[{"data":1,"prerenderedAt":125504},["ShallowReactive",2],{"post-\u002Fblog\u002F2026\u002F0015-the-bellman-equation-and-its-mathematical-properties":3,"i-ci:tag":-1,"i-mingcute:time-line":-1},{"post":4,"nextPost":72781,"prevPost":72974},{"id":5,"title":6,"body":7,"description":13,"draft":72767,"enableCodeHighlight":72767,"enableCodeHightlight":72767,"enableComment":72768,"enableD2":72767,"enableKatex":72768,"extension":72769,"image":72720,"important":72767,"location":72770,"meta":72771,"navigation":72768,"ogImage":72770,"path":72772,"seo":72773,"stem":72774,"summary":72775,"tags":72776,"time":72779,"__hash__":72780},"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":9,"toc":72719},"minimark",[10,14,17,20,25,30,1608,2077,2584,2588,3454,4159,5012,5591,5597,6359,6483,6486,6490,7414,7637,8391,8756,9913,9916,10036,10040,10044,10052,10823,10830,11681,11684,11746,11750,11817,12495,12498,12652,12926,12930,13350,14171,14354,14547,15194,15818,16122,16176,16421,16636,17185,17187,17328,17332,17411,17585,17774,17988,17991,18108,18111,18385,18388,18738,18741,19117,19120,19654,19741,19745,19748,20049,20140,20569,20830,21086,21229,21347,21351,21355,21432,21642,21804,22090,22615,22677,23476,23675,23679,24635,24788,25145,25673,25676,26441,26567,26629,26633,27446,27658,27914,27921,28202,28233,28596,28627,28916,29394,29398,30463,30587,30947,31251,31551,32562,32908,33592,33595,34078,34080,34084,34322,34520,34760,34763,35606,35772,36087,36416,36533,36537,36541,37246,37347,37351,37415,37934,38055,39566,39858,40122,40370,41149,41427,41994,42667,43159,43637,44048,44110,44232,45024,45344,45348,45518,45884,46550,46553,47101,47166,47170,48066,48412,48415,48419,48535,48838,49052,49141,49357,49417,49627,49820,50057,50289,50291,50418,50654,50956,51026,51030,51034,51145,51381,51529,52468,52473,53141,53172,53364,53368,53445,54744,54749,55892,55895,56284,56287,56863,56866,57437,57440,57866,57869,57924,58372,58581,58585,58704,59089,59092,59799,59854,60383,60438,60836,60953,61164,61433,61599,61603,61707,61877,62036,62143,62175,62381,62428,62631,62685,62689,62693,63056,63765,64013,64620,64962,65553,65848,67125,67707,68007,68011,68076,68295,68299,68302,68722,68795,68900,68903,68907,68911,68958,69020,69644,69735,69896,69900,70014,70602,71096,71201,71205,71381,71551,72082,72139,72174,72345,72348,72351,72627,72716],[11,12,13],"p",{},"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。",[11,15,16],{},"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。",[11,18,19],{},"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。",[21,22,24],"h2",{"id":23},"一形式化","一、形式化",[26,27,29],"h3",{"id":28},"_11-折扣马尔可夫决策过程","1.1 折扣马尔可夫决策过程",[31,32,35,743,1485],"theorem",{"id":33,"title":34},"definition-mdp","定义 1.1（折扣马尔可夫决策过程）",[11,36,37,38,86,87,116,117,264,265,294,295,323,324,386,387,516,517,599,600,683,684,713,714,742],{},"设 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为动作集，",[39,118,120,165],{"className":119,"translate":43},[42],[39,121,123],{"className":122},[47],[49,124,125],{"xmlns":51},[53,126,127,162],{},[56,128,129,132,137,148,151,153,156,159],{},[59,130,131],{},"P",[133,134,136],"mo",{"stretchy":135},"false","(",[138,139,140,143],"msup",{},[59,141,142],{},"s",[133,144,147],{"mathvariant":145,"lspace":146,"rspace":146},"normal","0em","′",[133,149,150],{},"∣",[59,152,142],{},[133,154,155],{"separator":70},",",[59,157,158],{},"a",[133,160,161],{"stretchy":135},")",[63,163,164],{"encoding":65},"P(s'\\mid s,a)",[39,166,168,239],{"className":167,"ariaHidden":70},[69],[39,169,171,175,179,183,227,232,236],{"className":170},[74],[39,172],{"className":173,"style":174},[78],"height:1.0019em;vertical-align:-0.25em;",[39,176,131],{"className":177,"style":178},[83,84],"margin-right:0.1389em;",[39,180,136],{"className":181},[182],"mopen",[39,184,186,189],{"className":185},[83],[39,187,142],{"className":188},[83,84],[39,190,193],{"className":191},[192],"msupsub",[39,194,197],{"className":195},[196],"vlist-t",[39,198,201],{"className":199},[200],"vlist-r",[39,202,206],{"className":203,"style":205},[204],"vlist","height:0.7519em;",[39,207,209,214],{"style":208},"top:-3.063em;margin-right:0.05em;",[39,210],{"className":211,"style":213},[212],"pstrut","height:2.7em;",[39,215,221],{"className":216},[217,218,219,220],"sizing","reset-size6","size3","mtight",[39,222,224],{"className":223},[83,220],[39,225,147],{"className":226},[83,220],[39,228],{"className":229,"style":231},[230],"mspace","margin-right:0.2778em;",[39,233,150],{"className":234},[235],"mrel",[39,237],{"className":238,"style":231},[230],[39,240,242,246,249,253,257,260],{"className":241},[74],[39,243],{"className":244,"style":245},[78],"height:1em;vertical-align:-0.25em;",[39,247,142],{"className":248},[83,84],[39,250,155],{"className":251},[252],"mpunct",[39,254],{"className":255,"style":256},[230],"margin-right:0.1667em;",[39,258,158],{"className":259},[83,84],[39,261,161],{"className":262},[263],"mclose"," 为在状态 ",[39,266,268,281],{"className":267,"translate":43},[42],[39,269,271],{"className":270},[47],[49,272,273],{"xmlns":51},[53,274,275,279],{},[56,276,277],{},[59,278,142],{},[63,280,142],{"encoding":65},[39,282,284],{"className":283,"ariaHidden":70},[69],[39,285,287,291],{"className":286},[74],[39,288],{"className":289,"style":290},[78],"height:0.4306em;",[39,292,142],{"className":293},[83,84]," 执行动作 ",[39,296,298,311],{"className":297,"translate":43},[42],[39,299,301],{"className":300},[47],[49,302,303],{"xmlns":51},[53,304,305,309],{},[56,306,307],{},[59,308,158],{},[63,310,158],{"encoding":65},[39,312,314],{"className":313,"ariaHidden":70},[69],[39,315,317,320],{"className":316},[74],[39,318],{"className":319,"style":290},[78],[39,321,158],{"className":322},[83,84]," 后转移到状态 ",[39,325,327,345],{"className":326,"translate":43},[42],[39,328,330],{"className":329},[47],[49,331,332],{"xmlns":51},[53,333,334,342],{},[56,335,336],{},[138,337,338,340],{},[59,339,142],{},[133,341,147],{"mathvariant":145,"lspace":146,"rspace":146},[63,343,344],{"encoding":65},"s'",[39,346,348],{"className":347,"ariaHidden":70},[69],[39,349,351,354],{"className":350},[74],[39,352],{"className":353,"style":205},[78],[39,355,357,360],{"className":356},[83],[39,358,142],{"className":359},[83,84],[39,361,363],{"className":362},[192],[39,364,366],{"className":365},[196],[39,367,369],{"className":368},[200],[39,370,372],{"className":371,"style":205},[204],[39,373,374,377],{"style":208},[39,375],{"className":376,"style":213},[212],[39,378,380],{"className":379},[217,218,219,220],[39,381,383],{"className":382},[83,220],[39,384,147],{"className":385},[83,220]," 的概率，",[39,388,390,426],{"className":389,"translate":43},[42],[39,391,393],{"className":392},[47],[49,394,395],{"xmlns":51},[53,396,397,423],{},[56,398,399,402,405,407,410,412,414,416,419],{},[59,400,401],{},"r",[133,403,404],{},":",[59,406,61],{},[133,408,409],{},"×",[59,411,101],{},[133,413,409],{},[59,415,61],{},[133,417,418],{},"→",[59,420,422],{"mathvariant":421},"double-struck","R",[63,424,425],{"encoding":65},"r:S\\times A\\times S\\to\\mathbb{R}",[39,427,429,448,469,487,505],{"className":428,"ariaHidden":70},[69],[39,430,432,435,439,442,445],{"className":431},[74],[39,433],{"className":434,"style":290},[78],[39,436,401],{"className":437,"style":438},[83,84],"margin-right:0.0278em;",[39,440],{"className":441,"style":231},[230],[39,443,404],{"className":444},[235],[39,446],{"className":447,"style":231},[230],[39,449,451,455,458,462,466],{"className":450},[74],[39,452],{"className":453,"style":454},[78],"height:0.7667em;vertical-align:-0.0833em;",[39,456,61],{"className":457,"style":85},[83,84],[39,459],{"className":460,"style":461},[230],"margin-right:0.2222em;",[39,463,409],{"className":464},[465],"mbin",[39,467],{"className":468,"style":461},[230],[39,470,472,475,478,481,484],{"className":471},[74],[39,473],{"className":474,"style":454},[78],[39,476,101],{"className":477},[83,84],[39,479],{"className":480,"style":461},[230],[39,482,409],{"className":483},[465],[39,485],{"className":486,"style":461},[230],[39,488,490,493,496,499,502],{"className":489},[74],[39,491],{"className":492,"style":79},[78],[39,494,61],{"className":495,"style":85},[83,84],[39,497],{"className":498,"style":231},[230],[39,500,418],{"className":501},[235],[39,503],{"className":504,"style":231},[230],[39,506,508,512],{"className":507},[74],[39,509],{"className":510,"style":511},[78],"height:0.6889em;",[39,513,422],{"className":514},[83,515],"mathbb"," 为奖励函数，",[39,518,520,552],{"className":519,"translate":43},[42],[39,521,523],{"className":522},[47],[49,524,525],{"xmlns":51},[53,526,527,549],{},[56,528,529,532,535,538,542,544,547],{},[59,530,531],{},"γ",[133,533,534],{},"∈",[133,536,537],{"stretchy":135},"[",[539,540,541],"mn",{},"0",[133,543,155],{"separator":70},[539,545,546],{},"1",[133,548,161],{"stretchy":135},[63,550,551],{"encoding":65},"\\gamma\\in[0,1)",[39,553,555,575],{"className":554,"ariaHidden":70},[69],[39,556,558,562,566,569,572],{"className":557},[74],[39,559],{"className":560,"style":561},[78],"height:0.7335em;vertical-align:-0.1944em;",[39,563,531],{"className":564,"style":565},[83,84],"margin-right:0.0556em;",[39,567],{"className":568,"style":231},[230],[39,570,534],{"className":571},[235],[39,573],{"className":574,"style":231},[230],[39,576,578,581,584,587,590,593,596],{"className":577},[74],[39,579],{"className":580,"style":245},[78],[39,582,537],{"className":583},[182],[39,585,541],{"className":586},[83],[39,588,155],{"className":589},[252],[39,591],{"className":592,"style":256},[230],[39,594,546],{"className":595},[83],[39,597,161],{"className":598},[263]," 为折扣因子。记 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为截至时刻 ",[39,685,687,700],{"className":686,"translate":43},[42],[39,688,690],{"className":689},[47],[49,691,692],{"xmlns":51},[53,693,694,698],{},[56,695,696],{},[59,697,621],{},[63,699,621],{"encoding":65},[39,701,703],{"className":702,"ariaHidden":70},[69],[39,704,706,710],{"className":705},[74],[39,707],{"className":708,"style":709},[78],"height:0.6151em;",[39,711,621],{"className":712},[83,84]," 的历史信息。若对任意 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Markov decision process，折扣 MDP），记作 ",[39,1494,1496,1535],{"className":1495,"translate":43},[42],[39,1497,1499],{"className":1498},[47],[49,1500,1501],{"xmlns":51},[53,1502,1503,1532],{},[56,1504,1505,1508,1510,1512,1514,1516,1518,1520,1522,1524,1526,1528,1530],{},[59,1506,1507],{"mathvariant":617},"M",[133,1509,793],{},[133,1511,136],{"stretchy":135},[59,1513,61],{},[133,1515,155],{"separator":70},[59,1517,101],{},[133,1519,155],{"separator":70},[59,1521,131],{},[133,1523,155],{"separator":70},[59,1525,401],{},[133,1527,155],{"separator":70},[59,1529,531],{},[133,1531,161],{"stretchy":135},[63,1533,1534],{"encoding":65},"\\mathcal{M}=(S,A,P,r,\\gamma)",[39,1536,1538,1556],{"className":1537,"ariaHidden":70},[69],[39,1539,1541,1544,1547,1550,1553],{"className":1540},[74],[39,1542],{"className":1543,"style":79},[78],[39,1545,1507],{"className":1546},[83,641],[39,1548],{"className":1549,"style":231},[230],[39,1551,793],{"className":1552},[235],[39,1554],{"className":1555,"style":231},[230],[39,1557,1559,1562,1565,1568,1571,1574,1577,1580,1583,1586,1589,1592,1595,1598,1601,1604],{"className":1558},[74],[39,1560],{"className":1561,"style":245},[78],[39,1563,136],{"className":1564},[182],[39,1566,61],{"className":1567,"style":85},[83,84],[39,1569,155],{"className":1570},[252],[39,1572],{"className":1573,"style":256},[230],[39,1575,101],{"className":1576},[83,84],[39,1578,155],{"className":1579},[252],[39,1581],{"className":1582,"style":256},[230],[39,1584,131],{"className":1585,"style":178},[83,84],[39,1587,155],{"className":1588},[252],[39,1590],{"className":1591,"style":256},[230],[39,1593,401],{"className":1594,"style":438},[83,84],[39,1596,155],{"className":1597},[252],[39,1599],{"className":1600,"style":256},[230],[39,1602,531],{"className":1603,"style":565},[83,84],[39,1605,161],{"className":1606},[263],"。",[11,1609,1610,1611,1639,1640,1668,1669,1785,1786,1977,1978,2076],{},"本文默认 ",[39,1612,1614,1627],{"className":1613,"translate":43},[42],[39,1615,1617],{"className":1616},[47],[49,1618,1619],{"xmlns":51},[53,1620,1621,1625],{},[56,1622,1623],{},[59,1624,61],{},[63,1626,61],{"encoding":65},[39,1628,1630],{"className":1629,"ariaHidden":70},[69],[39,1631,1633,1636],{"className":1632},[74],[39,1634],{"className":1635,"style":79},[78],[39,1637,61],{"className":1638,"style":85},[83,84]," 与 ",[39,1641,1643,1656],{"className":1642,"translate":43},[42],[39,1644,1646],{"className":1645},[47],[49,1647,1648],{"xmlns":51},[53,1649,1650,1654],{},[56,1651,1652],{},[59,1653,101],{},[63,1655,101],{"encoding":65},[39,1657,1659],{"className":1658,"ariaHidden":70},[69],[39,1660,1662,1665],{"className":1661},[74],[39,1663],{"className":1664,"style":79},[78],[39,1666,101],{"className":1667},[83,84]," 均为有限集，且存在常数 ",[39,1670,1672,1700],{"className":1671,"translate":43},[42],[39,1673,1675],{"className":1674},[47],[49,1676,1677],{"xmlns":51},[53,1678,1679,1697],{},[56,1680,1681,1691,1694],{},[613,1682,1683,1685,1688],{},[59,1684,422],{},[59,1686,1687],{},"max",[133,1689,1690],{},"⁡",[133,1692,1693],{},"\u003C",[59,1695,1696],{"mathvariant":145},"∞",[63,1698,1699],{"encoding":65},"R_{\\max}\u003C\\infty",[39,1701,1703,1776],{"className":1702,"ariaHidden":70},[69],[39,1704,1706,1709,1767,1770,1773],{"className":1705},[74],[39,1707],{"className":1708,"style":634},[78],[39,1710,1712,1716],{"className":1711},[83],[39,1713,422],{"className":1714,"style":1715},[83,84],"margin-right:0.0077em;",[39,1717,1719],{"className":1718},[192],[39,1720,1722,1759],{"className":1721},[196,649],[39,1723,1725,1756],{"className":1724},[200],[39,1726,1729],{"className":1727,"style":1728},[204],"height:0.1514em;",[39,1730,1732,1735],{"style":1731},"top:-2.55em;margin-left:-0.0077em;margin-right:0.05em;",[39,1733],{"className":1734,"style":213},[212],[39,1736,1738],{"className":1737},[217,218,219,220],[39,1739,1741],{"className":1740},[83,220],[39,1742,1745,1749,1752],{"className":1743},[1744,220],"mop",[39,1746,1748],{"className":1747},[220],"m",[39,1750,158],{"className":1751},[220],[39,1753,1755],{"className":1754},[220],"x",[39,1757,673],{"className":1758},[672],[39,1760,1762],{"className":1761},[200],[39,1763,1765],{"className":1764,"style":680},[204],[39,1766],{},[39,1768],{"className":1769,"style":231},[230],[39,1771,1693],{"className":1772},[235],[39,1774],{"className":1775,"style":231},[230],[39,1777,1779,1782],{"className":1778},[74],[39,1780],{"className":1781,"style":290},[78],[39,1783,1696],{"className":1784},[83]," 使得 ",[39,1787,1789,1836],{"className":1788,"translate":43},[42],[39,1790,1792],{"className":1791},[47],[49,1793,1794],{"xmlns":51},[53,1795,1796,1833],{},[56,1797,1798,1800,1802,1804,1806,1808,1810,1812,1818,1820,1822,1825],{},[59,1799,150],{"mathvariant":145},[59,1801,401],{},[133,1803,136],{"stretchy":135},[59,1805,142],{},[133,1807,155],{"separator":70},[59,1809,158],{},[133,1811,155],{"separator":70},[138,1813,1814,1816],{},[59,1815,142],{},[133,1817,147],{"mathvariant":145,"lspace":146,"rspace":146},[133,1819,161],{"stretchy":135},[59,1821,150],{"mathvariant":145},[133,1823,1824],{},"≤",[613,1826,1827,1829,1831],{},[59,1828,422],{},[59,1830,1687],{},[133,1832,1690],{},[63,1834,1835],{"encoding":65},"|r(s,a,s')|\\le R_{\\max}",[39,1837,1839,1919],{"className":1838,"ariaHidden":70},[69],[39,1840,1842,1845,1848,1851,1854,1857,1860,1863,1866,1869,1872,1904,1907,1910,1913,1916],{"className":1841},[74],[39,1843],{"className":1844,"style":174},[78],[39,1846,150],{"className":1847},[83],[39,1849,401],{"className":1850,"style":438},[83,84],[39,1852,136],{"className":1853},[182],[39,1855,142],{"className":1856},[83,84],[39,1858,155],{"className":1859},[252],[39,1861],{"className":1862,"style":256},[230],[39,1864,158],{"className":1865},[83,84],[39,1867,155],{"className":1868},[252],[39,1870],{"className":1871,"style":256},[230],[39,1873,1875,1878],{"className":1874},[83],[39,1876,142],{"className":1877},[83,84],[39,1879,1881],{"className":1880},[192],[39,1882,1884],{"className":1883},[196],[39,1885,1887],{"className":1886},[200],[39,1888,1890],{"className":1889,"style":205},[204],[39,1891,1892,1895],{"style":208},[39,1893],{"className":1894,"style":213},[212],[39,1896,1898],{"className":1897},[217,218,219,220],[39,1899,1901],{"className":1900},[83,220],[39,1902,147],{"className":1903},[83,220],[39,1905,161],{"className":1906},[263],[39,1908,150],{"className":1909},[83],[39,1911],{"className":1912,"style":231},[230],[39,1914,1824],{"className":1915},[235],[39,1917],{"className":1918,"style":231},[230],[39,1920,1922,1925],{"className":1921},[74],[39,1923],{"className":1924,"style":634},[78],[39,1926,1928,1931],{"className":1927},[83],[39,1929,422],{"className":1930,"style":1715},[83,84],[39,1932,1934],{"className":1933},[192],[39,1935,1937,1969],{"className":1936},[196,649],[39,1938,1940,1966],{"className":1939},[200],[39,1941,1943],{"className":1942,"style":1728},[204],[39,1944,1945,1948],{"style":1731},[39,1946],{"className":1947,"style":213},[212],[39,1949,1951],{"className":1950},[217,218,219,220],[39,1952,1954],{"className":1953},[83,220],[39,1955,1957,1960,1963],{"className":1956},[1744,220],[39,1958,1748],{"className":1959},[220],[39,1961,158],{"className":1962},[220],[39,1964,1755],{"className":1965},[220],[39,1967,673],{"className":1968},[672],[39,1970,1972],{"className":1971},[200],[39,1973,1975],{"className":1974,"style":680},[204],[39,1976],{}," 对一切 ",[39,1979,1981,2011],{"className":1980,"translate":43},[42],[39,1982,1984],{"className":1983},[47],[49,1985,1986],{"xmlns":51},[53,1987,1988,2008],{},[56,1989,1990,1992,1994,1996,1998,2000,2006],{},[133,1991,136],{"stretchy":135},[59,1993,142],{},[133,1995,155],{"separator":70},[59,1997,158],{},[133,1999,155],{"separator":70},[138,2001,2002,2004],{},[59,2003,142],{},[133,2005,147],{"mathvariant":145,"lspace":146,"rspace":146},[133,2007,161],{"stretchy":135},[63,2009,2010],{"encoding":65},"(s,a,s')",[39,2012,2014],{"className":2013,"ariaHidden":70},[69],[39,2015,2017,2020,2023,2026,2029,2032,2035,2038,2041,2073],{"className":2016},[74],[39,2018],{"className":2019,"style":174},[78],[39,2021,136],{"className":2022},[182],[39,2024,142],{"className":2025},[83,84],[39,2027,155],{"className":2028},[252],[39,2030],{"className":2031,"style":256},[230],[39,2033,158],{"className":2034},[83,84],[39,2036,155],{"className":2037},[252],[39,2039],{"className":2040,"style":256},[230],[39,2042,2044,2047],{"className":2043},[83],[39,2045,142],{"className":2046},[83,84],[39,2048,2050],{"className":2049},[192],[39,2051,2053],{"className":2052},[196],[39,2054,2056],{"className":2055},[200],[39,2057,2059],{"className":2058,"style":205},[204],[39,2060,2061,2064],{"style":208},[39,2062],{"className":2063,"style":213},[212],[39,2065,2067],{"className":2066},[217,218,219,220],[39,2068,2070],{"className":2069},[83,220],[39,2071,147],{"className":2072},[83,220],[39,2074,161],{"className":2075},[263]," 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。",[11,2078,2079,2080,2083,2084,2115,2116,2183,2184,2228,2229,2295,2296,2324,2325,2353,2354,742,2382,2447,2448,2477,2478,2506,2507,2510,2511,1607],{},"一个",[1489,2081,2082],{},"策略"," ",[39,2085,2087,2102],{"className":2086,"translate":43},[42],[39,2088,2090],{"className":2089},[47],[49,2091,2092],{"xmlns":51},[53,2093,2094,2099],{},[56,2095,2096],{},[59,2097,2098],{},"π",[63,2100,2101],{"encoding":65},"\\pi",[39,2103,2105],{"className":2104,"ariaHidden":70},[69],[39,2106,2108,2111],{"className":2107},[74],[39,2109],{"className":2110,"style":290},[78],[39,2112,2098],{"className":2113,"style":2114},[83,84],"margin-right:0.0359em;"," 是映射 ",[39,2117,2119,2144],{"className":2118,"translate":43},[42],[39,2120,2122],{"className":2121},[47],[49,2123,2124],{"xmlns":51},[53,2125,2126,2141],{},[56,2127,2128,2130,2132,2135,2137,2139],{},[59,2129,61],{},[133,2131,418],{},[59,2133,2134],{"mathvariant":145},"Δ",[133,2136,136],{"stretchy":135},[59,2138,101],{},[133,2140,161],{"stretchy":135},[63,2142,2143],{"encoding":65},"S\\to\\Delta(A)",[39,2145,2147,2165],{"className":2146,"ariaHidden":70},[69],[39,2148,2150,2153,2156,2159,2162],{"className":2149},[74],[39,2151],{"className":2152,"style":79},[78],[39,2154,61],{"className":2155,"style":85},[83,84],[39,2157],{"className":2158,"style":231},[230],[39,2160,418],{"className":2161},[235],[39,2163],{"className":2164,"style":231},[230],[39,2166,2168,2171,2174,2177,2180],{"className":2167},[74],[39,2169],{"className":2170,"style":245},[78],[39,2172,2134],{"className":2173},[83],[39,2175,136],{"className":2176},[182],[39,2178,101],{"className":2179},[83,84],[39,2181,161],{"className":2182},[263],"，其中 ",[39,2185,2187,2207],{"className":2186,"translate":43},[42],[39,2188,2190],{"className":2189},[47],[49,2191,2192],{"xmlns":51},[53,2193,2194,2204],{},[56,2195,2196,2198,2200,2202],{},[59,2197,2134],{"mathvariant":145},[133,2199,136],{"stretchy":135},[59,2201,101],{},[133,2203,161],{"stretchy":135},[63,2205,2206],{"encoding":65},"\\Delta(A)",[39,2208,2210],{"className":2209,"ariaHidden":70},[69],[39,2211,2213,2216,2219,2222,2225],{"className":2212},[74],[39,2214],{"className":2215,"style":245},[78],[39,2217,2134],{"className":2218},[83],[39,2220,136],{"className":2221},[182],[39,2223,101],{"className":2224},[83,84],[39,2226,161],{"className":2227},[263]," 是动作集上的概率单纯形。",[39,2230,2232,2256],{"className":2231,"translate":43},[42],[39,2233,2235],{"className":2234},[47],[49,2236,2237],{"xmlns":51},[53,2238,2239,2253],{},[56,2240,2241,2243,2245,2247,2249,2251],{},[59,2242,2098],{},[133,2244,136],{"stretchy":135},[59,2246,158],{},[133,2248,150],{},[59,2250,142],{},[133,2252,161],{"stretchy":135},[63,2254,2255],{"encoding":65},"\\pi(a\\mid s)",[39,2257,2259,2283],{"className":2258,"ariaHidden":70},[69],[39,2260,2262,2265,2268,2271,2274,2277,2280],{"className":2261},[74],[39,2263],{"className":2264,"style":245},[78],[39,2266,2098],{"className":2267,"style":2114},[83,84],[39,2269,136],{"className":2270},[182],[39,2272,158],{"className":2273},[83,84],[39,2275],{"className":2276,"style":231},[230],[39,2278,150],{"className":2279},[235],[39,2281],{"className":2282,"style":231},[230],[39,2284,2286,2289,2292],{"className":2285},[74],[39,2287],{"className":2288,"style":245},[78],[39,2290,142],{"className":2291},[83,84],[39,2293,161],{"className":2294},[263]," 表示在状态 ",[39,2297,2299,2312],{"className":2298,"translate":43},[42],[39,2300,2302],{"className":2301},[47],[49,2303,2304],{"xmlns":51},[53,2305,2306,2310],{},[56,2307,2308],{},[59,2309,142],{},[63,2311,142],{"encoding":65},[39,2313,2315],{"className":2314,"ariaHidden":70},[69],[39,2316,2318,2321],{"className":2317},[74],[39,2319],{"className":2320,"style":290},[78],[39,2322,142],{"className":2323},[83,84]," 选择动作 ",[39,2326,2328,2341],{"className":2327,"translate":43},[42],[39,2329,2331],{"className":2330},[47],[49,2332,2333],{"xmlns":51},[53,2334,2335,2339],{},[56,2336,2337],{},[59,2338,158],{},[63,2340,158],{"encoding":65},[39,2342,2344],{"className":2343,"ariaHidden":70},[69],[39,2345,2347,2350],{"className":2346},[74],[39,2348],{"className":2349,"style":290},[78],[39,2351,158],{"className":2352},[83,84]," 的概率。若对每个 ",[39,2355,2357,2370],{"className":2356,"translate":43},[42],[39,2358,2360],{"className":2359},[47],[49,2361,2362],{"xmlns":51},[53,2363,2364,2368],{},[56,2365,2366],{},[59,2367,142],{},[63,2369,142],{"encoding":65},[39,2371,2373],{"className":2372,"ariaHidden":70},[69],[39,2374,2376,2379],{"className":2375},[74],[39,2377],{"className":2378,"style":290},[78],[39,2380,142],{"className":2381},[83,84],[39,2383,2385,2408],{"className":2384,"translate":43},[42],[39,2386,2388],{"className":2387},[47],[49,2389,2390],{"xmlns":51},[53,2391,2392,2406],{},[56,2393,2394,2396,2398,2400,2402,2404],{},[59,2395,2098],{},[133,2397,136],{"stretchy":135},[59,2399,158],{},[133,2401,150],{},[59,2403,142],{},[133,2405,161],{"stretchy":135},[63,2407,2255],{"encoding":65},[39,2409,2411,2435],{"className":2410,"ariaHidden":70},[69],[39,2412,2414,2417,2420,2423,2426,2429,2432],{"className":2413},[74],[39,2415],{"className":2416,"style":245},[78],[39,2418,2098],{"className":2419,"style":2114},[83,84],[39,2421,136],{"className":2422},[182],[39,2424,158],{"className":2425},[83,84],[39,2427],{"className":2428,"style":231},[230],[39,2430,150],{"className":2431},[235],[39,2433],{"className":2434,"style":231},[230],[39,2436,2438,2441,2444],{"className":2437},[74],[39,2439],{"className":2440,"style":245},[78],[39,2442,142],{"className":2443},[83,84],[39,2445,161],{"className":2446},[263]," 只在某一个动作处取 ",[39,2449,2451,2464],{"className":2450,"translate":43},[42],[39,2452,2454],{"className":2453},[47],[49,2455,2456],{"xmlns":51},[53,2457,2458,2462],{},[56,2459,2460],{},[539,2461,546],{},[63,2463,546],{"encoding":65},[39,2465,2467],{"className":2466,"ariaHidden":70},[69],[39,2468,2470,2474],{"className":2469},[74],[39,2471],{"className":2472,"style":2473},[78],"height:0.6444em;",[39,2475,546],{"className":2476},[83],"，则称 ",[39,2479,2481,2494],{"className":2480,"translate":43},[42],[39,2482,2484],{"className":2483},[47],[49,2485,2486],{"xmlns":51},[53,2487,2488,2492],{},[56,2489,2490],{},[59,2491,2098],{},[63,2493,2101],{"encoding":65},[39,2495,2497],{"className":2496,"ariaHidden":70},[69],[39,2498,2500,2503],{"className":2499},[74],[39,2501],{"className":2502,"style":290},[78],[39,2504,2098],{"className":2505,"style":2114},[83,84]," 为",[1489,2508,2509],{},"确定性策略","，并可将其视为映射 ",[39,2512,2514,2536],{"className":2513,"translate":43},[42],[39,2515,2517],{"className":2516},[47],[49,2518,2519],{"xmlns":51},[53,2520,2521,2533],{},[56,2522,2523,2525,2527,2529,2531],{},[59,2524,2098],{},[133,2526,404],{},[59,2528,61],{},[133,2530,418],{},[59,2532,101],{},[63,2534,2535],{"encoding":65},"\\pi:S\\to A",[39,2537,2539,2557,2575],{"className":2538,"ariaHidden":70},[69],[39,2540,2542,2545,2548,2551,2554],{"className":2541},[74],[39,2543],{"className":2544,"style":290},[78],[39,2546,2098],{"className":2547,"style":2114},[83,84],[39,2549],{"className":2550,"style":231},[230],[39,2552,404],{"className":2553},[235],[39,2555],{"className":2556,"style":231},[230],[39,2558,2560,2563,2566,2569,2572],{"className":2559},[74],[39,2561],{"className":2562,"style":79},[78],[39,2564,61],{"className":2565,"style":85},[83,84],[39,2567],{"className":2568,"style":231},[230],[39,2570,418],{"className":2571},[235],[39,2573],{"className":2574,"style":231},[230],[39,2576,2578,2581],{"className":2577},[74],[39,2579],{"className":2580,"style":79},[78],[39,2582,101],{"className":2583},[83,84],[26,2585,2587],{"id":2586},"_12-回报与值函数","1.2 回报与值函数",[31,2589,2592,2844,3126,3162],{"id":2590,"title":2591},"definition-return","定义 1.2（折扣回报）",[11,2593,2594,2595,2843],{},"设轨迹为 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开始的",[1489,3159,3160],{},"折扣回报","定义为",[39,3163,3165],{"className":3164,"translate":43},[746],[39,3166,3168,3226],{"className":3167,"translate":43},[42],[39,3169,3171],{"className":3170},[47],[49,3172,3173],{"xmlns":51,"display":755},[53,3174,3175,3223],{},[56,3176,3177,3184,3186,3203,3209,3221],{},[613,3178,3179,3182],{},[59,3180,3181],{},"G",[59,3183,621],{},[133,3185,793],{},[3187,3188,3189,3192,3201],"munderover",{},[133,3190,3191],{},"∑",[56,3193,3194,3197,3199],{},[59,3195,3196],{},"k",[133,3198,793],{},[539,3200,541],{},[59,3202,1696],{"mathvariant":145},[138,3204,3205,3207],{},[59,3206,531],{},[59,3208,3196],{},[613,3210,3211,3213],{},[59,3212,401],{},[56,3214,3215,3217,3219],{},[59,3216,621],{},[133,3218,788],{},[59,3220,3196],{},[59,3222,2903],{"mathvariant":145},[63,3224,3225],{"encoding":65},"G_t=\\sum_{k=0}^{\\infty}\\gamma^k 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是在从 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转移到 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时获得的奖励。若奖励只依赖于 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成立，则对任意轨迹与任意 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由三角不等式与几何级数求和，",[39,5598,5600],{"className":5599,"translate":43},[746],[39,5601,5603,5747],{"className":5602,"translate":43},[42],[39,5604,5606],{"className":5605},[47],[49,5607,5608],{"xmlns":51,"display":755},[53,5609,5610,5744],{},[759,5611,5612,5676],{"rowspacing":761,"columnalign":762,"columnspacing":146},[764,5613,5614,5630],{},[767,5615,5616],{},[770,5617,5618],{"scriptlevel":541,"displaystyle":70},[56,5619,5620,5622,5628],{},[59,5621,150],{"mathvariant":145},[613,5623,5624,5626],{},[59,5625,3181],{},[59,5627,621],{},[59,5629,150],{"mathvariant":145},[767,5631,5632],{},[770,5633,5634],{"scriptlevel":541,"displaystyle":70},[56,5635,5636,5638,5640,5654,5660,5662,5674],{},[56,5637],{},[133,5639,1824],{},[3187,5641,5642,5644,5652],{},[133,5643,3191],{},[56,5645,5646,5648,5650],{},[59,5647,3196],{},[133,5649,793],{},[539,5651,541],{},[59,5653,1696],{"mathvariant":145},[138,5655,5656,5658],{},[59,5657,531],{},[59,5659,3196],{},[59,5661,150],{"mathvariant":145},[613,5663,5664,5666],{},[59,5665,401],{},[56,5667,5668,5670,5672],{},[59,5669,621],{},[133,5671,788],{},[59,5673,3196],{},[59,5675,150],{"mathvariant":145},[764,5677,5678,5684],{},[767,5679,5680],{},[770,5681,5682],{"scriptlevel":541,"displaystyle":70},[56,5683],{},[767,5685,5686],{},[770,5687,5688],{"scriptlevel":541,"displaystyle":70},[56,5689,5690,5692,5694,5702,5716,5722,5724,5742],{},[56,5691],{},[133,5693,1824],{},[613,5695,5696,5698,5700],{},[59,5697,422],{},[59,5699,1687],{},[133,5701,1690],{},[3187,5703,5704,5706,5714],{},[133,5705,3191],{},[56,5707,5708,5710,5712],{},[59,5709,3196],{},[133,5711,793],{},[539,5713,541],{},[59,5715,1696],{"mathvariant":145},[138,5717,5718,5720],{},[59,5719,531],{},[59,5721,3196],{},[133,5723,793],{},[5362,5725,5726,5734],{},[613,5727,5728,5730,5732],{},[59,5729,422],{},[59,5731,1687],{},[133,5733,1690],{},[56,5735,5736,5738,5740],{},[539,5737,546],{},[133,5739,5378],{},[59,5741,531],{},[59,5743,2903],{"mathvariant":145},[63,5745,5746],{"encoding":65},"\\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}",[39,5748,5750],{"className":5749,"ariaHidden":70},[69],[39,5751,5753,5757],{"className":5752},[74],[39,5754],{"className":5755,"style":5756},[78],"height:6.207em;vertical-align:-2.8535em;",[39,5758,5760],{"className":5759},[83],[39,5761,5763,5853],{"className":5762},[759],[39,5764,5766],{"className":5765},[918],[39,5767,5769,5844],{"className":5768},[196,649],[39,5770,5772,5841],{"className":5771},[200],[39,5773,5776,5832],{"className":5774,"style":5775},[204],"height:3.3535em;",[39,5777,5779,5783],{"style":5778},"top:-5.3535em;",[39,5780],{"className":5781,"style":5782},[212],"height:3.6514em;",[39,5784,5786,5789,5829],{"className":5785},[83],[39,5787,150],{"className":5788},[83],[39,5790,5792,5795],{"className":5791},[83],[39,5793,3181],{"className":5794},[83,84],[39,5796,5798],{"className":5797},[192],[39,5799,5801,5821],{"className":5800},[196,649],[39,5802,5804,5818],{"className":5803},[200],[39,5805,5807],{"className":5806,"style":656},[204],[39,5808,5809,5812],{"style":966},[39,5810],{"className":5811,"style":213},[212],[39,5813,5815],{"className":5814},[217,218,219,220],[39,5816,621],{"className":5817},[83,84,220],[39,5819,673],{"className":5820},[672],[39,5822,5824],{"className":5823},[200],[39,5825,5827],{"className":5826,"style":680},[204],[39,5828],{},[39,5830,150],{"className":5831},[83],[39,5833,5835,5838],{"style":5834},"top:-2.1em;",[39,5836],{"className":5837,"style":5782},[212],[39,5839],{"className":5840},[83],[39,5842,673],{"className":5843},[672],[39,5845,5847],{"className":5846},[200],[39,5848,5851],{"className":5849,"style":5850},[204],"height:2.8535em;",[39,5852],{},[39,5854,5856],{"className":5855},[1223],[39,5857,5859,6351],{"className":5858},[196,649],[39,5860,5862,6348],{"className":5861},[200],[39,5863,5865,6039],{"className":5864,"style":5775},[204],[39,5866,5867,5870],{"style":5778},[39,5868],{"className":5869,"style":5782},[212],[39,5871,5873,5876,5879,5882,5885,5952,5955,5984,5987,6036],{"className":5872},[83],[39,5874],{"className":5875},[83],[39,5877],{"className":5878,"style":231},[230],[39,5880,1824],{"className":5881},[235],[39,5883],{"className":5884,"style":231},[230],[39,5886,5888],{"className":5887},[1744,3294],[39,5889,5891,5944],{"className":5890},[196,649],[39,5892,5894,5941],{"className":5893},[200],[39,5895,5897,5917,5927],{"className":5896,"style":3304},[204],[39,5898,5899,5902],{"style":3307},[39,5900],{"className":5901,"style":3311},[212],[39,5903,5905],{"className":5904},[217,218,219,220],[39,5906,5908,5911,5914],{"className":5907},[83,220],[39,5909,3196],{"className":5910,"style":3321},[83,84,220],[39,5912,793],{"className":5913},[235,220],[39,5915,541],{"className":5916},[83,220],[39,5918,5919,5922],{"style":3330},[39,5920],{"className":5921,"style":3311},[212],[39,5923,5924],{},[39,5925,3191],{"className":5926},[1744,3339,3340],[39,5928,5929,5932],{"style":3343},[39,5930],{"className":5931,"style":3311},[212],[39,5933,5935],{"className":5934},[217,218,219,220],[39,5936,5938],{"className":5937},[83,220],[39,5939,1696],{"className":5940},[83,220],[39,5942,673],{"className":5943},[672],[39,5945,5947],{"className":5946},[200],[39,5948,5950],{"className":5949,"style":3365},[204],[39,5951],{},[39,5953],{"className":5954,"style":256},[230],[39,5956,5958,5961],{"className":5957},[83],[39,5959,531],{"className":5960,"style":565},[83,84],[39,5962,5964],{"className":5963},[192],[39,5965,5967],{"className":5966},[196],[39,5968,5970],{"className":5969},[200],[39,5971,5973],{"className":5972,"style":3389},[204],[39,5974,5975,5978],{"style":1027},[39,5976],{"className":5977,"style":213},[212],[39,5979,5981],{"className":5980},[217,218,219,220],[39,5982,3196],{"className":5983,"style":3321},[83,84,220],[39,5985,150],{"className":5986},[83],[39,5988,5990,5993],{"className":5989},[83],[39,5991,401],{"className":5992,"style":438},[83,84],[39,5994,5996],{"className":5995},[192],[39,5997,5999,6028],{"className":5998},[196,649],[39,6000,6002,6025],{"className":6001},[200],[39,6003,6005],{"className":6004,"style":3419},[204],[39,6006,6007,6010],{"style":2937},[39,6008],{"className":6009,"style":213},[212],[39,6011,6013],{"className":6012},[217,218,219,220],[39,6014,6016,6019,6022],{"className":6015},[83,220],[39,6017,621],{"className":6018},[83,84,220],[39,6020,788],{"className":6021},[465,220],[39,6023,3196],{"className":6024,"style":3321},[83,84,220],[39,6026,673],{"className":6027},[672],[39,6029,6031],{"className":6030},[200],[39,6032,6034],{"className":6033,"style":994},[204],[39,6035],{},[39,6037,150],{"className":6038},[83],[39,6040,6041,6044],{"style":5834},[39,6042],{"className":6043,"style":5782},[212],[39,6045,6047,6050,6053,6056,6059,6111,6114,6181,6184,6213,6216,6219,6222,6345],{"className":6046},[83],[39,6048],{"className":6049},[83],[39,6051],{"className":6052,"style":231},[230],[39,6054,1824],{"className":6055},[235],[39,6057],{"className":6058,"style":231},[230],[39,6060,6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",[39,6363,6365,6383],{"className":6364,"translate":43},[42],[39,6366,6368],{"className":6367},[47],[49,6369,6370],{"xmlns":51},[53,6371,6372,6380],{},[56,6373,6374],{},[138,6375,6376,6378],{},[59,6377,4230],{},[59,6379,2098],{},[63,6381,6382],{"encoding":65},"V^\\pi",[39,6384,6386],{"className":6385,"ariaHidden":70},[69],[39,6387,6389,6392],{"className":6388},[74],[39,6390],{"className":6391,"style":79},[78],[39,6393,6395,6398],{"className":6394},[83],[39,6396,4230],{"className":6397,"style":461},[83,84],[39,6399,6401],{"className":6400},[192],[39,6402,6404],{"className":6403},[196],[39,6405,6407],{"className":6406},[200],[39,6408,6411],{"className":6409,"style":6410},[204],"height:0.6644em;",[39,6412,6413,6416],{"style":208},[39,6414],{"className":6415,"style":213},[212],[39,6417,6419],{"className":6418},[217,218,219,220],[39,6420,2098],{"className":6421,"style":2114},[83,84,220],[39,6423,6425,6443],{"className":6424,"translate":43},[42],[39,6426,6428],{"className":6427},[47],[49,6429,6430],{"xmlns":51},[53,6431,6432,6440],{},[56,6433,6434],{},[138,6435,6436,6438],{},[59,6437,4295],{},[59,6439,2098],{},[63,6441,6442],{"encoding":65},"Q^\\pi",[39,6444,6446],{"className":6445,"ariaHidden":70},[69],[39,6447,6449,6453],{"className":6448},[74],[39,6450],{"className":6451,"style":6452},[78],"height:0.8778em;vertical-align:-0.1944em;",[39,6454,6456,6459],{"className":6455},[83],[39,6457,4295],{"className":6458},[83,84],[39,6460,6462],{"className":6461},[192],[39,6463,6465],{"className":6464},[196],[39,6466,6468],{"className":6467},[200],[39,6469,6471],{"className":6470,"style":6410},[204],[39,6472,6473,6476],{"style":208},[39,6474],{"className":6475,"style":213},[212],[39,6477,6479],{"className":6478},[217,218,219,220],[39,6480,2098],{"className":6481,"style":2114},[83,84,220]," 均为有界函数。证毕。",[11,6484,6485],{},"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。",[26,6487,6489],{"id":6488},"_13-两个基本恒等式","1.3 两个基本恒等式",[31,6491,6494,6526,6791,6794],{"id":6492,"title":6493},"proposition-basic-identities","命题 1.5（值函数的基本恒等式）",[11,6495,6496,6497,6525],{},"对任意策略 ",[39,6498,6500,6513],{"className":6499,"translate":43},[42],[39,6501,6503],{"className":6502},[47],[49,6504,6505],{"xmlns":51},[53,6506,6507,6511],{},[56,6508,6509],{},[59,6510,2098],{},[63,6512,2101],{"encoding":65},[39,6514,6516],{"className":6515,"ariaHidden":70},[69],[39,6517,6519,6522],{"className":6518},[74],[39,6520],{"className":6521,"style":290},[78],[39,6523,2098],{"className":6524,"style":2114},[83,84],"，有",[39,6527,6529],{"className":6528,"translate":43},[746],[39,6530,6532,6595],{"className":6531,"translate":43},[42],[39,6533,6535],{"className":6534},[47],[49,6536,6537],{"xmlns":51,"display":755},[53,6538,6539,6592],{},[56,6540,6541,6547,6549,6551,6553,6555,6562,6564,6566,6568,6570,6572,6574,6580,6582,6584,6586,6588,6590],{},[138,6542,6543,6545],{},[59,6544,4230],{},[59,6546,2098],{},[133,6548,136],{"stretchy":135},[59,6550,142],{},[133,6552,161],{"stretchy":135},[133,6554,793],{},[6556,6557,6558,6560],"munder",{},[133,6559,3191],{},[59,6561,158],{},[59,6563,2098],{},[133,6565,136],{"stretchy":135},[59,6567,158],{},[133,6569,150],{},[59,6571,142],{},[133,6573,161],{"stretchy":135},[138,6575,6576,6578],{},[59,6577,4295],{},[59,6579,2098],{},[133,6581,136],{"stretchy":135},[59,6583,142],{},[133,6585,155],{"separator":70},[59,6587,158],{},[133,6589,161],{"stretchy":135},[133,6591,155],{"separator":70},[63,6593,6594],{"encoding":65},"V^\\pi(s)=\\sum_{a}\\pi(a\\mid s)Q^\\pi(s,a),",[39,6596,6598,6651,6729],{"className":6597,"ariaHidden":70},[69],[39,6599,6601,6604,6633,6636,6639,6642,6645,6648],{"className":6600},[74],[39,6602],{"className":6603,"style":245},[78],[39,6605,6607,6610],{"className":6606},[83],[39,6608,4230],{"className":6609,"style":461},[83,84],[39,6611,6613],{"className":6612},[192],[39,6614,6616],{"className":6615},[196],[39,6617,6619],{"className":6618},[200],[39,6620,6622],{"className":6621,"style":4421},[204],[39,6623,6624,6627],{"style":1027},[39,6625],{"className":6626,"style":213},[212],[39,6628,6630],{"className":6629},[217,218,219,220],[39,6631,2098],{"className":6632,"style":2114},[83,84,220],[39,6634,136],{"className":6635},[182],[39,6637,142],{"className":6638},[83,84],[39,6640,161],{"className":6641},[263],[39,6643],{"className":6644,"style":231},[230],[39,6646,793],{"className":6647},[235],[39,6649],{"className":6650,"style":231},[230],[39,6652,6654,6658,6708,6711,6714,6717,6720,6723,6726],{"className":6653},[74],[39,6655],{"className":6656,"style":6657},[78],"height:2.3em;vertical-align:-1.25em;",[39,6659,6661],{"className":6660},[1744,3294],[39,6662,6664,6699],{"className":6663},[196,649],[39,6665,6667,6696],{"className":6666},[200],[39,6668,6671,6686],{"className":6669,"style":6670},[204],"height:1.05em;",[39,6672,6674,6677],{"style":6673},"top:-1.9em;margin-left:0em;",[39,6675],{"className":6676,"style":3311},[212],[39,6678,6680],{"className":6679},[217,218,219,220],[39,6681,6683],{"className":6682},[83,220],[39,6684,158],{"className":6685},[83,84,220],[39,6687,6688,6691],{"style":3330},[39,6689],{"className":6690,"style":3311},[212],[39,6692,6693],{},[39,6694,3191],{"className":6695},[1744,3339,3340],[39,6697,673],{"className":6698},[672],[39,6700,6702],{"className":6701},[200],[39,6703,6706],{"className":6704,"style":6705},[204],"height:1.25em;",[39,6707],{},[39,6709],{"className":6710,"style":256},[230],[39,6712,2098],{"className":6713,"style":2114},[83,84],[39,6715,136],{"className":6716},[182],[39,6718,158],{"className":6719},[83,84],[39,6721],{"className":6722,"style":231},[230],[39,6724,150],{"className":6725},[235],[39,6727],{"className":6728,"style":231},[230],[39,6730,6732,6735,6738,6741,6770,6773,6776,6779,6782,6785,6788],{"className":6731},[74],[39,6733],{"className":6734,"style":245},[78],[39,6736,142],{"className":6737},[83,84],[39,6739,161],{"className":6740},[263],[39,6742,6744,6747],{"className":6743},[83],[39,6745,4295],{"className":6746},[83,84],[39,6748,6750],{"className":6749},[192],[39,6751,6753],{"className":6752},[196],[39,6754,6756],{"className":6755},[200],[39,6757,6759],{"className":6758,"style":4421},[204],[39,6760,6761,6764],{"style":1027},[39,6762],{"className":6763,"style":213},[212],[39,6765,6767],{"className":6766},[217,218,219,220],[39,6768,2098],{"className":6769,"style":2114},[83,84,220],[39,6771,136],{"className":6772},[182],[39,6774,142],{"className":6775},[83,84],[39,6777,155],{"className":6778},[252],[39,6780],{"className":6781,"style":256},[230],[39,6783,158],{"className":6784},[83,84],[39,6786,161],{"className":6787},[263],[39,6789,155],{"className":6790},[252],[11,6792,6793],{},"以及",[39,6795,6797],{"className":6796,"translate":43},[746],[39,6798,6800,6945],{"className":6799,"translate":43},[42],[39,6801,6803],{"className":6802},[47],[49,6804,6805],{"xmlns":51,"display":755},[53,6806,6807,6942],{},[759,6808,6809,6873],{"rowspacing":761,"columnalign":762,"columnspacing":146},[764,6810,6811,6833],{},[767,6812,6813],{},[770,6814,6815],{"scriptlevel":541,"displaystyle":70},[56,6816,6817,6823,6825,6827,6829,6831],{},[138,6818,6819,6821],{},[59,6820,4295],{},[59,6822,2098],{},[133,6824,136],{"stretchy":135},[59,6826,142],{},[133,6828,155],{"separator":70},[59,6830,158],{},[133,6832,161],{"stretchy":135},[767,6834,6835],{},[770,6836,6837],{"scriptlevel":541,"displaystyle":70},[56,6838,6839,6841,6843,6853,6855,6857,6863,6865,6867,6869,6871],{},[56,6840],{},[133,6842,793],{},[6556,6844,6845,6847],{},[133,6846,3191],{},[138,6848,6849,6851],{},[59,6850,142],{},[133,6852,147],{"mathvariant":145,"lspace":146,"rspace":146},[59,6854,131],{},[133,6856,136],{"stretchy":135},[138,6858,6859,6861],{},[59,6860,142],{},[133,6862,147],{"mathvariant":145,"lspace":146,"rspace":146},[133,6864,150],{},[59,6866,142],{},[133,6868,155],{"separator":70},[59,6870,158],{},[133,6872,161],{"stretchy":135},[764,6874,6875,6881],{},[767,6876,6877],{},[770,6878,6879],{"scriptlevel":541,"displaystyle":70},[56,6880],{},[767,6882,6883],{},[770,6884,6885],{"scriptlevel":541,"displaystyle":70},[56,6886,6887,6889,6892,6894,6940],{},[56,6888],{},[230,6890],{"width":6891},"1em",[133,6893,409],{},[56,6895,6896,6898,6900,6902,6904,6906,6908,6910,6916,6918,6920,6922,6928,6930,6936,6938],{},[133,6897,537],{"fence":70},[59,6899,401],{},[133,6901,136],{"stretchy":135},[59,6903,142],{},[133,6905,155],{"separator":70},[59,6907,158],{},[133,6909,155],{"separator":70},[138,6911,6912,6914],{},[59,6913,142],{},[133,6915,147],{"mathvariant":145,"lspace":146,"rspace":146},[133,6917,161],{"stretchy":135},[133,6919,788],{},[59,6921,531],{},[138,6923,6924,6926],{},[59,6925,4230],{},[59,6927,2098],{},[133,6929,136],{"stretchy":135},[138,6931,6932,6934],{},[59,6933,142],{},[133,6935,147],{"mathvariant":145,"lspace":146,"rspace":146},[133,6937,161],{"stretchy":135},[133,6939,4280],{"fence":70},[59,6941,2903],{"mathvariant":145},[63,6943,6944],{"encoding":65},"\\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}",[39,6946,6948],{"className":6947,"ariaHidden":70},[69],[39,6949,6951,6955],{"className":6950},[74],[39,6952],{"className":6953,"style":6954},[78],"height:3.844em;vertical-align:-1.672em;",[39,6956,6958],{"className":6957},[83],[39,6959,6961,7051],{"className":6960},[759],[39,6962,6964],{"className":6963},[918],[39,6965,6967,7042],{"className":6966},[196,649],[39,6968,6970,7039],{"className":6969},[200],[39,6971,6974,7030],{"className":6972,"style":6973},[204],"height:2.172em;",[39,6975,6977,6980],{"style":6976},"top:-4.172em;",[39,6978],{"className":6979,"style":3311},[212],[39,6981,6983,7012,7015,7018,7021,7024,7027],{"className":6982},[83],[39,6984,6986,6989],{"className":6985},[83],[39,6987,4295],{"className":6988},[83,84],[39,6990,6992],{"className":6991},[192],[39,6993,6995],{"className":6994},[196],[39,6996,6998],{"className":6997},[200],[39,6999,7001],{"className":7000,"style":4421},[204],[39,7002,7003,7006],{"style":1027},[39,7004],{"className":7005,"style":213},[212],[39,7007,7009],{"className":7008},[217,218,219,220],[39,7010,2098],{"className":7011,"style":2114},[83,84,220],[39,7013,136],{"className":7014},[182],[39,7016,142],{"className":7017},[83,84],[39,7019,155],{"className":7020},[252],[39,7022],{"className":7023,"style":256},[230],[39,7025,158],{"className":7026},[83,84],[39,7028,161],{"className":7029},[263],[39,7031,7033,7036],{"style":7032},"top:-1.738em;",[39,7034],{"className":7035,"style":3311},[212],[39,7037],{"className":7038},[83],[39,7040,673],{"className":7041},[672],[39,7043,7045],{"className":7044},[200],[39,7046,7049],{"className":7047,"style":7048},[204],"height:1.672em;",[39,7050],{},[39,7052,7054],{"className":7053},[1223],[39,7055,7057,7406],{"className":7056},[196,649],[39,7058,7060,7403],{"className":7059},[200],[39,7061,7063,7226],{"className":7062,"style":6973},[204],[39,7064,7065,7068],{"style":6976},[39,7066],{"className":7067,"style":3311},[212],[39,7069,7071,7074,7077,7080,7083,7161,7164,7167,7170,7202,7205,7208,7211,7214,7217,7220,7223],{"className":7070},[83],[39,7072],{"className":7073},[83],[39,7075],{"className":7076,"style":231},[230],[39,7078,793],{"className":7079},[235],[39,7081],{"className":7082,"style":231},[230],[39,7084,7086],{"className":7085},[1744,3294],[39,7087,7089,7152],{"className":7088},[196,649],[39,7090,7092,7149],{"className":7091},[200],[39,7093,7095,7139],{"className":7094,"style":6670},[204],[39,7096,7098,7101],{"style":7097},"top:-1.856em;margin-left:0em;",[39,7099],{"className":7100,"style":3311},[212],[39,7102,7104],{"className":7103},[217,218,219,220],[39,7105,7107],{"className":7106},[83,220],[39,7108,7110,7113],{"className":7109},[83,220],[39,7111,142],{"className":7112},[83,84,220],[39,7114,7116],{"className":7115},[192],[39,7117,7119],{"className":7118},[196],[39,7120,7122],{"className":7121},[200],[39,7123,7125],{"className":7124,"style":3997},[204],[39,7126,7127,7130],{"style":4000},[39,7128],{"className":7129,"style":4004},[212],[39,7131,7133],{"className":7132},[217,4008,4009,220],[39,7134,7136],{"className":7135},[83,220],[39,7137,147],{"className":7138},[83,220],[39,7140,7141,7144],{"style":3330},[39,7142],{"className":7143,"style":3311},[212],[39,7145,7146],{},[39,7147,3191],{"className":7148},[1744,3339,3340],[39,7150,673],{"className":7151},[672],[39,7153,7155],{"className":7154},[200],[39,7156,7159],{"className":7157,"style":7158},[204],"height:1.294em;",[39,7160],{},[39,7162],{"className":7163,"style":256},[230],[39,7165,131],{"className":7166,"style":178},[83,84],[39,7168,136],{"className":7169},[182],[39,7171,7173,7176],{"className":7172},[83],[39,7174,142],{"className":7175},[83,84],[39,7177,7179],{"className":7178},[192],[39,7180,7182],{"className":7181},[196],[39,7183,7185],{"className":7184},[200],[39,7186,7188],{"className":7187,"style":1024},[204],[39,7189,7190,7193],{"style":1027},[39,7191],{"className":7192,"style":213},[212],[39,7194,7196],{"className":7195},[217,218,219,220],[39,7197,7199],{"className":7198},[83,220],[39,7200,147],{"className":7201},[83,220],[39,7203],{"className":7204,"style":231},[230],[39,7206,150],{"className":7207},[235],[39,7209],{"className":7210,"style":231},[230],[39,7212,142],{"className":7213},[83,84],[39,7215,155],{"className":7216},[252],[39,7218],{"className":7219,"style":256},[230],[39,7221,158],{"className":7222},[83,84],[39,7224,161],{"className":7225},[263],[39,7227,7228,7231],{"style":7032},[39,7229],{"className":7230,"style":3311},[212],[39,7232,7234,7237,7241,7244,7247,7250,7397,7400],{"className":7233},[83],[39,7235],{"className":7236},[83],[39,7238],{"className":7239,"style":7240},[230],"margin-right:1em;",[39,7242],{"className":7243,"style":461},[230],[39,7245,409],{"className":7246},[465],[39,7248],{"className":7249,"style":461},[230],[39,7251,7253,7256,7259,7262,7265,7268,7271,7274,7277,7280,7312,7315,7318,7321,7324,7327,7356,7359,7391,7394],{"className":7252},[2842],[39,7254,537],{"className":7255,"style":4592},[182,4591],[39,7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第一式由全期望公式直接得到。对于第二式，利用 ",[39,7420,7422,7464],{"className":7421,"translate":43},[42],[39,7423,7425],{"className":7424},[47],[49,7426,7427],{"xmlns":51},[53,7428,7429,7461],{},[56,7430,7431,7437,7439,7445,7447,7449],{},[613,7432,7433,7435],{},[59,7434,3181],{},[59,7436,621],{},[133,7438,793],{},[613,7440,7441,7443],{},[59,7442,401],{},[59,7444,621],{},[133,7446,788],{},[59,7448,531],{},[613,7450,7451,7453],{},[59,7452,3181],{},[56,7454,7455,7457,7459],{},[59,7456,621],{},[133,7458,788],{},[539,7460,546],{},[63,7462,7463],{"encoding":65},"G_t=r_t+\\gamma 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s,a)r(s,a,s')",[39,8460,8462,8590],{"className":8461,"ariaHidden":70},[69],[39,8463,8465,8468,8540,8543,8546,8549,8581,8584,8587],{"className":8464},[74],[39,8466],{"className":8467,"style":3945},[78],[39,8469,8471,8474],{"className":8470},[1744],[39,8472,3191],{"className":8473,"style":3953},[1744,3339,3952],[39,8475,8477],{"className":8476},[192],[39,8478,8480,8532],{"className":8479},[196,649],[39,8481,8483,8529],{"className":8482},[200],[39,8484,8486],{"className":8485,"style":3966},[204],[39,8487,8488,8491],{"style":3969},[39,8489],{"className":8490,"style":213},[212],[39,8492,8494],{"className":8493},[217,218,219,220],[39,8495,8497],{"className":8496},[83,220],[39,8498,8500,8503],{"className":8499},[83,220],[39,8501,142],{"className":8502},[83,84,220],[39,8504,8506],{"className":8505},[192],[39,8507,8509],{"className":8508},[196],[39,8510,8512],{"className":8511},[200],[39,8513,8515],{"className":8514,"style":3997},[204],[39,8516,8517,8520],{"style":4000},[39,8518],{"className":8519,"style":4004},[212],[39,8521,8523],{"className":8522},[217,4008,4009,220],[39,8524,8526],{"className":8525},[83,220],[39,8527,147],{"className":8528},[83,220],[39,8530,673],{"className":8531},[672],[39,8533,8535],{"className":8534},[200],[39,8536,8538],{"className":8537,"style":4025},[204],[39,8539],{},[39,8541],{"className":8542,"style":256},[230],[39,8544,131],{"className":8545,"style":178},[83,84],[39,8547,136],{"className":8548},[182],[39,8550,8552,8555],{"className":8551},[83],[39,8553,142],{"className":8554},[83,84],[39,8556,8558],{"className":8557},[192],[39,8559,8561],{"className":8560},[196],[39,8562,8564],{"className":8563},[200],[39,8565,8567],{"className":8566,"style":205},[204],[39,8568,8569,8572],{"style":208},[39,8570],{"className":8571,"style":213},[212],[39,8573,8575],{"className":8574},[217,218,219,220],[39,8576,8578],{"className":8577},[83,220],[39,8579,147],{"className":8580},[83,220],[39,8582],{"className":8583,"style":231},[230],[39,8585,150],{"className":8586},[235],[39,8588],{"className":8589,"style":231},[230],[39,8591,8593,8596,8599,8602,8605,8608,8611,8614,8617,8620,8623,8626,8629,8632,8635,8667],{"className":8592},[74],[39,8594],{"className":8595,"style":174},[78],[39,8597,142],{"className":8598},[83,84],[39,8600,155],{"className":8601},[252],[39,8603],{"className":8604,"style":256},[230],[39,8606,158],{"className":8607},[83,84],[39,8609,161],{"className":8610},[263],[39,8612,401],{"className":8613,"style":438},[83,84],[39,8615,136],{"className":8616},[182],[39,8618,142],{"className":8619},[83,84],[39,8621,155],{"className":8622},[252],[39,8624],{"className":8625,"style":256},[230],[39,8627,158],{"className":8628},[83,84],[39,8630,155],{"className":8631},[252],[39,8633],{"className":8634,"style":256},[230],[39,8636,8638,8641],{"className":8637},[83],[39,8639,142],{"className":8640},[83,84],[39,8642,8644],{"className":8643},[192],[39,8645,8647],{"className":8646},[196],[39,8648,8650],{"className":8649},[200],[39,8651,8653],{"className":8652,"style":205},[204],[39,8654,8655,8658],{"style":208},[39,8656],{"className":8657,"style":213},[212],[39,8659,8661],{"className":8660},[217,218,219,220],[39,8662,8664],{"className":8663},[83,220],[39,8665,147],{"className":8666},[83,220],[39,8668,161],{"className":8669},[263],"。第二项先用全期望公式对 ",[39,8672,8674,8697],{"className":8673,"translate":43},[42],[39,8675,8677],{"className":8676},[47],[49,8678,8679],{"xmlns":51},[53,8680,8681,8695],{},[56,8682,8683],{},[613,8684,8685,8687],{},[59,8686,142],{},[56,8688,8689,8691,8693],{},[59,8690,621],{},[133,8692,788],{},[539,8694,546],{},[63,8696,3625],{"encoding":65},[39,8698,8700],{"className":8699,"ariaHidden":70},[69],[39,8701,8703,8706],{"className":8702},[74],[39,8704],{"className":8705,"style":3635},[78],[39,8707,8709,8712],{"className":8708},[83],[39,8710,142],{"className":8711},[83,84],[39,8713,8715],{"className":8714},[192],[39,8716,8718,8747],{"className":8717},[196,649],[39,8719,8721,8744],{"className":8720},[200],[39,8722,8724],{"className":8723,"style":963},[204],[39,8725,8726,8729],{"style":966},[39,8727],{"className":8728,"style":213},[212],[39,8730,8732],{"className":8731},[217,218,219,220],[39,8733,8735,8738,8741],{"className":8734},[83,220],[39,8736,621],{"className":8737},[83,84,220],[39,8739,788],{"className":8740},[465,220],[39,8742,546],{"className":8743},[83,220],[39,8745,673],{"className":8746},[672],[39,8748,8750],{"className":8749},[200],[39,8751,8753],{"className":8752,"style":994},[204],[39,8754],{}," 分解，再利用马尔可夫性与策略的平稳性：",[39,8757,8759],{"className":8758,"translate":43},[746],[39,8760,8762,8999],{"className":8761,"translate":43},[42],[39,8763,8765],{"className":8764},[47],[49,8766,8767],{"xmlns":51,"display":755},[53,8768,8769,8996],{},[759,8770,8771,8865,8930],{"rowspacing":761,"columnalign":762,"columnspacing":146},[764,8772,8773,8825],{},[767,8774,8775],{},[770,8776,8777],{"scriptlevel":541,"displaystyle":70},[56,8778,8779,8785,8787,8799,8801,8807,8809,8811,8813,8819,8821,8823],{},[613,8780,8781,8783],{},[59,8782,4253],{"mathvariant":421},[59,8784,2098],{},[133,8786,537],{"stretchy":135},[613,8788,8789,8791],{},[59,8790,3181],{},[56,8792,8793,8795,8797],{},[59,8794,621],{},[133,8796,788],{},[539,8798,546],{},[133,8800,150],{},[613,8802,8803,8805],{},[59,8804,142],{},[59,8806,621],{},[133,8808,793],{},[59,8810,142],{},[133,8812,155],{"separator":70},[613,8814,8815,8817],{},[59,8816,158],{},[59,8818,621],{},[133,8820,793],{},[59,8822,158],{},[133,8824,4280],{"stretchy":135},[767,8826,8827],{},[770,8828,8829],{"scriptlevel":541,"displaystyle":70},[56,8830,8831,8833,8835,8845,8847,8849,8855,8857,8859,8861,8863],{},[56,8832],{},[133,8834,793],{},[6556,8836,8837,8839],{},[133,8838,3191],{},[138,8840,8841,8843],{},[59,8842,142],{},[133,8844,147],{"mathvariant":145,"lspace":146,"rspace":146},[59,8846,131],{},[133,8848,136],{"stretchy":135},[138,8850,8851,8853],{},[59,8852,142],{},[133,8854,147],{"mathvariant":145,"lspace":146,"rspace":146},[133,8856,150],{},[59,8858,142],{},[133,8860,155],{"separator":70},[59,8862,158],{},[133,8864,161],{"stretchy":135},[764,8866,8867,8873],{},[767,8868,8869],{},[770,8870,8871],{"scriptlevel":541,"displaystyle":70},[56,8872],{},[767,8874,8875],{},[770,8876,8877],{"scriptlevel":541,"displaystyle":70},[56,8878,8879,8881,8883,8886,8892,8894,8906,8908,8920,8922,8928],{},[56,8880],{},[230,8882],{"width":6891},[133,8884,8885],{},"⋅",[613,8887,8888,8890],{},[59,8889,4253],{"mathvariant":421},[59,8891,2098],{},[133,8893,537],{"stretchy":135},[613,8895,8896,8898],{},[59,8897,3181],{},[56,8899,8900,8902,8904],{},[59,8901,621],{},[133,8903,788],{},[539,8905,546],{},[133,8907,150],{},[613,8909,8910,8912],{},[59,8911,142],{},[56,8913,8914,8916,8918],{},[59,8915,621],{},[133,8917,788],{},[539,8919,546],{},[133,8921,793],{},[138,8923,8924,8926],{},[59,8925,142],{},[133,8927,147],{"mathvariant":145,"lspace":146,"rspace":146},[133,8929,4280],{"stretchy":135},[764,8931,8932,8938],{},[767,8933,8934],{},[770,8935,8936],{"scriptlevel":541,"displaystyle":70},[56,8937],{},[767,8939,8940],{},[770,8941,8942],{"scriptlevel":541,"displaystyle":70},[56,8943,8944,8946,8948,8958,8960,8962,8968,8970,8972,8974,8976,8978,8984,8986,8992,8994],{},[56,8945],{},[133,8947,793],{},[6556,8949,8950,8952],{},[133,8951,3191],{},[138,8953,8954,8956],{},[59,8955,142],{},[133,8957,147],{"mathvariant":145,"lspace":146,"rspace":146},[59,8959,131],{},[133,8961,136],{"stretchy":135},[138,8963,8964,8966],{},[59,8965,142],{},[133,8967,147],{"mathvariant":145,"lspace":146,"rspace":146},[133,8969,150],{},[59,8971,142],{},[133,8973,155],{"separator":70},[59,8975,158],{},[133,8977,161],{"stretchy":135},[138,8979,8980,8982],{},[59,8981,4230],{},[59,8983,2098],{},[133,8985,136],{"stretchy":135},[138,8987,8988,8990],{},[59,8989,142],{},[133,8991,147],{"mathvariant":145,"lspace":146,"rspace":146},[133,8993,161],{"stretchy":135},[59,8995,2903],{"mathvariant":145},[63,8997,8998],{"encoding":65},"\\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}",[39,9000,9002],{"className":9001,"ariaHidden":70},[69],[39,9003,9005,9009],{"className":9004},[74],[39,9006],{"className":9007,"style":9008},[78],"height:6.488em;vertical-align:-2.994em;",[39,9010,9012],{"className":9011},[83],[39,9013,9015,9281],{"className":9014},[759],[39,9016,9018],{"className":9017},[918],[39,9019,9021,9272],{"className":9020},[196,649],[39,9022,9024,9269],{"className":9023},[200],[39,9025,9028,9251,9260],{"className":9026,"style":9027},[204],"height:3.494em;",[39,9029,9031,9034],{"style":9030},"top:-5.494em;",[39,9032],{"className":9033,"style":3311},[212],[39,9035,9037,9077,9080,9129,9132,9135,9138,9178,9181,9184,9187,9190,9193,9196,9236,9239,9242,9245,9248],{"className":9036},[83],[39,9038,9040,9043],{"className":9039},[83],[39,9041,4253],{"className":9042},[83,515],[39,9044,9046],{"className":9045},[192],[39,9047,9049,9069],{"className":9048},[196,649],[39,9050,9052,9066],{"className":9051},[200],[39,9053,9055],{"className":9054,"style":1728},[204],[39,9056,9057,9060],{"style":966},[39,9058],{"className":9059,"style":213},[212],[39,9061,9063],{"className":9062},[217,218,219,220],[39,9064,2098],{"className":9065,"style":2114},[83,84,220],[39,9067,673],{"className":9068},[672],[39,9070,9072],{"className":9071},[200],[39,9073,9075],{"className":9074,"style":680},[204],[39,9076],{},[39,9078,537],{"className":9079},[182],[39,9081,9083,9086],{"className":9082},[83],[39,9084,3181],{"className":9085},[83,84],[39,9087,9089],{"className":9088},[192],[39,9090,9092,9121],{"className":9091},[196,649],[39,9093,9095,9118],{"className":9094},[200],[39,9096,9098],{"className":9097,"style":963},[204],[39,9099,9100,9103],{"style":966},[39,9101],{"className":9102,"style":213},[212],[39,9104,9106],{"className":9105},[217,218,219,220],[39,9107,9109,9112,9115],{"className":9108},[83,220],[39,9110,621],{"className":9111},[83,84,220],[39,9113,788],{"className":9114},[465,220],[39,9116,546],{"className":9117},[83,220],[39,9119,673],{"className":9120},[672],[39,9122,9124],{"className":9123},[200],[39,9125,9127],{"className":9126,"style":994},[204],[39,9128],{},[39,9130],{"className":9131,"style":231},[230],[39,9133,150],{"className":9134},[235],[39,9136],{"className":9137,"style":231},[230],[39,9139,9141,9144],{"className":9140},[83],[39,9142,142],{"className":9143},[83,84],[39,9145,9147],{"className":9146},[192],[39,9148,9150,9170],{"className":9149},[196,649],[39,9151,9153,9167],{"className":9152},[200],[39,9154,9156],{"className":9155,"style":656},[204],[39,9157,9158,9161],{"style":966},[39,9159],{"className":9160,"style":213},[212],[39,9162,9164],{"className":9163},[217,218,219,220],[39,9165,621],{"className":9166},[83,84,220],[39,9168,673],{"className":9169},[672],[39,9171,9173],{"className":9172},[200],[39,9174,9176],{"className":9175,"style":680},[204],[39,9177],{},[39,9179],{"className":9180,"style":231},[230],[39,9182,793],{"className":9183},[235],[39,9185],{"className":9186,"style":231},[230],[39,9188,142],{"className":9189},[83,84],[39,9191,155],{"className":9192},[252],[39,9194],{"className":9195,"style":256},[230],[39,9197,9199,9202],{"className":9198},[83],[39,9200,158],{"className":9201},[83,84],[39,9203,9205],{"className":9204},[192],[39,9206,9208,9228],{"className":9207},[196,649],[39,9209,9211,9225],{"className":9210},[200],[39,9212,9214],{"className":9213,"style":656},[204],[39,9215,9216,9219],{"style":966},[39,9217],{"className":9218,"style":213},[212],[39,9220,9222],{"className":9221},[217,218,219,220],[39,9223,621],{"className":9224},[83,84,220],[39,9226,673],{"className":9227},[672],[39,9229,9231],{"className":9230},[200],[39,9232,9234],{"className":9233,"style":680},[204],[39,9235],{},[39,9237],{"className":9238,"style":231},[230],[39,9240,793],{"className":9241},[235],[39,9243],{"className":9244,"style":231},[230],[39,9246,158],{"className":9247},[83,84],[39,9249,4280],{"className":9250},[263],[39,9252,9254,9257],{"style":9253},"top:-3.06em;",[39,9255],{"className":9256,"style":3311},[212],[39,9258],{"className":9259},[83],[39,9261,9263,9266],{"style":9262},"top:-1.35em;",[39,9264],{"className":9265,"style":3311},[212],[39,9267],{"className":9268},[83],[39,9270,673],{"className":9271},[672],[39,9273,9275],{"className":9274},[200],[39,9276,9279],{"className":9277,"style":9278},[204],"height:2.994em;",[39,9280],{},[39,9282,9284],{"className":9283},[1223],[39,9285,9287,9905],{"className":9286},[196,649],[39,9288,9290,9902],{"className":9289},[200],[39,9291,9293,9454,9671],{"className":9292,"style":9027},[204],[39,9294,9295,9298],{"style":9030},[39,9296],{"className":9297,"style":3311},[212],[39,9299,9301,9304,9307,9310,9313,9389,9392,9395,9398,9430,9433,9436,9439,9442,9445,9448,9451],{"className":9300},[83],[39,9302],{"className":9303},[83],[39,9305],{"className":9306,"style":231},[230],[39,9308,793],{"className":9309},[235],[39,9311],{"className":9312,"style":231},[230],[39,9314,9316],{"className":9315},[1744,3294],[39,9317,9319,9381],{"className":9318}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",[39,9920,9922,9943],{"className":9921,"translate":43},[42],[39,9923,9925],{"className":9924},[47],[49,9926,9927],{"xmlns":51},[53,9928,9929,9940],{},[56,9930,9931],{},[9932,9933,9934,9936,9938],"msubsup",{},[59,9935,4230],{},[59,9937,621],{},[59,9939,2098],{},[63,9941,9942],{"encoding":65},"V_t^\\pi",[39,9944,9946],{"className":9945,"ariaHidden":70},[69],[39,9947,9949,9953],{"className":9948},[74],[39,9950],{"className":9951,"style":9952},[78],"height:0.9303em;vertical-align:-0.247em;",[39,9954,9956,9959],{"className":9955},[83],[39,9957,4230],{"className":9958,"style":461},[83,84],[39,9960,9962],{"className":9961},[192],[39,9963,9965,9997],{"className":9964},[196,649],[39,9966,9968,9994],{"className":9967},[200],[39,9969,9971,9983],{"className":9970,"style":6410},[204],[39,9972,9974,9977],{"style":9973},"top:-2.453em;margin-left:-0.2222em;margin-right:0.05em;",[39,9975],{"className":9976,"style":213},[212],[39,9978,9980],{"className":9979},[217,218,219,220],[39,9981,621],{"className":9982},[83,84,220],[39,9984,9985,9988],{"style":208},[39,9986],{"className":9987,"style":213},[212],[39,9989,9991],{"className":9990},[217,218,219,220],[39,9992,2098],{"className":9993,"style":2114},[83,84,220],[39,9995,673],{"className":9996},[672],[39,9998,10000],{"className":9999},[200],[39,10001,10004],{"className":10002,"style":10003},[204],"height:0.247em;",[39,10005],{},"，递归形式仍然成立，但函数会随 ",[39,10008,10010,10023],{"className":10009,"translate":43},[42],[39,10011,10013],{"className":10012},[47],[49,10014,10015],{"xmlns":51},[53,10016,10017,10021],{},[56,10018,10019],{},[59,10020,621],{},[63,10022,621],{"encoding":65},[39,10024,10026],{"className":10025,"ariaHidden":70},[69],[39,10027,10029,10032],{"className":10028},[74],[39,10030],{"className":10031,"style":709},[78],[39,10033,621],{"className":10034},[83,84]," 变化。本文只讨论平稳策略，因此可以省略时间下标。",[21,10037,10039],{"id":10038},"二贝尔曼期望方程","二、贝尔曼期望方程",[26,10041,10043],{"id":10042},"_21-方程的推导","2.1 方程的推导",[11,10045,10046,10047,10051],{},"将",[158,10048,10050],{"href":10049},"#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。",[31,10053,10056,10088],{"id":10054,"title":10055},"theorem-bellman-expectation","定理 2.1（贝尔曼期望方程）",[11,10057,10058,10059,10087],{},"对任意平稳策略 ",[39,10060,10062,10075],{"className":10061,"translate":43},[42],[39,10063,10065],{"className":10064},[47],[49,10066,10067],{"xmlns":51},[53,10068,10069,10073],{},[56,10070,10071],{},[59,10072,2098],{},[63,10074,2101],{"encoding":65},[39,10076,10078],{"className":10077,"ariaHidden":70},[69],[39,10079,10081,10084],{"className":10080},[74],[39,10082],{"className":10083,"style":290},[78],[39,10085,2098],{"className":10086,"style":2114},[83,84],"，状态值函数满足",[39,10089,10091],{"className":10090,"translate":43},[746],[39,10092,10094,10266],{"className":10093,"translate":43},[42],[39,10095,10097],{"className":10096},[47],[49,10098,10099],{"xmlns":51,"display":755},[53,10100,10101,10263],{},[759,10102,10103,10181],{"rowspacing":761,"columnalign":762,"columnspacing":146},[764,10104,10105,10123],{},[767,10106,10107],{},[770,10108,10109],{"scriptlevel":541,"displaystyle":70},[56,10110,10111,10117,10119,10121],{},[138,10112,10113,10115],{},[59,10114,4230],{},[59,10116,2098],{},[133,10118,136],{"stretchy":135},[59,10120,142],{},[133,10122,161],{"stretchy":135},[767,10124,10125],{},[770,10126,10127],{"scriptlevel":541,"displaystyle":70},[56,10128,10129,10131,10133,10139,10141,10143,10145,10147,10149,10151,10161,10163,10165,10171,10173,10175,10177,10179],{},[56,10130],{},[133,10132,793],{},[6556,10134,10135,10137],{},[133,10136,3191],{},[59,10138,158],{},[59,10140,2098],{},[133,10142,136],{"stretchy":135},[59,10144,158],{},[133,10146,150],{},[59,10148,142],{},[133,10150,161],{"stretchy":135},[6556,10152,10153,10155],{},[133,10154,3191],{},[138,10156,10157,10159],{},[59,10158,142],{},[133,10160,147],{"mathvariant":145,"lspace":146,"rspace":146},[59,10162,131],{},[133,10164,136],{"stretchy":135},[138,10166,10167,10169],{},[59,10168,142],{},[133,10170,147],{"mathvariant":145,"lspace":146,"rspace":146},[133,10172,150],{},[59,10174,142],{},[133,10176,155],{"separator":70},[59,10178,158],{},[133,10180,161],{"stretchy":135},[764,10182,10183,10189],{},[767,10184,10185],{},[770,10186,10187],{"scriptlevel":541,"displaystyle":70},[56,10188],{},[767,10190,10191],{},[770,10192,10193],{"scriptlevel":541,"displaystyle":70},[56,10194,10195,10197,10199,10201,10247,10249,10252,10255,10257,10259,10261],{},[56,10196],{},[230,10198],{"width":6891},[133,10200,409],{},[56,10202,10203,10205,10207,10209,10211,10213,10215,10217,10223,10225,10227,10229,10235,10237,10243,10245],{},[133,10204,537],{"fence":70},[59,10206,401],{},[133,10208,136],{"stretchy":135},[59,10210,142],{},[133,10212,155],{"separator":70},[59,10214,158],{},[133,10216,155],{"separator":70},[138,10218,10219,10221],{},[59,10220,142],{},[133,10222,147],{"mathvariant":145,"lspace":146,"rspace":146},[133,10224,161],{"stretchy":135},[133,10226,788],{},[59,10228,531],{},[138,10230,10231,10233],{},[59,10232,4230],{},[59,10234,2098],{},[133,10236,136],{"stretchy":135},[138,10238,10239,10241],{},[59,10240,142],{},[133,10242,147],{"mathvariant":145,"lspace":146,"rspace":146},[133,10244,161],{"stretchy":135},[133,10246,4280],{"fence":70},[133,10248,155],{"separator":70},[230,10250],{"width":10251},"2em",[59,10253,10254],{"mathvariant":145},"∀",[59,10256,142],{},[133,10258,534],{},[59,10260,61],{},[59,10262,2903],{"mathvariant":145},[63,10264,10265],{"encoding":65},"\\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}",[39,10267,10269],{"className":10268,"ariaHidden":70},[69],[39,10270,10272,10275],{"className":10271},[74],[39,10273],{"className":10274,"style":6954},[78],[39,10276,10278],{"className":10277},[83],[39,10279,10281,10358],{"className":10280},[759],[39,10282,10284],{"className":10283},[918],[39,10285,10287,10350],{"className":10286},[196,649],[39,10288,10290,10347],{"className":10289},[200],[39,10291,10293,10339],{"className":10292,"style":6973},[204],[39,10294,10295,10298],{"style":6976},[39,10296],{"className":10297,"style":3311},[212],[39,10299,10301,10330,10333,10336],{"className":10300},[83],[39,10302,10304,10307],{"className":10303},[83],[39,10305,4230],{"className":10306,"style":461},[83,84],[39,10308,10310],{"className":10309},[192],[39,10311,10313],{"className":10312},[196],[39,10314,10316],{"className":10315},[200],[39,10317,10319],{"className":10318,"style":4421},[204],[39,10320,10321,10324],{"style":1027},[39,10322],{"className":10323,"style":213},[212],[39,10325,10327],{"className":10326},[217,218,219,220],[39,10328,2098],{"className":10329,"style":2114},[83,84,220],[39,10331,136],{"className":10332},[182],[39,10334,142],{"className":10335},[83,84],[39,10337,161],{"className":10338},[263],[39,10340,10341,10344],{"style":7032},[39,10342],{"className":10343,"style":3311},[212],[39,10345],{"className":10346},[83],[39,10348,673],{"className":10349},[672],[39,10351,10353],{"className":10352},[200],[39,10354,10356],{"className":10355,"style":7048},[204],[39,10357],{},[39,10359,10361],{"className":10360},[1223],[39,10362,10364,10815],{"className":10363},[196,649],[39,10365,10367,10812],{"className":10366},[200],[39,10368,10370,10608],{"className":10369,"style":6973},[204],[39,10371,10372,10375],{"style":6976},[39,10373],{"className":10374,"style":3311},[212],[39,10376,10378,10381,10384,10387,10390,10437,10440,10443,10446,10449,10452,10455,10458,10461,10464,10467,10543,10546,10549,10552,10584,10587,10590,10593,10596,10599,10602,10605],{"className":10377},[83],[39,10379],{"className":10380},[83],[39,10382],{"className":10383,"style":231},[230],[39,10385,793],{"className":10386},[235],[39,10388],{"className":10389,"style":231},[230],[39,10391,10393],{"className":10392},[1744,3294],[39,10394,10396,10429],{"className":10395},[196,649],[39,10397,10399,10426],{"className":10398},[200],[39,10400,10402,10416],{"className":10401,"style":6670},[204],[39,10403,10404,10407],{"style":6673},[39,10405],{"className":10406,"style":3311},[212],[39,10408,10410],{"className":10409},[217,218,219,220],[39,10411,10413],{"className":10412},[83,220],[39,10414,158],{"className":10415},[83,84,220],[39,10417,10418,10421],{"style":3330},[39,10419],{"className":10420,"style":3311},[212],[39,10422,10423],{},[39,10424,3191],{"className":10425},[1744,3339,3340],[39,10427,673],{"className":10428},[672],[39,10430,10432],{"className":10431},[200],[39,10433,10435],{"className":10434,"style":6705},[204],[39,10436],{},[39,10438],{"className":10439,"style":256},[230],[39,10441,2098],{"className":10442,"style":2114},[83,84],[39,10444,136],{"className":10445},[182],[39,10447,158],{"className":10448},[83,84],[39,10450],{"className":10451,"style":231},[230],[39,10453,150],{"className":10454},[235],[39,10456],{"className":10457,"style":231},[230],[39,10459,142],{"className":10460},[83,84],[39,10462,161],{"className":10463},[263],[39,10465],{"className":10466,"style":256},[230],[39,10468,10470],{"className":10469},[1744,3294],[39,10471,10473,10535],{"className":10472},[196,649],[39,10474,10476,10532],{"className":10475},[200],[39,10477,10479,10522],{"className":10478,"style":6670},[204],[39,10480,10481,10484],{"style":7097},[39,10482],{"className":10483,"style":3311},[212],[39,10485,10487],{"className":10486},[217,218,219,220],[39,10488,10490],{"className":10489},[83,220],[39,10491,10493,10496],{"className":10492},[83,220],[39,10494,142],{"className":10495},[83,84,220],[39,10497,10499],{"className":10498},[192],[39,10500,10502],{"className":10501},[196],[39,10503,10505],{"className":10504},[200],[39,10506,10508],{"className":10507,"style":3997},[204],[39,10509,10510,10513],{"style":4000},[39,10511],{"className":10512,"style":4004},[212],[39,10514,10516],{"className":10515},[217,4008,4009,220],[39,10517,10519],{"className":10518},[83,220],[39,10520,147],{"className":10521},[83,220],[39,10523,10524,10527],{"style":3330},[39,10525],{"className":10526,"style":3311},[212],[39,10528,10529],{},[39,10530,3191],{"className":10531},[1744,3339,3340],[39,10533,673],{"className":10534},[672],[39,10536,10538],{"className":10537},[200],[39,10539,10541],{"className":10540,"style":7158},[204],[39,10542],{},[39,10544],{"className":10545,"style":256},[230],[39,10547,131],{"className":10548,"style":178},[83,84],[39,10550,136],{"className":10551},[182],[39,10553,10555,10558],{"className":10554},[83],[39,10556,142],{"className":10557},[83,84],[39,10559,10561],{"className":10560},[192],[39,10562,10564],{"className":10563},[196],[39,10565,10567],{"className":10566},[200],[39,10568,10570],{"className":10569,"style":1024},[204],[39,10571,10572,10575],{"style":1027},[39,10573],{"className":10574,"style":213},[212],[39,10576,10578],{"className":10577},[217,218,219,220],[39,10579,10581],{"className":10580},[83,220],[39,10582,147],{"className":10583},[83,220],[39,10585],{"className":10586,"style":231},[230],[39,10588,150],{"className":10589},[235],[39,10591],{"className":10592,"style":231},[230],[39,10594,142],{"className":10595},[83,84],[39,10597,155],{"className":10598},[252],[39,10600],{"className":10601,"style":256},[230],[39,10603,158],{"className":10604},[83,84],[39,10606,161],{"className":10607},[263],[39,10609,10610,10613],{"style":7032},[39,10611],{"className":10612,"style":3311},[212],[39,10614,10616,10619,10622,10625,10628,10631,10778,10781,10784,10788,10791,10794,10797,10800,10803,10806,10809],{"className":10615},[83],[39,10617],{"className":10618},[83],[39,10620],{"className":10621,"style":7240},[230],[39,10623],{"className":10624,"style":461},[230],[39,10626,409],{"className":10627},[465],[39,10629],{"className":10630,"style":461},[230],[39,10632,10634,10637,10640,10643,10646,10649,10652,10655,10658,10661,10693,10696,10699,10702,10705,10708,10737,10740,10772,10775],{"className":10633},[2842],[39,10635,537],{"className":10636,"style":4592},[182,4591],[39,10638,401],{"className":10639,"style":438},[83,84],[39,10641,136],{"className":10642},[182],[39,10644,142],{"className":10645},[83,84],[39,10647,155],{"className":10648},[252],[39,10650],{"className":10651,"style":256},[230],[39,10653,158],{"className":10654},[83,84],[39,10656,155],{"className":10657},[252],[39,10659],{"className":10660,"style":256},[230],[39,10662,10664,10667],{"className":10663},[83],[39,10665,142],{"className":10666},[83,84],[39,10668,10670],{"className":10669},[192],[39,10671,10673],{"className":10672},[196],[39,10674,10676],{"className":10675},[200],[39,10677,10679],{"className":10678,"style":1024},[204],[39,10680,10681,10684],{"style":1027},[39,10682],{"className":10683,"style":213},[212],[39,10685,10687],{"className":10686},[217,218,219,220],[39,10688,10690],{"className":10689},[83,220],[39,10691,147],{"className":10692},[83,220],[39,10694,161],{"className":10695},[263],[39,10697],{"className":10698,"style":461},[230],[39,10700,788],{"className":10701},[465],[39,10703],{"className":10704,"style":461},[230],[39,10706,531],{"className":10707,"style":565},[83,84],[39,10709,10711,10714],{"className":10710},[83],[39,10712,4230],{"className":10713,"style":461},[83,84],[39,10715,10717],{"className":10716},[192],[39,10718,10720],{"className":10719},[196],[39,10721,10723],{"className":10722},[200],[39,10724,10726],{"className":10725,"style":4421},[204],[39,10727,10728,10731],{"style":1027},[39,10729],{"className":10730,"style":213},[212],[39,10732,10734],{"className":10733},[217,218,219,220],[39,10735,2098],{"className":10736,"style":2114},[83,84,220],[39,10738,136],{"className":10739},[182],[39,10741,10743,10746],{"className":10742},[83],[39,10744,142],{"className":10745},[83,84],[39,10747,10749],{"className":10748},[192],[39,10750,10752],{"className":10751},[196],[39,10753,10755],{"className":10754},[200],[39,10756,10758],{"className":10757,"style":1024},[204],[39,10759,10760,10763],{"style":1027},[39,10761],{"className":10762,"style":213},[212],[39,10764,10766],{"className":10765},[217,218,219,220],[39,10767,10769],{"className":10768},[83,220],[39,10770,147],{"className":10771},[83,220],[39,10773,161],{"className":10774},[263],[39,10776,4280],{"className":10777,"style":4592},[263,4591],[39,10779],{"className":10780,"style":256},[230],[39,10782,155],{"className":10783},[252],[39,10785],{"className":10786,"style":10787},[230],"margin-right:2em;",[39,10789],{"className":10790,"style":256},[230],[39,10792,10254],{"className":10793},[83],[39,10795,142],{"className":10796},[83,84],[39,10798],{"className":10799,"style":231},[230],[39,10801,534],{"className":10802},[235],[39,10804],{"className":10805,"style":231},[230],[39,10807,61],{"className":10808,"style":85},[83,84],[39,10810,2903],{"className":10811},[83],[39,10813,673],{"className":10814},[672],[39,10816,10818],{"className":10817},[200],[39,10819,10821],{"className":10820,"style":7048},[204],[39,10822],{},[11,10824,10825,10827,10828,742],{},[1489,10826,5595],{}," 由",[158,10829,10050],{"href":10049},[39,10831,10833],{"className":10832,"translate":43},[746],[39,10834,10836,11026],{"className":10835,"translate":43},[42],[39,10837,10839],{"className":10838},[47],[49,10840,10841],{"xmlns":51,"display":755},[53,10842,10843,11023],{},[759,10844,10845,10909],{"rowspacing":761,"columnalign":762,"columnspacing":146},[764,10846,10847,10865],{},[767,10848,10849],{},[770,10850,10851],{"scriptlevel":541,"displaystyle":70},[56,10852,10853,10859,10861,10863],{},[138,10854,10855,10857],{},[59,10856,4230],{},[59,10858,2098],{},[133,10860,136],{"stretchy":135},[59,10862,142],{},[133,10864,161],{"stretchy":135},[767,10866,10867],{},[770,10868,10869],{"scriptlevel":541,"displaystyle":70},[56,10870,10871,10873,10875,10881,10883,10885,10887,10889,10891,10893,10899,10901,10903,10905,10907],{},[56,10872],{},[133,10874,793],{},[6556,10876,10877,10879],{},[133,10878,3191],{},[59,10880,158],{},[59,10882,2098],{},[133,10884,136],{"stretchy":135},[59,10886,158],{},[133,10888,150],{},[59,10890,142],{},[133,10892,161],{"stretchy":135},[138,10894,10895,10897],{},[59,10896,4295],{},[59,10898,2098],{},[133,10900,136],{"stretchy":135},[59,10902,142],{},[133,10904,155],{"separator":70},[59,10906,158],{},[133,10908,161],{"stretchy":135},[764,10910,10911,10917],{},[767,10912,10913],{},[770,10914,10915],{"scriptlevel":541,"displaystyle":70},[56,10916],{},[767,10918,10919],{},[770,10920,10921],{"scriptlevel":541,"displaystyle":70},[56,10922,10923,10925,10927,10933,10935,10937,10939,10941,10943,10945,10955,10957,10959,10965,10967,10969,10971,10973,10975,11021],{},[56,10924],{},[133,10926,793],{},[6556,10928,10929,10931],{},[133,10930,3191],{},[59,10932,158],{},[59,10934,2098],{},[133,10936,136],{"stretchy":135},[59,10938,158],{},[133,10940,150],{},[59,10942,142],{},[133,10944,161],{"stretchy":135},[6556,10946,10947,10949],{},[133,10948,3191],{},[138,10950,10951,10953],{},[59,10952,142],{},[133,10954,147],{"mathvariant":145,"lspace":146,"rspace":146},[59,10956,131],{},[133,10958,136],{"stretchy":135},[138,10960,10961,10963],{},[59,10962,142],{},[133,10964,147],{"mathvariant":145,"lspace":146,"rspace":146},[133,10966,150],{},[59,10968,142],{},[133,10970,155],{"separator":70},[59,10972,158],{},[133,10974,161],{"stretchy":135},[56,10976,10977,10979,10981,10983,10985,10987,10989,10991,10997,10999,11001,11003,11009,11011,11017,11019],{},[133,10978,537],{"fence":70},[59,10980,401],{},[133,10982,136],{"stretchy":135},[59,10984,142],{},[133,10986,155],{"separator":70},[59,10988,158],{},[133,10990,155],{"separator":70},[138,10992,10993,10995],{},[59,10994,142],{},[133,10996,147],{"mathvariant":145,"lspace":146,"rspace":146},[133,10998,161],{"stretchy":135},[133,11000,788],{},[59,11002,531],{},[138,11004,11005,11007],{},[59,11006,4230],{},[59,11008,2098],{},[133,11010,136],{"stretchy":135},[138,11012,11013,11015],{},[59,11014,142],{},[133,11016,147],{"mathvariant":145,"lspace":146,"rspace":146},[133,11018,161],{"stretchy":135},[133,11020,4280],{"fence":70},[59,11022,2903],{"mathvariant":145},[63,11024,11025],{"encoding":65},"\\begin{aligned}\nV^\\pi(s)\n&=\\sum_{a}\\pi(a\\mid 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}",[39,11027,11029],{"className":11028,"ariaHidden":70},[69],[39,11030,11032,11036],{"className":11031},[74],[39,11033],{"className":11034,"style":11035},[78],"height:4.944em;vertical-align:-2.222em;",[39,11037,11039],{"className":11038},[83],[39,11040,11042,11123],{"className":11041},[759],[39,11043,11045],{"className":11044},[918],[39,11046,11048,11114],{"className":11047},[196,649],[39,11049,11051,11111],{"className":11050},[200],[39,11052,11055,11102],{"className":11053,"style":11054},[204],"height:2.722em;",[39,11056,11058,11061],{"style":11057},"top:-4.722em;",[39,11059],{"className":11060,"style":3311},[212],[39,11062,11064,11093,11096,11099],{"className":11063},[83],[39,11065,11067,11070],{"className":11066},[83],[39,11068,4230],{"className":11069,"style":461},[83,84],[39,11071,11073],{"className":11072},[192],[39,11074,11076],{"className":11075},[196],[39,11077,11079],{"className":11078},[200],[39,11080,11082],{"className":11081,"style":4421},[204],[39,11083,11084,11087],{"style":1027},[39,11085],{"className":11086,"style":213},[212],[39,11088,11090],{"className":11089},[217,218,219,220],[39,11091,2098],{"className":11092,"style":2114},[83,84,220],[39,11094,136],{"className":11095},[182],[39,11097,142],{"className":11098},[83,84],[39,11100,161],{"className":11101},[263],[39,11103,11105,11108],{"style":11104},"top:-2.122em;",[39,11106],{"className":11107,"style":3311},[212],[39,11109],{"className":11110},[83],[39,11112,673],{"className":11113},[672],[39,11115,11117],{"className":11116},[200],[39,11118,11121],{"className":11119,"style":11120},[204],"height:2.222em;",[39,11122],{},[39,11124,11126],{"className":11125},[1223],[39,11127,11129,11673],{"className":11128},[196,649],[39,11130,11132,11670],{"className":11131},[200],[39,11133,11135,11276],{"className":11134,"style":11054},[204],[39,11136,11137,11140],{"style":11057},[39,11138],{"className":11139,"style":3311},[212],[39,11141,11143,11146,11149,11152,11155,11202,11205,11208,11211,11214,11217,11220,11223,11226,11229,11258,11261,11264,11267,11270,11273],{"className":11142},[83],[39,11144],{"className":11145},[83],[39,11147],{"className":11148,"style":231},[230],[39,11150,793],{"className":11151},[235],[39,11153],{"className":11154,"style":231},[230],[39,11156,11158],{"className":11157},[1744,3294],[39,11159,11161,11194],{"className":11160},[196,649],[39,11162,11164,11191],{"className":11163},[200],[39,11165,11167,11181],{"className":11166,"style":6670},[204],[39,11168,11169,11172],{"style":6673},[39,11170],{"className":11171,"style":3311},[212],[39,11173,11175],{"className":11174},[217,218,219,220],[39,11176,11178],{"className":11177},[83,220],[39,11179,158],{"className":11180},[83,84,220],[39,11182,11183,11186],{"style":3330},[39,11184],{"className":11185,"style":3311},[212],[39,11187,11188],{},[39,11189,3191],{"className":11190},[1744,3339,3340],[39,11192,673],{"className":11193},[672],[39,11195,11197],{"className":11196},[200],[39,11198,11200],{"className":11199,"style":6705},[204],[39,11201],{},[39,11203],{"className":11204,"style":256},[230],[39,11206,2098],{"className":11207,"style":2114},[83,84],[39,11209,136],{"className":11210},[182],[39,11212,158],{"className":11213},[83,84],[39,11215],{"className":11216,"style":231},[230],[39,11218,150],{"className":11219},[235],[39,11221],{"className":11222,"style":231},[230],[39,11224,142],{"className":11225},[83,84],[39,11227,161],{"className":11228},[263],[39,11230,11232,11235],{"className":11231},[83],[39,11233,4295],{"className":11234},[83,84],[39,11236,11238],{"className":11237},[192],[39,11239,11241],{"className":11240},[196],[39,11242,11244],{"className":11243},[200],[39,11245,11247],{"className":11246,"style":4421},[204],[39,11248,11249,11252],{"style":1027},[39,11250],{"className":11251,"style":213},[212],[39,11253,11255],{"className":11254},[217,218,219,220],[39,11256,2098],{"className":11257,"style":2114},[83,84,220],[39,11259,136],{"className":11260},[182],[39,11262,142],{"className":11263},[83,84],[39,11265,155],{"className":11266},[252],[39,11268],{"className":11269,"style":256},[230],[39,11271,158],{"className":11272},[83,84],[39,11274,161],{"className":11275},[263],[39,11277,11278,11281],{"style":11104},[39,11279],{"className":11280,"style":3311},[212],[39,11282,11284,11287,11290,11293,11296,11343,11346,11349,11352,11355,11358,11361,11364,11367,11370,11373,11449,11452,11455,11458,11490,11493,11496,11499,11502,11505,11508,11511,11514,11517,11664,11667],{"className":11283},[83],[39,11285],{"className":11286},[83],[39,11288],{"className":11289,"style":231},[230],[39,11291,793],{"className":11292},[235],[39,11294],{"className":11295,"style":231},[230],[39,11297,11299],{"className":11298},[1744,3294],[39,11300,11302,11335],{"className":11301},[196,649],[39,11303,11305,11332],{"className":11304},[200],[39,11306,11308,11322],{"className":11307,"style":6670},[204],[39,11309,11310,11313],{"style":6673},[39,11311],{"className":11312,"style":3311},[212],[39,11314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tyle":2114},[83,84,220],[39,11624,136],{"className":11625},[182],[39,11627,11629,11632],{"className":11628},[83],[39,11630,142],{"className":11631},[83,84],[39,11633,11635],{"className":11634},[192],[39,11636,11638],{"className":11637},[196],[39,11639,11641],{"className":11640},[200],[39,11642,11644],{"className":11643,"style":1024},[204],[39,11645,11646,11649],{"style":1027},[39,11647],{"className":11648,"style":213},[212],[39,11650,11652],{"className":11651},[217,218,219,220],[39,11653,11655],{"className":11654},[83,220],[39,11656,147],{"className":11657},[83,220],[39,11659,161],{"className":11660},[263],[39,11662,4280],{"className":11663,"style":4592},[263,4591],[39,11665],{"className":11666,"style":256},[230],[39,11668,2903],{"className":11669},[83],[39,11671,673],{"className":11672},[672],[39,11674,11676],{"className":11675},[200],[39,11677,11679],{"className":11678,"style":11120},[204],[39,11680],{},[11,11682,11683],{},"证毕。",[11,11685,11686,11687,11745],{},"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 ",[39,11688,11690,11707],{"className":11689,"translate":43},[42],[39,11691,11693],{"className":11692},[47],[49,11694,11695],{"xmlns":51},[53,11696,11697,11705],{},[56,11698,11699],{},[138,11700,11701,11703],{},[59,11702,4230],{},[59,11704,2098],{},[63,11706,6382],{"encoding":65},[39,11708,11710],{"className":11709,"ariaHidden":70},[69],[39,11711,11713,11716],{"className":11712},[74],[39,11714],{"className":11715,"style":79},[78],[39,11717,11719,11722],{"className":11718},[83],[39,11720,4230],{"className":11721,"style":461},[83,84],[39,11723,11725],{"className":11724},[192],[39,11726,11728],{"className":11727},[196],[39,11729,11731],{"className":11730},[200],[39,11732,11734],{"className":11733,"style":6410},[204],[39,11735,11736,11739],{"style":208},[39,11737],{"className":11738,"style":213},[212],[39,11740,11742],{"className":11741},[217,218,219,220],[39,11743,2098],{"className":11744,"style":2114},[83,84,220],"，因此它不是普通的递推式，而是一个泛函方程。",[26,11747,11749],{"id":11748},"_22-算子形式","2.2 算子形式",[11,11751,11752,11753,2083,11756,2506],{},"为了把上述方程写成更紧凑的形式，定义",[1489,11754,11755],{},"贝尔曼期望算子",[39,11757,11759,11778],{"className":11758,"translate":43},[42],[39,11760,11762],{"className":11761},[47],[49,11763,11764],{"xmlns":51},[53,11765,11766,11775],{},[56,11767,11768],{},[138,11769,11770,11773],{},[59,11771,11772],{"mathvariant":617},"T",[59,11774,2098],{},[63,11776,11777],{"encoding":65},"\\mathcal{T}^\\pi",[39,11779,11781],{"className":11780,"ariaHidden":70},[69],[39,11782,11784,11787],{"className":11783},[74],[39,11785],{"className":11786,"style":79},[78],[39,11788,11790,11794],{"className":11789},[83],[39,11791,11772],{"className":11792,"style":11793},[83,641],"margin-right:0.2542em;",[39,11795,11797],{"className":11796},[192],[39,11798,11800],{"className":11799},[196],[39,11801,11803],{"className":11802},[200],[39,11804,11806],{"className":11805,"style":6410},[204],[39,11807,11808,11811],{"style":208},[39,11809],{"className":11810,"style":213},[212],[39,11812,11814],{"className":11813},[217,218,219,220],[39,11815,2098],{"className":11816,"style":2114},[83,84,220],[39,11818,11820],{"className":11819,"translate":43},[746],[39,11821,11823,11983],{"className":11822,"translate":43},[42],[39,11824,11826],{"className":11825},[47],[49,11827,11828],{"xmlns":51,"display":755},[53,11829,11830,11980],{},[759,11831,11832,11916],{"rowspacing":761,"columnalign":762,"columnspacing":146},[764,11833,11834,11858],{},[767,11835,11836],{},[770,11837,11838],{"scriptlevel":541,"displaystyle":70},[56,11839,11840,11842,11848,11850,11852,11854,11856],{},[133,11841,136],{"stretchy":135},[138,11843,11844,11846],{},[59,11845,11772],{"mathvariant":617},[59,11847,2098],{},[59,11849,4230],{},[133,11851,161],{"stretchy":135},[133,11853,136],{"stretchy":135},[59,11855,142],{},[133,11857,161],{"stretchy":135},[767,11859,11860],{},[770,11861,11862],{"scriptlevel":541,"displaystyle":70},[56,11863,11864,11866,11868,11874,11876,11878,11880,11882,11884,11886,11896,11898,11900,11906,11908,11910,11912,11914],{},[56,11865],{},[133,11867,793],{},[6556,11869,11870,11872],{},[133,11871,3191],{},[59,11873,158],{},[59,11875,2098],{},[133,11877,136],{"stretchy":135},[59,11879,158],{},[133,11881,150],{},[59,11883,142],{},[133,11885,161],{"stretchy":135},[6556,11887,11888,11890],{},[133,11889,3191],{},[138,11891,11892,11894],{},[59,11893,142],{},[133,11895,147],{"mathvariant":145,"lspace":146,"rspace":146},[59,11897,131],{},[133,11899,136],{"stretchy":135},[138,11901,11902,11904],{},[59,11903,142],{},[133,11905,147],{"mathvariant":145,"lspace":146,"rspace":146},[133,11907,150],{},[59,11909,142],{},[133,11911,155],{"separator":70},[59,11913,158],{},[133,11915,161],{"stretchy":135},[764,11917,11918,11924],{},[767,11919,11920],{},[770,11921,11922],{"scriptlevel":541,"displaystyle":70},[56,11923],{},[767,11925,11926],{},[770,11927,11928],{"scriptlevel":541,"displaystyle":70},[56,11929,11930,11932,11934,11936,11978],{},[56,11931],{},[230,11933],{"width":6891},[133,11935,409],{},[56,11937,11938,11940,11942,11944,11946,11948,11950,11952,11958,11960,11962,11964,11966,11968,11974,11976],{},[133,11939,537],{"fence":70},[59,11941,401],{},[133,11943,136],{"stretchy":135},[59,11945,142],{},[133,11947,155],{"separator":70},[59,11949,158],{},[133,11951,155],{"separator":70},[138,11953,11954,11956],{},[59,11955,142],{},[133,11957,147],{"mathvariant":145,"lspace":146,"rspace":146},[133,11959,161],{"stretchy":135},[133,11961,788],{},[59,11963,531],{},[59,11965,4230],{},[133,11967,136],{"stretchy":135},[138,11969,11970,11972],{},[59,11971,142],{},[133,11973,147],{"mathvariant":145,"lspace":146,"rspace":146},[133,11975,161],{"stretchy":135},[133,11977,4280],{"fence":70},[59,11979,2903],{"mathvariant":145},[63,11981,11982],{"encoding":65},"\\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}",[39,11984,11986],{"className":11985,"ariaHidden":70},[69],[39,11987,11989,11992],{"className":11988},[74],[39,11990],{"className":11991,"style":6954},[78],[39,11993,11995],{"className":11994},[83],[39,11996,11998,12084],{"className":11997},[759],[39,11999,12001],{"className":12000},[918],[39,12002,12004,12076],{"className":12003},[196,649],[39,12005,12007,12073],{"className":12006},[200],[39,12008,12010,12065],{"className":12009,"style":6973},[204],[39,12011,12012,12015],{"style":6976},[39,12013],{"className":12014,"style":3311},[212],[39,12016,12018,12021,12050,12053,12056,12059,12062],{"className":12017},[83],[39,12019,136],{"className":12020},[182],[39,12022,12024,12027],{"className":12023},[83],[39,12025,11772],{"className":12026,"style":11793},[83,641],[39,12028,12030],{"className":12029},[192],[39,12031,12033],{"className":12032},[196],[39,12034,12036],{"className":12035},[200],[39,12037,12039],{"className":12038,"style":4421},[204],[39,12040,12041,12044],{"style":1027},[39,12042],{"className":12043,"style":213},[212],[39,12045,12047],{"className":12046},[217,218,219,220],[39,12048,2098],{"className":12049,"style":2114},[83,84,220],[39,12051,4230],{"className":12052,"style":461},[83,84],[39,12054,161],{"className":12055},[263],[39,12057,136],{"className":12058},[182],[39,12060,142],{"className":12061},[83,84],[39,12063,161],{"className":12064},[263],[39,12066,12067,12070],{"style":7032},[39,12068],{"className":12069,"style":3311},[212],[39,12071],{"className":12072},[83],[39,12074,673],{"className":12075},[672],[39,12077,12079],{"className":12078},[200],[39,12080,12082],{"className":12081,"style":7048},[204],[39,12083],{},[39,12085,12087],{"className":12086},[1223],[39,12088,12090,12487],{"className":12089},[196,649],[39,12091,12093,12484],{"className":12092},[200],[39,12094,12096,12334],{"className":12095,"style":6973},[204],[39,12097,12098,12101],{"style":6976},[39,12099],{"className":12100,"style":3311},[212],[39,12102,12104,12107,12110,12113,12116,12163,12166,12169,12172,12175,12178,12181,12184,12187,12190,12193,12269,12272,12275,12278,12310,12313,12316,12319,12322,12325,12328,12331],{"className":12103},[83],[39,12105],{"className":12106},[83],[39,12108],{"className":12109,"style":231},[230],[39,12111,793],{"className":12112},[235],[39,12114],{"className":12115,"style":231},[230],[39,12117,12119],{"className":12118},[1744,3294],[39,12120,12122,12155],{"className":12121},[196,649],[39,12123,12125,12152],{"className":12124},[200],[39,12126,12128,12142],{"className":12127,"style":6670},[204],[39,12129,12130,12133],{"style":6673},[39,12131],{"className":12132,"style":3311},[212],[39,12134,12136],{"className":12135},[217,218,219,220],[39,12137,12139],{"className":12138},[83,220],[39,12140,158],{"className":12141},[83,84,220],[39,12143,12144,12147],{"style":3330},[39,12145],{"className":12146,"style":3311},[212],[39,12148,12149],{},[39,12150,3191],{"className":12151},[1744,3339,3340],[39,12153,673],{"className":12154},[672],[39,12156,12158],{"className":12157},[200],[39,12159,12161],{"className":12160,"style":6705},[204],[39,12162],{},[39,12164],{"className":12165,"style":256},[230],[39,12167,2098],{"className":12168,"style":2114},[83,84],[39,12170,136],{"className":12171},[182],[39,12173,158],{"className":12174},[83,84],[39,12176],{"className":12177,"style":231},[230],[39,12179,150],{"className":12180},[235],[39,12182],{"className":12183,"style":231},[230],[39,12185,142],{"className":12186},[83,84],[39,12188,161],{"className":12189},[263],[39,12191],{"className":12192,"style":256},[230],[39,12194,12196],{"className":12195},[1744,3294],[39,12197,12199,12261],{"className":12198},[196,649],[39,12200,12202,12258],{"className":12201},[200],[39,12203,12205,12248],{"className":12204,"style":6670},[204],[39,12206,12207,12210],{"style":7097},[39,12208],{"className":12209,"style":3311},[212],[39,12211,12213],{"className":12212},[217,218,219,220],[39,12214,12216],{"className":12215},[83,220],[39,12217,12219,12222],{"className":12218},[83,220],[39,12220,142],{"className":12221},[83,84,220],[39,12223,12225],{"className":12224},[192],[39,12226,12228],{"className":12227},[196],[39,12229,12231],{"className":12230},[200],[39,12232,12234],{"className":12233,"style":3997},[204],[39,12235,12236,12239],{"style":4000},[39,12237],{"className":12238,"style":4004},[212],[39,12240,12242],{"className":12241},[217,4008,4009,220],[39,12243,12245],{"className":12244},[83,220],[39,12246,147],{"className":12247},[83,220],[39,12249,12250,12253],{"style":3330},[39,12251],{"className":12252,"style":3311},[212],[39,12254,12255],{},[39,12256,3191],{"className":12257},[1744,3339,3340],[39,12259,673],{"className":12260},[672],[39,12262,12264],{"className":12263},[200],[39,12265,12267],{"className":12266,"style":7158},[204],[39,12268],{},[39,12270],{"className":12271,"style":256},[230],[39,12273,131],{"className":12274,"style":178},[83,84],[39,12276,136],{"className":12277},[182],[39,12279,12281,12284],{"className":12280},[83],[39,12282,142],{"className":12283},[83,84],[39,12285,12287],{"className":12286},[192],[39,12288,12290],{"className":12289},[196],[39,12291,12293],{"className":12292},[200],[39,12294,12296],{"className":12295,"style":1024},[204],[39,12297,12298,12301],{"style":1027},[39,12299],{"className":12300,"style":213},[212],[39,12302,12304],{"className":12303},[217,218,219,220],[39,12305,12307],{"className":12306},[83,220],[39,12308,147],{"className":12309},[83,220],[39,12311],{"className":12312,"style":231},[230],[39,12314,150],{"className":12315},[235],[39,12317],{"className":12318,"style":231},[230],[39,12320,142],{"className":12321},[83,84],[39,12323,155],{"className":12324},[252],[39,12326],{"className":12327,"style":256},[230],[39,12329,158],{"className":12330},[83,84],[39,12332,161],{"className":12333},[263],[39,12335,12336,12339],{"style":7032},[39,12337],{"className":12338,"style":3311},[212],[39,12340,12342,12345,12348,12351,12354,12357,12478,12481],{"className":12341},[83],[39,12343],{"className":12344},[83],[39,12346],{"className":12347,"style":7240},[230],[39,12349],{"className":12350,"style":461},[230],[39,12352,409],{"className":12353},[465],[39,12355],{"className":12356,"style":461},[230],[39,12358,12360,12363,12366,12369,12372,12375,12378,12381,12384,12387,12419,12422,12425,12428,12431,12434,12437,12440,12472,12475],{"className":12359},[2842],[39,12361,537],{"className":12362,"style":4592},[182,4591],[39,12364,401],{"className":12365,"style":438},[83,84],[39,12367,136],{"className":12368},[182],[39,12370,142],{"className":12371},[83,84],[39,12373,155],{"className":12374},[252],[39,12376],{"className":12377,"style":256},[230],[39,12379,158],{"className":12380},[83,84],[39,12382,155],{"className":12383},[252],[39,12385],{"className":12386,"style":256},[230],[39,12388,12390,12393],{"className":12389},[83],[39,12391,142],{"className":12392},[83,84],[39,12394,12396],{"className":12395},[192],[39,12397,12399],{"className":12398},[196],[39,12400,12402],{"className":12401},[200],[39,12403,12405],{"className":12404,"style":1024},[204],[39,12406,12407,12410],{"style":1027},[39,12408],{"className":12409,"style":213},[212],[39,12411,12413],{"className":12412},[217,218,219,220],[39,12414,12416],{"className":12415},[83,220],[39,12417,147],{"className":12418},[83,220],[39,12420,161],{"className":12421},[263],[39,12423],{"className":12424,"style":461},[230],[39,12426,788],{"className":12427},[465],[39,12429],{"className":12430,"style":461},[230],[39,12432,531],{"className":12433,"style":565},[83,84],[39,12435,4230],{"className":12436,"style":461},[83,84],[39,12438,136],{"className":12439},[182],[39,12441,12443,12446],{"className":12442},[83],[39,12444,142],{"className":12445},[83,84],[39,12447,12449],{"className":12448},[192],[39,12450,12452],{"className":12451},[196],[39,12453,12455],{"className":12454},[200],[39,12456,12458],{"className":12457,"style":1024},[204],[39,12459,12460,12463],{"style":1027},[39,12461],{"className":12462,"style":213},[212],[39,12464,12466],{"className":12465},[217,218,219,220],[39,12467,12469],{"className":12468},[83,220],[39,12470,147],{"className":12471},[83,220],[39,12473,161],{"className":12474},[263],[39,12476,4280],{"className":12477,"style":4592},[263,4591],[39,12479],{"className":12480,"style":256},[230],[39,12482,2903],{"className":12483},[83],[39,12485,673],{"className":12486},[672],[39,12488,12490],{"className":12489},[200],[39,12491,12493],{"className":12492,"style":7048},[204],[39,12494],{},[11,12496,12497],{},"于是贝尔曼期望方程可以写成不动点形式",[39,12499,12501],{"className":12500,"translate":43},[746],[39,12502,12504,12538],{"className":12503,"translate":43},[42],[39,12505,12507],{"className":12506},[47],[49,12508,12509],{"xmlns":51,"display":755},[53,12510,12511,12535],{},[56,12512,12513,12519,12521,12527,12533],{},[138,12514,12515,12517],{},[59,12516,4230],{},[59,12518,2098],{},[133,12520,793],{},[138,12522,12523,12525],{},[59,12524,11772],{"mathvariant":617},[59,12526,2098],{},[138,12528,12529,12531],{},[59,12530,4230],{},[59,12532,2098],{},[59,12534,2903],{"mathvariant":145},[63,12536,12537],{"encoding":65},"V^\\pi=\\mathcal{T}^\\pi V^\\pi.",[39,12539,12541,12585],{"className":12540,"ariaHidden":70},[69],[39,12542,12544,12547,12576,12579,12582],{"className":12543},[74],[39,12545],{"className":12546,"style":4421},[78],[39,12548,12550,12553],{"className":12549},[83],[39,12551,4230],{"className":12552,"style":461},[83,84],[39,12554,12556],{"className":12555},[192],[39,12557,12559],{"className":12558},[196],[39,12560,12562],{"className":12561},[200],[39,12563,12565],{"className":12564,"style":4421},[204],[39,12566,12567,12570],{"style":1027},[39,12568],{"className":12569,"style":213},[212],[39,12571,12573],{"className":12572},[217,218,219,220],[39,12574,2098],{"className":12575,"style":2114},[83,84,220],[39,12577],{"className":12578,"style":231},[230],[39,12580,793],{"className":12581},[235],[39,12583],{"className":12584,"style":231},[230],[39,12586,12588,12591,12620,12649],{"className":12587},[74],[39,12589],{"className":12590,"style":4421},[78],[39,12592,12594,12597],{"className":12593},[83],[39,12595,11772],{"className":12596,"style":11793},[83,641],[39,12598,12600],{"className":12599},[192],[39,12601,12603],{"className":12602},[196],[39,12604,12606],{"className":12605},[200],[39,12607,12609],{"className":12608,"style":4421},[204],[39,12610,12611,12614],{"style":1027},[39,12612],{"className":12613,"style":213},[212],[39,12615,12617],{"className":12616},[217,218,219,220],[39,12618,2098],{"className":12619,"style":2114},[83,84,220],[39,12621,12623,12626],{"className":12622},[83],[39,12624,4230],{"className":12625,"style":461},[83,84],[39,12627,12629],{"className":12628},[192],[39,12630,12632],{"className":12631},[196],[39,12633,12635],{"className":12634},[200],[39,12636,12638],{"className":12637,"style":4421},[204],[39,12639,12640,12643],{"style":1027},[39,12641],{"className":12642,"style":213},[212],[39,12644,12646],{"className":12645},[217,218,219,220],[39,12647,2098],{"className":12648,"style":2114},[83,84,220],[39,12650,2903],{"className":12651},[83],[11,12653,12654,12655,12713,12714,12772,12773,12801,12802,12866,12867,12925],{},"这里的 ",[39,12656,12658,12675],{"className":12657,"translate":43},[42],[39,12659,12661],{"className":12660},[47],[49,12662,12663],{"xmlns":51},[53,12664,12665,12673],{},[56,12666,12667],{},[138,12668,12669,12671],{},[59,12670,4230],{},[59,12672,2098],{},[63,12674,6382],{"encoding":65},[39,12676,12678],{"className":12677,"ariaHidden":70},[69],[39,12679,12681,12684],{"className":12680},[74],[39,12682],{"className":12683,"style":79},[78],[39,12685,12687,12690],{"className":12686},[83],[39,12688,4230],{"className":12689,"style":461},[83,84],[39,12691,12693],{"className":12692},[192],[39,12694,12696],{"className":12695},[196],[39,12697,12699],{"className":12698},[200],[39,12700,12702],{"className":12701,"style":6410},[204],[39,12703,12704,12707],{"style":208},[39,12705],{"className":12706,"style":213},[212],[39,12708,12710],{"className":12709},[217,218,219,220],[39,12711,2098],{"className":12712,"style":2114},[83,84,220]," 不是某个数，而是函数空间中的一个点。算子 ",[39,12715,12717,12734],{"className":12716,"translate":43},[42],[39,12718,12720],{"className":12719},[47],[49,12721,12722],{"xmlns":51},[53,12723,12724,12732],{},[56,12725,12726],{},[138,12727,12728,12730],{},[59,12729,11772],{"mathvariant":617},[59,12731,2098],{},[63,12733,11777],{"encoding":65},[39,12735,12737],{"className":12736,"ariaHidden":70},[69],[39,12738,12740,12743],{"className":12739},[74],[39,12741],{"className":12742,"style":79},[78],[39,12744,12746,12749],{"className":12745},[83],[39,12747,11772],{"className":12748,"style":11793},[83,641],[39,12750,12752],{"className":12751},[192],[39,12753,12755],{"className":12754},[196],[39,12756,12758],{"className":12757},[200],[39,12759,12761],{"className":12760,"style":6410},[204],[39,12762,12763,12766],{"style":208},[39,12764],{"className":12765,"style":213},[212],[39,12767,12769],{"className":12768},[217,218,219,220],[39,12770,2098],{"className":12771,"style":2114},[83,84,220]," 把一个有界函数 ",[39,12774,12776,12789],{"className":12775,"translate":43},[42],[39,12777,12779],{"className":12778},[47],[49,12780,12781],{"xmlns":51},[53,12782,12783,12787],{},[56,12784,12785],{},[59,12786,4230],{},[63,12788,4230],{"encoding":65},[39,12790,12792],{"className":12791,"ariaHidden":70},[69],[39,12793,12795,12798],{"className":12794},[74],[39,12796],{"className":12797,"style":79},[78],[39,12799,4230],{"className":12800,"style":461},[83,84]," 映射为另一个有界函数 ",[39,12803,12805,12825],{"className":12804,"translate":43},[42],[39,12806,12808],{"className":12807},[47],[49,12809,12810],{"xmlns":51},[53,12811,12812,12822],{},[56,12813,12814,12820],{},[138,12815,12816,12818],{},[59,12817,11772],{"mathvariant":617},[59,12819,2098],{},[59,12821,4230],{},[63,12823,12824],{"encoding":65},"\\mathcal{T}^\\pi V",[39,12826,12828],{"className":12827,"ariaHidden":70},[69],[39,12829,12831,12834,12863],{"className":12830},[74],[39,12832],{"className":12833,"style":79},[78],[39,12835,12837,12840],{"className":12836},[83],[39,12838,11772],{"className":12839,"style":11793},[83,641],[39,12841,12843],{"className":12842},[192],[39,12844,12846],{"className":12845},[196],[39,12847,12849],{"className":12848},[200],[39,12850,12852],{"className":12851,"style":6410},[204],[39,12853,12854,12857],{"style":208},[39,12855],{"className":12856,"style":213},[212],[39,12858,12860],{"className":12859},[217,218,219,220],[39,12861,2098],{"className":12862,"style":2114},[83,84,220],[39,12864,4230],{"className":12865,"style":461},[83,84],"。因此，求解贝尔曼方程等价于寻找算子 ",[39,12868,12870,12887],{"className":12869,"translate":43},[42],[39,12871,12873],{"className":12872},[47],[49,12874,12875],{"xmlns":51},[53,12876,12877,12885],{},[56,12878,12879],{},[138,12880,12881,12883],{},[59,12882,11772],{"mathvariant":617},[59,12884,2098],{},[63,12886,11777],{"encoding":65},[39,12888,12890],{"className":12889,"ariaHidden":70},[69],[39,12891,12893,12896],{"className":12892},[74],[39,12894],{"className":12895,"style":79},[78],[39,12897,12899,12902],{"className":12898},[83],[39,12900,11772],{"className":12901,"style":11793},[83,641],[39,12903,12905],{"className":12904},[192],[39,12906,12908],{"className":12907},[196],[39,12909,12911],{"className":12910},[200],[39,12912,12914],{"className":12913,"style":6410},[204],[39,12915,12916,12919],{"style":208},[39,12917],{"className":12918,"style":213},[212],[39,12920,12922],{"className":12921},[217,218,219,220],[39,12923,2098],{"className":12924,"style":2114},[83,84,220]," 的不动点。",[26,12927,12929],{"id":12928},"_23-有限状态下的矩阵解","2.3 有限状态下的矩阵解",[11,12931,12932,12933,13026,13027,1639,13109,13221,13222,13349],{},"当 ",[39,12934,12936,12969],{"className":12935,"translate":43},[42],[39,12937,12939],{"className":12938},[47],[49,12940,12941],{"xmlns":51},[53,12942,12943,12966],{},[56,12944,12945,12947,12949,12952,12954,12956,12958,12960,12963],{},[59,12946,61],{},[133,12948,793],{},[133,12950,12951],{"stretchy":135},"{",[539,12953,546],{},[133,12955,155],{"separator":70},[133,12957,2641],{},[133,12959,155],{"separator":70},[59,12961,12962],{},"n",[133,12964,12965],{"stretchy":135},"}",[63,12967,12968],{"encoding":65},"S=\\{1,\\dots,n\\}",[39,12970,12972,12990],{"className":12971,"ariaHidden":70},[69],[39,12973,12975,12978,12981,12984,12987],{"className":12974},[74],[39,12976],{"className":12977,"style":79},[78],[39,12979,61],{"className":12980,"style":85},[83,84],[39,12982],{"className":12983,"style":231},[230],[39,12985,793],{"className":12986},[235],[39,12988],{"className":12989,"style":231},[230],[39,12991,12993,12996,12999,13002,13005,13008,13011,13014,13017,13020,13023],{"className":12992},[74],[39,12994],{"className":12995,"style":245},[78],[39,12997,12951],{"className":12998},[182],[39,13000,546],{"className":13001},[83],[39,13003,155],{"className":13004},[252],[39,13006],{"className":13007,"style":256},[230],[39,13009,2641],{"className":13010},[2842],[39,13012],{"className":13013,"style":256},[230],[39,13015,155],{"className":13016},[252],[39,13018],{"className":13019,"style":256},[230],[39,13021,12962],{"className":13022},[83,84],[39,13024,12965],{"className":13025},[263]," 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 ",[39,13028,13030,13052],{"className":13029,"translate":43},[42],[39,13031,13033],{"className":13032},[47],[49,13034,13035],{"xmlns":51},[53,13036,13037,13049],{},[56,13038,13039,13041,13043],{},[59,13040,4230],{},[133,13042,534],{},[138,13044,13045,13047],{},[59,13046,422],{"mathvariant":421},[59,13048,12962],{},[63,13050,13051],{"encoding":65},"V\\in\\mathbb{R}^n",[39,13053,13055,13074],{"className":13054,"ariaHidden":70},[69],[39,13056,13058,13062,13065,13068,13071],{"className":13057},[74],[39,13059],{"className":13060,"style":13061},[78],"height:0.7224em;vertical-align:-0.0391em;",[39,13063,4230],{"className":13064,"style":461},[83,84],[39,13066],{"className":13067,"style":231},[230],[39,13069,534],{"className":13070},[235],[39,13072],{"className":13073,"style":231},[230],[39,13075,13077,13080],{"className":13076},[74],[39,13078],{"className":13079,"style":511},[78],[39,13081,13083,13086],{"className":13082},[83],[39,13084,422],{"className":13085},[83,515],[39,13087,13089],{"className":13088},[192],[39,13090,13092],{"className":13091},[196],[39,13093,13095],{"className":13094},[200],[39,13096,13098],{"className":13097,"style":6410},[204],[39,13099,13100,13103],{"style":208},[39,13101],{"className":13102,"style":213},[212],[39,13104,13106],{"className":13105},[217,218,219,220],[39,13107,12962],{"className":13108},[83,84,220],[39,13110,13112,13138],{"className":13111,"translate":43},[42],[39,13113,13115],{"className":13114},[47],[49,13116,13117],{"xmlns":51},[53,13118,13119,13135],{},[56,13120,13121,13127,13129],{},[138,13122,13123,13125],{},[59,13124,401],{},[59,13126,2098],{},[133,13128,534],{},[138,13130,13131,13133],{},[59,13132,422],{"mathvariant":421},[59,13134,12962],{},[63,13136,13137],{"encoding":65},"r^\\pi\\in\\mathbb{R}^n",[39,13139,13141,13186],{"className":13140,"ariaHidden":70},[69],[39,13142,13144,13148,13177,13180,13183],{"className":13143},[74],[39,13145],{"className":13146,"style":13147},[78],"height:0.7035em;vertical-align:-0.0391em;",[39,13149,13151,13154],{"className":13150},[83],[39,13152,401],{"className":13153,"style":438},[83,84],[39,13155,13157],{"className":13156},[192],[39,13158,13160],{"className":13159},[196],[39,13161,13163],{"className":13162},[200],[39,13164,13166],{"className":13165,"style":6410},[204],[39,13167,13168,13171],{"style":208},[39,13169],{"className":13170,"style":213},[212],[39,13172,13174],{"className":13173},[217,218,219,220],[39,13175,2098],{"className":13176,"style":2114},[83,84,220],[39,13178],{"className":13179,"style":231},[230],[39,13181,534],{"className":13182},[235],[39,13184],{"className":13185,"style":231},[230],[39,13187,13189,13192],{"className":13188},[74],[39,13190],{"className":13191,"style":511},[78],[39,13193,13195,13198],{"className":13194},[83],[39,13196,422],{"className":13197},[83,515],[39,13199,13201],{"className":13200},[192],[39,13202,13204],{"className":13203},[196],[39,13205,13207],{"className":13206},[200],[39,13208,13210],{"className":13209,"style":6410},[204],[39,13211,13212,13215],{"style":208},[39,13213],{"className":13214,"style":213},[212],[39,13216,13218],{"className":13217},[217,218,219,220],[39,13219,12962],{"className":13220},[83,84,220],"，以及矩阵 ",[39,13223,13225,13257],{"className":13224,"translate":43},[42],[39,13226,13228],{"className":13227},[47],[49,13229,13230],{"xmlns":51},[53,13231,13232,13254],{},[56,13233,13234,13240,13242],{},[138,13235,13236,13238],{},[59,13237,131],{},[59,13239,2098],{},[133,13241,534],{},[138,13243,13244,13246],{},[59,13245,422],{"mathvariant":421},[56,13247,13248,13250,13252],{},[59,13249,12962],{},[133,13251,409],{},[59,13253,12962],{},[63,13255,13256],{"encoding":65},"P^\\pi\\in\\mathbb{R}^{n\\times 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i)\\sum_{j}P(j\\mid i,a)r(i,a,j),\\\\\nP^\\pi_{ij}&=\\sum_{a}\\pi(a\\mid i)P(j\\mid 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P^\\pi)V^\\pi=r^\\pi.",[39,14406,14408,14430,14509],{"className":14407,"ariaHidden":70},[69],[39,14409,14411,14414,14417,14421,14424,14427],{"className":14410},[74],[39,14412],{"className":14413,"style":245},[78],[39,14415,136],{"className":14416},[182],[39,14418,14373],{"className":14419,"style":14420},[83,84],"margin-right:0.0785em;",[39,14422],{"className":14423,"style":461},[230],[39,14425,5378],{"className":14426},[465],[39,14428],{"className":14429,"style":461},[230],[39,14431,14433,14436,14439,14468,14471,14500,14503,14506],{"className":14432},[74],[39,14434],{"className":14435,"style":245},[78],[39,14437,531],{"className":14438,"style":565},[83,84],[39,14440,14442,14445],{"className":14441},[83],[39,14443,131],{"className":14444,"style":178},[83,84],[39,14446,14448],{"className":14447},[192],[39,14449,14451],{"className":14450},[196],[39,14452,14454],{"className":14453},[200],[39,14455,14457],{"className":14456,"style":4421},[204],[39,14458,14459,14462],{"style":1027},[39,14460],{"className":14461,"style":213},[212],[39,14463,14465],{"className":14464},[217,218,219,220],[39,14466,2098],{"className":14467,"style":2114},[83,84,220],[39,14469,161],{"className":14470},[263],[39,14472,14474,14477],{"className":14473},[83],[39,14475,4230],{"className":14476,"style":461},[83,84],[39,14478,14480],{"className":14479},[192],[39,14481,14483],{"className":14482},[196],[39,14484,14486],{"className":14485},[200],[39,14487,14489],{"className":14488,"style":4421},[204],[39,14490,14491,14494],{"style":1027},[39,14492],{"className":14493,"style":213},[212],[39,14495,14497],{"className":14496},[217,218,219,220],[39,14498,2098],{"className":14499,"style":2114},[83,84,220],[39,14501],{"className":14502,"style":231},[230],[39,14504,793],{"className":14505},[235],[39,14507],{"className":14508,"style":231},[230],[39,14510,14512,14515,14544],{"className":14511},[74],[39,14513],{"className":14514,"style":4421},[78],[39,14516,14518,14521],{"className":14517},[83],[39,14519,401],{"className":14520,"style":438},[83,84],[39,14522,14524],{"className":14523},[192],[39,14525,14527],{"className":14526},[196],[39,14528,14530],{"className":14529},[200],[39,14531,14533],{"className":14532,"style":4421},[204],[39,14534,14535,14538],{"style":1027},[39,14536],{"className":14537,"style":213},[212],[39,14539,14541],{"className":14540},[217,218,219,220],[39,14542,2098],{"className":14543,"style":2114},[83,84,220],[39,14545,2903],{"className":14546},[83],[31,14548,14551,14715],{"id":14549,"title":14550},"proposition-matrix-solution","命题 2.2（矩阵解与 Neumann 级数）",[11,14552,37,14553,14627,14628,14714],{},[39,14554,14556,14578],{"className":14555,"translate":43},[42],[39,14557,14559],{"className":14558},[47],[49,14560,14561],{"xmlns":51},[53,14562,14563,14575],{},[56,14564,14565,14567,14569,14571,14573],{},[539,14566,541],{},[133,14568,1824],{},[59,14570,531],{},[133,14572,1693],{},[539,14574,546],{},[63,14576,14577],{"encoding":65},"0\\le\\gamma\u003C1",[39,14579,14581,14600,14618],{"className":14580,"ariaHidden":70},[69],[39,14582,14584,14588,14591,14594,14597],{"className":14583},[74],[39,14585],{"className":14586,"style":14587},[78],"height:0.7804em;vertical-align:-0.136em;",[39,14589,541],{"className":14590},[83],[39,14592],{"className":14593,"style":231},[230],[39,14595,1824],{"className":14596},[235],[39,14598],{"className":14599,"style":231},[230],[39,14601,14603,14606,14609,14612,14615],{"className":14602},[74],[39,14604],{"className":14605,"style":561},[78],[39,14607,531],{"className":14608,"style":565},[83,84],[39,14610],{"className":14611,"style":231},[230],[39,14613,1693],{"className":14614},[235],[39,14616],{"className":14617,"style":231},[230],[39,14619,14621,14624],{"className":14620},[74],[39,14622],{"className":14623,"style":2473},[78],[39,14625,546],{"className":14626},[83],"。则 ",[39,14629,14631,14655],{"className":14630,"translate":43},[42],[39,14632,14634],{"className":14633},[47],[49,14635,14636],{"xmlns":51},[53,14637,14638,14652],{},[56,14639,14640,14642,14644,14646],{},[59,14641,14373],{},[133,14643,5378],{},[59,14645,531],{},[138,14647,14648,14650],{},[59,14649,131],{},[59,14651,2098],{},[63,14653,14654],{"encoding":65},"I-\\gamma P^\\pi",[39,14656,14658,14676],{"className":14657,"ariaHidden":70},[69],[39,14659,14661,14664,14667,14670,14673],{"className":14660},[74],[39,14662],{"className":14663,"style":454},[78],[39,14665,14373],{"className":14666,"style":14420},[83,84],[39,14668],{"className":14669,"style":461},[230],[39,14671,5378],{"className":14672},[465],[39,14674],{"className":14675,"style":461},[230],[39,14677,14679,14682,14685],{"className":14678},[74],[39,14680],{"className":14681,"style":6452},[78],[39,14683,531],{"className":14684,"style":565},[83,84],[39,14686,14688,14691],{"className":14687},[83],[39,14689,131],{"className":14690,"style":178},[83,84],[39,14692,14694],{"className":14693},[192],[39,14695,14697],{"className":14696},[196],[39,14698,14700],{"className":14699},[200],[39,14701,14703],{"className":14702,"style":6410},[204],[39,14704,14705,14708],{"style":208},[39,14706],{"className":14707,"style":213},[212],[39,14709,14711],{"className":14710},[217,218,219,220],[39,14712,2098],{"className":14713,"style":2114},[83,84,220]," 可逆，且",[39,14716,14718],{"className":14717,"translate":43},[746],[39,14719,14721,14815],{"className":14720,"translate":43},[42],[39,14722,14724],{"className":14723},[47],[49,14725,14726],{"xmlns":51,"display":755},[53,14727,14728,14812],{},[56,14729,14730,14736,14738,14740,14742,14744,14746,14752,14762,14768,14770,14784,14790,14792,14798,14804,14810],{},[138,14731,14732,14734],{},[59,14733,4230],{},[59,14735,2098],{},[133,14737,793],{},[133,14739,136],{"stretchy":135},[59,14741,14373],{},[133,14743,5378],{},[59,14745,531],{},[138,14747,14748,14750],{},[59,14749,131],{},[59,14751,2098],{},[138,14753,14754,14756],{},[133,14755,161],{"stretchy":135},[56,14757,14758,14760],{},[133,14759,5378],{},[539,14761,546],{},[138,14763,14764,14766],{},[59,14765,401],{},[59,14767,2098],{},[133,14769,793],{},[3187,14771,14772,14774,14782],{},[133,14773,3191],{},[56,14775,14776,14778,14780],{},[59,14777,3196],{},[133,14779,793],{},[539,14781,541],{},[59,14783,1696],{"mathvariant":145},[138,14785,14786,14788],{},[59,14787,531],{},[59,14789,3196],{},[133,14791,136],{"stretchy":135},[138,14793,14794,14796],{},[59,14795,131],{},[59,14797,2098],{},[138,14799,14800,14802],{},[133,14801,161],{"stretchy":135},[59,14803,3196],{},[138,14805,14806,14808],{},[59,14807,401],{},[59,14809,2098],{},[59,14811,2903],{"mathvariant":145},[63,14813,14814],{"encoding":65},"V^\\pi=(I-\\gamma P^\\pi)^{-1}r^\\pi\n=\\sum_{k=0}^{\\infty}\\gamma^k(P^\\pi)^k r^\\pi.",[39,14816,14818,14862,14883,14996],{"className":14817,"ariaHidden":70},[69],[39,14819,14821,14824,14853,14856,14859],{"className":14820},[74],[39,14822],{"className":14823,"style":4421},[78],[39,14825,14827,14830],{"className":14826},[83],[39,14828,4230],{"className":14829,"style":461},[83,84],[39,14831,14833],{"className":14832},[192],[39,14834,14836],{"className":14835},[196],[39,14837,14839],{"className":14838},[200],[39,14840,14842],{"className":14841,"style":4421},[204],[39,14843,14844,14847],{"style":1027},[39,14845],{"className":14846,"style":213},[212],[39,14848,14850],{"className":14849},[217,218,219,220],[39,14851,2098],{"className":14852,"style":2114},[83,84,220],[39,14854],{"className":14855,"style":231},[230],[39,14857,793],{"className":14858},[235],[39,14860],{"className":14861,"style":231},[230],[39,14863,14865,14868,14871,14874,14877,14880],{"className":14864},[74],[39,14866],{"className":14867,"style":245},[78],[39,14869,136],{"className":14870},[182],[39,14872,14373],{"className":14873,"style":14420},[83,84],[39,14875],{"className":14876,"style":461},[230],[39,14878,5378],{"className":14879},[465],[39,14881],{"className":14882,"style":461},[230],[39,14884,14886,14890,14893,14922,14958,14987,14990,14993],{"className":14885},[74],[39,14887],{"className":14888,"style":14889},[78],"height:1.1141em;vertical-align:-0.25em;",[39,14891,531],{"className":14892,"style":565},[83,84],[39,14894,14896,14899],{"className":14895},[83],[39,14897,131],{"className":14898,"style":178},[83,84],[39,14900,14902],{"className":14901},[192],[39,14903,14905],{"className":14904},[196],[39,14906,14908],{"className":14907},[200],[39,14909,14911],{"className":14910,"style":4421},[204],[39,14912,14913,14916],{"style":1027},[39,14914],{"className":14915,"style":213},[212],[39,14917,14919],{"className":14918},[217,218,219,220],[39,14920,2098],{"className":14921,"style":2114},[83,84,220],[39,14923,14925,14928],{"className":14924},[263],[39,14926,161],{"className":14927},[263],[39,14929,14931],{"className":14930},[192],[39,14932,14934],{"className":14933},[196],[39,14935,14937],{"className":14936},[200],[39,14938,14941],{"className":14939,"style":14940},[204],"height:0.8641em;",[39,14942,14943,14946],{"style":1027},[39,14944],{"className":14945,"style":213},[212],[39,14947,14949],{"className":14948},[217,218,219,220],[39,14950,14952,14955],{"className":14951},[83,220],[39,14953,5378],{"className":14954},[83,220],[39,14956,546],{"className":14957},[83,220],[39,14959,14961,14964],{"className":14960},[83],[39,14962,401],{"className":14963,"style":438},[83,84],[39,14965,14967],{"className":14966},[192],[39,14968,14970],{"className":14969},[196],[39,14971,14973],{"className":14972},[200],[39,14974,14976],{"className":14975,"style":4421},[204],[39,14977,14978,14981],{"style":1027},[39,14979],{"className":14980,"style":213},[212],[39,14982,14984],{"className":14983},[217,218,219,220],[39,14985,2098],{"className":14986,"style":2114},[83,84,220],[39,14988],{"className":14989,"style":231},[230],[39,14991,793],{"className":14992},[235],[39,14994],{"className":14995,"style":231},[230],[39,14997,14999,15002,15069,15072,15101,15104,15133,15162,15191],{"className":14998},[74],[39,15000],{"className":15001,"style":3290},[78],[39,15003,15005],{"className":15004},[1744,3294],[39,15006,15008,15061],{"className":15007},[196,649],[39,15009,15011,15058],{"className":15010},[200],[39,15012,15014,15034,15044],{"className":15013,"style":3304},[204],[39,15015,15016,15019],{"style":3307},[39,15017],{"className":15018,"style":3311},[212],[39,15020,15022],{"className":15021},[217,218,219,220],[39,15023,15025,15028,15031],{"className":15024},[83,220],[39,15026,3196],{"className":15027,"style":3321},[83,84,220],[39,15029,793],{"className":15030},[235,220],[39,15032,541],{"className":15033},[83,220],[39,15035,15036,15039],{"style":3330},[39,15037],{"className":15038,"style":3311},[212],[39,15040,15041],{},[39,15042,3191],{"className":15043},[1744,3339,3340],[39,15045,15046,15049],{"style":3343},[39,15047],{"className":15048,"style":3311},[212],[39,15050,15052],{"className":15051},[217,218,219,220],[39,15053,15055],{"className":15054},[83,220],[39,15056,1696],{"className":15057},[83,220],[39,15059,673],{"className":15060},[672],[39,15062,15064],{"className":15063},[200],[39,15065,15067],{"className":15066,"style":3365},[204],[39,15068],{},[39,15070],{"className":15071,"style":256},[230],[39,15073,15075,15078],{"className":15074},[83],[39,15076,531],{"className":15077,"style":565},[83,84],[39,15079,15081],{"className":15080},[192],[39,15082,15084],{"className":15083},[196],[39,15085,15087],{"className":15086},[200],[39,15088,15090],{"className":15089,"style":3389},[204],[39,15091,15092,15095],{"style":1027},[39,15093],{"className":15094,"style":213},[212],[39,15096,15098],{"className":15097},[217,218,219,220],[39,15099,3196],{"className":15100,"style":3321},[83,84,220],[39,15102,136],{"className":15103},[182],[39,15105,15107,15110],{"className":15106},[83],[39,15108,131],{"className":15109,"style":178},[83,84],[39,15111,15113],{"className":15112},[192],[39,15114,15116],{"className":15115},[196],[39,15117,15119],{"className":15118},[200],[39,15120,15122],{"className":15121,"style":4421},[204],[39,15123,15124,15127],{"style":1027},[39,15125],{"className":15126,"style":213},[212],[39,15128,15130],{"className":15129},[217,218,219,220],[39,15131,2098],{"className":15132,"style":2114},[83,84,220],[39,15134,15136,15139],{"className":15135},[263],[39,15137,161],{"className":15138},[263],[39,15140,15142],{"className":15141},[192],[39,15143,15145],{"className":15144},[196],[39,15146,15148],{"className":15147},[200],[39,15149,15151],{"className":15150,"style":3389},[204],[39,15152,15153,15156],{"style":1027},[39,15154],{"className":15155,"style":213},[212],[39,15157,15159],{"className":15158},[217,218,219,220],[39,15160,3196],{"className":15161,"style":3321},[83,84,220],[39,15163,15165,15168],{"className":15164},[83],[39,15166,401],{"className":15167,"style":438},[83,84],[39,15169,15171],{"className":15170},[192],[39,15172,15174],{"className":15173},[196],[39,15175,15177],{"className":15176},[200],[39,15178,15180],{"className":15179,"style":4421},[204],[39,15181,15182,15185],{"style":1027},[39,15183],{"className":15184,"style":213},[212],[39,15186,15188],{"className":15187},[217,218,219,220],[39,15189,2098],{"className":15190,"style":2114},[83,84,220],[39,15192,2903],{"className":15193},[83],[11,15195,15196,15198,15199,15285,15286,15345,15346,15374,15375,15508,15509,15674,15675,15817],{},[1489,15197,5595],{}," 记 ",[39,15200,15202,15226],{"className":15201,"translate":43},[42],[39,15203,15205],{"className":15204},[47],[49,15206,15207],{"xmlns":51},[53,15208,15209,15223],{},[56,15210,15211,15213,15215,15217],{},[59,15212,4295],{},[133,15214,793],{},[59,15216,531],{},[138,15218,15219,15221],{},[59,15220,131],{},[59,15222,2098],{},[63,15224,15225],{"encoding":65},"Q=\\gamma P^\\pi",[39,15227,15229,15247],{"className":15228,"ariaHidden":70},[69],[39,15230,15232,15235,15238,15241,15244],{"className":15231},[74],[39,15233],{"className":15234,"style":6452},[78],[39,15236,4295],{"className":15237},[83,84],[39,15239],{"className":15240,"style":231},[230],[39,15242,793],{"className":15243},[235],[39,15245],{"className":15246,"style":231},[230],[39,15248,15250,15253,15256],{"className":15249},[74],[39,15251],{"className":15252,"style":6452},[78],[39,15254,531],{"className":15255,"style":565},[83,84],[39,15257,15259,15262],{"className":15258},[83],[39,15260,131],{"className":15261,"style":178},[83,84],[39,15263,15265],{"className":15264},[192],[39,15266,15268],{"className":15267},[196],[39,15269,15271],{"className":15270},[200],[39,15272,15274],{"className":15273,"style":6410},[204],[39,15275,15276,15279],{"style":208},[39,15277],{"className":15278,"style":213},[212],[39,15280,15282],{"className":15281},[217,218,219,220],[39,15283,2098],{"className":15284,"style":2114},[83,84,220],"。