{"className":114233},[66],[31,114235],{"className":114236,"style":865},[70],[31,114238,53298],{"className":114239,"style":24157},[75,76],[31,114241],{"className":114242,"style":293},[119],[31,114244,223],{"className":114245},[297],[31,114247],{"className":114248,"style":293},[119],[31,114250,114252,114255],{"className":114251},[66],[31,114253],{"className":114254,"style":326},[70],[31,114256,205],{"className":114257},[75],"，贝尔曼算子一般不再是压缩映射，贝尔曼方程可能没有有限解，也可能有多个解。以下两个例子说明这两种失效方式。",[11,114260,114261,114262,114304,114305,114378,114379,70400],{},"第一个例子是零奖励的确定性循环。设两个状态 ",[31,114263,114265,114283],{"className":114264,"translate":35},[34],[31,114266,114268],{"className":114267},[39],[41,114269,114270],{"xmlns":43},[45,114271,114272,114280],{},[48,114273,114274,114276,114278],{},[91,114275,205],{},[95,114277,1210],{"separator":62},[91,114279,152],{},[55,114281,114282],{"encoding":57},"1,2",[31,114284,114286],{"className":114285,"ariaHidden":62},[61],[31,114287,114289,114292,114295,114298,114301],{"className":114288},[66],[31,114290],{"className":114291,"style":41982},[70],[31,114293,205],{"className":114294},[75],[31,114296,1210],{"className":114297},[1391],[31,114299],{"className":114300,"style":1467},[119],[31,114302,152],{"className":114303},[75],"，转移为 ",[31,114306,114308,114330],{"className":114307,"translate":35},[34],[31,114309,114311],{"className":114310},[39],[41,114312,114313],{"xmlns":43},[45,114314,114315,114327],{},[48,114316,114317,114319,114321,114323,114325],{},[91,114318,205],{},[95,114320,929],{},[91,114322,152],{},[95,114324,929],{},[91,114326,205],{},[55,114328,114329],{"encoding":57},"1\\to2\\to1",[31,114331,114333,114351,114369],{"className":114332,"ariaHidden":62},[61],[31,114334,114336,114339,114342,114345,114348],{"className":114335},[66],[31,114337],{"className":114338,"style":326},[70],[31,114340,205],{"className":114341},[75],[31,114343],{"className":114344,"style":293},[119],[31,114346,929],{"className":114347},[297],[31,114349],{"className":114350,"style":293},[119],[31,114352,114354,114357,114360,114363,114366],{"className":114353},[66],[31,114355],{"className":114356,"style":326},[70],[31,114358,152],{"className":114359},[75],[31,114361],{"className":114362,"style":293},[119],[31,114364,929],{"className":114365},[297],[31,114367],{"className":114368,"style":293},[119],[31,114370,114372,114375],{"className":114371},[66],[31,114373],{"className":114374,"style":326},[70],[31,114376,205],{"className":114377},[75],"，奖励恒为零，",[31,114380,114382,114399],{"className":114381,"translate":35},[34],[31,114383,114385],{"className":114384},[39],[41,114386,114387],{"xmlns":43},[45,114388,114389,114397],{},[48,114390,114391,114393,114395],{},[51,114392,53298],{},[95,114394,223],{},[91,114396,205],{},[55,114398,81593],{"encoding":57},[31,114400,114402,114420],{"className":114401,"ariaHidden":62},[61],[31,114403,114405,114408,114411,114414,114417],{"className":114404},[66],[31,114406],{"className":114407,"style":865},[70],[31,114409,53298],{"className":114410,"style":24157},[75,76],[31,114412],{"className":114413,"style":293},[119],[31,114415,223],{"className":114416},[297],[31,114418],{"className":114419,"style":293},[119],[31,114421,114423,114426],{"className":114422},[66],[31,114424],{"className":114425,"style":326},[70],[31,114427,205],{"className":114428},[75],[31,114430,114432],{"className":114431,"translate":35},[411],[31,114433,114435,114489],{"className":114434,"translate":35},[34],[31,114436,114438],{"className":114437},[39],[41,114439,114440],{"xmlns":43,"display":420},[45,114441,114442,114486],{},[48,114443,114444,114446,114448,114450,114452,114454,114456,114458,114460,114462,114464,114466,114468,114470,114472,114474,114476,114478,114480,114482,114484],{},[51,114445,43481],{},[95,114447,211],{"stretchy":210},[91,114449,205],{},[95,114451,220],{"stretchy":210},[95,114453,223],{},[51,114455,43481],{},[95,114457,211],{"stretchy":210},[91,114459,152],{},[95,114461,220],{"stretchy":210},[95,114463,1210],{"separator":62},[119,114465],{"width":20286},[51,114467,43481],{},[95,114469,211],{"stretchy":210},[91,114471,152],{},[95,114473,220],{"stretchy":210},[95,114475,223],{},[51,114477,43481],{},[95,114479,211],{"stretchy":210},[91,114481,205],{},[95,114483,220],{"stretchy":210},[51,114485,495],{"mathvariant":494},[55,114487,114488],{"encoding":57},"V(1)=V(2),\\qquad V(2)=V(1).",[31,114490,114492,114519,114567],{"className":114491,"ariaHidden":62},[61],[31,114493,114495,114498,114501,114504,114507,114510,114513,114516],{"className":114494},[66],[31,114496],{"className":114497,"style":260},[70],[31,114499,43481],{"className":114500,"style":120},[75,76],[31,114502,211],{"className":114503},[264],[31,114505,205],{"className":114506},[75],[31,114508,220],{"className":114509},[289],[31,114511],{"className":114512,"style":293},[119],[31,114514,223],{"className":114515},[297],[31,114517],{"className":114518,"style":293},[119],[31,114520,114522,114525,114528,114531,114534,114537,114540,114543,114546,114549,114552,114555,114558,114561,114564],{"className":114521},[66],[31,114523],{"className":114524,"style":260},[70],[31,114526,43481],{"className":114527,"style":120},[75,76],[31,114529,211],{"className":114530},[264],[31,114532,152],{"className":114533},[75],[31,114535,220],{"className":114536},[289],[31,114538,1210],{"className":114539},[1391],[31,114541],{"className":114542,"style":20525},[119],[31,114544],{"className":114545,"style":1467},[119],[31,114547,43481],{"className":114548,"style":120},[75,76],[31,114550,211],{"className":114551},[264],[31,114553,152],{"className":114554},[75],[31,114556,220],{"className":114557},[289],[31,114559],{"className":114560,"style":293},[119],[31,114562,223],{"className":114563},[297],[31,114565],{"className":114566,"style":293},[119],[31,114568,114570,114573,114576,114579,114582,114585],{"className":114569},[66],[31,114571],{"className":114572,"style":260},[70],[31,114574,43481],{"className":114575,"style":120},[75,76],[31,114577,211],{"className":114578},[264],[31,114580,205],{"className":114581},[75],[31,114583,220],{"className":114584},[289],[31,114586,495],{"className":114587},[75],[11,114589,114590,114591,114694],{},"任意常数函数 ",[31,114592,114594,114628],{"className":114593,"translate":35},[34],[31,114595,114597],{"className":114596},[39],[41,114598,114599],{"xmlns":43},[45,114600,114601,114625],{},[48,114602,114603,114605,114607,114609,114611,114613,114615,114617,114619,114621,114623],{},[51,114604,43481],{},[95,114606,211],{"stretchy":210},[91,114608,205],{},[95,114610,220],{"stretchy":210},[95,114612,223],{},[51,114614,43481],{},[95,114616,211],{"stretchy":210},[91,114618,152],{},[95,114620,220],{"stretchy":210},[95,114622,223],{},[51,114624,1412],{},[55,114626,114627],{"encoding":57},"V(1)=V(2)=c",[31,114629,114631,114658,114685],{"className":114630,"ariaHidden":62},[61],[31,114632,114634,114637,114640,114643,114646,114649,114652,114655],{"className":114633},[66],[31,114635],{"className":114636,"style":260},[70],[31,114638,43481],{"className":114639,"style":120},[75,76],[31,114641,211],{"className":114642},[264],[31,114644,205],{"className":114645},[75],[31,114647,220],{"className":114648},[289],[31,114650],{"className":114651,"style":293},[119],[31,114653,223],{"className":114654},[297],[31,114656],{"className":114657,"style":293},[119],[31,114659,114661,114664,114667,114670,114673,114676,114679,114682],{"className":114660},[66],[31,114662],{"className":114663,"style":260},[70],[31,114665,43481],{"className":114666,"style":120},[75,76],[31,114668,211],{"className":114669},[264],[31,114671,152],{"className":114672},[75],[31,114674,220],{"className":114675},[289],[31,114677],{"className":114678,"style":293},[119],[31,114680,223],{"className":114681},[297],[31,114683],{"className":114684,"style":293},[119],[31,114686,114688,114691],{"className":114687},[66],[31,114689],{"className":114690,"style":71},[70],[31,114692,1412],{"className":114693},[75,76]," 都是解，因此不动点不唯一。Banach 定理的唯一性结论失效。",[11,114696,114697,114698,114726],{},"第二个例子是每步奖励为 ",[31,114699,114701,114714],{"className":114700,"translate":35},[34],[31,114702,114704],{"className":114703},[39],[41,114705,114706],{"xmlns":43},[45,114707,114708,114712],{},[48,114709,114710],{},[91,114711,205],{},[55,114713,205],{"encoding":57},[31,114715,114717],{"className":114716,"ariaHidden":62},[61],[31,114718,114720,114723],{"className":114719},[66],[31,114721],{"className":114722,"style":326},[70],[31,114724,205],{"className":114725},[75]," 的同一循环。此时",[31,114728,114730],{"className":114729,"translate":35},[411],[31,114731,114733,114775],{"className":114732,"translate":35},[34],[31,114734,114736],{"className":114735},[39],[41,114737,114738],{"xmlns":43,"display":420},[45,114739,114740,114772],{},[48,114741,114742,114748,114750,114764,114766,114768,114770],{},[1004,114743,114744,114746],{},[51,114745,15297],{},[51,114747,12271],{},[95,114749,223],{},[42429,114751,114752,114754,114762],{},[95,114753,5625],{},[48,114755,114756,114758,114760],{},[51,114757,52065],{},[95,114759,223],{},[91,114761,1977],{},[51,114763,4587],{"mathvariant":494},[91,114765,205],{},[95,114767,223],{},[51,114769,4587],{"mathvariant":494},[95,114771,1210],{"separator":62},[55,114773,114774],{"encoding":57},"G_t=\\sum_{k=0}^{\\infty}1=\\infty,",[31,114776,114778,114833,114921],{"className":114777,"ariaHidden":62},[61],[31,114779,114781,114784,114824,114827,114830],{"className":114780},[66],[31,114782],{"className":114783,"style":5230},[70],[31,114785,114787,114790],{"className":114786},[75],[31,114788,15297],{"className":114789},[75,76],[31,114791,114793],{"className":114792},[1039],[31,114794,114796,114816],{"className":114795},[1043,1044],[31,114797,114799,114813],{"className":114798},[1048],[31,114800,114802],{"className":114801,"style":19220},[1052],[31,114803,114804,114807],{"style":2807},[31,114805],{"className":114806,"style":1061},[1060],[31,114808,114810],{"className":114809},[1065,1066,1067,1068],[31,114811,12271],{"className":114812},[75,76,1068],[31,114814,1076],{"className":114815},[1075],[31,114817,114819],{"className":114818},[1048],[31,114820,114822],{"className":114821,"style":2050},[1052],[31,114823],{},[31,114825],{"className":114826,"style":293},[119],[31,114828,223],{"className":114829},[297],[31,114831],{"className":114832,"style":293},[119],[31,114834,114836,114839,114906,114909,114912,114915,114918],{"className":114835},[66],[31,114837],{"className":114838,"style":55989},[70],[31,114840,114842],{"className":114841},[1460,1620],[31,114843,114845,114898],{"className":114844},[1043,1044],[31,114846,114848,114895],{"className":114847},[1048],[31,114849,114851,114871,114881],{"className":114850,"style":42702},[1052],[31,114852,114853,114856],{"style":56004},[31,114854],{"className":114855,"style":5907},[1060],[31,114857,114859],{"className":114858},[1065,1066,1067,1068],[31,114860,114862,114865,114868],{"className":114861},[75,1068],[31,114863,52065],{"className":114864,"style":52166},[75,76,1068],[31,114866,223],{"className":114867},[297,1068],[31,114869,1977],{"className":114870},[75,1068],[31,114872,114873,114876],{"style":5919},[31,114874],{"className":114875,"style":5907},[1060],[31,114877,114878],{},[31,114879,5625],{"className":114880},[1460,1461,1687],[31,114882,114883,114886],{"style":42735},[31,114884],{"className":114885,"style":5907},[1060],[31,114887,114889],{"className":114888},[1065,1066,1067,1068],[31,114890,114892],{"className":114891},[75,1068],[31,114893,4587],{"className":114894},[75,1068],[31,114896,1076],{"className":114897},[1075],[31,114899,114901],{"className":114900},[1048],[31,114902,114904],{"className":114903,"style":56056},[1052],[31,114905],{},[31,114907],{"className":114908,"style":1467},[119],[31,114910,205],{"className":114911},[75],[31,114913],{"className":114914,"style":293},[119],[31,114916,223],{"className":114917},[297],[31,114919],{"className":114920,"style":293},[119],[31,114922,114924,114927,114930],{"className":114923},[66],[31,114925],{"className":114926,"style":865},[70],[31,114928,4587],{"className":114929},[75],[31,114931,1210],{"className":114932},[1391],[11,114934,114935,114936,114979],{},"折扣回报不再是有限实数，值函数不属于 ",[31,114937,114939,114958],{"className":114938,"translate":35},[34],[31,114940,114942],{"className":114941},[39],[41,114943,114944],{"xmlns":43},[45,114945,114946,114956],{},[48,114947,114948,114950,114952,114954],{},[51,114949,1122],{},[95,114951,211],{"stretchy":210},[51,114953,21582],{},[95,114955,220],{"stretchy":210},[55,114957,73997],{"encoding":57},[31,114959,114961],{"className":114960,"ariaHidden":62},[61],[31,114962,114964,114967,114970,114973,114976],{"className":114963},[66],[31,114965],{"className":114966,"style":260},[70],[31,114968,1122],{"className":114969,"style":1137},[75,76],[31,114971,211],{"className":114972},[264],[31,114974,21582],{"className":114975,"style":21697},[75,76],[31,114977,220],{"className":114978},[289],"。对应的贝尔曼方程为",[31,114981,114983],{"className":114982,"translate":35},[411],[31,114984,114986,115048],{"className":114985,"translate":35},[34],[31,114987,114989],{"className":114988},[39],[41,114990,114991],{"xmlns":43,"display":420},[45,114992,114993,115045],{},[48,114994,114995,114997,114999,115001,115003,115005,115007,115009,115011,115013,115015,115017,115019,115021,115023,115025,115027,115029,115031,115033,115035,115037,115039,115041,115043],{},[51,114996,43481],{},[95,114998,211],{"stretchy":210},[91,115000,205],{},[95,115002,220],{"stretchy":210},[95,115004,223],{},[91,115006,205],{},[95,115008,434],{},[51,115010,43481],{},[95,115012,211],{"stretchy":210},[91,115014,152],{},[95,115016,220],{"stretchy":210},[95,115018,1210],{"separator":62},[119,115020],{"width":20286},[51,115022,43481],{},[95,115024,211],{"stretchy":210},[91,115026,152],{},[95,115028,220],{"stretchy":210},[95,115030,223],{},[91,115032,205],{},[95,115034,434],{},[51,115036,43481],{},[95,115038,211],{"stretchy":210},[91,115040,205],{},[95,115042,220],{"stretchy":210},[95,115044,1210],{"separator":62},[55,115046,115047],{"encoding":57},"V(1)=1+V(2),\\qquad V(2)=1+V(1),",[31,115049,115051,115078,115096,115144,115162],{"className":115050,"ariaHidden":62},[61],[31,115052,115054,115057,115060,115063,115066,115069,115072,115075],{"className":115053},[66],[31,115055],{"className":115056,"style":260},[70],[31,115058,43481],{"className":115059,"style":120},[75,76],[31,115061,211],{"className":115062},[264],[31,115064,205],{"className":115065},[75],[31,115067,220],{"className":115068},[289],[31,115070],{"className":115071,"style":293},[119],[31,115073,223],{"className":115074},[297],[31,115076],{"className":115077,"style":293},[119],[31,115079,115081,115084,115087,115090,115093],{"className":115080},[66],[31,115082],{"className":115083,"style":112},[70],[31,115085,205],{"className":115086},[75],[31,115088],{"className":115089,"style":120},[119],[31,115091,434],{"className":115092},[124],[31,115094],{"className":115095,"style":120},[119],[31,115097,115099,115102,115105,115108,115111,115114,115117,115120,115123,115126,115129,115132,115135,115138,115141],{"className":115098},[66],[31,115100],{"className":115101,"style":260},[70],[31,115103,43481],{"className":115104,"style":120},[75,76],[31,115106,211],{"className":115107},[264],[31,115109,152],{"className":115110},[75],[31,115112,220],{"className":115113},[289],[31,115115,1210],{"className":115116},[1391],[31,115118],{"className":115119,"style":20525},[119],[31,115121],{"className":115122,"style":1467},[119],[31,115124,43481],{"className":115125,"style":120},[75,76],[31,115127,211],{"className":115128},[264],[31,115130,152],{"className":115131},[75],[31,115133,220],{"className":115134},[289],[31,115136],{"className":115137,"style":293},[119],[31,115139,223],{"className":115140},[297],[31,115142],{"className":115143,"style":293},[119],[31,115145,115147,115150,115153,115156,115159],{"className":115146},[66],[31,115148],{"className":115149,"style":112},[70],[31,115151,205],{"className":115152},[75],[31,115154],{"className":115155,"style":120},[119],[31,115157,434],{"className":115158},[124],[31,115160],{"className":115161,"style":120},[119],[31,115163,115165,115168,115171,115174,115177,115180],{"className":115164},[66],[31,115166],{"className":115167,"style":260},[70],[31,115169,43481],{"className":115170,"style":120},[75,76],[31,115172,211],{"className":115173},[264],[31,115175,205],{"className":115176},[75],[31,115178,220],{"className":115179},[289],[31,115181,1210],{"className":115182},[1391],[11,115184,115185,115186,115236],{},"它也没有有限解。这说明 ",[31,115187,115189,115206],{"className":115188,"translate":35},[34],[31,115190,115192],{"className":115191},[39],[41,115193,115194],{"xmlns":43},[45,115195,115196,115204],{},[48,115197,115198,115200,115202],{},[51,115199,53298],{},[95,115201,9758],{},[91,115203,205],{},[55,115205,69926],{"encoding":57},[31,115207,115209,115227],{"className":115208,"ariaHidden":62},[61],[31,115210,115212,115215,115218,115221,115224],{"className":115211},[66],[31,115213],{"className":115214,"style":53323},[70],[31,115216,53298],{"className":115217,"style":24157},[75,76],[31,115219],{"className":115220,"style":293},[119],[31,115222,9758],{"className":115223},[297],[31,115225],{"className":115226,"style":293},[119],[31,115228,115230,115233],{"className":115229},[66],[31,115231],{"className":115232,"style":326},[70],[31,115234,205],{"className":115235},[75]," 不是技术上的方便假设，而是保证回报有界、算子压缩、解唯一的结构性条件。",[18,115238,115240],{"id":115239},"六最优子结构与最优性原理","六、最优子结构与最优性原理",[52860,115242,115244],{"id":115243},"_61-有限期情形","6.1 有限期情形",[11,115246,115247,115248,115319,115320,41892,115507,115577,115578,115606],{},"最优子结构最清晰的表述出现在有限期问题中。设时间范围为 ",[31,115249,115251,115277],{"className":115250,"translate":35},[34],[31,115252,115254],{"className":115253},[39],[41,115255,115256],{"xmlns":43},[45,115257,115258,115274],{},[48,115259,115260,115262,115264,115266,115268,115270,115272],{},[91,115261,1977],{},[95,115263,1210],{"separator":62},[91,115265,205],{},[95,115267,1210],{"separator":62},[95,115269,55349],{},[95,115271,1210],{"separator":62},[51,115273,919],{},[55,115275,115276],{"encoding":57},"0,1,\\dots,T",[31,115278,115280],{"className":115279,"ariaHidden":62},[61],[31,115281,115283,115286,115289,115292,115295,115298,115301,115304,115307,115310,115313,115316],{"className":115282},[66],[31,115284],{"className":115285,"style":59114},[70],[31,115287,1977],{"className":115288},[75],[31,115290,1210],{"className":115291},[1391],[31,115293],{"className":115294,"style":1467},[119],[31,115296,205],{"className":115297},[75],[31,115299,1210],{"className":115300},[1391],[31,115302],{"className":115303,"style":1467},[119],[31,115305,55349],{"className":115306},[3860],[31,115308],{"className":115309,"style":1467},[119],[31,115311,1210],{"className":115312},[1391],[31,115314],{"className":115315,"style":1467},[119],[31,115317,919],{"className":115318,"style":949},[75,76],"，策略为一个序列 ",[31,115321,115323,115367],{"className":115322,"translate":35},[34],[31,115324,115326],{"className":115325},[39],[41,115327,115328],{"xmlns":43},[45,115329,115330,115364],{},[48,115331,115332,115334,115336,115338,115344,115346,115348,115350,115362],{},[51,115333,2515],{},[95,115335,223],{},[95,115337,211],{"stretchy":210},[1004,115339,115340,115342],{},[51,115341,2515],{},[91,115343,1977],{},[95,115345,1210],{"separator":62},[95,115347,55349],{},[95,115349,1210],{"separator":62},[1004,115351,115352,115354],{},[51,115353,2515],{},[48,115355,115356,115358,115360],{},[51,115357,919],{},[95,115359,97],{},[91,115361,205],{},[95,115363,220],{"stretchy":210},[55,115365,115366],{"encoding":57},"\\pi=(\\pi_0,\\dots,\\pi_{T-1})",[31,115368,115370,115388],{"className":115369,"ariaHidden":62},[61],[31,115371,115373,115376,115379,115382,115385],{"className":115372},[66],[31,115374],{"className":115375,"style":71},[70],[31,115377,2515],{"className":115378,"style":2666},[75,76],[31,115380],{"className":115381,"style":293},[119],[31,115383,223],{"className":115384},[297],[31,115386],{"className":115387,"style":293},[119],[31,115389,115391,115394,115397,115437,115440,115443,115446,115449,115452,115455,115504],{"className":115390},[66],[31,115392],{"className":115393,"style":260},[70],[31,115395,211],{"className":115396},[264],[31,115398,115400,115403],{"className":115399},[75],[31,115401,2515],{"className":115402,"style":2666},[75,76],[31,115404,115406],{"className":115405},[1039],[31,115407,115409,115429],{"className":115408},[1043,1044],[31,115410,115412,115426],{"className":115411},[1048],[31,115413,115415],{"className":115414,"style":2028},[1052],[31,115416,115417,115420],{"style":5504},[31,115418],{"className":115419,"style":1061},[1060],[31,115421,115423],{"className":115422},[1065,1066,1067,1068],[31,115424,1977],{"className":115425},[75,1068],[31,115427,1076],{"className":115428},[1075],[31,115430,115432],{"className":115431},[1048],[31,115433,115435],{"className":115434,"style":2050},[1052],[31,115436],{},[31,115438,1210],{"className":115439},[1391],[31,115441],{"className":115442,"style":1467},[119],[31,115444,55349],{"className":115445},[3860],[31,115447],{"className":115448,"style":1467},[119],[31,115450,1210],{"className":115451},[1391],[31,115453],{"className":115454,"style":1467},[119],[31,115456,115458,115461],{"className":115457},[75],[31,115459,2515],{"className":115460,"style":2666},[75,76],[31,115462,115464],{"className":115463},[1039],[31,115465,115467,115496],{"className":115466},[1043,1044],[31,115468,115470,115493],{"className":115469},[1048],[31,115471,115473],{"className":115472,"style":27736},[1052],[31,115474,115475,115478],{"style":5504},[31,115476],{"className":115477,"style":1061},[1060],[31,115479,115481],{"className":115480},[1065,1066,1067,1068],[31,115482,115484,115487,115490],{"className":115483},[75,1068],[31,115485,919],{"className":115486,"style":949},[75,76,1068],[31,115488,97],{"className":115489},[124,1068],[31,115491,205],{"className":115492},[75,1068],[31,115494,1076],{"className":115495},[1075],[31,115497,115499],{"className":115498},[1048],[31,115500,115502],{"className":115501,"style":25082},[1052],[31,115503],{},[31,115505,220],{"className":115506},[289],[31,115508,115510,115528],{"className":115509,"translate":35},[34],[31,115511,115513],{"className":115512},[39],[41,115514,115515],{"xmlns":43},[45,115516,115517,115525],{},[48,115518,115519],{},[1004,115520,115521,115523],{},[51,115522,2515],{},[51,115524,12271],{},[55,115526,115527],{"encoding":57},"\\pi_t",[31,115529,115531],{"className":115530,"ariaHidden":62},[61],[31,115532,115534,115537],{"className":115533},[66],[31,115535],{"className":115536,"style":7090},[70],[31,115538,115540,115543],{"className":115539},[75],[31,115541,2515],{"className":115542,"style":2666},[75,76],[31,115544,115546],{"className":115545},[1039],[31,115547,115549,115569],{"className":115548},[1043,1044],[31,115550,115552,115566],{"className":115551},[1048],[31,115553,115555],{"className":115554,"style":19220},[1052],[31,115556,115557,115560],{"style":5504},[31,115558],{"className":115559,"style":1061},[1060],[31,115561,115563],{"className":115562},[1065,1066,1067,1068],[31,115564,12271],{"className":115565},[75,76,1068],[31,115567,1076],{"className":115568},[1075],[31,115570,115572],{"className":115571},[1048],[31,115573,115575],{"className":115574,"style":2050},[1052],[31,115576],{}," 是时刻 ",[31,115579,115581,115594],{"className":115580,"translate":35},[34],[31,115582,115584],{"className":115583},[39],[41,115585,115586],{"xmlns":43},[45,115587,115588,115592],{},[48,115589,115590],{},[51,115591,12271],{},[55,115593,12271],{"encoding":57},[31,115595,115597],{"className":115596,"ariaHidden":62},[61],[31,115598,115600,115603],{"className":115599},[66],[31,115601],{"className":115602,"style":12283},[70],[31,115604,12271],{"className":115605},[75,76]," 的策略。定义",[31,115608,115610],{"className":115609,"translate":35},[411],[31,115611,115613,115772],{"className":115612,"translate":35},[34],[31,115614,115616],{"className":115615},[39],[41,115617,115618],{"xmlns":43,"display":420},[45,115619,115620,115769],{},[5606,115621,115622,115701,115735],{"rowspacing":5608,"columnalign":5609,"columnspacing":5610},[5612,115623,115624,115644],{},[5615,115625,115626],{},[2071,115627,115628],{"scriptlevel":1977,"displaystyle":62},[48,115629,115630,115638,115640,115642],{},[12397,115631,115632,115634,115636],{},[51,115633,43481],{},[51,115635,12271],{},[51,115637,2515],{},[95,115639,211],{"stretchy":210},[51,115641,14551],{},[95,115643,220],{"stretchy":210},[5615,115645,115646],{},[2071,115647,115648],{"scriptlevel":1977,"displaystyle":62},[48,115649,115650,115652,115654,115660,115663,115683,115695],{},[48,115651],{},[95,115653,223],{},[1004,115655,115656,115658],{},[51,115657,22324],{"mathvariant":14789},[51,115659,2515],{},[95,115661,15175],{"fence":210,"stretchy":62,"minsize":115662,"maxsize":115662},"3em",[42429,115664,115665,115667,115675],{},[95,115666,5625],{},[48,115668,115669,115671,115673],{},[51,115670,52065],{},[95,115672,223],{},[51,115674,12271],{},[48,115676,115677,115679,115681],{},[51,115678,919],{},[95,115680,97],{},[91,115682,205],{},[1180,115684,115685,115687],{},[51,115686,53298],{},[48,115688,115689,115691,115693],{},[51,115690,52065],{},[95,115692,97],{},[51,115694,12271],{},[1004,115696,115697,115699],{},[51,115698,2549],{},[51,115700,52065],{},[5612,115702,115703,115709],{},[5615,115704,115705],{},[2071,115706,115707],{"scriptlevel":1977,"displaystyle":62},[48,115708],{},[5615,115710,115711],{},[2071,115712,115713],{"scriptlevel":1977,"displaystyle":62},[48,115714,115715,115717,115719,115721,115727,115729,115731,115733],{},[48,115716],{},[119,115718],{"width":20286},[95,115720,12680],{},[1004,115722,115723,115725],{},[51,115724,14551],{},[51,115726,12271],{},[95,115728,223],{},[51,115730,14551],{},[95,115732,15184],{"fence":210,"stretchy":62,"minsize":115662,"maxsize":115662},[95,115734,1210],{"separator":62},[5612,115736,115737,115757],{},[5615,115738,115739],{},[2071,115740,115741],{"scriptlevel":1977,"displaystyle":62},[48,115742,115743,115751,115753,115755],{},[12397,115744,115745,115747,115749],{},[51,115746,43481],{},[51,115748,919],{},[51,115750,2515],{},[95,115752,211],{"stretchy":210},[51,115754,14551],{},[95,115756,220],{"stretchy":210},[5615,115758,115759],{},[2071,115760,115761],{"scriptlevel":1977,"displaystyle":62},[48,115762,115763,115765,115767],{},[48,115764],{},[95,115766,223],{},[91,115768,81469],{},[55,115770,115771],{"encoding":57},"\\begin{aligned}\nV_t^\\pi(s)\n&=\\mathbb{E}_\\pi\\Bigg[\\sum_{k=t}^{T-1}\\gamma^{k-t}r_k\\\\\n&\\qquad\\mid s_t=s\\Bigg],\\\\\nV_T^\\pi(s)&=0.\n\\end{aligned}",[31,115773,115775],{"className":115774,"ariaHidden":62},[61],[31,115776,115778,115782],{"className":115777},[66],[31,115779],{"className":115780,"style":115781},[70],"height:7.9305em;vertical-align:-3.7152em;",[31,115783,115785],{"className":115784},[75],[31,115786,115788,115961],{"className":115787},[5606],[31,115789,115791],{"className":115790},[5867],[31,115792,115794,115952],{"className":115793},[1043,1044],[31,115795,115797,115949],{"className":115796},[1048],[31,115798,115801,115871,115880],{"className":115799,"style":115800},[1052],"height:4.2152em;",[31,115802,115804,115808],{"style":115803},"top:-6.2152em;",[31,115805],{"className":115806,"style":115807},[1060],"height:3.8283em;",[31,115809,115811,115862,115865,115868],{"className":115810},[75],[31,115812,115814,115817],{"className":115813},[75],[31,115815,43481],{"className":115816,"style":120},[75,76],[31,115818,115820],{"className":115819},[1039],[31,115821,115823,115854],{"className":115822},[1043,1044],[31,115824,115826,115851],{"className":115825},[1048],[31,115827,115829,115840],{"className":115828,"style":3987},[1052],[31,115830,115831,115834],{"style":62628},[31,115832],{"className":115833,"style":1061},[1060],[31,115835,115837],{"className":115836},[1065,1066,1067,1068],[31,115838,12271],{"className":115839},[75,76,1068],[31,115841,115842,115845],{"style":1330},[31,115843],{"className":115844,"style":1061},[1060],[31,115846,115848],{"className":115847},[1065,1066,1067,1068],[31,115849,2515],{"className":115850,"style":2666},[75,76,1068],[31,115852,1076],{"className":115853},[1075],[31,115855,115857],{"className":115856},[1048],[31,115858,115860],{"className":115859,"style":13499},[1052],[31,115861],{},[31,115863,211],{"className":115864},[264],[31,115866,14551],{"className":115867},[75,76],[31,115869,220],{"className":115870},[289],[31,115872,115874,115877],{"style":115873},"top:-2.8631em;",[31,115875],{"className":115876,"style":115807},[1060],[31,115878],{"className":115879},[75],[31,115881,115883,115886],{"style":115882},"top:-0.4731em;",[31,115884],{"className":115885,"style":115807},[1060],[31,115887,115889,115940,115943,115946],{"className":115888},[75],[31,115890,115892,115895],{"className":115891},[75],[31,115893,43481],{"className":115894,"style":120},[75,76],[31,115896,115898],{"className":115897},[1039],[31,115899,115901,115932],{"className":115900},[1043,1044],[31,115902,115904,115929],{"className":115903},[1048],[31,115905,115907,115918],{"className":115906,"style":3987},[1052],[31,115908,115909,115912],{"style":62628},[31,115910],{"className":115911,"style":1061},[1060],[31,115913,115915],{"className":115914},[1065,1066,1067,1068],[31,115916,919],{"className":115917,"style":949},[75,76,1068],[31,115919,115920,115923],{"style":1330},[31,115921],{"className":115922,"style":1061},[1060],[31,115924,115926],{"className":115925},[1065,1066,1067,1068],[31,115927,2515],{"className":115928,"style":2666},[75,76,1068],[31,115930,1076],{"className":115931},[1075],[31,115933,115935],{"className":115934},[1048],[31,115936,115938],{"className":115937,"style":13499},[1052],[31,115939],{},[31,115941,211],{"className":115942},[264],[31,115944,14551],{"className":115945},[75,76],[31,115947,220],{"className":115948},[289],[31,115950,1076],{"className":115951},[1075],[31,115953,115955],{"className":115954},[1048],[31,115956,115959],{"className":115957,"style":115958},[1052],"height:3.7152em;",[31,115960],{},[31,115962,115964],{"className":115963},[6062],[31,115965,115967,116308],{"className":115966},[1043,1044],[31,115968,115970,116305],{"className":115969},[1048],[31,115971,115973,116198,116282],{"className":115972,"style":115800},[1052],[31,115974,115975,115978],{"style":115803},[31,115976],{"className":115977,"style":115807},[1060],[31,115979,115981,115984,115987,115990,115993,116033,116040,116043,116117,116120,116158],{"className":115980},[75],[31,115982],{"className":115983},[75],[31,115985],{"className":115986,"style":293},[119],[31,115988,223],{"className":115989},[297],[31,115991],{"className":115992,"style":293},[119],[31,115994,115996,115999],{"className":115995},[75],[31,115997,22324],{"className":115998},[75,14947],[31,116000,116002],{"className":116001},[1039],[31,116003,116005,116025],{"className":116004},[1043,1044],[31,116006,116008,116022],{"className":116007},[1048],[31,116009,116011],{"className":116010,"style":3230},[1052],[31,116012,116013,116016],{"style":2807},[31,116014],{"className":116015,"style":1061},[1060],[31,116017,116019],{"className":116018},[1065,1066,1067,1068],[31,116020,2515],{"className":116021,"style":2666},[75,76,1068],[31,116023,1076],{"className":116024},[1075],[31,116026,116028],{"className":116027},[1048],[31,116029,116031],{"className":116030,"style":2050},[1052],[31,116032],{},[31,116034,116036],{"className":116035},[75],[31,116037,15175],{"className":116038},[1307,116039],"size4",[31,116041],{"className":116042,"style":1467},[119],[31,116044,116046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",[31,116320,116322,116374],{"className":116321,"translate":35},[34],[31,116323,116325],{"className":116324},[39],[41,116326,116327],{"xmlns":43},[45,116328,116329,116371],{},[48,116330,116331,116339,116341,116343,116345,116347,116357,116365,116367,116369],{},[12397,116332,116333,116335,116337],{},[51,116334,43481],{},[51,116336,12271],{},[95,116338,30648],{},[95,116340,211],{"stretchy":210},[51,116342,14551],{},[95,116344,220],{"stretchy":210},[95,116346,223],{},[1004,116348,116349,116355],{},[48,116350,116351,116353],{},[51,116352,10792],{},[95,116354,1540],{},[51,116356,2515],{},[12397,116358,116359,116361,116363],{},[51,116360,43481],{},[51,116362,12271],{},[51,116364,2515],{},[95,116366,211],{"stretchy":210},[51,116368,14551],{},[95,116370,220],{"stretchy":210},[55,116372,116373],{"encoding":57},"V_t^*(s)=\\sup_\\pi V_t^\\pi(s)",[31,116375,116377,116452],{"className":116376,"ariaHidden":62},[61],[31,116378,116380,116383,116434,116437,116440,116443,116446,116449],{"className":116379},[66],[31,116381],{"className":116382,"style":260},[70],[31,116384,116386,116389],{"className":116385},[75],[31,116387,43481],{"className":116388,"style":120},[75,76],[31,116390,116392],{"className":116391},[1039],[31,116393,116395,116426],{"className":116394},[1043,1044],[31,116396,116398,116423],{"className":116397},[1048],[31,116399,116401,116412],{"className":116400,"style":30719},[1052],[31,116402,116403,116406],{"style":62628},[31,116404],{"className":116405,"style":1061},[1060],[31,116407,116409],{"className":116408},[1065,1066,1067,1068],[31,116410,12271],{"className":116411},[75,76,1068],[31,116413,116414,116417],{"style":3631},[31,116415],{"className":116416,"style":1061},[1060],[31,116418,116420],{"className":116419},[1065,1066,1067,1068],[31,116421,30648],{"className":116422},[124,1068],[31,116424,1076],{"className":116425},[1075],[31,116427,116429],{"className":116428},[1048],[31,116430,116432],{"className":116431,"style":13499},[1052],[31,116433],{},[31,116435,211],{"className":116436},[264],[31,116438,14551],{"className":116439},[75,76],[31,116441,220],{"className":116442},[289],[31,116444],{"className":116445,"style":293},[119],[31,116447,223],{"className":116448},[297],[31,116450],{"className":116451,"style":293},[119],[31,116453,116455,116458,116499,116502,116553,116556,116559],{"className":116454},[66],[31,116456],{"className":116457,"style":260},[70],[31,116459,116461,116464],{"className":116460},[1460],[31,116462,10792],{"className":116463},[1460],[31,116465,116467],{"className":116466},[1039],[31,116468,116470,116491],{"className":116469},[1043,1044],[31,116471,116473,116488],{"className":116472},[1048],[31,116474,116477],{"className":116475,"style":116476},[1052],"height:0.0573em;",[31,116478,116479,116482],{"style":2762},[31,116480],{"className":116481,"style":1061},[1060],[31,116483,116485],{"className":116484},[1065,1066,1067,1068],[31,116486,2515],{"className":116487,"style":2666},[75,76,1068],[31,116489,1076],{"className":116490},[1075],[31,116492,116494],{"className":116493},[1048],[31,116495,116497],{"className":116496,"style":2784},[1052],[31,116498],{},[31,116500],{"className":116501,"style":1467},[119],[31,116503,116505,116508],{"className":116504},[75],[31,116506,43481],{"className":116507,"style":120},[75,76],[31,116509,116511],{"className":116510},[1039],[31,116512,116514,116545],{"className":116513},[1043,1044],[31,116515,116517,116542],{"className":116516},[1048],[31,116518,116520,116531],{"className":116519,"style":3628},[1052],[31,116521,116522,116525],{"style":62628},[31,116523],{"className":116524,"style":1061},[1060],[31,116526,116528],{"className":116527},[1065,1066,1067,1068],[31,116529,12271],{"className":116530},[75,76,1068],[31,116532,116533,116536],{"style":3631},[31,116534],{"className":116535,"style":1061},[1060],[31,116537,116539],{"className":116538},[1065,1066,1067,1068],[31,116540,2515],{"className":116541,"style":2666},[75,76,1068],[31,116543,1076],{"className":116544},[1075],[31,116546,116548],{"className":116547},[1048],[31,116549,116551],{"className":116550,"style":13499},[1052],[31,116552],{},[31,116554,211],{"className":116555},[264],[31,116557,14551],{"className":116558},[75,76],[31,116560,220],{"className":116561},[289],[52865,116563,116566,116705],{"id":116564,"title":116565},"theorem-finite-horizon","定理 6.1（有限期最优性原理）",[11,116567,9612,116568,59187],{},[31,116569,116571,116609],{"className":116570,"translate":35},[34],[31,116572,116574],{"className":116573},[39],[41,116575,116576],{"xmlns":43},[45,116577,116578,116606],{},[48,116579,116580,116582,116584,116586,116588,116590,116592,116594,116596,116598,116600,116602,116604],{},[51,116581,12271],{},[95,116583,223],{},[51,116585,919],{},[95,116587,97],{},[91,116589,205],{},[95,116591,1210],{"separator":62},[51,116593,919],{},[95,116595,97],{},[91,116597,152],{},[95,116599,1210],{"separator":62},[95,116601,55349],{},[95,116603,1210],{"separator":62},[91,116605,1977],{},[55,116607,116608],{"encoding":57},"t=T-1,T-2,\\dots,0",[31,116610,116612,116630,116648,116675],{"className":116611,"ariaHidden":62},[61],[31,116613,116615,116618,116621,116624,116627],{"className":116614},[66],[31,116616],{"className":116617,"style":12283},[70],[31,116619,12271],{"className":116620},[75,76],[31,116622],{"className":116623,"style":293},[119],[31,116625,223],{"className":116626},[297],[31,116628],{"className":116629,"style":293},[119],[31,116631,116633,116636,116639,116642,116645],{"className":116632},[66],[31,116634],{"className":116635,"style":53224},[70],[31,116637,919],{"className":116638,"style":949},[75,76],[31,116640],{"className":116641,"style":120},[119],[31,116643,97],{"className":116644},[124],[31,116646],{"className":116647,"style":120},[119],[31,116649,116651,116654,116657,116660,116663,116666,116669,116672],{"className":116650},[66],[31,116652],{"className":116653,"style":59114},[70],[31,116655,205],{"className":116656},[75],[31,116658,1210],{"className":116659},[1391],[31,116661],{"className":116662,"style":1467},[119],[31,116664,919],{"className":116665,"style":949},[75,76],[31,116667],{"className":116668,"style":120},[119],[31,116670,97],{"className":116671},[124],[31,116673],{"className":116674,"style":120},[119],[31,116676,116678,116681,116684,116687,116690,116693,116696,116699,116702],{"className":116677},[66],[31,116679],{"className":116680,"style":41982},[70],[31,116682,152],{"className":116683},[75],[31,116685,1210],{"className":116686},[1391],[31,116688],{"className":116689,"style":1467},[119],[31,116691,55349],{"className":116692},[3860],[31,116694],{"className":116695,"style":1467},[119],[31,116697,1210],{"className":116698},[1391],[31,116700],{"className":116701,"style":1467},[119],[31,116703,1977],{"className":116704},[75],[31,116706,116708],{"className":116707,"translate":35},[411],[31,116709,116711,116793],{"className":116710,"translate":35},[34],[31,116712,116714],{"className":116713},[39],[41,116715,116716],{"xmlns":43,"display":420},[45,116717,116718,116790],{},[48,116719,116720,116728,116730,116740,116754,116756,116758,116766,116768,116774,116788],{},[12397,116721,116722,116724,116726],{},[51,116723,43481],{},[51,116725,12271],{},[51,116727,2515],{},[95,116729,223],{},[1180,116731,116732,116734],{},[51,116733,919],{"mathvariant":925},[1004,116735,116736,116738],{},[51,116737,2515],{},[51,116739,12271],{},[12397,116741,116742,116744,116752],{},[51,116743,43481],{},[48,116745,116746,116748,116750],{},[51,116747,12271],{},[95,116749,434],{},[91,116751,205],{},[51,116753,2515],{},[95,116755,1210],{"separator":62},[119,116757],{"width":20286},[12397,116759,116760,116762,116764],{},[51,116761,43481],{},[51,116763,12271],{},[95,116765,30648],{},[95,116767,223],{},[1180,116769,116770,116772],{},[51,116771,919],{"mathvariant":925},[95,116773,30648],{},[12397,116775,116776,116778,116786],{},[51,116777,43481],{},[48,116779,116780,116782,116784],{},[51,116781,12271],{},[95,116783,434],{},[91,116785,205],{},[95,116787,30648],{},[51,116789,495],{"mathvariant":494},[55,116791,116792],{"encoding":57},"V_t^\\pi=\\mathcal{T}^{\\pi_t} V_{t+1}^\\pi,\n\\qquad\nV_t^*=\\mathcal{T}^* V_{t+1}^*.",[31,116794,116796,116863,117070],{"className":116795,"ariaHidden":62},[61],[31,116797,116799,116803,116854,116857,116860],{"className":116798},[66],[31,116800],{"className":116801,"style":116802},[70],"height:0.9614em;vertical-align:-0.247em;",[31,116804,116806,116809],{"className":116805},[75],[31,116807,43481],{"className":116808,"style":120},[75,76],[31,116810,116812],{"className":116811},[1039],[31,116813,116815,116846],{"className":116814},[1043,1044],[31,116816,116818,116843],{"className":116817},[1048],[31,116819,116821,116832],{"className":116820,"style":3987},[1052],[31,116822,116823,116826],{"style":62628},[31,116824],{"className":116825,"style":1061},[1060],[31,116827,116829],{"className":116828},[1065,1066,1067,1068],[31,116830,12271],{"className":116831},[75,76,1068],[31,116833,116834,116837],{"style":1330},[31,116835],{"className":116836,"style":1061},[1060],[31,116838,116840],{"className":116839},[1065,1066,1067,1068],[31,116841,2515],{"className":116842,"style":2666},[75,76,1068],[31,116844,1076],{"className":116845},[1075],[31,116847,116849],{"className":116848},[1048],[31,116850,116852],{"className":116851,"style":13499},[1052],[31,116853],{},[31,116855],{"className":116856,"style":293},[119],[31,116858,223],{"className":116859},[297],[31,116861],{"className":116862,"style":293},[119],[31,116864,116866,116870,116940,117001,117004,117007,117010,117061,117064,117067],{"className":116865},[66],[31,116867],{"className":116868,"style":116869},[70],"height:1.044em;vertical-align:-0.3053em;",[31,116871,116873,116876],{"className":116872},[75],[31,116874,919],{"className":116875,"style":64444},[75,968],[31,116877,116879],{"className":116878},[1039],[31,116880,116882],{"className":116881},[1043],[31,116883,116885],{"className":116884},[1048],[31,116886,116888],{"className":116887,"style":3987},[1052],[31,116889,116890,116893],{"style":1330},[31,116891],{"className":116892,"style":1061},[1060],[31,116894,116896],{"className":116895},[1065,1066,1067,1068],[31,116897,116899],{"className":116898},[75,1068],[31,116900,116902,116905],{"className":116901},[75,1068],[31,116903,2515],{"className":116904,"style":2666},[75,76,1068],[31,116906,116908],{"className":116907},[1039],[31,116909,116911,116932],{"className":116910},[1043,1044],[31,116912,116914,116929],{"className":116913},[1048],[31,116915,116918],{"className":116916,"style":116917},[1052],"height:0.2963em;",[31,116919,116920,116923],{"style":94680},[31,116921],{"className":116922,"style":11769},[1060],[31,116924,116926],{"className":116925},[1065,11773,1308,1068],[31,116927,12271],{"className":116928},[75,76,1068],[31,116930,1076],{"className":116931},[1075],[31,116933,116935],{"className":116934},[1048],[31,116936,116938],{"className":116937,"style":50036},[1052],[31,116939],{},[31,116941,116943,116946],{"className":116942},[75],[31,116944,43481],{"className":116945,"style":120},[75,76],[31,116947,116949],{"className":116948},[1039],[31,116950,116952,116992],{"className":116951},[1043,1044],[31,116953,116955,116989],{"className":116954},[1048],[31,116956,116958,116978],{"className":116957,"style":3987},[1052],[31,116959,116960,116963],{"style":62628},[31,116961],{"className":116962,"style":1061},[1060],[31,116964,116966],{"className":116965},[1065,1066,1067,1068],[31,116967,116969,116972,116975],{"className":116968},[75,1068],[31,116970,12271],{"className":116971},[75,76,1068],[31,116973,434],{"className":116974},[124,1068],[31,116976,205],{"className":116977},[75,1068],[31,116979,116980,116983],{"style":1330},[31,116981],{"className":116982,"style":1061},[1060],[31,116984,116986],{"className":116985},[1065,1066,1067,1068],[31,116987,2515],{"className":116988,"style":2666},[75,76,1068],[31,116990,1076],{"className":116991},[1075],[31,116993,116995],{"className":116994},[1048],[31,116996,116999],{"className":116997,"style":116998},[1052],"height:0.3053em;",[31,117000],{},[31,117002,1210],{"className":117003},[1391],[31,117005],{"className":117006,"style":20525},[119],[31,117008],{"className":117009,"style":1467},[119],[31,117011,117013,117016],{"className":117012},[75],[31,117014,43481],{"className":117015,"style":120},[75,76],[31,117017,117019],{"className":117018},[1039],[31,117020,117022,117053],{"className":117021},[1043,1044],[31,117023,117025,117050],{"className":117024},[1048],[31,117026,117028,117039],{"className":117027,"style":74855},[1052],[31,117029,117030,117033],{"style":62628},[31,117031],{"className":117032,"style":1061},[1060],[31,117034,117036],{"className":117035},[1065,1066,1067,1068],[31,117037,12271],{"className":117038},[75,76,1068],[31,117040,117041,117044],{"style":1330},[31,117042],{"className":117043,"style":1061},[1060],[31,117045,117047],{"className":117046},[1065,1066,1067,1068],[31,117048,30648],{"className":117049},[124,1068],[31,117051,1076],{"className":117052},[1075],[31,117054,117056],{"className":117055},[1048],[31,117057,117059],{"className":117058,"style":13499},[1052],[31,117060],{},[31,117062],{"className":117063,"style":293},[119],[31,117065,223],{"className":117066},[297],[31,117068],{"className":117069,"style":293},[119],[31,117071,117073,117076,117105,117165],{"className":117072},[66],[31,117074],{"className":117075,"style":116869},[70],[31,117077,117079,117082],{"className":117078},[75],[31,117080,919],{"className":117081,"style":64444},[75,968],[31,117083,117085],{"className":117084},[1039],[31,117086,117088],{"className":117087},[1043],[31,117089,117091],{"className":117090},[1048],[31,117092,117094],{"className":117093,"style":74855},[1052],[31,117095,117096,117099],{"style":1330},[31,117097],{"className":117098,"style":1061},[1060],[31,117100,117102],{"className":117101},[1065,1066,1067,1068],[31,117103,30648],{"className":117104},[124,1068],[31,117106,117108,117111],{"className":117107},[75],[31,117109,43481],{"className":117110,"style":120},[75,76],[31,117112,117114],{"className":117113},[1039],[31,117115,117117,117157],{"className":117116},[1043,1044],[31,117118,117120,117154],{"className":117119},[1048],[31,117121,117123,117143],{"className":117122,"style":74855},[1052],[31,117124,117125,117128],{"style":62628},[31,117126],{"className":117127,"style":1061},[1060],[31,117129,117131],{"className":117130},[1065,1066,1067,1068],[31,117132,117134,117137,117140],{"className":117133},[75,1068],[31,117135,12271],{"className":117136},[75,76,1068],[31,117138,434],{"className":117139},[124,1068],[31,117141,205],{"className":117142},[75,1068],[31,117144,117145,117148],{"style":1330},[31,117146],{"className":117147,"style":1061},[1060],[31,117149,117151],{"className":117150},[1065,1066,1067,1068],[31,117152,30648],{"className":117153},[124,1068],[31,117155,1076],{"className":117156},[1075],[31,117158,117160],{"className":117159},[1048],[31,117161,117163],{"className":117162,"style":116998},[1052],[31,117164],{},[31,117166,495],{"className":117167},[75],[11,117169,117170,117172,117173,117201,117202,117253,117254,117282,117283,117334,117335,117387,117388,117416,117417,117509],{},[7246,117171,7248],{}," 对 ",[31,117174,117176,117189],{"className":117175,"translate":35},[34],[31,117177,117179],{"className":117178},[39],[41,117180,117181],{"xmlns":43},[45,117182,117183,117187],{},[48,117184,117185],{},[51,117186,12271],{},[55,117188,12271],{"encoding":57},[31,117190,117192],{"className":117191,"ariaHidden":62},[61],[31,117193,117195,117198],{"className":117194},[66],[31,117196],{"className":117197,"style":12283},[70],[31,117199,12271],{"className":117200},[75,76]," 从 ",[31,117203,117205,117223],{"className":117204,"translate":35},[34],[31,117206,117208],{"className":117207},[39],[41,117209,117210],{"xmlns":43},[45,117211,117212,117220],{},[48,117213,117214,117216,117218],{},[51,117215,919],{},[95,117217,97],{},[91,117219,205],{},[55,117221,117222],{"encoding":57},"T-1",[31,117224,117226,117244],{"className":117225,"ariaHidden":62},[61],[31,117227,117229,117232,117235,117238,117241],{"className":117228},[66],[31,117230],{"className":117231,"style":53224},[70],[31,117233,919],{"className":117234,"style":949},[75,76],[31,117236],{"className":117237,"style":120},[119],[31,117239,97],{"className":117240},[124],[31,117242],{"className":117243,"style":120},[119],[31,117245,117247,117250],{"className":117246},[66],[31,117248],{"className":117249,"style":326},[70],[31,117251,205],{"className":117252},[75]," 到 ",[31,117255,117257,117270],{"className":117256,"translate":35},[34],[31,117258,117260],{"className":117259},[39],[41,117261,117262],{"xmlns":43},[45,117263,117264,117268],{},[48,117265,117266],{},[91,117267,1977],{},[55,117269,1977],{"encoding":57},[31,117271,117273],{"className":117272,"ariaHidden":62},[61],[31,117274,117276,117279],{"className":117275},[66],[31,117277],{"className":117278,"style":326},[70],[31,117280,1977],{"className":117281},[75]," 反向归纳。",[31,117284,117286,117304],{"className":117285,"translate":35},[34],[31,117287,117289],{"className":117288},[39],[41,117290,117291],{"xmlns":43},[45,117292,117293,117301],{},[48,117294,117295,117297,117299],{},[51,117296,12271],{},[95,117298,223],{},[51,117300,919],{},[55,117302,117303],{"encoding":57},"t=T",[31,117305,117307,117325],{"className":117306,"ariaHidden":62},[61],[31,117308,117310,117313,117316,117319,117322],{"className":117309},[66],[31,117311],{"className":117312,"style":12283},[70],[31,117314,12271],{"className":117315},[75,76],[31,117317],{"className":117318,"style":293},[119],[31,117320,223],{"className":117321},[297],[31,117323],{"className":117324,"style":293},[119],[31,117326,117328,117331],{"className":117327},[66],[31,117329],{"className":117330,"style":945},[70],[31,117332,919],{"className":117333,"style":949},[75,76]," 时两端均为零。设结论对 ",[31,117336,117338,117356],{"className":117337,"translate":35},[34],[31,117339,117341],{"className":117340},[39],[41,117342,117343],{"xmlns":43},[45,117344,117345,117353],{},[48,117346,117347,117349,117351],{},[51,117348,12271],{},[95,117350,434],{},[91,117352,205],{},[55,117354,117355],{"encoding":57},"t+1",[31,117357,117359,117378],{"className":117358,"ariaHidden":62},[61],[31,117360,117362,117366,117369,117372,117375],{"className":117361},[66],[31,117363],{"className":117364,"style":117365},[70],"height:0.6984em;vertical-align:-0.0833em;",[31,117367,12271],{"className":117368},[75,76],[31,117370],{"className":117371,"style":120},[119],[31,117373,434],{"className":117374},[124],[31,117376],{"className":117377,"style":120},[119],[31,117379,117381,117384],{"className":117380},[66],[31,117382],{"className":117383,"style":326},[70],[31,117385,205],{"className":117386},[75]," 成立。对任意策略 ",[31,117389,117391,117404],{"className":117390,"translate":35},[34],[31,117392,117394],{"className":117393},[39],[41,117395,117396],{"xmlns":43},[45,117397,117398,117402],{},[48,117399,117400],{},[51,117401,2515],{},[55,117403,3485],{"encoding":57},[31,117405,117407],{"className":117406,"ariaHidden":62},[61],[31,117408,117410,117413],{"className":117409},[66],[31,117411],{"className":117412,"style":71},[70],[31,117414,2515],{"className":117415,"style":2666},[75,76],"，在 ",[31,117418,117420,117442],{"className":117419,"translate":35},[34],[31,117421,117423],{"className":117422},[39],[41,117424,117425],{"xmlns":43},[45,117426,117427,117439],{},[48,117428,117429,117435,117437],{},[1004,117430,117431,117433],{},[51,117432,14551],{},[51,117434,12271],{},[95,117436,223],{},[51,117438,14551],{},[55,117440,117441],{"encoding":57},"s_t=s",[31,117443,117445,117500],{"className":117444,"ariaHidden":62},[61],[31,117446,117448,117451,117491,117494,117497],{"className":117447},[66],[31,117449],{"className":117450,"style":7090},[70],[31,117452,117454,117457],{"className":117453},[75],[31,117455,14551],{"className":117456},[75,76],[31,117458,117460],{"className":117459},[1039],[31,117461,117463,117483],{"className":117462},[1043,1044],[31,117464,117466,117480],{"className":117465},[1048],[31,117467,117469],{"className":117468,"style":19220},[1052],[31,117470,117471,117474],{"style":2807},[31,117472],{"className":117473,"style":1061},[1060],[31,117475,117477],{"className":117476},[1065,1066,1067,1068],[31,117478,12271],{"className":117479},[75,76,1068],[31,117481,1076],{"className":117482},[1075],[31,117484,117486],{"className":117485},[1048],[31,117487,117489],{"className":117488,"style":2050},[1052],[31,117490],{},[31,117492],{"className":117493,"style":293},[119],[31,117495,223],{"className":117496},[297],[31,117498],{"className":117499,"style":293},[119],[31,117501,117503,117506],{"className":117502},[66],[31,117504],{"className":117505,"style":71},[70],[31,117507,14551],{"className":117508},[75,76]," 处分解回报：",[31,117511,117513],{"className":117512,"translate":35},[411],[31,117514,117516,117642],{"className":117515,"translate":35},[34],[31,117517,117519],{"className":117518},[39],[41,117520,117521],{"xmlns":43,"display":420},[45,117522,117523,117639],{},[5606,117524,117525,117605],{"rowspacing":5608,"columnalign":5609,"columnspacing":5610},[5612,117526,117527,117547],{},[5615,117528,117529],{},[2071,117530,117531],{"scriptlevel":1977,"displaystyle":62},[48,117532,117533,117541,117543,117545],{},[12397,117534,117535,117537,117539],{},[51,117536,43481],{},[51,117538,12271],{},[51,117540,2515],{},[95,117542,211],{"stretchy":210},[51,117544,14551],{},[95,117546,220],{"stretchy":210},[5615,117548,117549],{},[2071,117550,117551],{"scriptlevel":1977,"displaystyle":62},[48,117552,117553,117555,117557,117563,117565,117571,117573,117575,117589,117591,117603],{},[48,117554],{},[95,117556,223],{},[1004,117558,117559,117561],{},[51,117560,22324],{"mathvariant":14789},[51,117562,2515],{},[95,117564,15175],{"fence":210,"stretchy":62,"minsize":2511,"maxsize":2511},[1004,117566,117567,117569],{},[51,117568,2549],{},[51,117570,12271],{},[95,117572,434],{},[51,117574,53298],{},[12397,117576,117577,117579,117587],{},[51,117578,43481],{},[48,117580,117581,117583,117585],{},[51,117582,12271],{},[95,117584,434],{},[91,117586,205],{},[51,117588,2515],{},[95,117590,211],{"stretchy":210},[1004,117592,117593,117595],{},[51,117594,14551],{},[48,117596,117597,117599,117601],{},[51,117598,12271],{},[95,117600,434],{},[91,117602,205],{},[95,117604,220],{"stretchy":210},[5612,117606,117607,117613],{},[5615,117608,117609],{},[2071,117610,117611],{"scriptlevel":1977,"displaystyle":62},[48,117612],{},[5615,117614,117615],{},[2071,117616,117617],{"scriptlevel":1977,"displaystyle":62},[48,117618,117619,117621,117623,117625,117631,117633,117635,117637],{},[48,117620],{},[119,117622],{"width":20286},[95,117624,12680],{},[1004,117626,117627,117629],{},[51,117628,14551],{},[51,117630,12271],{},[95,117632,223],{},[51,117634,14551],{},[95,117636,15184],{"fence":210,"stretchy":62,"minsize":2511,"maxsize":2511},[51,117638,495],{"mathvariant":494},[55,117640,117641],{"encoding":57},"\\begin{aligned}\nV_t^\\pi(s)\n&=\\mathbb{E}_\\pi\\Big[r_t+\\gamma V_{t+1}^\\pi(s_{t+1})\\\\\n&\\qquad\\mid s_t=s\\Big].\n\\end{aligned}",[31,117643,117645],{"className":117644,"ariaHidden":62},[61],[31,117646,117648,117652],{"className":117647},[66],[31,117649],{"className":117650,"style":117651},[70],"height:3.9em;vertical-align:-1.7em;",[31,117653,117655],{"className":117654},[75],[31,117656,117658,117760],{"className":117657},[5606],[31,117659,117661],{"className":117660},[5867],[31,117662,117664,117751],{"className":117663},[1043,1044],[31,117665,117667,117748],{"className":117666},[1048],[31,117668,117671,117740],{"className":117669,"style":117670},[1052],"height:2.2em;",[31,117672,117674,117677],{"style":117673},"top:-4.2em;",[31,117675],{"className":117676,"style":5884},[1060],[31,117678,117680,117731,117734,117737],{"className":117679},[75],[31,117681,117683,117686],{"className":117682},[75],[31,117684,43481],{"className":117685,"style":120},[75,76],[31,117687,117689],{"className":117688},[1039],[31,117690,117692,117723],{"className":117691},[1043,1044],[31,117693,117695,117720],{"className":117694},[1048],[31,117696,117698,117709],{"className":117697,"style":3987},[1052],[31,117699,117700,117703],{"style":62628},[31,117701],{"className":117702,"style":1061},[1060],[31,117704,117706],{"className":117705},[1065,1066,1067,1068],[31,117707,12271],{"className":117708},[75,76,1068],[31,117710,117711,117714],{"style":1330},[31,117712],{"className":117713,"style":1061},[1060],[31,117715,117717],{"className":117716},[1065,1066,1067,1068],[31,117718,2515],{"className":117719,"style":2666},[75,76,1068],[31,117721,1076],{"className":117722},[1075],[31,117724,117726],{"className":117725},[1048],[31,117727,117729],{"className":117728,"style":13499},[1052],[31,117730],{},[31,117732,211],{"className":117733},[264],[31,117735,14551],{"className":117736},[75,76],[31,117738,220],{"className":117739},[289],[31,117741,117742,117745],{"style":58497},[31,117743],{"className":117744,"style":5884},[1060],[31,117746],{"className":117747},[75],[31,117749,1076],{"className":117750},[1075],[31,117752,117754],{"className":117753},[1048],[31,117755,117758],{"className":117756,"style":117757},[1052],"height:1.7em;",[31,117759],{},[31,117761,117763],{"className":117762},[6062],[31,117764,117766,118092],{"className":117765},[1043,1044],[31,117767,117769,118089],{"className":117768},[1048],[31,117770,117772,118005],{"className":117771,"style":117670},[1052],[31,117773,117774,117777],{"style":117673},[31,117775],{"className":117776,"style":5884},[1060],[31,117778,117780,117783,117786,117789,117792,117832,117838,117878,117881,117884,117887,117890,117950,117953,118002],{"className":117779},[75],[31,117781],{"className":117782},[75],[31,117784],{"className":117785,"style":293},[119],[31,117787,223],{"className":117788},[297],[31,117790],{"className":117791,"style":293},[119],[31,117793,117795,117798],{"className":117794},[75],[31,117796,22324],{"className":117797},[75,14947],[31,117799,117801],{"className":117800},[1039],[31,117802,117804,117824],{"className":117803},[1043,1044],[31,117805,117807,117821],{"className":117806},[1048],[31,117808,117810],{"className":117809,"style":3230},[1052],[31,117811,117812,117815],{"style":2807},[31,117813],{"className":117814,"style":1061},[1060],[31,117816,117818],{"className":117817},[1065,1066,1067,1068],[31,117819,2515],{"className":117820,"style":2666},[75,76,1068],[31,117822,1076],{"className":117823},[1075],[31,117825,117827],{"className":117826},[1048],[31,117828,117830],{"className":117829,"style":2050},[1052],[31,117831],{},[31,117833,117835],{"className":117834},[75],[31,117836,15175],{"className":117837},[1307,2662],[31,117839,117841,117844],{"className":117840},[75],[31,117842,2549],{"className":117843,"style":19759},[75,76],[31,117845,117847],{"className":117846},[1039],[31,117848,117850,117870],{"className":117849},[1043,1044],[31,117851,117853,117867],{"className":117852},[1048],[31,117854,117856],{"className":117855,"style":19220},[1052],[31,117857,117858,117861],{"style":26726},[31,117859],{"className":117860,"style":1061},[1060],[31,117862,117864],{"className":117863},[1065,1066,1067,1068],[31,117865,12271],{"className":117866},[75,76,1068],[31,117868,1076],{"className":117869},[1075],[31,117871,117873],{"className":117872},[1048],[31,117874,117876],{"className":117875,"style":2050},[1052],[31,117877],{},[31,117879],{"className":117880,"style":120},[119],[31,117882,434],{"className":117883},[124],[31,117885],{"className":117886,"style":120},[119],[31,117888,53298],{"className":117889,"style":24157},[75,76],[31,117891,117893,117896],{"className":117892},[75],[31,117894,43481],{"className":117895,"style":120},[75,76],[31,117897,117899],{"className":117898},[1039],[31,117900,117902,117942],{"className":117901},[1043,1044],[31,117903,117905,117939],{"className":117904},[1048],[31,117906,117908,117928],{"className":117907,"style":3987},[1052],[31,117909,117910,117913],{"style":62628},[31,117911],{"className":117912,"style":1061},[1060],[31,117914,117916],{"className":117915},[1065,1066,1067,1068],[31,117917,117919,117922,117925],{"className":117918},[75,1068],[31,117920,12271],{"className":117921},[75,76,1068],[31,117923,434],{"className":117924},[124,1068],[31,117926,205],{"className":117927},[75,1068],[31,117929,117930,117933],{"style":1330},[31,117931],{"className":117932,"style":1061},[1060],[31,117934,117936],{"className":117935},[1065,1066,1067,1068],[31,117937,2515],{"className":117938,"style":2666},[75,76,1068],[31,117940,1076],{"className":117941},[1075],[31,117943,117945],{"className":117944},[1048],[31,117946,117948],{"className":117947,"style":116998},[1052],[31,117949],{},[31,117951,211],{"className":117952},[264],[31,117954,117956,117959],{"className":117955},[75],[31,117957,14551],{"className":117958},[75,76],[31,117960,117962],{"className":117961},[1039],[31,117963,117965,117994],{"className":117964},[1043,1044],[31,117966,117968,117991],{"className":117967},[1048],[31,117969,117971],{"className":117970,"style":2028},[1052],[31,117972,117973,117976],{"style":2807},[31,117974],{"className":117975,"style":1061},[1060],[31,117977,117979],{"className":117978},[1065,1066,1067,1068],[31,117980,117982,117985,117988],{"className":117981},[75,1068],[31,117983,12271],{"className":117984},[75,76,1068],[31,117986,434],{"className":117987},[124,1068],[31,117989,205],{"className":117990},[75,1068],[31,117992,1076],{"className":117993},[1075],[31,117995,117997],{"className":117996},[1048],[31,117998,118000],{"className":117999,"style":25082},[1052],[31,118001],{},[31,118003,220],{"className":118004},[289],[31,118006,118007,118010],{"style":58497},[31,118008],{"className":118009,"style":5884},[1060],[31,118011,118013,118016,118019,118022,118025,118028,118068,118071,118074,118077,118080,118086],{"className":118012},[75],[31,118014],{"className":118015},[75],[31,118017],{"className":118018,"style":20525},[119],[31,118020],{"className":118021,"style":293},[119],[31,118023,12680],{"className":118024},[297],[31,118026],{"className":118027,"style":293},[119],[31,118029,118031,118034],{"className":118030},[75],[31,118032,14551],{"className":118033},[75,76],[31,118035,118037],{"className":118036},[1039],[31,118038,118040,118060],{"className":118039},[1043,1044],[31,118041,118043,118057],{"className":118042},[1048],[31,118044,118046],{"className":118045,"style":19220},[1052],[31,118047,118048,118051],{"style":2807},[31,118049],{"className":118050,"style":1061},[1060],[31,118052,118054],{"className":118053},[1065,1066,1067,1068],[31,118055,12271],{"className":118056},[75,76,1068],[31,118058,1076],{"className":118059},[1075],[31,118061,118063],{"className":118062},[1048],[31,118064,118066],{"className":118065,"style":2050},[1052],[31,118067],{},[31,118069],{"className":118070,"style":293},[119],[31,118072,223],{"className":118073},[297],[31,118075],{"className":118076,"style":293},[119],[31,118078,14551],{"className":118079},[75,76],[31,118081,118083],{"className":118082},[75],[31,118084,15184],{"className":118085},[1307,2662],[31,118087,495],{"className":118088},[75],[31,118090,1076],{"className":118091},[1075],[31,118093,118095],{"className":118094},[1048],[31,118096,118098],{"className":118097,"style":117757},[1052],[31,118099],{},[11,118101,118102,118103,118204,118205,101979],{},"由 ",[31,118104,118106,118132],{"className":118105,"translate":35},[34],[31,118107,118109],{"className":118108},[39],[41,118110,118111],{"xmlns":43},[45,118112,118113,118129],{},[48,118114,118115],{},[12397,118116,118117,118119,118127],{},[51,118118,43481],{},[48,118120,118121,118123,118125],{},[51,118122,12271],{},[95,118124,434],{},[91,118126,205],{},[95,118128,30648],{},[55,118130,118131],{"encoding":57},"V_{t+1}^*",[31,118133,118135],{"className":118134,"ariaHidden":62},[61],[31,118136,118138,118142],{"className":118137},[66],[31,118139],{"className":118140,"style":118141},[70],"height:0.9951em;vertical-align:-0.3064em;",[31,118143,118145,118148],{"className":118144},[75],[31,118146,43481],{"className":118147,"style":120},[75,76],[31,118149,118151],{"className":118150},[1039],[31,118152,118154,118195],{"className":118153},[1043,1044],[31,118155,118157,118192],{"className":118156},[1048],[31,118158,118160,118181],{"className":118159,"style":30719},[1052],[31,118161,118163,118166],{"style":118162},"top:-2.4519em;margin-left:-0.2222em;margin-right:0.05em;",[31,118164],{"className":118165,"style":1061},[1060],[31,118167,118169],{"className":118168},[1065,1066,1067,1068],[31,118170,118172,118175,118178],{"className":118171},[75,1068],[31,118173,12271],{"className":118174},[75,76,1068],[31,118176,434],{"className":118177},[124,1068],[31,118179,205],{"className":118180},[75,1068],[31,118182,118183,118186],{"style":3631},[31,118184],{"className":118185,"style":1061},[1060],[31,118187,118189],{"className":118188},[1065,1066,1067,1068],[31,118190,30648],{"className":118191},[124,1068],[31,118193,1076],{"className":118194},[1075],[31,118196,118198],{"className":118197},[1048],[31,118199,118202],{"className":118200,"style":118201},[1052],"height:0.3064em;",[31,118203],{}," 的定义，",[31,118206,118208,118250],{"className":118207,"translate":35},[34],[31,118209,118211],{"className":118210},[39],[41,118212,118213],{"xmlns":43},[45,118214,118215,118247],{},[48,118216,118217,118231,118233],{},[12397,118218,118219,118221,118229],{},[51,118220,43481],{},[48,118222,118223,118225,118227],{},[51,118224,12271],{},[95,118226,434],{},[91,118228,205],{},[51,118230,2515],{},[95,118232,347],{},[12397,118234,118235,118237,118245],{},[51,118236,43481],{},[48,118238,118239,118241,118243],{},[51,118240,12271],{},[95,118242,434],{},[91,118244,205],{},[95,118246,30648],{},[55,118248,118249],{"encoding":57},"V_{t+1}^\\pi\\le 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s)\\sum_{s'}P(s'\\mid s,a)\n\\left[r(s,a,s')+\\gamma V_{t+1}^*(s')\\right]\\\\\n&\\le\\max_{a}\\sum_{s'}P(s'\\mid s,a)\n\\left[r(s,a,s')+\\gamma 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assName":119491},[1391],[31,119493],{"className":119494,"style":1467},[119],[31,119496,5813],{"className":119497},[75,76],[31,119499,1210],{"className":119500},[1391],[31,119502],{"className":119503,"style":1467},[119],[31,119505,119507,119510],{"className":119506},[75],[31,119508,14551],{"className":119509},[75,76],[31,119511,119513],{"className":119512},[1039],[31,119514,119516],{"className":119515},[1043],[31,119517,119519],{"className":119518},[1048],[31,119520,119522],{"className":119521,"style":14017},[1052],[31,119523,119524,119527],{"style":1330},[31,119525],{"className":119526,"style":1061},[1060],[31,119528,119530],{"className":119529},[1065,1066,1067,1068],[31,119531,119533],{"className":119532},[75,1068],[31,119534,13691],{"className":119535},[75,1068],[31,119537,220],{"className":119538},[289],[31,119540],{"className":119541,"style":120},[119],[31,119543,434],{"className":119544},[124],[31,119546],{"className":119547,"style":120},[119],[31,119549,53298],{"className":119550,"style":24157},[75,76],[31,119552,119554,119557],{"className":119553},[75],[31,119555,43481],{"className":119556,"style":120},[75,76],[31,119558,119560],{"className":119559},[1039],[31,119561,119563,119603],{"className":119562},[1043,1044],[31,119564,119566,119600],{"className":119565},[1048],[31,119567,119569,119589],{"className":119568,"style":74855},[1052],[31,119570,119571,119574],{"style":62628},[31,119572],{"className":119573,"style":1061},[1060],[31,119575,119577],{"className":119576},[1065,1066,1067,1068],[31,119578,119580,119583,119586],{"className":119579},[75,1068],[31,119581,12271],{"className":119582},[75,76,1068],[31,119584,434],{"className":119585},[124,1068],[31,119587,205],{"className":119588},[75,1068],[31,119590,119591,119594],{"style":1330},[31,119592],{"className":119593,"style":1061},[1060],[31,119595,119597],{"className":119596},[1065,1066,1067,1068],[31,119598,30648],{"className":119599},[124,1068],[31,119601,1076],{"className":119602},[1075],[31,119604,119606],{"className":119605},[1048],[31,119607,119609],{"className":119608,"style":116998},[1052],[31,119610],{},[31,119612,211],{"className":119613},[264],[31,119615,119617,119620],{"className":119616},[75],[31,119618,14551],{"className":119619},[75,76],[31,119621,119623],{"className":119622},[1039],[31,119624,119626],{"className":119625},[1043],[31,119627,119629],{"className":119628},[1048],[31,119630,119632],{"className":119631,"style":14017},[1052],[31,119633,119634,119637],{"style":1330},[31,119635],{"className":119636,"style":1061},[1060],[31,119638,119640],{"className":119639},[1065,1066,1067,1068],[31,119641,119643],{"className":119642},[75,1068],[31,119644,13691],{"className":119645},[75,1068],[31,119647,220],{"className":119648},[289],[31,119650,119652],{"className":119651,"style":3868},[289,3867],[31,119653,15184],{"className":119654},[1307,1308],[31,119656],{"className":119657,"style":1467},[119],[31,119659,495],{"className":119660},[75],[31,119662,1076],{"className":119663},[1075],[31,119665,119667],{"className":119666},[1048],[31,119668,119670],{"className":119669,"style":118780},[1052],[31,119671],{},[11,119673,119674,119675,4880,119772,119822,119823,119893,119894,119944,119945,120042,120043,120253],{},"另一方面，由归纳假设，",[31,119676,119678,119703],{"className":119677,"translate":35},[34],[31,119679,119681],{"className":119680},[39],[41,119682,119683],{"xmlns":43},[45,119684,119685,119701],{},[48,119686,119687],{},[12397,119688,119689,119691,119699],{},[51,119690,43481],{},[48,119692,119693,119695,119697],{},[51,119694,12271],{},[95,119696,434],{},[91,119698,205],{},[95,119700,30648],{},[55,119702,118131],{"encoding":57},[31,119704,119706],{"className":119705,"ariaHidden":62},[61],[31,119707,119709,119712],{"className":119708},[66],[31,119710],{"className":119711,"style":118141},[70],[31,119713,119715,119718],{"className":119714},[75],[31,119716,43481],{"className":119717,"style":120},[75,76],[31,119719,119721],{"className":119720},[1039],[31,119722,119724,119764],{"className":119723},[1043,1044],[31,119725,119727,119761],{"className":119726},[1048],[31,119728,119730,119750],{"className":119729,"style":30719},[1052],[31,119731,119732,119735],{"style":118162},[31,119733],{"className":119734,"style":1061},[1060],[31,119736,119738],{"className":119737},[1065,1066,1067,1068],[31,119739,119741,119744,119747],{"className":119740},[75,1068],[31,119742,12271],{"className":119743},[75,76,1068],[31,119745,434],{"className":119746},[124,1068],[31,119748,205],{"className":119749},[75,1068],[31,119751,119752,119755],{"style":3631},[31,119753],{"className":119754,"style":1061},[1060],[31,119756,119758],{"className":119757},[1065,1066,1067,1068],[31,119759,30648],{"className":119760},[124,1068],[31,119762,1076],{"className":119763},[1075],[31,119765,119767],{"className":119766},[1048],[31,119768,119770],{"className":119769,"style":118201},[1052],[31,119771],{},[31,119773,119775,119792],{"className":119774,"translate":35},[34],[31,119776,119778],{"className":119777},[39],[41,119779,119780],{"xmlns":43},[45,119781,119782,119790],{},[48,119783,119784,119786,119788],{},[51,119785,12271],{},[95,119787,434],{},[91,119789,205],{},[55,119791,117355],{"encoding":57},[31,119793,119795,119813],{"className":119794,"ariaHidden":62},[61],[31,119796,119798,119801,119804,119807,119810],{"className":119797},[66],[31,119799],{"className":119800,"style":117365},[70],[31,119802,12271],{"className":119803},[75,76],[31,119805],{"className":119806,"style":120},[119],[31,119808,434],{"className":119809},[124],[31,119811],{"className":119812,"style":120},[119],[31,119814,119816,119819],{"className":119815},[66],[31,119817],{"className":119818,"style":326},[70],[31,119820,205],{"className":119821},[75]," 时刻可达的最优值。