由于 ",[39,15287,15289,15307],{"className":15288,"translate":43},[42],[39,15290,15292],{"className":15291},[47],[49,15293,15294],{"xmlns":51},[53,15295,15296,15304],{},[56,15297,15298],{},[138,15299,15300,15302],{},[59,15301,131],{},[59,15303,2098],{},[63,15305,15306],{"encoding":65},"P^\\pi",[39,15308,15310],{"className":15309,"ariaHidden":70},[69],[39,15311,15313,15316],{"className":15312},[74],[39,15314],{"className":15315,"style":79},[78],[39,15317,15319,15322],{"className":15318},[83],[39,15320,131],{"className":15321,"style":178},[83,84],[39,15323,15325],{"className":15324},[192],[39,15326,15328],{"className":15327},[196],[39,15329,15331],{"className":15330},[200],[39,15332,15334],{"className":15333,"style":6410},[204],[39,15335,15336,15339],{"style":208},[39,15337],{"className":15338,"style":213},[212],[39,15340,15342],{"className":15341},[217,218,219,220],[39,15343,2098],{"className":15344,"style":2114},[83,84,220]," 的每一行之和为 ",[39,15347,15349,15362],{"className":15348,"translate":43},[42],[39,15350,15352],{"className":15351},[47],[49,15353,15354],{"xmlns":51},[53,15355,15356,15360],{},[56,15357,15358],{},[539,15359,546],{},[63,15361,546],{"encoding":65},[39,15363,15365],{"className":15364,"ariaHidden":70},[69],[39,15366,15368,15371],{"className":15367},[74],[39,15369],{"className":15370,"style":2473},[78],[39,15372,546],{"className":15373},[83],"，有 ",[39,15376,15378,15409],{"className":15377,"translate":43},[42],[39,15379,15381],{"className":15380},[47],[49,15382,15383],{"xmlns":51},[53,15384,15385,15406],{},[56,15386,15387,15390,15396,15402,15404],{},[59,15388,15389],{"mathvariant":145},"∥",[138,15391,15392,15394],{},[59,15393,131],{},[59,15395,2098],{},[613,15397,15398,15400],{},[59,15399,15389],{"mathvariant":145},[59,15401,1696],{"mathvariant":145},[133,15403,793],{},[539,15405,546],{},[63,15407,15408],{"encoding":65},"\\|P^\\pi\\|_\\infty=1",[39,15410,15412,15499],{"className":15411,"ariaHidden":70},[69],[39,15413,15415,15418,15421,15450,15490,15493,15496],{"className":15414},[74],[39,15416],{"className":15417,"style":245},[78],[39,15419,15389],{"className":15420},[83],[39,15422,15424,15427],{"className":15423},[83],[39,15425,131],{"className":15426,"style":178},[83,84],[39,15428,15430],{"className":15429},[192],[39,15431,15433],{"className":15432},[196],[39,15434,15436],{"className":15435},[200],[39,15437,15439],{"className":15438,"style":6410},[204],[39,15440,15441,15444],{"style":208},[39,15442],{"className":15443,"style":213},[212],[39,15445,15447],{"className":15446},[217,218,219,220],[39,15448,2098],{"className":15449,"style":2114},[83,84,220],[39,15451,15453,15456],{"className":15452},[83],[39,15454,15389],{"className":15455},[83],[39,15457,15459],{"className":15458},[192],[39,15460,15462,15482],{"className":15461},[196,649],[39,15463,15465,15479],{"className":15464},[200],[39,15466,15468],{"className":15467,"style":1728},[204],[39,15469,15470,15473],{"style":966},[39,15471],{"className":15472,"style":213},[212],[39,15474,15476],{"className":15475},[217,218,219,220],[39,15477,1696],{"className":15478},[83,220],[39,15480,673],{"className":15481},[672],[39,15483,15485],{"className":15484},[200],[39,15486,15488],{"className":15487,"style":680},[204],[39,15489],{},[39,15491],{"className":15492,"style":231},[230],[39,15494,793],{"className":15495},[235],[39,15497],{"className":15498,"style":231},[230],[39,15500,15502,15505],{"className":15501},[74],[39,15503],{"className":15504,"style":2473},[78],[39,15506,546],{"className":15507},[83],"，从而 ",[39,15510,15512,15546],{"className":15511,"translate":43},[42],[39,15513,15515],{"className":15514},[47],[49,15516,15517],{"xmlns":51},[53,15518,15519,15543],{},[56,15520,15521,15523,15529,15535,15537],{},[59,15522,15389],{"mathvariant":145},[138,15524,15525,15527],{},[59,15526,4295],{},[59,15528,3196],{},[613,15530,15531,15533],{},[59,15532,15389],{"mathvariant":145},[59,15534,1696],{"mathvariant":145},[133,15536,1824],{},[138,15538,15539,15541],{},[59,15540,531],{},[59,15542,3196],{},[63,15544,15545],{"encoding":65},"\\|Q^k\\|_\\infty\\le\\gamma^k",[39,15547,15549,15638],{"className":15548,"ariaHidden":70},[69],[39,15550,15552,15556,15559,15589,15629,15632,15635],{"className":15551},[74],[39,15553],{"className":15554,"style":15555},[78],"height:1.0991em;vertical-align:-0.25em;",[39,15557,15389],{"className":15558},[83],[39,15560,15562,15565],{"className":15561},[83],[39,15563,4295],{"className":15564},[83,84],[39,15566,15568],{"className":15567},[192],[39,15569,15571],{"className":15570},[196],[39,15572,15574],{"className":15573},[200],[39,15575,15578],{"className":15576,"style":15577},[204],"height:0.8491em;",[39,15579,15580,15583],{"style":208},[39,15581],{"className":15582,"style":213},[212],[39,15584,15586],{"className":15585},[217,218,219,220],[39,15587,3196],{"className":15588,"style":3321},[83,84,220],[39,15590,15592,15595],{"className":15591},[83],[39,15593,15389],{"className":15594},[83],[39,15596,15598],{"className":15597},[192],[39,15599,15601,15621],{"className":15600},[196,649],[39,15602,15604,15618],{"className":15603},[200],[39,15605,15607],{"className":15606,"style":1728},[204],[39,15608,15609,15612],{"style":966},[39,15610],{"className":15611,"style":213},[212],[39,15613,15615],{"className":15614},[217,218,219,220],[39,15616,1696],{"className":15617},[83,220],[39,15619,673],{"className":15620},[672],[39,15622,15624],{"className":15623},[200],[39,15625,15627],{"className":15626,"style":680},[204],[39,15628],{},[39,15630],{"className":15631,"style":231},[230],[39,15633,1824],{"className":15634},[235],[39,15636],{"className":15637,"style":231},[230],[39,15639,15641,15645],{"className":15640},[74],[39,15642],{"className":15643,"style":15644},[78],"height:1.0435em;vertical-align:-0.1944em;",[39,15646,15648,15651],{"className":15647},[83],[39,15649,531],{"className":15650,"style":565},[83,84],[39,15652,15654],{"className":15653},[192],[39,15655,15657],{"className":15656},[196],[39,15658,15660],{"className":15659},[200],[39,15661,15663],{"className":15662,"style":15577},[204],[39,15664,15665,15668],{"style":208},[39,15666],{"className":15667,"style":213},[212],[39,15669,15671],{"className":15670},[217,218,219,220],[39,15672,3196],{"className":15673,"style":3321},[83,84,220],"。因此 Neumann 级数 ",[39,15676,15678,15710],{"className":15677,"translate":43},[42],[39,15679,15681],{"className":15680},[47],[49,15682,15683],{"xmlns":51},[53,15684,15685,15707],{},[56,15686,15687,15701],{},[9932,15688,15689,15691,15699],{},[133,15690,3191],{},[56,15692,15693,15695,15697],{},[59,15694,3196],{},[133,15696,793],{},[539,15698,541],{},[59,15700,1696],{"mathvariant":145},[138,15702,15703,15705],{},[59,15704,4295],{},[59,15706,3196],{},[63,15708,15709],{"encoding":65},"\\sum_{k=0}^{\\infty}Q^k",[39,15711,15713],{"className":15712,"ariaHidden":70},[69],[39,15714,15716,15720,15785,15788],{"className":15715},[74],[39,15717],{"className":15718,"style":15719},[78],"height:1.1488em;vertical-align:-0.2997em;",[39,15721,15723,15726],{"className":15722},[1744],[39,15724,3191],{"className":15725,"style":3953},[1744,3339,3952],[39,15727,15729],{"className":15728},[192],[39,15730,15732,15777],{"className":15731},[196,649],[39,15733,15735,15774],{"className":15734},[200],[39,15736,15739,15759],{"className":15737,"style":15738},[204],"height:0.8043em;",[39,15740,15741,15744],{"style":3969},[39,15742],{"className":15743,"style":213},[212],[39,15745,15747],{"className":15746},[217,218,219,220],[39,15748,15750,15753,15756],{"className":15749},[83,220],[39,15751,3196],{"className":15752,"style":3321},[83,84,220],[39,15754,793],{"className":15755},[235,220],[39,15757,541],{"className":15758},[83,220],[39,15760,15762,15765],{"style":15761},"top:-3.2029em;margin-right:0.05em;",[39,15763],{"className":15764,"style":213},[212],[39,15766,15768],{"className":15767},[217,218,219,220],[39,15769,15771],{"className":15770},[83,220],[39,15772,1696],{"className":15773},[83,220],[39,15775,673],{"className":15776},[672],[39,15778,15780],{"className":15779},[200],[39,15781,15783],{"className":15782,"style":4025},[204],[39,15784],{},[39,15786],{"className":15787,"style":256},[230],[39,15789,15791,15794],{"className":15790},[83],[39,15792,4295],{"className":15793},[83,84],[39,15795,15797],{"className":15796},[192],[39,15798,15800],{"className":15799},[196],[39,15801,15803],{"className":15802},[200],[39,15804,15806],{"className":15805,"style":15577},[204],[39,15807,15808,15811],{"style":208},[39,15809],{"className":15810,"style":213},[212],[39,15812,15814],{"className":15813},[217,218,219,220],[39,15815,3196],{"className":15816,"style":3321},[83,84,220]," 在算子范数下绝对收敛。并且",[39,15819,15821],{"className":15820,"translate":43},[746],[39,15822,15824,15891],{"className":15823,"translate":43},[42],[39,15825,15827],{"className":15826},[47],[49,15828,15829],{"xmlns":51,"display":755},[53,15830,15831,15888],{},[56,15832,15833,15835,15837,15839,15841,15843,15857,15863,15865,15867,15869,15881,15884,15886],{},[133,15834,136],{"stretchy":135},[59,15836,14373],{},[133,15838,5378],{},[59,15840,4295],{},[133,15842,161],{"stretchy":135},[3187,15844,15845,15847,15855],{},[133,15846,3191],{},[56,15848,15849,15851,15853],{},[59,15850,3196],{},[133,15852,793],{},[539,15854,541],{},[59,15856,12962],{},[138,15858,15859,15861],{},[59,15860,4295],{},[59,15862,3196],{},[133,15864,793],{},[59,15866,14373],{},[133,15868,5378],{},[138,15870,15871,15873],{},[59,15872,4295],{},[56,15874,15875,15877,15879],{},[59,15876,12962],{},[133,15878,788],{},[539,15880,546],{},[133,15882,15883],{},"⟶",[59,15885,14373],{},[133,15887,155],{"separator":70},[63,15889,15890],{"encoding":65},"(I-Q)\\sum_{k=0}^{n}Q^k=I-Q^{n+1}\\longrightarrow I,",[39,15892,15894,15915,16038,16056,16110],{"className":15893,"ariaHidden":70},[69],[39,15895,15897,15900,15903,15906,15909,15912],{"className":15896},[74],[39,15898],{"className":15899,"style":245},[78],[39,15901,136],{"className":15902},[182],[39,15904,14373],{"className":15905,"style":14420},[83,84],[39,15907],{"className":15908,"style":461},[230],[39,15910,5378],{"className":15911},[465],[39,15913],{"className":15914,"style":461},[230],[39,15916,15918,15921,15924,15927,15930,15997,16000,16029,16032,16035],{"className":15917},[74],[39,15919],{"className":15920,"style":3290},[78],[39,15922,4295],{"className":15923},[83,84],[39,15925,161],{"className":15926},[263],[39,15928],{"className":15929,"style":256},[230],[39,15931,15933],{"className":15932},[1744,3294],[39,15934,15936,15989],{"className":15935},[196,649],[39,15937,15939,15986],{"className":15938},[200],[39,15940,15942,15962,15972],{"className":15941,"style":3304},[204],[39,15943,15944,15947],{"style":3307},[39,15945],{"className":15946,"style":3311},[212],[39,15948,15950],{"className":15949},[217,218,219,220],[39,15951,15953,15956,15959],{"className":15952},[83,220],[39,15954,3196],{"className":15955,"style":3321},[83,84,220],[39,15957,793],{"className":15958},[235,220],[39,15960,541],{"className":15961},[83,220],[39,15963,15964,15967],{"style":3330},[39,15965],{"className":15966,"style":3311},[212],[39,15968,15969],{},[39,15970,3191],{"className":15971},[1744,3339,3340],[39,15973,15974,15977],{"style":3343},[39,15975],{"className":15976,"style":3311},[212],[39,15978,15980],{"className":15979},[217,218,219,220],[39,15981,15983],{"className":15982},[83,220],[39,15984,12962],{"className":15985},[83,84,220],[39,15987,673],{"className":15988},[672],[39,15990,15992],{"className":15991},[200],[39,15993,15995],{"className":15994,"style":3365},[204],[39,15996],{},[39,15998],{"className":15999,"style":256},[230],[39,16001,16003,16006],{"className":16002},[83],[39,16004,4295],{"className":16005},[83,84],[39,16007,16009],{"className":16008},[192],[39,16010,16012],{"className":16011},[196],[39,16013,16015],{"className":16014},[200],[39,16016,16018],{"className":16017,"style":3389},[204],[39,16019,16020,16023],{"style":1027},[39,16021],{"className":16022,"style":213},[212],[39,16024,16026],{"className":16025},[217,218,219,220],[39,16027,3196],{"className":16028,"style":3321},[83,84,220],[39,16030],{"className":16031,"style":231},[230],[39,16033,793],{"className":16034},[235],[39,16036],{"className":16037,"style":231},[230],[39,16039,16041,16044,16047,16050,16053],{"className":16040},[74],[39,16042],{"className":16043,"style":454},[78],[39,16045,14373],{"className":16046,"style":14420},[83,84],[39,16048],{"className":16049,"style":461},[230],[39,16051,5378],{"className":16052},[465],[39,16054],{"className":16055,"style":461},[230],[39,16057,16059,16063,16101,16104,16107],{"className":16058},[74],[39,16060],{"className":16061,"style":16062},[78],"height:1.0585em;vertical-align:-0.1944em;",[39,16064,16066,16069],{"className":16065},[83],[39,16067,4295],{"className":16068},[83,84],[39,16070,16072],{"className":16071},[192],[39,16073,16075],{"className":16074},[196],[39,16076,16078],{"className":16077},[200],[39,16079,16081],{"className":16080,"style":14940},[204],[39,16082,16083,16086],{"style":1027},[39,16084],{"className":16085,"style":213},[212],[39,16087,16089],{"className":16088},[217,218,219,220],[39,16090,16092,16095,16098],{"className":16091},[83,220],[39,16093,12962],{"className":16094},[83,84,220],[39,16096,788],{"className":16097},[465,220],[39,16099,546],{"className":16100},[83,220],[39,16102],{"className":16103,"style":231},[230],[39,16105,15883],{"className":16106},[235],[39,16108],{"className":16109,"style":231},[230],[39,16111,16113,16116,16119],{"className":16112},[74],[39,16114],{"className":16115,"style":6452},[78],[39,16117,14373],{"className":16118,"style":14420},[83,84],[39,16120,155],{"className":16121},[252],[11,16123,16124,16125,14714],{},"故 ",[39,16126,16128,16146],{"className":16127,"translate":43},[42],[39,16129,16131],{"className":16130},[47],[49,16132,16133],{"xmlns":51},[53,16134,16135,16143],{},[56,16136,16137,16139,16141],{},[59,16138,14373],{},[133,16140,5378],{},[59,16142,4295],{},[63,16144,16145],{"encoding":65},"I-Q",[39,16147,16149,16167],{"className":16148,"ariaHidden":70},[69],[39,16150,16152,16155,16158,16161,16164],{"className":16151},[74],[39,16153],{"className":16154,"style":454},[78],[39,16156,14373],{"className":16157,"style":14420},[83,84],[39,16159],{"className":16160,"style":461},[230],[39,16162,5378],{"className":16163},[465],[39,16165],{"className":16166,"style":461},[230],[39,16168,16170,16173],{"className":16169},[74],[39,16171],{"className":16172,"style":6452},[78],[39,16174,4295],{"className":16175},[83,84],[39,16177,16179],{"className":16178,"translate":43},[746],[39,16180,16182,16236],{"className":16181,"translate":43},[42],[39,16183,16185],{"className":16184},[47],[49,16186,16187],{"xmlns":51,"display":755},[53,16188,16189,16233],{},[56,16190,16191,16193,16195,16197,16199,16209,16211,16225,16231],{},[133,16192,136],{"stretchy":135},[59,16194,14373],{},[133,16196,5378],{},[59,16198,4295],{},[138,16200,16201,16203],{},[133,16202,161],{"stretchy":135},[56,16204,16205,16207],{},[133,16206,5378],{},[539,16208,546],{},[133,16210,793],{},[3187,16212,16213,16215,16223],{},[133,16214,3191],{},[56,16216,16217,16219,16221],{},[59,16218,3196],{},[133,16220,793],{},[539,16222,541],{},[59,16224,1696],{"mathvariant":145},[138,16226,16227,16229],{},[59,16228,4295],{},[59,16230,3196],{},[59,16232,2903],{"mathvariant":145},[63,16234,16235],{"encoding":65},"(I-Q)^{-1}=\\sum_{k=0}^{\\infty}Q^k.",[39,16237,16239,16260,16313],{"className":16238,"ariaHidden":70},[69],[39,16240,16242,16245,16248,16251,16254,16257],{"className":16241},[74],[39,16243],{"className":16244,"style":245},[78],[39,16246,136],{"className":16247},[182],[39,16249,14373],{"className":16250,"style":14420},[83,84],[39,16252],{"className":16253,"style":461},[230],[39,16255,5378],{"className":16256},[465],[39,16258],{"className":16259,"style":461},[230],[39,16261,16263,16266,16269,16304,16307,16310],{"className":16262},[74],[39,16264],{"className":16265,"style":14889},[78],[39,16267,4295],{"className":16268},[83,84],[39,16270,16272,16275],{"className":16271},[263],[39,16273,161],{"className":16274},[263],[39,16276,16278],{"className":16277},[192],[39,16279,16281],{"className":16280},[196],[39,16282,16284],{"className":16283},[200],[39,16285,16287],{"className":16286,"style":14940},[204],[39,16288,16289,16292],{"style":1027},[39,16290],{"className":16291,"style":213},[212],[39,16293,16295],{"className":16294},[217,218,219,220],[39,16296,16298,16301],{"className":16297},[83,220],[39,16299,5378],{"className":16300},[83,220],[39,16302,546],{"className":16303},[83,220],[39,16305],{"className":16306,"style":231},[230],[39,16308,793],{"className":16309},[235],[39,16311],{"className":16312,"style":231},[230],[39,16314,16316,16319,16386,16389,16418],{"className":16315},[74],[39,16317],{"className":16318,"style":3290},[78],[39,16320,16322],{"className":16321},[1744,3294],[39,16323,16325,16378],{"className":16324},[196,649],[39,16326,16328,16375],{"className":16327},[200],[39,16329,16331,16351,16361],{"className":16330,"style":3304},[204],[39,16332,16333,16336],{"style":3307},[39,16334],{"className":16335,"style":3311},[212],[39,16337,16339],{"className":16338},[217,218,219,220],[39,16340,16342,16345,16348],{"className":16341},[83,220],[39,16343,3196],{"className":16344,"style":3321},[83,84,220],[39,16346,793],{"className":16347},[235,220],[39,16349,541],{"className":16350},[83,220],[39,16352,16353,16356],{"style":3330},[39,16354],{"className":16355,"style":3311},[212],[39,16357,16358],{},[39,16359,3191],{"className":16360},[1744,3339,3340],[39,16362,16363,16366],{"style":3343},[39,16364],{"className":16365,"style":3311},[212],[39,16367,16369],{"className":16368},[217,218,219,220],[39,16370,16372],{"className":16371},[83,220],[39,16373,1696],{"className":16374},[83,220],[39,16376,673],{"className":16377},[672],[39,16379,16381],{"className":16380},[200],[39,16382,16384],{"className":16383,"style":3365},[204],[39,16385],{},[39,16387],{"className":16388,"style":256},[230],[39,16390,16392,16395],{"className":16391},[83],[39,16393,4295],{"className":16394},[83,84],[39,16396,16398],{"className":16397},[192],[39,16399,16401],{"className":16400},[196],[39,16402,16404],{"className":16403},[200],[39,16405,16407],{"className":16406,"style":3389},[204],[39,16408,16409,16412],{"style":1027},[39,16410],{"className":16411,"style":213},[212],[39,16413,16415],{"className":16414},[217,218,219,220],[39,16416,3196],{"className":16417,"style":3321},[83,84,220],[39,16419,2903],{"className":16420},[83],[11,16422,16423,16424,742,16428,16486,16487,16635],{},"由",[158,16425,16427],{"href":16426},"#theorem-bellman-expectation","定理 2.1",[39,16429,16431,16448],{"className":16430,"translate":43},[42],[39,16432,16434],{"className":16433},[47],[49,16435,16436],{"xmlns":51},[53,16437,16438,16446],{},[56,16439,16440],{},[138,16441,16442,16444],{},[59,16443,4230],{},[59,16445,2098],{},[63,16447,6382],{"encoding":65},[39,16449,16451],{"className":16450,"ariaHidden":70},[69],[39,16452,16454,16457],{"className":16453},[74],[39,16455],{"className":16456,"style":79},[78],[39,16458,16460,16463],{"className":16459},[83],[39,16461,4230],{"className":16462,"style":461},[83,84],[39,16464,16466],{"className":16465},[192],[39,16467,16469],{"className":16468},[196],[39,16470,16472],{"className":16471},[200],[39,16473,16475],{"className":16474,"style":6410},[204],[39,16476,16477,16480],{"style":208},[39,16478],{"className":16479,"style":213},[212],[39,16481,16483],{"className":16482},[217,218,219,220],[39,16484,2098],{"className":16485,"style":2114},[83,84,220]," 满足 ",[39,16488,16490,16526],{"className":16489,"translate":43},[42],[39,16491,16493],{"className":16492},[47],[49,16494,16495],{"xmlns":51},[53,16496,16497,16523],{},[56,16498,16499,16501,16503,16505,16507,16509,16515,16517],{},[133,16500,136],{"stretchy":135},[59,16502,14373],{},[133,16504,5378],{},[59,16506,4295],{},[133,16508,161],{"stretchy":135},[138,16510,16511,16513],{},[59,16512,4230],{},[59,16514,2098],{},[133,16516,793],{},[138,16518,16519,16521],{},[59,16520,401],{},[59,16522,2098],{},[63,16524,16525],{"encoding":65},"(I-Q)V^\\pi=r^\\pi",[39,16527,16529,16550,16600],{"className":16528,"ariaHidden":70},[69],[39,16530,16532,16535,16538,16541,16544,16547],{"className":16531},[74],[39,16533],{"className":16534,"style":245},[78],[39,16536,136],{"className":16537},[182],[39,16539,14373],{"className":16540,"style":14420},[83,84],[39,16542],{"className":16543,"style":461},[230],[39,16545,5378],{"className":16546},[465],[39,16548],{"className":16549,"style":461},[230],[39,16551,16553,16556,16559,16562,16591,16594,16597],{"className":16552},[74],[39,16554],{"className":16555,"style":245},[78],[39,16557,4295],{"className":16558},[83,84],[39,16560,161],{"className":16561},[263],[39,16563,16565,16568],{"className":16564},[83],[39,16566,4230],{"className":16567,"style":461},[83,84],[39,16569,16571],{"className":16570},[192],[39,16572,16574],{"className":16573},[196],[39,16575,16577],{"className":16576},[200],[39,16578,16580],{"className":16579,"style":6410},[204],[39,16581,16582,16585],{"style":208},[39,16583],{"className":16584,"style":213},[212],[39,16586,16588],{"className":16587},[217,218,219,220],[39,16589,2098],{"className":16590,"style":2114},[83,84,220],[39,16592],{"className":16593,"style":231},[230],[39,16595,793],{"className":16596},[235],[39,16598],{"className":16599,"style":231},[230],[39,16601,16603,16606],{"className":16602},[74],[39,16604],{"className":16605,"style":6410},[78],[39,16607,16609,16612],{"className":16608},[83],[39,16610,401],{"className":16611,"style":438},[83,84],[39,16613,16615],{"className":16614},[192],[39,16616,16618],{"className":16617},[196],[39,16619,16621],{"className":16620},[200],[39,16622,16624],{"className":16623,"style":6410},[204],[39,16625,16626,16629],{"style":208},[39,16627],{"className":16628,"style":213},[212],[39,16630,16632],{"className":16631},[217,218,219,220],[39,16633,2098],{"className":16634,"style":2114},[83,84,220],"，于是",[39,16637,16639],{"className":16638,"translate":43},[746],[39,16640,16642,16760],{"className":16641,"translate":43},[42],[39,16643,16645],{"className":16644},[47],[49,16646,16647],{"xmlns":51,"display":755},[53,16648,16649,16757],{},[759,16650,16651,16697],{"rowspacing":761,"columnalign":762,"columnspacing":146},[764,16652,16653,16663],{},[767,16654,16655],{},[770,16656,16657],{"scriptlevel":541,"displaystyle":70},[138,16658,16659,16661],{},[59,16660,4230],{},[59,16662,2098],{},[767,16664,16665],{},[770,16666,16667],{"scriptlevel":541,"displaystyle":70},[56,16668,16669,16671,16673,16675,16677,16679,16681,16691],{},[56,16670],{},[133,16672,793],{},[133,16674,136],{"stretchy":135},[59,16676,14373],{},[133,16678,5378],{},[59,16680,4295],{},[138,16682,16683,16685],{},[133,16684,161],{"stretchy":135},[56,16686,16687,16689],{},[133,16688,5378],{},[539,16690,546],{},[138,16692,16693,16695],{},[59,16694,401],{},[59,16696,2098],{},[764,16698,16699,16705],{},[767,16700,16701],{},[770,16702,16703],{"scriptlevel":541,"displaystyle":70},[56,16704],{},[767,16706,16707],{},[770,16708,16709],{"scriptlevel":541,"displaystyle":70},[56,16710,16711,16713,16715,16729,16735,16737,16743,16749,16755],{},[56,16712],{},[133,16714,793],{},[3187,16716,16717,16719,16727],{},[133,16718,3191],{},[56,16720,16721,16723,16725],{},[59,16722,3196],{},[133,16724,793],{},[539,16726,541],{},[59,16728,1696],{"mathvariant":145},[138,16730,16731,16733],{},[59,16732,531],{},[59,16734,3196],{},[133,16736,136],{"stretchy":135},[138,16738,16739,16741],{},[59,16740,131],{},[59,16742,2098],{},[138,16744,16745,16747],{},[133,16746,161],{"stretchy":135},[59,16748,3196],{},[138,16750,16751,16753],{},[59,16752,401],{},[59,16754,2098],{},[59,16756,2903],{"mathvariant":145},[63,16758,16759],{"encoding":65},"\\begin{aligned}\nV^\\pi&=(I-Q)^{-1}r^\\pi\\\\\n&=\\sum_{k=0}^{\\infty}\\gamma^k(P^\\pi)^k r^\\pi.\n\\end{aligned}",[39,16761,16763],{"className":16762,"ariaHidden":70},[69],[39,16764,16766,16770],{"className":16765},[74],[39,16767],{"className":16768,"style":16769},[78],"height:4.4776em;vertical-align:-1.9888em;",[39,16771,16773],{"className":16772},[83],[39,16774,16776,16848],{"className":16775},[759],[39,16777,16779],{"className":16778},[918],[39,16780,16782,16839],{"className":16781},[196,649],[39,16783,16785,16836],{"className":16784},[200],[39,16786,16789,16827],{"className":16787,"style":16788},[204],"height:2.4888em;",[39,16790,16792,16795],{"style":16791},"top:-5.2761em;",[39,16793],{"className":16794,"style":5782},[212],[39,16796,16798],{"className":16797},[83],[39,16799,16801,16804],{"className":16800},[83],[39,16802,4230],{"className":16803,"style":461},[83,84],[39,16805,16807],{"className":16806},[192],[39,16808,16810],{"className":16809},[196],[39,16811,16813],{"className":16812},[200],[39,16814,16816],{"className":16815,"style":4421},[204],[39,16817,16818,16821],{"style":1027},[39,16819],{"className":16820,"style":213},[212],[39,16822,16824],{"className":16823},[217,218,219,220],[39,16825,2098],{"className":16826,"style":2114},[83,84,220],[39,16828,16830,16833],{"style":16829},"top:-2.9647em;",[39,16831],{"className":16832,"style":5782},[212],[39,16834],{"className":16835},[83],[39,16837,673],{"className":16838},[672],[39,16840,16842],{"className":16841},[200],[39,16843,16846],{"className":16844,"style":16845},[204],"height:1.9888em;",[39,16847],{},[39,16849,16851],{"className":16850},[1223],[39,16852,16854,17177],{"className":16853},[196,649],[39,16855,16857,17174],{"className":16856},[200],[39,16858,16860,16962],{"className":16859,"style":16788},[204],[39,16861,16862,16865],{"style":16791},[39,16863],{"className":16864,"style":5782},[212],[39,16866,16868,16871,16874,16877,16880,16883,16886,16889,16892,16895,16898,16933],{"className":16867},[83],[39,16869],{"className":16870},[83],[39,16872],{"className":16873,"style":231},[230],[39,16875,793],{"className":16876},[235],[39,16878],{"className":16879,"style":231},[230],[39,16881,136],{"className":16882},[182],[39,16884,14373],{"className":16885,"style":14420},[83,84],[39,16887],{"className":16888,"style":461},[230],[39,16890,5378],{"className":16891},[465],[39,16893],{"className":16894,"style":461},[230],[39,16896,4295],{"className":16897},[83,84],[39,16899,16901,16904],{"className":16900},[263],[39,16902,161],{"className":16903},[263],[39,16905,16907],{"className":16906},[192],[39,16908,16910],{"className":16909},[196],[39,16911,16913],{"className":16912},[200],[39,16914,16916],{"className":16915,"style":14940},[204],[39,16917,16918,16921],{"style":1027},[39,16919],{"className":16920,"style":213},[212],[39,16922,16924],{"className":16923},[217,218,219,220],[39,16925,16927,16930],{"className":16926},[83,220],[39,16928,5378],{"className":16929},[83,220],[39,16931,546],{"className":16932},[83,220],[39,16934,16936,16939],{"className":16935},[83],[39,16937,401],{"className":16938,"style":438},[83,84],[39,16940,16942],{"className":16941},[192],[39,16943,16945],{"className":16944},[196],[39,16946,16948],{"className":16947},[200],[39,16949,16951],{"className":16950,"style":4421},[204],[39,16952,16953,16956],{"style":1027},[39,16954],{"className":16955,"style":213},[212],[39,16957,16959],{"className":16958},[217,218,219,220],[39,16960,2098],{"className":16961,"style":2114},[83,84,220],[39,16963,16964,16967],{"style":16829},[39,16965],{"className":16966,"style":5782},[212],[39,16968,16970,16973,16976,16979,16982,17049,17052,17081,17084,17113,17142,17171],{"className":16969},[83],[39,16971],{"className":16972},[83],[39,16974],{"className":16975,"style":231},[230],[39,16977,793],{"className":16978},[235],[39,16980],{"className":16981,"style":231},[230],[39,16983,16985],{"className":16984},[1744,3294],[39,16986,16988,17041],{"className":16987},[196,649],[39,16989,16991,17038],{"className":16990},[200],[39,16992,16994,17014,17024],{"className":16993,"style":3304},[204],[39,16995,16996,16999],{"style":3307},[39,16997],{"className":16998,"style":3311},[212],[39,17000,17002],{"className":17001},[217,218,219,220],[39,17003,17005,17008,17011],{"className":17004},[83,220],[39,17006,3196],{"className":17007,"style":3321},[83,84,220],[39,17009,793],{"className":17010},[235,220],[39,17012,541],{"className":17013},[83,220],[39,17015,17016,17019],{"style":3330},[39,17017],{"className":17018,"style":3311},[212],[39,17020,17021],{},[39,17022,3191],{"className":17023},[1744,3339,3340],[39,17025,17026,17029],{"style":3343},[39,17027],{"className":17028,"style":3311},[212],[39,17030,17032],{"className":17031},[217,218,219,220],[39,17033,17035],{"className":17034},[83,220],[39,17036,1696],{"className":17037},[83,220],[39,17039,673],{"className":17040},[672],[39,17042,17044],{"className":17043},[200],[39,17045,17047],{"className":17046,"style":3365},[204],[39,17048],{},[39,17050],{"className":17051,"style":256},[230],[39,17053,17055,17058],{"className":17054},[83],[39,17056,531],{"className":17057,"style":565},[83,84],[39,17059,17061],{"className":17060},[192],[39,17062,17064],{"className":17063},[196],[39,17065,17067],{"className":17066},[200],[39,17068,17070],{"className":17069,"style":3389},[204],[39,17071,17072,17075],{"style":1027},[39,17073],{"className":17074,"style":213},[212],[39,17076,17078],{"className":17077},[217,218,219,220],[39,17079,3196],{"className":17080,"style":3321},[83,84,220],[39,17082,136],{"className":17083},[182],[39,17085,17087,17090],{"className":17086},[83],[39,17088,131],{"className":17089,"style":178},[83,84],[39,17091,17093],{"className":17092},[192],[39,17094,17096],{"className":17095},[196],[39,17097,17099],{"className":17098},[200],[39,17100,17102],{"className":17101,"style":4421},[204],[39,17103,17104,17107],{"style":1027},[39,17105],{"className":17106,"style":213},[212],[39,17108,17110],{"className":17109},[217,218,219,220],[39,17111,2098],{"className":17112,"style":2114},[83,84,220],[39,17114,17116,17119],{"className":17115},[263],[39,17117,161],{"className":17118},[263],[39,17120,17122],{"className":17121},[192],[39,17123,17125],{"className":17124},[196],[39,17126,17128],{"className":17127},[200],[39,17129,17131],{"className":17130,"style":3389},[204],[39,17132,17133,17136],{"style":1027},[39,17134],{"className":17135,"style":213},[212],[39,17137,17139],{"className":17138},[217,218,219,220],[39,17140,3196],{"className":17141,"style":3321},[83,84,220],[39,17143,17145,17148],{"className":17144},[83],[39,17146,401],{"className":17147,"style":438},[83,84],[39,17149,17151],{"className":17150},[192],[39,17152,17154],{"className":17153},[196],[39,17155,17157],{"className":17156},[200],[39,17158,17160],{"className":17159,"style":4421},[204],[39,17161,17162,17165],{"style":1027},[39,17163],{"className":17164,"style":213},[212],[39,17166,17168],{"className":17167},[217,218,219,220],[39,17169,2098],{"className":17170,"style":2114},[83,84,220],[39,17172,2903],{"className":17173},[83],[39,17175,673],{"className":17176},[672],[39,17178,17180],{"className":17179},[200],[39,17181,17183],{"className":17182,"style":16845},[204],[39,17184],{},[11,17186,11683],{},[11,17188,17189,17190,17275,17276,17327],{},"矩阵解给出了一个重要的结构性事实：在有限状态与固定策略下，贝尔曼方程是一个线性方程组，其系数矩阵为 ",[39,17191,17193,17216],{"className":17192,"translate":43},[42],[39,17194,17196],{"className":17195},[47],[49,17197,17198],{"xmlns":51},[53,17199,17200,17214],{},[56,17201,17202,17204,17206,17208],{},[59,17203,14373],{},[133,17205,5378],{},[59,17207,531],{},[138,17209,17210,17212],{},[59,17211,131],{},[59,17213,2098],{},[63,17215,14654],{"encoding":65},[39,17217,17219,17237],{"className":17218,"ariaHidden":70},[69],[39,17220,17222,17225,17228,17231,17234],{"className":17221},[74],[39,17223],{"className":17224,"style":454},[78],[39,17226,14373],{"className":17227,"style":14420},[83,84],[39,17229],{"className":17230,"style":461},[230],[39,17232,5378],{"className":17233},[465],[39,17235],{"className":17236,"style":461},[230],[39,17238,17240,17243,17246],{"className":17239},[74],[39,17241],{"className":17242,"style":6452},[78],[39,17244,531],{"className":17245,"style":565},[83,84],[39,17247,17249,17252],{"className":17248},[83],[39,17250,131],{"className":17251,"style":178},[83,84],[39,17253,17255],{"className":17254},[192],[39,17256,17258],{"className":17257},[196],[39,17259,17261],{"className":17260},[200],[39,17262,17264],{"className":17263,"style":6410},[204],[39,17265,17266,17269],{"style":208},[39,17267],{"className":17268,"style":213},[212],[39,17270,17272],{"className":17271},[217,218,219,220],[39,17273,2098],{"className":17274,"style":2114},[83,84,220],"。折扣因子 ",[39,17277,17279,17297],{"className":17278,"translate":43},[42],[39,17280,17282],{"className":17281},[47],[49,17283,17284],{"xmlns":51},[53,17285,17286,17294],{},[56,17287,17288,17290,17292],{},[59,17289,531],{},[133,17291,1693],{},[539,17293,546],{},[63,17295,17296],{"encoding":65},"\\gamma\u003C1",[39,17298,17300,17318],{"className":17299,"ariaHidden":70},[69],[39,17301,17303,17306,17309,17312,17315],{"className":17302},[74],[39,17304],{"className":17305,"style":561},[78],[39,17307,531],{"className":17308,"style":565},[83,84],[39,17310],{"className":17311,"style":231},[230],[39,17313,1693],{"className":17314},[235],[39,17316],{"className":17317,"style":231},[230],[39,17319,17321,17324],{"className":17320},[74],[39,17322],{"className":17323,"style":2473},[78],[39,17325,546],{"className":17326},[83]," 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。",[26,17329,17331],{"id":17330},"_24-例两状态马尔可夫链","2.4 例：两状态马尔可夫链",[11,17333,17334,17335,17410],{},"考虑两个状态 ",[39,17336,17338,17365],{"className":17337,"translate":43},[42],[39,17339,17341],{"className":17340},[47],[49,17342,17343],{"xmlns":51},[53,17344,17345,17362],{},[56,17346,17347,17349,17351,17353,17355,17357,17360],{},[59,17348,61],{},[133,17350,793],{},[133,17352,12951],{"stretchy":135},[539,17354,546],{},[133,17356,155],{"separator":70},[539,17358,17359],{},"2",[133,17361,12965],{"stretchy":135},[63,17363,17364],{"encoding":65},"S=\\{1,2\\}",[39,17366,17368,17386],{"className":17367,"ariaHidden":70},[69],[39,17369,17371,17374,17377,17380,17383],{"className":17370},[74],[39,17372],{"className":17373,"style":79},[78],[39,17375,61],{"className":17376,"style":85},[83,84],[39,17378],{"className":17379,"style":231},[230],[39,17381,793],{"className":17382},[235],[39,17384],{"className":17385,"style":231},[230],[39,17387,17389,17392,17395,17398,17401,17404,17407],{"className":17388},[74],[39,17390],{"className":17391,"style":245},[78],[39,17393,12951],{"className":17394},[182],[39,17396,546],{"className":17397},[83],[39,17399,155],{"className":17400},[252],[39,17402],{"className":17403,"style":256},[230],[39,17405,17359],{"className":17406},[83],[39,17408,12965],{"className":17409},[263],"，只有一个动作。设转移关系为",[39,17412,17414],{"className":17413,"translate":43},[746],[39,17415,17417,17467],{"className":17416,"translate":43},[42],[39,17418,17420],{"className":17419},[47],[49,17421,17422],{"xmlns":51,"display":755},[53,17423,17424,17464],{},[56,17425,17426,17428,17430,17432,17434,17436,17438,17440,17442,17444,17446,17448,17450,17452,17454,17456,17458,17460,17462],{},[59,17427,131],{},[133,17429,136],{"stretchy":135},[539,17431,17359],{},[133,17433,150],{},[539,17435,546],{},[133,17437,161],{"stretchy":135},[133,17439,793],{},[539,17441,546],{},[133,17443,155],{"separator":70},[230,17445],{"width":10251},[59,17447,131],{},[133,17449,136],{"stretchy":135},[539,17451,546],{},[133,17453,150],{},[539,17455,17359],{},[133,17457,161],{"stretchy":135},[133,17459,793],{},[539,17461,546],{},[133,17463,155],{"separator":70},[63,17465,17466],{"encoding":65},"P(2\\mid 1)=1,\\qquad P(1\\mid 2)=1,",[39,17468,17470,17494,17515,17551,17572],{"className":17469,"ariaHidden":70},[69],[39,17471,17473,17476,17479,17482,17485,17488,17491],{"className":17472},[74],[39,17474],{"className":17475,"style":245},[78],[39,17477,131],{"className":17478,"style":178},[83,84],[39,17480,136],{"className":17481},[182],[39,17483,17359],{"className":17484},[83],[39,17486],{"className":17487,"style":231},[230],[39,17489,150],{"className":17490},[235],[39,17492],{"className":17493,"style":231},[230],[39,17495,17497,17500,17503,17506,17509,17512],{"className":17496},[74],[39,17498],{"className":17499,"style":245},[78],[39,17501,546],{"className":17502},[83],[39,17504,161],{"className":17505},[263],[39,17507],{"className":17508,"style":231},[230],[39,17510,793],{"className":17511},[235],[39,17513],{"className":17514,"style":231},[230],[39,17516,17518,17521,17524,17527,17530,17533,17536,17539,17542,17545,17548],{"className":17517},[74],[39,17519],{"className":17520,"style":245},[78],[39,17522,546],{"className":17523},[83],[39,17525,155],{"className":17526},[252],[39,17528],{"className":17529,"style":10787},[230],[39,17531],{"className":17532,"style":256},[230],[39,17534,131],{"className":17535,"style":178},[83,84],[39,17537,136],{"className":17538},[182],[39,17540,546],{"className":17541},[83],[39,17543],{"className":17544,"style":231},[230],[39,17546,150],{"className":17547},[235],[39,17549],{"className":17550,"style":231},[230],[39,17552,17554,17557,17560,17563,17566,17569],{"className":17553},[74],[39,17555],{"className":17556,"style":245},[78],[39,17558,17359],{"className":17559},[83],[39,17561,161],{"className":17562},[263],[39,17564],{"className":17565,"style":231},[230],[39,17567,793],{"className":17568},[235],[39,17570],{"className":17571,"style":231},[230],[39,17573,17575,17579,17582],{"className":17574},[74],[39,17576],{"className":17577,"style":17578},[78],"height:0.8389em;vertical-align:-0.1944em;",[39,17580,546],{"className":17581},[83],[39,17583,155],{"className":17584},[252],[11,17586,17587,17588,742,17654,17720,17721,17773],{},"奖励为 ",[39,17589,17591,17615],{"className":17590,"translate":43},[42],[39,17592,17594],{"className":17593},[47],[49,17595,17596],{"xmlns":51},[53,17597,17598,17612],{},[56,17599,17600,17602,17604,17606,17608,17610],{},[59,17601,401],{},[133,17603,136],{"stretchy":135},[539,17605,546],{},[133,17607,161],{"stretchy":135},[133,17609,793],{},[539,17611,546],{},[63,17613,17614],{"encoding":65},"r(1)=1",[39,17616,17618,17645],{"className":17617,"ariaHidden":70},[69],[39,17619,17621,17624,17627,17630,17633,17636,17639,17642],{"className":17620},[74],[39,17622],{"className":17623,"style":245},[78],[39,17625,401],{"className":17626,"style":438},[83,84],[39,17628,136],{"className":17629},[182],[39,17631,546],{"className":17632},[83],[39,17634,161],{"className":17635},[263],[39,17637],{"className":17638,"style":231},[230],[39,17640,793],{"className":17641},[235],[39,17643],{"className":17644,"style":231},[230],[39,17646,17648,17651],{"className":17647},[74],[39,17649],{"className":17650,"style":2473},[78],[39,17652,546],{"className":17653},[83],[39,17655,17657,17681],{"className":17656,"translate":43},[42],[39,17658,17660],{"className":17659},[47],[49,17661,17662],{"xmlns":51},[53,17663,17664,17678],{},[56,17665,17666,17668,17670,17672,17674,17676],{},[59,17667,401],{},[133,17669,136],{"stretchy":135},[539,17671,17359],{},[133,17673,161],{"stretchy":135},[133,17675,793],{},[539,17677,541],{},[63,17679,17680],{"encoding":65},"r(2)=0",[39,17682,17684,17711],{"className":17683,"ariaHidden":70},[69],[39,17685,17687,17690,17693,17696,17699,17702,17705,17708],{"className":17686},[74],[39,17688],{"className":17689,"style":245},[78],[39,17691,401],{"className":17692,"style":438},[83,84],[39,17694,136],{"className":17695},[182],[39,17697,17359],{"className":17698},[83],[39,17700,161],{"className":17701},[263],[39,17703],{"className":17704,"style":231},[230],[39,17706,793],{"className":17707},[235],[39,17709],{"className":17710,"style":231},[230],[39,17712,17714,17717],{"className":17713},[74],[39,17715],{"className":17716,"style":2473},[78],[39,17718,541],{"className":17719},[83],"，折扣因子 ",[39,17722,17724,17743],{"className":17723,"translate":43},[42],[39,17725,17727],{"className":17726},[47],[49,17728,17729],{"xmlns":51},[53,17730,17731,17740],{},[56,17732,17733,17735,17737],{},[59,17734,531],{},[133,17736,793],{},[539,17738,17739],{},"0.9",[63,17741,17742],{"encoding":65},"\\gamma=0.9",[39,17744,17746,17764],{"className":17745,"ariaHidden":70},[69],[39,17747,17749,17752,17755,17758,17761],{"className":17748},[74],[39,17750],{"className":17751,"style":2654},[78],[39,17753,531],{"className":17754,"style":565},[83,84],[39,17756],{"className":17757,"style":231},[230],[39,17759,793],{"className":17760},[235],[39,17762],{"className":17763,"style":231},[230],[39,17765,17767,17770],{"className":17766},[74],[39,17768],{"className":17769,"style":2473},[78],[39,17771,17739],{"className":17772},[83],"。贝尔曼方程为",[39,17775,17777],{"className":17776,"translate":43},[746],[39,17778,17780,17846],{"className":17779,"translate":43},[42],[39,17781,17783],{"className":17782},[47],[49,17784,17785],{"xmlns":51,"display":755},[53,17786,17787,17843],{},[56,17788,17789,17791,17793,17795,17797,17799,17801,17803,17805,17807,17809,17811,17813,17815,17817,17819,17821,17823,17825,17827,17829,17831,17833,17835,17837,17839,17841],{},[59,17790,4230],{},[133,17792,136],{"stretchy":135},[539,17794,546],{},[133,17796,161],{"stretchy":135},[133,17798,793],{},[539,17800,546],{},[133,17802,788],{},[539,17804,17739],{},[59,17806,4230],{},[133,17808,136],{"stretchy":135},[539,17810,17359],{},[133,17812,161],{"stretchy":135},[133,17814,155],{"separator":70},[230,17816],{"width":10251},[59,17818,4230],{},[133,17820,136],{"stretchy":135},[539,17822,17359],{},[133,17824,161],{"stretchy":135},[133,17826,793],{},[539,17828,541],{},[133,17830,788],{},[539,17832,17739],{},[59,17834,4230],{},[133,17836,136],{"stretchy":135},[539,17838,546],{},[133,17840,161],{"stretchy":135},[59,17842,2903],{"mathvariant":145},[63,17844,17845],{"encoding":65},"V(1)=1+0.9V(2),\\qquad V(2)=0+0.9V(1).",[39,17847,17849,17876,17895,17946,17964],{"className":17848,"ariaHidden":70},[69],[39,17850,17852,17855,17858,17861,17864,17867,17870,17873],{"className":17851},[74],[39,17853],{"className":17854,"style":245},[78],[39,17856,4230],{"className":17857,"style":461},[83,84],[39,17859,136],{"className":17860},[182],[39,17862,546],{"className":17863},[83],[39,17865,161],{"className":17866},[263],[39,17868],{"className":17869,"style":231},[230],[39,17871,793],{"className":17872},[235],[39,17874],{"className":17875,"style":231},[230],[39,17877,17879,17883,17886,17889,17892],{"className":17878},[74],[39,17880],{"className":17881,"style":17882},[78],"height:0.7278em;vertical-align:-0.0833em;",[39,17884,546],{"className":17885},[83],[39,17887],{"className":17888,"style":461},[230],[39,17890,788],{"className":17891},[465],[39,17893],{"className":17894,"style":461},[230],[39,17896,17898,17901,17904,17907,17910,17913,17916,17919,17922,17925,17928,17931,17934,17937,17940,17943],{"className":17897},[74],[39,17899],{"className":17900,"style":245},[78],[39,17902,17739],{"className":17903},[83],[39,17905,4230],{"className":17906,"style":461},[83,84],[39,17908,136],{"className":17909},[182],[39,17911,17359],{"className":17912},[83],[39,17914,161],{"className":17915},[263],[39,17917,155],{"className":17918},[252],[39,17920],{"className":17921,"style":10787},[230],[39,17923],{"className":17924,"style":256},[230],[39,17926,4230],{"className":17927,"style":461},[83,84],[39,17929,136],{"className":17930},[182],[39,17932,17359],{"className":17933},[83],[39,17935,161],{"className":17936},[263],[39,17938],{"className":17939,"style":231},[230],[39,17941,793],{"className":17942},[235],[39,17944],{"className":17945,"style":231},[230],[39,17947,17949,17952,17955,17958,17961],{"className":17948},[74],[39,17950],{"className":17951,"style":17882},[78],[39,17953,541],{"className":17954},[83],[39,17956],{"className":17957,"style":461},[230],[39,17959,788],{"className":17960},[465],[39,17962],{"className":17963,"style":461},[230],[39,17965,17967,17970,17973,17976,17979,17982,17985],{"className":17966},[74],[39,17968],{"className":17969,"style":245},[78],[39,17971,17739],{"className":17972},[83],[39,17974,4230],{"className":17975,"style":461},[83,84],[39,17977,136],{"className":17978},[182],[39,17980,546],{"className":17981},[83],[39,17983,161],{"className":17984},[263],[39,17986,2903],{"className":17987},[83],[11,17989,17990],{},"将第二式代入第一式，得",[39,17992,17994],{"className":17993,"translate":43},[746],[39,17995,17997,18036],{"className":17996,"translate":43},[42],[39,17998,18000],{"className":17999},[47],[49,18001,18002],{"xmlns":51,"display":755},[53,18003,18004,18033],{},[56,18005,18006,18008,18010,18012,18014,18016,18018,18020,18023,18025,18027,18029,18031],{},[59,18007,4230],{},[133,18009,136],{"stretchy":135},[539,18011,546],{},[133,18013,161],{"stretchy":135},[133,18015,793],{},[539,18017,546],{},[133,18019,788],{},[539,18021,18022],{},"0.81",[59,18024,4230],{},[133,18026,136],{"stretchy":135},[539,18028,546],{},[133,18030,161],{"stretchy":135},[133,18032,155],{"separator":70},[63,18034,18035],{"encoding":65},"V(1)=1+0.81V(1),",[39,18037,18039,18066,18084],{"className":18038,"ariaHidden":70},[69],[39,18040,18042,18045,18048,18051,18054,18057,18060,18063],{"className":18041},[74],[39,18043],{"className":18044,"style":245},[78],[39,18046,4230],{"className":18047,"style":461},[83,84],[39,18049,136],{"className":18050},[182],[39,18052,546],{"className":18053},[83],[39,18055,161],{"className":18056},[263],[39,18058],{"className":18059,"style":231},[230],[39,18061,793],{"className":18062},[235],[39,18064],{"className":18065,"style":231},[230],[39,18067,18069,18072,18075,18078,18081],{"className":18068},[74],[39,18070],{"className":18071,"style":17882},[78],[39,18073,546],{"className":18074},[83],[39,18076],{"className":18077,"style":461},[230],[39,18079,788],{"className":18080},[465],[39,18082],{"className":18083,"style":461},[230],[39,18085,18087,18090,18093,18096,18099,18102,18105],{"className":18086},[74],[39,18088],{"className":18089,"style":245},[78],[39,18091,18022],{"className":18092},[83],[39,18094,4230],{"className":18095,"style":461},[83,84],[39,18097,136],{"className":18098},[182],[39,18100,546],{"className":18101},[83],[39,18103,161],{"className":18104},[263],[39,18106,155],{"className":18107},[252],[11,18109,18110],{},"因此",[39,18112,18114],{"className":18113,"translate":43},[746],[39,18115,18117,18185],{"className":18116,"translate":43},[42],[39,18118,18120],{"className":18119},[47],[49,18121,18122],{"xmlns":51,"display":755},[53,18123,18124,18182],{},[56,18125,18126,18128,18130,18132,18134,18136,18148,18150,18153,18155,18157,18159,18161,18163,18165,18167,18169,18171,18173,18175,18177,18179],{},[59,18127,4230],{},[133,18129,136],{"stretchy":135},[539,18131,546],{},[133,18133,161],{"stretchy":135},[133,18135,793],{},[5362,18137,18138,18140],{},[539,18139,546],{},[56,18141,18142,18144,18146],{},[539,18143,546],{},[133,18145,5378],{},[539,18147,18022],{},[133,18149,793],{},[539,18151,18152],{},"5.2632",[133,18154,155],{"separator":70},[230,18156],{"width":10251},[59,18158,4230],{},[133,18160,136],{"stretchy":135},[539,18162,17359],{},[133,18164,161],{"stretchy":135},[133,18166,793],{},[539,18168,17739],{},[59,18170,4230],{},[133,18172,136],{"stretchy":135},[539,18174,546],{},[133,18176,161],{"stretchy":135},[133,18178,793],{},[539,18180,18181],{},"4.7368.",[63,18183,18184],{"encoding":65},"V(1)=\\frac{1}{1-0.81}=5.2632,\\qquad\nV(2)=0.9V(1)=4.7368.",[39,18186,18188,18215,18307,18346,18376],{"className":18187,"ariaHidden":70},[69],[39,18189,18191,18194,18197,18200,18203,18206,18209,18212],{"className":18190},[74],[39,18192],{"className":18193,"style":245},[78],[39,18195,4230],{"className":18196,"style":461},[83,84],[39,18198,136],{"className":18199},[182],[39,18201,546],{"className":18202},[83],[39,18204,161],{"className":18205},[263],[39,18207],{"className":18208,"style":231},[230],[39,18210,793],{"className":18211},[235],[39,18213],{"className":18214,"style":231},[230],[39,18216,18218,18222,18298,18301,18304],{"className":18217},[74],[39,18219],{"className":18220,"style":18221},[78],"height:2.0908em;vertical-align:-0.7693em;",[39,18223,18225,18228,18295],{"className":18224},[83],[39,18226],{"className":18227},[182,5463],[39,18229,18231],{"className":18230},[5362],[39,18232,18234,18286],{"className":18233},[196,649],[39,18235,18237,18283],{"className":18236},[200],[39,18238,18241,18264,18272],{"className":18239,"style":18240},[204],"height:1.3214em;",[39,18242,18243,18246],{"style":5479},[39,18244],{"className":18245,"style":935},[212],[39,18247,18249,18252,18255,18258,18261],{"className":18248},[83],[39,18250,546],{"className":18251},[83],[39,18253],{"className":18254,"style":461},[230],[39,18256,5378],{"className":18257},[465],[39,18259],{"className":18260,"style":461},[230],[39,18262,18022],{"className":18263},[83],[39,18265,18266,18269],{"style":5503},[39,18267],{"className":18268,"style":935},[212],[39,18270],{"className":18271,"style":5511},[5510],[39,18273,18274,18277],{"style":5514},[39,18275],{"className":18276,"style":935},[212],[39,18278,18280],{"className":18279},[83],[39,18281,546],{"className":18282},[83],[39,18284,673],{"className":18285},[672],[39,18287,18289],{"className":18288},[200],[39,18290,18293],{"className":18291,"style":18292},[204],"height:0.7693em;",[39,18294],{},[39,18296],{"className":18297},[263,5463],[39,18299],{"className":18300,"style":231},[230],[39,18302,793],{"className":18303},[235],[39,18305],{"className":18306,"style":231},[230],[39,18308,18310,18313,18316,18319,18322,18325,18328,18331,18334,18337,18340,18343],{"className":18309},[74],[39,18311],{"className":18312,"style":245},[78],[39,18314,18152],{"className":18315},[83],[39,18317,155],{"className":18318},[252],[39,18320],{"className":18321,"style":10787},[230],[39,18323],{"className":18324,"style":256},[230],[39,18326,4230],{"className":18327,"style":461},[83,84],[39,18329,136],{"className":18330},[182],[39,18332,17359],{"className":18333},[83],[39,18335,161],{"className":18336},[263],[39,18338],{"className":18339,"style":231},[230],[39,18341,793],{"className":18342},[235],[39,18344],{"className":18345,"style":231},[230],[39,18347,18349,18352,18355,18358,18361,18364,18367,18370,18373],{"className":18348},[74],[39,18350],{"className":18351,"style":245},[78],[39,18353,17739],{"className":18354},[83],[39,18356,4230],{"className":18357,"style":461},[83,84],[39,18359,136],{"className":18360},[182],[39,18362,546],{"className":18363},[83],[39,18365,161],{"className":18366},[263],[39,18368],{"className":18369,"style":231},[230],[39,18371,793],{"className":18372},[235],[39,18374],{"className":18375,"style":231},[230],[39,18377,18379,18382],{"className":18378},[74],[39,18380],{"className":18381,"style":2473},[78],[39,18383,18181],{"className":18384},[83],[11,18386,18387],{},"用矩阵形式验证，转移矩阵与奖励向量为",[39,18389,18391],{"className":18390,"translate":43},[746],[39,18392,18394,18483],{"className":18393,"translate":43},[42],[39,18395,18397],{"className":18396},[47],[49,18398,18399],{"xmlns":51,"display":755},[53,18400,18401,18480],{},[56,18402,18403,18405,18407,18445,18447,18449,18451,18453,18478],{},[59,18404,131],{},[133,18406,793],{},[56,18408,18409,18411,18443],{},[133,18410,136],{"fence":70},[759,18412,18415,18429],{"rowspacing":18413,"columnalign":18414,"columnspacing":6891},"0.16em","center 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className":19321},[1223],[39,19323,19325,19646],{"className":19324},[196,649],[39,19326,19328,19643],{"className":19327},[200],[39,19329,19331,19410],{"className":19330,"style":19284},[204],[39,19332,19333,19336],{"style":19287},[39,19334],{"className":19335,"style":19291},[212],[39,19337,19339,19342,19345,19348,19351,19354,19357,19360,19363,19366,19369,19372,19407],{"className":19338},[83],[39,19340],{"className":19341},[83],[39,19343],{"className":19344,"style":231},[230],[39,19346,793],{"className":19347},[235],[39,19349],{"className":19350,"style":231},[230],[39,19352,136],{"className":19353},[182],[39,19355,14373],{"className":19356,"style":14420},[83,84],[39,19358],{"className":19359,"style":461},[230],[39,19361,5378],{"className":19362},[465],[39,19364],{"className":19365,"style":461},[230],[39,19367,17739],{"className":19368},[83],[39,19370,131],{"className":19371,"style":178},[83,84],[39,19373,19375,19378],{"className":19374},[263],[39,19376,161],{"className":19377},[263],[39,19379,19381],{"className":19380},[192],[39,19382,19384],{"className":19383},[196],[39,19385,19387],{"className":19386},[200],[39,19388,19390],{"className":19389,"style":14940},[204],[39,19391,19392,19395],{"style":1027},[39,19393],{"className":19394,"style":213},[212],[39,19396,19398],{"className":19397},[217,218,219,220],[39,19399,19401,19404],{"className":19400},[83,220],[39,19402,5378],{"className":19403},[83,220],[39,19405,546],{"className":19406},[83,220],[39,19408,401],{"className":19409,"style":438},[83,84],[39,19411,19412,19415],{"style":19300},[39,19413],{"className":19414,"style":19291},[212],[39,19416,19418,19421,19424,19427,19430,19493,19496,19562,19565,19568,19571,19637,19640],{"className":19417},[83],[39,19419],{"className":19420},[83],[39,19422],{"className":19423,"style":231},[230],[39,19425,793],{"className":19426},[235],[39,19428],{"className":19429,"style":231},[230],[39,19431,19433,19436,19490],{"className":19432},[83],[39,19434],{"className":19435},[182,5463],[39,19437,19439],{"className":19438},[5362],[39,19440,19442,19481],{"className":19441},[196,649],[39,19443,19445,19478],{"className":19444},[200],[39,19446,19448,19459,19467],{"className":19447,"style":18240},[204],[39,19449,19450,19453],{"style":5479},[39,19451],{"className":19452,"style":935},[212],[39,19454,19456],{"className":19455},[83],[39,19457,19199],{"className":19458},[83],[39,19460,19461,19464],{"style":5503},[39,19462],{"className":19463,"style":935},[212],[39,19465],{"className":19466,"style":5511},[5510],[39,19468,19469,19472],{"style":5514},[39,19470],{"className":19471,"style":935},[212],[39,19473,19475],{"className":19474},[83],[39,19476,546],{"className":19477},[83],[39,19479,673],{"className":19480},[672],[39,19482,19484],{"className":19483},[200],[39,19485,19488],{"className":19486,"style":19487},[204],"height:0.686em;",[39,19489],{},[39,19491],{"className":19492},[263,5463],[39,19494],{"className":19495,"style":256},[230],[39,19497,19499,19505,19556],{"className":19498},[2842],[39,19500,19502],{"className":19501,"style":4592},[182,4591],[39,19503,136],{"className":19504},[18520,219],[39,19506,19508],{"className":19507},[83],[39,19509,19511],{"className":19510},[759],[39,19512,19514],{"className":19513},[18530],[39,19515,19517,19548],{"className":19516},[196,649],[39,19518,19520,19545],{"className":19519},[200],[39,19521,19523,19534],{"className":19522,"style":18540},[204],[39,19524,19525,19528],{"style":18543},[39,19526],{"className":19527,"style":935},[212],[39,19529,19531],{"className":19530},[83],[39,19532,546],{"className":19533},[83],[39,19535,19536,19539],{"style":18555},[39,19537],{"className":19538,"style":935},[212],[39,19540,19542],{"className":19541},[83],[39,19543,17739],{"className":19544},[83],[39,19546,673],{"className":19547},[672],[39,19549,19551],{"className":19550},[200],[39,19552,19554],{"className":19553,"style":18574},[204],[39,19555],{},[39,19557,19559],{"className":19558,"style":4592},[263,4591],[39,19560,161],{"className":19561},[18520,219],[39,19563],{"className":19564,"style":231},[230],[39,19566,793],{"className":19567},[235],[39,19569],{"className":19570,"style":231},[230],[39,19572,19574,19580,19631],{"className":19573},[2842],[39,19575,19577],{"className":19576,"style":4592},[182,4591],[39,19578,136],{"className":19579},[18520,219],[39,19581,19583],{"className":19582},[83],[39,19584,19586],{"className":19585},[759],[39,19587,19589],{"className":19588},[18530],[39,19590,19592,19623],{"className":19591},[196,649],[39,19593,19595,19620],{"className":19594},[200],[39,19596,19598,19609],{"className":19597,"style":18540},[204],[39,19599,19600,19603],{"style":18543},[39,19601],{"className":19602,"style":935},[212],[39,19604,19606],{"className":19605},[83],[39,19607,18152],{"className":19608},[83],[39,19610,19611,19614],{"style":18555},[39,19612],{"className":19613,"style":935},[212],[39,19615,19617],{"className":19616},[83],[39,19618,19248],{"className":19619},[83],[39,19621,673],{"className":19622},[672],[39,19624,19626],{"className":19625},[200],[39,19627,19629],{"className":19628,"style":18574},[204],[39,19630],{},[39,19632,19634],{"className":19633,"style":4592},[263,4591],[39,19635,161],{"className":19636},[18520,219],[39,19638],{"className":19639,"style":256},[230],[39,19641,2903],{"className":19642},[83],[39,19644,673],{"className":19645},[672],[39,19647,19649],{"className":19648},[200],[39,19650,19652],{"className":19651,"style":19316},[204],[39,19653],{},[11,19655,19656,19657,19740],{},"这个例子说明，即使状态每两步就回到原处，只要 ",[39,19658,19660,19682],{"className":19659,"translate":43},[42],[39,19661,19663],{"className":19662},[47],[49,19664,19665],{"xmlns":51},[53,19666,19667,19679],{},[56,19668,19669,19675,19677],{},[138,19670,19671,19673],{},[59,19672,531],{},[539,19674,17359],{},[133,19676,1693],{},[539,19678,546],{},[63,19680,19681],{"encoding":65},"\\gamma^2\u003C1",[39,19683,19685,19731],{"className":19684,"ariaHidden":70},[69],[39,19686,19688,19692,19722,19725,19728],{"className":19687},[74],[39,19689],{"className":19690,"style":19691},[78],"height:1.0085em;vertical-align:-0.1944em;",[39,19693,19695,19698],{"className":19694},[83],[39,19696,531],{"className":19697,"style":565},[83,84],[39,19699,19701],{"className":19700},[192],[39,19702,19704],{"className":19703},[196],[39,19705,19707],{"className":19706},[200],[39,19708,19711],{"className":19709,"style":19710},[204],"height:0.8141em;",[39,19712,19713,19716],{"style":208},[39,19714],{"className":19715,"style":213},[212],[39,19717,19719],{"className":19718},[217,218,219,220],[39,19720,17359],{"className":19721},[83,220],[39,19723],{"className":19724,"style":231},[230],[39,19726,1693],{"className":19727},[235],[39,19729],{"className":19730,"style":231},[230],[39,19732,19734,19737],{"className":19733},[74],[39,19735],{"className":19736,"style":2473},[78],[39,19738,546],{"className":19739},[83],"，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。",[26,19742,19744],{"id":19743},"_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",[11,19746,19747],{},"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。",[31,19749,19752],{"id":19750,"title":19751},"proposition-policy-verification","命题 2.3（策略评估的验证定理）",[11,19753,37,19754,16486,19822,14627,19908,19989,19990,20048],{},[39,19755,19757,19782],{"className":19756,"translate":43},[42],[39,19758,19760],{"className":19759},[47],[49,19761,19762],{"xmlns":51},[53,19763,19764,19779],{},[56,19765,19766,19768,19770,19773,19775,19777],{},[59,19767,4230],{},[133,19769,534],{},[59,19771,19772],{},"B",[133,19774,136],{"stretchy":135},[59,19776,61],{},[133,19778,161],{"stretchy":135},[63,19780,19781],{"encoding":65},"V\\in B(S)",[39,19783,19785,19803],{"className":19784,"ariaHidden":70},[69],[39,19786,19788,19791,19794,19797,19800],{"className":19787},[74],[39,19789],{"className":19790,"style":13061},[78],[39,19792,4230],{"className":19793,"style":461},[83,84],[39,19795],{"className":19796,"style":231},[230],[39,19798,534],{"className":19799},[235],[39,19801],{"className":19802,"style":231},[230],[39,19804,19806,19809,19813,19816,19819],{"className":19805},[74],[39,19807],{"className":19808,"style":245},[78],[39,19810,19772],{"className":19811,"style":19812},[83,84],"margin-right:0.0502em;",[39,19814,136],{"className":19815},[182],[39,19817,61],{"className":19818,"style":85},[83,84],[39,19820,161],{"className":19821},[263],[39,19823,19825,19849],{"className":19824,"translate":43},[42],[39,19826,19828],{"className":19827},[47],[49,19829,19830],{"xmlns":51},[53,19831,19832,19846],{},[56,19833,19834,19836,19838,19844],{},[59,19835,4230],{},[133,19837,793],{},[138,19839,19840,19842],{},[59,19841,11772],{"mathvariant":617},[59,19843,2098],{},[59,19845,4230],{},[63,19847,19848],{"encoding":65},"V=\\mathcal{T}^\\pi V",[39,19850,19852,19870],{"className":19851,"ariaHidden":70},[69],[39,19853,19855,19858,19861,19864,19867],{"className":19854},[74],[39,19856],{"className":19857,"style":79},[78],[39,19859,4230],{"className":19860,"style":461},[83,84],[39,19862],{"className":19863,"style":231},[230],[39,19865,793],{"className":19866},[235],[39,19868],{"className":19869,"style":231},[230],[39,19871,19873,19876,19905],{"className":19872},[74],[39,19874],{"className":19875,"style":79},[78],[39,19877,19879,19882],{"className":19878},[83],[39,19880,11772],{"className":19881,"style":11793},[83,641],[39,19883,19885],{"className":19884},[192],[39,19886,19888],{"className":19887},[196],[39,19889,19891],{"className":19890},[200],[39,19892,19894],{"className":19893,"style":6410},[204],[39,19895,19896,19899],{"style":208},[39,19897],{"className":19898,"style":213},[212],[39,19900,19902],{"className":19901},[217,218,219,220],[39,19903,2098],{"className":19904,"style":2114},[83,84,220],[39,19906,4230],{"className":19907,"style":461},[83,84],[39,19909,19911,19933],{"className":19910,"translate":43},[42],[39,19912,19914],{"className":19913},[47],[49,19915,19916],{"xmlns":51},[53,19917,19918,19930],{},[56,19919,19920,19922,19924],{},[59,19921,4230],{},[133,19923,793],{},[138,19925,19926,19928],{},[59,19927,4230],{},[59,19929,2098],{},[63,19931,19932],{"encoding":65},"V=V^\\pi",[39,19934,19936,19954],{"className":19935,"ariaHidden":70},[69],[39,19937,19939,19942,19945,19948,19951],{"className":19938},[74],[39,19940],{"className":19941,"style":79},[78],[39,19943,4230],{"className":19944,"style":461},[83,84],[39,19946],{"className":19947,"style":231},[230],[39,19949,793],{"className":19950},[235],[39,19952],{"className":19953,"style":231},[230],[39,19955,19957,19960],{"className":19956},[74],[39,19958],{"className":19959,"style":79},[78],[39,19961,19963,19966],{"className":19962},[83],[39,19964,4230],{"className":19965,"style":461},[83,84],[39,19967,19969],{"className":19968},[192],[39,19970,19972],{"className":19971},[196],[39,19973,19975],{"className":19974},[200],[39,19976,19978],{"className":19977,"style":6410},[204],[39,19979,19980,19983],{"style":208},[39,19981],{"className":19982,"style":213},[212],[39,19984,19986],{"className":19985},[217,218,219,220],[39,19987,2098],{"className":19988,"style":2114},[83,84,220],"。特别地，",[39,19991,19993,20010],{"className":19992,"translate":43},[42],[39,19994,19996],{"className":19995},[47],[49,19997,19998],{"xmlns":51},[53,19999,20000,20008],{},[56,20001,20002],{},[138,20003,20004,20006],{},[59,20005,11772],{"mathvariant":617},[59,20007,2098],{},[63,20009,11777],{"encoding":65},[39,20011,20013],{"className":20012,"ariaHidden":70},[69],[39,20014,20016,20019],{"className":20015},[74],[39,20017],{"className":20018,"style":79},[78],[39,20020,20022,20025],{"className":20021},[83],[39,20023,11772],{"className":20024,"style":11793},[83,641],[39,20026,20028],{"className":20027},[192],[39,20029,20031],{"className":20030},[196],[39,20032,20034],{"className":20033},[200],[39,20035,20037],{"className":20036,"style":6410},[204],[39,20038,20039,20042],{"style":208},[39,20040],{"className":20041,"style":213},[212],[39,20043,20045],{"className":20044},[217,218,219,220],[39,20046,2098],{"className":20047,"style":2114},[83,84,220]," 在有界函数空间中至多有一个不动点。",[11,20050,20051,20053,20054,20139],{},[1489,20052,5595],{}," 设 ",[39,20055,20057,20080],{"className":20056,"translate":43},[42],[39,20058,20060],{"className":20059},[47],[49,20061,20062],{"xmlns":51},[53,20063,20064,20078],{},[56,20065,20066,20068,20070,20076],{},[59,20067,4230],{},[133,20069,793],{},[138,20071,20072,20074],{},[59,20073,11772],{"mathvariant":617},[59,20075,2098],{},[59,20077,4230],{},[63,20079,19848],{"encoding":65},[39,20081,20083,20101],{"className":20082,"ariaHidden":70},[69],[39,20084,20086,20089,20092,20095,20098],{"className":20085},[74],[39,20087],{"className":20088,"style":79},[78],[39,20090,4230],{"className":20091,"style":461},[83,84],[39,20093],{"className":20094,"style":231},[230],[39,20096,793],{"className":20097},[235],[39,20099],{"className":20100,"style":231},[230],[39,20102,20104,20107,20136],{"className":20103},[74],[39,20105],{"className":20106,"style":79},[78],[39,20108,20110,20113],{"className":20109},[83],[39,20111,11772],{"className":20112,"style":11793},[83,641],[39,20114,20116],{"className":20115},[192],[39,20117,20119],{"className":20118},[196],[39,20120,20122],{"className":20121},[200],[39,20123,20125],{"className":20124,"style":6410},[204],[39,20126,20127,20130],{"style":208},[39,20128],{"className":20129,"style":213},[212],[39,20131,20133],{"className":20132},[217,218,219,220],[39,20134,2098],{"className":20135,"style":2114},[83,84,220],[39,20137,4230],{"className":20138,"style":461},[83,84],"。反复代入得",[39,20141,20143],{"className":20142,"translate":43},[746],[39,20144,20146,20234],{"className":20145,"translate":43},[42],[39,20147,20149],{"className":20148},[47],[49,20150,20151],{"xmlns":51,"display":755},[53,20152,20153,20231],{},[56,20154,20155,20157,20159,20179,20185,20187,20193,20199,20205,20207,20213,20215,20221,20227,20229],{},[59,20156,4230],{},[133,20158,793],{},[3187,20160,20161,20163,20171],{},[133,20162,3191],{},[56,20164,20165,20167,20169],{},[59,20166,3196],{},[133,20168,793],{},[539,20170,541],{},[56,20172,20173,20175,20177],{},[59,20174,12962],{},[133,20176,5378],{},[539,20178,546],{},[138,20180,20181,20183],{},[59,20182,531],{},[59,20184,3196],{},[133,20186,136],{"stretchy":135},[138,20188,20189,20191],{},[59,20190,131],{},[59,20192,2098],{},[138,20194,20195,20197],{},[133,20196,161],{"stretchy":135},[59,20198,3196],{},[138,20200,20201,20203],{},[59,20202,401],{},[59,20204,2098],{},[133,20206,788],{},[138,20208,20209,20211],{},[59,20210,531],{},[59,20212,12962],{},[133,20214,136],{"stretchy":135},[138,20216,20217,20219],{},[59,20218,131],{},[59,20220,2098],{},[138,20222,20223,20225],{},[133,20224,161],{"stretchy":135},[59,20226,12962],{},[59,20228,4230],{},[59,20230,2903],{"mathvariant":145},[63,20232,20233],{"encoding":65},"V=\\sum_{k=0}^{n-1}\\gamma^k(P^\\pi)^k r^\\pi\n+\\gamma^n(P^\\pi)^n V.",[39,20235,20237,20255,20467],{"className":20236,"ariaHidden":70},[69],[39,20238,20240,20243,20246,20249,20252],{"className":20239},[74],[39,20241],{"className":20242,"style":79},[78],[39,20244,4230],{"className":20245,"style":461},[83,84],[39,20247],{"className":20248,"style":231},[230],[39,20250,793],{"className":20251},[235],[39,20253],{"className":20254,"style":231},[230],[39,20256,20258,20262,20336,20339,20368,20371,20400,20429,20458,20461,20464],{"className":20257},[74],[39,20259],{"className":20260,"style":20261},[78],"height:3.1032em;vertical-align:-1.3021em;",[39,20263,20265],{"className":20264},[1744,3294],[39,20266,20268,20328],{"className":20267},[196,649],[39,20269,20271,20325],{"className":20270},[200],[39,20272,20275,20295,20305],{"className":20273,"style":20274},[204],"height:1.8011em;",[39,20276,20277,20280],{"style":3307},[39,20278],{"className":20279,"style":3311},[212],[39,20281,20283],{"className":20282},[217,218,219,220],[39,20284,20286,20289,20292],{"className":20285},[83,220],[39,20287,3196],{"className":20288,"style":3321},[83,84,220],[39,20290,793],{"className":20291},[235,220],[39,20293,541],{"className":20294},[83,220],[39,20296,20297,20300],{"style":3330},[39,20298],{"className":20299,"style":3311},[212],[39,20301,20302],{},[39,20303,3191],{"className":20304},[1744,3339,3340],[39,20306,20307,20310],{"style":3343},[39,20308],{"className":20309,"style":3311},[212],[39,20311,20313],{"className":20312},[217,218,219,220],[39,20314,20316,20319,20322],{"className":20315},[83,220],[39,20317,12962],{"className":20318},[83,84,220],[39,20320,5378],{"className":20321},[465,220],[39,20323,546],{"className":20324},[83,220],[39,20326,673],{"className":20327},[672],[39,20329,20331],{"className":20330},[200],[39,20332,20334],{"className":20333,"style":3365},[204],[39,20335],{},[39,20337],{"className":20338,"style":256},[230],[39,20340,20342,20345],{"className":20341},[83],[39,20343,531],{"className":20344,"style":565},[83,84],[39,20346,20348],{"className":20347},[192],[39,20349,20351],{"className":20350},[196],[39,20352,20354],{"className":20353},[200],[39,20355,20357],{"className":20356,"style":3389},[204],[39,20358,20359,20362],{"style":1027},[39,20360],{"className":20361,"style":213},[212],[39,20363,20365],{"className":20364},[217,218,219,220],[39,20366,3196],{"className":20367,"style":3321},[83,84,220],[39,20369,136],{"className":20370},[182],[39,20372,20374,20377],{"className":20373},[83],[39,20375,131],{"className":20376,"style":178},[83,84],[39,20378,20380],{"className":20379},[192],[39,20381,20383],{"className":20382},[196],[39,20384,20386],{"className":20385},[200],[39,20387,20389],{"className":20388,"style":4421},[204],[39,20390,20391,20394],{"style":1027},[39,20392],{"className":20393,"style":213},[212],[39,20395,20397],{"className":20396},[217,218,219,220],[39,20398,2098],{"className":20399,"style":2114},[83,84,220],[39,20401,20403,20406],{"className":20402},[263],[39,20404,161],{"className":20405},[263],[39,20407,20409],{"className":20408},[192],[39,20410,20412],{"className":20411},[196],[39,20413,20415],{"className":20414},[200],[39,20416,20418],{"className":20417,"style":3389},[204],[39,20419,20420,20423],{"style":1027},[39,20421],{"className":20422,"style":213},[212],[39,20424,20426],{"className":20425},[217,218,219,220],[39,20427,3196],{"className":20428,"style":3321},[83,84,220],[39,20430,20432,20435],{"className":20431},[83],[39,20433,401],{"className":20434,"style":438},[83,84],[39,20436,20438],{"className":20437},[192],[39,20439,20441],{"className":20440},[196],[39,20442,20444],{"className":20443},[200],[39,20445,20447],{"className":20446,"style":4421},[204],[39,20448,20449,20452],{"style":1027},[39,20450],{"className":20451,"style":213},[212],[39,20453,20455],{"className":20454},[217,218,219,220],[39,20456,2098],{"className":20457,"style":2114},[83,84,220],[39,20459],{"className":20460,"style":461},[230],[39,20462,788],{"className":20463},[465],[39,20465],{"className":20466,"style":461},[230],[39,20468,20470,20473,20502,20505,20534,20563,20566],{"className":20469},[74],[39,20471],{"className":20472,"style":245},[78],[39,20474,20476,20479],{"className":20475},[83],[39,20477,531],{"className":20478,"style":565},[83,84],[39,20480,20482],{"className":20481},[192],[39,20483,20485],{"className":20484},[196],[39,20486,20488],{"className":20487},[200],[39,20489,20491],{"className":20490,"style":4421},[204],[39,20492,20493,20496],{"style":1027},[39,20494],{"className":20495,"style":213},[212],[39,20497,20499],{"className":20498},[217,218,219,220],[39,20500,12962],{"className":20501},[83,84,220],[39,20503,136],{"className":20504},[182],[39,20506,20508,20511],{"className":20507},[83],[39,20509,131],{"className":20510,"style":178},[83,84],[39,20512,20514],{"className":20513},[192],[39,20515,20517],{"className":20516},[196],[39,20518,20520],{"className":20519},[200],[39,20521,20523],{"className":20522,"style":4421},[204],[39,20524,20525,20528],{"style":1027},[39,20526],{"className":20527,"style":213},[212],[39,20529,20531],{"className":20530},[217,218,219,220],[39,20532,2098],{"className":20533,"style":2114},[83,84,220],[39,20535,20537,20540],{"className":20536},[263],[39,20538,161],{"className":20539},[263],[39,20541,20543],{"className":20542},[192],[39,20544,20546],{"className":20545},[196],[39,20547,20549],{"className":20548},[200],[39,20550,20552],{"className":20551,"style":4421},[204],[39,20553,20554,20557],{"style":1027},[39,20555],{"className":20556,"style":213},[212],[39,20558,20560],{"className":20559},[217,218,219,220],[39,20561,12962],{"className":20562},[83,84,220],[39,20564,4230],{"className":20565,"style":461},[83,84],[39,20567,2903],{"className":20568},[83],[11,20570,20571,20572,20800,20801,20829],{},"由于 ",[39,20573,20575,20623],{"className":20574,"translate":43},[42],[39,20576,20578],{"className":20577},[47],[49,20579,20580],{"xmlns":51},[53,20581,20582,20620],{},[56,20583,20584,20586,20588,20594,20600,20602,20608,20610,20612,20614],{},[59,20585,15389],{"mathvariant":145},[133,20587,136],{"stretchy":135},[138,20589,20590,20592],{},[59,20591,131],{},[59,20593,2098],{},[138,20595,20596,20598],{},[133,20597,161],{"stretchy":135},[59,20599,12962],{},[59,20601,4230],{},[613,20603,20604,20606],{},[59,20605,15389],{"mathvariant":145},[59,20607,1696],{"mathvariant":145},[133,20609,1824],{},[59,20611,15389],{"mathvariant":145},[59,20613,4230],{},[613,20615,20616,20618],{},[59,20617,15389],{"mathvariant":145},[59,20619,1696],{"mathvariant":145},[63,20621,20622],{"encoding":65},"\\|(P^\\pi)^n V\\|_\\infty\\le\\|V\\|_\\infty",[39,20624,20626,20748],{"className":20625,"ariaHidden":70},[69],[39,20627,20629,20632,20635,20638,20667,20696,20699,20739,20742,20745],{"className":20628},[74],[39,20630],{"className":20631,"style":245},[78],[39,20633,15389],{"className":20634},[83],[39,20636,136],{"className":20637},[182],[39,20639,20641,20644],{"className":20640},[83],[39,20642,131],{"className":20643,"style":178},[83,84],[39,20645,20647],{"className":20646},[192],[39,20648,20650],{"className":20649},[196],[39,20651,20653],{"className":20652},[200],[39,20654,20656],{"className":20655,"style":6410},[204],[39,20657,20658,20661],{"style":208},[39,20659],{"className":20660,"style":213},[212],[39,20662,20664],{"className":20663},[217,218,219,220],[39,20665,2098],{"className":20666,"style":2114},[83,84,220],[39,20668,20670,20673],{"className":20669},[263],[39,20671,161],{"className":20672},[263],[39,20674,20676],{"className":20675},[192],[39,20677,20679],{"className":20678},[196],[39,20680,20682],{"className":20681},[200],[39,20683,20685],{"className":20684,"style":6410},[204],[39,20686,20687,20690],{"style":208},[39,20688],{"className":20689,"style":213},[212],[39,20691,20693],{"className":20692},[217,218,219,220],[39,20694,12962],{"className":20695},[83,84,220],[39,20697,4230],{"className":20698,"style":461},[83,84],[39,20700,20702,20705],{"className":20701},[83],[39,20703,15389],{"className":20704},[83],[39,20706,20708],{"className":20707},[192],[39,20709,20711,20731],{"className":20710},[196,649],[39,20712,20714,20728],{"className":20713},[200],[39,20715,20717],{"className":20716,"style":1728},[204],[39,20718,20719,20722],{"style":966},[39,20720],{"className":20721,"style":213},[212],[39,20723,20725],{"className":20724},[217,218,219,220],[39,20726,1696],{"className":20727},[83,220],[39,20729,673],{"className":20730},[672],[39,20732,20734],{"className":20733},[200],[39,20735,20737],{"className":20736,"style":680},[204],[39,20738],{},[39,20740],{"className":20741,"style":231},[230],[39,20743,1824],{"className":20744},[235],[39,20746],{"className":20747,"style":231},[230],[39,20749,20751,20754,20757,20760],{"className":20750},[74],[39,20752],{"className":20753,"style":245},[78],[39,20755,15389],{"className":20756},[83],[39,20758,4230],{"className":20759,"style":461},[83,84],[39,20761,20763,20766],{"className":20762},[83],[39,20764,15389],{"className":20765},[83],[39,20767,20769],{"className":20768},[192],[39,20770,20772,20792],{"className":20771},[196,649],[39,20773,20775,20789],{"className":20774},[200],[39,20776,20778],{"className":20777,"style":1728},[204],[39,20779,20780,20783],{"style":966},[39,20781],{"className":20782,"style":213},[212],[39,20784,20786],{"className":20785},[217,218,219,220],[39,20787,1696],{"className":20788},[83,220],[39,20790,673],{"className":20791},[672],[39,20793,20795],{"className":20794},[200],[39,20796,20798],{"className":20797,"style":680},[204],[39,20799],{},"，最后一项在 sup 范数下趋于 ",[39,20802,20804,20817],{"className":20803,"translate":43},[42],[39,20805,20807],{"className":20806},[47],[49,20808,20809],{"xmlns":51},[53,20810,20811,20815],{},[56,20812,20813],{},[539,20814,541],{},[63,20816,541],{"encoding":65},[39,20818,20820],{"className":20819,"ariaHidden":70},[69],[39,20821,20823,20826],{"className":20822},[74],[39,20824],{"className":20825,"style":2473},[78],[39,20827,541],{"className":20828},[83],"。另一方面，",[39,20831,20833],{"className":20832,"translate":43},[746],[39,20834,20836,20888],{"className":20835,"translate":43},[42],[39,20837,20839],{"className":20838},[47],[49,20840,20841],{"xmlns":51,"display":755},[53,20842,20843,20885],{},[56,20844,20845,20859,20865,20867,20873,20879],{},[3187,20846,20847,20849,20857],{},[133,20848,3191],{},[56,20850,20851,20853,20855],{},[59,20852,3196],{},[133,20854,793],{},[539,20856,541],{},[59,20858,1696],{"mathvariant":145},[138,20860,20861,20863],{},[59,20862,531],{},[59,20864,3196],{},[133,20866,136],{"stretchy":135},[138,20868,20869,20871],{},[59,20870,131],{},[59,20872,2098],{},[138,20874,20875,20877],{},[133,20876,161],{"stretchy":135},[59,20878,3196],{},[138,20880,20881,20883],{},[59,20882,401],{},[59,20884,2098],{},[63,20886,20887],{"encoding":65},"\\sum_{k=0}^{\\infty}\\gamma^k(P^\\pi)^k r^\\pi",[39,20889,20891],{"className":20890,"ariaHidden":70},[69],[39,20892,20894,20897,20964,20967,20996,20999,21028,21057],{"className":20893},[74],[39,20895],{"className":20896,"style":3290},[78],[39,20898,20900],{"className":20899},[1744,3294],[39,20901,20903,20956],{"className":20902},[196,649],[39,20904,20906,20953],{"className":20905},[200],[39,20907,20909,20929,20939],{"className":20908,"style":3304},[204],[39,20910,20911,20914],{"style":3307},[39,20912],{"className":20913,"style":3311},[212],[39,20915,20917],{"className":20916},[217,218,219,220],[39,20918,20920,20923,20926],{"className":20919},[83,220],[39,20921,3196],{"className":20922,"style":3321},[83,84,220],[39,20924,793],{"className":20925},[235,220],[39,20927,541],{"className":20928},[83,220],[39,20930,20931,20934],{"style":3330},[39,20932],{"className":20933,"style":3311},[212],[39,20935,20936],{},[39,20937,3191],{"className":20938},[1744,3339,3340],[39,20940,20941,20944],{"style":3343},[39,20942],{"className":20943,"style":3311},[212],[39,20945,20947],{"className":20946},[217,218,219,220],[39,20948,20950],{"className":20949},[83,220],[39,20951,1696],{"className":20952},[83,220],[39,20954,673],{"className":20955},[672],[39,20957,20959],{"className":20958},[200],[39,20960,20962],{"className":20961,"style":3365},[204],[39,20963],{},[39,20965],{"className":20966,"style":256},[230],[39,20968,20970,20973],{"className":20969},[83],[39,20971,531],{"className":20972,"style":565},[83,84],[39,20974,20976],{"className":20975},[192],[39,20977,20979],{"className":20978},[196],[39,20980,20982],{"className":20981},[200],[39,20983,20985],{"className":20984,"style":3389},[204],[39,20986,20987,20990],{"style":1027},[39,20988],{"className":20989,"style":213},[212],[39,20991,20993],{"className":20992},[217,218,219,220],[39,20994,3196],{"className":20995,"style":3321},[83,84,220],[39,20997,136],{"className":20998},[182],[39,21000,21002,21005],{"className":21001},[83],[39,21003,131],{"className":21004,"style":178},[83,84],[39,21006,21008],{"className":21007},[192],[39,21009,21011],{"className":21010},[196],[39,21012,21014],{"className":21013},[200],[39,21015,21017],{"className":21016,"style":4421},[204],[39,21018,21019,21022],{"style":1027},[39,21020],{"className":21021,"style":213},[212],[39,21023,21025],{"className":21024},[217,218,219,220],[39,21026,2098],{"className":21027,"style":2114},[83,84,220],[39,21029,21031,21034],{"className":21030},[263],[39,21032,161],{"className":21033},[263],[39,21035,21037],{"className":21036},[192],[39,21038,21040],{"className":21039},[196],[39,21041,21043],{"className":21042},[200],[39,21044,21046],{"className":21045,"style":3389},[204],[39,21047,21048,21051],{"style":1027},[39,21049],{"className":21050,"style":213},[212],[39,21052,21054],{"className":21053},[217,218,219,220],[39,21055,3196],{"className":21056,"style":3321},[83,84,220],[39,21058,21060,21063],{"className":21059},[83],[39,21061,401],{"className":21062,"style":438},[83,84],[39,21064,21066],{"className":21065},[192],[39,21067,21069],{"className":21068},[196],[39,21070,21072],{"className":21071},[200],[39,21073,21075],{"className":21074,"style":4421},[204],[39,21076,21077,21080],{"style":1027},[39,21078],{"className":21079,"style":213},[212],[39,21081,21083],{"className":21082},[217,218,219,220],[39,21084,2098],{"className":21085,"style":2114},[83,84,220],[11,21087,21088,21089,21147,21148,21228],{},"正是折扣回报的期望，即 ",[39,21090,21092,21109],{"className":21091,"translate":43},[42],[39,21093,21095],{"className":21094},[47],[49,21096,21097],{"xmlns":51},[53,21098,21099,21107],{},[56,21100,21101],{},[138,21102,21103,21105],{},[59,21104,4230],{},[59,21106,2098],{},[63,21108,6382],{"encoding":65},[39,21110,21112],{"className":21111,"ariaHidden":70},[69],[39,21113,21115,21118],{"className":21114},[74],[39,21116],{"className":21117,"style":79},[78],[39,21119,21121,21124],{"className":21120},[83],[39,21122,4230],{"className":21123,"style":461},[83,84],[39,21125,21127],{"className":21126},[192],[39,21128,21130],{"className":21129},[196],[39,21131,21133],{"className":21132},[200],[39,21134,21136],{"className":21135,"style":6410},[204],[39,21137,21138,21141],{"style":208},[39,21139],{"className":21140,"style":213},[212],[39,21142,21144],{"className":21143},[217,218,219,220],[39,21145,2098],{"className":21146,"style":2114},[83,84,220],"。故 ",[39,21149,21151,21172],{"className":21150,"translate":43},[42],[39,21152,21154],{"className":21153},[47],[49,21155,21156],{"xmlns":51},[53,21157,21158,21170],{},[56,21159,21160,21162,21164],{},[59,21161,4230],{},[133,21163,793],{},[138,21165,21166,21168],{},[59,21167,4230],{},[59,21169,2098],{},[63,21171,19932],{"encoding":65},[39,21173,21175,21193],{"className":21174,"ariaHidden":70},[69],[39,21176,21178,21181,21184,21187,21190],{"className":21177},[74],[39,21179],{"className":21180,"style":79},[78],[39,21182,4230],{"className":21183,"style":461},[83,84],[39,21185],{"className":21186,"style":231},[230],[39,21188,793],{"className":21189},[235],[39,21191],{"className":21192,"style":231},[230],[39,21194,21196,21199],{"className":21195},[74],[39,21197],{"className":21198,"style":79},[78],[39,21200,21202,21205],{"className":21201},[83],[39,21203,4230],{"className":21204,"style":461},[83,84],[39,21206,21208],{"className":21207},[192],[39,21209,21211],{"className":21210},[196],[39,21212,21214],{"className":21213},[200],[39,21215,21217],{"className":21216,"style":6410},[204],[39,21218,21219,21222],{"style":208},[39,21220],{"className":21221,"style":213},[212],[39,21223,21225],{"className":21224},[217,218,219,220],[39,21226,2098],{"className":21227,"style":2114},[83,84,220],"。证毕。",[11,21230,21231,21232,21290,21291,21295,21296,21346],{},"这个证明不依赖矩阵的有限维结构，只用到 ",[39,21233,21235,21252],{"className":21234,"translate":43},[42],[39,21236,21238],{"className":21237},[47],[49,21239,21240],{"xmlns":51},[53,21241,21242,21250],{},[56,21243,21244],{},[138,21245,21246,21248],{},[59,21247,131],{},[59,21249,2098],{},[63,21251,15306],{"encoding":65},[39,21253,21255],{"className":21254,"ariaHidden":70},[69],[39,21256,21258,21261],{"className":21257},[74],[39,21259],{"className":21260,"style":79},[78],[39,21262,21264,21267],{"className":21263},[83],[39,21265,131],{"className":21266,"style":178},[83,84],[39,21268,21270],{"className":21269},[192],[39,21271,21273],{"className":21272},[196],[39,21274,21276],{"className":21275},[200],[39,21277,21279],{"className":21278,"style":6410},[204],[39,21280,21281,21284],{"style":208},[39,21282],{"className":21283,"style":213},[212],[39,21285,21287],{"className":21286},[217,218,219,220],[39,21288,2098],{"className":21289,"style":2114},[83,84,220]," 是随机核这一事实。因此它可以直接推广到一般状态空间。",[158,21292,21294],{"href":21293},"#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 ",[39,21297,21299,21316],{"className":21298,"translate":43},[42],[39,21300,21302],{"className":21301},[47],[49,21303,21304],{"xmlns":51},[53,21305,21306,21314],{},[56,21307,21308,21310,21312],{},[59,21309,531],{},[133,21311,1693],{},[539,21313,546],{},[63,21315,17296],{"encoding":65},[39,21317,21319,21337],{"className":21318,"ariaHidden":70},[69],[39,21320,21322,21325,21328,21331,21334],{"className":21321},[74],[39,21323],{"className":21324,"style":561},[78],[39,21326,531],{"className":21327,"style":565},[83,84],[39,21329],{"className":21330,"style":231},[230],[39,21332,1693],{"className":21333},[235],[39,21335],{"className":21336,"style":231},[230],[39,21338,21340,21343],{"className":21339},[74],[39,21341],{"className":21342,"style":2473},[78],[39,21344,546],{"className":21345},[83]," 且奖励有界，满足方程的函数就是值函数本身。",[21,21348,21350],{"id":21349},"三贝尔曼算子与压缩映射","三、贝尔曼算子与压缩映射",[26,21352,21354],{"id":21353},"_31-有界函数空间","3.1 有界函数空间",[11,21356,21357,21358,21402,21403,21431],{},"记 ",[39,21359,21361,21381],{"className":21360,"translate":43},[42],[39,21362,21364],{"className":21363},[47],[49,21365,21366],{"xmlns":51},[53,21367,21368,21378],{},[56,21369,21370,21372,21374,21376],{},[59,21371,19772],{},[133,21373,136],{"stretchy":135},[59,21375,61],{},[133,21377,161],{"stretchy":135},[63,21379,21380],{"encoding":65},"B(S)",[39,21382,21384],{"className":21383,"ariaHidden":70},[69],[39,21385,21387,21390,21393,21396,21399],{"className":21386},[74],[39,21388],{"className":21389,"style":245},[78],[39,21391,19772],{"className":21392,"style":19812},[83,84],[39,21394,136],{"className":21395},[182],[39,21397,61],{"className":21398,"style":85},[83,84],[39,21400,161],{"className":21401},[263]," 为 ",[39,21404,21406,21419],{"className":21405,"translate":43},[42],[39,21407,21409],{"className":21408},[47],[49,21410,21411],{"xmlns":51},[53,21412,21413,21417],{},[56,21414,21415],{},[59,21416,61],{},[63,21418,61],{"encoding":65},[39,21420,21422],{"className":21421,"ariaHidden":70},[69],[39,21423,21425,21428],{"className":21424},[74],[39,21426],{"className":21427,"style":79},[78],[39,21429,61],{"className":21430,"style":85},[83,84]," 上全体有界实值函数构成的空间，并赋予 sup 范数",[39,21433,21435],{"className":21434,"translate":43},[746],[39,21436,21438,21493],{"className":21437,"translate":43},[42],[39,21439,21441],{"className":21440},[47],[49,21442,21443],{"xmlns":51,"display":755},[53,21444,21445,21490],{},[56,21446,21447,21449,21451,21457,21459,21476,21478,21480,21482,21484,21486,21488],{},[59,21448,15389],{"mathvariant":145},[59,21450,4230],{},[613,21452,21453,21455],{},[59,21454,15389],{"mathvariant":145},[59,21456,1696],{"mathvariant":145},[133,21458,793],{},[6556,21460,21461,21468],{},[56,21462,21463,21466],{},[59,21464,21465],{},"sup",[133,21467,1690],{},[56,21469,21470,21472,21474],{},[59,21471,142],{},[133,21473,534],{},[59,21475,61],{},[59,21477,150],{"mathvariant":145},[59,21479,4230],{},[133,21481,136],{"stretchy":135},[59,21483,142],{},[133,21485,161],{"stretchy":135},[59,21487,150],{"mathvariant":145},[59,21489,2903],{"mathvariant":145},[63,21491,21492],{"encoding":65},"\\|V\\|_\\infty=\\sup_{s\\in S}|V(s)|.",[39,21494,21496,21557],{"className":21495,"ariaHidden":70},[69],[39,21497,21499,21502,21505,21508,21548,21551,21554],{"className":21498},[74],[39,21500],{"className":21501,"style":245},[78],[39,21503,15389],{"className":21504},[83],[39,21506,4230],{"className":21507,"style":461},[83,84],[39,21509,21511,21514],{"className":21510},[83],[39,21512,15389],{"className":21513},[83],[39,21515,21517],{"className":21516},[192],[39,21518,21520,21540],{"className":21519},[196,649],[39,21521,21523,21537],{"className":21522},[200],[39,21524,21526],{"className":21525,"style":1728},[204],[39,21527,21528,21531],{"style":966},[39,21529],{"className":21530,"style":213},[212],[39,21532,21534],{"className":21533},[217,218,219,220],[39,21535,1696],{"className":21536},[83,220],[39,21538,673],{"className":21539},[672],[39,21541,21543],{"className":21542},[200],[39,21544,21546],{"className":21545,"style":680},[204],[39,21547],{},[39,21549],{"className":21550,"style":231},[230],[39,21552,793],{"className":21553},[235],[39,21555],{"className":21556,"style":231},[230],[39,21558,21560,21564,21620,21623,21626,21629,21632,21635,21638],{"className":21559},[74],[39,21561],{"className":21562,"style":21563},[78],"height:1.7161em;vertical-align:-0.9661em;",[39,21565,21567],{"className":21566},[1744,3294],[39,21568,21570,21611],{"className":21569},[196,649],[39,21571,21573,21608],{"className":21572},[200],[39,21574,21576,21597],{"className":21575,"style":290},[204],[39,21577,21579,21582],{"style":21578},"top:-2.1612em;margin-left:0em;",[39,21580],{"className":21581,"style":935},[212],[39,21583,21585],{"className":21584},[217,218,219,220],[39,21586,21588,21591,21594],{"className":21587},[83,220],[39,21589,142],{"className":21590},[83,84,220],[39,21592,534],{"className":21593},[235,220],[39,21595,61],{"className":21596,"style":85},[83,84,220],[39,21598,21600,21603],{"style":21599},"top:-3em;",[39,21601],{"className":21602,"style":935},[212],[39,21604,21605],{},[39,21606,21465],{"className":21607},[1744],[39,21609,673],{"className":21610},[672],[39,21612,21614],{"className":21613},[200],[39,21615,21618],{"className":21616,"style":21617},[204],"height:0.9661em;",[39,21619],{},[39,21621],{"className":21622,"style":256},[230],[39,21624,150],{"className":21625},[83],[39,21627,4230],{"className":21628,"style":461},[83,84],[39,21630,136],{"className":21631},[182],[39,21633,142],{"className":21634},[83,84],[39,21636,161],{"className":21637},[263],[39,21639,21641],{"className":21640},[83],"∣.",[11,21643,12932,21644,21672,21673,21803],{},[39,21645,21647,21660],{"className":21646,"translate":43},[42],[39,21648,21650],{"className":21649},[47],[49,21651,21652],{"xmlns":51},[53,21653,21654,21658],{},[56,21655,21656],{},[59,21657,61],{},[63,21659,61],{"encoding":65},[39,21661,21663],{"className":21662,"ariaHidden":70},[69],[39,21664,21666,21669],{"className":21665},[74],[39,21667],{"className":21668,"style":79},[78],[39,21670,61],{"className":21671,"style":85},[83,84]," 为有限集时，上确界就是最大值。",[39,21674,21676,21712],{"className":21675,"translate":43},[42],[39,21677,21679],{"className":21678},[47],[49,21680,21681],{"xmlns":51},[53,21682,21683,21709],{},[56,21684,21685,21687,21689,21691,21693,21695,21697,21699,21701,21707],{},[133,21686,136],{"stretchy":135},[59,21688,19772],{},[133,21690,136],{"stretchy":135},[59,21692,61],{},[133,21694,161],{"stretchy":135},[133,21696,155],{"separator":70},[59,21698,15389],{"mathvariant":145},[133,21700,8885],{},[613,21702,21703,21705],{},[59,21704,15389],{"mathvariant":145},[59,21706,1696],{"mathvariant":145},[133,21708,161],{"stretchy":135},[63,21710,21711],{"encoding":65},"(B(S),\\|\\cdot\\|_\\infty)",[39,21713,21715,21754],{"className":21714,"ariaHidden":70},[69],[39,21716,21718,21721,21724,21727,21730,21733,21736,21739,21742,21745,21748,21751],{"className":21717},[74],[39,21719],{"className":21720,"style":245},[78],[39,21722,136],{"className":21723},[182],[39,21725,19772],{"className":21726,"style":19812},[83,84],[39,21728,136],{"className":21729},[182],[39,21731,61],{"className":21732,"style":85},[83,84],[39,21734,161],{"className":21735},[263],[39,21737,155],{"className":21738},[252],[39,21740],{"className":21741,"style":256},[230],[39,21743,15389],{"className":21744},[83],[39,21746],{"className":21747,"style":461},[230],[39,21749,8885],{"className":21750},[465],[39,21752],{"className":21753,"style":461},[230],[39,21755,21757,21760,21800],{"className":21756},[74],[39,21758],{"className":21759,"style":245},[78],[39,21761,21763,21766],{"className":21762},[83],[39,21764,15389],{"className":21765},[83],[39,21767,21769],{"className":21768},[192],[39,21770,21772,21792],{"className":21771},[196,649],[39,21773,21775,21789],{"className":21774},[200],[39,21776,21778],{"className":21777,"style":1728},[204],[39,21779,21780,21783],{"style":966},[39,21781],{"className":21782,"style":213},[212],[39,21784,21786],{"className":21785},[217,218,219,220],[39,21787,1696],{"className":21788},[83,220],[39,21790,673],{"className":21791},[672],[39,21793,21795],{"className":21794},[200],[39,21796,21798],{"className":21797,"style":680},[204],[39,21799],{},[39,21801,161],{"className":21802},[263]," 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。",[11,21805,21806,21807,21865,21866,21927,21928,21971,21972,22030,22031,22089],{},"贝尔曼期望算子 ",[39,21808,21810,21827],{"className":21809,"translate":43},[42],[39,21811,21813],{"className":21812},[47],[49,21814,21815],{"xmlns":51},[53,21816,21817,21825],{},[56,21818,21819],{},[138,21820,21821,21823],{},[59,21822,11772],{"mathvariant":617},[59,21824,2098],{},[63,21826,11777],{"encoding":65},[39,21828,21830],{"className":21829,"ariaHidden":70},[69],[39,21831,21833,21836],{"className":21832},[74],[39,21834],{"className":21835,"style":79},[78],[39,21837,21839,21842],{"className":21838},[83],[39,21840,11772],{"className":21841,"style":11793},[83,641],[39,21843,21845],{"className":21844},[192],[39,21846,21848],{"className":21847},[196],[39,21849,21851],{"className":21850},[200],[39,21852,21854],{"className":21853,"style":6410},[204],[39,21855,21856,21859],{"style":208},[39,21857],{"className":21858,"style":213},[212],[39,21860,21862],{"className":21861},[217,218,219,220],[39,21863,2098],{"className":21864,"style":2114},[83,84,220]," 与贝尔曼最优算子 ",[39,21867,21869,21888],{"className":21868,"translate":43},[42],[39,21870,21872],{"className":21871},[47],[49,21873,21874],{"xmlns":51},[53,21875,21876,21885],{},[56,21877,21878],{},[138,21879,21880,21882],{},[59,21881,11772],{"mathvariant":617},[133,21883,21884],{},"∗",[63,21886,21887],{"encoding":65},"\\mathcal{T}^*",[39,21889,21891],{"className":21890,"ariaHidden":70},[69],[39,21892,21894,21898],{"className":21893},[74],[39,21895],{"className":21896,"style":21897},[78],"height:0.6887em;",[39,21899,21901,21904],{"className":21900},[83],[39,21902,11772],{"className":21903,"style":11793},[83,641],[39,21905,21907],{"className":21906},[192],[39,21908,21910],{"className":21909},[196],[39,21911,21913],{"className":21912},[200],[39,21914,21916],{"className":21915,"style":21897},[204],[39,21917,21918,21921],{"style":208},[39,21919],{"className":21920,"style":213},[212],[39,21922,21924],{"className":21923},[217,218,219,220],[39,21925,21884],{"className":21926},[465,220]," 都将 ",[39,21929,21931,21950],{"className":21930,"translate":43},[42],[39,21932,21934],{"className":21933},[47],[49,21935,21936],{"xmlns":51},[53,21937,21938,21948],{},[56,21939,21940,21942,21944,21946],{},[59,21941,19772],{},[133,21943,136],{"stretchy":135},[59,21945,61],{},[133,21947,161],{"stretchy":135},[63,21949,21380],{"encoding":65},[39,21951,21953],{"className":21952,"ariaHidden":70},[69],[39,21954,21956,21959,21962,21965,21968],{"className":21955},[74],[39,21957],{"className":21958,"style":245},[78],[39,21960,19772],{"className":21961,"style":19812},[83,84],[39,21963,136],{"className":21964},[182],[39,21966,61],{"className":21967,"style":85},[83,84],[39,21969,161],{"className":21970},[263]," 映射到自身。",[39,21973,21975,21992],{"className":21974,"translate":43},[42],[39,21976,21978],{"className":21977},[47],[49,21979,21980],{"xmlns":51},[53,21981,21982,21990],{},[56,21983,21984],{},[138,21985,21986,21988],{},[59,21987,11772],{"mathvariant":617},[59,21989,2098],{},[63,21991,11777],{"encoding":65},[39,21993,21995],{"className":21994,"ariaHidden":70},[69],[39,21996,21998,22001],{"className":21997},[74],[39,21999],{"className":22000,"style":79},[78],[39,22002,22004,22007],{"className":22003},[83],[39,22005,11772],{"className":22006,"style":11793},[83,641],[39,22008,22010],{"className":22009},[192],[39,22011,22013],{"className":22012},[196],[39,22014,22016],{"className":22015},[200],[39,22017,22019],{"className":22018,"style":6410},[204],[39,22020,22021,22024],{"style":208},[39,22022],{"className":22023,"style":213},[212],[39,22025,22027],{"className":22026},[217,218,219,220],[39,22028,2098],{"className":22029,"style":2114},[83,84,220]," 的定义已在 2.2 节给出，",[39,22032,22034,22051],{"className":22033,"translate":43},[42],[39,22035,22037],{"className":22036},[47],[49,22038,22039],{"xmlns":51},[53,22040,22041,22049],{},[56,22042,22043],{},[138,22044,22045,22047],{},[59,22046,11772],{"mathvariant":617},[133,22048,21884],{},[63,22050,21887],{"encoding":65},[39,22052,22054],{"className":22053,"ariaHidden":70},[69],[39,22055,22057,22060],{"className":22056},[74],[39,22058],{"className":22059,"style":21897},[78],[39,22061,22063,22066],{"className":22062},[83],[39,22064,11772],{"className":22065,"style":11793},[83,641],[39,22067,22069],{"className":22068},[192],[39,22070,22072],{"className":22071},[196],[39,22073,22075],{"className":22074},[200],[39,22076,22078],{"className":22077,"style":21897},[204],[39,22079,22080,22083],{"style":208},[39,22081],{"className":22082,"style":213},[212],[39,22084,22086],{"className":22085},[217,218,219,220],[39,22087,21884],{"className":22088},[465,220]," 定义为",[39,22091,22093],{"className":22092,"translate":43},[746],[39,22094,22096,22212],{"className":22095,"translate":43},[42],[39,22097,22099],{"className":22098},[47],[49,22100,22101],{"xmlns":51,"display":755},[53,22102,22103,22209],{},[56,22104,22105,22107,22113,22115,22117,22119,22121,22123,22125,22135,22145,22147,22149,22155,22157,22159,22161,22163,22165,22207],{},[133,22106,136],{"stretchy":135},[138,22108,22109,22111],{},[59,22110,11772],{"mathvariant":617},[133,22112,21884],{},[59,22114,4230],{},[133,22116,161],{"stretchy":135},[133,22118,136],{"stretchy":135},[59,22120,142],{},[133,22122,161],{"stretchy":135},[133,22124,793],{},[6556,22126,22127,22133],{},[56,22128,22129,22131],{},[59,22130,1687],{},[133,22132,1690],{},[59,22134,158],{},[6556,22136,22137,22139],{},[133,22138,3191],{},[138,22140,22141,22143],{},[59,22142,142],{},[133,22144,147],{"mathvariant":145,"lspace":146,"rspace":146},[59,22146,131],{},[133,22148,136],{"stretchy":135},[138,22150,22151,22153],{},[59,22152,142],{},[133,22154,147],{"mathvariant":145,"lspace":146,"rspace":146},[133,22156,150],{},[59,22158,142],{},[133,22160,155],{"separator":70},[59,22162,158],{},[133,22164,161],{"stretchy":135},[56,22166,22167,22169,22171,22173,22175,22177,22179,22181,22187,22189,22191,22193,22195,22197,22203,22205],{},[133,22168,537],{"fence":70},[59,22170,401],{},[133,22172,136],{"stretchy":135},[59,22174,142],{},[133,22176,155],{"separator":70},[59,22178,158],{},[133,22180,155],{"separator":70},[138,22182,22183,22185],{},[59,22184,142],{},[133,22186,147],{"mathvariant":145,"lspace":146,"rspace":146},[133,22188,161],{"stretchy":135},[133,22190,788],{},[59,22192,531],{},[59,22194,4230],{},[133,22196,136],{"stretchy":135},[138,22198,22199,22201],{},[59,22200,142],{},[133,22202,147],{"mathvariant":145,"lspace":146,"rspace":146},[133,22204,161],{"stretchy":135},[133,22206,4280],{"fence":70},[59,22208,2903],{"mathvariant":145},[63,22210,22211],{"encoding":65},"(\\mathcal{T}^*V)(s)\n=\\max_{a}\\sum_{s'}P(s'\\mid s,a)\n\\left[r(s,a,s')+\\gamma V(s')\\right].",[39,22213,22215,22278,22463],{"className":22214,"ariaHidden":70},[69],[39,22216,22218,22221,22224,22254,22257,22260,22263,22266,22269,22272,22275],{"className":22217},[74],[39,22219],{"className":22220,"style":245},[78],[39,22222,136],{"className":22223},[182],[39,22225,22227,22230],{"className":22226},[83],[39,22228,11772],{"className":22229,"style":11793},[83,641],[39,22231,22233],{"className":22232},[192],[39,22234,22236],{"className":22235},[196],[39,22237,22239],{"className":22238},[200],[39,22240,22243],{"className":22241,"style":22242},[204],"height:0.7387em;",[39,22244,22245,22248],{"style":1027},[39,22246],{"className":22247,"style":213},[212],[39,22249,22251],{"className":22250},[217,218,219,220],[39,22252,21884],{"className":22253},[465,220],[39,22255,4230],{"className":22256,"style":461},[83,84],[39,22258,161],{"className":22259},[263],[39,22261,136],{"className":22262},[182],[39,22264,142],{"className":22265},[83,84],[39,22267,161],{"className":22268},[263],[39,22270],{"className":22271,"style":231},[230],[39,22273,793],{"className":22274},[235],[39,22276],{"className":22277,"style":231},[230],[39,22279,22281,22285,22334,22337,22413,22416,22419,22422,22454,22457,22460],{"className":22280},[74],[39,22282],{"className":22283,"style":22284},[78],"height:2.344em;vertical-align:-1.294em;",[39,22286,22288],{"className":22287},[1744,3294],[39,22289,22291,22325],{"className":22290},[196,649],[39,22292,22294,22322],{"className":22293},[200],[39,22295,22297,22312],{"className":22296,"style":290},[204],[39,22298,22300,22303],{"style":22299},"top:-2.4em;margin-left:0em;",[39,22301],{"className":22302,"style":935},[212],[39,22304,22306],{"className":22305},[217,218,219,220],[39,22307,22309],{"className":22308},[83,220],[39,22310,158],{"className":22311},[83,84,220],[39,22313,22314,22317],{"style":21599},[39,22315],{"className":22316,"style":935},[212],[39,22318,22319],{},[39,22320,1687],{"className":22321},[1744],[39,22323,673],{"className":22324},[672],[39,22326,22328],{"className":22327},[200],[39,22329,22332],{"className":22330,"style":22331},[204],"height:0.7em;",[39,22333],{},[39,22335],{"className":22336,"style":256},[230],[39,22338,22340],{"className":22339},[1744,3294],[39,22341,22343,22405],{"className":22342},[196,649],[39,22344,22346,22402],{"className":22345},[200],[39,22347,22349,22392],{"className":22348,"style":6670},[204],[39,22350,22351,22354],{"style":7097},[39,22352],{"className":22353,"style":3311},[212],[39,22355,22357],{"className":22356},[217,218,219,220],[39,22358,22360],{"className":22359},[83,220],[39,22361,22363,22366],{"className":22362},[83,220],[39,22364,142],{"className":22365},[83,84,220],[39,22367,22369],{"className":22368},[192],[39,22370,22372],{"className":22371},[196],[39,22373,22375],{"className":22374},[200],[39,22376,22378],{"className":22377,"style":3997},[204],[39,22379,22380,22383],{"style":4000},[39,22381],{"className":22382,"style":4004},[212],[39,22384,22386],{"className":22385},[217,4008,4009,220],[39,22387,22389],{"className":22388},[83,220],[39,22390,147],{"className":22391},[83,220],[39,22393,22394,22397],{"style":3330},[39,22395],{"className":22396,"style":3311},[212],[39,22398,22399],{},[39,22400,3191],{"className":22401},[1744,3339,3340],[39,22403,673],{"className":22404},[672],[39,22406,22408],{"className":22407},[200],[39,22409,22411],{"className":22410,"style":7158},[204],[39,22412],{},[39,22414],{"className":22415,"style":256},[230],[39,22417,131],{"className":22418,"style":178},[83,84],[39,22420,136],{"className":22421},[182],[39,22423,22425,22428],{"className":22424},[83],[39,22426,142],{"className":22427},[83,84],[39,22429,22431],{"className":22430},[192],[39,22432,22434],{"className":22433},[196],[39,22435,22437],{"className":22436},[200],[39,22438,22440],{"className":22439,"style":1024},[204],[39,22441,22442,22445],{"style":1027},[39,22443],{"className":22444,"style":213},[212],[39,22446,22448],{"className":22447},[217,218,219,220],[39,22449,22451],{"className":22450},[83,220],[39,22452,147],{"className":22453},[83,220],[39,22455],{"className":22456,"style":231},[230],[39,22458,150],{"className":22459},[235],[39,22461],{"className":22462,"style":231},[230],[39,22464,22466,22470,22473,22476,22479,22482,22485,22488,22609,22612],{"className":22465},[74],[39,22467],{"className":22468,"style":22469},[78],"height:1.0519em;vertical-align:-0.25em;",[39,22471,142],{"className":22472},[83,84],[39,22474,155],{"className":22475},[252],[39,22477],{"className":22478,"style":256},[230],[39,22480,158],{"className":22481},[83,84],[39,22483,161],{"className":22484},[263],[39,22486],{"className":22487,"style":256},[230],[39,22489,22491,22494,22497,22500,22503,22506,22509,22512,22515,22518,22550,22553,22556,22559,22562,22565,22568,22571,22603,22606],{"className":22490},[2842],[39,22492,537],{"className":22493,"style":4592},[182,4591],[39,22495,401],{"className":22496,"style":438},[83,84],[39,22498,136],{"className":22499},[182],[39,22501,142],{"className":22502},[83,84],[39,22504,155],{"className":22505},[252],[39,22507],{"className":22508,"style":256},[230],[39,22510,158],{"className":22511},[83,84],[39,22513,155],{"className":22514},[252],[39,22516],{"className":22517,"style":256},[230],[39,22519,22521,22524],{"className":22520},[83],[39,22522,142],{"className":22523},[83,84],[39,22525,22527],{"className":22526},[192],[39,22528,22530],{"className":22529},[196],[39,22531,22533],{"className":22532},[200],[39,22534,22536],{"className":22535,"style":1024},[204],[39,22537,22538,22541],{"style":1027},[39,22539],{"className":22540,"style":213},[212],[39,22542,22544],{"className":22543},[217,218,219,220],[39,22545,22547],{"className":22546},[83,220],[39,22548,147],{"className":22549},[83,220],[39,22551,161],{"className":22552},[263],[39,22554],{"className":22555,"style":461},[230],[39,22557,788],{"className":22558},[465],[39,22560],{"className":22561,"style":461},[230],[39,22563,531],{"className":22564,"style":565},[83,84],[39,22566,4230],{"className":22567,"style":461},[83,84],[39,22569,136],{"className":22570},[182],[39,22572,22574,22577],{"className":22573},[83],[39,22575,142],{"className":22576},[83,84],[39,22578,22580],{"className":22579},[192],[39,22581,22583],{"className":22582},[196],[39,22584,22586],{"className":22585},[200],[39,22587,22589],{"className":22588,"style":1024},[204],[39,22590,22591,22594],{"style":1027},[39,22592],{"className":22593,"style":213},[212],[39,22595,22597],{"className":22596},[217,218,219,220],[39,22598,22600],{"className":22599},[83,220],[39,22601,147],{"className":22602},[83,220],[39,22604,161],{"className":22605},[263],[39,22607,4280],{"className":22608,"style":4592},[263,4591],[39,22610],{"className":22611,"style":256},[230],[39,22613,2903],{"className":22614},[83],[11,22616,22617,22618,22676],{},"以 ",[39,22619,22621,22638],{"className":22620,"translate":43},[42],[39,22622,22624],{"className":22623},[47],[49,22625,22626],{"xmlns":51},[53,22627,22628,22636],{},[56,22629,22630],{},[138,22631,22632,22634],{},[59,22633,11772],{"mathvariant":617},[59,22635,2098],{},[63,22637,11777],{"encoding":65},[39,22639,22641],{"className":22640,"ariaHidden":70},[69],[39,22642,22644,22647],{"className":22643},[74],[39,22645],{"className":22646,"style":79},[78],[39,22648,22650,22653],{"className":22649},[83],[39,22651,11772],{"className":22652,"style":11793},[83,641],[39,22654,22656],{"className":22655},[192],[39,22657,22659],{"className":22658},[196],[39,22660,22662],{"className":22661},[200],[39,22663,22665],{"className":22664,"style":6410},[204],[39,22666,22667,22670],{"style":208},[39,22668],{"className":22669,"style":213},[212],[39,22671,22673],{"className":22672},[217,218,219,220],[39,22674,2098],{"className":22675,"style":2114},[83,84,220]," 为例，",[39,22678,22680],{"className":22679,"translate":43},[746],[39,22681,22683,22853],{"className":22682,"translate":43},[42],[39,22684,22686],{"className":22685},[47],[49,22687,22688],{"xmlns":51,"display":755},[53,22689,22690,22850],{},[759,22691,22692,22808],{"rowspacing":761,"columnalign":762,"columnspacing":146},[764,22693,22694,22722],{},[767,22695,22696],{},[770,22697,22698],{"scriptlevel":541,"displaystyle":70},[56,22699,22700,22702,22704,22710,22712,22714,22716,22718,22720],{},[59,22701,150],{"mathvariant":145},[133,22703,136],{"stretchy":135},[138,22705,22706,22708],{},[59,22707,11772],{"mathvariant":617},[59,22709,2098],{},[59,22711,4230],{},[133,22713,161],{"stretchy":135},[133,22715,136],{"stretchy":135},[59,22717,142],{},[133,22719,161],{"stretchy":135},[59,22721,150],{"mathvariant":145},[767,22723,22724],{},[770,22725,22726],{"scriptlevel":541,"displaystyle":70},[56,22727,22728,22730,22732,22738,22740,22742,22744,22746,22748,22750,22760,22762,22764,22770,22772,22774,22776,22778,22780],{},[56,22729],{},[133,22731,1824],{},[6556,22733,22734,22736],{},[133,22735,3191],{},[59,22737,158],{},[59,22739,2098],{},[133,22741,136],{"stretchy":135},[59,22743,158],{},[133,22745,150],{},[59,22747,142],{},[133,22749,161],{"stretchy":135},[6556,22751,22752,22754],{},[133,22753,3191],{},[138,22755,22756,22758],{},[59,22757,142],{},[133,22759,147],{"mathvariant":145,"lspace":146,"rspace":146},[59,22761,131],{},[133,22763,136],{"stretchy":135},[138,22765,22766,22768],{},[59,22767,142],{},[133,22769,147],{"mathvariant":145,"lspace":146,"rspace":146},[133,22771,150],{},[59,22773,142],{},[133,22775,155],{"separator":70},[59,22777,158],{},[133,22779,161],{"stretchy":135},[56,22781,22782,22784,22792,22794,22796,22798,22800,22806],{},[133,22783,136],{"fence":70},[613,22785,22786,22788,22790],{},[59,22787,422],{},[59,22789,1687],{},[133,22791,1690],{},[133,22793,788],{},[59,22795,531],{},[59,22797,15389],{"mathvariant":145},[59,22799,4230],{},[613,22801,22802,22804],{},[59,22803,15389],{"mathvariant":145},[59,22805,1696],{"mathvariant":145},[133,22807,161],{"fence":70},[764,22809,22810,22816],{},[767,22811,22812],{},[770,22813,22814],{"scriptlevel":541,"displaystyle":70},[56,22815],{},[767,22817,22818],{},[770,22819,22820],{"scriptlevel":541,"displaystyle":70},[56,22821,22822,22824,22826,22834,22836,22838,22840,22842,22848],{},[56,22823],{},[133,22825,793],{},[613,22827,22828,22830,22832],{},[59,22829,422],{},[59,22831,1687],{},[133,22833,1690],{},[133,22835,788],{},[59,22837,531],{},[59,22839,15389],{"mathvariant":145},[59,22841,4230],{},[613,22843,22844,22846],{},[59,22845,15389],{"mathvariant":145},[59,22847,1696],{"mathvariant":145},[133,22849,155],{"separator":70},[63,22851,22852],{"encoding":65},"\\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}",[39,22854,22856],{"className":22855,"ariaHidden":70},[69],[39,22857,22859,22862],{"className":22858},[74],[39,22860],{"className":22861,"style":6954},[78],[39,22863,22865],{"className":22864},[83],[39,22866,22868,22960],{"className":22867},[759],[39,22869,22871],{"className":22870},[918],[39,22872,22874,22952],{"className":22873},[196,649],[39,22875,22877,22949],{"className":22876},[200],[39,22878,22880,22941],{"className":22879,"style":6973},[204],[39,22881,22882,22885],{"style":6976},[39,22883],{"className":22884,"style":3311},[212],[39,22886,22888,22891,22894,22923,22926,22929,22932,22935,22938],{"className":22887},[83],[39,22889,150],{"className":22890},[83],[39,22892,136],{"className":22893},[182],[39,22895,22897,22900],{"className":22896},[83],[39,22898,11772],{"className":22899,"style":11793},[83,641],[39,22901,22903],{"className":22902},[192],[39,22904,22906],{"className":22905},[196],[39,22907,22909],{"className":22908},[200],[39,22910,22912],{"className":22911,"style":4421},[204],[39,22913,22914,22917],{"style":1027},[39,22915],{"className":22916,"style":213},[212],[39,22918,22920],{"className":22919},[217,218,219,220],[39,22921,2098],{"className":22922,"style":2114},[83,84,220],[39,22924,4230],{"className":22925,"style":461},[83,84],[39,22927,161],{"className":22928},[263],[39,22930,136],{"className":22931},[182],[39,22933,142],{"className":22934},[83,84],[39,22936,161],{"className":22937},[263],[39,22939,150],{"className":22940},[83],[39,22942,22943,22946],{"style":7032},[39,22944],{"className":22945,"style":3311},[212],[39,22947],{"className":22948},[83],[39,22950,673],{"className":22951},[672],[39,22953,22955],{"className":22954},[200],[39,22956,22958],{"className":22957,"style":7048},[204],[39,22959],{},[39,22961,22963],{"className":22962},[1223],[39,22964,22966,23468],{"className":22965},[196,649],[39,22967,22969,23465],{"className":22968},[200],[39,22970,22972,23332],{"className":22971,"style":6973},[204],[39,22973,22974,22977],{"style":6976},[39,22975],{"className":22976,"style":3311},[212],[39,22978,22980,22983,22986,22989,22992,23039,23042,23045,23048,23051,23054,23057,23060,23063,23066,23069,23145,23148,23151,23154,23186,23189,23192,23195,23198,23201,23204,23207,23210,23213],{"className":22979},[83],[39,22981],{"className":22982},[83],[39,22984],{"className":22985,"style":231},[230],[39,22987,1824],{"className":22988},[235],[39,22990],{"className":22991,"style":231},[230],[39,22993,22995],{"className":22994},[1744,3294],[39,22996,22998,23031],{"className":22997},[196,649],[39,22999,23001,23028],{"className":23000},[200],[39,23002,23004,23018],{"className":23003,"style":6670},[204],[39,23005,23006,23009],{"style":6673},[39,23007],{"className":23008,"style":3311},[212],[39,23010,23012],{"className":23011},[217,218,219,220],[39,23013,23015],{"className":23014},[83,220],[39,23016,158],{"className":23017},[83,84,220],[39,23019,23020,23023],{"style":3330},[39,23021],{"className":23022,"style":3311},[212],[39,23024,23025],{},[39,23026,3191],{"className":23027},[1744,3339,3340],[39,23029,673],{"className":23030},[672],[39,23032,23034],{"className":23033},[200],[39,23035,23037],{"className":23036,"style":6705},[204],[39,23038],{},[39,23040],{"className":23041,"style":256},[230],[39,23043,2098],{"className":23044,"style":2114},[83,84],[39,23046,136],{"className":23047},[182],[39,23049,158],{"className":23050},[83,84],[39,23052],{"className":23053,"style":231},[230],[39,23055,150],{"className":23056},[235],[39,23058],{"className":23059,"style":231},[230],[39,23061,142],{"className":23062},[83,84],[39,23064,161],{"className":23065},[263],[39,23067],{"className":23068,"style":256},[230],[39,23070,23072],{"className":23071},[1744,3294],[39,23073,23075,23137],{"className":23074},[196,649],[39,23076,23078,23134],{"className":23077},[200],[39,23079,23081,23124],{"className":23080,"style":6670},[204],[39,23082,23083,23086],{"style":7097},[39,23084],{"className":23085,"style":3311},[212],[39,23087,23089],{"className":23088},[217,218,219,220],[39,23090,23092],{"className":23091},[83,220],[39,23093,23095,23098],{"className":23094},[83,220],[39,23096,142],{"className":23097},[83,84,220],[39,23099,23101],{"className":23100},[192],[39,23102,23104],{"className":23103},[196],[39,23105,23107],{"className":23106},[200],[39,23108,23110],{"className":23109,"style":3997},[204],[39,23111,23112,23115],{"style":4000},[39,23113],{"className":23114,"style":4004},[212],[39,23116,23118],{"className":23117},[217,4008,4009,220],[39,23119,23121],{"className":23120},[83,220],[39,23122,147],{"className":23123},[83,220],[39,23125,23126,23129],{"style":3330},[39,23127],{"className":23128,"style":3311},[212],[39,23130,23131],{},[39,23132,3191],{"className":23133},[1744,3339,3340],[39,23135,673],{"className":23136},[672],[39,23138,23140],{"className":23139},[200],[39,23141,23143],{"className":23142,"style":7158},[204],[39,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V\\|_\\infty\u003C\\infty",[39,23514,23516,23606],{"className":23515,"ariaHidden":70},[69],[39,23517,23519,23522,23525,23554,23557,23597,23600,23603],{"className":23518},[74],[39,23520],{"className":23521,"style":245},[78],[39,23523,15389],{"className":23524},[83],[39,23526,23528,23531],{"className":23527},[83],[39,23529,11772],{"className":23530,"style":11793},[83,641],[39,23532,23534],{"className":23533},[192],[39,23535,23537],{"className":23536},[196],[39,23538,23540],{"className":23539},[200],[39,23541,23543],{"className":23542,"style":6410},[204],[39,23544,23545,23548],{"style":208},[39,23546],{"className":23547,"style":213},[212],[39,23549,23551],{"className":23550},[217,218,219,220],[39,23552,2098],{"className":23553,"style":2114},[83,84,220],[39,23555,4230],{"className":23556,"style":461},[83,84],[39,23558,23560,23563],{"className":23559},[83],[39,23561,15389],{"className":23562},[83],[39,23564,23566],{"className":23565},[192],[39,23567,23569,23589],{"className":23568},[196,649],[39,23570,23572,23586],{"className":23571},[200],[39,23573,23575],{"className":23574,"style":1728},[204],[39,23576,23577,23580],{"style":966},[39,23578],{"className":23579,"style":213},[212],[39,23581,23583],{"className":23582},[217,218,219,220],[39,23584,1696],{"className":23585},[83,220],[39,23587,673],{"className":23588},[672],[39,23590,23592],{"className":23591},[200],[39,23593,23595],{"className":23594,"style":680},[204],[39,23596],{},[39,23598],{"className":23599,"style":231},[230],[39,23601,1693],{"className":23602},[235],[39,23604],{"className":23605,"style":231},[230],[39,23607,23609,23612],{"className":23608},[74],[39,23610],{"className":23611,"style":290},[78],[39,23613,1696],{"className":23614},[83],"。对 ",[39,23617,23619,23636],{"className":23618,"translate":43},[42],[39,23620,23622],{"className":23621},[47],[49,23623,23624],{"xmlns":51},[53,23625,23626,23634],{},[56,23627,23628],{},[138,23629,23630,23632],{},[59,23631,11772],{"mathvariant":617},[133,23633,21884],{},[63,23635,21887],{"encoding":65},[39,23637,23639],{"className":23638,"ariaHidden":70},[69],[39,23640,23642,23645],{"className":23641},[74],[39,23643],{"className":23644,"style":21897},[78],[39,23646,23648,23651],{"className":23647},[83],[39,23649,11772],{"className":23650,"style":11793},[83,641],[39,23652,23654],{"className":23653},[192],[39,23655,23657],{"className":23656},[196],[39,23658,23660],{"className":23659},[200],[39,23661,23663],{"className":23662,"style":21897},[204],[39,23664,23665,23668],{"style":208},[39,23666],{"className":23667,"style":213},[212],[39,23669,23671],{"className":23670},[217,218,219,220],[39,23672,21884],{"className":23673},[465,220]," 有同样的界。因此算子迭代不会离开这个空间。",[26,23676,23678],{"id":23677},"_32-单调性与常数平移","3.2 单调性与常数平移",[31,23680,23683,23819,24060,24116,24573],{"id":23681,"title":23682},"proposition-monotonicity","命题 3.1（单调性与常数平移）",[11,23684,37,23685,23765,23766,23818],{},[39,23686,23688,23717],{"className":23687,"translate":43},[42],[39,23689,23691],{"className":23690},[47],[49,23692,23693],{"xmlns":51},[53,23694,23695,23714],{},[56,23696,23697,23699,23701,23704,23706,23708,23710,23712],{},[59,23698,4230],{},[133,23700,155],{"separator":70},[59,23702,23703],{},"W",[133,23705,534],{},[59,23707,19772],{},[133,23709,136],{"stretchy":135},[59,23711,61],{},[133,23713,161],{"stretchy":135},[63,23715,23716],{"encoding":65},"V,W\\in B(S)",[39,23718,23720,23747],{"className":23719,"ariaHidden":70},[69],[39,23721,23723,23726,23729,23732,23735,23738,23741,23744],{"className":23722},[74],[39,23724],{"className":23725,"style":6452},[78],[39,23727,4230],{"className":23728,"style":461},[83,84],[39,23730,155],{"className":23731},[252],[39,23733],{"className":23734,"style":256},[230],[39,23736,23703],{"className":23737,"style":178},[83,84],[39,23739],{"className":23740,"style":231},[230],[39,23742,534],{"className":23743},[235],[39,23745],{"className":23746,"style":231},[230],[39,23748,23750,23753,23756,23759,23762],{"className":23749},[74],[39,23751],{"className":23752,"style":245},[78],[39,23754,19772],{"className":23755,"style":19812},[83,84],[39,23757,136],{"className":23758},[182],[39,23760,61],{"className":23761,"style":85},[83,84],[39,23763,161],{"className":23764},[263]," 且 ",[39,23767,23769,23787],{"className":23768,"translate":43},[42],[39,23770,23772],{"className":23771},[47],[49,23773,23774],{"xmlns":51},[53,23775,23776,23784],{},[56,23777,23778,23780,23782],{},[59,23779,4230],{},[133,23781,1824],{},[59,23783,23703],{},[63,23785,23786],{"encoding":65},"V\\le W",[39,23788,23790,23809],{"className":23789,"ariaHidden":70},[69],[39,23791,23793,23797,23800,23803,23806],{"className":23792},[74],[39,23794],{"className":23795,"style":23796},[78],"height:0.8193em;vertical-align:-0.136em;",[39,23798,4230],{"className":23799,"style":461},[83,84],[39,23801],{"className":23802,"style":231},[230],[39,23804,1824],{"className":23805},[235],[39,23807],{"className":23808,"style":231},[230],[39,23810,23812,23815],{"className":23811},[74],[39,23813],{"className":23814,"style":79},[78],[39,23816,23703],{"className":23817,"style":178},[83,84]," 逐点成立。则",[39,23820,23822],{"className":23821,"translate":43},[746],[39,23823,23825,23879],{"className":23824,"translate":43},[42],[39,23826,23828],{"className":23827},[47],[49,23829,23830],{"xmlns":51,"display":755},[53,23831,23832,23876],{},[56,23833,23834,23840,23842,23844,23850,23852,23854,23856,23862,23864,23866,23872,23874],{},[138,23835,23836,23838],{},[59,23837,11772],{"mathvariant":617},[59,23839,2098],{},[59,23841,4230],{},[133,23843,1824],{},[138,23845,23846,23848],{},[59,23847,11772],{"mathvariant":617},[59,23849,2098],{},[59,23851,23703],{},[133,23853,155],{"separator":70},[230,23855],{"width":10251},[138,23857,23858,23860],{},[59,23859,11772],{"mathvariant":617},[133,23861,21884],{},[59,23863,4230],{},[133,23865,1824],{},[138,23867,23868,23870],{},[59,23869,11772],{"mathvariant":617},[133,23871,21884],{},[59,23873,23703],{},[59,23875,2903],{"mathvariant":145},[63,23877,23878],{"encoding":65},"\\mathcal{T}^\\pi V\\le\\mathcal{T}^\\pi W,\n\\qquad\n\\mathcal{T}^* V\\le\\mathcal{T}^* W.",[39,23880,23882,23930,24019],{"className":23881,"ariaHidden":70},[69],[39,23883,23885,23889,23918,23921,23924,23927],{"className":23884},[74],[39,23886],{"className":23887,"style":23888},[78],"height:0.8504em;vertical-align:-0.136em;",[39,23890,23892,23895],{"className":23891},[83],[39,23893,11772],{"className":23894,"style":11793},[83,641],[39,23896,23898],{"className":23897},[192],[39,23899,23901],{"className":23900},[196],[39,23902,23904],{"className":23903},[200],[39,23905,23907],{"className":23906,"style":4421},[204],[39,23908,23909,23912],{"style":1027},[39,23910],{"className":23911,"style":213},[212],[39,23913,23915],{"className":23914},[217,218,219,220],[39,23916,2098],{"className":23917,"style":2114},[83,84,220],[39,23919,4230],{"className":23920,"style":461},[83,84],[39,23922],{"className":23923,"style":231},[230],[39,23925,1824],{"className":23926},[235],[39,23928],{"className":23929,"style":231},[230],[39,23931,23933,23937,23966,23969,23972,23975,23978,24007,24010,24013,24016],{"className":23932},[74],[39,23934],{"className":23935,"style":23936},[78],"height:0.9331em;vertical-align:-0.1944em;",[39,23938,23940,23943],{"className":23939},[83],[39,23941,11772],{"className":23942,"style":11793},[83,641],[39,23944,23946],{"className":23945},[192],[39,23947,23949],{"className":23948},[196],[39,23950,23952],{"className":23951},[200],[39,23953,23955],{"className":23954,"style":4421},[204],[39,23956,23957,23960],{"style":1027},[39,23958],{"className":23959,"style":213},[212],[39,23961,23963],{"className":23962},[217,218,219,220],[39,23964,2098],{"className":23965,"style":2114},[83,84,220],[39,23967,23703],{"className":23968,"style":178},[83,84],[39,23970,155],{"className":23971},[252],[39,23973],{"className":23974,"style":10787},[230],[39,23976],{"className":23977,"style":256},[230],[39,23979,23981,23984],{"className":23980},[83],[39,23982,11772],{"className":23983,"style":11793},[83,641],[39,23985,23987],{"className":23986},[192],[39,23988,23990],{"className":23989},[196],[39,23991,23993],{"className":23992},[200],[39,23994,23996],{"className":23995,"style":22242},[204],[39,23997,23998,24001],{"style":1027},[39,23999],{"className":24000,"style":213},[212],[39,24002,24004],{"className":24003},[217,218,219,220],[39,24005,21884],{"className":24006},[465,220],[39,24008,4230],{"className":24009,"style":461},[83,84],[39,24011],{"className":24012,"style":231},[230],[39,24014,1824],{"className":24015},[235],[39,24017],{"className":24018,"style":231},[230],[39,24020,24022,24025,24054,24057],{"className":24021},[74],[39,24023],{"className":24024,"style":22242},[78],[39,24026,24028,24031],{"className":24027},[83],[39,24029,11772],{"className":24030,"style":11793},[83,641],[39,24032,24034],{"className":24033},[192],[39,24035,24037],{"className":24036},[196],[39,24038,24040],{"className":24039},[200],[39,24041,24043],{"className":24042,"style":22242},[204],[39,24044,24045,24048],{"style":1027},[39,24046],{"className":24047,"style":213},[212],[39,24049,24051],{"className":24050},[217,218,219,220],[39,24052,21884],{"className":24053},[465,220],[39,24055,23703],{"className":24056,"style":178},[83,84],[39,24058,2903],{"className":24059},[83],[11,24061,24062,24063,742],{},"此外，对任意常数 ",[39,24064,24066,24085],{"className":24065,"translate":43},[42],[39,24067,24069],{"className":24068},[47],[49,24070,24071],{"xmlns":51},[53,24072,24073,24082],{},[56,24074,24075,24078,24080],{},[59,24076,24077],{},"c",[133,24079,534],{},[59,24081,422],{"mathvariant":421},[63,24083,24084],{"encoding":65},"c\\in\\mathbb{R}",[39,24086,24088,24107],{"className":24087,"ariaHidden":70},[69],[39,24089,24091,24095,24098,24101,24104],{"className":24090},[74],[39,24092],{"className":24093,"style":24094},[78],"height:0.5782em;vertical-align:-0.0391em;",[39,24096,24077],{"className":24097},[83,84],[39,24099],{"className":24100,"style":231},[230],[39,24102,534],{"className":24103},[235],[39,24105],{"className":24106,"style":231},[230],[39,24108,24110,24113],{"className":24109},[74],[39,24111],{"className":24112,"style":511},[78],[39,24114,422],{"className":24115},[83,515],[39,24117,24119],{"className":24118,"translate":43},[746],[39,24120,24122,24243],{"className":24121,"translate":43},[42],[39,24123,24125],{"className":24124},[47],[49,24126,24127],{"xmlns":51,"display":755},[53,24128,24129,24240],{},[759,24130,24131,24186],{"rowspacing":761,"columnalign":762,"columnspacing":146},[764,24132,24133,24158],{},[767,24134,24135],{},[770,24136,24137],{"scriptlevel":541,"displaystyle":70},[56,24138,24139,24145,24147,24149,24151,24153,24156],{},[138,24140,24141,24143],{},[59,24142,11772],{"mathvariant":617},[59,24144,2098],{},[133,24146,136],{"stretchy":135},[59,24148,4230],{},[133,24150,788],{},[59,24152,24077],{},[539,24154,546],{"mathvariant":24155},"bold",[133,24157,161],{"stretchy":135},[767,24159,24160],{},[770,24161,24162],{"scriptlevel":541,"displaystyle":70},[56,24163,24164,24166,24168,24174,24176,24178,24180,24182,24184],{},[56,24165],{},[133,24167,793],{},[138,24169,24170,24172],{},[59,24171,11772],{"mathvariant":617},[59,24173,2098],{},[59,24175,4230],{},[133,24177,788],{},[59,24179,531],{},[59,24181,24077],{},[539,24183,546],{"mathvariant":24155},[133,24185,155],{"separator":70},[764,24187,24188,24212],{},[767,24189,24190],{},[770,24191,24192],{"scriptlevel":541,"displaystyle":70},[56,24193,24194,24200,24202,24204,24206,24208,24210],{},[138,24195,24196,24198],{},[59,24197,11772],{"mathvariant":617},[133,24199,21884],{},[133,24201,136],{"stretchy":135},[59,24203,4230],{},[133,24205,788],{},[59,24207,24077],{},[539,24209,546],{"mathvariant":24155},[133,24211,161],{"stretchy":135},[767,24213,24214],{},[770,24215,24216],{"scriptlevel":541,"displaystyle":70},[56,24217,24218,24220,24222,24228,24230,24232,24234,24236,24238],{},[56,24219],{},[133,24221,793],{},[138,24223,24224,24226],{},[59,24225,11772],{"mathvariant":617},[133,24227,21884],{},[59,24229,4230],{},[133,24231,788],{},[59,24233,531],{},[59,24235,24077],{},[539,24237,546],{"mathvariant":24155},[133,24239,155],{"separator":70},[63,24241,24242],{"encoding":65},"\\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}",[39,24244,24246],{"className":24245,"ariaHidden":70},[69],[39,24247,24249,24252],{"className":24248},[74],[39,24250],{"className":24251,"style":4374},[78],[39,24253,24255],{"className":24254},[83],[39,24256,24258,24404],{"className":24257},[759],[39,24259,24261],{"className":24260},[918],[39,24262,24264,24396],{"className":24263},[196,649],[39,24265,24267,24393],{"className":24266},[200],[39,24268,24270,24332],{"className":24269,"style":4393},[204],[39,24271,24272,24275],{"style":4396},[39,24273],{"className":24274,"style":935},[212],[39,24276,24278,24307,24310,24313,24316,24319,24322,24325,24329],{"className":24277},[83],[39,24279,24281,24284],{"className":24280},[83],[39,24282,11772],{"className":24283,"style":11793},[83,641],[39,24285,24287],{"className":24286},[192],[39,24288,24290],{"className":24289},[196],[39,24291,24293],{"className":24292},[200],[39,24294,24296],{"className":24295,"style":4421},[204],[39,24297,24298,24301],{"style":1027},[39,24299],{"className":24300,"style":213},[212],[39,24302,24304],{"className":24303},[217,218,219,220],[39,24305,2098],{"className":24306,"style":2114},[83,84,220],[39,24308,136],{"className":24309},[182],[39,24311,4230],{"className":24312,"style":461},[83,84],[39,24314],{"className":24315,"style":461},[230],[39,24317,788],{"className":24318},[465],[39,24320],{"className":24321,"style":461},[230],[39,24323,24077],{"className":24324},[83,84],[39,24326,546],{"className":24327},[83,24328],"mathbf",[39,24330,161],{"className":24331},[263],[39,24333,24334,24337],{"style":4444},[39,24335],{"className":24336,"style":935},[212],[39,24338,24340,24369,24372,24375,24378,24381,24384,24387,24390],{"className":24339},[83],[39,24341,24343,24346],{"className":24342},[83],[39,24344,11772],{"className":24345,"style":11793},[83,641],[39,24347,24349],{"className":24348},[192],[39,24350,24352],{"className":24351},[196],[39,24353,24355],{"className":24354},[200],[39,24356,24358],{"className":24357,"style":22242},[204],[39,24359,24360,24363],{"style":1027},[39,24361],{"className":24362,"style":213},[212],[39,24364,24366],{"className":24365},[217,218,219,220],[39,24367,21884],{"className":24368},[465,220],[39,24370,136],{"className":24371},[182],[39,24373,4230],{"className":24374,"style":461},[83,84],[39,24376],{"className":24377,"style":461},[230],[39,24379,788],{"className":24380},[465],[39,24382],{"className":24383,"style":461},[230],[39,24385,24077],{"className":24386},[83,84],[39,24388,546],{"className":24389},[83,24328],[39,24391,161],{"className":24392},[263],[39,24394,673],{"className":24395},[672],[39,24397,24399],{"className":24398},[200],[39,24400,24402],{"className":24401,"style":4507},[204],[39,24403],{},[39,24405,24407],{"className":24406},[1223],[39,24408,24410,24565],{"className":24409},[196,649],[39,24411,24413,24562],{"className":24412},[200],[39,24414,24416,24489],{"className":24415,"style":4393},[204],[39,24417,24418,24421],{"style":4396},[39,24419],{"className":24420,"style":935},[212],[39,24422,24424,24427,24430,24433,24436,24465,24468,24471,24474,24477,24480,24483,24486],{"className":24423},[83],[39,24425],{"className":24426},[83],[39,24428],{"className":24429,"style":231},[230],[39,24431,793],{"className":24432},[235],[39,24434],{"className":24435,"style":231},[230],[39,24437,24439,24442],{"className":24438},[83],[39,24440,11772],{"className":24441,"style":11793},[83,641],[39,24443,24445],{"className":24444},[192],[39,24446,24448],{"className":24447},[196],[39,24449,24451],{"className":24450},[200],[39,24452,24454],{"className":24453,"style":4421},[204],[39,24455,24456,24459],{"style":1027},[39,24457],{"className":24458,"style":213},[212],[39,24460,24462],{"className":24461},[217,218,219,220],[39,24463,2098],{"className":24464,"style":2114},[83,84,220],[39,24466,4230],{"className":24467,"style":461},[83,84],[39,24469],{"className":24470,"style":461},[230],[39,24472,788],{"className":24473},[465],[39,24475],{"className":24476,"style":461},[230],[39,24478,531],{"className":24479,"style":565},[83,84],[39,24481,24077],{"className":24482},[83,84],[39,24484,546],{"className":24485},[83,24328],[39,24487,155],{"className":24488},[252],[39,24490,24491,24494],{"style":4444},[39,24492],{"className":24493,"style":935},[212],[39,24495,24497,24500,24503,24506,24509,24538,24541,24544,24547,24550,24553,24556,24559],{"className":24496},[83],[39,24498],{"className":24499},[83],[39,24501],{"className":24502,"style":231},[230],[39,24504,793],{"className":24505},[235],[39,24507],{"className":24508,"style":231},[230],[39,24510,24512,24515],{"className":24511},[83],[39,24513,11772],{"className":24514,"style":11793},[83,641],[39,24516,24518],{"className":24517},[192],[39,24519,24521],{"className":24520},[196],[39,24522,24524],{"className":24523},[200],[39,24525,24527],{"className":24526,"style":22242},[204],[39,24528,24529,24532],{"style":1027},[39,24530],{"className":24531,"style":213},[212],[39,24533,24535],{"className":24534},[217,218,219,220],[39,24536,21884],{"className":24537},[465,220],[39,24539,4230],{"className":24540,"style":461},[83,84],[39,24542],{"className":24543,"style":461},[230],[39,24545,788],{"className":24546},[465],[39,24548],{"className":24549,"style":461},[230],[39,24551,531],{"className":24552,"style":565},[83,84],[39,24554,24077],{"className":24555},[83,84],[39,24557,546],{"className":24558},[83,24328],[39,24560,155],{"className":24561},[252],[39,24563,673],{"className":24564},[672],[39,24566,24568],{"className":24567},[200],[39,24569,24571],{"className":24570,"style":4507},[204],[39,24572],{},[11,24574,24575,24576,24605,24606,24634],{},"其中 ",[39,24577,24579,24593],{"className":24578,"translate":43},[42],[39,24580,24582],{"className":24581},[47],[49,24583,24584],{"xmlns":51},[53,24585,24586,24590],{},[56,24587,24588],{},[539,24589,546],{"mathvariant":24155},[63,24591,24592],{"encoding":65},"\\mathbf{1}",[39,24594,24596],{"className":24595,"ariaHidden":70},[69],[39,24597,24599,24602],{"className":24598},[74],[39,24600],{"className":24601,"style":2473},[78],[39,24603,546],{"className":24604},[83,24328]," 表示恒等于 ",[39,24607,24609,24622],{"className":24608,"translate":43},[42],[39,24610,24612],{"className":24611},[47],[49,24613,24614],{"xmlns":51},[53,24615,24616,24620],{},[56,24617,24618],{},[539,24619,546],{},[63,24621,546],{"encoding":65},[39,24623,24625],{"className":24624,"ariaHidden":70},[69],[39,24626,24628,24631],{"className":24627},[74],[39,24629],{"className":24630,"style":2473},[78],[39,24632,546],{"className":24633},[83]," 的函数。",[11,24636,24637,24639,24640,24690,24691,742],{},[1489,24638,5595],{}," 若 ",[39,24641,24643,24660],{"className":24642,"translate":43},[42],[39,24644,24646],{"className":24645},[47],[49,24647,24648],{"xmlns":51},[53,24649,24650,24658],{},[56,24651,24652,24654,24656],{},[59,24653,4230],{},[133,24655,1824],{},[59,24657,23703],{},[63,24659,23786],{"encoding":65},[39,24661,24663,24681],{"className":24662,"ariaHidden":70},[69],[39,24664,24666,24669,24672,24675,24678],{"className":24665},[74],[39,24667],{"className":24668,"style":23796},[78],[39,24670,4230],{"className":24671,"style":461},[83,84],[39,24673],{"className":24674,"style":231},[230],[39,24676,1824],{"className":24677},[235],[39,24679],{"className":24680,"style":231},[230],[39,24682,24684,24687],{"className":24683},[74],[39,24685],{"className":24686,"style":79},[78],[39,24688,23703],{"className":24689,"style":178},[83,84],"，则对每个 ",[39,24692,24694,24723],{"className":24693,"translate":43},[42],[39,24695,24697],{"className":24696},[47],[49,24698,24699],{"xmlns":51},[53,24700,24701,24721],{},[56,24702,24703,24705,24707,24709,24711,24713,24719],{},[133,24704,136],{"stretchy":135},[59,24706,142],{},[133,24708,155],{"separator":70},[59,24710,158],{},[133,24712,155],{"separator":70},[138,24714,24715,24717],{},[59,24716,142],{},[133,24718,147],{"mathvariant":145,"lspace":146,"rspace":146},[133,24720,161],{"stretchy":135},[63,24722,2010],{"encoding":65},[39,24724,24726],{"className":24725,"ariaHidden":70},[69],[39,24727,24729,24732,24735,24738,24741,24744,24747,24750,24753,24785],{"className":24728},[74],[39,24730],{"className":24731,"style":174},[78],[39,24733,136],{"className":24734},[182],[39,24736,142],{"className":24737},[83,84],[39,24739,155],{"className":24740},[252],[39,24742],{"className":24743,"style":256},[230],[39,24745,158],{"className":24746},[83,84],[39,24748,155],{"className":24749},[252],[39,24751],{"className":24752,"style":256},[230],[39,24754,24756,24759],{"className":24755},[83],[39,24757,142],{"className":24758},[83,84],[39,24760,24762],{"className":24761},[192],[39,24763,24765],{"className":24764},[196],[39,24766,24768],{"className":24767},[200],[39,24769,24771],{"className":24770,"style":205},[204],[39,24772,24773,24776],{"style":208},[39,24774],{"className":24775,"style":213},[212],[39,24777,24779],{"className":24778},[217,218,219,220],[39,24780,24782],{"className":24781},[83,220],[39,24783,147],{"className":24784},[83,220],[39,24786,161],{"className":24787},[263],[39,24789,24791],{"className":24790,"translate":43},[746],[39,24792,24794,24882],{"className":24793,"translate":43},[42],[39,24795,24797],{"className":24796},[47],[49,24798,24799],{"xmlns":51,"display":755},[53,24800,24801,24879],{},[56,24802,24803,24805,24807,24809,24811,24813,24815,24821,24823,24825,24827,24829,24831,24837,24839,24841,24843,24845,24847,24849,24851,24853,24859,24861,24863,24865,24867,24869,24875,24877],{},[59,24804,401],{},[133,24806,136],{"stretchy":135},[59,24808,142],{},[133,24810,155],{"separator":70},[59,24812,158],{},[133,24814,155],{"separator":70},[138,24816,24817,24819],{},[59,24818,142],{},[133,24820,147],{"mathvariant":145,"lspace":146,"rspace":146},[133,24822,161],{"stretchy":135},[133,24824,788],{},[59,24826,531],{},[59,24828,4230],{},[133,24830,136],{"stretchy":135},[138,24832,24833,24835],{},[59,24834,142],{},[133,24836,147],{"mathvariant":145,"lspace":146,"rspace":146},[133,24838,161],{"stretchy":135},[133,24840,1824],{},[59,24842,401],{},[133,24844,136],{"stretchy":135},[59,24846,142],{},[133,24848,155],{"separator":70},[59,24850,158],{},[133,24852,155],{"separator":70},[138,24854,24855,24857],{},[59,24856,142],{},[133,24858,147],{"mathvariant":145,"lspace":146,"rspace":146},[133,24860,161],{"stretchy":135},[133,24862,788],{},[59,24864,531],{},[59,24866,23703],{},[133,24868,136],{"stretchy":135},[138,24870,24871,24873],{},[59,24872,142],{},[133,24874,147],{"mathvariant":145,"lspace":146,"rspace":146},[133,24876,161],{"stretchy":135},[59,24878,2903],{"mathvariant":145},[63,24880,24881],{"encoding":65},"r(s,a,s')+\\gamma V(s')\\le r(s,a,s')+\\gamma W(s').",[39,24883,24885,24959,25018,25092],{"className":24884,"ariaHidden":70},[69],[39,24886,24888,24891,24894,24897,24900,24903,24906,24909,24912,24915,24947,24950,24953,24956],{"className":24887},[74],[39,24889],{"className":24890,"style":22469},[78],[39,24892,401],{"className":24893,"style":438},[83,84],[39,24895,136],{"className":24896},[182],[39,24898,142],{"className":24899},[83,84],[39,24901,155],{"className":24902},[252],[39,24904],{"className":24905,"style":256},[230],[39,24907,158],{"className":24908},[83,84],[39,24910,155],{"className":24911},[252],[39,24913],{"className":24914,"style":256},[230],[39,24916,24918,24921],{"className":24917},[83],[39,24919,142],{"className":24920},[83,84],[39,24922,24924],{"className":24923},[192],[39,24925,24927],{"className":24926},[196],[39,24928,24930],{"className":24929},[200],[39,24931,24933],{"className":24932,"style":1024},[204],[39,24934,24935,24938],{"style":1027},[39,24936],{"className":24937,"style":213},[212],[39,24939,24941],{"className":24940},[217,218,219,220],[39,24942,24944],{"className":24943},[83,220],[39,24945,147],{"className":24946},[83,220],[39,24948,161],{"className":24949},[263],[39,24951],{"className":24952,"style":461},[230],[39,24954,788],{"className":24955},[465],[39,24957],{"className":24958,"style":461},[230],[39,24960,24962,24965,24968,24971,24974,25006,25009,25012,25015],{"className":24961},[74],[39,24963],{"className":24964,"style":22469},[78],[39,24966,531],{"className":24967,"style":565},[83,84],[39,24969,4230],{"className":24970,"style":461},[83,84],[39,24972,136],{"className":24973},[182],[39,24975,24977,24980],{"className":24976},[83],[39,24978,142],{"className":24979},[83,84],[39,24981,24983],{"className":24982},[192],[39,24984,24986],{"className":24985},[196],[39,24987,24989],{"className":24988},[200],[39,24990,24992],{"className":24991,"style":1024},[204],[39,24993,24994,24997],{"style":1027},[39,24995],{"className":24996,"style":213},[212],[39,24998,25000],{"className":24999},[217,218,219,220],[39,25001,25003],{"className":25002},[83,220],[39,25004,147],{"className":25005},[83,220],[39,25007,161],{"className":25008},[263],[39,25010],{"className":25011,"style":231},[230],[39,25013,1824],{"className":25014},[235],[39,25016],{"className":25017,"style":231},[230],[39,25019,25021,25024,25027,25030,25033,25036,25039,25042,25045,25048,25080,25083,25086,25089],{"className":25020},[74],[39,25022],{"className":25023,"style":22469},[78],[39,25025,401],{"className":25026,"style":438},[83,84],[39,25028,136],{"className":25029},[182],[39,25031,142],{"className":25032},[83,84],[39,25034,155],{"className":25035},[252],[39,25037],{"className":25038,"style":256},[230],[39,25040,158],{"className":25041},[83,84],[39,25043,155],{"className":25044},[252],[39,25046],{"className":25047,"style":256},[230],[39,25049,25051,25054],{"className":25050},[83],[39,25052,142],{"className":25053},[83,84],[39,25055,25057],{"className":25056},[192],[39,25058,25060],{"className":25059},[196],[39,25061,25063],{"className":25062},[200],[39,25064,25066],{"className":25065,"style":1024},[204],[39,25067,25068,25071],{"style":1027},[39,25069],{"className":25070,"style":213},[212],[39,25072,25074],{"className":25073},[217,218,219,220],[39,25075,25077],{"className":25076},[83,220],[39,25078,147],{"className":25079},[83,220],[39,25081,161],{"className":25082},[263],[39,25084],{"className":25085,"style":461},[230],[39,25087,788],{"className":25088},[465],[39,25090],{"className":25091,"style":461},[230],[39,25093,25095,25098,25101,25104,25107,25139,25142],{"className":25094},[74],[39,25096],{"className":25097,"style":22469},[78],[39,25099,531],{"className":25100,"style":565},[83,84],[39,25102,23703],{"className":25103,"style":178},[83,84],[39,25105,136],{"className":25106},[182],[39,25108,25110,25113],{"className":25109},[83],[39,25111,142],{"className":25112},[83,84],[39,25114,25116],{"className":25115},[192],[39,25117,25119],{"className":25118},[196],[39,25120,25122],{"className":25121},[200],[39,25123,25125],{"className":25124,"style":1024},[204],[39,25126,25127,25130],{"style":1027},[39,25128],{"className":25129,"style":213},[212],[39,25131,25133],{"className":25132},[217,218,219,220],[39,25134,25136],{"className":25135},[83,220],[39,25137,147],{"className":25138},[83,220],[39,25140,161],{"className":25141},[263],[39,25143,2903],{"className":25144},[83],[11,25146,20571,25147,23765,25236,25370,25371,23615,25492,25550,25551,1607],{},[39,25148,25150,25179],{"className":25149,"translate":43},[42],[39,25151,25153],{"className":25152},[47],[49,25154,25155],{"xmlns":51},[53,25156,25157,25176],{},[56,25158,25159,25161,25163,25165,25167,25169,25171,25174],{},[59,25160,2098],{},[133,25162,136],{"stretchy":135},[59,25164,158],{},[133,25166,150],{},[59,25168,142],{},[133,25170,161],{"stretchy":135},[133,25172,25173],{},"≥",[539,25175,541],{},[63,25177,25178],{"encoding":65},"\\pi(a\\mid s)\\ge0",[39,25180,25182,25206,25227],{"className":25181,"ariaHidden":70},[69],[39,25183,25185,25188,25191,25194,25197,25200,25203],{"className":25184},[74],[39,25186],{"className":25187,"style":245},[78],[39,25189,2098],{"className":25190,"style":2114},[83,84],[39,25192,136],{"className":25193},[182],[39,25195,158],{"className":25196},[83,84],[39,25198],{"className":25199,"style":231},[230],[39,25201,150],{"className":25202},[235],[39,25204],{"className":25205,"style":231},[230],[39,25207,25209,25212,25215,25218,25221,25224],{"className":25208},[74],[39,25210],{"className":25211,"style":245},[78],[39,25213,142],{"className":25214},[83,84],[39,25216,161],{"className":25217},[263],[39,25219],{"className":25220,"style":231},[230],[39,25222,25173],{"className":25223},[235],[39,25225],{"className":25226,"style":231},[230],[39,25228,25230,25233],{"className":25229},[74],[39,25231],{"className":25232,"style":2473},[78],[39,25234,541],{"className":25235},[83],[39,25237,25239,25275],{"className":25238,"translate":43},[42],[39,25240,25242],{"className":25241},[47],[49,25243,25244],{"xmlns":51},[53,25245,25246,25272],{},[56,25247,25248,25250,25252,25258,25260,25262,25264,25266,25268,25270],{},[59,25249,131],{},[133,25251,136],{"stretchy":135},[138,25253,25254,25256],{},[59,25255,142],{},[133,25257,147],{"mathvariant":145,"lspace":146,"rspace":146},[133,25259,150],{},[59,25261,142],{},[133,25263,155],{"separator":70},[59,25265,158],{},[133,25267,161],{"stretchy":135},[133,25269,25173],{},[539,25271,541],{},[63,25273,25274],{"encoding":65},"P(s'\\mid s,a)\\ge0",[39,25276,25278,25331,25361],{"className":25277,"ariaHidden":70},[69],[39,25279,25281,25284,25287,25290,25322,25325,25328],{"className":25280},[74],[39,25282],{"className":25283,"style":174},[78],[39,25285,131],{"className":25286,"style":178},[83,84],[39,25288,136],{"className":25289},[182],[39,25291,25293,25296],{"className":25292},[83],[39,25294,142],{"className":25295},[83,84],[39,25297,25299],{"className":25298},[192],[39,25300,25302],{"className":25301},[196],[39,25303,25305],{"className":25304},[200],[39,25306,25308],{"className":25307,"style":205},[204],[39,25309,25310,25313],{"style":208},[39,25311],{"className":25312,"style":213},[212],[39,25314,25316],{"className":25315},[217,218,219,220],[39,25317,25319],{"className":25318},[83,220],[39,25320,147],{"className":25321},[83,220],[39,25323],{"className":25324,"style":231},[230],[39,25326,150],{"className":25327},[235],[39,25329],{"className":25330,"style":231},[230],[39,25332,25334,25337,25340,25343,25346,25349,25352,25355,25358],{"className":25333},[74],[39,25335],{"className":25336,"style":245},[78],[39,25338,142],{"className":25339},[83,84],[39,25341,155],{"className":25342},[252],[39,25344],{"className":25345,"style":256},[230],[39,25347,158],{"className":25348},[83,84],[39,25350,161],{"className":25351},[263],[39,25353],{"className":25354,"style":231},[230],[39,25356,25173],{"className":25357},[235],[39,25359],{"className":25360,"style":231},[230],[39,25362,25364,25367],{"className":25363},[74],[39,25365],{"className":25366,"style":2473},[78],[39,25368,541],{"className":25369},[83],"，加权求和保持不等号，故 ",[39,25372,25374,25404],{"className":25373,"translate":43},[42],[39,25375,25377],{"className":25376},[47],[49,25378,25379],{"xmlns":51},[53,25380,25381,25401],{},[56,25382,25383,25389,25391,25393,25399],{},[138,25384,25385,25387],{},[59,25386,11772],{"mathvariant":617},[59,25388,2098],{},[59,25390,4230],{},[133,25392,1824],{},[138,25394,25395,25397],{},[59,25396,11772],{"mathvariant":617},[59,25398,2098],{},[59,25400,23703],{},[63,25402,25403],{"encoding":65},"\\mathcal{T}^\\pi V\\le\\mathcal{T}^\\pi W",[39,25405,25407,25454],{"className":25406,"ariaHidden":70},[69],[39,25408,25410,25413,25442,25445,25448,25451],{"className":25409},[74],[39,25411],{"className":25412,"style":23796},[78],[39,25414,25416,25419],{"className":25415},[83],[39,25417,11772],{"className":25418,"style":11793},[83,641],[39,25420,25422],{"className":25421},[192],[39,25423,25425],{"className":25424},[196],[39,25426,25428],{"className":25427},[200],[39,25429,25431],{"className":25430,"style":6410},[204],[39,25432,25433,25436],{"style":208},[39,25434],{"className":25435,"style":213},[212],[39,25437,25439],{"className":25438},[217,218,219,220],[39,25440,2098],{"className":25441,"style":2114},[83,84,220],[39,25443,4230],{"className":25444,"style":461},[83,84],[39,25446],{"className":25447,"style":231},[230],[39,25449,1824],{"className":25450},[235],[39,25452],{"className":25453,"style":231},[230],[39,25455,25457,25460,25489],{"className":25456},[74],[39,25458],{"className":25459,"style":79},[78],[39,25461,25463,25466],{"className":25462},[83],[39,25464,11772],{"className":25465,"style":11793},[83,641],[39,25467,25469],{"className":25468},[192],[39,25470,25472],{"className":25471},[196],[39,25473,25475],{"className":25474},[200],[39,25476,25478],{"className":25477,"style":6410},[204],[39,25479,25480,25483],{"style":208},[39,25481],{"className":25482,"style":213},[212],[39,25484,25486],{"className":25485},[217,218,219,220],[39,25487,2098],{"className":25488,"style":2114},[83,84,220],[39,25490,23703],{"className":25491,"style":178},[83,84],[39,25493,25495,25512],{"className":25494,"translate":43},[42],[39,25496,25498],{"className":25497},[47],[49,25499,25500],{"xmlns":51},[53,25501,25502,25510],{},[56,25503,25504],{},[138,25505,25506,25508],{},[59,25507,11772],{"mathvariant":617},[133,25509,21884],{},[63,25511,21887],{"encoding":65},[39,25513,25515],{"className":25514,"ariaHidden":70},[69],[39,25516,25518,25521],{"className":25517},[74],[39,25519],{"className":25520,"style":21897},[78],[39,25522,25524,25527],{"className":25523},[83],[39,25525,11772],{"className":25526,"style":11793},[83,641],[39,25528,25530],{"className":25529},[192],[39,25531,25533],{"className":25532},[196],[39,25534,25536],{"className":25535},[200],[39,25537,25539],{"className":25538,"style":21897},[204],[39,25540,25541,25544],{"style":208},[39,25542],{"className":25543,"style":213},[212],[39,25545,25547],{"className":25546},[217,218,219,220],[39,25548,21884],{"className":25549},[465,220],"，逐点取最大值也保持逐点序，故 ",[39,25552,25554,25584],{"className":25553,"translate":43},[42],[39,25555,25557],{"className":25556},[47],[49,25558,25559],{"xmlns":51},[53,25560,25561,25581],{},[56,25562,25563,25569,25571,25573,25579],{},[138,25564,25565,25567],{},[59,25566,11772],{"mathvariant":617},[133,25568,21884],{},[59,25570,4230],{},[133,25572,1824],{},[138,25574,25575,25577],{},[59,25576,11772],{"mathvariant":617},[133,25578,21884],{},[59,25580,23703],{},[63,25582,25583],{"encoding":65},"\\mathcal{T}^*V\\le\\mathcal{T}^*W",[39,25585,25587,25635],{"className":25586,"ariaHidden":70},[69],[39,25588,25590,25594,25623,25626,25629,25632],{"className":25589},[74],[39,25591],{"className":25592,"style":25593},[78],"height:0.8247em;vertical-align:-0.136em;",[39,25595,25597,25600],{"className":25596},[83],[39,25598,11772],{"className":25599,"style":11793},[83,641],[39,25601,25603],{"className":25602},[192],[39,25604,25606],{"className":25605},[196],[39,25607,25609],{"className":25608},[200],[39,25610,25612],{"className":25611,"style":21897},[204],[39,25613,25614,25617],{"style":208},[39,25615],{"className":25616,"style":213},[212],[39,25618,25620],{"className":25619},[217,218,219,220],[39,25621,21884],{"className":25622},[465,220],[39,25624,4230],{"className":25625,"style":461},[83,84],[39,25627],{"className":25628,"style":231},[230],[39,25630,1824],{"className":25631},[235],[39,25633],{"className":25634,"style":231},[230],[39,25636,25638,25641,25670],{"className":25637},[74],[39,25639],{"className":25640,"style":21897},[78],[39,25642,25644,25647],{"className":25643},[83],[39,25645,11772],{"className":25646,"style":11793},[83,641],[39,25648,25650],{"className":25649},[192],[39,25651,25653],{"className":25652},[196],[39,25654,25656],{"className":25655},[200],[39,25657,25659],{"className":25658,"style":21897},[204],[39,25660,25661,25664],{"style":208},[39,25662],{"className":25663,"style":213},[212],[39,25665,25667],{"className":25666},[217,218,219,220],[39,25668,21884],{"className":25669},[465,220],[39,25671,23703],{"className":25672,"style":178},[83,84],[11,25674,25675],{},"常数平移式由直接计算得到：",[39,25677,25679],{"className":25678,"translate":43},[746],[39,25680,25682,25861],{"className":25681,"translate":43},[42],[39,25683,25685],{"className":25684},[47],[49,25686,25687],{"xmlns":51,"display":755},[53,25688,25689,25858],{},[759,25690,25691,25823],{"rowspacing":761,"columnalign":762,"columnspacing":146},[764,25692,25693,25717],{},[767,25694,25695],{},[770,25696,25697],{"scriptlevel":541,"displaystyle":70},[56,25698,25699,25705,25707,25709,25711,25713,25715],{},[138,25700,25701,25703],{},[59,25702,11772],{"mathvariant":617},[59,25704,2098],{},[133,25706,136],{"stretchy":135},[59,25708,4230],{},[133,25710,788],{},[59,25712,24077],{},[539,25714,546],{"mathvariant":24155},[133,25716,161],{"stretchy":135},[767,25718,25719],{},[770,25720,25721],{"scriptlevel":541,"displaystyle":70},[56,25722,25723,25725,25727,25733,25735,25737,25739,25741,25743,25745,25755,25757,25759,25765,25767,25769,25771,25773,25775],{},[56,25724],{},[133,25726,793],{},[6556,25728,25729,25731],{},[133,25730,3191],{},[59,25732,158],{},[59,25734,2098],{},[133,25736,136],{"stretchy":135},[59,25738,158],{},[133,25740,150],{},[59,25742,142],{},[133,25744,161],{"stretchy":135},[6556,25746,25747,25749],{},[133,25748,3191],{},[138,25750,25751,25753],{},[59,25752,142],{},[133,25754,147],{"mathvariant":145,"lspace":146,"rspace":146},[59,25756,131],{},[133,25758,136],{"stretchy":135},[138,25760,25761,25763],{},[59,25762,142],{},[133,25764,147],{"mathvariant":145,"lspace":146,"rspace":146},[133,25766,150],{},[59,25768,142],{},[133,25770,155],{"separator":70},[59,25772,158],{},[133,25774,161],{"stretchy":135},[56,25776,25777,25779,25781,25783,25785,25787,25789,25791,25797,25799,25801,25803,25805,25807,25813,25815,25817,25819,25821],{},[133,25778,537],{"fence":70},[59,25780,401],{},[133,25782,136],{"stretchy":135},[59,25784,142],{},[133,25786,155],{"separator":70},[59,25788,158],{},[133,25790,155],{"separator":70},[138,25792,25793,25795],{},[59,25794,142],{},[133,25796,147],{"mathvariant":145,"lspace":146,"rspace":146},[133,25798,161],{"stretchy":135},[133,25800,788],{},[59,25802,531],{},[59,25804,4230],{},[133,25806,136],{"stretchy":135},[138,25808,25809,25811],{},[59,25810,142],{},[133,25812,147],{"mathvariant":145,"lspace":146,"rspace":146},[133,25814,161],{"stretchy":135},[133,25816,788],{},[59,25818,531],{},[59,25820,24077],{},[133,25822,4280],{"fence":70},[764,25824,25825,25831],{},[767,25826,25827],{},[770,25828,25829],{"scriptlevel":541,"displaystyle":70},[56,25830],{},[767,25832,25833],{},[770,25834,25835],{"scriptlevel":541,"displaystyle":70},[56,25836,25837,25839,25841,25847,25849,25851,25853,25855],{},[56,25838],{},[133,25840,793],{},[138,25842,25843,25845],{},[59,25844,11772],{"mathvariant":617},[59,25846,2098],{},[59,25848,4230],{},[133,25850,788],{},[59,25852,531],{},[59,25854,24077],{},[539,25856,25857],{"mathvariant":24155},"1.",[63,25859,25860],{"encoding":65},"\\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}",[39,25862,25864],{"className":25863,"ariaHidden":70},[69],[39,25865,25867,25870],{"className":25866},[74],[39,25868],{"className":25869,"style":6954},[78],[39,25871,25873],{"className":25872},[83],[39,25874,25876,25968],{"className":25875},[759],[39,25877,25879],{"className":25878},[918],[39,25880,25882,25960],{"className":25881},[196,649],[39,25883,25885,25957],{"className":25884},[200],[39,25886,25888,25949],{"className":25887,"style":6973},[204],[39,25889,25890,25893],{"style":6976},[39,25891],{"className":25892,"style":3311},[212],[39,25894,25896,25925,25928,25931,25934,25937,25940,25943,25946],{"className":25895},[83],[39,25897,25899,25902],{"className":25898},[83],[39,25900,11772],{"className":25901,"style":11793},[83,641],[39,25903,25905],{"className":25904},[192],[39,25906,25908],{"className":25907},[196],[39,25909,25911],{"className":25910},[200],[39,25912,25914],{"className":25913,"style":4421},[204],[39,25915,25916,25919],{"style":1027},[39,25917],{"className":25918,"style":213},[212],[39,25920,25922],{"className":25921},[217,218,219,220],[39,25923,2098],{"className":25924,"style":2114},[83,84,220],[39,25926,136],{"className":25927},[182],[39,25929,4230],{"className":25930,"style":461},[83,84],[39,25932],{"className":25933,"style":461},[230],[39,25935,788],{"className":25936},[465],[39,25938],{"className":25939,"style":461},[230],[39,25941,24077],{"className":25942},[83,84],[39,25944,546],{"className":25945},[83,24328],[39,25947,161],{"className":25948},[263],[39,25950,25951,25954],{"style":7032},[39,25952],{"className":25953,"style":3311},[212],[39,25955],{"className":25956},[83],[39,25958,673],{"className":25959},[672],[39,25961,25963],{"className":25962},[200],[39,25964,25966],{"className":25965,"style":7048},[204],[39,25967],{},[39,25969,25971],{"className":25970},[1223],[39,25972,25974,26433],{"className":25973},[196,649],[39,25975,25977,26430],{"className":25976},[200],[39,25978,25980,26357],{"className":25979,"style":6973},[204],[39,25981,25982,25985],{"style":6976},[39,25983],{"className":25984,"style":3311},[212],[39,25986,25988,25991,25994,25997,26000,26047,26050,26053,26056,26059,26062,26065,26068,26071,26074,26077,26153,26156,26159,26162,26194,26197,26200,26203,26206,26209,26212,26215,26218,26221],{"className":25987},[83],[39,25989],{"className":25990},[83],[39,25992],{"className":25993,"style":231},[230],[39,25995,793],{"className":25996},[235],[39,25998],{"className":25999,"style":231},[230],[39,26001,26003],{"className":26002},[1744,3294],[39,26004,26006,26039],{"className":26005},[196,649],[39,26007,26009,26036],{"className":26008},[200],[39,26010,26012,26026],{"className":26011,"style":6670},[204],[39,26013,26014,26017],{"style":6673},[39,26015],{"className":26016,"style":3311},[212],[39,26018,26020],{"className":26019},[217,218,219,220],[39,26021,26023],{"className":26022},[83,220],[39,26024,158],{"className":26025},[83,84,220],[39,26027,26028,26031],{"style":3330},[39,26029],{"className":26030,"style":3311},[212],[39,26032,26033],{},[39,26034,3191],{"className":26035},[1744,3339,3340],[39,26037,673],{"className":26038},[672],[39,26040,26042],{"className":26041},[200],[39,26043,26045],{"className":26044,"style":6705},[204],[39,26046],{},[39,26048],{"className":26049,"style":256},[230],[39,26051,2098],{"className":26052,"style":2114},[83,84],[39,26054,136],{"className":26055},[182],[39,26057,158],{"className":26058},[83,84],[39,26060],{"className":26061,"style":231},[230],[39,26063,150],{"className":26064},[235],[39,26066],{"className":26067,"style":231},[230],[39,26069,142],{"className":26070},[83,84],[39,26072,161],{"className":26073},[263],[39,26075],{"className":26076,"style":256},[230],[39,26078,26080],{"className":26079},[1744,3294],[39,26081,26083,26145],{"className":26082},[196,649],[39,26084,26086,26142],{"className":26085},[200],[39,26087,26089,26132],{"className":26088,"style":6670},[204],[39,26090,26091,26094],{"style":7097},[39,26092],{"className":26093,"style":3311},[212],[39,26095,26097],{"className":26096},[217,218,219,220],[39,26098,26100],{"className":26099},[83,220],[39,26101,26103,26106],{"className":26102},[83,220],[39,26104,142],{"className":26105},[83,84,220],[39,26107,26109],{"className":26108},[192],[39,26110,26112],{"className":26111},[196],[39,26113,26115],{"className":26114},[200],[39,26116,26118],{"className":26117,"style":3997},[204],[39,26119,26120,26123],{"style":4000},[39,26121],{"className":26122,"style":4004},[212],[39,26124,26126],{"className":26125},[217,4008,4009,220],[39,26127,26129],{"className":26128},[83,220],[39,26130,147],{"className":26131},[83,220],[39,26133,26134,26137],{"style":3330},[39,26135],{"className":26136,"style":3311},[212],[39,26138,26139],{},[39,26140,3191],{"className":26141},[1744,3339,3340],[39,26143,673],{"className":26144},[672],[39,26146,26148],{"className":26147},[200],[39,26149,26151],{"className":26150,"style":7158},[204],[39,26152],{},[39,26154],{"className":26155,"style":256},[230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",[39,26445,26447,26464],{"className":26446,"translate":43},[42],[39,26448,26450],{"className":26449},[47],[49,26451,26452],{"xmlns":51},[53,26453,26454,26462],{},[56,26455,26456],{},[138,26457,26458,26460],{},[59,26459,11772],{"mathvariant":617},[133,26461,21884],{},[63,26463,21887],{"encoding":65},[39,26465,26467],{"className":26466,"ariaHidden":70},[69],[39,26468,26470,26473],{"className":26469},[74],[39,26471],{"className":26472,"style":21897},[78],[39,26474,26476,26479],{"className":26475},[83],[39,26477,11772],{"className":26478,"style":11793},[83,641],[39,26480,26482],{"className":26481},[192],[39,26483,26485],{"className":26484},[196],[39,26486,26488],{"className":26487},[200],[39,26489,26491],{"className":26490,"style":21897},[204],[39,26492,26493,26496],{"style":208},[39,26494],{"className":26495,"style":213},[212],[39,26497,26499],{"className":26498},[217,218,219,220],[39,26500,21884],{"className":26501},[465,220],"，常数 ",[39,26504,26506,26522],{"className":26505,"translate":43},[42],[39,26507,26509],{"className":26508},[47],[49,26510,26511],{"xmlns":51},[53,26512,26513,26519],{},[56,26514,26515,26517],{},[59,26516,531],{},[59,26518,24077],{},[63,26520,26521],{"encoding":65},"\\gamma c",[39,26523,26525],{"className":26524,"ariaHidden":70},[69],[39,26526,26528,26531,26534],{"className":26527},[74],[39,26529],{"className":26530,"style":2654},[78],[39,26532,531],{"className":26533,"style":565},[83,84],[39,26535,24077],{"className":26536},[83,84]," 与动作 ",[39,26539,26541,26554],{"className":26540,"translate":43},[42],[39,26542,26544],{"className":26543},[47],[49,26545,26546],{"xmlns":51},[53,26547,26548,26552],{},[56,26549,26550],{},[59,26551,158],{},[63,26553,158],{"encoding":65},[39,26555,26557],{"className":26556,"ariaHidden":70},[69],[39,26558,26560,26563],{"className":26559},[74],[39,26561],{"className":26562,"style":290},[78],[39,26564,158],{"className":26565},[83,84]," 无关，可以从最大值中提出。证毕。",[11,26568,26569,26570,26599,26600,26628],{},"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 ",[39,26571,26573,26587],{"className":26572,"translate":43},[42],[39,26574,26576],{"className":26575},[47],[49,26577,26578],{"xmlns":51},[53,26579,26580,26584],{},[56,26581,26582],{},[59,26583,531],{},[63,26585,26586],{"encoding":65},"\\gamma",[39,26588,26590],{"className":26589,"ariaHidden":70},[69],[39,26591,26593,26596],{"className":26592},[74],[39,26594],{"className":26595,"style":2654},[78],[39,26597,531],{"className":26598,"style":565},[83,84]," 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 ",[39,26601,26603,26616],{"className":26602,"translate":43},[42],[39,26604,26606],{"className":26605},[47],[49,26607,26608],{"xmlns":51},[53,26609,26610,26614],{},[56,26611,26612],{},[59,26613,531],{},[63,26615,26586],{"encoding":65},[39,26617,26619],{"className":26618,"ariaHidden":70},[69],[39,26620,26622,26625],{"className":26621},[74],[39,26623],{"className":26624,"style":2654},[78],[39,26626,531],{"className":26627,"style":565},[83,84]," 倍」这一结论。",[26,26630,26632],{"id":26631},"_33-压缩映射","3.3 压缩映射",[31,26634,26637,27062],{"id":26635,"title":26636},"theorem-contraction","定理 3.2（贝尔曼算子的压缩性）",[11,26638,37,26639,14627,26711,1639,26769,26827,26828,26871,26872,26900,26901,742],{},[39,26640,26642,26663],{"className":26641,"translate":43},[42],[39,26643,26645],{"className":26644},[47],[49,26646,26647],{"xmlns":51},[53,26648,26649,26661],{},[56,26650,26651,26653,26655,26657,26659],{},[539,26652,541],{},[133,26654,1824],{},[59,26656,531],{},[133,26658,1693],{},[539,26660,546],{},[63,26662,14577],{"encoding":65},[39,26664,26666,26684,26702],{"className":26665,"ariaHidden":70},[69],[39,26667,26669,26672,26675,26678,26681],{"className":26668},[74],[39,26670],{"className":26671,"style":14587},[78],[39,26673,541],{"className":26674},[83],[39,26676],{"className":26677,"style":231},[230],[39,26679,1824],{"className":26680},[235],[39,26682],{"className":26683,"style":231},[230],[39,26685,26687,26690,26693,26696,26699],{"className":26686},[74],[39,26688],{"className":26689,"style":561},[78],[39,26691,531],{"className":26692,"style":565},[83,84],[39,26694],{"className":26695,"style":231},[230],[39,26697,1693],{"className":26698},[235],[39,26700],{"className":26701,"style":231},[230],[39,26703,26705,26708],{"className":26704},[74],[39,26706],{"className":26707,"style":2473},[78],[39,26709,546],{"className":26710},[83],[39,26712,26714,26731],{"className":26713,"translate":43},[42],[39,26715,26717],{"className":26716},[47],[49,26718,26719],{"xmlns":51},[53,26720,26721,26729],{},[56,26722,26723],{},[138,26724,26725,26727],{},[59,26726,11772],{"mathvariant":617},[59,26728,2098],{},[63,26730,11777],{"encoding":65},[39,26732,26734],{"className":26733,"ariaHidden":70},[69],[39,26735,26737,26740],{"className":26736},[74],[39,26738],{"className":26739,"style":79},[78],[39,26741,26743,26746],{"className":26742},[83],[39,26744,11772],{"className":26745,"style":11793},[83,641],[39,26747,26749],{"className":26748},[192],[39,26750,26752],{"className":26751},[196],[39,26753,26755],{"className":26754},[200],[39,26756,26758],{"className":26757,"style":6410},[204],[39,26759,26760,26763],{"style":208},[39,26761],{"className":26762,"style":213},[212],[39,26764,26766],{"className":26765},[217,218,219,220],[39,26767,2098],{"className":26768,"style":2114},[83,84,220],[39,26770,26772,26789],{"className":26771,"translate":43},[42],[39,26773,26775],{"className":26774},[47],[49,26776,26777],{"xmlns":51},[53,26778,26779,26787],{},[56,26780,26781],{},[138,26782,26783,26785],{},[59,26784,11772],{"mathvariant":617},[133,26786,21884],{},[63,26788,21887],{"encoding":65},[39,26790,26792],{"className":26791,"ariaHidden":70},[69],[39,26793,26795,26798],{"className":26794},[74],[39,26796],{"className":26797,"style":21897},[78],[39,26799,26801,26804],{"className":26800},[83],[39,26802,11772],{"className":26803,"style":11793},[83,641],[39,26805,26807],{"className":26806},[192],[39,26808,26810],{"className":26809},[196],[39,26811,26813],{"className":26812},[200],[39,26814,26816],{"className":26815,"style":21897},[204],[39,26817,26818,26821],{"style":208},[39,26819],{"className":26820,"style":213},[212],[39,26822,26824],{"className":26823},[217,218,219,220],[39,26825,21884],{"className":26826},[465,220]," 都是 ",[39,26829,26831,26850],{"className":26830,"translate":43},[42],[39,26832,26834],{"className":26833},[47],[49,26835,26836],{"xmlns":51},[53,26837,26838,26848],{},[56,26839,26840,26842,26844,26846],{},[59,26841,19772],{},[133,26843,136],{"stretchy":135},[59,26845,61],{},[133,26847,161],{"stretchy":135},[63,26849,21380],{"encoding":65},[39,26851,26853],{"className":26852,"ariaHidden":70},[69],[39,26854,26856,26859,26862,26865,26868],{"className":26855},[74],[39,26857],{"className":26858,"style":245},[78],[39,26860,19772],{"className":26861,"style":19812},[83,84],[39,26863,136],{"className":26864},[182],[39,26866,61],{"className":26867,"style":85},[83,84],[39,26869,161],{"className":26870},[263]," 上的 ",[39,26873,26875,26888],{"className":26874,"translate":43},[42],[39,26876,26878],{"className":26877},[47],[49,26879,26880],{"xmlns":51},[53,26881,26882,26886],{},[56,26883,26884],{},[59,26885,531],{},[63,26887,26586],{"encoding":65},[39,26889,26891],{"className":26890,"ariaHidden":70},[69],[39,26892,26894,26897],{"className":26893},[74],[39,26895],{"className":26896,"style":2654},[78],[39,26898,531],{"className":26899,"style":565},[83,84],"-压缩映射，即对任意 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c\\mathbf{1}\n\\le\\mathcal{T}V_1\n\\le\\mathcal{T}V_2+\\gamma c\\mathbf{1}.",[39,27984,27986,28044,28068,28126,28184],{"className":27985,"ariaHidden":70},[69],[39,27987,27989,27992,27995,28035,28038,28041],{"className":27988},[74],[39,27990],{"className":27991,"style":634},[78],[39,27993,11772],{"className":27994,"style":11793},[83,641],[39,27996,27998,28001],{"className":27997},[83],[39,27999,4230],{"className":28000,"style":461},[83,84],[39,28002,28004],{"className":28003},[192],[39,28005,28007,28027],{"className":28006},[196,649],[39,28008,28010,28024],{"className":28009},[200],[39,28011,28013],{"className":28012,"style":963},[204],[39,28014,28015,28018],{"style":13607},[39,28016],{"className":28017,"style":213},[212],[39,28019,28021],{"className":28020},[217,218,219,220],[39,28022,17359],{"className":28023},[83,220],[39,28025,673],{"className":28026},[672],[39,28028,28030],{"className":28029},[200],[39,28031,28033],{"className":28032,"style":680},[204],[39,28034],{},[39,28036],{"className":28037,"style":461},[230],[39,28039,5378],{"className":28040},[465],[39,28042],{"className":28043,"style":461},[230],[39,28045,28047,28050,28053,28056,28059,28062,28065],{"className":28046},[74],[39,28048],{"className":28049,"style":17578},[78],[39,28051,531],{"className":28052,"style":565},[83,84],[39,28054,24077],{"className":28055},[83,84],[39,28057,546],{"className":28058},[83,24328],[39,28060],{"className":28061,"style":231},[230],[39,28063,1824],{"className":28064},[235],[39,28066],{"className":28067,"style":231},[230],[39,28069,28071,28074,28077,28117,28120,28123],{"className":28070},[74],[39,28072],{"className":28073,"style":634},[78],[39,28075,11772],{"className":28076,"style":11793},[83,641],[39,28078,28080,28083],{"className":28079},[83],[39,28081,4230],{"className":28082,"style":461},[83,84],[39,28084,28086],{"className":28085},[192],[39,28087,28089,28109],{"className":28088},[196,649],[39,28090,28092,28106],{"className":28091},[200],[39,28093,28095],{"className":28094,"style":963},[204],[39,28096,28097,28100],{"style":13607},[39,28098],{"className":28099,"style":213},[212],[39,28101,28103],{"className":28102},[217,218,219,220],[39,28104,546],{"className":28105},[83,220],[39,28107,673],{"className":28108},[672],[39,28110,28112],{"className":28111},[200],[39,28113,28115],{"className":28114,"style":680},[204],[39,28116],{},[39,28118],{"className":28119,"style":231},[230],[39,28121,1824],{"className":28122},[235],[39,28124],{"className":28125,"style":231},[230],[39,28127,28129,28132,28135,28175,28178,28181],{"className":28128},[74],[39,28130],{"className":28131,"style":634},[78],[39,28133,11772],{"className":28134,"style":11793},[83,641],[39,28136,28138,28141],{"className":28137},[83],[39,28139,4230],{"className":28140,"style":461},[83,84],[39,28142,28144],{"className":28143},[192],[39,28145,28147,28167],{"className":28146},[196,649],[39,28148,28150,28164],{"className":28149},[200],[39,28151,28153],{"className":28152,"style":963},[204],[39,28154,28155,28158],{"style":13607},[39,28156],{"className":28157,"style":213},[212],[39,28159,28161],{"className":28160},[217,218,219,220],[39,28162,17359],{"className":28163},[83,220],[39,28165,673],{"className":28166},[672],[39,28168,28170],{"className":28169},[200],[39,28171,28173],{"className":28172,"style":680},[204],[39,28174],{},[39,28176],{"className":28177,"style":461},[230],[39,28179,788],{"className":28180},[465],[39,28182],{"className":28183,"style":461},[230],[39,28185,28187,28190,28193,28196,28199],{"className":28186},[74],[39,28188],{"className":28189,"style":17578},[78],[39,28191,531],{"className":28192,"style":565},[83,84],[39,28194,24077],{"className":28195},[83,84],[39,28197,546],{"className":28198},[83,24328],[39,28200,2903],{"className":28201},[83],[11,28203,28204,28205,742],{},"因此对一切 ",[39,28206,28208,28221],{"className":28207,"translate":43},[42],[39,28209,28211],{"className":28210},[47],[49,28212,28213],{"xmlns":51},[53,28214,28215,28219],{},[56,28216,28217],{},[59,28218,142],{},[63,28220,142],{"encoding":65},[39,28222,28224],{"className":28223,"ariaHidden":70},[69],[39,28225,28227,28230],{"className":28226},[74],[39,28228],{"className":28229,"style":290},[78],[39,28231,142],{"className":28232},[83,84],[39,28234,28236],{"className":28235,"translate":43},[746],[39,28237,28239,28337],{"className":28238,"translate":43},[42],[39,28240,28242],{"className":28241},[47],[49,28243,28244],{"xmlns":51,"display":755},[53,28245,28246,28334],{},[759,28247,28248,28308],{"rowspacing":761,"columnalign":762,"columnspacing":146},[764,28249,28250,28256],{},[767,28251,28252],{},[770,28253,28254],{"scriptlevel":541,"displaystyle":70},[56,28255],{},[767,28257,28258],{},[770,28259,28260],{"scriptlevel":541,"displaystyle":70},[56,28261,28262,28264],{},[56,28263],{},[56,28265,28266,28268,28270,28272,28278,28280,28282,28284,28286,28288,28290,28292,28298,28300,28302,28304,28306],{},[133,28267,150],{"fence":70},[133,28269,136],{"stretchy":135},[59,28271,11772],{"mathvariant":617},[613,28273,28274,28276],{},[59,28275,4230],{},[539,28277,546],{},[133,28279,161],{"stretchy":135},[133,28281,136],{"stretchy":135},[59,28283,142],{},[133,28285,161],{"stretchy":135},[133,28287,5378],{},[133,28289,136],{"stretchy":135},[59,28291,11772],{"mathvariant":617},[613,28293,28294,28296],{},[59,28295,4230],{},[539,28297,17359],{},[133,28299,161],{"stretchy":135},[133,28301,136],{"stretchy":135},[59,28303,142],{},[133,28305,161],{"stretchy":135},[133,28307,150],{"fence":70},[764,28309,28310,28316],{},[767,28311,28312],{},[770,28313,28314],{"scriptlevel":541,"displaystyle":70},[56,28315],{},[767,28317,28318],{},[770,28319,28320],{"scriptlevel":541,"displaystyle":70},[56,28321,28322,28324,28326,28328,28330,28332],{},[56,28323],{},[230,28325],{"width":6891},[133,28327,1824],{},[59,28329,531],{},[59,28331,24077],{},[59,28333,2903],{"mathvariant":145},[63,28335,28336],{"encoding":65},"\\begin{aligned}\n&\\left|(\\mathcal{T}V_1)(s)-(\\mathcal{T}V_2)(s)\\right|\\\\\n&\\quad\\le\\gamma c.\n\\end{aligned}",[39,28338,28340],{"className":28339,"ariaHidden":70},[69],[39,28341,28343,28346],{"className":28342},[74],[39,28344],{"className":28345,"style":4374},[78],[39,28347,28349],{"className":28348},[83],[39,28350,28352,28393],{"className":28351},[759],[39,28353,28355],{"className":28354},[918],[39,28356,28358,28385],{"className":28357},[196,649],[39,28359,28361,28382],{"className":28360},[200],[39,28362,28364,28374],{"className":28363,"style":4393},[204],[39,28365,28367,28371],{"style":28366},"top:-3.6em;",[39,28368],{"className":28369,"style":28370},[212],"height:2.84em;",[39,28372],{"className":28373},[83],[39,28375,28376,28379],{"style":5834},[39,28377],{"className":28378,"style":28370},[212],[39,28380],{"className":28381},[83],[39,28383,673],{"className":28384},[672],[39,28386,28388],{"className":28387},[200],[39,28389,28391],{"className":28390,"style":4507},[204],[39,28392],{},[39,28394,28396],{"className":28395},[1223],[39,28397,28399,28588],{"className":28398},[196,649],[39,28400,28402,28585],{"className":28401},[200],[39,28403,28405,28553],{"className":28404,"style":4393},[204],[39,28406,28407,28410],{"style":4396},[39,28408],{"className":28409,"style":935},[212],[39,28411,28413,28416,28419],{"className":28412},[83],[39,28414],{"className":28415},[83],[39,28417],{"className":28418,"style":256},[230],[39,28420,28422,28425,28428,28431,28471,28474,28477,28480,28483,28486,28489,28492,28495,28498,28538,28541,28544,28547,28550],{"className":28421},[2842],[39,28423,150],{"className":28424,"style":4592},[182,4591],[39,28426,136],{"className":28427},[182],[39,28429,11772],{"className":28430,"style":11793},[83,641],[39,28432,28434,28437],{"className":28433},[83],[39,28435,4230],{"className":28436,"style":461},[83,84],[39,28438,28440],{"className":28439},[192],[39,28441,28443,28463],{"className":28442},[196,649],[39,28444,28446,28460],{"className":28445},[200],[39,28447,28449],{"className":28448,"style":963},[204],[39,28450,28451,28454],{"style":13607},[39,28452],{"className":28453,"style":213},[212],[39,28455,28457],{"className":28456},[217,218,219,220],[39,28458,546],{"className":28459},[83,220],[39,28461,673],{"className":28462},[672],[39,28464,28466],{"className":28465},[200],[39,28467,28469],{"className":28468,"style":680},[204],[39,28470],{},[39,28472,161],{"className":28473},[263],[39,28475,136],{"className":28476},[182],[39,28478,142],{"className":28479},[83,84],[39,28481,161],{"className":28482},[263],[39,28484],{"className":28485,"style":461},[230],[39,28487,5378],{"className":28488},[465],[39,28490],{"className":28491,"style":461},[230],[39,28493,136],{"className":28494},[182],[39,28496,11772],{"className":28497,"style":11793},[83,641],[39,28499,28501,28504],{"className":28500},[83],[39,28502,4230],{"className":28503,"style":461},[83,84],[39,28505,28507],{"className":28506},[192],[39,28508,28510,28530],{"className":28509},[196,649],[39,28511,28513,28527],{"className":28512},[200],[39,28514,28516],{"className":28515,"style":963},[204],[39,28517,28518,28521],{"style":13607},[39,28519],{"className":28520,"style":213},[212],[39,28522,28524],{"className":28523},[217,218,219,220],[39,28525,17359],{"className":28526},[83,220],[39,28528,673],{"className":28529},[672],[39,28531,28533],{"className":28532},[200],[39,28534,28536],{"className":28535,"style":680},[204],[39,28537],{},[39,28539,161],{"className":28540},[263],[39,28542,136],{"className":28543},[182],[39,28545,142],{"className":28546},[83,84],[39,28548,161],{"className":28549},[263],[39,28551,150],{"className":28552,"style":4592},[263,4591],[39,28554,28555,28558],{"style":4444},[39,28556],{"className":28557,"style":935},[212],[39,28559,28561,28564,28567,28570,28573,28576,28579,28582],{"className":28560},[83],[39,28562],{"className":28563},[83],[39,28565],{"className":28566,"style":7240},[230],[39,28568],{"className":28569,"style":231},[230],[39,28571,1824],{"className":28572},[235],[39,28574],{"className":28575,"style":231},[230],[39,28577,531],{"className":28578,"style":565},[83,84],[39,28580,24077],{"className":28581},[83,84],[39,28583,2903],{"className":28584},[83],[39,28586,673],{"className":28587},[672],[39,28589,28591],{"className":28590},[200],[39,28592,28594],{"className":28593,"style":4507},[204],[39,28595],{},[11,28597,26443,28598,28626],{},[39,28599,28601,28614],{"className":28600,"translate":43},[42],[39,28602,28604],{"className":28603},[47],[49,28605,28606],{"xmlns":51},[53,28607,28608,28612],{},[56,28609,28610],{},[59,28611,142],{},[63,28613,142],{"encoding":65},[39,28615,28617],{"className":28616,"ariaHidden":70},[69],[39,28618,28620,28623],{"className":28619},[74],[39,28621],{"className":28622,"style":290},[78],[39,28624,142],{"className":28625},[83,84]," 取上确界即得结论。证毕。",[11,28628,28629,28630,28658,28659,28709,28710,28739,28740,28769,28770,742,28799,28828,28829,28915],{},"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 ",[39,28631,28633,28646],{"className":28632,"translate":43},[42],[39,28634,28636],{"className":28635},[47],[49,28637,28638],{"xmlns":51},[53,28639,28640,28644],{},[56,28641,28642],{},[59,28643,531],{},[63,28645,26586],{"encoding":65},[39,28647,28649],{"className":28648,"ariaHidden":70},[69],[39,28650,28652,28655],{"className":28651},[74],[39,28653],{"className":28654,"style":2654},[78],[39,28656,531],{"className":28657,"style":565},[83,84]," 倍。若 ",[39,28660,28662,28679],{"className":28661,"translate":43},[42],[39,28663,28665],{"className":28664},[47],[49,28666,28667],{"xmlns":51},[53,28668,28669,28677],{},[56,28670,28671,28673,28675],{},[59,28672,531],{},[133,28674,793],{},[539,28676,17739],{},[63,28678,17742],{"encoding":65},[39,28680,28682,28700],{"className":28681,"ariaHidden":70},[69],[39,28683,28685,28688,28691,28694,28697],{"className":28684},[74],[39,28686],{"className":28687,"style":2654},[78],[39,28689,531],{"className":28690,"style":565},[83,84],[39,28692],{"className":28693,"style":231},[230],[39,28695,793],{"className":28696},[235],[39,28698],{"className":28699,"style":231},[230],[39,28701,28703,28706],{"className":28702},[74],[39,28704],{"className":28705,"style":2473},[78],[39,28707,17739],{"className":28708},[83],"，初始最大差异为 ",[39,28711,28713,28727],{"className":28712,"translate":43},[42],[39,28714,28716],{"className":28715},[47],[49,28717,28718],{"xmlns":51},[53,28719,28720,28725],{},[56,28721,28722],{},[539,28723,28724],{},"100",[63,28726,28724],{"encoding":65},[39,28728,28730],{"className":28729,"ariaHidden":70},[69],[39,28731,28733,28736],{"className":28732},[74],[39,28734],{"className":28735,"style":2473},[78],[39,28737,28724],{"className":28738},[83],"，则一次更新后差异不超过 ",[39,28741,28743,28757],{"className":28742,"translate":43},[42],[39,28744,28746],{"className":28745},[47],[49,28747,28748],{"xmlns":51},[53,28749,28750,28755],{},[56,28751,28752],{},[539,28753,28754],{},"90",[63,28756,28754],{"encoding":65},[39,28758,28760],{"className":28759,"ariaHidden":70},[69],[39,28761,28763,28766],{"className":28762},[74],[39,28764],{"className":28765,"style":2473},[78],[39,28767,28754],{"className":28768},[83],"，两次后不超过 ",[39,28771,28773,28787],{"className":28772,"translate":43},[42],[39,28774,28776],{"className":28775},[47],[49,28777,28778],{"xmlns":51},[53,28779,28780,28785],{},[56,28781,28782],{},[539,28783,28784],{},"81",[63,28786,28784],{"encoding":65},[39,28788,28790],{"className":28789,"ariaHidden":70},[69],[39,28791,28793,28796],{"className":28792},[74],[39,28794],{"className":28795,"style":2473},[78],[39,28797,28784],{"className":28798},[83],[39,28800,28802,28815],{"className":28801,"translate":43},[42],[39,28803,28805],{"className":28804},[47],[49,28806,28807],{"xmlns":51},[53,28808,28809,28813],{},[56,28810,28811],{},[59,28812,3196],{},[63,28814,3196],{"encoding":65},[39,28816,28818],{"className":28817,"ariaHidden":70},[69],[39,28819,28821,28825],{"className":28820},[74],[39,28822],{"className":28823,"style":28824},[78],"height:0.6944em;",[39,28826,3196],{"className":28827,"style":3321},[83,84]," 次后不超过 ",[39,28830,28832,28854],{"className":28831,"translate":43},[42],[39,28833,28835],{"className":28834},[47],[49,28836,28837],{"xmlns":51},[53,28838,28839,28851],{},[56,28840,28841,28847,28849],{},[138,28842,28843,28845],{},[539,28844,17739],{},[59,28846,3196],{},[133,28848,8885],{},[539,28850,28724],{},[63,28852,28853],{"encoding":65},"0.9^k\\cdot100",[39,28855,28857,28906],{"className":28856,"ariaHidden":70},[69],[39,28858,28860,28863,28867,28897,28900,28903],{"className":28859},[74],[39,28861],{"className":28862,"style":15577},[78],[39,28864,28866],{"className":28865},[83],"0.",[39,28868,28870,28874],{"className":28869},[83],[39,28871,28873],{"className":28872},[83],"9",[39,28875,28877],{"className":28876},[192],[39,28878,28880],{"className":28879},[196],[39,28881,28883],{"className":28882},[200],[39,28884,28886],{"className":28885,"style":15577},[204],[39,28887,28888,28891],{"style":208},[39,28889],{"className":28890,"style":213},[212],[39,28892,28894],{"className":28893},[217,218,219,220],[39,28895,3196],{"className":28896,"style":3321},[83,84,220],[39,28898],{"className":28899,"style":461},[230],[39,28901,8885],{"className":28902},[465],[39,28904],{"className":28905,"style":461},[230],[39,28907,28909,28912],{"className":28908},[74],[39,28910],{"className":28911,"style":2473},[78],[39,28913,28724],{"className":28914},[83],"。这正是后文几何收敛性的来源。",[11,28917,28918,28919,28969,28970,29021,29022,29393],{},"需要强调的是，压缩性依赖于 ",[39,28920,28922,28939],{"className":28921,"translate":43},[42],[39,28923,28925],{"className":28924},[47],[49,28926,28927],{"xmlns":51},[53,28928,28929,28937],{},[56,28930,28931,28933,28935],{},[59,28932,531],{},[133,28934,1693],{},[539,28936,546],{},[63,28938,17296],{"encoding":65},[39,28940,28942,28960],{"className":28941,"ariaHidden":70},[69],[39,28943,28945,28948,28951,28954,28957],{"className":28944},[74],[39,28946],{"className":28947,"style":561},[78],[39,28949,531],{"className":28950,"style":565},[83,84],[39,28952],{"className":28953,"style":231},[230],[39,28955,1693],{"className":28956},[235],[39,28958],{"className":28959,"style":231},[230],[39,28961,28963,28966],{"className":28962},[74],[39,28964],{"className":28965,"style":2473},[78],[39,28967,546],{"className":28968},[83],"。当 ",[39,28971,28973,28991],{"className":28972,"translate":43},[42],[39,28974,28976],{"className":28975},[47],[49,28977,28978],{"xmlns":51},[53,28979,28980,28988],{},[56,28981,28982,28984,28986],{},[59,28983,531],{},[133,28985,793],{},[539,28987,546],{},[63,28989,28990],{"encoding":65},"\\gamma=1",[39,28992,28994,29012],{"className":28993,"ariaHidden":70},[69],[39,28995,28997,29000,29003,29006,29009],{"className":28996},[74],[39,28998],{"className":28999,"style":2654},[78],[39,29001,531],{"className":29002,"style":565},[83,84],[39,29004],{"className":29005,"style":231},[230],[39,29007,793],{"className":29008},[235],[39,29010],{"className":29011,"style":231},[230],[39,29013,29015,29018],{"className":29014},[74],[39,29016],{"className":29017,"style":2473},[78],[39,29019,546],{"className":29020},[83]," 时，贝尔曼算子通常只是非扩张的，即只能得到 ",[39,29023,29025,29087],{"className":29024,"translate":43},[42],[39,29026,29028],{"className":29027},[47],[49,29029,29030],{"xmlns":51},[53,29031,29032,29084],{},[56,29033,29034,29036,29038,29044,29046,29048,29054,29060,29062,29064,29070,29072,29078],{},[59,29035,15389],{"mathvariant":145},[59,29037,11772],{"mathvariant":617},[613,29039,29040,29042],{},[59,29041,4230],{},[539,29043,546],{},[133,29045,5378],{},[59,29047,11772],{"mathvariant":617},[613,29049,29050,29052],{},[59,29051,4230],{},[539,29053,17359],{},[613,29055,29056,29058],{},[59,29057,15389],{"mathvariant":145},[59,29059,1696],{"mathvariant":145},[133,29061,1824],{},[59,29063,15389],{"mathvariant":145},[613,29065,29066,29068],{},[59,29067,4230],{},[539,29069,546],{},[133,29071,5378],{},[613,29073,29074,29076],{},[59,29075,4230],{},[539,29077,17359],{},[613,29079,29080,29082],{},[59,29081,15389],{"mathvariant":145},[59,29083,1696],{"mathvariant":145},[63,29085,29086],{"encoding":65},"\\|\\mathcal{T}V_1-\\mathcal{T}V_2\\|_\\infty\\le\\|V_1-V_2\\|_\\infty",[39,29088,29090,29151,29249,29307],{"className":29089,"ariaHidden":70},[69],[39,29091,29093,29096,29099,29102,29142,29145,29148],{"className":29092},[74],[39,29094],{"className":29095,"style":245},[78],[39,29097,15389],{"className":29098},[83],[39,29100,11772],{"className":29101,"style":11793},[83,641],[39,29103,29105,29108],{"className":29104},[83],[39,29106,4230],{"className":29107,"style":461},[83,84],[39,29109,29111],{"className":29110},[192],[39,29112,29114,29134],{"className":29113},[196,649],[39,29115,29117,29131],{"className":29116},[200],[39,29118,29120],{"className":29119,"style":963},[204],[39,29121,29122,29125],{"style":13607},[39,29123],{"className":29124,"style":213},[212],[39,29126,29128],{"className":29127},[217,218,219,220],[39,29129,546],{"className":29130},[83,220],[39,29132,673],{"className":29133},[672],[39,29135,29137],{"className":29136},[200],[39,29138,29140],{"className":29139,"style":680},[204],[39,29141],{},[39,29143],{"className":29144,"style":461},[230],[39,29146,5378],{"className":29147},[465],[39,29149],{"className":29150,"style":461},[230],[39,29152,29154,29157,29160,29200,29240,29243,29246],{"className":29153},[74],[39,29155],{"className":29156,"style":245},[78],[39,29158,11772],{"className":29159,"style":11793},[83,641],[39,29161,29163,29166],{"className":29162},[83],[39,29164,4230],{"className":29165,"style":461},[83,84],[39,29167,29169],{"className":29168},[192],[39,29170,29172,29192],{"className":29171},[196,649],[39,29173,29175,29189],{"className":29174},[200],[39,29176,29178],{"className":29177,"style":963},[204],[39,29179,29180,29183],{"style":13607},[39,29181],{"className":29182,"style":213},[212],[39,29184,29186],{"className":29185},[217,218,219,220],[39,29187,17359],{"className":29188},[83,220],[39,29190,673],{"className":29191},[672],[39,29193,29195],{"className":29194},[200],[39,29196,29198],{"className":29197,"style":680},[204],[39,29199],{},[39,29201,29203,29206],{"className":29202},[83],[39,29204,15389],{"className":29205},[83],[39,29207,29209],{"className":29208},[192],[39,29210,29212,29232],{"className":29211},[196,649],[39,29213,29215,29229],{"className":29214},[200],[39,29216,29218],{"className":29217,"style":1728},[204],[39,29219,29220,29223],{"style":966},[39,29221],{"className":29222,"style":213},[212],[39,29224,29226],{"className":29225},[217,218,219,220],[39,29227,1696],{"className":29228},[83,220],[39,29230,673],{"className":29231},[672],[39,29233,29235],{"className":29234},[200],[39,29236,29238],{"className":29237,"style":680},[204],[39,29239],{},[39,29241],{"className":29242,"style":231},[230],[39,29244,1824],{"className":29245},[235],[39,29247],{"className":29248,"style":231},[230],[39,29250,29252,29255,29258,29298,29301,29304],{"className":29251},[74],[39,29253],{"className":29254,"style":245},[78],[39,29256,15389],{"className":29257},[83],[39,29259,29261,29264],{"className":29260},[83],[39,29262,4230],{"className":29263,"style":461},[83,84],[39,29265,29267],{"className":29266},[192],[39,29268,29270,29290],{"className":29269},[196,649],[39,29271,29273,29287],{"className":29272},[200],[39,29274,29276],{"className":29275,"style":963},[204],[39,29277,29278,29281],{"style":13607},[39,29279],{"className":29280,"style":213},[212],[39,29282,29284],{"className":29283},[217,218,219,220],[39,29285,546],{"className":29286},[83,220],[39,29288,673],{"className":29289},[672],[39,29291,29293],{"className":29292},[200],[39,29294,29296],{"className":29295,"style":680},[204],[39,29297],{},[39,29299],{"className":29300,"style":461},[230],[39,29302,5378],{"className":29303},[465],[39,29305],{"className":29306,"style":461},[230],[39,29308,29310,29313,29353],{"className":29309},[74],[39,29311],{"className":29312,"style":245},[78],[39,29314,29316,29319],{"className":29315},[83],[39,29317,4230],{"className":29318,"style":461},[83,84],[39,29320,29322],{"className":29321},[192],[39,29323,29325,29345],{"className":29324},[196,649],[39,29326,29328,29342],{"className":29327},[200],[39,29329,29331],{"className":29330,"style":963},[204],[39,29332,29333,29336],{"style":13607},[39,29334],{"className":29335,"style":213},[212],[39,29337,29339],{"className":29338},[217,218,219,220],[39,29340,17359],{"className":29341},[83,220],[39,29343,673],{"className":29344},[672],[39,29346,29348],{"className":29347},[200],[39,29349,29351],{"className":29350,"style":680},[204],[39,29352],{},[39,29354,29356,29359],{"className":29355},[83],[39,29357,15389],{"className":29358},[83],[39,29360,29362],{"className":29361},[192],[39,29363,29365,29385],{"className":29364},[196,649],[39,29366,29368,29382],{"className":29367},[200],[39,29369,29371],{"className":29370,"style":1728},[204],[39,29372,29373,29376],{"style":966},[39,29374],{"className":29375,"style":213},[212],[39,29377,29379],{"className":29378},[217,218,219,220],[39,29380,1696],{"className":29381},[83,220],[39,29383,673],{"className":29384},[672],[39,29386,29388],{"className":29387},[200],[39,29389,29391],{"className":29390,"style":680},[204],[39,29392],{},"，不能保证距离严格缩小。第 5.4 节会给出反例。",[26,29395,29397],{"id":29396},"_34-banach-不动点定理","3.4 Banach 不动点定理",[31,29399,29402,29672,29802,30170],{"id":29400,"title":29401},"theorem-banach","定理 3.3（Banach 不动点定理）",[11,29403,37,29404,29458,29459,29532,29533,29606,29607,742],{},[39,29405,29407,29431],{"className":29406,"translate":43},[42],[39,29408,29410],{"className":29409},[47],[49,29411,29412],{"xmlns":51},[53,29413,29414,29428],{},[56,29415,29416,29418,29421,29423,29426],{},[133,29417,136],{"stretchy":135},[59,29419,29420],{},"X",[133,29422,155],{"separator":70},[59,29424,29425],{},"d",[133,29427,161],{"stretchy":135},[63,29429,29430],{"encoding":65},"(X,d)",[39,29432,29434],{"className":29433,"ariaHidden":70},[69],[39,29435,29437,29440,29443,29446,29449,29452,29455],{"className":29436},[74],[39,29438],{"className":29439,"style":245},[78],[39,29441,136],{"className":29442},[182],[39,29444,29420],{"className":29445,"style":14420},[83,84],[39,29447,155],{"className":29448},[252],[39,29450],{"className":29451,"style":256},[230],[39,29453,29425],{"className":29454},[83,84],[39,29456,161],{"className":29457},[263]," 是完备度量空间，",[39,29460,29462,29484],{"className":29461,"translate":43},[42],[39,29463,29465],{"className":29464},[47],[49,29466,29467],{"xmlns":51},[53,29468,29469,29481],{},[56,29470,29471,29473,29475,29477,29479],{},[59,29472,11772],{},[133,29474,404],{},[59,29476,29420],{},[133,29478,418],{},[59,29480,29420],{},[63,29482,29483],{"encoding":65},"T:X\\to X",[39,29485,29487,29505,29523],{"className":29486,"ariaHidden":70},[69],[39,29488,29490,29493,29496,29499,29502],{"className":29489},[74],[39,29491],{"className":29492,"style":79},[78],[39,29494,11772],{"className":29495,"style":178},[83,84],[39,29497],{"className":29498,"style":231},[230],[39,29500,404],{"className":29501},[235],[39,29503],{"className":29504,"style":231},[230],[39,29506,29508,29511,29514,29517,29520],{"className":29507},[74],[39,29509],{"className":29510,"style":79},[78],[39,29512,29420],{"className":29513,"style":14420},[83,84],[39,29515],{"className":29516,"style":231},[230],[39,29518,418],{"className":29519},[235],[39,29521],{"className":29522,"style":231},[230],[39,29524,29526,29529],{"className":29525},[74],[39,29527],{"className":29528,"style":79},[78],[39,29530,29420],{"className":29531,"style":14420},[83,84]," 满足：存在 ",[39,29534,29536,29561],{"className":29535,"translate":43},[42],[39,29537,29539],{"className":29538},[47],[49,29540,29541],{"xmlns":51},[53,29542,29543,29559],{},[56,29544,29545,29547,29549,29551,29553,29555,29557],{},[59,29546,531],{},[133,29548,534],{},[133,29550,537],{"stretchy":135},[539,29552,541],{},[133,29554,155],{"separator":70},[539,29556,546],{},[133,29558,161],{"stretchy":135},[63,29560,551],{"encoding":65},[39,29562,29564,29582],{"className":29563,"ariaHidden":70},[69],[39,29565,29567,29570,29573,29576,29579],{"className":29566},[74],[39,29568],{"className":29569,"style":561},[78],[39,29571,531],{"className":29572,"style":565},[83,84],[39,29574],{"className":29575,"style":231},[230],[39,29577,534],{"className":29578},[235],[39,29580],{"className":29581,"style":231},[230],[39,29583,29585,29588,29591,29594,29597,29600,29603],{"className":29584},[74],[39,29586],{"className":29587,"style":245},[78],[39,29589,537],{"className":29590},[182],[39,29592,541],{"className":29593},[83],[39,29595,155],{"className":29596},[252],[39,29598],{"className":29599,"style":256},[230],[39,29601,546],{"className":29602},[83],[39,29604,161],{"className":29605},[263],"，使得对任意 ",[39,29608,29610,29633],{"className":29609,"translate":43},[42],[39,29611,29613],{"className":29612},[47],[49,29614,29615],{"xmlns":51},[53,29616,29617,29630],{},[56,29618,29619,29621,29623,29626,29628],{},[59,29620,1755],{},[133,29622,155],{"separator":70},[59,29624,29625],{},"y",[133,29627,534],{},[59,29629,29420],{},[63,29631,29632],{"encoding":65},"x,y\\in X",[39,29634,29636,29663],{"className":29635,"ariaHidden":70},[69],[39,29637,29639,29642,29645,29648,29651,29654,29657,29660],{"className":29638},[74],[39,29640],{"className":29641,"style":561},[78],[39,29643,1755],{"className":29644},[83,84],[39,29646,155],{"className":29647},[252],[39,29649],{"className":29650,"style":256},[230],[39,29652,29625],{"className":29653,"style":2114},[83,84],[39,29655],{"className":29656,"style":231},[230],[39,29658,534],{"className":29659},[235],[39,29661],{"className":29662,"style":231},[230],[39,29664,29666,29669],{"className":29665},[74],[39,29667],{"className":29668,"style":79},[78],[39,29670,29420],{"className":29671,"style":14420},[83,84],[39,29673,29675],{"className":29674,"translate":43},[746],[39,29676,29678,29724],{"className":29677,"translate":43},[42],[39,29679,29681],{"className":29680},[47],[49,29682,29683],{"xmlns":51,"display":755},[53,29684,29685,29721],{},[56,29686,29687,29689,29691,29693,29695,29697,29699,29701,29703,29705,29707,29709,29711,29713,29715,29717,29719],{},[59,29688,29425],{},[133,29690,136],{"stretchy":135},[59,29692,11772],{},[59,29694,1755],{},[133,29696,155],{"separator":70},[59,29698,11772],{},[59,29700,29625],{},[133,29702,161],{"stretchy":135},[133,29704,1824],{},[59,29706,531],{},[59,29708,29425],{},[133,29710,136],{"stretchy":135},[59,29712,1755],{},[133,29714,155],{"separator":70},[59,29716,29625],{},[133,29718,161],{"stretchy":135},[59,29720,2903],{"mathvariant":145},[63,29722,29723],{"encoding":65},"d(Tx,Ty)\\le\\gamma d(x,y).",[39,29725,29727,29769],{"className":29726,"ariaHidden":70},[69],[39,29728,29730,29733,29736,29739,29742,29745,29748,29751,29754,29757,29760,29763,29766],{"className":29729},[74],[39,29731],{"className":29732,"style":245},[78],[39,29734,29425],{"className":29735},[83,84],[39,29737,136],{"className":29738},[182],[39,29740,11772],{"className":29741,"style":178},[83,84],[39,29743,1755],{"className":29744},[83,84],[39,29746,155],{"className":29747},[252],[39,29749],{"className":29750,"style":256},[230],[39,29752,11772],{"className":29753,"style":178},[83,84],[39,29755,29625],{"className":29756,"style":2114},[83,84],[39,29758,161],{"className":29759},[263],[39,29761],{"className":29762,"style":231},[230],[39,29764,1824],{"className":29765},[235],[39,29767],{"className":29768,"style":231},[230],[39,29770,29772,29775,29778,29781,29784,29787,29790,29793,29796,29799],{"className":29771},[74],[39,29773],{"className":29774,"style":245},[78],[39,29776,531],{"className":29777,"style":565},[83,84],[39,29779,29425],{"className":29780},[83,84],[39,29782,136],{"className":29783},[182],[39,29785,1755],{"className":29786},[83,84],[39,29788,155],{"className":29789},[252],[39,29791],{"className":29792,"style":256},[230],[39,29794,29625],{"className":29795,"style":2114},[83,84],[39,29797,161],{"className":29798},[263],[39,29800,2903],{"className":29801},[83],[11,29803,14173,29804,29832,29833,29861,29862,29921,29922,30015,30016,30169],{},[39,29805,29807,29820],{"className":29806,"translate":43},[42],[39,29808,29810],{"className":29809},[47],[49,29811,29812],{"xmlns":51},[53,29813,29814,29818],{},[56,29815,29816],{},[59,29817,11772],{},[63,29819,11772],{"encoding":65},[39,29821,29823],{"className":29822,"ariaHidden":70},[69],[39,29824,29826,29829],{"className":29825},[74],[39,29827],{"className":29828,"style":79},[78],[39,29830,11772],{"className":29831,"style":178},[83,84]," 在 ",[39,29834,29836,29849],{"className":29835,"translate":43},[42],[39,29837,29839],{"className":29838},[47],[49,29840,29841],{"xmlns":51},[53,29842,29843,29847],{},[56,29844,29845],{},[59,29846,29420],{},[63,29848,29420],{"encoding":65},[39,29850,29852],{"className":29851,"ariaHidden":70},[69],[39,29853,29855,29858],{"className":29854},[74],[39,29856],{"className":29857,"style":79},[78],[39,29859,29420],{"className":29860,"style":14420},[83,84]," 中存在唯一不动点 ",[39,29863,29865,29883],{"className":29864,"translate":43},[42],[39,29866,29868],{"className":29867},[47],[49,29869,29870],{"xmlns":51},[53,29871,29872,29880],{},[56,29873,29874],{},[138,29875,29876,29878],{},[59,29877,1755],{},[133,29879,21884],{},[63,29881,29882],{"encoding":65},"x^*",[39,29884,29886],{"className":29885,"ariaHidden":70},[69],[39,29887,29889,29892],{"className":29888},[74],[39,29890],{"className":29891,"style":21897},[78],[39,29893,29895,29898],{"className":29894},[83],[39,29896,1755],{"className":29897},[83,84],[39,29899,29901],{"className":29900},[192],[39,29902,29904],{"className":29903},[196],[39,29905,29907],{"className":29906},[200],[39,29908,29910],{"className":29909,"style":21897},[204],[39,29911,29912,29915],{"style":208},[39,29913],{"className":29914,"style":213},[212],[39,29916,29918],{"className":29917},[217,218,219,220],[39,29919,21884],{"className":29920},[465,220],"。此外，对任意初始点 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X",[39,29948,29950,30006],{"className":29949,"ariaHidden":70},[69],[39,29951,29953,29957,29997,30000,30003],{"className":29952},[74],[39,29954],{"className":29955,"style":29956},[78],"height:0.6891em;vertical-align:-0.15em;",[39,29958,29960,29963],{"className":29959},[83],[39,29961,1755],{"className":29962},[83,84],[39,29964,29966],{"className":29965},[192],[39,29967,29969,29989],{"className":29968},[196,649],[39,29970,29972,29986],{"className":29971},[200],[39,29973,29975],{"className":29974,"style":963},[204],[39,29976,29977,29980],{"style":966},[39,29978],{"className":29979,"style":213},[212],[39,29981,29983],{"className":29982},[217,218,219,220],[39,29984,541],{"className":29985},[83,220],[39,29987,673],{"className":29988},[672],[39,29990,29992],{"className":29991},[200],[39,29993,29995],{"className":29994,"style":680},[204],[39,29996],{},[39,29998],{"className":29999,"style":231},[230],[39,30001,534],{"className":30002},[235],[39,30004],{"className":30005,"style":231},[230],[39,30007,30009,30012],{"className":30008},[74],[39,30010],{"className":30011,"style":79},[78],[39,30013,29420],{"className":30014,"style":14420},[83,84],"，迭代 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满足",[39,30171,30173],{"className":30172,"translate":43},[746],[39,30174,30176,30238],{"className":30175,"translate":43},[42],[39,30177,30179],{"className":30178},[47],[49,30180,30181],{"xmlns":51,"display":755},[53,30182,30183,30235],{},[56,30184,30185,30187,30189,30195,30197,30203,30205,30207,30213,30215,30217,30223,30225,30231,30233],{},[59,30186,29425],{},[133,30188,136],{"stretchy":135},[613,30190,30191,30193],{},[59,30192,1755],{},[59,30194,3196],{},[133,30196,155],{"separator":70},[138,30198,30199,30201],{},[59,30200,1755],{},[133,30202,21884],{},[133,30204,161],{"stretchy":135},[133,30206,1824],{},[138,30208,30209,30211],{},[59,30210,531],{},[59,30212,3196],{},[59,30214,29425],{},[133,30216,136],{"stretchy":135},[613,30218,30219,30221],{},[59,30220,1755],{},[539,30222,541],{},[133,30224,155],{"separator":70},[138,30226,30227,30229],{},[59,30228,1755],{},[133,30230,21884],{},[133,30232,161],{"stretchy":135},[59,30234,2903],{"mathvariant":145},[63,30236,30237],{"encoding":65},"d(x_k,x^*)\\le\\gamma^k 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先证唯一性。若 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都是不动点，则",[39,30588,30590],{"className":30589,"translate":43},[746],[39,30591,30593,30677],{"className":30592,"translate":43},[42],[39,30594,30596],{"className":30595},[47],[49,30597,30598],{"xmlns":51,"display":755},[53,30599,30600,30674],{},[56,30601,30602,30604,30606,30612,30614,30620,30622,30624,30626,30628,30630,30636,30638,30640,30646,30648,30650,30652,30654,30656,30662,30664,30670,30672],{},[59,30603,29425],{},[133,30605,136],{"stretchy":135},[138,30607,30608,30610],{},[59,30609,1755],{},[133,30611,21884],{},[133,30613,155],{"separator":70},[138,30615,30616,30618],{},[59,30617,29625],{},[133,30619,21884],{},[133,30621,161],{"stretchy":135},[133,30623,793],{},[59,30625,29425],{},[133,30627,136],{"stretchy":135},[59,30629,11772],{},[138,30631,30632,30634],{},[59,30633,1755],{},[133,30635,21884],{},[133,30637,155],{"separator":70},[59,30639,11772],{},[138,30641,30642,30644],{},[59,30643,29625],{},[133,30645,21884],{},[133,30647,161],{"stretchy":135},[133,30649,1824],{},[59,30651,531],{},[59,30653,29425],{},[133,30655,136],{"stretchy":135},[138,30657,30658,30660],{},[59,30659,1755],{},[133,30661,21884],{},[133,30663,155],{"separator":70},[138,30665,30666,30668],{},[59,30667,29625],{},[133,30669,21884],{},[133,30671,161],{"stretchy":135},[59,30673,2903],{"mathvariant":145},[63,30675,30676],{"encoding":65},"d(x^*,y^*)=d(Tx^*,Ty^*)\\le\\gamma d(x^*,y^*).",[39,30678,30680,30768,30862],{"className":30679,"ariaHidden":70},[69],[39,30681,30683,30686,30689,30692,30721,30724,30727,30756,30759,30762,30765],{"className":30682},[74],[39,30684],{"className":30685,"style":245},[78],[39,30687,29425],{"className":30688},[83,84],[39,30690,136],{"className":30691},[182],[39,30693,30695,30698],{"className":30694},[83],[39,30696,1755],{"className":30697},[83,84],[39,30699,30701],{"className":30700},[192],[39,30702,30704],{"className":30703},[196],[39,30705,30707],{"className":30706},[200],[39,30708,30710],{"className":30709,"style":22242},[204],[39,30711,30712,30715],{"style":1027},[39,30713],{"className":30714,"style":213},[212],[39,30716,30718],{"className":30717},[217,218,219,220],[39,30719,21884],{"className":30720},[465,220],[39,30722,155],{"className":30723},[252],[39,30725],{"className":30726,"style":256},[230],[39,30728,30730,30733],{"className":30729},[83],[39,30731,29625],{"className":30732,"style":2114},[83,84],[39,30734,30736],{"className":30735},[192],[39,30737,30739],{"className":30738},[196],[39,30740,30742],{"className":30741},[200],[39,30743,30745],{"className":30744,"style":22242},[204],[39,30746,30747,30750],{"style":1027},[39,30748],{"className":30749,"style":213},[212],[39,30751,30753],{"className":30752},[217,218,219,220],[39,30754,21884],{"className":30755},[465,220],[39,30757,161],{"className":30758},[263],[39,30760],{"className":30761,"style":231},[230],[39,30763,793],{"className":30764},[235],[39,30766],{"className":30767,"style":231},[230],[39,30769,30771,30774,30777,30780,30783,30812,30815,30818,30821,30850,30853,30856,30859],{"className":30770},[74],[39,30772],{"className":30773,"style":245},[78],[39,30775,29425],{"className":30776},[83,84],[39,30778,136],{"className":30779},[182],[39,30781,11772],{"className":30782,"style":178},[83,84],[39,30784,30786,30789],{"className":30785},[83],[39,30787,1755],{"className":30788},[83,84],[39,30790,30792],{"className":30791},[192],[39,30793,30795],{"className":30794},[196],[39,30796,30798],{"className":30797},[200],[39,30799,30801],{"className":30800,"style":22242},[204],[39,30802,30803,30806],{"style":1027},[39,30804],{"className":30805,"style":213},[212],[39,30807,30809],{"className":30808},[217,218,219,220],[39,30810,21884],{"className":30811},[465,220],[39,30813,155],{"className":30814},[252],[39,30816],{"className":30817,"style":256},[230],[39,30819,11772],{"className":30820,"style":178},[83,84],[39,30822,30824,30827],{"className":30823},[83],[39,30825,29625],{"className":30826,"style":2114},[83,84],[39,30828,30830],{"className":30829},[192],[39,30831,30833],{"className":30832},[196],[39,30834,30836],{"className":30835},[200],[39,30837,30839],{"className":30838,"style":22242},[204],[39,30840,30841,30844],{"style":1027},[39,30842],{"className":30843,"style":213},[212],[39,30845,30847],{"className":30846},[217,218,219,220],[39,30848,21884],{"className":30849},[465,220],[39,30851,161],{"className":30852},[263],[39,30854],{"className":30855,"style":231},[230],[39,30857,1824],{"className":30858},[235],[39,30860],{"className":30861,"style":231},[230],[39,30863,30865,30868,30871,30874,30877,30906,30909,30912,30941,30944],{"className":30864},[74],[39,30866],{"className":30867,"style":245},[78],[39,30869,531],{"className":30870,"style":565},[83,84],[39,30872,29425],{"className":30873},[83,84],[39,30875,136],{"className":30876},[182],[39,30878,30880,30883],{"className":30879},[83],[39,30881,1755],{"className":30882},[83,84],[39,30884,30886],{"className":30885},[192],[39,30887,30889],{"className":30888},[196],[39,30890,30892],{"className":30891},[200],[39,30893,30895],{"className":30894,"style":22242},[204],[39,30896,30897,30900],{"style":1027},[39,30898],{"className":30899,"style":213},[212],[39,30901,30903],{"className":30902},[217,218,219,220],[39,30904,21884],{"className":30905},[465,220],[39,30907,155],{"className":30908},[252],[39,30910],{"className":30911,"style":256},[230],[39,30913,30915,30918],{"className":30914},[83],[39,30916,29625],{"className":30917,"style":2114},[83,84],[39,30919,30921],{"className":30920},[192],[39,30922,30924],{"className":30923},[196],[39,30925,30927],{"className":30926},[200],[39,30928,30930],{"className":30929,"style":22242},[204],[39,30931,30932,30935],{"style":1027},[39,30933],{"className":30934,"style":213},[212],[39,30936,30938],{"className":30937},[217,218,219,220],[39,30939,21884],{"className":30940},[465,220],[39,30942,161],{"className":30943},[263],[39,30945,2903],{"className":30946},[83],[11,30948,20571,30949,30999,31000,31139,31140,1607],{},[39,30950,30952,30969],{"className":30951,"translate":43},[42],[39,30953,30955],{"className":30954},[47],[49,30956,30957],{"xmlns":51},[53,30958,30959,30967],{},[56,30960,30961,30963,30965],{},[59,30962,531],{},[133,30964,1693],{},[539,30966,546],{},[63,30968,17296],{"encoding":65},[39,30970,30972,30990],{"className":30971,"ariaHidden":70},[69],[39,30973,30975,30978,30981,30984,30987],{"className":30974},[74],[39,30976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",[39,31001,31003,31039],{"className":31002,"translate":43},[42],[39,31004,31006],{"className":31005},[47],[49,31007,31008],{"xmlns":51},[53,31009,31010,31036],{},[56,31011,31012,31014,31016,31022,31024,31030,31032,31034],{},[59,31013,29425],{},[133,31015,136],{"stretchy":135},[138,31017,31018,31020],{},[59,31019,1755],{},[133,31021,21884],{},[133,31023,155],{"separator":70},[138,31025,31026,31028],{},[59,31027,29625],{},[133,31029,21884],{},[133,31031,161],{"stretchy":135},[133,31033,793],{},[539,31035,541],{},[63,31037,31038],{"encoding":65},"d(x^*,y^*)=0",[39,31040,31042,31130],{"className":31041,"ariaHidden":70},[69],[39,31043,31045,31048,31051,31054,31083,31086,31089,31118,31121,31124,31127],{"className":31044},[74],[39,31046],{"className":31047,"style":245},[78],[39,31049,29425],{"className":31050},[83,84],[39,31052,136],{"className":31053},[182],[39,31055,31057,31060],{"className":31056},[83],[39,31058,1755],{"className":31059},[83,84],[39,31061,31063],{"className":31062},[192],[39,31064,31066],{"className":31065},[196],[39,31067,31069],{"className":31068},[200],[39,31070,31072],{"className":31071,"style":21897},[204],[39,31073,31074,31077],{"style":208},[39,31075],{"className":31076,"style":213},[212],[39,31078,31080],{"className":31079},[217,218,219,220],[39,31081,21884],{"className":31082},[465,220],[39,31084,155],{"className":31085},[252],[39,31087],{"className":31088,"style":256},[230],[39,31090,31092,31095],{"className":31091},[83],[39,31093,29625],{"className":31094,"style":2114},[83,84],[39,31096,31098],{"className":31097},[192],[39,31099,31101],{"className":31100},[196],[39,31102,31104],{"className":31103},[200],[39,31105,31107],{"className":31106,"style":21897},[204],[39,31108,31109,31112],{"style":208},[39,31110],{"className":31111,"style":213},[212],[39,31113,31115],{"className":31114},[217,218,219,220],[39,31116,21884],{"className":31117},[465,220],[39,31119,161],{"className":31120},[263],[39,31122],{"className":31123,"style":231},[230],[39,31125,793],{"className":31126},[235],[39,31128],{"className":31129,"style":231},[230],[39,31131,31133,31136],{"className":31132},[74],[39,31134],{"className":31135,"style":2473},[78],[39,31137,541],{"className":31138},[83],"，即 ",[39,31141,31143,31169],{"className":31142,"translate":43},[42],[39,31144,31146],{"className":31145},[47],[49,31147,31148],{"xmlns":51},[53,31149,31150,31166],{},[56,31151,31152,31158,31160],{},[138,31153,31154,31156],{},[59,31155,1755],{},[133,31157,21884],{},[133,31159,793],{},[138,31161,31162,31164],{},[59,31163,29625],{},[133,31165,21884],{},[63,31167,31168],{"encoding":65},"x^*=y^*",[39,31170,31172,31216],{"className":31171,"ariaHidden":70},[69],[39,31173,31175,31178,31207,31210,31213],{"className":31174},[74],[39,31176],{"className":31177,"style":21897},[78],[39,31179,31181,31184],{"className":31180},[83],[39,31182,1755],{"className":31183},[83,84],[39,31185,31187],{"className":31186},[192],[39,31188,31190],{"className":31189},[196],[39,31191,31193],{"className":31192},[200],[39,31194,31196],{"className":31195,"style":21897},[204],[39,31197,31198,31201],{"style":208},[39,31199],{"className":31200,"style":213},[212],[39,31202,31204],{"className":31203},[217,218,219,220],[39,31205,21884],{"className":31206},[465,220],[39,31208],{"className":31209,"style":231},[230],[39,31211,793],{"className":31212},[235],[39,31214],{"className":31215,"style":231},[230],[39,31217,31219,31222],{"className":31218},[74],[39,31220],{"className":31221,"style":30556},[78],[39,31223,31225,31228],{"className":31224},[83],[39,31226,29625],{"className":31227,"style":2114},[83,84],[39,31229,31231],{"className":31230},[192],[39,31232,31234],{"className":31233},[196],[39,31235,31237],{"className":31236},[200],[39,31238,31240],{"className":31239,"style":21897},[204],[39,31241,31242,31245],{"style":208},[39,31243],{"className":31244,"style":213},[212],[39,31246,31248],{"className":31247},[217,218,219,220],[39,31249,21884],{"className":31250},[465,220],[11,31252,31253,31254,31345,31346,23615,31498,31550],{},"再证存在性。任取 ",[39,31255,31257,31278],{"className":31256,"translate":43},[42],[39,31258,31260],{"className":31259},[47],[49,31261,31262],{"xmlns":51},[53,31263,31264,31276],{},[56,31265,31266,31272,31274],{},[613,31267,31268,31270],{},[59,31269,1755],{},[539,31271,541],{},[133,31273,534],{},[59,31275,29420],{},[63,31277,29946],{"encoding":65},[39,31279,31281,31336],{"className":31280,"ariaHidden":70},[69],[39,31282,31284,31287,31327,31330,31333],{"className":31283},[74],[39,31285],{"className":31286,"style":29956},[78],[39,31288,31290,31293],{"className":31289},[83],[39,31291,1755],{"className":31292},[83,84],[39,31294,31296],{"className":31295},[192],[39,31297,31299,31319],{"className":31298},[196,649],[39,31300,31302,31316],{"className":31301},[200],[39,31303,31305],{"className":31304,"style":963},[204],[39,31306,31307,31310],{"style":966},[39,31308],{"className":31309,"style":213},[212],[39,31311,31313],{"className":31312},[217,218,219,220],[39,31314,541],{"className":31315},[83,220],[39,31317,673],{"className":31318},[672],[39,31320,31322],{"className":31321},[200],[39,31323,31325],{"className":31324,"style":680},[204],[39,31326],{},[39,31328],{"className":31329,"style":231},[230],[39,31331,534],{"className":31332},[235],[39,31334],{"className":31335,"style":231},[230],[39,31337,31339,31342],{"className":31338},[74],[39,31340],{"className":31341,"style":79},[78],[39,31343,29420],{"className":31344,"style":14420},[83,84],"，令 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是 Cauchy 列。由 ",[39,32646,32648,32661],{"className":32647,"translate":43},[42],[39,32649,32651],{"className":32650},[47],[49,32652,32653],{"xmlns":51},[53,32654,32655,32659],{},[56,32656,32657],{},[59,32658,29420],{},[63,32660,29420],{"encoding":65},[39,32662,32664],{"className":32663,"ariaHidden":70},[69],[39,32665,32667,32670],{"className":32666},[74],[39,32668],{"className":32669,"style":79},[78],[39,32671,29420],{"className":32672,"style":14420},[83,84]," 完备，存在 ",[39,32675,32677,32699],{"className":32676,"translate":43},[42],[39,32678,32680],{"className":32679},[47],[49,32681,32682],{"xmlns":51},[53,32683,32684,32696],{},[56,32685,32686,32692,32694],{},[138,32687,32688,32690],{},[59,32689,1755],{},[133,32691,21884],{},[133,32693,534],{},[59,32695,29420],{},[63,32697,32698],{"encoding":65},"x^*\\in 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x^*",[39,32786,32788,32843],{"className":32787,"ariaHidden":70},[69],[39,32789,32791,32794,32834,32837,32840],{"className":32790},[74],[39,32792],{"className":32793,"style":2916},[78],[39,32795,32797,32800],{"className":32796},[83],[39,32798,1755],{"className":32799},[83,84],[39,32801,32803],{"className":32802},[192],[39,32804,32806,32826],{"className":32805},[196,649],[39,32807,32809,32823],{"className":32808},[200],[39,32810,32812],{"className":32811,"style":1728},[204],[39,32813,32814,32817],{"style":966},[39,32815],{"className":32816,"style":213},[212],[39,32818,32820],{"className":32819},[217,218,219,220],[39,32821,12962],{"className":32822},[83,84,220],[39,32824,673],{"className":32825},[672],[39,32827,32829],{"className":32828},[200],[39,32830,32832],{"className":32831,"style":680},[204],[39,32833],{},[39,32835],{"className":32836,"style":231},[230],[39,32838,418],{"className":32839},[235],[39,32841],{"className":32842,"style":231},[230],[39,32844,32846,32849],{"className":32845},[74],[39,32847],{"className":32848,"style":21897},[78],[39,32850,32852,32855],{"className":32851},[83],[39,32853,1755],{"className":32854},[83,84],[39,32856,32858],{"className":32857},[192],[39,32859,32861],{"className":32860},[196],[39,32862,32864],{"className":32863},[200],[39,32865,32867],{"className":32866,"style":21897},[204],[39,32868,32869,32872],{"style":208},[39,32870],{"className":32871,"style":213},[212],[39,32873,32875],{"className":32874},[217,218,219,220],[39,32876,21884],{"className":32877},[465,220],"。算子 ",[39,32880,32882,32895],{"className":32881,"translate":43},[42],[39,32883,32885],{"className":32884},[47],[49,32886,32887],{"xmlns":51},[53,32888,32889,32893],{},[56,32890,32891],{},[59,32892,11772],{},[63,32894,11772],{"encoding":65},[39,32896,32898],{"className":32897,"ariaHidden":70},[69],[39,32899,32901,32904],{"className":32900},[74],[39,32902],{"className":32903,"style":79},[78],[39,32905,11772],{"className":32906,"style":178},[83,84]," 是 Lipschitz 连续的，故",[39,32909,32911],{"className":32910,"translate":43},[746],[39,32912,32914,33065],{"className":32913,"translate":43},[42],[39,32915,32917],{"className":32916},[47],[49,32918,32919],{"xmlns":51,"display":755},[53,32920,32921,33062],{},[759,32922,32923,33006],{"rowspacing":761,"columnalign":762,"columnspacing":146},[764,32924,32925,32939],{},[767,32926,32927],{},[770,32928,32929],{"scriptlevel":541,"displaystyle":70},[56,32930,32931,32933],{},[59,32932,11772],{},[138,32934,32935,32937],{},[59,32936,1755],{},[133,32938,21884],{},[767,32940,32941],{},[770,32942,32943],{"scriptlevel":541,"displaystyle":70},[56,32944,32945,32947,32949,32951,32980,32982,32998,33000],{},[56,32946],{},[133,32948,793],{},[59,32950,11772],{},[56,32952,32953,32955,32972,32978],{},[133,32954,136],{"fence":70},[6556,32956,32957,32964],{},[56,32958,32959,32962],{},[59,32960,32961],{},"lim",[133,32963,1690],{},[56,32965,32966,32968,32970],{},[59,32967,12962],{},[133,32969,418],{},[59,32971,1696],{"mathvariant":145},[613,32973,32974,32976],{},[59,32975,1755],{},[59,32977,12962],{},[133,32979,161],{"fence":70},[133,32981,793],{},[6556,32983,32984,32990],{},[56,32985,32986,32988],{},[59,32987,32961],{},[133,32989,1690],{},[56,32991,32992,32994,32996],{},[59,32993,12962],{},[133,32995,418],{},[59,32997,1696],{"mathvariant":145},[59,32999,11772],{},[613,33001,33002,33004],{},[59,33003,1755],{},[59,33005,12962],{},[764,33007,33008,33014],{},[767,33009,33010],{},[770,33011,33012],{"scriptlevel":541,"displaystyle":70},[56,33013],{},[767,33015,33016],{},[770,33017,33018],{"scriptlevel":541,"displaystyle":70},[56,33019,33020,33022,33024,33040,33052,33054,33060],{},[56,33021],{},[133,33023,793],{},[6556,33025,33026,33032],{},[56,33027,33028,33030],{},[59,33029,32961],{},[133,33031,1690],{},[56,33033,33034,33036,33038],{},[59,33035,12962],{},[133,33037,418],{},[59,33039,1696],{"mathvariant":145},[613,33041,33042,33044],{},[59,33043,1755],{},[56,33045,33046,33048,33050],{},[59,33047,12962],{},[133,33049,788],{},[539,33051,546],{},[133,33053,793],{},[138,33055,33056,33058],{},[59,33057,1755],{},[133,33059,21884],{},[59,33061,2903],{"mathvariant":145},[63,33063,33064],{"encoding":65},"\\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}",[39,33066,33068],{"className":33067,"ariaHidden":70},[69],[39,33069,33071,33075],{"className":33070},[74],[39,33072],{"className":33073,"style":33074},[78],"height:3.69em;vertical-align:-1.595em;",[39,33076,33078],{"className":33077},[83],[39,33079,33081,33157],{"className":33080},[759],[39,33082,33084],{"className":33083},[918],[39,33085,33087,33148],{"className":33086},[196,649],[39,33088,33090,33145],{"className":33089},[200],[39,33091,33094,33136],{"className":33092,"style":33093},[204],"height:2.095em;",[39,33095,33097,33101],{"style":33096},"top:-4.095em;",[39,33098],{"className":33099,"style":33100},[212],"height:3.15em;",[39,33102,33104,33107],{"className":33103},[83],[39,33105,11772],{"className":33106,"style":178},[83,84],[39,33108,33110,33113],{"className":33109},[83],[39,33111,1755],{"className":33112},[83,84],[39,33114,33116],{"className":33115},[192],[39,33117,33119],{"className":33118},[196],[39,33120,33122],{"className":33121},[200],[39,33123,33125],{"className":33124,"style":22242},[204],[39,33126,33127,33130],{"style":1027},[39,33128],{"className":33129,"style":213},[212],[39,33131,33133],{"className":33132},[217,218,219,220],[39,33134,21884],{"className":33135},[465,220],[39,33137,33139,33142],{"style":33138},"top:-2.255em;",[39,33140],{"className":33141,"style":33100},[212],[39,33143],{"className":33144},[83],[39,33146,673],{"className":33147},[672],[39,33149,33151],{"className":33150},[200],[39,33152,33155],{"className":33153,"style":33154},[204],"height:1.595em;",[39,33156],{},[39,33158,33160],{"className":33159},[1223],[39,33161,33163,33584],{"className":33162},[196,649],[39,33164,33166,33581],{"className":33165},[200],[39,33167,33169,33415],{"className":33168,"style":33093},[204],[39,33170,33171,33174],{"style":33096},[39,33172],{"className":33173,"style":33100},[212],[39,33175,33177,33180,33183,33186,33189,33192,33195,33307,33310,33313,33316,33369,33372,33375],{"className":33176},[83],[39,33178],{"className":33179},[83],[39,33181],{"className":33182,"style":231},[230],[39,33184,793],{"className":33185},[235],[39,33187],{"className":33188,"style":231},[230],[39,33190,11772],{"className":33191,"style":178},[83,84],[39,33193],{"className":33194,"style":256},[230],[39,33196,33198,33205,33258,33261,33301],{"className":33197},[2842],[39,33199,33201],{"className":33200,"style":4592},[182,4591],[39,33202,136],{"className":33203},[18520,33204],"size2",[39,33206,33208],{"className":33207},[1744,3294],[39,33209,33211,33250],{"className":33210},[196,649],[39,33212,33214,33247],{"className":33213},[200],[39,33215,33217,33237],{"className":33216,"style":28824},[204],[39,33218,33219,33222],{"style":22299},[39,33220],{"className":33221,"style":935},[212],[39,33223,33225],{"className":33224},[217,218,219,220],[39,33226,33228,33231,33234],{"className":33227},[83,220],[39,33229,12962],{"className":33230},[83,84,220],[39,33232,418],{"className":33233},[235,220],[39,33235,1696],{"className":33236},[83,220],[39,33238,33239,33242],{"style":21599},[39,33240],{"className":33241,"style":935},[212],[39,33243,33244],{},[39,33245,32961],{"className":33246},[1744],[39,33248,673],{"className":33249},[672],[39,33251,33253],{"className":33252},[200],[39,33254,33256],{"className":33255,"style":22331},[204],[39,33257],{},[39,33259],{"className":33260,"style":256},[230],[39,33262,33264,33267],{"className":33263},[83],[39,33265,1755],{"className":33266},[83,84],[39,33268,33270],{"className":33269},[192],[39,33271,33273,33293],{"className":33272},[196,649],[39,33274,33276,33290],{"className":33275},[200],[39,33277,33279],{"className":33278,"style":1728},[204],[39,33280,33281,33284],{"style":966},[39,33282],{"className":33283,"style":213},[212],[39,33285,33287],{"className":33286},[217,218,219,220],[39,33288,12962],{"className":33289},[83,84,220],[39,33291,673],{"className":33292},[672],[39,33294,33296],{"className":33295},[200],[39,33297,33299],{"className":33298,"style":680},[204],[39,33300],{},[39,33302,33304],{"className":33303,"style":4592},[263,4591],[39,33305,161],{"className":33306},[18520,33204],[39,33308],{"className":33309,"style":231},[230],[39,33311,793],{"className":33312},[235],[39,33314],{"className":33315,"style":231},[230],[39,33317,33319],{"className":33318},[1744,3294],[39,33320,33322,33361],{"className":33321},[196,649],[39,33323,33325,33358],{"className":33324},[200],[39,33326,33328,33348],{"className":33327,"style":28824},[204],[39,33329,33330,33333],{"style":22299},[39,33331],{"className":33332,"style":935},[212],[39,33334,33336],{"className":33335},[217,218,219,220],[39,33337,33339,33342,33345],{"className":33338},[83,220],[39,33340,12962],{"className":33341},[83,84,220],[39,33343,418],{"className":33344},[235,220],[39,33346,1696],{"className":33347},[83,220],[39,33349,33350,33353],{"style":21599},[39,33351],{"className":33352,"style":935},[212],[39,33354,33355],{},[39,33356,32961],{"className":33357},[1744],[39,33359,673],{"className":33360},[672],[39,33362,33364],{"className":33363},[200],[39,33365,33367],{"className":33366,"style":22331},[204],[39,33368],{},[39,33370],{"className":33371,"style":256},[230],[39,33373,11772],{"className":33374,"style":178},[83,84],[39,33376,33378,33381],{"className":33377},[83],[39,33379,1755],{"className":33380},[83,84],[39,33382,33384],{"className":33383},[192],[39,33385,33387,33407],{"className":33386},[196,649],[39,33388,33390,33404],{"className":33389},[200],[39,33391,33393],{"className":33392,"style":1728},[204],[39,33394,33395,33398],{"style":966},[39,33396],{"className":33397,"style":213},[212],[39,33399,33401],{"className":33400},[217,218,219,220],[39,33402,12962],{"className":33403},[83,84,220],[39,33405,673],{"className":33406},[672],[39,33408,33410],{"className":33409},[200],[39,33411,33413],{"className":33412,"style":680},[204],[39,33414],{},[39,33416,33417,33420],{"style":33138},[39,33418],{"className":33419,"style":33100},[212],[39,33421,33423,33426,33429,33432,33435,33488,33491,33540,33543,33546,33549,33578],{"className":33422},[83],[39,33424],{"className":33425},[83],[39,33427],{"className":33428,"style":231},[230],[39,33430,793],{"className":33431},[235],[39,33433],{"className":33434,"style":231},[230],[39,33436,33438],{"className":33437},[1744,3294],[39,33439,33441,33480],{"className":33440},[196,649],[39,33442,33444,33477],{"className":33443},[200],[39,33445,33447,33467],{"className":33446,"style":28824},[204],[39,33448,33449,33452],{"style":22299},[39,33450],{"className":33451,"style":935},[212],[39,33453,33455],{"className":33454},[217,218,219,220],[39,33456,33458,33461,33464],{"className":33457},[83,220],[39,33459,12962],{"className":33460},[83,84,220],[39,33462,418],{"className":33463},[235,220],[39,33465,1696],{"className":33466},[83,220],[39,33468,33469,33472],{"style":21599},[39,33470],{"className":33471,"style":935},[212],[39,33473,33474],{},[39,33475,32961],{"className":33476},[1744],[39,33478,673],{"className":33479},[672],[39,33481,33483],{"className":33482},[200],[39,33484,33486],{"className":33485,"style":22331},[204],[39,33487],{},[39,33489],{"className":33490,"style":256},[230],[39,33492,33494,33497],{"className":33493},[83],[39,33495,1755],{"className":33496},[83,84],[39,33498,33500],{"className":33499},[192],[39,33501,33503,33532],{"className":33502},[196,649],[39,33504,33506,33529],{"className":33505},[200],[39,33507,33509],{"className":33508,"style":963},[204],[39,33510,33511,33514],{"style":966},[39,33512],{"className":33513,"style":213},[212],[39,33515,33517],{"className":33516},[217,218,219,220],[39,33518,33520,33523,33526],{"className":33519},[83,220],[39,33521,12962],{"className":33522},[83,84,220],[39,33524,788],{"className":33525},[465,220],[39,33527,546],{"className":33528},[83,220],[39,33530,673],{"className":33531},[672],[39,33533,33535],{"className":33534},[200],[39,33536,33538],{"className":33537,"style":994},[204],[39,33539],{},[39,33541],{"className":33542,"style":231},[230],[39,33544,793],{"className":33545},[235],[39,33547],{"className":33548,"style":231},[230],[39,33550,33552,33555],{"className":33551},[83],[39,33553,1755],{"className":33554},[83,84],[39,33556,33558],{"className":33557},[192],[39,33559,33561],{"className":33560},[196],[39,33562,33564],{"className":33563},[200],[39,33565,33567],{"className":33566,"style":22242},[204],[39,33568,33569,33572],{"style":1027},[39,33570],{"className":33571,"style":213},[212],[39,33573,33575],{"className":33574},[217,218,219,220],[39,33576,21884],{"className":33577},[465,220],[39,33579,2903],{"className":33580},[83],[39,33582,673],{"className":33583},[672],[39,33585,33587],{"className":33586},[200],[39,33588,33590],{"className":33589,"style":33154},[204],[39,33591],{},[11,33593,33594],{},"最后，由压缩性直接得到",[39,33596,33598],{"className":33597,"translate":43},[746],[39,33599,33601,33697],{"className":33600,"translate":43},[42],[39,33602,33604],{"className":33603},[47],[49,33605,33606],{"xmlns":51,"display":755},[53,33607,33608,33694],{},[56,33609,33610,33612,33614,33620,33622,33628,33630,33632,33634,33636,33642,33648,33650,33656,33662,33664,33666,33672,33674,33676,33682,33684,33690,33692],{},[59,33611,29425],{},[133,33613,136],{"stretchy":135},[613,33615,33616,33618],{},[59,33617,1755],{},[59,33619,3196],{},[133,33621,155],{"separator":70},[138,33623,33624,33626],{},[59,33625,1755],{},[133,33627,21884],{},[133,33629,161],{"stretchy":135},[133,33631,793],{},[59,33633,29425],{},[133,33635,136],{"stretchy":135},[138,33637,33638,33640],{},[59,33639,11772],{},[59,33641,3196],{},[613,33643,33644,33646],{},[59,33645,1755],{},[539,33647,541],{},[133,33649,155],{"separator":70},[138,33651,33652,33654],{},[59,33653,11772],{},[59,33655,3196],{},[138,33657,33658,33660],{},[59,33659,1755],{},[133,33661,21884],{},[133,33663,161],{"stretchy":135},[133,33665,1824],{},[138,33667,33668,33670],{},[59,33669,531],{},[59,33671,3196],{},[59,33673,29425],{},[133,33675,136],{"stretchy":135},[613,33677,33678,33680],{},[59,33679,1755],{},[539,33681,541],{},[133,33683,155],{"separator":70},[138,33685,33686,33688],{},[59,33687,1755],{},[133,33689,21884],{},[133,33691,161],{"stretchy":135},[59,33693,2903],{"mathvariant":145},[63,33695,33696],{"encoding":65},"d(x_k,x^*)=d(T^kx_0,T^kx^*)\\le\\gamma^k d(x_0,x^*).",[39,33698,33700,33799,33956],{"className":33699,"ariaHidden":70},[69],[39,33701,33703,33706,33709,33712,33752,33755,33758,33787,33790,33793,33796],{"className":33702},[74],[39,33704],{"className":33705,"style":245},[78],[39,33707,29425],{"className":33708},[83,84],[39,33710,136],{"className":33711},[182],[39,33713,33715,33718],{"className":33714},[83],[39,33716,1755],{"className":33717},[83,84],[39,33719,33721],{"className":33720},[192],[39,33722,33724,33744],{"className":33723},[196,649],[39,33725,33727,33741],{"className":33726},[200],[39,33728,33730],{"className":33729,"style":3419},[204],[39,33731,33732,33735],{"style":966},[39,33733],{"className":33734,"style":213},[212],[39,33736,33738],{"className":33737},[217,218,219,220],[39,33739,3196],{"className":33740,"style":3321},[83,84,220],[39,33742,673],{"className":33743},[672],[39,33745,33747],{"className":33746},[200],[39,33748,33750],{"className":33749,"style":680},[204],[39,33751],{},[39,33753,155],{"className":33754},[252],[39,33756],{"className":33757,"style":256},[230],[39,33759,33761,33764],{"className":33760},[83],[39,33762,1755],{"className":33763},[83,84],[39,33765,33767],{"className":33766},[192],[39,33768,33770],{"className":33769},[196],[39,33771,33773],{"className":33772},[200],[39,33774,33776],{"className":33775,"style":22242},[204],[39,33777,33778,33781],{"style":1027},[39,33779],{"className":33780,"style":213},[212],[39,33782,33784],{"className":33783},[217,218,219,220],[39,33785,21884],{"className":33786},[465,220],[39,33788,161],{"className":33789},[263],[39,33791],{"className":33792,"style":231},[230],[39,33794,793],{"className":33795},[235],[39,33797],{"className":33798,"style":231},[230],[39,33800,33802,33805,33808,33811,33840,33880,33883,33886,33915,33944,33947,33950,33953],{"className":33801},[74],[39,33803],{"className":33804,"style":30346},[78],[39,33806,29425],{"className":33807},[83,84],[39,33809,136],{"className":33810},[182],[39,33812,33814,33817],{"className":33813},[83],[39,33815,11772],{"className":33816,"style":178},[83,84],[39,33818,33820],{"className":33819},[192],[39,33821,33823],{"className":33822},[196],[39,33824,33826],{"className":33825},[200],[39,33827,33829],{"className":33828,"style":3389},[204],[39,33830,33831,33834],{"style":1027},[39,33832],{"className":33833,"style":213},[212],[39,33835,33837],{"className":33836},[217,218,219,220],[39,33838,3196],{"className":33839,"style":3321},[83,84,220],[39,33841,33843,33846],{"className":33842},[83],[39,33844,1755],{"className":33845},[83,84],[39,33847,33849],{"className":33848},[192],[39,33850,33852,33872],{"className":33851},[196,649],[39,33853,33855,33869],{"className":33854},[200],[39,33856,33858],{"className":33857,"style":963},[204],[39,33859,33860,33863],{"style":966},[39,33861],{"className":33862,"style":213},[212],[39,33864,33866],{"className":33865},[217,218,219,220],[39,33867,541],{"className":33868},[83,220],[39,33870,673],{"className":33871},[672],[39,33873,33875],{"className":33874},[200],[39,33876,33878],{"className":33877,"style":680},[204],[39,33879],{},[39,33881,155],{"className":33882},[252],[39,33884],{"className":33885,"style":256},[230],[39,33887,33889,33892],{"className":33888},[83],[39,33890,11772],{"className":33891,"style":178},[83,84],[39,33893,33895],{"className":33894},[192],[39,33896,33898],{"className":33897},[196],[39,33899,33901],{"className":33900},[200],[39,33902,33904],{"className":33903,"style":3389},[204],[39,33905,33906,33909],{"style":1027},[39,33907],{"className":33908,"style":213},[212],[39,33910,33912],{"className":33911},[217,218,219,220],[39,33913,3196],{"className":33914,"style":3321},[83,84,220],[39,33916,33918,33921],{"className":33917},[83],[39,33919,1755],{"className":33920},[83,84],[39,33922,33924],{"className":33923},[192],[39,33925,33927],{"className":33926},[196],[39,33928,33930],{"className":33929},[200],[39,33931,33933],{"className":33932,"style":22242},[204],[39,33934,33935,33938],{"style":1027},[39,33936],{"className":33937,"style":213},[212],[39,33939,33941],{"className":33940},[217,218,219,220],[39,33942,21884],{"className":33943},[465,220],[39,33945,161],{"className":33946},[263],[39,33948],{"className":33949,"style":231},[230],[39,33951,1824],{"className":33952},[235],[39,33954],{"className":33955,"style":231},[230],[39,33957,33959,33962,33991,33994,33997,34037,34040,34043,34072,34075],{"className":33958},[74],[39,33960],{"className":33961,"style":30346},[78],[39,33963,33965,33968],{"className":33964},[83],[39,33966,531],{"className":33967,"style":565},[83,84],[39,33969,33971],{"className":33970},[192],[39,33972,33974],{"className":33973},[196],[39,33975,33977],{"className":33976},[200],[39,33978,33980],{"className":33979,"style":3389},[204],[39,33981,33982,33985],{"style":1027},[39,33983],{"className":33984,"style":213},[212],[39,33986,33988],{"className":33987},[217,218,219,220],[39,33989,3196],{"className":33990,"style":3321},[83,84,220],[39,33992,29425],{"className":33993},[83,84],[39,33995,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唯一性与上下界",[31,34085,34088],{"id":34086,"title":34087},"corollary-unique-fixed-point","推论 3.4（不动点的存在唯一性）",[11,34089,37,34090,14627,34162,1639,34220,29832,34278,34321],{},[39,34091,34093,34114],{"className":34092,"translate":43},[42],[39,34094,34096],{"className":34095},[47],[49,34097,34098],{"xmlns":51},[53,34099,34100,34112],{},[56,34101,34102,34104,34106,34108,34110],{},[539,34103,541],{},[133,34105,1824],{},[59,34107,531],{},[133,34109,1693],{},[539,34111,546],{},[63,34113,14577],{"encoding":65},[39,34115,34117,34135,34153],{"className":34116,"ariaHidden":70},[69],[39,34118,34120,34123,34126,34129,34132],{"className":34119},[74],[39,34121],{"className":34122,"style":14587},[78],[39,34124,541],{"className":34125},[83],[39,34127],{"className":34128,"style":231},[230],[39,34130,1824],{"className":34131},[235],[39,34133],{"className":34134,"style":231},[230],[39,34136,34138,34141,34144,34147,34150],{"className":34137},[74],[39,34139],{"className":34140,"style":561},[78],[39,34142,531],{"className":34143,"style":565},[83,84],[39,34145],{"className":34146,"style":231},[230],[39,34148,1693],{"className":34149},[235],[39,34151],{"className":34152,"style":231},[230],[39,34154,34156,34159],{"className":34155},[74],[39,34157],{"className":34158,"style":2473},[78],[39,34160,546],{"className":34161},[83],[39,34163,34165,34182],{"className":34164,"translate":43},[42],[39,34166,34168],{"className":34167},[47],[49,34169,34170],{"xmlns":51},[53,34171,34172,34180],{},[56,34173,34174],{},[138,34175,34176,34178],{},[59,34177,11772],{"mathvariant":617},[59,34179,2098],{},[63,34181,11777],{"encoding":65},[39,34183,34185],{"className":34184,"ariaHidden":70},[69],[39,34186,34188,34191],{"className":34187},[74],[39,34189],{"className":34190,"style":79},[78],[39,34192,34194,34197],{"className":34193},[83],[39,34195,11772],{"className":34196,"style":11793},[83,641],[39,34198,34200],{"className":34199},[192],[39,34201,34203],{"className":34202},[196],[39,34204,34206],{"className":34205},[200],[39,34207,34209],{"className":34208,"style":6410},[204],[39,34210,34211,34214],{"style":208},[39,34212],{"className":34213,"style":213},[212],[39,34215,34217],{"className":34216},[217,218,219,220],[39,34218,2098],{"className":34219,"style":2114},[83,84,220],[39,34221,34223,34240],{"className":34222,"translate":43},[42],[39,34224,34226],{"className":34225},[47],[49,34227,34228],{"xmlns":51},[53,34229,34230,34238],{},[56,34231,34232],{},[138,34233,34234,34236],{},[59,34235,11772],{"mathvariant":617},[133,34237,21884],{},[63,34239,21887],{"encoding":65},[39,34241,34243],{"className":34242,"ariaHidden":70},[69],[39,34244,34246,34249],{"className":34245},[74],[39,34247],{"className":34248,"style":21897},[78],[39,34250,34252,34255],{"className":34251},[83],[39,34253,11772],{"className":34254,"style":11793},[83,641],[39,34256,34258],{"className":34257},[192],[39,34259,34261],{"className":34260},[196],[39,34262,34264],{"className":34263},[200],[39,34265,34267],{"className":34266,"style":21897},[204],[39,34268,34269,34272],{"style":208},[39,34270],{"className":34271,"style":213},[212],[39,34273,34275],{"className":34274},[217,218,219,220],[39,34276,21884],{"className":34277},[465,220],[39,34279,34281,34300],{"className":34280,"translate":43},[42],[39,34282,34284],{"className":34283},[47],[49,34285,34286],{"xmlns":51},[53,34287,34288,34298],{},[56,34289,34290,34292,34294,34296],{},[59,34291,19772],{},[133,34293,136],{"stretchy":135},[59,34295,61],{},[133,34297,161],{"stretchy":135},[63,34299,21380],{"encoding":65},[39,34301,34303],{"className":34302,"ariaHidden":70},[69],[39,34304,34306,34309,34312,34315,34318],{"className":34305},[74],[39,34307],{"className":34308,"style":245},[78],[39,34310,19772],{"className":34311,"style":19812},[83,84],[39,34313,136],{"className":34314},[182],[39,34316,61],{"className":34317,"style":85},[83,84],[39,34319,161],{"className":34320},[263]," 中各自存在唯一不动点。",[11,34323,34324,2083,34326,34369,34370,1639,34428,26827,34486,34514,34515,34519],{},[1489,34325,5595],{},[39,34327,34329,34348],{"className":34328,"translate":43},[42],[39,34330,34332],{"className":34331},[47],[49,34333,34334],{"xmlns":51},[53,34335,34336,34346],{},[56,34337,34338,34340,34342,34344],{},[59,34339,19772],{},[133,34341,136],{"stretchy":135},[59,34343,61],{},[133,34345,161],{"stretchy":135},[63,34347,21380],{"encoding":65},[39,34349,34351],{"className":34350,"ariaHidden":70},[69],[39,34352,34354,34357,34360,34363,34366],{"className":34353},[74],[39,34355],{"className":34356,"style":245},[78],[39,34358,19772],{"className":34359,"style":19812},[83,84],[39,34361,136],{"className":34362},[182],[39,34364,61],{"className":34365,"style":85},[83,84],[39,34367,161],{"className":34368},[263]," 在 sup 范数下完备，",[39,34371,34373,34390],{"className":34372,"translate":43},[42],[39,34374,34376],{"className":34375},[47],[49,34377,34378],{"xmlns":51},[53,34379,34380,34388],{},[56,34381,34382],{},[138,34383,34384,34386],{},[59,34385,11772],{"mathvariant":617},[59,34387,2098],{},[63,34389,11777],{"encoding":65},[39,34391,34393],{"className":34392,"ariaHidden":70},[69],[39,34394,34396,34399],{"className":34395},[74],[39,34397],{"className":34398,"style":79},[78],[39,34400,34402,34405],{"className":34401},[83],[39,34403,11772],{"className":34404,"style":11793},[83,641],[39,34406,34408],{"className":34407},[192],[39,34409,34411],{"className":34410},[196],[39,34412,34414],{"className":34413},[200],[39,34415,34417],{"className":34416,"style":6410},[204],[39,34418,34419,34422],{"style":208},[39,34420],{"className":34421,"style":213},[212],[39,34423,34425],{"className":34424},[217,218,219,220],[39,34426,2098],{"className":34427,"style":2114},[83,84,220],[39,34429,34431,34448],{"className":34430,"translate":43},[42],[39,34432,34434],{"className":34433},[47],[49,34435,34436],{"xmlns":51},[53,34437,34438,34446],{},[56,34439,34440],{},[138,34441,34442,34444],{},[59,34443,11772],{"mathvariant":617},[133,34445,21884],{},[63,34447,21887],{"encoding":65},[39,34449,34451],{"className":34450,"ariaHidden":70},[69],[39,34452,34454,34457],{"className":34453},[74],[39,34455],{"className":34456,"style":21897},[78],[39,34458,34460,34463],{"className":34459},[83],[39,34461,11772],{"className":34462,"style":11793},[83,641],[39,34464,34466],{"className":34465},[192],[39,34467,34469],{"className":34468},[196],[39,34470,34472],{"className":34471},[200],[39,34473,34475],{"className":34474,"style":21897},[204],[39,34476,34477,34480],{"style":208},[39,34478],{"className":34479,"style":213},[212],[39,34481,34483],{"className":34482},[217,218,219,220],[39,34484,21884],{"className":34485},[465,220],[39,34487,34489,34502],{"className":34488,"translate":43},[42],[39,34490,34492],{"className":34491},[47],[49,34493,34494],{"xmlns":51},[53,34495,34496,34500],{},[56,34497,34498],{},[59,34499,531],{},[63,34501,26586],{"encoding":65},[39,34503,34505],{"className":34504,"ariaHidden":70},[69],[39,34506,34508,34511],{"className":34507},[74],[39,34509],{"className":34510,"style":2654},[78],[39,34512,531],{"className":34513,"style":565},[83,84],"-压缩映射。