取 ",[31,119824,119826,119844],{"className":119825,"translate":35},[34],[31,119827,119829],{"className":119828},[39],[41,119830,119831],{"xmlns":43},[45,119832,119833,119841],{},[48,119834,119835],{},[1004,119836,119837,119839],{},[51,119838,5813],{},[51,119840,12271],{},[55,119842,119843],{"encoding":57},"a_t",[31,119845,119847],{"className":119846,"ariaHidden":62},[61],[31,119848,119850,119853],{"className":119849},[66],[31,119851],{"className":119852,"style":7090},[70],[31,119854,119856,119859],{"className":119855},[75],[31,119857,5813],{"className":119858},[75,76],[31,119860,119862],{"className":119861},[1039],[31,119863,119865,119885],{"className":119864},[1043,1044],[31,119866,119868,119882],{"className":119867},[1048],[31,119869,119871],{"className":119870,"style":19220},[1052],[31,119872,119873,119876],{"style":2807},[31,119874],{"className":119875,"style":1061},[1060],[31,119877,119879],{"className":119878},[1065,1066,1067,1068],[31,119880,12271],{"className":119881},[75,76,1068],[31,119883,1076],{"className":119884},[1075],[31,119886,119888],{"className":119887},[1048],[31,119889,119891],{"className":119890,"style":2050},[1052],[31,119892],{}," 为使上式右端最大的动作，并从 ",[31,119895,119897,119914],{"className":119896,"translate":35},[34],[31,119898,119900],{"className":119899},[39],[41,119901,119902],{"xmlns":43},[45,119903,119904,119912],{},[48,119905,119906,119908,119910],{},[51,119907,12271],{},[95,119909,434],{},[91,119911,205],{},[55,119913,117355],{"encoding":57},[31,119915,119917,119935],{"className":119916,"ariaHidden":62},[61],[31,119918,119920,119923,119926,119929,119932],{"className":119919},[66],[31,119921],{"className":119922,"style":117365},[70],[31,119924,12271],{"className":119925},[75,76],[31,119927],{"className":119928,"style":120},[119],[31,119930,434],{"className":119931},[124],[31,119933],{"className":119934,"style":120},[119],[31,119936,119938,119941],{"className":119937},[66],[31,119939],{"className":119940,"style":326},[70],[31,119942,205],{"className":119943},[75]," 开始采用达到 ",[31,119946,119948,119973],{"className":119947,"translate":35},[34],[31,119949,119951],{"className":119950},[39],[41,119952,119953],{"xmlns":43},[45,119954,119955,119971],{},[48,119956,119957],{},[12397,119958,119959,119961,119969],{},[51,119960,43481],{},[48,119962,119963,119965,119967],{},[51,119964,12271],{},[95,119966,434],{},[91,119968,205],{},[95,119970,30648],{},[55,119972,118131],{"encoding":57},[31,119974,119976],{"className":119975,"ariaHidden":62},[61],[31,119977,119979,119982],{"className":119978},[66],[31,119980],{"className":119981,"style":118141},[70],[31,119983,119985,119988],{"className":119984},[75],[31,119986,43481],{"className":119987,"style":120},[75,76],[31,119989,119991],{"className":119990},[1039],[31,119992,119994,120034],{"className":119993},[1043,1044],[31,119995,119997,120031],{"className":119996},[1048],[31,119998,120000,120020],{"className":119999,"style":30719},[1052],[31,120001,120002,120005],{"style":118162},[31,120003],{"className":120004,"style":1061},[1060],[31,120006,120008],{"className":120007},[1065,1066,1067,1068],[31,120009,120011,120014,120017],{"className":120010},[75,1068],[31,120012,12271],{"className":120013},[75,76,1068],[31,120015,434],{"className":120016},[124,1068],[31,120018,205],{"className":120019},[75,1068],[31,120021,120022,120025],{"style":3631},[31,120023],{"className":120024,"style":1061},[1060],[31,120026,120028],{"className":120027},[1065,1066,1067,1068],[31,120029,30648],{"className":120030},[124,1068],[31,120032,1076],{"className":120033},[1075],[31,120035,120037],{"className":120036},[1048],[31,120038,120040],{"className":120039,"style":118201},[1052],[31,120041],{}," 的最优续策略，则等号可以达到。在有限状态与有限动作下，可以逐状态选取最优续策略，因此不需要额外的可测选择条件。故 ",[31,120044,120046,120088],{"className":120045,"translate":35},[34],[31,120047,120049],{"className":120048},[39],[41,120050,120051],{"xmlns":43},[45,120052,120053,120085],{},[48,120054,120055,120063,120065,120071],{},[12397,120056,120057,120059,120061],{},[51,120058,43481],{},[51,120060,12271],{},[95,120062,30648],{},[95,120064,223],{},[1180,120066,120067,120069],{},[51,120068,919],{"mathvariant":925},[95,120070,30648],{},[12397,120072,120073,120075,120083],{},[51,120074,43481],{},[48,120076,120077,120079,120081],{},[51,120078,12271],{},[95,120080,434],{},[91,120082,205],{},[95,120084,30648],{},[55,120086,120087],{"encoding":57},"V_t^*=\\mathcal{T}^*V_{t+1}^*",[31,120089,120091,120158],{"className":120090,"ariaHidden":62},[61],[31,120092,120094,120098,120149,120152,120155],{"className":120093},[66],[31,120095],{"className":120096,"style":120097},[70],"height:0.9357em;vertical-align:-0.247em;",[31,120099,120101,120104],{"className":120100},[75],[31,120102,43481],{"className":120103,"style":120},[75,76],[31,120105,120107],{"className":120106},[1039],[31,120108,120110,120141],{"className":120109},[1043,1044],[31,120111,120113,120138],{"className":120112},[1048],[31,120114,120116,120127],{"className":120115,"style":30719},[1052],[31,120117,120118,120121],{"style":62628},[31,120119],{"className":120120,"style":1061},[1060],[31,120122,120124],{"className":120123},[1065,1066,1067,1068],[31,120125,12271],{"className":120126},[75,76,1068],[31,120128,120129,120132],{"style":3631},[31,120130],{"className":120131,"style":1061},[1060],[31,120133,120135],{"className":120134},[1065,1066,1067,1068],[31,120136,30648],{"className":120137},[124,1068],[31,120139,1076],{"className":120140},[1075],[31,120142,120144],{"className":120143},[1048],[31,120145,120147],{"className":120146,"style":13499},[1052],[31,120148],{},[31,120150],{"className":120151,"style":293},[119],[31,120153,223],{"className":120154},[297],[31,120156],{"className":120157,"style":293},[119],[31,120159,120161,120164,120193],{"className":120160},[66],[31,120162],{"className":120163,"style":118141},[70],[31,120165,120167,120170],{"className":120166},[75],[31,120168,919],{"className":120169,"style":64444},[75,968],[31,120171,120173],{"className":120172},[1039],[31,120174,120176],{"className":120175},[1043],[31,120177,120179],{"className":120178},[1048],[31,120180,120182],{"className":120181,"style":30719},[1052],[31,120183,120184,120187],{"style":3631},[31,120185],{"className":120186,"style":1061},[1060],[31,120188,120190],{"className":120189},[1065,1066,1067,1068],[31,120191,30648],{"className":120192},[124,1068],[31,120194,120196,120199],{"className":120195},[75],[31,120197,43481],{"className":120198,"style":120},[75,76],[31,120200,120202],{"className":120201},[1039],[31,120203,120205,120245],{"className":120204},[1043,1044],[31,120206,120208,120242],{"className":120207},[1048],[31,120209,120211,120231],{"className":120210,"style":30719},[1052],[31,120212,120213,120216],{"style":118162},[31,120214],{"className":120215,"style":1061},[1060],[31,120217,120219],{"className":120218},[1065,1066,1067,1068],[31,120220,120222,120225,120228],{"className":120221},[75,1068],[31,120223,12271],{"className":120224},[75,76,1068],[31,120226,434],{"className":120227},[124,1068],[31,120229,205],{"className":120230},[75,1068],[31,120232,120233,120236],{"style":3631},[31,120234],{"className":120235,"style":1061},[1060],[31,120237,120239],{"className":120238},[1065,1066,1067,1068],[31,120240,30648],{"className":120241},[124,1068],[31,120243,1076],{"className":120244},[1075],[31,120246,120248],{"className":120247},[1048],[31,120249,120251],{"className":120250,"style":118201},[1052],[31,120252],{},"。策略评估式同理。证毕。",[11,120255,120256,120257,120285,120286,120383,120384,120468,120469,120553],{},"这个定理就是反向归纳的数学形式。它说明，在时刻 ",[31,120258,120260,120273],{"className":120259,"translate":35},[34],[31,120261,120263],{"className":120262},[39],[41,120264,120265],{"xmlns":43},[45,120266,120267,120271],{},[48,120268,120269],{},[51,120270,12271],{},[55,120272,12271],{"encoding":57},[31,120274,120276],{"className":120275,"ariaHidden":62},[61],[31,120277,120279,120282],{"className":120278},[66],[31,120280],{"className":120281,"style":12283},[70],[31,120283,12271],{"className":120284},[75,76],"，最优决策只需要知道当前状态与下一时刻的最优值函数 ",[31,120287,120289,120314],{"className":120288,"translate":35},[34],[31,120290,120292],{"className":120291},[39],[41,120293,120294],{"xmlns":43},[45,120295,120296,120312],{},[48,120297,120298],{},[12397,120299,120300,120302,120310],{},[51,120301,43481],{},[48,120303,120304,120306,120308],{},[51,120305,12271],{},[95,120307,434],{},[91,120309,205],{},[95,120311,30648],{},[55,120313,118131],{"encoding":57},[31,120315,120317],{"className":120316,"ariaHidden":62},[61],[31,120318,120320,120323],{"className":120319},[66],[31,120321],{"className":120322,"style":118141},[70],[31,120324,120326,120329],{"className":120325},[75],[31,120327,43481],{"className":120328,"style":120},[75,76],[31,120330,120332],{"className":120331},[1039],[31,120333,120335,120375],{"className":120334},[1043,1044],[31,120336,120338,120372],{"className":120337},[1048],[31,120339,120341,120361],{"className":120340,"style":30719},[1052],[31,120342,120343,120346],{"style":118162},[31,120344],{"className":120345,"style":1061},[1060],[31,120347,120349],{"className":120348},[1065,1066,1067,1068],[31,120350,120352,120355,120358],{"className":120351},[75,1068],[31,120353,12271],{"className":120354},[75,76,1068],[31,120356,434],{"className":120357},[124,1068],[31,120359,205],{"className":120360},[75,1068],[31,120362,120363,120366],{"style":3631},[31,120364],{"className":120365,"style":1061},[1060],[31,120367,120369],{"className":120368},[1065,1066,1067,1068],[31,120370,30648],{"className":120371},[124,1068],[31,120373,1076],{"className":120374},[1075],[31,120376,120378],{"className":120377},[1048],[31,120379,120381],{"className":120380,"style":118201},[1052],[31,120382],{},"。一旦后继状态 ",[31,120385,120387,120410],{"className":120386,"translate":35},[34],[31,120388,120390],{"className":120389},[39],[41,120391,120392],{"xmlns":43},[45,120393,120394,120408],{},[48,120395,120396],{},[1004,120397,120398,120400],{},[51,120399,14551],{},[48,120401,120402,120404,120406],{},[51,120403,12271],{},[95,120405,434],{},[91,120407,205],{},[55,120409,56314],{"encoding":57},[31,120411,120413],{"className":120412,"ariaHidden":62},[61],[31,120414,120416,120419],{"className":120415},[66],[31,120417],{"className":120418,"style":41068},[70],[31,120420,120422,120425],{"className":120421},[75],[31,120423,14551],{"className":120424},[75,76],[31,120426,120428],{"className":120427},[1039],[31,120429,120431,120460],{"className":120430},[1043,1044],[31,120432,120434,120457],{"className":120433},[1048],[31,120435,120437],{"className":120436,"style":2028},[1052],[31,120438,120439,120442],{"style":2807},[31,120440],{"className":120441,"style":1061},[1060],[31,120443,120445],{"className":120444},[1065,1066,1067,1068],[31,120446,120448,120451,120454],{"className":120447},[75,1068],[31,120449,12271],{"className":120450},[75,76,1068],[31,120452,434],{"className":120453},[124,1068],[31,120455,205],{"className":120456},[75,1068],[31,120458,1076],{"className":120459},[1075],[31,120461,120463],{"className":120462},[1048],[31,120464,120466],{"className":120465,"style":25082},[1052],[31,120467],{}," 确定，从 ",[31,120470,120472,120495],{"className":120471,"translate":35},[34],[31,120473,120475],{"className":120474},[39],[41,120476,120477],{"xmlns":43},[45,120478,120479,120493],{},[48,120480,120481],{},[1004,120482,120483,120485],{},[51,120484,14551],{},[48,120486,120487,120489,120491],{},[51,120488,12271],{},[95,120490,434],{},[91,120492,205],{},[55,120494,56314],{"encoding":57},[31,120496,120498],{"className":120497,"ariaHidden":62},[61],[31,120499,120501,120504],{"className":120500},[66],[31,120502],{"className":120503,"style":41068},[70],[31,120505,120507,120510],{"className":120506},[75],[31,120508,14551],{"className":120509},[75,76],[31,120511,120513],{"className":120512},[1039],[31,120514,120516,120545],{"className":120515},[1043,1044],[31,120517,120519,120542],{"className":120518},[1048],[31,120520,120522],{"className":120521,"style":2028},[1052],[31,120523,120524,120527],{"style":2807},[31,120525],{"className":120526,"style":1061},[1060],[31,120528,120530],{"className":120529},[1065,1066,1067,1068],[31,120531,120533,120536,120539],{"className":120532},[75,1068],[31,120534,12271],{"className":120535},[75,76,1068],[31,120537,434],{"className":120538},[124,1068],[31,120540,205],{"className":120541},[75,1068],[31,120543,1076],{"className":120544},[1075],[31,120546,120548],{"className":120547},[1048],[31,120549,120551],{"className":120550,"style":25082},[1052],[31,120552],{}," 开始的剩余决策必须构成该子问题的最优策略。否则，替换该续策略就可以严格改进原策略，与整体最优性矛盾。这就是贝尔曼最优性原理在最简单情形下的完整证明。",[52860,120555,120557],{"id":120556},"_62-无限期情形","6.2 无限期情形",[11,120559,120560,120561,120563,120564,120622],{},"无限期问题不能直接使用反向归纳，因为不存在一个最后的时刻。但第 4 节的压缩映射论证给出了同样的结论。",[5813,120562,89835],{"href":89834}," 表明，无限期最优贝尔曼方程就是 ",[31,120565,120567,120584],{"className":120566,"translate":35},[34],[31,120568,120570],{"className":120569},[39],[41,120571,120572],{"xmlns":43},[45,120573,120574,120582],{},[48,120575,120576],{},[1180,120577,120578,120580],{},[51,120579,919],{"mathvariant":925},[95,120581,30648],{},[55,120583,74501],{"encoding":57},[31,120585,120587],{"className":120586,"ariaHidden":62},[61],[31,120588,120590,120593],{"className":120589},[66],[31,120591],{"className":120592,"style":30719},[70],[31,120594,120596,120599],{"className":120595},[75],[31,120597,919],{"className":120598,"style":64444},[75,968],[31,120600,120602],{"className":120601},[1039],[31,120603,120605],{"className":120604},[1043],[31,120606,120608],{"className":120607},[1048],[31,120609,120611],{"className":120610,"style":30719},[1052],[31,120612,120613,120616],{"style":3631},[31,120614],{"className":120615,"style":1061},[1060],[31,120617,120619],{"className":120618},[1065,1066,1067,1068],[31,120620,30648],{"className":120621},[124,1068]," 的不动点方程，而最优值函数正是该方程的唯一有界解。",[11,120624,120625,120626,120841],{},"因此，最优子结构在无限期情形下不是一条需要额外假设的公理，而是折扣、马尔可夫性与可加效用三者共同作用的结果。折扣保证了算子的压缩性，马尔可夫性保证后继状态足以概括历史，可加效用保证了回报可以分解为 ",[31,120627,120629,120670],{"className":120628,"translate":35},[34],[31,120630,120632],{"className":120631},[39],[41,120633,120634],{"xmlns":43},[45,120635,120636,120668],{},[48,120637,120638,120644,120646,120652,120654,120656],{},[1004,120639,120640,120642],{},[51,120641,15297],{},[51,120643,12271],{},[95,120645,223],{},[1004,120647,120648,120650],{},[51,120649,2549],{},[51,120651,12271],{},[95,120653,434],{},[51,120655,53298],{},[1004,120657,120658,120660],{},[51,120659,15297],{},[48,120661,120662,120664,120666],{},[51,120663,12271],{},[95,120665,434],{},[91,120667,205],{},[55,120669,60121],{"encoding":57},[31,120671,120673,120728,120783],{"className":120672,"ariaHidden":62},[61],[31,120674,120676,120679,120719,120722,120725],{"className":120675},[66],[31,120677],{"className":120678,"style":5230},[70],[31,120680,120682,120685],{"className":120681},[75],[31,120683,15297],{"className":120684},[75,76],[31,120686,120688],{"className":120687},[1039],[31,120689,120691,120711],{"className":120690},[1043,1044],[31,120692,120694,120708],{"className":120693},[1048],[31,120695,120697],{"className":120696,"style":19220},[1052],[31,120698,120699,120702],{"style":2807},[31,120700],{"className":120701,"style":1061},[1060],[31,120703,120705],{"className":120704},[1065,1066,1067,1068],[31,120706,12271],{"className":120707},[75,76,1068],[31,120709,1076],{"className":120710},[1075],[31,120712,120714],{"className":120713},[1048],[31,120715,120717],{"className":120716,"style":2050},[1052],[31,120718],{},[31,120720],{"className":120721,"style":293},[119],[31,120723,223],{"className":120724},[297],[31,120726],{"className":120727,"style":293},[119],[31,120729,120731,120734,120774,120777,120780],{"className":120730},[66],[31,120732],{"className":120733,"style":5426},[70],[31,120735,120737,120740],{"className":120736},[75],[31,120738,2549],{"className":120739,"style":19759},[75,76],[31,120741,120743],{"className":120742},[1039],[31,120744,120746,120766],{"className":120745},[1043,1044],[31,120747,120749,120763],{"className":120748},[1048],[31,120750,120752],{"className":120751,"style":19220},[1052],[31,120753,120754,120757],{"style":26726},[31,120755],{"className":120756,"style":1061},[1060],[31,120758,120760],{"className":120759},[1065,1066,1067,1068],[31,120761,12271],{"className":120762},[75,76,1068],[31,120764,1076],{"className":120765},[1075],[31,120767,120769],{"className":120768},[1048],[31,120770,120772],{"className":120771,"style":2050},[1052],[31,120773],{},[31,120775],{"className":120776,"style":120},[119],[31,120778,434],{"className":120779},[124],[31,120781],{"className":120782,"style":120},[119],[31,120784,120786,120789,120792],{"className":120785},[66],[31,120787],{"className":120788,"style":60241},[70],[31,120790,53298],{"className":120791,"style":24157},[75,76],[31,120793,120795,120798],{"className":120794},[75],[31,120796,15297],{"className":120797},[75,76],[31,120799,120801],{"className":120800},[1039],[31,120802,120804,120833],{"className":120803},[1043,1044],[31,120805,120807,120830],{"className":120806},[1048],[31,120808,120810],{"className":120809,"style":2028},[1052],[31,120811,120812,120815],{"style":2807},[31,120813],{"className":120814,"style":1061},[1060],[31,120816,120818],{"className":120817},[1065,1066,1067,1068],[31,120819,120821,120824,120827],{"className":120820},[75,1068],[31,120822,12271],{"className":120823},[75,76,1068],[31,120825,434],{"className":120826},[124,1068],[31,120828,205],{"className":120829},[75,1068],[31,120831,1076],{"className":120832},[1075],[31,120834,120836],{"className":120835},[1048],[31,120837,120839],{"className":120838,"style":25082},[1052],[31,120840],{},"。缺少其中任何一个条件，贝尔曼方程的标准形式都可能失效。",[52860,120843,120845],{"id":120844},"_63-成立条件","6.3 成立条件",[11,120847,120848],{},"最优子结构的成立依赖以下三个条件。",[11,120850,120851,120852,121052,121053,121268],{},"第一，效用必须是可加且按常数折扣的。也就是说，回报必须能够写成 ",[31,120853,120855,120899],{"className":120854,"translate":35},[34],[31,120856,120858],{"className":120857},[39],[41,120859,120860],{"xmlns":43},[45,120861,120862,120896],{},[48,120863,120864,120878,120884],{},[12397,120865,120866,120868,120876],{},[95,120867,5625],{},[48,120869,120870,120872,120874],{},[51,120871,52065],{},[95,120873,223],{},[91,120875,1977],{},[51,120877,4587],{"mathvariant":494},[1180,120879,120880,120882],{},[51,120881,53298],{},[51,120883,52065],{},[1004,120885,120886,120888],{},[51,120887,2549],{},[48,120889,120890,120892,120894],{},[51,120891,12271],{},[95,120893,434],{},[51,120895,52065],{},[55,120897,120898],{"encoding":57},"\\sum_{k=0}^{\\infty}\\gamma^k r_{t+k}",[31,120900,120902],{"className":120901,"ariaHidden":62},[61],[31,120903,120905,120908,120971,120974,121003],{"className":120904},[66],[31,120906],{"className":120907,"style":68349},[70],[31,120909,120911,120914],{"className":120910},[1460],[31,120912,5625],{"className":120913,"style":21372},[1460,1461,1462],[31,120915,120917],{"className":120916},[1039],[31,120918,120920,120963],{"className":120919},[1043,1044],[31,120921,120923,120960],{"className":120922},[1048],[31,120924,120926,120946],{"className":120925,"style":68368},[1052],[31,120927,120928,120931],{"style":21388},[31,120929],{"className":120930,"style":1061},[1060],[31,120932,120934],{"className":120933},[1065,1066,1067,1068],[31,120935,120937,120940,120943],{"className":120936},[75,1068],[31,120938,52065],{"className":120939,"style":52166},[75,76,1068],[31,120941,223],{"className":120942},[297,1068],[31,120944,1977],{"className":120945},[75,1068],[31,120947,120948,120951],{"style":68391},[31,120949],{"className":120950,"style":1061},[1060],[31,120952,120954],{"className":120953},[1065,1066,1067,1068],[31,120955,120957],{"className":120956},[75,1068],[31,120958,4587],{"className":120959},[75,1068],[31,120961,1076],{"className":120962},[1075],[31,120964,120966],{"className":120965},[1048],[31,120967,120969],{"className":120968,"style":21407},[1052],[31,120970],{},[31,120972],{"className":120973,"style":1467},[119],[31,120975,120977,120980],{"className":120976},[75],[31,120978,53298],{"className":120979,"style":24157},[75,76],[31,120981,120983],{"className":120982},[1039],[31,120984,120986],{"className":120985},[1043],[31,120987,120989],{"className":120988},[1048],[31,120990,120992],{"className":120991,"style":20863},[1052],[31,120993,120994,120997],{"style":3631},[31,120995],{"className":120996,"style":1061},[1060],[31,120998,121000],{"className":120999},[1065,1066,1067,1068],[31,121001,52065],{"className":121002,"style":52166},[75,76,1068],[31,121004,121006,121009],{"className":121005},[75],[31,121007,2549],{"className":121008,"style":19759},[75,76],[31,121010,121012],{"className":121011},[1039],[31,121013,121015,121044],{"className":121014},[1043,1044],[31,121016,121018,121041],{"className":121017},[1048],[31,121019,121021],{"className":121020,"style":1053},[1052],[31,121022,121023,121026],{"style":26726},[31,121024],{"className":121025,"style":1061},[1060],[31,121027,121029],{"className":121028},[1065,1066,1067,1068],[31,121030,121032,121035,121038],{"className":121031},[75,1068],[31,121033,12271],{"className":121034},[75,76,1068],[31,121036,434],{"className":121037},[124,1068],[31,121039,52065],{"className":121040,"style":52166},[75,76,1068],[31,121042,1076],{"className":121043},[1075],[31,121045,121047],{"className":121046},[1048],[31,121048,121050],{"className":121049,"style":25082},[1052],[31,121051],{},"。这个形式保证了 ",[31,121054,121056,121097],{"className":121055,"translate":35},[34],[31,121057,121059],{"className":121058},[39],[41,121060,121061],{"xmlns":43},[45,121062,121063,121095],{},[48,121064,121065,121071,121073,121079,121081,121083],{},[1004,121066,121067,121069],{},[51,121068,15297],{},[51,121070,12271],{},[95,121072,223],{},[1004,121074,121075,121077],{},[51,121076,2549],{},[51,121078,12271],{},[95,121080,434],{},[51,121082,53298],{},[1004,121084,121085,121087],{},[51,121086,15297],{},[48,121088,121089,121091,121093],{},[51,121090,12271],{},[95,121092,434],{},[91,121094,205],{},[55,121096,60121],{"encoding":57},[31,121098,121100,121155,121210],{"className":121099,"ariaHidden":62},[61],[31,121101,121103,121106,121146,121149,121152],{"className":121102},[66],[31,121104],{"className":121105,"style":5230},[70],[31,121107,121109,121112],{"className":121108},[75],[31,121110,15297],{"className":121111},[75,76],[31,121113,121115],{"className":121114},[1039],[31,121116,121118,121138],{"className":121117},[1043,1044],[31,121119,121121,121135],{"className":121120},[1048],[31,121122,121124],{"className":121123,"style":19220},[1052],[31,121125,121126,121129],{"style":2807},[31,121127],{"className":121128,"style":1061},[1060],[31,121130,121132],{"className":121131},[1065,1066,1067,1068],[31,121133,12271],{"className":121134},[75,76,1068],[31,121136,1076],{"className":121137},[1075],[31,121139,121141],{"className":121140},[1048],[31,121142,121144],{"className":121143,"style":2050},[1052],[31,121145],{},[31,121147],{"className":121148,"style":293},[119],[31,121150,223],{"className":121151},[297],[31,121153],{"className":121154,"style":293},[119],[31,121156,121158,121161,121201,121204,121207],{"className":121157},[66],[31,121159],{"className":121160,"style":5426},[70],[31,121162,121164,121167],{"className":121163},[75],[31,121165,2549],{"className":121166,"style":19759},[75,76],[31,121168,121170],{"className":121169},[1039],[31,121171,121173,121193],{"className":121172},[1043,1044],[31,121174,121176,121190],{"className":121175},[1048],[31,121177,121179],{"className":121178,"style":19220},[1052],[31,121180,121181,121184],{"style":26726},[31,121182],{"className":121183,"style":1061},[1060],[31,121185,121187],{"className":121186},[1065,1066,1067,1068],[31,121188,12271],{"className":121189},[75,76,1068],[31,121191,1076],{"className":121192},[1075],[31,121194,121196],{"className":121195},[1048],[31,121197,121199],{"className":121198,"style":2050},[1052],[31,121200],{},[31,121202],{"className":121203,"style":120},[119],[31,121205,434],{"className":121206},[124],[31,121208],{"className":121209,"style":120},[119],[31,121211,121213,121216,121219],{"className":121212},[66],[31,121214],{"className":121215,"style":60241},[70],[31,121217,53298],{"className":121218,"style":24157},[75,76],[31,121220,121222,121225],{"className":121221},[75],[31,121223,15297],{"className":121224},[75,76],[31,121226,121228],{"className":121227},[1039],[31,121229,121231,121260],{"className":121230},[1043,1044],[31,121232,121234,121257],{"className":121233},[1048],[31,121235,121237],{"className":121236,"style":2028},[1052],[31,121238,121239,121242],{"style":2807},[31,121240],{"className":121241,"style":1061},[1060],[31,121243,121245],{"className":121244},[1065,1066,1067,1068],[31,121246,121248,121251,121254],{"className":121247},[75,1068],[31,121249,12271],{"className":121250},[75,76,1068],[31,121252,434],{"className":121253},[124,1068],[31,121255,205],{"className":121256},[75,1068],[31,121258,1076],{"className":121259},[1075],[31,121261,121263],{"className":121262},[1048],[31,121264,121266],{"className":121265,"style":25082},[1052],[31,121267],{},"。若效用是历史整体的非线性函数，例如某些非可加偏好，那么未来就不能被压缩成一个标量值函数。双曲折扣是一个典型例子。在双曲折扣下，今天的偏好与明天的偏好可能不一致，因而不存在一个与时间无关的递归值函数。",[11,121270,121271,121272,121341],{},"第二，状态必须满足马尔可夫性。若未来的转移依赖于更早的历史，当前状态 ",[31,121273,121275,121292],{"className":121274,"translate":35},[34],[31,121276,121278],{"className":121277},[39],[41,121279,121280],{"xmlns":43},[45,121281,121282,121290],{},[48,121283,121284],{},[1004,121285,121286,121288],{},[51,121287,14551],{},[51,121289,12271],{},[55,121291,56237],{"encoding":57},[31,121293,121295],{"className":121294,"ariaHidden":62},[61],[31,121296,121298,121301],{"className":121297},[66],[31,121299],{"className":121300,"style":7090},[70],[31,121302,121304,121307],{"className":121303},[75],[31,121305,14551],{"className":121306},[75,76],[31,121308,121310],{"className":121309},[1039],[31,121311,121313,121333],{"className":121312},[1043,1044],[31,121314,121316,121330],{"className":121315},[1048],[31,121317,121319],{"className":121318,"style":19220},[1052],[31,121320,121321,121324],{"style":2807},[31,121322],{"className":121323,"style":1061},[1060],[31,121325,121327],{"className":121326},[1065,1066,1067,1068],[31,121328,12271],{"className":121329},[75,76,1068],[31,121331,1076],{"className":121332},[1075],[31,121334,121336],{"className":121335},[1048],[31,121337,121339],{"className":121338,"style":2050},[1052],[31,121340],{}," 就不是充分的统计量。此时可以把状态扩充为包含必要历史信息的新状态，例如部分可观测问题中的信念状态。在扩充后的状态空间上，贝尔曼方程仍然可以保持相同的形式。",[11,121343,121344,121345,121395,121396,121446],{},"第三，回报必须有界，或者折扣必须足够强。有限期问题即使 ",[31,121346,121348,121365],{"className":121347,"translate":35},[34],[31,121349,121351],{"className":121350},[39],[41,121352,121353],{"xmlns":43},[45,121354,121355,121363],{},[48,121356,121357,121359,121361],{},[51,121358,53298],{},[95,121360,223],{},[91,121362,205],{},[55,121364,81593],{"encoding":57},[31,121366,121368,121386],{"className":121367,"ariaHidden":62},[61],[31,121369,121371,121374,121377,121380,121383],{"className":121370},[66],[31,121372],{"className":121373,"style":865},[70],[31,121375,53298],{"className":121376,"style":24157},[75,76],[31,121378],{"className":121379,"style":293},[119],[31,121381,223],{"className":121382},[297],[31,121384],{"className":121385,"style":293},[119],[31,121387,121389,121392],{"className":121388},[66],[31,121390],{"className":121391,"style":326},[70],[31,121393,205],{"className":121394},[75]," 也可以有有界回报，因为求和的项数是有限的。无限期问题则需要 ",[31,121397,121399,121416],{"className":121398,"translate":35},[34],[31,121400,121402],{"className":121401},[39],[41,121403,121404],{"xmlns":43},[45,121405,121406,121414],{},[48,121407,121408,121410,121412],{},[51,121409,53298],{},[95,121411,9758],{},[91,121413,205],{},[55,121415,69926],{"encoding":57},[31,121417,121419,121437],{"className":121418,"ariaHidden":62},[61],[31,121420,121422,121425,121428,121431,121434],{"className":121421},[66],[31,121423],{"className":121424,"style":53323},[70],[31,121426,53298],{"className":121427,"style":24157},[75,76],[31,121429],{"className":121430,"style":293},[119],[31,121432,9758],{"className":121433},[297],[31,121435],{"className":121436,"style":293},[119],[31,121438,121440,121443],{"className":121439},[66],[31,121441],{"className":121442,"style":326},[70],[31,121444,205],{"className":121445},[75],"，或者采用平均奖励、吸收态等替代准则。否则，值函数可能发散，或者贝尔曼方程的解不唯一。",[11,121448,121449],{},"这三个条件共同说明，最优子结构是一个有明确适用范围的定理，而不是一条无条件成立的哲学原则。",[18,121451,121453],{"id":121452},"七边界与常见误区","七、边界与常见误区",[52860,121455,121457],{"id":121456},"_71-连续状态空间","7.1 连续状态空间",[11,121459,121460,121461,121504],{},"在标准的可测性假设下，连续状态空间中的 ",[31,121462,121464,121483],{"className":121463,"translate":35},[34],[31,121465,121467],{"className":121466},[39],[41,121468,121469],{"xmlns":43},[45,121470,121471,121481],{},[48,121472,121473,121475,121477,121479],{},[51,121474,1122],{},[95,121476,211],{"stretchy":210},[51,121478,21582],{},[95,121480,220],{"stretchy":210},[55,121482,73997],{"encoding":57},[31,121484,121486],{"className":121485,"ariaHidden":62},[61],[31,121487,121489,121492,121495,121498,121501],{"className":121488},[66],[31,121490],{"className":121491,"style":260},[70],[31,121493,1122],{"className":121494,"style":1137},[75,76],[31,121496,211],{"className":121497},[264],[31,121499,21582],{"className":121500,"style":21697},[75,76],[31,121502,220],{"className":121503},[289]," 仍然是 Banach 空间，压缩映射的论证也仍然成立。因此，贝尔曼期望方程的唯一有界解依然存在，贝尔曼最优算子也依然有唯一有界不动点，值迭代的几何收敛性同样成立。在适当的可测选择条件下，该不动点等于最优值函数。",[11,121506,121507,121508,121566],{},"变化主要发生在 ",[31,121509,121511,121528],{"className":121510,"translate":35},[34],[31,121512,121514],{"className":121513},[39],[41,121515,121516],{"xmlns":43},[45,121517,121518,121526],{},[48,121519,121520],{},[1180,121521,121522,121524],{},[51,121523,919],{"mathvariant":925},[95,121525,30648],{},[55,121527,74501],{"encoding":57},[31,121529,121531],{"className":121530,"ariaHidden":62},[61],[31,121532,121534,121537],{"className":121533},[66],[31,121535],{"className":121536,"style":30719},[70],[31,121538,121540,121543],{"className":121539},[75],[31,121541,919],{"className":121542,"style":64444},[75,968],[31,121544,121546],{"className":121545},[1039],[31,121547,121549],{"className":121548},[1043],[31,121550,121552],{"className":121551},[1048],[31,121553,121555],{"className":121554,"style":30719},[1052],[31,121556,121557,121560],{"style":3631},[31,121558],{"className":121559,"style":1061},[1060],[31,121561,121563],{"className":121562},[1065,1066,1067,1068],[31,121564,30648],{"className":121565},[124,1068]," 的定义上。此时最大值未必取到，应将其写成",[31,121568,121570],{"className":121569,"translate":35},[411],[31,121571,121573,121723],{"className":121572,"translate":35},[34],[31,121574,121576],{"className":121575},[39],[41,121577,121578],{"xmlns":43,"display":420},[45,121579,121580,121720],{},[5606,121581,121582,121658],{"rowspacing":5608,"columnalign":5609,"columnspacing":5610},[5612,121583,121584,121608],{},[5615,121585,121586],{},[2071,121587,121588],{"scriptlevel":1977,"displaystyle":62},[48,121589,121590,121592,121598,121600,121602,121604,121606],{},[95,121591,211],{"stretchy":210},[1180,121593,121594,121596],{},[51,121595,919],{"mathvariant":925},[95,121597,30648],{},[51,121599,43481],{},[95,121601,220],{"stretchy":210},[95,121603,211],{"stretchy":210},[51,121605,14551],{},[95,121607,220],{"stretchy":210},[5615,121609,121610],{},[2071,121611,121612],{"scriptlevel":1977,"displaystyle":62},[48,121613,121614,121616,121618,121628,121630,121636,121638,121640,121642,121644,121646,121648,121650,121656],{},[48,121615],{},[95,121617,223],{},[1531,121619,121620,121626],{},[48,121621,121622,121624],{},[51,121623,10792],{},[95,121625,1540],{},[51,121627,5813],{},[95,121629,2512],{"fence":210,"stretchy":62,"minsize":115662,"maxsize":115662},[1004,121631,121632,121634],{},[95,121633,1409],{},[51,121635,21582],{},[95,121637,15175],{"fence":210,"stretchy":62,"minsize":2511,"maxsize":2511},[51,121639,2549],{},[95,121641,211],{"stretchy":210},[51,121643,14551],{},[95,121645,1210],{"separator":62},[51,121647,5813],{},[95,121649,1210],{"separator":62},[1180,121651,121652,121654],{},[51,121653,14551],{},[95,121655,13691],{"mathvariant":494,"lspace":5610,"rspace":5610},[95,121657,220],{"stretchy":210},[5612,121659,121660,121666],{},[5615,121661,121662],{},[2071,121663,121664],{"scriptlevel":1977,"displaystyle":62},[48,121665],{},[5615,121667,121668],{},[2071,121669,121670],{"scriptlevel":1977,"displaystyle":62},[48,121671,121672,121674,121676,121678,121680,121682,121684,121690,121692,121694,121696,121698,121700,121706,121708,121710,121712,121714,121716,121718],{},[48,121673],{},[119,121675],{"width":20286},[95,121677,434],{},[51,121679,53298],{},[51,121681,43481],{},[95,121683,211],{"stretchy":210},[1180,121685,121686,121688],{},[51,121687,14551],{},[95,121689,13691],{"mathvariant":494,"lspace":5610,"rspace":5610},[95,121691,220],{"stretchy":210},[95,121693,15184],{"fence":210,"stretchy":62,"minsize":2511,"maxsize":2511},[51,121695,2521],{},[95,121697,211],{"stretchy":210},[51,121699,1435],{"mathvariant":494},[1180,121701,121702,121704],{},[51,121703,14551],{},[95,121705,13691],{"mathvariant":494,"lspace":5610,"rspace":5610},[95,121707,12680],{},[51,121709,14551],{},[95,121711,1210],{"separator":62},[51,121713,5813],{},[95,121715,220],{"stretchy":210},[95,121717,2604],{"fence":210,"stretchy":62,"minsize":115662,"maxsize":115662},[51,121719,495],{"mathvariant":494},[55,121721,121722],{"encoding":57},"\\begin{aligned}\n(\\mathcal{T}^*V)(s)\n&=\\sup_{a}\\Bigg\\{\\int_S \\Big[r(s,a,s')\\\\\n&\\qquad+\\gamma V(s')\\Big]P(\\mathrm{d}s'\\mid s,a)\\Bigg\\}.\n\\end{aligned}",[31,121724,121726],{"className":121725,"ariaHidden":62},[61],[31,121727,121729,121733],{"className":121728},[66],[31,121730],{"className":121731,"style":121732},[70],"height:6.3001em;vertical-align:-2.9em;",[31,121734,121736],{"className":121735},[75],[31,121737,121739,121829],{"className":121738},[5606],[31,121740,121742],{"className":121741},[5867],[31,121743,121745,121820],{"className":121744},[1043,1044],[31,121746,121748,121817],{"className":121747},[1048],[31,121749,121752,121809],{"className":121750,"style":121751},[1052],"height:3.4em;",[31,121753,121755,121759],{"style":121754},"top:-5.4em;",[31,121756],{"className":121757,"style":121758},[1060],"height:3.75em;",[31,121760,121762,121765,121794,121797,121800,121803,121806],{"className":121761},[75],[31,121763,211],{"className":121764},[264],[31,121766,121768,121771],{"className":121767},[75],[31,121769,919],{"className":121770,"style":64444},[75,968],[31,121772,121774],{"className":121773},[1039],[31,121775,121777],{"className":121776},[1043],[31,121778,121780],{"className":121779},[1048],[31,121781,121783],{"className":121782,"style":74855},[1052],[31,121784,121785,121788],{"style":1330},[31,121786],{"className":121787,"style":1061},[1060],[31,121789,121791],{"className":121790},[1065,1066,1067,1068],[31,121792,30648],{"className":121793},[124,1068],[31,121795,43481],{"className":121796,"style":120},[75,76],[31,121798,220],{"className":121799},[289],[31,121801,211],{"className":121802},[264],[31,121804,14551],{"className":121805},[75,76],[31,121807,220],{"className":121808},[289],[31,121810,121811,121814],{"style":58497},[31,121812],{"className":121813,"style":121758},[1060],[31,121815],{"className":121816},[75],[31,121818,1076],{"className":121819},[1075],[31,121821,121823],{"className":121822},[1048],[31,121824,121827],{"className":121825,"style":121826},[1052],"height:2.9em;",[31,121828],{},[31,121830,121832],{"className":121831},[6062],[31,121833,121835,122178],{"className":121834},[1043,1044],[31,121836,121838,122175],{"className":121837},[1048],[31,121839,121841,122028],{"className":121840,"style":121751},[1052],[31,121842,121843,121846],{"style":121754},[31,121844],{"className":121845,"style":121758},[1060],[31,121847,121849,121852,121855,121858,121861,121908,121911,121917,121920,121960,121963,121969,121972,121975,121978,121981,121984,121987,121990,121993,122025],{"className":121848},[75],[31,121850],{"className":121851},[75],[31,121853],{"className":121854,"style":293},[119],[31,121856,223],{"className":121857},[297],[31,121859],{"className":121860,"style":293},[119],[31,121862,121864],{"className":121863},[1460,1620],[31,121865,121867,121900],{"className":121866},[1043,1044],[31,121868,121870,121897],{"className":121869},[1048],[31,121871,121873,121887],{"className":121872,"style":71},[1052],[31,121874,121875,121878],{"style":89312},[31,121876],{"className":121877,"style":1637},[1060],[31,121879,121881],{"className":121880},[1065,1066,1067,1068],[31,121882,121884],{"className":121883},[75,1068],[31,121885,5813],{"className":121886},[75,76,1068],[31,121888,121889,121892],{"style":1649},[31,121890],{"className":121891,"style":1637},[1060],[31,121893,121894],{},[31,121895,10792],{"className":121896},[1460],[31,121898,1076],{"className":121899},[1075],[31,121901,121903],{"className":121902},[1048],[31,121904,121906],{"className":121905,"style":89344},[1052],[31,121907],{},[31,121909],{"className":121910,"style":1467},[119],[31,121912,121914],{"className":121913},[75],[31,121915,2512],{"className":121916},[1307,116039],[31,121918],{"className":121919,"style":1467},[119],[31,121921,121923,121926],{"className":121922},[1460],[31,121924,1409],{"className":121925,"style":1688},[1460,1461,1687],[31,121927,121929],{"className":121928},[1039],[31,121930,121932,121952],{"className":121931},[1043,1044],[31,121933,121935,121949],{"className":121934},[1048],[31,121936,121938],{"className":121937,"style":1701},[1052],[31,121939,121940,121943],{"style":1704},[31,121941],{"className":121942,"style":1061},[1060],[31,121944,121946],{"className":121945},[1065,1066,1067,1068],[31,121947,21582],{"className":121948,"style":21697},[75,76,1068],[31,121950,1076],{"className":121951},[1075],[31,121953,121955],{"className":121954},[1048],[31,121956,121958],{"className":121957,"style":1726},[1052],[31,121959],{},[31,121961],{"className":121962,"style":1467},[119],[31,121964,121966],{"className":121965},[75],[31,121967,15175],{"className":121968},[1307,2662],[31,121970,2549],{"className":121971,"style":19759},[75,76],[31,121973,211],{"className":121974},[264],[31,121976,14551],{"className":121977},[75,76],[31,121979,1210],{"className":121980},[1391],[31,121982],{"className":121983,"style":1467},[119],[31,121985,5813],{"className":121986},[75,76],[31,121988,1210],{"className":121989},[1391],[31,121991],{"className":121992,"style":1467},[119],[31,121994,121996,121999],{"className":121995},[75],[31,121997,14551],{"className":121998},[75,76],[31,122000,122002],{"className":122001},[1039],[31,122003,122005],{"className":122004},[1043],[31,122006,122008],{"className":122007},[1048],[31,122009,122011],{"className":122010,"style":14017},[1052],[31,122012,122013,122016],{"style":1330},[31,122014],{"className":122015,"style":1061},[1060],[31,122017,122019],{"className":122018},[1065,1066,1067,1068],[31,122020,122022],{"className":122021},[75,1068],[31,122023,13691],{"className":122024},[75,1068],[31,122026,220],{"className":122027},[289],[31,122029,122030,122033],{"style":58497},[31,122031],{"className":122032,"style":121758},[1060],[31,122034,122036,122039,122042,122045,122048,122051,122054,122057,122060,122092,122095,122101,122104,122107,122110,122142,122145,122148,122151,122154,122157,122160,122163,122166,122172],{"className":122035},[75],[31,122037],{"className":122038},[75],[31,122040],{"className":122041,"style":20525},[119],[31,122043],{"className":122044,"style":120},[119],[31,122046,434],{"className":122047},[124],[31,122049],{"className":122050,"style":120},[119],[31,122052,53298],{"className":122053,"style":24157},[75,76],[31,122055,43481],{"className":122056,"style":120},[75,76],[31,122058,211],{"className":122059},[264],[31,122061,122063,122066],{"className":122062},[75],[31,122064,14551],{"className":122065},[75,76],[31,122067,122069],{"className":122068},[1039],[31,122070,122072],{"className":122071},[1043],[31,122073,122075],{"className":122074},[1048],[31,122076,122078],{"className":122077,"style":14017},[1052],[31,122079,122080,122083],{"style":1330},[31,122081],{"className":122082,"style":1061},[1060],[31,122084,122086],{"className":122085},[1065,1066,1067,1068],[31,122087,122089],{"className":122088},[75,1068],[31,122090,13691],{"className":122091},[75,1068],[31,122093,220],{"className":122094},[289],[31,122096,122098],{"className":122097},[75],[31,122099,15184],{"className":122100},[1307,2662],[31,122102,2521],{"className":122103,"style":949},[75,76],[31,122105,211],{"className":122106},[264],[31,122108,1435],{"className":122109},[75,2745],[31,122111,122113,122116],{"className":122112},[75],[31,122114,14551],{"className":122115},[75,76],[31,122117,122119],{"className":122118},[1039],[31,122120,122122],{"className":122121},[1043],[31,122123,122125],{"className":122124},[1048],[31,122126,122128],{"className":122127,"style":14017},[1052],[31,122129,122130,122133],{"style":1330},[31,122131],{"className":122132,"style":1061},[1060],[31,122134,122136],{"className":122135},[1065,1066,1067,1068],[31,122137,122139],{"className":122138},[75,1068],[31,122140,13691],{"className":122141},[75,1068],[31,122143],{"className":122144,"style":293},[119],[31,122146,12680],{"className":122147},[297],[31,122149],{"className":122150,"style":293},[119],[31,122152,14551],{"className":122153},[75,76],[31,122155,1210],{"className":122156},[1391],[31,122158],{"className":122159,"style":1467},[119],[31,122161,5813],{"className":122162},[75,76],[31,122164,220],{"className":122165},[289],[31,122167,122169],{"className":122168},[75],[31,122170,2604],{"className":122171},[1307,116039],[31,122173,495],{"className":122174},[75],[31,122176,1076],{"className":122177},[1075],[31,122179,122181],{"className":122180},[1048],[31,122182,122184],{"className":122183,"style":121826},[1052],[31,122185],{},[11,122187,122188,122189,122247,122248,122276],{},"压缩性证明中的「最大值保持逐点序」可以替换为「上确界保持逐点序」，因此 ",[31,122190,122192,122209],{"className":122191,"translate":35},[34],[31,122193,122195],{"className":122194},[39],[41,122196,122197],{"xmlns":43},[45,122198,122199,122207],{},[48,122200,122201],{},[1180,122202,122203,122205],{},[51,122204,919],{"mathvariant":925},[95,122206,30648],{},[55,122208,74501],{"encoding":57},[31,122210,122212],{"className":122211,"ariaHidden":62},[61],[31,122213,122215,122218],{"className":122214},[66],[31,122216],{"className":122217,"style":30719},[70],[31,122219,122221,122224],{"className":122220},[75],[31,122222,919],{"className":122223,"style":64444},[75,968],[31,122225,122227],{"className":122226},[1039],[31,122228,122230],{"className":122229},[1043],[31,122231,122233],{"className":122232},[1048],[31,122234,122236],{"className":122235,"style":30719},[1052],[31,122237,122238,122241],{"style":3631},[31,122239],{"className":122240,"style":1061},[1060],[31,122242,122244],{"className":122243},[1065,1066,1067,1068],[31,122245,30648],{"className":122246},[124,1068]," 仍然是 ",[31,122249,122251,122264],{"className":122250,"translate":35},[34],[31,122252,122254],{"className":122253},[39],[41,122255,122256],{"xmlns":43},[45,122257,122258,122262],{},[48,122259,122260],{},[51,122261,53298],{},[55,122263,79191],{"encoding":57},[31,122265,122267],{"className":122266,"ariaHidden":62},[61],[31,122268,122270,122273],{"className":122269},[66],[31,122271],{"className":122272,"style":865},[70],[31,122274,53298],{"className":122275,"style":24157},[75,76],"-压缩映射。唯一不动点仍然存在。",[11,122278,122279,122280,122338,122339,122435],{},"但是，最优策略的存在性需要额外条件。即使 ",[31,122281,122283,122300],{"className":122282,"translate":35},[34],[31,122284,122286],{"className":122285},[39],[41,122287,122288],{"xmlns":43},[45,122289,122290,122298],{},[48,122291,122292],{},[1180,122293,122294,122296],{},[51,122295,919],{"mathvariant":925},[95,122297,30648],{},[55,122299,74501],{"encoding":57},[31,122301,122303],{"className":122302,"ariaHidden":62},[61],[31,122304,122306,122309],{"className":122305},[66],[31,122307],{"className":122308,"style":30719},[70],[31,122310,122312,122315],{"className":122311},[75],[31,122313,919],{"className":122314,"style":64444},[75,968],[31,122316,122318],{"className":122317},[1039],[31,122319,122321],{"className":122320},[1043],[31,122322,122324],{"className":122323},[1048],[31,122325,122327],{"className":122326,"style":30719},[1052],[31,122328,122329,122332],{"style":3631},[31,122330],{"className":122331,"style":1061},[1060],[31,122333,122335],{"className":122334},[1065,1066,1067,1068],[31,122336,30648],{"className":122337},[124,1068]," 有唯一不动点，也不一定存在一个可测的映射 ",[31,122340,122342,122370],{"className":122341,"translate":35},[34],[31,122343,122345],{"className":122344},[39],[41,122346,122347],{"xmlns":43},[45,122348,122349,122367],{},[48,122350,122351,122353,122355,122361,122363,122365],{},[51,122352,14551],{},[95,122354,18339],{},[1180,122356,122357,122359],{},[51,122358,2515],{},[95,122360,30648],{},[95,122362,211],{"stretchy":210},[51,122364,14551],{},[95,122366,220],{"stretchy":210},[55,122368,122369],{"encoding":57},"s\\mapsto\\pi^*(s)",[31,122371,122373,122391],{"className":122372,"ariaHidden":62},[61],[31,122374,122376,122379,122382,122385,122388],{"className":122375},[66],[31,122377],{"className":122378,"style":18356},[70],[31,122380,14551],{"className":122381},[75,76],[31,122383],{"className":122384,"style":293},[119],[31,122386,18339],{"className":122387},[297],[31,122389],{"className":122390,"style":293},[119],[31,122392,122394,122397,122426,122429,122432],{"className":122393},[66],[31,122395],{"className":122396,"style":260},[70],[31,122398,122400,122403],{"className":122399},[75],[31,122401,2515],{"className":122402,"style":2666},[75,76],[31,122404,122406],{"className":122405},[1039],[31,122407,122409],{"className":122408},[1043],[31,122410,122412],{"className":122411},[1048],[31,122413,122415],{"className":122414,"style":30719},[1052],[31,122416,122417,122420],{"style":3631},[31,122418],{"className":122419,"style":1061},[1060],[31,122421,122423],{"className":122422},[1065,1066,1067,1068],[31,122424,30648],{"className":122425},[124,1068],[31,122427,211],{"className":122428},[264],[31,122430,14551],{"className":122431},[75,76],[31,122433,220],{"className":122434},[289]," 在每个状态都取到上确界。通常需要动作集紧、奖励与转移核连续，或者使用可测选择定理。本文的有限状态与有限动作假设避开了这些技术问题。这也是为什么在一般情形下，贝尔曼最优方程常写成上确界形式，而不是最大值形式。",[52860,122437,122439],{"id":122438},"_72-平均奖励与无折扣","7.2 平均奖励与无折扣",[11,122441,65583,122442,122492,122493,122523,122524,122552],{},[31,122443,122445,122462],{"className":122444,"translate":35},[34],[31,122446,122448],{"className":122447},[39],[41,122449,122450],{"xmlns":43},[45,122451,122452,122460],{},[48,122453,122454,122456,122458],{},[51,122455,53298],{},[95,122457,223],{},[91,122459,205],{},[55,122461,81593],{"encoding":57},[31,122463,122465,122483],{"className":122464,"ariaHidden":62},[61],[31,122466,122468,122471,122474,122477,122480],{"className":122467},[66],[31,122469],{"className":122470,"style":865},[70],[31,122472,53298],{"className":122473,"style":24157},[75,76],[31,122475],{"className":122476,"style":293},[119],[31,122478,223],{"className":122479},[297],[31,122481],{"className":122482,"style":293},[119],[31,122484,122486,122489],{"className":122485},[66],[31,122487],{"className":122488,"style":326},[70],[31,122490,205],{"className":122491},[75]," 且时间范围为无限时，折扣回报可能发散，贝尔曼方程的标准形式不再适用。此时常用的替代准则是平均奖励准则。