由 ",[158,34516,34518],{"href":34517},"#theorem-banach","定理 3.3"," 即得。证毕。",[11,34521,26443,34522,742,34580,34582,34583,23615,34641,34699,34700,34759],{},[39,34523,34525,34542],{"className":34524,"translate":43},[42],[39,34526,34528],{"className":34527},[47],[49,34529,34530],{"xmlns":51},[53,34531,34532,34540],{},[56,34533,34534],{},[138,34535,34536,34538],{},[59,34537,11772],{"mathvariant":617},[59,34539,2098],{},[63,34541,11777],{"encoding":65},[39,34543,34545],{"className":34544,"ariaHidden":70},[69],[39,34546,34548,34551],{"className":34547},[74],[39,34549],{"className":34550,"style":79},[78],[39,34552,34554,34557],{"className":34553},[83],[39,34555,11772],{"className":34556,"style":11793},[83,641],[39,34558,34560],{"className":34559},[192],[39,34561,34563],{"className":34562},[196],[39,34564,34566],{"className":34565},[200],[39,34567,34569],{"className":34568,"style":6410},[204],[39,34570,34571,34574],{"style":208},[39,34572],{"className":34573,"style":213},[212],[39,34575,34577],{"className":34576},[217,218,219,220],[39,34578,2098],{"className":34579,"style":2114},[83,84,220],[158,34581,21294],{"href":21293}," 已经指出这个唯一不动点就是 ",[39,34584,34586,34603],{"className":34585,"translate":43},[42],[39,34587,34589],{"className":34588},[47],[49,34590,34591],{"xmlns":51},[53,34592,34593,34601],{},[56,34594,34595],{},[138,34596,34597,34599],{},[59,34598,4230],{},[59,34600,2098],{},[63,34602,6382],{"encoding":65},[39,34604,34606],{"className":34605,"ariaHidden":70},[69],[39,34607,34609,34612],{"className":34608},[74],[39,34610],{"className":34611,"style":79},[78],[39,34613,34615,34618],{"className":34614},[83],[39,34616,4230],{"className":34617,"style":461},[83,84],[39,34619,34621],{"className":34620},[192],[39,34622,34624],{"className":34623},[196],[39,34625,34627],{"className":34626},[200],[39,34628,34630],{"className":34629,"style":6410},[204],[39,34631,34632,34635],{"style":208},[39,34633],{"className":34634,"style":213},[212],[39,34636,34638],{"className":34637},[217,218,219,220],[39,34639,2098],{"className":34640,"style":2114},[83,84,220],[39,34642,34644,34661],{"className":34643,"translate":43},[42],[39,34645,34647],{"className":34646},[47],[49,34648,34649],{"xmlns":51},[53,34650,34651,34659],{},[56,34652,34653],{},[138,34654,34655,34657],{},[59,34656,11772],{"mathvariant":617},[133,34658,21884],{},[63,34660,21887],{"encoding":65},[39,34662,34664],{"className":34663,"ariaHidden":70},[69],[39,34665,34667,34670],{"className":34666},[74],[39,34668],{"className":34669,"style":21897},[78],[39,34671,34673,34676],{"className":34672},[83],[39,34674,11772],{"className":34675,"style":11793},[83,641],[39,34677,34679],{"className":34678},[192],[39,34680,34682],{"className":34681},[196],[39,34683,34685],{"className":34684},[200],[39,34686,34688],{"className":34687,"style":21897},[204],[39,34689,34690,34693],{"style":208},[39,34691],{"className":34692,"style":213},[212],[39,34694,34696],{"className":34695},[217,218,219,220],[39,34697,21884],{"className":34698},[465,220],"，第 4 节将证明它的唯一不动点就是最优值函数 ",[39,34701,34703,34721],{"className":34702,"translate":43},[42],[39,34704,34706],{"className":34705},[47],[49,34707,34708],{"xmlns":51},[53,34709,34710,34718],{},[56,34711,34712],{},[138,34713,34714,34716],{},[59,34715,4230],{},[133,34717,21884],{},[63,34719,34720],{"encoding":65},"V^*",[39,34722,34724],{"className":34723,"ariaHidden":70},[69],[39,34725,34727,34730],{"className":34726},[74],[39,34728],{"className":34729,"style":21897},[78],[39,34731,34733,34736],{"className":34732},[83],[39,34734,4230],{"className":34735,"style":461},[83,84],[39,34737,34739],{"className":34738},[192],[39,34740,34742],{"className":34741},[196],[39,34743,34745],{"className":34744},[200],[39,34746,34748],{"className":34747,"style":21897},[204],[39,34749,34750,34753],{"style":208},[39,34751],{"className":34752,"style":213},[212],[39,34754,34756],{"className":34755},[217,218,219,220],[39,34757,21884],{"className":34758},[465,220],"。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。",[11,34761,34762],{},"单调性还给出了一个在理论上很有用的上下界判据。",[31,34764,34767],{"id":34765,"title":34766},"corollary-order-bounds","推论 3.5（上下界判据）",[11,34768,37,34769,21402,34798,34856,34857,34915,34916,34986,34987,35155,35156,34986,35301,35155,35463,1607],{},[39,34770,34772,34786],{"className":34771,"translate":43},[42],[39,34773,34775],{"className":34774},[47],[49,34776,34777],{"xmlns":51},[53,34778,34779,34783],{},[56,34780,34781],{},[59,34782,11772],{"mathvariant":617},[63,34784,34785],{"encoding":65},"\\mathcal{T}",[39,34787,34789],{"className":34788,"ariaHidden":70},[69],[39,34790,34792,34795],{"className":34791},[74],[39,34793],{"className":34794,"style":79},[78],[39,34796,11772],{"className":34797,"style":11793},[83,641],[39,34799,34801,34818],{"className":34800,"translate":43},[42],[39,34802,34804],{"className":34803},[47],[49,34805,34806],{"xmlns":51},[53,34807,34808,34816],{},[56,34809,34810],{},[138,34811,34812,34814],{},[59,34813,11772],{"mathvariant":617},[59,34815,2098],{},[63,34817,11777],{"encoding":65},[39,34819,34821],{"className":34820,"ariaHidden":70},[69],[39,34822,34824,34827],{"className":34823},[74],[39,34825],{"className":34826,"style":79},[78],[39,34828,34830,34833],{"className":34829},[83],[39,34831,11772],{"className":34832,"style":11793},[83,641],[39,34834,34836],{"className":34835},[192],[39,34837,34839],{"className":34838},[196],[39,34840,34842],{"className":34841},[200],[39,34843,34845],{"className":34844,"style":6410},[204],[39,34846,34847,34850],{"style":208},[39,34848],{"className":34849,"style":213},[212],[39,34851,34853],{"className":34852},[217,218,219,220],[39,34854,2098],{"className":34855,"style":2114},[83,84,220]," 或 ",[39,34858,34860,34877],{"className":34859,"translate":43},[42],[39,34861,34863],{"className":34862},[47],[49,34864,34865],{"xmlns":51},[53,34866,34867,34875],{},[56,34868,34869],{},[138,34870,34871,34873],{},[59,34872,11772],{"mathvariant":617},[133,34874,21884],{},[63,34876,21887],{"encoding":65},[39,34878,34880],{"className":34879,"ariaHidden":70},[69],[39,34881,34883,34886],{"className":34882},[74],[39,34884],{"className":34885,"style":21897},[78],[39,34887,34889,34892],{"className":34888},[83],[39,34890,11772],{"className":34891,"style":11793},[83,641],[39,34893,34895],{"className":34894},[192],[39,34896,34898],{"className":34897},[196],[39,34899,34901],{"className":34900},[200],[39,34902,34904],{"className":34903,"style":21897},[204],[39,34905,34906,34909],{"style":208},[39,34907],{"className":34908,"style":213},[212],[39,34910,34912],{"className":34911},[217,218,219,220],[39,34913,21884],{"className":34914},[465,220],"，其唯一不动点为 ",[39,34917,34919,34937],{"className":34918,"translate":43},[42],[39,34920,34922],{"className":34921},[47],[49,34923,34924],{"xmlns":51},[53,34925,34926,34934],{},[56,34927,34928],{},[613,34929,34930,34932],{},[59,34931,4230],{},[59,34933,1696],{"mathvariant":145},[63,34935,34936],{"encoding":65},"V_\\infty",[39,34938,34940],{"className":34939,"ariaHidden":70},[69],[39,34941,34943,34946],{"className":34942},[74],[39,34944],{"className":34945,"style":634},[78],[39,34947,34949,34952],{"className":34948},[83],[39,34950,4230],{"className":34951,"style":461},[83,84],[39,34953,34955],{"className":34954},[192],[39,34956,34958,34978],{"className":34957},[196,649],[39,34959,34961,34975],{"className":34960},[200],[39,34962,34964],{"className":34963,"style":1728},[204],[39,34965,34966,34969],{"style":13607},[39,34967],{"className":34968,"style":213},[212],[39,34970,34972],{"className":34971},[217,218,219,220],[39,34973,1696],{"className":34974},[83,220],[39,34976,673],{"className":34977},[672],[39,34979,34981],{"className":34980},[200],[39,34982,34984],{"className":34983,"style":680},[204],[39,34985],{},"。若 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V_\\infty",[39,36132,36134,36224],{"className":36133,"ariaHidden":70},[69],[39,36135,36137,36140,36169,36215,36218,36221],{"className":36136},[74],[39,36138],{"className":36139,"style":634},[78],[39,36141,36143,36146],{"className":36142},[83],[39,36144,11772],{"className":36145,"style":11793},[83,641],[39,36147,36149],{"className":36148},[192],[39,36150,36152],{"className":36151},[196],[39,36153,36155],{"className":36154},[200],[39,36156,36158],{"className":36157,"style":6410},[204],[39,36159,36160,36163],{"style":208},[39,36161],{"className":36162,"style":213},[212],[39,36164,36166],{"className":36165},[217,218,219,220],[39,36167,12962],{"className":36168},[83,84,220],[39,36170,36172,36175],{"className":36171},[83],[39,36173,4230],{"className":36174,"style":461},[83,84],[39,36176,36178],{"className":36177},[192],[39,36179,36181,36207],{"className":36180},[196,649],[39,36182,36184,36204],{"className":36183},[200],[39,36185,36187],{"className":36186,"style":3419},[204],[39,36188,36189,36192],{"style":13607},[39,36190],{"className":36191,"style":213},[212],[39,36193,36195],{"className":36194},[217,218,219,220],[39,36196,36198],{"className":36197},[83,220],[39,36199,36201],{"className":36200},[83,220],[39,36202,35079],{"className":36203,"style":35078},[83,35077,220],[39,36205,673],{"className":36206},[672],[39,36208,36210],{"className":36209},[200],[39,36211,36213],{"className":36212,"style":680},[204],[39,36214],{},[39,36216],{"className":36217,"style":231},[230],[39,36219,418],{"className":36220},[235],[39,36222],{"className":36223,"style":231},[230],[39,36225,36227,36230],{"className":36226},[74],[39,36228],{"className":36229,"style":634},[78],[39,36231,36233,36236],{"className":36232},[83],[39,36234,4230],{"className":36235,"style":461},[83,84],[39,36237,36239],{"className":36238},[192],[39,36240,36242,36262],{"className":36241},[196,649],[39,36243,36245,36259],{"className":36244},[200],[39,36246,36248],{"className":36247,"style":1728},[204],[39,36249,36250,36253],{"style":13607},[39,36251],{"className":36252,"style":213},[212],[39,36254,36256],{"className":36255},[217,218,219,220],[39,36257,1696],{"className":36258},[83,220],[39,36260,673],{"className":36261},[672],[39,36263,36265],{"className":36264},[200],[39,36266,36268],{"className":36267,"style":680},[204],[39,36269],{},"，故 ",[39,36272,36274,36305],{"className":36273,"translate":43},[42],[39,36275,36277],{"className":36276},[47],[49,36278,36279],{"xmlns":51},[53,36280,36281,36303],{},[56,36282,36283,36295,36297],{},[613,36284,36285,36287],{},[59,36286,4230],{},[56,36288,36289,36291,36293],{},[59,36290,35007],{"mathvariant":145},[59,36292,35010],{"mathvariant":145},[59,36294,35013],{"mathvariant":145},[133,36296,1824],{},[613,36298,36299,36301],{},[59,36300,4230],{},[59,36302,1696],{"mathvariant":145},[63,36304,35190],{"encoding":65},[39,36306,36308,36369],{"className":36307,"ariaHidden":70},[69],[39,36309,36311,36314,36360,36363,36366],{"className":36310},[74],[39,36312],{"className":36313,"style":634},[78],[39,36315,36317,36320],{"className":36316},[83],[39,36318,4230],{"className":36319,"style":461},[83,84],[39,36321,36323],{"className":36322},[192],[39,36324,36326,36352],{"className":36325},[196,649],[39,36327,36329,36349],{"className":36328},[200],[39,36330,36332],{"className":36331,"style":3419},[204],[39,36333,36334,36337],{"style":13607},[39,36335],{"className":36336,"style":213},[212],[39,36338,36340],{"className":36339},[217,218,219,220],[39,36341,36343],{"className":36342},[83,220],[39,36344,36346],{"className":36345},[83,220],[39,36347,35079],{"className":36348,"style":35078},[83,35077,220],[39,36350,673],{"className":36351},[672],[39,36353,36355],{"className":36354},[200],[39,36356,36358],{"className":36357,"style":680},[204],[39,36359],{},[39,36361],{"className":36362,"style":231},[230],[39,36364,1824],{"className":36365},[235],[39,36367],{"className":36368,"style":231},[230],[39,36370,36372,36375],{"className":36371},[74],[39,36373],{"className":36374,"style":634},[78],[39,36376,36378,36381],{"className":36377},[83],[39,36379,4230],{"className":36380,"style":461},[83,84],[39,36382,36384],{"className":36383},[192],[39,36385,36387,36407],{"className":36386},[196,649],[39,36388,36390,36404],{"className":36389},[200],[39,36391,36393],{"className":36392,"style":1728},[204],[39,36394,36395,36398],{"style":13607},[39,36396],{"className":36397,"style":213},[212],[39,36399,36401],{"className":36400},[217,218,219,220],[39,36402,1696],{"className":36403},[83,220],[39,36405,673],{"className":36406},[672],[39,36408,36410],{"className":36409},[200],[39,36411,36413],{"className":36412,"style":680},[204],[39,36414],{},"。上界情形同理，只是迭代序列单调下降。证毕。",[11,36417,36418,36419,36475,36476,36532],{},"这个判据说明，只要能够构造一个满足 ",[39,36420,36422,36442],{"className":36421,"translate":43},[42],[39,36423,36425],{"className":36424},[47],[49,36426,36427],{"xmlns":51},[53,36428,36429,36439],{},[56,36430,36431,36433,36435,36437],{},[59,36432,4230],{},[133,36434,1824],{},[59,36436,11772],{"mathvariant":617},[59,36438,4230],{},[63,36440,36441],{"encoding":65},"V\\le\\mathcal{T}V",[39,36443,36445,36463],{"className":36444,"ariaHidden":70},[69],[39,36446,36448,36451,36454,36457,36460],{"className":36447},[74],[39,36449],{"className":36450,"style":23796},[78],[39,36452,4230],{"className":36453,"style":461},[83,84],[39,36455],{"className":36456,"style":231},[230],[39,36458,1824],{"className":36459},[235],[39,36461],{"className":36462,"style":231},[230],[39,36464,36466,36469,36472],{"className":36465},[74],[39,36467],{"className":36468,"style":79},[78],[39,36470,11772],{"className":36471,"style":11793},[83,641],[39,36473,4230],{"className":36474,"style":461},[83,84]," 的下界函数，它就一直位于真实值函数之下。满足 ",[39,36477,36479,36499],{"className":36478,"translate":43},[42],[39,36480,36482],{"className":36481},[47],[49,36483,36484],{"xmlns":51},[53,36485,36486,36496],{},[56,36487,36488,36490,36492,36494],{},[59,36489,4230],{},[133,36491,25173],{},[59,36493,11772],{"mathvariant":617},[59,36495,4230],{},[63,36497,36498],{"encoding":65},"V\\ge\\mathcal{T}V",[39,36500,36502,36520],{"className":36501,"ariaHidden":70},[69],[39,36503,36505,36508,36511,36514,36517],{"className":36504},[74],[39,36506],{"className":36507,"style":23796},[78],[39,36509,4230],{"className":36510,"style":461},[83,84],[39,36512],{"className":36513,"style":231},[230],[39,36515,25173],{"className":36516},[235],[39,36518],{"className":36519,"style":231},[230],[39,36521,36523,36526,36529],{"className":36522},[74],[39,36524],{"className":36525,"style":79},[78],[39,36527,11772],{"className":36528,"style":11793},[83,641],[39,36530,4230],{"className":36531,"style":461},[83,84]," 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。",[21,36534,36536],{"id":36535},"四贝尔曼最优方程","四、贝尔曼最优方程",[26,36538,36540],{"id":36539},"_41-最优值函数","4.1 最优值函数",[31,36542,36545,36548,36976],{"id":36543,"title":36544},"definition-optimal-value","定义 4.1（最优值函数）",[11,36546,36547],{},"最优状态值函数与最优动作值函数分别定义为",[39,36549,36551],{"className":36550,"translate":43},[746],[39,36552,36554,36652],{"className":36553,"translate":43},[42],[39,36555,36557],{"className":36556},[47],[49,36558,36559],{"xmlns":51,"display":755},[53,36560,36561,36649],{},[56,36562,36563,36569,36571,36573,36575,36577,36587,36593,36595,36597,36599,36601,36603,36609,36611,36613,36615,36617,36619,36621,36631,36637,36639,36641,36643,36645,36647],{},[138,36564,36565,36567],{},[59,36566,4230],{},[133,36568,21884],{},[133,36570,136],{"stretchy":135},[59,36572,142],{},[133,36574,161],{"stretchy":135},[133,36576,793],{},[6556,36578,36579,36585],{},[56,36580,36581,36583],{},[59,36582,21465],{},[133,36584,1690],{},[59,36586,2098],{},[138,36588,36589,36591],{},[59,36590,4230],{},[59,36592,2098],{},[133,36594,136],{"stretchy":135},[59,36596,142],{},[133,36598,161],{"stretchy":135},[133,36600,155],{"separator":70},[230,36602],{"width":10251},[138,36604,36605,36607],{},[59,36606,4295],{},[133,36608,21884],{},[133,36610,136],{"stretchy":135},[59,36612,142],{},[133,36614,155],{"separator":70},[59,36616,158],{},[133,36618,161],{"stretchy":135},[133,36620,793],{},[6556,36622,36623,36629],{},[56,36624,36625,36627],{},[59,36626,21465],{},[133,36628,1690],{},[59,36630,2098],{},[138,36632,36633,36635],{},[59,36634,4295],{},[59,36636,2098],{},[133,36638,136],{"stretchy":135},[59,36640,142],{},[133,36642,155],{"separator":70},[59,36644,158],{},[133,36646,161],{"stretchy":135},[59,36648,2903],{"mathvariant":145},[63,36650,36651],{"encoding":65},"V^*(s)=\\sup_{\\pi}V^\\pi(s),\n\\qquad\nQ^*(s,a)=\\sup_{\\pi}Q^\\pi(s,a).",[39,36653,36655,36708,36870],{"className":36654,"ariaHidden":70},[69],[39,36656,36658,36661,36690,36693,36696,36699,36702,36705],{"className":36657},[74],[39,36659],{"className":36660,"style":245},[78],[39,36662,36664,36667],{"className":36663},[83],[39,36665,4230],{"className":36666,"style":461},[83,84],[39,36668,36670],{"className":36669},[192],[39,36671,36673],{"className":36672},[196],[39,36674,36676],{"className":36675},[200],[39,36677,36679],{"className":36678,"style":22242},[204],[39,36680,36681,36684],{"style":1027},[39,36682],{"className":36683,"style":213},[212],[39,36685,36687],{"className":36686},[217,218,219,220],[39,36688,21884],{"className":36689},[465,220],[39,36691,136],{"className":36692},[182],[39,36694,142],{"className":36695},[83,84],[39,36697,161],{"className":36698},[263],[39,36700],{"className":36701,"style":231},[230],[39,36703,793],{"className":36704},[235],[39,36706],{"className":36707,"style":231},[230],[39,36709,36711,36715,36764,36767,36796,36799,36802,36805,36808,36811,36814,36843,36846,36849,36852,36855,36858,36861,36864,36867],{"className":36710},[74],[39,36712],{"className":36713,"style":36714},[78],"height:1.6444em;vertical-align:-0.8944em;",[39,36716,36718],{"className":36717},[1744,3294],[39,36719,36721,36755],{"className":36720},[196,649],[39,36722,36724,36752],{"className":36723},[200],[39,36725,36727,36742],{"className":36726,"style":290},[204],[39,36728,36730,36733],{"style":36729},"top:-2.2056em;margin-left:0em;",[39,36731],{"className":36732,"style":935},[212],[39,36734,36736],{"className":36735},[217,218,219,220],[39,36737,36739],{"className":36738},[83,220],[39,36740,2098],{"className":36741,"style":2114},[83,84,220],[39,36743,36744,36747],{"style":21599},[39,36745],{"className":36746,"style":935},[212],[39,36748,36749],{},[39,36750,21465],{"className":36751},[1744],[39,36753,673],{"className":36754},[672],[39,36756,36758],{"className":36757},[200],[39,36759,36762],{"className":36760,"style":36761},[204],"height:0.8944em;",[39,36763],{},[39,36765],{"className":36766,"style":256},[230],[39,36768,36770,36773],{"className":36769},[83],[39,36771,4230],{"className":36772,"style":461},[83,84],[39,36774,36776],{"className":36775},[192],[39,36777,36779],{"className":36778},[196],[39,36780,36782],{"className":36781},[200],[39,36783,36785],{"className":36784,"style":4421},[204],[39,36786,36787,36790],{"style":1027},[39,36788],{"className":36789,"style":213},[212],[39,36791,36793],{"className":36792},[217,218,219,220],[39,36794,2098],{"className":36795,"style":2114},[83,84,220],[39,36797,136],{"className":36798},[182],[39,36800,142],{"className":36801},[83,84],[39,36803,161],{"className":36804},[263],[39,36806,155],{"className":36807},[252],[39,36809],{"className":36810,"style":10787},[230],[39,36812],{"className":36813,"style":256},[230],[39,36815,36817,36820],{"className":36816},[83],[39,36818,4295],{"className":36819},[83,84],[39,36821,36823],{"className":36822},[192],[39,36824,36826],{"className":36825},[196],[39,36827,36829],{"className":36828},[200],[39,36830,36832],{"className":36831,"style":22242},[204],[39,36833,36834,36837],{"style":1027},[39,36835],{"className":36836,"style":213},[212],[39,36838,36840],{"className":36839},[217,218,219,220],[39,36841,21884],{"className":36842},[465,220],[39,36844,136],{"className":36845},[182],[39,36847,142],{"className":36848},[83,84],[39,36850,155],{"className":36851},[252],[39,36853],{"className":36854,"style":256},[230],[39,36856,158],{"className":36857},[83,84],[39,36859,161],{"className":36860},[263],[39,36862],{"className":36863,"style":231},[230],[39,36865,793],{"className":36866},[235],[39,36868],{"className":36869,"style":231},[230],[39,36871,36873,36876,36923,36926,36955,36958,36961,36964,36967,36970,36973],{"className":36872},[74],[39,36874],{"className":36875,"style":36714},[78],[39,36877,36879],{"className":36878},[1744,3294],[39,36880,36882,36915],{"className":36881},[196,649],[39,36883,36885,36912],{"className":36884},[200],[39,36886,36888,36902],{"className":36887,"style":290},[204],[39,36889,36890,36893],{"style":36729},[39,36891],{"className":36892,"style":935},[212],[39,36894,36896],{"className":36895},[217,218,219,220],[39,36897,36899],{"className":36898},[83,220],[39,36900,2098],{"className":36901,"style":2114},[83,84,220],[39,36903,36904,36907],{"style":21599},[39,36905],{"className":36906,"style":935},[212],[39,36908,36909],{},[39,36910,21465],{"className":36911},[1744],[39,36913,673],{"className":36914},[672],[39,36916,36918],{"className":36917},[200],[39,36919,36921],{"className":36920,"style":36761},[204],[39,36922],{},[39,36924],{"className":36925,"style":256},[230],[39,36927,36929,36932],{"className":36928},[83],[39,36930,4295],{"className":36931},[83,84],[39,36933,36935],{"className":36934},[192],[39,36936,36938],{"className":36937},[196],[39,36939,36941],{"className":36940},[200],[39,36942,36944],{"className":36943,"style":4421},[204],[39,36945,36946,36949],{"style":1027},[39,36947],{"className":36948,"style":213},[212],[39,36950,36952],{"className":36951},[217,218,219,220],[39,36953,2098],{"className":36954,"style":2114},[83,84,220],[39,36956,136],{"className":36957},[182],[39,36959,142],{"className":36960},[83,84],[39,36962,155],{"className":36963},[252],[39,36965],{"className":36966,"style":256},[230],[39,36968,158],{"className":36969},[83,84],[39,36971,161],{"className":36972},[263],[39,36974,2903],{"className":36975},[83],[11,36977,36978,36979,16486,37038,2477,37185,2506,37243,1607],{},"若策略 ",[39,36980,36982,37000],{"className":36981,"translate":43},[42],[39,36983,36985],{"className":36984},[47],[49,36986,36987],{"xmlns":51},[53,36988,36989,36997],{},[56,36990,36991],{},[138,36992,36993,36995],{},[59,36994,2098],{},[133,36996,21884],{},[63,36998,36999],{"encoding":65},"\\pi^*",[39,37001,37003],{"className":37002,"ariaHidden":70},[69],[39,37004,37006,37009],{"className":37005},[74],[39,37007],{"className":37008,"style":21897},[78],[39,37010,37012,37015],{"className":37011},[83],[39,37013,2098],{"className":37014,"style":2114},[83,84],[39,37016,37018],{"className":37017},[192],[39,37019,37021],{"className":37020},[196],[39,37022,37024],{"className":37023},[200],[39,37025,37027],{"className":37026,"style":21897},[204],[39,37028,37029,37032],{"style":208},[39,37030],{"className":37031,"style":213},[212],[39,37033,37035],{"className":37034},[217,218,219,220],[39,37036,21884],{"className":37037},[465,220],[39,37039,37041,37071],{"className":37040,"translate":43},[42],[39,37042,37044],{"className":37043},[47],[49,37045,37046],{"xmlns":51},[53,37047,37048,37068],{},[56,37049,37050,37060,37062],{},[138,37051,37052,37054],{},[59,37053,4230],{},[138,37055,37056,37058],{},[59,37057,2098],{},[133,37059,21884],{},[133,37061,793],{},[138,37063,37064,37066],{},[59,37065,4230],{},[133,37067,21884],{},[63,37069,37070],{"encoding":65},"V^{\\pi^*}=V^*",[39,37072,37074,37150],{"className":37073,"ariaHidden":70},[69],[39,37075,37077,37081,37141,37144,37147],{"className":37076},[74],[39,37078],{"className":37079,"style":37080},[78],"height:0.8973em;",[39,37082,37084,37087],{"className":37083},[83],[39,37085,4230],{"className":37086,"style":461},[83,84],[39,37088,37090],{"className":37089},[192],[39,37091,37093],{"className":37092},[196],[39,37094,37096],{"className":37095},[200],[39,37097,37099],{"className":37098,"style":37080},[204],[39,37100,37101,37104],{"style":208},[39,37102],{"className":37103,"style":213},[212],[39,37105,37107],{"className":37106},[217,218,219,220],[39,37108,37110],{"className":37109},[83,220],[39,37111,37113,37116],{"className":37112},[83,220],[39,37114,2098],{"className":37115,"style":2114},[83,84,220],[39,37117,37119],{"className":37118},[192],[39,37120,37122],{"className":37121},[196],[39,37123,37125],{"className":37124},[200],[39,37126,37129],{"className":37127,"style":37128},[204],"height:0.7633em;",[39,37130,37132,37135],{"style":37131},"top:-2.931em;margin-right:0.0714em;",[39,37133],{"className":37134,"style":4004},[212],[39,37136,37138],{"className":37137},[217,4008,4009,220],[39,37139,21884],{"className":37140},[465,220],[39,37142],{"className":37143,"style":231},[230],[39,37145,793],{"className":37146},[235],[39,37148],{"className":37149,"style":231},[230],[39,37151,37153,37156],{"className":37152},[74],[39,37154],{"className":37155,"style":21897},[78],[39,37157,37159,37162],{"className":37158},[83],[39,37160,4230],{"className":37161,"style":461},[83,84],[39,37163,37165],{"className":37164},[192],[39,37166,37168],{"className":37167},[196],[39,37169,37171],{"className":37170},[200],[39,37172,37174],{"className":37173,"style":21897},[204],[39,37175,37176,37179],{"style":208},[39,37177],{"className":37178,"style":213},[212],[39,37180,37182],{"className":37181},[217,218,219,220],[39,37183,21884],{"className":37184},[465,220],[39,37186,37188,37205],{"className":37187,"translate":43},[42],[39,37189,37191],{"className":37190},[47],[49,37192,37193],{"xmlns":51},[53,37194,37195,37203],{},[56,37196,37197],{},[138,37198,37199,37201],{},[59,37200,2098],{},[133,37202,21884],{},[63,37204,36999],{"encoding":65},[39,37206,37208],{"className":37207,"ariaHidden":70},[69],[39,37209,37211,37214],{"className":37210},[74],[39,37212],{"className":37213,"style":21897},[78],[39,37215,37217,37220],{"className":37216},[83],[39,37218,2098],{"className":37219,"style":2114},[83,84],[39,37221,37223],{"className":37222},[192],[39,37224,37226],{"className":37225},[196],[39,37227,37229],{"className":37228},[200],[39,37230,37232],{"className":37231,"style":21897},[204],[39,37233,37234,37237],{"style":208},[39,37235],{"className":37236,"style":213},[212],[39,37238,37240],{"className":37239},[217,218,219,220],[39,37241,21884],{"className":37242},[465,220],[1489,37244,37245],{},"最优策略",[11,37247,37248,37249,37253,37254,37285,37286,37317,37318,37346],{},"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",[158,37250,37252],{"href":37251},"#theorem-bellman-optimality","定理 4.2"," 的证明一并给出。因此，本节的 ",[39,37255,37257,37273],{"className":37256,"translate":43},[42],[39,37258,37260],{"className":37259},[47],[49,37261,37262],{"xmlns":51},[53,37263,37264,37270],{},[56,37265,37266,37268],{},[59,37267,21465],{},[133,37269,1690],{},[63,37271,37272],{"encoding":65},"\\sup",[39,37274,37276],{"className":37275,"ariaHidden":70},[69],[39,37277,37279,37282],{"className":37278},[74],[39,37280],{"className":37281,"style":2654},[78],[39,37283,21465],{"className":37284},[1744]," 可以理解为 ",[39,37287,37289,37305],{"className":37288,"translate":43},[42],[39,37290,37292],{"className":37291},[47],[49,37293,37294],{"xmlns":51},[53,37295,37296,37302],{},[56,37297,37298,37300],{},[59,37299,1687],{},[133,37301,1690],{},[63,37303,37304],{"encoding":65},"\\max",[39,37306,37308],{"className":37307,"ariaHidden":70},[69],[39,37309,37311,37314],{"className":37310},[74],[39,37312],{"className":37313,"style":290},[78],[39,37315,1687],{"className":37316},[1744],"。本文的 ",[39,37319,37321,37334],{"className":37320,"translate":43},[42],[39,37322,37324],{"className":37323},[47],[49,37325,37326],{"xmlns":51},[53,37327,37328,37332],{},[56,37329,37330],{},[59,37331,2098],{},[63,37333,2101],{"encoding":65},[39,37335,37337],{"className":37336,"ariaHidden":70},[69],[39,37338,37340,37343],{"className":37339},[74],[39,37341],{"className":37342,"style":290},[78],[39,37344,2098],{"className":37345,"style":2114},[83,84]," 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。",[26,37348,37350],{"id":37349},"_42-最优方程的证明","4.2 最优方程的证明",[11,37352,37353,37354,2083,37357,13349],{},"第 3.1 节已经定义了",[1489,37355,37356],{},"贝尔曼最优算子",[39,37358,37360,37377],{"className":37359,"translate":43},[42],[39,37361,37363],{"className":37362},[47],[49,37364,37365],{"xmlns":51},[53,37366,37367,37375],{},[56,37368,37369],{},[138,37370,37371,37373],{},[59,37372,11772],{"mathvariant":617},[133,37374,21884],{},[63,37376,21887],{"encoding":65},[39,37378,37380],{"className":37379,"ariaHidden":70},[69],[39,37381,37383,37386],{"className":37382},[74],[39,37384],{"className":37385,"style":21897},[78],[39,37387,37389,37392],{"className":37388},[83],[39,37390,11772],{"className":37391,"style":11793},[83,641],[39,37393,37395],{"className":37394},[192],[39,37396,37398],{"className":37397},[196],[39,37399,37401],{"className":37400},[200],[39,37402,37404],{"className":37403,"style":21897},[204],[39,37405,37406,37409],{"style":208},[39,37407],{"className":37408,"style":213},[212],[39,37410,37412],{"className":37411},[217,218,219,220],[39,37413,21884],{"className":37414},[465,220],[39,37416,37418],{"className":37417,"translate":43},[746],[39,37419,37421,37536],{"className":37420,"translate":43},[42],[39,37422,37424],{"className":37423},[47],[49,37425,37426],{"xmlns":51,"display":755},[53,37427,37428,37534],{},[56,37429,37430,37432,37438,37440,37442,37444,37446,37448,37450,37460,37470,37472,37474,37480,37482,37484,37486,37488,37490,37532],{},[133,37431,136],{"stretchy":135},[138,37433,37434,37436],{},[59,37435,11772],{"mathvariant":617},[133,37437,21884],{},[59,37439,4230],{},[133,37441,161],{"stretchy":135},[133,37443,136],{"stretchy":135},[59,37445,142],{},[133,37447,161],{"stretchy":135},[133,37449,793],{},[6556,37451,37452,37458],{},[56,37453,37454,37456],{},[59,37455,1687],{},[133,37457,1690],{},[59,37459,158],{},[6556,37461,37462,37464],{},[133,37463,3191],{},[138,37465,37466,37468],{},[59,37467,142],{},[133,37469,147],{"mathvariant":145,"lspace":146,"rspace":146},[59,37471,131],{},[133,37473,136],{"stretchy":135},[138,37475,37476,37478],{},[59,37477,142],{},[133,37479,147],{"mathvariant":145,"lspace":146,"rspace":146},[133,37481,150],{},[59,37483,142],{},[133,37485,155],{"separator":70},[59,37487,158],{},[133,37489,161],{"stretchy":135},[56,37491,37492,37494,37496,37498,37500,37502,37504,37506,37512,37514,37516,37518,37520,37522,37528,37530],{},[133,37493,537],{"fence":70},[59,37495,401],{},[133,37497,136],{"stretchy":135},[59,37499,142],{},[133,37501,155],{"separator":70},[59,37503,158],{},[133,37505,155],{"separator":70},[138,37507,37508,37510],{},[59,37509,142],{},[133,37511,147],{"mathvariant":145,"lspace":146,"rspace":146},[133,37513,161],{"stretchy":135},[133,37515,788],{},[59,37517,531],{},[59,37519,4230],{},[133,37521,136],{"stretchy":135},[138,37523,37524,37526],{},[59,37525,142],{},[133,37527,147],{"mathvariant":145,"lspace":146,"rspace":146},[133,37529,161],{"stretchy":135},[133,37531,4280],{"fence":70},[59,37533,2903],{"mathvariant":145},[63,37535,22211],{"encoding":65},[39,37537,37539,37601,37783],{"className":37538,"ariaHidden":70},[69],[39,37540,37542,37545,37548,37577,37580,37583,37586,37589,37592,37595,37598],{"className":37541},[74],[39,37543],{"className":37544,"style":245},[78],[39,37546,136],{"className":37547},[182],[39,37549,37551,37554],{"className":37550},[83],[39,37552,11772],{"className":37553,"style":11793},[83,641],[39,37555,37557],{"className":37556},[192],[39,37558,37560],{"className":37559},[196],[39,37561,37563],{"className":37562},[200],[39,37564,37566],{"className":37565,"style":22242},[204],[39,37567,37568,37571],{"style":1027},[39,37569],{"className":37570,"style":213},[212],[39,37572,37574],{"className":37573},[217,218,219,220],[39,37575,21884],{"className":37576},[465,220],[39,37578,4230],{"className":37579,"style":461},[83,84],[39,37581,161],{"className":37582},[263],[39,37584,136],{"className":37585},[182],[39,37587,142],{"className":37588},[83,84],[39,37590,161],{"className":37591},[263],[39,37593],{"className":37594,"style":231},[230],[39,37596,793],{"className":37597},[235],[39,37599],{"className":37600,"style":231},[230],[39,37602,37604,37607,37654,37657,37733,37736,37739,37742,37774,37777,37780],{"className":37603},[74],[39,37605],{"className":37606,"style":22284},[78],[39,37608,37610],{"className":37609},[1744,3294],[39,37611,37613,37646],{"className":37612},[196,649],[39,37614,37616,37643],{"className":37615},[200],[39,37617,37619,37633],{"className":37618,"style":290},[204],[39,37620,37621,37624],{"style":22299},[39,37622],{"className":37623,"style":935},[212],[39,37625,37627],{"className":37626},[217,218,219,220],[39,37628,37630],{"className":37629},[83,220],[39,37631,158],{"className":37632},[83,84,220],[39,37634,37635,37638],{"style":21599},[39,37636],{"className":37637,"style":935},[212],[39,37639,37640],{},[39,37641,1687],{"className":37642},[1744],[39,37644,673],{"className":37645},[672],[39,37647,37649],{"className":37648},[200],[39,37650,37652],{"className":37651,"style":22331},[204],[39,37653],{},[39,37655],{"className":37656,"style":256},[230],[39,37658,37660],{"className":37659},[1744,3294],[39,37661,37663,37725],{"className":37662},[196,649],[39,37664,37666,37722],{"className":37665},[200],[39,37667,37669,37712],{"className":37668,"style":6670},[204],[39,37670,37671,37674],{"style":7097},[39,37672],{"className":37673,"style":3311},[212],[39,37675,37677],{"className":37676},[217,218,219,220],[39,37678,37680],{"className":37679},[83,220],[39,37681,37683,37686],{"className":37682},[83,220],[39,37684,142],{"className":37685},[83,84,220],[39,37687,37689],{"className":37688},[192],[39,37690,37692],{"className":37691},[196],[39,37693,37695],{"className":37694},[200],[39,37696,37698],{"className":37697,"style":3997},[204],[39,37699,37700,37703],{"style":4000},[39,37701],{"className":37702,"style":4004},[212],[39,37704,37706],{"className":37705},[217,4008,4009,220],[39,37707,37709],{"className":37708},[83,220],[39,37710,147],{"className":37711},[83,220],[39,37713,37714,37717],{"style":3330},[39,37715],{"className":37716,"style":3311},[212],[39,37718,37719],{},[39,37720,3191],{"className":37721},[1744,3339,3340],[39,37723,673],{"className":37724},[672],[39,37726,37728],{"className":37727},[200],[39,37729,37731],{"className":37730,"style":7158},[204],[39,37732],{},[39,37734],{"className":37735,"style":256},[230],[39,37737,131],{"className":37738,"style":178},[83,84],[39,37740,136],{"className":37741},[182],[39,37743,37745,37748],{"className":37744},[83],[39,37746,142],{"className":37747},[83,84],[39,37749,37751],{"className":37750},[192],[39,37752,37754],{"className":37753},[196],[39,37755,37757],{"className":37756},[200],[39,37758,37760],{"className":37759,"style":1024},[204],[39,37761,37762,37765],{"style":1027},[39,37763],{"className":37764,"style":213},[212],[39,37766,37768],{"className":37767},[217,218,219,220],[39,37769,37771],{"className":37770},[83,220],[39,37772,147],{"className":37773},[83,220],[39,37775],{"className":37776,"style":231},[230],[39,37778,150],{"className":37779},[235],[39,37781],{"className":37782,"style":231},[230],[39,37784,37786,37789,37792,37795,37798,37801,37804,37807,37928,37931],{"className":37785},[74],[39,37787],{"className":37788,"style":22469},[78],[39,37790,142],{"className":37791},[83,84],[39,37793,155],{"className":37794},[252],[39,37796],{"className":37797,"style":256},[230],[39,37799,158],{"className":37800},[83,84],[39,37802,161],{"className":37803},[263],[39,37805],{"className":37806,"style":256},[230],[39,37808,37810,37813,37816,37819,37822,37825,37828,37831,37834,37837,37869,37872,37875,37878,37881,37884,37887,37890,37922,37925],{"className":37809},[2842],[39,37811,537],{"className":37812,"style":4592},[182,4591],[39,37814,401],{"className":37815,"style":438},[83,84],[39,37817,136],{"className":37818},[182],[39,37820,142],{"className":37821},[83,84],[39,37823,155],{"className":37824},[252],[39,37826],{"className":37827,"style":256},[230],[39,37829,158],{"className":37830},[83,84],[39,37832,155],{"className":37833},[252],[39,37835],{"className":37836,"style":256},[230],[39,37838,37840,37843],{"className":37839},[83],[39,37841,142],{"className":37842},[83,84],[39,37844,37846],{"className":37845},[192],[39,37847,37849],{"className":37848},[196],[39,37850,37852],{"className":37851},[200],[39,37853,37855],{"className":37854,"style":1024},[204],[39,37856,37857,37860],{"style":1027},[39,37858],{"className":37859,"style":213},[212],[39,37861,37863],{"className":37862},[217,218,219,220],[39,37864,37866],{"className":37865},[83,220],[39,37867,147],{"className":37868},[83,220],[39,37870,161],{"className":37871},[263],[39,37873],{"className":37874,"style":461},[230],[39,37876,788],{"className":37877},[465],[39,37879],{"className":37880,"style":461},[230],[39,37882,531],{"className":37883,"style":565},[83,84],[39,37885,4230],{"className":37886,"style":461},[83,84],[39,37888,136],{"className":37889},[182],[39,37891,37893,37896],{"className":37892},[83],[39,37894,142],{"className":37895},[83,84],[39,37897,37899],{"className":37898},[192],[39,37900,37902],{"className":37901},[196],[39,37903,37905],{"className":37904},[200],[39,37906,37908],{"className":37907,"style":1024},[204],[39,37909,37910,37913],{"style":1027},[39,37911],{"className":37912,"style":213},[212],[39,37914,37916],{"className":37915},[217,218,219,220],[39,37917,37919],{"className":37918},[83,220],[39,37920,147],{"className":37921},[83,220],[39,37923,161],{"className":37924},[263],[39,37926,4280],{"className":37927,"style":4592},[263,4591],[39,37929],{"className":37930,"style":256},[230],[39,37932,2903],{"className":37933},[83],[11,37935,37936,37937,37995,37996,38054],{},"与 ",[39,37938,37940,37957],{"className":37939,"translate":43},[42],[39,37941,37943],{"className":37942},[47],[49,37944,37945],{"xmlns":51},[53,37946,37947,37955],{},[56,37948,37949],{},[138,37950,37951,37953],{},[59,37952,11772],{"mathvariant":617},[59,37954,2098],{},[63,37956,11777],{"encoding":65},[39,37958,37960],{"className":37959,"ariaHidden":70},[69],[39,37961,37963,37966],{"className":37962},[74],[39,37964],{"className":37965,"style":79},[78],[39,37967,37969,37972],{"className":37968},[83],[39,37970,11772],{"className":37971,"style":11793},[83,641],[39,37973,37975],{"className":37974},[192],[39,37976,37978],{"className":37977},[196],[39,37979,37981],{"className":37980},[200],[39,37982,37984],{"className":37983,"style":6410},[204],[39,37985,37986,37989],{"style":208},[39,37987],{"className":37988,"style":213},[212],[39,37990,37992],{"className":37991},[217,218,219,220],[39,37993,2098],{"className":37994,"style":2114},[83,84,220]," 不同，",[39,37997,37999,38016],{"className":37998,"translate":43},[42],[39,38000,38002],{"className":38001},[47],[49,38003,38004],{"xmlns":51},[53,38005,38006,38014],{},[56,38007,38008],{},[138,38009,38010,38012],{},[59,38011,11772],{"mathvariant":617},[133,38013,21884],{},[63,38015,21887],{"encoding":65},[39,38017,38019],{"className":38018,"ariaHidden":70},[69],[39,38020,38022,38025],{"className":38021},[74],[39,38023],{"className":38024,"style":21897},[78],[39,38026,38028,38031],{"className":38027},[83],[39,38029,11772],{"className":38030,"style":11793},[83,641],[39,38032,38034],{"className":38033},[192],[39,38035,38037],{"className":38036},[196],[39,38038,38040],{"className":38039},[200],[39,38041,38043],{"className":38042,"style":21897},[204],[39,38044,38045,38048],{"style":208},[39,38046],{"className":38047,"style":213},[212],[39,38049,38051],{"className":38050},[217,218,219,220],[39,38052,21884],{"className":38053},[465,220]," 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。",[31,38056,38059,38294,38926,39016],{"id":38057,"title":38058},"theorem-bellman-optimality","定理 4.2（贝尔曼最优方程）",[11,38060,37,38061,742,38134,1639,38162,38190,38191,38249,38250,38293],{},[39,38062,38064,38089],{"className":38063,"translate":43},[42],[39,38065,38067],{"className":38066},[47],[49,38068,38069],{"xmlns":51},[53,38070,38071,38087],{},[56,38072,38073,38075,38077,38079,38081,38083,38085],{},[59,38074,531],{},[133,38076,534],{},[133,38078,537],{"stretchy":135},[539,38080,541],{},[133,38082,155],{"separator":70},[539,38084,546],{},[133,38086,161],{"stretchy":135},[63,38088,551],{"encoding":65},[39,38090,38092,38110],{"className":38091,"ariaHidden":70},[69],[39,38093,38095,38098,38101,38104,38107],{"className":38094},[74],[39,38096],{"className":38097,"style":561},[78],[39,38099,531],{"className":38100,"style":565},[83,84],[39,38102],{"className":38103,"style":231},[230],[39,38105,534],{"className":38106},[235],[39,38108],{"className":38109,"style":231},[230],[39,38111,38113,38116,38119,38122,38125,38128,38131],{"className":38112},[74],[39,38114],{"className":38115,"style":245},[78],[39,38117,537],{"className":38118},[182],[39,38120,541],{"className":38121},[83],[39,38123,155],{"className":38124},[252],[39,38126],{"className":38127,"style":256},[230],[39,38129,546],{"className":38130},[83],[39,38132,161],{"className":38133},[263],[39,38135,38137,38150],{"className":38136,"translate":43},[42],[39,38138,38140],{"className":38139},[47],[49,38141,38142],{"xmlns":51},[53,38143,38144,38148],{},[56,38145,38146],{},[59,38147,61],{},[63,38149,61],{"encoding":65},[39,38151,38153],{"className":38152,"ariaHidden":70},[69],[39,38154,38156,38159],{"className":38155},[74],[39,38157],{"className":38158,"style":79},[78],[39,38160,61],{"className":38161,"style":85},[83,84],[39,38163,38165,38178],{"className":38164,"translate":43},[42],[39,38166,38168],{"className":38167},[47],[49,38169,38170],{"xmlns":51},[53,38171,38172,38176],{},[56,38173,38174],{},[59,38175,101],{},[63,38177,101],{"encoding":65},[39,38179,38181],{"className":38180,"ariaHidden":70},[69],[39,38182,38184,38187],{"className":38183},[74],[39,38185],{"className":38186,"style":79},[78],[39,38188,101],{"className":38189},[83,84]," 有限。则 ",[39,38192,38194,38211],{"className":38193,"translate":43},[42],[39,38195,38197],{"className":38196},[47],[49,38198,38199],{"xmlns":51},[53,38200,38201,38209],{},[56,38202,38203],{},[138,38204,38205,38207],{},[59,38206,4230],{},[133,38208,21884],{},[63,38210,34720],{"encoding":65},[39,38212,38214],{"className":38213,"ariaHidden":70},[69],[39,38215,38217,38220],{"className":38216},[74],[39,38218],{"className":38219,"style":21897},[78],[39,38221,38223,38226],{"className":38222},[83],[39,38224,4230],{"className":38225,"style":461},[83,84],[39,38227,38229],{"className":38228},[192],[39,38230,38232],{"className":38231},[196],[39,38233,38235],{"className":38234},[200],[39,38236,38238],{"className":38237,"style":21897},[204],[39,38239,38240,38243],{"style":208},[39,38241],{"className":38242,"style":213},[212],[39,38244,38246],{"className":38245},[217,218,219,220],[39,38247,21884],{"className":38248},[465,220]," 是 ",[39,38251,38253,38272],{"className":38252,"translate":43},[42],[39,38254,38256],{"className":38255},[47],[49,38257,38258],{"xmlns":51},[53,38259,38260,38270],{},[56,38261,38262,38264,38266,38268],{},[59,38263,19772],{},[133,38265,136],{"stretchy":135},[59,38267,61],{},[133,38269,161],{"stretchy":135},[63,38271,21380],{"encoding":65},[39,38273,38275],{"className":38274,"ariaHidden":70},[69],[39,38276,38278,38281,38284,38287,38290],{"className":38277},[74],[39,38279],{"className":38280,"style":245},[78],[39,38282,19772],{"className":38283,"style":19812},[83,84],[39,38285,136],{"className":38286},[182],[39,38288,61],{"className":38289,"style":85},[83,84],[39,38291,161],{"className":38292},[263]," 中满足",[39,38295,38297],{"className":38296,"translate":43},[746],[39,38298,38300,38452],{"className":38299,"translate":43},[42],[39,38301,38303],{"className":38302},[47],[49,38304,38305],{"xmlns":51,"display":755},[53,38306,38307,38449],{},[759,38308,38309,38375],{"rowspacing":761,"columnalign":762,"columnspacing":146},[764,38310,38311,38325],{},[767,38312,38313],{},[770,38314,38315],{"scriptlevel":541,"displaystyle":70},[56,38316,38317,38319,38321,38323],{},[59,38318,4230],{},[133,38320,136],{"stretchy":135},[59,38322,142],{},[133,38324,161],{"stretchy":135},[767,38326,38327],{},[770,38328,38329],{"scriptlevel":541,"displaystyle":70},[56,38330,38331,38333,38335,38345,38355,38357,38359,38365,38367,38369,38371,38373],{},[56,38332],{},[133,38334,793],{},[6556,38336,38337,38343],{},[56,38338,38339,38341],{},[59,38340,1687],{},[133,38342,1690],{},[59,38344,158],{},[6556,38346,38347,38349],{},[133,38348,3191],{},[138,38350,38351,38353],{},[59,38352,142],{},[133,38354,147],{"mathvariant":145,"lspace":146,"rspace":146},[59,38356,131],{},[133,38358,136],{"stretchy":135},[138,38360,38361,38363],{},[59,38362,142],{},[133,38364,147],{"mathvariant":145,"lspace":146,"rspace":146},[133,38366,150],{},[59,38368,142],{},[133,38370,155],{"separator":70},[59,38372,158],{},[133,38374,161],{"stretchy":135},[764,38376,38377,38383],{},[767,38378,38379],{},[770,38380,38381],{"scriptlevel":541,"displaystyle":70},[56,38382],{},[767,38384,38385],{},[770,38386,38387],{"scriptlevel":541,"displaystyle":70},[56,38388,38389,38391,38393,38395,38437,38439,38441,38443,38445,38447],{},[56,38390],{},[230,38392],{"width":6891},[133,38394,409],{},[56,38396,38397,38399,38401,38403,38405,38407,38409,38411,38417,38419,38421,38423,38425,38427,38433,38435],{},[133,38398,537],{"fence":70},[59,38400,401],{},[133,38402,136],{"stretchy":135},[59,38404,142],{},[133,38406,155],{"separator":70},[59,38408,158],{},[133,38410,155],{"separator":70},[138,38412,38413,38415],{},[59,38414,142],{},[133,38416,147],{"mathvariant":145,"lspace":146,"rspace":146},[133,38418,161],{"stretchy":135},[133,38420,788],{},[59,38422,531],{},[59,38424,4230],{},[133,38426,136],{"stretchy":135},[138,38428,38429,38431],{},[59,38430,142],{},[133,38432,147],{"mathvariant":145,"lspace":146,"rspace":146},[133,38434,161],{"stretchy":135},[133,38436,4280],{"fence":70},[133,38438,155],{"separator":70},[230,38440],{"width":10251},[59,38442,10254],{"mathvariant":145},[59,38444,142],{},[133,38446,534],{},[59,38448,61],{},[63,38450,38451],{"encoding":65},"\\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 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中存在唯一不动点，记作 ",[39,39677,39679,39699],{"className":39678,"translate":43},[42],[39,39680,39682],{"className":39681},[47],[49,39683,39684],{"xmlns":51},[53,39685,39686,39696],{},[56,39687,39688],{},[39689,39690,39691,39693],"mover",{"accent":70},[59,39692,4230],{},[133,39694,39695],{},"ˉ",[63,39697,39698],{"encoding":65},"\\bar 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取上确界得 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V=\\mathcal{T}^*\\bar V=\\bar V",[39,42050,42052,42170,42245],{"className":42051,"ariaHidden":70},[69],[39,42053,42055,42058,42130,42161,42164,42167],{"className":42054},[74],[39,42056],{"className":42057,"style":39708},[78],[39,42059,42061,42064],{"className":42060},[83],[39,42062,11772],{"className":42063,"style":11793},[83,641],[39,42065,42067],{"className":42066},[192],[39,42068,42070],{"className":42069},[196],[39,42071,42073],{"className":42072},[200],[39,42074,42076],{"className":42075,"style":6410},[204],[39,42077,42078,42081],{"style":208},[39,42079],{"className":42080,"style":213},[212],[39,42082,42084],{"className":42083},[217,218,219,220],[39,42085,42087],{"className":42086},[83,220],[39,42088,42090,42093],{"className":42089},[83,220],[39,42091,2098],{"className":42092,"style":2114},[83,84,220],[39,42094,42096],{"className":42095},[192],[39,42097,42099,42121],{"className":42098},[196,649],[39,42100,42102,42118],{"className":42101},[200],[39,42103,42106],{"className":42104,"style":42105},[204],"height:0.1645em;",[39,42107,42109,42112],{"style":42108},"top:-2.357em;margin-left:-0.0359em;margin-right:0.0714em;",[39,42110],{"className":42111,"style":4004},[212],[39,42113,42115],{"className":42114},[217,4008,4009,220],[39,42116,39209],{"className":42117,"style":2114},[83,84,220],[39,42119,673],{"className":42120},[672],[39,42122,42124],{"className":42123},[200],[39,42125,42128],{"className":42126,"style":42127},[204],"height:0.2819em;",[39,42129],{},[39,42131,42133],{"className":42132},[83,39712],[39,42134,42136],{"className":42135},[196],[39,42137,42139],{"className":42138},[200],[39,42140,42142,42150],{"className":42141,"style":39708},[204],[39,42143,42144,42147],{"style":21599},[39,42145],{"className":42146,"style":935},[212],[39,42148,4230],{"className":42149,"style":461},[83,84],[39,42151,42152,42155],{"style":39732},[39,42153],{"className":42154,"style":935},[212],[39,42156,42158],{"className":42157,"style":39740},[39739],[39,42159,39695],{"className":42160},[83],[39,42162],{"className":42163,"style":231},[230],[39,42165,793],{"className":42166},[235],[39,42168],{"className":42169,"style":231},[230],[39,42171,42173,42176,42205,42236,42239,42242],{"className":42172},[74],[39,42174],{"className":42175,"style":39708},[78],[39,42177,42179,42182],{"className":42178},[83],[39,42180,11772],{"className":42181,"style":11793},[83,641],[39,42183,42185],{"className":42184},[192],[39,42186,42188],{"className":42187},[196],[39,42189,42191],{"className":42190},[200],[39,42192,42194],{"className":42193,"style":21897},[204],[39,42195,42196,42199],{"style":208},[39,42197],{"className":42198,"style":213},[212],[39,42200,42202],{"className":42201},[217,218,219,220],[39,42203,21884],{"className":42204},[465,220],[39,42206,42208],{"className":42207},[83,39712],[39,42209,42211],{"className":42210},[196],[39,42212,42214],{"className":42213},[200],[39,42215,42217,42225],{"className":42216,"style":39708},[204],[39,42218,42219,42222],{"style":21599},[39,42220],{"className":42221,"style":935},[212],[39,42223,4230],{"className":42224,"style":461},[83,84],[39,42226,42227,42230],{"style":39732},[39,42228],{"className":42229,"style":935},[212],[39,42231,42233],{"className":42232,"style":39740},[39739],[39,42234,39695],{"className":42235},[83],[39,42237],{"className":42238,"style":231},[230],[39,42240,793],{"className":42241},[235],[39,42243],{"className":42244,"style":231},[230],[39,42246,42248,42251],{"className":42247},[74],[39,42249],{"className":42250,"style":39708},[78],[39,42252,42254],{"className":42253},[83,39712],[39,42255,42257],{"className":42256},[196],[39,42258,42260],{"className":42259},[200],[39,42261,42263,42271],{"className":42262,"style":39708},[204],[39,42264,42265,42268],{"style":21599},[39,42266],{"className":42267,"style":935},[212],[39,42269,4230],{"className":42270,"style":461},[83,84],[39,42272,42273,42276],{"style":39732},[39,42274],{"className":42275,"style":935},[212],[39,42277,42279],{"className":42278,"style":39740},[39739],[39,42280,39695],{"className":42281},[83],"。由",[158,42284,21294],{"href":21293},[39,42286,42288,42318],{"className":42287,"translate":43},[42],[39,42289,42291],{"className":42290},[47],[49,42292,42293],{"xmlns":51},[53,42294,42295,42315],{},[56,42296,42297,42303,42305],{},[39689,42298,42299,42301],{"accent":70},[59,42300,4230],{},[133,42302,39695],{},[133,42304,793],{},[138,42306,42307,42309],{},[59,42308,4230],{},[613,42310,42311,42313],{},[59,42312,2098],{},[59,42314,39209],{},[63,42316,42317],{"encoding":65},"\\bar V=V^{\\pi_g}",[39,42319,42321,42367],{"className":42320,"ariaHidden":70},[69],[39,42322,42324,42327,42358,42361,42364],{"className":42323},[74],[39,42325],{"className":42326,"style":39708},[78],[39,42328,42330],{"className":42329},[83,39712],[39,42331,42333],{"className":42332},[196],[39,42334,42336],{"className":42335},[200],[39,42337,42339,42347],{"className":42338,"style":39708},[204],[39,42340,42341,42344],{"style":21599},[39,42342],{"className":42343,"style":935},[212],[39,42345,4230],{"className":42346,"style":461},[83,84],[39,42348,42349,42352],{"style":39732},[39,42350],{"className":42351,"style":935},[212],[39,42353,42355],{"className":42354,"style":39740},[39739],[39,42356,39695],{"className":42357},[83],[39,42359],{"className":42360,"style":231},[230],[39,42362,793],{"className":42363},[235],[39,42365],{"className":42366,"style":231},[230],[39,42368,42370,42373],{"className":42369},[74],[39,42371],{"className":42372,"style":79},[78],[39,42374,42376,42379],{"className":42375},[83],[39,42377,4230],{"className":42378,"style":461},[83,84],[39,42380,42382],{"className":42381},[192],[39,42383,42385],{"className":42384},[196],[39,42386,42388],{"className":42387},[200],[39,42389,42391],{"className":42390,"style":6410},[204],[39,42392,42393,42396],{"style":208},[39,42394],{"className":42395,"style":213},[212],[39,42397,42399],{"className":42398},[217,218,219,220],[39,42400,42402],{"className":42401},[83,220],[39,42403,42405,42408],{"className":42404},[83,220],[39,42406,2098],{"className":42407,"style":2114},[83,84,220],[39,42409,42411],{"className":42410},[192],[39,42412,42414,42434],{"className":42413},[196,649],[39,42415,42417,42431],{"className":42416},[200],[39,42418,42420],{"className":42419,"style":42105},[204],[39,42421,42422,42425],{"style":42108},[39,42423],{"className":42424,"style":4004},[212],[39,42426,42428],{"className":42427},[217,4008,4009,220],[39,42429,39209],{"className":42430,"style":2114},[83,84,220],[39,42432,673],{"className":42433},[672],[39,42435,42437],{"className":42436},[200],[39,42438,42440],{"className":42439,"style":42127},[204],[39,42441],{},[39,42443,42445,42470],{"className":42444,"translate":43},[42],[39,42446,42448],{"className":42447},[47],[49,42449,42450],{"xmlns":51},[53,42451,42452,42468],{},[56,42453,42454,42460,42462],{},[39689,42455,42456,42458],{"accent":70},[59,42457,4230],{},[133,42459,39695],{},[133,42461,1824],{},[138,42463,42464,42466],{},[59,42465,4230],{},[133,42467,21884],{},[63,42469,41180],{"encoding":65},[39,42471,42473,42519],{"className":42472,"ariaHidden":70},[69],[39,42474,42476,42479,42510,42513,42516],{"className":42475},[74],[39,42477],{"className":42478,"style":40185},[78],[39,42480,42482],{"className":42481},[83,39712],[39,42483,42485],{"className":42484},[196],[39,42486,42488],{"className":42487},[200],[39,42489,42491,42499],{"className":42490,"style":39708},[204],[39,42492,42493,42496],{"style":21599},[39,42494],{"className":42495,"style":935},[212],[39,42497,4230],{"className":42498,"style":461},[83,84],[39,42500,42501,42504],{"style":39732},[39,42502],{"className":42503,"style":935},[212],[39,42505,42507],{"className":42506,"style":39740},[39739],[39,42508,39695],{"className":42509},[83],[39,42511],{"className":42512,"style":231},[230],[39,42514,1824],{"className":42515},[235],[39,42517],{"className":42518,"style":231},[230],[39,42520,42522,42525],{"className":42521},[74],[39,42523],{"className":42524,"style":21897},[78],[39,42526,42528,42531],{"className":42527},[83],[39,42529,4230],{"className":42530,"style":461},[83,84],[39,42532,42534],{"className":42533},[192],[39,42535,42537],{"className":42536},[196],[39,42538,42540],{"className":42539},[200],[39,42541,42543],{"className":42542,"style":21897},[204],[39,42544,42545,42548],{"style":208},[39,42546],{"className":42547,"style":213},[212],[39,42549,42551],{"className":42550},[217,218,219,220],[39,42552,21884],{"className":42553},[465,220],"。结合两个不等式得 ",[39,42556,42558,42583],{"className":42557,"translate":43},[42],[39,42559,42561],{"className":42560},[47],[49,42562,42563],{"xmlns":51},[53,42564,42565,42581],{},[56,42566,42567,42573,42575],{},[39689,42568,42569,42571],{"accent":70},[59,42570,4230],{},[133,42572,39695],{},[133,42574,793],{},[138,42576,42577,42579],{},[59,42578,4230],{},[133,42580,21884],{},[63,42582,39773],{"encoding":65},[39,42584,42586,42632],{"className":42585,"ariaHidden":70},[69],[39,42587,42589,42592,42623,42626,42629],{"className":42588},[74],[39,42590],{"className":42591,"style":39708},[78],[39,42593,42595],{"className":42594},[83,39712],[39,42596,42598],{"className":42597},[196],[39,42599,42601],{"className":42600},[200],[39,42602,42604,42612],{"className":42603,"style":39708},[204],[39,42605,42606,42609],{"style":21599},[39,42607],{"className":42608,"style":935},[212],[39,42610,4230],{"className":42611,"style":461},[83,84],[39,42613,42614,42617],{"style":39732},[39,42615],{"className":42616,"style":935},[212],[39,42618,42620],{"className":42619,"style":39740},[39739],[39,42621,39695],{"className":42622},[83],[39,42624],{"className":42625,"style":231},[230],[39,42627,793],{"className":42628},[235],[39,42630],{"className":42631,"style":231},[230],[39,42633,42635,42638],{"className":42634},[74],[39,42636],{"className":42637,"style":21897},[78],[39,42639,42641,42644],{"className":42640},[83],[39,42642,4230],{"className":42643,"style":461},[83,84],[39,42645,42647],{"className":42646},[192],[39,42648,42650],{"className":42649},[196],[39,42651,42653],{"className":42652},[200],[39,42654,42656],{"className":42655,"style":21897},[204],[39,42657,42658,42661],{"style":208},[39,42659],{"className":42660,"style":213},[212],[39,42662,42664],{"className":42663},[217,218,219,220],[39,42665,21884],{"className":42666},[465,220],[11,42668,42669,42670,42698,42699,42932,42933,42935,42936,43047,43048,43158],{},"最后证明策略的最优性判据。若确定性策略 ",[39,42671,42673,42686],{"className":42672,"translate":43},[42],[39,42674,42676],{"className":42675},[47],[49,42677,42678],{"xmlns":51},[53,42679,42680,42684],{},[56,42681,42682],{},[59,42683,2098],{},[63,42685,2101],{"encoding":65},[39,42687,42689],{"className":42688,"ariaHidden":70},[69],[39,42690,42692,42695],{"className":42691},[74],[39,42693],{"className":42694,"style":290},[78],[39,42696,2098],{"className":42697,"style":2114},[83,84]," 在每个状态都取到最大值，则 ",[39,42700,42702,42748],{"className":42701,"translate":43},[42],[39,42703,42705],{"className":42704},[47],[49,42706,42707],{"xmlns":51},[53,42708,42709,42745],{},[56,42710,42711,42717,42723,42725,42731,42737,42739],{},[138,42712,42713,42715],{},[59,42714,11772],{"mathvariant":617},[59,42716,2098],{},[138,42718,42719,42721],{},[59,42720,4230],{},[133,42722,21884],{},[133,42724,793],{},[138,42726,42727,42729],{},[59,42728,11772],{"mathvariant":617},[133,42730,21884],{},[138,42732,42733,42735],{},[59,42734,4230],{},[133,42736,21884],{},[133,42738,793],{},[138,42740,42741,42743],{},[59,42742,4230],{},[133,42744,21884],{},[63,42746,42747],{"encoding":65},"\\mathcal{T}^\\pi V^*=\\mathcal{T}^*V^*=V^*",[39,42749,42751,42824,42897],{"className":42750,"ariaHidden":70},[69],[39,42752,42754,42757,42786,42815,42818,42821],{"className":42753},[74],[39,42755],{"className":42756,"style":21897},[78],[39,42758,42760,42763],{"className":42759},[83],[39,42761,11772],{"className":42762,"style":11793},[83,641],[39,42764,42766],{"className":42765},[192],[39,42767,42769],{"className":42768},[196],[39,42770,42772],{"className":42771},[200],[39,42773,42775],{"className":42774,"style":6410},[204],[39,42776,42777,42780],{"style":208},[39,42778],{"className":42779,"style":213},[212],[39,42781,42783],{"className":42782},[217,218,219,220],[39,42784,2098],{"className":42785,"style":2114},[83,84,220],[39,42787,42789,42792],{"className":42788},[83],[39,42790,4230],{"className":42791,"style":461},[83,84],[39,42793,42795],{"className":42794},[192],[39,42796,42798],{"className":42797},[196],[39,42799,42801],{"className":42800},[200],[39,42802,42804],{"className":42803,"style":21897},[204],[39,42805,42806,42809],{"style":208},[39,42807],{"className":42808,"style":213},[212],[39,42810,42812],{"className":42811},[217,218,219,220],[39,42813,21884],{"className":42814},[465,220],[39,42816],{"className":42817,"style":231},[230],[39,42819,793],{"className":42820},[235],[39,42822],{"className":42823,"style":231},[230],[39,42825,42827,42830,42859,42888,42891,42894],{"className":42826},[74],[39,42828],{"className":42829,"style":21897},[78],[39,42831,42833,42836],{"className":42832},[83],[39,42834,11772],{"className":42835,"style":11793},[83,641],[39,42837,42839],{"className":42838},[192],[39,42840,42842],{"className":42841},[196],[39,42843,42845],{"className":42844},[200],[39,42846,42848],{"className":42847,"style":21897},[204],[39,42849,42850,42853],{"style":208},[39,42851],{"className":42852,"style":213},[212],[39,42854,42856],{"className":42855},[217,218,219,220],[39,42857,21884],{"className":42858},[465,220],[39,42860,42862,42865],{"className":42861},[83],[39,42863,4230],{"className":42864,"style":461},[83,84],[39,42866,42868],{"className":42867},[192],[39,42869,42871],{"className":42870},[196],[39,42872,42874],{"className":42873},[200],[39,42875,42877],{"className":42876,"style":21897},[204],[39,42878,42879,42882],{"style":208},[39,42880],{"className":42881,"style":213},[212],[39,42883,42885],{"className":42884},[217,218,219,220],[39,42886,21884],{"className":42887},[465,220],[39,42889],{"className":42890,"style":231},[230],[39,42892,793],{"className":42893},[235],[39,42895],{"className":42896,"style":231},[230],[39,42898,42900,42903],{"className":42899},[74],[39,42901],{"className":42902,"style":21897},[78],[39,42904,42906,42909],{"className":42905},[83],[39,42907,4230],{"className":42908,"style":461},[83,84],[39,42910,42912],{"className":42911},[192],[39,42913,42915],{"className":42914},[196],[39,42916,42918],{"className":42917},[200],[39,42919,42921],{"className":42920,"style":21897},[204],[39,42922,42923,42926],{"style":208},[39,42924],{"className":42925,"style":213},[212],[39,42927,42929],{"className":42928},[217,218,219,220],[39,42930,21884],{"className":42931},[465,220],"，由",[158,42934,21294],{"href":21293}," 得 ",[39,42937,42939,42965],{"className":42938,"translate":43},[42],[39,42940,42942],{"className":42941},[47],[49,42943,42944],{"xmlns":51},[53,42945,42946,42962],{},[56,42947,42948,42954,42956],{},[138,42949,42950,42952],{},[59,42951,4230],{},[59,42953,2098],{},[133,42955,793],{},[138,42957,42958,42960],{},[59,42959,4230],{},[133,42961,21884],{},[63,42963,42964],{"encoding":65},"V^\\pi=V^*",[39,42966,42968,43012],{"className":42967,"ariaHidden":70},[69],[39,42969,42971,42974,43003,43006,43009],{"className":42970},[74],[39,42972],{"className":42973,"style":79},[78],[39,42975,42977,42980],{"className":42976},[83],[39,42978,4230],{"className":42979,"style":461},[83,84],[39,42981,42983],{"className":42982},[192],[39,42984,42986],{"className":42985},[196],[39,42987,42989],{"className":42988},[200],[39,42990,42992],{"className":42991,"style":6410},[204],[39,42993,42994,42997],{"style":208},[39,42995],{"className":42996,"style":213},[212],[39,42998,43000],{"className":42999},[217,218,219,220],[39,43001,2098],{"className":43002,"style":2114},[83,84,220],[39,43004],{"className":43005,"style":231},[230],[39,43007,793],{"className":43008},[235],[39,43010],{"className":43011,"style":231},[230],[39,43013,43015,43018],{"className":43014},[74],[39,43016],{"className":43017,"style":21897},[78],[39,43019,43021,43024],{"className":43020},[83],[39,43022,4230],{"className":43023,"style":461},[83,84],[39,43025,43027],{"className":43026},[192],[39,43028,43030],{"className":43029},[196],[39,43031,43033],{"className":43032},[200],[39,43034,43036],{"className":43035,"style":21897},[204],[39,43037,43038,43041],{"style":208},[39,43039],{"className":43040,"style":213},[212],[39,43042,43044],{"className":43043},[217,218,219,220],[39,43045,21884],{"className":43046},[465,220],"。反之，若 ",[39,43049,43051,43076],{"className":43050,"translate":43},[42],[39,43052,43054],{"className":43053},[47],[49,43055,43056],{"xmlns":51},[53,43057,43058,43074],{},[56,43059,43060,43066,43068],{},[138,43061,43062,43064],{},[59,43063,4230],{},[59,43065,2098],{},[133,43067,793],{},[138,43069,43070,43072],{},[59,43071,4230],{},[133,43073,21884],{},[63,43075,42964],{"encoding":65},[39,43077,43079,43123],{"className":43078,"ariaHidden":70},[69],[39,43080,43082,43085,43114,43117,43120],{"className":43081},[74],[39,43083],{"className":43084,"style":79},[78],[39,43086,43088,43091],{"className":43087},[83],[39,43089,4230],{"className":43090,"style":461},[83,84],[39,43092,43094],{"className":43093},[192],[39,43095,43097],{"className":43096},[196],[39,43098,43100],{"className":43099},[200],[39,43101,43103],{"className":43102,"style":6410},[204],[39,43104,43105,43108],{"style":208},[39,43106],{"className":43107,"style":213},[212],[39,43109,43111],{"className":43110},[217,218,219,220],[39,43112,2098],{"className":43113,"style":2114},[83,84,220],[39,43115],{"className":43116,"style":231},[230],[39,43118,793],{"className":43119},[235],[39,43121],{"className":43122,"style":231},[230],[39,43124,43126,43129],{"className":43125},[74],[39,43127],{"className":43128,"style":21897},[78],[39,43130,43132,43135],{"className":43131},[83],[39,43133,4230],{"className":43134,"style":461},[83,84],[39,43136,43138],{"className":43137},[192],[39,43139,43141],{"className":43140},[196],[39,43142,43144],{"className":43143},[200],[39,43145,43147],{"className":43146,"style":21897},[204],[39,43148,43149,43152],{"style":208},[39,43150],{"className":43151,"style":213},[212],[39,43153,43155],{"className":43154},[217,218,219,220],[39,43156,21884],{"className":43157},[465,220],"，则",[39,43160,43162],{"className":43161,"translate":43},[746],[39,43163,43165,43267],{"className":43164,"translate":43},[42],[39,43166,43168],{"className":43167},[47],[49,43169,43170],{"xmlns":51,"display":755},[53,43171,43172,43264],{},[759,43173,43174,43224],{"rowspacing":761,"columnalign":762,"columnspacing":146},[764,43175,43176,43194],{},[767,43177,43178],{},[770,43179,43180],{"scriptlevel":541,"displaystyle":70},[56,43181,43182,43188],{},[138,43183,43184,43186],{},[59,43185,11772],{"mathvariant":617},[59,43187,2098],{},[138,43189,43190,43192],{},[59,43191,4230],{},[133,43193,21884],{},[767,43195,43196],{},[770,43197,43198],{"scriptlevel":541,"displaystyle":70},[56,43199,43200,43202,43204,43210,43216,43218],{},[56,43201],{},[133,43203,793],{},[138,43205,43206,43208],{},[59,43207,11772],{"mathvariant":617},[59,43209,2098],{},[138,43211,43212,43214],{},[59,43213,4230],{},[59,43215,2098],{},[133,43217,793],{},[138,43219,43220,43222],{},[59,43221,4230],{},[59,43223,2098],{},[764,43225,43226,43232],{},[767,43227,43228],{},[770,43229,43230],{"scriptlevel":541,"displaystyle":70},[56,43231],{},[767,43233,43234],{},[770,43235,43236],{"scriptlevel":541,"displaystyle":70},[56,43237,43238,43240,43242,43248,43250,43256,43262],{},[56,43239],{},[133,43241,793],{},[138,43243,43244,43246],{},[59,43245,4230],{},[133,43247,21884],{},[133,43249,793],{},[138,43251,43252,43254],{},[59,43253,11772],{"mathvariant":617},[133,43255,21884],{},[138,43257,43258,43260],{},[59,43259,4230],{},[133,43261,21884],{},[59,43263,2903],{"mathvariant":145},[63,43265,43266],{"encoding":65},"\\begin{aligned}\n\\mathcal{T}^\\pi V^*\n&=\\mathcal{T}^\\pi V^\\pi=V^\\pi\\\\\n&=V^*=\\mathcal{T}^*V^*.\n\\end{aligned}",[39,43268,43270],{"className":43269,"ariaHidden":70},[69],[39,43271,43273,43276],{"className":43272},[74],[39,43274],{"className":43275,"style":4374},[78],[39,43277,43279],{"className":43278},[83],[39,43280,43282,43379],{"className":43281},[759],[39,43283,43285],{"className":43284},[918],[39,43286,43288,43371],{"className":43287},[196,649],[39,43289,43291,43368],{"className":43290},[200],[39,43292,43294,43360],{"className":43293,"style":4393},[204],[39,43295,43296,43299],{"style":4396},[39,43297],{"className":43298,"style":935},[212],[39,43300,43302,43331],{"className":43301},[83],[39,43303,43305,43308],{"className":43304},[83],[39,43306,11772],{"className":43307,"style":11793},[83,641],[39,43309,43311],{"className":43310},[192],[39,43312,43314],{"className":43313},[196],[39,43315,43317],{"className":43316},[200],[39,43318,43320],{"className":43319,"style":4421},[204],[39,43321,43322,43325],{"style":1027},[39,43323],{"className":43324,"style":213},[212],[39,43326,43328],{"className":43327},[217,218,219,220],[39,43329,2098],{"className":43330,"style":2114},[83,84,220],[39,43332,43334,43337],{"className":43333},[83],[39,43335,4230],{"className":43336,"style":461},[83,84],[39,43338,43340],{"className":43339},[192],[39,43341,43343],{"className":43342},[196],[39,43344,43346],{"className":43345},[200],[39,43347,43349],{"className":43348,"style":22242},[204],[39,43350,43351,43354],{"style":1027},[39,43352],{"className":43353,"style":213},[212],[39,43355,43357],{"className":43356},[217,218,219,220],[39,43358,21884],{"className":43359},[465,220],[39,43361,43362,43365],{"style":4444},[39,43363],{"className":43364,"style":935},[212],[39,43366],{"className":43367},[83],[39,43369,673],{"className":43370},[672],[39,43372,43374],{"className":43373},[200],[39,43375,43377],{"className":43376,"style":4507},[204],[39,43378],{},[39,43380,43382],{"className":43381},[1223],[39,43383,43385,43629],{"className":43384},[196,649],[39,43386,43388,43626],{"className":43387},[200],[39,43389,43391,43507],{"className":43390,"style":4393},[204],[39,43392,43393,43396],{"style":4396},[39,43394],{"className":43395,"style":935},[212],[39,43397,43399,43402,43405,43408,43411,43440,43469,43472,43475,43478],{"className":43398},[83],[39,43400],{"className":43401},[83],[39,43403],{"className":43404,"style":231},[230],[39,43406,793],{"className":43407},[235],[39,43409],{"className":43410,"style":231},[230],[39,43412,43414,43417],{"className":43413},[83],[39,43415,11772],{"className":43416,"style":11793},[83,641],[39,43418,43420],{"className":43419},[192],[39,43421,43423],{"className":43422},[196],[39,43424,43426],{"className":43425},[200],[39,43427,43429],{"className":43428,"style":4421},[204],[39,43430,43431,43434],{"style":1027},[39,43432],{"className":43433,"style":213},[212],[39,43435,43437],{"className":43436},[217,218,219,220],[39,43438,2098],{"className":43439,"style":2114},[83,84,220],[39,43441,43443,43446],{"className":43442},[83],[39,43444,4230],{"className":43445,"style":461},[83,84],[39,43447,43449],{"className":43448},[192],[39,43450,43452],{"className":43451},[196],[39,43453,43455],{"className":43454},[200],[39,43456,43458],{"className":43457,"style":4421},[204],[39,43459,43460,43463],{"style":1027},[39,43461],{"className":43462,"style":213},[212],[39,43464,43466],{"className":43465},[217,218,219,220],[39,43467,2098],{"className":43468,"style":2114},[83,84,220],[39,43470],{"className":43471,"style":231},[230],[39,43473,793],{"className":43474},[235],[39,43476],{"className":43477,"style":231},[230],[39,43479,43481,43484],{"className":43480},[83],[39,43482,4230],{"className":43483,"style":461},[83,84],[39,43485,43487],{"className":43486},[192],[39,43488,43490],{"className":43489},[196],[39,43491,43493],{"className":43492},[200],[39,43494,43496],{"className":43495,"style":4421},[204],[39,43497,43498,43501],{"style":1027},[39,43499],{"className":43500,"style":213},[212],[39,43502,43504],{"className":43503},[217,218,219,220],[39,43505,2098],{"className":43506,"style":2114},[83,84,220],[39,43508,43509,43512],{"style":4444},[39,43510],{"className":43511,"style":935},[212],[39,43513,43515,43518,43521,43524,43527,43556,43559,43562,43565,43594,43623],{"className":43514},[83],[39,43516],{"className":43517},[83],[39,43519],{"className":43520,"style":231},[230],[39,43522,793],{"className":43523},[235],[39,43525],{"className":43526,"style":231},[230],[39,43528,43530,43533],{"className":43529},[83],[39,43531,4230],{"className":43532,"style":461},[83,84],[39,43534,43536],{"className":43535},[192],[39,43537,43539],{"className":43538},[196],[39,43540,43542],{"className":43541},[200],[39,43543,43545],{"className":43544,"style":22242},[204],[39,43546,43547,43550],{"style":1027},[39,43548],{"className":43549,"style":213},[212],[39,43551,43553],{"className":43552},[217,218,219,220],[39,43554,21884],{"className":43555},[465,220],[39,43557],{"className":43558,"style":231},[230],[39,43560,793],{"className":43561},[235],[39,43563],{"className":43564,"style":231},[230],[39,43566,43568,43571],{"className":43567},[83],[39,43569,11772],{"className":43570,"style":11793},[83,641],[39,43572,43574],{"className":43573},[192],[39,43575,43577],{"className":43576},[196],[39,43578,43580],{"className":43579},[200],[39,43581,43583],{"className":43582,"style":22242},[204],[39,43584,43585,43588],{"style":1027},[39,43586],{"className":43587,"style":213},[212],[39,43589,43591],{"className":43590},[217,218,219,220],[39,43592,21884],{"className":43593},[465,220],[39,43595,43597,43600],{"className":43596},[83],[39,43598,4230],{"className":43599,"style":461},[83,84],[39,43601,43603],{"className":43602},[192],[39,43604,43606],{"className":43605},[196],[39,43607,43609],{"className":43608},[200],[39,43610,43612],{"className":43611,"style":22242},[204],[39,43613,43614,43617],{"style":1027},[39,43615],{"className":43616,"style":213},[212],[39,43618,43620],{"className":43619},[217,218,219,220],[39,43621,21884],{"className":43622},[465,220],[39,43624,2903],{"className":43625},[83],[39,43627,673],{"className":43628},[672],[39,43630,43632],{"className":43631},[200],[39,43633,43635],{"className":43634,"style":4507},[204],[39,43636],{},[11,43638,20571,43639,40062,43748,43857,43858,43886,43887,43915,43916,21228],{},[39,43640,43642,43672],{"className":43641,"translate":43},[42],[39,43643,43645],{"className":43644},[47],[49,43646,43647],{"xmlns":51},[53,43648,43649,43669],{},[56,43650,43651,43657,43663,43665,43667],{},[138,43652,43653,43655],{},[59,43654,11772],{"mathvariant":617},[59,43656,2098],{},[138,43658,43659,43661],{},[59,43660,4230],{},[133,43662,21884],{},[133,43664,136],{"stretchy":135},[59,43666,142],{},[133,43668,161],{"stretchy":135},[63,43670,43671],{"encoding":65},"\\mathcal{T}^\\pi V^*(s)",[39,43673,43675],{"className":43674,"ariaHidden":70},[69],[39,43676,43678,43681,43710,43739,43742,43745],{"className":43677},[74],[39,43679],{"className":43680,"style":245},[78],[39,43682,43684,43687],{"className":43683},[83],[39,43685,11772],{"className":43686,"style":11793},[83,641],[39,43688,43690],{"className":43689},[192],[39,43691,43693],{"className":43692},[196],[39,43694,43696],{"className":43695},[200],[39,43697,43699],{"className":43698,"style":6410},[204],[39,43700,43701,43704],{"style":208},[39,43702],{"className":43703,"style":213},[212],[39,43705,43707],{"className":43706},[217,218,219,220],[39,43708,2098],{"className":43709,"style":2114},[83,84,220],[39,43711,43713,43716],{"className":43712},[83],[39,43714,4230],{"className":43715,"style":461},[83,84],[39,43717,43719],{"className":43718},[192],[39,43720,43722],{"className":43721},[196],[39,43723,43725],{"className":43724},[200],[39,43726,43728],{"className":43727,"style":21897},[204],[39,43729,43730,43733],{"style":208},[39,43731],{"className":43732,"style":213},[212],[39,43734,43736],{"className":43735},[217,218,219,220],[39,43737,21884],{"className":43738},[465,220],[39,43740,136],{"className":43741},[182],[39,43743,142],{"className":43744},[83,84],[39,43746,161],{"className":43747},[263],[39,43749,43751,43781],{"className":43750,"translate":43},[42],[39,43752,43754],{"className":43753},[47],[49,43755,43756],{"xmlns":51},[53,43757,43758,43778],{},[56,43759,43760,43766,43772,43774,43776],{},[138,43761,43762,43764],{},[59,43763,11772],{"mathvariant":617},[133,43765,21884],{},[138,43767,43768,43770],{},[59,43769,4230],{},[133,43771,21884],{},[133,43773,136],{"stretchy":135},[59,43775,142],{},[133,43777,161],{"stretchy":135},[63,43779,43780],{"encoding":65},"\\mathcal{T}^*V^*(s)",[39,43782,43784],{"className":43783,"ariaHidden":70},[69],[39,43785,43787,43790,43819,43848,43851,43854],{"className":43786},[74],[39,43788],{"className":43789,"style":245},[78],[39,43791,43793,43796],{"className":43792},[83],[39,43794,11772],{"className":43795,"style":11793},[83,641],[39,43797,43799],{"className":43798},[192],[39,43800,43802],{"className":43801},[196],[39,43803,43805],{"className":43804},[200],[39,43806,43808],{"className":43807,"style":21897},[204],[39,43809,43810,43813],{"style":208},[39,43811],{"className":43812,"style":213},[212],[39,43814,43816],{"className":43815},[217,218,219,220],[39,43817,21884],{"className":43818},[465,220],[39,43820,43822,43825],{"className":43821},[83],[39,43823,4230],{"className":43824,"style":461},[83,84],[39,43826,43828],{"className":43827},[192],[39,43829,43831],{"className":43830},[196],[39,43832,43834],{"className":43833},[200],[39,43835,43837],{"className":43836,"style":21897},[204],[39,43838,43839,43842],{"style":208},[39,43840],{"className":43841,"style":213},[212],[39,43843,43845],{"className":43844},[217,218,219,220],[39,43846,21884],{"className":43847},[465,220],[39,43849,136],{"className":43850},[182],[39,43852,142],{"className":43853},[83,84],[39,43855,161],{"className":43856},[263]," 是这些动作值的最大值，等号成立当且仅当 ",[39,43859,43861,43874],{"className":43860,"translate":43},[42],[39,43862,43864],{"className":43863},[47],[49,43865,43866],{"xmlns":51},[53,43867,43868,43872],{},[56,43869,43870],{},[59,43871,2098],{},[63,43873,2101],{"encoding":65},[39,43875,43877],{"className":43876,"ariaHidden":70},[69],[39,43878,43880,43883],{"className":43879},[74],[39,43881],{"className":43882,"style":290},[78],[39,43884,2098],{"className":43885,"style":2114},[83,84]," 在该状态以概率 ",[39,43888,43890,43903],{"className":43889,"translate":43},[42],[39,43891,43893],{"className":43892},[47],[49,43894,43895],{"xmlns":51},[53,43896,43897,43901],{},[56,43898,43899],{},[539,43900,546],{},[63,43902,546],{"encoding":65},[39,43904,43906],{"className":43905,"ariaHidden":70},[69],[39,43907,43909,43912],{"className":43908},[74],[39,43910],{"className":43911,"style":2473},[78],[39,43913,546],{"className":43914},[83]," 选择取得最大值的动作。对确定性策略，这正是 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",[39,44052,44054,44071],{"className":44053,"translate":43},[42],[39,44055,44057],{"className":44056},[47],[49,44058,44059],{"xmlns":51},[53,44060,44061,44069],{},[56,44062,44063],{},[138,44064,44065,44067],{},[59,44066,11772],{"mathvariant":617},[133,44068,21884],{},[63,44070,21887],{"encoding":65},[39,44072,44074],{"className":44073,"ariaHidden":70},[69],[39,44075,44077,44080],{"className":44076},[74],[39,44078],{"className":44079,"style":21897},[78],[39,44081,44083,44086],{"className":44082},[83],[39,44084,11772],{"className":44085,"style":11793},[83,641],[39,44087,44089],{"className":44088},[192],[39,44090,44092],{"className":44091},[196],[39,44093,44095],{"className":44094},[200],[39,44096,44098],{"className":44097,"style":21897},[204],[39,44099,44100,44103],{"style":208},[39,44101],{"className":44102,"style":213},[212],[39,44104,44106],{"className":44105},[217,218,219,220],[39,44107,21884],{"className":44108},[465,220]," 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。",[11,44111,16423,44112,1639,44114,44173,44174,44202,44203,44231],{},[158,44113,37252],{"href":37251},[39,44115,44117,44135],{"className":44116,"translate":43},[42],[39,44118,44120],{"className":44119},[47],[49,44121,44122],{"xmlns":51},[53,44123,44124,44132],{},[56,44125,44126],{},[138,44127,44128,44130],{},[59,44129,4295],{},[133,44131,21884],{},[63,44133,44134],{"encoding":65},"Q^*",[39,44136,44138],{"className":44137,"ariaHidden":70},[69],[39,44139,44141,44144],{"className":44140},[74],[39,44142],{"className":44143,"style":30556},[78],[39,44145,44147,44150],{"className":44146},[83],[39,44148,4295],{"className":44149},[83,84],[39,44151,44153],{"className":44152},[192],[39,44154,44156],{"className":44155},[196],[39,44157,44159],{"className":44158},[200],[39,44160,44162],{"className":44161,"style":21897},[204],[39,44163,44164,44167],{"style":208},[39,44165],{"className":44166,"style":213},[212],[39,44168,44170],{"className":44169},[217,218,219,220],[39,44171,21884],{"className":44172},[465,220]," 的定义，还可以得到最优动作值函数的表达式。若在状态 ",[39,44175,44177,44190],{"className":44176,"translate":43},[42],[39,44178,44180],{"className":44179},[47],[49,44181,44182],{"xmlns":51},[53,44183,44184,44188],{},[56,44185,44186],{},[59,44187,142],{},[63,44189,142],{"encoding":65},[39,44191,44193],{"className":44192,"ariaHidden":70},[69],[39,44194,44196,44199],{"className":44195},[74],[39,44197],{"className":44198,"style":290},[78],[39,44200,142],{"className":44201},[83,84]," 先固定动作 ",[39,44204,44206,44219],{"className":44205,"translate":43},[42],[39,44207,44209],{"className":44208},[47],[49,44210,44211],{"xmlns":51},[53,44212,44213,44217],{},[56,44214,44215],{},[59,44216,158],{},[63,44218,158],{"encoding":65},[39,44220,44222],{"className":44221,"ariaHidden":70},[69],[39,44223,44225,44228],{"className":44224},[74],[39,44226],{"className":44227,"style":290},[78],[39,44229,158],{"className":44230},[83,84],"，此后采用最优策略，则",[39,44233,44235],{"className":44234,"translate":43},[746],[39,44236,44238,44420],{"className":44237,"translate":43},[42],[39,44239,44241],{"className":44240},[47],[49,44242,44243],{"xmlns":51,"display":755},[53,44244,44245,44417],{},[759,44246,44247,44359],{"rowspacing":761,"columnalign":762,"columnspacing":146},[764,44248,44249,44271],{},[767,44250,44251],{},[770,44252,44253],{"scriptlevel":541,"displaystyle":70},[56,44254,44255,44261,44263,44265,44267,44269],{},[138,44256,44257,44259],{},[59,44258,4295],{},[133,44260,21884],{},[133,44262,136],{"stretchy":135},[59,44264,142],{},[133,44266,155],{"separator":70},[59,44268,158],{},[133,44270,161],{"stretchy":135},[767,44272,44273],{},[770,44274,44275],{"scriptlevel":541,"displaystyle":70},[56,44276,44277,44279,44281,44291,44293,44295,44301,44303,44305,44307,44309,44311,44357],{},[56,44278],{},[133,44280,793],{},[6556,44282,44283,44285],{},[133,44284,3191],{},[138,44286,44287,44289],{},[59,44288,142],{},[133,44290,147],{"mathvariant":145,"lspace":146,"rspace":146},[59,44292,131],{},[133,44294,136],{"stretchy":135},[138,44296,44297,44299],{},[59,44298,142],{},[133,44300,147],{"mathvariant":145,"lspace":146,"rspace":146},[133,44302,150],{},[59,44304,142],{},[133,44306,155],{"separator":70},[59,44308,158],{},[133,44310,161],{"stretchy":135},[56,44312,44313,44315,44317,44319,44321,44323,44325,44327,44333,44335,44337,44339,44345,44347,44353,44355],{},[133,44314,537],{"fence":70},[59,44316,401],{},[133,44318,136],{"stretchy":135},[59,44320,142],{},[133,44322,155],{"separator":70},[59,44324,158],{},[133,44326,155],{"separator":70},[138,44328,44329,44331],{},[59,44330,142],{},[133,44332,147],{"mathvariant":145,"lspace":146,"rspace":146},[133,44334,161],{"stretchy":135},[133,44336,788],{},[59,44338,531],{},[138,44340,44341,44343],{},[59,44342,4230],{},[133,44344,21884],{},[133,44346,136],{"stretchy":135},[138,44348,44349,44351],{},[59,44350,142],{},[133,44352,147],{"mathvariant":145,"lspace":146,"rspace":146},[133,44354,161],{"stretchy":135},[133,44356,4280],{"fence":70},[133,44358,155],{"separator":70},[764,44360,44361,44379],{},[767,44362,44363],{},[770,44364,44365],{"scriptlevel":541,"displaystyle":70},[56,44366,44367,44373,44375,44377],{},[138,44368,44369,44371],{},[59,44370,4230],{},[133,44372,21884],{},[133,44374,136],{"stretchy":135},[59,44376,142],{},[133,44378,161],{"stretchy":135},[767,44380,44381],{},[770,44382,44383],{"scriptlevel":541,"displaystyle":70},[56,44384,44385,44387,44389,44399,44405,44407,44409,44411,44413,44415],{},[56,44386],{},[133,44388,793],{},[6556,44390,44391,44397],{},[56,44392,44393,44395],{},[59,44394,1687],{},[133,44396,1690],{},[59,44398,158],{},[138,44400,44401,44403],{},[59,44402,4295],{},[133,44404,21884],{},[133,44406,136],{"stretchy":135},[59,44408,142],{},[133,44410,155],{"separator":70},[59,44412,158],{},[133,44414,161],{"stretchy":135},[59,44416,2903],{"mathvariant":145},[63,44418,44419],{"encoding":65},"\\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}",[39,44421,44423],{"className":44422,"ariaHidden":70},[69],[39,44424,44426,44430],{"className":44425},[74],[39,44427],{"className":44428,"style":44429},[78],"height:4.184em;vertical-align:-1.842em;",[39,44431,44433],{"className":44432},[83],[39,44434,44436,44564],{"className":44435},[759],[39,44437,44439],{"className":44438},[918],[39,44440,44442,44555],{"className":44441},[196,649],[39,44443,44445,44552],{"className":44444},[200],[39,44446,44449,44505],{"className":44447,"style":44448},[204],"height:2.342em;",[39,44450,44452,44455],{"style":44451},"top:-4.342em;",[39,44453],{"className":44454,"style":3311},[212],[39,44456,44458,44487,44490,44493,44496,44499,44502],{"className":44457},[83],[39,44459,44461,44464],{"className":44460},[83],[39,44462,4295],{"className":44463},[83,84],[39,44465,44467],{"className":44466},[192],[39,44468,44470],{"className":44469},[196],[39,44471,44473],{"className":44472},[200],[39,44474,44476],{"className":44475,"style":22242},[204],[39,44477,44478,44481],{"style":1027},[39,44479],{"className":44480,"style":213},[212],[39,44482,44484],{"className":44483},[217,218,219,220],[39,44485,21884],{"className":44486},[465,220],[39,44488,136],{"className":44489},[182],[39,44491,142],{"className":44492},[83,84],[39,44494,155],{"className":44495},[252],[39,44497],{"className":44498,"style":256},[230],[39,44500,158],{"className":44501},[83,84],[39,44503,161],{"className":44504},[263],[39,44506,44508,44511],{"style":44507},"top:-1.908em;",[39,44509],{"className":44510,"style":3311},[212],[39,44512,44514,44543,44546,44549],{"className":44513},[83],[39,44515,44517,44520],{"className":44516},[83],[39,44518,4230],{"className":44519,"style":461},[83,84],[39,44521,44523],{"className":44522},[192],[39,44524,44526],{"className":44525},[196],[39,44527,44529],{"className":44528},[200],[39,44530,44532],{"className":44531,"style":22242},[204],[39,44533,44534,44537],{"style":1027},[39,44535],{"className":44536,"style":213},[212],[39,44538,44540],{"className":44539},[217,218,219,220],[39,44541,21884],{"className":44542},[465,220],[39,44544,136],{"className":44545},[182],[39,44547,142],{"className":44548},[83,84],[39,44550,161],{"className":44551},[263],[39,44553,673],{"className":44554},[672],[39,44556,44558],{"className":44557},[200],[39,44559,44562],{"className":44560,"style":44561},[204],"height:1.842em;",[39,44563],{},[39,44565,44567],{"className":44566},[1223],[39,44568,44570,45016],{"className":44569},[196,649],[39,44571,44573,45013],{"className":44572},[200],[39,44574,44576,44893],{"className":44575,"style":44448},[204],[39,44577,44578,44581],{"style":44451},[39,44579],{"className":44580,"style":3311},[212],[39,44582,44584,44587,44590,44593,44596,44672,44675,44678,44681,44713,44716,44719,44722,44725,44728,44731,44734,44737,44740,44887,44890],{"className":44583},[83],[39,44585],{"className":44586},[83],[39,44588],{"className":44589,"style":231},[230],[39,44591,793],{"className":44592},[235],[39,44594],{"className":44595,"style":231},[230],[39,44597,44599],{"className":44598},[1744,3294],[39,44600,44602,44664],{"className":44601},[196,649],[39,44603,44605,44661],{"className":44604},[200],[39,44606,44608,44651],{"className":44607,"style":6670},[204],[39,44609,44610,44613],{"style":7097},[39,44611],{"className":44612,"style":3311},[212],[39,44614,44616],{"className":44615},[217,218,219,220],[39,44617,44619],{"className":44618},[83,220],[39,44620,44622,44625],{"className":44621},[83,220],[39,44623,142],{"className":44624},[83,84,220],[39,44626,44628],{"className":44627},[192],[39,44629,44631],{"className":44630},[196],[39,44632,44634],{"className":44633},[200],[39,44635,44637],{"className":44636,"style":3997},[204],[39,44638,44639,44642],{"style":4000},[39,44640],{"className":44641,"style":4004},[212],[39,44643,44645],{"className":44644},[217,4008,4009,220],[39,44646,44648],{"className":44647},[83,220],[39,44649,147],{"className":44650},[83,220],[39,44652,44653,44656],{"style":3330},[39,44654],{"className":44655,"style":3311},[212],[39,44657,44658],{},[39,44659,3191],{"className":44660},[1744,3339,3340],[39,44662,673],{"className":44663},[672],[39,44665,44667],{"className":44666},[200],[39,44668,44670],{"className":44669,"style":7158},[204],[39,44671],{},[39,44673],{"className":44674,"style":256},[230],[39,44676,131],{"className":44677,"style":178},[83,84],[39,44679,136],{"className":44680},[182],[39,44682,44684,44687],{"className":44683},[83],[39,44685,142],{"className":44686},[83,84],[39,44688,44690],{"className":44689},[192],[39,44691,44693],{"className":44692},[196],[39,44694,44696],{"className":44695},[200],[39,44697,44699],{"className":44698,"style":1024},[204],[39,44700,44701,44704],{"style":1027},[39,44702],{"className":44703,"style":213},[212],[39,44705,44707],{"className":44706},[217,218,219,220],[39,44708,44710],{"className":44709},[83,220],[39,44711,147],{"className":44712},[83,220],[39,44714],{"className":44715,"style":231},[230],[39,44717,150],{"className":44718},[235],[39,44720],{"className":44721,"style":231},[230],[39,44723,142],{"className":44724},[83,84],[39,44726,155],{"className":44727},[252],[39,44729],{"className":44730,"style":256},[230],[39,44732,158],{"className":44733},[83,84],[39,44735,161],{"className":44736},[263],[39,44738],{"className":44739,"style":256},[230],[39,44741,44743,44746,44749,44752,44755,44758,44761,44764,44767,44770,44802,44805,44808,44811,44814,44817,44846,44849,44881,44884],{"className":44742},[2842],[39,44744,537],{"className":44745,"style":4592},[182,4591],[39,44747,401],{"className":44748,"style":438},[83,84],[39,44750,136],{"className":44751},[182],[39,44753,142],{"className":44754},[83,84],[39,44756,155],{"className":44757},[252],[39,44759],{"className":44760,"style":256},[230],[39,44762,158],{"className":44763},[83,84],[39,44765,155],{"className":44766},[252],[39,44768],{"className":44769,"style":256},[230],[39,44771,44773,44776],{"className":44772},[83],[39,44774,142],{"className":44775},[83,84],[39,44777,44779],{"className":44778},[192],[39,44780,44782],{"className":44781},[196],[39,44783,44785],{"className":44784},[200],[39,44786,44788],{"className":44787,"style":1024},[204],[39,44789,44790,44793],{"style":1027},[39,44791],{"className":44792,"style":213},[212],[39,44794,44796],{"className":44795},[217,218,219,220],[39,44797,44799],{"className":44798},[83,220],[39,44800,147],{"className":44801},[83,220],[39,44803,161],{"className":44804},[263],[39,44806],{"className":44807,"style":461},[230],[39,44809,788],{"className":44810},[465],[39,44812],{"className":44813,"style":461},[230],[39,44815,531],{"className":44816,"style":565},[83,84],[39,44818,44820,44823],{"className":44819},[83],[39,44821,4230],{"className":44822,"style":461},[83,84],[39,44824,44826],{"className":44825},[192],[39,44827,44829],{"className":44828},[196],[39,44830,44832],{"className":44831},[200],[39,44833,44835],{"className":44834,"style":22242},[204],[39,44836,44837,44840],{"style":1027},[39,44838],{"className":44839,"style":213},[212],[39,44841,44843],{"className":44842},[217,218,219,220],[39,44844,21884],{"className":44845},[465,220],[39,44847,136],{"className":44848},[182],[39,44850,44852,44855],{"className":44851},[83],[39,44853,142],{"className":44854},[83,84],[39,44856,44858],{"className":44857},[192],[39,44859,44861],{"className":44860},[196],[39,44862,44864],{"className":44863},[200],[39,44865,44867],{"className":44866,"style":1024},[204],[39,44868,44869,44872],{"style":1027},[39,44870],{"className":44871,"style":213},[212],[39,44873,44875],{"className":44874},[217,218,219,220],[39,44876,44878],{"className":44877},[83,220],[39,44879,147],{"className":44880},[83,220],[39,44882,161],{"className":44883},[263],[39,44885,4280],{"className":44886,"style":4592},[263,4591],[39,44888],{"className":44889,"style":256},[230],[39,44891,155],{"className":44892},[252],[39,44894,44895,44898],{"style":44507},[39,44896],{"className":44897,"style":3311},[212],[39,44899,44901,44904,44907,44910,44913,44960,44963,44992,44995,44998,45001,45004,45007,45010],{"className":44900},[83],[39,44902],{"className":44903},[83],[39,44905],{"className":44906,"style":231},[230],[39,44908,793],{"className":44909},[235],[39,44911],{"className":44912,"style":231},[230],[39,44914,44916],{"className":44915},[1744,3294],[39,44917,44919,44952],{"className":44918},[196,649],[39,44920,44922,44949],{"className":44921},[200],[39,44923,44925,44939],{"className":44924,"style":290},[204],[39,44926,44927,44930],{"style":22299},[39,44928],{"className":44929,"style":935},[212],[39,44931,44933],{"className":44932},[217,218,219,220],[39,44934,44936],{"className":44935},[83,220],[39,44937,158],{"className":44938},[83,84,220],[39,44940,44941,44944],{"style":21599},[39,44942],{"className":44943,"style":935},[212],[39,44945,44946],{},[39,44947,1687],{"className":44948},[1744],[39,44950,673],{"className":44951},[672],[39,44953,44955],{"className":44954},[200],[39,44956,44958],{"className":44957,"style":22331},[204],[39,44959],{},[39,44961],{"className":44962,"style":256},[230],[39,44964,44966,44969],{"className":44965},[83],[39,44967,4295],{"className":44968},[83,84],[39,44970,44972],{"className":44971},[192],[39,44973,44975],{"className":44974},[196],[39,44976,44978],{"className":44977},[200],[39,44979,44981],{"className":44980,"style":22242},[204],[39,44982,44983,44986],{"style":1027},[39,44984],{"className":44985,"style":213},[212],[39,44987,44989],{"className":44988},[217,218,219,220],[39,44990,21884],{"className":44991},[465,220],[39,44993,136],{"className":44994},[182],[39,44996,142],{"className":44997},[83,84],[39,44999,155],{"className":45000},[252],[39,45002],{"className":45003,"style":256},[230],[39,45005,158],{"className":45006},[83,84],[39,45008,161],{"className":45009},[263],[39,45011,2903],{"className":45012},[83],[39,45014,673],{"className":45015},[672],[39,45017,45019],{"className":45018},[200],[39,45020,45022],{"className":45021,"style":44561},[204],[39,45023],{},[11,45025,45026,45027,45138,45139,45226,45227,1639,45285,45343],{},"第一式的上界由 ",[39,45028,45030,45056],{"className":45029,"translate":43},[42],[39,45031,45033],{"className":45032},[47],[49,45034,45035],{"xmlns":51},[53,45036,45037,45053],{},[56,45038,45039,45045,45047],{},[138,45040,45041,45043],{},[59,45042,4230],{},[59,45044,2098],{},[133,45046,1824],{},[138,45048,45049,45051],{},[59,45050,4230],{},[133,45052,21884],{},[63,45054,45055],{"encoding":65},"V^\\pi\\le V^*",[39,45057,45059,45103],{"className":45058,"ariaHidden":70},[69],[39,45060,45062,45065,45094,45097,45100],{"className":45061},[74],[39,45063],{"className":45064,"style":23796},[78],[39,45066,45068,45071],{"className":45067},[83],[39,45069,4230],{"className":45070,"style":461},[83,84],[39,45072,45074],{"className":45073},[192],[39,45075,45077],{"className":45076},[196],[39,45078,45080],{"className":45079},[200],[39,45081,45083],{"className":45082,"style":6410},[204],[39,45084,45085,45088],{"style":208},[39,45086],{"className":45087,"style":213},[212],[39,45089,45091],{"className":45090},[217,218,219,220],[39,45092,2098],{"className":45093,"style":2114},[83,84,220],[39,45095],{"className":45096,"style":231},[230],[39,45098,1824],{"className":45099},[235],[39,45101],{"className":45102,"style":231},[230],[39,45104,45106,45109],{"className":45105},[74],[39,45107],{"className":45108,"style":21897},[78],[39,45110,45112,45115],{"className":45111},[83],[39,45113,4230],{"className":45114,"style":461},[83,84],[39,45116,45118],{"className":45117},[192],[39,45119,45121],{"className":45120},[196],[39,45122,45124],{"className":45123},[200],[39,45125,45127],{"className":45126,"style":21897},[204],[39,45128,45129,45132],{"style":208},[39,45130],{"className":45131,"style":213},[212],[39,45133,45135],{"className":45134},[217,218,219,220],[39,45136,21884],{"className":45137},[465,220]," 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 ",[39,45140,45142,45170],{"className":45141,"translate":43},[42],[39,45143,45145],{"className":45144},[47],[49,45146,45147],{"xmlns":51},[53,45148,45149,45167],{},[56,45150,45151,45157,45159,45161,45163,45165],{},[138,45152,45153,45155],{},[59,45154,4295],{},[133,45156,21884],{},[133,45158,136],{"stretchy":135},[59,45160,142],{},[133,45162,155],{"separator":70},[59,45164,158],{},[133,45166,161],{"stretchy":135},[63,45168,45169],{"encoding":65},"Q^*(s,a)",[39,45171,45173],{"className":45172,"ariaHidden":70},[69],[39,45174,45176,45179,45208,45211,45214,45217,45220,45223],{"className":45175},[74],[39,45177],{"className":45178,"style":245},[78],[39,45180,45182,45185],{"className":45181},[83],[39,45183,4295],{"className":45184},[83,84],[39,45186,45188],{"className":45187},[192],[39,45189,45191],{"className":45190},[196],[39,45192,45194],{"className":45193},[200],[39,45195,45197],{"className":45196,"style":21897},[204],[39,45198,45199,45202],{"style":208},[39,45200],{"className":45201,"style":213},[212],[39,45203,45205],{"className":45204},[217,218,219,220],[39,45206,21884],{"className":45207},[465,220],[39,45209,136],{"className":45210},[182],[39,45212,142],{"className":45213},[83,84],[39,45215,155],{"className":45216},[252],[39,45218],{"className":45219,"style":256},[230],[39,45221,158],{"className":45222},[83,84],[39,45224,161],{"className":45225},[263]," 最大的动作。因此，",[39,45228,45230,45247],{"className":45229,"translate":43},[42],[39,45231,45233],{"className":45232},[47],[49,45234,45235],{"xmlns":51},[53,45236,45237,45245],{},[56,45238,45239],{},[138,45240,45241,45243],{},[59,45242,4295],{},[133,45244,21884],{},[63,45246,44134],{"encoding":65},[39,45248,45250],{"className":45249,"ariaHidden":70},[69],[39,45251,45253,45256],{"className":45252},[74],[39,45254],{"className":45255,"style":30556},[78],[39,45257,45259,45262],{"className":45258},[83],[39,45260,4295],{"className":45261},[83,84],[39,45263,45265],{"className":45264},[192],[39,45266,45268],{"className":45267},[196],[39,45269,45271],{"className":45270},[200],[39,45272,45274],{"className":45273,"style":21897},[204],[39,45275,45276,45279],{"style":208},[39,45277],{"className":45278,"style":213},[212],[39,45280,45282],{"className":45281},[217,218,219,220],[39,45283,21884],{"className":45284},[465,220],[39,45286,45288,45305],{"className":45287,"translate":43},[42],[39,45289,45291],{"className":45290},[47],[49,45292,45293],{"xmlns":51},[53,45294,45295,45303],{},[56,45296,45297],{},[138,45298,45299,45301],{},[59,45300,4230],{},[133,45302,21884],{},[63,45304,34720],{"encoding":65},[39,45306,45308],{"className":45307,"ariaHidden":70},[69],[39,45309,45311,45314],{"className":45310},[74],[39,45312],{"className":45313,"style":21897},[78],[39,45315,45317,45320],{"className":45316},[83],[39,45318,4230],{"className":45319,"style":461},[83,84],[39,45321,45323],{"className":45322},[192],[39,45324,45326],{"className":45325},[196],[39,45327,45329],{"className":45328},[200],[39,45330,45332],{"className":45331,"style":21897},[204],[39,45333,45334,45337],{"style":208},[39,45335],{"className":45336,"style":213},[212],[39,45338,45340],{"className":45339},[217,218,219,220],[39,45341,21884],{"className":45342},[465,220]," 的差别只在于第一步是否已经固定动作。",[26,45345,45347],{"id":45346},"_43-max-与期望不可交换","4.3 max 与期望不可交换",[11,45349,45350,45351,45434,45435,45517],{},"最优贝尔曼方程中是 ",[39,45352,45354,45378],{"className":45353,"translate":43},[42],[39,45355,45357],{"className":45356},[47],[49,45358,45359],{"xmlns":51},[53,45360,45361,45375],{},[56,45362,45363,45373],{},[613,45364,45365,45371],{},[56,45366,45367,45369],{},[59,45368,1687],{},[133,45370,1690],{},[59,45372,158],{},[59,45374,4253],{"mathvariant":421},[63,45376,45377],{"encoding":65},"\\max_a\\mathbb{E}",[39,45379,45381],{"className":45380,"ariaHidden":70},[69],[39,45382,45384,45388,45428,45431],{"className":45383},[74],[39,45385],{"className":45386,"style":45387},[78],"height:0.8389em;vertical-align:-0.15em;",[39,45389,45391,45394],{"className":45390},[1744],[39,45392,1687],{"className":45393},[1744],[39,45395,45397],{"className":45396},[192],[39,45398,45400,45420],{"className":45399},[196,649],[39,45401,45403,45417],{"className":45402},[200],[39,45404,45406],{"className":45405,"style":1728},[204],[39,45407,45408,45411],{"style":44021},[39,45409],{"className":45410,"style":213},[212],[39,45412,45414],{"className":45413},[217,218,219,220],[39,45415,158],{"className":45416},[83,84,220],[39,45418,673],{"className":45419},[672],[39,45421,45423],{"className":45422},[200],[39,45424,45426],{"className":45425,"style":680},[204],[39,45427],{},[39,45429],{"className":45430,"style":256},[230],[39,45432,4253],{"className":45433},[83,515],"，而不是 ",[39,45436,45438,45462],{"className":45437,"translate":43},[42],[39,45439,45441],{"className":45440},[47],[49,45442,45443],{"xmlns":51},[53,45444,45445,45459],{},[56,45446,45447,45449],{},[59,45448,4253],{"mathvariant":421},[613,45450,45451,45457],{},[56,45452,45453,45455],{},[59,45454,1687],{},[133,45456,1690],{},[59,45458,158],{},[63,45460,45461],{"encoding":65},"\\mathbb{E}\\max_a",[39,45463,45465],{"className":45464,"ariaHidden":70},[69],[39,45466,45468,45471,45474,45477],{"className":45467},[74],[39,45469],{"className":45470,"style":45387},[78],[39,45472,4253],{"className":45473},[83,515],[39,45475],{"className":45476,"style":256},[230],[39,45478,45480,45483],{"className":45479},[1744],[39,45481,1687],{"className":45482},[1744],[39,45484,45486],{"className":45485},[192],[39,45487,45489,45509],{"className":45488},[196,649],[39,45490,45492,45506],{"className":45491},[200],[39,45493,45495],{"className":45494,"style":1728},[204],[39,45496,45497,45500],{"style":44021},[39,45498],{"className":45499,"style":213},[212],[39,45501,45503],{"className":45502},[217,218,219,220],[39,45504,158],{"className":45505},[83,84,220],[39,45507,673],{"className":45508},[672],[39,45510,45512],{"className":45511},[200],[39,45513,45515],{"className":45514,"style":680},[204],[39,45516],{},"。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。",[11,45519,45520,45521,45549,45550,45674,45675,45709,45710,45780,45781,45709,45813,45883],{},"考虑一个状态 ",[39,45522,45524,45537],{"className":45523,"translate":43},[42],[39,45525,45527],{"className":45526},[47],[49,45528,45529],{"xmlns":51},[53,45530,45531,45535],{},[56,45532,45533],{},[59,45534,142],{},[63,45536,142],{"encoding":65},[39,45538,45540],{"className":45539,"ariaHidden":70},[69],[39,45541,45543,45546],{"className":45542},[74],[39,45544],{"className":45545,"style":290},[78],[39,45547,142],{"className":45548},[83,84],"，两个动作 ",[39,45551,45553,45579],{"className":45552,"translate":43},[42],[39,45554,45556],{"className":45555},[47],[49,45557,45558],{"xmlns":51},[53,45559,45560,45576],{},[56,45561,45562,45568,45570],{},[613,45563,45564,45566],{},[59,45565,158],{},[539,45567,546],{},[133,45569,155],{"separator":70},[613,45571,45572,45574],{},[59,45573,158],{},[539,45575,17359],{},[63,45577,45578],{"encoding":65},"a_1,a_2",[39,45580,45582],{"className":45581,"ariaHidden":70},[69],[39,45583,45585,45588,45628,45631,45634],{"className":45584},[74],[39,45586],{"className":45587,"style":2654},[78],[39,45589,45591,45594],{"className":45590},[83],[39,45592,158],{"className":45593},[83,84],[39,45595,45597],{"className":45596},[192],[39,45598,45600,45620],{"className":45599},[196,649],[39,45601,45603,45617],{"className":45602},[200],[39,45604,45606],{"className":45605,"style":963},[204],[39,45607,45608,45611],{"style":966},[39,45609],{"className":45610,"style":213},[212],[39,45612,45614],{"className":45613},[217,218,219,220],[39,45615,546],{"className":45616},[83,220],[39,45618,673],{"className":45619},[672],[39,45621,45623],{"className":45622},[200],[39,45624,45626],{"className":45625,"style":680},[204],[39,45627],{},[39,45629,155],{"className":45630},[252],[39,45632],{"className":45633,"style":256},[230],[39,45635,45637,45640],{"className":45636},[83],[39,45638,158],{"className":45639},[83,84],[39,45641,45643],{"className":45642},[192],[39,45644,45646,45666],{"className":45645},[196,649],[39,45647,45649,45663],{"className":45648},[200],[39,45650,45652],{"className":45651,"style":963},[204],[39,45653,45654,45657],{"style":966},[39,45655],{"className":45656,"style":213},[212],[39,45658,45660],{"className":45659},[217,218,219,220],[39,45661,17359],{"className":45662},[83,220],[39,45664,673],{"className":45665},[672],[39,45667,45669],{"className":45668},[200],[39,45670,45672],{"className":45671,"style":680},[204],[39,45673],{},"。下一状态以 ",[39,45676,45678,45697],{"className":45677,"translate":43},[42],[39,45679,45681],{"className":45680},[47],[49,45682,45683],{"xmlns":51},[53,45684,45685,45694],{},[56,45686,45687,45689,45692],{},[539,45688,546],{},[59,45690,45691],{"mathvariant":145},"\u002F",[539,45693,17359],{},[63,45695,45696],{"encoding":65},"1\u002F2",[39,45698,45700],{"className":45699,"ariaHidden":70},[69],[39,45701,45703,45706],{"className":45702},[74],[39,45704],{"className":45705,"style":245},[78],[39,45707,45696],{"className":45708},[83]," 的概率为 ",[39,45711,45713,45731],{"className":45712,"translate":43},[42],[39,45714,45716],{"className":45715},[47],[49,45717,45718],{"xmlns":51},[53,45719,45720,45728],{},[56,45721,45722],{},[613,45723,45724,45726],{},[59,45725,142],{},[539,45727,546],{},[63,45729,45730],{"encoding":65},"s_1",[39,45732,45734],{"className":45733,"ariaHidden":70},[69],[39,45735,45737,45740],{"className":45736},[74],[39,45738],{"className":45739,"style":2916},[78],[39,45741,45743,45746],{"className":45742},[83],[39,45744,142],{"className":45745},[83,84],[39,45747,45749],{"className":45748},[192],[39,45750,45752,45772],{"className":45751},[196,649],[39,45753,45755,45769],{"className":45754},[200],[39,45756,45758],{"className":45757,"style":963},[204],[39,45759,45760,45763],{"style":966},[39,45761],{"className":45762,"style":213},[212],[39,45764,45766],{"className":45765},[217,218,219,220],[39,45767,546],{"className":45768},[83,220],[39,45770,673],{"className":45771},[672],[39,45773,45775],{"className":45774},[200],[39,45776,45778],{"className":45777,"style":680},[204],[39,45779],{},"，以 ",[39,45782,45784,45801],{"className":45783,"translate":43},[42],[39,45785,45787],{"className":45786},[47],[49,45788,45789],{"xmlns":51},[53,45790,45791,45799],{},[56,45792,45793,45795,45797],{},[539,45794,546],{},[59,45796,45691],{"mathvariant":145},[539,45798,17359],{},[63,45800,45696],{"encoding":65},[39,45802,45804],{"className":45803,"ariaHidden":70},[69],[39,45805,45807,45810],{"className":45806},[74],[39,45808],{"className":45809,"style":245},[78],[39,45811,45696],{"className":45812},[83],[39,45814,45816,45834],{"className":45815,"translate":43},[42],[39,45817,45819],{"className":45818},[47],[49,45820,45821],{"xmlns":51},[53,45822,45823,45831],{},[56,45824,45825],{},[613,45826,45827,45829],{},[59,45828,142],{},[539,45830,17359],{},[63,45832,45833],{"encoding":65},"s_2",[39,45835,45837],{"className":45836,"ariaHidden":70},[69],[39,45838,45840,45843],{"className":45839},[74],[39,45841],{"className":45842,"style":2916},[78],[39,45844,45846,45849],{"className":45845},[83],[39,45847,142],{"className":45848},[83,84],[39,45850,45852],{"className":45851},[192],[39,45853,45855,45875],{"className":45854},[196,649],[39,45856,45858,45872],{"className":45857},[200],[39,45859,45861],{"className":45860,"style":963},[204],[39,45862,45863,45866],{"style":966},[39,45864],{"className":45865,"style":213},[212],[39,45867,45869],{"className":45868},[217,218,219,220],[39,45870,17359],{"className":45871},[83,220],[39,45873,673],{"className":45874},[672],[39,45876,45878],{"className":45877},[200],[39,45879,45881],{"className":45880,"style":680},[204],[39,45882],{},"，并设未来价值为零。奖励为",[39,45885,45887],{"className":45886,"translate":43},[746],[39,45888,45890,46026],{"className":45889,"translate":43},[42],[39,45891,45893],{"className":45892},[47],[49,45894,45895],{"xmlns":51,"display":755},[53,45896,45897,46023],{},[56,45898,45899,45901,45903,45905,45907,45913,45915,45921,45923,45925,45927,45929,45931,45933,45935,45937,45939,45945,45947,45953,45955,45957,45959,45961,45963,45965,45967,45969,45971,45977,45979,45985,45987,45989,45991,45993,45995,45997,45999,46001,46003,46009,46011,46017,46019,46021],{},[59,45900,401],{},[133,45902,136],{"stretchy":135},[59,45904,142],{},[133,45906,155],{"separator":70},[613,45908,45909,45911],{},[59,45910,158],{},[539,45912,546],{},[133,45914,155],{"separator":70},[613,45916,45917,45919],{},[59,45918,142],{},[539,45920,546],{},[133,45922,161],{"stretchy":135},[133,45924,793],{},[539,45926,546],{},[133,45928,155],{"separator":70},[230,45930],{"width":6891},[59,45932,401],{},[133,45934,136],{"stretchy":135},[59,45936,142],{},[133,45938,155],{"separator":70},[613,45940,45941,45943],{},[59,45942,158],{},[539,45944,546],{},[133,45946,155],{"separator":70},[613,45948,45949,45951],{},[59,45950,142],{},[539,45952,17359],{},[133,45954,161],{"stretchy":135},[133,45956,793],{},[539,45958,541],{},[133,45960,155],{"separator":70},[230,45962],{"width":10251},[59,45964,401],{},[133,45966,136],{"stretchy":135},[59,45968,142],{},[133,45970,155],{"separator":70},[613,45972,45973,45975],{},[59,45974,158],{},[539,45976,17359],{},[133,45978,155],{"separator":70},[613,45980,45981,45983],{},[59,45982,142],{},[539,45984,546],{},[133,45986,161],{"stretchy":135},[133,45988,793],{},[539,45990,541],{},[133,45992,155],{"separator":70},[230,45994],{"width":6891},[59,45996,401],{},[133,45998,136],{"stretchy":135},[59,46000,142],{},[133,46002,155],{"separator":70},[613,46004,46005,46007],{},[59,46006,158],{},[539,46008,17359],{},[133,46010,155],{"separator":70},[613,46012,46013,46015],{},[59,46014,142],{},[539,46016,17359],{},[133,46018,161],{"stretchy":135},[133,46020,793],{},[539,46022,25857],{},[63,46024,46025],{"encoding":65},"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.",[39,46027,46029,46148,46279,46410,46541],{"className":46028,"ariaHidden":70},[69],[39,46030,46032,46035,46038,46041,46044,46047,46050,46090,46093,46096,46136,46139,46142,46145],{"className":46031},[74],[39,46033],{"className":46034,"style":245},[78],[39,46036,401],{"className":46037,"style":438},[83,84],[39,46039,136],{"className":46040},[182],[39,46042,142],{"className":46043},[83,84],[39,46045,155],{"className":46046},[252],[39,46048],{"className":46049,"style":256},[230],[39,46051,46053,46056],{"className":46052},[83],[39,46054,158],{"className":46055},[83,84],[39,46057,46059],{"className":46058},[192],[39,46060,46062,46082],{"className":46061},[196,649],[39,46063,46065,46079],{"className":46064},[200],[39,46066,46068],{"className":46067,"style":963},[204],[39,46069,46070,46073],{"style":966},[39,46071],{"className":46072,"style":213},[212],[39,46074,46076],{"className":46075},[217,218,219,220],[39,46077,546],{"className":46078},[83,220],[39,46080,673],{"className":46081},[672],[39,46083,46085],{"className":46084},[200],[39,46086,46088],{"className":46087,"style":680},[204],[39,46089],{},[39,46091,155],{"className":46092},[252],[39,46094],{"className":46095,"style":256},[230],[39,46097,46099,46102],{"className":46098},[83],[39,46100,142],{"className":46101},[83,84],[39,46103,46105],{"className":46104},[192],[39,46106,46108,46128],{"className":46107},[196,649],[39,46109,46111,46125],{"className":46110},[200],[39,46112,46114],{"className":46113,"style":963},[204],[39,46115,46116,46119],{"style":966},[39,46117],{"className":46118,"style":213},[212],[39,46120,46122],{"className":46121},[217,218,219,220],[39,46123,546],{"className":46124},[83,220],[39,46126,673],{"className":46127},[672],[39,46129,46131],{"className":46130},[200],[39,46132,46134],{"className":46133,"style":680},[204],[39,46135],{},[39,46137,161],{"className":46138},[263],[39,46140],{"className":46141,"style":231},[230],[39,46143,793],{"className":46144},[235],[39,46146],{"className":46147,"style":231},[230],[39,46149,46151,46154,46157,46160,46163,46166,46169,46172,46175,46178,46181,46221,46224,46227,46267,46270,46273,46276],{"className":46150},[74],[39,46152],{"className":46153,"style":245},[78],[39,46155,546],{"className":46156},[83],[39,46158,155],{"className":46159},[252],[39,46161],{"className":46162,"style":7240},[230],[39,46164],{"className":46165,"style":256},[230],[39,46167,401],{"className":46168,"style":438},[83,84],[39,46170,136],{"className":46171},[182],[39,46173,142],{"className":46174},[83,84],[39,46176,155],{"className":46177},[252],[39,46179],{"className":46180,"style":256},[230],[39,46182,46184,46187],{"className":46183},[83],[39,46185,158],{"className":46186},[83,84],[39,46188,46190],{"className":46189},[192],[39,46191,46193,46213],{"className":46192},[196,649],[39,46194,46196,46210],{"className":46195},[200],[39,46197,46199],{"className":46198,"style":963},[204],[39,46200,46201,46204],{"style":966},[39,46202],{"className":46203,"style":213},[212],[39,46205,46207],{"className":46206},[217,218,219,220],[39,46208,546],{"className":46209},[83,220],[39,46211,673],{"className":46212},[672],[39,46214,46216],{"className":46215},[200],[39,46217,46219],{"className":46218,"style":680},[204],[39,46220],{},[39,46222,155],{"className":46223},[252],[39,46225],{"className":46226,"style":256},[230],[39,46228,46230,46233],{"className":46229},[83],[39,46231,142],{"className":46232},[83,84],[39,46234,46236],{"className":46235},[192],[39,46237,46239,46259],{"className":46238},[196,649],[39,46240,46242,46256],{"className":46241},[200],[39,46243,46245],{"className":46244,"style":963},[204],[39,46246,46247,46250],{"style":966},[39,46248],{"className":46249,"style":213},[212],[39,46251,46253],{"className":46252},[217,218,219,220],[39,46254,17359],{"className":46255},[83,220],[39,46257,673],{"className":46258},[672],[39,46260,46262],{"className":46261},[200],[39,46263,46265],{"className":46264,"style":680},[204],[39,46266],{},[39,46268,161],{"className":46269},[263],[39,46271],{"className":46272,"style":231},[230],[39,46274,793],{"className":46275},[235],[39,46277],{"className":46278,"style":231},[230],[39,46280,46282,46285,46288,46291,46294,46297,46300,46303,46306,46309,46312,46352,46355,46358,46398,46401,46404,46407],{"className":46281},[74],[39,46283],{"className":46284,"style":245},[78],[39,46286,541],{"className":46287},[83],[39,46289,155],{"className":46290},[252],[39,46292],{"className":46293,"style":10787},[230],[39,46295],{"className":46296,"style":256},[230],[39,46298,401],{"className":46299,"style":438},[83,84],[39,46301,136],{"className":46302},[182],[39,46304,142],{"className":46305},[83,84],[39,46307,155],{"className":46308},[252],[39,46310],{"className":46311,"style":256},[230],[39,46313,46315,46318],{"className":46314},[83],[39,46316,158],{"className":46317},[83,84],[39,46319,46321],{"className":46320},[192],[39,46322,46324,46344],{"className":46323},[196,649],[39,46325,46327,46341],{"className":46326},[200],[39,463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s,a]&=\\frac12,\\\\\n\\mathbb{E}\\left[\\max_{a}r(s,a,s')\\mid s\\right]&=1.\n\\end{aligned}",[39,46686,46688],{"className":46687,"ariaHidden":70},[69],[39,46689,46691,46695],{"className":46690},[74],[39,46692],{"className":46693,"style":46694},[78],"height:4.1714em;vertical-align:-1.8357em;",[39,46696,46698],{"className":46697},[83],[39,46699,46701,46970],{"className":46700},[759],[39,46702,46704],{"className":46703},[918],[39,46705,46707,46961],{"className":46706},[196,649],[39,46708,46710,46958],{"className":46709},[200],[39,46711,46714,46807],{"className":46712,"style":46713},[204],"height:2.3357em;",[39,46715,46717,46721],{"style":46716},"top:-4.3357em;",[39,46718],{"className":46719,"style":46720},[212],"height:3.3214em;",[39,46722,46724,46771,46774,46777,46780,46783,46786,46789,46792,46795,46798,46801,46804],{"className":46723},[83],[39,46725,46727],{"className":46726},[1744,3294],[39,46728,46730,46763],{"className":46729},[196,649],[39,46731,46733,46760],{"className":46732},[200],[39,46734,46736,46750],{"className":46735,"style":290},[204],[39,46737,46738,46741],{"style":22299},[39,46739],{"className":46740,"style":935},[212],[39,46742,46744],{"className":46743},[217,218,219,220],[39,46745,46747],{"className":46746},[83,220],[39,46748,158],{"className":46749},[83,84,220],[39,46751,46752,46755],{"style":21599},[39,46753],{"className":46754,"style":935},[212],[39,46756,46757],{},[39,46758,1687],{"className":46759},[1744],[39,46761,673],{"className":46762},[672],[39,46764,46766],{"className":46765},[200],[39,46767,46769],{"className":46768,"style":22331},[204],[39,46770],{},[39,46772],{"className":46773,"style":256},[230],[39,46775,4253],{"className":46776},[83,515],[39,46778,537],{"className":46779},[182],[39,46781,401],{"className":46782,"style":438},[83,84],[39,46784],{"className":46785,"style":231},[230],[39,46787,150],{"className":46788},[235],[39,46790],{"className":46791,"style":231},[230],[39,46793,142],{"className":46794},[83,84],[39,46796,155],{"className":46797},[252],[39,46799],{"className":46800,"style":256},[230],[39,46802,158],{"className":46803},[83,84],[39,46805,4280],{"className":46806},[263],[39,46808,46810,46813],{"style":46809},"top:-2.1857em;",[39,46811],{"className":46812,"style":46720},[212],[39,46814,46816,46819,46822],{"className":46815},[83],[39,46817,4253],{"className":46818},[83,515],[39,46820],{"className":46821,"style":256},[230],[39,46823,46825,46831,46878,46881,46884,46887,46890,46893,46896,46899,46902,46905,46937,46940,46943,46946,46949,46952],{"className":46824},[2842],[39,46826,46828],{"className":46827,"style":4592},[182,4591],[39,46829,537],{"className":46830},[18520,33204],[39,46832,46834],{"className":46833},[1744,3294],[39,46835,46837,46870],{"className":46836},[196,649],[39,46838,46840,46867],{"className":46839},[200],[39,46841,46843,46857],{"className":46842,"style":290},[204],[39,46844,46845,46848],{"style":22299},[39,46846],{"className":46847,"style":935},[212],[39,46849,46851],{"className":46850},[217,218,219,220],[39,46852,46854],{"className":46853},[83,220],[39,46855,158],{"className":46856},[83,84,220],[39,46858,46859,46862],{"style":21599},[39,46860],{"className":46861,"style":935},[212],[39,46863,46864],{},[39,46865,1687],{"className":46866},[1744],[39,46868,673],{"className":46869},[672],[39,46871,46873],{"className":46872},[200],[39,46874,46876],{"className":46875,"style":22331},[204],[39,46877],{},[39,46879],{"className":46880,"style":256},[230],[39,46882,401],{"className":46883,"style":438},[83,84],[39,46885,136],{"className":46886},[182],[39,46888,142],{"className":46889},[83,84],[39,46891,155],{"className":46892},[252],[39,46894],{"className":46895,"style":256},[230],[39,46897,158],{"className":46898},[83,84],[39,46900,155],{"className":46901},[252],[39,46903],{"className":46904,"style":256},[230],[39,46906,46908,46911],{"className":46907},[83],[39,46909,142],{"className":46910},[83,84],[39,46912,46914],{"className":46913},[192],[39,46915,46917],{"className":46916},[196],[39,46918,46920],{"className":46919},[200],[39,46921,46923],{"className":46922,"style":1024},[204],[39,46924,46925,46928],{"style":1027},[39,46926],{"className":46927,"style":213},[212],[39,46929,46931],{"className":46930},[217,218,219,220],[39,46932,46934],{"className":46933},[83,220],[39,46935,147],{"className":46936},[83,220],[39,46938,161],{"className":46939},[263],[39,46941],{"className":46942,"style":231},[230],[39,46944,150],{"className":46945},[235],[39,46947],{"className":46948,"style":231},[230],[39,46950,142],{"className":46951},[83,84],[39,46953,46955],{"className":46954,"style":4592},[263,4591],[39,46956,4280],{"className":46957},[18520,33204],[39,46959,673],{"className":46960},[672],[39,46962,46964],{"className":46963},[200],[39,46965,46968],{"className":46966,"style":46967},[204],"height:1.8357em;",[39,46969],{},[39,46971,46973],{"className":46972},[1223],[39,46974,46976,47093],{"className":46975},[196,649],[39,46977,46979,47090],{"className":46978},[200],[39,46980,46982,47067],{"className":46981,"style":46713},[204],[39,46983,46984,46987],{"style":46716},[39,46985],{"className":46986,"style":46720},[212],[39,46988,46990,46993,46996,46999,47002,47064],{"className":46989},[83],[39,46991],{"className":46992},[83],[39,46994],{"className":46995,"style":231},[230],[39,46997,793],{"className":46998},[235],[39,47000],{"className":47001,"style":231},[230],[39,47003,47005,47008,47061],{"className":47004},[83],[39,47006],{"className":47007},[182,5463],[39,47009,47011],{"className":47010},[5362],[39,47012,47014,47053],{"className":47013},[196,649],[39,47015,47017,47050],{"className":47016},[200],[39,47018,47020,47031,47039],{"className":47019,"style":18240},[204],[39,47021,47022,47025],{"style":5479},[39,47023],{"className":47024,"style":935},[212],[39,47026,47028],{"className":47027},[83],[39,47029,17359],{"className":47030},[83],[39,47032,47033,47036],{"style":5503},[39,47034],{"className":47035,"style":935},[212],[39,47037],{"className":47038,"style":5511},[5510],[39,47040,47041,47044],{"style":5514},[39,47042],{"className":47043,"style":935},[212],[39,47045,47047],{"className":47046},[83],[39,47048,546],{"className":47049},[83],[39,47051,673],{"className":47052},[672],[39,47054,47056],{"className":47055},[200],[39,47057,47059],{"className":47058,"style":19487},[204],[39,47060],{},[39,47062],{"className":47063},[263,5463],[39,47065,155],{"className":47066},[252],[39,47068,47069,47072],{"style":46809},[39,47070],{"className":47071,"style":46720},[212],[39,47073,47075,47078,47081,47084,47087],{"className":47074},[83],[39,47076],{"className":47077},[83],[39,47079],{"className":47080,"style":231},[230],[39,47082,793],{"className":47083},[235],[39,47085],{"className":47086,"style":231},[230],[39,47088,25857],{"className":47089},[83],[39,47091,673],{"className":47092},[672],[39,47094,47096],{"className":47095},[200],[39,47097,47099],{"className":47098,"style":46967},[204],[39,47100],{},[11,47102,47103,47104,47165],{},"后者相当于在观察到 ",[39,47105,47107,47124],{"className":47106,"translate":43},[42],[39,47108,47110],{"className":47109},[47],[49,47111,47112],{"xmlns":51},[53,47113,47114,47122],{},[56,47115,47116],{},[138,47117,47118,47120],{},[59,47119,142],{},[133,47121,147],{"mathvariant":145,"lspace":146,"rspace":146},[63,47123,344],{"encoding":65},[39,47125,47127],{"className":47126,"ariaHidden":70},[69],[39,47128,47130,47133],{"className":47129},[74],[39,47131],{"className":47132,"style":205},[78],[39,47134,47136,47139],{"className":47135},[83],[39,47137,142],{"className":47138},[83,84],[39,47140,47142],{"className":47141},[192],[39,47143,47145],{"className":47144},[196],[39,47146,47148],{"className":47147},[200],[39,47149,47151],{"className":47150,"style":205},[204],[39,47152,47153,47156],{"style":208},[39,47154],{"className":47155,"style":213},[212],[39,47157,47159],{"className":47158},[217,218,219,220],[39,47160,47162],{"className":47161},[83,220],[39,47163,147],{"className":47164},[83,220]," 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。",[26,47167,47169],{"id":47168},"_44-验证定理与最优策略","4.4 验证定理与最优策略",[31,47171,47174,47467,47954],{"id":47172,"title":47173},"corollary-optimal-verification","推论 4.3（最优性的验证定理）",[11,47175,37,47176,16486,47241,14627,47327,47408,47409,47437,47438,47466],{},[39,47177,47179,47202],{"className":47178,"translate":43},[42],[39,47180,47182],{"className":47181},[47],[49,47183,47184],{"xmlns":51},[53,47185,47186,47200],{},[56,47187,47188,47190,47192,47194,47196,47198],{},[59,47189,4230],{},[133,47191,534],{},[59,47193,19772],{},[133,47195,136],{"stretchy":135},[59,47197,61],{},[133,47199,161],{"stretchy":135},[63,47201,19781],{"encoding":65},[39,47203,47205,47223],{"className":47204,"ariaHidden":70},[69],[39,47206,47208,47211,47214,47217,47220],{"className":47207},[74],[39,47209],{"className":47210,"style":13061},[78],[39,47212,4230],{"className":47213,"style":461},[83,84],[39,47215],{"className":47216,"style":231},[230],[39,47218,534],{"className":47219},[235],[39,47221],{"className":47222,"style":231},[230],[39,47224,47226,47229,47232,47235,47238],{"className":47225},[74],[39,47227],{"className":47228,"style":245},[78],[39,47230,19772],{"className":47231,"style":19812},[83,84],[39,47233,136],{"className":47234},[182],[39,47236,61],{"className":47237,"style":85},[83,84],[39,47239,161],{"className":47240},[263],[39,47242,47244,47268],{"className":47243,"translate":43},[42],[39,47245,47247],{"className":47246},[47],[49,47248,47249],{"xmlns":51},[53,47250,47251,47265],{},[56,47252,47253,47255,47257,47263],{},[59,47254,4230],{},[133,47256,793],{},[138,47258,47259,47261],{},[59,47260,11772],{"mathvariant":617},[133,47262,21884],{},[59,47264,4230],{},[63,47266,47267],{"encoding":65},"V=\\mathcal{T}^*V",[39,47269,47271,47289],{"className":47270,"ariaHidden":70},[69],[39,47272,47274,47277,47280,47283,47286],{"className":47273},[74],[39,47275],{"className":47276,"style":79},[78],[39,47278,4230],{"className":47279,"style":461},[83,84],[39,47281],{"className":47282,"style":231},[230],[39,47284,793],{"className":47285},[235],[39,47287],{"className":47288,"style":231},[230],[39,47290,47292,47295,47324],{"className":47291},[74],[39,47293],{"className":47294,"style":21897},[78],[39,47296,47298,47301],{"className":47297},[83],[39,47299,11772],{"className":47300,"style":11793},[83,641],[39,47302,47304],{"className":47303},[192],[39,47305,47307],{"className":47306},[196],[39,47308,47310],{"className":47309},[200],[39,47311,47313],{"className":47312,"style":21897},[204],[39,47314,47315,47318],{"style":208},[39,47316],{"className":47317,"style":213},[212],[39,47319,47321],{"className":47320},[217,218,219,220],[39,47322,21884],{"className":47323},[465,220],[39,47325,4230],{"className":47326,"style":461},[83,84],[39,47328,47330,47352],{"className":47329,"translate":43},[42],[39,47331,47333],{"className":47332},[47],[49,47334,47335],{"xmlns":51},[53,47336,47337,47349],{},[56,47338,47339,47341,47343],{},[59,47340,4230],{},[133,47342,793],{},[138,47344,47345,47347],{},[59,47346,4230],{},[133,47348,21884],{},[63,47350,47351],{"encoding":65},"V=V^*",[39,47353,47355,47373],{"className":47354,"ariaHidden":70},[69],[39,47356,47358,47361,47364,47367,47370],{"className":47357},[74],[39,47359],{"className":47360,"style":79},[78],[39,47362,4230],{"className":47363,"style":461},[83,84],[39,47365],{"className":47366,"style":231},[230],[39,47368,793],{"className":47369},[235],[39,47371],{"className":47372,"style":231},[230],[39,47374,47376,47379],{"className":47375},[74],[39,47377],{"className":47378,"style":21897},[78],[39,47380,47382,47385],{"className":47381},[83],[39,47383,4230],{"className":47384,"style":461},[83,84],[39,47386,47388],{"className":47387},[192],[39,47389,47391],{"className":47390},[196],[39,47392,47394],{"className":47393},[200],[39,47395,47397],{"className":47396,"style":21897},[204],[39,47398,47399,47402],{"style":208},[39,47400],{"className":47401,"style":213},[212],[39,47403,47405],{"className":47404},[217,218,219,220],[39,47406,21884],{"className":47407},[465,220],"。