它引入增益 ",[31,122494,122496,122511],{"className":122495,"translate":35},[34],[31,122497,122499],{"className":122498},[39],[41,122500,122501],{"xmlns":43},[45,122502,122503,122508],{},[48,122504,122505],{},[51,122506,122507],{},"ρ",[55,122509,122510],{"encoding":57},"\\rho",[31,122512,122514],{"className":122513,"ariaHidden":62},[61],[31,122515,122517,122520],{"className":122516},[66],[31,122518],{"className":122519,"style":865},[70],[31,122521,122507],{"className":122522},[75,76]," 与偏差函数 ",[31,122525,122527,122540],{"className":122526,"translate":35},[34],[31,122528,122530],{"className":122529},[39],[41,122531,122532],{"xmlns":43},[45,122533,122534,122538],{},[48,122535,122536],{},[51,122537,29501],{},[55,122539,29501],{"encoding":57},[31,122541,122543],{"className":122542,"ariaHidden":62},[61],[31,122544,122546,122549],{"className":122545},[66],[31,122547],{"className":122548,"style":1630},[70],[31,122550,29501],{"className":122551},[75,76],"，其最优方程形如",[31,122554,122556],{"className":122555,"translate":35},[411],[31,122557,122559,122695],{"className":122558,"translate":35},[34],[31,122560,122562],{"className":122561},[39],[41,122563,122564],{"xmlns":43,"display":420},[45,122565,122566,122692],{},[5606,122567,122568,122626],{"rowspacing":5608,"columnalign":5609,"columnspacing":5610},[5612,122569,122570,122588],{},[5615,122571,122572],{},[2071,122573,122574],{"scriptlevel":1977,"displaystyle":62},[48,122575,122576,122578,122580,122582,122584,122586],{},[51,122577,122507],{},[95,122579,434],{},[51,122581,29501],{},[95,122583,211],{"stretchy":210},[51,122585,14551],{},[95,122587,220],{"stretchy":210},[5615,122589,122590],{},[2071,122591,122592],{"scriptlevel":1977,"displaystyle":62},[48,122593,122594,122596,122598,122608,122610,122616,122618,122620,122622,122624],{},[48,122595],{},[95,122597,223],{},[1531,122599,122600,122606],{},[48,122601,122602,122604],{},[51,122603,10154],{},[95,122605,1540],{},[51,122607,5813],{},[95,122609,2512],{"fence":210,"stretchy":62,"minsize":115662,"maxsize":115662},[42405,122611,122612,122614],{"accent":62},[51,122613,2549],{},[95,122615,92273],{},[95,122617,211],{"stretchy":210},[51,122619,14551],{},[95,122621,1210],{"separator":62},[51,122623,5813],{},[95,122625,220],{"stretchy":210},[5612,122627,122628,122634],{},[5615,122629,122630],{},[2071,122631,122632],{"scriptlevel":1977,"displaystyle":62},[48,122633],{},[5615,122635,122636],{},[2071,122637,122638],{"scriptlevel":1977,"displaystyle":62},[48,122639,122640,122642,122644,122646,122656,122658,122660,122666,122668,122670,122672,122674,122676,122678,122680,122686,122688,122690],{},[48,122641],{},[119,122643],{"width":20286},[95,122645,434],{},[1531,122647,122648,122650],{},[95,122649,5625],{},[1180,122651,122652,122654],{},[51,122653,14551],{},[95,122655,13691],{"mathvariant":494,"lspace":5610,"rspace":5610},[51,122657,2521],{},[95,122659,211],{"stretchy":210},[1180,122661,122662,122664],{},[51,122663,14551],{},[95,122665,13691],{"mathvariant":494,"lspace":5610,"rspace":5610},[95,122667,12680],{},[51,122669,14551],{},[95,122671,1210],{"separator":62},[51,122673,5813],{},[95,122675,220],{"stretchy":210},[51,122677,29501],{},[95,122679,211],{"stretchy":210},[1180,122681,122682,122684],{},[51,122683,14551],{},[95,122685,13691],{"mathvariant":494,"lspace":5610,"rspace":5610},[95,122687,220],{"stretchy":210},[95,122689,2604],{"fence":210,"stretchy":62,"minsize":115662,"maxsize":115662},[95,122691,1210],{"separator":62},[55,122693,122694],{"encoding":57},"\\begin{aligned}\n\\rho+h(s)&=\\max_{a}\\Bigg\\{\\bar r(s,a)\\\\\n&\\qquad+\\sum_{s'}P(s'\\mid s,a)h(s')\\Bigg\\},\n\\end{aligned}",[31,122696,122698],{"className":122697,"ariaHidden":62},[61],[31,122699,122701,122705],{"className":122700},[66],[31,122702],{"className":122703,"style":122704},[70],"height:6.344em;vertical-align:-2.922em;",[31,122706,122708],{"className":122707},[75],[31,122709,122711,122777],{"className":122710},[5606],[31,122712,122714],{"className":122713},[5867],[31,122715,122717,122768],{"className":122716},[1043,1044],[31,122718,122720,122765],{"className":122719},[1048],[31,122721,122724,122757],{"className":122722,"style":122723},[1052],"height:3.422em;",[31,122725,122727,122730],{"style":122726},"top:-5.422em;",[31,122728],{"className":122729,"style":121758},[1060],[31,122731,122733,122736,122739,122742,122745,122748,122751,122754],{"className":122732},[75],[31,122734,122507],{"className":122735},[75,76],[31,122737],{"className":122738,"style":120},[119],[31,122740,434],{"className":122741},[124],[31,122743],{"className":122744,"style":120},[119],[31,122746,29501],{"className":122747},[75,76],[31,122749,211],{"className":122750},[264],[31,122752,14551],{"className":122753},[75,76],[31,122755,220],{"className":122756},[289],[31,122758,122759,122762],{"style":63756},[31,122760],{"className":122761,"style":121758},[1060],[31,122763],{"className":122764},[75],[31,122766,1076],{"className":122767},[1075],[31,122769,122771],{"className":122770},[1048],[31,122772,122775],{"className":122773,"style":122774},[1052],"height:2.922em;",[31,122776],{},[31,122778,122780],{"className":122779},[6062],[31,122781,122783,123133],{"className":122782},[1043,1044],[31,122784,122786,123130],{"className":122785},[1048],[31,122787,122789,122916],{"className":122788,"style":122723},[1052],[31,122790,122791,122794],{"style":122726},[31,122792],{"className":122793,"style":121758},[1060],[31,122795,122797,122800,122803,122806,122809,122856,122859,122865,122898,122901,122904,122907,122910,122913],{"className":122796},[75],[31,122798],{"className":122799},[75],[31,122801],{"className":122802,"style":293},[119],[31,122804,223],{"className":122805},[297],[31,122807],{"className":122808,"style":293},[119],[31,122810,122812],{"className":122811},[1460,1620],[31,122813,122815,122848],{"className":122814},[1043,1044],[31,122816,122818,122845],{"className":122817},[1048],[31,122819,122821,122835],{"className":122820,"style":71},[1052],[31,122822,122823,122826],{"style":37361},[31,122824],{"className":122825,"style":1637},[1060],[31,122827,122829],{"className":122828},[1065,1066,1067,1068],[31,122830,122832],{"className":122831},[75,1068],[31,122833,5813],{"className":122834},[75,76,1068],[31,122836,122837,122840],{"style":1649},[31,122838],{"className":122839,"style":1637},[1060],[31,122841,122842],{},[31,122843,10154],{"className":122844},[1460],[31,122846,1076],{"className":122847},[1075],[31,122849,122851],{"className":122850},[1048],[31,122852,122854],{"className":122853,"style":37390},[1052],[31,122855],{},[31,122857],{"className":122858,"style":1467},[119],[31,122860,122862],{"className":122861},[75],[31,122863,2512],{"className":122864},[1307,116039],[31,122866,122868],{"className":122867},[75,42523],[31,122869,122871],{"className":122870},[1043],[31,122872,122874],{"className":122873},[1048],[31,122875,122878,122886],{"className":122876,"style":122877},[1052],"height:0.5678em;",[31,122879,122880,122883],{"style":1649},[31,122881],{"className":122882,"style":1637},[1060],[31,122884,2549],{"className":122885,"style":19759},[75,76],[31,122887,122888,122891],{"style":1649},[31,122889],{"className":122890,"style":1637},[1060],[31,122892,122895],{"className":122893,"style":122894},[42549],"left:-0.1944em;",[31,122896,92273],{"className":122897},[75],[31,122899,211],{"className":122900},[264],[31,122902,14551],{"className":122903},[75,76],[31,122905,1210],{"className":122906},[1391],[31,122908],{"className":122909,"style":1467},[119],[31,122911,5813],{"className":122912},[75,76],[31,122914,220],{"className":122915},[289],[31,122917,122918,122921],{"style":63756},[31,122919],{"className":122920,"style":121758},[1060],[31,122922,122924,122927,122930,122933,122936,122939,123015,123018,123021,123024,123056,123059,123062,123065,123068,123071,123074,123077,123080,123083,123086,123118,123121,123127],{"className":122923},[75],[31,122925],{"className":122926},[75],[31,122928],{"className":122929,"style":20525},[119],[31,122931],{"className":122932,"style":120},[119],[31,122934,434],{"className":122935},[124],[31,122937],{"className":122938,"style":120},[119],[31,122940,122942],{"className":122941},[1460,1620],[31,122943,122945,123007],{"className":122944},[1043,1044],[31,122946,122948,123004],{"className":122947},[1048],[31,122949,122951,122994],{"className":122950,"style":5900},[1052],[31,122952,122953,122956],{"style":59756},[31,122954],{"className":122955,"style":5907},[1060],[31,122957,122959],{"className":122958},[1065,1066,1067,1068],[31,122960,122962],{"className":122961},[75,1068],[31,122963,122965,122968],{"className":122964},[75,1068],[31,122966,14551],{"className":122967},[75,76,1068],[31,122969,122971],{"className":122970},[1039],[31,122972,122974],{"className":122973},[1043],[31,122975,122977],{"className":122976},[1048],[31,122978,122980],{"className":122979,"style":31478},[1052],[31,122981,122982,122985],{"style":31481},[31,122983],{"className":122984,"style":11769},[1060],[31,122986,122988],{"className":122987},[1065,11773,1308,1068],[31,122989,122991],{"className":122990},[75,1068],[31,122992,13691],{"className":122993},[75,1068],[31,122995,122996,122999],{"style":5919},[31,122997],{"className":122998,"style":5907},[1060],[31,123000,123001],{},[31,123002,5625],{"className":123003},[1460,1461,1687],[31,123005,1076],{"className":123006},[1075],[31,123008,123010],{"className":123009},[1048],[31,123011,123013],{"className":123012,"style":59817},[1052],[31,123014],{},[31,123016],{"className":123017,"style":1467},[119],[31,123019,2521],{"className":123020,"style":949},[75,76],[31,123022,211],{"className":123023},[264],[31,123025,123027,123030],{"className":123026},[75],[31,123028,14551],{"className":123029},[75,76],[31,123031,123033],{"className":123032},[1039],[31,123034,123036],{"className":123035},[1043],[31,123037,123039],{"className":123038},[1048],[31,123040,123042],{"className":123041,"style":14017},[1052],[31,123043,123044,123047],{"style":1330},[31,123045],{"className":123046,"style":1061},[1060],[31,123048,123050],{"className":123049},[1065,1066,1067,1068],[31,123051,123053],{"className":123052},[75,1068],[31,123054,13691],{"className":123055},[75,1068],[31,123057],{"className":123058,"style":293},[119],[31,123060,12680],{"className":123061},[297],[31,123063],{"className":123064,"style":293},[119],[31,123066,14551],{"className":123067},[75,76],[31,123069,1210],{"className":123070},[1391],[31,123072],{"className":123073,"style":1467},[119],[31,123075,5813],{"className":123076},[75,76],[31,123078,220],{"className":123079},[289],[31,123081,29501],{"className":123082},[75,76],[31,123084,211],{"className":123085},[264],[31,123087,123089,123092],{"className":123088},[75],[31,123090,14551],{"className":123091},[75,76],[31,123093,123095],{"className":123094},[1039],[31,123096,123098],{"className":123097},[1043],[31,123099,123101],{"className":123100},[1048],[31,123102,123104],{"className":123103,"style":14017},[1052],[31,123105,123106,123109],{"style":1330},[31,123107],{"className":123108,"style":1061},[1060],[31,123110,123112],{"className":123111},[1065,1066,1067,1068],[31,123113,123115],{"className":123114},[75,1068],[31,123116,13691],{"className":123117},[75,1068],[31,123119,220],{"className":123120},[289],[31,123122,123124],{"className":123123},[75],[31,123125,2604],{"className":123126},[1307,116039],[31,123128,1210],{"className":123129},[1391],[31,123131,1076],{"className":123132},[1075],[31,123134,123136],{"className":123135},[1048],[31,123137,123139],{"className":123138,"style":122774},[1052],[31,123140],{},[11,123142,50599,123143,123501,123502,123530,123531,123583,123584,123634],{},[31,123144,123146,123226],{"className":123145,"translate":35},[34],[31,123147,123149],{"className":123148},[39],[41,123150,123151],{"xmlns":43},[45,123152,123153,123223],{},[48,123154,123155,123161,123163,123165,123167,123169,123171,123173,123183,123185,123187,123193,123195,123197,123199,123201,123203,123205,123207,123209,123211,123213,123215,123221],{},[42405,123156,123157,123159],{"accent":62},[51,123158,2549],{},[95,123160,92273],{},[95,123162,211],{"stretchy":210},[51,123164,14551],{},[95,123166,1210],{"separator":62},[51,123168,5813],{},[95,123170,220],{"stretchy":210},[95,123172,223],{},[1004,123174,123175,123177],{},[95,123176,5625],{},[1180,123178,123179,123181],{},[51,123180,14551],{},[95,123182,13691],{"mathvariant":494,"lspace":5610,"rspace":5610},[51,123184,2521],{},[95,123186,211],{"stretchy":210},[1180,123188,123189,123191],{},[51,123190,14551],{},[95,123192,13691],{"mathvariant":494,"lspace":5610,"rspace":5610},[95,123194,12680],{},[51,123196,14551],{},[95,123198,1210],{"separator":62},[51,123200,5813],{},[95,123202,220],{"stretchy":210},[51,123204,2549],{},[95,123206,211],{"stretchy":210},[51,123208,14551],{},[95,123210,1210],{"separator":62},[51,123212,5813],{},[95,123214,1210],{"separator":62},[1180,123216,123217,123219],{},[51,123218,14551],{},[95,123220,13691],{"mathvariant":494,"lspace":5610,"rspace":5610},[95,123222,220],{"stretchy":210},[55,123224,123225],{"encoding":57},"\\bar r(s,a)=\\sum_{s'}P(s'\\mid s,a)r(s,a,s')",[31,123227,123229,123293,123421],{"className":123228,"ariaHidden":62},[61],[31,123230,123232,123235,123266,123269,123272,123275,123278,123281,123284,123287,123290],{"className":123231},[66],[31,123233],{"className":123234,"style":260},[70],[31,123236,123238],{"className":123237},[75,42523],[31,123239,123241],{"className":123240},[1043],[31,123242,123244],{"className":123243},[1048],[31,123245,123247,123255],{"className":123246,"style":122877},[1052],[31,123248,123249,123252],{"style":1649},[31,123250],{"className":123251,"style":1637},[1060],[31,123253,2549],{"className":123254,"style":19759},[75,76],[31,123256,123257,123260],{"style":1649},[31,123258],{"className":123259,"style":1637},[1060],[31,123261,123263],{"className":123262,"style":122894},[42549],[31,123264,92273],{"className":123265},[75],[31,123267,211],{"className":123268},[264],[31,123270,14551],{"className":123271},[75,76],[31,123273,1210],{"className":123274},[1391],[31,123276],{"className":123277,"style":1467},[119],[31,123279,5813],{"className":123280},[75,76],[31,123282,220],{"className":123283},[289],[31,123285],{"className":123286,"style":293},[119],[31,123288,223],{"className":123289},[297],[31,123291],{"className":123292,"style":293},[119],[31,123294,123296,123299,123371,123374,123377,123380,123412,123415,123418],{"className":123295},[66],[31,123297],{"className":123298,"style":56633},[70],[31,123300,123302,123305],{"className":123301},[1460],[31,123303,5625],{"className":123304,"style":21372},[1460,1461,1462],[31,123306,123308],{"className":123307},[1039],[31,123309,123311,123363],{"className":123310},[1043,1044],[31,123312,123314,123360],{"className":123313},[1048],[31,123315,123317],{"className":123316,"style":56652},[1052],[31,123318,123319,123322],{"style":21388},[31,123320],{"className":123321,"style":1061},[1060],[31,123323,123325],{"className":123324},[1065,1066,1067,1068],[31,123326,123328],{"className":123327},[75,1068],[31,123329,123331,123334],{"className":123330},[75,1068],[31,123332,14551],{"className":123333},[75,76,1068],[31,123335,123337],{"className":123336},[1039],[31,123338,123340],{"className":123339},[1043],[31,123341,123343],{"className":123342},[1048],[31,123344,123346],{"className":123345,"style":31478},[1052],[31,123347,123348,123351],{"style":31481},[31,123349],{"className":123350,"style":11769},[1060],[31,123352,123354],{"className":123353},[1065,11773,1308,1068],[31,123355,123357],{"className":123356},[75,1068],[31,123358,13691],{"className":123359},[75,1068],[31,123361,1076],{"className":123362},[1075],[31,123364,123366],{"className":123365},[1048],[31,123367,123369],{"className":123368,"style":21407},[1052],[31,123370],{},[31,123372],{"className":123373,"style":1467},[119],[31,123375,2521],{"className":123376,"style":949},[75,76],[31,123378,211],{"className":123379},[264],[31,123381,123383,123386],{"className":123382},[75],[31,123384,14551],{"className":123385},[75,76],[31,123387,123389],{"className":123388},[1039],[31,123390,123392],{"className":123391},[1043],[31,123393,123395],{"className":123394},[1048],[31,123396,123398],{"className":123397,"style":13732},[1052],[31,123399,123400,123403],{"style":3631},[31,123401],{"className":123402,"style":1061},[1060],[31,123404,123406],{"className":123405},[1065,1066,1067,1068],[31,123407,123409],{"className":123408},[75,1068],[31,123410,13691],{"className":123411},[75,1068],[31,123413],{"className":123414,"style":293},[119],[31,123416,12680],{"className":123417},[297],[31,123419],{"className":123420,"style":293},[119],[31,123422,123424,123427,123430,123433,123436,123439,123442,123445,123448,123451,123454,123457,123460,123463,123466,123498],{"className":123423},[66],[31,123425],{"className":123426,"style":13710},[70],[31,123428,14551],{"className":123429},[75,76],[31,123431,1210],{"className":123432},[1391],[31,123434],{"className":123435,"style":1467},[119],[31,123437,5813],{"className":123438},[75,76],[31,123440,220],{"className":123441},[289],[31,123443,2549],{"className":123444,"style":19759},[75,76],[31,123446,211],{"className":123447},[264],[31,123449,14551],{"className":123450},[75,76],[31,123452,1210],{"className":123453},[1391],[31,123455],{"className":123456,"style":1467},[119],[31,123458,5813],{"className":123459},[75,76],[31,123461,1210],{"className":123462},[1391],[31,123464],{"className":123465,"style":1467},[119],[31,123467,123469,123472],{"className":123468},[75],[31,123470,14551],{"className":123471},[75,76],[31,123473,123475],{"className":123474},[1039],[31,123476,123478],{"className":123477},[1043],[31,123479,123481],{"className":123480},[1048],[31,123482,123484],{"className":123483,"style":13732},[1052],[31,123485,123486,123489],{"style":3631},[31,123487],{"className":123488,"style":1061},[1060],[31,123490,123492],{"className":123491},[1065,1066,1067,1068],[31,123493,123495],{"className":123494},[75,1068],[31,123496,13691],{"className":123497},[75,1068],[31,123499,220],{"className":123500},[289]," 是期望即时奖励。这个方程与折扣贝尔曼方程在结构上相似，但数学性质不同。平均奖励问题通常只保证偏差函数在相差一个常数的意义下唯一，因为若 ",[31,123503,123505,123518],{"className":123504,"translate":35},[34],[31,123506,123508],{"className":123507},[39],[41,123509,123510],{"xmlns":43},[45,123511,123512,123516],{},[48,123513,123514],{},[51,123515,29501],{},[55,123517,29501],{"encoding":57},[31,123519,123521],{"className":123520,"ariaHidden":62},[61],[31,123522,123524,123527],{"className":123523},[66],[31,123525],{"className":123526,"style":1630},[70],[31,123528,29501],{"className":123529},[75,76]," 是解，则 ",[31,123532,123534,123552],{"className":123533,"translate":35},[34],[31,123535,123537],{"className":123536},[39],[41,123538,123539],{"xmlns":43},[45,123540,123541,123549],{},[48,123542,123543,123545,123547],{},[51,123544,29501],{},[95,123546,434],{},[51,123548,1412],{},[55,123550,123551],{"encoding":57},"h+c",[31,123553,123555,123574],{"className":123554,"ariaHidden":62},[61],[31,123556,123558,123562,123565,123568,123571],{"className":123557},[66],[31,123559],{"className":123560,"style":123561},[70],"height:0.7778em;vertical-align:-0.0833em;",[31,123563,29501],{"className":123564},[75,76],[31,123566],{"className":123567,"style":120},[119],[31,123569,434],{"className":123570},[124],[31,123572],{"className":123573,"style":120},[119],[31,123575,123577,123580],{"className":123576},[66],[31,123578],{"className":123579,"style":71},[70],[31,123581,1412],{"className":123582},[75,76]," 也是解。折扣因子 ",[31,123585,123587,123604],{"className":123586,"translate":35},[34],[31,123588,123590],{"className":123589},[39],[41,123591,123592],{"xmlns":43},[45,123593,123594,123602],{},[48,123595,123596,123598,123600],{},[51,123597,53298],{},[95,123599,9758],{},[91,123601,205],{},[55,123603,69926],{"encoding":57},[31,123605,123607,123625],{"className":123606,"ariaHidden":62},[61],[31,123608,123610,123613,123616,123619,123622],{"className":123609},[66],[31,123611],{"className":123612,"style":53323},[70],[31,123614,53298],{"className":123615,"style":24157},[75,76],[31,123617],{"className":123618,"style":293},[119],[31,123620,9758],{"className":123621},[297],[31,123623],{"className":123624,"style":293},[119],[31,123626,123628,123631],{"className":123627},[66],[31,123629],{"className":123630,"style":326},[70],[31,123632,205],{"className":123633},[75]," 消除了这种平移自由度，因此折扣情形的不动点是真正唯一的。",[11,123636,123637,123638,123688,123689,123739],{},"有限期问题是一个例外。即使 ",[31,123639,123641,123658],{"className":123640,"translate":35},[34],[31,123642,123644],{"className":123643},[39],[41,123645,123646],{"xmlns":43},[45,123647,123648,123656],{},[48,123649,123650,123652,123654],{},[51,123651,53298],{},[95,123653,223],{},[91,123655,205],{},[55,123657,81593],{"encoding":57},[31,123659,123661,123679],{"className":123660,"ariaHidden":62},[61],[31,123662,123664,123667,123670,123673,123676],{"className":123663},[66],[31,123665],{"className":123666,"style":865},[70],[31,123668,53298],{"className":123669,"style":24157},[75,76],[31,123671],{"className":123672,"style":293},[119],[31,123674,223],{"className":123675},[297],[31,123677],{"className":123678,"style":293},[119],[31,123680,123682,123685],{"className":123681},[66],[31,123683],{"className":123684,"style":326},[70],[31,123686,205],{"className":123687},[75],"，只要时间范围有限，回报就是有限项之和，值函数仍然有界，反向归纳仍然成立。因此，",[31,123690,123692,123709],{"className":123691,"translate":35},[34],[31,123693,123695],{"className":123694},[39],[41,123696,123697],{"xmlns":43},[45,123698,123699,123707],{},[48,123700,123701,123703,123705],{},[51,123702,53298],{},[95,123704,9758],{},[91,123706,205],{},[55,123708,69926],{"encoding":57},[31,123710,123712,123730],{"className":123711,"ariaHidden":62},[61],[31,123713,123715,123718,123721,123724,123727],{"className":123714},[66],[31,123716],{"className":123717,"style":53323},[70],[31,123719,53298],{"className":123720,"style":24157},[75,76],[31,123722],{"className":123723,"style":293},[119],[31,123725,9758],{"className":123726},[297],[31,123728],{"className":123729,"style":293},[119],[31,123731,123733,123736],{"className":123732},[66],[31,123734],{"className":123735,"style":326},[70],[31,123737,205],{"className":123738},[75]," 的必要性主要针对无限期问题。",[52860,123741,123743],{"id":123742},"_73-注意要点","7.3 注意要点",[11,123745,123746,123749,123750,123919],{},[7246,123747,123748],{},"把贝尔曼方程写成没有期望的形式。"," 在确定性转移的情形下，",[31,123751,123753,123803],{"className":123752,"translate":35},[34],[31,123754,123756],{"className":123755},[39],[41,123757,123758],{"xmlns":43},[45,123759,123760,123800],{},[48,123761,123762,123764,123766,123768,123770,123772,123774,123776,123778,123780,123782,123784,123786,123788,123790,123792,123798],{},[51,123763,43481],{},[95,123765,211],{"stretchy":210},[51,123767,14551],{},[95,123769,220],{"stretchy":210},[95,123771,223],{},[51,123773,2549],{},[95,123775,211],{"stretchy":210},[51,123777,14551],{},[95,123779,1210],{"separator":62},[51,123781,5813],{},[95,123783,220],{"stretchy":210},[95,123785,434],{},[51,123787,53298],{},[51,123789,43481],{},[95,123791,211],{"stretchy":210},[1180,123793,123794,123796],{},[51,123795,14551],{},[95,123797,13691],{"mathvariant":494,"lspace":5610,"rspace":5610},[95,123799,220],{"stretchy":210},[55,123801,123802],{"encoding":57},"V(s)=r(s,a)+\\gamma V(s')",[31,123804,123806,123833,123869],{"className":123805,"ariaHidden":62},[61],[31,123807,123809,123812,123815,123818,123821,123824,123827,123830],{"className":123808},[66],[31,123810],{"className":123811,"style":260},[70],[31,123813,43481],{"className":123814,"style":120},[75,76],[31,123816,211],{"className":123817},[264],[31,123819,14551],{"className":123820},[75,76],[31,123822,220],{"className":123823},[289],[31,123825],{"className":123826,"style":293},[119],[31,123828,223],{"className":123829},[297],[31,123831],{"className":123832,"style":293},[119],[31,123834,123836,123839,123842,123845,123848,123851,123854,123857,123860,123863,123866],{"className":123835},[66],[31,123837],{"className":123838,"style":260},[70],[31,123840,2549],{"className":123841,"style":19759},[75,76],[31,123843,211],{"className":123844},[264],[31,123846,14551],{"className":123847},[75,76],[31,123849,1210],{"className":123850},[1391],[31,123852],{"className":123853,"style":1467},[119],[31,123855,5813],{"className":123856},[75,76],[31,123858,220],{"className":123859},[289],[31,123861],{"className":123862,"style":120},[119],[31,123864,434],{"className":123865},[124],[31,123867],{"className":123868,"style":120},[119],[31,123870,123872,123875,123878,123881,123884,123916],{"className":123871},[66],[31,123873],{"className":123874,"style":13710},[70],[31,123876,53298],{"className":123877,"style":24157},[75,76],[31,123879,43481],{"className":123880,"style":120},[75,76],[31,123882,211],{"className":123883},[264],[31,123885,123887,123890],{"className":123886},[75],[31,123888,14551],{"className":123889},[75,76],[31,123891,123893],{"className":123892},[1039],[31,123894,123896],{"className":123895},[1043],[31,123897,123899],{"className":123898},[1048],[31,123900,123902],{"className":123901,"style":13732},[1052],[31,123903,123904,123907],{"style":3631},[31,123905],{"className":123906,"style":1061},[1060],[31,123908,123910],{"className":123909},[1065,1066,1067,1068],[31,123911,123913],{"className":123912},[75,1068],[31,123914,13691],{"className":123915},[75,1068],[31,123917,220],{"className":123918},[289]," 可以成立。但在随机转移下，必须对下一状态取期望。遗漏期望会把随机问题误写成确定性问题。",[11,123921,123922,123925,123926,124007,124008,124089],{},[7246,123923,123924],{},"把最大值放进期望内部。"," 最优方程是 ",[31,123927,123929,123952],{"className":123928,"translate":35},[34],[31,123930,123932],{"className":123931},[39],[41,123933,123934],{"xmlns":43},[45,123935,123936,123950],{},[48,123937,123938,123948],{},[1004,123939,123940,123946],{},[48,123941,123942,123944],{},[51,123943,10154],{},[95,123945,1540],{},[51,123947,5813],{},[51,123949,22324],{"mathvariant":14789},[55,123951,97946],{"encoding":57},[31,123953,123955],{"className":123954,"ariaHidden":62},[61],[31,123956,123958,123961,124001,124004],{"className":123957},[66],[31,123959],{"className":123960,"style":97956},[70],[31,123962,123964,123967],{"className":123963},[1460],[31,123965,10154],{"className":123966},[1460],[31,123968,123970],{"className":123969},[1039],[31,123971,123973,123993],{"className":123972},[1043,1044],[31,123974,123976,123990],{"className":123975},[1048],[31,123977,123979],{"className":123978,"style":3230},[1052],[31,123980,123981,123984],{"style":21790},[31,123982],{"className":123983,"style":1061},[1060],[31,123985,123987],{"className":123986},[1065,1066,1067,1068],[31,123988,5813],{"className":123989},[75,76,1068],[31,123991,1076],{"className":123992},[1075],[31,123994,123996],{"className":123995},[1048],[31,123997,123999],{"className":123998,"style":2050},[1052],[31,124000],{},[31,124002],{"className":124003,"style":1467},[119],[31,124005,22324],{"className":124006},[75,14947],"，不是 ",[31,124009,124011,124034],{"className":124010,"translate":35},[34],[31,124012,124014],{"className":124013},[39],[41,124015,124016],{"xmlns":43},[45,124017,124018,124032],{},[48,124019,124020,124022],{},[51,124021,22324],{"mathvariant":14789},[1004,124023,124024,124030],{},[48,124025,124026,124028],{},[51,124027,10154],{},[95,124029,1540],{},[51,124031,5813],{},[55,124033,98030],{"encoding":57},[31,124035,124037],{"className":124036,"ariaHidden":62},[61],[31,124038,124040,124043,124046,124049],{"className":124039},[66],[31,124041],{"className":124042,"style":97956},[70],[31,124044,22324],{"className":124045},[75,14947],[31,124047],{"className":124048,"style":1467},[119],[31,124050,124052,124055],{"className":124051},[1460],[31,124053,10154],{"className":124054},[1460],[31,124056,124058],{"className":124057},[1039],[31,124059,124061,124081],{"className":124060},[1043,1044],[31,124062,124064,124078],{"className":124063},[1048],[31,124065,124067],{"className":124066,"style":3230},[1052],[31,124068,124069,124072],{"style":21790},[31,124070],{"className":124071,"style":1061},[1060],[31,124073,124075],{"className":124074},[1065,1066,1067,1068],[31,124076,5813],{"className":124077},[75,76,1068],[31,124079,1076],{"className":124080},[1075],[31,124082,124084],{"className":124083},[1048],[31,124085,124087],{"className":124086,"style":2050},[1052],[31,124088],{},"。4.3 节的例子说明，二者一般相差很大。动作必须在观察到下一状态之前选定，这个时序约束正是最大值位于期望之外的原因。",[11,124091,124092,54797,124151,124225,124226,124312,124313,124465,124466,124575,124576,124620],{},[7246,124093,124094,124095,18171,124123,2372],{},"混淆 ",[31,124096,124098,124111],{"className":124097,"translate":35},[34],[31,124099,124101],{"className":124100},[39],[41,124102,124103],{"xmlns":43},[45,124104,124105,124109],{},[48,124106,124107],{},[51,124108,43481],{},[55,124110,43481],{"encoding":57},[31,124112,124114],{"className":124113,"ariaHidden":62},[61],[31,124115,124117,124120],{"className":124116},[66],[31,124118],{"className":124119,"style":945},[70],[31,124121,43481],{"className":124122,"style":120},[75,76],[31,124124,124126,124139],{"className":124125,"translate":35},[34],[31,124127,124129],{"className":124128},[39],[41,124130,124131],{"xmlns":43},[45,124132,124133,124137],{},[48,124134,124135],{},[51,124136,56971],{},[55,124138,56971],{"encoding":57},[31,124140,124142],{"className":124141,"ariaHidden":62},[61],[31,124143,124145,124148],{"className":124144},[66],[31,124146],{"className":124147,"style":59114},[70],[31,124149,56971],{"className":124150},[75,76],[31,124152,124154,124178],{"className":124153,"translate":35},[34],[31,124155,124157],{"className":124156},[39],[41,124158,124159],{"xmlns":43},[45,124160,124161,124175],{},[48,124162,124163,124169,124171,124173],{},[1180,124164,124165,124167],{},[51,124166,43481],{},[51,124168,2515],{},[95,124170,211],{"stretchy":210},[51,124172,14551],{},[95,124174,220],{"stretchy":210},[55,124176,124177],{"encoding":57},"V^\\pi(s)",[31,124179,124181],{"className":124180,"ariaHidden":62},[61],[31,124182,124184,124187,124216,124219,124222],{"className":124183},[66],[31,124185],{"className":124186,"style":260},[70],[31,124188,124190,124193],{"className":124189},[75],[31,124191,43481],{"className":124192,"style":120},[75,76],[31,124194,124196],{"className":124195},[1039],[31,124197,124199],{"className":124198},[1043],[31,124200,124202],{"className":124201},[1048],[31,124203,124205],{"className":124204,"style":3628},[1052],[31,124206,124207,124210],{"style":3631},[31,124208],{"className":124209,"style":1061},[1060],[31,124211,124213],{"className":124212},[1065,1066,1067,1068],[31,124214,2515],{"className":124215,"style":2666},[75,76,1068],[31,124217,211],{"className":124218},[264],[31,124220,14551],{"className":124221},[75,76],[31,124223,220],{"className":124224},[289]," 已经对策略的动作分布取了平均，因此它不再区分同一状态下的不同动作。要选择动作，应使用 ",[31,124227,124229,124256],{"className":124228,"translate":35},[34],[31,124230,124232],{"className":124231},[39],[41,124233,124234],{"xmlns":43},[45,124235,124236,124254],{},[48,124237,124238,124244,124246,124248,124250,124252],{},[1180,124239,124240,124242],{},[51,124241,56971],{},[95,124243,30648],{},[95,124245,211],{"stretchy":210},[51,124247,14551],{},[95,124249,1210],{"separator":62},[51,124251,5813],{},[95,124253,220],{"stretchy":210},[55,124255,97738],{"encoding":57},[31,124257,124259],{"className":124258,"ariaHidden":62},[61],[31,124260,124262,124265,124294,124297,124300,124303,124306,124309],{"className":124261},[66],[31,124263],{"className":124264,"style":260},[70],[31,124266,124268,124271],{"className":124267},[75],[31,124269,56971],{"className":124270},[75,76],[31,124272,124274],{"className":124273},[1039],[31,124275,124277],{"className":124276},[1043],[31,124278,124280],{"className":124279},[1048],[31,124281,124283],{"className":124282,"style":30719},[1052],[31,124284,124285,124288],{"style":3631},[31,124286],{"className":124287,"style":1061},[1060],[31,124289,124291],{"className":124290},[1065,1066,1067,1068],[31,124292,30648],{"className":124293},[124,1068],[31,124295,211],{"className":124296},[264],[31,124298,14551],{"className":124299},[75,76],[31,124301,1210],{"className":124302},[1391],[31,124304],{"className":124305,"style":1467},[119],[31,124307,5813],{"className":124308},[75,76],[31,124310,220],{"className":124311},[289],"，最优动作是 ",[31,124314,124316,124358],{"className":124315,"translate":35},[34],[31,124317,124319],{"className":124318},[39],[41,124320,124321],{"xmlns":43},[45,124322,124323,124355],{},[48,124324,124325,124327,124329,124339,124345,124347,124349,124351,124353],{},[51,124326,37263],{},[95,124328,1540],{},[1004,124330,124331,124337],{},[48,124332,124333,124335],{},[51,124334,10154],{},[95,124336,1540],{},[51,124338,5813],{},[1180,124340,124341,124343],{},[51,124342,56971],{},[95,124344,30648],{},[95,124346,211],{"stretchy":210},[51,124348,14551],{},[95,124350,1210],{"separator":62},[51,124352,5813],{},[95,124354,220],{"stretchy":210},[55,124356,124357],{"encoding":57},"\\arg\\max_a Q^*(s,a)",[31,124359,124361],{"className":124360,"ariaHidden":62},[61],[31,124362,124364,124367,124372,124375,124415,124418,124447,124450,124453,124456,124459,124462],{"className":124363},[66],[31,124365],{"className":124366,"style":260},[70],[31,124368,37341,124370],{"className":124369},[1460],[31,124371,14556],{"style":20180},[31,124373],{"className":124374,"style":1467},[119],[31,124376,124378,124381],{"className":124377},[1460],[31,124379,10154],{"className":124380},[1460],[31,124382,124384],{"className":124383},[1039],[31,124385,124387,124407],{"className":124386},[1043,1044],[31,124388,124390,124404],{"className":124389},[1048],[31,124391,124393],{"className":124392,"style":3230},[1052],[31,124394,124395,124398],{"style":21790},[31,124396],{"className":124397,"style":1061},[1060],[31,124399,124401],{"className":124400},[1065,1066,1067,1068],[31,124402,5813],{"className":124403},[75,76,1068],[31,124405,1076],{"className":124406},[1075],[31,124408,124410],{"className":124409},[1048],[31,124411,124413],{"className":124412,"style":2050},[1052],[31,124414],{},[31,124416],{"className":124417,"style":1467},[119],[31,124419,124421,124424],{"className":124420},[75],[31,124422,56971],{"className":124423},[75,76],[31,124425,124427],{"className":124426},[1039],[31,124428,124430],{"className":124429},[1043],[31,124431,124433],{"className":124432},[1048],[31,124434,124436],{"className":124435,"style":30719},[1052],[31,124437,124438,124441],{"style":3631},[31,124439],{"className":124440,"style":1061},[1060],[31,124442,124444],{"className":124443},[1065,1066,1067,1068],[31,124445,30648],{"className":124446},[124,1068],[31,124448,211],{"className":124449},[264],[31,124451,14551],{"className":124452},[75,76],[31,124454,1210],{"className":124455},[1391],[31,124457],{"className":124458,"style":1467},[119],[31,124460,5813],{"className":124461},[75,76],[31,124463,220],{"className":124464},[289],"。把 ",[31,124467,124469,124503],{"className":124468,"translate":35},[34],[31,124470,124472],{"className":124471},[39],[41,124473,124474],{"xmlns":43},[45,124475,124476,124500],{},[48,124477,124478,124480,124482,124492,124494,124496,124498],{},[51,124479,37263],{},[95,124481,1540],{},[1004,124483,124484,124490],{},[48,124485,124486,124488],{},[51,124487,10154],{},[95,124489,1540],{},[51,124491,5813],{},[51,124493,43481],{},[95,124495,211],{"stretchy":210},[51,124497,14551],{},[95,124499,220],{"stretchy":210},[55,124501,124502],{"encoding":57},"\\arg\\max_a V(s)",[31,124504,124506],{"className":124505,"ariaHidden":62},[61],[31,124507,124509,124512,124517,124520,124560,124563,124566,124569,124572],{"className":124508},[66],[31,124510],{"className":124511,"style":260},[70],[31,124513,37341,124515],{"className":124514},[1460],[31,124516,14556],{"style":20180},[31,124518],{"className":124519,"style":1467},[119],[31,124521,124523,124526],{"className":124522},[1460],[31,124524,10154],{"className":124525},[1460],[31,124527,124529],{"className":124528},[1039],[31,124530,124532,124552],{"className":124531},[1043,1044],[31,124533,124535,124549],{"className":124534},[1048],[31,124536,124538],{"className":124537,"style":3230},[1052],[31,124539,124540,124543],{"style":21790},[31,124541],{"className":124542,"style":1061},[1060],[31,124544,124546],{"className":124545},[1065,1066,1067,1068],[31,124547,5813],{"className":124548},[75,76,1068],[31,124550,1076],{"className":124551},[1075],[31,124553,124555],{"className":124554},[1048],[31,124556,124558],{"className":124557,"style":2050},[1052],[31,124559],{},[31,124561],{"className":124562,"style":1467},[119],[31,124564,43481],{"className":124565,"style":120},[75,76],[31,124567,211],{"className":124568},[264],[31,124570,14551],{"className":124571},[75,76],[31,124573,220],{"className":124574},[289]," 当作最优动作是没有意义的，因为 ",[31,124577,124579,124599],{"className":124578,"translate":35},[34],[31,124580,124582],{"className":124581},[39],[41,124583,124584],{"xmlns":43},[45,124585,124586,124596],{},[48,124587,124588,124590,124592,124594],{},[51,124589,43481],{},[95,124591,211],{"stretchy":210},[51,124593,14551],{},[95,124595,220],{"stretchy":210},[55,124597,124598],{"encoding":57},"V(s)",[31,124600,124602],{"className":124601,"ariaHidden":62},[61],[31,124603,124605,124608,124611,124614,124617],{"className":124604},[66],[31,124606],{"className":124607,"style":260},[70],[31,124609,43481],{"className":124610,"style":120},[75,76],[31,124612,211],{"className":124613},[264],[31,124615,14551],{"className":124616},[75,76],[31,124618,220],{"className":124619},[289]," 中已经没有动作变量。",[11,124622,124623,124626,124627,124677],{},[7246,124624,124625],{},"混淆必要性、充分性与唯一性。"," 在折扣有界 MDP 中，贝尔曼方程既必要又充分，而且解唯一。但在 ",[31,124628,124630,124647],{"className":124629,"translate":35},[34],[31,124631,124633],{"className":124632},[39],[41,124634,124635],{"xmlns":43},[45,124636,124637,124645],{},[48,124638,124639,124641,124643],{},[51,124640,53298],{},[95,124642,223],{},[91,124644,205],{},[55,124646,81593],{"encoding":57},[31,124648,124650,124668],{"className":124649,"ariaHidden":62},[61],[31,124651,124653,124656,124659,124662,124665],{"className":124652},[66],[31,124654],{"className":124655,"style":865},[70],[31,124657,53298],{"className":124658,"style":24157},[75,76],[31,124660],{"className":124661,"style":293},[119],[31,124663,223],{"className":124664},[297],[31,124666],{"className":124667,"style":293},[119],[31,124669,124671,124674],{"className":124670},[66],[31,124672],{"className":124673,"style":326},[70],[31,124675,205],{"className":124676},[75],"、奖励无界或状态空间不满足可测选择条件时，这些结论可能分别失效。讨论贝尔曼方程时，应明确所依赖的假设。",[11,124679,124680,124683,124684,124712],{},[7246,124681,124682],{},"认为最优策略唯一。"," 最优值函数是唯一的，但最优策略不一定唯一。例如只有一个状态、两个动作，两个动作都使状态保持不变且都获得奖励 ",[31,124685,124687,124700],{"className":124686,"translate":35},[34],[31,124688,124690],{"className":124689},[39],[41,124691,124692],{"xmlns":43},[45,124693,124694,124698],{},[48,124695,124696],{},[91,124697,205],{},[55,124699,205],{"encoding":57},[31,124701,124703],{"className":124702,"ariaHidden":62},[61],[31,124704,124706,124709],{"className":124705},[66],[31,124707],{"className":124708,"style":326},[70],[31,124710,205],{"className":124711},[75],"。此时",[31,124714,124716],{"className":124715,"translate":35},[411],[31,124717,124719,124753],{"className":124718,"translate":35},[34],[31,124720,124722],{"className":124721},[39],[41,124723,124724],{"xmlns":43,"display":420},[45,124725,124726,124750],{},[48,124727,124728,124734,124736,124748],{},[1180,124729,124730,124732],{},[51,124731,43481],{},[95,124733,30648],{},[95,124735,223],{},[2074,124737,124738,124740],{},[91,124739,205],{},[48,124741,124742,124744,124746],{},[91,124743,205],{},[95,124745,97],{},[51,124747,53298],{},[95,124749,1210],{"separator":62},[55,124751,124752],{"encoding":57},"V^*=\\frac{1}{1-\\gamma},",[31,124754,124756,124800],{"className":124755,"ariaHidden":62},[61],[31,124757,124759,124762,124791,124794,124797],{"className":124758},[66],[31,124760],{"className":124761,"style":74855},[70],[31,124763,124765,124768],{"className":124764},[75],[31,124766,43481],{"className":124767,"style":120},[75,76],[31,124769,124771],{"className":124770},[1039],[31,124772,124774],{"className":124773},[1043],[31,124775,124777],{"className":124776},[1048],[31,124778,124780],{"className":124779,"style":74855},[1052],[31,124781,124782,124785],{"style":1330},[31,124783],{"className":124784,"style":1061},[1060],[31,124786,124788],{"className":124787},[1065,1066,1067,1068],[31,124789,30648],{"className":124790},[124,1068],[31,124792],{"className":124793,"style":293},[119],[31,124795,223],{"className":124796},[297],[31,124798],{"className":124799,"style":293},[119],[31,124801,124803,124806,124880],{"className":124802},[66],[31,124804],{"className":124805,"style":41259},[70],[31,124807,124809,124812,124877],{"className":124808},[75],[31,124810],{"className":124811},[264,2139],[31,124813,124815],{"className":124814},[2074],[31,124816,124818,124869],{"className":124817},[1043,1044],[31,124819,124821,124866],{"className":124820},[1048],[31,124822,124824,124847,124855],{"className":124823,"style":26376},[1052],[31,124825,124826,124829],{"style":5090},[31,124827],{"className":124828,"style":1637},[1060],[31,124830,124832,124835,124838,124841,124844],{"className":124831},[75],[31,124833,205],{"className":124834},[75],[31,124836],{"className":124837,"style":120},[119],[31,124839,97],{"className":124840},[124],[31,124842],{"className":124843,"style":120},[119],[31,124845,53298],{"className":124846,"style":24157},[75,76],[31,124848,124849,124852],{"style":2170},[31,124850],{"className":124851,"style":1637},[1060],[31,124853],{"className":124854,"style":2178},[2177],[31,124856,124857,124860],{"style":5110},[31,124858],{"className":124859,"style":1637},[1060],[31,124861,124863],{"className":124862},[75],[31,124864,205],{"className":124865},[75],[31,124867,1076],{"className":124868},[1075],[31,124870,124872],{"className":124871},[1048],[31,124873,124875],{"className":124874,"style":41317},[1052],[31,124876],{},[31,124878],{"className":124879},[289,2139],[31,124881,1210],{"className":124882},[1391],[11,124884,124885],{},"两个动作在每个状态都达到最大值，因此都是最优的。唯一性属于值函数，而不属于策略。",[18,124887,124888],{"id":124888},"总结",[11,124890,124891,124892,125107,125108,125164],{},"贝尔曼方程的核心可以概括为三步。第一步，利用折扣回报的递归分解 ",[31,124893,124895,124936],{"className":124894,"translate":35},[34],[31,124896,124898],{"className":124897},[39],[41,124899,124900],{"xmlns":43},[45,124901,124902,124934],{},[48,124903,124904,124910,124912,124918,124920,124922],{},[1004,124905,124906,124908],{},[51,124907,15297],{},[51,124909,12271],{},[95,124911,223],{},[1004,124913,124914,124916],{},[51,124915,2549],{},[51,124917,12271],{},[95,124919,434],{},[51,124921,53298],{},[1004,124923,124924,124926],{},[51,124925,15297],{},[48,124927,124928,124930,124932],{},[51,124929,12271],{},[95,124931,434],{},[91,124933,205],{},[55,124935,60121],{"encoding":57},[31,124937,124939,124994,125049],{"className":124938,"ariaHidden":62},[61],[31,124940,124942,124945,124985,124988,124991],{"className":124941},[66],[31,124943],{"className":124944,"style":5230},[70],[31,124946,124948,124951],{"className":124947},[75],[31,124949,15297],{"className":124950},[75,76],[31,124952,124954],{"className":124953},[1039],[31,124955,124957,124977],{"className":124956},[1043,1044],[31,124958,124960,124974],{"className":124959},[1048],[31,124961,124963],{"className":124962,"style":19220},[1052],[31,124964,124965,124968],{"style":2807},[31,124966],{"className":124967,"style":1061},[1060],[31,124969,124971],{"className":124970},[1065,1066,1067,1068],[31,124972,12271],{"className":124973},[75,76,1068],[31,124975,1076],{"className":124976},[1075],[31,124978,124980],{"className":124979},[1048],[31,124981,124983],{"className":124982,"style":2050},[1052],[31,124984],{},[31,124986],{"className":124987,"style":293},[119],[31,124989,223],{"className":124990},[297],[31,124992],{"className":124993,"style":293},[119],[31,124995,124997,125000,125040,125043,125046],{"className":124996},[66],[31,124998],{"className":124999,"style":5426},[70],[31,125001,125003,125006],{"className":125002},[75],[31,125004,2549],{"className":125005,"style":19759},[75,76],[31,125007,125009],{"className":125008},[1039],[31,125010,125012,125032],{"className":125011},[1043,1044],[31,125013,125015,125029],{"className":125014},[1048],[31,125016,125018],{"className":125017,"style":19220},[1052],[31,125019,125020,125023],{"style":26726},[31,125021],{"className":125022,"style":1061},[1060],[31,125024,125026],{"className":125025},[1065,1066,1067,1068],[31,125027,12271],{"className":125028},[75,76,1068],[31,125030,1076],{"className":125031},[1075],[31,125033,125035],{"className":125034},[1048],[31,125036,125038],{"className":125037,"style":2050},[1052],[31,125039],{},[31,125041],{"className":125042,"style":120},[119],[31,125044,434],{"className":125045},[124],[31,125047],{"className":125048,"style":120},[119],[31,125050,125052,125055,125058],{"className":125051},[66],[31,125053],{"className":125054,"style":60241},[70],[31,125056,53298],{"className":125057,"style":24157},[75,76],[31,125059,125061,125064],{"className":125060},[75],[31,125062,15297],{"className":125063},[75,76],[31,125065,125067],{"className":125066},[1039],[31,125068,125070,125099],{"className":125069},[1043,1044],[31,125071,125073,125096],{"className":125072},[1048],[31,125074,125076],{"className":125075,"style":2028},[1052],[31,125077,125078,125081],{"style":2807},[31,125079],{"className":125080,"style":1061},[1060],[31,125082,125084],{"className":125083},[1065,1066,1067,1068],[31,125085,125087,125090,125093],{"className":125086},[75,1068],[31,125088,12271],{"className":125089},[75,76,1068],[31,125091,434],{"className":125092},[124,1068],[31,125094,205],{"className":125095},[75,1068],[31,125097,1076],{"className":125098},[1075],[31,125100,125102],{"className":125101},[1048],[31,125103,125105],{"className":125104,"style":25082},[1052],[31,125106],{}," 与马尔可夫性，把值函数写成当前奖励与后继值函数的期望。第二步，把该递归式视为有界函数空间上的算子方程 ",[31,125109,125111,125131],{"className":125110,"translate":35},[34],[31,125112,125114],{"className":125113},[39],[41,125115,125116],{"xmlns":43},[45,125117,125118,125128],{},[48,125119,125120,125122,125124,125126],{},[51,125121,43481],{},[95,125123,223],{},[51,125125,919],{"mathvariant":925},[51,125127,43481],{},[55,125129,125130],{"encoding":57},"V=\\mathcal{T}V",[31,125132,125134,125152],{"className":125133,"ariaHidden":62},[61],[31,125135,125137,125140,125143,125146,125149],{"className":125136},[66],[31,125138],{"className":125139,"style":945},[70],[31,125141,43481],{"className":125142,"style":120},[75,76],[31,125144],{"className":125145,"style":293},[119],[31,125147,223],{"className":125148},[297],[31,125150],{"className":125151,"style":293},[119],[31,125153,125155,125158,125161],{"className":125154},[66],[31,125156],{"className":125157,"style":945},[70],[31,125159,919],{"className":125160,"style":64444},[75,968],[31,125162,43481],{"className":125163,"style":120},[75,76],"。第三步，利用算子的单调性、常数平移性与压缩性，证明不动点存在唯一，并由 Banach 不动点定理得到几何收敛。",[11,125166,125167,125168,125253],{},"贝尔曼期望方程刻画固定策略的值函数，其有限状态形式是一个线性方程组，系数矩阵为 ",[31,125169,125171,125194],{"className":125170,"translate":35},[34],[31,125172,125174],{"className":125173},[39],[41,125175,125176],{"xmlns":43},[45,125177,125178,125192],{},[48,125179,125180,125182,125184,125186],{},[51,125181,34619],{},[95,125183,97],{},[51,125185,53298],{},[1180,125187,125188,125190],{},[51,125189,2521],{},[51,125191,2515],{},[55,125193,67289],{"encoding":57},[31,125195,125197,125215],{"className":125196,"ariaHidden":62},[61],[31,125198,125200,125203,125206,125209,125212],{"className":125199},[66],[31,125201],{"className":125202,"style":53224},[70],[31,125204,34619],{"className":125205,"style":30311},[75,76],[31,125207],{"className":125208,"style":120},[119],[31,125210,97],{"className":125211},[124],[31,125213],{"className":125214,"style":120},[119],[31,125216,125218,125221,125224],{"className":125217},[66],[31,125219],{"className":125220,"style":59114},[70],[31,125222,53298],{"className":125223,"style":24157},[75,76],[31,125225,125227,125230],{"className":125226},[75],[31,125228,2521],{"className":125229,"style":949},[75,76],[31,125231,125233],{"className":125232},[1039],[31,125234,125236],{"className":125235},[1043],[31,125237,125239],{"className":125238},[1048],[31,125240,125242],{"className":125241,"style":3628},[1052],[31,125243,125244,125247],{"style":3631},[31,125245],{"className":125246,"style":1061},[1060],[31,125248,125250],{"className":125249},[1065,1066,1067,1068],[31,125251,2515],{"className":125252,"style":2666},[75,76,1068],"。贝尔曼最优方程刻画最优值函数，其算子在动作上取最大值。两者的共同点是都把无限期问题转化为一个不动点问题。区别在于前者对策略取期望，因而在有限状态下是线性方程。后者对动作取最大值，因而含有非线性项。",[11,125255,125256],{},"最优子结构是这一整套理论的解释性核心。它说明，在可加折扣效用、马尔可夫转移与有界回报的条件下，整体最优决策可以沿着实际到达的后继状态分解为一系列子问题的最优决策。它不是一个无条件的哲学原则，而是一个由回报结构、状态表示与折扣条件共同保证的数学定理。",{"title":52812,"searchDepth":52813,"depth":52813,"links":125258},[125259,125265,125272,125279,125286,125292,125297,125302],{"id":52857,"depth":52813,"text":52858,"children":125260},[125261,125263,125264],{"id":52862,"depth":125262,"text":52863},3,{"id":55294,"depth":125262,"text":55295},{"id":59150,"depth":125262,"text":59151},{"id":62692,"depth":52813,"text":62693,"children":125266},[125267,125268,125269,125270,125271],{"id":62696,"depth":125262,"text":62697},{"id":64400,"depth":125262,"text":64401},{"id":65579,"depth":125262,"text":65580},{"id":69960,"depth":125262,"text":69961},{"id":72362,"depth":125262,"text":72363},{"id":73966,"depth":52813,"text":73967,"children":125273},[125274,125275,125276,125277,125278],{"id":73970,"depth":125262,"text":73971},{"id":76288,"depth":125262,"text":76289},{"id":79236,"depth":125262,"text":79237},{"id":81999,"depth":125262,"text":82000},{"id":86675,"depth":125262,"text":86676},{"id":89118,"depth":52813,"text":89119,"children":125280},[125281,125282,125283,125284,125285],{"id":89122,"depth":125262,"text":89123},{"id":89932,"depth":125262,"text":89933},{"id":97915,"depth":125262,"text":97916},{"id":99735,"depth":125262,"text":99736},{"id":100984,"depth":125262,"text":100985},{"id":103591,"depth":52813,"text":103592,"children":125287},[125288,125289,125290,125291],{"id":103595,"depth":125262,"text":103596},{"id":105928,"depth":125262,"text":105929},{"id":111142,"depth":125262,"text":111143},{"id":114153,"depth":125262,"text":114154},{"id":115239,"depth":52813,"text":115240,"children":125293},[125294,125295,125296],{"id":115243,"depth":125262,"text":115244},{"id":120556,"depth":125262,"text":120557},{"id":120844,"depth":125262,"text":120845},{"id":121452,"depth":52813,"text":121453,"children":125298},[125299,125300,125301],{"id":121456,"depth":125262,"text":121457},{"id":122438,"depth":125262,"text":122439},{"id":123742,"depth":125262,"text":123743},{"id":124888,"depth":52813,"text":124888},{},"\u002Fblog\u002F2026\u002F0015-the-bellman-equation-and-its-mathematical-properties",{"title":52843,"description":52848},"blog\u002F2026\u002F0015-the-bellman-equation-and-its-mathematical-properties","贝尔曼方程（Bellman equation）是动态规划的核心，本质是把一个多阶段的决策问题拆成当前决策 + 剩余子问题的递归关系。它由理查德·贝尔曼在 20 世纪 50 年代提出，是强化学习、最优控制、经济学等领域的基础工具。",[52837,52836],"2026-11-09T23:49:14+08:00","hpJs47-d_ZZWWr8USnYwX8xzwtvDogQjVGqetsjCV2g",{"id":125312,"title":125313,"body":125314,"description":125318,"draft":52826,"enableCodeHighlight":52827,"enableCodeHightlight":52826,"enableComment":52827,"enableD2":52827,"enableKatex":52827,"extension":52828,"image":52812,"important":52826,"location":52829,"meta":145000,"navigation":52827,"ogImage":52829,"path":145001,"seo":145002,"stem":145003,"summary":145004,"tags":145005,"time":145007,"__hash__":145008},"blog\u002Fblog\u002F2026\u002F0013-satisfiability-modulo-theories.md","可满足性模理论 (Satisfiability Modulo Theories)",{"type":8,"value":125315,"toc":144961},[125316,125319,125322,125463,125474,125477,125624,125627,125635,125657,125663,125667,125671,125706,125709,125712,125835,125897,126068,126072,126075,126401,126947,126950,127629,127995,128054,128065,128071,128075,128078,128286,128292,128324,128327,128330,128334,128338,128345,128391,128521,129144,129559,129562,129565,129593,129701,129919,129922,129926,129929,129932,130122,130125,130500,130690,130693,130696,130700,130703,130706,130877,130880,131307,131567,131703,131710,131714,131720,131723,131760,131763,132372,132375,132378,132382,132385,132399,132405,132409,132413,132420,132423,132522,132554,132573,132576,132580,132586,132769,132772,132942,132945,133377,133477,133484,133615,133619,133625,133632,133635,133805,134250,134253,134256,134440,134540,134546,134735,134739,134745,134821,134827,134831,134838,134841,134854,134857,134861,134865,134876,134883,134886,134907,134910,134914,134917,134920,135136,135139,135393,135396,135492,135699,135706,135862,135942,136250,136254,136257,136264,136267,136287,136290,136294,136330,136333,136338,136341,136481,136484,136710,136931,136934,136937,136942,137078,137080,137272,137396,137798,137801,137803,137808,137811,138010,138013,138020,138281,138424,138426,138434,138445,138447,138747,138966,138969,138975,138979,138986,138989,139144,139147,139376,139379,139490,139551,139554,139558,139561,139776,140229,140357,140513,140607,140729,140732,140739,140742,140779,140783,140789,140817,140820,140823,140827,140831,140834,140840,140843,141267,141353,141362,141368,141372,141375,141464,141481,141492,141623,141626,141818,141821,141853,141856,141862,141997,142000,142040,142043,142235,142241,142255,142293,142310,142314,142407,142485,142488,142596,142599,142990,143054,143057,143099,143102,143108,143111,143119,143315,143401,143411,143415,143418,143483,143494,143585,143591,143738,143744,143747,143813,143903,143960,143970,143977,143981,144055,144058,144167,144177,144180,144220,144223,144386,144391,144402,144722,144813,144817,144820,144827,144833,144840,144844,144947,144957],[11,125317,125318],{},"SMT 的出现，本质上是为了解决一个核心矛盾：很多实际问题用纯命题逻辑（SAT）表达时，要么根本无法表达，要么表达后规模爆炸；而用一般的一阶逻辑或高阶逻辑表达虽然自然，却无法自动高效求解。",[11,125320,125321],{},"看一段再普通不过的代码：",[125323,125324,125327],"pre",{"className":125325,"code":125326,"language":1412,"meta":52812,"style":52812},"language-c shiki shiki-themes github-light github-light github-dark","void f(int x, int y) {\n    int a[100] = {0};\n    a[0] = 5;\n    if (x + y > 10 && a[0] == 5) {\n        \u002F\u002F 目标分支\n    }\n}\n",[45608,125328,125329,125362,125388,125407,125444,125451,125457],{"__ignoreMap":52812},[31,125330,125333,125337,125341,125344,125347,125351,125354,125356,125359],{"class":125331,"line":125332},"line",1,[31,125334,125336],{"class":125335},"s8jYJ","void",[31,125338,125340],{"class":125339},"snPdu"," f",[31,125342,211],{"class":125343},"sxrX7",[31,125345,125346],{"class":125335},"int",[31,125348,125350],{"class":125349},"sP4rz"," x",[31,125352,125353],{"class":125343},", ",[31,125355,125346],{"class":125335},[31,125357,125358],{"class":125349}," y",[31,125360,125361],{"class":125343},") {\n",[31,125363,125364,125367,125370,125372,125375,125378,125380,125383,125385],{"class":125331,"line":52813},[31,125365,125366],{"class":125335},"    int",[31,125368,125369],{"class":125349}," a",[31,125371,15175],{"class":125343},[31,125373,22616],{"class":125374},"sBjJW",[31,125376,125377],{"class":125343},"] ",[31,125379,223],{"class":125335},[31,125381,125382],{"class":125343}," {",[31,125384,1977],{"class":125374},[31,125386,125387],{"class":125343},"};\n",[31,125389,125390,125393,125395,125397,125399,125401,125404],{"class":125331,"line":125262},[31,125391,125392],{"class":125349},"    a",[31,125394,15175],{"class":125343},[31,125396,1977],{"class":125374},[31,125398,125377],{"class":125343},[31,125400,223],{"class":125335},[31,125402,125403],{"class":125374}," 5",[31,125405,125406],{"class":125343},";\n",[31,125408,125410,125413,125416,125418,125421,125423,125426,125429,125431,125433,125435,125437,125440,125442],{"class":125331,"line":125409},4,[31,125411,125412],{"class":125335},"    if",[31,125414,125415],{"class":125343}," (x ",[31,125417,434],{"class":125335},[31,125419,125420],{"class":125343}," y ",[31,125422,16040],{"class":125335},[31,125424,125425],{"class":125374}," 10",[31,125427,125428],{"class":125335}," &&",[31,125430,125369],{"class":125349},[31,125432,15175],{"class":125343},[31,125434,1977],{"class":125374},[31,125436,125377],{"class":125343},[31,125438,125439],{"class":125335},"==",[31,125441,125403],{"class":125374},[31,125443,125361],{"class":125343},[31,125445,125447],{"class":125331,"line":125446},5,[31,125448,125450],{"class":125449},"sCsY4","        \u002F\u002F 目标分支\n",[31,125452,125454],{"class":125331,"line":125453},6,[31,125455,125456],{"class":125343},"    }\n",[31,125458,125460],{"class":125331,"line":125459},7,[31,125461,125462],{"class":125343},"}\n",[11,125464,125465,125466,15282,125468,125470,125471],{},"假设 ",[45608,125467,53],{},[45608,125469,3003],{}," 是来自外部的整数。我们想知道：",[7246,125472,125473],{},"有没有一组输入，能走进这个分支？",[11,125475,125476],{},"沿着这条路径，把进入分支需要满足的条件收集起来：",[31,125478,125480],{"className":125479,"translate":35},[411],[31,125481,125483,125525],{"className":125482,"translate":35},[34],[31,125484,125486],{"className":125485},[39],[41,125487,125488],{"xmlns":43,"display":420},[45,125489,125490,125522],{},[48,125491,125492,125494,125496,125498,125500,125502,125505,125508,125510,125512,125514,125516,125518,125520],{},[51,125493,53],{},[95,125495,434],{},[51,125497,3003],{},[95,125499,16040],{},[91,125501,19494],{},[1430,125503,125504],{},"  ",[95,125506,125507],{},"∧",[1430,125509,125504],{},[51,125511,5813],{},[95,125513,15175],{"stretchy":210},[91,125515,1977],{},[95,125517,15184],{"stretchy":210},[95,125519,223],{},[91,125521,7853],{},[55,125523,125524],{"encoding":57},"x + y > 10 \\;\\wedge\\; a[0] = 5",[31,125526,125528,125546,125564,125588,125615],{"className":125527,"ariaHidden":62},[61],[31,125529,125531,125534,125537,125540,125543],{"className":125530},[66],[31,125532],{"className":125533,"style":307},[70],[31,125535,53],{"className":125536},[75,76],[31,125538],{"className":125539,"style":120},[119],[31,125541,434],{"className":125542},[124],[31,125544],{"className":125545,"style":120},[119],[31,125547,125549,125552,125555,125558,125561],{"className":125548},[66],[31,125550],{"className":125551,"style":53323},[70],[31,125553,3003],{"className":125554,"style":2666},[75,76],[31,125556],{"className":125557,"style":293},[119],[31,125559,16040],{"className":125560},[297],[31,125562],{"className":125563,"style":293},[119],[31,125565,125567,125570,125573,125576,125579,125582,125585],{"className":125566},[66],[31,125568],{"className":125569,"style":326},[70],[31,125571,19494],{"className":125572},[75],[31,125574],{"className":125575,"style":293},[119],[31,125577],{"className":125578,"style":120},[119],[31,125580,125507],{"className":125581},[124],[31,125583],{"className":125584,"style":293},[119],[31,125586],{"className":125587,"style":120},[119],[31,125589,125591,125594,125597,125600,125603,125606,125609,125612],{"className":125590},[66],[31,125592],{"className":125593,"style":260},[70],[31,125595,5813],{"className":125596},[75,76],[31,125598,15175],{"className":125599},[264],[31,125601,1977],{"className":125602},[75],[31,125604,15184],{"className":125605},[289],[31,125607],{"className":125608,"style":293},[119],[31,125610,223],{"className":125611},[297],[31,125613],{"className":125614,"style":293},[119],[31,125616,125618,125621],{"className":125617},[66],[31,125619],{"className":125620,"style":326},[70],[31,125622,7853],{"className":125623},[75],[11,125625,125626],{},"问题于是变成：这组约束有没有解？",[11,125628,125629,125630,15282,125632,125634],{},"纯 SAT 求解器也能做，但你得先把 ",[45608,125631,53],{},[45608,125633,3003],{}," 各拆成 32 个布尔变量，把加法展开成加法器电路，把数组展开成 100 个元素，最后交给它几千条子句。它只认识真假，不认识整数，也不认识数组。",[11,125636,125637,125638,125641,125642,125645,125646,125649,125650,125652,125653,125656],{},"SMT 求解器则直接读原式：",[45608,125639,125640],{},"x + y > 10"," 交给算术，",[45608,125643,125644],{},"a[0] = 5"," 交给数组。算术说取 ",[45608,125647,125648],{},"x = 6, y = 5"," 就行，数组说 ",[45608,125651,125644],{}," 本来就成立。两边一对，返回 ",[7246,125654,125655],{},"sat","，并给出这组解。",[11,125658,125659,125660],{},"同一个问题，一个要把语义拆成门电路，一个能直接利用语义。",[7246,125661,125662],{},"SMT 要做的，就是后者。",[18,125664,125666],{"id":125665},"一smt-是什么","一、SMT 是什么",[52860,125668,125670],{"id":125669},"_11-定义","1.1 定义",[11,125672,125673,125674],{},"把上面的过程抽象一下。",[7246,125675,125676,125677,125705],{},"SMT 是「SAT + 背景理论」：在某个理论 ",[31,125678,125680,125693],{"className":125679,"translate":35},[34],[31,125681,125683],{"className":125682},[39],[41,125684,125685],{"xmlns":43},[45,125686,125687,125691],{},[48,125688,125689],{},[51,125690,919],{},[55,125692,919],{"encoding":57},[31,125694,125696],{"className":125695,"ariaHidden":62},[61],[31,125697,125699,125702],{"className":125698},[66],[31,125700],{"className":125701,"style":945},[70],[31,125703,919],{"className":125704,"style":949},[75,76]," 规定的语义下，判定一个一阶逻辑公式是否存在满足它的赋值。",[11,125707,125708],{},"这里的关键词是「模理论」（modulo theories）。「模」可以理解为「相对于」：我们不再问公式在全部数学结构里是否成立，而是问它在某个特定理论——比如整数算术、位向量、数组——的解释下是否成立。",[11,125710,125711],{},"形式化一点，SMT 要回答的是：",[31,125713,125715],{"className":125714,"translate":35},[411],[31,125716,125718,125752],{"className":125717,"translate":35},[34],[31,125719,125721],{"className":125720},[39],[41,125722,125723],{"xmlns":43,"display":420},[45,125724,125725,125749],{},[48,125726,125727,125730,125732,125734,125736,125738,125740,125747],{},[51,125728,125729],{"mathvariant":494},"∃",[1430,125731,1432],{},[51,125733,17342],{},[51,125735,495],{"mathvariant":494},[119,125737],{"width":1584},[51,125739,17342],{},[1004,125741,125742,125745],{},[95,125743,125744],{},"⊨",[51,125746,919],{},[51,125748,8059],{},[55,125750,125751],{"encoding":57},"\\exists\\, \\sigma.\\quad \\sigma \\models_T \\varphi",[31,125753,125755,125826],{"className":125754,"ariaHidden":62},[61],[31,125756,125758,125762,125765,125768,125771,125774,125777,125780,125783,125823],{"className":125757},[66],[31,125759],{"className":125760,"style":125761},[70],"height:0.999em;vertical-align:-0.249em;",[31,125763,125729],{"className":125764},[75],[31,125766],{"className":125767,"style":1467},[119],[31,125769,17342],{"className":125770,"style":2666},[75,76],[31,125772,495],{"className":125773},[75],[31,125775],{"className":125776,"style":1789},[119],[31,125778,17342],{"className":125779,"style":2666},[75,76],[31,125781],{"className":125782,"style":293},[119],[31,125784,125786,125789],{"className":125785},[297],[31,125787,125744],{"className":125788},[297],[31,125790,125792],{"className":125791},[1039],[31,125793,125795,125815],{"className":125794},[1043,1044],[31,125796,125798,125812],{"className":125797},[1048],[31,125799,125801],{"className":125800,"style":27736},[1052],[31,125802,125803,125806],{"style":2807},[31,125804],{"className":125805,"style":1061},[1060],[31,125807,125809],{"className":125808},[1065,1066,1067,1068],[31,125810,919],{"className":125811,"style":949},[75,76,1068],[31,125813,1076],{"className":125814},[1075],[31,125816,125818],{"className":125817},[1048],[31,125819,125821],{"className":125820,"style":2050},[1052],[31,125822],{},[31,125824],{"className":125825,"style":293},[119],[31,125827,125829,125832],{"className":125828},[66],[31,125830],{"className":125831,"style":865},[70],[31,125833,8059],{"className":125834},[75,76],[11,125836,125837,125838,125867,125868,125896],{},"也就是：是否存在一个赋值 ",[31,125839,125841,125855],{"className":125840,"translate":35},[34],[31,125842,125844],{"className":125843},[39],[41,125845,125846],{"xmlns":43},[45,125847,125848,125852],{},[48,125849,125850],{},[51,125851,17342],{},[55,125853,125854],{"encoding":57},"\\sigma",[31,125856,125858],{"className":125857,"ariaHidden":62},[61],[31,125859,125861,125864],{"className":125860},[66],[31,125862],{"className":125863,"style":71},[70],[31,125865,17342],{"className":125866,"style":2666},[75,76],"，把每个变量映射到理论域里的值，使得 ",[31,125869,125871,125884],{"className":125870,"translate":35},[34],[31,125872,125874],{"className":125873},[39],[41,125875,125876],{"xmlns":43},[45,125877,125878,125882],{},[48,125879,125880],{},[51,125881,8059],{},[55,125883,11503],{"encoding":57},[31,125885,125887],{"className":125886,"ariaHidden":62},[61],[31,125888,125890,125893],{"className":125889},[66],[31,125891],{"className":125892,"style":865},[70],[31,125894,8059],{"className":125895},[75,76]," 在这个理论的所有公理下为真。",[11,125898,125899,125900,125972,125973,126038,126039,126067],{},"回到开头的例子：",[31,125901,125903,125924],{"className":125902,"translate":35},[34],[31,125904,125906],{"className":125905},[39],[41,125907,125908],{"xmlns":43},[45,125909,125910,125922],{},[48,125911,125912,125914,125916,125918,125920],{},[51,125913,53],{},[95,125915,434],{},[51,125917,3003],{},[95,125919,16040],{},[91,125921,19494],{},[55,125923,125640],{"encoding":57},[31,125925,125927,125945,125963],{"className":125926,"ariaHidden":62},[61],[31,125928,125930,125933,125936,125939,125942],{"className":125929},[66],[31,125931],{"className":125932,"style":307},[70],[31,125934,53],{"className":125935},[75,76],[31,125937],{"className":125938,"style":120},[119],[31,125940,434],{"className":125941},[124],[31,125943],{"className":125944,"style":120},[119],[31,125946,125948,125951,125954,125957,125960],{"className":125947},[66],[31,125949],{"className":125950,"style":53323},[70],[31,125952,3003],{"className":125953,"style":2666},[75,76],[31,125955],{"className":125956,"style":293},[119],[31,125958,16040],{"className":125959},[297],[31,125961],{"className":125962,"style":293},[119],[31,125964,125966,125969],{"className":125965},[66],[31,125967],{"className":125968,"style":326},[70],[31,125970,19494],{"className":125971},[75]," 是算术原子，",[31,125974,125976,125999],{"className":125975,"translate":35},[34],[31,125977,125979],{"className":125978},[39],[41,125980,125981],{"xmlns":43},[45,125982,125983,125997],{},[48,125984,125985,125987,125989,125991,125993,125995],{},[51,125986,5813],{},[95,125988,15175],{"stretchy":210},[91,125990,1977],{},[95,125992,15184],{"stretchy":210},[95,125994,223],{},[91,125996,7853],{},[55,125998,125644],{"encoding":57},[31,126000,126002,126029],{"className":126001,"ariaHidden":62},[61],[31,126003,126005,126008,126011,126014,126017,126020,126023,126026],{"className":126004},[66],[31,126006],{"className":126007,"style":260},[70],[31,126009,5813],{"className":126010},[75,76],[31,126012,15175],{"className":126013},[264],[31,126015,1977],{"className":126016},[75],[31,126018,15184],{"className":126019},[289],[31,126021],{"className":126022,"style":293},[119],[31,126024,223],{"className":126025},[297],[31,126027],{"className":126028,"style":293},[119],[31,126030,126032,126035],{"className":126031},[66],[31,126033],{"className":126034,"style":326},[70],[31,126036,7853],{"className":126037},[75]," 是数组原子，理论 ",[31,126040,126042,126055],{"className":126041,"translate":35},[34],[31,126043,126045],{"className":126044},[39],[41,126046,126047],{"xmlns":43},[45,126048,126049,126053],{},[48,126050,126051],{},[51,126052,919],{},[55,126054,919],{"encoding":57},[31,126056,126058],{"className":126057,"ariaHidden":62},[61],[31,126059,126061,126064],{"className":126060},[66],[31,126062],{"className":126063,"style":945},[70],[31,126065,919],{"className":126066,"style":949},[75,76]," 就是整数算术与数组的组合。