若策略 ",[39,47410,47412,47425],{"className":47411,"translate":43},[42],[39,47413,47415],{"className":47414},[47],[49,47416,47417],{"xmlns":51},[53,47418,47419,47423],{},[56,47420,47421],{},[59,47422,2098],{},[63,47424,2101],{"encoding":65},[39,47426,47428],{"className":47427,"ariaHidden":70},[69],[39,47429,47431,47434],{"className":47430},[74],[39,47432],{"className":47433,"style":290},[78],[39,47435,2098],{"className":47436,"style":2114},[83,84]," 关于 ",[39,47439,47441,47454],{"className":47440,"translate":43},[42],[39,47442,47444],{"className":47443},[47],[49,47445,47446],{"xmlns":51},[53,47447,47448,47452],{},[56,47449,47450],{},[59,47451,4230],{},[63,47453,4230],{"encoding":65},[39,47455,47457],{"className":47456,"ariaHidden":70},[69],[39,47458,47460,47463],{"className":47459},[74],[39,47461],{"className":47462,"style":79},[78],[39,47464,4230],{"className":47465,"style":461},[83,84]," 贪婪，即",[39,47468,47470],{"className":47469,"translate":43},[746],[39,47471,47473,47583],{"className":47472,"translate":43},[42],[39,47474,47476],{"className":47475},[47],[49,47477,47478],{"xmlns":51,"display":755},[53,47479,47480,47580],{},[56,47481,47482,47484,47486,47488,47490,47492,47494,47496,47506,47516,47518,47520,47526,47528,47530,47532,47534,47536,47578],{},[59,47483,2098],{},[133,47485,136],{"stretchy":135},[59,47487,142],{},[133,47489,161],{"stretchy":135},[133,47491,534],{},[59,47493,39047],{},[133,47495,1690],{},[6556,47497,47498,47504],{},[56,47499,47500,47502],{},[59,47501,1687],{},[133,47503,1690],{},[59,47505,158],{},[6556,47507,47508,47510],{},[133,47509,3191],{},[138,47511,47512,47514],{},[59,47513,142],{},[133,47515,147],{"mathvariant":145,"lspace":146,"rspace":146},[59,47517,131],{},[133,47519,136],{"stretchy":135},[138,47521,47522,47524],{},[59,47523,142],{},[133,47525,147],{"mathvariant":145,"lspace":146,"rspace":146},[133,47527,150],{},[59,47529,142],{},[133,47531,155],{"separator":70},[59,47533,158],{},[133,47535,161],{"stretchy":135},[56,47537,47538,47540,47542,47544,47546,47548,47550,47552,47558,47560,47562,47564,47566,47568,47574,47576],{},[133,47539,537],{"fence":70},[59,47541,401],{},[133,47543,136],{"stretchy":135},[59,47545,142],{},[133,47547,155],{"separator":70},[59,47549,158],{},[133,47551,155],{"separator":70},[138,47553,47554,47556],{},[59,47555,142],{},[133,47557,147],{"mathvariant":145,"lspace":146,"rspace":146},[133,47559,161],{"stretchy":135},[133,47561,788],{},[59,47563,531],{},[59,47565,4230],{},[133,47567,136],{"stretchy":135},[138,47569,47570,47572],{},[59,47571,142],{},[133,47573,147],{"mathvariant":145,"lspace":146,"rspace":146},[133,47575,161],{"stretchy":135},[133,47577,4280],{"fence":70},[133,47579,155],{"separator":70},[63,47581,47582],{"encoding":65},"\\pi(s)\\in\\arg\\max_{a}\\sum_{s'}P(s'\\mid s,a)\n\\left[r(s,a,s')+\\gamma V(s')\\right],",[39,47584,47586,47613,47803],{"className":47585,"ariaHidden":70},[69],[39,47587,47589,47592,47595,47598,47601,47604,47607,47610],{"className":47588},[74],[39,47590],{"className":47591,"style":245},[78],[39,47593,2098],{"className":47594,"style":2114},[83,84],[39,47596,136],{"className":47597},[182],[39,47599,142],{"className":47600},[83,84],[39,47602,161],{"className":47603},[263],[39,47605],{"className":47606,"style":231},[230],[39,47608,534],{"className":47609},[235],[39,47611],{"className":47612,"style":231},[230],[39,47614,47616,47619,47624,47627,47674,47677,47753,47756,47759,47762,47794,47797,47800],{"className":47615},[74],[39,47617],{"className":47618,"style":22284},[78],[39,47620,39206,47622],{"className":47621},[1744],[39,47623,39209],{"style":35078},[39,47625],{"className":47626,"style":256},[230],[39,47628,47630],{"className":47629},[1744,3294],[39,47631,47633,47666],{"className":47632},[196,649],[39,47634,47636,47663],{"className":47635},[200],[39,47637,47639,47653],{"className":47638,"style":290},[204],[39,47640,47641,47644],{"style":22299},[39,47642],{"className":47643,"style":935},[212],[39,47645,47647],{"className":47646},[217,218,219,220],[39,47648,47650],{"className":47649},[83,220],[39,47651,158],{"className":47652},[83,84,220],[39,47654,47655,47658],{"style":21599},[39,47656],{"className":47657,"style":935},[212],[39,47659,47660],{},[39,47661,1687],{"className":47662},[1744],[39,47664,673],{"className":47665},[672],[39,47667,47669],{"className":47668},[200],[39,47670,47672],{"className":47671,"style":22331},[204],[39,47673],{},[39,47675],{"className":47676,"style":256},[230],[39,47678,47680],{"className":47679},[1744,3294],[39,47681,47683,47745],{"className":47682},[196,649],[39,47684,47686,47742],{"className":47685},[200],[39,47687,47689,47732],{"className":47688,"style":6670},[204],[39,47690,47691,47694],{"style":7097},[39,47692],{"className":47693,"style":3311},[212],[39,47695,47697],{"className":47696},[217,218,219,220],[39,47698,47700],{"className":47699},[83,220],[39,47701,47703,47706],{"className":47702},[83,220],[39,47704,142],{"className":47705},[83,84,220],[39,47707,47709],{"className":47708},[192],[39,47710,47712],{"className":47711},[196],[39,47713,47715],{"className":47714},[200],[39,47716,47718],{"className":47717,"style":3997},[204],[39,47719,47720,47723],{"style":4000},[39,47721],{"className":47722,"style":4004},[212],[39,47724,47726],{"className":47725},[217,4008,4009,220],[39,47727,47729],{"className":47728},[83,220],[39,47730,147],{"className":47731},[83,220],[39,47733,47734,47737],{"style":3330},[39,47735],{"className":47736,"style":3311},[212],[39,47738,47739],{},[39,47740,3191],{"className":47741},[1744,3339,3340],[39,47743,673],{"className":47744},[672],[39,47746,47748],{"className":47747},[200],[39,47749,47751],{"className":47750,"style":7158},[204],[39,47752],{},[39,47754],{"className":47755,"style":256},[230],[39,47757,131],{"className":47758,"style":178},[83,84],[39,47760,136],{"className":47761},[182],[39,47763,47765,47768],{"className":47764},[83],[39,47766,142],{"className":47767},[83,84],[39,47769,47771],{"className":47770},[192],[39,47772,47774],{"className":47773},[196],[39,47775,47777],{"className":47776},[200],[39,47778,47780],{"className":47779,"style":1024},[204],[39,47781,47782,47785],{"style":1027},[39,47783],{"className":47784,"style":213},[212],[39,47786,47788],{"className":47787},[217,218,219,220],[39,47789,47791],{"className":47790},[83,220],[39,47792,147],{"className":47793},[83,220],[39,47795],{"className":47796,"style":231},[230],[39,47798,150],{"className":47799},[235],[39,47801],{"className":47802,"style":231},[230],[39,47804,47806,47809,47812,47815,47818,47821,47824,47827,47948,47951],{"className":47805},[74],[39,47807],{"className":47808,"style":22469},[78],[39,47810,142],{"className":47811},[83,84],[39,47813,155],{"className":47814},[252],[39,47816],{"className":47817,"style":256},[230],[39,47819,158],{"className":47820},[83,84],[39,47822,161],{"className":47823},[263],[39,47825],{"className":47826,"style":256},[230],[39,47828,47830,47833,47836,47839,47842,47845,47848,47851,47854,47857,47889,47892,47895,47898,47901,47904,47907,47910,47942,47945],{"className":47829},[2842],[39,47831,537],{"className":47832,"style":4592},[182,4591],[39,47834,401],{"className":47835,"style":438},[83,84],[39,47837,136],{"className":47838},[182],[39,47840,142],{"className":47841},[83,84],[39,47843,155],{"className":47844},[252],[39,47846],{"className":47847,"style":256},[230],[39,47849,158],{"className":47850},[83,84],[39,47852,155],{"className":47853},[252],[39,47855],{"className":47856,"style":256},[230],[39,47858,47860,47863],{"className":47859},[83],[39,47861,142],{"className":47862},[83,84],[39,47864,47866],{"className":47865},[192],[39,47867,47869],{"className":47868},[196],[39,47870,47872],{"className":47871},[200],[39,47873,47875],{"className":47874,"style":1024},[204],[39,47876,47877,47880],{"style":1027},[39,47878],{"className":47879,"style":213},[212],[39,47881,47883],{"className":47882},[217,218,219,220],[39,47884,47886],{"className":47885},[83,220],[39,47887,147],{"className":47888},[83,220],[39,47890,161],{"className":47891},[263],[39,47893],{"className":47894,"style":461},[230],[39,47896,788],{"className":47897},[465],[39,47899],{"className":47900,"style":461},[230],[39,47902,531],{"className":47903,"style":565},[83,84],[39,47905,4230],{"className":47906,"style":461},[83,84],[39,47908,136],{"className":47909},[182],[39,47911,47913,47916],{"className":47912},[83],[39,47914,142],{"className":47915},[83,84],[39,47917,47919],{"className":47918},[192],[39,47920,47922],{"className":47921},[196],[39,47923,47925],{"className":47924},[200],[39,47926,47928],{"className":47927,"style":1024},[204],[39,47929,47930,47933],{"style":1027},[39,47931],{"className":47932,"style":213},[212],[39,47934,47936],{"className":47935},[217,218,219,220],[39,47937,47939],{"className":47938},[83,220],[39,47940,147],{"className":47941},[83,220],[39,47943,161],{"className":47944},[263],[39,47946,4280],{"className":47947,"style":4592},[263,4591],[39,47949],{"className":47950,"style":256},[230],[39,47952,155],{"className":47953},[252],[11,47955,14173,47956,15508,48037,48065],{},[39,47957,47959,47981],{"className":47958,"translate":43},[42],[39,47960,47962],{"className":47961},[47],[49,47963,47964],{"xmlns":51},[53,47965,47966,47978],{},[56,47967,47968,47974,47976],{},[138,47969,47970,47972],{},[59,47971,4230],{},[59,47973,2098],{},[133,47975,793],{},[59,47977,4230],{},[63,47979,47980],{"encoding":65},"V^\\pi=V",[39,47982,47984,48028],{"className":47983,"ariaHidden":70},[69],[39,47985,47987,47990,48019,48022,48025],{"className":47986},[74],[39,47988],{"className":47989,"style":79},[78],[39,47991,47993,47996],{"className":47992},[83],[39,47994,4230],{"className":47995,"style":461},[83,84],[39,47997,47999],{"className":47998},[192],[39,48000,48002],{"className":48001},[196],[39,48003,48005],{"className":48004},[200],[39,48006,48008],{"className":48007,"style":6410},[204],[39,48009,48010,48013],{"style":208},[39,48011],{"className":48012,"style":213},[212],[39,48014,48016],{"className":48015},[217,218,219,220],[39,48017,2098],{"className":48018,"style":2114},[83,84,220],[39,48020],{"className":48021,"style":231},[230],[39,48023,793],{"className":48024},[235],[39,48026],{"className":48027,"style":231},[230],[39,48029,48031,48034],{"className":48030},[74],[39,48032],{"className":48033,"style":79},[78],[39,48035,4230],{"className":48036,"style":461},[83,84],[39,48038,48040,48053],{"className":48039,"translate":43},[42],[39,48041,48043],{"className":48042},[47],[49,48044,48045],{"xmlns":51},[53,48046,48047,48051],{},[56,48048,48049],{},[59,48050,2098],{},[63,48052,2101],{"encoding":65},[39,48054,48056],{"className":48055,"ariaHidden":70},[69],[39,48057,48059,48062],{"className":48058},[74],[39,48060],{"className":48061,"style":290},[78],[39,48063,2098],{"className":48064,"style":2114},[83,84]," 是最优策略。",[11,48067,48068,48070,48071,48129,48130,47437,48158,48186,48187,42282,48330,742,48332,21228],{},[1489,48069,5595],{}," 第一式由 ",[39,48072,48074,48091],{"className":48073,"translate":43},[42],[39,48075,48077],{"className":48076},[47],[49,48078,48079],{"xmlns":51},[53,48080,48081,48089],{},[56,48082,48083],{},[138,48084,48085,48087],{},[59,48086,11772],{"mathvariant":617},[133,48088,21884],{},[63,48090,21887],{"encoding":65},[39,48092,48094],{"className":48093,"ariaHidden":70},[69],[39,48095,48097,48100],{"className":48096},[74],[39,48098],{"className":48099,"style":21897},[78],[39,48101,48103,48106],{"className":48102},[83],[39,48104,11772],{"className":48105,"style":11793},[83,641],[39,48107,48109],{"className":48108},[192],[39,48110,48112],{"className":48111},[196],[39,48113,48115],{"className":48114},[200],[39,48116,48118],{"className":48117,"style":21897},[204],[39,48119,48120,48123],{"style":208},[39,48121],{"className":48122,"style":213},[212],[39,48124,48126],{"className":48125},[217,218,219,220],[39,48127,21884],{"className":48128},[465,220]," 的不动点唯一性立即得到。若 ",[39,48131,48133,48146],{"className":48132,"translate":43},[42],[39,48134,48136],{"className":48135},[47],[49,48137,48138],{"xmlns":51},[53,48139,48140,48144],{},[56,48141,48142],{},[59,48143,2098],{},[63,48145,2101],{"encoding":65},[39,48147,48149],{"className":48148,"ariaHidden":70},[69],[39,48150,48152,48155],{"className":48151},[74],[39,48153],{"className":48154,"style":290},[78],[39,48156,2098],{"className":48157,"style":2114},[83,84],[39,48159,48161,48174],{"className":48160,"translate":43},[42],[39,48162,48164],{"className":48163},[47],[49,48165,48166],{"xmlns":51},[53,48167,48168,48172],{},[56,48169,48170],{},[59,48171,4230],{},[63,48173,4230],{"encoding":65},[39,48175,48177],{"className":48176,"ariaHidden":70},[69],[39,48178,48180,48183],{"className":48179},[74],[39,48181],{"className":48182,"style":79},[78],[39,48184,4230],{"className":48185,"style":461},[83,84]," 贪婪，则 ",[39,48188,48190,48224],{"className":48189,"translate":43},[42],[39,48191,48193],{"className":48192},[47],[49,48194,48195],{"xmlns":51},[53,48196,48197,48221],{},[56,48198,48199,48205,48207,48209,48215,48217,48219],{},[138,48200,48201,48203],{},[59,48202,11772],{"mathvariant":617},[59,48204,2098],{},[59,48206,4230],{},[133,48208,793],{},[138,48210,48211,48213],{},[59,48212,11772],{"mathvariant":617},[133,48214,21884],{},[59,48216,4230],{},[133,48218,793],{},[59,48220,4230],{},[63,48222,48223],{"encoding":65},"\\mathcal{T}^\\pi V=\\mathcal{T}^*V=V",[39,48225,48227,48274,48321],{"className":48226,"ariaHidden":70},[69],[39,48228,48230,48233,48262,48265,48268,48271],{"className":48229},[74],[39,48231],{"className":48232,"style":79},[78],[39,48234,48236,48239],{"className":48235},[83],[39,48237,11772],{"className":48238,"style":11793},[83,641],[39,48240,48242],{"className":48241},[192],[39,48243,48245],{"className":48244},[196],[39,48246,48248],{"className":48247},[200],[39,48249,48251],{"className":48250,"style":6410},[204],[39,48252,48253,48256],{"style":208},[39,48254],{"className":48255,"style":213},[212],[39,48257,48259],{"className":48258},[217,218,219,220],[39,48260,2098],{"className":48261,"style":2114},[83,84,220],[39,48263,4230],{"className":48264,"style":461},[83,84],[39,48266],{"className":48267,"style":231},[230],[39,48269,793],{"className":48270},[235],[39,48272],{"className":48273,"style":231},[230],[39,48275,48277,48280,48309,48312,48315,48318],{"className":48276},[74],[39,48278],{"className":48279,"style":21897},[78],[39,48281,48283,48286],{"className":48282},[83],[39,48284,11772],{"className":48285,"style":11793},[83,641],[39,48287,48289],{"className":48288},[192],[39,48290,48292],{"className":48291},[196],[39,48293,48295],{"className":48294},[200],[39,48296,48298],{"className":48297,"style":21897},[204],[39,48299,48300,48303],{"style":208},[39,48301],{"className":48302,"style":213},[212],[39,48304,48306],{"className":48305},[217,218,219,220],[39,48307,21884],{"className":48308},[465,220],[39,48310,4230],{"className":48311,"style":461},[83,84],[39,48313],{"className":48314,"style":231},[230],[39,48316,793],{"className":48317},[235],[39,48319],{"className":48320,"style":231},[230],[39,48322,48324,48327],{"className":48323},[74],[39,48325],{"className":48326,"style":79},[78],[39,48328,4230],{"className":48329,"style":461},[83,84],[158,48331,21294],{"href":21293},[39,48333,48335,48356],{"className":48334,"translate":43},[42],[39,48336,48338],{"className":48337},[47],[49,48339,48340],{"xmlns":51},[53,48341,48342,48354],{},[56,48343,48344,48350,48352],{},[138,48345,48346,48348],{},[59,48347,4230],{},[59,48349,2098],{},[133,48351,793],{},[59,48353,4230],{},[63,48355,47980],{"encoding":65},[39,48357,48359,48403],{"className":48358,"ariaHidden":70},[69],[39,48360,48362,48365,48394,48397,48400],{"className":48361},[74],[39,48363],{"className":48364,"style":79},[78],[39,48366,48368,48371],{"className":48367},[83],[39,48369,4230],{"className":48370,"style":461},[83,84],[39,48372,48374],{"className":48373},[192],[39,48375,48377],{"className":48376},[196],[39,48378,48380],{"className":48379},[200],[39,48381,48383],{"className":48382,"style":6410},[204],[39,48384,48385,48388],{"style":208},[39,48386],{"className":48387,"style":213},[212],[39,48389,48391],{"className":48390},[217,218,219,220],[39,48392,2098],{"className":48393,"style":2114},[83,84,220],[39,48395],{"className":48396,"style":231},[230],[39,48398,793],{"className":48399},[235],[39,48401],{"className":48402,"style":231},[230],[39,48404,48406,48409],{"className":48405},[74],[39,48407],{"className":48408,"style":79},[78],[39,48410,4230],{"className":48411,"style":461},[83,84],[11,48413,48414],{},"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。",[26,48416,48418],{"id":48417},"_45-例带决策的路径","4.5 例：带决策的路径",[11,48420,48421,48422,2183,48506,48534],{},"考虑如下路径选择问题。状态为 ",[39,48423,48425,48457],{"className":48424,"translate":43},[42],[39,48426,48428],{"className":48427},[47],[49,48429,48430],{"xmlns":51},[53,48431,48432,48454],{},[56,48433,48434,48436,48438,48440,48442,48445,48447,48450,48452],{},[59,48435,101],{},[133,48437,155],{"separator":70},[59,48439,19772],{},[133,48441,155],{"separator":70},[59,48443,48444],{},"C",[133,48446,155],{"separator":70},[59,48448,48449],{},"D",[133,48451,155],{"separator":70},[59,48453,4253],{},[63,48455,48456],{"encoding":65},"A,B,C,D,E",[39,48458,48460],{"className":48459,"ariaHidden":70},[69],[39,48461,48463,48466,48469,48472,48475,48478,48481,48484,48488,48491,48494,48497,48500,48503],{"className":48462},[74],[39,48464],{"className":48465,"style":6452},[78],[39,48467,101],{"className":48468},[83,84],[39,48470,155],{"className":48471},[252],[39,48473],{"className":48474,"style":256},[230],[39,48476,19772],{"className":48477,"style":19812},[83,84],[39,48479,155],{"className":48480},[252],[39,48482],{"className":48483,"style":256},[230],[39,48485,48444],{"className":48486,"style":48487},[83,84],"margin-right:0.0715em;",[39,48489,155],{"className":48490},[252],[39,48492],{"className":48493,"style":256},[230],[39,48495,48449],{"className":48496,"style":438},[83,84],[39,48498,155],{"className":48499},[252],[39,48501],{"className":48502,"style":256},[230],[39,48504,4253],{"className":48505,"style":85},[83,84],[39,48507,48509,48522],{"className":48508,"translate":43},[42],[39,48510,48512],{"className":48511},[47],[49,48513,48514],{"xmlns":51},[53,48515,48516,48520],{},[56,48517,48518],{},[59,48519,48444],{},[63,48521,48444],{"encoding":65},[39,48523,48525],{"className":48524,"ariaHidden":70},[69],[39,48526,48528,48531],{"className":48527},[74],[39,48529],{"className":48530,"style":79},[78],[39,48532,48444],{"className":48533,"style":48487},[83,84]," 为终止状态。转移关系为",[39,48536,48538],{"className":48537,"translate":43},[746],[39,48539,48541,48631],{"className":48540,"translate":43},[42],[39,48542,48544],{"className":48543},[47],[49,48545,48546],{"xmlns":51,"display":755},[53,48547,48548,48628],{},[759,48549,48550,48594],{"rowspacing":761,"columnalign":762,"columnspacing":146},[764,48551,48552,48558],{},[767,48553,48554],{},[770,48555,48556],{"scriptlevel":541,"displaystyle":70},[56,48557],{},[767,48559,48560],{},[770,48561,48562],{"scriptlevel":541,"displaystyle":70},[56,48563,48564,48566,48568,48570,48572,48574,48576,48578,48580,48582,48584,48586,48588,48590,48592],{},[56,48565],{},[59,48567,101],{},[133,48569,418],{},[59,48571,19772],{},[133,48573,155],{"separator":70},[230,48575],{"width":6891},[59,48577,19772],{},[133,48579,418],{},[59,48581,48444],{},[133,48583,155],{"separator":70},[230,48585],{"width":6891},[59,48587,101],{},[133,48589,418],{},[59,48591,48449],{},[133,48593,155],{"separator":70},[764,48595,48596,48602],{},[767,48597,48598],{},[770,48599,48600],{"scriptlevel":541,"displaystyle":70},[56,48601],{},[767,48603,48604],{},[770,48605,48606],{"scriptlevel":541,"displaystyle":70},[56,48607,48608,48610,48612,48614,48616,48618,48620,48622,48624,48626],{},[56,48609],{},[59,48611,48449],{},[133,48613,418],{},[59,48615,4253],{},[133,48617,155],{"separator":70},[230,48619],{"width":6891},[59,48621,4253],{},[133,48623,418],{},[59,48625,48444],{},[59,48627,2903],{"mathvariant":145},[63,48629,48630],{"encoding":65},"\\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}",[39,48632,48634],{"className":48633,"ariaHidden":70},[69],[39,48635,48637,48640],{"className":48636},[74],[39,48638],{"className":48639,"style":4374},[78],[39,48641,48643],{"className":48642},[83],[39,48644,48646,48685],{"className":48645},[759],[39,48647,48649],{"className":48648},[918],[39,48650,48652,48677],{"className":48651},[196,649],[39,48653,48655,48674],{"className":48654},[200],[39,48656,48658,48666],{"className":48657,"style":4393},[204],[39,48659,48660,48663],{"style":28366},[39,48661],{"className":48662,"style":28370},[212],[39,48664],{"className":48665},[83],[39,48667,48668,48671],{"style":5834},[39,48669],{"className":48670,"style":28370},[212],[39,48672],{"className":48673},[83],[39,48675,673],{"className":48676},[672],[39,48678,48680],{"className":48679},[200],[39,48681,48683],{"className":48682,"style":4507},[204],[39,48684],{},[39,48686,48688],{"className":48687},[1223],[39,48689,48691,48830],{"className":48690},[196,649],[39,48692,48694,48827],{"className":48693},[200],[39,48695,48697,48774],{"className":48696,"style":4393},[204],[39,48698,48699,48702],{"style":4396},[39,48700],{"className":48701,"style":935},[212],[39,48703,48705,48708,48711,48714,48717,48720,48723,48726,48729,48732,48735,48738,48741,48744,48747,48750,48753,48756,48759,48762,48765,48768,48771],{"className":48704},[83],[39,48706],{"className":48707},[83],[39,48709,101],{"className":48710},[83,84],[39,48712],{"className":48713,"style":231},[230],[39,48715,418],{"className":48716},[235],[39,48718],{"className":48719,"style":231},[230],[39,48721,19772],{"className":48722,"style":19812},[83,84],[39,48724,155],{"className":48725},[252],[39,48727],{"className":48728,"style":7240},[230],[39,48730],{"className":48731,"style":256},[230],[39,48733,19772],{"className":48734,"style":19812},[83,84],[39,48736],{"className":48737,"style":231},[230],[39,48739,418],{"className":48740},[235],[39,48742],{"className":48743,"style":231},[230],[39,48745,48444],{"className":48746,"style":48487},[83,84],[39,48748,155],{"className":48749},[252],[39,48751],{"className":48752,"style":7240},[230],[39,48754],{"className":48755,"style":256},[230],[39,48757,101],{"className":48758},[83,84],[39,48760],{"className":48761,"style":231},[230],[39,48763,418],{"className":48764},[235],[39,48766],{"className":48767,"style":231},[230],[39,48769,48449],{"className":48770,"style":438},[83,84],[39,48772,155],{"className":48773},[252],[39,48775,48776,48779],{"style":4444},[39,48777],{"className":48778,"style":935},[212],[39,48780,48782,48785,48788,48791,48794,48797,48800,48803,48806,48809,48812,48815,48818,48821,48824],{"className":48781},[83],[39,48783],{"className":48784},[83],[39,48786,48449],{"className":48787,"style":438},[83,84],[39,48789],{"className":48790,"style":231},[230],[39,48792,418],{"className":48793},[235],[39,48795],{"className":48796,"style":231},[230],[39,48798,4253],{"className":48799,"style":85},[83,84],[39,48801,155],{"className":48802},[252],[39,48804],{"className":48805,"style":7240},[230],[39,48807],{"className":48808,"style":256},[230],[39,48810,4253],{"className":48811,"style":85},[83,84],[39,48813],{"className":48814,"style":231},[230],[39,48816,418],{"className":48817},[235],[39,48819],{"className":48820,"style":231},[230],[39,48822,48444],{"className":48823,"style":48487},[83,84],[39,48825,2903],{"className":48826},[83],[39,48828,673],{"className":48829},[672],[39,48831,48833],{"className":48832},[200],[39,48834,48836],{"className":48835,"style":4507},[204],[39,48837],{},[11,48839,48840,48841,48875,48876,48904,48905,48934,48935,48985,48986,1607],{},"每次转移获得奖励 ",[39,48842,48844,48860],{"className":48843,"translate":43},[42],[39,48845,48847],{"className":48846},[47],[49,48848,48849],{"xmlns":51},[53,48850,48851,48857],{},[56,48852,48853,48855],{},[133,48854,5378],{},[539,48856,546],{},[63,48858,48859],{"encoding":65},"-1",[39,48861,48863],{"className":48862,"ariaHidden":70},[69],[39,48864,48866,48869,48872],{"className":48865},[74],[39,48867],{"className":48868,"style":17882},[78],[39,48870,5378],{"className":48871},[83],[39,48873,546],{"className":48874},[83],"，进入终止状态 ",[39,48877,48879,48892],{"className":48878,"translate":43},[42],[39,48880,48882],{"className":48881},[47],[49,48883,48884],{"xmlns":51},[53,48885,48886,48890],{},[56,48887,48888],{},[59,48889,48444],{},[63,48891,48444],{"encoding":65},[39,48893,48895],{"className":48894,"ariaHidden":70},[69],[39,48896,48898,48901],{"className":48897},[74],[39,48899],{"className":48900,"style":79},[78],[39,48902,48444],{"className":48903,"style":48487},[83,84]," 时额外获得奖励 ",[39,48906,48908,48922],{"className":48907,"translate":43},[42],[39,48909,48911],{"className":48910},[47],[49,48912,48913],{"xmlns":51},[53,48914,48915,48920],{},[56,48916,48917],{},[539,48918,48919],{},"10",[63,48921,48919],{"encoding":65},[39,48923,48925],{"className":48924,"ariaHidden":70},[69],[39,48926,48928,48931],{"className":48927},[74],[39,48929],{"className":48930,"style":2473},[78],[39,48932,48919],{"className":48933},[83],"。设 ",[39,48936,48938,48955],{"className":48937,"translate":43},[42],[39,48939,48941],{"className":48940},[47],[49,48942,48943],{"xmlns":51},[53,48944,48945,48953],{},[56,48946,48947,48949,48951],{},[59,48948,531],{},[133,48950,793],{},[539,48952,17739],{},[63,48954,17742],{"encoding":65},[39,48956,48958,48976],{"className":48957,"ariaHidden":70},[69],[39,48959,48961,48964,48967,48970,48973],{"className":48960},[74],[39,48962],{"className":48963,"style":2654},[78],[39,48965,531],{"className":48966,"style":565},[83,84],[39,48968],{"className":48969,"style":231},[230],[39,48971,793],{"className":48972},[235],[39,48974],{"className":48975,"style":231},[230],[39,48977,48979,48982],{"className":48978},[74],[39,48980],{"className":48981,"style":2473},[78],[39,48983,17739],{"className":48984},[83],"，并令 ",[39,48987,48989,49013],{"className":48988,"translate":43},[42],[39,48990,48992],{"className":48991},[47],[49,48993,48994],{"xmlns":51},[53,48995,48996,49010],{},[56,48997,48998,49000,49002,49004,49006,49008],{},[59,48999,4230],{},[133,49001,136],{"stretchy":135},[59,49003,48444],{},[133,49005,161],{"stretchy":135},[133,49007,793],{},[539,49009,541],{},[63,49011,49012],{"encoding":65},"V(C)=0",[39,49014,49016,49043],{"className":49015,"ariaHidden":70},[69],[39,49017,49019,49022,49025,49028,49031,49034,49037,49040],{"className":49018},[74],[39,49020],{"className":49021,"style":245},[78],[39,49023,4230],{"className":49024,"style":461},[83,84],[39,49026,136],{"className":49027},[182],[39,49029,48444],{"className":49030,"style":48487},[83,84],[39,49032,161],{"className":49033},[263],[39,49035],{"className":49036,"style":231},[230],[39,49038,793],{"className":49039},[235],[39,49041],{"className":49042,"style":231},[230],[39,49044,49046,49049],{"className":49045},[74],[39,49047],{"className":49048,"style":2473},[78],[39,49050,541],{"className":49051},[83],[11,49053,49054,49055,1639,49083,49111,49112,49140],{},"在 ",[39,49056,49058,49071],{"className":49057,"translate":43},[42],[39,49059,49061],{"className":49060},[47],[49,49062,49063],{"xmlns":51},[53,49064,49065,49069],{},[56,49066,49067],{},[59,49068,19772],{},[63,49070,19772],{"encoding":65},[39,49072,49074],{"className":49073,"ariaHidden":70},[69],[39,49075,49077,49080],{"className":49076},[74],[39,49078],{"className":49079,"style":79},[78],[39,49081,19772],{"className":49082,"style":19812},[83,84],[39,49084,49086,49099],{"className":49085,"translate":43},[42],[39,49087,49089],{"className":49088},[47],[49,49090,49091],{"xmlns":51},[53,49092,49093,49097],{},[56,49094,49095],{},[59,49096,4253],{},[63,49098,4253],{"encoding":65},[39,49100,49102],{"className":49101,"ariaHidden":70},[69],[39,49103,49105,49108],{"className":49104},[74],[39,49106],{"className":49107,"style":79},[78],[39,49109,4253],{"className":49110,"style":85},[83,84],"，只有一条通往 ",[39,49113,49115,49128],{"className":49114,"translate":43},[42],[39,49116,49118],{"className":49117},[47],[49,49119,49120],{"xmlns":51},[53,49121,49122,49126],{},[56,49123,49124],{},[59,49125,48444],{},[63,49127,48444],{"encoding":65},[39,49129,49131],{"className":49130,"ariaHidden":70},[69],[39,49132,49134,49137],{"className":49133},[74],[39,49135],{"className":49136,"style":79},[78],[39,49138,48444],{"className":49139,"style":48487},[83,84]," 的动作，因此",[39,49142,49144],{"className":49143,"translate":43},[746],[39,49145,49147,49210],{"className":49146,"translate":43},[42],[39,49148,49150],{"className":49149},[47],[49,49151,49152],{"xmlns":51,"display":755},[53,49153,49154,49207],{},[56,49155,49156,49158,49160,49162,49164,49166,49168,49170,49172,49174,49176,49178,49180,49182,49184,49186,49188,49190,49192,49194,49196,49198,49200,49202,49204],{},[59,49157,4230],{},[133,49159,136],{"stretchy":135},[59,49161,19772],{},[133,49163,161],{"stretchy":135},[133,49165,793],{},[133,49167,5378],{},[539,49169,546],{},[133,49171,788],{},[539,49173,48919],{},[133,49175,788],{},[539,49177,17739],{},[59,49179,4230],{},[133,49181,136],{"stretchy":135},[59,49183,48444],{},[133,49185,161],{"stretchy":135},[133,49187,793],{},[539,49189,28873],{},[133,49191,155],{"separator":70},[230,49193],{"width":10251},[59,49195,4230],{},[133,49197,136],{"stretchy":135},[59,49199,4253],{},[133,49201,161],{"stretchy":135},[133,49203,793],{},[539,49205,49206],{},"9.",[63,49208,49209],{"encoding":65},"V(B)=-1+10+0.9V(C)=9,\n\\qquad\nV(E)=9.",[39,49211,49213,49240,49261,49279,49309,49348],{"className":49212,"ariaHidden":70},[69],[39,49214,49216,49219,49222,49225,49228,49231,49234,49237],{"className":49215},[74],[39,49217],{"className":49218,"style":245},[78],[39,49220,4230],{"className":49221,"style":461},[83,84],[39,49223,136],{"className":49224},[182],[39,49226,19772],{"className":49227,"style":19812},[83,84],[39,49229,161],{"className":49230},[263],[39,49232],{"className":49233,"style":231},[230],[39,49235,793],{"className":49236},[235],[39,49238],{"className":49239,"style":231},[230],[39,49241,49243,49246,49249,49252,49255,49258],{"className":49242},[74],[39,49244],{"className":49245,"style":17882},[78],[39,49247,5378],{"className":49248},[83],[39,49250,546],{"className":49251},[83],[39,49253],{"className":49254,"style":461},[230],[39,49256,788],{"className":49257},[465],[39,49259],{"className":49260,"style":461},[230],[39,49262,49264,49267,49270,49273,49276],{"className":49263},[74],[39,49265],{"className":49266,"style":17882},[78],[39,49268,48919],{"className":49269},[83],[39,49271],{"className":49272,"style":461},[230],[39,49274,788],{"className":49275},[465],[39,49277],{"className":49278,"style":461},[230],[39,49280,49282,49285,49288,49291,49294,49297,49300,49303,49306],{"className":49281},[74],[39,49283],{"className":49284,"style":245},[78],[39,49286,17739],{"className":49287},[83],[39,49289,4230],{"className":49290,"style":461},[83,84],[39,49292,136],{"className":49293},[182],[39,49295,48444],{"className":49296,"style":48487},[83,84],[39,49298,161],{"className":49299},[263],[39,49301],{"className":49302,"style":231},[230],[39,49304,793],{"className":49305},[235],[39,49307],{"className":49308,"style":231},[230],[39,49310,49312,49315,49318,49321,49324,49327,49330,49333,49336,49339,49342,49345],{"className":49311},[74],[39,49313],{"className":49314,"style":245},[78],[39,49316,28873],{"className":49317},[83],[39,49319,155],{"className":49320},[252],[39,49322],{"className":49323,"style":10787},[230],[39,49325],{"className":49326,"style":256},[230],[39,49328,4230],{"className":49329,"style":461},[83,84],[39,49331,136],{"className":49332},[182],[39,49334,4253],{"className":49335,"style":85},[83,84],[39,49337,161],{"className":49338},[263],[39,49340],{"className":49341,"style":231},[230],[39,49343,793],{"className":49344},[235],[39,49346],{"className":49347,"style":231},[230],[39,49349,49351,49354],{"className":49350},[74],[39,49352],{"className":49353,"style":2473},[78],[39,49355,49206],{"className":49356},[83],[11,49358,49054,49359,49387,49388,49416],{},[39,49360,49362,49375],{"className":49361,"translate":43},[42],[39,49363,49365],{"className":49364},[47],[49,49366,49367],{"xmlns":51},[53,49368,49369,49373],{},[56,49370,49371],{},[59,49372,48449],{},[63,49374,48449],{"encoding":65},[39,49376,49378],{"className":49377,"ariaHidden":70},[69],[39,49379,49381,49384],{"className":49380},[74],[39,49382],{"className":49383,"style":79},[78],[39,49385,48449],{"className":49386,"style":438},[83,84],"，唯一动作通向 ",[39,49389,49391,49404],{"className":49390,"translate":43},[42],[39,49392,49394],{"className":49393},[47],[49,49395,49396],{"xmlns":51},[53,49397,49398,49402],{},[56,49399,49400],{},[59,49401,4253],{},[63,49403,4253],{"encoding":65},[39,49405,49407],{"className":49406,"ariaHidden":70},[69],[39,49408,49410,49413],{"className":49409},[74],[39,49411],{"className":49412,"style":79},[78],[39,49414,4253],{"className":49415,"style":85},[83,84],"，故",[39,49418,49420],{"className":49419,"translate":43},[746],[39,49421,49423,49480],{"className":49422,"translate":43},[42],[39,49424,49426],{"className":49425},[47],[49,49427,49428],{"xmlns":51,"display":755},[53,49429,49430,49477],{},[56,49431,49432,49434,49436,49438,49440,49442,49444,49446,49448,49450,49452,49454,49456,49458,49460,49462,49464,49466,49468,49470,49472,49474],{},[59,49433,4230],{},[133,49435,136],{"stretchy":135},[59,49437,48449],{},[133,49439,161],{"stretchy":135},[133,49441,793],{},[133,49443,5378],{},[539,49445,546],{},[133,49447,788],{},[539,49449,17739],{},[59,49451,4230],{},[133,49453,136],{"stretchy":135},[59,49455,4253],{},[133,49457,161],{"stretchy":135},[133,49459,793],{},[133,49461,5378],{},[539,49463,546],{},[133,49465,788],{},[539,49467,17739],{},[133,49469,409],{},[539,49471,28873],{},[133,49473,793],{},[539,49475,49476],{},"7.1.",[63,49478,49479],{"encoding":65},"V(D)=-1+0.9V(E)=-1+0.9\\times9=7.1.",[39,49481,49483,49510,49531,49561,49582,49600,49618],{"className":49482,"ariaHidden":70},[69],[39,49484,49486,49489,49492,49495,49498,49501,49504,49507],{"className":49485},[74],[39,49487],{"className":49488,"style":245},[78],[39,49490,4230],{"className":49491,"style":461},[83,84],[39,49493,136],{"className":49494},[182],[39,49496,48449],{"className":49497,"style":438},[83,84],[39,49499,161],{"className":49500},[263],[39,49502],{"className":49503,"style":231},[230],[39,49505,793],{"className":49506},[235],[39,49508],{"className":49509,"style":231},[230],[39,49511,49513,49516,49519,49522,49525,49528],{"className":49512},[74],[39,49514],{"className":49515,"style":17882},[78],[39,49517,5378],{"className":49518},[83],[39,49520,546],{"className":49521},[83],[39,49523],{"className":49524,"style":461},[230],[39,49526,788],{"className":49527},[465],[39,49529],{"className":49530,"style":461},[230],[39,49532,49534,49537,49540,49543,49546,49549,49552,49555,49558],{"className":49533},[74],[39,49535],{"className":49536,"style":245},[78],[39,49538,17739],{"className":49539},[83],[39,49541,4230],{"className":49542,"style":461},[83,84],[39,49544,136],{"className":49545},[182],[39,49547,4253],{"className":49548,"style":85},[83,84],[39,49550,161],{"className":49551},[263],[39,49553],{"className":49554,"style":231},[230],[39,49556,793],{"className":49557},[235],[39,49559],{"className":49560,"style":231},[230],[39,49562,49564,49567,49570,49573,49576,49579],{"className":49563},[74],[39,49565],{"className":49566,"style":17882},[78],[39,49568,5378],{"className":49569},[83],[39,49571,546],{"className":49572},[83],[39,49574],{"className":49575,"style":461},[230],[39,49577,788],{"className":49578},[465],[39,49580],{"className":49581,"style":461},[230],[39,49583,49585,49588,49591,49594,49597],{"className":49584},[74],[39,49586],{"className":49587,"style":17882},[78],[39,49589,17739],{"className":49590},[83],[39,49592],{"className":49593,"style":461},[230],[39,49595,409],{"className":49596},[465],[39,49598],{"className":49599,"style":461},[230],[39,49601,49603,49606,49609,49612,49615],{"className":49602},[74],[39,49604],{"className":49605,"style":2473},[78],[39,49607,28873],{"className":49608},[83],[39,49610],{"className":49611,"style":231},[230],[39,49613,793],{"className":49614},[235],[39,49616],{"className":49617,"style":231},[230],[39,49619,49621,49624],{"className":49620},[74],[39,49622],{"className":49623,"style":2473},[78],[39,49625,49476],{"className":49626},[83],[11,49628,49054,49629,49657,49658,1639,49724,49790,49791,49819],{},[39,49630,49632,49645],{"className":49631,"translate":43},[42],[39,49633,49635],{"className":49634},[47],[49,49636,49637],{"xmlns":51},[53,49638,49639,49643],{},[56,49640,49641],{},[59,49642,101],{},[63,49644,101],{"encoding":65},[39,49646,49648],{"className":49647,"ariaHidden":70},[69],[39,49649,49651,49654],{"className":49650},[74],[39,49652],{"className":49653,"style":79},[78],[39,49655,101],{"className":49656},[83,84],"，有两条动作。记 ",[39,49659,49661,49685],{"className":49660,"translate":43},[42],[39,49662,49664],{"className":49663},[47],[49,49665,49666],{"xmlns":51},[53,49667,49668,49682],{},[56,49669,49670,49672,49674,49676,49678,49680],{},[59,49671,4295],{},[133,49673,136],{"stretchy":135},[59,49675,101],{},[133,49677,418],{},[59,49679,19772],{},[133,49681,161],{"stretchy":135},[63,49683,49684],{"encoding":65},"Q(A\\to B)",[39,49686,49688,49712],{"className":49687,"ariaHidden":70},[69],[39,49689,49691,49694,49697,49700,49703,49706,49709],{"className":49690},[74],[39,49692],{"className":49693,"style":245},[78],[39,49695,4295],{"className":49696},[83,84],[39,49698,136],{"className":49699},[182],[39,49701,101],{"className":49702},[83,84],[39,49704],{"className":49705,"style":231},[230],[39,49707,418],{"className":49708},[235],[39,49710],{"className":49711,"style":231},[230],[39,49713,49715,49718,49721],{"className":49714},[74],[39,49716],{"className":49717,"style":245},[78],[39,49719,19772],{"className":49720,"style":19812},[83,84],[39,49722,161],{"className":49723},[263],[39,49725,49727,49751],{"className":49726,"translate":43},[42],[39,49728,49730],{"className":49729},[47],[49,49731,49732],{"xmlns":51},[53,49733,49734,49748],{},[56,49735,49736,49738,49740,49742,49744,49746],{},[59,49737,4295],{},[133,49739,136],{"stretchy":135},[59,49741,101],{},[133,49743,418],{},[59,49745,48449],{},[133,49747,161],{"stretchy":135},[63,49749,49750],{"encoding":65},"Q(A\\to D)",[39,49752,49754,49778],{"className":49753,"ariaHidden":70},[69],[39,49755,49757,49760,49763,49766,49769,49772,49775],{"className":49756},[74],[39,49758],{"className":49759,"style":245},[78],[39,49761,4295],{"className":49762},[83,84],[39,49764,136],{"className":49765},[182],[39,49767,101],{"className":49768},[83,84],[39,49770],{"className":49771,"style":231},[230],[39,49773,418],{"className":49774},[235],[39,49776],{"className":49777,"style":231},[230],[39,49779,49781,49784,49787],{"className":49780},[74],[39,49782],{"className":49783,"style":245},[78],[39,49785,48449],{"className":49786,"style":438},[83,84],[39,49788,161],{"className":49789},[263]," 分别为在 ",[39,49792,49794,49807],{"className":49793,"translate":43},[42],[39,49795,49797],{"className":49796},[47],[49,49798,49799],{"xmlns":51},[53,49800,49801,49805],{},[56,49802,49803],{},[59,49804,101],{},[63,49806,101],{"encoding":65},[39,49808,49810],{"className":49809,"ariaHidden":70},[69],[39,49811,49813,49816],{"className":49812},[74],[39,49814],{"className":49815,"style":79},[78],[39,49817,101],{"className":49818},[83,84]," 选择这两条动作的动作值，则",[39,49821,49823],{"className":49822,"translate":43},[746],[39,49824,49826,49889],{"className":49825,"translate":43},[42],[39,49827,49829],{"className":49828},[47],[49,49830,49831],{"xmlns":51,"display":755},[53,49832,49833,49886],{},[56,49834,49835,49837,49839,49841,49843,49845,49847,49849,49851,49853,49855,49857,49859,49861,49863,49865,49867,49869,49871,49873,49875,49877,49879,49881,49884],{},[59,49836,4295],{},[133,49838,136],{"stretchy":135},[59,49840,101],{},[133,49842,418],{},[59,49844,19772],{},[133,49846,161],{"stretchy":135},[133,49848,793],{},[133,49850,5378],{},[539,49852,546],{},[133,49854,788],{},[539,49856,17739],{},[59,49858,4230],{},[133,49860,136],{"stretchy":135},[59,49862,19772],{},[133,49864,161],{"stretchy":135},[133,49866,793],{},[133,49868,5378],{},[539,49870,546],{},[133,49872,788],{},[539,49874,17739],{},[133,49876,409],{},[539,49878,28873],{},[133,49880,793],{},[539,49882,49883],{},"7.1",[133,49885,155],{"separator":70},[63,49887,49888],{"encoding":65},"Q(A\\to B)=-1+0.9V(B)=-1+0.9\\times9=7.1,",[39,49890,49892,49916,49937,49958,49988,50009,50027,50045],{"className":49891,"ariaHidden":70},[69],[39,49893,49895,49898,49901,49904,49907,49910,49913],{"className":49894},[74],[39,49896],{"className":49897,"style":245},[78],[39,49899,4295],{"className":49900},[83,84],[39,49902,136],{"className":49903},[182],[39,49905,101],{"className":49906},[83,84],[39,49908],{"className":49909,"style":231},[230],[39,49911,418],{"className":49912},[235],[39,49914],{"className":49915,"style":231},[230],[39,49917,49919,49922,49925,49928,49931,49934],{"className":49918},[74],[39,49920],{"className":49921,"style":245},[78],[39,49923,19772],{"className":49924,"style":19812},[83,84],[39,49926,161],{"className":49927},[263],[39,49929],{"className":49930,"style":231},[230],[39,49932,793],{"className":49933},[235],[39,49935],{"className":49936,"style":231},[230],[39,49938,49940,49943,49946,49949,49952,49955],{"className":49939},[74],[39,49941],{"className":49942,"style":17882},[78],[39,49944,5378],{"className":49945},[83],[39,49947,546],{"className":49948},[83],[39,49950],{"className":49951,"style":461},[230],[39,49953,788],{"className":49954},[465],[39,49956],{"className":49957,"style":461},[230],[39,49959,49961,49964,49967,49970,49973,49976,49979,49982,49985],{"className":49960},[74],[39,49962],{"className":49963,"style":245},[78],[39,49965,17739],{"className":49966},[83],[39,49968,4230],{"className":49969,"style":461},[83,84],[39,49971,136],{"className":49972},[182],[39,49974,19772],{"className":49975,"style":19812},[83,84],[39,49977,161],{"className":49978},[263],[39,49980],{"className":49981,"style":231},[230],[39,49983,793],{"className":49984},[235],[39,49986],{"className":49987,"style":231},[230],[39,49989,49991,49994,49997,50000,50003,50006],{"className":49990},[74],[39,49992],{"className":49993,"style":17882},[78],[39,49995,5378],{"className":49996},[83],[39,49998,546],{"className":49999},[83],[39,50001],{"className":50002,"style":461},[230],[39,50004,788],{"className":50005},[465],[39,50007],{"className":50008,"style":461},[230],[39,50010,50012,50015,50018,50021,50024],{"className":50011},[74],[39,50013],{"className":50014,"style":17882},[78],[39,50016,17739],{"className":50017},[83],[39,50019],{"className":50020,"style":461},[230],[39,50022,409],{"className":50023},[465],[39,50025],{"className":50026,"style":461},[230],[39,50028,50030,50033,50036,50039,50042],{"className":50029},[74],[39,50031],{"className":50032,"style":2473},[78],[39,50034,28873],{"className":50035},[83],[39,50037],{"className":50038,"style":231},[230],[39,50040,793],{"className":50041},[235],[39,50043],{"className":50044,"style":231},[230],[39,50046,50048,50051,50054],{"className":50047},[74],[39,50049],{"className":50050,"style":17578},[78],[39,50052,49883],{"className":50053},[83],[39,50055,155],{"className":50056},[252],[39,50058,50060],{"className":50059,"translate":43},[746],[39,50061,50063,50124],{"className":50062,"translate":43},[42],[39,50064,50066],{"className":50065},[47],[49,50067,50068],{"xmlns":51,"display":755},[53,50069,50070,50121],{},[56,50071,50072,50074,50076,50078,50080,50082,50084,50086,50088,50090,50092,50094,50096,50098,50100,50102,50104,50106,50108,50110,50112,50114,50116,50118],{},[59,50073,4295],{},[133,50075,136],{"stretchy":135},[59,50077,101],{},[133,50079,418],{},[59,50081,48449],{},[133,50083,161],{"stretchy":135},[133,50085,793],{},[133,50087,5378],{},[539,50089,546],{},[133,50091,788],{},[539,50093,17739],{},[59,50095,4230],{},[133,50097,136],{"stretchy":135},[59,50099,48449],{},[133,50101,161],{"stretchy":135},[133,50103,793],{},[133,50105,5378],{},[539,50107,546],{},[133,50109,788],{},[539,50111,17739],{},[133,50113,409],{},[539,50115,49883],{},[133,50117,793],{},[539,50119,50120],{},"5.39.",[63,50122,50123],{"encoding":65},"Q(A\\to D)=-1+0.9V(D)=-1+0.9\\times7.1=5.39.",[39,50125,50127,50151,50172,50193,50223,50244,50262,50280],{"className":50126,"ariaHidden":70},[69],[39,50128,50130,50133,50136,50139,50142,50145,50148],{"className":50129},[74],[39,50131],{"className":50132,"style":245},[78],[39,50134,4295],{"className":50135},[83,84],[39,50137,136],{"className":50138},[182],[39,50140,101],{"className":50141},[83,84],[39,50143],{"className":50144,"style":231},[230],[39,50146,418],{"className":50147},[235],[39,50149],{"className":50150,"style":231},[230],[39,50152,50154,50157,50160,50163,50166,50169],{"className":50153},[74],[39,50155],{"className":50156,"style":245},[78],[39,50158,48449],{"className":50159,"style":438},[83,84],[39,50161,161],{"className":50162},[263],[39,50164],{"className":50165,"style":231},[230],[39,50167,793],{"className":50168},[235],[39,50170],{"className":50171,"style":231},[230],[39,50173,50175,50178,50181,50184,50187,50190],{"className":50174},[74],[39,50176],{"className":50177,"style":17882},[78],[39,50179,5378],{"className":50180},[83],[39,50182,546],{"className":50183},[83],[39,50185],{"className":50186,"style":461},[230],[39,50188,788],{"className":50189},[465],[39,50191],{"className":50192,"style":461},[230],[39,50194,50196,50199,50202,50205,50208,50211,50214,50217,50220],{"className":50195},[74],[39,50197],{"className":50198,"style":245},[78],[39,50200,17739],{"className":50201},[83],[39,50203,4230],{"className":50204,"style":461},[83,84],[39,50206,136],{"className":50207},[182],[39,50209,48449],{"className":50210,"style":438},[83,84],[39,50212,161],{"className":50213},[263],[39,50215],{"className":50216,"style":231},[230],[39,50218,793],{"className":50219},[235],[39,50221],{"className":50222,"style":231},[230],[39,50224,50226,50229,50232,50235,50238,50241],{"className":50225},[74],[39,50227],{"className":50228,"style":17882},[78],[39,50230,5378],{"className":50231},[83],[39,50233,546],{"className":50234},[83],[39,50236],{"className":50237,"style":461},[230],[39,50239,788],{"className":50240},[465],[39,50242],{"className":50243,"style":461},[230],[39,50245,50247,50250,50253,50256,50259],{"className":50246},[74],[39,50248],{"className":50249,"style":17882},[78],[39,50251,17739],{"className":50252},[83],[39,50254],{"className":50255,"style":461},[230],[39,50257,409],{"className":50258},[465],[39,50260],{"className":50261,"style":461},[230],[39,50263,50265,50268,50271,50274,50277],{"className":50264},[74],[39,50266],{"className":50267,"style":2473},[78],[39,50269,49883],{"className":50270},[83],[39,50272],{"className":50273,"style":231},[230],[39,50275,793],{"className":50276},[235],[39,50278],{"className":50279,"style":231},[230],[39,50281,50283,50286],{"className":50282},[74],[39,50284],{"className":50285,"style":2473},[78],[39,50287,50120],{"className":50288},[83],[11,50290,18110],{},[39,50292,50294],{"className":50293,"translate":43},[746],[39,50295,50297,50340],{"className":50296,"translate":43},[42],[39,50298,50300],{"className":50299},[47],[49,50301,50302],{"xmlns":51,"display":755},[53,50303,50304,50337],{},[56,50305,50306,50308,50310,50312,50314,50316,50318,50320,50322,50324,50326,50329,50331,50333,50335],{},[59,50307,4230],{},[133,50309,136],{"stretchy":135},[59,50311,101],{},[133,50313,161],{"stretchy":135},[133,50315,793],{},[59,50317,1687],{},[133,50319,1690],{},[133,50321,12951],{"stretchy":135},[539,50323,49883],{},[133,50325,155],{"separator":70},[539,50327,50328],{},"5.39",[133,50330,12965],{"stretchy":135},[133,50332,793],{},[539,50334,49883],{},[133,50336,155],{"separator":70},[63,50338,50339],{"encoding":65},"V(A)=\\max\\{7.1,5.39\\}=7.1,",[39,50341,50343,50370,50406],{"className":50342,"ariaHidden":70},[69],[39,50344,50346,50349,50352,50355,50358,50361,50364,50367],{"className":50345},[74],[39,50347],{"className":50348,"style":245},[78],[39,50350,4230],{"className":50351,"style":461},[83,84],[39,50353,136],{"className":50354},[182],[39,50356,101],{"className":50357},[83,84],[39,50359,161],{"className":50360},[263],[39,50362],{"className":50363,"style":231},[230],[39,50365,793],{"className":50366},[235],[39,50368],{"className":50369,"style":231},[230],[39,50371,50373,50376,50379,50382,50385,50388,50391,50394,50397,50400,50403],{"className":50372},[74],[39,50374],{"className":50375,"style":245},[78],[39,50377,1687],{"className":50378},[1744],[39,50380,12951],{"className":50381},[182],[39,50383,49883],{"className":50384},[83],[39,50386,155],{"className":50387},[252],[39,50389],{"className":50390,"style":256},[230],[39,50392,50328],{"className":50393},[83],[39,50395,12965],{"className":50396},[263],[39,50398],{"className":50399,"style":231},[230],[39,50401,793],{"className":50402},[235],[39,50404],{"className":50405,"style":231},[230],[39,50407,50409,50412,50415],{"className":50408},[74],[39,50410],{"className":50411,"style":17578},[78],[39,50413,49883],{"className":50414},[83],[39,50416,155],{"className":50417},[252],[11,50419,50420,50421,50449,50450,50523,50524,50619,50620,50653],{},"最优动作是走向 ",[39,50422,50424,50437],{"className":50423,"translate":43},[42],[39,50425,50427],{"className":50426},[47],[49,50428,50429],{"xmlns":51},[53,50430,50431,50435],{},[56,50432,50433],{},[59,50434,19772],{},[63,50436,19772],{"encoding":65},[39,50438,50440],{"className":50439,"ariaHidden":70},[69],[39,50441,50443,50446],{"className":50442},[74],[39,50444],{"className":50445,"style":79},[78],[39,50447,19772],{"className":50448,"style":19812},[83,84],"。路线 ",[39,50451,50453,50475],{"className":50452,"translate":43},[42],[39,50454,50456],{"className":50455},[47],[49,50457,50458],{"xmlns":51},[53,50459,50460,50472],{},[56,50461,50462,50464,50466,50468,50470],{},[59,50463,101],{},[133,50465,418],{},[59,50467,19772],{},[133,50469,418],{},[59,50471,48444],{},[63,50473,50474],{"encoding":65},"A\\to B\\to C",[39,50476,50478,50496,50514],{"className":50477,"ariaHidden":70},[69],[39,50479,50481,50484,50487,50490,50493],{"className":50480},[74],[39,50482],{"className":50483,"style":79},[78],[39,50485,101],{"className":50486},[83,84],[39,50488],{"className":50489,"style":231},[230],[39,50491,418],{"className":50492},[235],[39,50494],{"className":50495,"style":231},[230],[39,50497,50499,50502,50505,50508,50511],{"className":50498},[74],[39,50500],{"className":50501,"style":79},[78],[39,50503,19772],{"className":50504,"style":19812},[83,84],[39,50506],{"className":50507,"style":231},[230],[39,50509,418],{"className":50510},[235],[39,50512],{"className":50513,"style":231},[230],[39,50515,50517,50520],{"className":50516},[74],[39,50518],{"className":50519,"style":79},[78],[39,50521,48444],{"className":50522,"style":48487},[83,84]," 虽然与 ",[39,50525,50527,50553],{"className":50526,"translate":43},[42],[39,50528,50530],{"className":50529},[47],[49,50531,50532],{"xmlns":51},[53,50533,50534,50550],{},[56,50535,50536,50538,50540,50542,50544,50546,50548],{},[59,50537,101],{},[133,50539,418],{},[59,50541,48449],{},[133,50543,418],{},[59,50545,4253],{},[133,50547,418],{},[59,50549,48444],{},[63,50551,50552],{"encoding":65},"A\\to D\\to E\\to C",[39,50554,50556,50574,50592,50610],{"className":50555,"ariaHidden":70},[69],[39,50557,50559,50562,50565,50568,50571],{"className":50558},[74],[39,50560],{"className":50561,"style":79},[78],[39,50563,101],{"className":50564},[83,84],[39,50566],{"className":50567,"style":231},[230],[39,50569,418],{"className":50570},[235],[39,50572],{"className":50573,"style":231},[230],[39,50575,50577,50580,50583,50586,50589],{"className":50576},[74],[39,50578],{"className":50579,"style":79},[78],[39,50581,48449],{"className":50582,"style":438},[83,84],[39,50584],{"className":50585,"style":231},[230],[39,50587,418],{"className":50588},[235],[39,50590],{"className":50591,"style":231},[230],[39,50593,50595,50598,50601,50604,50607],{"className":50594},[74],[39,50596],{"className":50597,"style":79},[78],[39,50599,4253],{"className":50600,"style":85},[83,84],[39,50602],{"className":50603,"style":231},[230],[39,50605,418],{"className":50606},[235],[39,50608],{"className":50609,"style":231},[230],[39,50611,50613,50616],{"className":50612},[74],[39,50614],{"className":50615,"style":79},[78],[39,50617,48444],{"className":50618,"style":48487},[83,84]," 具有相同的终止奖励，但后者多走一步，多承担一次 ",[39,50621,50623,50638],{"className":50622,"translate":43},[42],[39,50624,50626],{"className":50625},[47],[49,50627,50628],{"xmlns":51},[53,50629,50630,50636],{},[56,50631,50632,50634],{},[133,50633,5378],{},[539,50635,546],{},[63,50637,48859],{"encoding":65},[39,50639,50641],{"className":50640,"ariaHidden":70},[69],[39,50642,50644,50647,50650],{"className":50643},[74],[39,50645],{"className":50646,"style":17882},[78],[39,50648,5378],{"className":50649},[83],[39,50651,546],{"className":50652},[83]," 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。",[11,50655,50656,50657,50685,50686,50714,50715,50743,50744,50795,50796,50824,50825,50743,50853,50926,50927,50955],{},"这个例子同时展示了最优子结构的两层含义。第一，从 ",[39,50658,50660,50673],{"className":50659,"translate":43},[42],[39,50661,50663],{"className":50662},[47],[49,50664,50665],{"xmlns":51},[53,50666,50667,50671],{},[56,50668,50669],{},[59,50670,101],{},[63,50672,101],{"encoding":65},[39,50674,50676],{"className":50675,"ariaHidden":70},[69],[39,50677,50679,50682],{"className":50678},[74],[39,50680],{"className":50681,"style":79},[78],[39,50683,101],{"className":50684},[83,84]," 出发的最优路线选择了 ",[39,50687,50689,50702],{"className":50688,"translate":43},[42],[39,50690,50692],{"className":50691},[47],[49,50693,50694],{"xmlns":51},[53,50695,50696,50700],{},[56,50697,50698],{},[59,50699,19772],{},[63,50701,19772],{"encoding":65},[39,50703,50705],{"className":50704,"ariaHidden":70},[69],[39,50706,50708,50711],{"className":50707},[74],[39,50709],{"className":50710,"style":79},[78],[39,50712,19772],{"className":50713,"style":19812},[83,84],"，而从 ",[39,50716,50718,50731],{"className":50717,"translate":43},[42],[39,50719,50721],{"className":50720},[47],[49,50722,50723],{"xmlns":51},[53,50724,50725,50729],{},[56,50726,50727],{},[59,50728,19772],{},[63,50730,19772],{"encoding":65},[39,50732,50734],{"className":50733,"ariaHidden":70},[69],[39,50735,50737,50740],{"className":50736},[74],[39,50738],{"className":50739,"style":79},[78],[39,50741,19772],{"className":50742,"style":19812},[83,84]," 出发的剩余路线 ",[39,50745,50747,50765],{"className":50746,"translate":43},[42],[39,50748,50750],{"className":50749},[47],[49,50751,50752],{"xmlns":51},[53,50753,50754,50762],{},[56,50755,50756,50758,50760],{},[59,50757,19772],{},[133,50759,418],{},[59,50761,48444],{},[63,50763,50764],{"encoding":65},"B\\to C",[39,50766,50768,50786],{"className":50767,"ariaHidden":70},[69],[39,50769,50771,50774,50777,50780,50783],{"className":50770},[74],[39,50772],{"className":50773,"style":79},[78],[39,50775,19772],{"className":50776,"style":19812},[83,84],[39,50778],{"className":50779,"style":231},[230],[39,50781,418],{"className":50782},[235],[39,50784],{"className":50785,"style":231},[230],[39,50787,50789,50792],{"className":50788},[74],[39,50790],{"className":50791,"style":79},[78],[39,50793,48444],{"className":50794,"style":48487},[83,84]," 本身是 ",[39,50797,50799,50812],{"className":50798,"translate":43},[42],[39,50800,50802],{"className":50801},[47],[49,50803,50804],{"xmlns":51},[53,50805,50806,50810],{},[56,50807,50808],{},[59,50809,19772],{},[63,50811,19772],{"encoding":65},[39,50813,50815],{"className":50814,"ariaHidden":70},[69],[39,50816,50818,50821],{"className":50817},[74],[39,50819],{"className":50820,"style":79},[78],[39,50822,19772],{"className":50823,"style":19812},[83,84]," 处子问题的最优解。第二，从 ",[39,50826,50828,50841],{"className":50827,"translate":43},[42],[39,50829,50831],{"className":50830},[47],[49,50832,50833],{"xmlns":51},[53,50834,50835,50839],{},[56,50836,50837],{},[59,50838,48449],{},[63,50840,48449],{"encoding":65},[39,50842,50844],{"className":50843,"ariaHidden":70},[69],[39,50845,50847,50850],{"className":50846},[74],[39,50848],{"className":50849,"style":79},[78],[39,50851,48449],{"className":50852,"style":438},[83,84],[39,50854,50856,50878],{"className":50855,"translate":43},[42],[39,50857,50859],{"className":50858},[47],[49,50860,50861],{"xmlns":51},[53,50862,50863,50875],{},[56,50864,50865,50867,50869,50871,50873],{},[59,50866,48449],{},[133,50868,418],{},[59,50870,4253],{},[133,50872,418],{},[59,50874,48444],{},[63,50876,50877],{"encoding":65},"D\\to E\\to C",[39,50879,50881,50899,50917],{"className":50880,"ariaHidden":70},[69],[39,50882,50884,50887,50890,50893,50896],{"className":50883},[74],[39,50885],{"className":50886,"style":79},[78],[39,50888,48449],{"className":50889,"style":438},[83,84],[39,50891],{"className":50892,"style":231},[230],[39,50894,418],{"className":50895},[235],[39,50897],{"className":50898,"style":231},[230],[39,50900,50902,50905,50908,50911,50914],{"className":50901},[74],[39,50903],{"className":50904,"style":79},[78],[39,50906,4253],{"className":50907,"style":85},[83,84],[39,50909],{"className":50910,"style":231},[230],[39,50912,418],{"className":50913},[235],[39,50915],{"className":50916,"style":231},[230],[39,50918,50920,50923],{"className":50919},[74],[39,50921],{"className":50922,"style":79},[78],[39,50924,48444],{"className":50925,"style":48487},[83,84]," 也是 ",[39,50928,50930,50943],{"className":50929,"translate":43},[42],[39,50931,50933],{"className":50932},[47],[49,50934,50935],{"xmlns":51},[53,50936,50937,50941],{},[56,50938,50939],{},[59,50940,48449],{},[63,50942,48449],{"encoding":65},[39,50944,50946],{"className":50945,"ariaHidden":70},[69],[39,50947,50949,50952],{"className":50948},[74],[39,50950],{"className":50951,"style":79},[78],[39,50953,48449],{"className":50954,"style":438},[83,84]," 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。",[11,50957,50958,50959,51025],{},"若改用 ",[39,50960,50962,50986],{"className":50961,"translate":43},[42],[39,50963,50965],{"className":50964},[47],[49,50966,50967],{"xmlns":51},[53,50968,50969,50983],{},[56,50970,50971,50973,50975,50977,50979,50981],{},[59,50972,4230],{},[133,50974,136],{"stretchy":135},[59,50976,48444],{},[133,50978,161],{"stretchy":135},[133,50980,793],{},[539,50982,48919],{},[63,50984,50985],{"encoding":65},"V(C)=10",[39,50987,50989,51016],{"className":50988,"ariaHidden":70},[69],[39,50990,50992,50995,50998,51001,51004,51007,51010,51013],{"className":50991},[74],[39,50993],{"className":50994,"style":245},[78],[39,50996,4230],{"className":50997,"style":461},[83,84],[39,50999,136],{"className":51000},[182],[39,51002,48444],{"className":51003,"style":48487},[83,84],[39,51005,161],{"className":51006},[263],[39,51008],{"className":51009,"style":231},[230],[39,51011,793],{"className":51012},[235],[39,51014],{"className":51015,"style":231},[230],[39,51017,51019,51022],{"className":51018},[74],[39,51020],{"className":51021,"style":2473},[78],[39,51023,48919],{"className":51024},[83]," 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。",[21,51027,51029],{"id":51028},"五收敛性与误差界","五、收敛性与误差界",[26,51031,51033],{"id":51032},"_51-不动点迭代","5.1 不动点迭代",[11,51035,51036,51037,51144],{},"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 ",[39,51038,51040,51068],{"className":51039,"translate":43},[42],[39,51041,51043],{"className":51042},[47],[49,51044,51045],{"xmlns":51},[53,51046,51047,51065],{},[56,51048,51049,51055,51057,51059,51061,51063],{},[613,51050,51051,51053],{},[59,51052,4230],{},[539,51054,541],{},[133,51056,534],{},[59,51058,19772],{},[133,51060,136],{"stretchy":135},[59,51062,61],{},[133,51064,161],{"stretchy":135},[63,51066,51067],{"encoding":65},"V_0\\in B(S)",[39,51069,51071,51126],{"className":51070,"ariaHidden":70},[69],[39,51072,51074,51077,51117,51120,51123],{"className":51073},[74],[39,51075],{"className":51076,"style":634},[78],[39,51078,51080,51083],{"className":51079},[83],[39,51081,4230],{"className":51082,"style":461},[83,84],[39,51084,51086],{"className":51085},[192],[39,51087,51089,51109],{"className":51088},[196,649],[39,51090,51092,51106],{"className":51091},[200],[39,51093,51095],{"className":51094,"style":963},[204],[39,51096,51097,51100],{"style":13607},[39,51098],{"className":51099,"style":213},[212],[39,51101,51103],{"className":51102},[217,218,219,220],[39,51104,541],{"className":51105},[83,220],[39,51107,673],{"className":51108},[672],[39,51110,51112],{"className":51111},[200],[39,51113,51115],{"className":51114,"style":680},[204],[39,51116],{},[39,51118],{"className":51119,"style":231},[230],[39,51121,534],{"className":51122},[235],[39,51124],{"className":51125,"style":231},[230],[39,51127,51129,51132,51135,51138,51141],{"className":51128},[74],[39,51130],{"className":51131,"style":245},[78],[39,51133,19772],{"className":51134,"style":19812},[83,84],[39,51136,136],{"className":51137},[182],[39,51139,61],{"className":51140,"style":85},[83,84],[39,51142,161],{"className":51143},[263],"，定义迭代",[39,51146,51148],{"className":51147,"translate":43},[746],[39,51149,51151,51207],{"className":51150,"translate":43},[42],[39,51152,51154],{"className":51153},[47],[49,51155,51156],{"xmlns":51,"display":755},[53,51157,51158,51204],{},[56,51159,51160,51172,51174,51176,51182,51184,51186,51188,51190,51192,51194,51196,51198,51200,51202],{},[613,51161,51162,51164],{},[59,51163,4230],{},[56,51165,51166,51168,51170],{},[59,51167,3196],{},[133,51169,788],{},[539,51171,546],{},[133,51173,793],{},[59,51175,11772],{"mathvariant":617},[613,51177,51178,51180],{},[59,51179,4230],{},[59,51181,3196],{},[133,51183,155],{"separator":70},[230,51185],{"width":10251},[59,51187,3196],{},[133,51189,793],{},[539,51191,541],{},[133,51193,155],{"separator":70},[539,51195,546],{},[133,51197,155],{"separator":70},[539,51199,17359],{},[133,51201,155],{"separator":70},[133,51203,2641],{},[63,51205,51206],{"encoding":65},"V_{k+1}=\\mathcal{T}V_k,\n\\qquad k=0,1,2,\\dots",[39,51208,51210,51274,51345],{"className":51209,"ariaHidden":70},[69],[39,51211,51213,51216,51265,51268,51271],{"className":51212},[74],[39,51214],{"className":51215,"style":7584},[78],[39,51217,51219,51222],{"className":51218},[83],[39,51220,4230],{"className":51221,"style":461},[83,84],[39,51223,51225],{"className":51224},[192],[39,51226,51228,51257],{"className":51227},[196,649],[39,51229,51231,51254],{"className":51230},[200],[39,51232,51234],{"className":51233,"style":3419},[204],[39,51235,51236,51239],{"style":13607},[39,51237],{"className":51238,"style":213},[212],[39,51240,51242],{"className":51241},[217,218,219,220],[39,51243,51245,51248,51251],{"className":51244},[83,220],[39,51246,3196],{"className":51247,"style":3321},[83,84,220],[39,51249,788],{"className":51250},[465,220],[39,51252,546],{"className":51253},[83,220],[39,51255,673],{"className":51256},[672],[39,51258,51260],{"className":51259},[200],[39,51261,51263],{"className":51262,"style":994},[204],[39,51264],{},[39,51266],{"className":51267,"style":231},[230],[39,51269,793],{"className":51270},[235],[39,51272],{"className":51273,"style":231},[230],[39,51275,51277,51281,51284,51324,51327,51330,51333,51336,51339,51342],{"className":51276},[74],[39,51278],{"className":51279,"style":51280},[78],"height:0.8889em;vertical-align:-0.1944em;",[39,51282,11772],{"className":51283,"style":11793},[83,641],[39,51285,51287,51290],{"className":51286},[83],[39,51288,4230],{"className":51289,"style":461},[83,84],[39,51291,51293],{"className":51292},[192],[39,51294,51296,51316],{"className":51295},[196,649],[39,51297,51299,51313],{"className":51298},[200],[39,51300,51302],{"className":51301,"style":3419},[204],[39,51303,51304,51307],{"style":13607},[39,51305],{"className":51306,"style":213},[212],[39,51308,51310],{"className":51309},[217,218,219,220],[39,51311,3196],{"className":51312,"style":3321},[83,84,220],[39,51314,673],{"className":51315},[672],[39,51317,51319],{"className":51318},[200],[39,51320,51322],{"className":51321,"style":680},[204],[39,51323],{},[39,51325,155],{"className":51326},[252],[39,51328],{"className":51329,"style":10787},[230],[39,51331],{"className":51332,"style":256},[230],[39,51334,3196],{"className":51335,"style":3321},[83,84],[39,51337],{"className":51338,"style":231},[230],[39,51340,793],{"className":51341},[235],[39,51343],{"className":51344,"style":231},[230],[39,51346,51348,51351,51354,51357,51360,51363,51366,51369,51372,51375,51378],{"className":51347},[74],[39,51349],{"className":51350,"style":17578},[78],[39,51352,541],{"className":51353},[83],[39,51355,155],{"className":51356},[252],[39,51358],{"className":51359,"style":256},[230],[39,51361,546],{"className":51362},[83],[39,51364,155],{"className":51365},[252],[39,51367],{"className":51368,"style":256},[230],[39,51370,17359],{"className":51371},[83],[39,51373,155],{"className":51374},[252],[39,51376],{"className":51377,"style":256},[230],[39,51379,2641],{"className":51380},[2842],[11,51382,24575,51383,51411,51412,34856,51470,51528],{},[39,51384,51386,51399],{"className":51385,"translate":43},[42],[39,51387,51389],{"className":51388},[47],[49,51390,51391],{"xmlns":51},[53,51392,51393,51397],{},[56,51394,51395],{},[59,51396,11772],{"mathvariant":617},[63,51398,34785],{"encoding":65},[39,51400,51402],{"className":51401,"ariaHidden":70},[69],[39,51403,51405,51408],{"className":51404},[74],[39,51406],{"className":51407,"style":79},[78],[39,51409,11772],{"className":51410,"style":11793},[83,641]," 可以是 ",[39,51413,51415,51432],{"className":51414,"translate":43},[42],[39,51416,51418],{"className":51417},[47],[49,51419,51420],{"xmlns":51},[53,51421,51422,51430],{},[56,51423,51424],{},[138,51425,51426,51428],{},[59,51427,11772],{"mathvariant":617},[59,51429,2098],{},[63,51431,11777],{"encoding":65},[39,51433,51435],{"className":51434,"ariaHidden":70},[69],[39,51436,51438,51441],{"className":51437},[74],[39,51439],{"className":51440,"style":79},[78],[39,51442,51444,51447],{"className":51443},[83],[39,51445,11772],{"className":51446,"style":11793},[83,641],[39,51448,51450],{"className":51449},[192],[39,51451,51453],{"className":51452},[196],[39,51454,51456],{"className":51455},[200],[39,51457,51459],{"className":51458,"style":6410},[204],[39,51460,51461,51464],{"style":208},[39,51462],{"className":51463,"style":213},[212],[39,51465,51467],{"className":51466},[217,218,219,220],[39,51468,2098],{"className":51469,"style":2114},[83,84,220],[39,51471,51473,51490],{"className":51472,"translate":43},[42],[39,51474,51476],{"className":51475},[47],[49,51477,51478],{"xmlns":51},[53,51479,51480,51488],{},[56,51481,51482],{},[138,51483,51484,51486],{},[59,51485,11772],{"mathvariant":617},[133,51487,21884],{},[63,51489,21887],{"encoding":65},[39,51491,51493],{"className":51492,"ariaHidden":70},[69],[39,51494,51496,51499],{"className":51495},[74],[39,51497],{"className":51498,"style":21897},[78],[39,51500,51502,51505],{"className":51501},[83],[39,51503,11772],{"className":51504,"style":11793},[83,641],[39,51506,51508],{"className":51507},[192],[39,51509,51511],{"className":51510},[196],[39,51512,51514],{"className":51513},[200],[39,51515,51517],{"className":51516,"style":21897},[204],[39,51518,51519,51522],{"style":208},[39,51520],{"className":51521,"style":213},[212],[39,51523,51525],{"className":51524},[217,218,219,220],[39,51526,21884],{"className":51527},[465,220],"。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。",[31,51530,51533,51927,52331],{"id":51531,"title":51532},"theorem-convergence","定理 5.1（几何收敛）",[11,51534,37,51535,742,51607,21402,51635,34856,51693,34915,51751,51820,51821,742],{},[39,51536,51538,51559],{"className":51537,"translate":43},[42],[39,51539,51541],{"className":51540},[47],[49,51542,51543],{"xmlns":51},[53,51544,51545,51557],{},[56,51546,51547,51549,51551,51553,51555],{},[539,51548,541],{},[133,51550,1824],{},[59,51552,531],{},[133,51554,1693],{},[539,51556,546],{},[63,51558,14577],{"encoding":65},[39,51560,51562,51580,51598],{"className":51561,"ariaHidden":70},[69],[39,51563,51565,51568,51571,51574,51577],{"className":51564},[74],[39,51566],{"className":51567,"style":14587},[78],[39,51569,541],{"className":51570},[83],[39,51572],{"className":51573,"style":231},[230],[39,51575,1824],{"className":51576},[235],[39,51578],{"className":51579,"style":231},[230],[39,51581,51583,51586,51589,51592,51595],{"className":51582},[74],[39,51584],{"className":51585,"style":561},[78],[39,51587,531],{"className":51588,"style":565},[83,84],[39,51590],{"className":51591,"style":231},[230],[39,51593,1693],{"className":51594},[235],[39,51596],{"className":51597,"style":231},[230],[39,51599,51601,51604],{"className":51600},[74],[39,51602],{"className":51603,"style":2473},[78],[39,51605,546],{"className":51606},[83],[39,51608,51610,51623],{"className":51609,"translate":43},[42],[39,51611,51613],{"className":51612},[47],[49,51614,51615],{"xmlns":51},[53,51616,51617,51621],{},[56,51618,51619],{},[59,51620,11772],{"mathvariant":617},[63,51622,34785],{"encoding":65},[39,51624,51626],{"className":51625,"ariaHidden":70},[69],[39,51627,51629,51632],{"className":51628},[74],[39,51630],{"className":51631,"style":79},[78],[39,51633,11772],{"className":51634,"style":11793},[83,641],[39,51636,51638,51655],{"className":51637,"translate":43},[42],[39,51639,51641],{"className":51640},[47],[49,51642,51643],{"xmlns":51},[53,51644,51645,51653],{},[56,51646,51647],{},[138,51648,51649,51651],{},[59,51650,11772],{"mathvariant":617},[59,51652,2098],{},[63,51654,11777],{"encoding":65},[39,51656,51658],{"className":51657,"ariaHidden":70},[69],[39,51659,51661,51664],{"className":51660},[74],[39,51662],{"className":51663,"style":79},[78],[39,51665,51667,51670],{"className":51666},[83],[39,51668,11772],{"className":51669,"style":11793},[83,641],[39,51671,51673],{"className":51672},[192],[39,51674,51676],{"className":51675},[196],[39,51677,51679],{"className":51678},[200],[39,51680,51682],{"className":51681,"style":6410},[204],[39,51683,51684,51687],{"style":208},[39,51685],{"className":51686,"style":213},[212],[39,51688,51690],{"className":51689},[217,218,219,220],[39,51691,2098],{"className":51692,"style":2114},[83,84,220],[39,51694,51696,51713],{"className":51695,"translate":43},[42],[39,51697,51699],{"className":51698},[47],[49,51700,51701],{"xmlns":51},[53,51702,51703,51711],{},[56,51704,51705],{},[138,51706,51707,51709],{},[59,51708,11772],{"mathvariant":617},[133,51710,21884],{},[63,51712,21887],{"encoding":65},[39,51714,51716],{"className":51715,"ariaHidden":70},[69],[39,51717,51719,51722],{"className":51718},[74],[39,51720],{"className":51721,"style":21897},[78],[39,51723,51725,51728],{"className":51724},[83],[39,51726,11772],{"className":51727,"style":11793},[83,641],[39,51729,51731],{"className":51730},[192],[39,51732,51734],{"className":51733},[196],[39,51735,51737],{"className":51736},[200],[39,51738,51740],{"className":51739,"style":21897},[204],[39,51741,51742,51745],{"style":208},[39,51743],{"className":51744,"style":213},[212],[39,51746,51748],{"className":51747},[217,218,219,220],[39,51749,21884],{"className":51750},[465,220],[39,51752,51754,51771],{"className":51753,"translate":43},[42],[39,51755,51757],{"className":51756},[47],[49,51758,51759],{"xmlns":51},[53,51760,51761,51769],{},[56,51762,51763],{},[613,51764,51765,51767],{},[59,51766,4230],{},[59,51768,1696],{"mathvariant":145},[63,51770,34936],{"encoding":65},[39,51772,51774],{"className":51773,"ariaHidden":70},[69],[39,51775,51777,51780],{"className":51776},[74],[39,51778],{"className":51779,"style":634},[78],[39,51781,51783,51786],{"className":51782},[83],[39,51784,4230],{"className":51785,"style":461},[83,84],[39,51787,51789],{"className":51788},[192],[39,51790,51792,51812],{"className":51791},[196,649],[39,51793,51795,51809],{"className":51794},[200],[39,51796,51798],{"className":51797,"style":1728},[204],[39,51799,51800,51803],{"style":13607},[39,51801],{"className":51802,"style":213},[212],[39,51804,51806],{"className":51805},[217,218,219,220],[39,51807,1696],{"className":51808},[83,220],[39,51810,673],{"className":51811},[672],[39,51813,51815],{"className":51814},[200],[39,51816,51818],{"className":51817,"style":680},[204],[39,51819],{},"。则对任意 ",[39,51822,51824,51851],{"className":51823,"translate":43},[42],[39,51825,51827],{"className":51826},[47],[49,51828,51829],{"xmlns":51},[53,51830,51831,51849],{},[56,51832,51833,51839,51841,51843,51845,51847],{},[613,51834,51835,51837],{},[59,51836,4230],{},[539,51838,541],{},[133,51840,534],{},[59,51842,19772],{},[133,51844,136],{"stretchy":135},[59,51846,61],{},[133,51848,161],{"stretchy":135},[63,51850,51067],{"encoding":65},[39,51852,51854,51909],{"className":51853,"ariaHidden":70},[69],[39,51855,51857,51860,51900,51903,51906],{"className":51856},[74],[39,51858],{"className":51859,"style":634},[78],[39,51861,51863,51866],{"className":51862},[83],[39,51864,4230],{"className":51865,"style":461},[83,84],[39,51867,51869],{"className":51868},[192],[39,51870,51872,51892],{"className":51871},[196,649],[39,51873,51875,51889],{"className":51874},[200],[39,51876,51878],{"className":51877,"style":963},[204],[39,51879,51880,51883],{"style":13607},[39,51881],{"className":51882,"style":213},[212],[39,51884,51886],{"className":51885},[217,218,219,220],[39,51887,541],{"className":51888},[83,220],[39,51890,673],{"className":51891},[672],[39,51893,51895],{"className":51894},[200],[39,51896,51898],{"className":51897,"style":680},[204],[39,51899],{},[39,51901],{"className":51902,"style":231},[230],[39,51904,534],{"className":51905},[235],[39,51907],{"className":51908,"style":231},[230],[39,51910,51912,51915,51918,51921,51924],{"className":51911},[74],[39,51913],{"className":51914,"style":245},[78],[39,51916,19772],{"className":51917,"style":19812},[83,84],[39,51919,136],{"className":51920},[182],[39,51922,61],{"className":51923,"style":85},[83,84],[39,51925,161],{"className":51926},[263],[39,51928,51930],{"className":51929,"translate":43},[746],[39,51931,51933,51999],{"className":51932,"translate":43},[42],[39,51934,51936],{"className":51935},[47],[49,51937,51938],{"xmlns":51,"display":755},[53,51939,51940,51996],{},[56,51941,51942,51944,51950,51952,51958,51964,51966,51972,51974,51980,51982,51988,51994],{},[59,51943,15389],{"mathvariant":145},[613,51945,51946,51948],{},[59,51947,4230],{},[59,51949,3196],{},[133,51951,5378],{},[613,51953,51954,51956],{},[59,51955,4230],{},[59,51957,1696],{"mathvariant":145},[613,51959,51960,51962],{},[59,51961,15389],{"mathvariant":145},[59,51963,1696],{"mathvariant":145},[133,51965,1824],{},[138,51967,51968,51970],{},[59,51969,531],{},[59,51971,3196],{},[59,51973,15389],{"mathvariant":145},[613,51975,51976,51978],{},[59,51977,4230],{},[539,51979,541],{},[133,51981,5378],{},[613,51983,51984,51986],{},[59,51985,4230],{},[59,51987,1696],{"mathvariant":145},[613,51989,51990,51992],{},[59,51991,15389],{"mathvariant":145},[59,51993,1696],{"mathvariant":145},[59,51995,2903],{"mathvariant":145},[63,51997,51998],{"encoding":65},"\\|V_k-V_\\infty\\|_\\infty\n\\le\\gamma^k\\|V_0-V_\\infty\\|_\\infty.",[39,52000,52002,52060,52155,52242],{"className":52001,"ariaHidden":70},[69],[39,52003,52005,52008,52011,52051,52054,52057],{"className":52004},[74],[39,52006],{"className":52007,"style":245},[78],[39,52009,15389],{"className":52010},[83],[39,52012,52014,52017],{"className":52013},[83],[39,52015,4230],{"className":52016,"style":461},[83,84],[39,52018,52020],{"className":52019},[192],[39,52021,52023,52043],{"className":52022},[196,649],[39,52024,52026,52040],{"className":52025},[200],[39,52027,52029],{"className":52028,"style":3419},[204],[39,52030,52031,52034],{"style":13607},[39,52032],{"className":52033,"style":213},[212],[39,52035,52037],{"className":52036},[217,218,219,220],[39,52038,3196],{"className":52039,"style":3321},[83,84,220],[39,52041,673],{"className":52042},[672],[39,52044,52046],{"className":52045},[200],[39,52047,52049],{"className":52048,"style":680},[204],[39,52050],{},[39,52052],{"className":52053,"style":461},[230],[39,52055,5378],{"className":52056},[465],[39,52058],{"className":52059,"style":461},[230],[39,52061,52063,52066,52106,52146,52149,52152],{"className":52062},[74],[39,52064],{"className":52065,"style":245},[78],[39,52067,52069,52072],{"className":52068},[83],[39,52070,4230],{"className":52071,"style":461},[83,84],[39,52073,52075],{"className":52074},[192],[39,52076,52078,52098],{"className":52077},[196,649],[39,52079,52081,52095],{"className":52080},[200],[39,52082,52084],{"className":52083,"style":1728},[204],[39,52085,52086,52089],{"style":13607},[39,52087],{"className":52088,"style":213},[212],[39,52090,52092],{"className":52091},[217,218,219,220],[39,52093,1696],{"className":52094},[83,220],[39,52096,673],{"className":52097},[672],[39,52099,52101],{"className":52100},[200],[39,52102,52104],{"className":52103,"style":680},[204],[39,52105],{},[39,52107,52109,52112],{"className":52108},[83],[39,52110,15389],{"className":52111},[83],[39,52113,52115],{"className":52114},[192],[39,52116,52118,52138],{"className":52117},[196,649],[39,52119,52121,52135],{"className":52120},[200],[39,52122,52124],{"className":52123,"style":1728},[204],[39,52125,52126,52129],{"style":966},[39,5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V_\\infty",[39,52364,52366,52421],{"className":52365,"ariaHidden":70},[69],[39,52367,52369,52372,52412,52415,52418],{"className":52368},[74],[39,52370],{"className":52371,"style":634},[78],[39,52373,52375,52378],{"className":52374},[83],[39,52376,4230],{"className":52377,"style":461},[83,84],[39,52379,52381],{"className":52380},[192],[39,52382,52384,52404],{"className":52383},[196,649],[39,52385,52387,52401],{"className":52386},[200],[39,52388,52390],{"className":52389,"style":3419},[204],[39,52391,52392,52395],{"style":13607},[39,52393],{"className":52394,"style":213},[212],[39,52396,52398],{"className":52397},[217,218,219,220],[39,52399,3196],{"className":52400,"style":3321},[83,84,220],[39,52402,673],{"className":52403},[672],[39,52405,52407],{"className":52406},[200],[39,52408,52410],{"className":52409,"style":680},[204],[39,52411],{},[39,52413],{"className":52414,"style":231},[230],[39,52416,418],{"className":52417},[235],[39,52419],{"className":52420,"style":231},[230],[39,52422,52424,52427],{"className":52423},[74],[39,52425],{"className":52426,"style":634},[78],[39,52428,52430,52433],{"className":52429},[83],[39,52431,4230],{"className":52432,"style":461},[83,84],[39,52434,52436],{"className":52435},[192],[39,52437,52439,52459],{"className":52438},[196,649],[39,52440,52442,52456],{"className":52441},[200],[39,52443,52445],{"className":52444,"style":1728},[204],[39,52446,52447,52450],{"style":13607},[39,52448],{"className":52449,"style":213},[212],[39,52451,52453],{"className":52452},[217,218,219,220],[39,52454,1696],{"className":52455},[83,220],[39,52457,673],{"className":52458},[672],[39,52460,52462],{"className":52461},[200],[39,52463,52465],{"className":52464,"style":680},[204],[39,52466],{}," 在 sup 范数下成立。",[11,52469,52470,52472],{},[1489,52471,5595],{}," 由压缩性，",[39,52474,52476],{"className":52475,"translate":43},[746],[39,52477,52479,52607],{"className":52478,"translate":43},[42],[39,52480,52482],{"className":52481},[47],[49,52483,52484],{"xmlns":51,"display":755},[53,52485,52486,52604],{},[759,52487,52488,52560],{"rowspacing":761,"columnalign":762,"columnspacing":146},[764,52489,52490,52524],{},[767,52491,52492],{},[770,52493,52494],{"scriptlevel":541,"displaystyle":70},[56,52495,52496,52498,52510,52512,52518],{},[59,52497,15389],{"mathvariant":145},[613,52499,52500,52502],{},[59,52501,4230],{},[56,52503,52504,52506,52508],{},[59,52505,3196],{},[133,52507,788],{},[539,52509,546],{},[133,52511,5378],{},[613,52513,52514,52516],{},[59,52515,4230],{},[59,52517,1696],{"mathvariant":145},[613,52519,52520,52522],{},[59,52521,15389],{"mathvariant":145},[59,52523,1696],{"mathvariant":145},[767,52525,52526],{},[770,52527,52528],{"scriptlevel":541,"displaystyle":70},[56,52529,52530,52532,52534,52536,52538,52544,52546,52548,52554],{},[56,52531],{},[133,52533,793],{},[59,52535,15389],{"mathvariant":145},[59,52537,11772],{"mathvariant":617},[613,52539,52540,52542],{},[59,52541,4230],{},[59,52543,3196],{},[133,52545,5378],{},[59,52547,11772],{"mathvariant":617},[613,52549,52550,52552],{},[59,52551,4230],{},[59,52553,1696],{"mathvariant":145},[613,52555,52556,52558],{},[59,52557,15389],{"mathvariant":145},[59,52559,1696],{"mathvariant":145},[764,52561,52562,52568],{},[767,52563,52564],{},[770,52565,52566],{"scriptlevel":541,"displaystyle":70},[56,52567],{},[767,52569,52570],{},[770,52571,52572],{"scriptlevel":541,"displaystyle":70},[56,52573,52574,52576,52578,52580,52582,52588,52590,52596,52602],{},[56,52575],{},[133,52577,1824],{},[59,52579,531],{},[59,52581,15389],{"mathvariant":145},[613,52583,52584,52586],{},[59,52585,4230],{},[59,52587,3196],{},[133,52589,5378],{},[613,52591,52592,52594],{},[59,52593,4230],{},[59,52595,1696],{"mathvariant":145},[613,52597,52598,52600],{},[59,52599,15389],{"mathvariant":145},[59,52601,1696],{"mathvariant":145},[59,52603,2903],{"mathvariant":145},[63,52605,52606],{"encoding":65},"\\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}",[39,52608,52610],{"className":52609,"ariaHidden":70},[69],[39,52611,52613,52616],{"className":52612},[74],[39,52614],{"className":52615,"style":4374},[78],[39,52617,52619],{"className":52618},[83],[39,52620,52622,52802],{"className":52621},[759],[39,52623,52625],{"className":52624},[918],[39,52626,52628,52794],{"className":52627},[196,649],[39,52629,52631,52791],{"className":52630},[200],[39,52632,52634,52783],{"className":52633,"style":4393},[204],[39,52635,52636,52639],{"style":4396},[39,52637],{"className":52638,"style":935},[212],[39,52640,52642,52645,52694,52697,52700,52703,52743],{"className":52641},[83],[39,52643,15389],{"className":52644},[83],[39,52646,52648,52651],{"className":52647},[83],[39,52649,4230],{"className":52650,"style":461},[83,84],[39,52652,52654],{"className":52653},[192],[39,52655,52657,52686],{"className":52656},[196,649],[39,52658,52660,52683],{"className":52659},[200],[39,52661,52663],{"className":52662,"style":3419},[204],[39,52664,52665,52668],{"style":13607},[39,52666],{"className":52667,"style":213},[212],[39,52669,52671],{"className":52670},[217,218,219,220],[39,52672,52674,52677,52680],{"className":52673},[83,220],[39,52675,3196],{"className":52676,"style":3321},[83,84,220],[39,52678,788],{"className":52679},[465,220],[39,52681,546],{"className":52682},[83,220],[39,52684,673],{"className":52685},[672],[39,52687,52689],{"className":52688},[200],[39,52690,52692],{"className":52691,"style":994},[204],[39,52693],{},[39,52695],{"className":52696,"style":461},[230],[39,52698,5378],{"className":52699},[465],[39,52701],{"className":52702,"style":461},[230],[39,52704,52706,52709],{"className":52705},[83],[39,52707,4230],{"className":52708,"style":461},[83,84],[39,52710,52712],{"className":52711},[192],[39,52713,52715,52735],{"className":52714},[196,649],[39,52716,52718,52732],{"className":52717},[200],[39,52719,52721],{"className":52720,"style":1728},[204],[39,52722,52723,52726],{"style":13607},[39,52724],{"className":52725,"style":213},[212],[39,52727,52729],{"className":52728},[217,218,219,220],[39,52730,1696],{"className":52731},[83,220],[39,52733,673],{"className":52734},[672],[39,52736,52738],{"className":52737},[200],[39,52739,52741],{"className":52740,"style":680},[204],[39,52742],{},[39,52744,52746,52749],{"className":52745},[83],[39,52747,15389],{"className":52748},[83],[39,52750,52752],{"className":52751},[192],[39,52753,52755,52775],{"className":52754},[196,649],[39,52756,52758,52772],{"className":52757},[200],[39,52759,52761],{"className":52760,"style":1728},[204],[39,52762,52763,52766],{"style":966},[39,52764],{"className":52765,"style":213},[212],[39,52767,52769],{"className":52768},[217,218,219,220],[39,52770,1696],{"className":52771},[83,220],[39,52773,673],{"className":52774},[672],[39,52776,52778],{"className":52777},[200],[39,52779,52781],{"className":52780,"style":680},[204],[39,52782],{},[39,52784,52785,52788],{"style":4444},[39,52786],{"className":52787,"style":935},[212],[39,52789],{"className":52790},[83],[39,52792,673],{"className":52793},[672],[39,52795,52797],{"className":52796},[200],[39,52798,52800],{"className":52799,"style":4507},[204],[39,52801],{},[39,52803,52805],{"className":52804},[1223],[39,52806,52808,53133],{"className":52807},[196,649],[39,52809,52811,53130],{"className":52810},[200],[39,52812,52814,52972],{"className":52813,"style":4393},[204],[39,52815,52816,52819],{"style":4396},[39,52817],{"className":52818,"style":935},[212],[39,52820,52822,52825,52828,52831,52834,52837,52840,52880,52883,52886,52889,52892,52932],{"className":52821},[83],[39,52823],{"className":52824},[83],[39,52826],{"className":52827,"style":231},[230],[39,52829,793],{"className":52830},[235],[39,52832],{"className":52833,"style":231},[230],[39,52835,15389],{"className":52836},[83],[39,52838,11772],{"className":52839,"style":11793},[83,641],[39,52841,52843,52846],{"className":52842},[83],[39,52844,4230],{"className":52845,"style":461},[83,84],[39,52847,52849],{"className":52848},[192],[39,52850,52852,52872],{"className":52851},[196,649],[39,52853,52855,52869],{"className":52854},[200],[39,52856,52858],{"className":52857,"style":3419},[204],[39,52859,52860,52863],{"style":13607},[39,52861],{"className":52862,"style":213},[212],[39,52864,52866],{"className":52865},[217,218,219,220],[39,52867,3196],{"className":52868,"style":3321},[83,84,220],[39,52870,673],{"className":52871},[672],[39,52873,52875],{"className":52874},[200],[39,52876,52878],{"className":52877,"style":680},[204],[39,52879],{},[39,52881],{"className":52882,"style":461},[230],[39,52884,5378],{"className":52885},[465],[39,52887],{"className":52888,"style":461},[230],[39,52890,11772],{"className":52891,"style":11793},[83,641],[39,52893,52895,52898],{"className":52894},[83],[39,52896,4230],{"className":52897,"style":461},[83,84],[39,52899,52901],{"className":52900},[192],[39,52902,52904,52924],{"className":52903},[196,649],[39,52905,52907,52921],{"className":52906},[200],[39,52908,52910],{"className":52909,"style":1728},[204],[39,52911,52912,52915],{"style":13607},[39,52913],{"className":52914,"style":213},[212],[39,52916,52918],{"className":52917},[217,218,219,220],[39,52919,1696],{"className":52920},[83,220],[39,52922,673],{"className":52923},[672],[39,52925,52927],{"className":52926},[200],[39,52928,52930],{"className":52929,"style":680},[204],[39,52931],{},[39,52933,52935,52938],{"className":52934},[83],[39,52936,15389],{"className":52937},[83],[39,52939,52941],{"className":52940},[192],[39,52942,52944,52964],{"className":52943},[196,649],[39,52945,52947,52961],{"className":52946},[200],[39,52948,52950],{"className":52949,"style":1728},[204],[39,52951,52952,52955],{"style":966},[39,52953],{"className":52954,"style":213},[212],[39,52956,52958],{"className":52957},[217,218,219,220],[39,52959,1696],{"className":52960},[83,220],[39,52962,673],{"className":52963},[672],[39,52965,52967],{"className":52966},[200],[39,52968,52970],{"className":52969,"style":680},[204],[39,52971],{},[39,52973,52974,52977],{"style":4444},[39,52975],{"className":52976,"style":935},[212],[39,52978,52980,52983,52986,52989,52992,52995,52998,53038,53041,53044,53047,53087,53127],{"className":52979},[83],[39,52981],{"className":52982},[83],[39,52984],{"className":52985,"style":231},[230],[39,52987,1824],{"className":52988},[235],[39,52990],{"className":52991,"style":231},[230],[39,52993,531],{"className":52994,"style":565},[83,84],[39,52996,15389],{"className":52997},[83],[39,52999,53001,53004],{"className":53000},[83],[39,53002,4230],{"className":53003,"style":461},[83,84],[39,53005,53007],{"className":53006},[192],[39,53008,53010,53030],{"className":53009},[196,649],[39,53011,53013,53027],{"className":53012},[200],[39,53014,53016],{"className":53015,"style":3419},[204],[39,53017,53018,53021],{"style":13607},[39,53019],{"className":53020,"style":213},[212],[39,53022,53024],{"className":53023},[217,218,219,220],[39,53025,3196],{"className":53026,"style":3321},[83,84,220],[39,53028,673],{"className":53029},[672],[39,53031,53033],{"className":53032},[200],[39,53034,53036],{"className":53035,"style":680},[204],[39,53037],{},[39,53039],{"className":53040,"style":461},[230],[39,53042,5378],{"className":53043},[465],[39,53045],{"className":53046,"style":461},[230],[39,53048,53050,53053],{"className":53049},[83],[39,53051,4230],{"className":53052,"style":461},[83,84],[39,53054,53056],{"className":53055},[192],[39,53057,53059,53079],{"className":53058},[196,649],[39,53060,53062,53076],{"className":53061},[200],[39,53063,53065],{"className":53064,"style":1728},[204],[39,53066,53067,53070],{"style":13607},[39,53068],{"className":53069,"style":213},[212],[39,53071,53073],{"className":53072},[217,218,219,220],[39,53074,1696],{"className":53075},[83,220],[39,53077,673],{"className":53078},[672],[39,53080,53082],{"className":53081},[200],[39,53083,53085],{"className":53084,"style":680},[204],[39,53086],{},[39,53088,53090,53093],{"className":53089},[83],[39,53091,15389],{"className":53092},[83],[39,53094,53096],{"className":53095},[192],[39,53097,53099,53119],{"className":53098},[196,649],[39,53100,53102,53116],{"className":53101},[200],[39,53103,53105],{"className":53104,"style":1728},[204],[39,53106,53107,53110],{"style":966},[39,53108],{"className":53109,"style":213},[212],[39,53111,53113],{"className":53112},[217,218,219,220],[39,53114,1696],{"className":53115},[83,220],[39,53117,673],{"className":53118},[672],[39,53120,53122],{"className":53121},[200],[39,53123,53125],{"className":53124,"style":680},[204],[39,53126],{},[39,53128,2903],{"className":53129},[83],[39,53131,673],{"className":53132},[672],[39,53134,53136],{"className":53135},[200],[39,53137,53139],{"className":53138,"style":4507},[204],[39,53140],{},[11,53142,26443,53143,53171],{},[39,53144,53146,53159],{"className":53145,"translate":43},[42],[39,53147,53149],{"className":53148},[47],[49,53150,53151],{"xmlns":51},[53,53152,53153,53157],{},[56,53154,53155],{},[59,53156,3196],{},[63,53158,3196],{"encoding":65},[39,53160,53162],{"className":53161,"ariaHidden":70},[69],[39,53163,53165,53168],{"className":53164},[74],[39,53166],{"className":53167,"style":28824},[78],[39,53169,3196],{"className":53170,"style":3321},[83,84]," 反复迭代即得。证毕。",[11,53173,53174,53175,53245,53246,53305,53306,53334,53335,53363],{},"这个定理说明，初始估计 ",[39,53176,53178,53196],{"className":53177,"translate":43},[42],[39,53179,53181],{"className":53180},[47],[49,53182,53183],{"xmlns":51},[53,53184,53185,53193],{},[56,53186,53187],{},[613,53188,53189,53191],{},[59,53190,4230],{},[539,53192,541],{},[63,53194,53195],{"encoding":65},"V_0",[39,53197,53199],{"className":53198,"ariaHidden":70},[69],[39,53200,53202,53205],{"className":53201},[74],[39,53203],{"className":53204,"style":634},[78],[39,53206,53208,53211],{"className":53207},[83],[39,53209,4230],{"className":53210,"style":461},[83,84],[39,53212,53214],{"className":53213},[192],[39,53215,53217,53237],{"className":53216},[196,649],[39,53218,53220,53234],{"className":53219},[200],[39,53221,53223],{"className":53222,"style":963},[204],[39,53224,53225,53228],{"style":13607},[39,53226],{"className":53227,"style":213},[212],[39,53229,53231],{"className":53230},[217,218,219,220],[39,53232,541],{"className":53233},[83,220],[39,53235,673],{"className":53236},[672],[39,53238,53240],{"className":53239},[200],[39,53241,53243],{"className":53242,"style":680},[204],[39,53244],{}," 不需要很准确。无论从哪个有界函数出发，误差都会按 ",[39,53247,53249,53267],{"className":53248,"translate":43},[42],[39,53250,53252],{"className":53251},[47],[49,53253,53254],{"xmlns":51},[53,53255,53256,53264],{},[56,53257,53258],{},[138,53259,53260,53262],{},[59,53261,531],{},[59,53263,3196],{},[63,53265,53266],{"encoding":65},"\\gamma^k",[39,53268,53270],{"className":53269,"ariaHidden":70},[69],[39,53271,53273,53276],{"className":53272},[74],[39,53274],{"className":53275,"style":15644},[78],[39,53277,53279,53282],{"className":53278},[83],[39,53280,531],{"className":53281,"style":565},[83,84],[39,53283,53285],{"className":53284},[192],[39,53286,53288],{"className":53287},[196],[39,53289,53291],{"className":53290},[200],[39,53292,53294],{"className":53293,"style":15577},[204],[39,53295,53296,53299],{"style":208},[39,53297],{"className":53298,"style":213},[212],[39,53300,53302],{"className":53301},[217,218,219,220],[39,53303,3196],{"className":53304,"style":3321},[83,84,220]," 的速度衰减。",[39,53307,53309,53322],{"className":53308,"translate":43},[42],[39,53310,53312],{"className":53311},[47],[49,53313,53314],{"xmlns":51},[53,53315,53316,53320],{},[56,53317,53318],{},[59,53319,531],{},[63,53321,26586],{"encoding":65},[39,53323,53325],{"className":53324,"ariaHidden":70},[69],[39,53326,53328,53331],{"className":53327},[74],[39,53329],{"className":53330,"style":2654},[78],[39,53332,531],{"className":53333,"style":565},[83,84]," 越接近 ",[39,53336,53338,53351],{"className":53337,"translate":43},[42],[39,53339,53341],{"className":53340},[47],[49,53342,53343],{"xmlns":51},[53,53344,53345,53349],{},[56,53346,53347],{},[539,53348,546],{},[63,53350,546],{"encoding":65},[39,53352,53354],{"className":53353,"ariaHidden":70},[69],[39,53355,53357,53360],{"className":53356},[74],[39,53358],{"className":53359,"style":2473},[78],[39,53361,546],{"className":53362},[83],"，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。",[26,53365,53367],{"id":53366},"_52-误差上界","5.2 误差上界",[11,53369,53370,53374,53375,53444],{},[158,53371,53373],{"href":53372},"#theorem-convergence","定理 5.1"," 给出的误差界含有未知的 ",[39,53376,53378,53395],{"className":53377,"translate":43},[42],[39,53379,53381],{"className":53380},[47],[49,53382,53383],{"xmlns":51},[53,53384,53385,53393],{},[56,53386,53387],{},[613,53388,53389,53391],{},[59,53390,4230],{},[59,53392,1696],{"mathvariant":145},[63,53394,34936],{"encoding":65},[39,53396,53398],{"className":53397,"ariaHidden":70},[69],[39,53399,53401,53404],{"className":53400},[74],[39,53402],{"className":53403,"style":634},[78],[39,53405,53407,53410],{"className":53406},[83],[39,53408,4230],{"className":53409,"style":461},[83,84],[39,53411,53413],{"className":53412},[192],[39,53414,53416,53436],{"className":53415},[196,649],[39,53417,53419,53433],{"className":53418},[200],[39,53420,53422],{"className":53421,"style":1728},[204],[39,53423,53424,53427],{"style":13607},[39,53425],{"className":53426,"style":213},[212],[39,53428,53430],{"className":53429},[217,218,219,220],[39,53431,1696],{"className":53432},[83,220],[39,53434,673],{"className":53435},[672],[39,53437,53439],{"className":53438},[200],[39,53440,53442],{"className":53441,"style":680},[204],[39,53443],{},"，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。",[31,53446,53449,53783,54270,54272],{"id":53447,"title":53448},"proposition-error-bounds","命题 5.2（先验与后验误差界）",[11,53450,37,53451,21402,53479,53507,53508,742,53661,53730,53731,742],{},[39,53452,53454,53467],{"className":53453,"translate":43},[42],[39,53455,53457],{"className":53456},[47],[49,53458,53459],{"xmlns":51},[53,53460,53461,53465],{},[56,53462,53463],{},[59,53464,11772],{"mathvariant":617},[63,53466,34785],{"encoding":65},[39,53468,53470],{"className":53469,"ariaHidden":70},[69],[39,53471,53473,53476],{"className":53472},[74],[39,53474],{"className":53475,"style":79},[78],[39,53477,11772],{"className":53478,"style":11793},[83,641],[39,53480,53482,53495],{"className":53481,"translate":43},[42],[39,53483,53485],{"className":53484},[47],[49,53486,53487],{"xmlns":51},[53,53488,53489,53493],{},[56,53490,53491],{},[59,53492,531],{},[63,53494,26586],{"encoding":65},[39,53496,53498],{"className":53497,"ariaHidden":70},[69],[39,53499,53501,53504],{"className":53500},[74],[39,53502],{"className":53503,"style":2654},[78],[39,53505,531],{"className":53506,"style":565},[83,84],"-压缩映射，",[39,53509,53511,53545],{"className":53510,"translate":43},[42],[39,53512,53514],{"className":53513},[47],[49,53515,53516],{"xmlns":51},[53,53517,53518,53542],{},[56,53519,53520,53532,53534,53536],{},[613,53521,53522,53524],{},[59,53523,4230],{},[56,53525,53526,53528,53530],{},[59,53527,3196],{},[133,53529,788],{},[539,53531,546],{},[133,53533,793],{},[59,53535,11772],{"mathvariant":617},[613,53537,53538,53540],{},[59,53539,4230],{},[59,53541,3196],{},[63,53543,53544],{"encoding":65},"V_{k+1}=\\mathcal{T}V_k",[39,53546,53548,53612],{"className":53547,"ariaHidden":70},[69],[39,53549,53551,53554,53603,53606,53609],{"className":53550},[74],[39,53552],{"className":53553,"style":7584},[78],[39,53555,53557,53560],{"className":53556},[83],[39,53558,4230],{"className":53559,"style":461},[83,84],[39,53561,53563],{"className":53562},[192],[39,53564,53566,53595],{"className":53565},[196,649],[39,53567,53569,53592],{"className":53568},[200],[39,53570,53572],{"className":53571,"style":3419},[204],[39,53573,53574,53577],{"style":13607},[39,53575],{"className":53576,"style":213},[212],[39,53578,53580],{"className":53579},[217,218,219,220],[39,53581,53583,53586,53589],{"className":53582},[83,220],[39,53584,3196],{"className":53585,"style":3321},[83,84,220],[39,53587,788],{"className":53588},[465,220],[39,53590,546],{"className":53591},[83,220],[39,53593,673],{"className":53594},[672],[39,53596,53598],{"className":53597},[200],[39,53599,53601],{"className":53600,"style":994},[204],[39,53602],{},[39,53604],{"className":53605,"style":231},[230],[39,53607,793],{"className":53608},[235],[39,53610],{"className":53611,"style":231},[230],[39,53613,53615,53618,53621],{"className":53614},[74],[39,53616],{"className":53617,"style":634},[78],[39,53619,11772],{"className":53620,"style":11793},[83,641],[39,53622,53624,53627],{"className":53623},[83],[39,53625,4230],{"className":53626,"style":461},[83,84],[39,53628,53630],{"className":53629},[192],[39,53631,53633,53653],{"className":53632},[196,649],[39,53634,53636,53650],{"className":53635},[200],[39,53637,53639],{"className":53638,"style":3419},[204],[39,53640,53641,53644],{"style":13607},[39,53642],{"className":53643,"style":213},[212],[39,53645,53647],{"className":53646},[217,218,219,220],[39,53648,3196],{"className":53649,"style":3321},[83,84,220],[39,53651,673],{"className":53652},[672],[39,53654,53656],{"className":53655},[200],[39,53657,53659],{"className":53658,"style":680},[204],[39,53660],{},[39,53662,53664,53681],{"className":53663,"translate":43},[42],[39,53665,53667],{"className":53666},[47],[49,53668,53669],{"xmlns":51},[53,53670,53671,53679],{},[56,53672,53673],{},[613,53674,53675,53677],{},[59,53676,4230],{},[59,53678,1696],{"mathvariant":145},[63,53680,34936],{"encoding":65},[39,53682,53684],{"className":53683,"ariaHidden":70},[69],[39,53685,53687,53690],{"className":53686},[74],[39,53688],{"className":53689,"style":634},[78],[39,53691,53693,53696],{"className":53692},[83],[39,53694,4230],{"className":53695,"style":461},[83,84],[39,53697,53699],{"className":53698},[192],[39,53700,53702,53722],{"className":53701},[196,649],[39,53703,53705,53719],{"className":53704},[200],[39,53706,53708],{"className":53707,"style":1728},[204],[39,53709,53710,53713],{"style":13607},[39,53711],{"className":53712,"style":213},[212],[39,53714,53716],{"className":53715},[217,218,219,220],[39,53717,1696],{"className":53718},[83,220],[39,53720,673],{"className":53721},[672],[39,53723,53725],{"className":53724},[200],[39,53726,53728],{"className":53727,"style":680},[204],[39,53729],{}," 为不动点。则对任意 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先证第一式。由三角不等式与压缩性，",[39,54750,54752],{"className":54751,"translate":43},[746],[39,54753,54755,54951],{"className":54754,"translate":43},[42],[39,54756,54758],{"className":54757},[47],[49,54759,54760],{"xmlns":51,"display":755},[53,54761,54762,54948],{},[759,54763,54764,54846],{"rowspacing":761,"columnalign":762,"columnspacing":146},[764,54765,54766,54794],{},[767,54767,54768],{},[770,54769,54770],{"scriptlevel":541,"displaystyle":70},[56,54771,54772,54774,54780,54782,54788],{},[59,54773,15389],{"mathvariant":145},[613,54775,54776,54778],{},[59,54777,4230],{},[59,54779,3196],{},[133,54781,5378],{},[613,54783,54784,54786],{},[59,54785,4230],{},[59,54787,1696],{"mathvariant":145},[613,54789,54790,54792],{},[59,54791,15389],{"mathvariant":145},[59,54793,1696],{"mathvariant":145},[767,54795,54796],{},[770,54797,54798],{"scriptlevel":541,"displaystyle":70},[56,54799,54800,54802,54804,54818,54820,54832,54834,54840],{},[56,54801],{},[133,54803,1824],{},[3187,54805,54806,54808,54816],{},[133,54807,3191],{},[56,54809,54810,54812,54814],{},[59,54811,13446],{},[133,54813,793],{},[59,54815,3196],{},[59,54817,1696],{"mathvariant":145},[59,54819,15389],{"mathvariant":145},[613,54821,54822,54824],{},[59,54823,4230],{},[56,54825,54826,54828,54830],{},[59,54827,13446],{},[133,54829,788],{},[539,54831,546],{},[133,54833,5378],{},[613,54835,54836,54838],{},[59,54837,4230],{},[59,54839,13446],{},[613,54841,54842,54844],{},[59,54843,15389],{"mathvariant":145},[59,54845,1696],{"mathvariant":145},[764,54847,54848,54854],{},[767,54849,54850],{},[770,54851,54852],{"scriptlevel":541,"displaystyle":70},[56,54853],{},[767,54855,54856],{},[770,54857,54858],{"scriptlevel":541,"displaystyle":70},[56,54859,54860,54862,54864,54878,54884,54886,54892,54894,54900,54906,54908,54924,54926,54932,54934,54940,54946],{},[56,54861],{},[133,54863,1824],{},[3187,54865,54866,54868,54876],{},[133,54867,3191],{},[56,54869,54870,54872,54874],{},[59,54871,13446],{},[133,54873,793],{},[59,54875,3196],{},[59,54877,1696],{"mathvariant":145},[138,54879,54880,54882],{},[59,54881,531],{},[59,54883,13446],{},[59,54885,15389],{"mathvariant":145},[613,54887,54888,54890],{},[59,54889,4230],{},[539,54891,546],{},[133,54893,5378],{},[613,54895,54896,54898],{},[59,54897,4230],{},[539,54899,541],{},[613,54901,54902,54904],{},[59,54903,15389],{"mathvariant":145},[59,54905,1696],{"mathvariant":145},[133,54907,793],{},[5362,54909,54910,54916],{},[138,54911,54912,54914],{},[59,54913,531],{},[59,54915,3196],{},[56,54917,54918,54920,54922],{},[539,54919,546],{},[133,54921,5378],{},[59,54923,531],{},[59,54925,15389],{"mathvariant":145},[613,54927,54928,54930],{},[59,54929,4230],{},[539,54931,546],{},[133,54933,5378],{},[613,54935,54936,54938],{},[59,54937,4230],{},[539,54939,541],{},[613,54941,54942,54944],{},[59,54943,15389],{"mathvariant":145},[59,54945,1696],{"mathvariant":145},[59,54947,2903],{"mathvariant":145},[63,54949,54950],{"encoding":65},"\\begin{aligned}\n\\|V_k-V_\\infty\\|_\\infty\n&\\le\\sum_{j=k}^{\\infty}\\|V_{j+1}-V_j\\|_\\infty\\\\\n&\\le\\sum_{j=k}^{\\infty}\\gamma^j\\|V_1-V_0\\|_\\infty\n=\\frac{\\gamma^k}{1-\\gamma}\\|V_1-V_0\\|_\\infty.\n\\end{aligned}",[39,54952,54954],{"className":54953,"ariaHidden":70},[69],[39,54955,54957,54961],{"className":54956},[74],[39,54958],{"className":54959,"style":54960},[78],"height:6.4792em;vertical-align:-2.9896em;",[39,54962,54964],{"className":54963},[83],[39,54965,54967,55141],{"className":54966},[759],[39,54968,54970],{"className":54969},[918],[39,54971,54973,55132],{"className":54972},[196,649],[39,54974,54976,55129],{"className":54975},[200],[39,54977,54980,55121],{"className":54978,"style":54979},[204],"height:3.4896em;",[39,54981,54983,54986],{"style":54982},"top:-5.4896em;",[39,54984],{"className":54985,"style":5782},[212],[39,54987,54989,54992,55032,55035,55038,55041,55081],{"className":54988},[83],[39,54990,15389],{"className":54991},[83],[39,54993,54995,54998],{"className":54994},[83],[39,54996,4230],{"className":54997,"style":461},[83,84],[39,54999,55001],{"className":55000},[192],[39,55002,55004,55024],{"className":55003},[196,649],[39,55005,55007,55021],{"className":55006},[200],[39,55008,55010],{"className":55009,"style":3419},[204],[39,55011,55012,55015],{"style":13607},[39,55013],{"className":55014,"style":213},[212],[39,55016,55018],{"className":55017},[217,218,219,220],[39,55019,3196],{"className":55020,"style":3321},[83,84,220],[39,55022,673],{"className":55023},[672],[39,55025,55027],{"className":55026},[200],[39,55028,55030],{"className":55029,"style":680},[204],[39,55031],{},[39,55033],{"className":55034,"style":461},[230],[39,55036,5378],{"className":55037},[465],[39,55039],{"className":55040,"style":461},[230],[39,55042,55044,55047],{"className":55043},[83],[39,55045,4230],{"className":55046,"style":461},[83,84],[39,55048,55050],{"className":55049},[192],[39,55051,55053,55073],{"className":55052},[196,649],[39,55054,55056,55070],{"className":55055},[200],[39,55057,55059],{"className":55058,"style"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220],[39,57810,3196],{"className":57811,"style":3321},[83,84,220],[39,57813,673],{"className":57814},[672],[39,57816,57818],{"className":57817},[200],[39,57819,57821],{"className":57820,"style":680},[204],[39,57822],{},[39,57824,57826,57829],{"className":57825},[83],[39,57827,15389],{"className":57828},[83],[39,57830,57832],{"className":57831},[192],[39,57833,57835,57855],{"className":57834},[196,649],[39,57836,57838,57852],{"className":57837},[200],[39,57839,57841],{"className":57840,"style":1728},[204],[39,57842,57843,57846],{"style":966},[39,57844],{"className":57845,"style":213},[212],[39,57847,57849],{"className":57848},[217,218,219,220],[39,57850,1696],{"className":57851},[83,220],[39,57853,673],{"className":57854},[672],[39,57856,57858],{"className":57857},[200],[39,57859,57861],{"className":57860,"style":680},[204],[39,57862],{},[39,57864,155],{"className":57865},[252],[11,57867,57868],{},"即第二式。证毕。",[11,57870,57871,57872,57923],{},"第一式称为先验界，因为它在迭代之前就可以用来估计所需步数。第二式称为后验界，因为它使用当前迭代的实际差分，可以在计算过程中判断是否已经足够接近不动点。由第二式，当 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时，若",[39,57925,57927],{"className":57926,"translate":43},[746],[39,57928,57930,58027],{"className":57929,"translate":43},[42],[39,57931,57933],{"className":57932},[47],[49,57934,57935],{"xmlns":51,"display":755},[53,57936,57937,58024],{},[759,57938,57939,57983],{"rowspacing":761,"columnalign":762,"columnspacing":146},[764,57940,57941,57947],{},[767,57942,57943],{},[770,57944,57945],{"scriptlevel":541,"displaystyle":70},[56,57946],{},[767,57948,57949],{},[770,57950,57951],{"scriptlevel":541,"displaystyle":70},[56,57952,57953,57955,57957,57963,57965,57977],{},[56,57954],{},[59,57956,15389],{"mathvariant":145},[613,57958,57959,57961],{},[59,57960,4230],{},[59,57962,3196],{},[133,57964,5378],{},[613,57966,57967,57969],{},[59,57968,4230],{},[56,57970,57971,57973,57975],{},[59,57972,3196],{},[133,57974,5378],{},[539,57976,546],{},[613,57978,57979,57981],{},[59,57980,15389],{"mathvariant":145},[59,57982,1696],{"mathvariant":145},[764,57984,57985,57991],{},[767,57986,57987],{},[770,57988,57989],{"scriptlevel":541,"displaystyle":70},[56,57990],{},[767,57992,57993],{},[770,57994,57995],{"scriptlevel":541,"displaystyle":70},[56,57996,57997,57999,58001,58003,58022],{},[56,57998],{},[230,58000],{"width":6891},[133,58002,1824],{},[5362,58004,58005,58020],{},[56,58006,58007,58010,58012,58014,58016,58018],{},[59,58008,58009],{},"ε",[133,58011,136],{"stretchy":135},[539,58013,546],{},[133,58015,5378],{},[59,58017,531],{},[133,58019,161],{"stretchy":135},[59,58021,531],{},[133,58023,155],{"separator":70},[63,58025,58026],{"encoding":65},"\\begin{aligned}\n&\\|V_k-V_{k-1}\\|_\\infty\\\\\n&\\quad\\le\\frac{\\varepsilon(1-\\gamma)}{\\gamma},\n\\end{aligned}",[39,58028,58030],{"className":58029,"ariaHidden":70},[69],[39,58031,58033,58037],{"className":58032},[74],[39,58034],{"className":58035,"style":58036},[78],"height:3.8074em;vertical-align:-1.6537em;",[39,58038,58040],{"className":58039},[83],[39,58041,58043,58087],{"className":58042},[759],[39,58044,58046],{"className":58045},[918],[39,58047,58049,58078],{"className":58048},[196,649],[39,58050,58052,58075],{"className":58051},[200],[39,58053,58056,58066],{"className":58054,"style":58055},[204],"height:2.1537em;",[39,58057,58059,58063],{"style":58058},"top:-4.7407em;",[39,58060],{"className":58061,"style":58062},[212],"height:3.427em;",[39,58064],{"className":58065},[83],[39,58067,58069,58072],{"style":58068},"top:-2.6537em;",[39,58070],{"className":58071,"style":58062},[212],[39,58073],{"className":58074},[83],[39,58076,673],{"className":58077},[672],[39,58079,58081],{"className":58080},[200],[39,58082,58085],{"className":58083,"style":58084},[204],"height:1.6537em;",[39,58086],{},[39,58088,58090],{"className":58089},[1223],[39,58091,58093,58364],{"className":58092},[196,649],[39,58094,58096,58361],{"className":58095},[200],[39,58097,58099,58251],{"className":58098,"style":58055},[204],[39,58100,58101,58104],{"style":58058},[39,58102],{"className":58103,"style":58062},[212],[39,58105,58107,58110,58113,58153,58156,58159,58162,58211],{"className":58106},[83],[39,58108],{"className":58109},[83],[39,58111,15389],{"className":58112},[83],[39,58114,58116,58119],{"className":58115},[83],[39,58117,4230],{"className":58118,"style":461},[83,84],[39,58120,58122],{"className":58121},[192],[39,58123,58125,58145],{"className":58124},[196,649],[39,58126,58128,58142],{"className":58127},[200],[39,58129,58131],{"className":58130,"style":3419},[204],[39,58132,58133,58136],{"style":13607},[39,58134],{"className":58135,"style":213},[212],[39,58137,58139],{"className":58138},[217,218,219,220],[39,58140,3196],{"className":58141,"style":3321},[83,84,220],[39,58143,673],{"className":58144},[672],[39,58146,58148],{"className":58147},[200],[39,58149,58151],{"className":58150,"style":680},[204],[39,58152],{},[39,58154],{"className":58155,"style":461},[230],[39,58157,5378],{"className":58158},[465],[39,58160],{"className":58161,"style":461},[230],[39,58163,58165,58168],{"className":58164},[83],[39,58166,4230],{"className":58167,"style":461},[83,84],[39,58169,58171],{"className":58170},[192],[39,58172,58174,58203],{"className":58173},[196,649],[39,58175,58177,58200],{"className":58176},[200],[39,58178,58180],{"className":58179,"style":3419},[204],[39,58181,58182,58185],{"style":13607},[39,58183],{"className":58184,"style":213},[212],[39,58186,58188],{"className":58187},[217,218,219,220],[39,58189,58191,58194,58197],{"className":58190},[83,220],[39,58192,3196],{"className":58193,"style":3321},[83,84,220],[39,58195,5378],{"className":58196},[465,220],[39,58198,546],{"className":58199},[83,220],[39,58201,673],{"className":58202},[672],[39,58204,58206],{"className":58205},[200],[39,58207,58209],{"className":58208,"style":994},[204],[39,58210],{},[39,58212,58214,58217],{"className":58213},[83],[39,58215,15389],{"className":58216},[83],[39,58218,58220],{"className":58219},[192],[39,58221,58223,58243],{"className":58222},[196,649],[39,58224,58226,58240],{"className":58225},[200],[39,58227,58229],{"className":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",[39,58376,58378,58416],{"className":58377,"translate":43},[42],[39,58379,58381],{"className":58380},[47],[49,58382,58383],{"xmlns":51},[53,58384,58385,58413],{},[56,58386,58387,58389,58395,58397,58403,58409,58411],{},[59,58388,15389],{"mathvariant":145},[613,58390,58391,58393],{},[59,58392,4230],{},[59,58394,3196],{},[133,58396,5378],{},[613,58398,58399,58401],{},[59,58400,4230],{},[59,58402,1696],{"mathvariant":145},[613,58404,58405,58407],{},[59,58406,15389],{"mathvariant":145},[59,58408,1696],{"mathvariant":145},[133,58410,1824],{},[59,58412,58009],{},[63,58414,58415],{"encoding":65},"\\|V_k-V_\\infty\\|_\\infty\\le\\varepsilon",[39,58417,58419,58477,58572],{"className":58418,"ariaHidden":70},[69],[39,58420,58422,58425,58428,58468,58471,58474],{"className":58421},[74],[39,58423],{"className":58424,"style":245},[78],[39,58426,15389],{"className":58427},[83],[39,58429,58431,58434],{"className":58430},[83],[39,58432,4230],{"className":58433,"style":461},[83,84],[39,58435,58437],{"className":58436},[192],[39,58438,58440,58460],{"className":58439},[196,649],[39,58441,58443,58457],{"className":58442},[200],[39,58444,58446],{"className":58445,"style":3419},[204],[39,58447,58448,58451],{"style":13607},[39,58449],{"className":58450,"style":213},[212],[39,58452,58454],{"className":58453},[217,218,219,220],[39,58455,3196],{"className":58456,"style":3321},[83,84,220],[39,58458,673],{"className":58459},[672],[39,58461,58463],{"className":58462},[200],[39,58464,58466],{"className":58465,"style":680},[204],[39,58467],{},[39,58469],{"className":58470,"style":461},[230],[39,58472,5378],{"className":58473},[465],[39,58475],{"className":58476,"style":461},[230],[39,58478,58480,58483,58523,58563,58566,58569],{"className":58479},[74],[39,58481],{"className":58482,"style":245},[78],[39,58484,58486,58489],{"className":58485},[83],[39,58487,4230],{"className":58488,"style":461},[83,84],[39,58490,58492],{"className":58491},[192],[39,58493,58495,58515],{"className":58494},[196,649],[39,58496,58498,58512],{"className":58497},[200],[39,58499,58501],{"className":58500,"style":1728},[204],[39,58502,58503,58506],{"style":13607},[39,58504],{"className":58505,"style":213},[212],[39,58507,58509],{"className":58508},[217,218,219,220],[39,58510,1696],{"className":58511},[83,220],[39,58513,673],{"className":58514},[672],[39,58516,58518],{"className":58517},[200],[39,58519,58521],{"className":58520,"style":680},[204],[39,58522],{},[39,58524,58526,58529],{"className":58525},[83],[39,58527,15389],{"className":58528},[83],[39,58530,58532],{"className":58531},[192],[39,58533,58535,58555],{"className":58534},[196,649],[39,58536,58538,58552],{"className":58537},[200],[39,58539,58541],{"className":58540,"style":1728},[204],[39,58542,58543,58546],{"style":966},[39,58544],{"className":58545,"style":213},[212],[39,58547,58549],{"className":58548},[217,218,219,220],[39,58550,1696],{"className":58551},[83,220],[39,58553,673],{"className":58554},[672],[39,58556,58558],{"className":58557},[200],[39,58559,58561],{"className":58560,"style":680},[204],[39,58562],{},[39,58564],{"className":58565,"style":231},[230],[39,58567,1824],{"className":58568},[235],[39,58570],{"className":58571,"style":231},[230],[39,58573,58575,58578],{"className":58574},[74],[39,58576],{"className":58577,"style":290},[78],[39,58579,58009],{"className":58580},[83,84],[26,58582,58584],{"id":58583},"_53-例显式迭代","5.3 例：显式迭代",[11,58586,58587,58588,58703],{},"回到 2.4 节的两状态马尔可夫链。