SMT 要回答的，正是这组原子能不能同时为真。",[52860,126069,126071],{"id":126070},"_12-公式的样子","1.2 公式的样子",[11,126073,126074],{},"SMT 公式由布尔结构和理论原子组成：",[31,126076,126078],{"className":126077,"translate":35},[411],[31,126079,126081,126168],{"className":126080,"translate":35},[34],[31,126082,126084],{"className":126083},[39],[41,126085,126086],{"xmlns":43,"display":420},[45,126087,126088,126165],{},[48,126089,126090,126092,126094,126096,126098,126100,126102,126104,126106,126109,126111,126113,126115,126117,126119,126121,126123,126126,126128,126130,126132,126134,126136,126138,126140,126142,126144,126146,126148,126151,126153,126155,126157,126159,126161,126163],{},[51,126091,8059],{},[95,126093,922],{},[95,126095,922],{},[95,126097,223],{},[51,126099,20800],{"mathvariant":494},[1430,126101,125504],{},[95,126103,12680],{},[1430,126105,125504],{},[51,126107,126108],{"mathvariant":494},"⊥",[1430,126110,125504],{},[95,126112,12680],{},[1430,126114,125504],{},[51,126116,18446],{},[1430,126118,125504],{},[95,126120,12680],{},[1430,126122,125504],{},[51,126124,126125],{"mathvariant":494},"¬",[51,126127,8059],{},[1430,126129,125504],{},[95,126131,12680],{},[1430,126133,125504],{},[51,126135,8059],{},[95,126137,125507],{},[51,126139,8059],{},[1430,126141,125504],{},[95,126143,12680],{},[1430,126145,125504],{},[51,126147,8059],{},[95,126149,126150],{},"∨",[51,126152,8059],{},[1430,126154,125504],{},[95,126156,12680],{},[1430,126158,125504],{},[51,126160,8059],{},[95,126162,929],{},[51,126164,8059],{},[55,126166,126167],{"encoding":57},"\\varphi ::= \\top \\;\\mid\\; \\bot \\;\\mid\\; A \\;\\mid\\; \\neg\\varphi \\;\\mid\\; \\varphi \\wedge \\varphi \\;\\mid\\; \\varphi \\vee \\varphi \\;\\mid\\; \\varphi \\rightarrow \\varphi",[31,126169,126171,126190,126214,126238,126262,126289,126308,126332,126350,126374,126392],{"className":126170,"ariaHidden":62},[61],[31,126172,126174,126177,126180,126183,126187],{"className":126173},[66],[31,126175],{"className":126176,"style":865},[70],[31,126178,8059],{"className":126179},[75,76],[31,126181],{"className":126182,"style":293},[119],[31,126184,126186],{"className":126185},[297],"::=",[31,126188],{"className":126189,"style":293},[119],[31,126191,126193,126196,126199,126202,126205,126208,126211],{"className":126192},[66],[31,126194],{"className":126195,"style":260},[70],[31,126197,20800],{"className":126198},[75],[31,126200],{"className":126201,"style":293},[119],[31,126203],{"className":126204,"style":293},[119],[31,126206,12680],{"className":126207},[297],[31,126209],{"className":126210,"style":293},[119],[31,126212],{"className":126213,"style":293},[119],[31,126215,126217,126220,126223,126226,126229,126232,126235],{"className":126216},[66],[31,126218],{"className":126219,"style":260},[70],[31,126221,126108],{"className":126222},[75],[31,126224],{"className":126225,"style":293},[119],[31,126227],{"className":126228,"style":293},[119],[31,126230,12680],{"className":126231},[297],[31,126233],{"className":126234,"style":293},[119],[31,126236],{"className":126237,"style":293},[119],[31,126239,126241,126244,126247,126250,126253,126256,126259],{"className":126240},[66],[31,126242],{"className":126243,"style":260},[70],[31,126245,18446],{"className":126246},[75,76],[31,126248],{"className":126249,"style":293},[119],[31,126251],{"className":126252,"style":293},[119],[31,126254,12680],{"className":126255},[297],[31,126257],{"className":126258,"style":293},[119],[31,126260],{"className":126261,"style":293},[119],[31,126263,126265,126268,126271,126274,126277,126280,126283,126286],{"className":126264},[66],[31,126266],{"className":126267,"style":260},[70],[31,126269,126125],{"className":126270},[75],[31,126272,8059],{"className":126273},[75,76],[31,126275],{"className":126276,"style":293},[119],[31,126278],{"className":126279,"style":293},[119],[31,126281,12680],{"className":126282},[297],[31,126284],{"className":126285,"style":293},[119],[31,126287],{"className":126288,"style":293},[119],[31,126290,126292,126296,126299,126302,126305],{"className":126291},[66],[31,126293],{"className":126294,"style":126295},[70],"height:0.75em;vertical-align:-0.1944em;",[31,126297,8059],{"className":126298},[75,76],[31,126300],{"className":126301,"style":120},[119],[31,126303,125507],{"className":126304},[124],[31,126306],{"className":126307,"style":120},[119],[31,126309,126311,126314,126317,126320,126323,126326,126329],{"className":126310},[66],[31,126312],{"className":126313,"style":260},[70],[31,126315,8059],{"className":126316},[75,76],[31,126318],{"className":126319,"style":293},[119],[31,126321],{"className":126322,"style":293},[119],[31,126324,12680],{"className":126325},[297],[31,126327],{"className":126328,"style":293},[119],[31,126330],{"className":126331,"style":293},[119],[31,126333,126335,126338,126341,126344,126347],{"className":126334},[66],[31,126336],{"className":126337,"style":126295},[70],[31,126339,8059],{"className":126340},[75,76],[31,126342],{"className":126343,"style":120},[119],[31,126345,126150],{"className":126346},[124],[31,126348],{"className":126349,"style":120},[119],[31,126351,126353,126356,126359,126362,126365,126368,126371],{"className":126352},[66],[31,126354],{"className":126355,"style":260},[70],[31,126357,8059],{"className":126358},[75,76],[31,126360],{"className":126361,"style":293},[119],[31,126363],{"className":126364,"style":293},[119],[31,126366,12680],{"className":126367},[297],[31,126369],{"className":126370,"style":293},[119],[31,126372],{"className":126373,"style":293},[119],[31,126375,126377,126380,126383,126386,126389],{"className":126376},[66],[31,126378],{"className":126379,"style":865},[70],[31,126381,8059],{"className":126382},[75,76],[31,126384],{"className":126385,"style":293},[119],[31,126387,929],{"className":126388},[297],[31,126390],{"className":126391,"style":293},[119],[31,126393,126395,126398],{"className":126394},[66],[31,126396],{"className":126397,"style":865},[70],[31,126399,8059],{"className":126400},[75,76],[11,126402,50599,126403,126431,126432,126435,126436,15282,126570,15282,126703,15282,126851,126946],{},[31,126404,126406,126419],{"className":126405,"translate":35},[34],[31,126407,126409],{"className":126408},[39],[41,126410,126411],{"xmlns":43},[45,126412,126413,126417],{},[48,126414,126415],{},[51,126416,18446],{},[55,126418,18446],{"encoding":57},[31,126420,126422],{"className":126421,"ariaHidden":62},[61],[31,126423,126425,126428],{"className":126424},[66],[31,126426],{"className":126427,"style":945},[70],[31,126429,18446],{"className":126430},[75,76]," 是",[7246,126433,126434],{},"理论原子","，形如 ",[31,126437,126439,126465],{"className":126438,"translate":35},[34],[31,126440,126442],{"className":126441},[39],[41,126443,126444],{"xmlns":43},[45,126445,126446,126462],{},[48,126447,126448,126454,126456],{},[1004,126449,126450,126452],{},[51,126451,12271],{},[91,126453,205],{},[95,126455,223],{},[1004,126457,126458,126460],{},[51,126459,12271],{},[91,126461,152],{},[55,126463,126464],{"encoding":57},"t_1 = t_2",[31,126466,126468,126524],{"className":126467,"ariaHidden":62},[61],[31,126469,126471,126475,126515,126518,126521],{"className":126470},[66],[31,126472],{"className":126473,"style":126474},[70],"height:0.7651em;vertical-align:-0.15em;",[31,126476,126478,126481],{"className":126477},[75],[31,126479,12271],{"className":126480},[75,76],[31,126482,126484],{"className":126483},[1039],[31,126485,126487,126507],{"className":126486},[1043,1044],[31,126488,126490,126504],{"className":126489},[1048],[31,126491,126493],{"className":126492,"style":2028},[1052],[31,126494,126495,126498],{"style":2807},[31,126496],{"className":126497,"style":1061},[1060],[31,126499,126501],{"className":126500},[1065,1066,1067,1068],[31,126502,205],{"className":126503},[75,1068],[31,126505,1076],{"className":126506},[1075],[31,126508,126510],{"className":126509},[1048],[31,126511,126513],{"className":126512,"style":2050},[1052],[31,126514],{},[31,126516],{"className":126517,"style":293},[119],[31,126519,223],{"className":126520},[297],[31,126522],{"className":126523,"style":293},[119],[31,126525,126527,126530],{"className":126526},[66],[31,126528],{"className":126529,"style":126474},[70],[31,126531,126533,126536],{"className":126532},[75],[31,126534,12271],{"className":126535},[75,76],[31,126537,126539],{"className":126538},[1039],[31,126540,126542,126562],{"className":126541},[1043,1044],[31,126543,126545,126559],{"className":126544},[1048],[31,126546,126548],{"className":126547,"style":2028},[1052],[31,126549,126550,126553],{"style":2807},[31,126551],{"className":126552,"style":1061},[1060],[31,126554,126556],{"className":126555},[1065,1066,1067,1068],[31,126557,152],{"className":126558},[75,1068],[31,126560,1076],{"className":126561},[1075],[31,126563,126565],{"className":126564},[1048],[31,126566,126568],{"className":126567,"style":2050},[1052],[31,126569],{},[31,126571,126573,126599],{"className":126572,"translate":35},[34],[31,126574,126576],{"className":126575},[39],[41,126577,126578],{"xmlns":43},[45,126579,126580,126596],{},[48,126581,126582,126588,126590],{},[1004,126583,126584,126586],{},[51,126585,12271],{},[91,126587,205],{},[95,126589,9758],{},[1004,126591,126592,126594],{},[51,126593,12271],{},[91,126595,152],{},[55,126597,126598],{"encoding":57},"t_1 \u003C t_2",[31,126600,126602,126657],{"className":126601,"ariaHidden":62},[61],[31,126603,126605,126608,126648,126651,126654],{"className":126604},[66],[31,126606],{"className":126607,"style":126474},[70],[31,126609,126611,126614],{"className":126610},[75],[31,126612,12271],{"className":126613},[75,76],[31,126615,126617],{"className":126616},[1039],[31,126618,126620,126640],{"className":126619},[1043,1044],[31,126621,126623,126637],{"className":126622},[1048],[31,126624,126626],{"className":126625,"style":2028},[1052],[31,126627,126628,126631],{"style":2807},[31,126629],{"className":126630,"style":1061},[1060],[31,126632,126634],{"className":126633},[1065,1066,1067,1068],[31,126635,205],{"className":126636},[75,1068],[31,126638,1076],{"className":126639},[1075],[31,126641,126643],{"className":126642},[1048],[31,126644,126646],{"className":126645,"style":2050},[1052],[31,126647],{},[31,126649],{"className":126650,"style":293},[119],[31,126652,9758],{"className":126653},[297],[31,126655],{"className":126656,"style":293},[119],[31,126658,126660,126663],{"className":126659},[66],[31,126661],{"className":126662,"style":126474},[70],[31,126664,126666,126669],{"className":126665},[75],[31,126667,12271],{"className":126668},[75,76],[31,126670,126672],{"className":126671},[1039],[31,126673,126675,126695],{"className":126674},[1043,1044],[31,126676,126678,126692],{"className":126677},[1048],[31,126679,126681],{"className":126680,"style":2028},[1052],[31,126682,126683,126686],{"style":2807},[31,126684],{"className":126685,"style":1061},[1060],[31,126687,126689],{"className":126688},[1065,1066,1067,1068],[31,126690,152],{"className":126691},[75,1068],[31,126693,1076],{"className":126694},[1075],[31,126696,126698],{"className":126697},[1048],[31,126699,126701],{"className":126700,"style":2050},[1052],[31,126702],{},[31,126704,126706,126738],{"className":126705,"translate":35},[34],[31,126707,126709],{"className":126708},[39],[41,126710,126711],{"xmlns":43},[45,126712,126713,126735],{},[48,126714,126715,126717,126719,126725,126727,126729],{},[51,126716,1662],{},[95,126718,211],{"stretchy":210},[1004,126720,126721,126723],{},[51,126722,12271],{},[91,126724,205],{},[95,126726,220],{"stretchy":210},[95,126728,223],{},[1004,126730,126731,126733],{},[51,126732,12271],{},[91,126734,152],{},[55,126736,126737],{"encoding":57},"f(t_1) = 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属于",[7246,127709,127710],{},"线性整数算术","：它知道 ",[31,127713,127715,127728],{"className":127714,"translate":35},[34],[31,127716,127718],{"className":127717},[39],[41,127719,127720],{"xmlns":43},[45,127721,127722,127726],{},[48,127723,127724],{},[95,127725,434],{},[55,127727,434],{"encoding":57},[31,127729,127731],{"className":127730,"ariaHidden":62},[61],[31,127732,127734,127737],{"className":127733},[66],[31,127735],{"className":127736,"style":307},[70],[31,127738,434],{"className":127739},[75]," 是加法、",[31,127742,127744,127757],{"className":127743,"translate":35},[34],[31,127745,127747],{"className":127746},[39],[41,127748,127749],{"xmlns":43},[45,127750,127751,127755],{},[48,127752,127753],{},[95,127754,16040],{},[55,127756,16040],{"encoding":57},[31,127758,127760],{"className":127759,"ariaHidden":62},[61],[31,127761,127763,127766],{"className":127762},[66],[31,127764],{"className":127765,"style":21113},[70],[31,127767,16040],{"className":127768},[297]," 是全序；",[127633,127771,127772,127707,127837,127711,127840,2446,127842,127845],{},[31,127773,127775,127798],{"className":127774,"translate":35},[34],[31,127776,127778],{"className":127777},[39],[41,127779,127780],{"xmlns":43},[45,127781,127782,127796],{},[48,127783,127784,127786,127788,127790,127792,127794],{},[51,127785,5813],{},[95,127787,15175],{"stretchy":210},[91,127789,1977],{},[95,127791,15184],{"stretchy":210},[95,127793,223],{},[91,127795,7853],{},[55,127797,125644],{"encoding":57},[31,127799,127801,127828],{"className":127800,"ariaHidden":62},[61],[31,127802,127804,127807,127810,127813,127816,127819,127822,127825],{"className":127803},[66],[31,127805],{"className":127806,"style":260},[70],[31,127808,5813],{"className":127809},[75,76],[31,127811,15175],{"className":127812},[264],[31,127814,1977],{"className":127815},[75],[31,127817,15184],{"className":127818},[289],[31,127820],{"className":127821,"style":293},[119],[31,127823,223],{"className":127824},[297],[31,127826],{"className":127827,"style":293},[119],[31,127829,127831,127834],{"className":127830},[66],[31,127832],{"className":127833,"style":326},[70],[31,127835,7853],{"className":127836},[75],[7246,127838,127839],{},"数组理论",[45608,127841,126909],{},[45608,127843,127844],{},"store"," 的读写语义；",[127633,127847,127848,127707,127962,127965,127966,127994],{},[31,127849,127851,127881],{"className":127850,"translate":35},[34],[31,127852,127854],{"className":127853},[39],[41,127855,127856],{"xmlns":43},[45,127857,127858,127878],{},[48,127859,127860,127862,127864,127866,127868,127870,127872,127874,127876],{},[51,127861,1662],{},[95,127863,211],{"stretchy":210},[51,127865,53],{},[95,127867,220],{"stretchy":210},[95,127869,21271],{"mathvariant":494},[51,127871,1662],{},[95,127873,211],{"stretchy":210},[51,127875,3003],{},[95,127877,220],{"stretchy":210},[55,127879,127880],{"encoding":57},"f(x) \\ne f(y)",[31,127882,127884,127944],{"className":127883,"ariaHidden":62},[61],[31,127885,127887,127890,127893,127896,127899,127902,127905,127941],{"className":127886},[66],[31,127888],{"className":127889,"style":260},[70],[31,127891,1662],{"className":127892,"style":35196},[75,76],[31,127894,211],{"className":127895},[264],[31,127897,53],{"className":127898},[75,76],[31,127900,220],{"className":127901},[289],[31,127903],{"className":127904,"style":293},[119],[31,127906,127908,127935,127938],{"className":127907},[297],[31,127909,127911],{"className":127910},[297],[31,127912,127914],{"className":127913},[75,21516],[31,127915,127917],{"className":127916},[21520],[31,127918,127920,127923,127932],{"className":127919},[21524],[31,127921],{"className":127922,"style":4202},[70],[31,127924,127926],{"className":127925},[21531],[31,127927,127929],{"className":127928},[75],[31,127930,21538],{"className":127931},[297],[31,127933],{"className":127934},[21542],[31,127936],{"className":127937},[119,21546],[31,127939,223],{"className":127940},[297],[31,127942],{"className":127943,"style":293},[119],[31,127945,127947,127950,127953,127956,127959],{"className":127946},[66],[31,127948],{"className":127949,"style":260},[70],[31,127951,1662],{"className":127952,"style":35196},[75,76],[31,127954,211],{"className":127955},[264],[31,127957,3003],{"className":127958,"style":2666},[75,76],[31,127960,220],{"className":127961},[289],[7246,127963,127964],{},"未解释函数","：它不知道 ",[31,127967,127969,127982],{"className":127968,"translate":35},[34],[31,127970,127972],{"className":127971},[39],[41,127973,127974],{"xmlns":43},[45,127975,127976,127980],{},[48,127977,127978],{},[51,127979,1662],{},[55,127981,1662],{"encoding":57},[31,127983,127985],{"className":127984,"ariaHidden":62},[61],[31,127986,127988,127991],{"className":127987},[66],[31,127989],{"className":127990,"style":4202},[70],[31,127992,1662],{"className":127993,"style":35196},[75,76]," 具体算什么，只知道「同一个输入必须得到同一个输出」。",[11,127996,127997,127998,128053],{},"对 SAT 来说，这三个原子只是三个裸的布尔变量 ",[31,127999,128001,128023],{"className":128000,"translate":35},[34],[31,128002,128004],{"className":128003},[39],[41,128005,128006],{"xmlns":43},[45,128007,128008,128020],{},[48,128009,128010,128012,128014,128016,128018],{},[51,128011,11],{},[95,128013,1210],{"separator":62},[51,128015,27666],{},[95,128017,1210],{"separator":62},[51,128019,2549],{},[55,128021,128022],{"encoding":57},"p, q, r",[31,128024,128026],{"className":128025,"ariaHidden":62},[61],[31,128027,128029,128032,128035,128038,128041,128044,128047,128050],{"className":128028},[66],[31,128030],{"className":128031,"style":865},[70],[31,128033,11],{"className":128034},[75,76],[31,128036,1210],{"className":128037},[1391],[31,128039],{"className":128040,"style":1467},[119],[31,128042,27666],{"className":128043,"style":2666},[75,76],[31,128045,1210],{"className":128046},[1391],[31,128048],{"className":128049,"style":1467},[119],[31,128051,2549],{"className":128052,"style":19759},[75,76],"。对 SMT 来说，它们各自带着一整套内部结构。",[128055,128056,128057],"div",{},[128058,128059],"img",{"height":128060,"width":128061,"style":128062,"data-d2-theme":128063,"alt":52812,"src":128064},614,552,"max-width:100%;height:auto","default","data:image\u002Fsvg+xml;base64,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\u002FmgKfnrpfmnK88L3RzcGFuPjwvdGV4dD48L2c+PGcgY2xhc3M9IllUST0gYXRvbSI+PHBhdGggZmlsbD0iI2ZmZjhlNiIgc3Ryb2tlPSIjYzlhMjI3IiBkPSJNMjI1IDM2NGg5NnY4MmgtOTZ6IiBjbGFzcz0ic2hhcGUiIHN0eWxlPSJzdHJva2Utd2lkdGg6MiIvPjx0ZXh0IHg9IjI3MyIgeT0iNDAyLjUiIGZpbGw9IiMwYTBmMjUiIGNsYXNzPSJ0ZXh0LWJvbGQgZmlsbC1OMSIgc3R5bGU9InRleHQtYW5jaG9yOm1pZGRsZTtmb250LXNpemU6MTZweCI+PHRzcGFuIHg9IjI3MyIgZHk9IjAiPmFbMF0gPSA1PC90c3Bhbj48dHNwYW4geD0iMjczIiBkeT0iMTguNSI+5pWw57uEPC90c3Bhbj48L3RleHQ+PC9nPjxnIGNsYXNzPSJZVE09IGF0b20iPjxwYXRoIGZpbGw9IiNmZmY4ZTYiIHN0cm9rZT0iI2M5YTIyNyIgZD0iTTAgMzY0aDE0NXY4MkgweiIgY2xhc3M9InNoYXBlIiBzdHlsZT0ic3Ryb2tlLXdpZHRoOjIiLz48dGV4dCB4PSI3Mi41IiB5PSI0MDIuNSIgZmlsbD0iIzBhMGYyNSIgY2xhc3M9InRleHQtYm9sZCBmaWxsLU4xIiBzdHlsZT0idGV4dC1hbmNob3I6bWlkZGxlO2ZvbnQtc2l6ZToxNnB4Ij48dHNwYW4geD0iNzIuNSIgZHk9IjAiPmYoeCkg4omgIGYoeSk8L3RzcGFuPjx0c3BhbiB4PSI3Mi41IiBkeT0iMTguNSI+5pyq6Kej6YeK5Ye95pWwPC90c3Bhbj48L3RleHQ+PC9nPjxnIGNsYXNzPSJjekU9IHNvbHZlciI+PHBhdGggZmlsbD0iI2VhZjdlZSIgc3Ryb2tlPSIjNGE5ZDZiIiBkPSJNNDA1IDU0NmgxNDV2NjZINDA1eiIgY2xhc3M9InNoYXBlIiBzdHlsZT0ic3Ryb2tlLXdpZHRoOjIiLz48dGV4dCB4PSI0NzcuNSIgeT0iNTg0LjUiIGZpbGw9IiMwYTBmMjUiIGNsYXNzPSJ0ZXh0LWJvbGQgZmlsbC1OMSIgc3R5bGU9InRleHQtYW5jaG9yOm1pZGRsZTtmb250LXNpemU6MTZweCI+566X5pyv5rGC6Kej5ZmoPC90ZXh0PjwvZz48ZyBjbGFzcz0iY3pJPSBzb2x2ZXIiPjxwYXRoIGZpbGw9IiNlYWY3ZWUiIHN0cm9rZT0iIzRhOWQ2YiIgZD0iTTIwMCA1NDZoMTQ1djY2SDIwMHoiIGNsYXNzPSJzaGFwZSIgc3R5bGU9InN0cm9rZS13aWR0aDoyIi8+PHRleHQgeD0iMjcyLjUiIHk9IjU4NC41IiBmaWxsPSIjMGEwZjI1IiBjbGFzcz0idGV4dC1ib2xkIGZpbGwtTjEiIHN0eWxlPSJ0ZXh0LWFuY2hvcjptaWRkbGU7Zm9udC1zaXplOjE2cHgiPuaVsOe7hOaxguino+WZqDwvdGV4dD48L2c+PGcgY2xhc3M9ImN6TT0gc29sdmVyIj48cGF0aCBmaWxsPSIjZWFmN2VlIiBzdHJva2U9IiM0YTlkNmIiIGQ9Ik01IDU0NmgxMzV2NjZINXoiIGNsYXNzPSJzaGFwZSIgc3R5bGU9InN0cm9rZS13aWR0aDoyIi8+PHRleHQgeD0iNzIuNSIgeT0iNTg0LjUiIGZpbGw9IiMwYTBmMjUiIGNsYXNzPSJ0ZXh0LWJvbGQgZmlsbC1OMSIgc3R5bGU9InRleHQtYW5jaG9yOm1pZGRsZTtmb250LXNpemU6MTZweCI+RVVGIOaxguino+WZqDwvdGV4dD48L2c+PGcgY2xhc3M9IktHWnZjbTExYkdFZ0xTWm5kRHNnYzJ0bGJHVjBiMjRwV3pCZCI+PG1hcmtlciBpZD0ibWstZDItMTQ1MDQ0NDU2Ni0zNDg4Mzc4MTM0IiBtYXJrZXJIZWlnaHQ9IjEyIiBtYXJrZXJVbml0cz0idXNlclNwYWNlT25Vc2UiIG1hcmtlcldpZHRoPSIxMCIgb3JpZW50PSJhdXRvIiByZWZYPSI3IiByZWZZPSI2IiB2aWV3Qm94PSIwIDAgMTAgMTIiPjxwYXRoIHN0cm9rZS13aWR0aD0iMiIgZD0ibTAgMCAxMCA2LTEwIDZ6IiBjbGFzcz0iY29ubmVjdGlvbiIgc3R5bGU9ImZpbGw6IzBkMzJiMiIvPjwvbWFya2VyPjxwYXRoIGZpbGw9Im5vbmUiIHN0cm9rZT0iIzBkMzJiMiIgbWFya2VyLWVuZD0idXJsKCNtay1kMi0xNDUwNDQ0NTY2LTM0ODgzNzgxMzQpIiBkPSJNMjcyLjUgODR2OTQiIGNsYXNzPSJjb25uZWN0aW9uIHN0cm9rZS1CMSIgbWFzaz0idXJsKCNkMi0xNDUwNDQ0NTY2KSIgc3R5bGU9InN0cm9rZS13aWR0aDoyIi8+PC9nPjxnIGNsYXNzPSJLSE5yWld4bGRHOXVJQzBtWjNRN0lHRXhLVnN3WFE9PSI+PHBhdGggZmlsbD0ibm9uZSIgc3Ryb2tlPSIjMGQzMmIyIiBtYXJrZXItZW5kPSJ1cmwoI21rLWQyLTE0NTA0NDQ1NjYtMzQ4ODM3ODEzNCkiIGQ9Ik0zNTguMzI1IDI2MC42NkM0NTMuMyAzMDMuMTY4IDQ3Ny41IDMyNCA0NzcuNSAzNjAiIGNsYXNzPSJjb25uZWN0aW9uIHN0cm9rZS1CMSIgbWFzaz0idXJsKCNkMi0xNDUwNDQ0NTY2KSIgc3R5bGU9InN0cm9rZS13aWR0aDoyIi8+PC9nPjxnIGNsYXNzPSJLSE5yWld4bGRHOXVJQzBtWjNRN0lHRXlLVnN3WFE9PSI+PHBhdGggZmlsbD0ibm9uZSIgc3Ryb2tlPSIjMGQzMmIyIiBtYXJrZXItZW5kPSJ1cmwoI21rLWQyLTE0NTA0NDQ1NjYtMzQ4ODM3ODEzNCkiIGQ9Ik0yNzIuNSAyNjZ2OTQiIGNsYXNzPSJjb25uZWN0aW9uIHN0cm9rZS1CMSIgbWFzaz0idXJsKCNkMi0xNDUwNDQ0NTY2KSIgc3R5bGU9InN0cm9rZS13aWR0aDoyIi8+PC9nPjxnIGNsYXNzPSJLSE5yWld4bGRHOXVJQzBtWjNRN0lHRXpLVnN3WFE9PSI+PHBhdGggZmlsbD0ibm9uZSIgc3Ryb2tlPSIjMGQzMmIyIiBtYXJrZXItZW5kPSJ1cmwoI21rLWQyLTE0NTA0NDQ1NjYtMzQ4ODM3ODEzNCkiIGQ9Ik0xODcuNjggMjYxLjU5M0M5NS44OTkgMzAzLjM1MyA3Mi41IDMyNCA3Mi41IDM2MCIgY2xhc3M9ImNvbm5lY3Rpb24gc3Ryb2tlLUIxIiBtYXNrPSJ1cmwoI2QyLTE0NTA0NDQ1NjYpIiBzdHlsZT0ic3Ryb2tlLXdpZHRoOjIiLz48L2c+PGcgY2xhc3M9IktHRXhJQzBtWjNRN0lITXhLVnN3WFE9PSI+PHBhdGggZmlsbD0ibm9uZSIgc3Ryb2tlPSIjMGQzMmIyIiBtYXJrZXItZW5kPSJ1cmwoI21rLWQyLTE0NTA0NDQ1NjYtMzQ4ODM3ODEzNCkiIGQ9Ik00NzcuNSA0NDh2OTQiIGNsYXNzPSJjb25uZWN0aW9uIHN0cm9rZS1CMSIgbWFzaz0idXJsKCNkMi0xNDUwNDQ0NTY2KSIgc3R5bGU9InN0cm9rZS13aWR0aDoyIi8+PC9nPjxnIGNsYXNzPSJLR0V5SUMwbVozUTdJSE15S1Zzd1hRPT0iPjxwYXRoIGZpbGw9Im5vbmUiIHN0cm9rZT0iIzBkMzJiMiIgbWFya2VyLWVuZD0idXJsKCNtay1kMi0xNDUwNDQ0NTY2LTM0ODgzNzgxMzQpIiBkPSJNMjcyLjUgNDQ4djk0IiBjbGFzcz0iY29ubmVjdGlvbiBzdHJva2UtQjEiIG1hc2s9InVybCgjZDItMTQ1MDQ0NDU2NikiIHN0eWxlPSJzdHJva2Utd2lkdGg6MiIvPjwvZz48ZyBjbGFzcz0iS0dFeklDMG1aM1E3SUhNektWc3dYUT09Ij48cGF0aCBmaWxsPSJub25lIiBzdHJva2U9IiMwZDMyYjIiIG1hcmtlci1lbmQ9InVybCgjbWstZDItMTQ1MDQ0NDU2Ni0zNDg4Mzc4MTM0KSIgZD0iTTcyLjUgNDQ4djk0IiBjbGFzcz0iY29ubmVjdGlvbiBzdHJva2UtQjEiIG1hc2s9InVybCgjZDItMTQ1MDQ0NDU2NikiIHN0eWxlPSJzdHJva2Utd2lkdGg6MiIvPjwvZz48bWFzayBpZD0iZDItMTQ1MDQ0NDU2NiIgd2lkdGg9IjU1MiIgaGVpZ2h0PSI2MTQiIHg9Ii0xIiB5PSItMSIgbWFza1VuaXRzPSJ1c2VyU3BhY2VPblVzZSI+PHBhdGggZmlsbD0iI2ZmZiIgZD0iTS0xLTFoNTUydjYxNEgtMXoiLz48L21hc2s+PC9zdmc+PC9zdmc+",[11,128066,128067,128068,2372],{},"这张图就是 SMT 的骨架：",[7246,128069,128070],{},"布尔层负责「怎么组合」，理论层负责「每个原子是什么意思」",[52860,128072,128074],{"id":128073},"_13-背景理论","1.3 背景理论",[11,128076,128077],{},"SMT 里常用的理论有这么几类：",[127630,128079,128080,128090,128166,128178,128247,128264,128273],{},[127633,128081,128082,128089],{},[7246,128083,128084,128085,128088],{},"EUF（",[45608,128086,128087],{},"QF_UF","）","：只处理相等和未解释函数，适合描述对象相等、指针别名。",[127633,128091,128092,128102,128103,15282,128131,128160,128161,128163,128164,2372],{},[7246,128093,128094,128095,128098,128099,128088],{},"线性算术（",[45608,128096,128097],{},"QF_LRA"," \u002F ",[45608,128100,128101],{},"QF_LIA","：处理 ",[31,128104,128106,128119],{"className":128105,"translate":35},[34],[31,128107,128109],{"className":128108},[39],[41,128110,128111],{"xmlns":43},[45,128112,128113,128117],{},[48,128114,128115],{},[95,128116,434],{},[55,128118,434],{"encoding":57},[31,128120,128122],{"className":128121,"ariaHidden":62},[61],[31,128123,128125,128128],{"className":128124},[66],[31,128126],{"className":128127,"style":307},[70],[31,128129,434],{"className":128130},[75],[31,128132,128134,128148],{"className":128133,"translate":35},[34],[31,128135,128137],{"className":128136},[39],[41,128138,128139],{"xmlns":43},[45,128140,128141,128145],{},[48,128142,128143],{},[95,128144,347],{},[55,128146,128147],{"encoding":57},"\\le",[31,128149,128151],{"className":128150,"ariaHidden":62},[61],[31,128152,128154,128157],{"className":128153},[66],[31,128155],{"className":128156,"style":385},[70],[31,128158,347],{"className":128159},[297]," 和常数；实数版是 ",[45608,128162,128097],{},"，整数版是 ",[45608,128165,128101],{},[127633,128167,128168,128177],{},[7246,128169,128170,128171,128098,128174,128088],{},"非线性算术（",[45608,128172,128173],{},"QF_NRA",[45608,128175,128176],{},"QF_NIA","：允许乘法；实数版可判定，整数版不可判定。",[127633,128179,128180,128186,128187,128246],{},[7246,128181,128182,128183,128088],{},"位向量（",[45608,128184,128185],{},"QF_BV","：描述机器整数和硬件电路，加法是模 ",[31,128188,128190,128208],{"className":128189,"translate":35},[34],[31,128191,128193],{"className":128192},[39],[41,128194,128195],{"xmlns":43},[45,128196,128197,128205],{},[48,128198,128199],{},[1180,128200,128201,128203],{},[91,128202,152],{},[51,128204,14559],{},[55,128206,128207],{"encoding":57},"2^n",[31,128209,128211],{"className":128210,"ariaHidden":62},[61],[31,128212,128214,128217],{"className":128213},[66],[31,128215],{"className":128216,"style":3628},[70],[31,128218,128220,128223],{"className":128219},[75],[31,128221,152],{"className":128222},[75],[31,128224,128226],{"className":128225},[1039],[31,128227,128229],{"className":128228},[1043],[31,128230,128232],{"className":128231},[1048],[31,128233,128235],{"className":128234,"style":3628},[1052],[31,128236,128237,128240],{"style":3631},[31,128238],{"className":128239,"style":1061},[1060],[31,128241,128243],{"className":128242},[1065,1066,1067,1068],[31,128244,14559],{"className":128245},[75,76,1068]," 的。",[127633,128248,128249,128258,128259,2446,128261,128263],{},[7246,128250,128251,128252,128098,128255,128088],{},"数组（",[45608,128253,128254],{},"QF_AX",[45608,128256,128257],{},"QF_AUFBV","：用 ",[45608,128260,126909],{},[45608,128262,127844],{}," 描述内存读写。",[127633,128265,128266,128272],{},[7246,128267,128268,128269,128088],{},"浮点（",[45608,128270,128271],{},"QF_FP","：描述 IEEE 754 浮点运算。",[127633,128274,128275,128278,128279,15282,128282,128285],{},[7246,128276,128277],{},"字符串","：描述 ",[45608,128280,128281],{},"concat",[45608,128283,128284],{},"contains"," 等操作，工业实现常常不完整。",[11,128287,128288,128291],{},[45608,128289,128290],{},"QF"," 是 quantifier-free 的缩写，意思是「无量词」。绝大多数 SMT 求解器面对的都是这个片段。",[11,128293,128294,128295,128323],{},"可以把 SAT 看成 SMT 的一个特例：当理论 ",[31,128296,128298,128311],{"className":128297,"translate":35},[34],[31,128299,128301],{"className":128300},[39],[41,128302,128303],{"xmlns":43},[45,128304,128305,128309],{},[48,128306,128307],{},[51,128308,919],{},[55,128310,919],{"encoding":57},[31,128312,128314],{"className":128313,"ariaHidden":62},[61],[31,128315,128317,128320],{"className":128316},[66],[31,128318],{"className":128319,"style":945},[70],[31,128321,919],{"className":128322,"style":949},[75,76]," 为空、所有原子都是裸命题时，SMT 就退化成 SAT。",[11,128325,128326],{},"反过来，SMT 也可以看成 SAT 的一次「语义升级」：布尔骨架不变，但每个命题不再是黑盒，而是有内部结构、可以被理论求解器检查和反驳的对象。",[11,128328,128329],{},"这个升级带来的好处，要放到具体的历史困境里才看得清楚。",[18,128331,128333],{"id":128332},"二smt-之前","二、SMT 之前",[52860,128335,128337],{"id":128336},"_21-位爆破","2.1 位爆破",[11,128339,128340,128341,128344],{},"SAT 只认布尔变量。要用它处理整数，标准做法叫",[7246,128342,128343],{},"位爆破","（bit-blasting）：把整数拆成一个个布尔变量，再把算术运算展开成逻辑门电路，最后用 Tseitin 变换把电路转成 CNF 子句。",[11,128346,128347,128348,128390],{},"看一个具体的小例子。取两个 4 位无符号整数 ",[31,128349,128351,128369],{"className":128350,"translate":35},[34],[31,128352,128354],{"className":128353},[39],[41,128355,128356],{"xmlns":43},[45,128357,128358,128366],{},[48,128359,128360,128362,128364],{},[51,128361,53],{},[95,128363,1210],{"separator":62},[51,128365,3003],{},[55,128367,128368],{"encoding":57},"x, y",[31,128370,128372],{"className":128371,"ariaHidden":62},[61],[31,128373,128375,128378,128381,128384,128387],{"className":128374},[66],[31,128376],{"className":128377,"style":865},[70],[31,128379,53],{"className":128380},[75,76],[31,128382,1210],{"className":128383},[1391],[31,128385],{"className":128386,"style":1467},[119],[31,128388,3003],{"className":128389,"style":2666},[75,76],"，约束是：",[31,128392,128394],{"className":128393,"translate":35},[411],[31,128395,128397,128431],{"className":128396,"translate":35},[34],[31,128398,128400],{"className":128399},[39],[41,128401,128402],{"xmlns":43,"display":420},[45,128403,128404,128428],{},[48,128405,128406,128408,128410,128412,128414,128416,128418,128420,128422,128424,128426],{},[51,128407,53],{},[95,128409,434],{},[51,128411,3003],{},[95,128413,223],{},[91,128415,19494],{},[1430,128417,125504],{},[95,128419,125507],{},[1430,128421,125504],{},[51,128423,53],{},[95,128425,16040],{},[51,128427,3003],{},[55,128429,128430],{"encoding":57},"x + y = 10 \\;\\wedge\\; x > y",[31,128432,128434,128452,128470,128494,128512],{"className":128433,"ariaHidden":62},[61],[31,128435,128437,128440,128443,128446,128449],{"className":128436},[66],[31,128438],{"className":128439,"style":307},[70],[31,128441,53],{"className":128442},[75,76],[31,128444],{"className":128445,"style":120},[119],[31,128447,434],{"className":128448},[124],[31,128450],{"className":128451,"style":120},[119],[31,128453,128455,128458,128461,128464,128467],{"className":128454},[66],[31,128456],{"className":128457,"style":865},[70],[31,128459,3003],{"className":128460,"style":2666},[75,76],[31,128462],{"className":128463,"style":293},[119],[31,128465,223],{"className":128466},[297],[31,128468],{"className":128469,"style":293},[119],[31,128471,128473,128476,128479,128482,128485,128488,128491],{"className":128472},[66],[31,128474],{"className":128475,"style":326},[70],[31,128477,19494],{"className":128478},[75],[31,128480],{"className":128481,"style":293},[119],[31,128483],{"className":128484,"style":120},[119],[31,128486,125507],{"className":128487},[124],[31,128489],{"className":128490,"style":293},[119],[31,128492],{"className":128493,"style":120},[119],[31,128495,128497,128500,128503,128506,128509],{"className":128496},[66],[31,128498],{"className":128499,"style":21113},[70],[31,128501,53],{"className":128502},[75,76],[31,128504],{"className":128505,"style":293},[119],[31,128507,16040],{"className":128508},[297],[31,128510],{"className":128511,"style":293},[119],[31,128513,128515,128518],{"className":128514},[66],[31,128516],{"className":128517,"style":865},[70],[31,128519,3003],{"className":128520,"style":2666},[75,76],[11,128522,128523,128524,2446,128732,128940,128941,129044,129045,129143],{},"SAT 看到的是 8 个布尔变量 ",[31,128525,128527,128563],{"className":128526,"translate":35},[34],[31,128528,128530],{"className":128529},[39],[41,128531,128532],{"xmlns":43},[45,128533,128534,128560],{},[48,128535,128536,128542,128548,128554],{},[1004,128537,128538,128540],{},[51,128539,53],{},[91,128541,93],{},[1004,128543,128544,128546],{},[51,128545,53],{},[91,128547,152],{},[1004,128549,128550,128552],{},[51,128551,53],{},[91,128553,205],{},[1004,128555,128556,128558],{},[51,128557,53],{},[91,128559,1977],{},[55,128561,128562],{"encoding":57},"x_3 x_2 x_1 x_0",[31,128564,128566],{"className":128565,"ariaHidden":62},[61],[31,128567,128569,128572,128612,128652,128692],{"className":128568},[66],[31,128570],{"className":128571,"style":7090},[70],[31,128573,128575,128578],{"className":128574},[75],[31,128576,53],{"className":128577},[75,76],[31,128579,128581],{"className":128580},[1039],[31,128582,128584,128604],{"className":128583},[1043,1044],[31,128585,128587,128601],{"className":128586},[1048],[31,128588,128590],{"className":128589,"style":2028},[1052],[31,128591,128592,128595],{"style":2807},[31,128593],{"className":128594,"style":1061},[1060],[31,128596,128598],{"className":128597},[1065,1066,1067,1068],[31,128599,93],{"className":128600},[75,1068],[31,128602,1076],{"className":128603},[1075],[31,128605,128607],{"className":128606},[1048],[31,128608,128610],{"className":128609,"style":2050},[1052],[31,128611],{},[31,128613,128615,128618],{"className":128614},[75],[31,128616,53],{"className":128617},[75,76],[31,128619,128621],{"className":128620},[1039],[31,128622,128624,128644],{"className":128623},[1043,1044],[31,128625,128627,128641],{"className":128626},[1048],[31,128628,128630],{"className":128629,"style":2028},[1052],[31,128631,128632,128635],{"style":2807},[31,128633],{"className":128634,"style":1061},[1060],[31,128636,128638],{"className":128637},[1065,1066,1067,1068],[31,128639,152],{"className":128640},[75,1068],[31,128642,1076],{"className":128643},[1075],[31,128645,128647],{"className":128646},[1048],[31,128648,128650],{"className":128649,"style":2050},[1052],[31,128651],{},[31,128653,128655,128658],{"className":128654},[75],[31,128656,53],{"className":128657},[75,76],[31,128659,128661],{"className":128660},[1039],[31,128662,128664,128684],{"className":128663},[1043,1044],[31,128665,128667,128681],{"className":128666},[1048],[31,128668,128670],{"className":128669,"style":2028},[1052],[31,128671,128672,128675],{"style":2807},[31,128673],{"className":128674,"style":1061},[1060],[31,128676,128678],{"className":128677},[1065,1066,1067,1068],[31,128679,205],{"className":128680},[75,1068],[31,128682,1076],{"className":128683},[1075],[31,128685,128687],{"className":128686},[1048],[31,128688,128690],{"className":128689,"style":2050},[1052],[31,128691],{},[31,128693,128695,128698],{"className":128694},[75],[31,128696,53],{"className":128697},[75,76],[31,128699,128701],{"className":128700},[1039],[31,128702,128704,128724],{"className":128703},[1043,1044],[31,128705,128707,128721],{"className":128706},[1048],[31,128708,128710],{"className":128709,"style":2028},[1052],[31,128711,128712,128715],{"style":2807},[31,128713],{"className":128714,"style":1061},[1060],[31,128716,128718],{"className":128717},[1065,1066,1067,1068],[31,128719,1977],{"className":128720},[75,1068],[31,128722,1076],{"className":128723},[1075],[31,128725,128727],{"className":128726},[1048],[31,128728,128730],{"className":128729,"style":2050},[1052],[31,128731],{},[31,128733,128735,128771],{"className":128734,"translate":35},[34],[31,128736,128738],{"className":128737},[39],[41,128739,128740],{"xmlns":43},[45,128741,128742,128768],{},[48,128743,128744,128750,128756,128762],{},[1004,128745,128746,128748],{},[51,128747,3003],{},[91,128749,93],{},[1004,128751,128752,128754],{},[51,128753,3003],{},[91,128755,152],{},[1004,128757,128758,128760],{},[51,128759,3003],{},[91,128761,205],{},[1004,128763,128764,128766],{},[51,128765,3003],{},[91,128767,1977],{},[55,128769,128770],{"encoding":57},"y_3 y_2 y_1 y_0",[31,128772,128774],{"className":128773,"ariaHidden":62},[61],[31,128775,128777,128780,128820,128860,128900],{"className":128776},[66],[31,128778],{"className":128779,"style":865},[70],[31,128781,128783,128786],{"className":128782},[75],[31,128784,3003],{"className":128785,"style":2666},[75,76],[31,128787,128789],{"className":128788},[1039],[31,128790,128792,128812],{"className":128791},[1043,1044],[31,128793,128795,128809],{"className":128794},[1048],[31,128796,128798],{"className":128797,"style":2028},[1052],[31,128799,128800,128803],{"style":5504},[31,128801],{"className":128802,"style":1061},[1060],[31,128804,128806],{"className":128805},[1065,1066,1067,1068],[31,128807,93],{"className":128808},[75,1068],[31,128810,1076],{"className":128811},[1075],[31,128813,128815],{"className":128814},[1048],[31,128816,128818],{"className":128817,"style":2050},[1052],[31,128819],{},[31,128821,128823,128826],{"className":128822},[75],[31,128824,3003],{"className":128825,"style":2666},[75,76],[31,128827,128829],{"className":128828},[1039],[31,128830,128832,128852],{"className":128831},[1043,1044],[31,128833,128835,128849],{"className":128834},[1048],[31,128836,128838],{"className":128837,"style":2028},[1052],[31,128839,128840,128843],{"style":5504},[31,128841],{"className":128842,"style":1061},[1060],[31,128844,128846],{"className":128845},[1065,1066,1067,1068],[31,128847,152],{"className":128848},[75,1068],[31,128850,1076],{"className":128851},[1075],[31,128853,128855],{"className":128854},[1048],[31,128856,128858],{"className":128857,"style":2050},[1052],[31,128859],{},[31,128861,128863,128866],{"className":128862},[75],[31,128864,3003],{"className":128865,"style":2666},[75,76],[31,128867,128869],{"className":128868},[1039],[31,128870,128872,128892],{"className":128871},[1043,1044],[31,128873,128875,128889],{"className":128874},[1048],[31,128876,128878],{"className":128877,"style":2028},[1052],[31,128879,128880,128883],{"style":5504},[31,128881],{"className":128882,"style":1061},[1060],[31,128884,128886],{"className":128885},[1065,1066,1067,1068],[31,128887,205],{"className":128888},[75,1068],[31,128890,1076],{"className":128891},[1075],[31,128893,128895],{"className":128894},[1048],[31,128896,128898],{"className":128897,"style":2050},[1052],[31,128899],{},[31,128901,128903,128906],{"className":128902},[75],[31,128904,3003],{"className":128905,"style":2666},[75,76],[31,128907,128909],{"className":128908},[1039],[31,128910,128912,128932],{"className":128911},[1043,1044],[31,128913,128915,128929],{"className":128914},[1048],[31,128916,128918],{"className":128917,"style":2028},[1052],[31,128919,128920,128923],{"style":5504},[31,128921],{"className":128922,"style":1061},[1060],[31,128924,128926],{"className":128925},[1065,1066,1067,1068],[31,128927,1977],{"className":128928},[75,1068],[31,128930,1076],{"className":128931},[1075],[31,128933,128935],{"className":128934},[1048],[31,128936,128938],{"className":128937,"style":2050},[1052],[31,128939],{},"。为了表达加法，需要 4 个全加器。每个全加器有三个输入 ",[31,128942,128944,128974],{"className":128943,"translate":35},[34],[31,128945,128947],{"className":128946},[39],[41,128948,128949],{"xmlns":43},[45,128950,128951,128971],{},[48,128952,128953,128955,128957,128959,128961],{},[51,128954,5813],{},[95,128956,1210],{"separator":62},[51,128958,5834],{},[95,128960,1210],{"separator":62},[1004,128962,128963,128965],{},[51,128964,1412],{},[48,128966,128967,128969],{},[51,128968,5391],{},[51,128970,14559],{},[55,128972,128973],{"encoding":57},"a, b, c_{in}",[31,128975,128977],{"className":128976,"ariaHidden":62},[61],[31,128978,128980,128983,128986,128989,128992,128995,128998,129001],{"className":128979},[66],[31,128981],{"className":128982,"style":4202},[70],[31,128984,5813],{"className":128985},[75,76],[31,128987,1210],{"className":128988},[1391],[31,128990],{"className":128991,"style":1467},[119],[31,128993,5834],{"className":128994},[75,76],[31,128996,1210],{"className":128997},[1391],[31,128999],{"className":129000,"style":1467},[119],[31,129002,129004,129007],{"className":129003},[75],[31,129005,1412],{"className":129006},[75,76],[31,129008,129010],{"className":129009},[1039],[31,129011,129013,129036],{"className":129012},[1043,1044],[31,129014,129016,129033],{"className":129015},[1048],[31,129017,129019],{"className":129018,"style":5445},[1052],[31,129020,129021,129024],{"style":2807},[31,129022],{"className":129023,"style":1061},[1060],[31,129025,129027],{"className":129026},[1065,1066,1067,1068],[31,129028,129030],{"className":129029},[75,1068],[31,129031,1658],{"className":129032},[75,76,1068],[31,129034,1076],{"className":129035},[1075],[31,129037,129039],{"className":129038},[1048],[31,129040,129042],{"className":129041,"style":2050},[1052],[31,129043],{},"、两个输出 ",[31,129046,129048,129076],{"className":129047,"translate":35},[34],[31,129049,129051],{"className":129050},[39],[41,129052,129053],{"xmlns":43},[45,129054,129055,129073],{},[48,129056,129057,129059,129061],{},[51,129058,14551],{},[95,129060,1210],{"separator":62},[1004,129062,129063,129065],{},[51,129064,1412],{},[48,129066,129067,129069,129071],{},[51,129068,2552],{},[51,129070,5324],{},[51,129072,12271],{},[55,129074,129075],{"encoding":57},"s, c_{out}",[31,129077,129079],{"className":129078,"ariaHidden":62},[61],[31,129080,129082,129085,129088,129091,129094],{"className":129081},[66],[31,129083],{"className":129084,"style":865},[70],[31,129086,14551],{"className":129087},[75,76],[31,129089,1210],{"className":129090},[1391],[31,129092],{"className":129093,"style":1467},[119],[31,129095,129097,129100],{"className":129096},[75],[31,129098,1412],{"className":129099},[75,76],[31,129101,129103],{"className":129102},[1039],[31,129104,129106,129135],{"className":129105},[1043,1044],[31,129107,129109,129132],{"className":129108},[1048],[31,129110,129112],{"className":129111,"style":19220},[1052],[31,129113,129114,129117],{"style":2807},[31,129115],{"className":129116,"style":1061},[1060],[31,129118,129120],{"className":129119},[1065,1066,1067,1068],[31,129121,129123,129126,129129],{"className":129122},[75,1068],[31,129124,2552],{"className":129125},[75,76,1068],[31,129127,5324],{"className":129128},[75,76,1068],[31,129130,12271],{"className":129131},[75,76,1068],[31,129133,1076],{"className":129134},[1075],[31,129136,129138],{"className":129137},[1048],[31,129139,129141],{"className":129140,"style":2050},[1052],[31,129142],{},"，满足：",[31,129145,129147],{"className":129146,"translate":35},[411],[31,129148,129150,129241],{"className":129149,"translate":35},[34],[31,129151,129153],{"className":129152},[39],[41,129154,129155],{"xmlns":43,"display":420},[45,129156,129157,129238],{},[48,129158,129159,129161,129163,129165,129168,129170,129172,129182,129184,129186,129198,129200,129202,129204,129206,129208,129210,129212,129214,129224,129226,129228,129230,129232,129234,129236],{},[51,129160,14551],{},[95,129162,223],{},[51,129164,5813],{},[95,129166,129167],{},"⊕",[51,129169,5834],{},[95,129171,129167],{},[1004,129173,129174,129176],{},[51,129175,1412],{},[48,129177,129178,129180],{},[51,129179,5391],{},[51,129181,14559],{},[95,129183,1210],{"separator":62},[119,129185],{"width":20286},[1004,129187,129188,129190],{},[51,129189,1412],{},[48,129191,129192,129194,129196],{},[51,129193,2552],{},[51,129195,5324],{},[51,129197,12271],{},[95,129199,223],{},[95,129201,211],{"stretchy":210},[51,129203,5813],{},[95,129205,125507],{},[51,129207,5834],{},[95,129209,220],{"stretchy":210},[95,129211,126150],{},[95,129213,211],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[1004,129215,129216,129218],{},[51,129217,1412],{},[48,129219,129220,129222],{},[51,129221,5391],{},[51,129223,14559],{},[95,129225,125507],{},[95,129227,211],{"stretchy":210},[51,129229,5813],{},[95,129231,129167],{},[51,129233,5834],{},[95,129235,220],{"stretchy":210},[95,129237,220],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[55,129239,129240],{"encoding":57},"s = a \\oplus b \\oplus c_{in}, \\qquad c_{out} = (a \\wedge b) \\vee \\bigl(c_{in} \\wedge (a \\oplus b)\\bigr)",[31,129242,129244,129262,129280,129298,129414,129435,129456,129520,129541],{"className":129243,"ariaHidden":62},[61],[31,129245,129247,129250,129253,129256,129259],{"className":129246},[66],[31,129248],{"className":129249,"style":71},[70],[31,129251,14551],{"className":129252},[75,76],[31,129254],{"className":129255,"style":293},[119],[31,129257,223],{"className":129258},[297],[31,129260],{"className":129261,"style":293},[119],[31,129263,129265,129268,129271,129274,129277],{"className":129264},[66],[31,129266],{"className":129267,"style":307},[70],[31,129269,5813],{"className":129270},[75,76],[31,129272],{"className":129273,"style":120},[119],[31,129275,129167],{"className":129276},[124],[31,129278],{"className":129279,"style":120},[119],[31,129281,129283,129286,129289,129292,129295],{"className":129282},[66],[31,129284],{"className":129285,"style":123561},[70],[31,129287,5834],{"className":129288},[75,76],[31,129290],{"className":129291,"style":120},[119],[31,129293,129167],{"className":129294},[124],[31,129296],{"className":129297,"style":120},[119],[31,129299,129301,129304,129347,129350,129353,129356,129405,129408,129411],{"className":129300},[66],[31,129302],{"className":129303,"style":865},[70],[31,129305,129307,129310],{"className":129306},[75],[31,129308,1412],{"className":129309},[75,76],[31,129311,129313],{"className":129312},[1039],[31,129314,129316,129339],{"className":129315},[1043,1044],[31,129317,129319,129336],{"className":129318},[1048],[31,129320,129322],{"className":129321,"style":5445},[1052],[31,129323,129324,129327],{"style":2807},[31,129325],{"className":129326,"style":1061},[1060],[31,129328,129330],{"className":129329},[1065,1066,1067,1068],[31,129331,129333],{"className":129332},[75,1068],[31,129334,1658],{"className":129335},[75,76,1068],[31,129337,1076],{"className":129338},[1075],[31,129340,129342],{"className":129341},[1048],[31,129343,129345],{"className":129344,"style":2050},[1052],[31,129346],{},[31,129348,1210],{"className":129349},[1391],[31,129351],{"className":129352,"style":20525},[119],[31,129354],{"className":129355,"style":1467},[119],[31,129357,129359,129362],{"className":129358},[75],[31,129360,1412],{"className":129361},[75,76],[31,129363,129365],{"className":129364},[1039],[31,129366,129368,129397],{"className":129367},[1043,1044],[31,129369,129371,129394],{"className":129370},[1048],[31,129372,129374],{"className":129373,"style":19220},[1052],[31,129375,129376,129379],{"style":2807},[31,129377],{"className":129378,"style":1061},[1060],[31,129380,129382],{"className":129381},[1065,1066,1067,1068],[31,129383,129385,129388,129391],{"className":129384},[75,1068],[31,129386,2552],{"className":129387},[75,76,1068],[31,129389,5324],{"className":129390},[75,76,1068],[31,129392,12271],{"className":129393},[75,76,1068],[31,129395,1076],{"className":129396},[1075],[31,129398,129400],{"className":129399},[1048],[31,129401,129403],{"className":129402,"style":2050},[1052],[31,129404],{},[31,129406],{"className":129407,"style":293},[119],[31,129409,223],{"className":129410},[297],[31,129412],{"className":129413,"style":293},[119],[31,129415,129417,129420,129423,129426,129429,129432],{"className":129416},[66],[31,129418],{"className":129419,"style":260},[70],[31,129421,211],{"className":129422},[264],[31,129424,5813],{"className":129425},[75,76],[31,129427],{"className":129428,"style":120},[119],[31,129430,125507],{"className":129431},[124],[31,129433],{"className":129434,"style":120},[119],[31,129436,129438,129441,129444,129447,129450,129453],{"className":129437},[66],[31,129439],{"className":129440,"style":260},[70],[31,129442,5834],{"className":129443},[75,76],[31,129445,220],{"className":129446},[289],[31,129448],{"className":129449,"style":120},[119],[31,129451,126150],{"className":129452},[124],[31,129454],{"className":129455,"style":120},[119],[31,129457,129459,129462,129468,129511,129514,129517],{"className":129458},[66],[31,129460],{"className":129461,"style":4988},[70],[31,129463,129465],{"className":129464},[264],[31,129466,211],{"className":129467},[1307,1308],[31,129469,129471,129474],{"className":129470},[75],[31,129472,1412],{"className":129473},[75,76],[31,129475,129477],{"className":129476},[1039],[31,129478,129480,129503],{"className":129479},[1043,1044],[31,129481,129483,129500],{"className":129482},[1048],[31,129484,129486],{"className":129485,"style":5445},[1052],[31,129487,129488,129491],{"style":2807},[31,129489],{"className":129490,"style":1061},[1060],[31,129492,129494],{"className":129493},[1065,1066,1067,1068],[31,129495,129497],{"className":129496},[75,1068],[31,129498,1658],{"className":129499},[75,76,1068],[31,129501,1076],{"className":129502},[1075],[31,129504,129506],{"className":129505},[1048],[31,129507,129509],{"className":129508,"style":2050},[1052],[31,129510],{},[31,129512],{"className":129513,"style":120},[119],[31,129515,125507],{"className":129516},[124],[31,129518],{"className":129519,"style":120},[119],[31,129521,129523,129526,129529,129532,129535,129538],{"className":129522},[66],[31,129524],{"className":129525,"style":260},[70],[31,129527,211],{"className":129528},[264],[31,129530,5813],{"className":129531},[75,76],[31,129533],{"className":129534,"style":120},[119],[31,129536,129167],{"className":129537},[124],[31,129539],{"className":129540,"style":120},[119],[31,129542,129544,129547,129550,129553],{"className":129543},[66],[31,129545],{"className":129546,"style":4988},[70],[31,129548,5834],{"className":129549},[75,76],[31,129551,220],{"className":129552},[289],[31,129554,129556],{"className":129555},[289],[31,129557,220],{"className":129558},[1307,1308],[11,129560,129561],{},"把这两个等式转成 CNF，一个全加器大约 17 条子句。4 位加法器就是 4 个全加器，约 70 条子句。再加上「和等于 1010、进位为 0」的 5 条单元子句，以及一个 4 位比较器的几十条子句，整个约束变成一百多条子句、几十个辅助变量。",[11,129563,129564],{},"而原始约束只有一行：",[125323,129566,129568],{"className":125325,"code":129567,"language":1412,"meta":52812,"style":52812},"x + y == 10 && x > y\n",[45608,129569,129570],{"__ignoreMap":52812},[31,129571,129572,129575,129577,129579,129581,129583,129585,129588,129590],{"class":125331,"line":125332},[31,129573,129574],{"class":125343},"x ",[31,129576,434],{"class":125335},[31,129578,125420],{"class":125343},[31,129580,125439],{"class":125335},[31,129582,125425],{"class":125374},[31,129584,125428],{"class":125335},[31,129586,129587],{"class":125343}," x ",[31,129589,16040],{"class":125335},[31,129591,129592],{"class":125343}," y\n",[11,129594,129595,129596,15282,129648,129700],{},"求解器可能给出 ",[31,129597,129599,129618],{"className":129598,"translate":35},[34],[31,129600,129602],{"className":129601},[39],[41,129603,129604],{"xmlns":43},[45,129605,129606,129615],{},[48,129607,129608,129610,129612],{},[51,129609,53],{},[95,129611,223],{},[91,129613,129614],{},"0110",[55,129616,129617],{"encoding":57},"x = 0110",[31,129619,129621,129639],{"className":129620,"ariaHidden":62},[61],[31,129622,129624,129627,129630,129633,129636],{"className":129623},[66],[31,129625],{"className":129626,"style":71},[70],[31,129628,53],{"className":129629},[75,76],[31,129631],{"className":129632,"style":293},[119],[31,129634,223],{"className":129635},[297],[31,129637],{"className":129638,"style":293},[119],[31,129640,129642,129645],{"className":129641},[66],[31,129643],{"className":129644,"style":326},[70],[31,129646,129614],{"className":129647},[75],[31,129649,129651,129670],{"className":129650,"translate":35},[34],[31,129652,129654],{"className":129653},[39],[41,129655,129656],{"xmlns":43},[45,129657,129658,129667],{},[48,129659,129660,129662,129664],{},[51,129661,3003],{},[95,129663,223],{},[91,129665,129666],{},"0100",[55,129668,129669],{"encoding":57},"y = 0100",[31,129671,129673,129691],{"className":129672,"ariaHidden":62},[61],[31,129674,129676,129679,129682,129685,129688],{"className":129675},[66],[31,129677],{"className":129678,"style":865},[70],[31,129680,3003],{"className":129681,"style":2666},[75,76],[31,129683],{"className":129684,"style":293},[119],[31,129686,223],{"className":129687},[297],[31,129689],{"className":129690,"style":293},[119],[31,129692,129694,129697],{"className":129693},[66],[31,129695],{"className":129696,"style":326},[70],[31,129698,129666],{"className":129699},[75],"，但它并不知道这是 6 和 4。它只知道这一组真假值恰好让所有子句为真。",[11,129702,129703,129704,129707,129708,129803,129804,129832,129833,129861,129862,129832,129890,129918],{},"规模还不是最麻烦的，",[7246,129705,129706],{},"语义丢失","才是。假设我们想证明 ",[31,129709,129711,129737],{"className":129710,"translate":35},[34],[31,129712,129714],{"className":129713},[39],[41,129715,129716],{"xmlns":43},[45,129717,129718,129734],{},[48,129719,129720,129722,129724,129726,129728,129730,129732],{},[51,129721,53],{},[95,129723,434],{},[51,129725,3003],{},[95,129727,223],{},[51,129729,3003],{},[95,129731,434],{},[51,129733,53],{},[55,129735,129736],{"encoding":57},"x + y = y + x",[31,129738,129740,129758,129776,129794],{"className":129739,"ariaHidden":62},[61],[31,129741,129743,129746,129749,129752,129755],{"className":129742},[66],[31,129744],{"className":129745,"style":307},[70],[31,129747,53],{"className":129748},[75,76],[31,129750],{"className":129751,"style":120},[119],[31,129753,434],{"className":129754},[124],[31,129756],{"className":129757,"style":120},[119],[31,129759,129761,129764,129767,129770,129773],{"className":129760},[66],[31,129762],{"className":129763,"style":865},[70],[31,129765,3003],{"className":129766,"style":2666},[75,76],[31,129768],{"className":129769,"style":293},[119],[31,129771,223],{"className":129772},[297],[31,129774],{"className":129775,"style":293},[119],[31,129777,129779,129782,129785,129788,129791],{"className":129778},[66],[31,129780],{"className":129781,"style":9831},[70],[31,129783,3003],{"className":129784,"style":2666},[75,76],[31,129786],{"className":129787,"style":120},[119],[31,129789,434],{"className":129790},[124],[31,129792],{"className":129793,"style":120},[119],[31,129795,129797,129800],{"className":129796},[66],[31,129798],{"className":129799,"style":71},[70],[31,129801,53],{"className":129802},[75,76],"。在算术里这是一条公理，一秒都不用。但在 SAT 里，左边是一个「先加 ",[31,129805,129807,129820],{"className":129806,"translate":35},[34],[31,129808,129810],{"className":129809},[39],[41,129811,129812],{"xmlns":43},[45,129813,129814,129818],{},[48,129815,129816],{},[51,129817,53],{},[55,129819,53],{"encoding":57},[31,129821,129823],{"className":129822,"ariaHidden":62},[61],[31,129824,129826,129829],{"className":129825},[66],[31,129827],{"className":129828,"style":71},[70],[31,129830,53],{"className":129831},[75,76]," 再加 ",[31,129834,129836,129849],{"className":129835,"translate":35},[34],[31,129837,129839],{"className":129838},[39],[41,129840,129841],{"xmlns":43},[45,129842,129843,129847],{},[48,129844,129845],{},[51,129846,3003],{},[55,129848,3003],{"encoding":57},[31,129850,129852],{"className":129851,"ariaHidden":62},[61],[31,129853,129855,129858],{"className":129854},[66],[31,129856],{"className":129857,"style":865},[70],[31,129859,3003],{"className":129860,"style":2666},[75,76],"」的电路，右边是一个「先加 ",[31,129863,129865,129878],{"className":129864,"translate":35},[34],[31,129866,129868],{"className":129867},[39],[41,129869,129870],{"xmlns":43},[45,129871,129872,129876],{},[48,129873,129874],{},[51,129875,3003],{},[55,129877,3003],{"encoding":57},[31,129879,129881],{"className":129880,"ariaHidden":62},[61],[31,129882,129884,129887],{"className":129883},[66],[31,129885],{"className":129886,"style":865},[70],[31,129888,3003],{"className":129889,"style":2666},[75,76],[31,129891,129893,129906],{"className":129892,"translate":35},[34],[31,129894,129896],{"className":129895},[39],[41,129897,129898],{"xmlns":43},[45,129899,129900,129904],{},[48,129901,129902],{},[51,129903,53],{},[55,129905,53],{"encoding":57},[31,129907,129909],{"className":129908,"ariaHidden":62},[61],[31,129910,129912,129915],{"className":129911},[66],[31,129913],{"className":129914,"style":71},[70],[31,129916,53],{"className":129917},[75,76],"」的电路，两个电路内部结构不同，SAT 只能把两边都展开，再逐位证明输出相等。它不知道「加法可交换」，只知道一堆门。",[11,129920,129921],{},"打个比方：这就像为了判断一首诗是否押韵，先把它拆成一个个字母，再让机器从字母的排列里重新发现什么是韵脚。信息没有丢，但结构没了。",[52860,129923,129925],{"id":129924},"_22-不可判定","2.2 不可判定",[11,129927,129928],{},"那干脆用表达力最强的一阶逻辑，让自动定理证明器去证，不行吗？",[11,129930,129931],{},"不行。看一个具体方程：",[31,129933,129935],{"className":129934,"translate":35},[411],[31,129936,129938,129977],{"className":129937,"translate":35},[34],[31,129939,129941],{"className":129940},[39],[41,129942,129943],{"xmlns":43,"display":420},[45,129944,129945,129974],{},[48,129946,129947,129953,129955,129961,129963,129969,129971],{},[1180,129948,129949,129951],{},[51,129950,53],{},[91,129952,93],{},[95,129954,434],{},[1180,129956,129957,129959],{},[51,129958,3003],{},[91,129960,93],{},[95,129962,434],{},[1180,129964,129965,129967],{},[51,129966,46458],{},[91,129968,93],{},[95,129970,223],{},[91,129972,129973],{},"33",[55,129975,129976],{"encoding":57},"x^3 + y^3 + z^3 = 33",[31,129978,129980,130025,130069,130113],{"className":129979,"ariaHidden":62},[61],[31,129981,129983,129987,130016,130019,130022],{"className":129982},[66],[31,129984],{"className":129985,"style":129986},[70],"height:0.9474em;vertical-align:-0.0833em;",[31,129988,129990,129993],{"className":129989},[75],[31,129991,53],{"className":129992},[75,76],[31,129994,129996],{"className":129995},[1039],[31,129997,129999],{"className":129998},[1043],[31,130000,130002],{"className":130001},[1048],[31,130003,130005],{"className":130004,"style":1327},[1052],[31,130006,130007,130010],{"style":1330},[31,130008],{"className":130009,"style":1061},[1060],[31,130011,130013],{"className":130012},[1065,1066,1067,1068],[31,130014,93],{"className":130015},[75,1068],[31,130017],{"className":130018,"style":120},[119],[31,130020,434],{"className":130021},[124],[31,130023],{"className":130024,"style":120},[119],[31,130026,130028,130031,130060,130063,130066],{"className":130027},[66],[31,130029],{"className":130030,"style":68692},[70],[31,130032,130034,130037],{"className":130033},[75],[31,130035,3003],{"className":130036,"style":2666},[75,76],[31,130038,130040],{"className":130039},[1039],[31,130041,130043],{"className":130042},[1043],[31,130044,130046],{"className":130045},[1048],[31,130047,130049],{"className":130048,"style":1327},[1052],[31,130050,130051,130054],{"style":1330},[31,130052],{"className":130053,"style":1061},[1060],[31,130055,130057],{"className":130056},[1065,1066,1067,1068],[31,130058,93],{"className":130059},[75,1068],[31,130061],{"className":130062,"style":120},[119],[31,130064,434],{"className":130065},[124],[31,130067],{"className":130068,"style":120},[119],[31,130070,130072,130075,130104,130107,130110],{"className":130071},[66],[31,130073],{"className":130074,"style":1327},[70],[31,130076,130078,130081],{"className":130077},[75],[31,130079,46458],{"className":130080,"style":46500},[75,76],[31,130082,130084],{"className":130083},[1039],[31,130085,130087],{"className":130086},[1043],[31,130088,130090],{"className":130089},[1048],[31,130091,130093],{"className":130092,"style":1327},[1052],[31,130094,130095,130098],{"style":1330},[31,130096],{"className":130097,"style":1061},[1060],[31,130099,130101],{"className":130100},[1065,1066,1067,1068],[31,130102,93],{"className":130103},[75,1068],[31,130105],{"className":130106,"style":293},[119],[31,130108,223],{"className":130109},[297],[31,130111],{"className":130112,"style":293},[119],[31,130114,130116,130119],{"className":130115},[66],[31,130117],{"className":130118,"style":326},[70],[31,130120,129973],{"className":130121},[75],[11,130123,130124],{},"它看起来很简单。它有整数解，一个已知解是：",[31,130126,130128],{"className":130127,"translate":35},[411],[31,130129,130131,130234],{"className":130130,"translate":35},[34],[31,130132,130134],{"className":130133},[39],[41,130135,130136],{"xmlns":43,"display":420},[45,130137,130138,130231],{},[5606,130139,130140,130169,130200],{"rowspacing":5608,"columnalign":5609,"columnspacing":5610},[5612,130141,130142,130154],{},[5615,130143,130144],{},[2071,130145,130146],{"scriptlevel":1977,"displaystyle":62},[48,130147,130148,130150,130152],{},[91,130149,129973],{},[95,130151,223],{},[48,130153],{},[5615,130155,130156],{},[2071,130157,130158],{"scriptlevel":1977,"displaystyle":62},[48,130159,130160,130162],{},[48,130161],{},[1180,130163,130164,130167],{},[91,130165,130166],{},"8866128975287528",[91,130168,93],{},[5612,130170,130171,130177],{},[5615,130172,130173],{},[2071,130174,130175],{"scriptlevel":1977,"displaystyle":62},[48,130176],{},[5615,130178,130179],{},[2071,130180,130181],{"scriptlevel":1977,"displaystyle":62},[48,130182,130183,130185,130187,130189,130191,130194],{},[48,130184],{},[95,130186,434],{},[95,130188,211],{"stretchy":210},[95,130190,97],{},[91,130192,130193],{},"8778405442862239",[1180,130195,130196,130198],{},[95,130197,220],{"stretchy":210},[91,130199,93],{},[5612,130201,130202,130208],{},[5615,130203,130204],{},[2071,130205,130206],{"scriptlevel":1977,"displaystyle":62},[48,130207],{},[5615,130209,130210],{},[2071,130211,130212],{"scriptlevel":1977,"displaystyle":62},[48,130213,130214,130216,130218,130220,130222,130225],{},[48,130215],{},[95,130217,434],{},[95,130219,211],{"stretchy":210},[95,130221,97],{},[91,130223,130224],{},"2736111468807040",[1180,130226,130227,130229],{},[95,130228,220],{"stretchy":210},[91,130230,93],{},[55,130232,130233],{"encoding":57},"\\begin{aligned}\n33 ={}& 8866128975287528^3 \\\\\n      &+ (-8778405442862239)^3 \\\\\n      &+ (-2736111468807040)^3\n\\end{aligned}",[31,130235,130237],{"className":130236,"ariaHidden":62},[61],[31,130238,130240,130244],{"className":130239},[66],[31,130241],{"className":130242,"style":130243},[70],"height:4.2723em;vertical-align:-1.8862em;",[31,130245,130247],{"className":130246},[75],[31,130248,130250,130317],{"className":130249},[5606],[31,130251,130253],{"className":130252},[5867],[31,130254,130256,130308],{"className":130255},[1043,1044],[31,130257,130259,130305],{"className":130258},[1048],[31,130260,130263,130287,130296],{"className":130261,"style":130262},[1052],"height:2.3862em;",[31,130264,130266,130269],{"style":130265},"top:-4.5221em;",[31,130267],{"className":130268,"style":1637},[1060],[31,130270,130272,130275,130278,130281,130284],{"className":130271},[75],[31,130273,129973],{"className":130274},[75],[31,130276],{"className":130277,"style":293},[119],[31,130279,223],{"className":130280},[297],[31,130282],{"className":130283,"style":293},[119],[31,130285],{"className":130286},[75],[31,130288,130290,130293],{"style":130289},"top:-2.9979em;",[31,130291],{"className":130292,"style":1637},[1060],[31,130294],{"className":130295},[75],[31,130297,130299,130302],{"style":130298},"top:-1.4738em;",[31,130300],{"className":130301,"style":1637},[1060],[31,130303],{"className":130304},[75],[31,130306,1076],{"className":130307},[1075],[31,130309,130311],{"className":130310},[1048],[31,130312,130315],{"className":130313,"style":130314},[1052],"height:1.8862em;",[31,130316],{},[31,130318,130320],{"className":130319},[6062],[31,130321,130323,130492],{"className":130322},[1043,1044],[31,130324,130326,130489],{"className":130325},[1048],[31,130327,130329,130373,130431],{"className":130328,"style":130262},[1052],[31,130330,130331,130334],{"style":130265},[31,130332],{"className":130333,"style":1637},[1060],[31,130335,130337,130340,130344],{"className":130336},[75],[31,130338],{"className":130339},[75],[31,130341,130343],{"className":130342},[75],"886612897528752",[31,130345,130347,130350],{"className":130346},[75],[31,130348,19641],{"className":130349},[75],[31,130351,130353],{"className":130352},[1039],[31,130354,130356],{"className":130355},[1043],[31,130357,130359],{"className":130358},[1048],[31,130360,130362],{"className":130361,"style":1327},[1052],[31,130363,130364,130367],{"style":1330},[31,130365],{"className":130366,"style":1061},[1060],[31,130368,130370],{"className":130369},[1065,1066,1067,1068],[31,130371,93],{"className":130372},[75,1068],[31,130374,130375,130378],{"style":130289},[31,130376],{"className":130377,"style":1637},[1060],[31,130379,130381,130384,130387,130390,130393,130396,130399,130402],{"className":130380},[75],[31,130382],{"className":130383},[75],[31,130385],{"className":130386,"style":120},[119],[31,130388,434],{"className":130389},[124],[31,130391],{"className":130392,"style":120},[119],[31,130394,211],{"className":130395},[264],[31,130397,97],{"className":130398},[75],[31,130400,130193],{"className":130401},[75],[31,130403,130405,130408],{"className":130404},[289],[31,130406,220],{"className":130407},[289],[31,130409,130411],{"className":130410},[1039],[31,130412,130414],{"className":130413},[1043],[31,130415,130417],{"className":130416},[1048],[31,130418,130420],{"className":130419,"style":1327},[1052],[31,130421,130422,130425],{"style":1330},[31,130423],{"className":130424,"style":1061},[1060],[31,130426,130428],{"className":130427},[1065,1066,1067,1068],[31,130429,93],{"className":130430},[75,1068],[31,130432,130433,130436],{"style":130298},[31,130434],{"className":130435,"style":1637},[1060],[31,130437,130439,130442,130445,130448,130451,130454,130457,130460],{"className":130438},[75],[31,130440],{"className":130441},[75],[31,130443],{"className":130444,"style":120},[119],[31,130446,434],{"className":130447},[124],[31,130449],{"className":130450,"style":120},[119],[31,130452,211],{"className":130453},[264],[31,130455,97],{"className":130456},[75],[31,130458,130224],{"className":130459},[75],[31,130461,130463,130466],{"className":130462},[289],[31,130464,220],{"className":130465},[289],[31,130467,130469],{"className":130468},[1039],[31,130470,130472],{"className":130471},[1043],[31,130473,130475],{"className":130474},[1048],[31,130476,130478],{"className":130477,"style":1327},[1052],[31,130479,130480,130483],{"style":1330},[31,130481],{"className":130482,"style":1061},[1060],[31,130484,130486],{"className":130485},[1065,1066,1067,1068],[31,130487,93],{"className":130488},[75,1068],[31,130490,1076],{"className":130491},[1075],[31,130493,130495],{"className":130494},[1048],[31,130496,130498],{"className":130497,"style":130314},[1052],[31,130499],{},[11,130501,130502,130503,130558,130559,130622,130623,130689],{},"这三个数都是 16 位数，直到 2019 年才被找到。如果把 ",[31,130504,130506,130528],{"className":130505,"translate":35},[34],[31,130507,130509],{"className":130508},[39],[41,130510,130511],{"xmlns":43},[45,130512,130513,130525],{},[48,130514,130515,130517,130519,130521,130523],{},[51,130516,53],{},[95,130518,1210],{"separator":62},[51,130520,3003],{},[95,130522,1210],{"separator":62},[51,130524,46458],{},[55,130526,130527],{"encoding":57},"x, y, z",[31,130529,130531],{"className":130530,"ariaHidden":62},[61],[31,130532,130534,130537,130540,130543,130546,130549,130552,130555],{"className":130533},[66],[31,130535],{"className":130536,"style":865},[70],[31,130538,53],{"className":130539},[75,76],[31,130541,1210],{"className":130542},[1391],[31,130544],{"className":130545,"style":1467},[119],[31,130547,3003],{"className":130548,"style":2666},[75,76],[31,130550,1210],{"className":130551},[1391],[31,130553],{"className":130554,"style":1467},[119],[31,130556,46458],{"className":130557,"style":46500},[75,76]," 都编码成 64 位整数，搜索空间是 ",[31,130560,130562,130581],{"className":130561,"translate":35},[34],[31,130563,130565],{"className":130564},[39],[41,130566,130567],{"xmlns":43},[45,130568,130569,130578],{},[48,130570,130571],{},[1180,130572,130573,130575],{},[91,130574,152],{},[91,130576,130577],{},"192",[55,130579,130580],{"encoding":57},"2^{192}",[31,130582,130584],{"className":130583,"ariaHidden":62},[61],[31,130585,130587,130590],{"className":130586},[66],[31,130588],{"className":130589,"style":5135},[70],[31,130591,130593,130596],{"className":130592},[75],[31,130594,152],{"className":130595},[75],[31,130597,130599],{"className":130598},[1039],[31,130600,130602],{"className":130601},[1043],[31,130603,130605],{"className":130604},[1048],[31,130606,130608],{"className":130607,"style":5135},[1052],[31,130609,130610,130613],{"style":3631},[31,130611],{"className":130612,"style":1061},[1060],[31,130614,130616],{"className":130615},[1065,1066,1067,1068],[31,130617,130619],{"className":130618},[75,1068],[31,130620,130577],{"className":130621},[75,1068],"；按每秒试十亿个赋值来算，也要 ",[31,130624,130626,130645],{"className":130625,"translate":35},[34],[31,130627,130629],{"className":130628},[39],[41,130630,130631],{"xmlns":43},[45,130632,130633,130642],{},[48,130634,130635],{},[1180,130636,130637,130639],{},[91,130638,19494],{},[91,130640,130641],{},"41",[55,130643,130644],{"encoding":57},"10^{41}",[31,130646,130648],{"className":130647,"ariaHidden":62},[61],[31,130649,130651,130654,130657],{"className":130650},[66],[31,130652],{"className":130653,"style":5135},[70],[31,130655,205],{"className":130656},[75],[31,130658,130660,130663],{"className":130659},[75],[31,130661,1977],{"className":130662},[75],[31,130664,130666],{"className":130665},[1039],[31,130667,130669],{"className":130668},[1043],[31,130670,130672],{"className":130671},[1048],[31,130673,130675],{"className":130674,"style":5135},[1052],[31,130676,130677,130680],{"style":3631},[31,130678],{"className":130679,"style":1061},[1060],[31,130681,130683],{"className":130682},[1065,1066,1067,1068],[31,130684,130686],{"className":130685},[75,1068],[31,130687,130641],{"className":130688},[75,1068]," 年量级。位爆破在这里基本不可行。",[11,130691,130692],{},"更根本的是，这不是「这一个方程特别难」的问题。希尔伯特第十问题问的是「是否存在算法判定整系数多项式方程有没有整数解」，这个问题在 1970 年被马蒂亚谢维奇证明为否。证明的思路很巧妙：任给一台图灵机和一段输入，都能构造出一个整系数多项式方程，使方程有解当且仅当图灵机停机。既然停机问题不可判定，一般的方程判定问题也不可判定。换句话说，想写一个能判定所有这类方程的通用算法，这件事本身就是不可能的。",[11,130694,130695],{},"工程上的后果很直接：自动定理证明器可能跑了一整天，最后告诉你「我不知道」。对于一个需要每秒判定成千上万条路径条件的程序分析工具来说，这是不可接受的。",[52860,130697,130699],{"id":130698},"_23-难以组合","2.3 难以组合",[11,130701,130702],{},"另一条路是「一个问题配一个专用算法」：线性规划用单纯形法，只懂实数线性约束；并查集只懂「相等」；BDD 只懂布尔函数；自动机只懂正则语言和字符串匹配。",[11,130704,130705],{},"这些算法在各自领域都很好用，但真实的程序约束往往是混合的。