令 ",[39,58589,58591,58621],{"className":58590,"translate":43},[42],[39,58592,58594],{"className":58593},[47],[49,58595,58596],{"xmlns":51},[53,58597,58598,58618],{},[56,58599,58600,58606,58608,58610,58612,58614,58616],{},[613,58601,58602,58604],{},[59,58603,4230],{},[539,58605,541],{},[133,58607,793],{},[133,58609,136],{"stretchy":135},[539,58611,541],{},[133,58613,155],{"separator":70},[539,58615,541],{},[133,58617,161],{"stretchy":135},[63,58619,58620],{"encoding":65},"V_0=(0,0)",[39,58622,58624,58679],{"className":58623,"ariaHidden":70},[69],[39,58625,58627,58630,58670,58673,58676],{"className":58626},[74],[39,58628],{"className":58629,"style":634},[78],[39,58631,58633,58636],{"className":58632},[83],[39,58634,4230],{"className":58635,"style":461},[83,84],[39,58637,58639],{"className":58638},[192],[39,58640,58642,58662],{"className":58641},[196,649],[39,58643,58645,58659],{"className":58644},[200],[39,58646,58648],{"className":58647,"style":963},[204],[39,58649,58650,58653],{"style":13607},[39,58651],{"className":58652,"style":213},[212],[39,58654,58656],{"className":58655},[217,218,219,220],[39,58657,541],{"className":58658},[83,220],[39,58660,673],{"className":58661},[672],[39,58663,58665],{"className":58664},[200],[39,58666,58668],{"className":58667,"style":680},[204],[39,58669],{},[39,58671],{"className":58672,"style":231},[230],[39,58674,793],{"className":58675},[235],[39,58677],{"className":58678,"style":231},[230],[39,58680,58682,58685,58688,58691,58694,58697,58700],{"className":58681},[74],[39,58683],{"className":58684,"style":245},[78],[39,58686,136],{"className":58687},[182],[39,58689,541],{"className":58690},[83],[39,58692,155],{"className":58693},[252],[39,58695],{"className":58696,"style":256},[230],[39,58698,541],{"className":58699},[83],[39,58701,161],{"className":58702},[263],"，迭代为",[39,58705,58707],{"className":58706,"translate":43},[746],[39,58708,58710,58800],{"className":58709,"translate":43},[42],[39,58711,58713],{"className":58712},[47],[49,58714,58715],{"xmlns":51,"display":755},[53,58716,58717,58797],{},[56,58718,58719,58731,58733,58735,58737,58739,58741,58743,58745,58751,58753,58755,58757,58759,58761,58773,58775,58777,58779,58781,58783,58789,58791,58793,58795],{},[613,58720,58721,58723],{},[59,58722,4230],{},[56,58724,58725,58727,58729],{},[59,58726,3196],{},[133,58728,788],{},[539,58730,546],{},[133,58732,136],{"stretchy":135},[539,58734,546],{},[133,58736,161],{"stretchy":135},[133,58738,793],{},[539,58740,546],{},[133,58742,788],{},[539,58744,17739],{},[613,58746,58747,58749],{},[59,58748,4230],{},[59,58750,3196],{},[133,58752,136],{"stretchy":135},[539,58754,17359],{},[133,58756,161],{"stretchy":135},[133,58758,155],{"separator":70},[230,58760],{"width":10251},[613,58762,58763,58765],{},[59,58764,4230],{},[56,58766,58767,58769,58771],{},[59,58768,3196],{},[133,58770,788],{},[539,58772,546],{},[133,58774,136],{"stretchy":135},[539,58776,17359],{},[133,58778,161],{"stretchy":135},[133,58780,793],{},[539,58782,17739],{},[613,58784,58785,58787],{},[59,58786,4230],{},[59,58788,3196],{},[133,58790,136],{"stretchy":135},[539,58792,546],{},[133,58794,161],{"stretchy":135},[59,58796,2903],{"mathvariant":145},[63,58798,58799],{"encoding":65},"V_{k+1}(1)=1+0.9V_k(2),\n\\qquad\nV_{k+1}(2)=0.9V_k(1).",[39,58801,58803,58876,58894,59028],{"className":58802,"ariaHidden":70},[69],[39,58804,58806,58809,58858,58861,58864,58867,58870,58873],{"className":58805},[74],[39,58807],{"className":58808,"style":245},[78],[39,58810,58812,58815],{"className":58811},[83],[39,58813,4230],{"className":58814,"style":461},[83,84],[39,58816,58818],{"className":58817},[192],[39,58819,58821,58850],{"className":58820},[196,649],[39,58822,58824,58847],{"className":58823},[200],[39,58825,58827],{"className":58826,"style":3419},[204],[39,58828,58829,58832],{"style":13607},[39,58830],{"className":58831,"style":213},[212],[39,58833,58835],{"className":58834},[217,218,219,220],[39,58836,58838,58841,58844],{"className":58837},[83,220],[39,58839,3196],{"className":58840,"style":3321},[83,84,220],[39,58842,788],{"className":58843},[465,220],[39,58845,546],{"className":58846},[83,220],[39,58848,673],{"className":58849},[672],[39,58851,58853],{"className":58852},[200],[39,58854,58856],{"className":58855,"style":994},[204],[39,58857],{},[39,58859,136],{"className":58860},[182],[39,58862,546],{"className":58863},[83],[39,58865,161],{"className":58866},[263],[39,58868],{"className":58869,"style":231},[230],[39,58871,793],{"className":58872},[235],[39,58874],{"className":58875,"style":231},[230],[39,58877,58879,58882,58885,58888,58891],{"className":58878},[74],[39,58880],{"className":58881,"style":17882},[78],[39,58883,546],{"className":58884},[83],[39,58886],{"className":58887,"style":461},[230],[39,58889,788],{"className":58890},[465],[39,58892],{"className":58893,"style":461},[230],[39,58895,58897,58900,58903,58943,58946,58949,58952,58955,58958,58961,59010,59013,59016,59019,59022,59025],{"className":58896},[74],[39,58898],{"className":58899,"style":245},[78],[39,58901,17739],{"className":58902},[83],[39,58904,58906,58909],{"className":58905},[83],[39,58907,4230],{"className":58908,"style":461},[83,84],[39,58910,58912],{"className":58911},[192],[39,58913,58915,58935],{"className":58914},[196,649],[39,58916,58918,58932],{"className":58917},[200],[39,58919,58921],{"className":58920,"style":3419},[204],[39,58922,58923,58926],{"style":13607},[39,58924],{"className":58925,"style":213},[212],[39,58927,58929],{"className":58928},[217,218,219,220],[39,58930,3196],{"className":58931,"style":3321},[83,84,220],[39,58933,673],{"className":58934},[672],[39,58936,58938],{"className":58937},[200],[39,58939,58941],{"className":58940,"style":680},[204],[39,58942],{},[39,58944,136],{"className":58945},[182],[39,58947,17359],{"className":58948},[83],[39,58950,161],{"className":58951},[263],[39,58953,155],{"className":58954},[252],[39,58956],{"className":58957,"style":10787},[230],[39,58959],{"className":58960,"style":256},[230],[39,58962,58964,58967],{"className":58963},[83],[39,58965,4230],{"className":58966,"style":461},[83,84],[39,58968,58970],{"className":58969},[192],[39,58971,58973,59002],{"className":58972},[196,649],[39,58974,58976,58999],{"className":58975},[200],[39,58977,58979],{"className":58978,"style":3419},[204],[39,58980,58981,58984],{"style":13607},[39,58982],{"className":58983,"style":213},[212],[39,58985,58987],{"className":58986},[217,218,219,220],[39,58988,58990,58993,58996],{"className":58989},[83,220],[39,58991,3196],{"className":58992,"style":3321},[83,84,220],[39,58994,788],{"className":58995},[465,220],[39,58997,546],{"className":58998},[83,220],[39,59000,673],{"className":59001},[672],[39,59003,59005],{"className":59004},[200],[39,59006,59008],{"className":59007,"style":994},[204],[39,59009],{},[39,59011,136],{"className":59012},[182],[39,59014,17359],{"className":59015},[83],[39,59017,161],{"className":59018},[263],[39,59020],{"className":59021,"style":231},[230],[39,59023,793],{"className":59024},[235],[39,59026],{"className":59027,"style":231},[230],[39,59029,59031,59034,59037,59077,59080,59083,59086],{"className":59030},[74],[39,59032],{"className":59033,"style":245},[78],[39,59035,17739],{"className":59036},[83],[39,59038,59040,59043],{"className":59039},[83],[39,59041,4230],{"className":59042,"style":461},[83,84],[39,59044,59046],{"className":59045},[192],[39,59047,59049,59069],{"className":59048},[196,649],[39,59050,59052,59066],{"className":59051},[200],[39,59053,59055],{"className":59054,"style":3419},[204],[39,59056,59057,59060],{"style":13607},[39,59058],{"className":59059,"style":213},[212],[39,59061,59063],{"className":59062},[217,218,219,220],[39,59064,3196],{"className":59065,"style":3321},[83,84,220],[39,59067,673],{"className":59068},[672],[39,59070,59072],{"className":59071},[200],[39,59073,59075],{"className":59074,"style":680},[204],[39,59076],{},[39,59078,136],{"className":59079},[182],[39,59081,546],{"className":59082},[83],[39,59084,161],{"className":59085},[263],[39,59087,2903],{"className":59088},[83],[11,59090,59091],{},"前几步为",[39,59093,59095],{"className":59094,"translate":43},[746],[39,59096,59098,59286],{"className":59097,"translate":43},[42],[39,59099,59101],{"className":59100},[47],[49,59102,59103],{"xmlns":51,"display":755},[53,59104,59105,59283],{},[759,59106,59107,59141,59175,59211,59247],{"rowspacing":761,"columnalign":762,"columnspacing":146},[764,59108,59109,59119],{},[767,59110,59111],{},[770,59112,59113],{"scriptlevel":541,"displaystyle":70},[613,59114,59115,59117],{},[59,59116,4230],{},[539,59118,546],{},[767,59120,59121],{},[770,59122,59123],{"scriptlevel":541,"displaystyle":70},[56,59124,59125,59127,59129,59131,59133,59135,59137,59139],{},[56,59126],{},[133,59128,793],{},[133,59130,136],{"stretchy":135},[539,59132,546],{},[133,59134,155],{"separator":70},[539,59136,541],{},[133,59138,161],{"stretchy":135},[133,59140,155],{"separator":70},[764,59142,59143,59153],{},[767,59144,59145],{},[770,59146,59147],{"scriptlevel":541,"displaystyle":70},[613,59148,59149,59151],{},[59,59150,4230],{},[539,59152,17359],{},[767,59154,59155],{},[770,59156,59157],{"scriptlevel":541,"displaystyle":70},[56,59158,59159,59161,59163,59165,59167,59169,59171,59173],{},[56,59160],{},[133,59162,793],{},[133,59164,136],{"stretchy":135},[539,59166,546],{},[133,59168,155],{"separator":70},[539,59170,17739],{},[133,59172,161],{"stretchy":135},[133,59174,155],{"separator":70},[764,59176,59177,59188],{},[767,59178,59179],{},[770,59180,59181],{"scriptlevel":541,"displaystyle":70},[613,59182,59183,59185],{},[59,59184,4230],{},[539,59186,59187],{},"3",[767,59189,59190],{},[770,59191,59192],{"scriptlevel":541,"displaystyle":70},[56,59193,59194,59196,59198,59200,59203,59205,59207,59209],{},[56,59195],{},[133,59197,793],{},[133,59199,136],{"stretchy":135},[539,59201,59202],{},"1.81",[133,59204,155],{"separator":70},[539,59206,17739],{},[133,59208,161],{"stretchy":135},[133,59210,155],{"separator":70},[764,59212,59213,59224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",[39,60387,60389,60407],{"className":60388,"translate":43},[42],[39,60390,60392],{"className":60391},[47],[49,60393,60394],{"xmlns":51},[53,60395,60396,60404],{},[56,60397,60398,60400,60402],{},[59,60399,1748],{},[133,60401,418],{},[59,60403,1696],{"mathvariant":145},[63,60405,60406],{"encoding":65},"m\\to\\infty",[39,60408,60410,60428],{"className":60409,"ariaHidden":70},[69],[39,60411,60413,60416,60419,60422,60425],{"className":60412},[74],[39,60414],{"className":60415,"style":290},[78],[39,60417,1748],{"className":60418},[83,84],[39,60420],{"className":60421,"style":231},[230],[39,60423,418],{"className":60424},[235],[39,60426],{"className":60427,"style":231},[230],[39,60429,60431,60434],{"className":60430},[74],[39,60432],{"className":60433,"style":290},[78],[39,60435,1696],{"className":60436},[83],"，得到",[39,60439,60441],{"className":60440,"translate":43},[746],[39,60442,60444,60520],{"className":60443,"translate":43},[42],[39,60445,60447],{"className":60446},[47],[49,60448,60449],{"xmlns":51,"display":755},[53,60450,60451,60517],{},[56,60452,60453,60455,60457,60459,60461,60463,60479,60481,60483,60485,60487,60489,60491,60493,60495,60497,60513,60515],{},[59,60454,4230],{},[133,60456,136],{"stretchy":135},[539,60458,546],{},[133,60460,161],{"stretchy":135},[133,60462,793],{},[5362,60464,60465,60467],{},[539,60466,546],{},[56,60468,60469,60471,60473],{},[539,60470,546],{},[133,60472,5378],{},[138,60474,60475,60477],{},[539,60476,17739],{},[539,60478,17359],{},[133,60480,793],{},[539,60482,18152],{},[133,60484,155],{"separator":70},[230,60486],{"width":10251},[59,60488,4230],{},[133,60490,136],{"stretchy":135},[539,60492,17359],{},[133,60494,161],{"stretchy":135},[133,60496,793],{},[5362,60498,60499,60501],{},[539,60500,17739],{},[56,60502,60503,60505,60507],{},[539,60504,546],{},[133,60506,5378],{},[138,60508,60509,60511],{},[539,60510,17739],{},[539,60512,17359],{},[133,60514,793],{},[539,60516,18181],{},[63,60518,60519],{"encoding":65},"V(1)=\\frac{1}{1-0.9^2}=5.2632,\n\\qquad\nV(2)=\\frac{0.9}{1-0.9^2}=4.7368.",[39,60521,60523,60550,60670,60709,60827],{"className":60522,"ariaHidden":70},[69],[39,60524,60526,60529,60532,60535,60538,60541,60544,60547],{"className":60525},[74],[39,60527],{"className":60528,"style":245},[78],[39,60530,4230],{"className":60531,"style":461},[83,84],[39,60533,136],{"className":60534},[182],[39,60536,546],{"className":60537},[83],[39,60539,161],{"className":60540},[263],[39,60542],{"className":60543,"style":231},[230],[39,60545,793],{"className":60546},[235],[39,60548],{"className":60549,"style":231},[230],[39,60551,60553,60556,60661,60664,60667],{"className":60552},[74],[39,60554],{"className":60555,"style":18221},[78],[39,60557,60559,60562,60658],{"className":60558},[83],[39,60560],{"className":60561},[182,5463],[39,60563,60565],{"className":60564},[5362],[39,60566,60568,60650],{"className":60567},[196,649],[39,60569,60571,60647],{"className":60570},[200],[39,60572,60574,60628,60636],{"className":60573,"style":18240},[204],[39,60575,60576,60579],{"style":5479},[39,60577],{"className":60578,"style":935},[212],[39,60580,60582,60585,60588,60591,60594,60597],{"className":60581},[83],[39,60583,546],{"className":60584},[83],[39,60586],{"className":60587,"style":461},[230],[39,60589,5378],{"className":60590},[465],[39,60592],{"className":60593,"style":461},[230],[39,60595,28866],{"className":60596},[83],[39,60598,60600,60603],{"className":60599},[83],[39,60601,28873],{"className":60602},[83],[39,60604,60606],{"className":60605},[192],[39,60607,60609],{"className":60608},[196],[39,60610,60612],{"className":60611},[200],[39,60613,60616],{"className":60614,"style":60615},[204],"height:0.7401em;",[39,60617,60619,60622],{"style":60618},"top:-2.989em;margin-right:0.05em;",[39,60620],{"className":60621,"style":213},[212],[39,60623,60625],{"className":60624},[217,218,219,220],[39,60626,17359],{"className":60627},[83,220],[39,60629,60630,60633],{"style":5503},[39,60631],{"className":60632,"style":935},[212],[39,60634],{"className":60635,"style":5511},[5510],[39,60637,60638,60641],{"style":5514},[39,60639],{"className":60640,"style":935},[212],[39,60642,60644],{"className":60643},[83],[39,60645,546],{"className":60646},[83],[39,60648,673],{"className":60649},[672],[39,60651,60653],{"className":60652},[200],[39,60654,60656],{"className":60655,"style":18292},[204],[39,60657],{},[39,60659],{"className":60660},[263,5463],[39,60662],{"className":60663,"style":231},[230],[39,60665,793],{"className":60666},[235],[39,60668],{"className":60669,"style":231},[230],[39,60671,60673,60676,60679,60682,60685,60688,60691,60694,60697,60700,60703,60706],{"className":60672},[74],[39,60674],{"className":60675,"style":245},[78],[39,60677,18152],{"className":60678},[83],[39,60680,155],{"className":60681},[252],[39,60683],{"className":60684,"style":10787},[230],[39,60686],{"className":60687,"style":256},[230],[39,60689,4230],{"className":60690,"style":461},[83,84],[39,60692,136],{"className":60693},[182],[39,60695,17359],{"className":60696},[83],[39,60698,161],{"className":60699},[263],[39,60701],{"className":60702,"style":231},[230],[39,60704,793],{"className":60705},[235],[39,60707],{"className":60708,"style":231},[230],[39,60710,60712,60715,60818,60821,60824],{"className":60711},[74],[39,60713],{"className":60714,"style":18221},[78],[39,60716,60718,60721,60815],{"className":60717},[83],[39,60719],{"className":60720},[182,5463],[39,60722,60724],{"className":60723},[5362],[39,60725,60727,60807],{"className":60726},[196,649],[39,60728,60730,60804],{"className":60729},[200],[39,60731,60733,60785,60793],{"className":60732,"style":18240},[204],[39,60734,60735,60738],{"style":5479},[39,60736],{"className":60737,"style":935},[212],[39,60739,60741,60744,60747,60750,60753,60756],{"className":60740},[83],[39,60742,546],{"className":60743},[83],[39,60745],{"className":60746,"style":461},[230],[39,60748,5378],{"className":60749},[465],[39,60751],{"className":60752,"style":461},[230],[39,60754,28866],{"className":60755},[83],[39,60757,60759,60762],{"className":60758},[83],[39,60760,28873],{"className":60761},[83],[39,60763,60765],{"className":60764},[192],[39,60766,60768],{"className":60767},[196],[39,60769,60771],{"className":60770},[200],[39,60772,60774],{"className":60773,"style":60615},[204],[39,60775,60776,60779],{"style":60618},[39,60777],{"className":60778,"style":213},[212],[39,60780,60782],{"className":60781},[217,218,219,220],[39,60783,17359],{"className":60784},[83,220],[39,60786,60787,60790],{"style":5503},[39,60788],{"className":60789,"style":935},[212],[39,60791],{"className":60792,"style":5511},[5510],[39,60794,60795,60798],{"style":5514},[39,60796],{"className":60797,"style":935},[212],[39,60799,60801],{"className":60800},[83],[39,60802,17739],{"className":60803},[83],[39,60805,673],{"className":60806},[672],[39,60808,60810],{"className":60809},[200],[39,60811,60813],{"className":60812,"style":18292},[204],[39,60814],{},[39,60816],{"className":60817},[263,5463],[39,60819],{"className":60820,"style":231},[230],[39,60822,793],{"className":60823},[235],[39,60825],{"className":60826,"style":231},[230],[39,60828,60830,60833],{"className":60829},[74],[39,60831],{"className":60832,"style":2473},[78],[39,60834,18181],{"className":60835},[83],[11,60837,60838,60839,60923,60924,60952],{},"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 ",[39,60840,60842,60864],{"className":60841,"translate":43},[42],[39,60843,60845],{"className":60844},[47],[49,60846,60847],{"xmlns":51},[53,60848,60849,60861],{},[56,60850,60851,60857,60859],{},[138,60852,60853,60855],{},[539,60854,17739],{},[539,60856,17359],{},[133,60858,793],{},[539,60860,18022],{},[63,60862,60863],{"encoding":65},"0.9^2=0.81",[39,60865,60867,60914],{"className":60866,"ariaHidden":70},[69],[39,60868,60870,60873,60876,60905,60908,60911],{"className":60869},[74],[39,60871],{"className":60872,"style":19710},[78],[39,60874,28866],{"className":60875},[83],[39,60877,60879,60882],{"className":60878},[83],[39,60880,28873],{"className":60881},[83],[39,60883,60885],{"className":60884},[192],[39,60886,60888],{"className":60887},[196],[39,60889,60891],{"className":60890},[200],[39,60892,60894],{"className":60893,"style":19710},[204],[39,60895,60896,60899],{"style":208},[39,60897],{"className":60898,"style":213},[212],[39,60900,60902],{"className":60901},[217,218,219,220],[39,60903,17359],{"className":60904},[83,220],[39,60906],{"className":60907,"style":231},[230],[39,60909,793],{"className":60910},[235],[39,60912],{"className":60913,"style":231},[230],[39,60915,60917,60920],{"className":60916},[74],[39,60918],{"className":60919,"style":2473},[78],[39,60921,18022],{"className":60922},[83]," 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 ",[39,60925,60927,60940],{"className":60926,"translate":43},[42],[39,60928,60930],{"className":60929},[47],[49,60931,60932],{"xmlns":51},[53,60933,60934,60938],{},[56,60935,60936],{},[539,60937,17739],{},[63,60939,17739],{"encoding":65},[39,60941,60943],{"className":60942,"ariaHidden":70},[69],[39,60944,60946,60949],{"className":60945},[74],[39,60947],{"className":60948,"style":2473},[78],[39,60950,17739],{"className":60951},[83]," 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。",[11,60954,16423,60955,60957,60958,49416],{},[158,60956,53373],{"href":53372},"，初始误差为 ",[39,60959,60961,60999],{"className":60960,"translate":43},[42],[39,60962,60964],{"className":60963},[47],[49,60965,60966],{"xmlns":51},[53,60967,60968,60996],{},[56,60969,60970,60972,60978,60980,60986,60992,60994],{},[59,60971,15389],{"mathvariant":145},[613,60973,60974,60976],{},[59,60975,4230],{},[539,60977,541],{},[133,60979,5378],{},[613,60981,60982,60984],{},[59,60983,4230],{},[59,60985,1696],{"mathvariant":145},[613,60987,60988,60990],{},[59,60989,15389],{"mathvariant":145},[59,60991,1696],{"mathvariant":145},[133,60993,793],{},[539,60995,18152],{},[63,60997,60998],{"encoding":65},"\\|V_0-V_\\infty\\|_\\infty=5.2632",[39,61000,61002,61060,61155],{"className":61001,"ariaHidden":70},[69],[39,61003,61005,61008,61011,61051,61054,61057],{"className":61004},[74],[39,61006],{"className":61007,"style":245},[78],[39,61009,15389],{"className":61010},[83],[39,61012,61014,61017],{"className":61013},[83],[39,61015,4230],{"className":61016,"style":461},[83,84],[39,61018,61020],{"className":61019},[192],[39,61021,61023,61043],{"className":61022},[196,649],[39,61024,61026,61040],{"className":61025},[200],[39,61027,61029],{"className":61028,"style":963},[204],[39,61030,61031,61034],{"style":13607},[39,61032],{"className":61033,"style":213},[212],[39,61035,61037],{"className":61036},[217,218,219,220],[39,61038,541],{"className":61039},[83,220],[39,61041,673],{"className":61042},[672],[39,61044,61046],{"className":61045},[200],[39,61047,61049],{"className":61048,"style":680},[204],[39,61050],{},[39,61052],{"className":61053,"style":461},[230],[39,61055,5378],{"className":61056},[465],[39,61058],{"className":61059,"style":461},[230],[39,61061,61063,61066,61106,61146,61149,61152],{"className":61062},[74],[39,61064],{"className":61065,"style":245},[78],[39,61067,61069,61072],{"className":61068},[83],[39,61070,4230],{"className":61071,"style":461},[83,84],[39,61073,61075],{"className":61074},[192],[39,61076,61078,61098],{"className":61077},[196,649],[39,61079,61081,61095],{"className":61080},[200],[39,61082,61084],{"className":61083,"style":1728},[204],[39,61085,61086,61089],{"style":13607},[39,61087],{"className":61088,"style":213},[212],[39,61090,61092],{"className":61091},[217,218,219,220],[39,61093,1696],{"className":61094},[83,220],[39,61096,673],{"className":61097},[672],[39,61099,61101],{"className":61100},[200],[39,61102,61104],{"className":61103,"style":680},[204],[39,61105],{},[39,61107,61109,61112],{"className":61108},[83],[39,61110,15389],{"className":61111},[83],[39,61113,61115],{"className":61114},[192],[39,61116,61118,61138],{"className":61117},[196,649],[39,61119,61121,61135],{"className":61120},[200],[39,61122,61124],{"className":61123,"style":1728},[204],[39,61125,61126,61129],{"style":966},[39,61127],{"className":61128,"style":213},[212],[39,61130,61132],{"className":61131},[217,218,219,220],[39,61133,1696],{"className":61134},[83,220],[39,61136,673],{"className":61137},[672],[39,61139,61141],{"className":61140},[200],[39,61142,61144],{"className":61143,"style":680},[204],[39,61145],{},[39,61147],{"className":61148,"style":231},[230],[39,61150,793],{"className":61151},[235],[39,61153],{"className":61154,"style":231},[230],[39,61156,61158,61161],{"className":61157},[74],[39,61159],{"className":61160,"style":2473},[78],[39,61162,18152],{"className":61163},[83],[39,61165,61167],{"className":61166,"translate":43},[746],[39,61168,61170,61218],{"className":61169,"translate":43},[42],[39,61171,61173],{"className":61172},[47],[49,61174,61175],{"xmlns":51,"display":755},[53,61176,61177,61215],{},[56,61178,61179,61181,61187,61189,61195,61201,61203,61205,61207,61213],{},[59,61180,15389],{"mathvariant":145},[613,61182,61183,61185],{},[59,61184,4230],{},[59,61186,3196],{},[133,61188,5378],{},[613,61190,61191,61193],{},[59,61192,4230],{},[59,61194,1696],{"mathvariant":145},[613,61196,61197,61199],{},[59,61198,15389],{"mathvariant":145},[59,61200,1696],{"mathvariant":145},[133,61202,1824],{},[539,61204,18152],{},[133,61206,409],{},[138,61208,61209,61211],{},[539,61210,17739],{},[59,61212,3196],{},[59,61214,2903],{"mathvariant":145},[63,61216,61217],{"encoding":65},"\\|V_k-V_\\infty\\|_\\infty\\le5.2632\\times0.9^k.",[39,61219,61221,61279,61374,61392],{"className":61220,"ariaHidden":70},[69],[39,61222,61224,61227,61230,61270,61273,61276],{"className":61223},[74],[39,61225],{"className":61226,"style":245},[78],[39,61228,15389],{"className":61229},[83],[39,61231,61233,61236],{"className":61232},[83],[39,61234,4230],{"className":61235,"style":461},[83,84],[39,61237,61239],{"className":61238},[192],[39,61240,61242,61262],{"className":61241},[196,649],[39,61243,61245,61259],{"className":61244},[200],[39,61246,61248],{"className":61247,"style":3419},[204],[39,61249,61250,61253],{"style":13607},[39,61251],{"className":61252,"style":213},[212],[39,61254,61256],{"className":61255},[217,218,219,220],[39,61257,3196],{"className":61258,"style":3321},[83,84,220],[39,61260,673],{"className":61261},[672],[39,61263,61265],{"className":61264},[200],[39,61266,61268],{"className":61267,"style":680},[204],[39,61269],{},[39,61271],{"className":61272,"style":461},[230],[39,61274,5378],{"className":61275},[465],[39,61277],{"className":61278,"style":461},[230],[39,61280,61282,61285,61325,61365,61368,61371],{"className":61281},[74],[39,61283],{"className":61284,"style":245},[78],[39,61286,61288,61291],{"className":61287},[83],[39,61289,4230],{"className":61290,"style":461},[83,84],[39,61292,61294],{"className":61293},[192],[39,61295,61297,61317],{"className":61296},[196,649],[39,61298,61300,61314],{"className":61299},[200],[39,61301,61303],{"className":61302,"style":1728},[204],[39,61304,61305,61308],{"style":13607},[39,61306],{"className":61307,"style":213},[212],[39,61309,61311],{"className":61310},[217,218,219,220],[39,61312,1696],{"className":61313},[83,220],[39,61315,673],{"className":61316},[672],[39,61318,61320],{"className":61319},[200],[39,61321,61323],{"className":61322,"style":680},[204],[39,61324],{},[39,61326,61328,61331],{"className":61327},[83],[39,61329,15389],{"className":61330},[83],[39,61332,61334],{"className":61333},[192],[39,61335,61337,61357],{"className":61336},[196,649],[39,61338,61340,61354],{"className":61339},[200],[39,61341,61343],{"className":61342,"style":1728},[204],[39,61344,61345,61348],{"style":966},[39,61346],{"className":61347,"style":213},[212],[39,61349,61351],{"className":61350},[217,218,219,220],[39,61352,1696],{"className":61353},[83,220],[39,61355,673],{"className":61356},[672],[39,61358,61360],{"className":61359},[200],[39,61361,61363],{"className":61362,"style":680},[204],[39,61364],{},[39,61366],{"className":61367,"style":231},[230],[39,61369,1824],{"className":61370},[235],[39,61372],{"className":61373,"style":231},[230],[39,61375,61377,61380,61383,61386,61389],{"className":61376},[74],[39,61378],{"className":61379,"style":17882},[78],[39,61381,18152],{"className":61382},[83],[39,61384],{"className":61385,"style":461},[230],[39,61387,409],{"className":61388},[465],[39,61390],{"className":61391,"style":461},[230],[39,61393,61395,61398,61401,61430],{"className":61394},[74],[39,61396],{"className":61397,"style":3389},[78],[39,61399,28866],{"className":61400},[83],[39,61402,61404,61407],{"className":61403},[83],[39,61405,28873],{"className":61406},[83],[39,61408,61410],{"className":61409},[192],[39,61411,61413],{"className":61412},[196],[39,61414,61416],{"className":61415},[200],[39,61417,61419],{"className":61418,"style":3389},[204],[39,61420,61421,61424],{"style":1027},[39,61422],{"className":61423,"style":213},[212],[39,61425,61427],{"className":61426},[217,218,219,220],[39,61428,3196],{"className":61429,"style":3321},[83,84,220],[39,61431,2903],{"className":61432},[83],[11,61434,12932,61435,61487,61488,28969,61517,61487,61569,61598],{},[39,61436,61438,61457],{"className":61437,"translate":43},[42],[39,61439,61441],{"className":61440},[47],[49,61442,61443],{"xmlns":51},[53,61444,61445,61454],{},[56,61446,61447,61449,61451],{},[59,61448,3196],{},[133,61450,793],{},[539,61452,61453],{},"20",[63,61455,61456],{"encoding":65},"k=20",[39,61458,61460,61478],{"className":61459,"ariaHidden":70},[69],[39,61461,61463,61466,61469,61472,61475],{"className":61462},[74],[39,61464],{"className":61465,"style":28824},[78],[39,61467,3196],{"className":61468,"style":3321},[83,84],[39,61470],{"className":61471,"style":231},[230],[39,61473,793],{"className":61474},[235],[39,61476],{"className":61477,"style":231},[230],[39,61479,61481,61484],{"className":61480},[74],[39,61482],{"className":61483,"style":2473},[78],[39,61485,61453],{"className":61486},[83]," 时，右端约为 ",[39,61489,61491,61505],{"className":61490,"translate":43},[42],[39,61492,61494],{"className":61493},[47],[49,61495,61496],{"xmlns":51},[53,61497,61498,61503],{},[56,61499,61500],{},[539,61501,61502],{},"0.64",[63,61504,61502],{"encoding":65},[39,61506,61508],{"className":61507,"ariaHidden":70},[69],[39,61509,61511,61514],{"className":61510},[74],[39,61512],{"className":61513,"style":2473},[78],[39,61515,61502],{"className":61516},[83],[39,61518,61520,61539],{"className":61519,"translate":43},[42],[39,61521,61523],{"className":61522},[47],[49,61524,61525],{"xmlns":51},[53,61526,61527,61536],{},[56,61528,61529,61531,61533],{},[59,61530,3196],{},[133,61532,793],{},[539,61534,61535],{},"50",[63,61537,61538],{"encoding":65},"k=50",[39,61540,61542,61560],{"className":61541,"ariaHidden":70},[69],[39,61543,61545,61548,61551,61554,61557],{"className":61544},[74],[39,61546],{"className":61547,"style":28824},[78],[39,61549,3196],{"className":61550,"style":3321},[83,84],[39,61552],{"className":61553,"style":231},[230],[39,61555,793],{"className":61556},[235],[39,61558],{"className":61559,"style":231},[230],[39,61561,61563,61566],{"className":61562},[74],[39,61564],{"className":61565,"style":2473},[78],[39,61567,61535],{"className":61568},[83],[39,61570,61572,61586],{"className":61571,"translate":43},[42],[39,61573,61575],{"className":61574},[47],[49,61576,61577],{"xmlns":51},[53,61578,61579,61584],{},[56,61580,61581],{},[539,61582,61583],{},"0.027",[63,61585,61583],{"encoding":65},[39,61587,61589],{"className":61588,"ariaHidden":70},[69],[39,61590,61592,61595],{"className":61591},[74],[39,61593],{"className":61594,"style":2473},[78],[39,61596,61583],{"className":61597},[83],"。这与几何收敛的预期一致。",[26,61600,61602],{"id":61601},"_54-折扣因子的必要性","5.4 折扣因子的必要性",[11,61604,61605,61606,34986,61656,61706],{},"压缩性依赖 ",[39,61607,61609,61626],{"className":61608,"translate":43},[42],[39,61610,61612],{"className":61611},[47],[49,61613,61614],{"xmlns":51},[53,61615,61616,61624],{},[56,61617,61618,61620,61622],{},[59,61619,531],{},[133,61621,1693],{},[539,61623,546],{},[63,61625,17296],{"encoding":65},[39,61627,61629,61647],{"className":61628,"ariaHidden":70},[69],[39,61630,61632,61635,61638,61641,61644],{"className":61631},[74],[39,61633],{"className":61634,"style":561},[78],[39,61636,531],{"className":61637,"style":565},[83,84],[39,61639],{"className":61640,"style":231},[230],[39,61642,1693],{"className":61643},[235],[39,61645],{"className":61646,"style":231},[230],[39,61648,61650,61653],{"className":61649},[74],[39,61651],{"className":61652,"style":2473},[78],[39,61654,546],{"className":61655},[83],[39,61657,61659,61676],{"className":61658,"translate":43},[42],[39,61660,61662],{"className":61661},[47],[49,61663,61664],{"xmlns":51},[53,61665,61666,61674],{},[56,61667,61668,61670,61672],{},[59,61669,531],{},[133,61671,793],{},[539,61673,546],{},[63,61675,28990],{"encoding":65},[39,61677,61679,61697],{"className":61678,"ariaHidden":70},[69],[39,61680,61682,61685,61688,61691,61694],{"className":61681},[74],[39,61683],{"className":61684,"style":2654},[78],[39,61686,531],{"className":61687,"style":565},[83,84],[39,61689],{"className":61690,"style":231},[230],[39,61692,793],{"className":61693},[235],[39,61695],{"className":61696,"style":231},[230],[39,61698,61700,61703],{"className":61699},[74],[39,61701],{"className":61702,"style":2473},[78],[39,61704,546],{"className":61705},[83],"，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。",[11,61708,61709,61710,61752,61753,61826,61827,17773],{},"第一个例子是零奖励的确定性循环。设两个状态 ",[39,61711,61713,61731],{"className":61712,"translate":43},[42],[39,61714,61716],{"className":61715},[47],[49,61717,61718],{"xmlns":51},[53,61719,61720,61728],{},[56,61721,61722,61724,61726],{},[539,61723,546],{},[133,61725,155],{"separator":70},[539,61727,17359],{},[63,61729,61730],{"encoding":65},"1,2",[39,61732,61734],{"className":61733,"ariaHidden":70},[69],[39,61735,61737,61740,61743,61746,61749],{"className":61736},[74],[39,61738],{"className":61739,"style":17578},[78],[39,61741,546],{"className":61742},[83],[39,61744,155],{"className":61745},[252],[39,61747],{"className":61748,"style":256},[230],[39,61750,17359],{"className":61751},[83],"，转移为 ",[39,61754,61756,61778],{"className":61755,"translate":43},[42],[39,61757,61759],{"className":61758},[47],[49,61760,61761],{"xmlns":51},[53,61762,61763,61775],{},[56,61764,61765,61767,61769,61771,61773],{},[539,61766,546],{},[133,61768,418],{},[539,61770,17359],{},[133,61772,418],{},[539,61774,546],{},[63,61776,61777],{"encoding":65},"1\\to2\\to1",[39,61779,61781,61799,61817],{"className":61780,"ariaHidden":70},[69],[39,61782,61784,61787,61790,61793,61796],{"className":61783},[74],[39,61785],{"className":61786,"style":2473},[78],[39,61788,546],{"className":61789},[83],[39,61791],{"className":61792,"style":231},[230],[39,61794,418],{"className":61795},[235],[39,61797],{"className":61798,"style":231},[230],[39,61800,61802,61805,61808,61811,61814],{"className":61801},[74],[39,61803],{"className":61804,"style":2473},[78],[39,61806,17359],{"className":61807},[83],[39,61809],{"className":61810,"style":231},[230],[39,61812,418],{"className":61813},[235],[39,61815],{"className":61816,"style":231},[230],[39,61818,61820,61823],{"className":61819},[74],[39,61821],{"className":61822,"style":2473},[78],[39,61824,546],{"className":61825},[83],"，奖励恒为零，",[39,61828,61830,61847],{"className":61829,"translate":43},[42],[39,61831,61833],{"className":61832},[47],[49,61834,61835],{"xmlns":51},[53,61836,61837,61845],{},[56,61838,61839,61841,61843],{},[59,61840,531],{},[133,61842,793],{},[539,61844,546],{},[63,61846,28990],{"encoding":65},[39,61848,61850,61868],{"className":61849,"ariaHidden":70},[69],[39,61851,61853,61856,61859,61862,61865],{"className":61852},[74],[39,61854],{"className":61855,"style":2654},[78],[39,61857,531],{"className":61858,"style":565},[83,84],[39,61860],{"className":61861,"style":231},[230],[39,61863,793],{"className":61864},[235],[39,61866],{"className":61867,"style":231},[230],[39,61869,61871,61874],{"className":61870},[74],[39,61872],{"className":61873,"style":2473},[78],[39,61875,546],{"className":61876},[83],[39,61878,61880],{"className":61879,"translate":43},[746],[39,61881,61883,61937],{"className":61882,"translate":43},[42],[39,61884,61886],{"className":61885},[47],[49,61887,61888],{"xmlns":51,"display":755},[53,61889,61890,61934],{},[56,61891,61892,61894,61896,61898,61900,61902,61904,61906,61908,61910,61912,61914,61916,61918,61920,61922,61924,61926,61928,61930,61932],{},[59,61893,4230],{},[133,61895,136],{"stretchy":135},[539,61897,546],{},[133,61899,161],{"stretchy":135},[133,61901,793],{},[59,61903,4230],{},[133,61905,136],{"stretchy":135},[539,61907,17359],{},[133,61909,161],{"stretchy":135},[133,61911,155],{"separator":70},[230,61913],{"width":10251},[59,61915,4230],{},[133,61917,136],{"stretchy":135},[539,61919,17359],{},[133,61921,161],{"stretchy":135},[133,61923,793],{},[59,61925,4230],{},[133,61927,136],{"stretchy":135},[539,61929,546],{},[133,61931,161],{"stretchy":135},[59,61933,2903],{"mathvariant":145},[63,61935,61936],{"encoding":65},"V(1)=V(2),\\qquad V(2)=V(1).",[39,61938,61940,61967,62015],{"className":61939,"ariaHidden":70},[69],[39,61941,61943,61946,61949,61952,61955,61958,61961,61964],{"className":61942},[74],[39,61944],{"className":61945,"style":245},[78],[39,61947,4230],{"className":61948,"style":461},[83,84],[39,61950,136],{"className":61951},[182],[39,61953,546],{"className":61954},[83],[39,61956,161],{"className":61957},[263],[39,61959],{"className":61960,"style":231},[230],[39,61962,793],{"className":61963},[235],[39,61965],{"className":61966,"style":231},[230],[39,61968,61970,61973,61976,61979,61982,61985,61988,61991,61994,61997,62000,62003,62006,62009,62012],{"className":61969},[74],[39,61971],{"className":61972,"style":245},[78],[39,61974,4230],{"className":61975,"style":461},[83,84],[39,61977,136],{"className":61978},[182],[39,61980,17359],{"className":61981},[83],[39,61983,161],{"className":61984},[263],[39,61986,155],{"className":61987},[252],[39,61989],{"className":61990,"style":10787},[230],[39,61992],{"className":61993,"style":256},[230],[39,61995,4230],{"className":61996,"style":461},[83,84],[39,61998,136],{"className":61999},[182],[39,62001,17359],{"className":62002},[83],[39,62004,161],{"className":62005},[263],[39,62007],{"className":62008,"style":231},[230],[39,62010,793],{"className":62011},[235],[39,62013],{"className":62014,"style":231},[230],[39,62016,62018,62021,62024,62027,62030,62033],{"className":62017},[74],[39,62019],{"className":62020,"style":245},[78],[39,62022,4230],{"className":62023,"style":461},[83,84],[39,62025,136],{"className":62026},[182],[39,62028,546],{"className":62029},[83],[39,62031,161],{"className":62032},[263],[39,62034,2903],{"className":62035},[83],[11,62037,62038,62039,62142],{},"任意常数函数 ",[39,62040,62042,62076],{"className":62041,"translate":43},[42],[39,62043,62045],{"className":62044},[47],[49,62046,62047],{"xmlns":51},[53,62048,62049,62073],{},[56,62050,62051,62053,62055,62057,62059,62061,62063,62065,62067,62069,62071],{},[59,62052,4230],{},[133,62054,136],{"stretchy":135},[539,62056,546],{},[133,62058,161],{"stretchy":135},[133,62060,793],{},[59,62062,4230],{},[133,62064,136],{"stretchy":135},[539,62066,17359],{},[133,62068,161],{"stretchy":135},[133,62070,793],{},[59,62072,24077],{},[63,62074,62075],{"encoding":65},"V(1)=V(2)=c",[39,62077,62079,62106,62133],{"className":62078,"ariaHidden":70},[69],[39,62080,62082,62085,62088,62091,62094,62097,62100,62103],{"className":62081},[74],[39,62083],{"className":62084,"style":245},[78],[39,62086,4230],{"className":62087,"style":461},[83,84],[39,62089,136],{"className":62090},[182],[39,62092,546],{"className":62093},[83],[39,62095,161],{"className":62096},[263],[39,62098],{"className":62099,"style":231},[230],[39,62101,793],{"className":62102},[235],[39,62104],{"className":62105,"style":231},[230],[39,62107,62109,62112,62115,62118,62121,62124,62127,62130],{"className":62108},[74],[39,62110],{"className":62111,"style":245},[78],[39,62113,4230],{"className":62114,"style":461},[83,84],[39,62116,136],{"className":62117},[182],[39,62119,17359],{"className":62120},[83],[39,62122,161],{"className":62123},[263],[39,62125],{"className":62126,"style":231},[230],[39,62128,793],{"className":62129},[235],[39,62131],{"className":62132,"style":231},[230],[39,62134,62136,62139],{"className":62135},[74],[39,62137],{"className":62138,"style":290},[78],[39,62140,24077],{"className":62141},[83,84]," 都是解，因此不动点不唯一。Banach 定理的唯一性结论失效。",[11,62144,62145,62146,62174],{},"第二个例子是每步奖励为 ",[39,62147,62149,62162],{"className":62148,"translate":43},[42],[39,62150,62152],{"className":62151},[47],[49,62153,62154],{"xmlns":51},[53,62155,62156,62160],{},[56,62157,62158],{},[539,62159,546],{},[63,62161,546],{"encoding":65},[39,62163,62165],{"className":62164,"ariaHidden":70},[69],[39,62166,62168,62171],{"className":62167},[74],[39,62169],{"className":62170,"style":2473},[78],[39,62172,546],{"className":62173},[83]," 的同一循环。此时",[39,62176,62178],{"className":62177,"translate":43},[746],[39,62179,62181,62223],{"className":62180,"translate":43},[42],[39,62182,62184],{"className":62183},[47],[49,62185,62186],{"xmlns":51,"display":755},[53,62187,62188,62220],{},[56,62189,62190,62196,62198,62212,62214,62216,62218],{},[613,62191,62192,62194],{},[59,62193,3181],{},[59,62195,621],{},[133,62197,793],{},[3187,62199,62200,62202,62210],{},[133,62201,3191],{},[56,62203,62204,62206,62208],{},[59,62205,3196],{},[133,62207,793],{},[539,62209,541],{},[59,62211,1696],{"mathvariant":145},[539,62213,546],{},[133,62215,793],{},[59,62217,1696],{"mathvariant":145},[133,62219,155],{"separator":70},[63,62221,62222],{"encoding":65},"G_t=\\sum_{k=0}^{\\infty}1=\\infty,",[39,62224,62226,62281,62369],{"className":62225,"ariaHidden":70},[69],[39,62227,62229,62232,62272,62275,62278],{"className":62228},[74],[39,62230],{"className":62231,"style":634},[78],[39,62233,62235,62238],{"className":62234},[83],[39,62236,3181],{"className":62237},[83,84],[39,62239,62241],{"className":62240},[192],[39,62242,62244,62264],{"className":62243},[196,649],[39,62245,62247,62261],{"className":62246},[200],[39,62248,62250],{"className":62249,"style":656},[204],[39,62251,62252,62255],{"style":966},[39,62253],{"className":62254,"style":213},[212],[39,62256,62258],{"className":62257},[217,218,219,220],[39,62259,621],{"className":62260},[83,84,220],[39,62262,673],{"className":62263},[672],[39,62265,62267],{"className":62266},[200],[39,62268,62270],{"className":62269,"style":680},[204],[39,62271],{},[39,62273],{"className":62274,"style":231},[230],[39,62276,793],{"className":62277},[235],[39,62279],{"className":62280,"style":231},[230],[39,62282,62284,62287,62354,62357,62360,62363,62366],{"className":62283},[74],[39,62285],{"className":62286,"style":3290},[78],[39,62288,62290],{"className":62289},[1744,3294],[39,62291,62293,62346],{"className":62292},[196,649],[39,62294,62296,62343],{"className":62295},[200],[39,62297,62299,62319,62329],{"className":62298,"style":3304},[204],[39,62300,62301,62304],{"style":3307},[39,62302],{"className":62303,"style":3311},[212],[39,62305,62307],{"className":62306},[217,218,219,220],[39,62308,62310,62313,62316],{"className":62309},[83,220],[39,62311,3196],{"className":62312,"style":3321},[83,84,220],[39,62314,793],{"className":62315},[235,220],[39,62317,541],{"className":62318},[83,220],[39,62320,62321,62324],{"style":3330},[39,62322],{"className":62323,"style":3311},[212],[39,62325,62326],{},[39,62327,3191],{"className":62328},[1744,3339,3340],[39,62330,62331,62334],{"style":3343},[39,62332],{"className":62333,"style":3311},[212],[39,62335,62337],{"className":62336},[217,218,219,220],[39,62338,62340],{"className":62339},[83,220],[39,62341,1696],{"className":62342},[83,220],[39,62344,673],{"className":62345},[672],[39,62347,62349],{"className":62348},[200],[39,62350,62352],{"className":62351,"style":3365},[204],[39,62353],{},[39,62355],{"className":62356,"style":256},[230],[39,62358,546],{"className":62359},[83],[39,62361],{"className":62362,"style":231},[230],[39,62364,793],{"className":62365},[235],[39,62367],{"className":62368,"style":231},[230],[39,62370,62372,62375,62378],{"className":62371},[74],[39,62373],{"className":62374,"style":2654},[78],[39,62376,1696],{"className":62377},[83],[39,62379,155],{"className":62380},[252],[11,62382,62383,62384,62427],{},"折扣回报不再是有限实数，值函数不属于 ",[39,62385,62387,62406],{"className":62386,"translate":43},[42],[39,62388,62390],{"className":62389},[47],[49,62391,62392],{"xmlns":51},[53,62393,62394,62404],{},[56,62395,62396,62398,62400,62402],{},[59,62397,19772],{},[133,62399,136],{"stretchy":135},[59,62401,61],{},[133,62403,161],{"stretchy":135},[63,62405,21380],{"encoding":65},[39,62407,62409],{"className":62408,"ariaHidden":70},[69],[39,62410,62412,62415,62418,62421,62424],{"className":62411},[74],[39,62413],{"className":62414,"style":245},[78],[39,62416,19772],{"className":62417,"style":19812},[83,84],[39,62419,136],{"className":62420},[182],[39,62422,61],{"className":62423,"style":85},[83,84],[39,62425,161],{"className":62426},[263],"。对应的贝尔曼方程为",[39,62429,62431],{"className":62430,"translate":43},[746],[39,62432,62434,62496],{"className":62433,"translate":43},[42],[39,62435,62437],{"className":62436},[47],[49,62438,62439],{"xmlns":51,"display":755},[53,62440,62441,62493],{},[56,62442,62443,62445,62447,62449,62451,62453,62455,62457,62459,62461,62463,62465,62467,62469,62471,62473,62475,62477,62479,62481,62483,62485,62487,62489,62491],{},[59,62444,4230],{},[133,62446,136],{"stretchy":135},[539,62448,546],{},[133,62450,161],{"stretchy":135},[133,62452,793],{},[539,62454,546],{},[133,62456,788],{},[59,62458,4230],{},[133,62460,136],{"stretchy":135},[539,62462,17359],{},[133,62464,161],{"stretchy":135},[133,62466,155],{"separator":70},[230,62468],{"width":10251},[59,62470,4230],{},[133,62472,136],{"stretchy":135},[539,62474,17359],{},[133,62476,161],{"stretchy":135},[133,62478,793],{},[539,62480,546],{},[133,62482,788],{},[59,62484,4230],{},[133,62486,136],{"stretchy":135},[539,62488,546],{},[133,62490,161],{"stretchy":135},[133,62492,155],{"separator":70},[63,62494,62495],{"encoding":65},"V(1)=1+V(2),\\qquad V(2)=1+V(1),",[39,62497,62499,62526,62544,62592,62610],{"className":62498,"ariaHidden":70},[69],[39,62500,62502,62505,62508,62511,62514,62517,62520,62523],{"className":62501},[74],[39,62503],{"className":62504,"style":245},[78],[39,62506,4230],{"className":62507,"style":461},[83,84],[39,62509,136],{"className":62510},[182],[39,62512,546],{"className":62513},[83],[39,62515,161],{"className":62516},[263],[39,62518],{"className":62519,"style":231},[230],[39,62521,793],{"className":62522},[235],[39,62524],{"className":62525,"style":231},[230],[39,62527,62529,62532,62535,62538,62541],{"className":62528},[74],[39,62530],{"className":62531,"style":17882},[78],[39,62533,546],{"className":62534},[83],[39,62536],{"className":62537,"style":461},[230],[39,62539,788],{"className":62540},[465],[39,62542],{"className":62543,"style":461},[230],[39,62545,62547,62550,62553,62556,62559,62562,62565,62568,62571,62574,62577,62580,62583,62586,62589],{"className":62546},[74],[39,62548],{"className":62549,"style":245},[78],[39,62551,4230],{"className":62552,"style":461},[83,84],[39,62554,136],{"className":62555},[182],[39,62557,17359],{"className":62558},[83],[39,62560,161],{"className":62561},[263],[39,62563,155],{"className":62564},[252],[39,62566],{"className":62567,"style":10787},[230],[39,62569],{"className":62570,"style":256},[230],[39,62572,4230],{"className":62573,"style":461},[83,84],[39,62575,136],{"className":62576},[182],[39,62578,17359],{"className":62579},[83],[39,62581,161],{"className":62582},[263],[39,62584],{"className":62585,"style":231},[230],[39,62587,793],{"className":62588},[235],[39,62590],{"className":62591,"style":231},[230],[39,62593,62595,62598,62601,62604,62607],{"className":62594},[74],[39,62596],{"className":62597,"style":17882},[78],[39,62599,546],{"className":62600},[83],[39,62602],{"className":62603,"style":461},[230],[39,62605,788],{"className":62606},[465],[39,62608],{"className":62609,"style":461},[230],[39,62611,62613,62616,62619,62622,62625,62628],{"className":62612},[74],[39,62614],{"className":62615,"style":245},[78],[39,62617,4230],{"className":62618,"style":461},[83,84],[39,62620,136],{"className":62621},[182],[39,62623,546],{"className":62624},[83],[39,62626,161],{"className":62627},[263],[39,62629,155],{"className":62630},[252],[11,62632,62633,62634,62684],{},"它也没有有限解。这说明 ",[39,62635,62637,62654],{"className":62636,"translate":43},[42],[39,62638,62640],{"className":62639},[47],[49,62641,62642],{"xmlns":51},[53,62643,62644,62652],{},[56,62645,62646,62648,62650],{},[59,62647,531],{},[133,62649,1693],{},[539,62651,546],{},[63,62653,17296],{"encoding":65},[39,62655,62657,62675],{"className":62656,"ariaHidden":70},[69],[39,62658,62660,62663,62666,62669,62672],{"className":62659},[74],[39,62661],{"className":62662,"style":561},[78],[39,62664,531],{"className":62665,"style":565},[83,84],[39,62667],{"className":62668,"style":231},[230],[39,62670,1693],{"className":62671},[235],[39,62673],{"className":62674,"style":231},[230],[39,62676,62678,62681],{"className":62677},[74],[39,62679],{"className":62680,"style":2473},[78],[39,62682,546],{"className":62683},[83]," 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。",[21,62686,62688],{"id":62687},"六最优子结构与最优性原理","六、最优子结构与最优性原理",[26,62690,62692],{"id":62691},"_61-有限期情形","6.1 有限期情形",[11,62694,62695,62696,62767,62768,2183,62956,63026,63027,63055],{},"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 ",[39,62697,62699,62725],{"className":62698,"translate":43},[42],[39,62700,62702],{"className":62701},[47],[49,62703,62704],{"xmlns":51},[53,62705,62706,62722],{},[56,62707,62708,62710,62712,62714,62716,62718,62720],{},[539,62709,541],{},[133,62711,155],{"separator":70},[539,62713,546],{},[133,62715,155],{"separator":70},[133,62717,2641],{},[133,62719,155],{"separator":70},[59,62721,11772],{},[63,62723,62724],{"encoding":65},"0,1,\\dots,T",[39,62726,62728],{"className":62727,"ariaHidden":70},[69],[39,62729,62731,62734,62737,62740,62743,62746,62749,62752,62755,62758,62761,62764],{"className":62730},[74],[39,62732],{"className":62733,"style":6452},[78],[39,62735,541],{"className":62736},[83],[39,62738,155],{"className":62739},[252],[39,62741],{"className":62742,"style":256},[230],[39,62744,546],{"className":62745},[83],[39,62747,155],{"className":62748},[252],[39,62750],{"className":62751,"style":256},[230],[39,62753,2641],{"className":62754},[2842],[39,62756],{"className":62757,"style":256},[230],[39,62759,155],{"className":62760},[252],[39,62762],{"className":62763,"style":256},[230],[39,62765,11772],{"className":62766,"style":178},[83,84],"，策略为一个序列 ",[39,62769,62771,62815],{"className":62770,"translate":43},[42],[39,62772,62774],{"className":62773},[47],[49,62775,62776],{"xmlns":51},[53,62777,62778,62812],{},[56,62779,62780,62782,62784,62786,62792,62794,62796,62798,62810],{},[59,62781,2098],{},[133,62783,793],{},[133,62785,136],{"stretchy":135},[613,62787,62788,62790],{},[59,62789,2098],{},[539,62791,541],{},[133,62793,155],{"separator":70},[133,62795,2641],{},[133,62797,155],{"separator":70},[613,62799,62800,62802],{},[59,62801,2098],{},[56,62803,62804,62806,62808],{},[59,62805,11772],{},[133,62807,5378],{},[539,62809,546],{},[133,62811,161],{"stretchy":135},[63,62813,62814],{"encoding":65},"\\pi=(\\pi_0,\\dots,\\pi_{T-1})",[39,62816,62818,62836],{"className":62817,"ariaHidden":70},[69],[39,62819,62821,62824,62827,62830,62833],{"className":62820},[74],[39,62822],{"className":62823,"style":290},[78],[39,62825,2098],{"className":62826,"style":2114},[83,84],[39,62828],{"className":62829,"style":231},[230],[39,62831,793],{"className":62832},[235],[39,62834],{"className":62835,"style":231},[230],[39,62837,62839,62842,62845,62885,62888,62891,62894,62897,62900,62903,62953],{"className":62838},[74],[39,62840],{"className":62841,"style":245},[78],[39,62843,136],{"className":62844},[182],[39,62846,62848,62851],{"className":62847},[83],[39,62849,2098],{"className":62850,"style":2114},[83,84],[39,62852,62854],{"className":62853},[192],[39,62855,62857,62877],{"className":62856},[196,649],[39,62858,62860,62874],{"className":62859},[200],[39,62861,62863],{"className":62862,"style":963},[204],[39,62864,62865,62868],{"style":41316},[39,62866],{"className":62867,"style":213},[212],[39,62869,62871],{"className":62870},[217,218,219,220],[39,62872,541],{"className":62873},[83,220],[39,62875,673],{"className":62876},[672],[39,62878,62880],{"className":62879},[200],[39,62881,62883],{"className":62882,"style":680},[204],[39,62884],{},[39,62886,155],{"className":62887},[252],[39,62889],{"className":62890,"style":256},[230],[39,62892,2641],{"className":62893},[2842],[39,62895],{"className":62896,"style":256},[230],[39,62898,155],{"className":62899},[252],[39,62901],{"className":62902,"style":256},[230],[39,62904,62906,62909],{"className":62905},[83],[39,62907,2098],{"className":62908,"style":2114},[83,84],[39,62910,62912],{"className":62911},[192],[39,62913,62915,62945],{"className":62914},[196,649],[39,62916,62918,62942],{"className":62917},[200],[39,62919,62922],{"className":62920,"style":62921},[204],"height:0.3283em;",[39,62923,62924,62927],{"style":41316},[39,62925],{"className":62926,"style":213},[212],[39,62928,62930],{"className":62929},[217,218,219,220],[39,62931,62933,62936,62939],{"className":62932},[83,220],[39,62934,11772],{"className":62935,"style":178},[83,84,220],[39,62937,5378],{"className":62938},[465,220],[39,62940,546],{"className":62941},[83,220],[39,62943,673],{"className":62944},[672],[39,62946,62948],{"className":62947},[200],[39,62949,62951],{"className":62950,"style":994},[204],[39,62952],{},[39,62954,161],{"className":62955},[263],[39,62957,62959,62977],{"className":62958,"translate":43},[42],[39,62960,62962],{"className":62961},[47],[49,62963,62964],{"xmlns":51},[53,62965,62966,62974],{},[56,62967,62968],{},[613,62969,62970,62972],{},[59,62971,2098],{},[59,62973,621],{},[63,62975,62976],{"encoding":65},"\\pi_t",[39,62978,62980],{"className":62979,"ariaHidden":70},[69],[39,62981,62983,62986],{"className":62982},[74],[39,62984],{"className":62985,"style":2916},[78],[39,62987,62989,62992],{"className":62988},[83],[39,62990,2098],{"className":62991,"style":2114},[83,84],[39,62993,62995],{"className":62994},[192],[39,62996,62998,63018],{"className":62997},[196,649],[39,62999,63001,63015],{"className":63000},[200],[39,63002,63004],{"className":63003,"style":656},[204],[39,63005,63006,63009],{"style":41316},[39,63007],{"className":63008,"style":213},[212],[39,63010,63012],{"className":63011},[217,218,219,220],[39,63013,621],{"className":63014},[83,84,220],[39,63016,673],{"className":63017},[672],[39,63019,63021],{"className":63020},[200],[39,63022,63024],{"className":63023,"style":680},[204],[39,63025],{}," 是时刻 ",[39,63028,63030,63043],{"className":63029,"translate":43},[42],[39,63031,63033],{"className":63032},[47],[49,63034,63035],{"xmlns":51},[53,63036,63037,63041],{},[56,63038,63039],{},[59,63040,621],{},[63,63042,621],{"encoding":65},[39,63044,63046],{"className":63045,"ariaHidden":70},[69],[39,63047,63049,63052],{"className":63048},[74],[39,63050],{"className":63051,"style":709},[78],[39,63053,621],{"className":63054},[83,84]," 的策略。定义",[39,63057,63059],{"className":63058,"translate":43},[746],[39,63060,63062,63221],{"className":63061,"translate":43},[42],[39,63063,63065],{"className":63064},[47],[49,63066,63067],{"xmlns":51,"display":755},[53,63068,63069,63218],{},[759,63070,63071,63150,63184],{"rowspacing":761,"columnalign":762,"columnspacing":146},[764,63072,63073,63093],{},[767,63074,63075],{},[770,63076,63077],{"scriptlevel":541,"displaystyle":70},[56,63078,63079,63087,63089,63091],{},[9932,63080,63081,63083,63085],{},[59,63082,4230],{},[59,63084,621],{},[59,63086,2098],{},[133,63088,136],{"stretchy":135},[59,63090,142],{},[133,63092,161],{"stretchy":135},[767,63094,63095],{},[770,63096,63097],{"scriptlevel":541,"displaystyle":70},[56,63098,63099,63101,63103,63109,63112,63132,63144],{},[56,63100],{},[133,63102,793],{},[613,63104,63105,63107],{},[59,63106,4253],{"mathvariant":421},[59,63108,2098],{},[133,63110,537],{"fence":135,"stretchy":70,"minsize":63111,"maxsize":63111},"3em",[3187,63113,63114,63116,63124],{},[133,63115,3191],{},[56,63117,63118,63120,63122],{},[59,63119,3196],{},[133,63121,793],{},[59,63123,621],{},[56,63125,63126,63128,63130],{},[59,63127,11772],{},[133,63129,5378],{},[539,63131,546],{},[138,63133,63134,63136],{},[59,63135,531],{},[56,63137,63138,63140,63142],{},[59,63139,3196],{},[133,63141,5378],{},[59,63143,621],{},[613,63145,63146,63148],{},[59,63147,401],{},[59,63149,3196],{},[764,63151,63152,63158],{},[767,63153,63154],{},[770,63155,63156],{"scriptlevel":541,"displaystyle":70},[56,63157],{},[767,63159,63160],{},[770,63161,63162],{"scriptlevel":541,"displaystyle":70},[56,63163,63164,63166,63168,63170,63176,63178,63180,63182],{},[56,63165],{},[230,63167],{"width":10251},[133,63169,150],{},[613,63171,63172,63174],{},[59,63173,142],{},[59,63175,621],{},[133,63177,793],{},[59,63179,142],{},[133,63181,4280],{"fence":135,"stretchy":70,"minsize":63111,"maxsize":63111},[133,63183,155],{"separator":70},[764,63185,63186,63206],{},[767,63187,63188],{},[770,63189,63190],{"scriptlevel":541,"displaystyle":70},[56,63191,63192,63200,63202,63204],{},[9932,63193,63194,63196,63198],{},[59,63195,4230],{},[59,63197,11772],{},[59,63199,2098],{},[133,63201,136],{"stretchy":135},[59,63203,142],{},[133,63205,161],{"stretchy":135},[767,63207,63208],{},[770,63209,63210],{"scriptlevel":541,"displaystyle":70},[56,63211,63212,63214,63216],{},[56,63213],{},[133,63215,793],{},[539,63217,28866],{},[63,63219,63220],{"encoding":65},"\\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}",[39,63222,63224],{"className":63223,"ariaHidden":70},[69],[39,63225,63227,63231],{"className":63226},[74],[39,63228],{"className":63229,"style":63230},[78],"height:7.9305em;vertical-align:-3.7152em;",[39,63232,63234],{"className":63233},[83],[39,63235,63237,63410],{"className":63236},[759],[39,63238,63240],{"className":63239},[918],[39,63241,63243,63401],{"className":63242},[196,649],[39,63244,63246,63398],{"className":63245},[200],[39,63247,63250,63320,63329],{"className":63248,"style":63249},[204],"height:4.2152em;",[39,63251,63253,63257],{"style":63252},"top:-6.2152em;",[39,63254],{"className":63255,"style":63256},[212],"height:3.8283em;",[39,63258,63260,63311,63314,63317],{"className":63259},[83],[39,63261,63263,63266],{"className":63262},[83],[39,63264,4230],{"className":63265,"style":461},[83,84],[39,63267,63269],{"className":63268},[192],[39,63270,63272,63303],{"className":63271},[196,649],[39,63273,63275,63300],{"className":63274},[200],[39,63276,63278,63289],{"className":63277,"style":4421},[204],[39,63279,63280,63283],{"style":9973},[39,63281],{"className":63282,"style":213},[212],[39,63284,63286],{"className":63285},[217,218,219,220],[39,63287,621],{"className":63288},[83,84,220],[39,63290,63291,63294],{"style":1027},[39,63292],{"className":63293,"style":213},[212],[39,63295,63297],{"className":63296},[217,218,219,220],[39,63298,2098],{"className":63299,"style":2114},[83,84,220],[39,63301,673],{"className":63302},[672],[39,63304,63306],{"className":63305},[200],[39,63307,63309],{"className":63308,"style":10003},[204],[39,63310],{},[39,63312,136],{"className":63313},[182],[39,63315,142],{"className":63316},[83,84],[39,63318,161],{"className":63319},[263],[39,63321,63323,63326],{"style":63322},"top:-2.8631em;",[39,63324],{"className":63325,"style":63256},[212],[39,63327],{"className":63328},[83],[39,63330,63332,63335],{"style":63331},"top:-0.4731em;",[39,63333],{"className":63334,"style":63256},[212],[39,63336,63338,63389,63392,63395],{"className":63337},[83],[39,63339,63341,63344],{"className":63340},[83],[39,63342,4230],{"className":63343,"style":461},[83,84],[39,63345,63347],{"className":63346},[192],[39,63348,63350,63381],{"className":63349},[196,649],[39,63351,63353,63378],{"className":63352},[200],[39,63354,63356,63367],{"className":63355,"style":4421},[204],[39,63357,63358,63361],{"style":9973},[39,63359],{"className":63360,"style":213},[212],[39,63362,63364],{"className":63363},[217,218,219,220],[39,63365,11772],{"className":63366,"style":178},[83,84,220],[39,63368,63369,63372],{"style":1027},[39,63370],{"className":63371,"style":213},[212],[39,63373,63375],{"className":63374},[217,218,219,220],[39,63376,2098],{"className":63377,"style":2114},[83,84,220],[39,63379,673],{"className":63380},[672],[39,63382,63384],{"className":63383},[200],[39,63385,63387],{"className":63386,"style":10003},[204],[39,63388],{},[39,63390,136],{"className":63391},[182],[39,63393,142],{"className":63394},[83,84],[39,63396,161],{"className":63397},[263],[39,63399,673],{"className":63400},[672],[39,63402,63404],{"className":63403},[200],[39,63405,63408],{"className":63406,"style":63407},[204],"height:3.7152em;",[39,63409],{},[39,63411,63413],{"className":63412},[1223],[39,63414,63416,63757],{"className":63415},[196,649],[39,63417,63419,63754],{"className":63418},[200],[39,63420,63422,63647,63731],{"className":63421,"style":63249},[204],[39,63423,63424,63427],{"style":63252},[39,63425],{"className":63426,"style":63256},[212],[39,63428,63430,63433,63436,63439,63442,63482,63489,63492,63566,63569,63607],{"className":63429},[83],[39,63431],{"className":63432},[83],[39,63434],{"className":63435,"style":231},[230],[39,63437,793],{"className":63438},[235],[39,63440],{"className":63441,"style":231},[230],[39,63443,63445,63448],{"className":63444},[83],[39,63446,4253],{"className":63447},[83,515],[39,63449,63451],{"className":63450},[192],[39,63452,63454,63474],{"className":63453},[196,649],[39,63455,63457,63471],{"className":63456},[200],[39,63458,63460],{"className":63459,"style":1728},[204],[39,63461,63462,63465],{"style":966},[39,63463],{"className":63464,"style":213},[212],[39,63466,63468],{"className":63467},[217,218,219,220],[39,63469,2098],{"className":63470,"style":2114},[83,84,220],[39,63472,673],{"className":63473},[672],[39,63475,63477],{"className":63476},[200],[39,63478,63480],{"className":63479,"style":680},[204],[39,63481],{},[39,63483,63485],{"className":63484},[83],[39,63486,537],{"className":63487},[18520,63488],"size4",[39,63490],{"className":63491,"style":256},[230],[39,63493,63495],{"className":63494},[1744,3294],[39,63496,63498,63558],{"className":63497},[196,649],[39,63499,63501,63555],{"className":63500},[200],[39,63502,63505,63525,63535],{"className":63503,"style":63504},[204],"height:1.8283em;",[39,63506,63507,63510],{"style":3307},[39,63508],{"className":63509,"style":3311},[212],[39,63511,63513],{"className":63512},[217,218,219,220],[39,63514,635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V_t^\\pi(s)",[39,63824,63826,63901],{"className":63825,"ariaHidden":70},[69],[39,63827,63829,63832,63883,63886,63889,63892,63895,63898],{"className":63828},[74],[39,63830],{"className":63831,"style":245},[78],[39,63833,63835,63838],{"className":63834},[83],[39,63836,4230],{"className":63837,"style":461},[83,84],[39,63839,63841],{"className":63840},[192],[39,63842,63844,63875],{"className":63843},[196,649],[39,63845,63847,63872],{"className":63846},[200],[39,63848,63850,63861],{"className":63849,"style":21897},[204],[39,63851,63852,63855],{"style":9973},[39,63853],{"className":63854,"style":213},[212],[39,63856,63858],{"className":63857},[217,218,219,220],[39,63859,621],{"className":63860},[83,84,220],[39,63862,63863,63866],{"style":208},[39,63864],{"className":63865,"style":213},[212],[39,63867,63869],{"className":63868},[217,218,219,220],[39,63870,21884],{"className":63871},[465,220],[39,63873,673],{"className":63874},[672],[39,63876,63878],{"className":63877},[200],[39,63879,63881],{"className":63880,"style":10003},[204],[39,63882],{},[39,63884,136],{"className":63885},[182],[39,63887,142],{"className":63888},[83,84],[39,63890,161],{"className":63891},[263],[39,63893],{"className":63894,"style":231},[230],[39,63896,793],{"className":63897},[235],[39,63899],{"className":63900,"style":231},[230],[39,63902,63904,63907,63950,63953,64004,64007,64010],{"className":63903},[74],[39,63905],{"className":63906,"style":245},[78],[39,63908,63910,63913],{"className":63909},[1744],[39,63911,21465],{"className":63912},[1744],[39,63914,63916],{"className":63915},[192],[39,63917,63919,63941],{"className":63918},[196,649],[39,63920,63922,63938],{"className":63921},[200],[39,63923,63926],{"className":63924,"style":63925},[204],"height:0.0573em;",[39,63927,63929,63932],{"style":63928},"top:-2.4559em;margin-right:0.05em;",[39,63930],{"className":63931,"style":213},[212],[39,63933,63935],{"className":63934},[217,218,219,220],[39,63936,2098],{"className":63937,"style":2114},[83,84,220],[39,63939,673],{"className":63940},[672],[39,63942,63944],{"className":63943},[200],[39,63945,63948],{"className":63946,"style":63947},[204],"height:0.2441em;",[39,63949],{},[39,63951],{"className":63952,"style":256},[230],[39,63954,63956,63959],{"className":63955},[83],[39,63957,4230],{"className":63958,"style":461},[83,84],[39,63960,63962],{"className":63961},[192],[39,63963,63965,63996],{"className":63964},[196,649],[39,63966,63968,63993],{"className":63967},[200],[39,63969,63971,63982],{"className":63970,"style":6410},[204],[39,63972,63973,63976],{"style":9973},[39,63974],{"className":63975,"style":213},[212],[39,63977,63979],{"className":63978},[217,218,219,220],[39,63980,621],{"className":63981},[83,84,220],[39,63983,63984,63987],{"style":208},[39,63985],{"className":63986,"style":213},[212],[39,63988,63990],{"className":63989},[217,218,219,220],[39,63991,2098],{"className":63992,"style":2114},[83,84,220],[39,63994,673],{"className":63995},[672],[39,63997,63999],{"className":63998},[200],[39,64000,64002],{"className":64001,"style":10003},[204],[39,64003],{},[39,64005,136],{"className":64006},[182],[39,64008,142],{"className":64009},[83,84],[39,64011,161],{"className":64012},[263],[31,64014,64017,64156],{"id":64015,"title":64016},"theorem-finite-horizon","定理 6.1（有限期最优性原理）",[11,64018,26443,64019,6525],{},[39,64020,64022,64060],{"className":64021,"translate":43},[42],[39,64023,64025],{"className":64024},[47],[49,64026,64027],{"xmlns":51},[53,64028,64029,64057],{},[56,64030,64031,64033,64035,64037,64039,64041,64043,64045,64047,64049,64051,64053,64055],{},[59,64032,621],{},[133,64034,793],{},[59,64036,11772],{},[133,64038,5378],{},[539,64040,546],{},[133,64042,155],{"separator":70},[59,64044,11772],{},[133,64046,5378],{},[539,64048,17359],{},[133,64050,155],{"separator":70},[133,64052,2641],{},[133,64054,155],{"separator":70},[539,64056,541],{},[63,64058,64059],{"encoding":65},"t=T-1,T-2,\\dots,0",[39,64061,64063,64081,64099,64126],{"className":64062,"ariaHidden":70},[69],[39,64064,64066,64069,64072,64075,64078],{"className":64065},[74],[39,64067],{"className":64068,"style":709},[78],[39,64070,621],{"className":64071},[83,84],[39,64073],{"className":64074,"style":231},[230],[39,64076,793],{"className":64077},[235],[39,64079],{"className":64080,"style":231},[230],[39,64082,64084,64087,64090,64093,64096],{"className":64083},[74],[39,64085],{"className":64086,"style":454},[78],[39,64088,11772],{"className":64089,"style":178},[83,84],[39,64091],{"className":64092,"style":461},[230],[39,64094,5378],{"className":64095},[465],[39,64097],{"className":64098,"style":461},[230],[39,64100,64102,64105,64108,64111,64114,64117,64120,64123],{"className":64101},[74],[39,64103],{"className":64104,"style":6452},[78],[39,64106,546],{"className":64107},[83],[39,64109,155],{"className":64110},[252],[39,64112],{"className":64113,"style":256},[230],[39,64115,11772],{"className":64116,"style":178},[83,84],[39,64118],{"className":64119,"style":461},[230],[39,64121,5378],{"className":64122},[465],[39,64124],{"className":64125,"style":461},[230],[39,64127,64129,64132,64135,64138,64141,64144,64147,64150,64153],{"className":64128},[74],[39,64130],{"className":64131,"style":17578},[78],[39,64133,17359],{"className":64134},[83],[39,64136,155],{"className":64137},[252],[39,64139],{"className":64140,"style":256},[230],[39,64142,2641],{"className":64143},[2842],[39,64145],{"className":64146,"style":256},[230],[39,64148,155],{"className":64149},[252],[39,64151],{"className":64152,"style":256},[230],[39,64154,541],{"className":64155},[83],[39,64157,64159],{"className":64158,"translate":43},[746],[39,64160,64162,64244],{"className":64161,"translate":43},[42],[39,64163,64165],{"className":64164},[47],[49,64166,64167],{"xmlns":51,"display":755},[53,64168,64169,64241],{},[56,64170,64171,64179,64181,64191,64205,64207,64209,64217,64219,64225,64239],{},[9932,64172,64173,64175,64177],{},[59,64174,4230],{},[59,64176,621],{},[59,64178,2098],{},[133,64180,793],{},[138,64182,64183,64185],{},[59,64184,11772],{"mathvariant":617},[613,64186,64187,64189],{},[59,64188,2098],{},[59,64190,621],{},[9932,64192,64193,64195,64203],{},[59,64194,4230],{},[56,64196,64197,64199,64201],{},[59,64198,621],{},[133,64200,788],{},[539,64202,546],{},[59,64204,2098],{},[133,64206,155],{"separator":70},[230,64208],{"width":10251},[9932,64210,64211,64213,64215],{},[59,64212,4230],{},[59,64214,621],{},[133,64216,21884],{},[133,64218,793],{},[138,64220,64221,64223],{},[59,64222,11772],{"mathvariant":617},[133,64224,21884],{},[9932,64226,64227,64229,64237],{},[59,64228,4230],{},[56,64230,64231,64233,64235],{},[59,64232,621],{},[133,64234,788],{},[539,64236,546],{},[133,64238,21884],{},[59,64240,2903],{"mathvariant":145},[63,64242,64243],{"encoding":65},"V_t^\\pi=\\mathcal{T}^{\\pi_t} V_{t+1}^\\pi,\n\\qquad\nV_t^*=\\mathcal{T}^* V_{t+1}^*.",[39,64245,64247,64314,64522],{"className":64246,"ariaHidden":70},[69],[39,64248,64250,64254,64305,64308,64311],{"className":64249},[74],[39,64251],{"className":64252,"style":64253},[78],"height:0.9614em;vertical-align:-0.247em;",[39,64255,64257,64260],{"className":64256},[83],[39,64258,4230],{"className":64259,"style":461},[83,84],[39,64261,64263],{"className":64262},[192],[39,64264,64266,64297],{"className":64265},[196,649],[39,64267,64269,64294],{"className":64268},[200],[39,64270,64272,64283],{"className":64271,"style":4421},[204],[39,64273,64274,64277],{"style":9973},[39,64275],{"className":64276,"style":213},[212],[39,64278,64280],{"className":64279},[217,218,219,220],[39,64281,621],{"className":64282},[83,84,220],[39,64284,64285,64288],{"style":1027},[39,64286],{"className":64287,"style":213},[212],[39,64289,64291],{"className":64290},[217,218,219,220],[39,64292,2098],{"className":64293,"style":2114},[83,84,220],[39,64295,673],{"className":64296},[672],[39,64298,64300],{"className":64299},[200],[39,64301,64303],{"className":64302,"style":10003},[204],[39,64304],{},[39,64306],{"className":64307,"style":231},[230],[39,64309,793],{"className":64310},[235],[39,64312],{"className":64313,"style":231},[230],[39,64315,64317,64321,64392,64453,64456,64459,64462,64513,64516,64519],{"className":64316},[74],[39,64318],{"className":64319,"style":64320},[78],"height:1.044em;vertical-align:-0.3053em;",[39,64322,64324,64327],{"className":64323},[83],[39,64325,11772],{"className":64326,"style":11793},[83,641],[39,64328,64330],{"className":64329},[192],[39,64331,64333],{"className":64332},[196],[39,64334,64336],{"className":64335},[200],[39,64337,64339],{"className":64338,"style":4421},[204],[39,64340,64341,64344],{"style":1027},[39,64342],{"className":64343,"style":213},[212],[39,64345,64347],{"className":64346},[217,218,219,220],[39,64348,64350],{"className":64349},[83,220],[39,64351,64353,64356],{"className":64352},[83,220],[39,64354,2098],{"className":64355,"style":2114},[83,84,220],[39,64357,64359],{"className":64358},[192],[39,64360,64362,64383],{"className":64361},[196,649],[39,64363,64365,64380],{"className":64364},[200],[39,64366,64369],{"className":64367,"style":64368},[204],"height:0.2963em;",[39,64370,64371,64374],{"style":42108},[39,64372],{"className":64373,"style":4004},[212],[39,64375,64377],{"className":64376},[217,4008,4009,220],[39,64378,621],{"className":64379},[83,84,220],[39,64381,673],{"className":64382},[672],[39,64384,64386],{"className":64385},[200],[39,64387,64390],{"className":64388,"style":64389},[204],"height:0.143em;",[39,64391],{},[39,64393,64395,64398],{"className":64394},[83],[39,64396,4230],{"className":64397,"style":461},[83,84],[39,64399,64401],{"className":64400},[192],[39,64402,64404,64444],{"className":64403},[196,649],[39,64405,64407,64441],{"className":64406},[200],[39,64408,64410,64430],{"className":64409,"style":4421},[204],[39,64411,64412,64415],{"style":9973},[39,64413],{"className":64414,"style":213},[212],[39,64416,64418],{"className":64417},[217,218,219,220],[39,64419,64421,64424,64427],{"className":64420},[83,220],[39,64422,621],{"className":64423},[83,84,220],[39,64425,788],{"className":64426},[465,220],[39,64428,546],{"className":64429},[83,220],[39,64431,64432,64435],{"style":1027},[39,64433],{"className":64434,"style":213},[212],[39,64436,64438],{"className":64437},[217,218,219,220],[39,64439,2098],{"className":64440,"style":2114},[83,84,220],[39,64442,673],{"className":64443},[672],[39,64445,64447],{"className":64446},[200],[39,64448,64451],{"className":64449,"style":64450},[204],"height:0.3053em;",[39,64452],{},[39,64454,155],{"className":64455},[252],[39,64457],{"className":64458,"style":10787},[230],[39,64460],{"className":64461,"style":256},[230],[39,64463,64465,64468],{"className":64464},[83],[39,64466,4230],{"className":64467,"style":461},[83,84],[39,64469,64471],{"className":64470},[192],[39,64472,64474,64505],{"className":64473},[196,649],[39,64475,64477,64502],{"className":64476},[200],[39,64478,64480,64491],{"className":64479,"style":22242},[204],[39,64481,64482,64485],{"style":9973},[39,64483],{"className":64484,"style":213},[212],[39,64486,64488],{"className":64487},[217,218,219,220],[39,64489,621],{"className":64490},[83,84,220],[39,64492,64493,64496],{"style":1027},[39,64494],{"className":64495,"style":213},[212],[39,64497,64499],{"className":64498},[217,218,219,220],[39,64500,21884],{"className":64501},[465,220],[39,64503,673],{"className":64504},[672],[39,64506,64508],{"className":64507},[200],[39,64509,64511],{"className":64510,"style":10003},[204],[39,64512],{},[39,64514],{"className":64515,"style":231},[230],[39,64517,793],{"className":64518},[235],[39,64520],{"className":64521,"style":231},[230],[39,64523,64525,64528,64557,64617],{"className":64524},[74],[39,64526],{"className":64527,"style":64320},[78],[39,64529,64531,64534],{"className":64530},[83],[39,64532,11772],{"className":64533,"style":11793},[83,641],[39,64535,64537],{"classN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对 ",[39,64626,64628,64641],{"className":64627,"translate":43},[42],[39,64629,64631],{"className":64630},[47],[49,64632,64633],{"xmlns":51},[53,64634,64635,64639],{},[56,64636,64637],{},[59,64638,621],{},[63,64640,621],{"encoding":65},[39,64642,64644],{"className":64643,"ariaHidden":70},[69],[39,64645,64647,64650],{"className":64646},[74],[39,64648],{"className":64649,"style":709},[78],[39,64651,621],{"className":64652},[83,84]," 从 ",[39,64655,64657,64675],{"className":64656,"translate":43},[42],[39,64658,64660],{"className":64659},[47],[49,64661,64662],{"xmlns":51},[53,64663,64664,64672],{},[56,64665,64666,64668,64670],{},[59,64667,11772],{},[133,64669,5378],{},[539,64671,546],{},[63,64673,64674],{"encoding":65},"T-1",[39,64676,64678,64696],{"className":64677,"ariaHidden":70},[69],[39,64679,64681,64684,64687,64690,64693],{"className":64680},[74],[39,64682],{"className":64683,"style":454},[78],[39,64685,11772],{"className":64686,"style":178},[83,84],[39,64688],{"className":64689,"style":461},[230],[39,64691,5378],{"className":64692},[465],[39,64694],{"className":64695,"style":461},[230],[39,64697,64699,64702],{"className":64698},[74],[39,64700],{"className":64701,"style":2473},[78],[39,64703,546],{"className":64704},[83]," 到 ",[39,64707,64709,64722],{"className":64708,"translate":43},[42],[39,64710,64712],{"className":64711},[47],[49,64713,64714],{"xmlns":51},[53,64715,64716,64720],{},[56,64717,64718],{},[539,64719,541],{},[63,64721,541],{"encoding":65},[39,64723,64725],{"className":64724,"ariaHidden":70},[69],[39,64726,64728,64731],{"className":64727},[74],[39,64729],{"className":64730,"style":2473},[78],[39,64732,541],{"className":64733},[83]," 反向归纳。",[39,64736,64738,64756],{"className":64737,"translate":43},[42],[39,64739,64741],{"className":64740},[47],[49,64742,64743],{"xmlns":51},[53,64744,64745,64753],{},[56,64746,64747,64749,64751],{},[59,64748,621],{},[133,64750,793],{},[59,64752,11772],{},[63,64754,64755],{"encoding":65},"t=T",[39,64757,64759,64777],{"className":64758,"ariaHidden":70},[69],[39,64760,64762,64765,64768,64771,64774],{"className":64761},[74],[39,64763],{"className":64764,"style":709},[78],[39,64766,621],{"className":64767},[83,84],[39,64769],{"className":64770,"style":231},[230],[39,64772,793],{"className":64773},[235],[39,64775],{"className":64776,"style":231},[230],[39,64778,64780,64783],{"className":64779},[74],[39,64781],{"className":64782,"style":79},[78],[39,64784,11772],{"className":64785,"style":178},[83,84]," 时两端均为零。设结论对 ",[39,64788,64790,64808],{"className":64789,"translate":43},[42],[39,64791,64793],{"className":64792},[47],[49,64794,64795],{"xmlns":51},[53,64796,64797,64805],{},[56,64798,64799,64801,64803],{},[59,64800,621],{},[133,64802,788],{},[539,64804,546],{},[63,64806,64807],{"encoding":65},"t+1",[39,64809,64811,64830],{"className":64810,"ariaHidden":70},[69],[39,64812,64814,64818,64821,64824,64827],{"className":64813},[74],[39,64815],{"className":64816,"style":64817},[78],"height:0.6984em;vertical-align:-0.0833em;",[39,64819,621],{"className":64820},[83,84],[39,64822],{"className":64823,"style":461},[230],[39,64825,788],{"className":64826},[465],[39,64828],{"className":64829,"style":461},[230],[39,64831,64833,64836],{"className":64832},[74],[39,64834],{"className":64835,"style":2473},[78],[39,64837,546],{"className":64838},[83]," 成立。对任意策略 ",[39,64841,64843,64856],{"className":64842,"translate":43},[42],[39,64844,64846],{"className":64845},[47],[49,64847,64848],{"xmlns":51},[53,64849,64850,64854],{},[56,64851,64852],{},[59,64853,2098],{},[63,64855,2101],{"encoding":65},[39,64857,64859],{"className":64858,"ariaHidden":70},[69],[39,64860,64862,64865],{"className":64861},[74],[39,64863],{"className":64864,"style":290},[78],[39,64866,2098],{"className":64867,"style":2114},[83,84],"，在 ",[39,64870,64872,64894],{"className":64871,"translate":43},[42],[39,64873,64875],{"className":64874},[47],[49,64876,64877],{"xmlns":51},[53,64878,64879,64891],{},[56,64880,64881,64887,64889],{},[613,64882,64883,64885],{},[59,64884,142],{},[59,64886,621],{},[133,64888,793],{},[59,64890,142],{},[63,64892,64893],{"encoding":65},"s_t=s",[39,64895,64897,64952],{"className":64896,"ariaHidden":70},[69],[39,64898,64900,64903,64943,64946,64949],{"className":64899},[74],[39,64901],{"className":64902,"style":2916},[78],[39,64904,64906,64909],{"className":64905},[83],[39,64907,142],{"className":64908},[83,84],[39,64910,64912],{"className":64911},[192],[39,64913,64915,64935],{"className":64914},[196,649],[39,64916,64918,64932],{"className":64917},[200],[39,64919,64921],{"className":64920,"style":656},[204],[39,64922,64923,64926],{"style":966},[39,64924],{"className":64925,"style":213},[212],[39,64927,64929],{"className":64928},[217,218,219,220],[39,64930,621],{"className":64931},[83,84,220],[39,64933,673],{"className":64934},[672],[39,64936,64938],{"className":64937},[200],[39,64939,64941],{"className":64940,"style":680},[204],[39,64942],{},[39,64944],{"className":64945,"style":231},[230],[39,64947,793],{"className":64948},[235],[39,64950],{"className":64951,"style":231},[230],[39,64953,64955,64958],{"className":64954},[74],[39,64956],{"className":64957,"style":290},[78],[39,64959,142],{"className":64960},[83,84]," 处分解回报：",[39,64963,64965],{"className":64964,"translate":43},[746],[39,64966,64968,65095],{"className":64967,"translate":43},[42],[39,64969,64971],{"className":64970},[47],[49,64972,64973],{"xmlns":51,"display":755},[53,64974,64975,65092],{},[759,64976,64977,65058],{"rowspacing":761,"columnalign":762,"columnspacing":146},[764,64978,64979,64999],{},[767,64980,64981],{},[770,64982,64983],{"scriptlevel":541,"displaystyle":70},[56,64984,64985,64993,64995,64997],{},[9932,64986,64987,64989,64991],{},[59,64988,4230],{},[59,64990,621],{},[59,64992,2098],{},[133,64994,136],{"stretchy":135},[59,64996,142],{},[133,64998,161],{"stretchy":135},[767,65000,65001],{},[770,65002,65003],{"scriptlevel":541,"displaystyle":70},[56,65004,65005,65007,65009,65015,65018,65024,65026,65028,65042,65044,65056],{},[56,65006],{},[133,65008,793],{},[613,65010,65011,65013],{},[59,65012,4253],{"mathvariant":421},[59,65014,2098],{},[133,65016,537],{"fence":135,"stretchy":70,"minsize":65017,"maxsize":65017},"1.8em",[613,65019,65020,65022],{},[59,65021,401],{},[59,65023,621],{},[133,65025,788],{},[59,65027,531],{},[9932,65029,65030,65032,65040],{},[59,65031,4230],{},[56,65033,65034,65036,65038],{},[59,65035,621],{},[133,65037,788],{},[539,65039,546],{},[59,65041,2098],{},[133,65043,136],{"stretchy":135},[613,65045,65046,65048],{},[59,65047,142],{},[56,65049,65050,65052,65054],{},[59,65051,621],{},[133,65053,788],{},[539,65055,546],{},[133,65057,161],{"stretchy":135},[764,65059,65060,65066],{},[767,65061,65062],{},[770,65063,65064],{"scriptlevel":541,"displaystyle":70},[56,65065],{},[767,65067,65068],{},[770,65069,65070],{"scriptlevel":541,"displaystyle":70},[56,65071,65072,65074,65076,65078,65084,65086,65088,65090],{},[56,65073],{},[230,65075],{"width":10251},[133,65077,150],{},[613,65079,65080,65082],{},[59,65081,142],{},[59,65083,621],{},[133,65085,793],{},[59,65087,142],{},[133,65089,4280],{"fence":135,"stretchy":70,"minsize":65017,"maxsize":65017},[59,65091,2903],{"mathvariant":145},[63,65093,65094],{"encoding":65},"\\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}",[39,65096,65098],{"className":65097,"ariaHidden":70},[69],[39,65099,65101,65105],{"className":65100},[74],[39,65102],{"className":65103,"style":65104},[78],"height:3.9em;vertical-align:-1.7em;",[39,65106,65108],{"className":65107},[83],[39,65109,65111,65213],{"className":65110},[759],[39,65112,65114],{"className":65113},[918],[39,65115,65117,65204],{"className":65116},[196,649],[39,65118,65120,65201],{"className":65119},[200],[39,65121,65124,65193],{"className":65122,"style":65123},[204],"height:2.2em;",[39,65125,65127,65130],{"style":65126},"top:-4.2em;",[39,65128],{"className":65129,"style":33100},[212],[39,65131,65133,65184,65187,65190],{"className":65132},[83],[39,65134,65136,65139],{"className":65135},[83],[39,65137,4230],{"className":65138,"style":461},[83,84],[39,65140,65142],{"className":65141},[192],[39,65143,65145,65176],{"className":65144},[196,649],[39,65146,65148,65173],{"className":65147},[200],[39,65149,65151,65162],{"className":65150,"style":4421},[204],[39,65152,65153,65156],{"style":9973},[39,65154],{"className":65155,"style":213},[212],[39,65157,65159],{"className":65158},[217,218,219,220],[39,65160,621],{"className":65161},[83,84,220],[39,65163,65164,65167],{"style":1027},[39,65165],{"className":65166,"style":213},[212],[39,65168,65170],{"className":65169},[217,218,219,220],[39,65171,2098],{"className":65172,"style":2114},[83,84,220],[39,65174,673],{"className":65175},[672],[39,65177,65179],{"className":65178},[200],[39,65180,65182],{"className":65181,"style":10003},[204],[39,65183],{},[39,65185,136],{"className":65186},[182],[39,65188,142],{"className":65189},[83,84],[39,65191,161],{"className":65192},[263],[39,65194,65195,65198],{"style":5834},[39,65196],{"className":65197,"style":33100},[212],[39,65199],{"className":65200},[83],[39,65202,673],{"className":65203},[672],[39,65205,65207],{"className":65206},[200],[39,65208,65211],{"className":65209,"style":65210},[204],"height:1.7em;",[39,65212],{},[39,65214,65216],{"className":65215},[1223],[39,65217,65219,65545],{"className":65218},[196,649],[39,65220,65222,65542],{"className":65221},[200],[39,65223,65225,65458],{"className":65224,"style":65123},[204],[39,65226,65227,65230],{"style":65126},[39,65228],{"className":65229,"style":33100},[212],[39,65231,65233,65236,65239,65242,65245,65285,65291,65331,65334,65337,65340,65343,65403,65406,65455],{"className":65232},[83],[39,65234],{"className":65235},[83],[39,65237],{"className":65238,"style":231},[230],[39,65240,793],{"className":65241},[235],[39,65243],{"className":65244,"style":231},[230],[39,65246,65248,65251],{"className":65247},[83],[39,65249,4253],{"className":65250},[83,515],[39,65252,65254],{"className":65253},[192],[39,65255,65257,65277],{"className":65256},[196,649],[39,65258,65260,65274],{"className":65259},[200],[39,65261,65263],{"className":65262,"style":1728},[204],[39,65264,65265,65268],{"style":966},[39,65266],{"className":65267,"style":213},[212],[39,65269,65271],{"className":65270},[217,218,219,220],[39,65272,2098],{"className":65273,"style":2114},[83,84,220],[39,65275,673],{"className":65276},[672],[39,65278,65280],{"className":65279},[200],[39,65281,65283],{"className":65282,"style":680},[204],[39,65284],{},[39,65286,65288],{"className":65287},[83],[39,65289,537],{"className":65290},[18520,33204],[39,65292,65294,65297],{"className":65293},[83],[39,65295,401],{"className":65296,"style":438},[83,84],[39,65298,65300],{"className":65299},[192],[39,65301,65303,65323],{"className":65302},[196,649],[39,65304,65306,65320],{"className":65305},[200],[39,65307,65309],{"className":65308,"style":656},[204],[39,65310,65311,65314],{"style":2937},[39,65312],{"className":65313,"style":213},[212],[39,65315,65317],{"className":65316},[217,218,219,220],[39,65318,621],{"className":65319},[83,84,220],[39,65321,673],{"className":65322},[672],[39,65324,65326],{"className":65325},[200],[39,65327,65329],{"className":65328,"style":680},[204],[39,65330],{},[39,65332],{"className":65333,"style":461},[230],[39,65335,788],{"className":65336},[465],[39,65338],{"className":65339,"style":461},[230],[39,65341,531],{"className":65342,"style":565},[83,84],[39,65344,65346,65349],{"className":65345},[83],[39,65347,4230],{"className":65348,"style":461},[83,84],[39,65350,65352],{"className":65351},[192],[39,65353,65355,65395],{"className":65354},[196,649],[39,65356,65358,65392],{"className":65357},[200],[39,65359,65361,65381],{"className":65360,"style":4421},[204],[39,65362,65363,65366],{"style":9973},[39,65364],{"className":65365,"style":213},[212],[39,65367,65369],{"className":65368},[217,218,219,220],[39,65370,65372,65375,65378],{"className":65371},[83,220],[39,65373,621],{"className":65374},[83,84,220],[39,65376,788],{"className":65377},[465,220],[39,65379,546],{"className":65380},[83,220],[39,65382,65383,65386],{"style":1027},[39,65384],{"className":65385,"style":213},[212],[39,65387,65389],{"className":65388},[217,218,219,220],[39,65390,2098],{"className":65391,"style":2114},[83,84,220],[39,65393,673],{"className":65394},[672],[39,65396,65398],{"className":65397},[200],[39,65399,65401],{"className":65400,"style":64450},[204],[39,65402],{},[39,65404,136],{"className":65405},[182],[39,65407,65409,65412],{"className":65408},[83],[39,65410,142],{"className":65411},[83,84],[39,65413,65415],{"className":65414},[192],[39,65416,65418,65447],{"className":65417},[196,649],[39,65419,65421,65444],{"className":65420},[200],[39,65422,65424],{"className":65423,"style":963},[204],[39,65425,65426,65429],{"style":966},[39,65427],{"className":65428,"style":213},[212],[39,65430,65432],{"className":65431},[217,218,219,220],[39,65433,65435,65438,65441],{"className":65434},[83,220],[39,65436,621],{"className":65437},[83,84,220],[39,65439,788],{"className":65440},[465,220],[39,65442,546],{"className":65443},[83,220],[39,65445,673],{"className":65446},[672],[39,65448,65450],{"className":65449},[200],[39,65451,65453],{"className":65452,"style":994},[204],[39,65454],{},[39,65456,161],{"className":65457},[263],[39,65459,65460,65463],{"style":5834},[39,65461],{"className":65462,"style":33100},[212],[39,65464,65466,65469,65472,65475,65478,65481,65521,65524,65527,65530,65533,65539],{"className":65465},[83],[39,65467],{"className":65468},[83],[39,65470],{"className":65471,"style":10787},[230],[39,65473],{"className":65474,"style":231},[230],[39,65476,150],{"className":65477},[235],[39,65479],{"className":65480,"style":231},[230],[39,65482,65484,65487],{"className":65483},[83],[39,65485,142],{"className":65486},[83,84],[39,65488,65490],{"className":65489},[192],[39,65491,65493,65513],{"className":65492},[196,649],[39,65494,65496,65510],{"className":65495},[200],[39,65497,65499],{"className":65498,"style":656},[204],[39,65500,65501,65504],{"style":966},[39,65502],{"className":65503,"style":213},[212],[39,65505,65507],{"className":65506},[217,218,219,220],[39,65508,621],{"className":65509},[83,84,220],[39,65511,673],{"className":65512},[672],[39,65514,65516],{"className":65515},[200],[39,65517,65519],{"className":65518,"style":680},[204],[39,65520],{},[39,65522],{"className":65523,"style":231},[230],[39,65525,793],{"className":65526},[235],[39,65528],{"className":65529,"style":231},[230],[39,65531,142],{"className":65532},[83,84],[39,65534,65536],{"className":65535},[83],[39,65537,4280],{"className":65538},[18520,33204],[39,65540,2903],{"className":65541},[83],[39,65543,673],{"className":65544},[672],[39,65546,65548],{"className":65547},[200],[39,65549,65551],{"className":65550,"style":65210},[204],[39,65552],{},[11,65554,65555,65556,65657,65658,49416],{},"由 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的定义，",[39,65659,65661,65703],{"className":65660,"translate":43},[42],[39,65662,65664],{"className":65663},[47],[49,65665,65666],{"xmlns":51},[53,65667,65668,65700],{},[56,65669,65670,65684,65686],{},[9932,65671,65672,65674,65682],{},[59,65673,4230],{},[56,65675,65676,65678,65680],{},[59,65677,621],{},[133,65679,788],{},[539,65681,546],{},[59,65683,2098],{},[133,65685,1824],{},[9932,65687,65688,65690,65698],{},[59,65689,4230],{},[56,65691,65692,65694,65696],{},[59,65693,621],{},[133,65695,788],{},[539,65697,546],{},[133,65699,21884],{},[63,65701,65702],{"encoding":65},"V_{t+1}^\\pi\\le 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时刻可达的最优值。取 ",[39,67277,67279,67297],{"className":67278,"translate":43},[42],[39,67280,67282],{"className":67281},[47],[49,67283,67284],{"xmlns":51},[53,67285,67286,67294],{},[56,67287,67288],{},[613,67289,67290,67292],{},[59,67291,158],{},[59,67293,621],{},[63,67295,67296],{"encoding":65},"a_t",[39,67298,67300],{"className":67299,"ariaHidden":70},[69],[39,67301,67303,67306],{"className":67302},[74],[39,67304],{"className":67305,"style":2916},[78],[39,67307,67309,67312],{"className":67308},[83],[39,67310,158],{"className":67311},[83,84],[39,67313,67315],{"className":67314},[192],[39,67316,67318,67338],{"className":67317},[196,649],[39,67319,67321,67335],{"className":67320},[200],[39,67322,67324],{"className":67323,"style":656},[204],[39,67325,67326,67329],{"style":966},[39,67327],{"className":67328,"style":213},[212],[39,67330,67332],{"className":67331},[217,218,219,220],[39,67333,621],{"className":67334},[83,84,220],[39,67336,673],{"className":67337},[672],[39,67339,67341],{"className":67340},[200],[39,67342,67344],{"className":67343,"style":680},[204],[39,67345],{}," 为使上式右端最大的动作，并从 ",[39,67348,67350,67367],{"className":67349,"translate":43},[42],[39,67351,67353],{"className":67352},[47],[49,67354,67355],{"xmlns":51},[53,67356,67357,67365],{},[56,67358,67359,67361,67363],{},[59,67360,621],{},[133,67362,788],{},[539,67364,546],{},[63,67366,64807],{"encoding":65},[39,67368,67370,67388],{"className":67369,"ariaHidden":70},[69],[39,67371,67373,67376,67379,67382,67385],{"className":67372},[74],[39,67374],{"className":67375,"style":64817},[78],[39,67377,621],{"className":67378},[83,84],[39,67380],{"className":67381,"style":461},[230],[39,67383,788],{"className":67384},[465],[39,67386],{"className":67387,"style":461},[230],[39,67389,67391,67394],{"className":67390},[74],[39,67392],{"className":67393,"style":2473},[78],[39,67395,546],{"className":67396},[83]," 开始采用达到 ",[39,67399,67401,67426],{"className":67400,"translate":43},[42],[39,67402,67404],{"className":67403},[47],[49,67405,67406],{"xmlns":51},[53,67407,67408,67424],{},[56,67409,67410],{},[9932,67411,67412,67414,67422],{},[59,67413,4230],{},[56,67415,67416,67418,67420],{},[59,67417,621],{},[133,67419,788],{},[539,67421,546],{},[133,67423,21884],{},[63,67425,65584],{"encoding":65},[39,67427,67429],{"className":67428,"ariaHidden":70},[69],[39,67430,67432,67435],{"className":67431},[74],[39,67433],{"className":67434,"style":65594},[78],[39,67436,67438,67441],{"className":67437},[83],[39,67439,4230],{"className":67440,"style":461},[83,84],[39,67442,67444],{"className":67443},[192],[39,67445,67447,67487],{"className":67446},[196,649],[39,67448,67450,67484],{"className":67449},[200],[39,67451,67453,67473],{"className":67452,"style":21897},[204],[39,67454,67455,67458],{"style":65615},[39,67456],{"className":67457,"style":213},[212],[39,67459,67461],{"className":67460},[217,218,219,220],[39,67462,67464,67467,67470],{"className":67463},[83,220],[39,67465,621],{"className":67466},[83,84,220],[39,67468,788],{"className":67469},[465,220],[39,67471,546],{"className":67472},[83,220],[39,67474,67475,67478],{"style":208},[39,67476],{"className":67477,"style":213},[212],[39,67479,67481],{"className":67480},[217,218,219,220],[39,67482,21884],{"className":67483},[465,220],[39,67485,673],{"className":67486},[672],[39,67488,67490],{"className":67489},[200],[39,67491,67493],{"className":67492,"style":65654},[204],[39,67494],{}," 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 ",[39,67497,67499,67541],{"className":67498,"translate":43},[42],[39,67500,67502],{"className":67501},[47],[49,67503,67504],{"xmlns":51},[53,67505,67506,67538],{},[56,67507,67508,67516,67518,67524],{},[9932,67509,67510,67512,67514],{},[59,67511,4230],{},[59,67513,621],{},[133,67515,21884],{},[133,67517,793],{},[138,67519,67520,67522],{},[59,67521,11772],{"mathvariant":617},[133,67523,21884],{},[9932,67525,67526,67528,67536],{},[59,67527,4230],{},[56,67529,67530,67532,67534],{},[59,67531,621],{},[133,67533,788],{},[539,67535,546],{},[133,67537,21884],{},[63,67539,67540],{"encoding":65},"V_t^*=\\mathcal{T}^*V_{t+1}^*",[39,67542,67544,67611],{"className":67543,"ariaHidden":70},[69],[39,67545,67547,67551,67602,67605,67608],{"className":67546},[74],[39,67548],{"className":67549,"style":67550},[78],"height:0.9357em;vertical-align:-0.247em;",[39,67552,67554,67557],{"className":67553},[83],[39,67555,4230],{"className":67556,"style":461},[83,84],[39,67558,67560],{"className":67559},[192],[39,67561,67563,67594],{"className":67562},[196,649],[39,67564,67566,67591],{"className":67565},[200],[39,67567,67569,67580],{"className":67568,"style":21897},[204],[39,67570,67571,67574],{"style":9973},[39,67572],{"className":67573,"style":213},[212],[39,67575,67577],{"className":67576},[217,218,219,220],[39,67578,621],{"className":67579},[83,84,220],[39,67581,67582,67585],{"style":208},[39,67583],{"className":67584,"style":213},[212],[39,67586,67588],{"className":67587},[217,218,219,220],[39,67589,21884],{"className":67590},[465,220],[39,67592,673],{"className":67593},[672],[39,67595,67597],{"className":67596},[200],[39,67598,67600],{"className":67599,"style":10003},[204],[39,67601],{},[39,67603],{"className":67604,"style":231},[230],[39,67606,793],{"className":67607},[235],[39,67609],{"className":67610,"style":231},[230],[39,67612,67614,67617,67646],{"className":67613},[74],[39,67615],{"className":67616,"style":65594},[78],[39,67618,67620,67623],{"className":67619},[83],[39,67621,11772],{"className":67622,"style":11793},[83,641],[39,67624,67626],{"className":67625},[192],[39,67627,67629],{"className":67628},[196],[39,67630,67632],{"className":67631},[200],[39,67633,67635],{"className":67634,"style":21897},[204],[39,67636,67637,67640],{"style":208},[39,67638],{"className":67639,"style":213},[212],[39,67641,67643],{"className":67642},[217,218,219,220],[39,67644,21884],{"className":67645},[465,220],[39,67647,67649,67652],{"className":67648},[83],[39,67650,4230],{"className":67651,"style":461},[83,84],[39,67653,67655],{"className":67654},[192],[39,67656,67658,67698],{"className":67657},[196,649],[39,67659,67661,67695],{"className":67660},[200],[39,67662,67664,67684],{"className":67663,"style":21897},[204],[39,67665,67666,67669],{"style":65615},[39,67667],{"className":67668,"style":213},[212],[39,67670,67672],{"className":67671},[217,218,219,220],[39,67673,67675,67678,67681],{"className":67674},[83,220],[39,67676,621],{"className":67677},[83,84,220],[39,67679,788],{"className":67680},[465,220],[39,67682,546],{"className":67683},[83,220],[39,67685,67686,67689],{"style":208},[39,67687],{"className":67688,"style":213},[212],[39,67690,67692],{"className":67691},[217,218,219,220],[39,67693,21884],{"className":67694},[465,220],[39,67696,673],{"className":67697},[672],[39,67699,67701],{"className":67700},[200],[39,67702,67704],{"className":67703,"style":65654},[204],[39,67705],{},"。策略评估式同理。证毕。",[11,67708,67709,67710,67738,67739,67836,67837,67921,67922,68006],{},"这个定理就是反向归纳的数学形式。它说明，在时刻 ",[39,67711,67713,67726],{"className":67712,"translate":43},[42],[39,67714,67716],{"className":67715},[47],[49,67717,67718],{"xmlns":51},[53,67719,67720,67724],{},[56,67721,67722],{},[59,67723,621],{},[63,67725,621],{"encoding":65},[39,67727,67729],{"className":67728,"ariaHidden":70},[69],[39,67730,67732,67735],{"className":67731},[74],[39,67733],{"className":67734,"style":709},[78],[39,67736,621],{"className":67737},[83,84],"，最优决策只需要知道当前状态与下一时刻的最优值函数 ",[39,67740,67742,67767],{"className":67741,"translate":43},[42],[39,67743,67745],{"className":67744},[47],[49,67746,67747],{"xmlns":51},[53,67748,67749,67765],{},[56,67750,67751],{},[9932,67752,67753,67755,67763],{},[59,67754,4230],{},[56,67756,67757,67759,67761],{},[59,67758,621],{},[133,67760,788],{},[539,67762,546],{},[133,67764,21884],{},[63,67766,65584],{"encoding":65},[39,67768,67770],{"className":67769,"ariaHidden":70},[69],[39,67771,67773,67776],{"className":67772},[74],[39,67774],{"className":67775,"style":65594},[78],[39,67777,67779,67782],{"className":67778},[83],[39,67780,4230],{"className":67781,"style":461},[83,84],[39,67783,67785],{"className":67784},[192],[39,67786,67788,67828],{"className":67787},[196,649],[39,67789,67791,67825],{"className":67790},[200],[39,67792,67794,67814],{"className":67793,"style":21897},[204],[39,67795,67796,67799],{"style":65615},[39,67797],{"className":67798,"style":213},[212],[39,67800,67802],{"className":67801},[217,218,219,220],[39,67803,67805,67808,67811],{"className":67804},[83,220],[39,67806,621],{"className":67807},[83,84,220],[39,67809,788],{"className":67810},[465,220],[39,67812,546],{"className":67813},[83,220],[39,67815,67816,67819],{"style":208},[39,67817],{"className":67818,"style":213},[212],[39,67820,67822],{"className":67821},[217,218,219,220],[39,67823,21884],{"className":67824},[465,220],[39,67826,673],{"className":67827},[672],[39,67829,67831],{"className":67830},[200],[39,67832,67834],{"className":67833,"style":65654},[204],[39,67835],{},"。一旦后继状态 ",[39,67838,67840,67863],{"className":67839,"translate":43},[42],[39,67841,67843],{"className":67842},[47],[49,67844,67845],{"xmlns":51},[53,67846,67847,67861],{},[56,67848,67849],{},[613,67850,67851,67853],{},[59,67852,142],{},[56,67854,67855,67857,67859],{},[59,67856,621],{},[133,67858,788],{},[539,67860,546],{},[63,67862,3625],{"encoding":65},[39,67864,67866],{"className":67865,"ariaHidden":70},[69],[39,67867,67869,67872],{"className":67868},[74],[39,67870],{"className":67871,"style":3635},[78],[39,67873,67875,67878],{"className":67874},[83],[39,67876,142],{"className":67877},[83,84],[39,67879,67881],{"className":67880},[192],[39,67882,67884,67913],{"className":67883},[196,649],[39,67885,67887,67910],{"className":67886},[200],[39,67888,67890],{"className":67889,"style":963},[204],[39,67891,67892,67895],{"style":966},[39,67893],{"className":67894,"style":213},[212],[39,67896,67898],{"className":67897},[217,218,219,220],[39,67899,67901,67904,67907],{"className":67900},[83,220],[39,67902,621],{"className":67903},[83,84,220],[39,67905,788],{"className":67906},[465,220],[39,67908,546],{"className":67909},[83,220],[39,67911,673],{"className":67912},[672],[39,67914,67916],{"className":67915},[200],[39,67917,67919],{"className":67918,"style":994},[204],[39,67920],{}," 确定，从 ",[39,67923,67925,67948],{"className":67924,"translate":43},[42],[39,67926,67928],{"className":67927},[47],[49,67929,67930],{"xmlns":51},[53,67931,67932,67946],{},[56,67933,67934],{},[613,67935,67936,67938],{},[59,67937,142],{},[56,67939,67940,67942,67944],{},[59,67941,621],{},[133,67943,788],{},[539,67945,546],{},[63,67947,3625],{"encoding":65},[39,67949,67951],{"className":67950,"ariaHidden":70},[69],[39,67952,67954,67957],{"className":67953},[74],[39,67955],{"className":67956,"style":3635},[78],[39,67958,67960,67963],{"className":67959},[83],[39,67961,142],{"className":67962},[83,84],[39,67964,67966],{"className":67965},[192],[39,67967,67969,67998],{"className":67968},[196,649],[39,67970,67972,67995],{"className":67971},[200],[39,67973,67975],{"className":67974,"style":963},[204],[39,67976,67977,67980],{"style":966},[39,67978],{"className":67979,"style":213},[212],[39,67981,67983],{"className":67982},[217,218,219,220],[39,67984,67986,67989,67992],{"className":67985},[83,220],[39,67987,621],{"className":67988},[83,84,220],[39,67990,788],{"className":67991},[465,220],[39,67993,546],{"className":67994},[83,220],[39,67996,673],{"className":67997},[672],[39,67999,68001],{"className":68000},[200],[39,68002,68004],{"className":68003,"style":994},[204],[39,68005],{}," 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。",[26,68008,68010],{"id":68009},"_62-无限期情形","6.2 无限期情形",[11,68012,68013,68014,68016,68017,68075],{},"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。",[158,68015,37252],{"href":37251}," 表明，无限期最优贝尔曼方程就是 ",[39,68018,68020,68037],{"className":68019,"translate":43},[42],[39,68021,68023],{"className":68022},[47],[49,68024,68025],{"xmlns":51},[53,68026,68027,68035],{},[56,68028,68029],{},[138,68030,68031,68033],{},[59,68032,11772],{"mathvariant":617},[133,68034,21884],{},[63,68036,21887],{"encoding":65},[39,68038,68040],{"className":68039,"ariaHidden":70},[69],[39,68041,68043,68046],{"className":68042},[74],[39,68044],{"className":68045,"style":21897},[78],[39,68047,68049,68052],{"className":68048},[83],[39,68050,11772],{"className":68051,"style":11793},[83,641],[39,68053,68055],{"className":68054},[192],[39,68056,68058],{"className":68057},[196],[39,68059,68061],{"className":68060},[200],[39,68062,68064],{"className":68063,"style":21897},[204],[39,68065,68066,68069],{"style":208},[39,68067],{"className":68068,"style":213},[212],[39,68070,68072],{"className":68071},[217,218,219,220],[39,68073,21884],{"className":68074},[465,220]," 的不动点方程，而最优值函数正是该方程的唯一有界解。",[11,68077,68078,68079,68294],{},"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 ",[39,68080,68082,68123],{"className":68081,"translate":43},[42],[39,68083,68085],{"className":68084},[47],[49,68086,68087],{"xmlns":51},[53,68088,68089,68121],{},[56,68090,68091,68097,68099,68105,68107,68109],{},[613,68092,68093,68095],{},[59,68094,3181],{},[59,68096,621],{},[133,68098,793],{},[613,68100,68101,68103],{},[59,68102,401],{},[59,68104,621],{},[133,68106,788],{},[59,68108,531],{},[613,68110,68111,68113],{},[59,68112,3181],{},[56,68114,68115,68117,68119],{},[59,68116,621],{},[133,68118,788],{},[539,68120,546],{},[63,68122,7463],{"encoding":65},[39,68124,68126,68181,68236],{"className":68125,"ariaHidden":70},[69],[39,68127,68129,68132,68172,68175,68178],{"className":68128},[74],[39,68130],{"className":68131,"style":634},[78],[39,68133,68135,68138],{"className":68134},[83],[39,68136,3181],{"className":68137},[83,84],[39,68139,68141],{"className":68140},[192],[39,68142,68144,68164],{"className":68143},[196,649],[39,68145,68147,68161],{"className":68146},[200],[39,68148,68150],{"className":68149,"style":656},[204],[39,68151,68152,68155],{"style":966},[39,68153],{"className":68154,"style":213},[212],[39,68156,68158],{"className":68157},[217,218,219,220],[39,68159,621],{"className":68160},[83,84,220],[39,68162,673],{"className":68163},[672],[39,68165,68167],{"className":68166},[200],[39,68168,68170],{"className":68169,"style":680},[204],[39,68171],{},[39,68173],{"className":68174,"style":231},[230],[39,68176,793],{"className":68177},[235],[39,68179],{"className":68180,"style":231},[230],[39,68182,68184,68187,68227,68230,68233],{"className":68183},[74],[39,68185],{"className":68186,"style":7528},[78],[39,68188,68190,68193],{"className":68189},[83],[39,68191,401],{"className":68192,"style":438},[83,84],[39,68194,68196],{"className":68195},[192],[39,68197,68199,68219],{"className":68198},[196,649],[39,68200,68202,68216],{"className":68201},[200],[39,68203,68205],{"className":68204,"style":656},[204],[39,68206,68207,68210],{"style":2937},[39,68208],{"className":68209,"style":213},[212],[39,68211,68213],{"className":68212},[217,218,219,220],[39,68214,621],{"className":68215},[83,84,220],[39,68217,673],{"className":68218},[672],[39,68220,68222],{"className":68221},[200],[39,68223,68225],{"className":68224,"style":680},[204],[39,68226],{},[39,68228],{"className":68229,"style":461},[230],[39,68231,788],{"className":68232},[465],[39,68234],{"className":68235,"style":461},[230],[39,68237,68239,68242,68245],{"className":68238},[74],[39,68240],{"className":68241,"style":7584},[78],[39,68243,531],{"className":68244,"style":565},[83,84],[39,68246,68248,68251],{"className":68247},[83],[39,68249,3181],{"className":68250},[83,84],[39,68252,68254],{"className":68253},[192],[39,68255,68257,68286],{"className":68256},[196,649],[39,68258,68260,68283],{"className":68259},[200],[39,68261,68263],{"className":68262,"style":963},[204],[39,68264,68265,68268],{"style":966},[39,68266],{"className":68267,"style":213},[212],[39,68269,68271],{"className":68270},[217,218,219,220],[39,68272,68274,68277,68280],{"className":68273},[83,220],[39,68275,621],{"className":68276},[83,84,220],[39,68278,788],{"className":68279},[465,220],[39,68281,546],{"className":68282},[83,220],[39,68284,673],{"className":68285},[672],[39,68287,68289],{"className":68288},[200],[39,68290,68292],{"className":68291,"style":994},[204],[39,68293],{},"。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。",[26,68296,68298],{"id":68297},"_63-成立条件","6.3 成立条件",[11,68300,68301],{},"最优子结构的成立依赖以下三个条件。",[11,68303,68304,68305,68505,68506,68721],{},"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 ",[39,68306,68308,68352],{"className":68307,"translate":43},[42],[39,68309,68311],{"className":68310},[47],[49,68312,68313],{"xmlns":51},[53,68314,68315,68349],{},[56,68316,68317,68331,68337],{},[9932,68318,68319,68321,68329],{},[133,68320,3191],{},[56,68322,68323,68325,68327],{},[59,68324,3196],{},[133,68326,793],{},[539,68328,541],{},[59,68330,1696],{"mathvariant":145},[138,68332,68333,68335],{},[59,68334,531],{},[59,68336,3196],{},[613,68338,68339,68341],{},[59,68340,401],{},[56,68342,68343,68345,68347],{},[59,68344,621],{},[133,68346,788],{},[59,68348,3196],{},[63,68350,68351],{"encoding":65},"\\sum_{k=0}^{\\infty}\\gamma^k r_{t+k}",[39,68353,68355],{"className":68354,"ariaHidden":70},[69],[39,68356,68358,68361,68424,68427,68456],{"className":68357},[74],[39,68359],{"className":68360,"style":15719},[78],[39,68362,68364,68367],{"className":68363},[1744],[39,68365,3191],{"className":68366,"style":3953},[1744,3339,3952],[39,68368,68370],{"className":68369},[192],[39,68371,68373,68416],{"className":68372},[196,649],[39,68374,68376,68413],{"className":68375},[200],[39,68377,68379,68399],{"className":68378,"style":15738},[204],[39,68380,68381,68384],{"style":3969},[39,68382],{"className":68383,"style":213},[212],[39,68385,68387],{"className":68386},[217,218,219,220],[39,68388,68390,68393,68396],{"className":68389},[83,220],[39,68391,3196],{"className":68392,"style":3321},[83,84,220],[39,68394,793],{"className":68395},[235,220],[39,68397,541],{"className":68398},[83,220],[39,68400,68401,68404],{"style":15761},[39,68402],{"className":68403,"style":213},[212],[39,68405,68407],{"className":68406},[217,218,219,220],[39,68408,68410],{"className":68409},[83,220],[39,68411,1696],{"className":68412},[83,220],[39,68414,673],{"className":68415},[672],[39,68417,68419],{"className":68418},[200],[39,68420,68422],{"className":68421,"style":4025},[204],[39,68423],{},[39,68425],{"className":68426,"style":256},[230],[39,68428,68430,68433],{"className":68429},[83],[39,68431,531],{"className":68432,"style":565},[83,84],[39,68434,68436],{"className":68435},[192],[39,68437,68439],{"className":68438},[196],[39,68440,68442],{"className":68441},[200],[39,68443,68445],{"className":68444,"style":15577},[204],[39,68446,68447,68450],{"style":208},[39,68448],{"className":68449,"style":213},[212],[39,68451,68453],{"className":68452},[217,218,219,220],[39,68454,3196],{"className":68455,"style":3321},[83,84,220],[39,68457,68459,68462],{"className":68458},[83],[39,68460,401],{"className":68461,"style":438},[83,84],[39,68463,68465],{"className":68464},[192],[39,68466,68468,68497],{"className":68467},[196,649],[39,68469,68471,68494],{"className":68470},[200],[39,68472,68474],{"className":68473,"style":3419},[204],[39,68475,68476,68479],{"style":2937},[39,68477],{"className":68478,"style":213},[212],[39,68480,68482],{"className":68481},[217,218,219,220],[39,68483,68485,68488,68491],{"className":68484},[83,220],[39,68486,621],{"className":68487},[83,84,220],[39,68489,788],{"className":68490},[465,220],[39,68492,3196],{"className":68493,"style":3321},[83,84,220],[39,68495,673],{"className":68496},[672],[39,68498,68500],{"className":68499},[200],[39,68501,68503],{"className":68502,"style":994},[204],[39,68504],{},"。这个形式保证了 ",[39,68507,68509,68550],{"className":68508,"translate":43},[42],[39,68510,68512],{"className":68511},[47],[49,68513,68514],{"xmlns":51},[53,68515,68516,68548],{},[56,68517,68518,68524,68526,68532,68534,68536],{},[613,68519,68520,68522],{},[59,68521,3181],{},[59,68523,621],{},[133,68525,793],{},[613,68527,68528,68530],{},[59,68529,401],{},[59,68531,621],{},[133,68533,788],{},[59,68535,531],{},[613,68537,68538,68540],{},[59,68539,3181],{},[56,68541,68542,68544,68546],{},[59,68543,621],{},[133,68545,788],{},[539,68547,546],{},[63,68549,7463],{"encoding":65},[39,68551,68553,68608,68663],{"className":68552,"ariaHidden":70},[69],[39,68554,68556,68559,68599,68602,68605],{"className":68555},[74],[39,68557],{"className":68558,"style":634},[78],[39,68560,68562,68565],{"className":68561},[83],[39,68563,3181],{"className":68564},[83,84],[39,68566,68568],{"className":68567},[192],[39,68569,68571,68591],{"className":68570},[196,649],[39,68572,68574,68588],{"className":68573},[200],[39,68575,68577],{"className":68576,"style":656},[204],[39,68578,68579,68582],{"style":966},[39,68580],{"className":68581,"style":213},[212],[39,68583,68585],{"className":68584},[217,218,219,220],[39,68586,621],{"className":68587},[83,84,220],[39,68589,673],{"className":68590},[672],[39,68592,68594],{"className":68593},[200],[39,68595,68597],{"className":68596,"style":680},[204],[39,68598],{},[39,68600],{"className":68601,"style":231},[230],[39,68603,793],{"className":68604},[235],[39,68606],{"className":68607,"style":231},[230],[39,68609,68611,68614,68654,68657,68660],{"className":68610},[74],[39,68612],{"className":68613,"style":7528},[78],[39,68615,68617,68620],{"className":68616},[83],[39,68618,401],{"className":68619,"style":438},[83,84],[39,68621,68623],{"className":68622},[192],[39,68624,68626,68646],{"className":68625},[196,649],[39,68627,68629,68643],{"className":68628},[200],[39,68630,68632],{"className":68631,"style":656},[204],[39,68633,68634,68637],{"style":2937},[39,68635],{"className":68636,"style":213},[212],[39,68638,68640],{"className":68639},[217,218,219,220],[39,68641,621],{"className":68642},[83,84,220],[39,68644,673],{"className":68645},[672],[39,68647,68649],{"className":68648},[200],[39,68650,68652],{"className":68651,"style":680},[204],[39,68653],{},[39,68655],{"className":68656,"style":461},[230],[39,68658,788],{"className":68659},[465],[39,68661],{"className":68662,"style":461},[230],[39,68664,68666,68669,68672],{"className":68665},[74],[39,68667],{"className":68668,"style":7584},[78],[39,68670,531],{"className":68671,"style":565},[83,84],[39,68673,68675,68678],{"className":68674},[83],[39,68676,3181],{"className":68677},[83,84],[39,68679,68681],{"className":68680},[192],[39,68682,68684,68713],{"className":68683},[196,649],[39,68685,68687,68710],{"className":68686},[200],[39,68688,68690],{"className":68689,"style":963},[204],[39,68691,68692,68695],{"style":966},[39,68693],{"className":68694,"style":213},[212],[39,68696,68698],{"className":68697},[217,218,219,220],[39,68699,68701,68704,68707],{"className":68700},[83,220],[39,68702,621],{"className":68703},[83,84,220],[39,68705,788],{"className":68706},[465,220],[39,68708,546],{"className":68709},[83,220],[39,68711,673],{"className":68712},[672],[39,68714,68716],{"className":68715},[200],[39,68717,68719],{"className":68718,"style":994},[204],[39,68720],{},"。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。",[11,68723,68724,68725,68794],{},"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 ",[39,68726,68728,68745],{"className":68727,"translate":43},[42],[39,68729,68731],{"className":68730},[47],[49,68732,68733],{"xmlns":51},[53,68734,68735,68743],{},[56,68736,68737],{},[613,68738,68739,68741],{},[59,68740,142],{},[59,68742,621],{},[63,68744,3548],{"encoding":65},[39,68746,68748],{"className":68747,"ariaHidden":70},[69],[39,68749,68751,68754],{"className":68750},[74],[39,68752],{"className":68753,"style":2916},[78],[39,68755,68757,68760],{"className":68756},[83],[39,68758,142],{"className":68759},[83,84],[39,68761,68763],{"className":68762},[192],[39,68764,68766,68786],{"className":68765},[196,649],[39,68767,68769,68783],{"className":68768},[200],[39,68770,68772],{"className":68771,"style":656},[204],[39,68773,68774,68777],{"style":966},[39,68775],{"className":68776,"style":213},[212],[39,68778,68780],{"className":68779},[217,218,219,220],[39,68781,621],{"className":68782},[83,84,220],[39,68784,673],{"className":68785},[672],[39,68787,68789],{"className":68788},[200],[39,68790,68792],{"className":68791,"style":680},[204],[39,68793],{}," 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。",[11,68796,68797,68798,68848,68849,68899],{},"第三，回报必须有界，或者折扣必须足够强。有限期问题即使 ",[39,68799,68801,68818],{"className":68800,"translate":43},[42],[39,68802,68804],{"className":68803},[47],[49,68805,68806],{"xmlns":51},[53,68807,68808,68816],{},[56,68809,68810,68812,68814],{},[59,68811,531],{},[133,68813,793],{},[539,68815,546],{},[63,68817,28990],{"encoding":65},[39,68819,68821,68839],{"className":68820,"ariaHidden":70},[69],[39,68822,68824,68827,68830,68833,68836],{"className":68823},[74],[39,68825],{"className":68826,"style":2654},[78],[39,68828,531],{"className":68829,"style":565},[83,84],[39,68831],{"className":68832,"style":231},[230],[39,68834,793],{"className":68835},[235],[39,68837],{"className":68838,"style":231},[230],[39,68840,68842,68845],{"className":68841},[74],[39,68843],{"className":68844,"style":2473},[78],[39,68846,546],{"className":68847},[83]," 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 ",[39,68850,68852,68869],{"className":68851,"translate":43},[42],[39,68853,68855],{"className":68854},[47],[49,68856,68857],{"xmlns":51},[53,68858,68859,68867],{},[56,68860,68861,68863,68865],{},[59,68862,531],{},[133,68864,1693],{},[539,68866,546],{},[63,68868,17296],{"encoding":65},[39,68870,68872,68890],{"className":68871,"ariaHidden":70},[69],[39,68873,68875,68878,68881,68884,68887],{"className":68874},[74],[39,68876],{"className":68877,"style":561},[78],[39,68879,531],{"className":68880,"style":565},[83,84],[39,68882],{"className":68883,"style":231},[230],[39,68885,1693],{"className":68886},[235],[39,68888],{"className":68889,"style":231},[230],[39,68891,68893,68896],{"className":68892},[74],[39,68894],{"className":68895,"style":2473},[78],[39,68897,546],{"className":68898},[83],"，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。",[11,68901,68902],{},"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。",[21,68904,68906],{"id":68905},"七边界与常见误区","七、边界与常见误区",[26,68908,68910],{"id":68909},"_71-连续状态空间","7.1 连续状态空间",[11,68912,68913,68914,68957],{},"在标准的可测性假设下，连续状态空间中的 ",[39,68915,68917,68936],{"className":68916,"translate":43},[42],[39,68918,68920],{"className":68919},[47],[49,68921,68922],{"xmlns":51},[53,68923,68924,68934],{},[56,68925,68926,68928,68930,68932],{},[59,68927,19772],{},[133,68929,136],{"stretchy":135},[59,68931,61],{},[133,68933,161],{"stretchy":135},[63,68935,21380],{"encoding":65},[39,68937,68939],{"className":68938,"ariaHidden":70},[69],[39,68940,68942,68945,68948,68951,68954],{"className":68941},[74],[39,68943],{"className":68944,"style":245},[78],[39,68946,19772],{"className":68947,"style":19812},[83,84],[39,68949,136],{"className":68950},[182],[39,68952,61],{"className":68953,"style":85},[83,84],[39,68955,161],{"className":68956},[263]," 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。",[11,68959,68960,68961,69019],{},"变化主要发生在 ",[39,68962,68964,68981],{"className":68963,"translate":43},[42],[39,68965,68967],{"className":68966},[47],[49,68968,68969],{"xmlns":51},[53,68970,68971,68979],{},[56,68972,68973],{},[138,68974,68975,68977],{},[59,68976,11772],{"mathvariant":617},[133,68978,21884],{},[63,68980,21887],{"encoding":65},[39,68982,68984],{"className":68983,"ariaHidden":70},[69],[39,68985,68987,68990],{"className":68986},[74],[39,68988],{"className":68989,"style":21897},[78],[39,68991,68993,68996],{"className":68992},[83],[39,68994,11772],{"className":68995,"style":11793},[83,641],[39,68997,68999],{"className":68998},[192],[39,69000,69002],{"className":69001},[196],[39,69003,69005],{"className":69004},[200],[39,69006,69008],{"className":69007,"style":21897},[204],[39,69009,69010,69013],{"style":208},[39,69011],{"className":69012,"style":213},[212],[39,69014,69016],{"className":69015},[217,218,219,220],[39,69017,21884],{"className":69018},[465,220]," 的定义上。此时最大值未必取到，应将其写成",[39,69021,69023],{"className":69022,"translate":43},[746],[39,69024,69026,69177],{"className":69025,"translate":43},[42],[39,69027,69029],{"className":69028},[47],[49,69030,69031],{"xmlns":51,"display":755},[53,69032,69033,69174],{},[759,69034,69035,69112],{"rowspacing":761,"columnalign":762,"columnspacing":146},[764,69036,69037,69061],{},[767,69038,69039],{},[770,69040,69041],{"scriptlevel":541,"displaystyle":70},[56,69042,69043,69045,69051,69053,69055,69057,69059],{},[133,69044,136],{"stretchy":135},[138,69046,69047,69049],{},[59,69048,11772],{"mathvariant":617},[133,69050,21884],{},[59,69052,4230],{},[133,69054,161],{"stretchy":135},[133,69056,136],{"stretchy":135},[59,69058,142],{},[133,69060,161],{"stretchy":135},[767,69062,69063],{},[770,69064,69065],{"scriptlevel":541,"displaystyle":70},[56,69066,69067,69069,69071,69081,69083,69090,69092,69094,69096,69098,69100,69102,69104,69110],{},[56,69068],{},[133,69070,793],{},[6556,69072,69073,69079],{},[56,69074,69075,69077],{},[59,69076,21465],{},[133,69078,1690],{},[59,69080,158],{},[133,69082,12951],{"fence":135,"stretchy":70,"minsize":63111,"maxsize":63111},[613,69084,69085,69088],{},[133,69086,69087],{},"∫",[59,69089,61],{},[133,69091,537],{"fence":135,"stretchy":70,"minsize":65017,"maxsize":65017},[59,69093,401],{},[133,69095,136],{"stretchy":135},[59,69097,142],{},[133,69099,155],{"separator":70},[59,69101,158],{},[133,69103,155],{"separator":70},[138,69105,69106,69108],{},[59,69107,142],{},[133,69109,147],{"mathvariant":145,"lspace":146,"rspace":146},[133,69111,161],{"stretchy":135},[764,69113,69114,69120],{},[767,69115,69116],{},[770,69117,69118],{"scriptlevel":541,"displaystyle":70},[56,69119],{},[767,69121,69122],{},[770,69123,69124],{"scriptlevel":541,"displaystyle":70},[56,69125,69126,69128,69130,69132,69134,69136,69138,69144,69146,69148,69150,69152,69154,69160,69162,69164,69166,69168,69170,69172],{},[56,69127],{},[230,69129],{"width":10251},[133,69131,788],{},[59,69133,531],{},[59,69135,4230],{},[133,69137,136],{"stretchy":135},[138,69139,69140,69142],{},[59,69141,142],{},[133,69143,147],{"mathvariant":145,"lspace":146,"rspace":146},[133,69145,161],{"stretchy":135},[133,69147,4280],{"fence":135,"stretchy":70,"minsize":65017,"maxsize":65017},[59,69149,131],{},[133,69151,136],{"stretchy":135},[59,69153,29425],{"mathvariant":145},[138,69155,69156,69158],{},[59,69157,142],{},[133,69159,147],{"mathvariant":145,"lspace":146,"rspace":146},[133,69161,150],{},[59,69163,142],{},[133,69165,155],{"separator":70},[59,69167,158],{},[133,69169,161],{"stretchy":135},[133,69171,12965],{"fence":135,"stretchy":70,"minsize":63111,"maxsize":63111},[59,69173,2903],{"mathvariant":145},[63,69175,69176],{"encoding":65},"\\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}",[39,69178,69180],{"className":69179,"ariaHidden":70},[69],[39,69181,69183,69187],{"className":69182},[74],[39,69184],{"className":69185,"style":69186},[78],"height:6.3001em;vertical-align:-2.9em;",[39,69188,69190],{"className":69189},[83],[39,69191,69193,69283],{"className":69192},[759],[39,69194,69196],{"className":69195},[918],[39,69197,69199,69274],{"className":69198},[196,649],[39,69200,69202,69271],{"className":69201},[200],[39,69203,69206,69263],{"className":69204,"style":69205},[204],"height:3.4em;",[39,69207,69209,69213],{"style":69208},"top:-5.4em;",[39,69210],{"className":69211,"style":69212},[212],"height:3.75em;",[39,69214,69216,69219,69248,69251,69254,69257,69260],{"className":69215},[83],[39,69217,136],{"className":69218},[182],[39,69220,69222,69225],{"className":69221},[83],[39,69223,11772],{"className":69224,"style":11793},[83,641],[39,69226,69228],{"className":69227},[192],[39,69229,69231],{"className":69230},[196],[39,69232,69234],{"className":69233},[200],[39,69235,69237],{"className":69236,"style":22242},[204],[39,69238,69239,69242],{"style":1027},[39,69240],{"className":69241,"style":213},[212],[39,69243,69245],{"className":69244},[217,218,219,220],[39,69246,21884],{"className":69247},[465,220],[39,69249,4230],{"className":69250,"style":461},[83,84],[39,69252,161],{"className":69253},[263],[39,69255,136],{"className":69256},[182],[39,69258,142],{"className":69259},[83,84],[39,69261,161],{"className":69262},[263],[39,69264,69265,69268],{"style":5834},[39,69266],{"className":69267,"style":69212},[212],[39,69269],{"className":69270},[83],[39,69272,673],{"className":69273},[672],[39,69275,69277],{"className":69276},[200],[39,69278,69281],{"className":69279,"style":69280},[204],"height:2.9em;",[39,69282],{},[39,69284,69286],{"className":69285},[1223],[39,69287,69289,69636],{"className":69288},[196,649],[39,69290,69292,69633],{"className":69291},[200],[39,69293,69295,69486],{"className":69294,"style":69205},[204],[39,69296,69297,69300],{"style":69208},[39,69298],{"className":69299,"style":69212},[212],[39,69301,69303,69306,69309,69312,69315,69362,69365,69371,69374,69418,69421,69427,69430,69433,69436,69439,69442,69445,69448,69451,69483],{"className":69302},[83],[39,69304],{"className":69305},[83],[39,69307],{"className":69308,"style":231},[230],[39,69310,793],{"className":69311},[235],[39,69313],{"className":69314,"style":231},[230],[39,69316,69318],{"className":69317},[1744,3294],[39,69319,69321,69354],{"className":69320},[196,649],[39,69322,69324,69351],{"className":69323},[200],[39,69325,69327,69341],{"className":69326,"style":290},[204],[39,69328,69329,69332],{"style":36729},[39,69330],{"className":69331,"style":935},[212],[39,69333,69335],{"className":69334},[217,218,219,220],[39,69336,69338],{"className":69337},[83,220],[39,69339,158],{"className":69340},[83,84,220],[39,69342,69343,69346],{"style":21599},[39,69344],{"className":69345,"style":935},[212],[39,69347,69348],{},[39,69349,21465],{"className":69350},[1744],[39,69352,673],{"className":69353},[672],[39,69355,69357],{"className":69356},[200],[39,69358,69360],{"className":69359,"style":36761},[204],[39,69361],{},[39,69363],{"className":69364,"style":256},[230],[39,69366,69368],{"className":69367},[83],[39,69369,12951],{"className":69370},[18520,63488],[39,69372],{"className":69373,"style":256},[230],[39,69375,69377,69381],{"className":69376},[1744],[39,69378,69087],{"className":69379,"style":69380},[1744,3339,3340],"margin-right:0.4445em;position:relative;top:-0.0011em;",[39,69382,69384],{"className":69383},[192],[39,69385,69387,69409],{"className":69386},[196,649],[39,69388,69390,69406],{"className":69389},[200],[39,69391,69394],{"className":69392,"style":69393},[204],"height:-0.4336em;",[39,69395,69397,69400],{"style":69396},"top:-1.7881em;margin-left:-0.4445em;margin-right:0.05em;",[39,69398],{"className":69399,"style":213},[212],[39,69401,69403],{"className":69402},[217,218,219,220],[39,69404,61],{"className":69405,"style":85},[83,84,220],[39,69407,673],{"className":69408},[672],[39,69410,69412],{"className":69411},[200],[39,69413,69416],{"className":69414,"style":69415},[204],"height:0.9119em;",[39,69417],{},[39,69419],{"className":69420,"style":256},[230],[39,69422,69424],{"className":69423},[83],[39,69425,537],{"className":69426},[18520,33204],[39,69428,401],{"className":69429,"style":438},[83,84],[39,69431,136],{"className":69432},[182],[39,69434,142],{"className":69435},[83,84],[39,69437,155],{"className":69438},[252],[39,69440],{"className":69441,"style":256},[230],[39,69443,158],{"className":69444},[83,84],[39,69446,155],{"className":69447},[252],[39,69449],{"className":69450,"style":256},[230],[39,69452,69454,69457],{"className":69453},[83],[39,69455,142],{"className":69456},[83,84],[39,69458,69460],{"className":69459},[192],[39,69461,69463],{"className":69462},[196],[39,69464,69466],{"className":69465},[200],[39,69467,69469],{"className":69468,"style":1024},[204],[39,69470,69471,69474],{"style":1027},[39,69472],{"className":69473,"style":213},[212],[39,69475,69477],{"className":69476},[217,218,219,220],[39,69478,69480],{"className":69479},[83,220],[39,69481,147],{"className":69482},[83,220],[39,69484,161],{"className":69485},[263],[39,69487,69488,69491],{"style":5834},[39,69489],{"className":69490,"style":69212},[212],[39,69492,69494,69497,69500,69503,69506,69509,69512,69515,69518,69550,69553,69559,69562,69565,69568,69600,69603,69606,69609,69612,69615,69618,69621,69624,69630],{"className":69493},[83],[39,69495],{"className":69496},[83],[39,69498],{"className":69499,"style":10787},[230],[39,69501],{"className":69502,"style":461},[230],[39,69504,788],{"className":69505},[465],[39,69507],{"className":69508,"style":461},[230],[39,69510,531],{"className":69511,"style":565},[83,84],[39,69513,4230],{"className":69514,"style":461},[83,84],[39,69516,136],{"className":69517},[182],[39,69519,69521,69524],{"className":69520},[83],[39,69522,142],{"className":69523},[83,84],[39,69525,69527],{"className":69526},[192],[39,69528,69530],{"className":69529},[196],[39,69531,69533],{"className":69532},[200],[39,69534,69536],{"className":69535,"style":1024},[204],[39,69537,69538,69541],{"style":1027},[39,69539],{"className":69540,"style":213},[212],[39,69542,69544],{"className":69543},[217,218,219,220],[39,69545,69547],{"className":69546},[83,220],[39,69548,147],{"className":69549},[83,220],[39,69551,161],{"className":69552},[263],[39,69554,69556],{"className":69555},[83],[39,69557,4280],{"className":69558},[18520,33204],[39,69560,131],{"className":69561,"style":178},[83,84],[39,69563,136],{"className":69564},[182],[39,69566,29425],{"className":69567},[83,35077],[39,69569,69571,69574],{"className":69570},[83],[39,69572,142],{"className":69573},[83,84],[39,69575,69577],{"className":69576},[192],[39,69578,69580],{"className":69579},[196],[39,69581,69583],{"className":69582},[200],[39,69584,69586],{"className":69585,"style":1024},[204],[39,69587,69588,69591],{"style":1027},[39,69589],{"className":69590,"style":213},[212],[39,69592,69594],{"className":69593},[217,218,219,220],[39,69595,69597],{"className":69596},[83,220],[39,69598,147],{"className":69599},[83,220],[39,69601],{"className":69602,"style":231},[230],[39,69604,150],{"className":69605},[235],[39,69607],{"className":69608,"style":231},[230],[39,69610,142],{"className":69611},[83,84],[39,69613,155],{"className":69614},[252],[39,69616],{"className":69617,"style":256},[230],[39,69619,158],{"className":69620},[83,84],[39,69622,161],{"className":69623},[263],[39,69625,69627],{"className":69626},[83],[39,69628,12965],{"className":69629},[18520,63488],[39,69631,2903],{"className":69632},[83],[39,69634,673],{"className":69635},[672],[39,69637,69639],{"className":69638},[200],[39,69640,69642],{"className":69641,"style":69280},[204],[39,69643],{},[11,69645,69646,69647,69705,69706,69734],{},"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 ",[39,69648,69650,69667],{"className":69649,"translate":43},[42],[39,69651,69653],{"className":69652},[47],[49,69654,69655],{"xmlns":51},[53,69656,69657,69665],{},[56,69658,69659],{},[138,69660,69661,69663],{},[59,69662,11772],{"mathvariant":617},[133,69664,21884],{},[63,69666,21887],{"encoding":65},[39,69668,69670],{"className":69669,"ariaHidden":70},[69],[39,69671,69673,69676],{"className":69672},[74],[39,69674],{"className":69675,"style":21897},[78],[39,69677,69679,69682],{"className":69678},[83],[39,69680,11772],{"className":69681,"style":11793},[83,641],[39,69683,69685],{"className":69684},[192],[39,69686,69688],{"className":69687},[196],[39,69689,69691],{"className":69690},[200],[39,69692,69694],{"className":69693,"style":21897},[204],[39,69695,69696,69699],{"style":208},[39,69697],{"className":69698,"style":213},[212],[39,69700,69702],{"className":69701},[217,218,219,220],[39,69703,21884],{"className":69704},[465,220]," 仍然是 ",[39,69707,69709,69722],{"className":69708,"translate":43},[42],[39,69710,69712],{"className":69711},[47],[49,69713,69714],{"xmlns":51},[53,69715,69716,69720],{},[56,69717,69718],{},[59,69719,531],{},[63,69721,26586],{"encoding":65},[39,69723,69725],{"className":69724,"ariaHidden":70},[69],[39,69726,69728,69731],{"className":69727},[74],[39,69729],{"className":69730,"style":2654},[78],[39,69732,531],{"className":69733,"style":565},[83,84],"-压缩映射。唯一不动点仍然存在。",[11,69736,69737,69738,69796,69797,69895],{},"但是，最优策略的存在性需要额外条件。即使 ",[39,69739,69741,69758],{"className":69740,"translate":43},[42],[39,69742,69744],{"className":69743},[47],[49,69745,69746],{"xmlns":51},[53,69747,69748,69756],{},[56,69749,69750],{},[138,69751,69752,69754],{},[59,69753,11772],{"mathvariant":617},[133,69755,21884],{},[63,69757,21887],{"encoding":65},[39,69759,69761],{"className":69760,"ariaHidden":70},[69],[39,69762,69764,69767],{"className":69763},[74],[39,69765],{"className":69766,"style":21897},[78],[39,69768,69770,69773],{"className":69769},[83],[39,69771,11772],{"className":69772,"style":11793},[83,641],[39,69774,69776],{"className":69775},[192],[39,69777,69779],{"className":69778},[196],[39,69780,69782],{"className":69781},[200],[39,69783,69785],{"className":69784,"style":21897},[204],[39,69786,69787,69790],{"style":208},[39,69788],{"className":69789,"style":213},[212],[39,69791,69793],{"className":69792},[217,218,219,220],[39,69794,21884],{"className":69795},[465,220]," 有唯一不动点，也不一定存在一个可测的映射 ",[39,69798,69800,69829],{"className":69799,"translate":43},[42],[39,69801,69803],{"className":69802},[47],[49,69804,69805],{"xmlns":51},[53,69806,69807,69826],{},[56,69808,69809,69811,69814,69820,69822,69824],{},[59,69810,142],{},[133,69812,69813],{},"↦",[138,69815,69816,69818],{},[59,69817,2098],{},[133,69819,21884],{},[133,69821,136],{"stretchy":135},[59,69823,142],{},[133,69825,161],{"stretchy":135},[63,69827,69828],{"encoding":65},"s\\mapsto\\pi^*(s)",[39,69830,69832,69851],{"className":69831,"ariaHidden":70},[69],[39,69833,69835,69839,69842,69845,69848],{"className":69834},[74],[39,69836],{"className":69837,"style":69838},[78],"height:0.522em;vertical-align:-0.011em;",[39,69840,142],{"className":69841},[83,84],[39,69843],{"className":69844,"style":231},[230],[39,69846,69813],{"className":69847},[235],[39,69849],{"className":69850,"style":231},[230],[39,69852,69854,69857,69886,69889,69892],{"className":69853},[74],[39,69855],{"className":69856,"style":245},[78],[39,69858,69860,69863],{"className":69859},[83],[39,69861,2098],{"className":69862,"style":2114},[83,84],[39,69864,69866],{"className":69865},[192],[39,69867,69869],{"className":69868},[196],[39,69870,69872],{"className":69871},[200],[39,69873,69875],{"className":69874,"style":21897},[204],[39,69876,69877,69880],{"style":208},[39,69878],{"className":69879,"style":213},[212],[39,69881,69883],{"className":69882},[217,218,219,220],[39,69884,21884],{"className":69885},[465,220],[39,69887,136],{"className":69888},[182],[39,69890,142],{"className":69891},[83,84],[39,69893,161],{"className":69894},[263]," 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。",[26,69897,69899],{"id":69898},"_72-平均奖励与无折扣","7.2 平均奖励与无折扣",[11,69901,12932,69902,69952,69953,69983,69984,70013],{},[39,69903,69905,69922],{"className":69904,"translate":43},[42],[39,69906,69908],{"className":69907},[47],[49,69909,69910],{"xmlns":51},[53,69911,69912,69920],{},[56,69913,69914,69916,69918],{},[59,69915,531],{},[133,69917,793],{},[539,69919,546],{},[63,69921,28990],{"encoding":65},[39,69923,69925,69943],{"className":69924,"ariaHidden":70},[69],[39,69926,69928,69931,69934,69937,69940],{"className":69927},[74],[39,69929],{"className":69930,"style":2654},[78],[39,69932,531],{"className":69933,"style":565},[83,84],[39,69935],{"className":69936,"style":231},[230],[39,69938,793],{"className":69939},[235],[39,69941],{"className":69942,"style":231},[230],[39,69944,69946,69949],{"className":69945},[74],[39,69947],{"className":69948,"style":2473},[78],[39,69950,546],{"className":69951},[83]," 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 ",[39,69954,69956,69971],{"className":69955,"translate":43},[42],[39,69957,69959],{"className":69958},[47],[49,69960,69961],{"xmlns":51},[53,69962,69963,69968],{},[56,69964,69965],{},[59,69966,69967],{},"ρ",[63,69969,69970],{"encoding":65},"\\rho",[39,69972,69974],{"className":69973,"ariaHidden":70},[69],[39,69975,69977,69980],{"className":69976},[74],[39,69978],{"className":69979,"style":2654},[78],[39,69981,69967],{"className":69982},[83,84]," 与偏差函数 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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 ",[39,70964,70966,70979],{"className":70965,"translate":43},[42],[39,70967,70969],{"className":70968},[47],[49,70970,70971],{"xmlns":51},[53,70972,70973,70977],{},[56,70974,70975],{},[59,70976,69998],{},[63,70978,69998],{"encoding":65},[39,70980,70982],{"className":70981,"ariaHidden":70},[69],[39,70983,70985,70988],{"className":70984},[74],[39,70986],{"className":70987,"style":28824},[78],[39,70989,69998],{"className":70990},[83,84]," 是解，则 ",[39,70993,70995,71013],{"className":70994,"translate":43},[42],[39,70996,70998],{"className":70997},[47],[49,70999,71000],{"xmlns":51},[53,71001,71002,71010],{},[56,71003,71004,71006,71008],{},[59,71005,69998],{},[133,71007,788],{},[59,71009,24077],{},[63,71011,71012],{"encoding":65},"h+c",[39,71014,71016,71035],{"className":71015,"ariaHidden":70},[69],[39,71017,71019,71023,71026,71029,71032],{"className":71018},[74],[39,71020],{"className":71021,"style":71022},[78],"height:0.7778em;vertical-align:-0.0833em;",[39,71024,69998],{"className":71025},[83,84],[39,71027],{"className":71028,"style":461},[230],[39,71030,788],{"className":71031},[465],[39,71033],{"className":71034,"style":461},[230],[39,71036,71038,71041],{"className":71037},[74],[39,71039],{"className":71040,"style":290},[78],[39,71042,24077],{"className":71043},[83,84]," 也是解。折扣因子 ",[39,71046,71048,71065],{"className":71047,"translate":43},[42],[39,71049,71051],{"className":71050},[47],[49,71052,71053],{"xmlns":51},[53,71054,71055,71063],{},[56,71056,71057,71059,71061],{},[59,71058,531],{},[133,71060,1693],{},[539,71062,546],{},[63,71064,17296],{"encoding":65},[39,71066,71068,71086],{"className":71067,"ariaHidden":70},[69],[39,71069,71071,71074,71077,71080,71083],{"className":71070},[74],[39,71072],{"className":71073,"style":561},[78],[39,71075,531],{"className":71076,"style":565},[83,84],[39,71078],{"className":71079,"style":231},[230],[39,71081,1693],{"className":71082},[235],[39,71084],{"className":71085,"style":231},[230],[39,71087,71089,71092],{"className":71088},[74],[39,71090],{"className":71091,"style":2473},[78],[39,71093,546],{"className":71094},[83]," 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。",[11,71097,71098,71099,71149,71150,71200],{},"有限期问题是一个例外。即使 ",[39,71100,71102,71119],{"className":71101,"translate":43},[42],[39,71103,71105],{"className":71104},[47],[49,71106,71107],{"xmlns":51},[53,71108,71109,71117],{},[56,71110,71111,71113,71115],{},[59,71112,531],{},[133,71114,793],{},[539,71116,546],{},[63,71118,28990],{"encoding":65},[39,71120,71122,71140],{"className":71121,"ariaHidden":70},[69],[39,71123,71125,71128,71131,71134,71137],{"className":71124},[74],[39,71126],{"className":71127,"style":2654},[78],[39,71129,531],{"className":71130,"style":565},[83,84],[39,71132],{"className":71133,"style":231},[230],[39,71135,793],{"className":71136},[235],[39,71138],{"className":71139,"style":231},[230],[39,71141,71143,71146],{"className":71142},[74],[39,71144],{"className":71145,"style":2473},[78],[39,71147,546],{"className":71148},[83],"，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，",[39,71151,71153,71170],{"className":71152,"translate":43},[42],[39,71154,71156],{"className":71155},[47],[49,71157,71158],{"xmlns":51},[53,71159,71160,71168],{},[56,71161,71162,71164,71166],{},[59,71163,531],{},[133,71165,1693],{},[539,71167,546],{},[63,71169,17296],{"encoding":65},[39,71171,71173,71191],{"className":71172,"ariaHidden":70},[69],[39,71174,71176,71179,71182,71185,71188],{"className":71175},[74],[39,71177],{"className":71178,"style":561},[78],[39,71180,531],{"className":71181,"style":565},[83,84],[39,71183],{"className":71184,"style":231},[230],[39,71186,1693],{"className":71187},[235],[39,71189],{"className":71190,"style":231},[230],[39,71192,71194,71197],{"className":71193},[74],[39,71195],{"className":71196,"style":2473},[78],[39,71198,546],{"className":71199},[83]," 的必要性主要针对无限期问题。",[26,71202,71204],{"id":71203},"_73-注意要点","7.3 注意要点",[11,71206,71207,71210,71211,71380],{},[1489,71208,71209],{},"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，",[39,71212,71214,71264],{"className":71213,"translate":43},[42],[39,71215,71217],{"className":71216},[47],[49,71218,71219],{"xmlns":51},[53,71220,71221,71261],{},[56,71222,71223,71225,71227,71229,71231,71233,71235,71237,71239,71241,71243,71245,71247,71249,71251,71253,71259],{},[59,71224,4230],{},[133,71226,136],{"stretchy":135},[59,71228,142],{},[133,71230,161],{"stretchy":135},[133,71232,793],{},[59,71234,401],{},[133,71236,136],{"stretchy":135},[59,71238,142],{},[133,71240,155],{"separator":70},[59,71242,158],{},[133,71244,161],{"stretchy":135},[133,71246,788],{},[59,71248,531],{},[59,71250,4230],{},[133,71252,136],{"stretchy":135},[138,71254,71255,71257],{},[59,71256,142],{},[133,71258,147],{"mathvariant":145,"lspace":146,"rspace":146},[133,71260,161],{"stretchy":135},[63,71262,71263],{"encoding":65},"V(s)=r(s,a)+\\gamma V(s')",[39,71265,71267,71294,71330],{"className":71266,"ariaHidden":70},[69],[39,71268,71270,71273,71276,71279,71282,71285,71288,71291],{"className":71269},[74],[39,71271],{"className":71272,"style":245},[78],[39,71274,4230],{"className":71275,"style":461},[83,84],[39,71277,136],{"className":71278},[182],[39,71280,142],{"className":71281},[83,84],[39,71283,161],{"className":71284},[263],[39,71286],{"className":71287,"style":231},[230],[39,71289,793],{"className":71290},[235],[39,71292],{"className":71293,"style":231},[230],[39,71295,71297,71300,71303,71306,71309,71312,71315,71318,71321,71324,71327],{"className":71296},[74],[39,71298],{"className":71299,"style":245},[78],[39,71301,401],{"className":71302,"style":438},[83,84],[39,71304,136],{"className":71305},[182],[39,71307,142],{"className":71308},[83,84],[39,71310,155],{"className":71311},[252],[39,71313],{"className":71314,"style":256},[230],[39,71316,158],{"className":71317},[83,84],[39,71319,161],{"className":71320},[263],[39,71322],{"className":71323,"style":461},[230],[39,71325,788],{"className":71326},[465],[39,71328],{"className":71329,"style":461},[230],[39,71331,71333,71336,71339,71342,71345,71377],{"className":71332},[74],[39,71334],{"className":71335,"style":174},[78],[39,71337,531],{"className":71338,"style":565},[83,84],[39,71340,4230],{"className":71341,"style":461},[83,84],[39,71343,136],{"className":71344},[182],[39,71346,71348,71351],{"className":71347},[83],[39,71349,142],{"className":71350},[83,84],[39,71352,71354],{"className":71353},[192],[39,71355,71357],{"className":71356},[196],[39,71358,71360],{"className":71359},[200],[39,71361,71363],{"className":71362,"style":205},[204],[39,71364,71365,71368],{"style":208},[39,71366],{"className":71367,"style":213},[212],[39,71369,71371],{"className":71370},[217,218,219,220],[39,71372,71374],{"className":71373},[83,220],[39,71375,147],{"className":71376},[83,220],[39,71378,161],{"className":71379},[263]," 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。",[11,71382,71383,71386,71387,71468,71469,71550],{},[1489,71384,71385],{},"把最大值放进期望内部。"," 最优方程是 ",[39,71388,71390,71413],{"className":71389,"translate":43},[42],[39,71391,71393],{"className":71392},[47],[49,71394,71395],{"xmlns":51},[53,71396,71397,71411],{},[56,71398,71399,71409],{},[613,71400,71401,71407],{},[56,71402,71403,71405],{},[59,71404,1687],{},[133,71406,1690],{},[59,71408,158],{},[59,71410,4253],{"mathvariant":421},[63,71412,45377],{"encoding":65},[39,71414,71416],{"className":71415,"ariaHidden":70},[69],[39,71417,71419,71422,71462,71465],{"className":71418},[74],[39,71420],{"className":71421,"style":45387},[78],[39,71423,71425,71428],{"className":71424},[1744],[39,71426,1687],{"className":71427},[1744],[39,71429,71431],{"className":71430},[192],[39,71432,71434,71454],{"className":71433},[196,649],[39,71435,71437,71451],{"className":71436},[200],[39,71438,71440],{"className":71439,"style":1728},[204],[39,71441,71442,71445],{"style":44021},[39,71443],{"className":71444,"style":213},[212],[39,71446,71448],{"className":71447},[217,218,219,220],[39,71449,158],{"className":71450},[83,84,220],[39,71452,673],{"className":71453},[672],[39,71455,71457],{"className":71456},[200],[39,71458,71460],{"className":71459,"style":680},[204],[39,71461],{},[39,71463],{"className":71464,"style":256},[230],[39,71466,4253],{"className":71467},[83,515],"，不是 ",[39,71470,71472,71495],{"className":71471,"translate":43},[42],[39,71473,71475],{"className":71474},[47],[49,71476,71477],{"xmlns":51},[53,71478,71479,71493],{},[56,71480,71481,71483],{},[59,71482,4253],{"mathvariant":421},[613,71484,71485,71491],{},[56,71486,71487,71489],{},[59,71488,1687],{},[133,71490,1690],{},[59,71492,158],{},[63,71494,45461],{"encoding":65},[39,71496,71498],{"className":71497,"ariaHidden":70},[69],[39,71499,71501,71504,71507,71510],{"className":71500},[74],[39,71502],{"className":71503,"style":45387},[78],[39,71505,4253],{"className":71506},[83,515],[39,71508],{"className":71509,"style":256},[230],[39,71511,71513,71516],{"className":71512},[1744],[39,71514,1687],{"className":71515},[1744],[39,71517,71519],{"className":71518},[192],[39,71520,71522,71542],{"className":71521},[196,649],[39,71523,71525,71539],{"className":71524},[200],[39,71526,71528],{"className":71527,"style":1728},[204],[39,71529,71530,71533],{"style":44021},[39,71531],{"className":71532,"style":213},[212],[39,71534,71536],{"className":71535},[217,218,219,220],[39,71537,158],{"className":71538},[83,84,220],[39,71540,673],{"className":71541},[672],[39,71543,71545],{"className":71544},[200],[39,71546,71548],{"className":71547,"style":680},[204],[39,71549],{},"。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。",[11,71552,71553,2083,71612,71686,71687,71773,71774,71926,71927,72036,72037,72081],{},[1489,71554,71555,71556,1639,71584,1607],{},"混淆 ",[39,71557,71559,71572],{"className":71558,"translate":43},[42],[39,71560,71562],{"className":71561},[47],[49,71563,71564],{"xmlns":51},[53,71565,71566,71570],{},[56,71567,71568],{},[59,71569,4230],{},[63,71571,4230],{"encoding":65},[39,71573,71575],{"className":71574,"ariaHidden":70},[69],[39,71576,71578,71581],{"className":71577},[74],[39,71579],{"className":71580,"style":79},[78],[39,71582,4230],{"className":71583,"style":461},[83,84],[39,71585,71587,71600],{"className":71586,"translate":43},[42],[39,71588,71590],{"className":71589},[47],[49,71591,71592],{"xmlns":51},[53,71593,71594,71598],{},[56,71595,71596],{},[59,71597,4295],{},[63,71599,4295],{"encoding":65},[39,71601,71603],{"className":71602,"ariaHidden":70},[69],[39,71604,71606,71609],{"className":71605},[74],[39,71607],{"className":71608,"style":6452},[78],[39,71610,4295],{"className":71611},[83,84],[39,71613,71615,71639],{"className":71614,"translate":43},[42],[39,71616,71618],{"className":71617},[47],[49,71619,71620],{"xmlns":51},[53,71621,71622,71636],{},[56,71623,71624,71630,71632,71634],{},[138,71625,71626,71628],{},[59,71627,4230],{},[59,71629,2098],{},[133,71631,136],{"stretchy":135},[59,71633,142],{},[133,71635,161],{"stretchy":135},[63,71637,71638],{"encoding":65},"V^\\pi(s)",[39,71640,71642],{"className":71641,"ariaHidden":70},[69],[39,71643,71645,71648,71677,71680,71683],{"className":71644},[74],[39,71646],{"className":71647,"style":245},[78],[39,71649,71651,71654],{"className":71650},[83],[39,71652,4230],{"className":71653,"style":461},[83,84],[39,71655,71657],{"className":71656},[192],[39,71658,71660],{"className":71659},[196],[39,71661,71663],{"className":71662},[200],[39,71664,71666],{"className":71665,"style":6410},[204],[39,71667,71668,71671],{"style":208},[39,71669],{"className":71670,"style":213},[212],[39,71672,71674],{"className":71673},[217,218,219,220],[39,71675,2098],{"className":71676,"style":2114},[83,84,220],[39,71678,136],{"className":71679},[182],[39,71681,142],{"className":71682},[83,84],[39,71684,161],{"className":71685},[263]," 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 ",[39,71688,71690,71717],{"className":71689,"translate":43},[42],[39,71691,71693],{"className":71692},[47],[49,71694,71695],{"xmlns":51},[53,71696,71697,71715],{},[56,71698,71699,71705,71707,71709,71711,71713],{},[138,71700,71701,71703],{},[59,71702,4295],{},[133,71704,21884],{},[133,71706,136],{"stretchy":135},[59,71708,142],{},[133,71710,155],{"separator":70},[59,71712,158],{},[133,71714,161],{"stretchy":135},[63,71716,45169],{"encoding":65},[39,71718,71720],{"className":71719,"ariaHidden":70},[69],[39,71721,71723,71726,71755,71758,71761,71764,71767,71770],{"className":71722},[74],[39,71724],{"className":71725,"style":245},[78],[39,71727,71729,71732],{"className":71728},[83],[39,71730,4295],{"className":71731},[83,84],[39,71733,71735],{"className":71734},[192],[39,71736,71738],{"className":71737},[196],[39,71739,71741],{"className":71740},[200],[39,71742,71744],{"className":71743,"style":21897},[204],[39,71745,71746,71749],{"style":208},[39,71747],{"className":71748,"style":213},[212],[39,71750,71752],{"className":71751},[217,218,219,220],[39,71753,21884],{"className":71754},[465,220],[39,71756,136],{"className":71757},[182],[39,71759,142],{"className":71760},[83,84],[39,71762,155],{"className":71763},[252],[39,71765],{"className":71766,"style":256},[230],[39,71768,158],{"className":71769},[83,84],[39,71771,161],{"className":71772},[263],"，最优动作是 ",[39,71775,71777,71819],{"className":71776,"translate":43},[42],[39,71778,71780],{"className":71779},[47],[49,71781,71782],{"xmlns":51},[53,71783,71784,71816],{},[56,71785,71786,71788,71790,71800,71806,71808,71810,71812,71814],{},[59,71787,39047],{},[133,71789,1690],{},[613,71791,71792,71798],{},[56,71793,71794,71796],{},[59,71795,1687],{},[133,71797,1690],{},[59,71799,158],{},[138,71801,71802,71804],{},[59,71803,4295],{},[133,71805,21884],{},[133,71807,136],{"stretchy":135},[59,71809,142],{},[133,71811,155],{"separator":70},[59,71813,158],{},[133,71815,161],{"stretchy":135},[63,71817,71818],{"encoding":65},"\\arg\\max_a Q^*(s,a)",[39,71820,71822],{"className":71821,"ariaHidden":70},[69],[39,71823,71825,71828,71833,71836,71876,71879,71908,71911,71914,71917,71920,71923],{"className":71824},[74],[39,71826],{"className":71827,"style":245},[78],[39,71829,39206,71831],{"className":71830},[1744],[39,71832,39209],{"style":35078},[39,71834],{"className":71835,"style":256},[230],[39,71837,71839,71842],{"className":71838},[1744],[39,71840,1687],{"className":71841},[1744],[39,71843,71845],{"className":71844},[192],[39,71846,71848,71868],{"className":71847},[196,649],[39,71849,71851,71865],{"className":71850},[200],[39,71852,71854],{"className":71853,"style":1728},[204],[39,71855,71856,71859],{"style":44021},[39,71857],{"className":71858,"style":213},[212],[39,71860,71862],{"className":71861},[217,218,219,220],[39,71863,158],{"className":71864},[83,84,220],[39,71866,673],{"className":71867},[672],[39,71869,71871],{"className":71870},[200],[39,71872,71874],{"className":71873,"style":680},[204],[39,71875],{},[39,71877],{"className":71878,"style":256},[230],[39,71880,71882,71885],{"className":71881},[83],[39,71883,4295],{"className":71884},[83,84],[39,71886,71888],{"className":71887},[192],[39,71889,71891],{"className":71890},[196],[39,71892,71894],{"className":71893},[200],[39,71895,71897],{"className":71896,"style":21897},[204],[39,71898,71899,71902],{"style":208},[39,71900],{"className":71901,"style":213},[212],[39,71903,71905],{"className":71904},[217,218,219,220],[39,71906,21884],{"className":71907},[465,220],[39,71909,136],{"className":71910},[182],[39,71912,142],{"className":71913},[83,84],[39,71915,155],{"className":71916},[252],[39,71918],{"className":71919,"style":256},[230],[39,71921,158],{"className":71922},[83,84],[39,71924,161],{"className":71925},[263],"。把 ",[39,71928,71930,71964],{"className":71929,"translate":43},[42],[39,71931,71933],{"className":71932},[47],[49,71934,71935],{"xmlns":51},[53,71936,71937,71961],{},[56,71938,71939,71941,71943,71953,71955,71957,71959],{},[59,71940,39047],{},[133,71942,1690],{},[613,71944,71945,71951],{},[56,71946,71947,71949],{},[59,71948,1687],{},[133,71950,1690],{},[59,71952,158],{},[59,71954,4230],{},[133,71956,136],{"stretchy":135},[59,71958,142],{},[133,71960,161],{"stretchy":135},[63,71962,71963],{"encoding":65},"\\arg\\max_a V(s)",[39,71965,71967],{"className":71966,"ariaHidden":70},[69],[39,71968,71970,71973,71978,71981,72021,72024,72027,72030,72033],{"className":71969},[74],[39,71971],{"className":71972,"style":245},[78],[39,71974,39206,71976],{"className":71975},[1744],[39,71977,39209],{"style":35078},[39,71979],{"className":71980,"style":256},[230],[39,71982,71984,71987],{"className":71983},[1744],[39,71985,1687],{"className":71986},[1744],[39,71988,71990],{"className":71989},[192],[39,71991,71993,72013],{"className":71992},[196,649],[39,71994,71996,72010],{"className":71995},[200],[39,71997,71999],{"className":71998,"style":1728},[204],[39,72000,72001,72004],{"style":44021},[39,72002],{"className":72003,"style":213},[212],[39,72005,72007],{"className":72006},[217,218,219,220],[39,72008,158],{"className":72009},[83,84,220],[39,72011,673],{"className":72012},[672],[39,72014,72016],{"className":72015},[200],[39,72017,72019],{"className":72018,"style":680},[204],[39,72020],{},[39,72022],{"className":72023,"style":256},[230],[39,72025,4230],{"className":72026,"style":461},[83,84],[39,72028,136],{"className":72029},[182],[39,72031,142],{"className":72032},[83,84],[39,72034,161],{"className":72035},[263]," 当作最优动作是没有意义的，因为 ",[39,72038,72040,72060],{"className":72039,"translate":43},[42],[39,72041,72043],{"className":72042},[47],[49,72044,72045],{"xmlns":51},[53,72046,72047,72057],{},[56,72048,72049,72051,72053,72055],{},[59,72050,4230],{},[133,72052,136],{"stretchy":135},[59,72054,142],{},[133,72056,161],{"stretchy":135},[63,72058,72059],{"encoding":65},"V(s)",[39,72061,72063],{"className":72062,"ariaHidden":70},[69],[39,72064,72066,72069,72072,72075,72078],{"className":72065},[74],[39,72067],{"className":72068,"style":245},[78],[39,72070,4230],{"className":72071,"style":461},[83,84],[39,72073,136],{"className":72074},[182],[39,72076,142],{"className":72077},[83,84],[39,72079,161],{"className":72080},[263]," 中已经没有动作变量。",[11,72083,72084,72087,72088,72138],{},[1489,72085,72086],{},"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 ",[39,72089,72091,72108],{"className":72090,"translate":43},[42],[39,72092,72094],{"className":72093},[47],[49,72095,72096],{"xmlns":51},[53,72097,72098,72106],{},[56,72099,72100,72102,72104],{},[59,72101,531],{},[133,72103,793],{},[539,72105,546],{},[63,72107,28990],{"encoding":65},[39,72109,72111,72129],{"className":72110,"ariaHidden":70},[69],[39,72112,72114,72117,72120,72123,72126],{"className":72113},[74],[39,72115],{"className":72116,"style":2654},[78],[39,72118,531],{"className":72119,"style":565},[83,84],[39,72121],{"className":72122,"style":231},[230],[39,72124,793],{"className":72125},[235],[39,72127],{"className":72128,"style":231},[230],[39,72130,72132,72135],{"className":72131},[74],[39,72133],{"className":72134,"style":2473},[78],[39,72136,546],{"className":72137},[83],"、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。",[11,72140,72141,72144,72145,72173],{},[1489,72142,72143],{},"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 ",[39,72146,72148,72161],{"className":72147,"translate":43},[42],[39,72149,72151],{"className":72150},[47],[49,72152,72153],{"xmlns":51},[53,72154,72155,72159],{},[56,72156,72157],{},[539,72158,546],{},[63,72160,546],{"encoding":65},[39,72162,72164],{"className":72163,"ariaHidden":70},[69],[39,72165,72167,72170],{"className":72166},[74],[39,72168],{"className":72169,"style":2473},[78],[39,72171,546],{"className":72172},[83],"。此时",[39,72175,72177],{"className":72176,"translate":43},[746],[39,72178,72180,72214],{"className":72179,"translate":43},[42],[39,72181,72183],{"className":72182},[47],[49,72184,72185],{"xmlns":51,"display":755},[53,72186,72187,72211],{},[56,72188,72189,72195,72197,72209],{},[138,72190,72191,72193],{},[59,72192,4230],{},[133,72194,21884],{},[133,72196,793],{},[5362,72198,72199,72201],{},[539,72200,546],{},[56,72202,72203,72205,72207],{},[539,72204,546],{},[133,72206,5378],{},[59,72208,531],{},[133,72210,155],{"separator":70},[63,72212,72213],{"encoding":65},"V^*=\\frac{1}{1-\\gamma},",[39,72215,72217,72261],{"className":72216,"ariaHidden":70},[69],[39,72218,72220,72223,72252,72255,72258],{"className":72219},[74],[39,72221],{"className":72222,"style":22242},[78],[39,72224,72226,72229],{"className":72225},[83],[39,72227,4230],{"className":72228,"style":461},[83,84],[39,72230,72232],{"className":72231},[192],[39,72233,72235],{"className":72234},[196],[39,72236,72238],{"className":72237},[200],[39,72239,72241],{"className":72240,"style":22242},[204],[39,72242,72243,72246],{"style":1027},[39,72244],{"className":72245,"style":213},[212],[39,72247,72249],{"className":72248},[217,218,219,220],[39,72250,21884],{"className":72251},[465,220],[39,72253],{"className":72254,"style":231},[230],[39,72256,793],{"className":72257},[235],[39,72259],{"className":72260,"style":231},[230],[39,72262,72264,72268,72342],{"className":72263},[74],[39,72265],{"className":72266,"style":72267},[78],"height:2.2019em;vertical-align:-0.8804em;",[39,72269,72271,72274,72339],{"className":72270},[83],[39,72272],{"className":72273},[182,5463],[39,72275,72277],{"className":72276},[5362],[39,72278,72280,72331],{"className":72279},[196,649],[39,72281,72283,72328],{"className":72282},[200],[39,72284,72286,72309,72317],{"className":72285,"style":18240},[204],[39,72287,72288,72291],{"style":5479},[39,72289],{"className":72290,"style":935},[212],[39,72292,72294,72297,72300,72303,72306],{"className":72293},[83],[39,72295,546],{"className":72296},[83],[39,72298],{"className":72299,"style":461},[230],[39,72301,5378],{"className":72302},[465],[39,72304],{"className":72305,"style":461},[230],[39,72307,531],{"className":72308,"style":565},[83,84],[39,72310,72311,72314],{"style":5503},[39,72312],{"className":72313,"style":935},[212],[39,72315],{"className":72316,"style":5511},[5510],[39,72318,72319,72322],{"style":5514},[39,72320],{"className":72321,"style":935},[212],[39,72323,72325],{"className":72324},[83],[39,72326,546],{"className":72327},[83],[39,72329,673],{"className":72330},[672],[39,72332,72334],{"className":72333},[200],[39,72335,72337],{"className":72336,"style":5582},[204],[39,72338],{},[39,72340],{"className":72341},[263,5463],[39,72343,155],{"className":72344},[252],[11,72346,72347],{},"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。",[21,72349,72350],{"id":72350},"总结",[11,72352,72353,72354,72569,72570,72626],{},"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 ",[39,72355,72357,72398],{"className":72356,"translate":43},[42],[39,72358,72360],{"className":72359},[47],[49,72361,72362],{"xmlns":51},[53,72363,72364,72396],{},[56,72365,72366,72372,72374,72380,72382,72384],{},[613,72367,72368,72370],{},[59,72369,3181],{},[59,72371,621],{},[133,72373,793],{},[613,72375,72376,72378],{},[59,72377,401],{},[59,72379,621],{},[133,72381,788],{},[59,72383,531],{},[613,72385,72386,72388],{},[59,72387,3181],{},[56,72389,72390,72392,72394],{},[59,72391,621],{},[133,72393,788],{},[539,72395,546],{},[63,72397,7463],{"encoding":65},[39,72399,72401,72456,72511],{"className":72400,"ariaHidden":70},[69],[39,72402,72404,72407,72447,72450,72453],{"className":72403},[74],[39,72405],{"className":72406,"style":634},[78],[39,72408,72410,72413],{"className":72409},[83],[39,72411,3181],{"className":72412},[83,84],[39,72414,72416],{"className":72415},[192],[39,72417,72419,72439],{"className":72418},[196,649],[39,72420,72422,72436],{"className":72421},[200],[39,72423,72425],{"className":72424,"style":656},[204],[39,72426,72427,72430],{"style":966},[39,72428],{"className":72429,"style":213},[212],[39,72431,72433],{"className":72432},[217,218,219,220],[39,72434,621],{"className":72435},[83,84,220],[39,72437,673],{"className":72438},[672],[39,72440,72442],{"className":72441},[200],[39,72443,72445],{"className":72444,"style":680},[204],[39,72446],{},[39,72448],{"className":72449,"style":231},[230],[39,72451,793],{"className":72452},[235],[39,72454],{"className":72455,"style":231},[230],[39,72457,72459,72462,72502,72505,72508],{"className":72458},[74],[39,72460],{"className":72461,"style":7528},[78],[39,72463,72465,72468],{"className":72464},[83],[39,72466,401],{"className":72467,"style":438},[83,84],[39,72469,72471],{"className":72470},[192],[39,72472,72474,72494],{"className":72473},[196,649],[39,72475,72477,72491],{"className":72476},[200],[39,72478,72480],{"className":72479,"style":656},[204],[39,72481,72482,72485],{"style":2937},[39,72483],{"className":72484,"style":213},[212],[39,72486,72488],{"className":72487},[217,218,219,220],[39,72489,621],{"className":72490},[83,84,220],[39,72492,673],{"className":72493},[672],[39,72495,72497],{"className":72496},[200],[39,72498,72500],{"className":72499,"style":680},[204],[39,72501],{},[39,72503],{"className":72504,"style":461},[230],[39,72506,788],{"className":72507},[465],[39,72509],{"className":72510,"style":461},[230],[39,72512,72514,72517,72520],{"className":72513},[74],[39,72515],{"className":72516,"style":7584},[78],[39,72518,531],{"className":72519,"style":565},[83,84],[39,72521,72523,72526],{"className":72522},[83],[39,72524,3181],{"className":72525},[83,84],[39,72527,72529],{"className":72528},[192],[39,72530,72532,72561],{"className":72531},[196,649],[39,72533,72535,72558],{"className":72534},[200],[39,72536,72538],{"className":72537,"style":963},[204],[39,72539,72540,72543],{"style":966},[39,72541],{"className":72542,"style":213},[212],[39,72544,72546],{"className":72545},[217,218,219,220],[39,72547,72549,72552,72555],{"className":72548},[83,220],[39,72550,621],{"className":72551},[83,84,220],[39,72553,788],{"className":72554},[465,220],[39,72556,546],{"className":72557},[83,220],[39,72559,673],{"className":72560},[672],[39,72562,72564],{"className":72563},[200],[39,72565,72567],{"className":72566,"style":994},[204],[39,72568],{}," 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 ",[39,72571,72573,72593],{"className":72572,"translate":43},[42],[39,72574,72576],{"className":72575},[47],[49,72577,72578],{"xmlns":51},[53,72579,72580,72590],{},[56,72581,72582,72584,72586,72588],{},[59,72583,4230],{},[133,72585,793],{},[59,72587,11772],{"mathvariant":617},[59,72589,4230],{},[63,72591,72592],{"encoding":65},"V=\\mathcal{T}V",[39,72594,72596,72614],{"className":72595,"ariaHidden":70},[69],[39,72597,72599,72602,72605,72608,72611],{"className":72598},[74],[39,72600],{"className":72601,"style":79},[78],[39,72603,4230],{"className":72604,"style":461},[83,84],[39,72606],{"className":72607,"style":231},[230],[39,72609,793],{"className":72610},[235],[39,72612],{"className":72613,"style":231},[230],[39,72615,72617,72620,72623],{"className":72616},[74],[39,72618],{"className":72619,"style":79},[78],[39,72621,11772],{"className":72622,"style":11793},[83,641],[39,72624,4230],{"className":72625,"style":461},[83,84],"。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。",[11,72628,72629,72630,72715],{},"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 ",[39,72631,72633,72656],{"className":72632,"translate":43},[42],[39,72634,72636],{"className":72635},[47],[49,72637,72638],{"xmlns":51},[53,72639,72640,72654],{},[56,72641,72642,72644,72646,72648],{},[59,72643,14373],{},[133,72645,5378],{},[59,72647,531],{},[138,72649,72650,72652],{},[59,72651,131],{},[59,72653,2098],{},[63,72655,14654],{"encoding":65},[39,72657,72659,72677],{"className":72658,"ariaHidden":70},[69],[39,72660,72662,72665,72668,72671,72674],{"className":72661},[74],[39,72663],{"className":72664,"style":454},[78],[39,72666,14373],{"className":72667,"style":14420},[83,84],[39,72669],{"className":72670,"style":461},[230],[39,72672,5378],{"className":72673},[465],[39,72675],{"className":72676,"style":461},[230],[39,72678,72680,72683,72686],{"className":72679},[74],[39,72681],{"className":72682,"style":6452},[78],[39,72684,531],{"className":72685,"style":565},[83,84],[39,72687,72689,72692],{"className":72688},[83],[39,72690,131],{"className":72691,"style":178},[83,84],[39,72693,72695],{"className":72694},[192],[39,72696,72698],{"className":72697},[196],[39,72699,72701],{"className":72700},[200],[39,72702,72704],{"className":72703,"style":6410},[204],[39,72705,72706,72709],{"style":208},[39,72707],{"className":72708,"style":213},[212],[39,72710,72712],{"className":72711},[217,218,219,220],[39,72713,2098],{"className":72714,"style":2114},[83,84,220],"。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。",[11,72717,72718],{},"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":72720,"searchDepth":72721,"depth":72721,"links":72722},"",2,[72723,72729,72736,72743,72750,72756,72761,72766],{"id":23,"depth":72721,"text":24,"children":72724},[72725,72727,72728],{"id":28,"depth":72726,"text":29},3,{"id":2586,"depth":72726,"text":2587},{"id":6488,"depth":72726,"text":6489},{"id":10038,"depth":72721,"text":10039,"children":72730},[72731,72732,72733,72734,72735],{"id":10042,"depth":72726,"text":10043},{"id":11748,"depth":72726,"text":11749},{"id":12928,"depth":72726,"text":12929},{"id":17330,"depth":72726,"text":17331},{"id":19743,"depth":72726,"text":19744},{"id":21349,"depth":72721,"text":21350,"children":72737},[72738,72739,72740,72741,72742],{"id":21353,"depth":72726,"text":21354},{"id":23677,"depth":72726,"text":23678},{"id":26631,"depth":72726,"text":26632},{"id":29396,"depth":72726,"text":29397},{"id":34082,"depth":72726,"text":34083},{"id":36535,"depth":72721,"text":36536,"children":72744},[72745,72746,72747,72748,72749],{"id":36539,"depth":72726,"text":36540},{"id":37349,"depth":72726,"text":37350},{"id":45346,"depth":72726,"text":45347},{"id":47168,"depth":72726,"text":47169},{"id":48417,"depth":72726,"text":48418},{"id":51028,"depth":72721,"text":51029,"children":72751},[72752,72753,72754,72755],{"id":51032,"depth":72726,"text":51033},{"id":53366,"depth":72726,"text":53367},{"id":58583,"depth":72726,"text":58584},{"id":61601,"depth":72726,"text":61602},{"id":62687,"depth":72721,"text":62688,"children":72757},[72758,72759,72760],{"id":62691,"depth":72726,"text":62692},{"id":68009,"depth":72726,"text":68010},{"id":68297,"depth":72726,"text":68298},{"id":68905,"depth":72721,"text":68906,"children":72762},[72763,72764,72765],{"id":68909,"depth":72726,"text":68910},{"id":69898,"depth":72726,"text":69899},{"id":71203,"depth":72726,"text":71204},{"id":72350,"depth":72721,"text":72350},false,true,"md",null,{},"\u002Fblog\u002F2026\u002F0015-the-bellman-equation-and-its-mathematical-properties",{"title":6,"description":13},"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。",[72777,72778],"machine-learning","mathematics","2026-11-09T23:49:14+08:00","hpJs47-d_ZZWWr8USnYwX8xzwtvDogQjVGqetsjCV2g",{"id":72782,"title":72783,"body":72784,"description":72788,"draft":72767,"enableCodeHighlight":72767,"enableCodeHightlight":72767,"enableComment":72768,"enableD2":72767,"enableKatex":72767,"extension":72769,"image":72720,"important":72767,"location":72770,"meta":72965,"navigation":72768,"ogImage":72770,"path":72966,"seo":72967,"stem":72968,"summary":72969,"tags":72970,"time":72972,"__hash__":72973},"blog\u002Fblog\u002F2026\u002F0016-yu-zhu-de-fei-xu.md","预筑的废墟：苏联美学里永恒与必朽",{"type":8,"value":72785,"toc":72963},[72786,72789,72795,72798,72801,72804,72807,72810,72815,72818,72821,72824,72827,72833,72836,72839,72842,72845,72848,72854,72857,72862,72865,72868,72873,72876,72879,72882,72885,72888,72891,72894,72897,72900,72903,72906,72909,72914,72917,72920,72923,72925,72928,72931,72941,72947,72950,72953,72958],[11,72787,72788],{},"苏联美学，永远具有一种冷冽的悲伤。它最擅长制造宏大的未来，也最清楚未来无法归还什么。它用巨像对抗死亡，用纪念碑对抗遗忘，用胜利解释牺牲，用不断向上的线条告诉所有人，继续向前。这种美学之所以永远带着寒意，是因为它最擅长的事情，恰恰是用永恒的尺度去丈量那些注定消逝的生命。",[11,72790,72791],{},[72792,72793],"img",{"alt":72720,"src":72794},"https:\u002F\u002Fimage-assets.dreams.plus\u002F202611112205632.png",[11,72796,72797],{},"尤其是构成主义与砼核风格，痴迷于巨大。那些混凝土浇筑的几何体，直插云霄的纪念碑，压迫感十足的工业建筑群，本质上是一种对抗遗忘的视觉宣言。列宁时期的纪念碑宣传计划，就明确将雕塑与建筑视为向大众传达政治理念的沟通工具，视觉符号在某种程度上被当作现实的替代品，用以确认理想已经实现。",[11,72799,72800],{},"但这里的悲剧性在于，巨像越是庞大，站在它面前的人就越显得渺小与短暂。在《雁南飞》等电影的构图中，人物常常被置于宏伟建筑之下，成为景观的附属品。冰冷的钢筋水泥占据了画面的绝大部分，传递出一种个人在集体面前的无能与身不由己。",[11,72802,72803],{},"巨物的压迫感和崇高感本就是一体。人站在纪念碑前，被收进宏大叙事里，你的牺牲、痛苦、失去，会被打磨成勋章，放进集体通往未来的序列。在这套语法里，个体的消逝不再是纯粹的湮灭，而是通向那个光明前途的耗材。肖斯塔科维奇第五交响曲那个著名的分歧刚好是绝佳注脚：外人听见胜利的凯歌，亲历者听见的是强行压下去的哀恸，是不得不乐观。曲调明亮，底色是吞下去的苦难。",[11,72805,72806],{},"这种美学在赋予观者崇高的使命感的同时，也让他们感受到了冰冷的压迫感。悲伤，就藏在那种你为之奉献一切的东西，比你活得更久的预感里。​用胜利解释牺牲，而解释本身需要反复讲述。",[11,72808,72809],{},"苏联美学最擅长将苦难转化为勋章。卫国战争题材的电影总是悲壮的，因为那场战争本身就是悲壮的，数千万人的伤亡，每一场胜利的代价都过于艰涩。但宏大的叙事要求它必须壮：烈士的血不能白流，复仇必须理直气壮。",[11,72811,72812],{},[72792,72813],{"alt":72720,"src":72814},"https:\u002F\u002Fimage-assets.dreams.plus\u002F202611112147906.png",[11,72816,72817],{},"于是，艺术在这里承担了一个近乎残酷的职责，为无法挽回的损失赋予意义。肖斯塔科维奇的音乐就体现了这种撕裂感。他的第五交响曲末乐章，美国指挥家往往演奏得激昂欢腾，而苏联指挥家穆拉文斯基却处理得更慢更沉重。",[11,72819,72820],{},"那种拖滞感，像是在漫长而不安宁的等待中，被迫咽下所有说不出口的东西。音乐里的乐观与悲观同时存在，就像一个人用最阳光的曲调唱着最凄冷的歌词。",[11,72822,72823],{},"​那些不断向上的线条，例如，尖塔、直插天际的楼宇，仰望星空的人物姿态，本质上是一种方向性的命令。它们宣告未来是向上的、进步的，必然到来的。",[11,72825,72826],{},"尤其是苏联航天纪念碑。它们有一种非常迷人的矛盾：人类终于离开地球，向宇宙进发；但承载这种理想的，却是沉重、冰冷、几乎不朽的钢铁和混凝土。仿佛人类试图用最坚硬的物质，表达最遥远的梦想。",[11,72828,72829],{},[72792,72830],{"alt":72831,"src":72832},"苏联航天纪念碑","https:\u002F\u002Fimage-assets.dreams.plus\u002F202611112146379.png",[11,72834,72835],{},"而今天，当我们重新观看这些建筑时，它们又有了另一层意义：它们证明了人类曾经如此认真地相信未来。",[11,72837,72838],{},"然而，悲剧的种子就埋在这个必然的种子里。\n当未来的必然性一旦失去，所有的向上线条都变成了悬在半空的讽刺。就像那首《最美好的前途》，曲调中满是经典的俄罗斯式忧伤，歌词却期盼着最美好的前途。可这首歌诞生后仅仅几年，苏联就解体了。",[11,72840,72841],{},"曲调是斯拉夫骨血里的小调忧伤，歌词是滚烫，笃定的未来许诺。它诞生的时候，所有人认真地相信那个前途。没过多久，许诺的体系崩塌。于是旋律和歌词之间的裂痕，不再只是艺术手法，变成历史本身的反讽。它提前歌颂的未来，没有如约而至。那些向上的线条，一下子悬在半空，不再指向天国一样的明天，只剩下孤零零的、凝固的期待。",[11,72843,72844],{},"曾经从零点起步奔向光明未来的承诺，变成了对个人命运无可奈何的隐喻。用最凄冷的曲调歌唱最阳光的歌词，这种形式与内容的错位，本身就是一种悲剧。苏联美学最擅长将苦难转化为勋章，却忘了勋章本身也是冰冷的。",[11,72846,72847],{},"所以，冷冽，它来自地域与历史的双重寒冷。斯拉夫民族的音乐本就多用自然小调，寒带的广袤与严酷塑造了沉郁的底色。\n更深一层，它来自一个把未来揉和了东正传统宗教来信奉的文明，最终不得不直面未来并未如期归还的结局。苏联美学用巨像对抗死亡，但巨像终将沦为废墟。用纪念碑对抗遗忘。",[11,72849,72850],{},[72792,72851],{"alt":72852,"src":72853},"切博克萨雷的楚瓦什国立歌剧芭蕾舞剧院（Чувашский государственный театр оперы и балета）","https:\u002F\u002Fimage-assets.dreams.plus\u002F202611112206635.png",[11,72855,72856],{},"普通的崇高，是人和宏大之物一起奔赴希望。而这里的崇高，暗含一条无声的伏笔。它用永恒的尺度包裹转瞬即逝的生命，许诺你的不朽，是消融在集体叙事里的不朽。所以站在那些砼核墙体，巨大雕塑之下，压迫感和感动是一体两面，被赋予使命，同时渺小，痛苦、无法被补偿的失去，被宏大叙事收编，变成勋章。",[11,72858,72859],{},[72792,72860],{"alt":72720,"src":72861},"https:\u002F\u002Fimage-assets.dreams.plus\u002F202611112213917.png",[11,72863,72864],{},"但纪念碑的主人已被遗忘。用胜利解释牺牲，但牺牲者带走了解释权。",[11,72866,72867],{},"​​用永恒的尺度去丈量那些注定消逝的生命，是苏联美学最残酷也最迷人的地方。斯拉夫的寒带，东正的传统，解体的结局，让那些直插云霄的混凝土几何体，成了对抗遗忘的视觉宣言，也成了个人命运无可奈何的隐喻。",[11,72869,72870],{},[72792,72871],{"alt":72720,"src":72872},"https:\u002F\u002Fimage-assets.dreams.plus\u002F202611112146414.png",[11,72874,72875],{},"这本书《二手时间》写的真很棒，虽然我是 00 后，从未见识过苏联时代。但对苏联的感受，不像是二手怀旧而更像共情。\n令人扼腕的是，那种用永恒丈量必朽的动作本身，在别处也见过这个动作，甚至可能被这个动作丈量过。曾经相信过的人，后来怎么面对信念的消散。宏大巨像崩塌之后，留下来的是一个个被抛下的人，他们曾经把自己的生命押给那个未来。因为这套语法后来被无数地方，无数机构，无数广告反复使用，用到它变成一种通用的修辞外壳。",[11,72877,72878],{},"《二手时间》里那些老兵家属、理想主义者的后代，很多人不是在恨苏联，而是在为\"那个值得牺牲的东西究竟存不存在\"而痛苦——这比单纯的批判深刻得多。这本书里最残酷的地方，在于它没有简单地把苏联解体写成一个政治制度的终结。阿列克谢耶维奇写的根本不只是政权解体，而是一套意义语言的死亡。",[11,72880,72881],{},"它写的是人如何失去自己赖以理解世界的语言。",[11,72883,72884],{},"对于很多经历过苏联时代的人来说，苏联不只是一个国家，也不只是一个政治制度。",[11,72886,72887],{},"它可能是：父母年轻时相信的东西、一生奉献的事业、战争中牺牲的亲人的意义、对公平、平等和人类共同未来的想象、一套关于什么值得追求、什么值得牺牲的价值体系。",[11,72889,72890],{},"当这个体系消失时，真正令人痛苦的问题并不是谁赢了、谁输了。而是：如果我曾经为之奉献一生的东西最终没有实现，那么我的一生究竟意味着什么？",[11,72892,72893],{},"通常我们理解废墟，是因为某种已经存在的文明走向衰亡。罗马遗迹，古希腊神庙，被遗弃的工业设施，都属于这种工业风废土审美。",[11,72895,72896],{},"它和古罗马古希腊废墟不一样。古典废墟是文明的余烬，是曾经有过。苏联的废墟，是一场没能兑现的预演。它建造的时候，就已经在想象胜利的终点；最后留给后世的，是一群认真相信过终点的人，以及一堆静静矗立，等待未来到来，却再也等不到的混凝土巨像。",[11,72898,72899],{},"但苏联美学有一种非常特殊的时间错位。许多建筑、雕塑和艺术作品，本就为了纪念一个尚未完全到来的未来而建造的。不是在怀念过去，而是在提前纪念未来，而且它曾经如此认真地相信未来。",[11,72901,72902],{},"建筑可以保存，制度可以更替，口号可以遗忘，但那些曾经相信过它们的人，却只有一次生命。",[11,72904,72905],{},"而且，他们无法提前知道自己所相信的未来最终会变成什么。",[11,72907,72908],{},"有些废墟之所以令人悲伤，不是因为它们曾经辉煌，而是因为它们曾经如此确信自己不会成为废墟。",[11,72910,72911],{},[72792,72912],{"alt":72720,"src":72913},"https:\u002F\u002Fimage-assets.dreams.plus\u002F202611112148277.png",[11,72915,72916],{},"人类文明似乎一直在重复同一种行为：我们建造比自己寿命更长的东西，试图让自己的信念获得永恒。金字塔、神庙、帝国大道、革命纪念碑、摩天大楼，乃至今天各种代表技术进步的宏伟工程，都有这种冲动。",[11,72918,72919],{},"人类知道自己会死，于是希望自己的劳动、信仰和理想能够活得更久。这本身并没有错。",[11,72921,72922],{},"真正悲伤的是，建筑可以保存，制度可以更替，口号可以遗忘，但那些曾经相信过它们的人，却只有一次生命。",[11,72924,72905],{},[11,72926,72927],{},"这大概就是苏联美学最独特的时间感。",[11,72929,72930],{},"用宏大、恒久的载体，收纳个体短暂的付出，许诺一个更远的、更好的明天”，这个冲动是人类共通的。我们总能在别的地方再次认出它。所以我们共情，不只是隔着历史看另一个遥远国度的废墟，也是看见人永恒的困境：",[72932,72933,72934],"blockquote",{},[11,72935,72936,72937,72940],{},"我们总想造出比自己活得更久的东西，想让自己短暂的存在，借由某种宏大之物获得不朽。  ",[72938,72939],"br",{},"\n可永恒的尺度，永远会温柔又残酷地丈量所有必朽的生命。",[11,72942,72943],{},[72792,72944],{"alt":72945,"src":72946},"《再见，列宁》剧照","https:\u002F\u002Fimage-assets.dreams.plus\u002F202611112152351.png",[11,72948,72949],{},"它的悲伤不是简单的怀旧，也不是对某个政治时代的浪漫化。",[11,72951,72952],{},"而是站在一个已经知道结局的时代，回望那些尚不知道结局的人。他们仰望天空。他们建造纪念碑。他们歌唱最美好的前途。而我们知道，未来并没有按照他们想象的方式到来。但我不认为这就意味着他们的相信毫无价值。",[72932,72954,72955],{},[11,72956,72957],{},"一个未来最终没有实现，并不意味着曾经为它而活的人生，从来没有意义。",[11,72959,72960],{},[72792,72961],{"alt":72720,"src":72962},"https:\u002F\u002Fimage-assets.dreams.plus\u002F202611112213350.png",{"title":72720,"searchDepth":72721,"depth":72721,"links":72964},[],{},"\u002Fblog\u002F2026\u002F0016-yu-zhu-de-fei-xu",{"title":72783,"description":72788},"blog\u002F2026\u002F0016-yu-zhu-de-fei-xu","苏联美学最核心的那层悲剧，不是衰败之后的挽歌，而是预先写给一个落空未来的纪念碑。那些向上收束的线条、厚重的混凝土体量、巨人般的雕塑，不是追忆过往荣光，而是以物质的永恒，为尚未兑现的理想确权。",[72971],"LHPA","2026-11-11T22:15:41+08:00","vnX5uH_SGvoL6cAq_8jfaZHozYxtlD19uK94W97o_P4",{"id":72975,"title":72976,"body":72977,"description":72981,"draft":72767,"enableCodeHighlight":72767,"enableCodeHightlight":72767,"enableComment":72768,"enableD2":72767,"enableKatex":72768,"extension":72769,"image":72720,"important":72767,"location":72770,"meta":125495,"navigation":72768,"ogImage":72770,"path":125496,"seo":125497,"stem":125498,"summary":125499,"tags":125500,"time":125502,"__hash__":125503},"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":72978,"toc":125482},[72979,72982,72985,72988,72991,72994,73333,73671,73754,73820,74029,74274,74388,74724,75326,75329,75813,75931,76297,76329,76499,76869,77567,77727,77977,78009,78112,78115,78118,78180,78392,78395,79771,80033,80805,80808,82390,82910,83527,83530,83533,83887,84320,84645,84648,85021,85024,85027,85152,85376,85570,85761,85834,86351,86529,87114,87506,87911,88625,89107,89110,90041,91152,91363,91801,91804,92177,92180,92211,92454,92591,92594,92966,92969,93385,93614,93813,94291,94845,94900,95032,95319,95508,95615,95618,95836,95869,96093,96096,96099,96492,96495,96580,97319,97694,97697,97832,98777,98780,98783,98786,98989,99161,99331,99804,100244,100347,100711,101293,101296,102113,102116,102119,102180,102183,102413,102855,103235,103727,103730,105085,105208,105366,105842,106610,106613,106617,106770,106971,107176,107299,107864,107867,107928,108214,108385,109379,109382,109868,109944,109947,110406,110639,111040,112863,112939,112942,113170,113661,113664,114112,114615,114618,114756,114986,115076,115079,115678,116082,116812,117152,117494,118274,118277,118280,118283,118286,118351,118517,118667,119108,119111,119275,119730,119733,120106,120168,120795,121744,121847,122205,122406,122477,123273,123480,123483,123486,123489,123604,124090,124122,124125,124701,124704,124707,124859,125111,125114,125117,125120,125123,125155,125158,125476,125479],[11,72980,72981],{},"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”。",[11,72983,72984],{},"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。",[21,72986,72987],{"id":72987},"搬土",[11,72989,72990],{},"先算一道运输题。",[11,72992,72993],{},"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？",[11,72995,72996,72997,73025,73026,73077,73078,73129,73130,73258,73259,73332],{},"设甲运往 A 的吨数是 ",[39,72998,73000,73013],{"className":72999,"translate":43},[42],[39,73001,73003],{"className":73002},[47],[49,73004,73005],{"xmlns":51},[53,73006,73007,73011],{},[56,73008,73009],{},[59,73010,1755],{},[63,73012,1755],{"encoding":65},[39,73014,73016],{"className":73015,"ariaHidden":70},[69],[39,73017,73019,73022],{"className":73018},[74],[39,73020],{"className":73021,"style":290},[78],[39,73023,1755],{"className":73024},[83,84],"。甲的 3 吨要分完，所以甲到 B 是 ",[39,73027,73029,73047],{"className":73028,"translate":43},[42],[39,73030,73032],{"className":73031},[47],[49,73033,73034],{"xmlns":51},[53,73035,73036,73044],{},[56,73037,73038,73040,73042],{},[539,73039,59187],{},[133,73041,5378],{},[59,73043,1755],{},[63,73045,73046],{"encoding":65},"3-x",[39,73048,73050,73068],{"className":73049,"ariaHidden":70},[69],[39,73051,73053,73056,73059,73062,73065],{"className":73052},[74],[39,73054],{"className":73055,"style":17882},[78],[39,73057,59187],{"className":73058},[83],[39,73060],{"className":73061,"style":461},[230],[39,73063,5378],{"className":73064},[465],[39,73066],{"className":73067,"style":461},[230],[39,73069,73071,73074],{"className":73070},[74],[39,73072],{"className":73073,"style":290},[78],[39,73075,1755],{"className":73076},[83,84],"；A 要 2 吨，所以乙到 A 是 ",[39,73079,73081,73099],{"className":73080,"translate":43},[42],[39,73082,73084],{"className":73083},[47],[49,73085,73086],{"xmlns":51},[53,73087,73088,73096],{},[56,73089,73090,73092,73094],{},[539,73091,17359],{},[133,73093,5378],{},[59,73095,1755],{},[63,73097,73098],{"encoding":65},"2-x",[39,73100,73102,73120],{"className":73101,"ariaHidden":70},[69],[39,73103,73105,73108,73111,73114,73117],{"className":73104},[74],[39,73106],{"className":73107,"style":17882},[78],[39,73109,17359],{"className":73110},[83],[39,73112],{"className":73113,"style":461},[230],[39,73115,5378],{"className":73116},[465],[39,73118],{"className":73119,"style":461},[230],[39,73121,73123,73126],{"className":73122},[74],[39,73124],{"className":73125,"style":290},[78],[39,73127,1755],{"className":73128},[83,84],"；乙只有 1 吨，所以乙到 B 是 ",[39,73131,73133,73167],{"className":73132,"translate":43},[42],[39,73134,73136],{"className":73135},[47],[49,73137,73138],{"xmlns":51},[53,73139,73140,73164],{},[56,73141,73142,73144,73146,73148,73150,73152,73154,73156,73158,73160,73162],{},[539,73143,546],{},[133,73145,5378],{},[133,73147,136],{"stretchy":135},[539,73149,17359],{},[133,73151,5378],{},[59,73153,1755],{},[133,73155,161],{"stretchy":135},[133,73157,793],{},[59,73159,1755],{},[133,73161,5378],{},[539,73163,546],{},[63,73165,73166],{"encoding":65},"1-(2-x)=x-1",[39,73168,73170,73188,73209,73230,73249],{"className":73169,"ariaHidden":70},[69],[39,73171,73173,73176,73179,73182,73185],{"className":73172},[74],[39,73174],{"className":73175,"style":17882},[78],[39,73177,546],{"className":73178},[83],[39,73180],{"className":73181,"style":461},[230],[39,73183,5378],{"className":73184},[465],[39,73186],{"className":73187,"style":461},[230],[39,73189,73191,73194,73197,73200,73203,73206],{"className":73190},[74],[39,73192],{"className":73193,"style":245},[78],[39,73195,136],{"className":73196},[182],[39,73198,17359],{"className":73199},[83],[39,73201],{"className":73202,"style":461},[230],[39,73204,5378],{"className":73205},[465],[39,73207],{"className":73208,"style":461},[230],[39,73210,73212,73215,73218,73221,73224,73227],{"className":73211},[74],[39,73213],{"className":73214,"style":245},[78],[39,73216,1755],{"className":73217},[83,84],[39,73219,161],{"className":73220},[263],[39,73222],{"className":73223,"style":231},[230],[39,73225,793],{"className":73226},[235],[39,73228],{"className":73229,"style":231},[230],[39,73231,73233,73237,73240,73243,73246],{"className":73232},[74],[39,73234],{"className":73235,"style":73236},[78],"height:0.6667em;vertical-align:-0.0833em;",[39,73238,1755],{"className":73239},[83,84],[39,73241],{"className":73242,"style":461},[230],[39,73244,5378],{"className":73245},[465],[39,73247],{"className":73248,"style":461},[230],[39,73250,73252,73255],{"className":73251},[74],[39,73253],{"className":73254,"style":2473},[78],[39,73256,546],{"className":73257},[83],"。四个数都得非负，于是 ",[39,73260,73262,73284],{"className":73261,"translate":43},[42],[39,73263,73265],{"className":73264},[47],[49,73266,73267],{"xmlns":51},[53,73268,73269,73281],{},[56,73270,73271,73273,73275,73277,73279],{},[539,73272,546],{},[133,73274,1824],{},[59,73276,1755],{},[133,73278,1824],{},[539,73280,17359],{},[63,73282,73283],{"encoding":65},"1\\le x\\le 2",[39,73285,73287,73305,73323],{"className":73286,"ariaHidden":70},[69],[39,73288,73290,73293,73296,73299,73302],{"className":73289},[74],[39,73291],{"className":73292,"style":14587},[78],[39,73294,546],{"className":73295},[83],[39,73297],{"className":73298,"style":231},[230],[39,73300,1824],{"className":73301},[235],[39,73303],{"className":73304,"style":231},[230],[39,73306,73308,73311,73314,73317,73320],{"className":73307},[74],[39,73309],{"className":73310,"style":59832},[78],[39,73312,1755],{"className":73313},[83,84],[39,73315],{"className":73316,"style":231},[230],[39,73318,1824],{"className":73319},[235],[39,73321],{"className":73322,"style":231},[230],[39,73324,73326,73329],{"className":73325},[74],[39,73327],{"className":73328,"style":2473},[78],[39,73330,17359],{"className":73331},[83],"。总运费",[39,73334,73336],{"className":73335,"translate":43},[746],[39,73337,73339,73418],{"className":73338,"translate":43},[42],[39,73340,73342],{"className":73341},[47],[49,73343,73344],{"xmlns":51,"display":755},[53,73345,73346,73415],{},[56,73347,73348,73350,73352,73354,73356,73358,73360,73362,73364,73366,73368,73370,73372,73374,73376,73378,73380,73382,73384,73386,73388,73390,73392,73394,73396,73398,73400,73402,73404,73407,73409,73411,73413],{},[59,73349,1755],{},[133,73351,8885],{},[539,73353,546],{},[133,73355,788],{},[133,73357,136],{"stretchy":135},[539,73359,59187],{},[133,73361,5378],{},[59,73363,1755],{},[133,73365,161],{"stretchy":135},[133,73367,8885],{},[539,73369,17359],{},[133,73371,788],{},[133,73373,136],{"stretchy":135},[539,73375,17359],{},[133,73377,5378],{},[59,73379,1755],{},[133,73381,161],{"stretchy":135},[133,73383,8885],{},[539,73385,59187],{},[133,73387,788],{},[133,73389,136],{"stretchy":135},[59,73391,1755],{},[133,73393,5378],{},[539,73395,546],{},[133,73397,161],{"stretchy":135},[133,73399,8885],{},[539,73401,546],{},[133,73403,793],{},[539,73405,73406],{},"11",[133,73408,5378],{},[539,73410,59187],{},[59,73412,1755],{},[59,73414,2903],{"mathvariant":145},[63,73416,73417],{"encoding":65},"x\\cdot1+(3-x)\\cdot2+(2-x)\\cdot3+(x-1)\\cdot1=11-3x .",[39,73419,73421,73440,73458,73479,73500,73518,73539,73560,73578,73599,73620,73638,73656],{"className":73420,"ariaHidden":70},[69],[39,73422,73424,73428,73431,73434,73437],{"className":73423},[74],[39,73425],{"className":73426,"style":73427},[78],"height:0.4445em;",[39,73429,1755],{"className":73430},[83,84],[39,73432],{"className":73433,"style":461},[230],[39,73435,8885],{"className":73436},[465],[39,73438],{"className":73439,"style":461},[230],[39,73441,73443,73446,73449,73452,73455],{"className":73442},[74],[39,73444],{"className":73445,"style":17882},[78],[39,73447,546],{"className":73448},[83],[39,73450],{"className":73451,"style":461},[230],[39,73453,788],{"className":73454},[465],[39,73456],{"className":73457,"style":461},[230],[39,73459,73461,73464,73467,73470,73473,73476],{"className":73460},[74],[39,73462],{"className":73463,"style":245},[78],[39,73465,136],{"className":73466},[182],[39,73468,59187],{"className":73469},[83],[39,73471],{"className":73472,"style":461},[230],[39,73474,5378],{"className":73475},[465],[39,73477],{"className":73478,"style":461},[230],[39,73480,73482,73485,73488,73491,73494,73497],{"className":73481},[74],[39,73483],{"className":73484,"style":245},[78],[39,73486,1755],{"className":73487},[83,84],[39,73489,161],{"className":73490},[263],[39,73492],{"className":73493,"style":461},[230],[39,73495,8885],{"className":73496},[465],[39,73498],{"className":73499,"style":461},[230],[39,73501,73503,73506,73509,73512,73515],{"className":73502},[74],[39,73504],{"className":73505,"style":17882},[78],[39,73507,17359],{"className":73508},[83],[39,73510],{"className":73511,"style":461},[230],[39,73513,788],{"className":73514},[465],[39,73516],{"className":73517,"style":461},[230],[39,73519,73521,73524,73527,73530,73533,73536],{"className":73520},[74],[39,73522],{"className":73523,"style":245},[78],[39,73525,136],{"className":73526},[182],[39,73528,17359],{"className":73529},[83],[39,73531],{"className":73532,"style":461},[230],[39,73534,5378],{"className":73535},[465],[39,73537],{"className":73538,"style":461},[230],[39,73540,73542,73545,73548,73551,73554,73557],{"className":73541},[74],[39,73543],{"className":73544,"style":245},[78],[39,73546,1755],{"className":73547},[83,84],[39,73549,161],{"className":73550},[263],[39,73552],{"className":73553,"style":461},[230],[39,73555,8885],{"className":73556},[465],[39,73558],{"className":73559,"style":461},[230],[39,73561,73563,73566,73569,73572,73575],{"className":73562},[74],[39,73564],{"className":73565,"style":17882},[78],[39,73567,59187],{"className":73568},[83],[39,73570],{"className":73571,"style":461},[230],[39,73573,788],{"className":73574},[465],[39,73576],{"className":73577,"style":461},[230],[39,73579,73581,73584,73587,73590,73593,73596],{"className":73580},[74],[39,73582],{"className":73583,"style":245},[78],[39,73585,136],{"className":73586},[182],[39,73588,1755],{"className":73589},[83,84],[39,73591],{"className":73592,"style":461},[230],[39,73594,5378],{"className":73595},[465],[39,73597],{"className":73598,"style":461},[230],[39,73600,73602,73605,73608,73611,73614,73617],{"className":73601},[74],[39,73603],{"className":73604,"style":245},[78],[39,73606,546],{"className":73607},[83],[39,73609,161],{"className":73610},[263],[39,73612],{"className":73613,"style":461},[230],[39,73615,8885],{"className":73616},[465],[39,73618],{"className":73619,"style":461},[230],[39,73621,73623,73626,73629,73632,73635],{"className":73622},[74],[39,73624],{"className":73625,"style":2473},[78],[39,73627,546],{"className":73628},[83],[39,73630],{"className":73631,"style":231},[230],[39,73633,793],{"className":73634},[235],[39,73636],{"className":73637,"style":231},[230],[39,73639,73641,73644,73647,73650,73653],{"className":73640},[74],[39,73642],{"className":73643,"style":17882},[78],[39,73645,73406],{"className":73646},[83],[39,73648],{"className":73649,"style":461},[230],[39,73651,5378],{"className":73652},[465],[39,73654],{"className":73655,"style":461},[230],[39,73657,73659,73662,73665,73668],{"className":73658},[74],[39,73660],{"className":73661,"style":2473},[78],[39,73663,59187],{"className":73664},[83],[39,73666,1755],{"className":73667},[83,84],[39,73669,2903],{"className":73670},[83],[11,73672,73673,73701,73702,73753],{},[39,73674,73676,73689],{"className":73675,"translate":43},[42],[39,73677,73679],{"className":73678},[47],[49,73680,73681],{"xmlns":51},[53,73682,73683,73687],{},[56,73684,73685],{},[59,73686,1755],{},[63,73688,1755],{"encoding":65},[39,73690,73692],{"className":73691,"ariaHidden":70},[69],[39,73693,73695,73698],{"className":73694},[74],[39,73696],{"className":73697,"style":290},[78],[39,73699,1755],{"className":73700},[83,84]," 越大越省，取 ",[39,73703,73705,73723],{"className":73704,"translate":43},[42],[39,73706,73708],{"className":73707},[47],[49,73709,73710],{"xmlns":51},[53,73711,73712,73720],{},[56,73713,73714,73716,73718],{},[59,73715,1755],{},[133,73717,793],{},[539,73719,17359],{},[63,73721,73722],{"encoding":65},"x=2",[39,73724,73726,73744],{"className":73725,"ariaHidden":70},[69],[39,73727,73729,73732,73735,73738,73741],{"className":73728},[74],[39,73730],{"className":73731,"style":290},[78],[39,73733,1755],{"className":73734},[83,84],[39,73736],{"className":73737,"style":231},[230],[39,73739,793],{"className":73740},[235],[39,73742],{"className":73743,"style":231},[230],[39,73745,73747,73750],{"className":73746},[74],[39,73748],{"className":73749,"style":2473},[78],[39,73751,17359],{"className":73752},[83],"，总运费 5 块：甲到 A 运 2 吨，甲到 B 运 1 吨，乙到 B 运 1 吨。",[11,73755,73756,73757,73787,73788,73819],{},"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 ",[39,73758,73760,73775],{"className":73759,"translate":43},[42],[39,73761,73763],{"className":73762},[47],[49,73764,73765],{"xmlns":51},[53,73766,73767,73772],{},[56,73768,73769],{},[59,73770,73771],{},"μ",[63,73773,73774],{"encoding":65},"\\mu",[39,73776,73778],{"className":73777,"ariaHidden":70},[69],[39,73779,73781,73784],{"className":73780},[74],[39,73782],{"className":73783,"style":2654},[78],[39,73785,73771],{"className":73786},[83,84]," 和一个坑 ",[39,73789,73791,73806],{"className":73790,"translate":43},[42],[39,73792,73794],{"className":73793},[47],[49,73795,73796],{"xmlns":51},[53,73797,73798,73803],{},[56,73799,73800],{},[59,73801,73802],{},"ν",[63,73804,73805],{"encoding":65},"\\nu",[39,73807,73809],{"className":73808,"ariaHidden":70},[69],[39,73810,73812,73815],{"className":73811},[74],[39,73813],{"className":73814,"style":290},[78],[39,73816,73802],{"className":73817,"style":73818},[83,84],"margin-right:0.0637em;","，怎么搬最省。",[11,73821,73822,73823,73900,73901,74000,74001,742],{},"他的写法是找一个映射 ",[39,73824,73826,73849],{"className":73825,"translate":43},[42],[39,73827,73829],{"className":73828},[47],[49,73830,73831],{"xmlns":51},[53,73832,73833,73846],{},[56,73834,73835,73837,73839,73841,73843],{},[59,73836,11772],{},[133,73838,404],{},[59,73840,29420],{"mathvariant":617},[133,73842,418],{},[59,73844,73845],{"mathvariant":617},"Y",[63,73847,73848],{"encoding":65},"T:\\mathcal X\\to\\mathcal Y",[39,73850,73852,73870,73889],{"className":73851,"ariaHidden":70},[69],[39,73853,73855,73858,73861,73864,73867],{"className":73854},[74],[39,73856],{"className":73857,"style":79},[78],[39,73859,11772],{"className":73860,"style":178},[83,84],[39,73862],{"className":73863,"style":231},[230],[39,73865,404],{"className":73866},[235],[39,73868],{"className":73869,"style":231},[230],[39,73871,73873,73876,73880,73883,73886],{"className":73872},[74],[39,73874],{"className":73875,"style":79},[78],[39,73877,29420],{"className":73878,"style":73879},[83,641],"margin-right:0.1464em;",[39,73881],{"className":73882,"style":231},[230],[39,73884,418],{"className":73885},[235],[39,73887],{"className":73888,"style":231},[230],[39,73890,73892,73896],{"className":73891},[74],[39,73893],{"className":73894,"style":73895},[78],"height:0.7805em;vertical-align:-0.0972em;",[39,73897,73845],{"className":73898,"style":73899},[83,641],"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 ",[39,73902,73904,73929],{"className":73903,"translate":43},[42],[39,73905,73907],{"className":73906},[47],[49,73908,73909],{"xmlns":51},[53,73910,73911,73926],{},[56,73912,73913,73920,73922,73924],{},[613,73914,73915,73917],{},[59,73916,11772],{},[59,73918,73919],{"mathvariant":145},"#",[59,73921,73771],{},[133,73923,793],{},[59,73925,73802],{},[63,73927,73928],{"encoding":65},"T_\\#\\mu=\\nu",[39,73930,73932,73991],{"className":73931,"ariaHidden":70},[69],[39,73933,73935,73938,73979,73982,73985,73988],{"className":73934},[74],[39,73936],{"className":73937,"style":35350},[78],[39,73939,73941,73944],{"className":73940},[83],[39,73942,11772],{"className":73943,"style":178},[83,84],[39,73945,73947],{"className":73946},[192],[39,73948,73950,73971],{"className":73949},[196,649],[39,73951,73953,73968],{"className":73952},[200],[39,73954,73956],{"className":73955,"style":3419},[204],[39,73957,73959,73962],{"style":73958},"top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;",[39,73960],{"className":73961,"style":213},[212],[39,73963,73965],{"className":73964},[217,218,219,220],[39,73966,73919],{"className":73967},[83,220],[39,73969,673],{"className":73970},[672],[39,73972,73974],{"className":73973},[200],[39,73975,73977],{"className":73976,"style":35396},[204],[39,73978],{},[39,73980,73771],{"className":73981},[83,84],[39,73983],{"className":73984,"style":231},[230],[39,73986,793],{"className":73987},[235],[39,73989],{"className":73990,"style":231},[230],[39,73992,73994,73997],{"className":73993},[74],[39,73995],{"className":73996,"style":290},[78],[39,73998,73802],{"className":73999,"style":73818},[83,84],"，即对任意可测集 ",[39,74002,74004,74017],{"className":74003,"translate":43},[42],[39,74005,74007],{"className":74006},[47],[49,74008,74009],{"xmlns":51},[53,74010,74011,74015],{},[56,74012,74013],{},[59,74014,19772],{},[63,74016,19772],{"encoding":65},[39,74018,74020],{"className":74019,"ariaHidden":70},[69],[39,74021,74023,74026],{"className":74022},[74],[39,74024],{"className":74025,"style":79},[78],[39,74027,19772],{"className":74028,"style":19812},[83,84],[39,74030,74032],{"className":74031,"translate":43},[746],[39,74033,74035,74102],{"className":74034,"translate":43},[42],[39,74036,74038],{"className":74037},[47],[49,74039,74040],{"xmlns":51,"display":755},[53,74041,74042,74099],{},[56,74043,74044,74046,74052,74054,74056,74058,74060,74062,74064,74066,74069,74079,74081,74083,74085,74087,74089,74091,74093,74095,74097],{},[133,74045,136],{"stretchy":135},[613,74047,74048,74050],{},[59,74049,11772],{},[59,74051,73919],{"mathvariant":145},[59,74053,73771],{},[133,74055,161],{"stretchy":135},[133,74057,136],{"stretchy":135},[59,74059,19772],{},[133,74061,161],{"stretchy":135},[133,74063,793],{},[59,74065,73771],{},[133,74067,136],{"fence":70,"stretchy":70,"minsize":74068,"maxsize":74068},"1.2em",[138,74070,74071,74073],{},[59,74072,11772],{},[56,74074,74075,74077],{},[133,74076,5378],{},[539,74078,546],{},[133,74080,136],{"stretchy":135},[59,74082,19772],{},[133,74084,161],{"stretchy":135},[133,74086,161],{"fence":70,"stretchy":70,"minsize":74068,"maxsize":74068},[133,74088,793],{},[59,74090,73802],{},[133,74092,136],{"stretchy":135},[59,74094,19772],{},[133,74096,161],{"stretchy":135},[133,74098,155],{"separator":70},[63,74100,74101],{"encoding":65},"(T_\\#\\mu)(B)=\\mu\\bigl(T^{-1}(B)\\bigr)=\\nu(B),",[39,74103,74105,74178,74253],{"className":74104,"ariaHidden":70},[69],[39,74106,74108,74111,74114,74154,74157,74160,74163,74166,74169,74172,74175],{"className":74107},[74],[39,74109],{"className":74110,"style":41560},[78],[39,74112,136],{"className":74113},[182],[39,74115,74117,74120],{"className":74116},[83],[39,74118,11772],{"className":74119,"style":178},[83,84],[39,74121,74123],{"className":74122},[192],[39,74124,74126,74146],{"className":74125},[196,649],[39,74127,74129,74143],{"className":74128},[200],[39,74130,74132],{"className":74131,"style":3419},[204],[39,74133,74134,74137],{"style":73958},[39,74135],{"className":74136,"style":213},[212],[39,74138,74140],{"className":74139},[217,218,219,220],[39,74141,73919],{"className":74142},[83,220],[39,74144,673],{"className":74145},[672],[39,74147,74149],{"className":74148},[200],[39,74150,74152],{"className":74151,"style":35396},[204],[39,74153],{},[39,74155,73771],{"className":74156},[83,84],[39,74158,161],{"className":74159},[263],[39,74161,136],{"className":74162},[182],[39,74164,19772],{"className":74165,"style":19812},[83,84],[39,74167,161],{"className":74168},[263],[39,74170],{"className":74171,"style":231},[230],[39,74173,793],{"className":74174},[235],[39,74176],{"className":74177,"style":231},[230],[39,74179,74181,74185,74188,74194,74229,74232,74235,74238,74244,74247,74250],{"className":74180},[74],[39,74182],{"className":74183,"style":74184},[78],"height:1.2141em;vertical-align:-0.35em;",[39,74186,73771],{"className":74187},[83,84],[39,74189,74191],{"className":74190},[182],[39,74192,136],{"className":74193},[18520,4009],[39,74195,74197,74200],{"className":74196},[83],[39,74198,11772],{"className":74199,"style":178},[83,84],[39,74201,74203],{"className":74202},[192],[39,74204,74206],{"className":74205},[196],[39,74207,74209],{"className":74208},[200],[39,74210,74212],{"className":74211,"style":14940},[204],[39,74213,74214,74217],{"style":1027},[39,74215],{"className":74216,"style":213},[212],[39,74218,74220],{"className":74219},[217,218,219,220],[39,74221,74223,74226],{"className":74222},[83,220],[39,74224,5378],{"className":74225},[83,220],[39,74227,546],{"className":74228},[83,220],[39,74230,136],{"className":74231},[182],[39,74233,19772],{"className":74234,"style":19812},[83,84],[39,74236,161],{"className":74237},[263],[39,74239,74241],{"className":74240},[263],[39,74242,161],{"className":74243},[18520,4009],[39,74245],{"className":74246,"style":231},[230],[39,74248,793],{"className":74249},[235],[39,74251],{"className":74252,"style":231},[230],[39,74254,74256,74259,74262,74265,74268,74271],{"className":74255},[74],[39,74257],{"className":74258,"style":245},[78],[39,74260,73802],{"className":74261,"style":73818},[83,84],[39,74263,136],{"className":74264},[182],[39,74266,19772],{"className":74267,"style":19812},[83,84],[39,74269,161],{"className":74270},[263],[39,74272,155],{"className":74273},[252],[11,74275,74276,74277,74387],{},"代价是 ",[39,74278,74280,74324],{"className":74279,"translate":43},[42],[39,74281,74283],{"className":74282},[47],[49,74284,74285],{"xmlns":51},[53,74286,74287,74321],{},[56,74288,74289,74291,74293,74295,74297,74299,74301,74303,74305,74307,74309,74311,74313,74315,74317,74319],{},[133,74290,69087],{},[59,74292,24077],{},[133,74294,136],{"stretchy":135},[59,74296,1755],{},[133,74298,155],{"separator":70},[59,74300,11772],{},[133,74302,136],{"stretchy":135},[59,74304,1755],{},[133,74306,161],{"stretchy":135},[133,74308,161],{"stretchy":135},[35841,74310,35843],{},[59,74312,29425],{},[59,74314,73771],{},[133,74316,136],{"stretchy":135},[59,74318,1755],{},[133,74320,161],{"stretchy":135},[63,74322,74323],{"encoding":65},"\\int c(x,T(x))\\,d\\mu(x)",[39,74325,74327],{"className":74326,"ariaHidden":70},[69],[39,74328,74330,74334,74338,74341,74344,74347,74350,74353,74356,74359,74362,74365,74369,74372,74375,74378,74381,74384],{"className":74329},[74],[39,74331],{"className":74332,"style":74333},[78],"height:1.1111em;vertical-align:-0.3061em;",[39,74335,69087],{"className":74336,"style":74337},[1744,3339,3952],"margin-right:0.1945em;position:relative;top:-0.0006em;",[39,74339],{"className":74340,"style":256},[230],[39,74342,24077],{"className":74343},[83,84],[39,74345,136],{"className":74346},[182],[39,74348,1755],{"className":74349},[83,84],[39,74351,155],{"className":74352},[252],[39,74354],{"className":74355,"style":256},[230],[39,74357,11772],{"className":74358,"style":178},[83,84],[39,74360,136],{"className":74361},[182],[39,74363,1755],{"className":74364},[83,84],[39,74366,74368],{"className":74367},[263],"))",[39,74370],{"className":74371,"style":256},[230],[39,74373,29425],{"className":74374},[83,84],[39,74376,73771],{"className":74377},[83,84],[39,74379,136],{"className":74380},[182],[39,74382,1755],{"className":74383},[83,84],[39,74385,161],{"className":74386},[263],"。于是 Monge 问题就是",[39,74389,74391],{"className":74390,"translate":43},[746],[39,74392,74394,74477],{"className":74393,"translate":43},[42],[39,74395,74397],{"className":74396},[47],[49,74398,74399],{"xmlns":51,"display":755},[53,74400,74401,74474],{},[56,74402,74403,74414,74417,74423,74425,74427,74429,74431,74433,74435,74437,74439,74441,74443,74445,74447,74449,74451,74453,74455,74458,74460,74466,74468,74470,74472],{},[6556,74404,74405,74412],{},[56,74406,74407,74410],{},[59,74408,74409],{},"inf",[133,74411,1690],{},[59,74413,11772],{},[35841,74415,74416],{}," ",[613,74418,74419,74421],{},[133,74420,69087],{},[59,74422,29420],{"mathvariant":617},[59,74424,24077],{},[133,74426,136],{"fence":70,"stretchy":70,"minsize":74068,"maxsize":74068},[59,74428,1755],{},[133,74430,155],{"separator":70},[59,74432,11772],{},[133,74434,136],{"stretchy":135},[59,74436,1755],{},[133,74438,161],{"stretchy":135},[133,74440,161],{"fence":70,"stretchy":70,"minsize":74068,"maxsize":74068},[35841,74442,35843],{},[59,74444,29425],{},[59,74446,73771],{},[133,74448,136],{"stretchy":135},[59,74450,1755],{},[133,74452,161],{"stretchy":135},[230,74454],{"width":6891},[35841,74456,74457],{},"s.t.",[230,74459],{"width":6891},[613,74461,74462,74464],{},[59,74463,11772],{},[59,74465,73919],{"mathvariant":145},[59,74467,73771],{},[133,74469,793],{},[59,74471,73802],{},[59,74473,2903],{"mathvariant":145},[63,74475,74476],{"encoding":65},"\\inf_{T}\\ \\int_{\\mathcal X} c\\bigl(x,T(x)\\bigr)\\,d\\mu(x)\n\\quad\\text{s.t.}\\quad T_\\#\\mu=\\nu .",[39,74478,74480,74712],{"className":74479,"ariaHidden":70},[69],[39,74481,74483,74487,74541,74544,74547,74590,74593,74596,74602,74605,74608,74611,74614,74617,74620,74623,74629,74632,74635,74638,74641,74644,74647,74650,74657,74660,74700,74703,74706,74709],{"className":74482},[74],[39,74484],{"className":74485,"style":74486},[78],"height:2.2719em;vertical-align:-0.9119em;",[39,74488,74490],{"className":74489},[1744,3294],[39,74491,74493,74532],{"className":74492},[196,649],[39,74494,74496,74529],{"className":74495},[200],[39,74497,74499,74514],{"className":74498,"style":28824},[204],[39,74500,74502,74505],{"style":74501},"top:-2.3557em;margin-left:0em;",[39,74503],{"className":74504,"style":935},[212],[39,74506,74508],{"className":74507},[217,218,219,220],[39,74509,74511],{"className":74510},[83,220],[39,74512,11772],{"className":74513,"style":178},[83,84,220],[39,74515,74516,74519],{"style":21599},[39,74517],{"className":74518,"style":935},[212],[39,74520,74521],{},[39,74522,74524,74525],{"className":74523},[1744],"in",[39,74526,74528],{"style":74527},"margin-right:0.0778em;","f",[39,74530,673],{"className":74531},[672],[39,74533,74535],{"className":74534},[200],[39,74536,74539],{"className":74537,"style":74538},[204],"height:0.7443em;",[39,74540],{},[39,74542,74416],{"className":74543},[230],[39,74545],{"className":74546,"style":256},[230],[39,74548,74550,74553],{"className":74549},[1744],[39,74551,69087],{"className":74552,"style":69380},[1744,3339,3340],[39,74554,74556],{"className":74555},[192],[39,74557,74559,74582],{"className":74558},[196,649],[39,74560,74562,74579],{"className":74561},[200],[39,74563,74565],{"className":74564,"style":69393},[204],[39,74566,74567,74570],{"style":69396},[39,74568],{"className":74569,"style":213},[212],[39,74571,74573],{"className":74572},[217,218,219,220],[39,74574,74576],{"className":74575},[83,220],[39,74577,29420],{"className":74578,"style":73879},[83,641,220],[39,74580,673],{"className":74581},[672],[39,74583,74585],{"className":74584},[200],[39,74586,74588],{"className":74587,"style":69415},[204],[39,74589],{},[39,74591],{"className":74592,"style":256},[230],[39,74594,24077],{"className":74595},[83,84],[39,74597,74599],{"className":74598},[182],[39,74600,136],{"className":74601},[18520,4009],[39,74603,1755],{"className":74604},[83,84],[39,74606,155],{"className":74607},[252],[39,74609],{"className":74610,"style":256},[230],[39,74612,11772],{"className":74613,"style":178},[83,84],[39,74615,136],{"className":74616},[182],[39,74618,1755],{"className":74619},[83,84],[39,74621,161],{"className":74622},[263],[39,74624,74626],{"className":74625},[263],[39,74627,161],{"className":74628},[18520,4009],[39,74630],{"className":74631,"style":256},[230],[39,74633,29425],{"className":74634},[83,84],[39,74636,73771],{"className":74637},[83,84],[39,74639,136],{"className":74640},[182],[39,74642,1755],{"className":74643},[83,84],[39,74645,161],{"className":74646},[263],[39,74648],{"className":74649,"style":7240},[230],[39,74651,74654],{"className":74652},[83,74653],"text",[39,74655,74457],{"className":74656},[83],[39,74658],{"className":74659,"style":7240},[230],[39,74661,74663,74666],{"className":74662},[83],[39,74664,11772],{"className":74665,"style":178},[83,84],[39,74667,74669],{"className":74668},[192],[39,74670,74672,74692],{"className":74671},[196,649],[39,74673,74675,74689],{"className":74674},[200],[39,74676,74678],{"className":74677,"style":3419},[204],[39,74679,74680,74683],{"style":73958},[39,74681],{"className":74682,"style":213},[212],[39,74684,74686],{"className":74685},[217,218,219,220],[39,74687,73919],{"className":74688},[83,220],[39,74690,673],{"className":74691},[672],[39,74693,74695],{"className":74694},[200],[39,74696,74698],{"className":74697,"style":35396},[204],[39,74699],{},[39,74701,73771],{"className":74702},[83,84],[39,74704],{"className":74705,"style":231},[230],[39,74707,793],{"className":74708},[235],[39,74710],{"className":74711,"style":231},[230],[39,74713,74715,74718,74721],{"className":74714},[74],[39,74716],{"className":74717,"style":290},[78],[39,74719,73802],{"className":74720,"style":73818},[83,84],[39,74722,2903],{"className":74723},[83],[11,74725,74726,74727,74755,74756,74784,74785,74813,74814,74910,74911,1607,75223,75267,75268,75296,75297,75325],{},"这个提法有一个硬伤：",[39,74728,74730,74743],{"className":74729,"translate":43},[42],[39,74731,74733],{"className":74732},[47],[49,74734,74735],{"xmlns":51},[53,74736,74737,74741],{},[56,74738,74739],{},[59,74740,11772],{},[63,74742,11772],{"encoding":65},[39,74744,74746],{"className":74745,"ariaHidden":70},[69],[39,74747,74749,74752],{"className":74748},[74],[39,74750],{"className":74751,"style":79},[78],[39,74753,11772],{"className":74754,"style":178},[83,84]," 必须是确定性的，一个产地只能有一条固定去向。如果 ",[39,74757,74759,74772],{"className":74758,"translate":43},[42],[39,74760,74762],{"className":74761},[47],[49,74763,74764],{"xmlns":51},[53,74765,74766,74770],{},[56,74767,74768],{},[59,74769,73771],{},[63,74771,73774],{"encoding":65},[39,74773,74775],{"className":74774,"ariaHidden":70},[69],[39,74776,74778,74781],{"className":74777},[74],[39,74779],{"className":74780,"style":2654},[78],[39,74782,73771],{"className":74783},[83,84]," 在某个点上有原子，而 ",[39,74786,74788,74801],{"className":74787,"translate":43},[42],[39,74789,74791],{"className":74790},[47],[49,74792,74793],{"xmlns":51},[53,74794,74795,74799],{},[56,74796,74797],{},[59,74798,73802],{},[63,74800,73805],{"encoding":65},[39,74802,74804],{"className":74803,"ariaHidden":70},[69],[39,74805,74807,74810],{"className":74806},[74],[39,74808],{"className":74809,"style":290},[78],[39,74811,73802],{"className":74812,"style":73818},[83,84]," 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，",[39,74815,74817,74840],{"className":74816,"translate":43},[42],[39,74818,74820],{"className":74819},[47],[49,74821,74822],{"xmlns":51},[53,74823,74824,74837],{},[56,74825,74826,74828,74830],{},[59,74827,73771],{},[133,74829,793],{},[613,74831,74832,74835],{},[59,74833,74834],{},"δ",[539,74836,541],{},[63,74838,74839],{"encoding":65},"\\mu=\\delta_0",[39,74841,74843,74861],{"className":74842,"ariaHidden":70},[69],[39,74844,74846,74849,74852,74855,74858],{"className":74845},[74],[39,74847],{"className":74848,"style":2654},[78],[39,74850,73771],{"className":74851},[83,84],[39,74853],{"className":74854,"style":231},[230],[39,74856,793],{"className":74857},[235],[39,74859],{"className":74860,"style":231},[230],[39,74862,74864,74868],{"className":74863},[74],[39,74865],{"className":74866,"style":74867},[78],"height:0.8444em;vertical-align:-0.15em;",[39,74869,74871,74875],{"className":74870},[83],[39,74872,74834],{"className":74873,"style":74874},[83,84],"margin-right:0.0379em;",[39,74876,74878],{"className":74877},[192],[39,74879,74881,74902],{"className":74880},[196,649],[39,74882,74884,74899],{"className":74883},[200],[39,74885,74887],{"className":74886,"style":963},[204],[39,74888,74890,74893],{"style":74889},"top:-2.55em;margin-left:-0.0379em;margin-right:0.05em;",[39,74891],{"className":74892,"style":213},[212],[39,74894,74896],{"className":74895},[217,218,219,220],[39,74897,541],{"className":74898},[83,220],[39,74900,673],{"className":74901},[672],[39,74903,74905],{"className":74904},[200],[39,74906,74908],{"className":74907,"style":680},[204],[39,74909],{},"；而两个厂各要一半，",[39,74912,74914,74960],{"className":74913,"translate":43},[42],[39,74915,74917],{"className":74916},[47],[49,74918,74919],{"xmlns":51},[53,74920,74921,74957],{},[56,74922,74923,74925,74927,74935,74941,74943,74951],{},[59,74924,73802],{},[133,74926,793],{},[770,74928,74929],{"scriptlevel":541,"displaystyle":135},[5362,74930,74931,74933],{},[539,74932,546],{},[539,74934,17359],{},[613,74936,74937,74939],{},[59,74938,74834],{},[539,74940,546],{},[133,74942,788],{},[770,74944,74945],{"scriptlevel":541,"displaystyle":135},[5362,74946,74947,74949],{},[539,74948,546],{},[539,74950,17359],{},[613,74952,74953,74955],{},[59,74954,74834],{},[539,74956,17359],{},[63,74958,74959],{"encoding":65},"\\nu=\\tfrac12\\delta_1+\\tfrac12\\delta_2",[39,74961,74963,74981,75109],{"className":74962,"ariaHidden":70},[69],[39,74964,74966,74969,74972,74975,74978],{"className":74965},[74],[39,74967],{"className":74968,"style":290},[78],[39,74970,73802],{"className":74971,"style":73818},[83,84],[39,74973],{"className":74974,"style":231},[230],[39,74976,793],{"className":74977},[235],[39,74979],{"className":74980,"style":231},[230],[39,74982,74984,74988,75060,75100,75103,75106],{"className":74983},[74],[39,74985],{"className":74986,"style":74987},[78],"height:1.1901em;vertical-align:-0.345em;",[39,74989,74991,74994,75057],{"className":74990},[83],[39,74992],{"className":74993},[182,5463],[39,74995,74997],{"className":74996},[5362],[39,74998,75000,75048],{"className":74999},[196,649],[39,75001,75003,75045],{"className":75002},[200],[39,75004,75007,75022,75030],{"className":75005,"style":75006},[204],"height:0.8451em;",[39,75008,75010,75013],{"style":75009},"top:-2.655em;",[39,75011],{"className":75012,"style":935},[212],[39,75014,75016],{"className":75015},[217,218,219,220],[39,75017,75019],{"className":75018},[83,220],[39,75020,17359],{"className":75021},[83,220],[39,75023,75024,75027],{"style":5503},[39,75025],{"className":75026,"style":935},[212],[39,75028],{"className":75029,"style":5511},[5510],[39,75031,75033,75036],{"style":75032},"top:-3.394em;",[39,75034],{"className":75035,"style":935},[212],[39,75037,75039],{"className":75038},[217,218,219,220],[39,75040,75042],{"className":75041},[83,220],[39,75043,546],{"className":75044},[83,220],[39,75046,673],{"className":75047},[672],[39,75049,75051],{"className":75050},[200],[39,75052,75055],{"className":75053,"style":75054},[204],"height:0.345em;",[39,75056],{},[39,75058],{"className":75059},[263,5463],[39,75061,75063,75066],{"className":75062},[83],[39,75064,74834],{"className":75065,"style":74874},[83,84],[39,75067,75069],{"className":75068},[192],[39,75070,75072,75092],{"className":75071},[196,649],[39,75073,75075,75089],{"className":75074},[200],[39,75076,75078],{"className":75077,"style":963},[204],[39,75079,75080,75083],{"style":74889},[39,75081],{"className":75082,"style":213},[212],[39,75084,75086],{"className":75085},[217,218,219,220],[39,75087,546],{"className":75088},[83,220],[39,75090,673],{"className":75091},[672],[39,75093,75095],{"className":75094},[200],[39,75096,75098],{"className":75097,"style":680},[204],[39,75099],{},[39,75101],{"className":75102,"style":461},[230],[39,75104,788],{"className":75105},[465],[39,75107],{"className":75108,"style":461},[230],[39,75110,75112,75115,75183],{"className":75111},[74],[39,75113],{"className":75114,"style":74987},[78],[39,75116,75118,75121,75180],{"className":75117},[83],[39,75119],{"className":75120},[182,5463],[39,75122,75124],{"className":75123},[5362],[39,75125,75127,75172],{"className":75126},[196,649],[39,75128,75130,75169],{"className":75129},[200],[39,75131,75133,75147,75155],{"className":75132,"style":75006},[204],[39,75134,75135,75138],{"style":75009},[39,75136],{"className":75137,"style":935},[212],[39,75139,75141],{"className":75140},[217,218,219,220],[39,75142,75144],{"className":75143},[83,220],[39,75145,17359],{"className":75146},[83,220],[39,75148,75149,75152],{"style":5503},[39,75150],{"className":75151,"style":935},[212],[39,75153],{"className":75154,"style":5511},[5510],[39,75156,75157,75160],{"style":75032},[39,75158],{"className":75159,"style":935},[212],[39,75161,75163],{"className":75162},[217,218,219,220],[39,75164,75166],{"className":75165},[83,220],[39,75167,546],{"className":75168},[83,220],[39,75170,673],{"className":75171},[672],[39,75173,75175],{"className":75174},[200],[39,75176,75178],{"className":75177,"style":75054},[204],[39,75179],{},[39,75181],{"className":75182},[263,5463],[39,75184,75186,75189],{"className":75185},[83],[39,75187,74834],{"className":75188,"style":74874},[83,84],[39,75190,75192],{"className":75191},[192],[39,75193,75195,75215],{"className":75194},[196,649],[39,75196,75198,75212],{"className":75197},[200],[39,75199,75201],{"className":75200,"style":963},[204],[39,75202,75203,75206],{"style":74889},[39,75204],{"className":75205,"style":213},[212],[39,75207,75209],{"className":75208},[217,218,219,220],[39,75210,17359],{"className":75211},[83,220],[39,75213,673],{"className":75214},[672],[39,75216,75218],{"className":75217},[200],[39,75219,75221],{"className":75220,"style":680},[204],[39,75222],{},[39,75224,75226,75246],{"className":75225,"translate":43},[42],[39,75227,75229],{"className":75228},[47],[49,75230,75231],{"xmlns":51},[53,75232,75233,75243],{},[56,75234,75235,75237,75239,75241],{},[59,75236,11772],{},[133,75238,136],{"stretchy":135},[539,75240,541],{},[133,75242,161],{"stretchy":135},[63,75244,75245],{"encoding":65},"T(0)",[39,75247,75249],{"className":75248,"ariaHidden":70},[69],[39,75250,75252,75255,75258,75261,75264],{"className":75251},[74],[39,75253],{"className":75254,"style":245},[78],[39,75256,11772],{"className":75257,"style":178},[83,84],[39,75259,136],{"className":75260},[182],[39,75262,541],{"className":75263},[83],[39,75265,161],{"className":75266},[263]," 必须同时等于 ",[39,75269,75271,75284],{"className":75270,"translate":43},[42],[39,75272,75274],{"className":75273},[47],[49,75275,75276],{"xmlns":51},[53,75277,75278,75282],{},[56,75279,75280],{},[539,75281,546],{},[63,75283,546],{"encoding":65},[39,75285,75287],{"className":75286,"ariaHidden":70},[69],[39,75288,75290,75293],{"className":75289},[74],[39,75291],{"className":75292,"style":2473},[78],[39,75294,546],{"className":75295},[83]," 和 ",[39,75298,75300,75313],{"className":75299,"translate":43},[42],[39,75301,75303],{"className":75302},[47],[49,75304,75305],{"xmlns":51},[53,75306,75307,75311],{},[56,75308,75309],{},[539,75310,17359],{},[63,75312,17359],{"encoding":65},[39,75314,75316],{"className":75315,"ariaHidden":70},[69],[39,75317,75319,75322],{"className":75318},[74],[39,75320],{"className":75321,"style":2473},[78],[39,75323,17359],{"className":75324},[83],"，函数做不到。",[11,75327,75328],{},"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",[39,75330,75332],{"className":75331,"translate":43},[746],[39,75333,75335,75450],{"className":75334,"translate":43},[42],[39,75336,75338],{"className":75337},[47],[49,75339,75340],{"xmlns":51,"display":755},[53,75341,75342,75447],{},[56,75343,75344,75347,75349,75351,75353,75355,75357,75359,75361,75363,75365,75367,75369,75371,75373,75375,75377,75379,75381,75383,75385,75399,75405,75407,75409,75411,75413,75415,75417,75431,75437,75439,75441,75443,75445],{},[59,75345,75346],{"mathvariant":145},"Π",[133,75348,136],{"stretchy":135},[59,75350,73771],{},[133,75352,155],{"separator":70},[59,75354,73802],{},[133,75356,161],{"stretchy":135},[133,75358,793],{},[133,75360,12951],{"fence":70,"stretchy":70,"minsize":65017,"maxsize":65017},[59,75362,2098],{},[133,75364,534],{},[59,75366,131],{"mathvariant":617},[133,75368,136],{"stretchy":135},[59,75370,29420],{"mathvariant":617},[133,75372,409],{},[59,75374,73845],{"mathvariant":617},[133,75376,161],{"stretchy":135},[35841,75378,74416],{},[133,75380,404],{},[35841,75382,74416],{},[133,75384,136],{"stretchy":135},[613,75386,75387,75397],{},[56,75388,75389,75391,75393,75395],{},[59,75390,11],{"mathvariant":145},[59,75392,401],{"mathvariant":145},[59,75394,35010],{"mathvariant":145},[59,75396,13446],{"mathvariant":145},[59,75398,29420],{"mathvariant":617},[613,75400,75401,75403],{},[133,75402,161],{"stretchy":135},[59,75404,73919],{"mathvariant":145},[59,75406,2098],{},[133,75408,793],{},[59,75410,73771],{},[133,75412,155],{"separator":70},[35841,75414,74416],{},[133,75416,136],{"stretchy":135},[613,75418,75419,75429],{},[56,75420,75421,75423,75425,75427],{},[59,75422,11],{"mathvariant":145},[59,75424,401],{"mathvariant":145},[59,75426,35010],{"mathvariant":145},[59,75428,13446],{"mathvariant":145},[59,75430,73845],{"mathvariant":617},[613,75432,75433,75435],{},[133,75434,161],{"stretchy":135},[59,75436,73919],{"mathvariant":145},[59,75438,2098],{},[133,75440,793],{},[59,75442,73802],{},[133,75444,12965],{"fence":70,"stretchy":70,"minsize":65017,"maxsize":65017},[133,75446,155],{"separator":70},[63,75448,75449],{"encoding":65},"\\Pi(\\mu,\\nu)=\\Bigl\\{\\pi\\in\\mathcal P(\\mathcal X\\times\\mathcal Y)\\ :\\ (\\mathrm{proj}_{\\mathcal X})_\\#\\pi=\\mu,\\ (\\mathrm{proj}_{\\mathcal 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读作“从 ",[39,75874,75876,75889],{"className":75875,"translate":43},[42],[39,75877,75879],{"className":75878},[47],[49,75880,75881],{"xmlns":51},[53,75882,75883,75887],{},[56,75884,75885],{},[59,75886,1755],{},[63,75888,1755],{"encoding":65},[39,75890,75892],{"className":75891,"ariaHidden":70},[69],[39,75893,75895,75898],{"className":75894},[74],[39,75896],{"className":75897,"style":290},[78],[39,75899,1755],{"className":75900},[83,84]," 搬到 ",[39,75903,75905,75918],{"className":75904,"translate":43},[42],[39,75906,75908],{"className":75907},[47],[49,75909,75910],{"xmlns":51},[53,75911,75912,75916],{},[56,75913,75914],{},[59,75915,29625],{},[63,75917,29625],{"encoding":65},[39,75919,75921],{"className":75920,"ariaHidden":70},[69],[39,75922,75924,75927],{"className":75923},[74],[39,75925],{"className":75926,"style":2654},[78],[39,75928,29625],{"className":75929,"style":2114},[83,84]," 的那部分质量”，于是",[39,75932,75934],{"className":75933,"translate":43},[746],[39,75935,75937,76035],{"className":75936,"translate":43},[42],[39,75938,75940],{"className":75939},[47],[49,75941,75942],{"xmlns":51,"display":755},[53,75943,75944,76032],{},[56,75945,75946,75952,75954,75956,75958,75960,75962,75964,75990,76002,76004,76006,76008,76010,76012,76014,76016,76018,76020,76022,76024,76026,76028,76030],{},[613,75947,75948,75950],{},[59,75949,23703],{},[59,75951,24077],{},[133,75953,136],{"stretchy":135},[59,75955,73771],{},[133,75957,155],{"separator":70},[59,75959,73802],{},[133,75961,161],{"stretchy":135},[133,75963,793],{},[6556,75965,75966,75972],{},[56,75967,75968,75970],{},[59,75969,74409],{},[133,75971,1690],{},[56,75973,75974,75976,75978,75980,75982,75984,75986,75988],{},[59,75975,2098],{},[133,75977,534],{},[59,75979,75346],{"mathvariant":145},[133,75981,136],{"stretchy":135},[59,75983,73771],{},[133,75985,155],{"separator":70},[59,75987,73802],{},[133,75989,161],{"stretchy":135},[613,75991,75992,75994],{},[133,75993,69087],{},[56,75995,75996,75998,76000],{},[59,75997,29420],{"mathvariant":617},[133,75999,409],{},[59,76001,73845],{"mathvariant":617},[59,76003,24077],{},[133,76005,136],{"stretchy":135},[59,76007,1755],{},[133,76009,155],{"separator":70},[59,76011,29625],{},[133,76013,161],{"stretchy":135},[35841,76015,35843],{},[59,76017,29425],{},[59,76019,2098],{},[133,76021,136],{"stretchy":135},[59,76023,1755],{},[133,76025,155],{"separator":70},[59,76027,29625],{},[133,76029,161],{"stretchy":135},[59,76031,2903],{"mathvariant":145},[63,76033,76034],{"encoding":65},"W_c(\\mu,\\nu)=\\inf_{\\pi\\in\\Pi(\\mu,\\nu)}\\int_{\\mathcal 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就是那张运输表。",[11,76330,76331,76332,76469,76470,76498],{},"取 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是度量），则",[39,76500,76502],{"className":76501,"translate":43},[746],[39,76503,76505,76601],{"className":76504,"translate":43},[42],[39,76506,76508],{"className":76507},[47],[49,76509,76510],{"xmlns":51,"display":755},[53,76511,76512,76598],{},[56,76513,76514,76520,76522,76524,76526,76528,76530,76532],{},[613,76515,76516,76518],{},[59,76517,23703],{},[59,76519,11],{},[133,76521,136],{"stretchy":135},[59,76523,73771],{},[133,76525,155],{"separator":70},[59,76527,73802],{},[133,76529,161],{"stretchy":135},[133,76531,793],{},[138,76533,76534,76590],{},[56,76535,76536,76538,76564,76566,76568,76570,76572,76574,76576,76582,76584,76586,76588],{},[133,76537,136],{"fence":70},[6556,76539,76540,76546],{},[56,76541,76542,76544],{},[59,76543,74409],{},[133,76545,1690],{},[56,76547,76548,76550,76552,76554,76556,76558,76560,76562],{},[59,76549,2098],{},[133,76551,534],{},[59,76553,75346],{"mathvariant":145},[133,76555,136],{"stretchy":135},[59,76557,73771],{},[133,76559,155],{"separator":70},[59,76561,73802],{},[133,76563,161],{"stretchy":135},[133,76565,69087],{},[59,76567,29425],{},[133,76569,136],{"stretchy":135},[59,76571,1755],{},[133,76573,155],{"separator":70},[59,76575,29625],{},[138,76577,76578,76580],{},[133,76579,161],{"stretchy":135},[59,76581,11],{},[35841,76583,35843],{},[59,76585,29425],{},[59,76587,2098],{},[133,76589,161],{"fence":70},[56,76591,76592,76594,76596],{},[539,76593,546],{},[59,76595,45691],{"mathvariant":145},[59,76597,11],{},[63,76599,76600],{"encoding":65},"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 ",[39,76960,76962,76998],{"className":76961,"translate":43},[42],[39,76963,76965],{"className":76964},[47],[49,76966,76967],{"xmlns":51},[53,76968,76969,76995],{},[56,76970,76971,76973,76975,76981,76983,76985,76987,76989],{},[59,76972,73771],{},[133,76974,793],{},[613,76976,76977,76979],{},[59,76978,74834],{},[59,76980,1755],{},[133,76982,155],{"separator":70},[35841,76984,74416],{},[59,76986,73802],{},[133,76988,793],{},[613,76990,76991,76993],{},[59,76992,74834],{},[59,76994,29625],{},[63,76996,76997],{"encoding":65},"\\mu=\\delta_x,\\ 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是个有限的数，而 ",[39,77365,77367,77411],{"className":77366,"translate":43},[42],[39,77368,77370],{"className":77369},[47],[49,77371,77372],{"xmlns":51},[53,77373,77374,77408],{},[56,77375,77376,77384,77386,77392,77394,77400,77402,77404,77406],{},[56,77377,77378,77381],{},[59,77379,77380],{"mathvariant":145},"K",[59,77382,77383],{"mathvariant":145},"L",[133,77385,136],{"stretchy":135},[613,77387,77388,77390],{},[59,77389,74834],{},[59,77391,1755],{},[59,77393,15389],{"mathvariant":145},[613,77395,77396,77398],{},[59,77397,74834],{},[59,77399,29625],{},[133,77401,161],{"stretchy":135},[133,77403,793],{},[133,77405,788],{},[59,77407,1696],{"mathvariant":145},[63,77409,77410],{"encoding":65},"\\mathrm{KL}(\\delta_x\\|\\delta_y)=+\\infty",[39,77412,77414,77525],{"className":77413,"ariaHidden":70},[69],[39,77415,77417,77420,77427,77430,77470,77473,77513,77516,77519,77522],{"className":77416},[74],[39,77418],{"className":77419,"style":41560},[78],[39,77421,77423],{"className":77422},[83],[39,77424,77426],{"className":77425},[83,35077],"KL",[39,77428,136],{"className":77429},[182],[39,77431,77433,77436],{"className":77432},[83],[39,77434,74834],{"className":77435,"style":74874},[83,84],[39,77437,77439],{"className":77438},[192],[39,77440,77442,77462],{"className":77441},[196,649],[39,77443,77445,77459],{"className":77444},[200],[39,77446,77448],{"className":77447,"style":1728},[204],[39,77449,77450,77453],{"style":74889},[39,77451],{"className":77452,"style":213},[212],[39,77454,77456],{"className":77455},[217,218,219,220],[39,77457,1755],{"className":77458},[83,84,220],[39,77460,673],{"className":77461},[672],[39,77463,77465],{"className":77464},[200],[39,77466,77468],{"className":77467,"style":680},[204],[39,77469],{},[39,77471,15389],{"className":77472},[83],[39,77474,77476,77479],{"className":77475},[83],[39,77477,74834],{"className":77478,"style":74874},[83,84],[39,77480,77482],{"className":77481},[192],[39,77483,77485,77505],{"className":77484},[196,649],[39,77486,77488,77502],{"className":77487},[200],[39,77489,77491],{"className":77490,"style":1728},[204],[39,77492,77493,77496],{"style":74889},[39,77494],{"className":77495,"style":213},[212],[39,77497,77499],{"className":77498},[217,218,219,220],[39,77500,29625],{"className":77501,"style":2114},[83,84,220],[39,77503,673],{"className":77504},[672],[39,77506,77508],{"className":77507},[200],[39,77509,77511],{"className":77510,"style":35396},[204],[39,77512],{},[39,77514,161],{"className":77515},[263],[39,77517],{"className":77518,"style":231},[230],[39,77520,793],{"className":77521},[235],[39,77523],{"className":77524,"style":231},[230],[39,77526,77528,77531,77534],{"className":77527},[74],[39,77529],{"className":77530,"style":73236},[78],[39,77532,788],{"className":77533},[83],[39,77535,1696],{"className":77536},[83],"。支撑不交的时候，KL 要么发散要么给出零梯度，",[39,77539,77541,77554],{"className":77540,"translate":43},[42],[39,77542,77544],{"className":77543},[47],[49,77545,77546],{"xmlns":51},[53,77547,77548,77552],{},[56,77549,77550],{},[59,77551,23703],{},[63,77553,23703],{"encoding":65},[39,77555,77557],{"className":77556,"ariaHidden":70},[69],[39,77558,77560,77563],{"className":77559},[74],[39,77561],{"className":77562,"style":79},[78],[39,77564,23703],{"className":77565,"style":178},[83,84]," 却仍然告诉你“隔了多远”。",[11,77568,77569,77570,77598,77599,13349,77628,38249,77698,77726],{},"平移的例子更直白。把 ",[39,77571,77573,77586],{"className":77572,"translate":43},[42],[39,77574,77576],{"className":77575},[47],[49,77577,77578],{"xmlns":51},[53,77579,77580,77584],{},[56,77581,77582],{},[59,77583,73771],{},[63,77585,73774],{"encoding":65},[39,77587,77589],{"className":77588,"ariaHidden":70},[69],[39,77590,77592,77595],{"className":77591},[74],[39,77593],{"className":77594,"style":2654},[78],[39,77596,73771],{"className":77597},[83,84]," 整体挪 ",[39,77600,77602,77616],{"className":77601,"translate":43},[42],[39,77603,77605],{"className":77604},[47],[49,77606,77607],{"xmlns":51},[53,77608,77609,77613],{},[56,77610,77611],{},[59,77612,58009],{},[63,77614,77615],{"encoding":65},"\\varepsilon",[39,77617,77619],{"className":77618,"ariaHidden":70},[69],[39,77620,77622,77625],{"className":77621},[74],[39,77623],{"className":77624,"style":290},[78],[39,77626,58009],{"className":77627},[83,84],[39,77629,77631,77649],{"className":77630,"translate":43},[42],[39,77632,77634],{"className":77633},[47],[49,77635,77636],{"xmlns":51},[53,77637,77638,77646],{},[56,77639,77640],{},[613,77641,77642,77644],{},[59,77643,23703],{},[59,77645,11],{},[63,77647,77648],{"encoding":65},"W_p",[39,77650,77652],{"className":77651,"ariaHidden":70},[69],[39,77653,77655,77658],{"className":77654},[74],[39,77656],{"className":77657,"style":35350},[78],[39,77659,77661,77664],{"className":77660},[83],[39,77662,23703],{"className":77663,"style":178},[83,84],[39,77665,77667],{"className":77666},[192],[39,77668,77670,77690],{"className":77669},[196,649],[39,77671,77673,77687],{"className":77672},[200],[39,77674,77676],{"className":77675,"style":1728},[204],[39,77677,77678,77681],{"style":73958},[39,77679],{"className":77680,"style":213},[212],[39,77682,77684],{"className":77683},[217,218,219,220],[39,77685,11],{"className":77686},[83,84,220],[39,77688,673],{"className":77689},[672],[39,77691,77693],{"className":77692},[200],[39,77694,77696],{"className":77695,"style":35396},[204],[39,77697],{},[39,77699,77701,77714],{"className":77700,"translate":43},[42],[39,77702,77704],{"className":77703},[47],[49,77705,77706],{"xmlns":51},[53,77707,77708,77712],{},[56,77709,77710],{},[59,77711,58009],{},[63,77713,77615],{"encoding":65},[39,77715,77717],{"className":77716,"ariaHidden":70},[69],[39,77718,77720,77723],{"className":77719},[74],[39,77721],{"className":77722,"style":290},[78],[39,77724,58009],{"className":77725},[83,84]," 量级，而",[39,77728,77730],{"className":77729,"translate":43},[746],[39,77731,77733,77796],{"className":77732,"translate":43},[42],[39,77734,77736],{"className":77735},[47],[49,77737,77738],{"xmlns":51,"display":755},[53,77739,77740,77793],{},[56,77741,77742,77748,77750,77753,77755,77757,77759,77761,77763,77765,77767,77769,77771,77773,77775,77777,77779,77781,77791],{},[56,77743,77744,77746],{},[59,77745,77380],{"mathvariant":145},[59,77747,77383],{"mathvariant":145},[133,77749,136],{"fence":70,"stretchy":70,"minsize":74068,"maxsize":74068},[59,77751,77752],{"mathvariant":617},"N",[133,77754,136],{"stretchy":135},[539,77756,541],{},[133,77758,155],{"separator":70},[539,77760,546],{},[133,77762,161],{"stretchy":135},[59,77764,15389],{"mathvariant":145},[59,77766,77752],{"mathvariant":617},[133,77768,136],{"stretchy":135},[59,77770,58009],{},[133,77772,155],{"separator":70},[539,77774,546],{},[133,77776,161],{"stretchy":135},[133,77778,161],{"fence":70,"stretchy":70,"minsize":74068,"maxsize":74068},[133,77780,793],{},[5362,77782,77783,77789],{},[138,77784,77785,77787],{},[59,77786,58009],{},[539,77788,17359],{},[539,77790,17359],{},[133,77792,155],{"separator":70},[63,77794,77795],{"encoding":65},"\\mathrm{KL}\\bigl(\\mathcal N(0,1)\\|\\mathcal N(\\varepsilon,1)\\bigr)=\\frac{\\varepsilon^2}{2},",[39,77797,77799,77878],{"className":77798,"ariaHidden":70},[69],[39,77800,77802,77805,77811,77817,77821,77824,77827,77830,77833,77836,77839,77842,77845,77848,77851,77854,77857,77860,77863,77869,77872,77875],{"className":77801},[74],[39,77803],{"className":77804,"style":908},[78],[39,77806,77808],{"className":77807},[83],[39,77809,77426],{"className":77810},[83,35077],[39,77812,77814],{"className":77813},[182],[39,77815,136],{"className":77816},[18520,4009],[39,77818,77752],{"className":77819,"style":77820},[83,641],"margin-right:0.1474em;",[39,77822,136],{"className":77823},[182],[39,77825,541],{"className":77826},[83],[39,77828,155],{"className":77829},[252],[39,77831],{"className":77832,"style":256},[230],[39,77834,546],{"className":77835},[83],[39,77837,161],{"className":77838},[263],[39,77840,15389],{"className":77841},[83],[39,77843,77752],{"className":77844,"style":77820},[83,641],[39,77846,136],{"className":77847},[182],[39,77849,58009],{"className":77850},[83,84],[39,77852,155],{"className":77853},[252],[39,77855],{"className":77856,"style":256},[230],[39,77858,546],{"className":77859},[83],[39,77861,161],{"className":77862},[263],[39,77864,77866],{"className":77865},[263],[39,77867,161],{"className":77868},[18520,4009],[39,77870],{"className":77871,"style":231},[230],[39,77873,793],{"className":77874},[235],[39,77876],{"className":77877,"style":231},[230],[39,77879,77881,77885,77974],{"className":77880},[74],[39,77882],{"className":77883,"style":77884},[78],"height:2.1771em;vertical-align:-0.686em;",[39,77886,77888,77891,77971],{"className":77887},[83],[39,77889],{"className":77890},[182,5463],[39,77892,77894],{"className":77893},[5362],[39,77895,77897,77963],{"className":77896},[196,649],[39,77898,77900,77960],{"className":77899},[200],[39,77901,77904,77915,77923],{"className":77902,"style":77903},[204],"height:1.4911em;",[39,77905,77906,77909],{"style":5479},[39,77907],{"className":77908,"style":935},[212],[39,77910,77912],{"className":77911},[83],[39,77913,17359],{"className":77914},[83],[39,77916,77917,77920],{"style":5503},[39,77918],{"className":77919,"style":935},[212],[39,77921],{"className":77922,"style":5511},[5510],[39,77924,77925,77928],{"style":5514},[39,77926],{"className":77927,"style":935},[212],[39,77929,77931],{"className":77930},[83],[39,77932,77934,77937],{"className":77933},[83],[39,77935,58009],{"className":77936},[83,84],[39,77938,77940],{"className":77939},[192],[39,77941,77943],{"className":77942},[196],[39,77944,77946],{"className":77945},[200],[39,77947,77949],{"className":77948,"style":19710},[204],[39,77950,77951,77954],{"style":208},[39,77952],{"className":77953,"style":213},[212],[39,77955,77957],{"className":77956},[217,218,219,220],[39,77958,17359],{"className":77959},[83,220],[39,77961,673],{"className":77962},[672],[39,77964,77966],{"className":77965},[200],[39,77967,77969],{"className":77968,"style":19487},[204],[39,77970],{},[39,77972],{"className":77973},[263,5463],[39,77975,155],{"className":77976},[252],[11,77978,77979,77980,78008],{},"是二阶的。所以小位移时 KL 的梯度趋近于零，",[39,77981,77983,77996],{"className":77982,"translate":43},[42],[39,77984,77986],{"className":77985},[47],[49,77987,77988],{"xmlns":51},[53,77989,77990,77994],{},[56,77991,77992],{},[59,77993,23703],{},[63,77995,23703],{"encoding":65},[39,77997,77999],{"className":77998,"ariaHidden":70},[69],[39,78000,78002,78005],{"className":78001},[74],[39,78003],{"className":78004,"style":79},[78],[39,78006,23703],{"className":78007,"style":178},[83,84]," 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。",[11,78010,78011,78012,78082,78083,78111],{},"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 ",[39,78013,78015,78033],{"className":78014,"translate":43},[42],[39,78016,78018],{"className":78017},[47],[49,78019,78020],{"xmlns":51},[53,78021,78022,78030],{},[56,78023,78024],{},[613,78025,78026,78028],{},[59,78027,23703],{},[539,78029,546],{},[63,78031,78032],{"encoding":65},"W_1",[39,78034,78036],{"className":78035,"ariaHidden":70},[69],[39,78037,78039,78042],{"className":78038},[74],[39,78040],{"className":78041,"style":634},[78],[39,78043,78045,78048],{"className":78044},[83],[39,78046,23703],{"className":78047,"style":178},[83,84],[39,78049,78051],{"className":78050},[192],[39,78052,78054,78074],{"className":78053},[196,649],[39,78055,78057,78071],{"className":78056},[200],[39,78058,78060],{"className":78059,"style":963},[204],[39,78061,78062,78065],{"style":73958},[39,78063],{"className":78064,"style":213},[212],[39,78066,78068],{"className":78067},[217,218,219,220],[39,78069,546],{"className":78070},[83,220],[39,78072,673],{"className":78073},[672],[39,78075,78077],{"className":78076},[200],[39,78078,78080],{"className":78079,"style":680},[204],[39,78081],{},"，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，",[39,78084,78086,78099],{"className":78085,"translate":43},[42],[39,78087,78089],{"className":78088},[47],[49,78090,78091],{"xmlns":51},[53,78092,78093,78097],{},[56,78094,78095],{},[59,78096,23703],{},[63,78098,23703],{"encoding":65},[39,78100,78102],{"className":78101,"ariaHidden":70},[69],[39,78103,78105,78108],{"className":78104},[74],[39,78106],{"className":78107,"style":79},[78],[39,78109,23703],{"className":78110,"style":178},[83,84]," 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。",[21,78113,78114],{"id":78114},"对偶",[11,78116,78117],{},"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？",[11,78119,78120,78121,78149,78150,78179],{},"不去枚举所有运输表。我给每个矿定一个“提货加价” ",[39,78122,78124,78137],{"className":78123,"translate":43},[42],[39,78125,78127],{"className":78126},[47],[49,78128,78129],{"xmlns":51},[53,78130,78131,78135],{},[56,78132,78133],{},[59,78134,35321],{},[63,78136,35321],{"encoding":65},[39,78138,78140],{"className":78139,"ariaHidden":70},[69],[39,78141,78143,78146],{"className":78142},[74],[39,78144],{"className":78145,"style":290},[78],[39,78147,35321],{"className":78148},[83,84],"，每个厂定一个“到货减价” ",[39,78151,78153,78167],{"className":78152,"translate":43},[42],[39,78154,78156],{"className":78155},[47],[49,78157,78158],{"xmlns":51},[53,78159,78160,78165],{},[56,78161,78162],{},[59,78163,78164],{},"v",[63,78166,78164],{"encoding":65},[39,78168,78170],{"className":78169,"ariaHidden":70},[69],[39,78171,78173,78176],{"className":78172},[74],[39,78174],{"className":78175,"style":290},[78],[39,78177,78164],{"className":78178,"style":2114},[83,84],"，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，",[39,78181,78183],{"className":78182,"translate":43},[746],[39,78184,78186,78226],{"className":78185,"translate":43},[42],[39,78187,78189],{"className":78188},[47],[49,78190,78191],{"xmlns":51,"display":755},[53,78192,78193,78223],{},[56,78194,78195,78201,78203,78209,78211,78221],{},[613,78196,78197,78199],{},[59,78198,35321],{},[59,78200,13377],{},[133,78202,788],{},[613,78204,78205,78207],{},[59,78206,78164],{},[59,78208,13446],{},[133,78210,1824],{},[613,78212,78213,78215],{},[59,78214,24077],{},[56,78216,78217,78219],{},[59,78218,13377],{},[59,78220,13446],{},[59,78222,2903],{"mathvariant":145},[63,78224,78225],{"encoding":65},"u_i+v_j\\le c_{ij}.",[39,78227,78229,78284,78340],{"className":78228,"ariaHidden":70},[69],[39,78230,78232,78235,78275,78278,78281],{"className":78231},[74],[39,78233],{"className":78234,"style":7528},[78],[39,78236,78238,78241],{"className":78237},[83],[39,78239,35321],{"className":78240},[83,84],[39,78242,78244],{"className":78243},[192],[39,78245,78247,78267],{"className":78246},[196,649],[39,78248,78250,78264],{"className":78249},[200],[39,78251,78253],{"className":78252,"style":13604},[204],[39,78254,78255,78258],{"style":966},[39,78256],{"className":78257,"style":213},[212],[39,78259,78261],{"className":78260},[217,218,219,220],[39,78262,13377],{"className":78263},[83,84,220],[39,78265,673],{"className":78266},[672],[39,78268,78270],{"className":78269},[200],[39,78271,78273],{"className":78272,"style":680},[204],[39,78274],{},[39,78276],{"className":78277,"style":461},[230],[39,78279,788],{"className":78280},[465],[39,78282],{"className":78283,"style":461},[230],[39,78285,78287,78291,78331,78334,78337],{"className":78286},[74],[39,78288],{"className":78289,"style":78290},[78],"height:0.9221em;vertical-align:-0.2861em;",[39,78292,78294,78297],{"className":78293},[83],[39,78295,78164],{"className":78296,"style":2114},[83,84],[39,78298,78300],{"className":78299},[192],[39,78301,78303,78323],{"className":78302},[196,649],[39,78304,78306,78320],{"className":78305},[200],[39,78307,78309],{"className":78308,"style":13604},[204],[39,78310,78311,78314],{"style":41316},[39,78312],{"className":78313,"style":213},[212],[39,78315,78317],{"className":78316},[217,218,219,220],[39,78318,13446],{"className":78319,"style":13731},[83,84,220],[39,78321,673],{"className":78322},[672],[39,78324,78326],{"className":78325},[200],[39,78327,78329],{"className":78328,"style":35396},[204],[39,78330],{},[39,78332],{"className":78333,"style":231},[230],[39,78335,1824],{"className":78336},[235],[39,78338],{"className":78339,"style":231},[230],[39,78341,78343,78346,78389],{"className":78342},[74],[39,78344],{"className":78345,"style":41295},[78],[39,78347,78349,78352],{"className":78348},[83],[39,78350,24077],{"className":78351},[83,84],[39,78353,78355],{"className":78354},[192],[39,78356,78358,78381],{"className":78357},[196,649],[39,78359,78361,78378],{"className":78360},[200],[39,78362,78364],{"className":78363,"style":13604},[204],[39,78365,78366,78369],{"style":966},[39,78367],{"className":78368,"style":213},[212],[39,78370,78372],{"className":78371},[217,218,219,220],[39,78373,78375],{"className":78374},[83,220],[39,78376,13732],{"className":78377,"style":13731},[83,84,220],[39,78379,673],{"className":78380},[672],[39,78382,78384],{"className":78383},[200],[39,78385,78387],{"className":78386,"style":35396},[204],[39,78388],{},[39,78390,2903],{"className":78391},[83],[11,78393,78394],{},"那么任何一张运单的总价都满足",[39,78396,78398],{"className":78397,"translate":43},[746],[39,78399,78401,78642],{"className":78400,"translate":43},[42],[39,78402,78404],{"className":78403},[47],[49,78405,78406],{"xmlns":51,"display":755},[53,78407,78408,78639],{},[759,78409,78410,78496,78580],{"rowspacing":761,"columnalign":762,"columnspacing":146},[764,78411,78412,78448],{},[767,78413,78414],{},[770,78415,78416],{"scriptlevel":541,"displaystyle":70},[56,78417,78418,78428,78438],{},[6556,78419,78420,78422],{},[133,78421,3191],{},[56,78423,78424,78426],{},[59,78425,13377],{},[59,78427,13446],{},[613,78429,78430,78432],{},[59,78431,2098],{},[56,78433,78434,78436],{},[59,78435,13377],{},[59,78437,13446],{},[613,78439,78440,78442],{},[59,78441,24077],{},[56,78443,78444,78446],{},[59,78445,13377],{},[59,78447,13446],{},[767,78449,78450],{},[770,78451,78452],{"scriptlevel":541,"displaystyle":70},[56,78453,78454,78456,78458,78468,78478,78480,78486,78488,78494],{},[56,78455],{},[133,78457,25173],{},[6556,78459,78460,78462],{},[133,78461,3191],{},[56,78463,78464,78466],{},[59,78465,13377],{},[59,78467,13446],{},[613,78469,78470,78472],{},[59,78471,2098],{},[56,78473,78474,78476],{},[59,78475,13377],{},[59,78477,13446],{},[133,78479,136],{"stretchy":135},[613,78481,78482,78484],{},[59,78483,35321],{},[59,78485,13377],{},[133,78487,788],{},[613,78489,78490,78492],{},[59,78491,78164],{},[59,78493,13446],{},[133,78495,161],{"stretchy":135},[764,78497,78498,78504],{},[767,78499,78500],{},[770,78501,78502],{"scriptlevel":541,"displaystyle":70},[56,78503],{},[767,78505,78506],{},[770,78507,78508],{"scriptlevel":541,"displaystyle":70},[56,78509,78510,78512,78514,78520,78526,78528,78534,78544,78546,78548,78554,78560,78562,78568,78578],{},[56,78511],{},[133,78513,793],{},[6556,78515,78516,78518],{},[133,78517,3191],{},[59,78519,13377],{},[613,78521,78522,78524],{},[59,78523,35321],{},[59,78525,13377],{},[133,78527,136],{"fence":70,"stretchy":70,"minsize":65017,"maxsize":65017},[6556,78529,78530,78532],{},[133,78531,3191],{},[59,78533,13446],{},[613,78535,78536,78538],{},[59,78537,2098],{},[56,78539,78540,78542],{},[59,78541,13377],{},[59,78543,13446],{},[133,78545,161],{"fence":70,"stretch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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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",[39,79775,79777,79795],{"className":79776,"translate":43},[42],[39,79778,79780],{"className":79779},[47],[49,79781,79782],{"xmlns":51},[53,79783,79784,79792],{},[56,79785,79786,79788,79790],{},[59,79787,35321],{},[133,79789,155],{"separator":70},[59,79791,78164],{},[63,79793,79794],{"encoding":65},"u,v",[39,79796,79798],{"className":79797,"ariaHidden":70},[69],[39,79799,79801,79804,79807,79810,79813],{"className":79800},[74],[39,79802],{"className":79803,"style":2654},[78],[39,79805,35321],{"className":79806},[83,84],[39,79808,155],{"className":79809},[252],[39,79811],{"className":79812,"style":256},[230],[39,79814,78164],{"className":79815,"style":2114},[83,84]," 有关，跟运单无关；而中间那步把 ",[39,79818,79820,79833],{"className":79819,"translate":43},[42],[39,79821,79823],{"className":79822},[47],[49,79824,79825],{"xmlns":51},[53,79826,79827,79831],{},[56,79828,79829],{},[59,79830,2098],{},[63,79832,2101],{"encoding":65},[39,79834,79836],{"className":79835,"ariaHidden":70},[69],[39,79837,79839,79842],{"className":79838},[74],[39,79840],{"className":79841,"style":290},[78],[39,79843,2098],{"className":79844,"style":2114},[83,84]," 的边缘和换成了产量 ",[39,79847,79849,79867],{"className":79848,"translate":43},[42],[39,79850,79852],{"className":79851},[47],[49,79853,79854],{"xmlns":51},[53,79855,79856,79864],{},[56,79857,79858],{},[613,79859,79860,79862],{},[59,79861,158],{},[59,79863,13377],{},[63,79865,79866],{"encoding":65},"a_i",[39,79868,79870],{"className":79869,"ariaHidden":70},[69],[39,79871,79873,79876],{"className":79872},[74],[39,79874],{"className":79875,"style":2916},[78],[39,79877,79879,79882],{"className":79878},[83],[39,79880,158],{"className":79881},[83,84],[39,79883,79885],{"className":79884},[192],[39,79886,79888,79908],{"className":79887},[196,649],[39,79889,79891,79905],{"className":79890},[200],[39,79892,79894],{"className":79893,"style":13604},[204],[39,79895,79896,79899],{"style":966},[39,79897],{"className":79898,"style":213},[212],[39,79900,79902],{"className":79901},[217,218,219,220],[39,79903,13377],{"className":79904},[83,84,220],[39,79906,673],{"className":79907},[672],[39,79909,79911],{"className":79910},[200],[39,79912,79914],{"className":79913,"style":680},[204],[39,79915],{},"、需求 ",[39,79918,79920,79938],{"className":79919,"translate":43},[42],[39,79921,79923],{"className":79922},[47],[49,79924,79925],{"xmlns":51},[53,79926,79927,79935],{},[56,79928,79929],{},[613,79930,79931,79933],{},[59,79932,78628],{},[59,79934,13446],{},[63,79936,79937],{"encoding":65},"b_j",[39,79939,79941],{"className":79940,"ariaHidden":70},[69],[39,79942,79944,79947],{"className":79943},[74],[39,79945],{"className":79946,"style":77092},[78],[39,79948,79950,79953],{"className":79949},[83],[39,79951,78628],{"className":79952},[83,84],[39,79954,79956],{"className":79955},[192],[39,79957,79959,79979],{"className":79958},[196,649],[39,79960,79962,79976],{"className":79961},[200],[39,79963,79965],{"className":79964,"style":13604},[204],[39,79966,79967,79970],{"style":966},[39,79968],{"className":79969,"style":213},[212],[39,79971,79973],{"className":79972},[217,218,219,220],[39,79974,13446],{"className":79975,"style":13731},[83,84,220],[39,79977,673],{"className":79978},[672],[39,79980,79982],{"className":79981},[200],[39,79983,79985],{"className":79984,"style":35396},[204],[39,79986],{},"。所以只要我找到一组 ",[39,79989,79991,80008],{"className":79990,"translate":43},[42],[39,79992,79994],{"className":79993},[47],[49,79995,79996],{"xmlns":51},[53,79997,79998,80006],{},[56,79999,80000,80002,80004],{},[59,80001,35321],{},[133,80003,155],{"separator":70},[59,80005,78164],{},[63,80007,79794],{"encoding":65},[39,80009,80011],{"className":80010,"ariaHidden":70},[69],[39,80012,80014,80017,80020,80023,80026],{"className":80013},[74],[39,80015],{"className":80016,"style":2654},[78],[39,80018,35321],{"className":80019},[83,84],[39,80021,155],{"className":80022},[252],[39,80024],{"className":80025,"style":256},[230],[39,80027,78164],{"className":80028,"style":2114},[83,84],"，使这个和等于 5，我就",[1489,80031,5595],{},"了运费不可能低于 5 块。",[11,80034,80035,80036,742,80115,80189,80190,80285,80286,80381,80382,80482,80483,80583,80584,80804],{},"试一试：取 ",[39,80037,80039,80067],{"className":80038,"translate":43},[42],[39,80040,80042],{"className":80041},[47],[49,80043,80044],{"xmlns":51},[53,80045,80046,80064],{},[56,80047,80048,80050,80052,80054,80056,80058,80060,80062],{},[59,80049,35321],{},[133,80051,793],{},[133,80053,136],{"stretchy":135},[539,80055,541],{},[133,80057,155],{"separator":70},[133,80059,5378],{},[539,80061,546],{},[133,80063,161],{"stretchy":135},[63,80065,80066],{"encoding":65},"u=(0,-1)",[39,80068,80070,80088],{"className":80069,"ariaHidden":70},[69],[39,80071,80073,80076,80079,80082,80085],{"className":80072},[74],[39,80074],{"className":80075,"style":290},[78],[39,80077,35321],{"className":80078},[83,84],[39,80080],{"className":80081,"style":231},[230],[39,80083,793],{"className":80084},[235],[39,80086],{"className":80087,"style":231},[230],[39,80089,80091,80094,80097,80100,80103,80106,80109,80112],{"className":80090},[74],[39,80092],{"className":80093,"style":245},[78],[39,80095,136],{"className":80096},[182],[39,80098,541],{"className":80099},[83],[39,80101,155],{"className":80102},[252],[39,80104],{"className":80105,"style":256},[230],[39,80107,5378],{"className":80108},[83],[39,80110,546],{"className":80111},[83],[39,80113,161],{"className":80114},[263],[39,80116,80118,80144],{"className":80117,"translate":43},[42],[39,80119,80121],{"className":80120},[47],[49,80122,80123],{"xmlns":51},[53,80124,80125,80141],{},[56,80126,80127,80129,80131,80133,80135,80137,80139],{},[59,80128,78164],{},[133,80130,793],{},[133,80132,136],{"stretchy":135},[539,80134,546],{},[133,80136,155],{"separator":70},[539,80138,17359],{},[133,80140,161],{"stretchy":135},[63,80142,80143],{"encoding":65},"v=(1,2)",[39,80145,80147,80165],{"className":80146,"ariaHidden":70},[69],[39,80148,80150,80153,80156,80159,80162],{"className":80149},[74],[39,80151],{"className":80152,"style":290},[78],[39,80154,78164],{"className":80155,"style":2114},[83,84],[39,80157],{"className":80158,"style":231},[230],[39,80160,793],{"className":80161},[235],[39,80163],{"className":80164,"style":231},[230],[39,80166,80168,80171,80174,80177,80180,80183,80186],{"className":80167},[74],[39,80169],{"className":80170,"style":245},[78],[39,80172,136],{"className":80173},[182],[39,80175,546],{"className":80176},[83],[39,80178,155],{"className":80179},[252],[39,80181],{"className":80182,"style":256},[230],[39,80184,17359],{"className":80185},[83],[39,80187,161],{"className":80188},[263],"。四条路线逐一检查：甲到 A 是 ",[39,80191,80193,80219],{"className":80192,"translate":43},[42],[39,80194,80196],{"className":80195},[47],[49,80197,80198],{"xmlns":51},[53,80199,80200,80216],{},[56,80201,80202,80204,80206,80208,80210,80212,80214],{},[539,80203,541],{},[133,80205,788],{},[539,80207,546],{},[133,80209,793],{},[539,80211,546],{},[133,80213,1824],{},[539,80215,546],{},[63,80217,80218],{"encoding":65},"0+1=1\\le 1",[39,80220,80222,80240,80258,80276],{"className":80221,"ariaHidden":70},[69],[39,80223,80225,80228,80231,80234,80237],{"className":80224},[74],[39,80226],{"className":80227,"style":17882},[78],[39,80229,541],{"className":80230},[83],[39,80232],{"className":80233,"style":461},[230],[39,80235,788],{"className":80236},[465],[39,80238],{"className":80239,"style":461},[230],[39,80241,80243,80246,80249,80252,80255],{"className":80242},[74],[39,80244],{"className":80245,"style":2473},[78],[39,80247,546],{"className":80248},[83],[39,80250],{"className":80251,"style":231},[230],[39,80253,793],{"className":80254},[235],[39,80256],{"className":80257,"style":231},[230],[39,80259,80261,80264,80267,80270,80273],{"className":80260},[74],[39,80262],{"className":80263,"style":14587},[78],[39,80265,546],{"className":80266},[83],[39,80268],{"className":80269,"style":231},[230],[39,80271,1824],{"className":80272},[235],[39,80274],{"className":80275,"style":231},[230],[39,80277,80279,80282],{"className":80278},[74],[39,80280],{"className":80281,"style":2473},[78],[39,80283,546],{"className":80284},[83],"，甲到 B 是 ",[39,80287,80289,80315],{"className":80288,"translate":43},[42],[39,80290,80292],{"className":80291},[47],[49,80293,80294],{"xmlns":51},[53,80295,80296,80312],{},[56,80297,80298,80300,80302,80304,80306,80308,80310],{},[539,80299,541],{},[133,80301,788],{},[539,80303,17359],{},[133,80305,793],{},[539,80307,17359],{},[133,80309,1824],{},[539,80311,17359],{},[63,80313,80314],{"encoding":65},"0+2=2\\le 2",[39,80316,80318,80336,80354,80372],{"className":80317,"ariaHidden":70},[69],[39,80319,80321,80324,80327,80330,80333],{"className":80320},[74],[39,80322],{"className":80323,"style":17882},[78],[39,80325,541],{"className":80326},[83],[39,80328],{"className":80329,"style":461},[230],[39,80331,788],{"className":80332},[465],[39,80334],{"className":80335,"style":461},[230],[39,80337,80339,80342,80345,80348,80351],{"className":80338},[74],[39,80340],{"className":80341,"style":2473},[78],[39,80343,17359],{"className":80344},[83],[39,80346],{"className":80347,"style":231},[230],[39,80349,793],{"className":80350},[235],[39,80352],{"className":80353,"style":231},[230],[39,80355,80357,80360,80363,80366,80369],{"className":80356},[74],[39,80358],{"className":80359,"style":14587},[78],[39,80361,17359],{"className":80362},[83],[39,80364],{"className":80365,"style":231},[230],[39,80367,1824],{"className":80368},[235],[39,80370],{"className":80371,"style":231},[230],[39,80373,80375,80378],{"className":80374},[74],[39,80376],{"className":80377,"style":2473},[78],[39,80379,17359],{"className":80380},[83],"，乙到 A 是 ",[39,80383,80385,80413],{"className":80384,"translate":43},[42],[39,80386,80388],{"className":80387},[47],[49,80389,80390],{"xmlns":51},[53,80391,80392,80410],{},[56,80393,80394,80396,80398,80400,80402,80404,80406,80408],{},[133,80395,5378],{},[539,80397,546],{},[133,80399,788],{},[539,80401,546],{},[133,80403,793],{},[539,80405,541],{},[133,80407,1824],{},[539,80409,59187],{},[63,80411,80412],{"encoding":65},"-1+1=0\\le 3",[39,80414,80416,80437,80455,80473],{"className":80415,"ariaHidden":70},[69],[39,80417,80419,80422,80425,80428,80431,80434],{"className":80418},[74],[39,80420],{"className":80421,"style":17882},[78],[39,80423,5378],{"className":80424},[83],[39,80426,546],{"className":80427},[83],[39,80429],{"className":80430,"style":461},[230],[39,80432,788],{"className":80433},[465],[39,80435],{"className":80436,"style":461},[230],[39,80438,80440,80443,80446,80449,80452],{"className":80439},[74],[39,80441],{"className":80442,"style":2473},[78],[39,80444,546],{"className":80445},[83],[39,80447],{"className":80448,"style":231},[230],[39,80450,793],{"className":80451},[235],[39,80453],{"className":80454,"style":231},[230],[39,80456,80458,80461,80464,80467,80470],{"className":80457},[74],[39,80459],{"className":80460,"style":14587},[78],[39,80462,541],{"className":80463},[83],[39,80465],{"className":80466,"style":231},[230],[39,80468,1824],{"className":80469},[235],[39,80471],{"className":80472,"style":231},[230],[39,80474,80476,80479],{"className":80475},[74],[39,80477],{"className":80478,"style":2473},[78],[39,80480,59187],{"className":80481},[83],"，乙到 B 是 ",[39,80484,80486,80514],{"className":80485,"translate":43},[42],[39,80487,80489],{"className":80488},[47],[49,80490,80491],{"xmlns":51},[53,80492,80493,80511],{},[56,80494,80495,80497,80499,80501,80503,80505,80507,80509],{},[133,80496,5378],{},[539,80498,546],{},[133,80500,788],{},[539,80502,17359],{},[133,80504,793],{},[539,80506,546],{},[133,80508,1824],{},[539,80510,546],{},[63,80512,80513],{"encoding":65},"-1+2=1\\le 1",[39,80515,80517,80538,80556,80574],{"className":80516,"ariaHidden":70},[69],[39,80518,80520,80523,80526,80529,80532,80535],{"className":80519},[74],[39,80521],{"className":80522,"style":17882},[78],[39,80524,5378],{"className":80525},[83],[39,80527,546],{"className":80528},[83],[39,80530],{"className":80531,"style":461},[230],[39,80533,788],{"className":80534},[465],[39,80536],{"className":80537,"style":461},[230],[39,80539,80541,80544,80547,80550,80553],{"className":80540},[74],[39,80542],{"className":80543,"style":2473},[78],[39,80545,17359],{"className":80546},[83],[39,80548],{"className":80549,"style":231},[230],[39,80551,793],{"className":80552},[235],[39,80554],{"className":80555,"style":231},[230],[39,80557,80559,80562,80565,80568,80571],{"className":80558},[74],[39,80560],{"className":80561,"style":14587},[78],[39,80563,546],{"className":80564},[83],[39,80566],{"className":80567,"style":231},[230],[39,80569,1824],{"className":80570},[235],[39,80572],{"className":80573,"style":231},[230],[39,80575,80577,80580],{"className":80576},[74],[39,80578],{"className":80579,"style":2473},[78],[39,80581,546],{"className":80582},[83],"，全部成立。