看一个只有两个原子的例子：",[31,130707,130709],{"className":130708,"translate":35},[411],[31,130710,130712,130754],{"className":130711,"translate":35},[34],[31,130713,130715],{"className":130714},[39],[41,130716,130717],{"xmlns":43,"display":420},[45,130718,130719,130751],{},[48,130720,130721,130723,130725,130727,130729,130731,130733,130735,130737,130739,130741,130743,130745,130747,130749],{},[51,130722,53],{},[95,130724,223],{},[51,130726,3003],{},[1430,130728,125504],{},[95,130730,125507],{},[1430,130732,125504],{},[51,130734,1662],{},[95,130736,211],{"stretchy":210},[51,130738,53],{},[95,130740,220],{"stretchy":210},[95,130742,21271],{"mathvariant":494},[51,130744,1662],{},[95,130746,211],{"stretchy":210},[51,130748,3003],{},[95,130750,220],{"stretchy":210},[55,130752,130753],{"encoding":57},"x = y \\;\\wedge\\; f(x) \\ne f(y)",[31,130755,130757,130775,130799,130859],{"className":130756,"ariaHidden":62},[61],[31,130758,130760,130763,130766,130769,130772],{"className":130759},[66],[31,130761],{"className":130762,"style":71},[70],[31,130764,53],{"className":130765},[75,76],[31,130767],{"className":130768,"style":293},[119],[31,130770,223],{"className":130771},[297],[31,130773],{"className":130774,"style":293},[119],[31,130776,130778,130781,130784,130787,130790,130793,130796],{"className":130777},[66],[31,130779],{"className":130780,"style":126295},[70],[31,130782,3003],{"className":130783,"style":2666},[75,76],[31,130785],{"className":130786,"style":293},[119],[31,130788],{"className":130789,"style":120},[119],[31,130791,125507],{"className":130792},[124],[31,130794],{"className":130795,"style":293},[119],[31,130797],{"className":130798,"style":120},[119],[31,130800,130802,130805,130808,130811,130814,130817,130820,130856],{"className":130801},[66],[31,130803],{"className":130804,"style":260},[70],[31,130806,1662],{"className":130807,"style":35196},[75,76],[31,130809,211],{"className":130810},[264],[31,130812,53],{"className":130813},[75,76],[31,130815,220],{"className":130816},[289],[31,130818],{"className":130819,"style":293},[119],[31,130821,130823,130850,130853],{"className":130822},[297],[31,130824,130826],{"className":130825},[297],[31,130827,130829],{"className":130828},[75,21516],[31,130830,130832],{"className":130831},[21520],[31,130833,130835,130838,130847],{"className":130834},[21524],[31,130836],{"className":130837,"style":4202},[70],[31,130839,130841],{"className":130840},[21531],[31,130842,130844],{"className":130843},[75],[31,130845,21538],{"className":130846},[297],[31,130848],{"className":130849},[21542],[31,130851],{"className":130852},[119,21546],[31,130854,223],{"className":130855},[297],[31,130857],{"className":130858,"style":293},[119],[31,130860,130862,130865,130868,130871,130874],{"className":130861},[66],[31,130863],{"className":130864,"style":260},[70],[31,130866,1662],{"className":130867,"style":35196},[75,76],[31,130869,211],{"className":130870},[264],[31,130872,3003],{"className":130873,"style":2666},[75,76],[31,130875,220],{"className":130876},[289],[11,130878,130879],{},"假设我们分别去问两个求解器：",[127630,130881,130882,131013],{},[127633,130883,130884,130885,130936,130937,130939,130940,36037],{},"算术求解器只看 ",[31,130886,130888,130906],{"className":130887,"translate":35},[34],[31,130889,130891],{"className":130890},[39],[41,130892,130893],{"xmlns":43},[45,130894,130895,130903],{},[48,130896,130897,130899,130901],{},[51,130898,53],{},[95,130900,223],{},[51,130902,3003],{},[55,130904,130905],{"encoding":57},"x = y",[31,130907,130909,130927],{"className":130908,"ariaHidden":62},[61],[31,130910,130912,130915,130918,130921,130924],{"className":130911},[66],[31,130913],{"className":130914,"style":71},[70],[31,130916,53],{"className":130917},[75,76],[31,130919],{"className":130920,"style":293},[119],[31,130922,223],{"className":130923},[297],[31,130925],{"className":130926,"style":293},[119],[31,130928,130930,130933],{"className":130929},[66],[31,130931],{"className":130932,"style":865},[70],[31,130934,3003],{"className":130935,"style":2666},[75,76],"，回答 ",[7246,130938,125655],{},"，比如 ",[31,130941,130943,130965],{"className":130942,"translate":35},[34],[31,130944,130946],{"className":130945},[39],[41,130947,130948],{"xmlns":43},[45,130949,130950,130962],{},[48,130951,130952,130954,130956,130958,130960],{},[51,130953,53],{},[95,130955,223],{},[51,130957,3003],{},[95,130959,223],{},[91,130961,1977],{},[55,130963,130964],{"encoding":57},"x = y = 0",[31,130966,130968,130986,131004],{"className":130967,"ariaHidden":62},[61],[31,130969,130971,130974,130977,130980,130983],{"className":130970},[66],[31,130972],{"className":130973,"style":71},[70],[31,130975,53],{"className":130976},[75,76],[31,130978],{"className":130979,"style":293},[119],[31,130981,223],{"className":130982},[297],[31,130984],{"className":130985,"style":293},[119],[31,130987,130989,130992,130995,130998,131001],{"className":130988},[66],[31,130990],{"className":130991,"style":865},[70],[31,130993,3003],{"className":130994,"style":2666},[75,76],[31,130996],{"className":130997,"style":293},[119],[31,130999,223],{"className":131000},[297],[31,131002],{"className":131003,"style":293},[119],[31,131005,131007,131010],{"className":131006},[66],[31,131008],{"className":131009,"style":326},[70],[31,131011,1977],{"className":131012},[75],[127633,131014,131015,131016,131129,131130,131132,131133,131174,131175,15282,131241,2372],{},"EUF 求解器只看 ",[31,131017,131019,131048],{"className":131018,"translate":35},[34],[31,131020,131022],{"className":131021},[39],[41,131023,131024],{"xmlns":43},[45,131025,131026,131046],{},[48,131027,131028,131030,131032,131034,131036,131038,131040,131042,131044],{},[51,131029,1662],{},[95,131031,211],{"stretchy":210},[51,131033,53],{},[95,131035,220],{"stretchy":210},[95,131037,21271],{"mathvariant":494},[51,131039,1662],{},[95,131041,211],{"stretchy":210},[51,131043,3003],{},[95,131045,220],{"stretchy":210},[55,131047,127880],{"encoding":57},[31,131049,131051,131111],{"className":131050,"ariaHidden":62},[61],[31,131052,131054,131057,131060,131063,131066,131069,131072,131108],{"className":131053},[66],[31,131055],{"className":131056,"style":260},[70],[31,131058,1662],{"className":131059,"style":35196},[75,76],[31,131061,211],{"className":131062},[264],[31,131064,53],{"className":131065},[75,76],[31,131067,220],{"className":131068},[289],[31,131070],{"className":131071,"style":293},[119],[31,131073,131075,131102,131105],{"className":131074},[297],[31,131076,131078],{"className":131077},[297],[31,131079,131081],{"className":131080},[75,21516],[31,131082,131084],{"className":131083},[21520],[31,131085,131087,131090,131099],{"className":131086},[21524],[31,131088],{"className":131089,"style":4202},[70],[31,131091,131093],{"className":131092},[21531],[31,131094,131096],{"className":131095},[75],[31,131097,21538],{"className":131098},[297],[31,131100],{"className":131101},[21542],[31,131103],{"className":131104},[119,21546],[31,131106,223],{"className":131107},[297],[31,131109],{"className":131110,"style":293},[119],[31,131112,131114,131117,131120,131123,131126],{"className":131113},[66],[31,131115],{"className":131116,"style":260},[70],[31,131118,1662],{"className":131119,"style":35196},[75,76],[31,131121,211],{"className":131122},[264],[31,131124,3003],{"className":131125,"style":2666},[75,76],[31,131127,220],{"className":131128},[289],"，也回答 ",[7246,131131,125655],{},"：它把 ",[31,131134,131136,131153],{"className":131135,"translate":35},[34],[31,131137,131139],{"className":131138},[39],[41,131140,131141],{"xmlns":43},[45,131142,131143,131151],{},[48,131144,131145,131147,131149],{},[51,131146,53],{},[95,131148,1210],{"separator":62},[51,131150,3003],{},[55,131152,128368],{"encoding":57},[31,131154,131156],{"className":131155,"ariaHidden":62},[61],[31,131157,131159,131162,131165,131168,131171],{"className":131158},[66],[31,131160],{"className":131161,"style":865},[70],[31,131163,53],{"className":131164},[75,76],[31,131166,1210],{"className":131167},[1391],[31,131169],{"className":131170,"style":1467},[119],[31,131172,3003],{"className":131173,"style":2666},[75,76]," 当成两个不同的常量，令 ",[31,131176,131178,131202],{"className":131177,"translate":35},[34],[31,131179,131181],{"className":131180},[39],[41,131182,131183],{"xmlns":43},[45,131184,131185,131199],{},[48,131186,131187,131189,131191,131193,131195,131197],{},[51,131188,1662],{},[95,131190,211],{"stretchy":210},[51,131192,53],{},[95,131194,220],{"stretchy":210},[95,131196,223],{},[91,131198,1977],{},[55,131200,131201],{"encoding":57},"f(x) = 0",[31,131203,131205,131232],{"className":131204,"ariaHidden":62},[61],[31,131206,131208,131211,131214,131217,131220,131223,131226,131229],{"className":131207},[66],[31,131209],{"className":131210,"style":260},[70],[31,131212,1662],{"className":131213,"style":35196},[75,76],[31,131215,211],{"className":131216},[264],[31,131218,53],{"className":131219},[75,76],[31,131221,220],{"className":131222},[289],[31,131224],{"className":131225,"style":293},[119],[31,131227,223],{"className":131228},[297],[31,131230],{"className":131231,"style":293},[119],[31,131233,131235,131238],{"className":131234},[66],[31,131236],{"className":131237,"style":326},[70],[31,131239,1977],{"className":131240},[75],[31,131242,131244,131268],{"className":131243,"translate":35},[34],[31,131245,131247],{"className":131246},[39],[41,131248,131249],{"xmlns":43},[45,131250,131251,131265],{},[48,131252,131253,131255,131257,131259,131261,131263],{},[51,131254,1662],{},[95,131256,211],{"stretchy":210},[51,131258,3003],{},[95,131260,220],{"stretchy":210},[95,131262,223],{},[91,131264,205],{},[55,131266,131267],{"encoding":57},"f(y) = 1",[31,131269,131271,131298],{"className":131270,"ariaHidden":62},[61],[31,131272,131274,131277,131280,131283,131286,131289,131292,131295],{"className":131273},[66],[31,131275],{"className":131276,"style":260},[70],[31,131278,1662],{"className":131279,"style":35196},[75,76],[31,131281,211],{"className":131282},[264],[31,131284,3003],{"className":131285,"style":2666},[75,76],[31,131287,220],{"className":131288},[289],[31,131290],{"className":131291,"style":293},[119],[31,131293,223],{"className":131294},[297],[31,131296],{"className":131297,"style":293},[119],[31,131299,131301,131304],{"className":131300},[66],[31,131302],{"className":131303,"style":326},[70],[31,131305,205],{"className":131306},[75],[11,131308,131309,131310,131360,131361,131404,131405,131449,131450,131563,131564,2372],{},"如果把两个答案一拼，说「两边都可满足，所以整体可满足」，就错了。因为 ",[31,131311,131313,131330],{"className":131312,"translate":35},[34],[31,131314,131316],{"className":131315},[39],[41,131317,131318],{"xmlns":43},[45,131319,131320,131328],{},[48,131321,131322,131324,131326],{},[51,131323,53],{},[95,131325,223],{},[51,131327,3003],{},[55,131329,130905],{"encoding":57},[31,131331,131333,131351],{"className":131332,"ariaHidden":62},[61],[31,131334,131336,131339,131342,131345,131348],{"className":131335},[66],[31,131337],{"className":131338,"style":71},[70],[31,131340,53],{"className":131341},[75,76],[31,131343],{"className":131344,"style":293},[119],[31,131346,223],{"className":131347},[297],[31,131349],{"className":131350,"style":293},[119],[31,131352,131354,131357],{"className":131353},[66],[31,131355],{"className":131356,"style":865},[70],[31,131358,3003],{"className":131359,"style":2666},[75,76]," 意味着 ",[31,131362,131364,131383],{"className":131363,"translate":35},[34],[31,131365,131367],{"className":131366},[39],[41,131368,131369],{"xmlns":43},[45,131370,131371,131381],{},[48,131372,131373,131375,131377,131379],{},[51,131374,1662],{},[95,131376,211],{"stretchy":210},[51,131378,53],{},[95,131380,220],{"stretchy":210},[55,131382,40190],{"encoding":57},[31,131384,131386],{"className":131385,"ariaHidden":62},[61],[31,131387,131389,131392,131395,131398,131401],{"className":131388},[66],[31,131390],{"className":131391,"style":260},[70],[31,131393,1662],{"className":131394,"style":35196},[75,76],[31,131396,211],{"className":131397},[264],[31,131399,53],{"className":131400},[75,76],[31,131402,220],{"className":131403},[289]," 必须等于 ",[31,131406,131408,131428],{"className":131407,"translate":35},[34],[31,131409,131411],{"className":131410},[39],[41,131412,131413],{"xmlns":43},[45,131414,131415,131425],{},[48,131416,131417,131419,131421,131423],{},[51,131418,1662],{},[95,131420,211],{"stretchy":210},[51,131422,3003],{},[95,131424,220],{"stretchy":210},[55,131426,131427],{"encoding":57},"f(y)",[31,131429,131431],{"className":131430,"ariaHidden":62},[61],[31,131432,131434,131437,131440,131443,131446],{"className":131433},[66],[31,131435],{"className":131436,"style":260},[70],[31,131438,1662],{"className":131439,"style":35196},[75,76],[31,131441,211],{"className":131442},[264],[31,131444,3003],{"className":131445,"style":2666},[75,76],[31,131447,220],{"className":131448},[289],"，与 ",[31,131451,131453,131482],{"className":131452,"translate":35},[34],[31,131454,131456],{"className":131455},[39],[41,131457,131458],{"xmlns":43},[45,131459,131460,131480],{},[48,131461,131462,131464,131466,131468,131470,131472,131474,131476,131478],{},[51,131463,1662],{},[95,131465,211],{"stretchy":210},[51,131467,53],{},[95,131469,220],{"stretchy":210},[95,131471,21271],{"mathvariant":494},[51,131473,1662],{},[95,131475,211],{"stretchy":210},[51,131477,3003],{},[95,131479,220],{"stretchy":210},[55,131481,127880],{"encoding":57},[31,131483,131485,131545],{"className":131484,"ariaHidden":62},[61],[31,131486,131488,131491,131494,131497,131500,131503,131506,131542],{"className":131487},[66],[31,131489],{"className":131490,"style":260},[70],[31,131492,1662],{"className":131493,"style":35196},[75,76],[31,131495,211],{"className":131496},[264],[31,131498,53],{"className":131499},[75,76],[31,131501,220],{"className":131502},[289],[31,131504],{"className":131505,"style":293},[119],[31,131507,131509,131536,131539],{"className":131508},[297],[31,131510,131512],{"className":131511},[297],[31,131513,131515],{"className":131514},[75,21516],[31,131516,131518],{"className":131517},[21520],[31,131519,131521,131524,131533],{"className":131520},[21524],[31,131522],{"className":131523,"style":4202},[70],[31,131525,131527],{"className":131526},[21531],[31,131528,131530],{"className":131529},[75],[31,131531,21538],{"className":131532},[297],[31,131534],{"className":131535},[21542],[31,131537],{"className":131538},[119,21546],[31,131540,223],{"className":131541},[297],[31,131543],{"className":131544,"style":293},[119],[31,131546,131548,131551,131554,131557,131560],{"className":131547},[66],[31,131549],{"className":131550,"style":260},[70],[31,131552,1662],{"className":131553,"style":35196},[75,76],[31,131555,211],{"className":131556},[264],[31,131558,3003],{"className":131559,"style":2666},[75,76],[31,131561,220],{"className":131562},[289]," 直接矛盾。整体其实是 ",[7246,131565,131566],{},"unsat",[11,131568,131569,131570,131620,131621,131702],{},"要得到正确答案，两个求解器必须交换信息：算术把等式 ",[31,131571,131573,131590],{"className":131572,"translate":35},[34],[31,131574,131576],{"className":131575},[39],[41,131577,131578],{"xmlns":43},[45,131579,131580,131588],{},[48,131581,131582,131584,131586],{},[51,131583,53],{},[95,131585,223],{},[51,131587,3003],{},[55,131589,130905],{"encoding":57},[31,131591,131593,131611],{"className":131592,"ariaHidden":62},[61],[31,131594,131596,131599,131602,131605,131608],{"className":131595},[66],[31,131597],{"className":131598,"style":71},[70],[31,131600,53],{"className":131601},[75,76],[31,131603],{"className":131604,"style":293},[119],[31,131606,223],{"className":131607},[297],[31,131609],{"className":131610,"style":293},[119],[31,131612,131614,131617],{"className":131613},[66],[31,131615],{"className":131616,"style":865},[70],[31,131618,3003],{"className":131619,"style":2666},[75,76]," 告诉 EUF，EUF 立刻推出 ",[31,131622,131624,131654],{"className":131623,"translate":35},[34],[31,131625,131627],{"className":131626},[39],[41,131628,131629],{"xmlns":43},[45,131630,131631,131651],{},[48,131632,131633,131635,131637,131639,131641,131643,131645,131647,131649],{},[51,131634,1662],{},[95,131636,211],{"stretchy":210},[51,131638,53],{},[95,131640,220],{"stretchy":210},[95,131642,223],{},[51,131644,1662],{},[95,131646,211],{"stretchy":210},[51,131648,3003],{},[95,131650,220],{"stretchy":210},[55,131652,131653],{"encoding":57},"f(x) = f(y)",[31,131655,131657,131684],{"className":131656,"ariaHidden":62},[61],[31,131658,131660,131663,131666,131669,131672,131675,131678,131681],{"className":131659},[66],[31,131661],{"className":131662,"style":260},[70],[31,131664,1662],{"className":131665,"style":35196},[75,76],[31,131667,211],{"className":131668},[264],[31,131670,53],{"className":131671},[75,76],[31,131673,220],{"className":131674},[289],[31,131676],{"className":131677,"style":293},[119],[31,131679,223],{"className":131680},[297],[31,131682],{"className":131683,"style":293},[119],[31,131685,131687,131690,131693,131696,131699],{"className":131686},[66],[31,131688],{"className":131689,"style":260},[70],[31,131691,1662],{"className":131692,"style":35196},[75,76],[31,131694,211],{"className":131695},[264],[31,131697,3003],{"className":131698,"style":2666},[75,76],[31,131700,220],{"className":131701},[289],"，冲突。谁来负责交换？交换哪些事实？什么时候交换？在 SMT 之前，这些都是需要人工设计的事情。",[11,131704,131705,131706,131709],{},"数组和算术之间也有同样的坑。",[45608,131707,131708],{},"i = j ∧ a[i] ≠ a[j]"," 单独看两边都可满足，合起来却不可满足，因为下标相等就意味着读到的值相等。没有任何一个专用算法能独自看出这一点。",[52860,131711,131713],{"id":131712},"_24-门槛太高","2.4 门槛太高",[11,131715,131716,131717,2372],{},"还有一条路是 Coq、Isabelle、ACL2 这类交互式定理证明器。它们的表达力极强，能处理带量词、带归纳的复杂性质，但代价是",[7246,131718,131719],{},"需要人写证明脚本",[11,131721,131722],{},"比如在 Coq 里证明「任何自然数加零都等于它自己」，要写：",[125323,131724,131728],{"className":131725,"code":131726,"language":131727,"meta":52812,"style":52812},"language-coq shiki shiki-themes github-light github-light github-dark","Theorem plus_0_r : forall n : nat, n + 0 = n.\nProof.\n  induction n as [| n' IH].\n  - simpl. reflexivity.\n  - simpl. rewrite IH. reflexivity.\nQed.\n","coq",[45608,131729,131730,131735,131740,131745,131750,131755],{"__ignoreMap":52812},[31,131731,131732],{"class":125331,"line":125332},[31,131733,131734],{},"Theorem plus_0_r : forall n : nat, n + 0 = n.\n",[31,131736,131737],{"class":125331,"line":52813},[31,131738,131739],{},"Proof.\n",[31,131741,131742],{"class":125331,"line":125262},[31,131743,131744],{},"  induction n as [| n' IH].\n",[31,131746,131747],{"class":125331,"line":125409},[31,131748,131749],{},"  - simpl. reflexivity.\n",[31,131751,131752],{"class":125331,"line":125446},[31,131753,131754],{},"  - simpl. rewrite IH. reflexivity.\n",[31,131756,131757],{"class":125331,"line":125453},[31,131758,131759],{},"Qed.\n",[11,131761,131762],{},"这几行不是装饰：",[127630,131764,131765,131921,132012],{},[127633,131766,131767,131770,131771,2446,131822,36037],{},[45608,131768,131769],{},"induction n"," 把目标拆成两种情况：",[31,131772,131774,131792],{"className":131773,"translate":35},[34],[31,131775,131777],{"className":131776},[39],[41,131778,131779],{"xmlns":43},[45,131780,131781,131789],{},[48,131782,131783,131785,131787],{},[51,131784,14559],{},[95,131786,223],{},[91,131788,1977],{},[55,131790,131791],{"encoding":57},"n = 0",[31,131793,131795,131813],{"className":131794,"ariaHidden":62},[61],[31,131796,131798,131801,131804,131807,131810],{"className":131797},[66],[31,131799],{"className":131800,"style":71},[70],[31,131802,14559],{"className":131803},[75,76],[31,131805],{"className":131806,"style":293},[119],[31,131808,223],{"className":131809},[297],[31,131811],{"className":131812,"style":293},[119],[31,131814,131816,131819],{"className":131815},[66],[31,131817],{"className":131818,"style":326},[70],[31,131820,1977],{"className":131821},[75],[31,131823,131825,131853],{"className":131824,"translate":35},[34],[31,131826,131828],{"className":131827},[39],[41,131829,131830],{"xmlns":43},[45,131831,131832,131850],{},[48,131833,131834,131836,131838,131840,131842,131848],{},[51,131835,14559],{},[95,131837,223],{},[51,131839,21582],{},[95,131841,211],{"stretchy":210},[1180,131843,131844,131846],{},[51,131845,14559],{},[95,131847,13691],{"mathvariant":494,"lspace":5610,"rspace":5610},[95,131849,220],{"stretchy":210},[55,131851,131852],{"encoding":57},"n = S(n')",[31,131854,131856,131874],{"className":131855,"ariaHidden":62},[61],[31,131857,131859,131862,131865,131868,131871],{"className":131858},[66],[31,131860],{"className":131861,"style":71},[70],[31,131863,14559],{"className":131864},[75,76],[31,131866],{"className":131867,"style":293},[119],[31,131869,223],{"className":131870},[297],[31,131872],{"className":131873,"style":293},[119],[31,131875,131877,131880,131883,131886,131918],{"className":131876},[66],[31,131878],{"className":131879,"style":13710},[70],[31,131881,21582],{"className":131882,"style":21697},[75,76],[31,131884,211],{"className":131885},[264],[31,131887,131889,131892],{"className":131888},[75],[31,131890,14559],{"className":131891},[75,76],[31,131893,131895],{"className":131894},[1039],[31,131896,131898],{"className":131897},[1043],[31,131899,131901],{"className":131900},[1048],[31,131902,131904],{"className":131903,"style":13732},[1052],[31,131905,131906,131909],{"style":3631},[31,131907],{"className":131908,"style":1061},[1060],[31,131910,131912],{"className":131911},[1065,1066,1067,1068],[31,131913,131915],{"className":131914},[75,1068],[31,131916,13691],{"className":131917},[75,1068],[31,131919,220],{"className":131920},[289],[127633,131922,131923,131924,131927,131928,131979,131980,1138,132008,132011],{},"第一种情况，",[45608,131925,131926],{},"simpl"," 把 ",[31,131929,131931,131949],{"className":131930,"translate":35},[34],[31,131932,131934],{"className":131933},[39],[41,131935,131936],{"xmlns":43},[45,131937,131938,131946],{},[48,131939,131940,131942,131944],{},[91,131941,1977],{},[95,131943,434],{},[91,131945,1977],{},[55,131947,131948],{"encoding":57},"0 + 0",[31,131950,131952,131970],{"className":131951,"ariaHidden":62},[61],[31,131953,131955,131958,131961,131964,131967],{"className":131954},[66],[31,131956],{"className":131957,"style":112},[70],[31,131959,1977],{"className":131960},[75],[31,131962],{"className":131963,"style":120},[119],[31,131965,434],{"className":131966},[124],[31,131968],{"className":131969,"style":120},[119],[31,131971,131973,131976],{"className":131972},[66],[31,131974],{"className":131975,"style":326},[70],[31,131977,1977],{"className":131978},[75]," 化简成 ",[31,131981,131983,131996],{"className":131982,"translate":35},[34],[31,131984,131986],{"className":131985},[39],[41,131987,131988],{"xmlns":43},[45,131989,131990,131994],{},[48,131991,131992],{},[91,131993,1977],{},[55,131995,1977],{"encoding":57},[31,131997,131999],{"className":131998,"ariaHidden":62},[61],[31,132000,132002,132005],{"className":132001},[66],[31,132003],{"className":132004,"style":326},[70],[31,132006,1977],{"className":132007},[75],[45608,132009,132010],{},"reflexivity"," 确认两边相等；",[127633,132013,132014,132015,131927,132017,132116,132117,132216,132217,131927,132220,132305,132306,132368,132369,132371],{},"第二种情况，",[45608,132016,131926],{},[31,132018,132020,132048],{"className":132019,"translate":35},[34],[31,132021,132023],{"className":132022},[39],[41,132024,132025],{"xmlns":43},[45,132026,132027,132045],{},[48,132028,132029,132031,132033,132039,132041,132043],{},[51,132030,21582],{},[95,132032,211],{"stretchy":210},[1180,132034,132035,132037],{},[51,132036,14559],{},[95,132038,13691],{"mathvariant":494,"lspace":5610,"rspace":5610},[95,132040,220],{"stretchy":210},[95,132042,434],{},[91,132044,1977],{},[55,132046,132047],{"encoding":57},"S(n') + 0",[31,132049,132051,132107],{"className":132050,"ariaHidden":62},[61],[31,132052,132054,132057,132060,132063,132095,132098,132101,132104],{"className":132053},[66],[31,132055],{"className":132056,"style":13710},[70],[31,132058,21582],{"className":132059,"style":21697},[75,76],[31,132061,211],{"className":132062},[264],[31,132064,132066,132069],{"className":132065},[75],[31,132067,14559],{"className":132068},[75,76],[31,132070,132072],{"className":132071},[1039],[31,132073,132075],{"className":132074},[1043],[31,132076,132078],{"className":132077},[1048],[31,132079,132081],{"className":132080,"style":13732},[1052],[31,132082,132083,132086],{"style":3631},[31,132084],{"className":132085,"style":1061},[1060],[31,132087,132089],{"className":132088},[1065,1066,1067,1068],[31,132090,132092],{"className":132091},[75,1068],[31,132093,13691],{"className":132094},[75,1068],[31,132096,220],{"className":132097},[289],[31,132099],{"className":132100,"style":120},[119],[31,132102,434],{"className":132103},[124],[31,132105],{"className":132106,"style":120},[119],[31,132108,132110,132113],{"className":132109},[66],[31,132111],{"className":132112,"style":326},[70],[31,132114,1977],{"className":132115},[75]," 化成 ",[31,132118,132120,132148],{"className":132119,"translate":35},[34],[31,132121,132123],{"className":132122},[39],[41,132124,132125],{"xmlns":43},[45,132126,132127,132145],{},[48,132128,132129,132131,132133,132139,132141,132143],{},[51,132130,21582],{},[95,132132,211],{"stretchy":210},[1180,132134,132135,132137],{},[51,132136,14559],{},[95,132138,13691],{"mathvariant":494,"lspace":5610,"rspace":5610},[95,132140,434],{},[91,132142,1977],{},[95,132144,220],{"stretchy":210},[55,132146,132147],{"encoding":57},"S(n' + 0)",[31,132149,132151,132204],{"className":132150,"ariaHidden":62},[61],[31,132152,132154,132157,132160,132163,132195,132198,132201],{"className":132153},[66],[31,132155],{"className":132156,"style":13710},[70],[31,132158,21582],{"className":132159,"style":21697},[75,76],[31,132161,211],{"className":132162},[264],[31,132164,132166,132169],{"className":132165},[75],[31,132167,14559],{"className":132168},[75,76],[31,132170,132172],{"className":132171},[1039],[31,132173,132175],{"className":132174},[1043],[31,132176,132178],{"className":132177},[1048],[31,132179,132181],{"className":132180,"style":13732},[1052],[31,132182,132183,132186],{"style":3631},[31,132184],{"className":132185,"style":1061},[1060],[31,132187,132189],{"className":132188},[1065,1066,1067,1068],[31,132190,132192],{"className":132191},[75,1068],[31,132193,13691],{"className":132194},[75,1068],[31,132196],{"className":132197,"style":120},[119],[31,132199,434],{"className":132200},[124],[31,132202],{"className":132203,"style":120},[119],[31,132205,132207,132210,132213],{"className":132206},[66],[31,132208],{"className":132209,"style":260},[70],[31,132211,1977],{"className":132212},[75],[31,132214,220],{"className":132215},[289],"，再用归纳假设 ",[45608,132218,132219],{},"IH",[31,132221,132223,132245],{"className":132222,"translate":35},[34],[31,132224,132226],{"className":132225},[39],[41,132227,132228],{"xmlns":43},[45,132229,132230,132242],{},[48,132231,132232,132238,132240],{},[1180,132233,132234,132236],{},[51,132235,14559],{},[95,132237,13691],{"mathvariant":494,"lspace":5610,"rspace":5610},[95,132239,434],{},[91,132241,1977],{},[55,132243,132244],{"encoding":57},"n' + 0",[31,132246,132248,132296],{"className":132247,"ariaHidden":62},[61],[31,132249,132251,132255,132287,132290,132293],{"className":132250},[66],[31,132252],{"className":132253,"style":132254},[70],"height:0.8352em;vertical-align:-0.0833em;",[31,132256,132258,132261],{"className":132257},[75],[31,132259,14559],{"className":132260},[75,76],[31,132262,132264],{"className":132263},[1039],[31,132265,132267],{"className":132266},[1043],[31,132268,132270],{"className":132269},[1048],[31,132271,132273],{"className":132272,"style":13732},[1052],[31,132274,132275,132278],{"style":3631},[31,132276],{"className":132277,"style":1061},[1060],[31,132279,132281],{"className":132280},[1065,1066,1067,1068],[31,132282,132284],{"className":132283},[75,1068],[31,132285,13691],{"className":132286},[75,1068],[31,132288],{"className":132289,"style":120},[119],[31,132291,434],{"className":132292},[124],[31,132294],{"className":132295,"style":120},[119],[31,132297,132299,132302],{"className":132298},[66],[31,132300],{"className":132301,"style":326},[70],[31,132303,1977],{"className":132304},[75]," 换成 ",[31,132307,132309,132327],{"className":132308,"translate":35},[34],[31,132310,132312],{"className":132311},[39],[41,132313,132314],{"xmlns":43},[45,132315,132316,132324],{},[48,132317,132318],{},[1180,132319,132320,132322],{},[51,132321,14559],{},[95,132323,13691],{"mathvariant":494,"lspace":5610,"rspace":5610},[55,132325,132326],{"encoding":57},"n'",[31,132328,132330],{"className":132329,"ariaHidden":62},[61],[31,132331,132333,132336],{"className":132332},[66],[31,132334],{"className":132335,"style":13732},[70],[31,132337,132339,132342],{"className":132338},[75],[31,132340,14559],{"className":132341},[75,76],[31,132343,132345],{"className":132344},[1039],[31,132346,132348],{"className":132347},[1043],[31,132349,132351],{"className":132350},[1048],[31,132352,132354],{"className":132353,"style":13732},[1052],[31,132355,132356,132359],{"style":3631},[31,132357],{"className":132358,"style":1061},[1060],[31,132360,132362],{"className":132361},[1065,1066,1067,1068],[31,132363,132365],{"className":132364},[75,1068],[31,132366,13691],{"className":132367},[75,1068],"，最后 ",[45608,132370,132010],{}," 收尾。",[11,132373,132374],{},"每一步往哪走、用哪条归纳假设，都是人选的。这还只是自然数加零。真实的程序验证要证循环不变式、数组不变式、指针分离，脚本可能几百上千行。",[11,132376,132377],{},"程序分析场景里，工具每秒要生成并判定几千条路径条件。不可能每条都停下来等工程师写一段证明。自动化程度不够，就无法工业部署。",[52860,132379,132381],{"id":132380},"_25-小结","2.5 小结",[11,132383,132384],{},"几条路，各有各的难处：",[127630,132386,132387,132390,132393,132396],{},[127633,132388,132389],{},"纯 SAT 快，但把高层语义压成门电路之后，规模爆炸、语义丢失；",[127633,132391,132392],{},"一阶逻辑表达力强，但一般情形不可判定，自动证明器可能永远给不出答案；",[127633,132394,132395],{},"专用算法在自己的理论里很好用，碰到混合约束就各自为政；",[127633,132397,132398],{},"交互式定理证明能处理复杂性质，但每条路径都等人写脚本，工业化无从谈起。",[11,132400,132401,132402],{},"SMT 要做的，就是在这几条路的夹缝里找一个中间地带：",[7246,132403,132404],{},"表达力足够描述程序，判定过程保证自动，还能把多个理论组合起来。",[18,132406,132408],{"id":132407},"三smt-的解法","三、SMT 的解法",[52860,132410,132412],{"id":132411},"_31-收缩片段","3.1 收缩片段",[11,132414,132415,132416,132419],{},"SMT 的第一个策略是",[7246,132417,132418],{},"主动放弃一部分表达力","，只保留可判定的片段。",[11,132421,132422],{},"具体做法是：把主战场放在无量词片段，只允许特定理论的组合。这样既避开了 2.2 节里的不可判定性，又保留了描述程序行为所需的大部分能力。",[11,132424,132425,132426,1138,132455,132521],{},"为什么可以放弃量词？因为程序分析产生的验证条件，大多数天然就是无量词的。符号执行收集到的路径条件，是对「这一条路径上各个变量的具体取值」的约束，不需要「对所有 ",[31,132427,132429,132442],{"className":132428,"translate":35},[34],[31,132430,132432],{"className":132431},[39],[41,132433,132434],{"xmlns":43},[45,132435,132436,132440],{},[48,132437,132438],{},[51,132439,5391],{},[55,132441,5391],{"encoding":57},[31,132443,132445],{"className":132444,"ariaHidden":62},[61],[31,132446,132448,132452],{"className":132447},[66],[31,132449],{"className":132450,"style":132451},[70],"height:0.6595em;",[31,132453,5391],{"className":132454},[75,76],[31,132456,132458,132482],{"className":132457,"translate":35},[34],[31,132459,132461],{"className":132460},[39],[41,132462,132463],{"xmlns":43},[45,132464,132465,132479],{},[48,132466,132467,132469,132471,132473,132475,132477],{},[51,132468,5813],{},[95,132470,15175],{"stretchy":210},[51,132472,5391],{},[95,132474,15184],{"stretchy":210},[95,132476,16040],{},[91,132478,1977],{},[55,132480,132481],{"encoding":57},"a[i] > 0",[31,132483,132485,132512],{"className":132484,"ariaHidden":62},[61],[31,132486,132488,132491,132494,132497,132500,132503,132506,132509],{"className":132487},[66],[31,132489],{"className":132490,"style":260},[70],[31,132492,5813],{"className":132493},[75,76],[31,132495,15175],{"className":132496},[264],[31,132498,5391],{"className":132499},[75,76],[31,132501,15184],{"className":132502},[289],[31,132504],{"className":132505,"style":293},[119],[31,132507,16040],{"className":132508},[297],[31,132510],{"className":132511,"style":293},[119],[31,132513,132515,132518],{"className":132514},[66],[31,132516],{"className":132517,"style":326},[70],[31,132519,1977],{"className":132520},[75],"」这种全称陈述。",[11,132523,132524,132525,132553],{},"需要量词的地方——循环不变式、数组的整体性质、归纳证明——通常由上层工具负责：抽象解释负责生成不变式，",[31,132526,132528,132541],{"className":132527,"translate":35},[34],[31,132529,132531],{"className":132530},[39],[41,132532,132533],{"xmlns":43},[45,132534,132535,132539],{},[48,132536,132537],{},[51,132538,52065],{},[55,132540,52065],{"encoding":57},[31,132542,132544],{"className":132543,"ariaHidden":62},[61],[31,132545,132547,132550],{"className":132546},[66],[31,132548],{"className":132549,"style":1630},[70],[31,132551,52065],{"className":132552,"style":52166},[75,76],"-induction 负责把归纳化归为有限次查询，SMT 只作为底层的判定引擎。这是一个分工，而不是缺陷。",[11,132555,132556,132557,15282,132559,15282,132561,132563,132564,132566,132567,132569,132570,132572],{},"当然，「可判定」不等于「容易」。",[45608,132558,128087],{},[45608,132560,128101],{},[45608,132562,128185],{}," 都是 NP-complete；",[45608,132565,128097],{}," 是多项式时间；",[45608,132568,128173],{}," 可判定，但最坏情况是双指数；",[45608,132571,128176],{}," 则直接不可判定。",[11,132574,132575],{},"SMT 在选定的片段内保证了「答案一定会来」，但不保证「答案来得快」。工程上的成败，很大程度上取决于启发式和问题编码。",[52860,132577,132579],{"id":132578},"_32-直接表达","3.2 直接表达",[11,132581,132582,132583,2372],{},"第二个策略是",[7246,132584,132585],{},"让公式直接说人话",[11,132587,132588,132589,76375,132661,132726,132727,132768],{},"同样是「",[31,132590,132592,132613],{"className":132591,"translate":35},[34],[31,132593,132595],{"className":132594},[39],[41,132596,132597],{"xmlns":43},[45,132598,132599,132611],{},[48,132600,132601,132603,132605,132607,132609],{},[51,132602,53],{},[95,132604,434],{},[51,132606,3003],{},[95,132608,16040],{},[91,132610,19494],{},[55,132612,125640],{"encoding":57},[31,132614,132616,132634,132652],{"className":132615,"ariaHidden":62},[61],[31,132617,132619,132622,132625,132628,132631],{"className":132618},[66],[31,132620],{"className":132621,"style":307},[70],[31,132623,53],{"className":132624},[75,76],[31,132626],{"className":132627,"style":120},[119],[31,132629,434],{"className":132630},[124],[31,132632],{"className":132633,"style":120},[119],[31,132635,132637,132640,132643,132646,132649],{"className":132636},[66],[31,132638],{"className":132639,"style":53323},[70],[31,132641,3003],{"className":132642,"style":2666},[75,76],[31,132644],{"className":132645,"style":293},[119],[31,132647,16040],{"className":132648},[297],[31,132650],{"className":132651,"style":293},[119],[31,132653,132655,132658],{"className":132654},[66],[31,132656],{"className":132657,"style":326},[70],[31,132659,19494],{"className":132660},[75],[31,132662,132664,132687],{"className":132663,"translate":35},[34],[31,132665,132667],{"className":132666},[39],[41,132668,132669],{"xmlns":43},[45,132670,132671,132685],{},[48,132672,132673,132675,132677,132679,132681,132683],{},[51,132674,5813],{},[95,132676,15175],{"stretchy":210},[91,132678,1977],{},[95,132680,15184],{"stretchy":210},[95,132682,223],{},[91,132684,7853],{},[55,132686,125644],{"encoding":57},[31,132688,132690,132717],{"className":132689,"ariaHidden":62},[61],[31,132691,132693,132696,132699,132702,132705,132708,132711,132714],{"className":132692},[66],[31,132694],{"className":132695,"style":260},[70],[31,132697,5813],{"className":132698},[75,76],[31,132700,15175],{"className":132701},[264],[31,132703,1977],{"className":132704},[75],[31,132706,15184],{"className":132707},[289],[31,132709],{"className":132710,"style":293},[119],[31,132712,223],{"className":132713},[297],[31,132715],{"className":132716,"style":293},[119],[31,132718,132720,132723],{"className":132719},[66],[31,132721],{"className":132722,"style":326},[70],[31,132724,7853],{"className":132725},[75],"」，SMT 不需要把 ",[31,132728,132730,132747],{"className":132729,"translate":35},[34],[31,132731,132733],{"className":132732},[39],[41,132734,132735],{"xmlns":43},[45,132736,132737,132745],{},[48,132738,132739,132741,132743],{},[51,132740,53],{},[95,132742,1210],{"separator":62},[51,132744,3003],{},[55,132746,128368],{"encoding":57},[31,132748,132750],{"className":132749,"ariaHidden":62},[61],[31,132751,132753,132756,132759,132762,132765],{"className":132752},[66],[31,132754],{"className":132755,"style":865},[70],[31,132757,53],{"className":132758},[75,76],[31,132760,1210],{"className":132761},[1391],[31,132763],{"className":132764,"style":1467},[119],[31,132766,3003],{"className":132767,"style":2666},[75,76]," 拆成 64 个布尔变量，也不需要把数组展开成 100 个元素。它把这两个条件原样保留为两个理论原子，交给对应的求解器。",[11,132770,132771],{},"关键在于「按需展开」：",[127630,132773,132774,132780,132786],{},[127633,132775,132776,132779],{},[7246,132777,132778],{},"算术","：直接用单纯形法或分支定界，在整数\u002F实数域上推理，完全不经过布尔编码。",[127633,132781,132782,132785],{},[7246,132783,132784],{},"数组","：用 read-over-write 公理，只在公式里真的出现某个下标时才实例化。",[127633,132787,132788,132791,132792,132795,132796,15282,132868,132941],{},[7246,132789,132790],{},"位向量","：这一条要诚实一点——大多数求解器内部",[7246,132793,132794],{},"仍然会做位爆破","，因为位级编码对硬件电路太有效了。但这件事发生在求解器内部，只针对相关子表达式，而且会配合字级重写（比如 ",[31,132797,132799,132826],{"className":132798,"translate":35},[34],[31,132800,132802],{"className":132801},[39],[41,132803,132804],{"xmlns":43},[45,132805,132806,132823],{},[48,132807,132808,132810,132812,132815,132817,132819,132821],{},[51,132809,53],{},[1430,132811,1432],{},[51,132813,132814],{"mathvariant":494},"&",[1430,132816,1432],{},[91,132818,1977],{},[95,132820,223],{},[91,132822,1977],{},[55,132824,132825],{"encoding":57},"x \\,\\&\\, 0 = 0",[31,132827,132829,132859],{"className":132828,"ariaHidden":62},[61],[31,132830,132832,132835,132838,132841,132844,132847,132850,132853,132856],{"className":132831},[66],[31,132833],{"className":132834,"style":1630},[70],[31,132836,53],{"className":132837},[75,76],[31,132839],{"className":132840,"style":1467},[119],[31,132842,132814],{"className":132843},[75],[31,132845],{"className":132846,"style":1467},[119],[31,132848,1977],{"className":132849},[75],[31,132851],{"className":132852,"style":293},[119],[31,132854,223],{"className":132855},[297],[31,132857],{"className":132858,"style":293},[119],[31,132860,132862,132865],{"className":132861},[66],[31,132863],{"className":132864,"style":326},[70],[31,132866,1977],{"className":132867},[75],[31,132869,132871,132893],{"className":132870,"translate":35},[34],[31,132872,132874],{"className":132873},[39],[41,132875,132876],{"xmlns":43},[45,132877,132878,132890],{},[48,132879,132880,132882,132884,132886,132888],{},[51,132881,53],{},[95,132883,97],{},[51,132885,53],{},[95,132887,223],{},[91,132889,1977],{},[55,132891,132892],{"encoding":57},"x - x = 0",[31,132894,132896,132914,132932],{"className":132895,"ariaHidden":62},[61],[31,132897,132899,132902,132905,132908,132911],{"className":132898},[66],[31,132900],{"className":132901,"style":307},[70],[31,132903,53],{"className":132904},[75,76],[31,132906],{"className":132907,"style":120},[119],[31,132909,97],{"className":132910},[124],[31,132912],{"className":132913,"style":120},[119],[31,132915,132917,132920,132923,132926,132929],{"className":132916},[66],[31,132918],{"className":132919,"style":71},[70],[31,132921,53],{"className":132922},[75,76],[31,132924],{"className":132925,"style":293},[119],[31,132927,223],{"className":132928},[297],[31,132930],{"className":132931,"style":293},[119],[31,132933,132935,132938],{"className":132934},[66],[31,132936],{"className":132937,"style":326},[70],[31,132939,1977],{"className":132940},[75],"）和等价类合并。对使用者来说，它仍然是透明的。",[11,132943,132944],{},"数组的 read-over-write 公理长这样：",[31,132946,132948],{"className":132947,"translate":35},[411],[31,132949,132951,133091],{"className":132950,"translate":35},[34],[31,132952,132954],{"className":132953},[39],[41,132955,132956],{"xmlns":43,"display":420},[45,132957,132958,133088],{},[48,132959,132960,132974,132976,132988,132990,132992,132994,132996,132998,133000,133002,133004,133006,133008,133010],{},[48,132961,132962,132964,132966,132968,132970,132972],{},[51,132963,14551],{"mathvariant":494},[51,132965,30042],{"mathvariant":494},[51,132967,87599],{"mathvariant":494},[51,132969,30042],{"mathvariant":494},[51,132971,1412],{"mathvariant":494},[51,132973,12271],{"mathvariant":494},[95,132975,211],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[48,132977,132978,132980,132982,132984,132986],{},[51,132979,14551],{"mathvariant":494},[51,132981,12271],{"mathvariant":494},[51,132983,2552],{"mathvariant":494},[51,132985,2549],{"mathvariant":494},[51,132987,30042],{"mathvariant":494},[95,132989,211],{"stretchy":210},[51,132991,5813],{},[95,132993,1210],{"separator":62},[51,132995,5391],{},[95,132997,1210],{"separator":62},[51,132999,5354],{},[95,133001,220],{"stretchy":210},[95,133003,1210],{"separator":62},[51,133005,2555],{},[95,133007,220],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[95,133009,223],{},[48,133011,133012,133014],{},[95,133013,2512],{"fence":62},[5606,133015,133018,133042],{"rowspacing":133016,"columnalign":133017,"columnspacing":1584},"0.36em","left left",[5612,133019,133020,133030],{},[5615,133021,133022],{},[2071,133023,133024],{"scriptlevel":1977,"displaystyle":210},[48,133025,133026,133028],{},[51,133027,5354],{},[95,133029,1210],{"separator":62},[5615,133031,133032],{},[2071,133033,133034],{"scriptlevel":1977,"displaystyle":210},[48,133035,133036,133038,133040],{},[51,133037,5391],{},[95,133039,223],{},[51,133041,2555],{},[5612,133043,133044,133076],{},[5615,133045,133046],{},[2071,133047,133048],{"scriptlevel":1977,"displaystyle":210},[48,133049,133050,133064,133066,133068,133070,133072,133074],{},[48,133051,133052,133054,133056,133058,133060,133062],{},[51,133053,14551],{"mathvariant":494},[51,133055,30042],{"mathvariant":494},[51,133057,87599],{"mathvariant":494},[51,133059,30042],{"mathvariant":494},[51,133061,1412],{"mathvariant":494},[51,133063,12271],{"mathvariant":494},[95,133065,211],{"stretchy":210},[51,133067,5813],{},[95,133069,1210],{"separator":62},[51,133071,2555],{},[95,133073,220],{"stretchy":210},[95,133075,1210],{"separator":62},[5615,133077,133078],{},[2071,133079,133080],{"scriptlevel":1977,"displaystyle":210},[48,133081,133082,133084,133086],{},[51,133083,5391],{},[95,133085,21271],{"mathvariant":494},[51,133087,2555],{},[55,133089,133090],{"encoding":57},"\\mathrm{select}\\bigl(\\mathrm{store}(a, i, v), j\\bigr) =\n\\begin{cases}\nv, & i = j \\\\\n\\mathrm{select}(a, j), & i \\ne j\n\\end{cases}",[31,133092,133094,133169],{"className":133093,"ariaHidden":62},[61],[31,133095,133097,133100,133106,133112,133118,133121,133124,133127,133130,133133,133136,133139,133142,133145,133148,133151,133154,133160,133163,133166],{"className":133096},[66],[31,133098],{"className":133099,"style":4988},[70],[31,133101,133103],{"className":133102},[75],[31,133104,126909],{"className":133105},[75,2745],[31,133107,133109],{"className":133108},[264],[31,133110,211],{"className":133111},[1307,1308],[31,133113,133115],{"className":133114},[75],[31,133116,127844],{"className":133117},[75,2745],[31,133119,211],{"className":133120},[264],[31,133122,5813],{"className":133123},[75,76],[31,133125,1210],{"className":133126},[1391],[31,133128],{"className":133129,"style":1467},[119],[31,133131,5391],{"className":133132},[75,76],[31,133134,1210],{"className":133135},[1391],[31,133137],{"className":133138,"style":1467},[119],[31,133140,5354],{"className":133141,"style":2666},[75,76],[31,133143,220],{"className":133144},[289],[31,133146,1210],{"className":133147},[1391],[31,133149],{"className":133150,"style":1467},[119],[31,133152,2555],{"className":133153,"style":5514},[75,76],[31,133155,133157],{"className":133156},[289],[31,133158,220],{"className":133159},[1307,1308],[31,133161],{"className":133162,"style":293},[119],[31,133164,223],{"className":133165},[297],[31,133167],{"className":133168,"style":293},[119],[31,133170,133172,133176],{"className":133171},[66],[31,133173],{"className":133174,"style":133175},[70],"height:3em;vertical-align:-1.25em;",[31,133177,133179,133185,133374],{"className":133178},[3860],[31,133180,133182],{"className":133181,"style":3868},[264,3867],[31,133183,2512],{"className":133184},[1307,116039],[31,133186,133188],{"className":133187},[75],[31,133189,133191,133268,133272],{"className":133190},[5606],[31,133192,133194],{"className":133193},[6062],[31,133195,133197,133259],{"className":133196},[1043,1044],[31,133198,133200,133256],{"className":133199},[1048],[31,133201,133204,133220],{"className":133202,"style":133203},[1052],"height:1.69em;",[31,133205,133207,133211],{"style":133206},"top:-3.69em;",[31,133208],{"className":133209,"style":133210},[1060],"height:3.008em;",[31,133212,133214,133217],{"className":133213},[75],[31,133215,5354],{"className":133216,"style":2666},[75,76],[31,133218,1210],{"className":133219},[1391],[31,133221,133223,133226],{"style":133222},"top:-2.25em;",[31,133224],{"className":133225,"style":133210},[1060],[31,133227,133229,133235,133238,133241,133244,133247,133250,133253],{"className":133228},[75],[31,133230,133232],{"className":133231},[75],[31,133233,126909],{"className":133234},[75,2745],[31,133236,211],{"className":133237},[264],[31,133239,5813],{"className":133240},[75,76],[31,133242,1210],{"className":133243},[1391],[31,133245],{"className":133246,"style":1467},[119],[31,133248,2555],{"className":133249,"style":5514},[75,76],[31,133251,220],{"className":133252},[289],[31,133254,1210],{"className":133255},[1391],[31,133257,1076],{"className":133258},[1075],[31,133260,133262],{"className":133261},[1048],[31,133263,133266],{"className":133264,"style":133265},[1052],"height:1.19em;",[31,133267],{},[31,133269],{"className":133270,"style":133271},[71201],"width:1em;",[31,133273,133275],{"className":133274},[6062],[31,133276,133278,133366],{"className":133277},[1043,1044],[31,133279,133281,133363],{"className":133280},[1048],[31,133282,133284,133307],{"className":133283,"style":133203},[1052],[31,133285,133286,133289],{"style":133206},[31,133287],{"className":133288,"style":133210},[1060],[31,133290,133292,133295,133298,133301,133304],{"className":133291},[75],[31,133293,5391],{"className":133294},[75,76],[31,133296],{"className":133297,"style":293},[119],[31,133299,223],{"className":133300},[297],[31,133302],{"className":133303,"style":293},[119],[31,133305,2555],{"className":133306,"style":5514},[75,76],[31,133308,133309,133312],{"style":133222},[31,133310],{"className":133311,"style":133210},[1060],[31,133313,133315,133318,133321,133357,133360],{"className":133314},[75],[31,133316,5391],{"className":133317},[75,76],[31,133319],{"className":133320,"style":293},[119],[31,133322,133324,133351,133354],{"className":133323},[297],[31,133325,133327],{"className":133326},[297],[31,133328,133330],{"className":133329},[75,21516],[31,133331,133333],{"className":133332},[21520],[31,133334,133336,133339,133348],{"className":133335},[21524],[31,133337],{"className":133338,"style":4202},[70],[31,133340,133342],{"className":133341},[21531],[31,133343,133345],{"className":133344},[75],[31,133346,21538],{"className":133347},[297],[31,133349],{"className":133350},[21542],[31,133352],{"className":133353},[119,21546],[31,133355,223],{"className":133356},[297],[31,133358],{"className":133359,"style":293},[119],[31,133361,2555],{"className":133362,"style":5514},[75,76],[31,133364,1076],{"className":133365},[1075],[31,133367,133369],{"className":133368},[1048],[31,133370,133372],{"className":133371,"style":133265},[1052],[31,133373],{},[31,133375],{"className":133376},[289,2139],[11,133378,133379,133380,133424,133425,133476],{},"如果公式里只关心 ",[31,133381,133383,133403],{"className":133382,"translate":35},[34],[31,133384,133386],{"className":133385},[39],[41,133387,133388],{"xmlns":43},[45,133389,133390,133400],{},[48,133391,133392,133394,133396,133398],{},[51,133393,5813],{},[95,133395,15175],{"stretchy":210},[91,133397,1977],{},[95,133399,15184],{"stretchy":210},[55,133401,133402],{"encoding":57},"a[0]",[31,133404,133406],{"className":133405,"ariaHidden":62},[61],[31,133407,133409,133412,133415,133418,133421],{"className":133408},[66],[31,133410],{"className":133411,"style":260},[70],[31,133413,5813],{"className":133414},[75,76],[31,133416,15175],{"className":133417},[264],[31,133419,1977],{"className":133420},[75],[31,133422,15184],{"className":133423},[289],"，求解器就只实例化 ",[31,133426,133428,133446],{"className":133427,"translate":35},[34],[31,133429,133431],{"className":133430},[39],[41,133432,133433],{"xmlns":43},[45,133434,133435,133443],{},[48,133436,133437,133439,133441],{},[51,133438,5391],{},[95,133440,223],{},[91,133442,1977],{},[55,133444,133445],{"encoding":57},"i = 0",[31,133447,133449,133467],{"className":133448,"ariaHidden":62},[61],[31,133450,133452,133455,133458,133461,133464],{"className":133451},[66],[31,133453],{"className":133454,"style":132451},[70],[31,133456,5391],{"className":133457},[75,76],[31,133459],{"className":133460,"style":293},[119],[31,133462,223],{"className":133463},[297],[31,133465],{"className":133466,"style":293},[119],[31,133468,133470,133473],{"className":133469},[66],[31,133471],{"className":133472,"style":326},[70],[31,133474,1977],{"className":133475},[75]," 这一条，剩下的 99 个元素根本不会出现。这是「按需」和「全局」的区别，也是 SMT 能在数组上跑赢位爆破的原因。",[11,133478,133479,133480,133483],{},"所以更准确的说法不是「SMT 消灭了底层编码」，而是：",[7246,133481,133482],{},"SMT 把「怎么编码」的决定权从用户手里交给了求解器","。求解器可以根据理论特性选择最合适的内部表示，而不是让用户先做一次不可逆的全局展开。",[11,133485,133486,133487,133528,133529,2372],{},"用同一个约束对比一下。纯 SAT 的版本要生成数百到上千个布尔变量、上千条子句；SMT 的版本只保留两个理论原子。SAT 只能做布尔剪枝，SMT 能利用算术和数组的语义；SAT 的答案是一组布尔赋值，还要反解出 ",[31,133488,133490,133507],{"className":133489,"translate":35},[34],[31,133491,133493],{"className":133492},[39],[41,133494,133495],{"xmlns":43},[45,133496,133497,133505],{},[48,133498,133499,133501,133503],{},[51,133500,53],{},[95,133502,1210],{"separator":62},[51,133504,3003],{},[55,133506,128368],{"encoding":57},[31,133508,133510],{"className":133509,"ariaHidden":62},[61],[31,133511,133513,133516,133519,133522,133525],{"className":133512},[66],[31,133514],{"className":133515,"style":865},[70],[31,133517,53],{"className":133518},[75,76],[31,133520,1210],{"className":133521},[1391],[31,133523],{"className":133524,"style":1467},[119],[31,133526,3003],{"className":133527,"style":2666},[75,76],"，SMT 直接给出 ",[31,133530,133532,133558],{"className":133531,"translate":35},[34],[31,133533,133535],{"className":133534},[39],[41,133536,133537],{"xmlns":43},[45,133538,133539,133556],{},[48,133540,133541,133543,133545,133548,133550,133552,133554],{},[51,133542,53],{},[95,133544,223],{},[91,133546,133547],{},"6",[95,133549,1210],{"separator":62},[51,133551,3003],{},[95,133553,223],{},[91,133555,7853],{},[55,133557,125648],{"encoding":57},[31,133559,133561,133579,133606],{"className":133560,"ariaHidden":62},[61],[31,133562,133564,133567,133570,133573,133576],{"className":133563},[66],[31,133565],{"className":133566,"style":71},[70],[31,133568,53],{"className":133569},[75,76],[31,133571],{"className":133572,"style":293},[119],[31,133574,223],{"className":133575},[297],[31,133577],{"className":133578,"style":293},[119],[31,133580,133582,133585,133588,133591,133594,133597,133600,133603],{"className":133581},[66],[31,133583],{"className":133584,"style":41982},[70],[31,133586,133547],{"className":133587},[75],[31,133589,1210],{"className":133590},[1391],[31,133592],{"className":133593,"style":1467},[119],[31,133595,3003],{"className":133596,"style":2666},[75,76],[31,133598],{"className":133599,"style":293},[119],[31,133601,223],{"className":133602},[297],[31,133604],{"className":133605,"style":293},[119],[31,133607,133609,133612],{"className":133608},[66],[31,133610],{"className":133611,"style":326},[70],[31,133613,7853],{"className":133614},[75],[52860,133616,133618],{"id":133617},"_33-组合理论","3.3 组合理论",[11,133620,133621,133622,2372],{},"第三个策略是",[7246,133623,133624],{},"让不同理论通过共享等式来协作",[11,133626,133627,133628,133631],{},"最经典的方法是 Nelson–Oppen 组合。它的前提是：两个理论的签名不相交（各自管各自的符号），并且都满足「稳定无限」性质（任何可满足的公式都有无穷模型）。在这个前提下，两个理论只需要交换",[7246,133629,133630],{},"变量之间的等式","，就能组合成一个完整的判定过程。",[11,133633,133634],{},"看一个例子：",[31,133636,133638],{"className":133637,"translate":35},[411],[31,133639,133641,133682],{"className":133640,"translate":35},[34],[31,133642,133644],{"className":133643},[39],[41,133645,133646],{"xmlns":43,"display":420},[45,133647,133648,133680],{},[48,133649,133650,133652,133654,133656,133658,133660,133662,133664,133666,133668,133670,133672,133674,133676,133678],{},[51,133651,53],{},[95,133653,223],{},[51,133655,3003],{},[1430,133657,125504],{},[95,133659,125507],{},[1430,133661,125504],{},[51,133663,1662],{},[95,133665,211],{"stretchy":210},[51,133667,53],{},[95,133669,220],{"stretchy":210},[95,133671,21271],{"mathvariant":494},[51,133673,1662],{},[95,133675,211],{"stretchy":210},[51,133677,3003],{},[95,133679,220],{"stretchy":210},[55,133681,130753],{"encoding":57},[31,133683,133685,133703,133727,133787],{"className":133684,"ariaHidden":62},[61],[31,133686,133688,133691,133694,133697,133700],{"className":133687},[66],[31,133689],{"className":133690,"style":71},[70],[31,133692,53],{"className":133693},[75,76],[31,133695],{"className":133696,"style":293},[119],[31,133698,223],{"className":133699},[297],[31,133701],{"className":133702,"style":293},[119],[31,133704,133706,133709,133712,133715,133718,133721,133724],{"className":133705},[66],[31,133707],{"className":133708,"style":126295},[70],[31,133710,3003],{"className":133711,"style":2666},[75,76],[31,133713],{"className":133714,"style":293},[119],[31,133716],{"className":133717,"style":120},[119],[31,133719,125507],{"className":133720},[124],[31,133722],{"className":133723,"style":293},[119],[31,133725],{"className":133726,"style":120},[119],[31,133728,133730,133733,133736,133739,133742,133745,133748,133784],{"className":133729},[66],[31,133731],{"className":133732,"style":260},[70],[31,133734,1662],{"className":133735,"style":35196},[75,76],[31,133737,211],{"className":133738},[264],[31,133740,53],{"className":133741},[75,76],[31,133743,220],{"className":133744},[289],[31,133746],{"className":133747,"style":293},[119],[31,133749,133751,133778,133781],{"className":133750},[297],[31,133752,133754],{"className":133753},[297],[31,133755,133757],{"className":133756},[75,21516],[31,133758,133760],{"className":133759},[21520],[31,133761,133763,133766,133775],{"className":133762},[21524],[31,133764],{"className":133765,"style":4202},[70],[31,133767,133769],{"className":133768},[21531],[31,133770,133772],{"className":133771},[75],[31,133773,21538],{"className":133774},[297],[31,133776],{"className":133777},[21542],[31,133779],{"className":133780},[119,21546],[31,133782,223],{"className":133783},[297],[31,133785],{"className":133786,"style":293},[119],[31,133788,133790,133793,133796,133799,133802],{"className":133789},[66],[31,133791],{"className":133792,"style":260},[70],[31,133794,1662],{"className":133795,"style":35196},[75,76],[31,133797,211],{"className":133798},[264],[31,133800,3003],{"className":133801,"style":2666},[75,76],[31,133803,220],{"className":133804},[289],[127630,133806,133807,133861,134002],{},[127633,133808,133809,133810,133860],{},"算术部分看到 ",[31,133811,133813,133830],{"className":133812,"translate":35},[34],[31,133814,133816],{"className":133815},[39],[41,133817,133818],{"xmlns":43},[45,133819,133820,133828],{},[48,133821,133822,133824,133826],{},[51,133823,53],{},[95,133825,223],{},[51,133827,3003],{},[55,133829,130905],{"encoding":57},[31,133831,133833,133851],{"className":133832,"ariaHidden":62},[61],[31,133834,133836,133839,133842,133845,133848],{"className":133835},[66],[31,133837],{"className":133838,"style":71},[70],[31,133840,53],{"className":133841},[75,76],[31,133843],{"className":133844,"style":293},[119],[31,133846,223],{"className":133847},[297],[31,133849],{"className":133850,"style":293},[119],[31,133852,133854,133857],{"className":133853},[66],[31,133855],{"className":133856,"style":865},[70],[31,133858,3003],{"className":133859,"style":2666},[75,76],"，本身没有矛盾；",[127633,133862,133863,133864,133914,133915,2446,133958,134001],{},"EUF 部分如果不知道 ",[31,133865,133867,133884],{"className":133866,"translate":35},[34],[31,133868,133870],{"className":133869},[39],[41,133871,133872],{"xmlns":43},[45,133873,133874,133882],{},[48,133875,133876,133878,133880],{},[51,133877,53],{},[95,133879,223],{},[51,133881,3003],{},[55,133883,130905],{"encoding":57},[31,133885,133887,133905],{"className":133886,"ariaHidden":62},[61],[31,133888,133890,133893,133896,133899,133902],{"className":133889},[66],[31,133891],{"className":133892,"style":71},[70],[31,133894,53],{"className":133895},[75,76],[31,133897],{"className":133898,"style":293},[119],[31,133900,223],{"className":133901},[297],[31,133903],{"className":133904,"style":293},[119],[31,133906,133908,133911],{"className":133907},[66],[31,133909],{"className":133910,"style":865},[70],[31,133912,3003],{"className":133913,"style":2666},[75,76],"，就会认为 ",[31,133916,133918,133937],{"className":133917,"translate":35},[34],[31,133919,133921],{"className":133920},[39],[41,133922,133923],{"xmlns":43},[45,133924,133925,133935],{},[48,133926,133927,133929,133931,133933],{},[51,133928,1662],{},[95,133930,211],{"stretchy":210},[51,133932,53],{},[95,133934,220],{"stretchy":210},[55,133936,40190],{"encoding":57},[31,133938,133940],{"className":133939,"ariaHidden":62},[61],[31,133941,133943,133946,133949,133952,133955],{"className":133942},[66],[31,133944],{"className":133945,"style":260},[70],[31,133947,1662],{"className":133948,"style":35196},[75,76],[31,133950,211],{"className":133951},[264],[31,133953,53],{"className":133954},[75,76],[31,133956,220],{"className":133957},[289],[31,133959,133961,133980],{"className":133960,"translate":35},[34],[31,133962,133964],{"className":133963},[39],[41,133965,133966],{"xmlns":43},[45,133967,133968,133978],{},[48,133969,133970,133972,133974,133976],{},[51,133971,1662],{},[95,133973,211],{"stretchy":210},[51,133975,3003],{},[95,133977,220],{"stretchy":210},[55,133979,131427],{"encoding":57},[31,133981,133983],{"className":133982,"ariaHidden":62},[61],[31,133984,133986,133989,133992,133995,133998],{"className":133985},[66],[31,133987],{"className":133988,"style":260},[70],[31,133990,1662],{"className":133991,"style":35196},[75,76],[31,133993,211],{"className":133994},[264],[31,133996,3003],{"className":133997,"style":2666},[75,76],[31,133999,220],{"className":134000},[289]," 可以不同，也不会报错；",[127633,134003,134004,134005,134055,134056,131449,134136,134249],{},"一旦算术把等式 ",[31,134006,134008,134025],{"className":134007,"translate":35},[34],[31,134009,134011],{"className":134010},[39],[41,134012,134013],{"xmlns":43},[45,134014,134015,134023],{},[48,134016,134017,134019,134021],{},[51,134018,53],{},[95,134020,223],{},[51,134022,3003],{},[55,134024,130905],{"encoding":57},[31,134026,134028,134046],{"className":134027,"ariaHidden":62},[61],[31,134029,134031,134034,134037,134040,134043],{"className":134030},[66],[31,134032],{"className":134033,"style":71},[70],[31,134035,53],{"className":134036},[75,76],[31,134038],{"className":134039,"style":293},[119],[31,134041,223],{"className":134042},[297],[31,134044],{"className":134045,"style":293},[119],[31,134047,134049,134052],{"className":134048},[66],[31,134050],{"className":134051,"style":865},[70],[31,134053,3003],{"className":134054,"style":2666},[75,76]," 传给 EUF，EUF 立刻推出 ",[31,134057,134059,134088],{"className":134058,"translate":35},[34],[31,134060,134062],{"className":134061},[39],[41,134063,134064],{"xmlns":43},[45,134065,134066,134086],{},[48,134067,134068,134070,134072,134074,134076,134078,134080,134082,134084],{},[51,134069,1662],{},[95,134071,211],{"stretchy":210},[51,134073,53],{},[95,134075,220],{"stretchy":210},[95,134077,223],{},[51,134079,1662],{},[95,134081,211],{"stretchy":210},[51,134083,3003],{},[95,134085,220],{"stretchy":210},[55,134087,131653],{"encoding":57},[31,134089,134091,134118],{"className":134090,"ariaHidden":62},[61],[31,134092,134094,134097,134100,134103,134106,134109,134112,134115],{"className":134093},[66],[31,134095],{"className":134096,"style":260},[70],[31,134098,1662],{"className":134099,"style":35196},[75,76],[31,134101,211],{"className":134102},[264],[31,134104,53],{"className":134105},[75,76],[31,134107,220],{"className":134108},[289],[31,134110],{"className":134111,"style":293},[119],[31,134113,223],{"className":134114},[297],[31,134116],{"className":134117,"style":293},[119],[31,134119,134121,134124,134127,134130,134133],{"className":134120},[66],[31,134122],{"className":134123,"style":260},[70],[31,134125,1662],{"className":134126,"style":35196},[75,76],[31,134128,211],{"className":134129},[264],[31,134131,3003],{"className":134132,"style":2666},[75,76],[31,134134,220],{"className":134135},[289],[31,134137,134139,134168],{"className":134138,"translate":35},[34],[31,134140,134142],{"className":134141},[39],[41,134143,134144],{"xmlns":43},[45,134145,134146,134166],{},[48,134147,134148,134150,134152,134154,134156,134158,134160,134162,134164],{},[51,134149,1662],{},[95,134151,211],{"stretchy":210},[51,134153,53],{},[95,134155,220],{"stretchy":210},[95,134157,21271],{"mathvariant":494},[51,134159,1662],{},[95,134161,211],{"stretchy":210},[51,134163,3003],{},[95,134165,220],{"stretchy":210},[55,134167,127880],{"encoding":57},[31,134169,134171,134231],{"className":134170,"ariaHidden":62},[61],[31,134172,134174,134177,134180,134183,134186,134189,134192,134228],{"className":134173},[66],[31,134175],{"className":134176,"style":260},[70],[31,134178,1662],{"className":134179,"style":35196},[75,76],[31,134181,211],{"className":134182},[264],[31,134184,53],{"className":134185},[75,76],[31,134187,220],{"className":134188},[289],[31,134190],{"className":134191,"style":293},[119],[31,134193,134195,134222,134225],{"className":134194},[297],[31,134196,134198],{"className":134197},[297],[31,134199,134201],{"className":134200},[75,21516],[31,134202,134204],{"className":134203},[21520],[31,134205,134207,134210,134219],{"className":134206},[21524],[31,134208],{"className":134209,"style":4202},[70],[31,134211,134213],{"className":134212},[21531],[31,134214,134216],{"className":134215},[75],[31,134217,21538],{"className":134218},[297],[31,134220],{"className":134221},[21542],[31,134223],{"className":134224},[119,21546],[31,134226,223],{"className":134227},[297],[31,134229],{"className":134230,"style":293},[119],[31,134232,134234,134237,134240,134243,134246],{"className":134233},[66],[31,134235],{"className":134236,"style":260},[70],[31,134238,1662],{"className":134239,"style":35196},[75,76],[31,134241,211],{"className":134242},[264],[31,134244,3003],{"className":134245,"style":2666},[75,76],[31,134247,220],{"className":134248},[289]," 冲突。",[11,134251,134252],{},"于是整个公式被判为不可满足。这个结论不是某一个理论单独得出的，而是两个理论「对了一下账」才对出来的。",[11,134254,134255],{},"再看一个反方向的例子：",[31,134257,134259],{"className":134258,"translate":35},[411],[31,134260,134262,134308],{"className":134261,"translate":35},[34],[31,134263,134265],{"className":134264},[39],[41,134266,134267],{"xmlns":43,"display":420},[45,134268,134269,134305],{},[48,134270,134271,134273,134275,134277,134279,134281,134283,134285,134287,134289,134291,134293,134295,134297,134299,134301,134303],{},[51,134272,53],{},[95,134274,434],{},[51,134276,3003],{},[95,134278,16040],{},[91,134280,19494],{},[1430,134282,125504],{},[95,134284,125507],{},[1430,134286,125504],{},[51,134288,53],{},[95,134290,223],{},[91,134292,93],{},[1430,134294,125504],{},[95,134296,125507],{},[1430,134298,125504],{},[51,134300,3003],{},[95,134302,223],{},[91,134304,15416],{},[55,134306,134307],{"encoding":57},"x + y > 10 \\;\\wedge\\; x = 3 \\;\\wedge\\; y = 4",[31,134309,134311,134329,134347,134371,134389,134413,134431],{"className":134310,"ariaHidden":62},[61],[31,134312,134314,134317,134320,134323,134326],{"className":134313},[66],[31,134315],{"className":134316,"style":307},[70],[31,134318,53],{"className":134319},[75,76],[31,134321],{"className":134322,"style":120},[119],[31,134324,434],{"className":134325},[124],[31,134327],{"className":134328,"style":120},[119],[31,134330,134332,134335,134338,134341,134344],{"className":134331},[66],[31,134333],{"className":134334,"style":53323},[70],[31,134336,3003],{"className":134337,"style":2666},[75,76],[31,134339],{"className":134340,"style":293},[119],[31,134342,16040],{"className":134343},[297],[31,134345],{"className":134346,"style":293},[119],[31,134348,134350,134353,134356,134359,134362,134365,134368],{"className":134349},[66],[31,134351],{"className":134352,"style":326},[70],[31,134354,19494],{"className":134355},[75],[31,134357],{"className":134358,"style":293},[119],[31,134360],{"className":134361,"style":120},[119],[31,134363,125507],{"className":134364},[124],[31,134366],{"className":134367,"style":293},[119],[31,134369],{"className":134370,"style":120},[119],[31,134372,134374,134377,134380,134383,134386],{"className":134373},[66],[31,134375],{"className":134376,"style":71},[70],[31,134378,53],{"className":134379},[75,76],[31,134381],{"className":134382,"style":293},[119],[31,134384,223],{"className":134385},[297],[31,134387],{"className":134388,"style":293},[119],[31,134390,134392,134395,134398,134401,134404,134407,134410],{"className":134391},[66],[31,134393],{"className":134394,"style":326},[70],[31,134396,93],{"className":134397},[75],[31,134399],{"className":134400,"style":293},[119],[31,134402],{"className":134403,"style":120},[119],[31,134405,125507],{"className":134406},[124],[31,134408],{"className":134409,"style":293},[119],[31,134411],{"className":134412,"style":120},[119],[31,134414,134416,134419,134422,134425,134428],{"className":134415},[66],[31,134417],{"className":134418,"style":865},[70],[31,134420,3003],{"className":134421,"style":2666},[75,76],[31,134423],{"className":134424,"style":293},[119],[31,134426,223],{"className":134427},[297],[31,134429],{"className":134430,"style":293},[119],[31,134432,134434,134437],{"className":134433},[66],[31,134435],{"className":134436,"style":326},[70],[31,134438,15416],{"className":134439},[75],[11,134441,134442,134443,134539],{},"这里算术部分自己就能算出 ",[31,134444,134446,134472],{"className":134445,"translate":35},[34],[31,134447,134449],{"className":134448},[39],[41,134450,134451],{"xmlns":43},[45,134452,134453,134469],{},[48,134454,134455,134457,134459,134461,134463,134465,134467],{},[91,134456,93],{},[95,134458,434],{},[91,134460,15416],{},[95,134462,223],{},[91,134464,44675],{},[95,134466,9758],{},[91,134468,19494],{},[55,134470,134471],{"encoding":57},"3 + 4 = 7 \u003C 10",[31,134473,134475,134493,134511,134530],{"className":134474,"ariaHidden":62},[61],[31,134476,134478,134481,134484,134487,134490],{"className":134477},[66],[31,134479],{"className":134480,"style":112},[70],[31,134482,93],{"className":134483},[75],[31,134485],{"className":134486,"style":120},[119],[31,134488,434],{"className":134489},[124],[31,134491],{"className":134492,"style":120},[119],[31,134494,134496,134499,134502,134505,134508],{"className":134495},[66],[31,134497],{"className":134498,"style":326},[70],[31,134500,15416],{"className":134501},[75],[31,134503],{"className":134504,"style":293},[119],[31,134506,223],{"className":134507},[297],[31,134509],{"className":134510,"style":293},[119],[31,134512,134514,134518,134521,134524,134527],{"className":134513},[66],[31,134515],{"className":134516,"style":134517},[70],"height:0.6835em;vertical-align:-0.0391em;",[31,134519,44675],{"className":134520},[75],[31,134522],{"className":134523,"style":293},[119],[31,134525,9758],{"className":134526},[297],[31,134528],{"className":134529,"style":293},[119],[31,134531,134533,134536],{"className":134532},[66],[31,134534],{"className":134535,"style":326},[70],[31,134537,19494],{"className":134538},[75],"，直接判 unsat，根本不需要 EUF 参与。",[11,134541,134542,134543],{},"现代求解器（Z3、cvc5 等）通常不严格照搬经典 Nelson–Oppen，而是在 DPLL(T) 框架下做「理论传播 + 冲突解释」。但核心思想没变：",[7246,134544,134545],{},"每个理论只懂自己那一摊，通过交换少量关键事实来达成全局一致。",[11,134547,134548,134549,134633,134634,134734],{},"有一个细节值得注意：如果理论是「非凸」的，光交换等式还不够。比如在整数算术里，",[31,134550,134552,134570],{"className":134551,"translate":35},[34],[31,134553,134555],{"className":134554},[39],[41,134556,134557],{"xmlns":43},[45,134558,134559,134567],{},[48,134560,134561,134563,134565],{},[51,134562,53],{},[95,134564,21271],{"mathvariant":494},[91,134566,1977],{},[55,134568,134569],{"encoding":57},"x \\ne 0",[31,134571,134573,134624],{"className":134572,"ariaHidden":62},[61],[31,134574,134576,134579,134582,134585,134621],{"className":134575},[66],[31,134577],{"className":134578,"style":4202},[70],[31,134580,53],{"className":134581},[75,76],[31,134583],{"className":134584,"style":293},[119],[31,134586,134588,134615,134618],{"className":134587},[297],[31,134589,134591],{"className":134590},[297],[31,134592,134594],{"className":134593},[75,21516],[31,134595,134597],{"className":134596},[21520],[31,134598,134600,134603,134612],{"className":134599},[21524],[31,134601],{"className":134602,"style":4202},[70],[31,134604,134606],{"className":134605},[21531],[31,134607,134609],{"className":134608},[75],[31,134610,21538],{"className":134611},[297],[31,134613],{"className":134614},[21542],[31,134616],{"className":134617},[119,21546],[31,134619,223],{"className":134620},[297],[31,134622],{"className":134623,"style":293},[119],[31,134625,134627,134630],{"className":134626},[66],[31,134628],{"className":134629,"style":326},[70],[31,134631,1977],{"className":134632},[75]," 等价于 ",[31,134635,134637,134665],{"className":134636,"translate":35},[34],[31,134638,134640],{"className":134639},[39],[41,134641,134642],{"xmlns":43},[45,134643,134644,134662],{},[48,134645,134646,134648,134650,134652,134654,134656,134658,134660],{},[51,134647,53],{},[95,134649,347],{},[95,134651,97],{},[91,134653,205],{},[95,134655,126150],{},[51,134657,53],{},[95,134659,5662],{},[91,134661,205],{},[55,134663,134664],{"encoding":57},"x \\le -1 \\vee x \\ge 1",[31,134666,134668,134686,134707,134725],{"className":134667,"ariaHidden":62},[61],[31,134669,134671,134674,134677,134680,134683],{"className":134670},[66],[31,134672],{"className":134673,"style":385},[70],[31,134675,53],{"className":134676},[75,76],[31,134678],{"className":134679,"style":293},[119],[31,134681,347],{"className":134682},[297],[31,134684],{"className":134685,"style":293},[119],[31,134687,134689,134692,134695,134698,134701,134704],{"className":134688},[66],[31,134690],{"className":134691,"style":112},[70],[31,134693,97],{"className":134694},[75],[31,134696,205],{"className":134697},[75],[31,134699],{"className":134700,"style":120},[119],[31,134702,126150],{"className":134703},[124],[31,134705],{"className":134706,"style":120},[119],[31,134708,134710,134713,134716,134719,134722],{"className":134709},[66],[31,134711],{"className":134712,"style":385},[70],[31,134714,53],{"className":134715},[75,76],[31,134717],{"className":134718,"style":293},[119],[31,134720,5662],{"className":134721},[297],[31,134723],{"className":134724,"style":293},[119],[31,134726,134728,134731],{"className":134727},[66],[31,134729],{"className":134730,"style":326},[70],[31,134732,205],{"className":134733},[75],"，这是一个析取，没法用一组等式表达。上层只能做情况分裂（case split），分别假设每一支再检查。这会让组合复杂度上升，也是混合理论求解有时会突然变慢的原因之一。",[52860,134736,134738],{"id":134737},"_34-给出解释","3.4 给出解释",[11,134740,134741,134742,2372],{},"第四个策略是",[7246,134743,134744],{},"不只回答是或否，还给出可操作的结果",[127630,134746,134747,134752,134757,134816],{},[127633,134748,134749,134751],{},[7246,134750,125655],{},"：求解器返回一个模型，给出每个变量的具体取值。对符号执行来说，这就是一条能触发目标路径的测试输入。",[127633,134753,134754,134756],{},[7246,134755,131566],{},"：求解器可以返回一个 unsat core，也就是一组不可满足的子约束，告诉你「是这几条打架了」。",[127633,134758,134759,134762,134763,134815],{},[7246,134760,134761],{},"插值","：从 ",[31,134764,134766,134785],{"className":134765,"translate":35},[34],[31,134767,134769],{"className":134768},[39],[41,134770,134771],{"xmlns":43},[45,134772,134773,134782],{},[48,134774,134775,134777,134780],{},[51,134776,18446],{},[95,134778,134779],{},"⇒",[51,134781,1122],{},[55,134783,134784],{"encoding":57},"A \\Rightarrow B",[31,134786,134788,134806],{"className":134787,"ariaHidden":62},[61],[31,134789,134791,134794,134797,134800,134803],{"className":134790},[66],[31,134792],{"className":134793,"style":945},[70],[31,134795,18446],{"className":134796},[75,76],[31,134798],{"className":134799,"style":293},[119],[31,134801,134779],{"className":134802},[297],[31,134804],{"className":134805,"style":293},[119],[31,134807,134809,134812],{"className":134808},[66],[31,134810],{"className":134811,"style":945},[70],[31,134813,1122],{"className":134814,"style":1137},[75,76]," 的不可满足性中提取一个只含共同变量的中间公式（Craig interpolant），用于程序验证中的不变式合成和抽象精化。",[127633,134817,134818,134820],{},[7246,134819,7248],{},"：部分求解器能输出 resolution 证明，供独立检查，避免「求解器说有漏洞，但你不知道为什么」的黑盒困境。",[11,134822,134823,134824,134826],{},"这一点经常被忽略，但它决定了 SMT 能不能真正嵌进工具链。一个只会输出 ",[45608,134825,131566],{}," 的求解器，在调试时几乎没用；一个能告诉你「第 3、7、9 条约束冲突」的求解器，才能帮工程师定位问题。",[52860,134828,134830],{"id":134829},"_35-增量求解","3.5 增量求解",[11,134832,134833,134834,134837],{},"第五个策略是",[7246,134835,134836],{},"增量求解","（incremental solving）。",[11,134839,134840],{},"符号执行的路径是一棵树：父路径的约束是子路径约束的前缀。如果每换一条路径都从头求解，大量工作会被重复浪费。",[11,134842,134843,134844,128098,134847,2446,134850,134853],{},"SMT-LIB 提供了 ",[45608,134845,134846],{},"push",[45608,134848,134849],{},"pop",[45608,134851,134852],{},"check-sat-assuming","，允许在已有约束的基础上临时加入假设、求解、再回退。求解器会保留之前学到的子句和理论状态。一条路径判 unsat 后回退到分叉点，下一条路径可以接着用。",[11,134855,134856],{},"KLEE、angr 这类工具都重度依赖这个特性。没有增量求解，符号执行的性能会差一个数量级。",[18,134858,134860],{"id":134859},"四dpllt","四、DPLL(T)",[52860,134862,134864],{"id":134863},"_41-总体架构","4.1 总体架构",[11,134866,134867,134868,134871,134872,134875],{},"现代 SMT 求解器的标准架构叫 ",[7246,134869,134870],{},"DPLL(T)","，更准确的名字是 ",[7246,134873,134874],{},"CDCL(T)","：外层是带冲突学习的 SAT 求解器，内层挂一个或多个理论求解器。",[128055,134877,134878],{},[128058,134879],{"height":134880,"width":134881,"style":128062,"data-d2-theme":128063,"alt":52812,"src":134882},320,2578,"data:image\u002Fsvg+xml;base64,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\u002Fmu6HotrM\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",[11,134884,134885],{},"用一句话概括这个循环：",[134887,134888,134889,134892,134895,134898,134901,134904],"ol",{},[127633,134890,134891],{},"把公式里的每个理论原子替换成一个命题变量，得到一个纯布尔公式；",[127633,134893,134894],{},"让 SAT 求解器找一个布尔模型；",[127633,134896,134897],{},"把布尔模型翻译回理论原子的真假，交给理论求解器检查；",[127633,134899,134900],{},"如果理论层说「一致」，就得到 sat 和模型；",[127633,134902,134903],{},"如果理论层说「冲突」，就把冲突总结成一条新的布尔子句，塞回 SAT 层，让它换个方向再搜；",[127633,134905,134906],{},"如果 SAT 层把所有可能性都穷尽了，就得到 unsat。",[11,134908,134909],{},"可以把它想象成一个总承包商和几个专业工种的协作：总承包商（SAT）负责排期和组合，电工、水管工（理论求解器）负责检查自己那部分是否可行。电工说「这个排期下电线会短路」，总承包商就把这条经验记下来，换一个排期重新试。",[52860,134911,134913],{"id":134912},"_42-布尔抽象","4.2 布尔抽象",[11,134915,134916],{},"布尔抽象（也叫 propositional abstraction）是两层之间的接口。",[11,134918,134919],{},"以这个公式为例：",[31,134921,134923],{"className":134922,"translate":35},[411],[31,134924,134926,134980],{"className":134925,"translate":35},[34],[31,134927,134929],{"className":134928},[39],[41,134930,134931],{"xmlns":43,"display":420},[45,134932,134933,134977],{},[48,134934,134935,134937,134939,134941,134943,134945,134947,134949,134951,134953,134955,134957,134959,134961,134963,134965,134967,134969,134971,134973,134975],{},[51,134936,8059],{},[95,134938,223],{},[95,134940,211],{"stretchy":210},[51,134942,53],{},[95,134944,16040],{},[91,134946,1977],{},[95,134948,126150],{},[51,134950,3003],{},[95,134952,16040],{},[91,134954,1977],{},[95,134956,220],{"stretchy":210},[1430,134958,125504],{},[95,134960,125507],{},[1430,134962,125504],{},[95,134964,211],{"stretchy":210},[51,134966,53],{},[95,134968,434],{},[51,134970,3003],{},[95,134972,9758],{},[91,134974,1977],{},[95,134976,220],{"stretchy":210},[55,134978,134979],{"encoding":57},"\\varphi = (x > 0 \\vee y > 0) \\;\\wedge\\; (x + y \u003C 0)",[31,134981,134983,135001,135022,135040,135058,135085,135106,135124],{"className":134982,"ariaHidden":62},[61],[31,134984,134986,134989,134992,134995,134998],{"className":134985},[66],[31,134987],{"className":134988,"style":865},[70],[31,134990,8059],{"className":134991},[75,76],[31,134993],{"className":134994,"style":293},[119],[31,134996,223],{"className":134997},[297],[31,134999],{"className":135000,"style":293},[119],[31,135002,135004,135007,135010,135013,135016,135019],{"className":135003},[66],[31,135005],{"className":135006,"style":260},[70],[31,135008,211],{"className":135009},[264],[31,135011,53],{"className":135012},[75,76],[31,135014],{"className":135015,"style":293},[119],[31,135017,16040],{"className":135018},[297],[31,135020],{"className":135021,"style":293},[119],[31,135023,135025,135028,135031,135034,135037],{"className":135024},[66],[31,135026],{"className":135027,"style":326},[70],[31,135029,1977],{"className":135030},[75],[31,135032],{"className":135033,"style":120},[119],[31,135035,126150],{"className":135036},[124],[31,135038],{"className":135039,"style":120},[119],[31,135041,135043,135046,135049,135052,135055],{"className":135042},[66],[31,135044],{"className":135045,"style":53323},[70],[31,135047,3003],{"className":135048,"style":2666},[75,76],[31,135050],{"className":135051,"style":293},[119],[31,135053,16040],{"className":135054},[297],[31,135056],{"className":135057,"style":293},[119],[31,135059,135061,135064,135067,135070,135073,135076,135079,135082],{"className":135060},[66],[31,135062],{"className":135063,"style":260},[70],[31,135065,1977],{"className":135066},[75],[31,135068,220],{"className":135069},[289],[31,135071],{"className":135072,"style":293},[119],[31,135074],{"className":135075,"style":120},[119],[31,135077,125507],{"className":135078},[124],[31,135080],{"className":135081,"style":293},[119],[31,135083],{"className":135084,"style":120},[119],[31,135086,135088,135091,135094,135097,135100,135103],{"className":135087},[66],[31,135089],{"className":135090,"style":260},[70],[31,135092,211],{"className":135093},[264],[31,135095,53],{"className":135096},[75,76],[31,135098],{"className":135099,"style":120},[119],[31,135101,434],{"className":135102},[124],[31,135104],{"className":135105,"style":120},[119],[31,135107,135109,135112,135115,135118,135121],{"className":135108},[66],[31,135110],{"className":135111,"style":53323},[70],[31,135113,3003],{"className":135114,"style":2666},[75,76],[31,135116],{"className":135117,"style":293},[119],[31,135119,9758],{"className":135120},[297],[31,135122],{"className":135123,"style":293},[119],[31,135125,135127,135130,135133],{"className":135126},[66],[31,135128],{"className":135129,"style":260},[70],[31,135131,1977],{"className":135132},[75],[31,135134,220],{"className":135135},[289],[11,135137,135138],{},"引入三个命题变量：",[31,135140,135142],{"className":135141,"translate":35},[411],[31,135143,135145,135212],{"className":135144,"translate":35},[34],[31,135146,135148],{"className":135147},[39],[41,135149,135150],{"xmlns":43,"display":420},[45,135151,135152,135209],{},[48,135153,135154,135156,135159,135161,135163,135165,135167,135169,135171,135173,135175,135177,135179,135181,135183,135185,135187,135189,135191,135193,135195,135197,135199,135201,135203,135205,135207],{},[51,135155,11],{},[95,135157,135158],{},"≡",[95,135160,211],{"stretchy":210},[51,135162,53],{},[95,135164,16040],{},[91,135166,1977],{},[95,135168,220],{"stretchy":210},[95,135170,1210],{"separator":62},[119,135172],{"width":20286},[51,135174,27666],{},[95,135176,135158],{},[95,135178,211],{"stretchy":210},[51,135180,3003],{},[95,135182,16040],{},[91,135184,1977],{},[95,135186,220],{"stretchy":210},[95,135188,1210],{"separator":62},[119,135190],{"width":20286},[51,135192,2549],{},[95,135194,135158],{},[95,135196,211],{"stretchy":210},[51,135198,53],{},[95,135200,434],{},[51,135202,3003],{},[95,135204,9758],{},[91,135206,1977],{},[95,135208,220],{"stretchy":210},[55,135210,135211],{"encoding":57},"p \\equiv (x > 0), \\qquad q \\equiv (y > 0), \\qquad r \\equiv (x + y \u003C 0)",[31,135213,135215,135234,135255,135288,135309,135342,135363,135381],{"className":135214,"ariaHidden":62},[61],[31,135216,135218,135222,135225,135228,135231],{"className":135217},[66],[31,135219],{"className":135220,"style":135221},[70],"height:0.6582em;vertical-align:-0.1944em;",[31,135223,11],{"className":135224},[75,76],[31,135226],{"className":135227,"style":293},[119],[31,135229,135158],{"className":135230},[297],[31,135232],{"className":135233,"style":293},[119],[31,135235,135237,135240,135243,135246,135249,135252],{"className":135236},[66],[31,135238],{"className":135239,"style":260},[70],[31,135241,211],{"className":135242},[264],[31,135244,53],{"className":135245},[75,76],[31,135247],{"className":135248,"style":293},[119],[31,135250,16040],{"className":135251},[297],[31,135253],{"className":135254,"style":293},[119],[31,135256,135258,135261,135264,135267,135270,135273,135276,135279,135282,135285],{"className":135257},[66],[31,135259],{"className":135260,"style":260},[70],[31,135262,1977],{"className":135263},[75],[31,135265,220],{"className":135266},[289],[31,135268,1210],{"className":135269},[1391],[31,135271],{"className":135272,"style":20525},[119],[31,135274],{"className":135275,"style":1467},[119],[31,135277,27666],{"className":135278,"style":2666},[75,76],[31,135280],{"className":135281,"style":293},[119],[31,135283,135158],{"className":135284},[297],[31,135286],{"className":135287,"style":293},[119],[31,135289,135291,135294,135297,135300,135303,135306],{"className":135290},[66],[31,135292],{"className":135293,"style":260},[70],[31,135295,211],{"className":135296},[264],[31,135298,3003],{"className":135299,"style":2666},[75,76],[31,135301],{"className":135302,"style":293},[119],[31,135304,16040],{"className":135305},[297],[31,135307],{"className":135308,"style":293},[119],[31,135310,135312,135315,135318,135321,135324,135327,135330,135333,135336,135339],{"className":135311},[66],[31,135313],{"className":135314,"style":260},[70],[31,135316,1977],{"className":135317},[75],[31,135319,220],{"className":135320},[289],[31,135322,1210],{"className":135323},[1391],[31,135325],{"className":135326,"style":20525},[119],[31,135328],{"className":135329,"style":1467},[119],[31,135331,2549],{"className":135332,"style":19759},[75,76],[31,135334],{"className":135335,"style":293},[119],[31,135337,135158],{"className":135338},[297],[31,135340],{"className":135341,"style":293},[119],[31,135343,135345,135348,135351,135354,135357,135360],{"className":135344},[66],[31,135346],{"className":135347,"style":260},[70],[31,135349,211],{"className":135350},[264],[31,135352,53],{"className":135353},[75,76],[31,135355],{"className":135356,"style":120},[119],[31,135358,434],{"className":135359},[124],[31,135361],{"className":135362,"style":120},[119],[31,135364,135366,135369,135372,135375,135378],{"className":135365},[66],[31,135367],{"className":135368,"style":53323},[70],[31,135370,3003],{"className":135371,"style":2666},[75,76],[31,135373],{"className":135374,"style":293},[119],[31,135376,9758],{"className":135377},[297],[31,135379],{"className":135380,"style":293},[119],[31,135382,135384,135387,135390],{"className":135383},[66],[31,135385],{"className":135386,"style":260},[70],[31,135388,1977],{"className":135389},[75],[31,135391,220],{"className":135392},[289],[11,135394,135395],{},"布尔骨架变成：",[31,135397,135399],{"className":135398,"translate":35},[411],[31,135400,135402,135432],{"className":135401,"translate":35},[34],[31,135403,135405],{"className":135404},[39],[41,135406,135407],{"xmlns":43,"display":420},[45,135408,135409,135429],{},[48,135410,135411,135413,135415,135417,135419,135421,135423,135425,135427],{},[95,135412,211],{"stretchy":210},[51,135414,11],{},[95,135416,126150],{},[51,135418,27666],{},[95,135420,220],{"stretchy":210},[1430,135422,125504],{},[95,135424,125507],{},[1430,135426,125504],{},[51,135428,2549],{},[55,135430,135431],{"encoding":57},"(p \\vee q) \\;\\wedge\\; r",[31,135433,135435,135456,135483],{"className":135434,"ariaHidden":62},[61],[31,135436,135438,135441,135444,135447,135450,135453],{"className":135437},[66],[31,135439],{"className":135440,"style":260},[70],[31,135442,211],{"className":135443},[264],[31,135445,11],{"className":135446},[75,76],[31,135448],{"className":135449,"style":120},[119],[31,135451,126150],{"className":135452},[124],[31,135454],{"className":135455,"style":120},[119],[31,135457,135459,135462,135465,135468,135471,135474,135477,135480],{"className":135458},[66],[31,135460],{"className":135461,"style":260},[70],[31,135463,27666],{"className":135464,"style":2666},[75,76],[31,135466,220],{"className":135467},[289],[31,135469],{"className":135470,"style":293},[119],[31,135472],{"className":135473,"style":120},[119],[31,135475,125507],{"className":135476},[124],[31,135478],{"className":135479,"style":293},[119],[31,135481],{"className":135482,"style":120},[119],[31,135484,135486,135489],{"className":135485},[66],[31,135487],{"className":135488,"style":71},[70],[31,135490,2549],{"className":135491,"style":19759},[75,76],[11,135493,135494,135495,135576,135577,135698],{},"SAT 求解器在这个骨架上的搜索空间只有 ",[31,135496,135498,135520],{"className":135497,"translate":35},[34],[31,135499,135501],{"className":135500},[39],[41,135502,135503],{"xmlns":43},[45,135504,135505,135517],{},[48,135506,135507,135513,135515],{},[1180,135508,135509,135511],{},[91,135510,152],{},[91,135512,93],{},[95,135514,223],{},[91,135516,19641],{},[55,135518,135519],{"encoding":57},"2^3 = 8",[31,135521,135523,135567],{"className":135522,"ariaHidden":62},[61],[31,135524,135526,135529,135558,135561,135564],{"className":135525},[66],[31,135527],{"className":135528,"style":5135},[70],[31,135530,135532,135535],{"className":135531},[75],[31,135533,152],{"className":135534},[75],[31,135536,135538],{"className":135537},[1039],[31,135539,135541],{"className":135540},[1043],[31,135542,135544],{"className":135543},[1048],[31,135545,135547],{"className":135546,"style":5135},[1052],[31,135548,135549,135552],{"style":3631},[31,135550],{"className":135551,"style":1061},[1060],[31,135553,135555],{"className":135554},[1065,1066,1067,1068],[31,135556,93],{"className":135557},[75,1068],[31,135559],{"className":135560,"style":293},[119],[31,135562,223],{"className":135563},[297],[31,135565],{"className":135566,"style":293},[119],[31,135568,135570,135573],{"className":135569},[66],[31,135571],{"className":135572,"style":326},[70],[31,135574,19641],{"className":135575},[75]," 种赋值。它先试 ",[31,135578,135580,135614],{"className":135579,"translate":35},[34],[31,135581,135583],{"className":135582},[39],[41,135584,135585],{"xmlns":43},[45,135586,135587,135611],{},[48,135588,135589,135591,135593,135595,135597,135599,135601,135603,135605,135607,135609],{},[51,135590,11],{},[95,135592,223],{},[51,135594,20800],{"mathvariant":494},[95,135596,1210],{"separator":62},[51,135598,27666],{},[95,135600,223],{},[51,135602,20800],{"mathvariant":494},[95,135604,1210],{"separator":62},[51,135606,2549],{},[95,135608,223],{},[51,135610,20800],{"mathvariant":494},[55,135612,135613],{"encoding":57},"p = \\top, q = \\top, r = \\top",[31,135615,135617,135635,135662,135689],{"className":135616,"ariaHidden":62},[61],[31,135618,135620,135623,135626,135629,135632],{"className":135619},[66],[31,135621],{"className":135622,"style":865},[70],[31,135624,11],{"className":135625},[75,76],[31,135627],{"className":135628,"style":293},[119],[31,135630,223],{"className":135631},[297],[31,135633],{"className":135634,"style":293},[119],[31,135636,135638,135641,135644,135647,135650,135653,135656,135659],{"className":135637},[66],[31,135639],{"className":135640,"style":4202},[70],[31,135642,20800],{"className":135643},[75],[31,135645,1210],{"className":135646},[1391],[31,135648],{"className":135649,"style":1467},[119],[31,135651,27666],{"className":135652,"style":2666},[75,76],[31,135654],{"className":135655,"style":293},[119],[31,135657,223],{"className":135658},[297],[31,135660],{"className":135661,"style":293},[119],[31,135663,135665,135668,135671,135674,135677,135680,135683,135686],{"className":135664},[66],[31,135666],{"className":135667,"style":4202},[70],[31,135669,20800],{"className":135670},[75],[31,135672,1210],{"className":135673},[1391],[31,135675],{"className":135676,"style":1467},[119],[31,135678,2549],{"className":135679,"style":19759},[75,76],[31,135681],{"className":135682,"style":293},[119],[31,135684,223],{"className":135685},[297],[31,135687],{"className":135688,"style":293},[119],[31,135690,135692,135695],{"className":135691},[66],[31,135693],{"className":135694,"style":1630},[70],[31,135696,20800],{"className":135697},[75],"，把这三个真值交回理论层。",[11,135700,135701,135702,135705],{},"理论层看到的是三个",[7246,135703,135704],{},"文字","（literal）：",[31,135707,135709],{"className":135708,"translate":35},[411],[31,135710,135712,135754],{"className":135711,"translate":35},[34],[31,135713,135715],{"className":135714},[39],[41,135716,135717],{"xmlns":43,"display":420},[45,135718,135719,135751],{},[48,135720,135721,135723,135725,135727,135729,135731,135733,135735,135737,135739,135741,135743,135745,135747,135749],{},[51,135722,53],{},[95,135724,16040],{},[91,135726,1977],{},[95,135728,1210],{"separator":62},[119,135730],{"width":20286},[51,135732,3003],{},[95,135734,16040],{},[91,135736,1977],{},[95,135738,1210],{"separator":62},[119,135740],{"width":20286},[51,135742,53],{},[95,135744,434],{},[51,135746,3003],{},[95,135748,9758],{},[91,135750,1977],{},[55,135752,135753],{"encoding":57},"x > 0, \\qquad y > 0, \\qquad x + y \u003C 0",[31,135755,135757,135775,135805,135835,135853],{"className":135756,"ariaHidden":62},[61],[31,135758,135760,135763,135766,135769,135772],{"className":135759},[66],[31,135761],{"className":135762,"style":21113},[70],[31,135764,53],{"className":135765},[75,76],[31,135767],{"className":135768,"style":293},[119],[31,135770,16040],{"className":135771},[297],[31,135773],{"className":135774,"style":293},[119],[31,135776,135778,135781,135784,135787,135790,135793,135796,135799,135802],{"className":135777},[66],[31,135779],{"className":135780,"style":41982},[70],[31,135782,1977],{"className":135783},[75],[31,135785,1210],{"className":135786},[1391],[31,135788],{"className":135789,"style":20525},[119],[31,135791],{"className":135792,"style":1467},[119],[31,135794,3003],{"className":135795,"style":2666},[75,76],[31,135797],{"className":135798,"style":293},[119],[31,135800,16040],{"className":135801},[297],[31,135803],{"className":135804,"style":293},[119],[31,135806,135808,135811,135814,135817,135820,135823,135826,135829,135832],{"className":135807},[66],[31,135809],{"className":135810,"style":41982},[70],[31,135812,1977],{"className":135813},[75],[31,135815,1210],{"className":135816},[1391],[31,135818],{"className":135819,"style":20525},[119],[31,135821],{"className":135822,"style":1467},[119],[31,135824,53],{"className":135825},[75,76],[31,135827],{"className":135828,"style":120},[119],[31,135830,434],{"className":135831},[124],[31,135833],{"className":135834,"style":120},[119],[31,135836,135838,135841,135844,135847,135850],{"className":135837},[66],[31,135839],{"className":135840,"style":53323},[70],[31,135842,3003],{"className":135843,"style":2666},[75,76],[31,135845],{"className":135846,"style":293},[119],[31,135848,9758],{"className":135849},[297],[31,135851],{"className":135852,"style":293},[119],[31,135854,135856,135859],{"className":135855},[66],[31,135857],{"className":135858,"style":326},[70],[31,135860,1977],{"className":135861},[75],[11,135863,135864,135865,135938,135939,2372],{},"线性算术求解器立刻发现：前两个推出 ",[31,135866,135868,135890],{"className":135867,"translate":35},[34],[31,135869,135871],{"className":135870},[39],[41,135872,135873],{"xmlns":43},[45,135874,135875,135887],{},[48,135876,135877,135879,135881,135883,135885],{},[51,135878,53],{},[95,135880,434],{},[51,135882,3003],{},[95,135884,16040],{},[91,135886,1977],{},[55,135888,135889],{"encoding":57},"x + y > 0",[31,135891,135893,135911,135929],{"className":135892,"ariaHidden":62},[61],[31,135894,135896,135899,135902,135905,135908],{"className":135895},[66],[31,135897],{"className":135898,"style":307},[70],[31,135900,53],{"className":135901},[75,76],[31,135903],{"className":135904,"style":120},[119],[31,135906,434],{"className":135907},[124],[31,135909],{"className":135910,"style":120},[119],[31,135912,135914,135917,135920,135923,135926],{"className":135913},[66],[31,135915],{"className":135916,"style":53323},[70],[31,135918,3003],{"className":135919,"style":2666},[75,76],[31,135921],{"className":135922,"style":293},[119],[31,135924,16040],{"className":135925},[297],[31,135927],{"className":135928,"style":293},[119],[31,135930,135932,135935],{"className":135931},[66],[31,135933],{"className":135934,"style":326},[70],[31,135936,1977],{"className":135937},[75],"，与第三个矛盾。这是一次",[7246,135940,135941],{},"理论冲突",[11,135943,135944,135945,135996,135997,11486,136063,136114,136115,136249],{},"注意，理论层收到的不是「原子为真」，而是「带正负号的文字」。",[31,135946,135948,135966],{"className":135947,"translate":35},[34],[31,135949,135951],{"className":135950},[39],[41,135952,135953],{"xmlns":43},[45,135954,135955,135963],{},[48,135956,135957,135959,135961],{},[51,135958,27666],{},[95,135960,223],{},[51,135962,126108],{"mathvariant":494},[55,135964,135965],{"encoding":57},"q = \\bot",[31,135967,135969,135987],{"className":135968,"ariaHidden":62},[61],[31,135970,135972,135975,135978,135981,135984],{"className":135971},[66],[31,135973],{"className":135974,"style":865},[70],[31,135976,27666],{"className":135977,"style":2666},[75,76],[31,135979],{"className":135980,"style":293},[119],[31,135982,223],{"className":135983},[297],[31,135985],{"className":135986,"style":293},[119],[31,135988,135990,135993],{"className":135989},[66],[31,135991],{"className":135992,"style":1630},[70],[31,135994,126108],{"className":135995},[75]," 对应的是 ",[31,135998,136000,136024],{"className":135999,"translate":35},[34],[31,136001,136003],{"className":136002},[39],[41,136004,136005],{"xmlns":43},[45,136006,136007,136021],{},[48,136008,136009,136011,136013,136015,136017,136019],{},[51,136010,126125],{"mathvariant":494},[95,136012,211],{"stretchy":210},[51,136014,3003],{},[95,136016,16040],{},[91,136018,1977],{},[95,136020,220],{"stretchy":210},[55,136022,136023],{"encoding":57},"\\neg(y > 0)",[31,136025,136027,136051],{"className":136026,"ariaHidden":62},[61],[31,136028,136030,136033,136036,136039,136042,136045,136048],{"className":136029},[66],[31,136031],{"className":136032,"style":260},[70],[31,136034,126125],{"className":136035},[75],[31,136037,211],{"className":136038},[264],[31,136040,3003],{"className":136041,"style":2666},[75,76],[31,136043],{"className":136044,"style":293},[119],[31,136046,16040],{"className":136047},[297],[31,136049],{"className":136050,"style":293},[119],[31,136052,136054,136057,136060],{"className":136053},[66],[31,136055],{"className":136056,"style":260},[70],[31,136058,1977],{"className":136059},[75],[31,136061,220],{"className":136062},[289],[31,136064,136066,136084],{"className":136065,"translate":35},[34],[31,136067,136069],{"className":136068},[39],[41,136070,136071],{"xmlns":43},[45,136072,136073,136081],{},[48,136074,136075,136077,136079],{},[51,136076,3003],{},[95,136078,347],{},[91,136080,1977],{},[55,136082,136083],{"encoding":57},"y \\le 0",[31,136085,136087,136105],{"className":136086,"ariaHidden":62},[61],[31,136088,136090,136093,136096,136099,136102],{"className":136089},[66],[31,136091],{"className":136092,"style":23593},[70],[31,136094,3003],{"className":136095,"style":2666},[75,76],[31,136097],{"className":136098,"style":293},[119],[31,136100,347],{"className":136101},[297],[31,136103],{"className":136104,"style":293},[119],[31,136106,136108,136111],{"className":136107},[66],[31,136109],{"className":136110,"style":326},[70],[31,136112,1977],{"className":136113},[75],"。理论求解器必须能处理这种否定形式，这也是为什么它通常直接操作形如 ",[31,136116,136118,136144],{"className":136117,"translate":35},[34],[31,136119,136121],{"className":136120},[39],[41,136122,136123],{"xmlns":43},[45,136124,136125,136141],{},[48,136126,136127,136133,136135],{},[1004,136128,136129,136131],{},[51,136130,12271],{},[91,136132,205],{},[95,136134,347],{},[1004,136136,136137,136139],{},[51,136138,12271],{},[91,136140,152],{},[55,136142,136143],{"encoding":57},"t_1 \\le t_2",[31,136145,136147,136203],{"className":136146,"ariaHidden":62},[61],[31,136148,136150,136154,136194,136197,136200],{"className":136149},[66],[31,136151],{"className":136152,"style":136153},[70],"height:0.786em;vertical-align:-0.15em;",[31,136155,136157,136160],{"className":136156},[75],[31,136158,12271],{"className":136159},[75,76],[31,136161,136163],{"className":136162},[1039],[31,136164,136166,136186],{"className":136165},[1043,1044],[31,136167,136169,136183],{"className":136168},[1048],[31,136170,136172],{"className":136171,"style":2028},[1052],[31,136173,136174,136177],{"style":2807},[31,136175],{"className":136176,"style":1061},[1060],[31,136178,136180],{"className":136179},[1065,1066,1067,1068],[31,136181,205],{"className":136182},[75,1068],[31,136184,1076],{"className":136185},[1075],[31,136187,136189],{"className":136188},[1048],[31,136190,136192],{"className":136191,"style":2050},[1052],[31,136193],{},[31,136195],{"className":136196,"style":293},[119],[31,136198,347],{"className":136199},[297],[31,136201],{"className":136202,"style":293},[119],[31,136204,136206,136209],{"className":136205},[66],[31,136207],{"className":136208,"style":126474},[70],[31,136210,136212,136215],{"className":136211},[75],[31,136213,12271],{"className":136214},[75,76],[31,136216,136218],{"className":136217},[1039],[31,136219,136221,136241],{"className":136220},[1043,1044],[31,136222,136224,136238],{"className":136223},[1048],[31,136225,136227],{"className":136226,"style":2028},[1052],[31,136228,136229,136232],{"style":2807},[31,136230],{"className":136231,"style":1061},[1060],[31,136233,136235],{"className":136234},[1065,1066,1067,1068],[31,136236,152],{"className":136237},[75,1068],[31,136239,1076],{"className":136240},[1075],[31,136242,136244],{"className":136243},[1048],[31,136245,136247],{"className":136246,"style":2050},[1052],[31,136248],{}," 的不等式，而不是只处理等式。",[52860,136251,136253],{"id":136252},"_43-cdcl","4.3 CDCL",[11,136255,136256],{},"SAT 层用的是 CDCL（冲突驱动子句学习）。它的核心循环大致如下：",[125323,136258,136262],{"className":136259,"code":136261,"language":1793},[136260],"language-text","loop:\n    conflict = unit_propagate()      # 单元传播\n    if conflict:\n        if decision_level == 0: return UNSAT\n        lemma = analyze(conflict)    # 冲突分析\n        backtrack()                  # 非时序回跳\n        add_clause(lemma)            # 学习子句\n    else:\n        if all_variables_assigned(): return SAT\n        decide()                     # 选一个未赋值的变量\n",[45608,136263,136261],{"__ignoreMap":52812},[11,136265,136266],{},"三个关键机制：",[127630,136268,136269,136275,136281],{},[127633,136270,136271,136274],{},[7246,136272,136273],{},"单元传播","：一条子句里只剩一个未赋值的文字时，强制它取能让子句为真的值。",[127633,136276,136277,136280],{},[7246,136278,136279],{},"冲突分析","：从冲突出发，反向找出是哪几个决策导致的，生成一条新的子句。这条子句是原公式的逻辑推论，加进去不会改变解集。",[127633,136282,136283,136286],{},[7246,136284,136285],{},"非时序回跳","：不退回上一层，而是直接跳回冲突的真正源头，跳过大量无用的搜索。",[11,136288,136289],{},"CDCL 是过去二十年 SAT 求解器性能提升的主要来源。SMT 做的事情，基本上是把这套成熟的工程能力「复用」到理论层。",[52860,136291,136293],{"id":136292},"_44-理论求解器","4.4 理论求解器",[11,136295,136296,136297,136300,136301,136329],{},"理论层要做的事情叫 ",[7246,136298,136299],{},"T-一致性检查","：给定一组带正负号的原子，判断它们在理论 ",[31,136302,136304,136317],{"className":136303,"translate":35},[34],[31,136305,136307],{"className":136306},[39],[41,136308,136309],{"xmlns":43},[45,136310,136311,136315],{},[48,136312,136313],{},[51,136314,919],{},[55,136316,919],{"encoding":57},[31,136318,136320],{"className":136319,"ariaHidden":62},[61],[31,136321,136323,136326],{"className":136322},[66],[31,136324],{"className":136325,"style":945},[70],[31,136327,919],{"className":136328,"style":949},[75,76]," 下是否可能同时为真。",[11,136331,136332],{},"不同理论有不同的算法。",[136334,136335,136337],"h4",{"id":136336},"euf","EUF",[11,136339,136340],{},"EUF（Equality with Uninterpreted Functions）只关心相等关系和函数的一致性。核心是两件事：",[134887,136342,136343,136346],{},[127633,136344,136345],{},"用并查集维护变量的等价类；",[127633,136347,136348,136349,40580,136400,2372],{},"用同余闭包（congruence closure）保证：如果 ",[31,136350,136352,136370],{"className":136351,"translate":35},[34],[31,136353,136355],{"className":136354},[39],[41,136356,136357],{"xmlns":43},[45,136358,136359,136367],{},[48,136360,136361,136363,136365],{},[51,136362,5813],{},[95,136364,223],{},[51,136366,5834],{},[55,136368,136369],{"encoding":57},"a = b",[31,136371,136373,136391],{"className":136372,"ariaHidden":62},[61],[31,136374,136376,136379,136382,136385,136388],{"className":136375},[66],[31,136377],{"className":136378,"style":71},[70],[31,136380,5813],{"className":136381},[75,76],[31,136383],{"className":136384,"style":293},[119],[31,136386,223],{"className":136387},[297],[31,136389],{"className":136390,"style":293},[119],[31,136392,136394,136397],{"className":136393},[66],[31,136395],{"className":136396,"style":1630},[70],[31,136398,5834],{"className":136399},[75,76],[31,136401,136403,136433],{"className":136402,"translate":35},[34],[31,136404,136406],{"className":136405},[39],[41,136407,136408],{"xmlns":43},[45,136409,136410,136430],{},[48,136411,136412,136414,136416,136418,136420,136422,136424,136426,136428],{},[51,136413,1662],{},[95,136415,211],{"stretchy":210},[51,136417,5813],{},[95,136419,220],{"stretchy":210},[95,136421,223],{},[51,136423,1662],{},[95,136425,211],{"stretchy":210},[51,136427,5834],{},[95,136429,220],{"stretchy":210},[55,136431,136432],{"encoding":57},"f(a) = f(b)",[31,136434,136436,136463],{"className":136435,"ariaHidden":62},[61],[31,136437,136439,136442,136445,136448,136451,136454,136457,136460],{"className":136438},[66],[31,136440],{"className":136441,"style":260},[70],[31,136443,1662],{"className":136444,"style":35196},[75,76],[31,136446,211],{"className":136447},[264],[31,136449,5813],{"className":136450},[75,76],[31,136452,220],{"className":136453},[289],[31,136455],{"className":136456,"style":293},[119],[31,136458,223],{"className":136459},[297],[31,136461],{"className":136462,"style":293},[119],[31,136464,136466,136469,136472,136475,136478],{"className":136465},[66],[31,136467],{"className":136468,"style":260},[70],[31,136470,1662],{"className":136471,"style":35196},[75,76],[31,136473,211],{"className":136474},[264],[31,136476,5834],{"className":136477},[75,76],[31,136479,220],{"className":136480},[289],[11,136482,136483],{},"例子：",[31,136485,136487],{"className":136486,"translate":35},[411],[31,136488,136490,136544],{"className":136489,"translate":35},[34],[31,136491,136493],{"className":136492},[39],[41,136494,136495],{"xmlns":43,"display":420},[45,136496,136497,136541],{},[48,136498,136499,136501,136503,136505,136507,136509,136511,136513,136515,136517,136519,136521,136523,136525,136527,136529,136531,136533,136535,136537,136539],{},[51,136500,5813],{},[95,136502,223],{},[51,136504,5834],{},[1430,136506,125504],{},[95,136508,125507],{},[1430,136510,125504],{},[51,136512,5834],{},[95,136514,223],{},[51,136516,1412],{},[1430,136518,125504],{},[95,136520,125507],{},[1430,136522,125504],{},[51,136524,1662],{},[95,136526,211],{"stretchy":210},[51,136528,5813],{},[95,136530,220],{"stretchy":210},[95,136532,21271],{"mathvariant":494},[51,136534,1662],{},[95,136536,211],{"stretchy":210},[51,136538,1412],{},[95,136540,220],{"stretchy":210},[55,136542,136543],{"encoding":57},"a = b \\;\\wedge\\; b = c \\;\\wedge\\; f(a) \\ne f(c)",[31,136545,136547,136565,136589,136607,136632,136692],{"className":136546,"ariaHidden":62},[61],[31,136548,136550,136553,136556,136559,136562],{"className":136549},[66],[31,136551],{"className":136552,"style":71},[70],[31,136554,5813],{"className":136555},[75,76],[31,136557],{"className":136558,"style":293},[119],[31,136560,223],{"className":136561},[297],[31,136563],{"className":136564,"style":293},[119],[31,136566,136568,136571,136574,136577,136580,136583,136586],{"className":136567},[66],[31,136569],{"className":136570,"style":1630},[70],[31,136572,5834],{"className":136573},[75,76],[31,136575],{"className":136576,"style":293},[119],[31,136578],{"className":136579,"style":120},[119],[31,136581,125507],{"className":136582},[124],[31,136584],{"className":136585,"style":293},[119],[31,136587],{"className":136588,"style":120},[119],[31,136590,136592,136595,136598,136601,136604],{"className":136591},[66],[31,136593],{"className":136594,"style":1630},[70],[31,136596,5834],{"className":136597},[75,76],[31,136599],{"className":136600,"style":293},[119],[31,136602,223],{"className":136603},[297],[31,136605],{"className":136606,"style":293},[119],[31,136608,136610,136614,136617,136620,136623,136626,136629],{"className":136609},[66],[31,136611],{"className":136612,"style":136613},[70],"height:0.5556em;",[31,136615,1412],{"className":136616},[75,76],[31,136618],{"className":136619,"style":293},[119],[31,136621],{"className":136622,"style":120},[119],[31,136624,125507],{"className":136625},[124],[31,136627],{"className":136628,"style":293},[119],[31,136630],{"className":136631,"style":120},[119],[31,136633,136635,136638,136641,136644,136647,136650,136653,136689],{"className":136634},[66],[31,136636],{"className":136637,"style":260},[70],[31,136639,1662],{"className":136640,"style":35196},[75,76],[31,136642,211],{"className":136643},[264],[31,136645,5813],{"className":136646},[75,76],[31,136648,220],{"className":136649},[289],[31,136651],{"className":136652,"style":293},[119],[31,136654,136656,136683,136686],{"className":136655},[297],[31,136657,136659],{"className":136658},[297],[31,136660,136662],{"className":136661},[75,21516],[31,136663,136665],{"className":136664},[21520],[31,136666,136668,136671,136680],{"className":136667},[21524],[31,136669],{"className":136670,"style":4202},[70],[31,136672,136674],{"className":136673},[21531],[31,136675,136677],{"className":136676},[75],[31,136678,21538],{"className":136679},[297],[31,136681],{"className":136682},[21542],[31,136684],{"className":136685},[119,21546],[31,136687,223],{"className":136688},[297],[31,136690],{"className":136691,"style":293},[119],[31,136693,136695,136698,136701,136704,136707],{"className":136694},[66],[31,136696],{"className":136697,"style":260},[70],[31,136699,1662],{"className":136700,"style":35196},[75,76],[31,136702,211],{"className":136703},[264],[31,136705,1412],{"className":136706},[75,76],[31,136708,220],{"className":136709},[289],[11,136711,136712,136713,136768,136769,2446,136813,136857,136858,136930],{},"并查集把 ",[31,136714,136716,136738],{"className":136715,"translate":35},[34],[31,136717,136719],{"className":136718},[39],[41,136720,136721],{"xmlns":43},[45,136722,136723,136735],{},[48,136724,136725,136727,136729,136731,136733],{},[51,136726,5813],{},[95,136728,1210],{"separator":62},[51,136730,5834],{},[95,136732,1210],{"separator":62},[51,136734,1412],{},[55,136736,136737],{"encoding":57},"a, b, c",[31,136739,136741],{"className":136740,"ariaHidden":62},[61],[31,136742,136744,136747,136750,136753,136756,136759,136762,136765],{"className":136743},[66],[31,136745],{"className":136746,"style":4202},[70],[31,136748,5813],{"className":136749},[75,76],[31,136751,1210],{"className":136752},[1391],[31,136754],{"className":136755,"style":1467},[119],[31,136757,5834],{"className":136758},[75,76],[31,136760,1210],{"className":136761},[1391],[31,136763],{"className":136764,"style":1467},[119],[31,136766,1412],{"className":136767},[75,76]," 合并成一个类；同余闭包发现 ",[31,136770,136772,136792],{"className":136771,"translate":35},[34],[31,136773,136775],{"className":136774},[39],[41,136776,136777],{"xmlns":43},[45,136778,136779,136789],{},[48,136780,136781,136783,136785,136787],{},[51,136782,1662],{},[95,136784,211],{"stretchy":210},[51,136786,5813],{},[95,136788,220],{"stretchy":210},[55,136790,136791],{"encoding":57},"f(a)",[31,136793,136795],{"className":136794,"ariaHidden":62},[61],[31,136796,136798,136801,136804,136807,136810],{"className":136797},[66],[31,136799],{"className":136800,"style":260},[70],[31,136802,1662],{"className":136803,"style":35196},[75,76],[31,136805,211],{"className":136806},[264],[31,136808,5813],{"className":136809},[75,76],[31,136811,220],{"className":136812},[289],[31,136814,136816,136836],{"className":136815,"translate":35},[34],[31,136817,136819],{"className":136818},[39],[41,136820,136821],{"xmlns":43},[45,136822,136823,136833],{},[48,136824,136825,136827,136829,136831],{},[51,136826,1662],{},[95,136828,211],{"stretchy":210},[51,136830,1412],{},[95,136832,220],{"stretchy":210},[55,136834,136835],{"encoding":57},"f(c)",[31,136837,136839],{"className":136838,"ariaHidden":62},[61],[31,136840,136842,136845,136848,136851,136854],{"className":136841},[66],[31,136843],{"className":136844,"style":260},[70],[31,136846,1662],{"className":136847,"style":35196},[75,76],[31,136849,211],{"className":136850},[264],[31,136852,1412],{"className":136853},[75,76],[31,136855,220],{"className":136856},[289]," 的参数相等，于是把 ",[31,136859,136861,136891],{"className":136860,"translate":35},[34],[31,136862,136864],{"className":136863},[39],[41,136865,136866],{"xmlns":43},[45,136867,136868,136888],{},[48,136869,136870,136872,136874,136876,136878,136880,136882,136884,136886],{},[51,136871,1662],{},[95,136873,211],{"stretchy":210},[51,136875,5813],{},[95,136877,220],{"stretchy":210},[95,136879,1210],{"separator":62},[51,136881,1662],{},[95,136883,211],{"stretchy":210},[51,136885,1412],{},[95,136887,220],{"stretchy":210},[55,136889,136890],{"encoding":57},"f(a), f(c)",[31,136892,136894],{"className":136893,"ariaHidden":62},[61],[31,136895,136897,136900,136903,136906,136909,136912,136915,136918,136921,136924,136927],{"className":136896},[66],[31,136898],{"className":136899,"style":260},[70],[31,136901,1662],{"className":136902,"style":35196},[75,76],[31,136904,211],{"className":136905},[264],[31,136907,5813],{"className":136908},[75,76],[31,136910,220],{"className":136911},[289],[31,136913,1210],{"className":136914},[1391],[31,136916],{"className":136917,"style":1467},[119],[31,136919,1662],{"className":136920,"style":35196},[75,76],[31,136922,211],{"className":136923},[264],[31,136925,1412],{"className":136926},[75,76],[31,136928,220],{"className":136929},[289]," 也合并；但公式又要求它们不等，冲突，判 unsat。",[11,136932,136933],{},"EUF 在程序分析里非常有用，因为指针别名、对象身份、类型相等，本质上都是「未解释函数的相等性」。",[136334,136935,136936],{"id":136936},"线性算术",[11,136938,136939,136941],{},[45608,136940,128097],{}," 用单纯形法（更准确地说，是 SMT 里常用的 general simplex）。它先维护一组满足所有等式的赋值，再通过枢轴变换（pivot）调整变量的取值，直到所有上下界约束也被满足；如果怎么调都满足不了，就报告冲突。",[11,136943,136944,136946,136947,136950,136951,2446,137014,137077],{},[45608,136945,128101],{}," 在此基础上加",[7246,136948,136949],{},"分支定界","：先按实数求解，如果解里某个整数变量取了小数，就分裂成两支，比如 ",[31,136952,136954,136978],{"className":136953,"translate":35},[34],[31,136955,136957],{"className":136956},[39],[41,136958,136959],{"xmlns":43},[45,136960,136961,136975],{},[48,136962,136963,136965,136967,136970,136972],{},[51,136964,53],{},[95,136966,347],{},[95,136968,136969],{"stretchy":210},"⌊",[51,136971,5354],{},[95,136973,136974],{"stretchy":210},"⌋",[55,136976,136977],{"encoding":57},"x \\le \\lfloor v \\rfloor",[31,136979,136981,136999],{"className":136980,"ariaHidden":62},[61],[31,136982,136984,136987,136990,136993,136996],{"className":136983},[66],[31,136985],{"className":136986,"style":385},[70],[31,136988,53],{"className":136989},[75,76],[31,136991],{"className":136992,"style":293},[119],[31,136994,347],{"className":136995},[297],[31,136997],{"className":136998,"style":293},[119],[31,137000,137002,137005,137008,137011],{"className":137001},[66],[31,137003],{"className":137004,"style":260},[70],[31,137006,136969],{"className":137007},[264],[31,137009,5354],{"className":137010,"style":2666},[75,76],[31,137012,136974],{"className":137013},[289],[31,137015,137017,137041],{"className":137016,"translate":35},[34],[31,137018,137020],{"className":137019},[39],[41,137021,137022],{"xmlns":43},[45,137023,137024,137038],{},[48,137025,137026,137028,137030,137033,137035],{},[51,137027,53],{},[95,137029,5662],{},[95,137031,137032],{"stretchy":210},"⌈",[51,137034,5354],{},[95,137036,137037],{"stretchy":210},"⌉",[55,137039,137040],{"encoding":57},"x \\ge \\lceil v \\rceil",[31,137042,137044,137062],{"className":137043,"ariaHidden":62},[61],[31,137045,137047,137050,137053,137056,137059],{"className":137046},[66],[31,137048],{"className":137049,"style":385},[70],[31,137051,53],{"className":137052},[75,76],[31,137054],{"className":137055,"style":293},[119],[31,137057,5662],{"className":137058},[297],[31,137060],{"className":137061,"style":293},[119],[31,137063,137065,137068,137071,137074],{"className":137064},[66],[31,137066],{"className":137067,"style":260},[70],[31,137069,137032],{"className":137070},[264],[31,137072,5354],{"className":137073,"style":2666},[75,76],[31,137075,137037],{"className":137076},[289],"，分别再解。",[11,137079,136483],{},[31,137081,137083],{"className":137082,"translate":35},[411],[31,137084,137086,137140],{"className":137085,"translate":35},[34],[31,137087,137089],{"className":137088},[39],[41,137090,137091],{"xmlns":43,"display":420},[45,137092,137093,137137],{},[48,137094,137095,137097,137099,137101,137103,137105,137107,137109,137111,137113,137115,137117,137119,137121,137123,137125,137127,137129,137131,137133,137135],{},[91,137096,93],{},[51,137098,53],{},[95,137100,434],{},[91,137102,152],{},[51,137104,3003],{},[95,137106,223],{},[91,137108,7853],{},[95,137110,1210],{"separator":62},[119,137112],{"width":20286},[51,137114,53],{},[95,137116,1210],{"separator":62},[51,137118,3003],{},[95,137120,5662],{},[91,137122,1977],{},[95,137124,1210],{"separator":62},[119,137126],{"width":20286},[51,137128,53],{},[95,137130,1210],{"separator":62},[51,137132,3003],{},[95,137134,2518],{},[51,137136,28644],{"mathvariant":14789},[55,137138,137139],{"encoding":57},"3x + 2y = 5, \\qquad x, y \\ge 0, \\qquad x, y \\in \\mathbb{Z}",[31,137141,137143,137164,137185,137224,137263],{"className":137142,"ariaHidden":62},[61],[31,137144,137146,137149,137152,137155,137158,137161],{"className":137145},[66],[31,137147],{"className":137148,"style":112},[70],[31,137150,93],{"className":137151},[75],[31,137153,53],{"className":137154},[75,76],[31,137156],{"className":137157,"style":120},[119],[31,137159,434],{"className":137160},[124],[31,137162],{"className":137163,"style":120},[119],[31,137165,137167,137170,137173,137176,137179,137182],{"className":137166},[66],[31,137168],{"className":137169,"style":41982},[70],[31,137171,152],{"className":137172},[75],[31,137174,3003],{"className":137175,"style":2666},[75,76],[31,137177],{"className":137178,"style":293},[119],[31,137180,223],{"className":137181},[297],[31,137183],{"className":137184,"style":293},[119],[31,137186,137188,137191,137194,137197,137200,137203,137206,137209,137212,137215,137218,137221],{"className":137187},[66],[31,137189],{"className":137190,"style":41982},[70],[31,137192,7853],{"className":137193},[75],[31,137195,1210],{"className":137196},[1391],[31,137198],{"className":137199,"style":20525},[119],[31,137201],{"className":137202,"style":1467},[119],[31,137204,53],{"className":137205},[75,76],[31,137207,1210],{"className":137208},[1391],[31,137210],{"className":137211,"style":1467},[119],[31,137213,3003],{"className":137214,"style":2666},[75,76],[31,137216],{"className":137217,"style":293},[119],[31,137219,5662],{"className":137220},[297],[31,137222],{"className":137223,"style":293},[119],[31,137225,137227,137230,137233,137236,137239,137242,137245,137248,137251,137254,137257,137260],{"className":137226},[66],[31,137228],{"className":137229,"style":41982},[70],[31,137231,1977],{"className":137232},[75],[31,137234,1210],{"className":137235},[1391],[31,137237],{"className":137238,"style":20525},[119],[31,137240],{"className":137241,"style":1467},[119],[31,137243,53],{"className":137244},[75,76],[31,137246,1210],{"className":137247},[1391],[31,137249],{"className":137250,"style":1467},[119],[31,137252,3003],{"className":137253,"style":2666},[75,76],[31,137255],{"className":137256,"style":293},[119],[31,137258,2518],{"className":137259},[297],[31,137261],{"className":137262,"style":293},[119],[31,137264,137266,137269],{"className":137265},[66],[31,137267],{"className":137268,"style":53279},[70],[31,137270,28644],{"className":137271},[75,14947],[11,137273,137274,137275,137366,137367,137395],{},"实数松弛下，",[31,137276,137278,137308],{"className":137277,"translate":35},[34],[31,137279,137281],{"className":137280},[39],[41,137282,137283],{"xmlns":43},[45,137284,137285,137305],{},[48,137286,137287,137289,137291,137293,137295,137297,137299,137301,137303],{},[51,137288,53],{},[95,137290,223],{},[91,137292,7853],{},[51,137294,3768],{"mathvariant":494},[91,137296,93],{},[95,137298,1210],{"separator":62},[51,137300,3003],{},[95,137302,223],{},[91,137304,1977],{},[55,137306,137307],{"encoding":57},"x = 5\u002F3, y = 0",[31,137309,137311,137329,137357],{"className":137310,"ariaHidden":62},[61],[31,137312,137314,137317,137320,137323,137326],{"className":137313},[66],[31,137315],{"className":137316,"style":71},[70],[31,137318,53],{"className":137319},[75,76],[31,137321],{"className":137322,"style":293},[119],[31,137324,223],{"className":137325},[297],[31,137327],{"className":137328,"style":293},[119],[31,137330,137332,137335,137339,137342,137345,137348,137351,137354],{"className":137331},[66],[31,137333],{"className":137334,"style":260},[70],[31,137336,137338],{"className":137337},[75],"5\u002F3",[31,137340,1210],{"className":137341},[1391],[31,137343],{"className":137344,"style":1467},[119],[31,137346,3003],{"className":137347,"style":2666},[75,76],[31,137349],{"className":137350,"style":293},[119],[31,137352,223],{"className":137353},[297],[31,137355],{"className":137356,"style":293},[119],[31,137358,137360,137363],{"className":137359},[66],[31,137361],{"className":137362,"style":326},[70],[31,137364,1977],{"className":137365},[75]," 是一个解，但 ",[31,137368,137370,137383],{"className":137369,"translate":35},[34],[31,137371,137373],{"className":137372},[39],[41,137374,137375],{"xmlns":43},[45,137376,137377,137381],{},[48,137378,137379],{},[51,137380,53],{},[55,137382,53],{"encoding":57},[31,137384,137386],{"className":137385,"ariaHidden":62},[61],[31,137387,137389,137392],{"className":137388},[66],[31,137390],{"className":137391,"style":71},[70],[31,137393,53],{"className":137394},[75,76]," 不是整数。分支：",[127630,137397,137398,137743],{},[127633,137399,125465,137400,4809,137451,137479,137480,23342,137508,36037,137536,21051,137587,137639,137640,21051,137691,137742],{},[31,137401,137403,137421],{"className":137402,"translate":35},[34],[31,137404,137406],{"className":137405},[39],[41,137407,137408],{"xmlns":43},[45,137409,137410,137418],{},[48,137411,137412,137414,137416],{},[51,137413,53],{},[95,137415,347],{},[91,137417,205],{},[55,137419,137420],{"encoding":57},"x \\le 1",[31,137422,137424,137442],{"className":137423,"ariaHidden":62},[61],[31,137425,137427,137430,137433,137436,137439],{"className":137426},[66],[31,137428],{"className":137429,"style":385},[70],[31,137431,53],{"className":137432},[75,76],[31,137434],{"className":137435,"style":293},[119],[31,137437,347],{"className":137438},[297],[31,137440],{"className":137441,"style":293},[119],[31,137443,137445,137448],{"className":137444},[66],[31,137446],{"className":137447,"style":326},[70],[31,137449,205],{"className":137450},[75],[31,137452,137454,137467],{"className":137453,"translate":35},[34],[31,137455,137457],{"className":137456},[39],[41,137458,137459],{"xmlns":43},[45,137460,137461,137465],{},[48,137462,137463],{},[51,137464,53],{},[55,137466,53],{"encoding":57},[31,137468,137470],{"className":137469,"ariaHidden":62},[61],[31,137471,137473,137476],{"className":137472},[66],[31,137474],{"className":137475,"style":71},[70],[31,137477,53],{"className":137478},[75,76]," 只能取 ",[31,137481,137483,137496],{"className":137482,"translate":35},[34],[31,137484,137486],{"className":137485},[39],[41,137487,137488],{"xmlns":43},[45,137489,137490,137494],{},[48,137491,137492],{},[91,137493,1977],{},[55,137495,1977],{"encoding":57},[31,137497,137499],{"className":137498,"ariaHidden":62},[61],[31,137500,137502,137505],{"className":137501},[66],[31,137503],{"className":137504,"style":326},[70],[31,137506,1977],{"className":137507},[75],[31,137509,137511,137524],{"className":137510,"translate":35},[34],[31,137512,137514],{"className":137513},[39],[41,137515,137516],{"xmlns":43},[45,137517,137518,137522],{},[48,137519,137520],{},[91,137521,205],{},[55,137523,205],{"encoding":57},[31,137525,137527],{"className":137526,"ariaHidden":62},[61],[31,137528,137530,137533],{"className":137529},[66],[31,137531],{"className":137532,"style":326},[70],[31,137534,205],{"className":137535},[75],[31,137537,137539,137557],{"className":137538,"translate":35},[34],[31,137540,137542],{"className":137541},[39],[41,137543,137544],{"xmlns":43},[45,137545,137546,137554],{},[48,137547,137548,137550,137552],{},[51,137549,53],{},[95,137551,223],{},[91,137553,1977],{},[55,137555,137556],{"encoding":57},"x = 0",[31,137558,137560,137578],{"className":137559,"ariaHidden":62},[61],[31,137561,137563,137566,137569,137572,137575],{"className":137562},[66],[31,137564],{"className":137565,"style":71},[70],[31,137567,53],{"className":137568},[75,76],[31,137570],{"className":137571,"style":293},[119],[31,137573,223],{"className":137574},[297],[31,137576],{"className":137577,"style":293},[119],[31,137579,137581,137584],{"className":137580},[66],[31,137582],{"className":137583,"style":326},[70],[31,137585,1977],{"className":137586},[75],[31,137588,137590,137609],{"className":137589,"translate":35},[34],[31,137591,137593],{"className":137592},[39],[41,137594,137595],{"xmlns":43},[45,137596,137597,137606],{},[48,137598,137599,137601,137603],{},[51,137600,3003],{},[95,137602,223],{},[91,137604,137605],{},"2.5",[55,137607,137608],{"encoding":57},"y = 2.5",[31,137610,137612,137630],{"className":137611,"ariaHidden":62},[61],[31,137613,137615,137618,137621,137624,137627],{"className":137614},[66],[31,137616],{"className":137617,"style":865},[70],[31,137619,3003],{"className":137620,"style":2666},[75,76],[31,137622],{"className":137623,"style":293},[119],[31,137625,223],{"className":137626},[297],[31,137628],{"className":137629,"style":293},[119],[31,137631,137633,137636],{"className":137632},[66],[31,137634],{"className":137635,"style":326},[70],[31,137637,137605],{"className":137638},[75]," 仍不是整数，",[31,137641,137643,137661],{"className":137642,"translate":35},[34],[31,137644,137646],{"className":137645},[39],[41,137647,137648],{"xmlns":43},[45,137649,137650,137658],{},[48,137651,137652,137654,137656],{},[51,137653,53],{},[95,137655,223],{},[91,137657,205],{},[55,137659,137660],{"encoding":57},"x = 1",[31,137662,137664,137682],{"className":137663,"ariaHidden":62},[61],[31,137665,137667,137670,137673,137676,137679],{"className":137666},[66],[31,137668],{"className":137669,"style":71},[70],[31,137671,53],{"className":137672},[75,76],[31,137674],{"className":137675,"style":293},[119],[31,137677,223],{"className":137678},[297],[31,137680],{"className":137681,"style":293},[119],[31,137683,137685,137688],{"className":137684},[66],[31,137686],{"className":137687,"style":326},[70],[31,137689,205],{"className":137690},[75],[31,137692,137694,137712],{"className":137693,"translate":35},[34],[31,137695,137697],{"className":137696},[39],[41,137698,137699],{"xmlns":43},[45,137700,137701,137709],{},[48,137702,137703,137705,137707],{},[51,137704,3003],{},[95,137706,223],{},[91,137708,205],{},[55,137710,137711],{"encoding":57},"y = 1",[31,137713,137715,137733],{"className":137714,"ariaHidden":62},[61],[31,137716,137718,137721,137724,137727,137730],{"className":137717},[66],[31,137719],{"className":137720,"style":865},[70],[31,137722,3003],{"className":137723,"style":2666},[75,76],[31,137725],{"className":137726,"style":293},[119],[31,137728,223],{"className":137729},[297],[31,137731],{"className":137732,"style":293},[119],[31,137734,137736,137739],{"className":137735},[66],[31,137737],{"className":137738,"style":326},[70],[31,137740,205],{"className":137741},[75],"，得到整数解，sat；",[127633,137744,137745,137746,137797],{},"另一支 ",[31,137747,137749,137767],{"className":137748,"translate":35},[34],[31,137750,137752],{"className":137751},[39],[41,137753,137754],{"xmlns":43},[45,137755,137756,137764],{},[48,137757,137758,137760,137762],{},[51,137759,53],{},[95,137761,5662],{},[91,137763,152],{},[55,137765,137766],{"encoding":57},"x \\ge 2",[31,137768,137770,137788],{"className":137769,"ariaHidden":62},[61],[31,137771,137773,137776,137779,137782,137785],{"className":137772},[66],[31,137774],{"className":137775,"style":385},[70],[31,137777,53],{"className":137778},[75,76],[31,137780],{"className":137781,"style":293},[119],[31,137783,5662],{"className":137784},[297],[31,137786],{"className":137787,"style":293},[119],[31,137789,137791,137794],{"className":137790},[66],[31,137792],{"className":137793,"style":326},[70],[31,137795,152],{"className":137796},[75]," 就无须再看了。",[11,137799,137800],{},"如果两支都不可行，就判 unsat。实际求解器还会加上割平面（cut）和更强的预处理，避免分支树爆炸。",[136334,137802,132790],{"id":132790},[11,137804,137805,137807],{},[45608,137806,128185],{}," 最常用的方法还是位爆破：把位向量运算展开成门电路，再交给 SAT 层。但因为位向量有固定宽度，这种展开是有限且可预测的。",[11,137809,137810],{},"现代求解器会先做字级重写，把明显恒真的部分消掉，比如：",[31,137812,137814],{"className":137813,"translate":35},[411],[31,137815,137817,137872],{"className":137816,"translate":35},[34],[31,137818,137820],{"className":137819},[39],[41,137821,137822],{"xmlns":43,"display":420},[45,137823,137824,137869],{},[48,137825,137826,137828,137830,137832,137834,137836,137838,137840,137842,137844,137846,137848,137850,137852,137854,137856,137858,137860,137863,137865,137867],{},[51,137827,53],{},[1430,137829,1432],{},[51,137831,132814],{"mathvariant":494},[1430,137833,1432],{},[91,137835,1977],{},[95,137837,223],{},[91,137839,1977],{},[95,137841,1210],{"separator":62},[119,137843],{"width":20286},[51,137845,53],{},[95,137847,97],{},[51,137849,53],{},[95,137851,223],{},[91,137853,1977],{},[95,137855,1210],{"separator":62},[119,137857],{"width":20286},[51,137859,53],{},[95,137861,137862],{},"≪",[91,137864,1977],{},[95,137866,223],{},[51,137868,53],{},[55,137870,137871],{"encoding":57},"x \\,\\&\\, 0 = 0, \\qquad x - x = 0, \\qquad x \\ll 0 = x",[31,137873,137875,137905,137935,137953,137983,138001],{"className":137874,"ariaHidden":62},[61],[31,137876,137878,137881,137884,137887,137890,137893,137896,137899,137902],{"className":137877},[66],[31,137879],{"className":137880,"style":1630},[70],[31,137882,53],{"className":137883},[75,76],[31,137885],{"className":137886,"style":1467},[119],[31,137888,132814],{"className":137889},[75],[31,137891],{"className":137892,"style":1467},[119],[31,137894,1977],{"className":137895},[75],[31,137897],{"className":137898,"style":293},[119],[31,137900,223],{"className":137901},[297],[31,137903],{"className":137904,"style":293},[119],[31,137906,137908,137911,137914,137917,137920,137923,137926,137929,137932],{"className":137907},[66],[31,137909],{"className":137910,"style":41982},[70],[31,137912,1977],{"className":137913},[75],[31,137915,1210],{"className":137916},[1391],[31,137918],{"className":137919,"style":20525},[119],[31,137921],{"className":137922,"style":1467},[119],[31,137924,53],{"className":137925},[75,76],[31,137927],{"className":137928,"style":120},[119],[31,137930,97],{"className":137931},[124],[31,137933],{"className":137934,"style":120},[119],[31,137936,137938,137941,137944,137947,137950],{"className":137937},[66],[31,137939],{"className":137940,"style":71},[70],[31,137942,53],{"className":137943},[75,76],[31,137945],{"className":137946,"style":293},[119],[31,137948,223],{"className":137949},[297],[31,137951],{"className":137952,"style":293},[119],[31,137954,137956,137959,137962,137965,137968,137971,137974,137977,137980],{"className":137955},[66],[31,137957],{"className":137958,"style":41982},[70],[31,137960,1977],{"className":137961},[75],[31,137963,1210],{"className":137964},[1391],[31,137966],{"className":137967,"style":20525},[119],[31,137969],{"className":137970,"style":1467},[119],[31,137972,53],{"className":137973},[75,76],[31,137975],{"className":137976,"style":293},[119],[31,137978,137862],{"className":137979},[297],[31,137981],{"className":137982,"style":293},[119],[31,137984,137986,137989,137992,137995,137998],{"className":137985},[66],[31,137987],{"className":137988,"style":326},[70],[31,137990,1977],{"className":137991},[75],[31,137993],{"className":137994,"style":293},[119],[31,137996,223],{"className":137997},[297],[31,137999],{"className":138000,"style":293},[119],[31,138002,138004,138007],{"className":138003},[66],[31,138005],{"className":138006,"style":71},[70],[31,138008,53],{"className":138009},[75,76],[11,138011,138012],{},"然后再决定哪些子表达式值得展开。乘法是位向量的难点，因为乘法器电路规模大，求解器通常会配合专门的乘法推理和等价类传播。",[11,138014,138015,138016,138019],{},"位向量的语义是",[7246,138017,138018],{},"模运算","，这一点非常重要：",[31,138021,138023],{"className":138022,"translate":35},[411],[31,138024,138026,138088],{"className":138025,"translate":35},[34],[31,138027,138029],{"className":138028},[39],[41,138030,138031],{"xmlns":43,"display":420},[45,138032,138033,138085],{},[48,138034,138035,138037,138039,138041,138043,138045,138047,138049,138051,138059,138062,138069,138071,138073,138075,138077,138079,138081],{},[51,138036,53],{},[95,138038,434],{},[91,138040,205],{},[95,138042,135158],{},[91,138044,1977],{},[119,138046],{},[119,138048],{"width":1584},[95,138050,211],{"stretchy":210},[48,138052,138053,138055,138057],{},[51,138054,16396],{"mathvariant":494},[51,138056,2552],{"mathvariant":494},[51,138058,1435],{"mathvariant":494},[119,138060],{"width":138061},"0.3333em",[1180,138063,138064,138066],{},[91,138065,152],{},[91,138067,138068],{},"32",[95,138070,220],{"stretchy":210},[1430,138072,125504],{},[95,138074,125507],{},[1430,138076,125504],{},[51,138078,53],{},[95,138080,21271],{"mathvariant":494},[1430,138082,138084],{"mathvariant":138083},"monospace","0xFFFFFFFF",[55,138086,138087],{"encoding":57},"x + 1 \\equiv 0 \\pmod{2^{32}} \\;\\wedge\\; x \\ne \\texttt{0xFFFFFFFF}",[31,138089,138091,138109,138127,138143,138216,138267],{"className":138090,"ariaHidden":62},[61],[31,138092,138094,138097,138100,138103,138106],{"className":138093},[66],[31,138095],{"className":138096,"style":307},[70],[31,138098,53],{"className":138099},[75,76],[31,138101],{"className":138102,"style":120},[119],[31,138104,434],{"className":138105},[124],[31,138107],{"className":138108,"style":120},[119],[31,138110,138112,138115,138118,138121,138124],{"className":138111},[66],[31,138113],{"className":138114,"style":326},[70],[31,138116,205],{"className":138117},[75],[31,138119],{"className":138120,"style":293},[119],[31,138122,135158],{"className":138123},[297],[31,138125],{"className":138126,"style":293},[119],[31,138128,138130,138133,138136,138140],{"className":138129},[66],[31,138131],{"className":138132,"style":326},[70],[31,138134,1977],{"className":138135},[75],[31,138137],{"className":138138},[119,138139],"allowbreak",[31,138141],{"className":138142,"style":1789},[119],[31,138144,138146,138149,138152,138162,138166,138198,138201,138204,138207,138210,138213],{"className":138145},[66],[31,138147],{"className":138148,"style":16882},[70],[31,138150,211],{"className":138151},[264],[31,138153,138155],{"className":138154},[75],[31,138156,138158],{"className":138157},[75],[31,138159,138161],{"className":138160},[75,2745],"mod",[31,138163],{"className":138164,"style":138165},[119],"margin-right:0.3333em;",[31,138167,138169,138172],{"className":138168},[75],[31,138170,152],{"className":138171},[75],[31,138173,138175],{"className":138174},[1039],[31,138176,138178],{"className":138177},[1043],[31,138179,138181],{"className":138180},[1048],[31,138182,138184],{"className":138183,"style":1327},[1052],[31,138185,138186,138189],{"style":1330},[31,138187],{"className":138188,"style":1061},[1060],[31,138190,138192],{"className":138191},[1065,1066,1067,1068],[31,138193,138195],{"className":138194},[75,1068],[31,138196,138068],{"className":138197},[75,1068],[31,138199,220],{"className":138200},[289],[31,138202],{"className":138203,"style":293},[119],[31,138205],{"className":138206,"style":120},[119],[31,138208,125507],{"className":138209},[124],[31,138211],{"className":138212,"style":293},[119],[31,138214],{"className":138215,"style":120},[119],[31,138217,138219,138222,138225,138228,138264],{"className":138218},[66],[31,138220],{"className":138221,"style":4202},[70],[31,138223,53],{"className":138224},[75,76],[31,138226],{"className":138227,"style":293},[119],[31,138229,138231,138258,138261],{"className":138230},[297],[31,138232,138234],{"className":138233},[297],[31,138235,138237],{"className":138236},[75,21516],[31,138238,138240],{"className":138239},[21520],[31,138241,138243,138246,138255],{"className":138242},[21524],[31,138244],{"className":138245,"style":4202},[70],[31,138247,138249],{"className":138248},[21531],[31,138250,138252],{"className":138251},[75],[31,138253,21538],{"className":138254},[297],[31,138256],{"className":138257},[21542],[31,138259],{"className":138260},[119,21546],[31,138262,223],{"className":138263},[297],[31,138265],{"className":138266,"style":293},[119],[31,138268,138270,138274],{"className":138269},[66],[31,138271],{"className":138272,"style":138273},[70],"height:0.6111em;",[31,138275,138277],{"className":138276},[75,1793],[31,138278,138084],{"className":138279},[75,138280],"texttt",[11,138282,138283,138284,138357,138358,138360,138361,138417,138418,138420,138421],{},"在 32 位无符号整数里，唯一满足 ",[31,138285,138287,138309],{"className":138286,"translate":35},[34],[31,138288,138290],{"className":138289},[39],[41,138291,138292],{"xmlns":43},[45,138293,138294,138306],{},[48,138295,138296,138298,138300,138302,138304],{},[51,138297,53],{},[95,138299,434],{},[91,138301,205],{},[95,138303,135158],{},[91,138305,1977],{},[55,138307,138308],{"encoding":57},"x + 1 \\equiv 0",[31,138310,138312,138330,138348],{"className":138311,"ariaHidden":62},[61],[31,138313,138315,138318,138321,138324,138327],{"className":138314},[66],[31,138316],{"className":138317,"style":307},[70],[31,138319,53],{"className":138320},[75,76],[31,138322],{"className":138323,"style":120},[119],[31,138325,434],{"className":138326},[124],[31,138328],{"className":138329,"style":120},[119],[31,138331,138333,138336,138339,138342,138345],{"className":138332},[66],[31,138334],{"className":138335,"style":326},[70],[31,138337,205],{"className":138338},[75],[31,138340],{"className":138341,"style":293},[119],[31,138343,135158],{"className":138344},[297],[31,138346],{"className":138347,"style":293},[119],[31,138349,138351,138354],{"className":138350},[66],[31,138352],{"className":138353,"style":326},[70],[31,138355,1977],{"className":138356},[75]," 的值就是 ",[45608,138359,138084],{},"，所以这个公式是 unsat。如果把它当成数学整数，",[31,138362,138364,138384],{"className":138363,"translate":35},[34],[31,138365,138367],{"className":138366},[39],[41,138368,138369],{"xmlns":43},[45,138370,138371,138381],{},[48,138372,138373,138375,138377,138379],{},[51,138374,53],{},[95,138376,223],{},[95,138378,97],{},[91,138380,205],{},[55,138382,138383],{"encoding":57},"x = -1",[31,138385,138387,138405],{"className":138386,"ariaHidden":62},[61],[31,138388,138390,138393,138396,138399,138402],{"className":138389},[66],[31,138391],{"className":138392,"style":71},[70],[31,138394,53],{"className":138395},[75,76],[31,138397],{"className":138398,"style":293},[119],[31,138400,223],{"className":138401},[297],[31,138403],{"className":138404,"style":293},[119],[31,138406,138408,138411,138414],{"className":138407},[66],[31,138409],{"className":138410,"style":112},[70],[31,138412,97],{"className":138413},[75],[31,138415,205],{"className":138416},[75]," 又不等于 ",[45608,138419,138084],{},"，结论就完全不同。",[7246,138422,138423],{},"选错理论，漏洞检测就会漏报或误报。",[136334,138425,132784],{"id":132784},[11,138427,138428,138429,2446,138431,138433],{},"数组理论的核心是 ",[45608,138430,126909],{},[45608,138432,127844],{}," 两条操作，以及前面提到的 read-over-write 公理。",[11,138435,138436,138437,138440,138441,138444],{},"求解器不会一次性展开所有下标，而是采用",[7246,138438,138439],{},"惰性公理实例化","（lazy axiom instantiation）：先假设数组是一致的，当检查中发现某个 ",[45608,138442,138443],{},"select(store(...))"," 需要用到公理时，才生成对应的引理加入公式。这叫「按需生成」，也叫 lemma-on-demand。",[11,138446,136483],{},[31,138448,138450],{"className":138449,"translate":35},[411],[31,138451,138453,138541],{"className":138452,"translate":35},[34],[31,138454,138456],{"className":138455},[39],[41,138457,138458],{"xmlns":43,"display":420},[45,138459,138460,138538],{},[48,138461,138462,138476,138478,138490,138492,138494,138496,138498,138500,138502,138504,138506,138508,138510,138512,138514,138516,138518,138520,138522,138524,138526,138528,138530,138532,138534,138536],{},[48,138463,138464,138466,138468,138470,138472,138474],{},[51,138465,14551],{"mathvariant":494},[51,138467,30042],{"mathvariant":494},[51,138469,87599],{"mathvariant":494},[51,138471,30042],{"mathvariant":494},[51,138473,1412],{"mathvariant":494},[51,138475,12271],{"mathvariant":494},[95,138477,211],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[48,138479,138480,138482,138484,138486,138488],{},[51,138481,14551],{"mathvariant":494},[51,138483,12271],{"mathvariant":494},[51,138485,2552],{"mathvariant":494},[51,138487,2549],{"mathvariant":494},[51,138489,30042],{"mathvariant":494},[95,138491,211],{"stretchy":210},[51,138493,5813],{},[95,138495,1210],{"separator":62},[51,138497,5391],{},[95,138499,1210],{"separator":62},[51,138501,5354],{},[95,138503,220],{"stretchy":210},[95,138505,1210],{"separator":62},[51,138507,2555],{},[95,138509,220],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[95,138511,223],{},[51,138513,87604],{},[1430,138515,125504],{},[95,138517,125507],{},[1430,138519,125504],{},[51,138521,5391],{},[95,138523,223],{},[51,138525,2555],{},[1430,138527,125504],{},[95,138529,125507],{},[1430,138531,125504],{},[51,138533,87604],{},[95,138535,21271],{"mathvariant":494},[51,138537,5354],{},[55,138539,138540],{"encoding":57},"\\mathrm{select}\\bigl(\\mathrm{store}(a, i, v), j\\bigr) = w \\;\\wedge\\; i = j \\;\\wedge\\; w \\ne v",[31,138542,138544,138619,138644,138662,138687,138738],{"className":138543,"ariaHidden":62},[61],[31,138545,138547,138550,138556,138562,138568,138571,138574,138577,138580,138583,138586,138589,138592,138595,138598,138601,138604,138610,138613,138616],{"className":138546},[66],[31,138548],{"className":138549,"style":4988},[70],[31,138551,138553],{"className":138552},[75],[31,138554,126909],{"className":138555},[75,2745],[31,138557,138559],{"className":138558},[264],[31,138560,211],{"className":138561},[1307,1308],[31,138563,138565],{"className":138564},[75],[31,138566,127844],{"className":138567},[75,2745],[31,138569,211],{"className":138570},[264],[31,138572,5813],{"className":138573},[75,76],[31,138575,1210],{"className":138576},[1391],[31,138578],{"className":138579,"style":1467},[119],[31,138581,5391],{"className":138582},[75,76],[31,138584,1210],{"className":138585},[1391],[31,138587],{"className":138588,"style":1467},[119],[31,138590,5354],{"className":138591,"style":2666},[75,76],[31,138593,220],{"className":138594},[289],[31,138596,1210],{"className":138597},[1391],[31,138599],{"className":138600,"style":1467},[119],[31,138602,2555],{"className":138603,"style":5514},[75,76],[31,138605,138607],{"className":138606},[289],[31,138608,220],{"className":138609},[1307,1308],[31,138611],{"className":138612,"style":293},[119],[31,138614,223],{"className":138615},[297],[31,138617],{"className":138618,"style":293},[119],[31,138620,138622,138625,138629,138632,138635,138638,138641],{"className":138621},[66],[31,138623],{"className":138624,"style":136613},[70],[31,138626,87604],{"className":138627,"style":138628},[75,76],"margin-right:0.0269em;",[31,138630],{"className":138631,"style":293},[119],[31,138633],{"className":138634,"style":120},[119],[31,138636,125507],{"className":138637},[124],[31,138639],{"className":138640,"style":293},[119],[31,138642],{"className":138643,"style":120},[119],[31,138645,138647,138650,138653,138656,138659],{"className":138646},[66],[31,138648],{"className":138649,"style":132451},[70],[31,138651,5391],{"className":138652},[75,76],[31,138654],{"className":138655,"style":293},[119],[31,138657,223],{"className":138658},[297],[31,138660],{"className":138661,"style":293},[119],[31,138663,138665,138669,138672,138675,138678,138681,138684],{"className":138664},[66],[31,138666],{"className":138667,"style":138668},[70],"height:0.854em;vertical-align:-0.1944em;",[31,138670,2555],{"className":138671,"style":5514},[75,76],[31,138673],{"className":138674,"style":293},[119],[31,138676],{"className":138677,"style":120},[119],[31,138679,125507],{"className":138680},[124],[31,138682],{"className":138683,"style":293},[119],[31,138685],{"className":138686,"style":120},[119],[31,138688,138690,138693,138696,138699,138735],{"className":138689},[66],[31,138691],{"className":138692,"style":4202},[70],[31,138694,87604],{"className":138695,"style":138628},[75,76],[31,138697],{"className":138698,"style":293},[119],[31,138700,138702,138729,138732],{"className":138701},[297],[31,138703,138705],{"className":138704},[297],[31,138706,138708],{"className":138707},[75,21516],[31,138709,138711],{"className":138710},[21520],[31,138712,138714,138717,138726],{"className":138713},[21524],[31,138715],{"className":138716,"style":4202},[70],[31,138718,138720],{"className":138719},[21531],[31,138721,138723],{"className":138722},[75],[31,138724,21538],{"className":138725},[297],[31,138727],{"className":138728},[21542],[31,138730],{"className":138731},[119,21546],[31,138733,223],{"className":138734},[297],[31,138736],{"className":138737,"style":293},[119],[31,138739,138741,138744],{"className":138740},[66],[31,138742],{"className":138743,"style":71},[70],[31,138745,5354],{"className":138746,"style":2666},[75,76],[11,138748,118102,138749,138800,138801,138829,138830,131449,138881,138965],{},[31,138750,138752,138770],{"className":138751,"translate":35},[34],[31,138753,138755],{"className":138754},[39],[41,138756,138757],{"xmlns":43},[45,138758,138759,138767],{},[48,138760,138761,138763,138765],{},[51,138762,5391],{},[95,138764,223],{},[51,138766,2555],{},[55,138768,138769],{"encoding":57},"i = j",[31,138771,138773,138791],{"className":138772,"ariaHidden":62},[61],[31,138774,138776,138779,138782,138785,138788],{"className":138775},[66],[31,138777],{"className":138778,"style":132451},[70],[31,138780,5391],{"className":138781},[75,76],[31,138783],{"className":138784,"style":293},[119],[31,138786,223],{"className":138787},[297],[31,138789],{"className":138790,"style":293},[119],[31,138792,138794,138797],{"className":138793},[66],[31,138795],{"className":138796,"style":138668},[70],[31,138798,2555],{"className":138799,"style":5514},[75,76]," 和 read-over-write 公理，左边等于 ",[31,138802,138804,138817],{"className":138803,"translate":35},[34],[31,138805,138807],{"className":138806},[39],[41,138808,138809],{"xmlns":43},[45,138810,138811,138815],{},[48,138812,138813],{},[51,138814,5354],{},[55,138816,5354],{"encoding":57},[31,138818,138820],{"className":138819,"ariaHidden":62},[61],[31,138821,138823,138826],{"className":138822},[66],[31,138824],{"className":138825,"style":71},[70],[31,138827,5354],{"className":138828,"style":2666},[75,76],"；于是 ",[31,138831,138833,138851],{"className":138832,"translate":35},[34],[31,138834,138836],{"className":138835},[39],[41,138837,138838],{"xmlns":43},[45,138839,138840,138848],{},[48,138841,138842,138844,138846],{},[51,138843,5354],{},[95,138845,223],{},[51,138847,87604],{},[55,138849,138850],{"encoding":57},"v = w",[31,138852,138854,138872],{"className":138853,"ariaHidden":62},[61],[31,138855,138857,138860,138863,138866,138869],{"className":138856},[66],[31,138858],{"className":138859,"style":71},[70],[31,138861,5354],{"className":138862,"style":2666},[75,76],[31,138864],{"className":138865,"style":293},[119],[31,138867,223],{"className":138868},[297],[31,138870],{"className":138871,"style":293},[119],[31,138873,138875,138878],{"className":138874},[66],[31,138876],{"className":138877,"style":71},[70],[31,138879,87604],{"className":138880,"style":138628},[75,76],[31,138882,138884,138902],{"className":138883,"translate":35},[34],[31,138885,138887],{"className":138886},[39],[41,138888,138889],{"xmlns":43},[45,138890,138891,138899],{},[48,138892,138893,138895,138897],{},[51,138894,87604],{},[95,138896,21271],{"mathvariant":494},[51,138898,5354],{},[55,138900,138901],{"encoding":57},"w \\ne v",[31,138903,138905,138956],{"className":138904,"ariaHidden":62},[61],[31,138906,138908,138911,138914,138917,138953],{"className":138907},[66],[31,138909],{"className":138910,"style":4202},[70],[31,138912,87604],{"className":138913,"style":138628},[75,76],[31,138915],{"className":138916,"style":293},[119],[31,138918,138920,138947,138950],{"className":138919},[297],[31,138921,138923],{"className":138922},[297],[31,138924,138926],{"className":138925},[75,21516],[31,138927,138929],{"className":138928},[21520],[31,138930,138932,138935,138944],{"className":138931},[21524],[31,138933],{"className":138934,"style":4202},[70],[31,138936,138938],{"className":138937},[21531],[31,138939,138941],{"className":138940},[75],[31,138942,21538],{"className":138943},[297],[31,138945],{"className":138946},[21542],[31,138948],{"className":138949},[119,21546],[31,138951,223],{"className":138952},[297],[31,138954],{"className":138955,"style":293},[119],[31,138957,138959,138962],{"className":138958},[66],[31,138960],{"className":138961,"style":71},[70],[31,138963,5354],{"className":138964,"style":2666},[75,76]," 冲突，unsat。整个过程中，数组的长度、其他下标的值，全都不需要知道。",[136334,138967,138968],{"id":138968},"其他理论",[11,138970,138971,138972,138974],{},"字符串理论通常用自动机或序列理论处理，word equation 本身可判定，但加上长度约束、正则约束后很多片段会变得不可判定，工业求解器往往采用不完整但实用的策略。浮点理论通常把 IEEE 754 的位级语义编码成位向量，再交给 ",[45608,138973,128185],{}," 后端。",[52860,138976,138978],{"id":138977},"_45-冲突学习","4.5 冲突学习",[11,138980,138981,138982,138985],{},"理论冲突不会只让当前搜索失败一次，它会变成一条",[7246,138983,138984],{},"理论引理","（theory lemma），永久加入布尔公式。",[11,138987,138988],{},"前面 4.2 节的例子：",[31,138990,138992],{"className":138991,"translate":35},[411],[31,138993,138995,139036],{"className":138994,"translate":35},[34],[31,138996,138998],{"className":138997},[39],[41,138999,139000],{"xmlns":43,"display":420},[45,139001,139002,139034],{},[48,139003,139004,139006,139008,139010,139012,139014,139016,139018,139020,139022,139024,139026,139028,139030,139032],{},[51,139005,53],{},[95,139007,16040],{},[91,139009,1977],{},[95,139011,1210],{"separator":62},[119,139013],{"width":20286},[51,139015,3003],{},[95,139017,16040],{},[91,139019,1977],{},[95,139021,1210],{"separator":62},[119,139023],{"width":20286},[51,139025,53],{},[95,139027,434],{},[51,139029,3003],{},[95,139031,9758],{},[91,139033,1977],{},[55,139035,135753],{"encoding":57},[31,139037,139039,139057,139087,139117,139135],{"className":139038,"ariaHidden":62},[61],[31,139040,139042,139045,139048,139051,139054],{"className":139041},[66],[31,139043],{"className":139044,"style":21113},[70],[31,139046,53],{"className":139047},[75,76],[31,139049],{"className":139050,"style":293},[119],[31,139052,16040],{"className":139053},[297],[31,139055],{"className":139056,"style":293},[119],[31,139058,139060,139063,139066,139069,139072,139075,139078,139081,139084],{"className":139059},[66],[31,139061],{"className":139062,"style":41982},[70],[31,139064,1977],{"className":139065},[75],[31,139067,1210],{"className":139068},[1391],[31,139070],{"className":139071,"style":20525},[119],[31,139073],{"className":139074,"style":1467},[119],[31,139076,3003],{"className":139077,"style":2666},[75,76],[31,139079],{"className":139080,"style":293},[119],[31,139082,16040],{"className":139083},[297],[31,139085],{"className":139086,"style":293},[119],[31,139088,139090,139093,139096,139099,139102,139105,139108,139111,139114],{"className":139089},[66],[31,139091],{"className":139092,"style":41982},[70],[31,139094,1977],{"className":139095},[75],[31,139097,1210],{"className":139098},[1391],[31,139100],{"className":139101,"style":20525},[119],[31,139103],{"className":139104,"style":1467},[119],[31,139106,53],{"className":139107},[75,76],[31,139109],{"className":139110,"style":120},[119],[31,139112,434],{"className":139113},[124],[31,139115],{"className":139116,"style":120},[119],[31,139118,139120,139123,139126,139129,139132],{"className":139119},[66],[31,139121],{"className":139122,"style":53323},[70],[31,139124,3003],{"className":139125,"style":2666},[75,76],[31,139127],{"className":139128,"style":293},[119],[31,139130,9758],{"className":139131},[297],[31,139133],{"className":139134,"style":293},[119],[31,139136,139138,139141],{"className":139137},[66],[31,139139],{"className":139140,"style":326},[70],[31,139142,1977],{"className":139143},[75],[11,139145,139146],{},"这三条文字不可同时为真，所以下面这条子句是理论有效的：",[31,139148,139150],{"className":139149,"translate":35},[411],[31,139151,139153,139217],{"className":139152,"translate":35},[34],[31,139154,139156],{"className":139155},[39],[41,139157,139158],{"xmlns":43,"display":420},[45,139159,139160,139214],{},[48,139161,139162,139164,139166,139168,139170,139172,139174,139176,139178,139180,139182,139184,139186,139188,139190,139192,139194,139196,139198,139200,139202,139204,139206,139208,139210,139212],{},[51,139163,126125],{"mathvariant":494},[95,139165,211],{"stretchy":210},[51,139167,53],{},[95,139169,16040],{},[91,139171,1977],{},[95,139173,220],{"stretchy":210},[1430,139175,125504],{},[95,139177,126150],{},[1430,139179,125504],{},[51,139181,126125],{"mathvariant":494},[95,139183,211],{"stretchy":210},[51,139185,3003],{},[95,139187,16040],{},[91,139189,1977],{},[95,139191,220],{"stretchy":210},[1430,139193,125504],{},[95,139195,126150],{},[1430,139197,125504],{},[51,139199,126125],{"mathvariant":494},[95,139201,211],{"stretchy":210},[51,139203,53],{},[95,139205,434],{},[51,139207,3003],{},[95,139209,9758],{},[91,139211,1977],{},[95,139213,220],{"stretchy":210},[55,139215,139216],{"encoding":57},"\\neg(x > 0) \\;\\vee\\; \\neg(y > 0) \\;\\vee\\; \\neg(x + y \u003C 0)",[31,139218,139220,139244,139271,139295,139322,139346,139364],{"className":139219,"ariaHidden":62},[61],[31,139221,139223,139226,139229,139232,139235,139238,139241],{"className":139222},[66],[31,139224],{"className":139225,"style":260},[70],[31,139227,126125],{"className":139228},[75],[31,139230,211],{"className":139231},[264],[31,139233,53],{"className":139234},[75,76],[31,139236],{"className":139237,"style":293},[119],[31,139239,16040],{"className":139240},[297],[31,139242],{"className":139243,"style":293},[119],[31,139245,139247,139250,139253,139256,139259,139262,139265,139268],{"className":139246},[66],[31,139248],{"className":139249,"style":260},[70],[31,139251,1977],{"className":139252},[75],[31,139254,220],{"className":139255},[289],[31,139257],{"className":139258,"style":293},[119],[31,139260],{"className":139261,"style":120},[119],[31,139263,126150],{"className":139264},[124],[31,139266],{"className":139267,"style":293},[119],[31,139269],{"className":139270,"style":120},[119],[31,139272,139274,139277,139280,139283,139286,139289,139292],{"className":139273},[66],[31,139275],{"className":139276,"style":260},[70],[31,139278,126125],{"className":139279},[75],[31,139281,211],{"className":139282},[264],[31,139284,3003],{"className":139285,"style":2666},[75,76],[31,139287],{"className":139288,"style":293},[119],[31,139290,16040],{"className":139291},[297],[31,139293],{"className":139294,"style":293},[119],[31,139296,139298,139301,139304,139307,139310,139313,139316,139319],{"className":139297},[66],[31,139299],{"className":139300,"style":260},[70],[31,139302,1977],{"className":139303},[75],[31,139305,220],{"className":139306},[289],[31,139308],{"className":139309,"style":293},[119],[31,139311],{"className":139312,"style":120},[119],[31,139314,126150],{"className":139315},[124],[31,139317],{"className":139318,"style":293},[119],[31,139320],{"className":139321,"style":120},[119],[31,139323,139325,139328,139331,139334,139337,139340,139343],{"className":139324},[66],[31,139326],{"className":139327,"style":260},[70],[31,139329,126125],{"className":139330},[75],[31,139332,211],{"className":139333},[264],[31,139335,53],{"className":139336},[75,76],[31,139338],{"className":139339,"style":120},[119],[31,139341,434],{"className":139342},[124],[31,139344],{"className":139345,"style":120},[119],[31,139347,139349,139352,139355,139358,139361],{"className":139348},[66],[31,139350],{"className":139351,"style":53323},[70],[31,139353,3003],{"className":139354,"style":2666},[75,76],[31,139356],{"className":139357,"style":293},[119],[31,139359,9758],{"className":139360},[297],[31,139362],{"className":139363,"style":293},[119],[31,139365,139367,139370,139373],{"className":139366},[66],[31,139368],{"className":139369,"style":260},[70],[31,139371,1977],{"className":139372},[75],[31,139374,220],{"className":139375},[289],[11,139377,139378],{},"把它翻译回命题变量，就是：",[31,139380,139382],{"className":139381,"translate":35},[411],[31,139383,139385,139421],{"className":139384,"translate":35},[34],[31,139386,139388],{"className":139387},[39],[41,139389,139390],{"xmlns":43,"display":420},[45,139391,139392,139418],{},[48,139393,139394,139396,139398,139400,139402,139404,139406,139408,139410,139412,139414,139416],{},[51,139395,126125],{"mathvariant":494},[51,139397,11],{},[1430,139399,125504],{},[95,139401,126150],{},[1430,139403,125504],{},[51,139405,126125],{"mathvariant":494},[51,139407,27666],{},[1430,139409,125504],{},[95,139411,126150],{},[1430,139413,125504],{},[51,139415,126125],{"mathvariant":494},[51,139417,2549],{},[55,139419,139420],{"encoding":57},"\\neg p \\;\\vee\\; \\neg q \\;\\vee\\; \\neg r",[31,139422,139424,139451,139478],{"className":139423,"ariaHidden":62},[61],[31,139425,139427,139430,139433,139436,139439,139442,139445,139448],{"className":139426},[66],[31,139428],{"className":139429,"style":126295},[70],[31,139431,126125],{"className":139432},[75],[31,139434,11],{"className":139435},[75,76],[31,139437],{"className":139438,"style":293},[119],[31,139440],{"className":139441,"style":120},[119],[31,139443,126150],{"className":139444},[124],[31,139446],{"className":139447,"style":293},[119],[31,139449],{"className":139450,"style":120},[119],[31,139452,139454,139457,139460,139463,139466,139469,139472,139475],{"className":139453},[66],[31,139455],{"className":139456,"style":126295},[70],[31,139458,126125],{"className":139459},[75],[31,139461,27666],{"className":139462,"style":2666},[75,76],[31,139464],{"className":139465,"style":293},[119],[31,139467],{"className":139468,"style":120},[119],[31,139470,126150],{"className":139471},[124],[31,139473],{"className":139474,"style":293},[119],[31,139476],{"className":139477,"style":120},[119],[31,139479,139481,139484,139487],{"className":139480},[66],[31,139482],{"className":139483,"style":71},[70],[31,139485,126125],{"className":139486},[75],[31,139488,2549],{"className":139489,"style":19759},[75,76],[11,139491,139492,139493,139547,139548,2372],{},"SAT 层学到这条子句之后，就再也不会同时选 ",[31,139494,139496,139517],{"className":139495,"translate":35},[34],[31,139497,139499],{"className":139498},[39],[41,139500,139501],{"xmlns":43},[45,139502,139503,139515],{},[48,139504,139505,139507,139509,139511,139513],{},[51,139506,11],{},[95,139508,1210],{"separator":62},[51,139510,27666],{},[95,139512,1210],{"separator":62},[51,139514,2549],{},[55,139516,128022],{"encoding":57},[31,139518,139520],{"className":139519,"ariaHidden":62},[61],[31,139521,139523,139526,139529,139532,139535,139538,139541,139544],{"className":139522},[66],[31,139524],{"className":139525,"style":865},[70],[31,139527,11],{"className":139528},[75,76],[31,139530,1210],{"className":139531},[1391],[31,139533],{"className":139534,"style":1467},[119],[31,139536,27666],{"className":139537,"style":2666},[75,76],[31,139539,1210],{"className":139540},[1391],[31,139542],{"className":139543,"style":1467},[119],[31,139545,2549],{"className":139546,"style":19759},[75,76]," 都为真了。这一步是 DPLL(T) 效率的关键：",[7246,139549,139550],{},"理论层的「一次性检查」变成了布尔层的「永久知识」",[11,139552,139553],{},"如果没有冲突学习，SAT 层会在同一个理论冲突上反复撞墙；有了它，搜索空间被真正地剪掉了。",[52860,139555,139557],{"id":139556},"_46-完整例子","4.6 完整例子",[11,139559,139560],{},"还是用这个公式：",[31,139562,139564],{"className":139563,"translate":35},[411],[31,139565,139567,139620],{"className":139566,"translate":35},[34],[31,139568,139570],{"className":139569},[39],[41,139571,139572],{"xmlns":43,"display":420},[45,139573,139574,139618],{},[48,139575,139576,139578,139580,139582,139584,139586,139588,139590,139592,139594,139596,139598,139600,139602,139604,139606,139608,139610,139612,139614,139616],{},[51,139577,8059],{},[95,139579,223],{},[95,139581,211],{"stretchy":210},[51,139583,53],{},[95,139585,16040],{},[91,139587,1977],{},[95,139589,126150],{},[51,139591,3003],{},[95,139593,16040],{},[91,139595,1977],{},[95,139597,220],{"stretchy":210},[1430,139599,125504],{},[95,139601,125507],{},[1430,139603,125504],{},[95,139605,211],{"stretchy":210},[51,139607,53],{},[95,139609,434],{},[51,139611,3003],{},[95,139613,9758],{},[91,139615,1977],{},[95,139617,220],{"stretchy":210},[55,139619,134979],{"encoding":57},[31,139621,139623,139641,139662,139680,139698,139725,139746,139764],{"className":139622,"ariaHidden":62},[61],[31,139624,139626,139629,139632,139635,139638],{"className":139625},[66],[31,139627],{"className":139628,"style":865},[70],[31,139630,8059],{"className":139631},[75,76],[31,139633],{"className":139634,"style":293},[119],[31,139636,223],{"className":139637},[297],[31,139639],{"className":139640,"style":293},[119],[31,139642,139644,139647,139650,139653,139656,139659],{"className":139643},[66],[31,139645],{"className":139646,"style":260},[70],[31,139648,211],{"className":139649},[264],[31,139651,53],{"className":139652},[75,76],[31,139654],{"className":139655,"style":293},[119],[31,139657,16040],{"className":139658},[297],[31,139660],{"className":139661,"style":293},[119],[31,139663,139665,139668,139671,139674,139677],{"className":139664},[66],[31,139666],{"className":139667,"style":326},[70],[31,139669,1977],{"className":139670},[75],[31,139672],{"className":139673,"style":120},[119],[31,139675,126150],{"className":139676},[124],[31,139678],{"className":139679,"style":120},[119],[31,139681,139683,139686,139689,139692,139695],{"className":139682},[66],[31,139684],{"className":139685,"style":53323},[70],[31,139687,3003],{"className":139688,"style":2666},[75,76],[31,139690],{"className":139691,"style":293},[119],[31,139693,16040],{"className":139694},[297],[31,139696],{"className":139697,"style":293},[119],[31,139699,139701,139704,139707,139710,139713,139716,139719,139722],{"className":139700},[66],[31,139702],{"className":139703,"style":260},[70],[31,139705,1977],{"className":139706},[75],[31,139708,220],{"className":139709},[289],[31,139711],{"className":139712,"style":293},[119],[31,139714],{"className":139715,"style":120},[119],[31,139717,125507],{"className":139718},[124],[31,139720],{"className":139721,"style":293},[119],[31,139723],{"className":139724,"style":120},[119],[31,139726,139728,139731,139734,139737,139740,139743],{"className":139727},[66],[31,139729],{"className":139730,"style":260},[70],[31,139732,211],{"className":139733},[264],[31,139735,53],{"className":139736},[75,76],[31,139738],{"className":139739,"style":120},[119],[31,139741,434],{"className":139742},[124],[31,139744],{"className":139745,"style":120},[119],[31,139747,139749,139752,139755,139758,139761],{"className":139748},[66],[31,139750],{"className":139751,"style":53323},[70],[31,139753,3003],{"className":139754,"style":2666},[75,76],[31,139756],{"className":139757,"style":293},[119],[31,139759,9758],{"className":139760},[297],[31,139762],{"className":139763,"style":293},[119],[31,139765,139767,139770,139773],{"className":139766},[66],[31,139768],{"className":139769,"style":260},[70],[31,139771,1977],{"className":139772},[75],[31,139774,220],{"className":139775},[289],[11,139777,139778,139781,139782,139902,139903,140066,140067,140139,140140,140228],{},[7246,139779,139780],{},"第 1 轮","：SAT 层选择 ",[31,139783,139785,139818],{"className":139784,"translate":35},[34],[31,139786,139788],{"className":139787},[39],[41,139789,139790],{"xmlns":43},[45,139791,139792,139816],{},[48,139793,139794,139796,139798,139800,139802,139804,139806,139808,139810,139812,139814],{},[51,139795,11],{},[95,139797,223],{},[51,139799,20800],{"mathvariant":494},[95,139801,1210],{"separator":62},[51,139803,27666],{},[95,139805,223],{},[51,139807,20800],{"mathvariant":494},[95,139809,1210],{"separator":62},[51,139811,2549],{},[95,139813,223],{},[51,139815,20800],{"mathvariant":494},[55,139817,135613],{"encoding":57},[31,139819,139821,139839,139866,139893],{"className":139820,"ariaHidden":62},[61],[31,139822,139824,139827,139830,139833,139836],{"className":139823},[66],[31,139825],{"className":139826,"style":865},[70],[31,139828,11],{"className":139829},[75,76],[31,139831],{"className":139832,"style":293},[119],[31,139834,223],{"className":139835},[297],[31,139837],{"className":139838,"style":293},[119],[31,139840,139842,139845,139848,139851,139854,139857,139860,139863],{"className":139841},[66],[31,139843],{"className":139844,"style":4202},[70],[31,139846,20800],{"className":139847},[75],[31,139849,1210],{"className":139850},[1391],[31,139852],{"className":139853,"style":1467},[119],[31,139855,27666],{"className":139856,"style":2666},[75,76],[31,139858],{"className":139859,"style":293},[119],[31,139861,223],{"className":139862},[297],[31,139864],{"className":139865,"style":293},[119],[31,139867,139869,139872,139875,139878,139881,139884,139887,139890],{"className":139868},[66],[31,139870],{"className":139871,"style":4202},[70],[31,139873,20800],{"className":139874},[75],[31,139876,1210],{"className":139877},[1391],[31,139879],{"className":139880,"style":1467},[119],[31,139882,2549],{"className":139883,"style":19759},[75,76],[31,139885],{"className":139886,"style":293},[119],[31,139888,223],{"className":139889},[297],[31,139891],{"className":139892,"style":293},[119],[31,139894,139896,139899],{"className":139895},[66],[31,139897],{"className":139898,"style":1630},[70],[31,139900,20800],{"className":139901},[75],"。理论层检查 ",[31,139904,139906,139952],{"className":139905,"translate":35},[34],[31,139907,139909],{"className":139908},[39],[41,139910,139911],{"xmlns":43},[45,139912,139913,139949],{},[48,139914,139915,139917,139919,139921,139923,139925,139927,139929,139931,139933,139935,139937,139939,139941,139943,139945,139947],{},[95,139916,2512],{"stretchy":210},[51,139918,53],{},[95,139920,16040],{},[91,139922,1977],{},[95,139924,1210],{"separator":62},[1430,139926,1545],{},[51,139928,3003],{},[95,139930,16040],{},[91,139932,1977],{},[95,139934,1210],{"separator":62},[1430,139936,1545],{},[51,139938,53],{},[95,139940,434],{},[51,139942,3003],{},[95,139944,9758],{},[91,139946,1977],{},[95,139948,2604],{"stretchy":210},[55,139950,139951],{"encoding":57},"\\{x > 0,\\ y > 0,\\ x + y \u003C 0\\}",[31,139953,139955,139976,140006,140036,140054],{"className":139954,"ariaHidden":62},[61],[31,139956,139958,139961,139964,139967,139970,139973],{"className":139957},[66],[31,139959],{"className":139960,"style":260},[70],[31,139962,2512],{"className":139963},[264],[31,139965,53],{"className":139966},[75,76],[31,139968],{"className":139969,"style":293},[119],[31,139971,16040],{"className":139972},[297],[31,139974],{"className":139975,"style":293},[119],[31,139977,139979,139982,139985,139988,139991,139994,139997,140000,140003],{"className":139978},[66],[31,139980],{"className":139981,"style":41982},[70],[31,139983,1977],{"className":139984},[75],[31,139986,1210],{"className":139987},[1391],[31,139989,1545],{"className":139990},[119],[31,139992],{"className":139993,"style":1467},[119],[31,139995,3003],{"className":139996,"style":2666},[75,76],[31,139998],{"className":139999,"style":293},[119],[31,140001,16040],{"className":140002},[297],[31,140004],{"className":140005,"style":293},[119],[31,140007,140009,140012,140015,140018,140021,140024,140027,140030,140033],{"className":140008},[66],[31,140010],{"className":140011,"style":41982},[70],[31,140013,1977],{"className":140014},[75],[31,140016,1210],{"className":140017},[1391],[31,140019,1545],{"className":140020},[119],[31,140022],{"className":140023,"style":1467},[119],[31,140025,53],{"className":140026},[75,76],[31,140028],{"className":140029,"style":120},[119],[31,140031,434],{"className":140032},[124],[31,140034],{"className":140035,"style":120},[119],[31,140037,140039,140042,140045,140048,140051],{"className":140038},[66],[31,140040],{"className":140041,"style":53323},[70],[31,140043,3003],{"className":140044,"style":2666},[75,76],[31,140046],{"className":140047,"style":293},[119],[31,140049,9758],{"className":140050},[297],[31,140052],{"className":140053,"style":293},[119],[31,140055,140057,140060,140063],{"className":140056},[66],[31,140058],{"className":140059,"style":260},[70],[31,140061,1977],{"className":140062},[75],[31,140064,2604],{"className":140065},[289],"，发现 ",[31,140068,140070,140091],{"className":140069,"translate":35},[34],[31,140071,140073],{"className":140072},[39],[41,140074,140075],{"xmlns":43},[45,140076,140077,140089],{},[48,140078,140079,140081,140083,140085,140087],{},[51,140080,53],{},[95,140082,434],{},[51,140084,3003],{},[95,140086,16040],{},[91,140088,1977],{},[55,140090,135889],{"encoding":57},[31,140092,140094,140112,140130],{"className":140093,"ariaHidden":62},[61],[31,140095,140097,140100,140103,140106,140109],{"className":140096},[66],[31,140098],{"className":140099,"style":307},[70],[31,140101,53],{"className":140102},[75,76],[31,140104],{"className":140105,"style":120},[119],[31,140107,434],{"className":140108},[124],[31,140110],{"className":140111,"style":120},[119],[31,140113,140115,140118,140121,140124,140127],{"className":140114},[66],[31,140116],{"className":140117,"style":53323},[70],[31,140119,3003],{"className":140120,"style":2666},[75,76],[31,140122],{"className":140123,"style":293},[119],[31,140125,16040],{"className":140126},[297],[31,140128],{"className":140129,"style":293},[119],[31,140131,140133,140136],{"className":140132},[66],[31,140134],{"className":140135,"style":326},[70],[31,140137,1977],{"className":140138},[75],"，冲突。生成引理 ",[31,140141,140143,140171],{"className":140142,"translate":35},[34],[31,140144,140146],{"className":140145},[39],[41,140147,140148],{"xmlns":43},[45,140149,140150,140168],{},[48,140151,140152,140154,140156,140158,140160,140162,140164,140166],{},[51,140153,126125],{"mathvariant":494},[51,140155,11],{},[95,140157,126150],{},[51,140159,126125],{"mathvariant":494},[51,140161,27666],{},[95,140163,126150],{},[51,140165,126125],{"mathvariant":494},[51,140167,2549],{},[55,140169,140170],{"encoding":57},"\\neg p \\vee \\neg q \\vee \\neg r",[31,140172,140174,140195,140216],{"className":140173,"ariaHidden":62},[61],[31,140175,140177,140180,140183,140186,140189,140192],{"className":140176},[66],[31,140178],{"className":140179,"style":126295},[70],[31,140181,126125],{"className":140182},[75],[31,140184,11],{"className":140185},[75,76],[31,140187],{"className":140188,"style":120},[119],[31,140190,126150],{"className":140191},[124],[31,140193],{"className":140194,"style":120},[119],[31,140196,140198,140201,140204,140207,140210,140213],{"className":140197},[66],[31,140199],{"className":140200,"style":126295},[70],[31,140202,126125],{"className":140203},[75],[31,140205,27666],{"className":140206,"style":2666},[75,76],[31,140208],{"className":140209,"style":120},[119],[31,140211,126150],{"className":140212},[124],[31,140214],{"className":140215,"style":120},[119],[31,140217,140219,140222,140225],{"className":140218},[66],[31,140220],{"className":140221,"style":71},[70],[31,140223,126125],{"className":140224},[75],[31,140226,2549],{"className":140227,"style":19759},[75,76]," 并回跳。",[11,140230,140231,140234,140235,140356],{},[7246,140232,140233],{},"第 2 轮","：SAT 层在引理约束下重新搜索，可能选择 ",[31,140236,140238,140272],{"className":140237,"translate":35},[34],[31,140239,140241],{"className":140240},[39],[41,140242,140243],{"xmlns":43},[45,140244,140245,140269],{},[48,140246,140247,140249,140251,140253,140255,140257,140259,140261,140263,140265,140267],{},[51,140248,11],{},[95,140250,223],{},[51,140252,20800],{"mathvariant":494},[95,140254,1210],{"separator":62},[51,140256,27666],{},[95,140258,223],{},[51,140260,126108],{"mathvariant":494},[95,140262,1210],{"separator":62},[51,140264,2549],{},[95,140266,223],{},[51,140268,20800],{"mathvariant":494},[55,140270,140271],{"encoding":57},"p = \\top, q = \\bot, r = \\top",[31,140273,140275,140293,140320,140347],{"className":140274,"ariaHidden":62},[61],[31,140276,140278,140281,140284,140287,140290],{"className":140277},[66],[31,140279],{"className":140280,"style":865},[70],[31,140282,11],{"className":140283},[75,76],[31,140285],{"className":140286,"style":293},[119],[31,140288,223],{"className":140289},[297],[31,140291],{"className":140292,"style":293},[119],[31,140294,140296,140299,140302,140305,140308,140311,140314,140317],{"className":140295},[66],[31,140297],{"className":140298,"style":4202},[70],[31,140300,20800],{"className":140301},[75],[31,140303,1210],{"className":140304},[1391],[31,140306],{"className":140307,"style":1467},[119],[31,140309,27666],{"className":140310,"style":2666},[75,76],[31,140312],{"className":140313,"style":293},[119],[31,140315,223],{"className":140316},[297],[31,140318],{"className":140319,"style":293},[119],[31,140321,140323,140326,140329,140332,140335,140338,140341,140344],{"className":140322},[66],[31,140324],{"className":140325,"style":4202},[70],[31,140327,126108],{"className":140328},[75],[31,140330,1210],{"className":140331},[1391],[31,140333],{"className":140334,"style":1467},[119],[31,140336,2549],{"className":140337,"style":19759},[75,76],[31,140339],{"className":140340,"style":293},[119],[31,140342,223],{"className":140343},[297],[31,140345],{"className":140346,"style":293},[119],[31,140348,140350,140353],{"className":140349},[66],[31,140351],{"className":140352,"style":1630},[70],[31,140354,20800],{"className":140355},[75],"。理论层检查：",[31,140358,140360],{"className":140359,"translate":35},[411],[31,140361,140363,140405],{"className":140362,"translate":35},[34],[31,140364,140366],{"className":140365},[39],[41,140367,140368],{"xmlns":43,"display":420},[45,140369,140370,140402],{},[48,140371,140372,140374,140376,140378,140380,140382,140384,140386,140388,140390,140392,140394,140396,140398,140400],{},[51,140373,53],{},[95,140375,16040],{},[91,140377,1977],{},[95,140379,1210],{"separator":62},[119,140381],{"width":20286},[51,140383,3003],{},[95,140385,347],{},[91,140387,1977],{},[95,140389,1210],{"separator":62},[119,140391],{"width":20286},[51,140393,53],{},[95,140395,434],{},[51,140397,3003],{},[95,140399,9758],{},[91,140401,1977],{},[55,140403,140404],{"encoding":57},"x > 0, \\qquad y \\le 0, \\qquad x + y \u003C 0",[31,140406,140408,140426,140456,140486,140504],{"className":140407,"ariaHidden":62},[61],[31,140409,140411,140414,140417,140420,140423],{"className":140410},[66],[31,140412],{"className":140413,"style":21113},[70],[31,140415,53],{"className":140416},[75,76],[31,140418],{"className":140419,"style":293},[119],[31,140421,16040],{"className":140422},[297],[31,140424],{"className":140425,"style":293},[119],[31,140427,140429,140432,140435,140438,140441,140444,140447,140450,140453],{"className":140428},[66],[31,140430],{"className":140431,"style":41982},[70],[31,140433,1977],{"className":140434},[75],[31,140436,1210],{"className":140437},[1391],[31,140439],{"className":140440,"style":20525},[119],[31,140442],{"className":140443,"style":1467},[119],[31,140445,3003],{"className":140446,"style":2666},[75,76],[31,140448],{"className":140449,"style":293},[119],[31,140451,347],{"className":140452},[297],[31,140454],{"className":140455,"style":293},[119],[31,140457,140459,140462,140465,140468,140471,140474,140477,140480,140483],{"className":140458},[66],[31,140460],{"className":140461,"style":41982},[70],[31,140463,1977],{"className":140464},[75],[31,140466,1210],{"className":140467},[1391],[31,140469],{"className":140470,"style":20525},[119],[31,140472],{"className":140473,"style":1467},[119],[31,140475,53],{"className":140476},[75,76],[31,140478],{"className":140479,"style":120},[119],[31,140481,434],{"className":140482},[124],[31,140484],{"className":140485,"style":120},[119],[31,140487,140489,140492,140495,140498,140501],{"className":140488},[66],[31,140490],{"className":140491,"style":53323},[70],[31,140493,3003],{"className":140494,"style":2666},[75,76],[31,140496],{"className":140497,"style":293},[119],[31,140499,9758],{"className":140500},[297],[31,140502],{"className":140503,"style":293},[119],[31,140505,140507,140510],{"className":140506},[66],[31,140508],{"className":140509,"style":326},[70],[31,140511,1977],{"className":140512},[75],[11,140514,3501,140515,140606],{},[31,140516,140518,140546],{"className":140517,"translate":35},[34],[31,140519,140521],{"className":140520},[39],[41,140522,140523],{"xmlns":43},[45,140524,140525,140543],{},[48,140526,140527,140529,140531,140533,140535,140537,140539,140541],{},[51,140528,53],{},[95,140530,223],{},[91,140532,205],{},[95,140534,1210],{"separator":62},[51,140536,3003],{},[95,140538,223],{},[95,140540,97],{},[91,140542,152],{},[55,140544,140545],{"encoding":57},"x = 1, y = -2",[31,140547,140549,140567,140594],{"className":140548,"ariaHidden":62},[61],[31,140550,140552,140555,140558,140561,140564],{"className":140551},[66],[31,140553],{"className":140554,"style":71},[70],[31,140556,53],{"className":140557},[75,76],[31,140559],{"className":140560,"style":293},[119],[31,140562,223],{"className":140563},[297],[31,140565],{"className":140566,"style":293},[119],[31,140568,140570,140573,140576,140579,140582,140585,140588,140591],{"className":140569},[66],[31,140571],{"className":140572,"style":41982},[70],[31,140574,205],{"className":140575},[75],[31,140577,1210],{"className":140578},[1391],[31,140580],{"className":140581,"style":1467},[119],[31,140583,3003],{"className":140584,"style":2666},[75,76],[31,140586],{"className":140587,"style":293},[119],[31,140589,223],{"className":140590},[297],[31,140592],{"className":140593,"style":293},[119],[31,140595,140597,140600,140603],{"className":140596},[66],[31,140598],{"className":140599,"style":112},[70],[31,140601,97],{"className":140602},[75],[31,140604,152],{"className":140605},[75],"，三条同时成立。理论层返回一致，求解器输出：",[31,140608,140610],{"className":140609,"translate":35},[411],[31,140611,140613,140649],{"className":140612,"translate":35},[34],[31,140614,140616],{"className":140615},[39],[41,140617,140618],{"xmlns":43,"display":420},[45,140619,140620,140646],{},[48,140621,140622,140624,140626,140628,140630,140632,140634,140636,140638,140640,140642,140644],{},[1430,140623,125655],{"mathvariant":76763},[95,140625,1210],{"separator":62},[119,140627],{"width":20286},[51,140629,53],{},[95,140631,223],{},[91,140633,205],{},[95,140635,1210],{"separator":62},[1430,140637,1545],{},[51,140639,3003],{},[95,140641,223],{},[95,140643,97],{},[91,140645,152],{},[55,140647,140648],{"encoding":57},"\\textbf{sat}, \\qquad x = 1,\\ y = -2",[31,140650,140652,140687,140717],{"className":140651,"ariaHidden":62},[61],[31,140653,140655,140659,140666,140669,140672,140675,140678,140681,140684],{"className":140654},[66],[31,140656],{"className":140657,"style":140658},[70],"height:0.8294em;vertical-align:-0.1944em;",[31,140660,140662],{"className":140661},[75,1793],[31,140663,125655],{"className":140664},[75,140665],"textbf",[31,140667,1210],{"className":140668},[1391],[31,140670],{"className":140671,"style":20525},[119],[31,140673],{"className":140674,"style":1467},[119],[31,140676,53],{"className":140677},[75,76],[31,140679],{"className":140680,"style":293},[119],[31,140682,223],{"className":140683},[297],[31,140685],{"className":140686,"style":293},[119],[31,140688,140690,140693,140696,140699,140702,140705,140708,140711,140714],{"className":140689},[66],[31,140691],{"className":140692,"style":41982},[70],[31,140694,205],{"className":140695},[75],[31,140697,1210],{"className":140698},[1391],[31,140700,1545],{"className":140701},[119],[31,140703],{"className":140704,"style":1467},[119],[31,140706,3003],{"className":140707,"style":2666},[75,76],[31,140709],{"className":140710,"style":293},[119],[31,140712,223],{"className":140713},[297],[31,140715],{"className":140716,"style":293},[119],[31,140718,140720,140723,140726],{"className":140719},[66],[31,140721],{"className":140722,"style":112},[70],[31,140724,97],{"className":140725},[75],[31,140727,152],{"className":140728},[75],[11,140730,140731],{},"这就是一个完整的 DPLL(T) 循环：一次冲突、一次学习、一次成功。",[11,140733,140734,140735,140738],{},"在真实求解器里，理论层还会做",[7246,140736,140737],{},"理论传播","：不等 SAT 层给出完整赋值，就提前推出某些文字必须为真或为假，从而更早剪枝。这让循环通常只需要很少几轮。",[11,140740,140741],{},"理论正确不代表跑得快。工业级求解器还依赖大量工程优化：",[127630,140743,140744,140750,140755,140761,140767,140773],{},[127633,140745,140746,140749],{},[7246,140747,140748],{},"预处理","：化简公式、消除冗余变量、把公式规范化成求解器偏好的形式。",[127633,140751,140752,140754],{},[7246,140753,140737],{},"：理论层主动推导文字，而不是被动等待完整赋值。",[127633,140756,140757,140760],{},[7246,140758,140759],{},"重启策略","：按 Luby 序列或指数策略重启，避免陷入坏的搜索区域。",[127633,140762,140763,140766],{},[7246,140764,140765],{},"子句删除","：学习子句太多会拖慢传播，需要定期清理。",[127633,140768,140769,140772],{},[7246,140770,140771],{},"多引擎组合","：同一个问题用不同策略并行求解（portfolio），谁先出结果用谁。",[127633,140774,140775,140778],{},[7246,140776,140777],{},"专用推理","：针对乘法、数组、字符串等难点，加入专门的推理规则。",[52860,140780,140782],{"id":140781},"_47-复杂度边界","4.7 复杂度边界",[11,140784,140785,140786,2372],{},"必须强调：",[7246,140787,140788],{},"可判定不等于可解",[127630,140790,140791,140794,140803,140808,140812],{},[127633,140792,140793],{},"SAT 是 NP-complete；",[127633,140795,140796,15282,140798,15282,140800,140802],{},[45608,140797,128185],{},[45608,140799,128101],{},[45608,140801,128087],{}," 也都是 NP-complete；",[127633,140804,140805,140807],{},[45608,140806,128097],{}," 虽然是多项式时间，但实际常数很大；",[127633,140809,140810,132569],{},[45608,140811,128173],{},[127633,140813,140814,140816],{},[45608,140815,128176],{}," 直接不可判定。",[11,140818,140819],{},"这意味着，工业问题的最坏情况仍然是指数级的。求解器的成功，很大程度上依赖问题结构、编码方式和启发式。遇到超时，工具通常只能返回「未知」，而不是「安全」。",[11,140821,140822],{},"这也是为什么实际系统里，SMT 很少单独使用，而是和抽象解释、fuzzing、测试生成等方法组合。",[18,140824,140826],{"id":140825},"五漏洞检测","五、漏洞检测",[52860,140828,140830],{"id":140829},"_51-问题转化","5.1 问题转化",[11,140832,140833],{},"软件漏洞检测的核心问题可以写成一句话：",[140835,140836,140837],"blockquote",{},[11,140838,140839],{},"是否存在一个输入，让程序走到某个危险位置，并且违反某条安全性质？",[11,140841,140842],{},"形式化之后，它就是一个存在性查询：",[31,140844,140846],{"className":140845,"translate":35},[411],[31,140847,140849,140963],{"className":140848,"translate":35},[34],[31,140850,140852],{"className":140851},[39],[41,140853,140854],{"xmlns":43,"display":420},[45,140855,140856,140960],{},[48,140857,140858,140860,140862,140874,140876,140878,140911,140913,140915,140917],{},[51,140859,125729],{"mathvariant":494},[1430,140861,1432],{},[48,140863,140864,140866,140868,140870,140872],{},[51,140865,5391],{},[51,140867,14559],{},[51,140869,11],{},[51,140871,5324],{},[51,140873,12271],{},[51,140875,495],{"mathvariant":494},[119,140877],{"width":1584},[1531,140879,140880,140908],{},[1531,140881,140882,140906],{},[48,140883,140884,140890,140892,140904],{},[48,140885,140886,140888],{},[51,140887,2521],{"mathvariant":494},[51,140889,8101],{"mathvariant":494},[95,140891,211],{"stretchy":210},[48,140893,140894,140896,140898,140900,140902],{},[51,140895,5391],{},[51,140897,14559],{},[51,140899,11],{},[51,140901,5324],{},[51,140903,12271],{},[95,140905,220],{"stretchy":210},[95,140907,38662],{"stretchy":62},[1430,140909,140910],{},"路径条件",[1430,140912,125504],{},[95,140914,125507],{},[1430,140916,125504],{},[1531,140918,140919,140957],{},[1531,140920,140921,140955],{},[48,140922,140923,140939,140941,140953],{},[48,140924,140925,140927,140929,140931,140933,140935,140937],{},[51,140926,43481],{"mathvariant":494},[51,140928,5391],{"mathvariant":494},[51,140930,2552],{"mathvariant":494},[51,140932,87599],{"mathvariant":494},[51,140934,5813],{"mathvariant":494},[51,140936,12271],{"mathvariant":494},[51,140938,30042],{"mathvariant":494},[95,140940,211],{"stretchy":210},[48,140942,140943,140945,140947,140949,140951],{},[51,140944,5391],{},[51,140946,14559],{},[51,140948,11],{},[51,140950,5324],{},[51,140952,12271],{},[95,140954,220],{"stretchy":210},[95,140956,38662],{"stretchy":62},[1430,140958,140959],{},"安全性质取反",[55,140961,140962],{"encoding":57},"\\exists\\, \\mathit{input}.\\quad \\underbrace{\\mathrm{PC}(\\mathit{input})}_{\\text{路径条件}} \\;\\wedge\\; \\underbrace{\\mathrm{Violate}(\\mathit{input})}_{\\text{安全性质取反}}",[31,140964,140966,141136],{"className":140965,"ariaHidden":62},[61],[31,140967,140969,140973,140976,140979,140987,140990,140993,140996,141121,141124,141127,141130,141133],{"className":140968},[66],[31,140970],{"className":140971,"style":140972},[70],"height:2.3263em;vertical-align:-1.5763em;",[31,140974,125729],{"className":140975},[75],[31,140977],{"className":140978,"style":1467},[119],[31,140980,140982],{"className":140981},[75],[31,140983,140986],{"className":140984},[75,140985],"mathit","input",[31,140988,495],{"className":140989},[75],[31,140991],{"className":140992,"style":1789},[119],[31,140994],{"className":140995,"style":1467},[119],[31,140997,140999],{"className":140998},[3860,1531],[31,141000,141002,141113],{"className":141001},[1043,1044],[31,141003,141005,141110],{"className":141004},[1048],[31,141006,141008,141025],{"className":141007,"style":127094},[1052],[31,141009,141010,141013],{"style":39515},[31,141011],{"className":141012,"style":1637},[1060],[31,141014,141016],{"className":141015},[1065,1066,1067,1068],[31,141017,141019],{"className":141018},[75,1068],[31,141020,141022],{"className":141021},[75,1793,1068],[31,141023,140910],{"className":141024},[75,12601,1068],[31,141026,141027,141030],{"style":1649},[31,141028],{"className":141029,"style":1637},[1060],[31,141031,141033],{"className":141032},[3860,1531],[31,141034,141036,141102],{"className":141035},[1043,1044],[31,141037,141039,141099],{"className":141038},[1048],[31,141040,141042,141072],{"className":141041,"style":127094},[1052],[31,141043,141045,141048],{"className":141044,"style":39551},[26550],[31,141046],{"className":141047,"style":1637},[1060],[31,141049,141051,141058,141065],{"className":141050,"style":39559},[39558],[31,141052,141054],{"className":141053,"style":39564},[39563],[13341,141055,141056],{"xmlns":13343,"width":26596,"height":39567,"viewBox":39568,"preserveAspectRatio":26599},[13347,141057],{"d":39571},[31,141059,141061],{"className":141060,"style":39564},[39575],[13341,141062,141063],{"xmlns":13343,"width":26596,"height":39567,"viewBox":39568,"preserveAspectRatio":39578},[13347,141064],{"d":39581},[31,141066,141068],{"className":141067,"style":39564},[39585],[13341,141069,141070],{"xmlns":13343,"width":26596,"height":39567,"viewBox":39568,"preserveAspectRatio":39588},[13347,141071],{"d":39591},[31,141073,141074,141077],{"style":1649},[31,141075],{"className":141076,"style":1637},[1060],[31,141078,141080,141087,141090,141096],{"className":141079},[75],[31,141081,141083],{"className":141082},[75],[31,141084,141086],{"className":141085},[75,2745],"PC",[31,141088,211],{"className":141089},[264],[31,141091,141093],{"className":141092},[75],[31,141094,140986],{"className":141095},[75,140985],[31,141097,220],{"className":141098},[289],[31,141100,1076],{"className":141101},[1075],[31,141103,141105],{"className":141104},[1048],[31,141106,141108],{"className":141107,"style":39973},[1052],[31,141109],{},[31,141111,1076],{"className":141112},[1075],[31,141114,141116],{"className":141115},[1048],[31,141117,141119],{"className":141118,"style":39985},[1052],[31,141120],{},[31,141122],{"className":141123,"style":293},[119],[31,141125],{"className":141126,"style":120},[119],[31,141128,125507],{"className":141129},[124],[31,141131],{"className":141132,"style":293},[119],[31,141134],{"className":141135,"style":120},[119],[31,141137,141139,141142],{"className":141138},[66],[31,141140],{"className":141141,"style":140972},[70],[31,141143,141145],{"className":141144},[3860,1531],[31,141146,141148,141259],{"className":141147},[1043,1044],[31,141149,141151,141256],{"className":141150},[1048],[31,141152,141154,141171],{"className":141153,"style":127094},[1052],[31,141155,141156,141159],{"style":39515},[31,141157],{"className":141158,"style":1637},[1060],[31,141160,141162],{"className":141161},[1065,1066,1067,1068],[31,141163,141165],{"className":141164},[75,1068],[31,141166,141168],{"className":141167},[75,1793,1068],[31,141169,140959],{"className":141170},[75,12601,1068],[31,141172,141173,141176],{"style":1649},[31,141174],{"className":141175,"style":1637},[1060],[31,141177,141179],{"className":141178},[3860,1531],[31,141180,141182,141248],{"className":141181},[1043,1044],[31,141183,141185,141245],{"className":141184},[1048],[31,141186,141188,141218],{"className":141187,"style":127094},[1052],[31,141189,141191,141194],{"className":141190,"style":39551},[26550],[31,141192],{"className":141193,"style":1637},[1060],[31,141195,141197,141204,141211],{"className":141196,"style":39559},[39558],[31,141198,141200],{"className":141199,"style":39564},[39563],[13341,141201,141202],{"xmlns":13343,"width":26596,"height":39567,"viewBox":39568,"preserveAspectRatio":26599},[13347,141203],{"d":39571},[31,141205,141207],{"className":141206,"style":39564},[39575],[13341,141208,141209],{"xmlns":13343,"width":26596,"height":39567,"viewBox":39568,"preserveAspectRatio":39578},[13347,141210],{"d":39581},[31,141212,141214],{"className":141213,"style":39564},[39585],[13341,141215,141216],{"xmlns":13343,"width":26596,"height":39567,"viewBox":39568,"preserveAspectRatio":39588},[13347,141217],{"d":39591},[31,141219,141220,141223],{"style":1649},[31,141221],{"className":141222,"style":1637},[1060],[31,141224,141226,141233,141236,141242],{"className":141225},[75],[31,141227,141229],{"className":141228},[75],[31,141230,141232],{"className":141231},[75,2745],"Violate",[31,141234,211],{"className":141235},[264],[31,141237,141239],{"className":141238},[75],[31,141240,140986],{"className":141241},[75,140985],[31,141243,220],{"className":141244},[289],[31,141246,1076],{"className":141247},[1075],[31,141249,141251],{"className":141250},[1048],[31,141252,141254],{"className":141253,"style":39973},[1052],[31,141255],{},[31,141257,1076],{"className":141258},[1075],[31,141260,141262],{"className":141261},[1048],[31,141263,141265],{"className":141264,"style":39985},[1052],[31,141266],{},[127630,141268,141269,141306],{},[127633,141270,141271,141305],{},[31,141272,141274,141290],{"className":141273,"translate":35},[34],[31,141275,141277],{"className":141276},[39],[41,141278,141279],{"xmlns":43},[45,141280,141281,141287],{},[48,141282,141283,141285],{},[51,141284,2521],{"mathvariant":494},[51,141286,8101],{"mathvariant":494},[55,141288,141289],{"encoding":57},"\\mathrm{PC}",[31,141291,141293],{"className":141292,"ariaHidden":62},[61],[31,141294,141296,141299],{"className":141295},[66],[31,141297],{"className":141298,"style":945},[70],[31,141300,141302],{"className":141301},[75],[31,141303,141086],{"className":141304},[75,2745]," 是路径条件，描述「走到这里需要满足哪些约束」；",[127633,141307,141308,141352],{},[31,141309,141311,141337],{"className":141310,"translate":35},[34],[31,141312,141314],{"className":141313},[39],[41,141315,141316],{"xmlns":43},[45,141317,141318,141334],{},[48,141319,141320,141322,141324,141326,141328,141330,141332],{},[51,141321,43481],{"mathvariant":494},[51,141323,5391],{"mathvariant":494},[51,141325,2552],{"mathvariant":494},[51,141327,87599],{"mathvariant":494},[51,141329,5813],{"mathvariant":494},[51,141331,12271],{"mathvariant":494},[51,141333,30042],{"mathvariant":494},[55,141335,141336],{"encoding":57},"\\mathrm{Violate}",[31,141338,141340],{"className":141339,"ariaHidden":62},[61],[31,141341,141343,141346],{"className":141342},[66],[31,141344],{"className":141345,"style":1630},[70],[31,141347,141349],{"className":141348},[75],[31,141350,141232],{"className":141351},[75,2745]," 是安全性质的否定，描述「怎样才算越界\u002F溢出\u002F空指针解引用」。",[11,141354,141355,141356,141358,141359,141361],{},"如果这个公式 ",[7246,141357,125655],{},"，求解器给出的模型就是一条能触发漏洞的输入，漏洞可达；如果 ",[7246,141360,131566],{},"，说明在这条路径上不可能违反性质，可以安全剪枝。",[11,141363,141364,141365,2372],{},"这个转化是整个应用的基石：",[7246,141366,141367],{},"「找漏洞」变成了「解约束」",[52860,141369,141371],{"id":141370},"_52-缓冲区溢出","5.2 缓冲区溢出",[11,141373,141374],{},"先看一个最直观的例子：",[125323,141376,141378],{"className":125325,"code":141377,"language":1412,"meta":52812,"style":52812},"void process(char *input, int len) {\n    char buf[8];\n    if (len > 4 && len \u003C 12) {\n        memcpy(buf, input, len);   \u002F\u002F 可疑点\n    }\n}\n",[45608,141379,141380,141406,141421,141445,141456,141460],{"__ignoreMap":52812},[31,141381,141382,141384,141387,141389,141392,141395,141397,141399,141401,141404],{"class":125331,"line":125332},[31,141383,125336],{"class":125335},[31,141385,141386],{"class":125339}," process",[31,141388,211],{"class":125343},[31,141390,141391],{"class":125335},"char",[31,141393,141394],{"class":125335}," *",[31,141396,140986],{"class":125349},[31,141398,125353],{"class":125343},[31,141400,125346],{"class":125335},[31,141402,141403],{"class":125349}," len",[31,141405,125361],{"class":125343},[31,141407,141408,141411,141414,141416,141418],{"class":125331,"line":52813},[31,141409,141410],{"class":125335},"    char",[31,141412,141413],{"class":125349}," buf",[31,141415,15175],{"class":125343},[31,141417,19641],{"class":125374},[31,141419,141420],{"class":125343},"];\n",[31,141422,141423,141425,141428,141430,141433,141435,141438,141440,141443],{"class":125331,"line":125262},[31,141424,125412],{"class":125335},[31,141426,141427],{"class":125343}," (len ",[31,141429,16040],{"class":125335},[31,141431,141432],{"class":125374}," 4",[31,141434,125428],{"class":125335},[31,141436,141437],{"class":125343}," len ",[31,141439,9758],{"class":125335},[31,141441,141442],{"class":125374}," 12",[31,141444,125361],{"class":125343},[31,141446,141447,141450,141453],{"class":125331,"line":125409},[31,141448,141449],{"class":125339},"        memcpy",[31,141451,141452],{"class":125343},"(buf, input, len);",[31,141454,141455],{"class":125449},"   \u002F\u002F 可疑点\n",[31,141457,141458],{"class":125331,"line":125446},[31,141459,125456],{"class":125343},[31,141461,141462],{"class":125331,"line":125453},[31,141463,125462],{"class":125343},[11,141465,141466,141469,141470,141473,141474,141477,141478,141480],{},[45608,141467,141468],{},"buf"," 只有 8 字节，而 ",[45608,141471,141472],{},"memcpy"," 拷贝 ",[45608,141475,141476],{},"len"," 字节。如果 ",[45608,141479,141476],{}," 能大于 8，就会发生栈溢出。",[11,141482,141483,141484,2446,141486,141488,141489,141491],{},"符号执行引擎把 ",[45608,141485,140986],{},[45608,141487,141476],{}," 当作符号变量，沿路径走到 ",[45608,141490,141472],{}," 时，收集到的路径条件是：",[31,141493,141495],{"className":141494,"translate":35},[411],[31,141496,141498,141537],{"className":141497,"translate":35},[34],[31,141499,141501],{"className":141500},[39],[41,141502,141503],{"xmlns":43,"display":420},[45,141504,141505,141534],{},[48,141506,141507,141509,141511,141513,141515,141517,141519,141521,141523,141525,141527,141529,141531],{},[51,141508,87599],{},[51,141510,30042],{},[51,141512,14559],{},[95,141514,16040],{},[91,141516,15416],{},[1430,141518,125504],{},[95,141520,125507],{},[1430,141522,125504],{},[51,141524,87599],{},[51,141526,30042],{},[51,141528,14559],{},[95,141530,9758],{},[91,141532,141533],{},"12",[55,141535,141536],{"encoding":57},"len > 4 \\;\\wedge\\; len \u003C 12",[31,141538,141540,141566,141590,141614],{"className":141539,"ariaHidden":62},[61],[31,141541,141543,141547,141551,141554,141557,141560,141563],{"className":141542},[66],[31,141544],{"className":141545,"style":141546},[70],"height:0.7335em;vertical-align:-0.0391em;",[31,141548,87599],{"className":141549,"style":141550},[75,76],"margin-right:0.0197em;",[31,141552,30042],{"className":141553},[75,76],[31,141555,14559],{"className":141556},[75,76],[31,141558],{"className":141559,"style":293},[119],[31,141561,16040],{"className":141562},[297],[31,141564],{"className":141565,"style":293},[119],[31,141567,141569,141572,141575,141578,141581,141584,141587],{"className":141568},[66],[31,141570],{"className":141571,"style":326},[70],[31,141573,15416],{"className":141574},[75],[31,141576],{"className":141577,"style":293},[119],[31,141579],{"className":141580,"style":120},[119],[31,141582,125507],{"className":141583},[124],[31,141585],{"className":141586,"style":293},[119],[31,141588],{"className":141589,"style":120},[119],[31,141591,141593,141596,141599,141602,141605,141608,141611],{"className":141592},[66],[31,141594],{"className":141595,"style":141546},[70],[31,141597,87599],{"className":141598,"style":141550},[75,76],[31,141600,30042],{"className":141601},[75,76],[31,141603,14559],{"className":141604},[75,76],[31,141606],{"className":141607,"style":293},[119],[31,141609,9758],{"className":141610},[297],[31,141612],{"className":141613,"style":293},[119],[31,141615,141617,141620],{"className":141616},[66],[31,141618],{"className":141619,"style":326},[70],[31,141621,141533],{"className":141622},[75],[11,141624,141625],{},"要判断溢出是否可达，就在路径条件上再加一条「写入长度超过缓冲区大小」：",[31,141627,141629],{"className":141628,"translate":35},[411],[31,141630,141632,141686],{"className":141631,"translate":35},[34],[31,141633,141635],{"className":141634},[39],[41,141636,141637],{"xmlns":43,"display":420},[45,141638,141639,141683],{},[48,141640,141641,141643,141645,141647,141649,141651,141653,141655,141657,141659,141661,141663,141665,141667,141669,141671,141673,141675,141677,141679,141681],{},[51,141642,87599],{},[51,141644,30042],{},[51,141646,14559],{},[95,141648,16040],{},[91,141650,15416],{},[1430,141652,125504],{},[95,141654,125507],{},[1430,141656,125504],{},[51,141658,87599],{},[51,141660,30042],{},[51,141662,14559],{},[95,141664,9758],{},[91,141666,141533],{},[1430,141668,125504],{},[95,141670,125507],{},[1430,141672,125504],{},[51,141674,87599],{},[51,141676,30042],{},[51,141678,14559],{},[95,141680,16040],{},[91,141682,19641],{},[55,141684,141685],{"encoding":57},"len > 4 \\;\\wedge\\; len \u003C 12 \\;\\wedge\\; len > 8",[31,141687,141689,141713,141737,141761,141785,141809],{"className":141688,"ariaHidden":62},[61],[31,141690,141692,141695,141698,141701,141704,141707,141710],{"className":141691},[66],[31,141693],{"className":141694,"style":141546},[70],[31,141696,87599],{"className":141697,"style":141550},[75,76],[31,141699,30042],{"className":141700},[75,76],[31,141702,14559],{"className":141703},[75,76],[31,141705],{"className":141706,"style":293},[119],[31,141708,16040],{"className":141709},[297],[31,141711],{"className":141712,"style":293},[119],[31,141714,141716,141719,141722,141725,141728,141731,141734],{"className":141715},[66],[31,141717],{"className":141718,"style":326},[70],[31,141720,15416],{"className":141721},[75],[31,141723],{"className":141724,"style":293},[119],[31,141726],{"className":141727,"style":120},[119],[31,141729,125507],{"className":141730},[124],[31,141732],{"className":141733,"style":293},[119],[31,141735],{"className":141736,"style":120},[119],[31,141738,141740,141743,141746,141749,141752,141755,141758],{"className":141739},[66],[31,141741],{"className":141742,"style":141546},[70],[31,141744,87599],{"className":141745,"style":141550},[75,76],[31,141747,30042],{"className":141748},[75,76],[31,141750,14559],{"className":141751},[75,76],[31,141753],{"className":141754,"style":293},[119],[31,141756,9758],{"className":141757},[297],[31,141759],{"className":141760,"style":293},[119],[31,141762,141764,141767,141770,141773,141776,141779,141782],{"className":141763},[66],[31,141765],{"className":141766,"style":326},[70],[31,141768,141533],{"className":141769},[75],[31,141771],{"className":141772,"style":293},[119],[31,141774],{"className":141775,"style":120},[119],[31,141777,125507],{"className":141778},[124],[31,141780],{"className":141781,"style":293},[119],[31,141783],{"className":141784,"style":120},[119],[31,141786,141788,141791,141794,141797,141800,141803,141806],{"className":141787},[66],[31,141789],{"className":141790,"style":141546},[70],[31,141792,87599],{"className":141793,"style":141550},[75,76],[31,141795,30042],{"className":141796},[75,76],[31,141798,14559],{"className":141799},[75,76],[31,141801],{"className":141802,"style":293},[119],[31,141804,16040],{"className":141805},[297],[31,141807],{"className":141808,"style":293},[119],[31,141810,141812,141815],{"className":141811},[66],[31,141813],{"className":141814,"style":326},[70],[31,141816,19641],{"className":141817},[75],[11,141819,141820],{},"写成 SMT-LIB：",[125323,141822,141826],{"className":141823,"code":141824,"language":141825,"meta":52812,"style":52812},"language-smt shiki shiki-themes github-light github-light github-dark","(set-logic QF_LIA)\n(declare-const len Int)\n(assert (and (> len 4) (\u003C len 12) (> len 8)))\n(check-sat)\n(get-model)\n","smt",[45608,141827,141828,141833,141838,141843,141848],{"__ignoreMap":52812},[31,141829,141830],{"class":125331,"line":125332},[31,141831,141832],{},"(set-logic QF_LIA)\n",[31,141834,141835],{"class":125331,"line":52813},[31,141836,141837],{},"(declare-const len Int)\n",[31,141839,141840],{"class":125331,"line":125262},[31,141841,141842],{},"(assert (and (> len 4) (\u003C len 12) (> len 8)))\n",[31,141844,141845],{"class":125331,"line":125409},[31,141846,141847],{},"(check-sat)\n",[31,141849,141850],{"class":125331,"line":125446},[31,141851,141852],{},"(get-model)\n",[11,141854,141855],{},"求解器返回：",[125323,141857,141860],{"className":141858,"code":141859,"language":1793},[136260],"sat\n(model\n  (define-fun len () Int\n    9)\n)\n",[45608,141861,141859],{"__ignoreMap":52812},[11,141863,141864,141867,141868,141941,141942,141993,141994,141996],{},[45608,141865,141866],{},"len = 9"," 是一个合法解：它满足 ",[31,141869,141871,141893],{"className":141870,"translate":35},[34],[31,141872,141874],{"className":141873},[39],[41,141875,141876],{"xmlns":43},[45,141877,141878,141890],{},[48,141879,141880,141882,141884,141886,141888],{},[91,141881,15416],{},[95,141883,9758],{},[91,141885,81476],{},[95,141887,9758],{},[91,141889,141533],{},[55,141891,141892],{"encoding":57},"4 \u003C 9 \u003C 12",[31,141894,141896,141914,141932],{"className":141895,"ariaHidden":62},[61],[31,141897,141899,141902,141905,141908,141911],{"className":141898},[66],[31,141900],{"className":141901,"style":134517},[70],[31,141903,15416],{"className":141904},[75],[31,141906],{"className":141907,"style":293},[119],[31,141909,9758],{"className":141910},[297],[31,141912],{"className":141913,"style":293},[119],[31,141915,141917,141920,141923,141926,141929],{"className":141916},[66],[31,141918],{"className":141919,"style":134517},[70],[31,141921,81476],{"className":141922},[75],[31,141924],{"className":141925,"style":293},[119],[31,141927,9758],{"className":141928},[297],[31,141930],{"className":141931,"style":293},[119],[31,141933,141935,141938],{"className":141934},[66],[31,141936],{"className":141937,"style":326},[70],[31,141939,141533],{"className":141940},[75],"，同时 ",[31,141943,141945,141963],{"className":141944,"translate":35},[34],[31,141946,141948],{"className":141947},[39],[41,141949,141950],{"xmlns":43},[45,141951,141952,141960],{},[48,141953,141954,141956,141958],{},[91,141955,81476],{},[95,141957,16040],{},[91,141959,19641],{},[55,141961,141962],{"encoding":57},"9 > 8",[31,141964,141966,141984],{"className":141965,"ariaHidden":62},[61],[31,141967,141969,141972,141975,141978,141981],{"className":141968},[66],[31,141970],{"className":141971,"style":134517},[70],[31,141973,81476],{"className":141974},[75],[31,141976],{"className":141977,"style":293},[119],[31,141979,16040],{"className":141980},[297],[31,141982],{"className":141983,"style":293},[119],[31,141985,141987,141990],{"className":141986},[66],[31,141988],{"className":141989,"style":326},[70],[31,141991,19641],{"className":141992},[75],"。这条路径可达，",[45608,141995,141472],{}," 会往 8 字节的缓冲区里写 9 字节，溢出 1 字节。是否真的覆盖到返回地址取决于栈布局，但「越界写」这个事实本身已经成立。",[11,141998,141999],{},"如果把代码改成：",[125323,142001,142003],{"className":125325,"code":142002,"language":1412,"meta":52812,"style":52812},"if (len > 4 && len \u003C= 8) {\n    memcpy(buf, input, len);\n}\n",[45608,142004,142005,142028,142036],{"__ignoreMap":52812},[31,142006,142007,142010,142012,142014,142016,142018,142020,142023,142026],{"class":125331,"line":125332},[31,142008,142009],{"class":125335},"if",[31,142011,141427],{"class":125343},[31,142013,16040],{"class":125335},[31,142015,141432],{"class":125374},[31,142017,125428],{"class":125335},[31,142019,141437],{"class":125343},[31,142021,142022],{"class":125335},"\u003C=",[31,142024,142025],{"class":125374}," 8",[31,142027,125361],{"class":125343},[31,142029,142030,142033],{"class":125331,"line":52813},[31,142031,142032],{"class":125339},"    memcpy",[31,142034,142035],{"class":125343},"(buf, input, len);\n",[31,142037,142038],{"class":125331,"line":125262},[31,142039,125462],{"class":125343},[11,142041,142042],{},"约束就变成：",[31,142044,142046],{"className":142045,"translate":35},[411],[31,142047,142049,142103],{"className":142048,"translate":35},[34],[31,142050,142052],{"className":142051},[39],[41,142053,142054],{"xmlns":43,"display":420},[45,142055,142056,142100],{},[48,142057,142058,142060,142062,142064,142066,142068,142070,142072,142074,142076,142078,142080,142082,142084,142086,142088,142090,142092,142094,142096,142098],{},[51,142059,87599],{},[51,142061,30042],{},[51,142063,14559],{},[95,142065,16040],{},[91,142067,15416],{},[1430,142069,125504],{},[95,142071,125507],{},[1430,142073,125504],{},[51,142075,87599],{},[51,142077,30042],{},[51,142079,14559],{},[95,142081,347],{},[91,142083,19641],{},[1430,142085,125504],{},[95,142087,125507],{},[1430,142089,125504],{},[51,142091,87599],{},[51,142093,30042],{},[51,142095,14559],{},[95,142097,16040],{},[91,142099,19641],{},[55,142101,142102],{"encoding":57},"len > 4 \\;\\wedge\\; len \\le 8 \\;\\wedge\\; len > 8",[31,142104,142106,142130,142154,142178,142202,142226],{"className":142105,"ariaHidden":62},[61],[31,142107,142109,142112,142115,142118,142121,142124,142127],{"className":142108},[66],[31,142110],{"className":142111,"style":141546},[70],[31,142113,87599],{"className":142114,"style":141550},[75,76],[31,142116,30042],{"className":142117},[75,76],[31,142119,14559],{"className":142120},[75,76],[31,142122],{"className":142123,"style":293},[119],[31,142125,16040],{"className":142126},[297],[31,142128],{"className":142129,"style":293},[119],[31,142131,142133,142136,142139,142142,142145,142148,142151],{"className":142132},[66],[31,142134],{"className":142135,"style":326},[70],[31,142137,15416],{"className":142138},[75],[31,142140],{"className":142141,"style":293},[119],[31,142143],{"className":142144,"style":120},[119],[31,142146,125507],{"className":142147},[124],[31,142149],{"className":142150,"style":293},[119],[31,142152],{"className":142153,"style":120},[119],[31,142155,142157,142160,142163,142166,142169,142172,142175],{"className":142156},[66],[31,142158],{"className":142159,"style":106323},[70],[31,142161,87599],{"className":142162,"style":141550},[75,76],[31,142164,30042],{"className":142165},[75,76],[31,142167,14559],{"className":142168},[75,76],[31,142170],{"className":142171,"style":293},[119],[31,142173,347],{"className":142174},[297],[31,142176],{"className":142177,"style":293},[119],[31,142179,142181,142184,142187,142190,142193,142196,142199],{"className":142180},[66],[31,142182],{"className":142183,"style":326},[70],[31,142185,19641],{"className":142186},[75],[31,142188],{"className":142189,"style":293},[119],[31,142191],{"className":142192,"style":120},[119],[31,142194,125507],{"className":142195},[124],[31,142197],{"className":142198,"style":293},[119],[31,142200],{"className":142201,"style":120},[119],[31,142203,142205,142208,142211,142214,142217,142220,142223],{"className":142204},[66],[31,142206],{"className":142207,"style":141546},[70],[31,142209,87599],{"className":142210,"style":141550},[75,76],[31,142212,30042],{"className":142213},[75,76],[31,142215,14559],{"className":142216},[75,76],[31,142218],{"className":142219,"style":293},[119],[31,142221,16040],{"className":142222},[297],[31,142224],{"className":142225,"style":293},[119],[31,142227,142229,142232],{"className":142228},[66],[31,142230],{"className":142231,"style":326},[70],[31,142233,19641],{"className":142234},[75],[11,142236,142237,142238,142240],{},"求解器返回 ",[7246,142239,131566],{},"。这条溢出路径不可达，工具不会报告漏洞。",[11,142242,142243,142244,142246,142247,142249,142250,142252,142253,4809],{},"注意这里用的是 ",[45608,142245,128101],{},"（整数线性算术）。真实的 C 代码里 ",[45608,142248,141476],{}," 是 32 位 ",[45608,142251,125346],{},"，更精确的建模应该用 ",[45608,142254,128185],{},[125323,142256,142258],{"className":141823,"code":142257,"language":141825,"meta":52812,"style":52812},"(set-logic QF_BV)\n(declare-const len (_ BitVec 32))\n(assert (and (bvsgt len (_ bv4 32))\n             (bvslt len (_ bv12 32))\n             (bvsgt len (_ bv8 32))))\n(check-sat)\n(get-model)\n",[45608,142259,142260,142265,142270,142275,142280,142285,142289],{"__ignoreMap":52812},[31,142261,142262],{"class":125331,"line":125332},[31,142263,142264],{},"(set-logic QF_BV)\n",[31,142266,142267],{"class":125331,"line":52813},[31,142268,142269],{},"(declare-const len (_ BitVec 32))\n",[31,142271,142272],{"class":125331,"line":125262},[31,142273,142274],{},"(assert (and (bvsgt len (_ bv4 32))\n",[31,142276,142277],{"class":125331,"line":125409},[31,142278,142279],{},"             (bvslt len (_ bv12 32))\n",[31,142281,142282],{"class":125331,"line":125446},[31,142283,142284],{},"             (bvsgt len (_ bv8 32))))\n",[31,142286,142287],{"class":125331,"line":125453},[31,142288,141847],{},[31,142290,142291],{"class":125331,"line":125459},[31,142292,141852],{},[11,142294,142295,142296,128098,142299,142302,142303,142305,142306,142309],{},"注意这里用的是有符号比较（",[45608,142297,142298],{},"bvsgt",[45608,142300,142301],{},"bvslt","），因为 C 里的 ",[45608,142304,125346],{}," 是有符号的。一个可能的模型是 ",[45608,142307,142308],{},"len = #x00000009","。对这个例子来说，两种建模的结论一样；但下一个例子就会看到，理论选错，结论可能完全相反。",[52860,142311,142313],{"id":142312},"_53-整数溢出","5.3 整数溢出",[125323,142315,142317],{"className":125325,"code":142316,"language":1412,"meta":52812,"style":52812},"void copy(char *src, unsigned int off, unsigned int len) {\n    char buf[64];\n    if (off + len \u003C= 64) {\n        memcpy(buf + off, src, len);\n    }\n}\n",[45608,142318,142319,142356,142369,142387,142399,142403],{"__ignoreMap":52812},[31,142320,142321,142323,142326,142328,142330,142332,142335,142337,142340,142343,142346,142348,142350,142352,142354],{"class":125331,"line":125332},[31,142322,125336],{"class":125335},[31,142324,142325],{"class":125339}," copy",[31,142327,211],{"class":125343},[31,142329,141391],{"class":125335},[31,142331,141394],{"class":125335},[31,142333,142334],{"class":125349},"src",[31,142336,125353],{"class":125343},[31,142338,142339],{"class":125335},"unsigned",[31,142341,142342],{"class":125335}," int",[31,142344,142345],{"class":125349}," off",[31,142347,125353],{"class":125343},[31,142349,142339],{"class":125335},[31,142351,142342],{"class":125335},[31,142353,141403],{"class":125349},[31,142355,125361],{"class":125343},[31,142357,142358,142360,142362,142364,142367],{"class":125331,"line":52813},[31,142359,141410],{"class":125335},[31,142361,141413],{"class":125349},[31,142363,15175],{"class":125343},[31,142365,142366],{"class":125374},"64",[31,142368,141420],{"class":125343},[31,142370,142371,142373,142376,142378,142380,142382,142385],{"class":125331,"line":125262},[31,142372,125412],{"class":125335},[31,142374,142375],{"class":125343}," (off ",[31,142377,434],{"class":125335},[31,142379,141437],{"class":125343},[31,142381,142022],{"class":125335},[31,142383,142384],{"class":125374}," 64",[31,142386,125361],{"class":125343},[31,142388,142389,142391,142394,142396],{"class":125331,"line":125409},[31,142390,141449],{"class":125339},[31,142392,142393],{"class":125343},"(buf ",[31,142395,434],{"class":125335},[31,142397,142398],{"class":125343}," off, src, len);\n",[31,142400,142401],{"class":125331,"line":125446},[31,142402,125456],{"class":125343},[31,142404,142405],{"class":125331,"line":125453},[31,142406,125462],{"class":125343},[11,142408,142409,142410,142413,142414,2446,142417,142419,142420,128246],{},"看起来检查很严：只有 ",[45608,142411,142412],{},"off + len"," 不超过 64 才拷贝。但 ",[45608,142415,142416],{},"off",[45608,142418,141476],{}," 都是 32 位无符号整数，加法是",[7246,142421,142422,142423],{},"模 ",[31,142424,142426,142444],{"className":142425,"translate":35},[34],[31,142427,142429],{"className":142428},[39],[41,142430,142431],{"xmlns":43},[45,142432,142433,142441],{},[48,142434,142435],{},[1180,142436,142437,142439],{},[91,142438,152],{},[91,142440,138068],{},[55,142442,142443],{"encoding":57},"2^{32}",[31,142445,142447],{"className":142446,"ariaHidden":62},[61],[31,142448,142450,142453],{"className":142449},[66],[31,142451],{"className":142452,"style":5135},[70],[31,142454,142456,142459],{"className":142455},[75],[31,142457,152],{"className":142458},[75],[31,142460,142462],{"className":142461},[1039],[31,142463,142465],{"className":142464},[1043],[31,142466,142468],{"className":142467},[1048],[31,142469,142471],{"className":142470,"style":5135},[1052],[31,142472,142473,142476],{"style":3631},[31,142474],{"className":142475,"style":1061},[1060],[31,142477,142479],{"className":142478},[1065,1066,1067,1068],[31,142480,142482],{"className":142481},[75,1068],[31,142483,138068],{"className":142484},[75,1068],[11,142486,142487],{},"攻击者可以取：",[31,142489,142491],{"className":142490,"translate":35},[411],[31,142492,142494,142524],{"className":142493,"translate":35},[34],[31,142495,142497],{"className":142496},[39],[41,142498,142499],{"xmlns":43,"display":420},[45,142500,142501,142521],{},[48,142502,142503,142505,142507,142510,142512,142514,142516,142518],{},[1430,142504,142416],{},[95,142506,223],{},[1430,142508,142509],{"mathvariant":138083},"0xFFFFFFF0",[95,142511,1210],{"separator":62},[119,142513],{"width":20286},[1430,142515,141476],{},[95,142517,223],{},[1430,142519,142520],{"mathvariant":138083},"0x20",[55,142522,142523],{"encoding":57},"\\text{off} = \\texttt{0xFFFFFFF0}, \\qquad \\text{len} = \\texttt{0x20}",[31,142525,142527,142548,142584],{"className":142526,"ariaHidden":62},[61],[31,142528,142530,142533,142539,142542,142545],{"className":142529},[66],[31,142531],{"className":142532,"style":1630},[70],[31,142534,142536],{"className":142535},[75,1793],[31,142537,142416],{"className":142538},[75],[31,142540],{"className":142541,"style":293},[119],[31,142543,223],{"className":142544},[297],[31,142546],{"className":142547,"style":293},[119],[31,142549,142551,142554,142560,142563,142566,142569,142575,142578,142581],{"className":142550},[66],[31,142552],{"className":142553,"style":4202},[70],[31,142555,142557],{"className":142556},[75,1793],[31,142558,142509],{"className":142559},[75,138280],[31,142561,1210],{"className":142562},[1391],[31,142564],{"className":142565,"style":20525},[119],[31,142567],{"className":142568,"style":1467},[119],[31,142570,142572],{"className":142571},[75,1793],[31,142573,141476],{"className":142574},[75],[31,142576],{"className":142577,"style":293},[119],[31,142579,223],{"className":142580},[297],[31,142582],{"className":142583,"style":293},[119],[31,142585,142587,142590],{"className":142586},[66],[31,142588],{"className":142589,"style":138273},[70],[31,142591,142593],{"className":142592},[75,1793],[31,142594,142520],{"className":142595},[75,138280],[11,142597,142598],{},"计算一下：",[31,142600,142602],{"className":142601,"translate":35},[411],[31,142603,142605,142718],{"className":142604,"translate":35},[34],[31,142606,142608],{"className":142607},[39],[41,142609,142610],{"xmlns":43,"display":420},[45,142611,142612,142715],{},[5606,142613,142614,142644,142665],{"rowspacing":5608,"columnalign":5609,"columnspacing":5610},[5612,142615,142616,142628],{},[5615,142617,142618],{},[2071,142619,142620],{"scriptlevel":1977,"displaystyle":62},[48,142621,142622,142624,142626],{},[1430,142623,142416],{},[95,142625,434],{},[1430,142627,141476],{},[5615,142629,142630],{},[2071,142631,142632],{"scriptlevel":1977,"displaystyle":62},[48,142633,142634,142636,142638,142640,142642],{},[48,142635],{},[95,142637,223],{},[1430,142639,142509],{"mathvariant":138083},[95,142641,434],{},[1430,142643,142520],{"mathvariant":138083},[5612,142645,142646,142652],{},[5615,142647,142648],{},[2071,142649,142650],{"scriptlevel":1977,"displaystyle":62},[48,142651],{},[5615,142653,142654],{},[2071,142655,142656],{"scriptlevel":1977,"displaystyle":62},[48,142657,142658,142660,142662],{},[48,142659],{},[95,142661,223],{},[1430,142663,142664],{"mathvariant":138083},"0x100000010",[5612,142666,142667,142673],{},[5615,142668,142669],{},[2071,142670,142671],{"scriptlevel":1977,"displaystyle":62},[48,142672],{},[5615,142674,142675],{},[2071,142676,142677],{"scriptlevel":1977,"displaystyle":62},[48,142678,142679,142681,142683,142686,142688,142691,142693,142695,142697,142705,142707,142713],{},[48,142680],{},[95,142682,135158],{},[1430,142684,142685],{"mathvariant":138083},"0x10",[95,142687,223],{},[91,142689,142690],{},"16",[119,142692],{},[119,142694],{"width":1584},[95,142696,211],{"stretchy":210},[48,142698,142699,142701,142703],{},[51,142700,16396],{"mathvariant":494},[51,142702,2552],{"mathvariant":494},[51,142704,1435],{"mathvariant":494},[119,142706],{"width":138061},[1180,142708,142709,142711],{},[91,142710,152],{},[91,142712,138068],{},[95,142714,220],{"stretchy":210},[55,142716,142717],{"encoding":57},"\\begin{aligned}\n\\text{off} + \\text{len}\n&= \\texttt{0xFFFFFFF0} + \\texttt{0x20} \\\\\n&= \\texttt{0x100000010} \\\\\n&\\equiv \\texttt{0x10} = 16 \\pmod{2^{32}}\n\\end{aligned}",[31,142719,142721],{"className":142720,"ariaHidden":62},[61],[31,142722,142724,142728],{"className":142723},[66],[31,142725],{"className":142726,"style":142727},[70],"height:4.2241em;vertical-align:-1.8621em;",[31,142729,142731],{"className":142730},[75],[31,142732,142734,142806],{"className":142733},[5606],[31,142735,142737],{"className":142736},[5867],[31,142738,142740,142797],{"className":142739},[1043,1044],[31,142741,142743,142794],{"className":142742},[1048],[31,142744,142747,142776,142785],{"className":142745,"style":142746},[1052],"height:2.3621em;",[31,142748,142749,142752],{"style":130265},[31,142750],{"className":142751,"style":1637},[1060],[31,142753,142755,142761,142764,142767,142770],{"className":142754},[75],[31,142756,142758],{"className":142757},[75,1793],[31,142759,142416],{"className":142760},[75],[31,142762],{"className":142763,"style":120},[119],[31,142765,434],{"className":142766},[124],[31,142768],{"className":142769,"style":120},[119],[31,142771,142773],{"className":142772},[75,1793],[31,142774,141476],{"className":142775},[75],[31,142777,142779,142782],{"style":142778},"top:-3.0221em;",[31,142780],{"className":142781,"style":1637},[1060],[31,142783],{"className":142784},[75],[31,142786,142788,142791],{"style":142787},"top:-1.4979em;",[31,142789],{"className":142790,"style":1637},[1060],[31,142792],{"className":142793},[75],[31,142795,1076],{"className":142796},[1075],[31,142798,142800],{"className":142799},[1048],[31,142801,142804],{"className":142802,"style":142803},[1052],"height:1.8621em;",[31,142805],{},[31,142807,142809],{"className":142808},[6062],[31,142810,142812,142982],{"className":142811},[1043,1044],[31,142813,142815,142979],{"className":142814},[1048],[31,142816,142818,142859,142885],{"className":142817,"style":142746},[1052],[31,142819,142820,142823],{"style":130265},[31,142821],{"className":142822,"style":1637},[1060],[31,142824,142826,142829,142832,142835,142838,142844,142847,142850,142853],{"className":142825},[75],[31,142827],{"className":142828},[75],[31,142830],{"className":142831,"style":293},[119],[31,142833,223],{"className":142834},[297],[31,142836],{"className":142837,"style":293},[119],[31,142839,142841],{"className":142840},[75,1793],[31,142842,142509],{"className":142843},[75,138280],[31,142845],{"className":142846,"style":120},[119],[31,142848,434],{"className":142849},[124],[31,142851],{"className":142852,"style":120},[119],[31,142854,142856],{"className":142855},[75,1793],[31,142857,142520],{"className":142858},[75,138280],[31,142860,142861,142864],{"style":142778},[31,142862],{"className":142863,"style":1637},[1060],[31,142865,142867,142870,142873,142876,142879],{"className":142866},[75],[31,142868],{"className":142869},[75],[31,142871],{"className":142872,"style":293},[119],[31,142874,223],{"className":142875},[297],[31,142877],{"className":142878,"style":293},[119],[31,142880,142882],{"className":142881},[75,1793],[31,142883,142664],{"className":142884},[75,138280],[31,142886,142887,142890],{"style":142787},[31,142888],{"className":142889,"style":1637},[1060],[31,142891,142893,142896,142899,142902,142905,142911,142914,142917,142920,142923,142926,142929,142932,142941,142944,142976],{"className":142892},[75],[31,142894],{"className":142895},[75],[31,142897],{"className":142898,"style":293},[119],[31,142900,135158],{"className":142901},[297],[31,142903],{"className":142904,"style":293},[119],[31,142906,142908],{"className":142907},[75,1793],[31,142909,142685],{"className":142910},[75,138280],[31,142912],{"className":142913,"style":293},[119],[31,142915,223],{"className":142916},[297],[31,142918],{"className":142919,"style":293},[119],[31,142921,142690],{"className":142922},[75],[31,142924],{"className":142925},[119,138139],[31,142927],{"className":142928,"style":1789},[119],[31,142930,211],{"className":142931},[264],[31,142933,142935],{"className":142934},[75],[31,142936,142938],{"className":142937},[75],[31,142939,138161],{"className":142940},[75,2745],[31,142942],{"className":142943,"style":138165},[119],[31,142945,142947,142950],{"className":142946},[75],[31,142948,152],{"className":142949},[75],[31,142951,142953],{"className":142952},[1039],[31,142954,142956],{"className":142955},[1043],[31,142957,142959],{"className":142958},[1048],[31,142960,142962],{"className":142961,"style":1327},[1052],[31,142963,142964,142967],{"style":1330},[31,142965],{"className":142966,"style":1061},[1060],[31,142968,142970],{"className":142969},[1065,1066,1067,1068],[31,142971,142973],{"className":142972},[75,1068],[31,142974,138068],{"className":142975},[75,1068],[31,142977,220],{"className":142978},[289],[31,142980,1076],{"className":142981},[1075],[31,142983,142985],{"className":142984},[1048],[31,142986,142988],{"className":142987,"style":142803},[1052],[31,142989],{},[11,142991,142992,143043,143044,143047,143048,143050,143051,2372],{},[31,142993,142995,143013],{"className":142994,"translate":35},[34],[31,142996,142998],{"className":142997},[39],[41,142999,143000],{"xmlns":43},[45,143001,143002,143010],{},[48,143003,143004,143006,143008],{},[91,143005,142690],{},[95,143007,347],{},[91,143009,142366],{},[55,143011,143012],{"encoding":57},"16 \\le 64",[31,143014,143016,143034],{"className":143015,"ariaHidden":62},[61],[31,143017,143019,143022,143025,143028,143031],{"className":143018},[66],[31,143020],{"className":143021,"style":366},[70],[31,143023,142690],{"className":143024},[75],[31,143026],{"className":143027,"style":293},[119],[31,143029,347],{"className":143030},[297],[31,143032],{"className":143033,"style":293},[119],[31,143035,143037,143040],{"className":143036},[66],[31,143038],{"className":143039,"style":326},[70],[31,143041,142366],{"className":143042},[75],"，检查通过。但 ",[45608,143045,143046],{},"buf + off"," 指向的是缓冲区外大约 4GB 的位置，",[45608,143049,141472],{}," 会往一个完全无关的地址写 32 字节。这是一个典型的",[7246,143052,143053],{},"整数溢出导致的任意地址写",[11,143055,143056],{},"用位向量理论来问：",[125323,143058,143060],{"className":141823,"code":143059,"language":141825,"meta":52812,"style":52812},"(set-logic QF_BV)\n(declare-const off (_ BitVec 32))\n(declare-const len (_ BitVec 32))\n(assert (bvule (bvadd off len) (_ bv64 32)))\n(assert (bvugt off (_ bv64 32)))\n(assert (bvule len (_ bv64 32)))\n(check-sat)\n(get-model)\n",[45608,143061,143062,143066,143071,143075,143080,143085,143090,143094],{"__ignoreMap":52812},[31,143063,143064],{"class":125331,"line":125332},[31,143065,142264],{},[31,143067,143068],{"class":125331,"line":52813},[31,143069,143070],{},"(declare-const off (_ BitVec 32))\n",[31,143072,143073],{"class":125331,"line":125262},[31,143074,142269],{},[31,143076,143077],{"class":125331,"line":125409},[31,143078,143079],{},"(assert (bvule (bvadd off len) (_ bv64 32)))\n",[31,143081,143082],{"class":125331,"line":125446},[31,143083,143084],{},"(assert (bvugt off (_ bv64 32)))\n",[31,143086,143087],{"class":125331,"line":125453},[31,143088,143089],{},"(assert (bvule len (_ bv64 32)))\n",[31,143091,143092],{"class":125331,"line":125459},[31,143093,141847],{},[31,143095,143097],{"class":125331,"line":143096},8,[31,143098,141852],{},[11,143100,143101],{},"求解器返回 sat，模型可能是：",[125323,143103,143106],{"className":143104,"code":143105,"language":1793},[136260],"off = #xfffffff0\nlen = #x00000020\n",[45608,143107,143105],{"__ignoreMap":52812},[11,143109,143110],{},"漏洞可达。",[11,143112,143113,143114,2446,143116,143118],{},"如果换一种建模，把 ",[45608,143115,142416],{},[45608,143117,141476],{}," 当成数学整数：",[31,143120,143122],{"className":143121,"translate":35},[411],[31,143123,143125,143171],{"className":143124,"translate":35},[34],[31,143126,143128],{"className":143127},[39],[41,143129,143130],{"xmlns":43,"display":420},[45,143131,143132,143168],{},[48,143133,143134,143136,143138,143140,143142,143144,143146,143148,143150,143152,143154,143156,143158,143160,143162,143164,143166],{},[1430,143135,142416],{},[95,143137,434],{},[1430,143139,141476],{},[95,143141,347],{},[91,143143,142366],{},[1430,143145,125504],{},[95,143147,125507],{},[1430,143149,125504],{},[1430,143151,142416],{},[95,143153,16040],{},[91,143155,142366],{},[1430,143157,125504],{},[95,143159,125507],{},[1430,143161,125504],{},[1430,143163,141476],{},[95,143165,5662],{},[91,143167,1977],{},[55,143169,143170],{"encoding":57},"\\text{off} + \\text{len} \\le 64 \\;\\wedge\\; \\text{off} > 64 \\;\\wedge\\; \\text{len} \\ge 0",[31,143172,143174,143195,143216,143240,143261,143285,143306],{"className":143173,"ariaHidden":62},[61],[31,143175,143177,143180,143186,143189,143192],{"className":143176},[66],[31,143178],{"className":143179,"style":123561},[70],[31,143181,143183],{"className":143182},[75,1793],[31,143184,142416],{"className":143185},[75],[31,143187],{"className":143188,"style":120},[119],[31,143190,434],{"className":143191},[124],[31,143193],{"className":143194,"style":120},[119],[31,143196,143198,143201,143207,143210,143213],{"className":143197},[66],[31,143199],{"className":143200,"style":106323},[70],[31,143202,143204],{"className":143203},[75,1793],[31,143205,141476],{"className":143206},[75],[31,143208],{"className":143209,"style":293},[119],[31,143211,347],{"className":143212},[297],[31,143214],{"className":143215,"style":293},[119],[31,143217,143219,143222,143225,143228,143231,143234,143237],{"className":143218},[66],[31,143220],{"className":143221,"style":326},[70],[31,143223,142366],{"className":143224},[75],[31,143226],{"className":143227,"style":293},[119],[31,143229],{"className":143230,"style":120},[119],[31,143232,125507],{"className":143233},[124],[31,143235],{"className":143236,"style":293},[119],[31,143238],{"className":143239,"style":120},[119],[31,143241,143243,143246,143252,143255,143258],{"className":143242},[66],[31,143244],{"className":143245,"style":141546},[70],[31,143247,143249],{"className":143248},[75,1793],[31,143250,142416],{"className":143251},[75],[31,143253],{"className":143254,"style":293},[119],[31,143256,16040],{"className":143257},[297],[31,143259],{"className":143260,"style":293},[119],[31,143262,143264,143267,143270,143273,143276,143279,143282],{"className":143263},[66],[31,143265],{"className":143266,"style":326},[70],[31,143268,142366],{"className":143269},[75],[31,143271],{"className":143272,"style":293},[119],[31,143274],{"className":143275,"style":120},[119],[31,143277,125507],{"className":143278},[124],[31,143280],{"className":143281,"style":293},[119],[31,143283],{"className":143284,"style":120},[119],[31,143286,143288,143291,143297,143300,143303],{"className":143287},[66],[31,143289],{"className":143290,"style":106323},[70],[31,143292,143294],{"className":143293},[75,1793],[31,143295,141476],{"className":143296},[75],[31,143298],{"className":143299,"style":293},[119],[31,143301,5662],{"className":143302},[297],[31,143304],{"className":143305,"style":293},[119],[31,143307,143309,143312],{"className":143308},[66],[31,143310],{"className":143311,"style":326},[70],[31,143313,1977],{"className":143314},[75],[11,143316,143317,143318,143320,143321,143400],{},"这个公式是 ",[7246,143319,131566],{},"，因为 ",[31,143322,143324,143346],{"className":143323,"translate":35},[34],[31,143325,143327],{"className":143326},[39],[41,143328,143329],{"xmlns":43},[45,143330,143331,143343],{},[48,143332,143333,143335,143337,143339,143341],{},[1430,143334,142416],{},[95,143336,434],{},[1430,143338,141476],{},[95,143340,16040],{},[91,143342,142366],{},[55,143344,143345],{"encoding":57},"\\text{off} + \\text{len} > 64",[31,143347,143349,143370,143391],{"className":143348,"ariaHidden":62},[61],[31,143350,143352,143355,143361,143364,143367],{"className":143351},[66],[31,143353],{"className":143354,"style":123561},[70],[31,143356,143358],{"className":143357},[75,1793],[31,143359,142416],{"className":143360},[75],[31,143362],{"className":143363,"style":120},[119],[31,143365,434],{"className":143366},[124],[31,143368],{"className":143369,"style":120},[119],[31,143371,143373,143376,143382,143385,143388],{"className":143372},[66],[31,143374],{"className":143375,"style":141546},[70],[31,143377,143379],{"className":143378},[75,1793],[31,143380,141476],{"className":143381},[75],[31,143383],{"className":143384,"style":293},[119],[31,143386,16040],{"className":143387},[297],[31,143389],{"className":143390,"style":293},[119],[31,143392,143394,143397],{"className":143393},[66],[31,143395],{"className":143396,"style":326},[70],[31,143398,142366],{"className":143399},[75]," 直接和第一条矛盾。工具会得出结论「没有漏洞」，而实际上漏洞就在那里。",[11,143402,143403,143406,143407,143410],{},[7246,143404,143405],{},"同一个程序，两种理论，两个相反的答案。"," 这就是为什么漏洞检测工具必须精确建模目标语言的整数语义：C 的 ",[45608,143408,143409],{},"unsigned int"," 是模运算，不是数学整数。",[52860,143412,143414],{"id":143413},"_54-空指针","5.4 空指针",[11,143416,143417],{},"静态分析经常报「可能的空指针解引用」。这类告警里有很多是误报，SMT 可以用来过滤。",[125323,143419,143421],{"className":125325,"code":143420,"language":1412,"meta":52812,"style":52812},"int read_flag(struct node *p) {\n    if (p == NULL) return 0;\n    return p->flag;     \u002F\u002F 静态分析：p 可能为空？\n}\n",[45608,143422,143423,143445,143468,143479],{"__ignoreMap":52812},[31,143424,143425,143427,143430,143432,143435,143438,143441,143443],{"class":125331,"line":125332},[31,143426,125346],{"class":125335},[31,143428,143429],{"class":125339}," read_flag",[31,143431,211],{"class":125343},[31,143433,143434],{"class":125335},"struct",[31,143436,143437],{"class":125343}," node ",[31,143439,143440],{"class":125335},"*",[31,143442,11],{"class":125349},[31,143444,125361],{"class":125343},[31,143446,143447,143449,143452,143454,143457,143460,143463,143466],{"class":125331,"line":52813},[31,143448,125412],{"class":125335},[31,143450,143451],{"class":125343}," (p ",[31,143453,125439],{"class":125335},[31,143455,143456],{"class":125374}," NULL",[31,143458,143459],{"class":125343},") ",[31,143461,143462],{"class":125335},"return",[31,143464,143465],{"class":125374}," 0",[31,143467,125406],{"class":125343},[31,143469,143470,143473,143476],{"class":125331,"line":125262},[31,143471,143472],{"class":125335},"    return",[31,143474,143475],{"class":125343}," p->flag;",[31,143477,143478],{"class":125449},"     \u002F\u002F 静态分析：p 可能为空？\n",[31,143480,143481],{"class":125331,"line":125409},[31,143482,125462],{"class":125343},[11,143484,143485,143486,143489,143490,143493],{},"工具在 ",[45608,143487,143488],{},"p->flag"," 处收集路径条件。走到这一行，必然经过了 ",[45608,143491,143492],{},"p == NULL"," 为假的分支，所以路径条件里包含：",[31,143495,143497],{"className":143496,"translate":35},[411],[31,143498,143500,143519],{"className":143499,"translate":35},[34],[31,143501,143503],{"className":143502},[39],[41,143504,143505],{"xmlns":43,"display":420},[45,143506,143507,143516],{},[48,143508,143509,143511,143513],{},[51,143510,11],{},[95,143512,21271],{"mathvariant":494},[1430,143514,143515],{"mathvariant":138083},"NULL",[55,143517,143518],{"encoding":57},"p \\ne \\texttt{NULL}",[31,143520,143522,143573],{"className":143521,"ariaHidden":62},[61],[31,143523,143525,143528,143531,143534,143570],{"className":143524},[66],[31,143526],{"className":143527,"style":4202},[70],[31,143529,11],{"className":143530},[75,76],[31,143532],{"className":143533,"style":293},[119],[31,143535,143537,143564,143567],{"className":143536},[297],[31,143538,143540],{"className":143539},[297],[31,143541,143543],{"className":143542},[75,21516],[31,143544,143546],{"className":143545},[21520],[31,143547,143549,143552,143561],{"className":143548},[21524],[31,143550],{"className":143551,"style":4202},[70],[31,143553,143555],{"className":143554},[21531],[31,143556,143558],{"className":143557},[75],[31,143559,21538],{"className":143560},[297],[31,143562],{"className":143563},[21542],[31,143565],{"className":143566},[119,21546],[31,143568,223],{"className":143569},[297],[31,143571],{"className":143572,"style":293},[119],[31,143574,143576,143579],{"className":143575},[66],[31,143577],{"className":143578,"style":138273},[70],[31,143580,143582],{"className":143581},[75,1793],[31,143583,143515],{"className":143584},[75,138280],[11,143586,143587,143588,143590],{},"静态分析如果只做局部启发式，可能会说「",[45608,143589,11],{}," 可能是空指针」。这时问 SMT：",[31,143592,143594],{"className":143593,"translate":35},[411],[31,143595,143597,143627],{"className":143596,"translate":35},[34],[31,143598,143600],{"className":143599},[39],[41,143601,143602],{"xmlns":43,"display":420},[45,143603,143604,143624],{},[48,143605,143606,143608,143610,143612,143614,143616,143618,143620,143622],{},[51,143607,11],{},[95,143609,223],{},[1430,143611,143515],{"mathvariant":138083},[1430,143613,125504],{},[95,143615,125507],{},[1430,143617,125504],{},[51,143619,11],{},[95,143621,21271],{"mathvariant":494},[1430,143623,143515],{"mathvariant":138083},[55,143625,143626],{"encoding":57},"p = \\texttt{NULL} \\;\\wedge\\; p \\ne \\texttt{NULL}",[31,143628,143630,143648,143675,143726],{"className":143629,"ariaHidden":62},[61],[31,143631,143633,143636,143639,143642,143645],{"className":143632},[66],[31,143634],{"className":143635,"style":865},[70],[31,143637,11],{"className":143638},[75,76],[31,143640],{"className":143641,"style":293},[119],[31,143643,223],{"className":143644},[297],[31,143646],{"className":143647,"style":293},[119],[31,143649,143651,143654,143660,143663,143666,143669,143672],{"className":143650},[66],[31,143652],{"className":143653,"style":138273},[70],[31,143655,143657],{"className":143656},[75,1793],[31,143658,143515],{"className":143659},[75,138280],[31,143661],{"className":143662,"style":293},[119],[31,143664],{"className":143665,"style":120},[119],[31,143667,125507],{"className":143668},[124],[31,143670],{"className":143671,"style":293},[119],[31,143673],{"className":143674,"style":120},[119],[31,143676,143678,143681,143684,143687,143723],{"className":143677},[66],[31,143679],{"className":143680,"style":4202},[70],[31,143682,11],{"className":143683},[75,76],[31,143685],{"className":143686,"style":293},[119],[31,143688,143690,143717,143720],{"className":143689},[297],[31,143691,143693],{"className":143692},[297],[31,143694,143696],{"className":143695},[75,21516],[31,143697,143699],{"className":143698},[21520],[31,143700,143702,143705,143714],{"className":143701},[21524],[31,143703],{"className":143704,"style":4202},[70],[31,143706,143708],{"className":143707},[21531],[31,143709,143711],{"className":143710},[75],[31,143712,21538],{"className":143713},[297],[31,143715],{"className":143716},[21542],[31,143718],{"className":143719},[119,21546],[31,143721,223],{"className":143722},[297],[31,143724],{"className":143725,"style":293},[119],[31,143727,143729,143732],{"className":143728},[66],[31,143730],{"className":143731,"style":138273},[70],[31,143733,143735],{"className":143734},[75,1793],[31,143736,143515],{"className":143737},[75,138280],[11,143739,143740,143741,143743],{},"这是明显的矛盾，返回 ",[7246,143742,131566],{},"，告警被排除。",[11,143745,143746],{},"反过来，如果代码是：",[125323,143748,143750],{"className":125325,"code":143749,"language":1412,"meta":52812,"style":52812},"int read_flag(struct node *p) {\n    if (p->flag == 1) {   \u002F\u002F 这里没有检查 p\n        return 1;\n    }\n    return 0;\n}\n",[45608,143751,143752,143770,143788,143797,143801,143809],{"__ignoreMap":52812},[31,143753,143754,143756,143758,143760,143762,143764,143766,143768],{"class":125331,"line":125332},[31,143755,125346],{"class":125335},[31,143757,143429],{"class":125339},[31,143759,211],{"class":125343},[31,143761,143434],{"class":125335},[31,143763,143437],{"class":125343},[31,143765,143440],{"class":125335},[31,143767,11],{"class":125349},[31,143769,125361],{"class":125343},[31,143771,143772,143774,143777,143779,143782,143785],{"class":125331,"line":52813},[31,143773,125412],{"class":125335},[31,143775,143776],{"class":125343}," (p->flag ",[31,143778,125439],{"class":125335},[31,143780,143781],{"class":125374}," 1",[31,143783,143784],{"class":125343},") {",[31,143786,143787],{"class":125449},"   \u002F\u002F 这里没有检查 p\n",[31,143789,143790,143793,143795],{"class":125331,"line":125262},[31,143791,143792],{"class":125335},"        return",[31,143794,143781],{"class":125374},[31,143796,125406],{"class":125343},[31,143798,143799],{"class":125331,"line":125409},[31,143800,125456],{"class":125343},[31,143802,143803,143805,143807],{"class":125331,"line":125446},[31,143804,143472],{"class":125335},[31,143806,143465],{"class":125374},[31,143808,125406],{"class":125343},[31,143810,143811],{"class":125331,"line":125453},[31,143812,125462],{"class":125343},[11,143814,143815,143816,143902],{},"路径条件里没有 ",[31,143817,143819,143836],{"className":143818,"translate":35},[34],[31,143820,143822],{"className":143821},[39],[41,143823,143824],{"xmlns":43},[45,143825,143826,143834],{},[48,143827,143828,143830,143832],{},[51,143829,11],{},[95,143831,21271],{"mathvariant":494},[1430,143833,143515],{"mathvariant":138083},[55,143835,143518],{"encoding":57},[31,143837,143839,143890],{"className":143838,"ariaHidden":62},[61],[31,143840,143842,143845,143848,143851,143887],{"className":143841},[66],[31,143843],{"className":143844,"style":4202},[70],[31,143846,11],{"className":143847},[75,76],[31,143849],{"className":143850,"style":293},[119],[31,143852,143854,143881,143884],{"className":143853},[297],[31,143855,143857],{"className":143856},[297],[31,143858,143860],{"className":143859},[75,21516],[31,143861,143863],{"className":143862},[21520],[31,143864,143866,143869,143878],{"className":143865},[21524],[31,143867],{"className":143868,"style":4202},[70],[31,143870,143872],{"className":143871},[21531],[31,143873,143875],{"className":143874},[75],[31,143876,21538],{"className":143877},[297],[31,143879],{"className":143880},[21542],[31,143882],{"className":143883},[119,21546],[31,143885,223],{"className":143886},[297],[31,143888],{"className":143889,"style":293},[119],[31,143891,143893,143896],{"className":143892},[66],[31,143894],{"className":143895,"style":138273},[70],[31,143897,143899],{"className":143898},[75,1793],[31,143900,143515],{"className":143901},[75,138280],"，查询：",[31,143904,143906],{"className":143905,"translate":35},[411],[31,143907,143909,143927],{"className":143908,"translate":35},[34],[31,143910,143912],{"className":143911},[39],[41,143913,143914],{"xmlns":43,"display":420},[45,143915,143916,143924],{},[48,143917,143918,143920,143922],{},[51,143919,11],{},[95,143921,223],{},[1430,143923,143515],{"mathvariant":138083},[55,143925,143926],{"encoding":57},"p = \\texttt{NULL}",[31,143928,143930,143948],{"className":143929,"ariaHidden":62},[61],[31,143931,143933,143936,143939,143942,143945],{"className":143932},[66],[31,143934],{"className":143935,"style":865},[70],[31,143937,11],{"className":143938},[75,76],[31,143940],{"className":143941,"style":293},[119],[31,143943,223],{"className":143944},[297],[31,143946],{"className":143947,"style":293},[119],[31,143949,143951,143954],{"className":143950},[66],[31,143952],{"className":143953,"style":138273},[70],[31,143955,143957],{"className":143956},[75,1793],[31,143958,143515],{"className":143959},[75,138280],[11,143961,143962,143963,143965,143966,143969],{},"返回 ",[7246,143964,125655],{},"，模型给出 ",[45608,143967,143968],{},"p = NULL","。这是一条真实可达的空指针路径，告警保留。",[11,143971,143972,143973,143976],{},"这个用法体现了一个重要原则：",[7246,143974,143975],{},"SMT 不负责发现漏洞，它负责判断「这条路径到底可不可能」","。发现可疑点靠前端分析，排除误报靠 SMT。",[52860,143978,143980],{"id":143979},"_55-数组越界","5.5 数组越界",[125323,143982,143984],{"className":125325,"code":143983,"language":1412,"meta":52812,"style":52812},"int get(int *a, int i) {\n    if (i \u003C 100) {\n        return a[i];     \u002F\u002F i 可能为负\n    }\n    return -1;\n}\n",[45608,143985,143986,144010,144024,144036,144040,144051],{"__ignoreMap":52812},[31,143987,143988,143990,143993,143995,143997,143999,144001,144003,144005,144008],{"class":125331,"line":125332},[31,143989,125346],{"class":125335},[31,143991,143992],{"class":125339}," get",[31,143994,211],{"class":125343},[31,143996,125346],{"class":125335},[31,143998,141394],{"class":125335},[31,144000,5813],{"class":125349},[31,144002,125353],{"class":125343},[31,144004,125346],{"class":125335},[31,144006,144007],{"class":125349}," i",[31,144009,125361],{"class":125343},[31,144011,144012,144014,144017,144019,144022],{"class":125331,"line":52813},[31,144013,125412],{"class":125335},[31,144015,144016],{"class":125343}," (i ",[31,144018,9758],{"class":125335},[31,144020,144021],{"class":125374}," 100",[31,144023,125361],{"class":125343},[31,144025,144026,144028,144030,144033],{"class":125331,"line":125262},[31,144027,143792],{"class":125335},[31,144029,125369],{"class":125349},[31,144031,144032],{"class":125343},"[i];",[31,144034,144035],{"class":125449},"     \u002F\u002F i 可能为负\n",[31,144037,144038],{"class":125331,"line":125409},[31,144039,125456],{"class":125343},[31,144041,144042,144044,144047,144049],{"class":125331,"line":125446},[31,144043,143472],{"class":125335},[31,144045,144046],{"class":125335}," -",[31,144048,205],{"class":125374},[31,144050,125406],{"class":125343},[31,144052,144053],{"class":125331,"line":125453},[31,144054,125462],{"class":125343},[11,144056,144057],{},"只检查了上界，没检查下界。问 SMT：",[31,144059,144061],{"className":144060,"translate":35},[411],[31,144062,144064,144094],{"className":144063,"translate":35},[34],[31,144065,144067],{"className":144066},[39],[41,144068,144069],{"xmlns":43,"display":420},[45,144070,144071,144091],{},[48,144072,144073,144075,144077,144079,144081,144083,144085,144087,144089],{},[51,144074,5391],{},[95,144076,9758],{},[91,144078,22616],{},[1430,144080,125504],{},[95,144082,125507],{},[1430,144084,125504],{},[51,144086,5391],{},[95,144088,9758],{},[91,144090,1977],{},[55,144092,144093],{"encoding":57},"i \u003C 100 \\;\\wedge\\; i \u003C 0",[31,144095,144097,144116,144140,144158],{"className":144096,"ariaHidden":62},[61],[31,144098,144100,144104,144107,144110,144113],{"className":144099},[66],[31,144101],{"className":144102,"style":144103},[70],"height:0.6986em;vertical-align:-0.0391em;",[31,144105,5391],{"className":144106},[75,76],[31,144108],{"className":144109,"style":293},[119],[31,144111,9758],{"className":144112},[297],[31,144114],{"className":144115,"style":293},[119],[31,144117,144119,144122,144125,144128,144131,144134,144137],{"className":144118},[66],[31,144120],{"className":144121,"style":326},[70],[31,144123,22616],{"className":144124},[75],[31,144126],{"className":144127,"style":293},[119],[31,144129],{"className":144130,"style":120},[119],[31,144132,125507],{"className":144133},[124],[31,144135],{"className":144136,"style":293},[119],[31,144138],{"className":144139,"style":120},[119],[31,144141,144143,144146,144149,144152,144155],{"className":144142},[66],[31,144144],{"className":144145,"style":144103},[70],[31,144147,5391],{"className":144148},[75,76],[31,144150],{"className":144151,"style":293},[119],[31,144153,9758],{"className":144154},[297],[31,144156],{"className":144157,"style":293},[119],[31,144159,144161,144164],{"className":144160},[66],[31,144162],{"className":144163,"style":326},[70],[31,144165,1977],{"className":144166},[75],[11,144168,144169,144170,2372,144173,144176],{},"返回 sat，例如 ",[45608,144171,144172],{},"i = -1",[45608,144174,144175],{},"a[-1]"," 是一次越界读。",[11,144178,144179],{},"如果加上下界检查：",[125323,144181,144183],{"className":125325,"code":144182,"language":1412,"meta":52812,"style":52812},"if (i >= 0 && i \u003C 100) {\n    return a[i];\n}\n",[45608,144184,144185,144207,144216],{"__ignoreMap":52812},[31,144186,144187,144189,144191,144194,144196,144198,144201,144203,144205],{"class":125331,"line":125332},[31,144188,142009],{"class":125335},[31,144190,144016],{"class":125343},[31,144192,144193],{"class":125335},">=",[31,144195,143465],{"class":125374},[31,144197,125428],{"class":125335},[31,144199,144200],{"class":125343}," i ",[31,144202,9758],{"class":125335},[31,144204,144021],{"class":125374},[31,144206,125361],{"class":125343},[31,144208,144209,144211,144213],{"class":125331,"line":52813},[31,144210,143472],{"class":125335},[31,144212,125369],{"class":125349},[31,144214,144215],{"class":125343},"[i];\n",[31,144217,144218],{"class":125331,"line":125262},[31,144219,125462],{"class":125343},[11,144221,144222],{},"约束变成：",[31,144224,144226],{"className":144225,"translate":35},[411],[31,144227,144229,144271],{"className":144228,"translate":35},[34],[31,144230,144232],{"className":144231},[39],[41,144233,144234],{"xmlns":43,"display":420},[45,144235,144236,144268],{},[48,144237,144238,144240,144242,144244,144246,144248,144250,144252,144254,144256,144258,144260,144262,144264,144266],{},[51,144239,5391],{},[95,144241,5662],{},[91,144243,1977],{},[1430,144245,125504],{},[95,144247,125507],{},[1430,144249,125504],{},[51,144251,5391],{},[95,144253,9758],{},[91,144255,22616],{},[1430,144257,125504],{},[95,144259,125507],{},[1430,144261,125504],{},[51,144263,5391],{},[95,144265,9758],{},[91,144267,1977],{},[55,144269,144270],{"encoding":57},"i \\ge 0 \\;\\wedge\\; i \u003C 100 \\;\\wedge\\; i \u003C 0",[31,144272,144274,144293,144317,144335,144359,144377],{"className":144273,"ariaHidden":62},[61],[31,144275,144277,144281,144284,144287,144290],{"className":144276},[66],[31,144278],{"className":144279,"style":144280},[70],"height:0.7955em;vertical-align:-0.136em;",[31,144282,5391],{"className":144283},[75,76],[31,144285],{"className":144286,"style":293},[119],[31,144288,5662],{"className":144289},[297],[31,144291],{"className":144292,"style":293},[119],[31,144294,144296,144299,144302,144305,144308,144311,144314],{"className":144295},[66],[31,144297],{"className":144298,"style":326},[70],[31,144300,1977],{"className":144301},[75],[31,144303],{"className":144304,"style":293},[119],[31,144306],{"className":144307,"style":120},[119],[31,144309,125507],{"className":144310},[124],[31,144312],{"className":144313,"style":293},[119],[31,144315],{"className":144316,"style":120},[119],[31,144318,144320,144323,144326,144329,144332],{"className":144319},[66],[31,144321],{"className":144322,"style":144103},[70],[31,144324,5391],{"className":144325},[75,76],[31,144327],{"className":144328,"style":293},[119],[31,144330,9758],{"className":144331},[297],[31,144333],{"className":144334,"style":293},[119],[31,144336,144338,144341,144344,144347,144350,144353,144356],{"className":144337},[66],[31,144339],{"className":144340,"style":326},[70],[31,144342,22616],{"className":144343},[75],[31,144345],{"className":144346,"style":293},[119],[31,144348],{"className":144349,"style":120},[119],[31,144351,125507],{"className":144352},[124],[31,144354],{"className":144355,"style":293},[119],[31,144357],{"className":144358,"style":120},[119],[31,144360,144362,144365,144368,144371,144374],{"className":144361},[66],[31,144363],{"className":144364,"style":144103},[70],[31,144366,5391],{"className":144367},[75,76],[31,144369],{"className":144370,"style":293},[119],[31,144372,9758],{"className":144373},[297],[31,144375],{"className":144376,"style":293},[119],[31,144378,144380,144383],{"className":144379},[66],[31,144381],{"className":144382,"style":326},[70],[31,144384,1977],{"className":144385},[75],[11,144387,143962,144388,144390],{},[7246,144389,131566],{},"，越界路径被排除。",[11,144392,144393,144394,144397,144398,144401],{},"这个例子还可以扩展成带数组理论的版本。比如要验证「下标不同时，写入 ",[45608,144395,144396],{},"a[j]"," 不会影响对 ",[45608,144399,144400],{},"a[i]"," 的读取」，可以把「影响了」写成公式，再问它是否可满足：",[31,144403,144405],{"className":144404,"translate":35},[411],[31,144406,144408,144506],{"className":144407,"translate":35},[34],[31,144409,144411],{"className":144410},[39],[41,144412,144413],{"xmlns":43,"display":420},[45,144414,144415,144503],{},[48,144416,144417,144419,144421,144423,144425,144427,144429,144443,144445,144457,144459,144461,144463,144465,144467,144469,144471,144473,144475,144477,144479,144493,144495,144497,144499,144501],{},[51,144418,5391],{},[95,144420,21271],{"mathvariant":494},[51,144422,2555],{},[1430,144424,125504],{},[95,144426,125507],{},[1430,144428,125504],{},[48,144430,144431,144433,144435,144437,144439,144441],{},[51,144432,14551],{"mathvariant":494},[51,144434,30042],{"mathvariant":494},[51,144436,87599],{"mathvariant":494},[51,144438,30042],{"mathvariant":494},[51,144440,1412],{"mathvariant":494},[51,144442,12271],{"mathvariant":494},[95,144444,211],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[48,144446,144447,144449,144451,144453,144455],{},[51,144448,14551],{"mathvariant":494},[51,144450,12271],{"mathvariant":494},[51,144452,2552],{"mathvariant":494},[51,144454,2549],{"mathvariant":494},[51,144456,30042],{"mathvariant":494},[95,144458,211],{"stretchy":210},[51,144460,5813],{},[95,144462,1210],{"separator":62},[51,144464,2555],{},[95,144466,1210],{"separator":62},[51,144468,5354],{},[95,144470,220],{"stretchy":210},[95,144472,1210],{"separator":62},[51,144474,5391],{},[95,144476,220],{"fence":62,"stretchy":62,"minsize":1178,"maxsize":1178},[95,144478,21271],{"mathvariant":494},[48,144480,144481,144483,144485,144487,144489,144491],{},[51,144482,14551],{"mathvariant":494},[51,144484,30042],{"mathvariant":494},[51,144486,87599],{"mathvariant":494},[51,144488,30042],{"mathvariant":494},[51,144490,1412],{"mathvariant":494},[51,144492,12271],{"mathvariant":494},[95,144494,211],{"stretchy":210},[51,144496,5813],{},[95,144498,1210],{"separator":62},[51,144500,5391],{},[95,144502,220],{"stretchy":210},[55,144504,144505],{"encoding":57},"i \\ne j \\;\\wedge\\; \\mathrm{select}\\bigl(\\mathrm{store}(a, j, v), i\\bigr) \\ne \\mathrm{select}(a, i)",[31,144507,144509,144560,144584,144692],{"className":144508,"ariaHidden":62},[61],[31,144510,144512,144515,144518,144521,144557],{"className":144511},[66],[31,144513],{"className":144514,"style":4202},[70],[31,144516,5391],{"className":144517},[75,76],[31,144519],{"className":144520,"style":293},[119],[31,144522,144524,144551,144554],{"className":144523},[297],[31,144525,144527],{"className":144526},[297],[31,144528,144530],{"className":144529},[75,21516],[31,144531,144533],{"className":144532},[21520],[31,144534,144536,144539,144548],{"className":144535},[21524],[31,144537],{"className":144538,"style":4202},[70],[31,144540,144542],{"className":144541},[21531],[31,144543,144545],{"className":144544},[75],[31,144546,21538],{"className":144547},[297],[31,144549],{"className":144550},[21542],[31,144552],{"className":144553},[119,21546],[31,144555,223],{"className":144556},[297],[31,144558],{"className":144559,"style":293},[119],[31,144561,144563,144566,144569,144572,144575,144578,144581],{"className":144562},[66],[31,144564],{"className":144565,"style":138668},[70],[31,144567,2555],{"className":144568,"style":5514},[75,76],[31,144570],{"className":144571,"style":293},[119],[31,144573],{"className":144574,"style":120},[119],[31,144576,125507],{"className":144577},[124],[31,144579],{"className":144580,"style":293},[119],[31,144582],{"className":144583,"style":120},[119],[31,144585,144587,144590,144596,144602,144608,144611,144614,144617,144620,144623,144626,144629,144632,144635,144638,144641,144644,144650,144653,144689],{"className":144586},[66],[31,144588],{"className":144589,"style":4988},[70],[31,144591,144593],{"className":144592},[75],[31,144594,126909],{"className":144595},[75,2745],[31,144597,144599],{"className":144598},[264],[31,144600,211],{"className":144601},[1307,1308],[31,144603,144605],{"className":144604},[75],[31,144606,127844],{"className":144607},[75,2745],[31,144609,211],{"className":144610},[264],[31,144612,5813],{"className":144613},[75,76],[31,144615,1210],{"className":144616},[1391],[31,144618],{"className":144619,"style":1467},[119],[31,144621,2555],{"className":144622,"style":5514},[75,76],[31,144624,1210],{"className":144625},[1391],[31,144627],{"className":144628,"style":1467},[119],[31,144630,5354],{"className":144631,"style":2666},[75,76],[31,144633,220],{"className":144634},[289],[31,144636,1210],{"className":144637},[1391],[31,144639],{"className":144640,"style":1467},[119],[31,144642,5391],{"className":144643},[75,76],[31,144645,144647],{"className":144646},[289],[31,144648,220],{"className":144649},[1307,1308],[31,144651],{"className":144652,"style":293},[119],[31,144654,144656,144683,144686],{"className":144655},[297],[31,144657,144659],{"className":144658},[297],[31,144660,144662],{"className":144661},[75,21516],[31,144663,144665],{"className":144664},[21520],[31,144666,144668,144671,144680],{"className":144667},[21524],[31,144669],{"className":144670,"style":4202},[70],[31,144672,144674],{"className":144673},[21531],[31,144675,144677],{"className":144676},[75],[31,144678,21538],{"className":144679},[297],[31,144681],{"className":144682},[21542],[31,144684],{"className":144685},[119,21546],[31,144687,223],{"className":144688},[297],[31,144690],{"className":144691,"style":293},[119],[31,144693,144695,144698,144704,144707,144710,144713,144716,144719],{"className":144694},[66],[31,144696],{"className":144697,"style":260},[70],[31,144699,144701],{"className":144700},[75],[31,144702,126909],{"className":144703},[75,2745],[31,144705,211],{"className":144706},[264],[31,144708,5813],{"className":144709},[75,76],[31,144711,1210],{"className":144712},[1391],[31,144714],{"className":144715,"style":1467},[119],[31,144717,5391],{"className":144718},[75,76],[31,144720,220],{"className":144721},[289],[11,144723,144724,144725,144809,144810,144812],{},"数组理论会展开 read-over-write 公理，算术理论会处理 ",[31,144726,144728,144746],{"className":144727,"translate":35},[34],[31,144729,144731],{"className":144730},[39],[41,144732,144733],{"xmlns":43},[45,144734,144735,144743],{},[48,144736,144737,144739,144741],{},[51,144738,5391],{},[95,144740,21271],{"mathvariant":494},[51,144742,2555],{},[55,144744,144745],{"encoding":57},"i \\ne j",[31,144747,144749,144800],{"className":144748,"ariaHidden":62},[61],[31,144750,144752,144755,144758,144761,144797],{"className":144751},[66],[31,144753],{"className":144754,"style":4202},[70],[31,144756,5391],{"className":144757},[75,76],[31,144759],{"className":144760,"style":293},[119],[31,144762,144764,144791,144794],{"className":144763},[297],[31,144765,144767],{"className":144766},[297],[31,144768,144770],{"className":144769},[75,21516],[31,144771,144773],{"className":144772},[21520],[31,144774,144776,144779,144788],{"className":144775},[21524],[31,144777],{"className":144778,"style":4202},[70],[31,144780,144782],{"className":144781},[21531],[31,144783,144785],{"className":144784},[75],[31,144786,21538],{"className":144787},[297],[31,144789],{"className":144790},[21542],[31,144792],{"className":144793},[119,21546],[31,144795,223],{"className":144796},[297],[31,144798],{"className":144799,"style":293},[119],[31,144801,144803,144806],{"className":144802},[66],[31,144804],{"className":144805,"style":138668},[70],[31,144807,2555],{"className":144808,"style":5514},[75,76],"，两个理论协作给出 ",[7246,144811,131566],{},"——也就是说，「写入影响了另一个下标」的情况不可能发生。这正是 3.3 节里理论组合的实际用武之地。",[52860,144814,144816],{"id":144815},"_56-完整流程","5.6 完整流程",[11,144818,144819],{},"把上面的步骤串起来，一个典型的漏洞检测流程是这样的：",[128055,144821,144822],{},[128058,144823],{"height":144824,"width":144825,"style":128062,"data-d2-theme":128063,"alt":52812,"src":144826},240,2053,"data:image\u002Fsvg+xml;base64,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\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\u002FlvoTkuI3lj6\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只负责中间那个方框。前面「怎么把程序变成公式」是符号执行引擎的事，后面「怎么确认漏洞」是测试和人工的事。",[7246,144831,144832],{},"公式建得对不对，直接决定 SMT 的结论有没有意义。",[11,144834,144835,144836,144839],{},"所以，SMT 在漏洞检测里的正确定位是：",[7246,144837,144838],{},"一个强大但需要被正确使用的判定引擎","。它能把「这条路径可不可能」回答得很准，但「该问哪些问题」「怎么把程序翻译成问题」，仍然依赖前端工具和工程师的判断。",[18,144841,144843],{"id":144842},"六小结","六、小结",[11,144845,144846,144847,144942,144943,144946],{},"回到开头那道小学题。SMT 的价值，不是把 ",[31,144848,144850,144876],{"className":144849,"translate":35},[34],[31,144851,144853],{"className":144852},[39],[41,144854,144855],{"xmlns":43},[45,144856,144857,144873],{},[48,144858,144859,144861,144863,144865,144867,144869,144871],{},[51,144860,53],{},[95,144862,16040],{},[91,144864,93],{},[95,144866,125507],{},[51,144868,53],{},[95,144870,9758],{},[91,144872,7853],{},[55,144874,144875],{"encoding":57},"x > 3 \\wedge x \u003C 5",[31,144877,144879,144897,144915,144933],{"className":144878,"ariaHidden":62},[61],[31,144880,144882,144885,144888,144891,144894],{"className":144881},[66],[31,144883],{"className":144884,"style":21113},[70],[31,144886,53],{"className":144887},[75,76],[31,144889],{"className":144890,"style":293},[119],[31,144892,16040],{"className":144893},[297],[31,144895],{"className":144896,"style":293},[119],[31,144898,144900,144903,144906,144909,144912],{"className":144899},[66],[31,144901],{"className":144902,"style":326},[70],[31,144904,93],{"className":144905},[75],[31,144907],{"className":144908,"style":120},[119],[31,144910,125507],{"className":144911},[124],[31,144913],{"className":144914,"style":120},[119],[31,144916,144918,144921,144924,144927,144930],{"className":144917},[66],[31,144919],{"className":144920,"style":21113},[70],[31,144922,53],{"className":144923},[75,76],[31,144925],{"className":144926,"style":293},[119],[31,144928,9758],{"className":144929},[297],[31,144931],{"className":144932,"style":293},[119],[31,144934,144936,144939],{"className":144935},[66],[31,144937],{"className":144938,"style":326},[70],[31,144940,7853],{"className":144941},[75]," 算得更快，而是让机器",[7246,144944,144945],{},"在保留「整数」「数组」「位向量」这些概念的前提下","，仍然能自动地回答「有没有解」。",[11,144948,144949,144952,144953,144956],{},[7246,144950,144951],{},"SMT = SAT + 背景理论。"," 布尔层负责组合，理论层负责语义，两者通过「原子真值」这一层接口通信。 ",[7246,144954,144955],{},"它的核心取舍是用表达力换可判定性。"," 把主战场放在无量词的可判定片段，换来一个在片段内保证终止、且能处理工业规模混合约束的自动判定过程。",[144958,144959,144960],"style",{},"html pre.shiki code .s8jYJ, html code.shiki .s8jYJ{--shiki-light:#D73A49;--shiki-default:#D73A49;--shiki-dark:#F97583}html pre.shiki code .snPdu, html code.shiki .snPdu{--shiki-light:#6F42C1;--shiki-default:#6F42C1;--shiki-dark:#B392F0}html pre.shiki code .sxrX7, html code.shiki .sxrX7{--shiki-light:#24292E;--shiki-default:#24292E;--shiki-dark:#E1E4E8}html pre.shiki code .sP4rz, html code.shiki .sP4rz{--shiki-light:#E36209;--shiki-default:#E36209;--shiki-dark:#FFAB70}html pre.shiki code .sBjJW, html code.shiki .sBjJW{--shiki-light:#005CC5;--shiki-default:#005CC5;--shiki-dark:#79B8FF}html pre.shiki code .sCsY4, html code.shiki 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把 SAT 的布尔搜索与多个理论求解器组合起来，使得混合理论约束可以被统一、自动地判定。",[145006],"logic","2026-10-29T09:03:03+08:00","1C-2B31uaFHKLW1GJ3_3j1Cbaf3yP10u6PKSP7bBHZ8",1794483690791]