而 ",[39,80585,80587,80639],{"className":80586,"translate":43},[42],[39,80588,80590],{"className":80589},[47],[49,80591,80592],{"xmlns":51},[53,80593,80594,80636],{},[56,80595,80596,80598,80600,80602,80604,80606,80608,80610,80612,80614,80616,80618,80620,80622,80624,80626,80628,80630,80632,80634],{},[539,80597,59187],{},[133,80599,8885],{},[539,80601,541],{},[133,80603,788],{},[539,80605,546],{},[133,80607,8885],{},[133,80609,136],{"stretchy":135},[133,80611,5378],{},[539,80613,546],{},[133,80615,161],{"stretchy":135},[133,80617,788],{},[539,80619,17359],{},[133,80621,8885],{},[539,80623,546],{},[133,80625,788],{},[539,80627,17359],{},[133,80629,8885],{},[539,80631,17359],{},[133,80633,793],{},[539,80635,59259],{},[63,80637,80638],{"encoding":65},"3\\cdot0+1\\cdot(-1)+2\\cdot1+2\\cdot2=5",[39,80640,80642,80660,80678,80696,80723,80741,80759,80777,80795],{"className":80641,"ariaHidden":70},[69],[39,80643,80645,80648,80651,80654,80657],{"className":80644},[74],[39,80646],{"className":80647,"style":2473},[78],[39,80649,59187],{"className":80650},[83],[39,80652],{"className":80653,"style":461},[230],[39,80655,8885],{"className":80656},[465],[39,80658],{"className":80659,"style":461},[230],[39,80661,80663,80666,80669,80672,80675],{"className":80662},[74],[39,80664],{"className":80665,"style":17882},[78],[39,80667,541],{"className":80668},[83],[39,80670],{"className":80671,"style":461},[230],[39,80673,788],{"className":80674},[465],[39,80676],{"className":80677,"style":461},[230],[39,80679,80681,80684,80687,80690,80693],{"className":80680},[74],[39,80682],{"className":80683,"style":2473},[78],[39,80685,546],{"className":80686},[83],[39,80688],{"className":80689,"style":461},[230],[39,80691,8885],{"className":80692},[465],[39,80694],{"className":80695,"style":461},[230],[39,80697,80699,80702,80705,80708,80711,80714,80717,80720],{"className":80698},[74],[39,80700],{"className":80701,"style":245},[78],[39,80703,136],{"className":80704},[182],[39,80706,5378],{"className":80707},[83],[39,80709,546],{"className":80710},[83],[39,80712,161],{"className":80713},[263],[39,80715],{"className":80716,"style":461},[230],[39,80718,788],{"className":80719},[465],[39,80721],{"className":80722,"style":461},[230],[39,80724,80726,80729,80732,80735,80738],{"className":80725},[74],[39,80727],{"className":80728,"style":2473},[78],[39,80730,17359],{"className":80731},[83],[39,80733],{"className":80734,"style":461},[230],[39,80736,8885],{"className":80737},[465],[39,80739],{"className":80740,"style":461},[230],[39,80742,80744,80747,80750,80753,80756],{"className":80743},[74],[39,80745],{"className":80746,"style":17882},[78],[39,80748,546],{"className":80749},[83],[39,80751],{"className":80752,"style":461},[230],[39,80754,788],{"className":80755},[465],[39,80757],{"className":80758,"style":461},[230],[39,80760,80762,80765,80768,80771,80774],{"className":80761},[74],[39,80763],{"className":80764,"style":2473},[78],[39,80766,17359],{"className":80767},[83],[39,80769],{"className":80770,"style":461},[230],[39,80772,8885],{"className":80773},[465],[39,80775],{"className":80776,"style":461},[230],[39,80778,80780,80783,80786,80789,80792],{"className":80779},[74],[39,80781],{"className":80782,"style":2473},[78],[39,80784,17359],{"className":80785},[83],[39,80787],{"className":80788,"style":231},[230],[39,80790,793],{"className":80791},[235],[39,80793],{"className":80794,"style":231},[230],[39,80796,80798,80801],{"className":80797},[74],[39,80799],{"className":80800,"style":2473},[78],[39,80802,59259],{"className":80803},[83],"。证书找到了。",[11,80806,80807],{},"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：",[39,80809,80811],{"className":80810,"translate":43},[746],[39,80812,80814,81086],{"className":80813,"translate":43},[42],[39,80815,80817],{"className":80816},[47],[49,80818,80819],{"xmlns":51,"display":755},[53,80820,80821,81083],{},[759,80822,80823,80973],{"rowspacing":761,"columnalign":762,"columnspacing":146},[764,80824,80825,80849],{},[767,80826,80827],{},[770,80828,80829],{"scriptlevel":541,"displaystyle":70},[56,80830,80831,80833,80835,80837,80839,80842,80844,80847],{},[59,80832,77383],{},[133,80834,136],{"stretchy":135},[59,80836,2098],{},[133,80838,155],{"separator":70},[59,80840,80841],{},"φ",[133,80843,155],{"separator":70},[59,80845,80846],{},"ψ",[133,80848,161],{"stretchy":135},[767,80850,80851],{},[770,80852,80853],{"scriptlevel":541,"displaystyle":70},[56,80854,80855,80857,80859,80869,80879,80889,80891,80897,80903,80905,80911,80913,80919,80929,80931,80933,80939,80945,80947,80953,80955,80961,80971],{},[56,80856],{},[133,80858,793],{},[6556,80860,80861,80863],{},[133,80862,3191],{},[56,80864,80865,80867],{},[59,80866,13377],{},[59,80868,13446],{},[613,80870,80871,80873],{},[59,80872,2098],{},[56,80874,80875,80877],{},[59,80876,13377],{},[59,80878,13446],{},[613,80880,80881,80883],{},[59,80882,48444],{},[56,80884,80885,80887],{},[59,80886,13377],{},[59,80888,13446],{},[133,80890,788],{},[6556,80892,80893,80895],{},[133,80894,3191],{},[59,80896,13377],{},[613,80898,80899,80901],{},[59,80900,80841],{},[59,80902,13377],{},[133,80904,136],{"fence":70,"stretchy":70,"minsize":65017,"maxsize":65017},[613,80906,80907,80909],{},[59,80908,158],{},[59,80910,13377],{},[133,80912,5378],{},[6556,80914,80915,80917],{},[133,80916,3191],{},[59,80918,13446],{},[613,80920,80921,80923],{},[59,80922,2098],{},[56,80924,80925,80927],{},[59,80926,13377],{},[59,80928,13446],{},[133,80930,161],{"fence":70,"stretchy":70,"minsize":65017,"maxsize":65017},[133,80932,788],{},[6556,80934,80935,80937],{},[133,80936,3191],{},[59,80938,13446],{},[613,80940,80941,80943],{},[59,80942,80846],{},[59,80944,13446],{},[133,80946,136],{"fence":70,"stretchy":70,"minsize":65017,"maxsize":65017},[613,80948,80949,80951],{},[59,80950,78628],{},[59,80952,13446],{},[133,80954,5378],{},[6556,80956,80957,80959],{},[133,80958,3191],{},[59,80960,13377],{},[613,80962,80963,80965],{},[59,80964,2098],{},[56,80966,80967,80969],{},[59,80968,13377],{},[59,80970,13446],{},[133,80972,161],{"fence":70,"stretchy":70,"minsize":65017,"maxsize":65017},[764,80974,80975,80981],{},[767,80976,80977],{},[770,80978,80979],{"scriptlevel":541,"displaystyle":70},[56,80980],{},[767,80982,80983],{},[770,80984,80985],{"scriptlevel":541,"displaystyle":70},[56,80986,80987,80989,80991,80997,81003,81009,81011,81017,81023,81029,81031,81041,81051,81053,81063,81065,81071,81073,81079,81081],{},[56,80988],{},[133,80990,793],{},[6556,80992,80993,80995],{},[133,80994,3191],{},[59,80996,13377],{},[613,80998,80999,81001],{},[59,81000,158],{},[59,81002,13377],{},[613,81004,81005,81007],{},[59,81006,80841],{},[59,81008,13377],{},[133,81010,788],{},[6556,81012,81013,81015],{},[133,81014,3191],{},[59,81016,13446],{},[613,81018,81019,81021],{},[59,81020,78628],{},[59,81022,13446],{},[613,81024,81025,81027],{},[59,81026,80846],{},[59,81028,13446],{},[133,81030,788],{},[6556,81032,81033,81035],{},[133,81034,3191],{},[56,81036,81037,81039],{},[59,81038,13377],{},[59,81040,13446],{},[613,81042,81043,81045],{},[59,81044,2098],{},[56,81046,81047,81049],{},[59,81048,13377],{},[59,81050,13446],{},[133,81052,136],{"fence":70,"stretchy":70,"minsize":74068,"maxsize":74068},[613,81054,81055,81057],{},[59,81056,48444],{},[56,81058,81059,81061],{},[59,81060,13377],{},[59,81062,13446],{},[133,81064,5378],{},[613,81066,81067,81069],{},[59,81068,80841],{},[59,81070,13377],{},[133,81072,5378],{},[613,81074,81075,81077],{},[59,81076,80846],{},[59,81078,13446],{},[133,81080,161],{"fence":70,"stretchy":70,"minsize":74068,"maxsize":74068},[59,81082,2903],{"mathvariant":145},[63,81084,81085],{"encoding":65},"\\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}",[39,81087,81089],{"className":81088,"ariaHidden":70},[69],[39,81090,81092,81096],{"className":81091},[74],[39,81093],{"className":81094,"style":81095},[78],"height:5.3276em;vertical-align:-2.4138em;",[39,81097,81099],{"className":81098},[83],[39,81100,81102,81175],{"className":81101},[759],[39,81103,81105],{"className":81104},[918],[39,81106,81108,81166],{"className":81107},[196,649],[39,81109,81111,81163],{"className":81110},[200],[39,81112,81115,81154],{"className":81113,"style":81114},[204],"height:2.9138em;",[39,81116,81118,81121],{"style":81117},"top:-4.9138em;",[39,81119],{"className":81120,"style":33100},[212],[39,81122,81124,81127,81130,81133,81136,81139,81142,81145,81148,81151],{"className":81123},[83],[39,81125,77383],{"className":81126},[83,84],[39,81128,136],{"className":81129},[182],[39,81131,2098],{"className":81132,"style":2114},[83,84],[39,81134,155],{"className":81135},[252],[39,81137],{"className":81138,"style":256},[230],[39,81140,80841],{"className":81141},[83,84],[39,81143,155],{"className":81144},[252],[39,81146],{"className":81147,"style":256},[230],[39,81149,80846],{"className":81150,"style":2114},[83,84],[39,81152,161],{"className":81153},[263],[39,81155,81157,81160],{"style":81156},"top:-2.15em;",[39,81158],{"className":81159,"style":33100},[212],[39,81161],{"className":81162},[83],[39,81164,673],{"className":81165},[672],[39,81167,81169],{"className":81168},[200],[39,81170,81173],{"className":81171,"style":81172},[204],"height:2.4138em;",[39,81174],{},[39,81176,81178],{"className":81177},[1223],[39,81179,81181,82382],{"className":81180},[196,649],[39,81182,81184,82379],{"className":81183},[200],[39,81185,81187,81838],{"className":81186,"style":81114},[204],[39,81188,81189,81192],{"style":81117},[39,81190],{"className":81191,"style":33100},[212],[39,81193,81195,81198,81201,81204,81207,81254,81257,81300,81344,81347,81350,81353,81397,81400,81440,81446,81486,81489,81492,81495,81539,81542,81585,81591,81594,81597,81600,81644,81647,81687,81693,81733,81736,81739,81742,81786,81789,81832],{"className":81194},[83],[39,81196],{"className":81197},[83],[39,81199],{"className":81200,"style":231},[230],[39,81202,793],{"className":81203},[235],[39,81205],{"className":81206,"style":231},[230],[39,81208,81210],{"className":81209},[1744,3294],[39,81211,81213,81246],{"className":81212},[196,649],[39,81214,81216,81243],{"className":81215},[200],[39,81217,81219,81233],{"className":81218,"style":6670},[204],[39,81220,81221,81224],{"style":13926},[39,81222],{"className":81223,"style":3311},[212],[39,81225,81227],{"className":81226},[217,218,219,220],[39,81228,81230],{"className":81229},[83,220],[39,81231,13732],{"className":81232,"style":13731},[83,84,220],[39,81234,81235,81238],{"style":3330},[39,81236],{"className":81237,"style":3311},[212],[39,81239,81240],{},[39,81241,3191],{"className":81242},[1744,3339,3340],[39,81244,673],{"className":81245},[672],[39,81247,81249],{"className":81248},[200],[39,81250,81252],{"className":81251,"style":13958},[204],[39,81253],{},[39,81255],{"className":81256,"style":256},[230],[39,81258,81260,81263],{"className":81259},[83],[39,81261,2098],{"className":81262,"style":2114},[83,84],[39,81264,81266],{"className":81265},[192],[39,81267,81269,81292],{"className":81268},[196,649],[39,81270,81272,81289],{"className":81271},[200],[39,81273,81275],{"className":81274,"style":13604},[204],[39,81276,81277,81280],{"style":41316},[39,81278],{"className":81279,"style":213},[212],[39,81281,81283],{"className":81282},[217,218,219,220],[39,81284,81286],{"className":81285},[83,220],[39,81287,13732],{"className":81288,"style":13731},[83,84,220],[39,81290,673],{"className":81291},[672],[39,81293,81295],{"className":81294},[200],[39,81296,81298],{"className":81297,"style":35396},[204],[39,81299],{},[39,81301,81303,81306],{"className":81302},[83],[39,81304,48444],{"className":81305,"style":48487},[83,84],[39,81307,81309],{"className":81308},[192],[39,81310,81312,81336],{"className":81311},[196,649],[39,81313,81315,81333],{"className":81314},[200],[39,81316,81318],{"className":81317,"style":13604},[204],[39,81319,81321,81324],{"style":81320},"top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;",[39,81322],{"className":81323,"style":213},[212],[39,81325,81327],{"className":81326},[217,218,219,220],[39,81328,81330],{"className":81329},[83,220],[39,81331,13732],{"className":81332,"style":13731},[83,84,220],[39,81334,673],{"className":81335},[672],[39,81337,81339],{"className":81338},[200],[39,81340,81342],{"className":81341,"style":35396},[204],[39,81343],{},[39,81345],{"className":81346,"style":461},[230],[39,81348,788],{"className":81349},[465],[39,81351],{"className":81352,"style":461},[230],[39,81354,81356],{"className":81355},[1744,3294],[39,81357,81359,81389],{"className":81358},[196,649],[39,81360,81362,81386],{"className":81361},[200],[39,81363,81365,81376],{"className":81364,"style":6670},[204],[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取下确界：只要有一对 ",[39,82445,82447,82469],{"className":82446,"translate":43},[42],[39,82448,82450],{"className":82449},[47],[49,82451,82452],{"xmlns":51},[53,82453,82454,82466],{},[56,82455,82456,82458,82460,82462,82464],{},[133,82457,136],{"stretchy":135},[59,82459,13377],{},[133,82461,155],{"separator":70},[59,82463,13446],{},[133,82465,161],{"stretchy":135},[63,82467,82468],{"encoding":65},"(i,j)",[39,82470,82472],{"className":82471,"ariaHidden":70},[69],[39,82473,82475,82478,82481,82484,82487,82490,82493],{"className":82474},[74],[39,82476],{"className":82477,"style":245},[78],[39,82479,136],{"className":82480},[182],[39,82482,13377],{"className":82483},[83,84],[39,82485,155],{"className":82486},[252],[39,82488],{"className":82489,"style":256},[230],[39,82491,13446],{"className":82492,"style":13731},[83,84],[39,82494,161],{"className":82495},[263]," 使 ",[39,82498,82500,82542],{"className":82499,"translate":43},[42],[39,82501,82503],{"className":82502},[47],[49,82504,82505],{"xmlns":51},[53,82506,82507,82539],{},[56,82508,82509,82519,82521,82527,82529,82535,82537],{},[613,82510,82511,82513],{},[59,82512,48444],{},[56,82514,82515,82517],{},[59,82516,13377],{},[59,82518,13446],{},[133,82520,5378],{},[613,82522,82523,82525],{},[59,82524,80841],{},[59,82526,13377],{},[133,82528,5378],{},[613,82530,82531,82533],{},[59,82532,80846],{},[59,82534,13446],{},[133,82536,1693],{},[539,82538,541],{},[63,82540,82541],{"encoding":65},"C_{ij}-\\varphi_i-\\psi_j\u003C0",[39,82543,82545,82603,82659,82714],{"className":82544,"ariaHidden":70},[69],[39,82546,82548,82551,82594,82597,82600],{"className":82547},[74],[39,82549],{"className":82550,"style":35350},[78],[39,82552,82554,82557],{"className":82553},[83],[39,82555,48444],{"className":82556,"style":48487},[83,84],[39,82558,82560],{"className":82559},[192],[39,82561,82563,82586],{"className":82562},[196,649],[39,82564,82566,82583],{"className":82565},[200],[39,82567,82569],{"className":82568,"style":13604},[204],[39,82570,82571,82574],{"style":81320},[39,82572],{"className":82573,"style":213},[212],[39,82575,82577],{"className":82576},[217,218,219,220],[39,82578,82580],{"className":82579},[83,220],[39,82581,13732],{"className":82582,"style":13731},[83,84,220],[39,82584,673],{"className":82585},[672],[39,82587,82589],{"className":82588},[200],[39,82590,82592],{"className":82591,"style":35396},[204],[39,82593],{},[39,82595],{"className":82596,"style":461},[230],[39,82598,5378],{"className":82599},[465],[39,82601],{"className":82602,"style":461},[230],[39,82604,82606,82610,82650,82653,82656],{"className":82605},[74],[39,82607],{"className":82608,"style":82609},[78],"height:0.7778em;vertical-align:-0.1944em;",[39,82611,82613,82616],{"className":82612},[83],[39,82614,80841],{"className":82615},[83,84],[39,82617,82619],{"className":82618},[192],[39,82620,82622,82642],{"className":82621},[196,649],[39,82623,82625,82639],{"className":82624},[200],[39,82626,82628],{"className":82627,"style":13604},[204],[39,82629,82630,82633],{"style":966},[39,82631],{"className":82632,"style":213},[212],[39,82634,82636],{"className":82635},[217,218,219,220],[39,82637,13377],{"className":82638},[83,84,220],[39,82640,673],{"className":82641},[672],[39,82643,82645],{"className":82644},[200],[39,82646,82648],{"className":82647,"style":680},[204],[39,82649],{},[39,82651],{"className":82652,"style":461},[230],[39,82654,5378],{"className":82655},[465],[39,82657],{"className":82658,"style":461},[230],[39,82660,82662,82665,82705,82708,82711],{"className":82661},[74],[39,82663],{"className":82664,"style":77092},[78],[39,82666,82668,82671],{"className":82667},[83],[39,82669,80846],{"className":82670,"style":2114},[83,84],[39,82672,82674],{"className":82673},[192],[39,82675,82677,82697],{"className":82676},[196,649],[39,82678,82680,82694],{"className":82679},[200],[39,82681,82683],{"className":82682,"style":13604},[204],[39,82684,82685,82688],{"style":41316},[39,82686],{"className":82687,"style":213},[212],[39,82689,82691],{"className":82690},[217,218,219,220],[39,82692,13446],{"className":82693,"style":13731},[83,84,220],[39,82695,673],{"className":82696},[672],[39,82698,82700],{"className":82699},[200],[39,82701,82703],{"className":82702,"style":35396},[204],[39,82704],{},[39,82706],{"className":82707,"style":231},[230],[39,82709,1693],{"className":82710},[235],[39,82712],{"className":82713,"style":231},[230],[39,82715,82717,82720],{"className":82716},[74],[39,82718],{"className":82719,"style":2473},[78],[39,82721,541],{"className":82722},[83],[39,82724,82726,82752],{"className":82725,"translate":43},[42],[39,82727,82729],{"className":82728},[47],[49,82730,82731],{"xmlns":51},[53,82732,82733,82749],{},[56,82734,82735,82745,82747],{},[613,82736,82737,82739],{},[59,82738,2098],{},[56,82740,82741,82743],{},[59,82742,13377],{},[59,82744,13446],{},[133,82746,418],{},[59,82748,1696],{"mathvariant":145},[63,82750,82751],{"encoding":65},"\\pi_{ij}\\to\\infty",[39,82753,82755,82813],{"className":82754,"ariaHidden":70},[69],[39,82756,82758,82761,82804,82807,82810],{"className":82757},[74],[39,82759],{"className":82760,"style":41295},[78],[39,82762,82764,82767],{"className":82763},[83],[39,82765,2098],{"className":82766,"style":2114},[83,84],[39,82768,82770],{"className":82769},[192],[39,82771,82773,82796],{"className":82772},[196,649],[39,82774,82776,82793],{"className":82775},[200],[39,82777,82779],{"className":82778,"style":13604},[204],[39,82780,82781,82784],{"style":41316},[39,82782],{"className":82783,"style":213},[212],[39,82785,82787],{"className":82786},[217,218,219,220],[39,82788,82790],{"className":82789},[83,220],[39,82791,13732],{"className":82792,"style":13731},[83,84,220],[39,82794,673],{"className":82795},[672],[39,82797,82799],{"className":82798},[200],[39,82800,82802],{"className":82801,"style":35396},[204],[39,82803],{},[39,82805],{"className":82806,"style":231},[230],[39,82808,418],{"className":82809},[235],[39,82811],{"className":82812,"style":231},[230],[39,82814,82816,82819],{"className":82815},[74],[39,82817],{"className":82818,"style":290},[78],[39,82820,1696],{"className":82821},[83]," 就得到 ",[39,82824,82826,82842],{"className":82825,"translate":43},[42],[39,82827,82829],{"className":82828},[47],[49,82830,82831],{"xmlns":51},[53,82832,82833,82839],{},[56,82834,82835,82837],{},[133,82836,5378],{},[59,82838,1696],{"mathvariant":145},[63,82840,82841],{"encoding":65},"-\\infty",[39,82843,82845],{"className":82844,"ariaHidden":70},[69],[39,82846,82848,82851,82854],{"className":82847},[74],[39,82849],{"className":82850,"style":73236},[78],[39,82852,5378],{"className":82853},[83],[39,82855,1696],{"className":82856},[83],"；否则下确界在 ",[39,82859,82861,82879],{"className":82860,"translate":43},[42],[39,82862,82864],{"className":82863},[47],[49,82865,82866],{"xmlns":51},[53,82867,82868,82876],{},[56,82869,82870,82872,82874],{},[59,82871,2098],{},[133,82873,793],{},[539,82875,541],{},[63,82877,82878],{"encoding":65},"\\pi=0",[39,82880,82882,82900],{"className":82881,"ariaHidden":70},[69],[39,82883,82885,82888,82891,82894,82897],{"className":82884},[74],[39,82886],{"className":82887,"style":290},[78],[39,82889,2098],{"className":82890,"style":2114},[83,84],[39,82892],{"className":82893,"style":231},[230],[39,82895,793],{"className":82896},[235],[39,82898],{"className":82899,"style":231},[230],[39,82901,82903,82906],{"className":82902},[74],[39,82904],{"className":82905,"style":2473},[78],[39,82907,541],{"className":82908},[83]," 处取到。于是对偶问题就是",[39,82911,82913],{"className":82912,"translate":43},[746],[39,82914,82916,83018],{"className":82915,"translate":43},[42],[39,82917,82919],{"className":82918},[47],[49,82920,82921],{"xmlns":51,"display":755},[53,82922,82923,83015],{},[56,82924,82925,82941,82943,82949,82955,82961,82963,82969,82975,82981,82983,82985,82987,82993,82995,83001,83003,83013],{},[6556,82926,82927,82933],{},[56,82928,82929,82931],{},[59,82930,1687],{},[133,82932,1690],{},[56,82934,82935,82937,82939],{},[59,82936,80841],{},[133,82938,155],{"separator":70},[59,82940,80846],{},[35841,82942,74416],{},[6556,82944,82945,82947],{},[133,82946,3191],{},[59,82948,13377],{},[613,82950,82951,82953],{},[59,82952,158],{},[59,82954,13377],{},[613,82956,82957,82959],{},[59,82958,80841],{},[59,82960,13377],{},[133,82962,788],{},[6556,82964,82965,82967],{},[133,82966,3191],{},[59,82968,13446],{},[613,82970,82971,82973],{},[59,82972,78628],{},[59,82974,13446],{},[613,82976,82977,82979],{},[59,82978,80846],{},[59,82980,13446],{},[230,82982],{"width":6891},[35841,82984,74457],{},[230,82986],{"width":6891},[613,82988,82989,82991],{},[59,82990,80841],{},[59,82992,13377],{},[133,82994,788],{},[613,82996,82997,82999],{},[59,82998,80846],{},[59,83000,13446],{},[133,83002,1824],{},[613,83004,83005,83007],{},[59,83006,48444],{},[56,83008,83009,83011],{},[59,83010,13377],{},[59,83012,13446],{},[133,83014,155],{"separator":70},[63,83016,83017],{"encoding":65},"\\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 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允许我取 ",[39,84074,84076,84096],{"className":84075,"translate":43},[42],[39,84077,84079],{"className":84078},[47],[49,84080,84081],{"xmlns":51},[53,84082,84083,84093],{},[56,84084,84085,84087,84089,84091],{},[59,84086,80846],{},[133,84088,793],{},[133,84090,5378],{},[59,84092,80841],{},[63,84094,84095],{"encoding":65},"\\psi=-\\varphi",[39,84097,84099,84117],{"className":84098,"ariaHidden":70},[69],[39,84100,84102,84105,84108,84111,84114],{"className":84101},[74],[39,84103],{"className":84104,"style":51280},[78],[39,84106,80846],{"className":84107,"style":2114},[83,84],[39,84109],{"className":84110,"style":231},[230],[39,84112,793],{"className":84113},[235],[39,84115],{"className":84116,"style":231},[230],[39,84118,84120,84123,84126],{"className":84119},[74],[39,84121],{"className":84122,"style":82609},[78],[39,84124,5378],{"className":84125},[83],[39,84127,80841],{"className":84128},[83,84],"，此时它变成 ",[39,84131,84133,84177],{"className":84132,"translate":43},[42],[39,84134,84136],{"className":84135},[47],[49,84137,84138],{"xmlns":51},[53,84139,84140,84174],{},[56,84141,84142,84144,84146,84148,84150,84152,84154,84156,84158,84160,84162,84164,84166,84168,84170,84172],{},[59,84143,80841],{},[133,84145,136],{"stretchy":135},[59,84147,1755],{},[133,84149,161],{"stretchy":135},[133,84151,5378],{},[59,84153,80841],{},[133,84155,136],{"stretchy":135},[59,84157,29625],{},[133,84159,161],{"stretchy":135},[133,84161,1824],{},[59,84163,29425],{},[133,84165,136],{"stretchy":135},[59,84167,1755],{},[133,84169,155],{"separator":70},[59,84171,29625],{},[133,84173,161],{"stretchy":135},[63,84175,84176],{"encoding":65},"\\varphi(x)-\\varphi(y)\\le d(x,y)",[39,84178,84180,84207,84234],{"className":84179,"ariaHidden":70},[69],[39,84181,84183,84186,84189,84192,84195,84198,84201,84204],{"className":84182},[74],[39,84184],{"className":84185,"style":245},[78],[39,84187,80841],{"className":84188},[83,84],[39,84190,136],{"className":84191},[182],[39,84193,1755],{"className":84194},[83,84],[39,84196,161],{"className":84197},[263],[39,84199],{"className":84200,"style":461},[230],[39,84202,5378],{"className":84203},[465],[39,84205],{"className":84206,"style":461},[230],[39,84208,84210,84213,84216,84219,84222,84225,84228,84231],{"className":84209},[74],[39,84211],{"className":84212,"style":245},[78],[39,84214,80841],{"className":84215},[83,84],[39,84217,136],{"className":84218},[182],[39,84220,29625],{"className":84221,"style":2114},[83,84],[39,84223,161],{"className":84224},[263],[39,84226],{"className":84227,"style":231},[230],[39,84229,1824],{"className":84230},[235],[39,84232],{"className":84233,"style":231},[230],[39,84235,84237,84240,84243,84246,84249,84252,84255,84258],{"className":84236},[74],[39,84238],{"className":84239,"style":245},[78],[39,84241,29425],{"className":84242},[83,84],[39,84244,136],{"className":84245},[182],[39,84247,1755],{"className":84248},[83,84],[39,84250,155],{"className":84251},[252],[39,84253],{"className":84254,"style":256},[230],[39,84256,29625],{"className":84257,"style":2114},[83,84],[39,84259,161],{"className":84260},[263],"，也就是 ",[39,84263,84265,84279],{"className":84264,"translate":43},[42],[39,84266,84268],{"className":84267},[47],[49,84269,84270],{"xmlns":51},[53,84271,84272,84276],{},[56,84273,84274],{},[59,84275,80841],{},[63,84277,84278],{"encoding":65},"\\varphi",[39,84280,84282],{"className":84281,"ariaHidden":70},[69],[39,84283,84285,84288],{"className":84284},[74],[39,84286],{"className":84287,"style":2654},[78],[39,84289,80841],{"className":84290},[83,84],[39,84292,84294,84307],{"className":84293,"translate":43},[42],[39,84295,84297],{"className":84296},[47],[49,84298,84299],{"xmlns":51},[53,84300,84301,84305],{},[56,84302,84303],{},[539,84304,546],{},[63,84306,546],{"encoding":65},[39,84308,84310],{"className":84309,"ariaHidden":70},[69],[39,84311,84313,84316],{"className":84312},[74],[39,84314],{"className":84315,"style":2473},[78],[39,84317,546],{"className":84318},[83],"-Lipschitz 的。所以",[39,84321,84323],{"className":84322,"translate":43},[746],[39,84324,84326,84406],{"className":84325,"translate":43},[42],[39,84327,84329],{"className":84328},[47],[49,84330,84331],{"xmlns":51,"display":755},[53,84332,84333,84403],{},[56,84334,84335,84341,84343,84345,84347,84349,84351,84353,84383,84385,84387,84389,84391,84393,84395,84397,84399,84401],{},[613,84336,84337,84339],{},[59,84338,23703],{},[539,84340,546],{},[133,84342,136],{"stretchy":135},[59,84344,73771],{},[133,84346,155],{"separator":70},[59,84348,73802],{},[133,84350,161],{"stretchy":135},[133,84352,793],{},[6556,84354,84355,84361],{},[56,84356,84357,84359],{},[59,84358,21465],{},[133,84360,1690],{},[56,84362,84363,84365,84367,84379,84381],{},[59,84364,15389],{"mathvariant":145},[59,84366,80841],{},[613,84368,84369,84371],{},[59,84370,15389],{"mathvariant":145},[56,84372,84373,84375,84377],{},[59,84374,77383],{"mathvariant":145},[59,84376,13377],{"mathvariant":145},[59,84378,11],{"mathvariant":145},[133,84380,1824],{},[539,84382,546],{},[133,84384,69087],{},[59,84386,80841],{},[35841,84388,35843],{},[59,84390,29425],{},[133,84392,136],{"stretchy":135},[59,84394,73771],{},[133,84396,5378],{},[59,84398,73802],{},[133,84400,161],{"stretchy":135},[59,84402,2903],{"mathvariant":145},[63,84404,84405],{"encoding":65},"W_1(\\mu,\\nu)=\\sup_{\\|\\varphi\\|_{\\mathrm{Lip}}\\le 1}\\int\\varphi\\,d(\\mu-\\nu).",[39,84407,84409,84482,84630],{"className":84408,"ariaHidden":70},[69],[39,84410,84412,84415,84455,84458,84461,84464,84467,84470,84473,84476,84479],{"className":84411},[74],[39,84413],{"className":84414,"style":245},[78],[39,84416,84418,84421],{"className":84417},[83],[39,84419,23703],{"className":84420,"style":178},[83,84],[39,84422,84424],{"className":84423},[192],[39,84425,84427,84447],{"className":84426},[196,649],[39,84428,84430,84444],{"className":84429},[200],[39,84431,84433],{"className":84432,"style":963},[204],[39,84434,84435,84438],{"style":73958},[39,84436],{"className":84437,"style":213},[212],[39,84439,84441],{"className":84440},[217,218,219,220],[39,84442,546],{"className":84443},[83,220],[39,84445,673],{"className":84446},[672],[39,84448,84450],{"className":84449},[200],[39,84451,84453],{"className":84452,"style":680},[204],[39,84454],{},[39,84456,136],{"className":84457},[182],[39,84459,73771],{"className":84460},[83,84],[39,84462,155],{"className":84463},[252],[39,84465],{"className":84466,"style":256},[230],[39,84468,73802],{"className":84469,"style":73818},[83,84],[39,84471,161],{"className":84472},[263],[39,84474],{"className":84475,"style":231},[230],[39,84477,793],{"className":84478},[235],[39,84480],{"className":84481,"style":231},[230],[39,84483,84485,84489,84597,84600,84603,84606,84609,84612,84615,84618,84621,84624,84627],{"className":84484},[74],[39,84486],{"className":84487,"style":84488},[78],"height:2.543em;vertical-align:-1.183em;",[39,84490,84492],{"className":84491},[1744,3294],[39,84493,84495,84588],{"className":84494},[196,649],[39,84496,84498,84585],{"className":84497},[200],[39,84499,84501,84575],{"className":84500,"style":290},[204],[39,84502,84504,84507],{"style":84503},"top:-2.1146em;margin-left:0em;",[39,84505],{"className":84506,"style":935},[212],[39,84508,84510],{"className":84509},[217,218,219,220],[39,84511,84513,84516,84519,84569,84572],{"className":84512},[83,220],[39,84514,15389],{"className":84515},[83,220],[39,84517,80841],{"className":84518},[83,84,220],[39,84520,84522,84525],{"className":84521},[83,220],[39,84523,15389],{"className":84524},[83,220],[39,84526,84528],{"className":84527},[192],[39,84529,84531,84560],{"className":84530},[196,649],[39,84532,84534,84557],{"className":84533},[200],[39,84535,84538],{"className":84536,"style":84537},[204],"height:0.3448em;",[39,84539,84541,84544],{"style":84540},"top:-2.3567em;margin-left:0em;margin-right:0.0714em;",[39,84542],{"className":84543,"style":4004},[212],[39,84545,84547],{"className":84546},[217,4008,4009,220],[39,84548,84550],{"className":84549},[83,220],[39,84551,84553],{"className":84552},[83,220],[39,84554,84556],{"className":84555},[83,35077,220],"Lip",[39,84558,673],{"className":84559},[672],[39,84561,84563],{"className":84562},[200],[39,84564,84567],{"className":84565,"style":84566},[204],"height:0.2822em;",[39,84568],{},[39,84570,1824],{"className":84571},[235,220],[39,84573,546],{"className":84574},[83,220],[39,84576,84577,84580],{"style":21599},[39,84578],{"className":84579,"style":935},[212],[39,84581,84582],{},[39,84583,21465],{"className":84584},[1744],[39,84586,673],{"className":84587},[672],[39,84589,84591],{"className":84590},[200],[39,84592,84595],{"className":84593,"style":84594},[204],"height:1.183em;",[39,84596],{},[39,84598],{"className":84599,"style":256},[230],[39,84601,69087],{"className":84602,"style":69380},[1744,3339,3340],[39,84604],{"className":84605,"style":256},[230],[39,84607,80841],{"className":84608},[83,84],[39,84610],{"className":84611,"style":256},[230],[39,84613,29425],{"className":84614},[83,84],[39,84616,136],{"className":84617},[182],[39,84619,73771],{"className":84620},[83,84],[39,84622],{"className":84623,"style":461},[230],[39,84625,5378],{"className":84626},[465],[39,84628],{"className":84629,"style":461},[230],[39,84631,84633,84636,84639,84642],{"className":84632},[74],[39,84634],{"className":84635,"style":245},[78],[39,84637,73802],{"className":84638,"style":73818},[83,84],[39,84640,161],{"className":84641},[263],[39,84643,2903],{"className":84644},[83],[11,84646,84647],{},"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。",[11,84649,84650,84651,84841,84842,84870,84871,84899,84900,84957,84958,84986,84987,1607],{},"二次代价 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时还有一个 Brenier 定理：若 ",[39,84843,84845,84858],{"className":84844,"translate":43},[42],[39,84846,84848],{"className":84847},[47],[49,84849,84850],{"xmlns":51},[53,84851,84852,84856],{},[56,84853,84854],{},[59,84855,73771],{},[63,84857,73774],{"encoding":65},[39,84859,84861],{"className":84860,"ariaHidden":70},[69],[39,84862,84864,84867],{"className":84863},[74],[39,84865],{"className":84866,"style":2654},[78],[39,84868,73771],{"className":84869},[83,84]," 关于 Lebesgue 测度绝对连续，最优的 ",[39,84872,84874,84887],{"className":84873,"translate":43},[42],[39,84875,84877],{"className":84876},[47],[49,84878,84879],{"xmlns":51},[53,84880,84881,84885],{},[56,84882,84883],{},[59,84884,2098],{},[63,84886,2101],{"encoding":65},[39,84888,84890],{"className":84889,"ariaHidden":70},[69],[39,84891,84893,84896],{"className":84892},[74],[39,84894],{"className":84895,"style":290},[78],[39,84897,2098],{"className":84898,"style":2114},[83,84]," 由映射 ",[39,84901,84903,84924],{"className":84902,"translate":43},[42],[39,84904,84906],{"className":84905},[47],[49,84907,84908],{"xmlns":51},[53,84909,84910,84921],{},[56,84911,84912,84914,84916,84919],{},[59,84913,11772],{},[133,84915,793],{},[59,84917,84918],{"mathvariant":145},"∇",[59,84920,35321],{},[63,84922,84923],{"encoding":65},"T=\\nabla u",[39,84925,84927,84945],{"className":84926,"ariaHidden":70},[69],[39,84928,84930,84933,84936,84939,84942],{"className":84929},[74],[39,84931],{"className":84932,"style":79},[78],[39,84934,11772],{"className":84935,"style":178},[83,84],[39,84937],{"className":84938,"style":231},[230],[39,84940,793],{"className":84941},[235],[39,84943],{"className":84944,"style":231},[230],[39,84946,84948,84951,84954],{"className":84947},[74],[39,84949],{"className":84950,"style":79},[78],[39,84952,84918],{"className":84953},[83],[39,84955,35321],{"className":84956},[83,84]," 给出，其中 ",[39,84959,84961,84974],{"className":84960,"translate":43},[42],[39,84962,84964],{"className":84963},[47],[49,84965,84966],{"xmlns":51},[53,84967,84968,84972],{},[56,84969,84970],{},[59,84971,35321],{},[63,84973,35321],{"encoding":65},[39,84975,84977],{"className":84976,"ariaHidden":70},[69],[39,84978,84980,84983],{"className":84979},[74],[39,84981],{"className":84982,"style":290},[78],[39,84984,35321],{"className":84985},[83,84]," 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 ",[39,84988,84990,85006],{"className":84989,"translate":43},[42],[39,84991,84993],{"className":84992},[47],[49,84994,84995],{"xmlns":51},[53,84996,84997,85003],{},[56,84998,84999,85001],{},[59,85000,84918],{"mathvariant":145},[59,85002,35321],{},[63,85004,85005],{"encoding":65},"\\nabla u",[39,85007,85009],{"className":85008,"ariaHidden":70},[69],[39,85010,85012,85015,85018],{"className":85011},[74],[39,85013],{"className":85014,"style":79},[78],[39,85016,84918],{"className":85017},[83],[39,85019,35321],{"className":85020},[83,84],[21,85022,85023],{"id":85023},"一维与高斯",[11,85025,85026],{},"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？",[11,85028,85029,85030,85058,85059,85087,85088,85122,85123,85151],{},"“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 ",[39,85031,85033,85046],{"className":85032,"translate":43},[42],[39,85034,85036],{"className":85035},[47],[49,85037,85038],{"xmlns":51},[53,85039,85040,85044],{},[56,85041,85042],{},[59,85043,621],{},[63,85045,621],{"encoding":65},[39,85047,85049],{"className":85048,"ariaHidden":70},[69],[39,85050,85052,85055],{"className":85051},[74],[39,85053],{"className":85054,"style":709},[78],[39,85056,621],{"className":85057},[83,84]," 的人从 ",[39,85060,85062,85075],{"className":85061,"translate":43},[42],[39,85063,85065],{"className":85064},[47],[49,85066,85067],{"xmlns":51},[53,85068,85069,85073],{},[56,85070,85071],{},[59,85072,621],{},[63,85074,621],{"encoding":65},[39,85076,85078],{"className":85077,"ariaHidden":70},[69],[39,85079,85081,85084],{"className":85080},[74],[39,85082],{"className":85083,"style":709},[78],[39,85085,621],{"className":85086},[83,84]," 万变成 ",[39,85089,85091,85107],{"className":85090,"translate":43},[42],[39,85092,85094],{"className":85093},[47],[49,85095,85096],{"xmlns":51},[53,85097,85098,85104],{},[56,85099,85100,85102],{},[539,85101,17359],{},[59,85103,621],{},[63,85105,85106],{"encoding":65},"2t",[39,85108,85110],{"className":85109,"ariaHidden":70},[69],[39,85111,85113,85116,85119],{"className":85112},[74],[39,85114],{"className":85115,"style":2473},[78],[39,85117,17359],{"className":85118},[83],[39,85120,621],{"className":85121},[83,84]," 万，涨幅是 ",[39,85124,85126,85139],{"className":85125,"translate":43},[42],[39,85127,85129],{"className":85128},[47],[49,85130,85131],{"xmlns":51},[53,85132,85133,85137],{},[56,85134,85135],{},[59,85136,621],{},[63,85138,621],{"encoding":65},[39,85140,85142],{"className":85141,"ariaHidden":70},[69],[39,85143,85145,85148],{"className":85144},[74],[39,85146],{"className":85147,"style":709},[78],[39,85149,621],{"className":85150},[83,84]," 万，平均",[39,85153,85155],{"className":85154,"translate":43},[746],[39,85156,85158,85201],{"className":85157,"translate":43},[42],[39,85159,85161],{"className":85160},[47],[49,85162,85163],{"xmlns":51,"display":755},[53,85164,85165,85198],{},[56,85166,85167,85175,85177,85179,85181,85183,85185,85193,85196],{},[9932,85168,85169,85171,85173],{},[133,85170,69087],{},[539,85172,541],{},[539,85174,546],{},[59,85176,621],{},[35841,85178,35843],{},[59,85180,29425],{},[59,85182,621],{},[133,85184,793],{},[770,85186,85187],{"scriptlevel":541,"displaystyle":135},[5362,85188,85189,85191],{},[539,85190,546],{},[539,85192,17359],{},[35841,85194,85195],{}," 万",[59,85197,2903],{"mathvariant":145},[63,85199,85200],{"encoding":65},"\\int_0^1 t\\,dt=\\tfrac12\\ \\text{万}.",[39,85202,85204,85288],{"className":85203,"ariaHidden":70},[69],[39,85205,85207,85211,85264,85267,85270,85273,85276,85279,85282,85285],{"className":85206},[74],[39,85208],{"className":85209,"style":85210},[78],"height:2.476em;vertical-align:-0.9119em;",[39,85212,85214,85217],{"className":85213},[1744],[39,85215,69087],{"className":85216,"style":69380},[1744,3339,3340],[39,85218,85220],{"className":85219},[192],[39,85221,85223,85256],{"className":85222},[196,649],[39,85224,85226,85253],{"className":85225},[200],[39,85227,85230,85241],{"className":85228,"style":85229},[204],"height:1.564em;",[39,85231,85232,85235],{"style":69396},[39,85233],{"className":85234,"style":213},[212],[39,85236,85238],{"className":85237},[217,218,219,220],[39,85239,541],{"className":85240},[83,220],[39,85242,85244,85247],{"style":85243},"top:-3.8129em;margin-right:0.05em;",[39,85245],{"className":85246,"style":213},[212],[39,85248,85250],{"className":85249},[217,218,219,220],[39,85251,546],{"className":85252},[83,220],[39,85254,673],{"className":85255},[672],[39,85257,85259],{"className":85258},[200],[39,85260,85262],{"className":85261,"style":69415},[204],[39,85263],{},[39,85265],{"className":85266,"style":256},[230],[39,85268,621],{"className":85269},[83,84],[39,85271],{"className":85272,"style":256},[230],[39,85274,29425],{"className":85275},[83,84],[39,85277,621],{"className":85278},[83,84],[39,85280],{"className":85281,"style":231},[230],[39,85283,793],{"className":85284},[235],[39,85286],{"className":85287,"style":231},[230],[39,85289,85291,85294,85362,85365,85373],{"className":85290},[74],[39,85292],{"className":85293,"style":74987},[78],[39,85295,85297,85300,85359],{"className":85296},[83],[39,85298],{"className":85299},[182,5463],[39,85301,85303],{"className":85302},[5362],[39,85304,85306,85351],{"className":85305},[196,649],[39,85307,85309,85348],{"className":85308},[200],[39,85310,85312,85326,85334],{"className":85311,"style":75006},[204],[39,85313,85314,85317],{"style":75009},[39,85315],{"className":85316,"style":935},[212],[39,85318,85320],{"className":85319},[217,218,219,220],[39,85321,85323],{"className":85322},[83,220],[39,85324,17359],{"className":85325},[83,220],[39,85327,85328,85331],{"style":5503},[39,85329],{"className":85330,"style":935},[212],[39,85332],{"className":85333,"style":5511},[5510],[39,85335,85336,85339],{"style":75032},[39,85337],{"className":85338,"style":935},[212],[39,85340,85342],{"className":85341},[217,218,219,220],[39,85343,85345],{"className":85344},[83,220],[39,85346,546],{"className":85347},[83,220],[39,85349,673],{"className":85350},[672],[39,85352,85354],{"className":85353},[200],[39,85355,85357],{"className":85356,"style":75054},[204],[39,85358],{},[39,85360],{"className":85361},[263,5463],[39,85363,74416],{"className":85364},[230],[39,85366,85368],{"className":85367},[83,74653],[39,85369,85372],{"className":85370},[83,85371],"cjk_fallback","万",[39,85374,2903],{"className":85375},[83],[11,85377,85378,85379,85435,85436,85500,85501,1607],{},"这里可以直接用 ",[39,85380,85382,85402],{"className":85381,"translate":43},[42],[39,85383,85385],{"className":85384},[47],[49,85386,85387],{"xmlns":51},[53,85388,85389,85399],{},[56,85390,85391,85393,85395,85397],{},[539,85392,17359],{},[59,85394,621],{},[133,85396,5378],{},[59,85398,621],{},[63,85400,85401],{"encoding":65},"2t-t",[39,85403,85405,85426],{"className":85404,"ariaHidden":70},[69],[39,85406,85408,85411,85414,85417,85420,85423],{"className":85407},[74],[39,85409],{"className":85410,"style":17882},[78],[39,85412,17359],{"className":85413},[83],[39,85415,621],{"className":85416},[83,84],[39,85418],{"className":85419,"style":461},[230],[39,85421,5378],{"className":85422},[465],[39,85424],{"className":85425,"style":461},[230],[39,85427,85429,85432],{"className":85428},[74],[39,85430],{"className":85431,"style":709},[78],[39,85433,621],{"className":85434},[83,84]," 而不是 ",[39,85437,85439,85463],{"className":85438,"translate":43},[42],[39,85440,85442],{"className":85441},[47],[49,85443,85444],{"xmlns":51},[53,85445,85446,85460],{},[56,85447,85448,85450,85452,85454,85456,85458],{},[59,85449,150],{"mathvariant":145},[539,85451,17359],{},[59,85453,621],{},[133,85455,5378],{},[59,85457,621],{},[59,85459,150],{"mathvariant":145},[63,85461,85462],{"encoding":65},"|2t-t|",[39,85464,85466,85488],{"className":85465,"ariaHidden":70},[69],[39,85467,85469,85472,85476,85479,85482,85485],{"className":85468},[74],[39,85470],{"className":85471,"style":245},[78],[39,85473,85475],{"className":85474},[83],"∣2",[39,85477,621],{"className":85478},[83,84],[39,85480],{"className":85481,"style":461},[230],[39,85483,5378],{"className":85484},[465],[39,85486],{"className":85487,"style":461},[230],[39,85489,85491,85494,85497],{"className":85490},[74],[39,85492],{"className":85493,"style":245},[78],[39,85495,621],{"className":85496},[83,84],[39,85498,150],{"className":85499},[83],"，因为所有人都在往同一个方向涨。而这个“平均涨幅”正是 ",[39,85502,85504,85521],{"className":85503,"translate":43},[42],[39,85505,85507],{"className":85506},[47],[49,85508,85509],{"xmlns":51},[53,85510,85511,85519],{},[56,85512,85513],{},[613,85514,85515,85517],{},[59,85516,23703],{},[539,85518,546],{},[63,85520,78032],{"encoding":65},[39,85522,85524],{"className":85523,"ariaHidden":70},[69],[39,85525,85527,85530],{"className":85526},[74],[39,85528],{"className":85529,"style":634},[78],[39,85531,85533,85536],{"className":85532},[83],[39,85534,23703],{"className":85535,"style":178},[83,84],[39,85537,85539],{"className":85538},[192],[39,85540,85542,85562],{"className":85541},[196,649],[39,85543,85545,85559],{"className":85544},[200],[39,85546,85548],{"className":85547,"style":963},[204],[39,85549,85550,85553],{"style":73958},[39,85551],{"className":85552,"style":213},[212],[39,85554,85556],{"className":85555},[217,218,219,220],[39,85557,546],{"className":85558},[83,220],[39,85560,673],{"className":85561},[672],[39,85563,85565],{"className":85564},[200],[39,85566,85568],{"className":85567,"style":680},[204],[39,85569],{},[11,85571,85572,85573,85632,85633,85690,85691,85760],{},"如果反过来，A 是 ",[39,85574,85576,85601],{"className":85575,"translate":43},[42],[39,85577,85579],{"className":85578},[47],[49,85580,85581],{"xmlns":51},[53,85582,85583,85598],{},[56,85584,85585,85588,85590,85592,85594,85596],{},[59,85586,85587],{},"U",[133,85589,136],{"stretchy":135},[539,85591,541],{},[133,85593,155],{"separator":70},[539,85595,17359],{},[133,85597,161],{"stretchy":135},[63,85599,85600],{"encoding":65},"U(0,2)",[39,85602,85604],{"className":85603,"ariaHidden":70},[69],[39,85605,85607,85610,85614,85617,85620,85623,85626,85629],{"className":85606},[74],[39,85608],{"className":85609,"style":245},[78],[39,85611,85587],{"className":85612,"style":85613},[83,84],"margin-right:0.109em;",[39,85615,136],{"className":85616},[182],[39,85618,541],{"className":85619},[83],[39,85621,155],{"className":85622},[252],[39,85624],{"className":85625,"style":256},[230],[39,85627,17359],{"className":85628},[83],[39,85630,161],{"className":85631},[263],"、B 是 ",[39,85634,85636,85660],{"className":85635,"translate":43},[42],[39,85637,85639],{"className":85638},[47],[49,85640,85641],{"xmlns":51},[53,85642,85643,85657],{},[56,85644,85645,85647,85649,85651,85653,85655],{},[59,85646,85587],{},[133,85648,136],{"stretchy":135},[539,85650,541],{},[133,85652,155],{"separator":70},[539,85654,546],{},[133,85656,161],{"stretchy":135},[63,85658,85659],{"encoding":65},"U(0,1)",[39,85661,85663],{"className":85662,"ariaHidden":70},[69],[39,85664,85666,85669,85672,85675,85678,85681,85684,85687],{"className":85665},[74],[39,85667],{"className":85668,"style":245},[78],[39,85670,85587],{"className":85671,"style":85613},[83,84],[39,85673,136],{"className":85674},[182],[39,85676,541],{"className":85677},[83],[39,85679,155],{"className":85680},[252],[39,85682],{"className":85683,"style":256},[230],[39,85685,546],{"className":85686},[83],[39,85688,161],{"className":85689},[263],"，那就是平均降 0.5 万，",[39,85692,85694,85711],{"className":85693,"translate":43},[42],[39,85695,85697],{"className":85696},[47],[49,85698,85699],{"xmlns":51},[53,85700,85701,85709],{},[56,85702,85703],{},[613,85704,85705,85707],{},[59,85706,23703],{},[539,85708,546],{},[63,85710,78032],{"encoding":65},[39,85712,85714],{"className":85713,"ariaHidden":70},[69],[39,85715,85717,85720],{"className":85716},[74],[39,85718],{"className":85719,"style":634},[78],[39,85721,85723,85726],{"className":85722},[83],[39,85724,23703],{"className":85725,"style":178},[83,84],[39,85727,85729],{"className":85728},[192],[39,85730,85732,85752],{"className":85731},[196,649],[39,85733,85735,85749],{"className":85734},[200],[39,85736,85738],{"className":85737,"style":963},[204],[39,85739,85740,85743],{"style":73958},[39,85741],{"className":85742,"style":213},[212],[39,85744,85746],{"className":85745},[217,218,219,220],[39,85747,546],{"className":85748},[83,220],[39,85750,673],{"className":85751},[672],[39,85753,85755],{"className":85754},[200],[39,85756,85758],{"className":85757,"style":680},[204],[39,85759],{}," 仍然是 0.5——它量的是“搬了多少”，不管涨还是跌。",[11,85762,85763,85764,85833],{},"一般地，一维的 ",[39,85765,85767,85784],{"className":85766,"translate":43},[42],[39,85768,85770],{"className":85769},[47],[49,85771,85772],{"xmlns":51},[53,85773,85774,85782],{},[56,85775,85776],{},[613,85777,85778,85780],{},[59,85779,23703],{},[59,85781,11],{},[63,85783,77648],{"encoding":65},[39,85785,85787],{"className":85786,"ariaHidden":70},[69],[39,85788,85790,85793],{"className":85789},[74],[39,85791],{"className":85792,"style":35350},[78],[39,85794,85796,85799],{"className":85795},[83],[39,85797,23703],{"className":85798,"style":178},[83,84],[39,85800,85802],{"className":85801},[192],[39,85803,85805,85825],{"className":85804},[196,649],[39,85806,85808,85822],{"className":85807},[200],[39,85809,85811],{"className":85810,"style":1728},[204],[39,85812,85813,85816],{"style":73958},[39,85814],{"className":85815,"style":213},[212],[39,85817,85819],{"className":85818},[217,218,219,220],[39,85820,11],{"className":85821},[83,84,220],[39,85823,673],{"className":85824},[672],[39,85826,85828],{"className":85827},[200],[39,85829,85831],{"className":85830,"style":35396},[204],[39,85832],{}," 有闭式解：",[39,85835,85837],{"className":85836,"translate":43},[746],[39,85838,85840,85934],{"className":85839,"translate":43},[42],[39,85841,85843],{"className":85842},[47],[49,85844,85845],{"xmlns":51,"display":755},[53,85846,85847,85931],{},[56,85848,85849,85857,85859,85861,85863,85865,85867,85869,85877,85879,85891,85893,85895,85897,85899,85911,85913,85915,85917,85923,85925,85927,85929],{},[9932,85850,85851,85853,85855],{},[59,85852,23703],{},[59,85854,11],{},[59,85856,11],{},[133,85858,136],{"stretchy":135},[59,85860,73771],{},[133,85862,155],{"separator":70},[59,85864,73802],{},[133,85866,161],{"stretchy":135},[133,85868,793],{},[9932,85870,85871,85873,85875],{},[133,85872,69087],{},[539,85874,541],{},[539,85876,546],{},[133,85878,150],{"fence":70,"stretchy":70,"minsize":74068,"maxsize":74068},[9932,85880,85881,85883,85885],{},[59,85882,618],{},[59,85884,73771],{},[56,85886,85887,85889],{},[133,85888,5378],{},[539,85890,546],{},[133,85892,136],{"stretchy":135},[59,85894,621],{},[133,85896,161],{"stretchy":135},[133,85898,5378],{},[9932,85900,85901,85903,85905],{},[59,85902,618],{},[59,85904,73802],{},[56,85906,85907,85909],{},[133,85908,5378],{},[539,85910,546],{},[133,85912,136],{"stretchy":135},[59,85914,621],{},[133,85916,161],{"stretchy":135},[138,85918,85919,85921],{},[133,85920,150],{"fence":70,"stretchy":70,"minsize":74068,"maxsize":74068},[59,85922,11],{},[35841,85924,35843],{},[59,85926,29425],{},[59,85928,621],{},[133,85930,155],{"separator":70},[63,85932,85933],{"encoding":65},"W_p^p(\\mu,\\nu)=\\int_0^1\\bigl|F_\\mu^{-1}(t)-F_\\nu^{-1}(t)\\bigr|^p\\,dt 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时有更漂亮的形式。用上一节的对偶，对绝对连续的 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时等号成立。于是",[39,87507,87509],{"className":87508,"translate":43},[746],[39,87510,87512,87586],{"className":87511,"translate":43},[42],[39,87513,87515],{"className":87514},[47],[49,87516,87517],{"xmlns":51,"display":755},[53,87518,87519,87583],{},[56,87520,87521,87527,87529,87531,87533,87535,87537,87539,87545,87547,87553,87555,87557,87559,87561,87567,87569,87571,87573,87575,87577,87579,87581],{},[613,87522,87523,87525],{},[59,87524,23703],{},[539,87526,546],{},[133,87528,136],{"stretchy":135},[59,87530,73771],{},[133,87532,155],{"separator":70},[59,87534,73802],{},[133,87536,161],{"stretchy":135},[133,87538,793],{},[613,87540,87541,87543],{},[133,87542,69087],{},[59,87544,422],{"mathvariant":421},[133,87546,150],{"fence":70,"stretchy":70,"minsize":74068,"maxsize":74068},[613,87548,87549,87551],{},[59,87550,618],{},[59,87552,73771],{},[133,87554,136],{"stretchy":135},[59,87556,1755],{},[133,87558,161],{"stretchy":135},[133,87560,5378],{},[613,87562,87563,87565],{},[59,87564,618],{},[59,87566,73802],{},[133,87568,136],{"stretchy":135},[59,87570,1755],{},[133,87572,161],{"stretchy":135},[133,87574,150],{"fence":70,"stretchy":70,"minsize":74068,"maxsize":74068},[35841,87576,35843],{},[59,87578,29425],{},[59,87580,1755],{},[59,87582,2903],{"mathvariant":145},[63,87584,87585],{"encoding":65},"W_1(\\mu,\\nu)=\\int_{\\mathbb 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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 ",[39,87915,87917,87939],{"className":87916,"translate":43},[42],[39,87918,87920],{"className":87919},[47],[49,87921,87922],{"xmlns":51},[53,87923,87924,87936],{},[56,87925,87926,87928,87930,87932,87934],{},[133,87927,537],{"stretchy":135},[539,87929,541],{},[133,87931,155],{"separator":70},[539,87933,546],{},[133,87935,4280],{"stretchy":135},[63,87937,87938],{"encoding":65},"[0,1]",[39,87940,87942],{"className":87941,"ariaHidden":70},[69],[39,87943,87945,87948,87951,87954,87957,87960,87963],{"className":87944},[74],[39,87946],{"className":87947,"style":245},[78],[39,87949,537],{"className":87950},[182],[39,87952,541],{"className":87953},[83],[39,87955,155],{"className":87956},[252],[39,87958],{"className":87959,"style":256},[230],[39,87961,546],{"className":87962},[83],[39,87964,4280],{"className":87965},[263]," 上 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",[39,88110,88112,88130],{"className":88111,"translate":43},[42],[39,88113,88115],{"className":88114},[47],[49,88116,88117],{"xmlns":51},[53,88118,88119,88127],{},[56,88120,88121,88123,88125],{},[59,88122,1755],{},[59,88124,45691],{"mathvariant":145},[539,88126,17359],{},[63,88128,88129],{"encoding":65},"x\u002F2",[39,88131,88133],{"className":88132,"ariaHidden":70},[69],[39,88134,88136,88139,88142],{"className":88135},[74],[39,88137],{"className":88138,"style":245},[78],[39,88140,1755],{"className":88141},[83,84],[39,88143,88107],{"className":88144},[83],"，积出 ",[39,88147,88149,88169],{"className":88148,"translate":43},[42],[39,88150,88152],{"className":88151},[47],[49,88153,88154],{"xmlns":51},[53,88155,88156,88166],{},[56,88157,88158],{},[770,88159,88160],{"scriptlevel":541,"displaystyle":135},[5362,88161,88162,88164],{},[539,88163,546],{},[539,88165,59223],{},[63,88167,88168],{"encoding":65},"\\tfrac14",[39,88170,88172],{"className":88171,"ariaHidden":70},[69],[39,88173,88175,88178],{"className":88174},[74],[39,88176],{"className":88177,"style":74987},[78],[39,88179,88181,88184,88243],{"className":88180},[83],[39,88182],{"className":88183},[182,5463],[39,88185,88187],{"className":88186},[5362],[39,88188,88190,88235],{"className":88189},[196,649],[39,88191,88193,88232],{"className":88192},[200],[39,88194,88196,88210,88218],{"className":88195,"style":75006},[204],[39,88197,88198,88201],{"style":75009},[39,88199],{"className":88200,"style":935},[212],[39,88202,88204],{"className":88203},[217,218,219,220],[39,88205,88207],{"className":88206},[83,220],[39,88208,59223],{"className":88209},[83,220],[39,88211,88212,88215],{"style":5503},[39,88213],{"className":88214,"style":935},[212],[39,88216],{"className":88217,"style":5511},[5510],[39,88219,88220,88223],{"style":75032},[39,88221],{"className":88222,"style":935},[212],[39,88224,88226],{"className":88225},[217,218,219,220],[39,88227,88229],{"className":88228},[83,220],[39,88230,546],{"className":88231},[83,220],[39,88233,673],{"className":88234},[672],[39,88236,88238],{"className":88237},[200],[39,88239,88241],{"className":88240,"style":75054},[204],[39,88242],{},[39,88244],{"className":88245},[263,5463],"；在 ",[39,88248,88250,88272],{"className":88249,"translate":43},[42],[39,88251,88253],{"className":88252},[47],[49,88254,88255],{"xmlns":51},[53,88256,88257,88269],{},[56,88258,88259,88261,88263,88265,88267],{},[133,88260,537],{"stretchy":135},[539,88262,546],{},[133,88264,155],{"separator":70},[539,88266,17359],{},[133,88268,4280],{"stretchy":135},[63,88270,88271],{"encoding":65},"[1,2]",[39,88273,88275],{"className":88274,"ariaHidden":70},[69],[39,88276,88278,88281,88284,88287,88290,88293,88296],{"className":88277},[74],[39,88279],{"className":88280,"style":245},[78],[39,88282,537],{"className":88283},[182],[39,88285,546],{"className":88286},[83],[39,88288,155],{"className":88289},[252],[39,88291],{"className":88292,"style":256},[230],[39,88294,17359],{"className":88295},[83],[39,88297,4280],{"className":88298},[263],[39,88300,88302,88326],{"className":88301,"translate":43},[42],[39,88303,88305],{"className":88304},[47],[49,88306,88307],{"xmlns":51},[53,88308,88309,88323],{},[56,88310,88311,88313,88315,88317,88319,88321],{},[59,88312,618],{},[133,88314,136],{"stretchy":135},[59,88316,1755],{},[133,88318,161],{"stretchy":135},[133,88320,793],{},[539,88322,546],{},[63,88324,88325],{"encoding":65},"F(x)=1",[39,88327,88329,88356],{"className":88328,"ariaHidden":70},[69],[39,88330,88332,88335,88338,88341,88344,88347,88350,88353],{"className":88331},[74],[39,88333],{"className":88334,"style":245},[78],[39,88336,618],{"className":88337,"style":178},[83,84],[39,88339,136],{"className":88340},[182],[39,88342,1755],{"className":88343},[83,84],[39,88345,161],{"className":88346},[263],[39,88348],{"className":88349,"style":231},[230],[39,88351,793],{"className":88352},[235],[39,88354],{"className":88355,"style":231},[230],[39,88357,88359,88362],{"className":88358},[74],[39,88360],{"className":88361,"style":2473},[78],[39,88363,546],{"className":88364},[83],[39,88366,88368,88390],{"className":88367,"translate":43},[42],[39,88369,88371],{"className":88370},[47],[49,88372,88373],{"xmlns":51},[53,88374,88375,88387],{},[56,88376,88377,88379,88381,88383,88385],{},[539,88378,546],{},[133,88380,5378],{},[59,88382,1755],{},[59,88384,45691],{"mathvariant":145},[539,88386,17359],{},[63,88388,88389],{"encoding":65},"1-x\u002F2",[39,88391,88393,88411],{"className":88392,"ariaHidden":70},[69],[39,88394,88396,88399,88402,88405,88408],{"className":88395},[74],[39,88397],{"className":88398,"style":17882},[78],[39,88400,546],{"className":88401},[83],[39,88403],{"className":88404,"style":461},[230],[39,88406,5378],{"className":88407},[465],[39,88409],{"className":88410,"style":461},[230],[39,88412,88414,88417,88420],{"className":88413},[74],[39,88415],{"className":88416,"style":245},[78],[39,88418,1755],{"className":88419},[83,84],[39,88421,88107],{"className":88422},[83],"，也积出 ",[39,88425,88427,88446],{"className":88426,"translate":43},[42],[39,88428,88430],{"className":88429},[47],[49,88431,88432],{"xmlns":51},[53,88433,88434,88444],{},[56,88435,88436],{},[770,88437,88438],{"scriptlevel":541,"displaystyle":135},[5362,88439,88440,88442],{},[539,88441,546],{},[539,88443,59223],{},[63,88445,88168],{"encoding":65},[39,88447,88449],{"className":88448,"ariaHidden":70},[69],[39,88450,88452,88455],{"className":88451},[74],[39,88453],{"className":88454,"style":74987},[78],[39,88456,88458,88461,88520],{"className":88457},[83],[39,88459],{"className":88460},[182,5463],[39,88462,88464],{"className":88463},[5362],[39,88465,88467,88512],{"className":88466},[196,649],[39,88468,88470,88509],{"className":88469},[200],[39,88471,88473,88487,88495],{"className":88472,"style":75006},[204],[39,88474,88475,88478],{"style":75009},[39,88476],{"className":88477,"style":935},[212],[39,88479,88481],{"className":88480},[217,218,219,220],[39,88482,88484],{"className":88483},[83,220],[39,88485,59223],{"className":88486},[83,220],[39,88488,88489,88492],{"style":5503},[39,88490],{"className":88491,"style":935},[212],[39,88493],{"className":88494,"style":5511},[5510],[39,88496,88497,88500],{"style":75032},[39,88498],{"className":88499,"style":935},[212],[39,88501,88503],{"className":88502},[217,218,219,220],[39,88504,88506],{"className":88505},[83,220],[39,88507,546],{"className":88508},[83,220],[39,88510,673],{"className":88511},[672],[39,88513,88515],{"className":88514},[200],[39,88516,88518],{"className":88517,"style":75054},[204],[39,88519],{},[39,88521],{"className":88522},[263,5463],"。合计 ",[39,88525,88527,88547],{"className":88526,"translate":43},[42],[39,88528,88530],{"className":88529},[47],[49,88531,88532],{"xmlns":51},[53,88533,88534,88544],{},[56,88535,88536],{},[770,88537,88538],{"scriptlevel":541,"displaystyle":135},[5362,88539,88540,88542],{},[539,88541,546],{},[539,88543,17359],{},[63,88545,88546],{"encoding":65},"\\tfrac12",[39,88548,88550],{"className":88549,"ariaHidden":70},[69],[39,88551,88553,88556],{"className":88552},[74],[39,88554],{"className":88555,"style":74987},[78],[39,88557,88559,88562,88621],{"className":88558},[83],[39,88560],{"className":88561},[182,5463],[39,88563,88565],{"className":88564},[5362],[39,88566,88568,88613],{"className":88567},[196,649],[39,88569,88571,88610],{"className":88570},[200],[39,88572,88574,88588,88596],{"className":88573,"style":75006},[204],[39,88575,88576,88579],{"style":75009},[39,88577],{"className":88578,"style":935},[212],[39,88580,88582],{"className":88581},[217,218,219,220],[39,88583,88585],{"className":88584},[83,220],[39,88586,17359],{"className":88587},[83,220],[39,88589,88590,88593],{"style":5503},[39,88591],{"className":88592,"style":935},[212],[39,88594],{"className":88595,"style":5511},[5510],[39,88597,88598,88601],{"style":75032},[39,88599],{"className":88600,"style":935},[212],[39,88602,88604],{"className":88603},[217,218,219,220],[39,88605,88607],{"className":88606},[83,220],[39,88608,546],{"className":88609},[83,220],[39,88611,673],{"className":88612},[672],[39,88614,88616],{"className":88615},[200],[39,88617,88619],{"className":88618,"style":75054},[204],[39,88620],{},[39,88622],{"className":88623},[263,5463],"，和分位数算法的结果一致。",[11,88626,88627,88628,88761,88762,88925,88926,89106],{},"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 ",[39,88629,88631,88657],{"className":88630,"translate":43},[42],[39,88632,88634],{"className":88633},[47],[49,88635,88636],{"xmlns":51},[53,88637,88638,88654],{},[56,88639,88640,88646,88648],{},[613,88641,88642,88644],{},[59,88643,1755],{},[539,88645,546],{},[133,88647,1693],{},[613,88649,88650,88652],{},[59,88651,1755],{},[539,88653,17359],{},[63,88655,88656],{"encoding":65},"x_1\u003Cx_2",[39,88658,88660,88715],{"className":88659,"ariaHidden":70},[69],[39,88661,88663,88666,88706,88709,88712],{"className":88662},[74],[39,88664],{"className":88665,"style":29956},[78],[39,88667,88669,88672],{"className":88668},[83],[39,88670,1755],{"className":88671},[83,84],[39,88673,88675],{"className":88674},[192],[39,88676,88678,88698],{"className":88677},[196,649],[39,88679,88681,88695],{"className":88680},[200],[39,88682,88684],{"className":88683,"style":963},[204],[39,88685,88686,88689],{"style":966},[39,88687],{"className":88688,"style":213},[212],[39,88690,88692],{"className":88691},[217,218,219,220],[39,88693,546],{"className":88694},[83,220],[39,88696,673],{"className":88697},[672],[39,88699,88701],{"className":88700},[200],[39,88702,88704],{"className":88703,"style":680},[204],[39,88705],{},[39,88707],{"className":88708,"style":231},[230],[39,88710,1693],{"className":88711},[235],[39,88713],{"className":88714,"style":231},[230],[39,88716,88718,88721],{"className":88717},[74],[39,88719],{"className":88720,"style":2916},[78],[39,88722,88724,88727],{"className":88723},[83],[39,88725,1755],{"className":88726},[83,84],[39,88728,88730],{"className":88729},[192],[39,88731,88733,88753],{"className":88732},[196,649],[39,88734,88736,88750],{"className":88735},[200],[39,88737,88739],{"className":88738,"style":963},[204],[39,88740,88741,88744],{"style":966},[39,88742],{"className":88743,"style":213},[212],[39,88745,88747],{"className":88746},[217,218,219,220],[39,88748,17359],{"className":88749},[83,220],[39,88751,673],{"className":88752},[672],[39,88754,88756],{"className":88755},[200],[39,88757,88759],{"className":88758,"style":680},[204],[39,88760],{}," 却 ",[39,88763,88765,88803],{"className":88764,"translate":43},[42],[39,88766,88768],{"className":88767},[47],[49,88769,88770],{"xmlns":51},[53,88771,88772,88800],{},[56,88773,88774,88776,88778,88784,88786,88788,88790,88792,88798],{},[59,88775,11772],{},[133,88777,136],{"stretchy":135},[613,88779,88780,88782],{},[59,88781,1755],{},[539,88783,546],{},[133,88785,161],{"stretchy":135},[133,88787,31514],{},[59,88789,11772],{},[133,88791,136],{"stretchy":135},[613,88793,88794,88796],{},[59,88795,1755],{},[539,88797,17359],{},[133,88799,161],{"stretchy":135},[63,88801,88802],{"encoding":65},"T(x_1)>T(x_2)",[39,88804,88806,88870],{"className":88805,"ariaHidden":70},[69],[39,88807,88809,88812,88815,88818,88858,88861,88864,88867],{"className":88808},[74],[39,88810],{"className":88811,"style":245},[78],[39,88813,11772],{"className":88814,"style":178},[83,84],[39,88816,136],{"className":88817},[182],[39,88819,88821,88824],{"className":88820},[83],[39,88822,1755],{"className":88823},[83,84],[39,88825,88827],{"className":88826},[192],[39,88828,88830,88850],{"className":88829},[196,649],[39,88831,88833,88847],{"className":88832},[200],[39,88834,88836],{"className":88835,"style":963},[204],[39,88837,88838,88841],{"style":966},[39,88839],{"className":88840,"style":213},[212],[39,88842,88844],{"className":88843},[217,218,219,220],[39,88845,546],{"className":88846},[83,220],[39,88848,673],{"className":88849},[672],[39,88851,88853],{"className":88852},[200],[39,88854,88856],{"className":88855,"style":680},[204],[39,88857],{},[39,88859,161],{"className":88860},[263],[39,88862],{"className":88863,"style":231},[230],[39,88865,31514],{"className":88866},[235],[39,88868],{"className":88869,"style":231},[230],[39,88871,88873,88876,88879,88882,88922],{"className":88872},[74],[39,88874],{"className":88875,"style":245},[78],[39,88877,11772],{"className":88878,"style":178},[83,84],[39,88880,136],{"className":88881},[182],[39,88883,88885,88888],{"className":88884},[83],[39,88886,1755],{"className":88887},[83,84],[39,88889,88891],{"className":88890},[192],[39,88892,88894,88914],{"className":88893},[196,649],[39,88895,88897,88911],{"className":88896},[200],[39,88898,88900],{"className":88899,"style":963},[204],[39,88901,88902,88905],{"style":966},[39,88903],{"className":88904,"style":213},[212],[39,88906,88908],{"className":88907},[217,218,219,220],[39,88909,17359],{"className":88910},[83,220],[39,88912,673],{"className":88913},[672],[39,88915,88917],{"className":88916},[200],[39,88918,88920],{"className":88919,"style":680},[204],[39,88921],{},[39,88923,161],{"className":88924},[263],"，否则交换一下就更省。所以 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F_\\mu",[39,88967,88969,88987,89060],{"className":88968,"ariaHidden":70},[69],[39,88970,88972,88975,88978,88981,88984],{"className":88971},[74],[39,88973],{"className":88974,"style":79},[78],[39,88976,11772],{"className":88977,"style":178},[83,84],[39,88979],{"className":88980,"style":231},[230],[39,88982,793],{"className":88983},[235],[39,88985],{"className":88986,"style":231},[230],[39,88988,88990,88994,89051,89054,89057],{"className":88989},[74],[39,88991],{"className":88992,"style":88993},[78],"height:1.0611em;vertical-align:-0.247em;",[39,88995,88997,89000],{"className":88996},[83],[39,88998,618],{"className":88999,"style":178},[83,84],[39,89001,89003],{"className":89002},[192],[39,89004,89006,89043],{"className":89005},[196,649],[39,89007,89009,89040],{"className":89008},[200],[39,89010,89012,89023],{"className":89011,"style":19710},[204],[39,89013,89014,89017],{"style":13718},[39,89015],{"className":89016,"style":213},[212],[39,89018,89020],{"className":89019},[217,218,219,220],[39,89021,73802],{"className":89022,"style":73818},[83,84,220],[39,89024,89025,89028],{"style":208},[39,89026],{"className":89027,"style":213},[212],[39,89029,89031],{"className":89030},[217,218,219,220],[39,89032,89034,89037],{"className":89033},[83,220],[39,89035,5378],{"className":89036},[83,220],[39,89038,546],{"className":89039},[83,220],[39,89041,673],{"className":89042},[672],[39,89044,89046],{"className":89045},[200],[39,89047,89049],{"className":89048,"style":10003},[204],[39,89050],{},[39,89052],{"className":89053,"style":461},[230],[39,89055,88956],{"className":89056},[465],[39,89058],{"className":89059,"style":461},[230],[39,89061,89063,89066],{"className":89062},[74],[39,89064],{"className":89065,"style":35350},[78],[39,89067,89069,89072],{"className":89068},[83],[39,89070,618],{"className":89071,"style":178},[83,84],[39,89073,89075],{"className":89074},[192],[39,89076,89078,89098],{"className":89077},[196,649],[39,89079,89081,89095],{"className":89080},[200],[39,89082,89084],{"className":89083,"style":1728},[204],[39,89085,89086,89089],{"style":73958},[39,89087],{"className":89088,"style":213},[212],[39,89090,89092],{"className":89091},[217,218,219,220],[39,89093,73771],{"className":89094},[83,84,220],[39,89096,673],{"className":89097},[672],[39,89099,89101],{"className":89100},[200],[39,89102,89104],{"className":89103,"style":35396},[204],[39,89105],{}," 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ame":90663},[200],[39,90665,90667],{"className":90666,"style":680},[204],[39,90668],{},[39,90670,90672,90675],{"className":90671},[83],[39,90673,90086],{"className":90674,"style":2114},[83,84],[39,90676,90678],{"className":90677},[192],[39,90679,90681,90701],{"className":90680},[196,649],[39,90682,90684,90698],{"className":90683},[200],[39,90685,90687],{"className":90686,"style":963},[204],[39,90688,90689,90692],{"style":41316},[39,90690],{"className":90691,"style":213},[212],[39,90693,90695],{"className":90694},[217,218,219,220],[39,90696,17359],{"className":90697},[83,220],[39,90699,673],{"className":90700},[672],[39,90702,90704],{"className":90703},[200],[39,90705,90707],{"className":90706,"style":680},[204],[39,90708],{},[39,90710,150],{"className":90711},[83],[39,90713,161],{"className":90714},[263],[39,90716],{"className":90717,"style":231},[230],[39,90719,793],{"className":90720},[235],[39,90722],{"className":90723,"style":231},[230],[39,90725,90727,90730,90733,90773,90776,90779],{"className":90726},[74],[39,90728],{"className":90729,"style":245},[78],[39,90731,136],{"className":90732},[182],[39,90734,90736,90739],{"className":90735},[83],[39,90737,90086],{"className":90738,"style":2114},[83,84],[39,90740,90742],{"className":90741},[192],[39,90743,90745,90765],{"className":90744},[196,649],[39,90746,90748,90762],{"className":90747},[200],[39,90749,90751],{"className":90750,"style":963},[204],[39,90752,90753,90756],{"style":41316},[39,90754],{"className":90755,"style":213},[212],[39,90757,90759],{"className":90758},[217,218,219,220],[39,90760,546],{"className":90761},[83,220],[39,90763,673],{"className":90764},[672],[39,90766,90768],{"className":90767},[200],[39,90769,90771],{"className":90770,"style":680},[204],[39,90772],{},[39,90774],{"className":90775,"style":461},[230],[39,90777,5378],{"className":90778},[465],[39,90780],{"className":90781,"style":461},[230],[39,90783,90785,90788,90828],{"className":90784},[74],[39,90786],{"className":90787,"style":84808},[78],[39,90789,90791,90794],{"className":90790},[83],[39,90792,90086],{"className":90793,"style":2114},[83,84],[39,90795,90797],{"className":90796},[192],[39,90798,90800,90820],{"className":90799},[196,649],[39,90801,90803,90817],{"className":90802},[200],[39,90804,90806],{"className":90805,"style":963},[204],[39,90807,90808,90811],{"style":41316},[39,90809],{"className":90810,"style":213},[212],[39,90812,90814],{"className":90813},[217,218,219,220],[39,90815,17359],{"className":90816},[83,220],[39,90818,673],{"className":90819},[672],[39,90821,90823],{"className":90822},[200],[39,90824,90826],{"className":90825,"style":680},[204],[39,90827],{},[39,90829,90831,90834],{"className":90830},[263],[39,90832,161],{"className":90833},[263],[39,90835,90837],{"className":90836},[192],[39,90838,90840],{"className":90839},[196],[39,90841,90843],{"className":90842},[200],[39,90844,90846],{"className":90845,"style":19710},[204],[39,90847,90848,90851],{"style":208},[39,90849],{"className":90850,"style":213},[212],[39,90852,90854],{"className":90853},[217,218,219,220],[39,90855,17359],{"className":90856},[83,220],"，对得上。取 ",[39,90859,90861,90885],{"className":90860,"translate":43},[42],[39,90862,90864],{"className":90863},[47],[49,90865,90866],{"xmlns":51},[53,90867,90868,90882],{},[56,90869,90870,90872,90874,90876,90878,90880],{},[59,90871,77752],{"mathvariant":617},[133,90873,136],{"stretchy":135},[539,90875,541],{},[133,90877,155],{"separator":70},[539,90879,546],{},[133,90881,161],{"stretchy":135},[63,90883,90884],{"encoding":65},"\\mathcal N(0,1)",[39,90886,90888],{"className":90887,"ariaHidden":70},[69],[39,90889,90891,90894,90897,90900,90903,90906,90909,90912],{"className":90890},[74],[39,90892],{"className":90893,"style":245},[78],[39,90895,77752],{"className":90896,"style":77820},[83,641],[39,90898,136],{"className":90899},[182],[39,90901,541],{"className":90902},[83],[39,90904,155],{"className":90905},[252],[39,90907],{"className":90908,"style":256},[230],[39,90910,546],{"className":90911},[83],[39,90913,161],{"className":90914},[263],[39,90916,90918,90942],{"className":90917,"translate":43},[42],[39,90919,90921],{"className":90920},[47],[49,90922,90923],{"xmlns":51},[53,90924,90925,90939],{},[56,90926,90927,90929,90931,90933,90935,90937],{},[59,90928,77752],{"mathvariant":617},[133,90930,136],{"stretchy":135},[539,90932,541],{},[133,90934,155],{"separator":70},[539,90936,59223],{},[133,90938,161],{"stretchy":135},[63,90940,90941],{"encoding":65},"\\mathcal N(0,4)",[39,90943,90945],{"className":90944,"ariaHidden":70},[69],[39,90946,90948,90951,90954,90957,90960,90963,90966,90969],{"className":90947},[74],[39,90949],{"className":90950,"style":245},[78],[39,90952,77752],{"className":90953,"style":77820},[83,641],[39,90955,136],{"className":90956},[182],[39,90958,541],{"className":90959},[83],[39,90961,155],{"className":90962},[252],[39,90964],{"className":90965,"style":256},[230],[39,90967,59223],{"className":90968},[83],[39,90970,161],{"className":90971},[263],"，得 ",[39,90974,90976,90998],{"className":90975,"translate":43},[42],[39,90977,90979],{"className":90978},[47],[49,90980,90981],{"xmlns":51},[53,90982,90983,90995],{},[56,90984,90985,90991,90993],{},[613,90986,90987,90989],{},[59,90988,23703],{},[539,90990,17359],{},[133,90992,793],{},[539,90994,546],{},[63,90996,90997],{"encoding":65},"W_2=1",[39,90999,91001,91056],{"className":91000,"ariaHidden":70},[69],[39,91002,91004,91007,91047,91050,91053],{"className":91003},[74],[39,91005],{"className":91006,"style":634},[78],[39,91008,91010,91013],{"className":91009},[83],[39,91011,23703],{"className":91012,"style":178},[83,84],[39,91014,91016],{"className":91015},[192],[39,91017,91019,91039],{"className":91018},[196,649],[39,91020,91022,91036],{"className":91021},[200],[39,91023,91025],{"className":91024,"style":963},[204],[39,91026,91027,91030],{"style":73958},[39,91028],{"className":91029,"style":213},[212],[39,91031,91033],{"className":91032},[217,218,219,220],[39,91034,17359],{"className":91035},[83,220],[39,91037,673],{"className":91038},[672],[39,91040,91042],{"className":91041},[200],[39,91043,91045],{"className":91044,"style":680},[204],[39,91046],{},[39,91048],{"className":91049,"style":231},[230],[39,91051,793],{"className":91052},[235],[39,91054],{"className":91055,"style":231},[230],[39,91057,91059,91062],{"className":91058},[74],[39,91060],{"className":91061,"style":2473},[78],[39,91063,546],{"className":91064},[83],"，正好是标准差之差；而最优映射是 ",[39,91067,91069,91089],{"className":91068,"translate":43},[42],[39,91070,91072],{"className":91071},[47],[49,91073,91074],{"xmlns":51},[53,91075,91076,91086],{},[56,91077,91078,91080,91082,91084],{},[59,91079,1755],{},[133,91081,69813],{},[539,91083,17359],{},[59,91085,1755],{},[63,91087,91088],{"encoding":65},"x\\mapsto2x",[39,91090,91092,91110],{"className":91091,"ariaHidden":70},[69],[39,91093,91095,91098,91101,91104,91107],{"className":91094},[74],[39,91096],{"className":91097,"style":69838},[78],[39,91099,1755],{"className":91100},[83,84],[39,91102],{"className":91103,"style":231},[230],[39,91105,69813],{"className":91106},[235],[39,91108],{"className":91109,"style":231},[230],[39,91111,91113,91116,91119],{"className":91112},[74],[39,91114],{"className":91115,"style":2473},[78],[39,91117,17359],{"className":91118},[83],[39,91120,1755],{"className":91121},[83,84],"，位移的均方根也正好是 ",[39,91124,91126,91139],{"className":91125,"translate":43},[42],[39,91127,91129],{"className":91128},[47],[49,91130,91131],{"xmlns":51},[53,91132,91133,91137],{},[56,91134,91135],{},[539,91136,546],{},[63,91138,546],{"encoding":65},[39,91140,91142],{"className":91141,"ariaHidden":70},[69],[39,91143,91145,91148],{"className":91144},[74],[39,91146],{"className":91147,"style":2473},[78],[39,91149,546],{"className":91150},[83],"。两个数在这里是同一个数。",[11,91153,91154,91155,91362],{},"最优映射还是仿射的：",[39,91156,91158,91204],{"className":91157,"translate":43},[42],[39,91159,91161],{"className":91160},[47],[49,91162,91163],{"xmlns":51},[53,91164,91165,91201],{},[56,91166,91167,91169,91171,91173,91175,91177,91183,91185,91187,91189,91191,91193,91199],{},[59,91168,11772],{},[133,91170,136],{"stretchy":135},[59,91172,1755],{},[133,91174,161],{"stretchy":135},[133,91176,793],{},[613,91178,91179,91181],{},[59,91180,1748],{},[539,91182,17359],{},[133,91184,788],{},[59,91186,101],{},[133,91188,136],{"stretchy":135},[59,91190,1755],{},[133,91192,5378],{},[613,91194,91195,91197],{},[59,91196,1748],{},[539,91198,546],{},[133,91200,161],{"stretchy":135},[63,91202,91203],{"encoding":65},"T(x)=m_2+A(x-m_1)",[39,91205,91207,91234,91289,91313],{"className":91206,"ariaHidden":70},[69],[39,91208,91210,91213,91216,91219,91222,91225,91228,91231],{"className":91209},[74],[39,91211],{"className":91212,"style":245},[78],[39,91214,11772],{"className":91215,"style":178},[83,84],[39,91217,136],{"className":91218},[182],[39,91220,1755],{"className":91221},[83,84],[39,91223,161],{"className":91224},[263],[39,91226],{"className":91227,"style":231},[230],[39,91229,793],{"className":91230},[235],[39,91232],{"className":91233,"style":231},[230],[39,91235,91237,91240,91280,91283,91286],{"className":91236},[74],[39,91238],{"className":91239,"style":7528},[78],[39,91241,91243,91246],{"className":91242},[83],[39,91244,1748],{"className":91245},[83,84],[39,91247,91249],{"className":91248},[192],[39,91250,91252,91272],{"className":91251},[196,649],[39,91253,91255,91269],{"className":91254},[200],[39,91256,91258],{"className":91257,"style":963},[204],[39,91259,91260,91263],{"style":966},[39,91261],{"className":91262,"style":213},[212],[39,91264,91266],{"className":91265},[217,218,219,220],[39,91267,17359],{"className":91268},[83,220],[39,91270,673],{"className":91271},[672],[39,91273,91275],{"className":91274},[200],[39,91276,91278],{"className":91277,"style":680},[204],[39,91279],{},[39,91281],{"className":91282,"style":461},[230],[39,91284,788],{"className":91285},[465],[39,91287],{"className":91288,"style":461},[230],[39,91290,91292,91295,91298,91301,91304,91307,91310],{"className":91291},[74],[39,91293],{"className":91294,"style":245},[78],[39,91296,101],{"className":91297},[83,84],[39,91299,136],{"className":91300},[182],[39,91302,1755],{"className":91303},[83,84],[39,91305],{"className":91306,"style":461},[230],[39,91308,5378],{"className":91309},[465],[39,91311],{"className":91312,"style":461},[230],[39,91314,91316,91319,91359],{"className":91315},[74],[39,91317],{"className":91318,"style":245},[78],[39,91320,91322,91325],{"className":91321},[83],[39,91323,1748],{"className":91324},[83,84],[39,91326,91328],{"className":91327},[192],[39,91329,91331,91351],{"className":91330},[196,649],[39,91332,91334,91348],{"className":91333},[200],[39,91335,91337],{"className":91336,"style":963},[204],[39,91338,91339,91342],{"style":966},[39,91340],{"className":91341,"style":213},[212],[39,91343,91345],{"className":91344},[217,218,219,220],[39,91346,546],{"className":91347},[83,220],[39,91349,673],{"className":91350},[672],[39,91352,91354],{"className":91353},[200],[39,91355,91357],{"className":91356,"style":680},[204],[39,91358],{},[39,91360,161],{"className":91361},[263],"，其中",[39,91364,91366],{"className":91365,"translate":43},[746],[39,91367,91369,91467],{"className":91368,"translate":43},[42],[39,91370,91372],{"className":91371},[47],[49,91373,91374],{"xmlns":51,"display":755},[53,91375,91376,91464],{},[56,91377,91378,91380,91382,91398,91400,91414,91420,91434,91446,91462],{},[59,91379,101],{},[133,91381,793],{},[9932,91383,91384,91386,91388],{},[59,91385,89151],{"mathvariant":145},[539,91387,546],{},[56,91389,91390,91392,91394,91396],{},[133,91391,5378],{},[539,91393,546],{},[59,91395,45691],{"mathvariant":145},[539,91397,17359],{},[133,91399,136],{"fence":70,"stretchy":70,"minsize":74068,"maxsize":74068},[9932,91401,91402,91404,91406],{},[59,91403,89151],{"mathvariant":145},[539,91405,546],{},[56,91407,91408,91410,91412],{},[539,91409,546],{},[59,91411,45691],{"mathvariant":145},[539,91413,17359],{},[613,91415,91416,91418],{},[59,91417,89151],{"mathvariant":145},[539,91419,17359],{},[9932,91421,91422,91424,91426],{},[59,91423,89151],{"mathvariant":145},[539,91425,546],{},[56,91427,91428,91430,91432],{},[539,91429,546],{},[59,91431,45691],{"mathvariant":145},[539,91433,17359],{},[138,91435,91436,91438],{},[133,91437,161],{"fence":70,"stretchy":70,"minsize":74068,"maxsize":74068},[56,91439,91440,91442,91444],{},[539,91441,546],{},[59,91443,45691],{"mathvariant":145},[539,91445,17359],{},[9932,91447,91448,91450,91452],{},[59,91449,89151],{"mathvariant":145},[539,91451,546],{},[56,91453,91454,91456,91458,91460],{},[133,91455,5378],{},[539,91457,546],{},[59,91459,45691],{"mathvariant":145},[539,91461,17359],{},[59,91463,2903],{"mathvariant":145},[63,91465,91466],{"encoding":65},"A=\\Sigma_1^{-1\u002F2}\\bigl(\\Sigma_1^{1\u002F2}\\Sigma_2\\Sigma_1^{1\u002F2}\\bigr)^{1\u002F2}\\Sigma_1^{-1\u002F2}.",[39,91468,91470,91488],{"className":91469,"ariaHidden":70},[69],[39,91471,91473,91476,91479,91482,91485],{"className":91472},[74],[39,91474],{"className":91475,"style":79},[78],[39,91477,101],{"className":91478},[83,84],[39,91480],{"className":91481,"style":231},[230],[39,91483,793],{"className":91484},[235],[39,91486],{"className":91487,"style":231},[230],[39,91489,91491,91495,91552,91558,91612,91652,91706,91741,91798],{"className":91490},[74],[39,91492],{"className":91493,"style":91494},[78],"height:1.4779em;vertical-align:-0.35em;",[39,91496,91498,91501],{"className":91497},[83],[39,91499,89151],{"className":91500},[83],[39,91502,91504],{"className":91503},[192],[39,91505,91507,91544],{"className":91506},[196,649],[39,91508,91510,91541],{"className":91509},[200],[39,91511,91513,91524],{"className":91512,"style":89862},[204],[39,91514,91515,91518],{"style":89865},[39,91516],{"className":91517,"style":213},[212],[39,91519,91521],{"className":91520},[217,218,219,220],[39,91522,546],{"className":91523},[83,220],[39,91525,91526,91529],{"style":89877},[39,91527],{"className":91528,"style":213},[212],[39,91530,91532],{"className":91531},[217,218,219,220],[39,91533,91535,91538],{"className":91534},[83,220],[39,91536,5378],{"className":91537},[83,220],[39,91539,45696],{"className":91540},[83,220],[39,91542,673],{"className":91543},[672],[39,91545,91547],{"className":91546},[200],[39,91548,91550],{"className":91549,"style":89899},[204],[39,91551],{},[39,91553,91555],{"className":91554},[182],[39,91556,136],{"className":91557},[18520,4009],[39,91559,91561,91564],{"className":91560},[83],[39,91562,89151],{"className":91563},[83],[39,91565,91567],{"className":91566},[192],[39,91568,91570,91604],{"className":91569},[196,649],[39,91571,91573,91601],{"className":91572},[200],[39,91574,91576,91587],{"className":91575,"style":89862},[204],[39,91577,91578,91581],{"style":89865},[39,91579],{"className":91580,"style":213},[212],[39,91582,91584],{"className":91583},[217,218,219,220],[39,91585,546],{"className":91586},[83,220],[39,91588,91589,91592],{"style":89877},[39,91590],{"className":91591,"style":213},[212],[39,91593,91595],{"className":91594},[217,218,219,220],[39,91596,91598],{"className":91597},[83,220],[39,91599,45696],{"className":91600},[83,220],[39,91602,673],{"className":91603},[672],[39,91605,91607],{"className":91606},[200],[39,91608,91610],{"className":91609,"style":89899},[204],[39,91611],{},[39,91613,91615,91618],{"className":91614},[83],[39,91616,89151],{"className":91617},[83],[39,91619,91621],{"className":91620},[192],[39,91622,91624,91644],{"className":91623},[196,649],[39,91625,91627,91641],{"className":91626},[200],[39,91628,91630],{"className":91629,"style":963},[204],[39,91631,91632,91635],{"style":966},[39,91633],{"className":91634,"style":213},[212],[39,91636,91638],{"className":91637},[217,218,219,220],[39,91639,17359],{"className":91640},[83,220],[39,91642,673],{"className":91643},[672],[39,91645,91647],{"className":91646},[200],[39,91648,91650],{"className":91649,"style":680},[204],[39,91651],{},[39,91653,91655,91658],{"className":91654},[83],[39,91656,89151],{"className":91657},[83],[39,91659,91661],{"className":91660},[192],[39,91662,91664,91698],{"className":91663},[196,649],[39,91665,91667,91695],{"className":91666},[200],[39,91668,91670,91681],{"className":91669,"style":89862},[204],[39,91671,91672,91675],{"style":89865},[39,91673],{"className":91674,"style":213},[212],[39,91676,91678],{"className":91677},[217,218,219,220],[39,91679,546],{"className":91680},[83,220],[39,91682,91683,91686],{"style":89877},[39,91684],{"className":91685,"style":213},[212],[39,91687,91689],{"className":91688},[217,218,219,220],[39,91690,91692],{"className":91691},[83,220],[39,91693,45696],{"className":91694},[83,220],[39,91696,673],{"className":91697},[672],[39,91699,91701],{"className":91700},[200],[39,91702,91704],{"className":91703,"style":89899},[204],[39,91705],{},[39,91707,91709,91715],{"className":91708},[263],[39,91710,91712],{"className":91711},[263],[39,91713,161],{"className":91714},[18520,4009],[39,91716,91718],{"className":91717},[192],[39,91719,91721],{"className":91720},[196],[39,91722,91724],{"className":91723},[200],[39,91725,91727],{"className":91726,"style":90017},[204],[39,91728,91729,91732],{"style":86329},[39,91730],{"className":91731,"style":213},[212],[39,91733,91735],{"className":91734},[217,218,219,220],[39,91736,91738],{"className":91737},[83,220],[39,91739,45696],{"className":91740},[83,220],[39,91742,91744,91747],{"className":91743},[83],[39,91745,89151],{"className":91746},[83],[39,91748,91750],{"className":91749},[192],[39,91751,91753,91790],{"className":91752},[196,649],[39,91754,91756,91787],{"className":91755},[200],[39,91757,91759,91770],{"className":91758,"style":89862},[204],[39,91760,91761,91764],{"style":89865},[39,91762],{"className":91763,"style":213},[212],[39,91765,91767],{"className":91766},[217,218,219,220],[39,91768,546],{"className":91769},[83,220],[39,91771,91772,91775],{"style":89877},[39,91773],{"className":91774,"style":213},[212],[39,91776,91778],{"className":91777},[217,218,219,220],[39,91779,91781,91784],{"className":91780},[83,220],[39,91782,5378],{"className":91783},[83,220],[39,91785,45696],{"className":91786},[83,220],[39,91788,673],{"className":91789},[672],[39,91791,91793],{"className":91792},[200],[39,91794,91796],{"className":91795,"style":89899},[204],[39,91797],{},[39,91799,2903],{"className":91800},[83],[11,91802,91803],{},"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。",[11,91805,91806,91807,91877,91878,92106,92107,92176],{},"高斯集合配上 ",[39,91808,91810,91828],{"className":91809,"translate":43},[42],[39,91811,91813],{"className":91812},[47],[49,91814,91815],{"xmlns":51},[53,91816,91817,91825],{},[56,91818,91819],{},[613,91820,91821,91823],{},[59,91822,23703],{},[539,91824,17359],{},[63,91826,91827],{"encoding":65},"W_2",[39,91829,91831],{"className":91830,"ariaHidden":70},[69],[39,91832,91834,91837],{"className":91833},[74],[39,91835],{"className":91836,"style":634},[78],[39,91838,91840,91843],{"className":91839},[83],[39,91841,23703],{"className":91842,"style":178},[83,84],[39,91844,91846],{"className":91845},[192],[39,91847,91849,91869],{"className":91848},[196,649],[39,91850,91852,91866],{"className":91851},[200],[39,91853,91855],{"className":91854,"style":963},[204],[39,91856,91857,91860],{"style":73958},[39,91858],{"className":91859,"style":213},[212],[39,91861,91863],{"className":91862},[217,218,219,220],[39,91864,17359],{"className":91865},[83,220],[39,91867,673],{"className":91868},[672],[39,91870,91872],{"className":91871},[200],[39,91873,91875],{"className":91874,"style":680},[204],[39,91876],{}," 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 ",[39,91879,91881,91933],{"className":91880,"translate":43},[42],[39,91882,91884],{"className":91883},[47],[49,91885,91886],{"xmlns":51},[53,91887,91888,91930],{},[56,91889,91890,91896,91898,91900,91902,91904,91906,91908,91910,91916,91918,91920,91922,91928],{},[613,91891,91892,91894],{},[59,91893,73771],{},[59,91895,621],{},[133,91897,793],{},[133,91899,136],{"fence":70,"stretchy":70,"minsize":74068,"maxsize":74068},[133,91901,136],{"stretchy":135},[539,91903,546],{},[133,91905,5378],{},[59,91907,621],{},[133,91909,161],{"stretchy":135},[56,91911,91912,91914],{},[59,91913,13377],{"mathvariant":145},[59,91915,29425],{"mathvariant":145},[133,91917,788],{},[59,91919,621],{},[59,91921,11772],{},[613,91923,91924,91926],{},[133,91925,161],{"fence":70,"stretchy":70,"minsize":74068,"maxsize":74068},[59,91927,73919],{"mathvariant":145},[59,91929,73771],{},[63,91931,91932],{"encoding":65},"\\mu_t=\\bigl((1-t)\\mathrm{id}+tT\\bigr)_\\#\\mu",[39,91934,91936,91991,92018,92046],{"className":91935,"ariaHidden":70},[69],[39,91937,91939,91942,91982,91985,91988],{"className":91938},[74],[39,91940],{"className":91941,"style":2654},[78],[39,91943,91945,91948],{"className":91944},[83],[39,91946,73771],{"className":91947},[83,84],[39,91949,91951],{"className":91950},[192],[39,91952,91954,91974],{"className":91953},[196,649],[39,91955,91957,91971],{"className":91956},[200],[39,91958,91960],{"className":91959,"style":656},[204],[39,91961,91962,91965],{"style":966},[39,91963],{"className":91964,"style":213},[212],[39,91966,91968],{"className":91967},[217,218,219,220],[39,91969,621],{"className":91970},[83,84,220],[39,91972,673],{"className":91973},[672],[39,91975,91977],{"className":91976},[200],[39,91978,91980],{"className":91979,"style":680},[204],[39,91981],{},[39,91983],{"className":91984,"style":231},[230],[39,91986,793],{"className":91987},[235],[39,91989],{"className":91990,"style":231},[230],[39,91992,91994,91997,92003,92006,92009,92012,92015],{"className":91993},[74],[39,91995],{"className":91996,"style":908},[78],[39,91998,92000],{"className":91999},[182],[39,92001,136],{"className":92002},[18520,4009],[39,92004,136],{"className":92005},[182],[39,92007,546],{"className":92008},[83],[39,92010],{"className":92011,"style":461},[230],[39,92013,5378],{"className":92014},[465],[39,92016],{"className":92017,"style":461},[230],[39,92019,92021,92024,92027,92030,92037,92040,92043],{"className":92020},[74],[39,92022],{"className":92023,"style":245},[78],[39,92025,621],{"className":92026},[83,84],[39,92028,161],{"className":92029},[263],[39,92031,92033],{"className":92032},[83],[39,92034,92036],{"className":92035},[83,35077],"id",[39,92038],{"className":92039,"style":461},[230],[39,92041,788],{"className":92042},[465],[39,92044],{"className":92045,"style":461},[230],[39,92047,92049,92053,92057,92103],{"className":92048},[74],[39,92050],{"className":92051,"style":92052},[78],"height:1.3858em;vertical-align:-0.5358em;",[39,92054,92056],{"className":92055,"style":178},[83,84],"tT",[39,92058,92060,92066],{"className":92059},[263],[39,92061,92063],{"className":92062},[263],[39,92064,161],{"className":92065},[18520,4009],[39,92067,92069],{"className":92068},[192],[39,92070,92072,92094],{"className":92071},[196,649],[39,92073,92075,92091],{"className":92074},[200],[39,92076,92079],{"className":92077,"style":92078},[204],"height:0.0864em;",[39,92080,92082,92085],{"style":92081},"top:-2.3003em;margin-right:0.05em;",[39,92083],{"className":92084,"style":213},[212],[39,92086,92088],{"className":92087},[217,218,219,220],[39,92089,73919],{"className":92090},[83,220],[39,92092,673],{"className":92093},[672],[39,92095,92097],{"className":92096},[200],[39,92098,92101],{"className":92099,"style":92100},[204],"height:0.5358em;",[39,92102],{},[39,92104,73771],{"className":92105},[83,84],"。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；",[39,92108,92110,92127],{"className":92109,"translate":43},[42],[39,92111,92113],{"className":92112},[47],[49,92114,92115],{"xmlns":51},[53,92116,92117,92125],{},[56,92118,92119],{},[613,92120,92121,92123],{},[59,92122,23703],{},[539,92124,17359],{},[63,92126,91827],{"encoding":65},[39,92128,92130],{"className":92129,"ariaHidden":70},[69],[39,92131,92133,92136],{"className":92132},[74],[39,92134],{"className":92135,"style":634},[78],[39,92137,92139,92142],{"className":92138},[83],[39,92140,23703],{"className":92141,"style":178},[83,84],[39,92143,92145],{"className":92144},[192],[39,92146,92148,92168],{"className":92147},[196,649],[39,92149,92151,92165],{"className":92150},[200],[39,92152,92154],{"className":92153,"style":963},[204],[39,92155,92156,92159],{"style":73958},[39,92157],{"className":92158,"style":213},[212],[39,92160,92162],{"className":92161},[217,218,219,220],[39,92163,17359],{"className":92164},[83,220],[39,92166,673],{"className":92167},[672],[39,92169,92171],{"className":92170},[200],[39,92172,92174],{"className":92173,"style":680},[204],[39,92175],{}," 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。",[21,92178,92179],{"id":92179},"计算的代价",[11,92181,92182,92210],{},[39,92183,92185,92198],{"className":92184,"translate":43},[42],[39,92186,92188],{"className":92187},[47],[49,92189,92190],{"xmlns":51},[53,92191,92192,92196],{},[56,92193,92194],{},[59,92195,23703],{},[63,92197,23703],{"encoding":65},[39,92199,92201],{"className":92200,"ariaHidden":70},[69],[39,92202,92204,92207],{"className":92203},[74],[39,92205],{"className":92206,"style":79},[78],[39,92208,23703],{"className":92209,"style":178},[83,84]," 定义了“是什么”，但没告诉我“怎么算”。",[11,92212,92213,92297,92298,92360,92361,92424,92425,92453],{},[39,92214,92216,92238],{"className":92215,"translate":43},[42],[39,92217,92219],{"className":92218},[47],[49,92220,92221],{"xmlns":51},[53,92222,92223,92235],{},[56,92224,92225,92227,92229],{},[59,92226,12962],{},[133,92228,793],{},[138,92230,92231,92233],{},[539,92232,48919],{},[539,92234,59223],{},[63,92236,92237],{"encoding":65},"n=10^4",[39,92239,92241,92259],{"className":92240,"ariaHidden":70},[69],[39,92242,92244,92247,92250,92253,92256],{"className":92243},[74],[39,92245],{"className":92246,"style":290},[78],[39,92248,12962],{"className":92249},[83,84],[39,92251],{"className":92252,"style":231},[230],[39,92254,793],{"className":92255},[235],[39,92257],{"className":92258,"style":231},[230],[39,92260,92262,92265,92268],{"className":92261},[74],[39,92263],{"className":92264,"style":19710},[78],[39,92266,546],{"className":92267},[83],[39,92269,92271,92274],{"className":92270},[83],[39,92272,541],{"className":92273},[83],[39,92275,92277],{"className":92276},[192],[39,92278,92280],{"className":92279},[196],[39,92281,92283],{"className":92282},[200],[39,92284,92286],{"className":92285,"style":19710},[204],[39,92287,92288,92291],{"style":208},[39,92289],{"className":92290,"style":213},[212],[39,92292,92294],{"className":92293},[217,218,219,220],[39,92295,59223],{"className":92296},[83,220]," 个样本对 ",[39,92299,92301,92319],{"className":92300,"translate":43},[42],[39,92302,92304],{"className":92303},[47],[49,92305,92306],{"xmlns":51},[53,92307,92308,92316],{},[56,92309,92310],{},[138,92311,92312,92314],{},[539,92313,48919],{},[539,92315,59223],{},[63,92317,92318],{"encoding":65},"10^4",[39,92320,92322],{"className":92321,"ariaHidden":70},[69],[39,92323,92325,92328,92331],{"className":92324},[74],[39,92326],{"className":92327,"style":19710},[78],[39,92329,546],{"className":92330},[83],[39,92332,92334,92337],{"className":92333},[83],[39,92335,541],{"className":92336},[83],[39,92338,92340],{"className":92339},[192],[39,92341,92343],{"className":92342},[196],[39,92344,92346],{"className":92345},[200],[39,92347,92349],{"className":92348,"style":19710},[204],[39,92350,92351,92354],{"style":208},[39,92352],{"className":92353,"style":213},[212],[39,92355,92357],{"className":92356},[217,218,219,220],[39,92358,59223],{"className":92359},[83,220]," 个样本，代价矩阵就是 ",[39,92362,92364,92383],{"className":92363,"translate":43},[42],[39,92365,92367],{"className":92366},[47],[49,92368,92369],{"xmlns":51},[53,92370,92371,92380],{},[56,92372,92373],{},[138,92374,92375,92377],{},[539,92376,48919],{},[539,92378,92379],{},"8",[63,92381,92382],{"encoding":65},"10^8",[39,92384,92386],{"className":92385,"ariaHidden":70},[69],[39,92387,92389,92392,92395],{"className":92388},[74],[39,92390],{"className":92391,"style":19710},[78],[39,92393,546],{"className":92394},[83],[39,92396,92398,92401],{"className":92397},[83],[39,92399,541],{"className":92400},[83],[39,92402,92404],{"className":92403},[192],[39,92405,92407],{"className":92406},[196],[39,92408,92410],{"className":92409},[200],[39,92411,92413],{"className":92412,"style":19710},[204],[39,92414,92415,92418],{"style":208},[39,92416],{"className":92417,"style":213},[212],[39,92419,92421],{"className":92420},[217,218,219,220],[39,92422,92379],{"className":92423},[83,220]," 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 ",[39,92426,92428,92441],{"className":92427,"translate":43},[42],[39,92429,92431],{"className":92430},[47],[49,92432,92433],{"xmlns":51},[53,92434,92435,92439],{},[56,92436,92437],{},[59,92438,12962],{},[63,92440,12962],{"encoding":65},[39,92442,92444],{"className":92443,"ariaHidden":70},[69],[39,92445,92447,92450],{"className":92446},[74],[39,92448],{"className":92449,"style":290},[78],[39,92451,12962],{"className":92452},[83,84]," 开到 5000 就被内存劝退了。",[11,92455,92456,92457,92532,92533,92561,92562,92590],{},"离散 OT 是线性规划，实践上是 ",[39,92458,92460,92485],{"className":92459,"translate":43},[42],[39,92461,92463],{"className":92462},[47],[49,92464,92465],{"xmlns":51},[53,92466,92467,92482],{},[56,92468,92469,92472,92474,92480],{},[59,92470,92471],{},"O",[133,92473,136],{"stretchy":135},[138,92475,92476,92478],{},[59,92477,12962],{},[539,92479,59187],{},[133,92481,161],{"stretchy":135},[63,92483,92484],{"encoding":65},"O(n^3)",[39,92486,92488],{"className":92487,"ariaHidden":70},[69],[39,92489,92491,92494,92497,92500,92529],{"className":92490},[74],[39,92492],{"className":92493,"style":84808},[78],[39,92495,92471],{"className":92496,"style":438},[83,84],[39,92498,136],{"className":92499},[182],[39,92501,92503,92506],{"className":92502},[83],[39,92504,12962],{"className":92505},[83,84],[39,92507,92509],{"className":92508},[192],[39,92510,92512],{"className":92511},[196],[39,92513,92515],{"className":92514},[200],[39,92516,92518],{"className":92517,"style":19710},[204],[39,92519,92520,92523],{"style":208},[39,92521],{"className":92522,"style":213},[212],[39,92524,92526],{"className":92525},[217,218,219,220],[39,92527,59187],{"className":92528},[83,220],[39,92530,161],{"className":92531},[263]," 量级的算法。更麻烦的是解落在多面体的顶点上，",[39,92534,92536,92549],{"className":92535,"translate":43},[42],[39,92537,92539],{"className":92538},[47],[49,92540,92541],{"xmlns":51},[53,92542,92543,92547],{},[56,92544,92545],{},[59,92546,2098],{},[63,92548,2101],{"encoding":65},[39,92550,92552],{"className":92551,"ariaHidden":70},[69],[39,92553,92555,92558],{"className":92554},[74],[39,92556],{"className":92557,"style":290},[78],[39,92559,2098],{"className":92560,"style":2114},[83,84]," 很稀疏，而 ",[39,92563,92565,92578],{"className":92564,"translate":43},[42],[39,92566,92568],{"className":92567},[47],[49,92569,92570],{"xmlns":51},[53,92571,92572,92576],{},[56,92573,92574],{},[59,92575,23703],{},[63,92577,23703],{"encoding":65},[39,92579,92581],{"className":92580,"ariaHidden":70},[69],[39,92582,92584,92587],{"className":92583},[74],[39,92585],{"className":92586,"style":79},[78],[39,92588,23703],{"className":92589,"style":178},[83,84]," 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。",[11,92592,92593],{},"Cuturi 在 2013 年的做法是加熵正则：",[39,92595,92597],{"className":92596,"translate":43},[746],[39,92598,92600,92694],{"className":92599,"translate":43},[42],[39,92601,92603],{"className":92602},[47],[49,92604,92605],{"xmlns":51,"display":755},[53,92606,92607,92691],{},[56,92608,92609,92636,92638,92641,92643,92645,92647,92650,92652,92654,92664,92674,92677,92679,92689],{},[6556,92610,92611,92618],{},[56,92612,92613,92616],{},[59,92614,92615],{},"min",[133,92617,1690],{},[56,92619,92620,92622,92624,92626,92628,92630,92632,92634],{},[59,92621,2098],{},[133,92623,534],{},[59,92625,75346],{"mathvariant":145},[133,92627,136],{"stretchy":135},[59,92629,73771],{},[133,92631,155],{"separator":70},[59,92633,73802],{},[133,92635,161],{"stretchy":135},[35841,92637,74416],{},[133,92639,92640],{"stretchy":135},"⟨",[59,92642,48444],{},[133,92644,155],{"separator":70},[59,92646,2098],{},[133,92648,92649],{"stretchy":135},"⟩",[133,92651,788],{},[59,92653,58009],{},[6556,92655,92656,92658],{},[133,92657,3191],{},[56,92659,92660,92662],{},[59,92661,13377],{},[59,92663,13446],{},[613,92665,92666,92668],{},[59,92667,2098],{},[56,92669,92670,92672],{},[59,92671,13377],{},[59,92673,13446],{},[59,92675,92676],{},"log",[133,92678,1690],{},[613,92680,92681,92683],{},[59,92682,2098],{},[56,92684,92685,92687],{},[59,92686,13377],{},[59,92688,13446],{},[59,92690,2903],{"mathvariant":145},[63,92692,92693],{"encoding":65},"\\min_{\\pi\\in\\Pi(\\mu,\\nu)}\\ \\langle C,\\pi\\rangle+\\varepsilon\\sum_{ij}\\pi_{ij}\\log\\pi_{ij}.",[39,92695,92697,92803],{"className":92696,"ariaHidden":70},[69],[39,92698,92700,92704,92773,92776,92779,92782,92785,92788,92791,92794,92797,92800],{"className":92699},[74],[39,92701],{"className":92702,"style":92703},[78],"height:1.716em;vertical-align:-0.966em;",[39,92705,92707],{"className":92706},[1744,3294],[39,92708,92710,92765],{"className":92709},[196,649],[39,92711,92713,92762],{"className":92712},[200],[39,92714,92717,92752],{"className":92715,"style":92716},[204],"height:0.6679em;",[39,92718,92719,92722],{"style":76132},[39,92720],{"className":92721,"style":935},[212],[39,92723,92725],{"className":92724},[217,218,219,220],[39,92726,92728,92731,92734,92737,92740,92743,92746,92749],{"className":92727},[83,220],[39,92729,2098],{"className":92730,"style":2114},[83,84,220],[39,92732,534],{"className":92733},[235,220],[39,92735,75346],{"className":92736},[83,220],[39,92738,136],{"className":92739},[182,220],[39,92741,73771],{"className":92742},[83,84,220],[39,92744,155],{"className":92745},[252,220],[39,92747,73802],{"className":92748,"style":73818},[83,84,220],[39,92750,161],{"className":92751},[263,220],[39,92753,92754,92757],{"style":21599},[39,92755],{"className":92756,"style":935},[212],[39,92758,92759],{},[39,92760,92615],{"className":92761},[1744],[39,92763,673],{"className":92764},[672],[39,92766,92768],{"className":92767},[200],[39,92769,92771],{"className":92770,"style":76187},[204],[39,92772],{},[39,92774,74416],{"className":92775},[230],[39,92777,92640],{"className":92778},[182],[39,92780,48444],{"className":92781,"style":48487},[83,84],[39,92783,155],{"className":92784},[252],[39,92786],{"className":92787,"style":256},[230],[39,92789,2098],{"className":92790,"style":2114},[83,84],[39,92792,92649],{"className":92793},[263],[39,92795],{"className":92796,"style":461},[230],[39,92798,788],{"className":92799},[465],[39,92801],{"className":92802,"style":461},[230],[39,92804,92806,92809,92812,92815,92862,92865,92908,92911,92917,92920,92963],{"className":92805},[74],[39,92807],{"className":92808,"style":83231},[78],[39,92810,58009],{"className":92811},[83,84],[39,92813],{"className":92814,"style":256},[230],[39,92816,92818],{"className":92817},[1744,3294],[39,92819,92821,92854],{"className":92820},[196,649],[39,92822,92824,92851],{"className":92823},[200],[39,92825,92827,92841],{"className":92826,"style":6670},[204],[39,92828,92829,92832],{"style":13926},[39,92830],{"className":92831,"style":3311},[212],[39,92833,92835],{"className":92834},[217,218,219,220],[39,92836,92838],{"className":92837},[83,220],[39,92839,13732],{"className":92840,"style":13731},[83,84,220],[39,92842,92843,92846],{"style":3330},[39,92844],{"className":92845,"style":3311},[212],[39,92847,92848],{},[39,92849,3191],{"className":92850},[1744,3339,3340],[39,92852,673],{"className":92853},[672],[39,92855,92857],{"className":92856},[200],[39,92858,92860],{"className":92859,"style":13958},[204],[39,92861],{},[39,92863],{"className":92864,"style":256},[230],[39,92866,92868,92871],{"className":92867},[83],[39,92869,2098],{"className":92870,"style":2114},[83,84],[39,92872,92874],{"className":92873},[192],[39,92875,92877,92900],{"className":92876},[196,649],[39,92878,92880,92897],{"className":92879},[200],[39,92881,92883],{"className":92882,"style":13604},[204],[39,92884,92885,92888],{"style":41316},[39,92886],{"className":92887,"style":213},[212],[39,92889,92891],{"className":92890},[217,218,219,220],[39,92892,92894],{"className":92893},[83,220],[39,92895,13732],{"className":92896,"style":13731},[83,84,220],[39,92898,673],{"className":92899},[672],[39,92901,92903],{"className":92902},[200],[39,92904,92906],{"className":92905,"style":35396},[204],[39,92907],{},[39,92909],{"className":92910,"style":256},[230],[39,92912,92914,92915],{"className":92913},[1744],"lo",[39,92916,39209],{"style":35078},[39,92918],{"className":92919,"style":256},[230],[39,92921,92923,92926],{"className":92922},[83],[39,92924,2098],{"className":92925,"style":2114},[83,84],[39,92927,92929],{"className":92928},[192],[39,92930,92932,92955],{"className":92931},[196,649],[39,92933,92935,92952],{"className":92934},[200],[39,92936,92938],{"className":92937,"style":13604},[204],[39,92939,92940,92943],{"style":41316},[39,92941],{"className":92942,"style":213},[212],[39,92944,92946],{"className":92945},[217,218,219,220],[39,92947,92949],{"className":92948},[83,220],[39,92950,13732],{"className":92951,"style":13731},[83,84,220],[39,92953,673],{"className":92954},[672],[39,92956,92958],{"className":92957},[200],[39,92959,92961],{"className":92960,"style":35396},[204],[39,92962],{},[39,92964,2903],{"className":92965},[83],[11,92967,92968],{},"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",[39,92970,92972],{"className":92971,"translate":43},[746],[39,92973,92975,93064],{"className":92974,"translate":43},[42],[39,92976,92978],{"className":92977},[47],[49,92979,92980],{"xmlns":51,"display":755},[53,92981,92982,93061],{},[56,92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K_{ij}=\\exp(-C_{ij}\u002F\\varepsilon),",[39,93065,93067,93125,93315],{"className":93066,"ariaHidden":70},[69],[39,93068,93070,93073,93116,93119,93122],{"className":93069},[74],[39,93071],{"className":93072,"style":41295},[78],[39,93074,93076,93079],{"className":93075},[83],[39,93077,2098],{"className":93078,"style":2114},[83,84],[39,93080,93082],{"className":93081},[192],[39,93083,93085,93108],{"className":93084},[196,649],[39,93086,93088,93105],{"className":93087},[200],[39,93089,93091],{"className":93090,"style":13604},[204],[39,93092,93093,93096],{"style":41316},[39,93094],{"className":93095,"style":213},[212],[39,93097,93099],{"className":93098},[217,218,219,220],[39,93100,93102],{"className":93101},[83,220],[39,93103,13732],{"className":93104,"style":13731},[83,84,220],[39,93106,673],{"className":93107},[672],[39,93109,93111],{"className":93110},[200],[39,93112,93114],{"className":93113,"style":35396},[204],[39,93115],{},[39,93117],{"className":93118,"style":231},[230],[39,93120,793],{"className":93121},[235],[39,93123],{"className":93124,"style":231},[230],[39,93126,93128,93131,93171,93214,93254,93257,93260,93263,93306,93309,93312],{"className":93127},[74],[39,93129],{"className":93130,"style":35350},[78],[39,93132,93134,93137],{"className":93133},[83],[39,93135,35321],{"className":93136},[83,84],[39,93138,93140],{"className":93139},[192],[39,93141,93143,93163],{"className":93142},[196,649],[39,93144,93146,93160],{"className":93145},[200],[39,93147,93149],{"className":93148,"style":13604},[204],[39,93150,93151,93154],{"style":966},[39,93152],{"className":93153,"style":213},[212],[39,93155,93157],{"className":93156},[217,218,219,220],[39,93158,13377],{"className":93159},[83,84,220],[39,93161,673],{"className":93162},[672],[39,93164,93166],{"className":93165},[200],[39,93167,93169],{"className":93168,"style":680},[204],[39,93170],{},[39,93172,93174,93177],{"className":93173},[83],[39,93175,77380],{"className":93176,"style":48487},[83,84],[39,93178,93180],{"className":93179},[192],[39,93181,93183,93206],{"className":93182},[196,649],[39,93184,93186,93203],{"className":93185},[200],[39,93187,93189],{"className":93188,"style":13604},[204],[39,93190,93191,93194],{"style":81320},[39,93192],{"className":93193,"style":213},[212],[39,93195,93197],{"className":93196},[217,218,219,220],[39,93198,93200],{"className":93199},[83,220],[39,93201,13732],{"className":93202,"style":13731},[83,84,220],[39,93204,673],{"className":93205},[672],[39,93207,93209],{"className":93208},[200],[39,93210,93212],{"className":93211,"style":35396},[204],[39,93213],{},[39,93215,93217,93220],{"className":93216},[83],[39,93218,78164],{"className":93219,"style":2114},[83,84],[39,93221,93223],{"className":93222},[192],[39,93224,93226,93246],{"className":93225},[196,649],[39,93227,93229,93243],{"className":93228},[200],[39,93230,93232],{"className":93231,"style":13604},[204],[39,93233,93234,93237],{"style":41316},[39,93235],{"className":93236,"style":213},[212],[39,93238,93240],{"className":93239},[217,218,219,220],[39,93241,13446],{"className":93242,"style":13731},[83,84,220],[39,93244,673],{"className":93245},[672],[39,93247,93249],{"className":93248},[200],[39,93250,93252],{"className":93251,"style":35396},[204],[39,93253],{},[39,93255,155],{"className":93256},[252],[39,93258],{"className":93259,"style":10787},[230],[39,93261],{"className":93262,"style":256},[230],[39,93264,93266,93269],{"className":93265},[83],[39,93267,77380],{"className":93268,"style":48487},[83,84],[39,93270,93272],{"className":93271},[192],[39,93273,93275,93298],{"className":93274},[196,649],[39,93276,93278,93295],{"className":93277},[200],[39,93279,93281],{"className":93280,"style":13604},[204],[39,93282,93283,93286],{"style":81320},[39,93284],{"className":93285,"style":213},[212],[39,93287,93289],{"className":93288},[217,218,219,220],[39,93290,93292],{"className":93291},[83,220],[39,93293,13732],{"className":93294,"style":13731},[83,84,220],[39,93296,673],{"className":93297},[672],[39,93299,93301],{"className":93300},[200],[39,93302,93304],{"className":93303,"style":35396},[204],[39,93305],{},[39,93307],{"className":93308,"style":231},[230],[39,93310,793],{"className":93311},[235],[39,93313],{"className":93314,"style":231},[230],[39,93316,93318,93321,93324,93327,93330,93373,93376,93379,93382],{"className":93317},[74],[39,93319],{"className":93320,"style":41560},[78],[39,93322,93036],{"className":93323},[1744],[39,93325,136],{"className":93326},[182],[39,93328,5378],{"className":93329},[83],[39,93331,93333,93336],{"className":93332},[83],[39,93334,48444],{"className":93335,"style":48487},[83,84],[39,93337,93339],{"className":93338},[192],[39,93340,93342,93365],{"className":93341},[196,649],[39,93343,93345,93362],{"className":93344},[200],[39,93346,93348],{"className":93347,"style":13604},[204],[39,93349,93350,93353],{"style":81320},[39,93351],{"className":93352,"style":213},[212],[39,93354,93356],{"className":93355},[217,218,219,220],[39,93357,93359],{"className":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由边缘约束定出，交替更新 ",[39,93430,93432,93461],{"className":93431,"translate":43},[42],[39,93433,93435],{"className":93434},[47],[49,93436,93437],{"xmlns":51},[53,93438,93439,93458],{},[56,93440,93441,93443,93446,93448,93450,93452,93454,93456],{},[59,93442,35321],{},[133,93444,93445],{},"←",[59,93447,158],{},[59,93449,45691],{"mathvariant":145},[133,93451,136],{"stretchy":135},[59,93453,77380],{},[59,93455,78164],{},[133,93457,161],{"stretchy":135},[63,93459,93460],{"encoding":65},"u\\leftarrow a\u002F(Kv)",[39,93462,93464,93482],{"className":93463,"ariaHidden":70},[69],[39,93465,93467,93470,93473,93476,93479],{"className":93466},[74],[39,93468],{"className":93469,"style":290},[78],[39,93471,35321],{"className":93472},[83,84],[39,93474],{"className":93475,"style":231},[230],[39,93477,93445],{"className":93478},[235],[39,93480],{"className":93481,"style":231},[230],[39,93483,93485,93488,93491,93494,93497,93500,93503],{"className":93484},[74],[39,93486],{"className":93487,"style":245},[78],[39,93489,158],{"className":93490},[83,84],[39,93492,45691],{"className":93493},[83],[39,93495,136],{"className":93496},[182],[39,93498,77380],{"className":93499,"style":48487},[83,84],[39,93501,78164],{"className":93502,"style":2114},[83,84],[39,93504,161],{"className":93505},[263],[39,93507,93509,93542],{"className":93508,"translate":43},[42],[39,93510,93512],{"className":93511},[47],[49,93513,93514],{"xmlns":51},[53,93515,93516,93539],{},[56,93517,93518,93520,93522,93524,93526,93528,93535,93537],{},[59,93519,78164],{},[133,93521,93445],{},[59,93523,78628],{},[59,93525,45691],{"mathvariant":145},[133,93527,136],{"stretchy":135},[138,93529,93530,93532],{},[59,93531,77380],{},[59,93533,93534],{"mathvariant":145},"⊤",[59,93536,35321],{},[133,93538,161],{"stretchy":135},[63,93540,93541],{"encoding":65},"v\\leftarrow b\u002F(K^\\top u)",[39,93543,93545,93563],{"className":93544,"ariaHidden":70},[69],[39,93546,93548,93551,93554,93557,93560],{"className":93547},[74],[39,93549],{"className":93550,"style":290},[78],[39,93552,78164],{"className":93553,"style":2114},[83,84],[39,93555],{"className":93556,"style":231},[230],[39,93558,93445],{"className":93559},[235],[39,93561],{"className":93562,"style":231},[230],[39,93564,93566,93569,93572,93575,93578,93607,93610],{"className":93565},[74],[39,93567],{"className":93568,"style":15555},[78],[39,93570,78628],{"className":93571},[83,84],[39,93573,45691],{"className":93574},[83],[39,93576,136],{"className":93577},[182],[39,93579,93581,93584],{"className":93580},[83],[39,93582,77380],{"className":93583,"style":48487},[83,84],[39,93585,93587],{"className":93586},[192],[39,93588,93590],{"className":93589},[196],[39,93591,93593],{"className":93592},[200],[39,93594,93596],{"className":93595,"style":15577},[204],[39,93597,93598,93601],{"style":208},[39,93599],{"className":93600,"style":213},[212],[39,93602,93604],{"className":93603},[217,218,219,220],[39,93605,93534],{"className":93606},[83,220],[39,93608,35321],{"className":93609},[83,84],[39,93611,161],{"className":93612},[263],"，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。",[11,93615,93616,93644,93645,93673,93674,93702,93703,93731,93732,93783,93784,93812],{},[39,93617,93619,93632],{"className":93618,"translate":43},[42],[39,93620,93622],{"className":93621},[47],[49,93623,93624],{"xmlns":51},[53,93625,93626,93630],{},[56,93627,93628],{},[59,93629,58009],{},[63,93631,77615],{"encoding":65},[39,93633,93635],{"className":93634,"ariaHidden":70},[69],[39,93636,93638,93641],{"className":93637},[74],[39,93639],{"className":93640,"style":290},[78],[39,93642,58009],{"className":93643},[83,84]," 可以读作一个旋钮。取三个点搬到三个点：",[39,93646,93648,93661],{"className":93647,"translate":43},[42],[39,93649,93651],{"className":93650},[47],[49,93652,93653],{"xmlns":51},[53,93654,93655,93659],{},[56,93656,93657],{},[59,93658,58009],{},[63,93660,77615],{"encoding":65},[39,93662,93664],{"className":93663,"ariaHidden":70},[69],[39,93665,93667,93670],{"className":93666},[74],[39,93668],{"className":93669,"style":290},[78],[39,93671,58009],{"className":93672},[83,84]," 很大时，",[39,93675,93677,93690],{"className":93676,"translate":43},[42],[39,93678,93680],{"className":93679},[47],[49,93681,93682],{"xmlns":51},[53,93683,93684,93688],{},[56,93685,93686],{},[59,93687,77380],{},[63,93689,77380],{"encoding":65},[39,93691,93693],{"className":93692,"ariaHidden":70},[69],[39,93694,93696,93699],{"className":93695},[74],[39,93697],{"className":93698,"style":79},[78],[39,93700,77380],{"className":93701,"style":48487},[83,84]," 接近全 ",[39,93704,93706,93719],{"className":93705,"translate":43},[42],[39,93707,93709],{"className":93708},[47],[49,93710,93711],{"xmlns":51},[53,93712,93713,93717],{},[56,93714,93715],{},[539,93716,546],{},[63,93718,546],{"encoding":65},[39,93720,93722],{"className":93721,"ariaHidden":70},[69],[39,93723,93725,93728],{"className":93724},[74],[39,93726],{"className":93727,"style":2473},[78],[39,93729,546],{"className":93730},[83]," 矩阵，每个点都往每处送一点，解是稠密的、模糊的；",[39,93733,93735,93753],{"className":93734,"translate":43},[42],[39,93736,93738],{"className":93737},[47],[49,93739,93740],{"xmlns":51},[53,93741,93742,93750],{},[56,93743,93744,93746,93748],{},[59,93745,58009],{},[133,93747,418],{},[539,93749,541],{},[63,93751,93752],{"encoding":65},"\\varepsilon\\to0",[39,93754,93756,93774],{"className":93755,"ariaHidden":70},[69],[39,93757,93759,93762,93765,93768,93771],{"className":93758},[74],[39,93760],{"className":93761,"style":290},[78],[39,93763,58009],{"className":93764},[83,84],[39,93766],{"className":93767,"style":231},[230],[39,93769,418],{"className":93770},[235],[39,93772],{"className":93773,"style":231},[230],[39,93775,93777,93780],{"className":93776},[74],[39,93778],{"className":93779,"style":2473},[78],[39,93781,541],{"className":93782},[83]," 时 ",[39,93785,93787,93800],{"className":93786,"translate":43},[42],[39,93788,93790],{"className":93789},[47],[49,93791,93792],{"xmlns":51},[53,93793,93794,93798],{},[56,93795,93796],{},[59,93797,77380],{},[63,93799,77380],{"encoding":65},[39,93801,93803],{"className":93802,"ariaHidden":70},[69],[39,93804,93806,93809],{"className":93805},[74],[39,93807],{"className":93808,"style":79},[78],[39,93810,77380],{"className":93811,"style":48487},[83,84]," 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。",[11,93814,74276,93815,93866,93867,93945,93946,94290],{},[39,93816,93818,93836],{"className":93817,"translate":43},[42],[39,93819,93821],{"className":93820},[47],[49,93822,93823],{"xmlns":51},[53,93824,93825,93833],{},[56,93826,93827,93829,93831],{},[59,93828,58009],{},[133,93830,31514],{},[539,93832,541],{},[63,93834,93835],{"encoding":65},"\\varepsilon>0",[39,93837,93839,93857],{"className":93838,"ariaHidden":70},[69],[39,93840,93842,93845,93848,93851,93854],{"className":93841},[74],[39,93843],{"className":93844,"style":24094},[78],[39,93846,58009],{"className":93847},[83,84],[39,93849],{"className":93850,"style":231},[230],[39,93852,31514],{"className":93853},[235],[39,93855],{"className":93856,"style":231},[230],[39,93858,93860,93863],{"className":93859},[74],[39,93861],{"className":93862,"style":2473},[78],[39,93864,541],{"className":93865},[83]," 时它不再是度量。最直接的一点：把 ",[39,93868,93870,93902],{"className":93869,"translate":43},[42],[39,93871,93873],{"className":93872},[47],[49,93874,93875],{"xmlns":51},[53,93876,93877,93899],{},[56,93878,93879,93881,93883,93893,93895,93897],{},[59,93880,2098],{},[133,93882,793],{},[56,93884,93885,93887,93889,93891],{},[59,93886,29425],{"mathvariant":145},[59,93888,13377],{"mathvariant":145},[59,93890,158],{"mathvariant":145},[59,93892,39209],{"mathvariant":145},[133,93894,136],{"stretchy":135},[59,93896,158],{},[133,93898,161],{"stretchy":135},[63,93900,93901],{"encoding":65},"\\pi=\\mathrm{diag}(a)",[39,93903,93905,93923],{"className":93904,"ariaHidden":70},[69],[39,93906,93908,93911,93914,93917,93920],{"className":93907},[74],[39,93909],{"className":93910,"style":290},[78],[39,93912,2098],{"className":93913,"style":2114},[83,84],[39,93915],{"className":93916,"style":231},[230],[39,93918,793],{"className":93919},[235],[39,93921],{"className":93922,"style":231},[230],[39,93924,93926,93929,93936,93939,93942],{"className":93925},[74],[39,93927],{"className":93928,"style":245},[78],[39,93930,93932],{"className":93931},[83],[39,93933,93935],{"className":93934,"style":35078},[83,35077],"diag",[39,93937,136],{"className":93938},[182],[39,93940,158],{"className":93941},[83,84],[39,93943,161],{"className":93944},[263]," 代入，",[39,93947,93949,94008],{"className":93948,"translate":43},[42],[39,93950,93952],{"className":93951},[47],[49,93953,93954],{"xmlns":51},[53,93955,93956,94005],{},[56,93957,93958,93964,93966,93968,93970,93972,93974,93976,93978,93984,93990,93992,93994,94000,94003],{},[613,93959,93960,93962],{},[59,93961,23703],{},[59,93963,58009],{},[133,93965,136],{"stretchy":135},[59,93967,73771],{},[133,93969,155],{"separator":70},[59,93971,73771],{},[133,93973,161],{"stretchy":135},[133,93975,793],{},[59,93977,58009],{},[613,93979,93980,93982],{},[133,93981,3191],{},[59,93983,13377],{},[613,93985,93986,93988],{},[59,93987,158],{},[59,93989,13377],{},[59,93991,92676],{},[133,93993,1690],{},[613,93995,93996,93998],{},[59,93997,158],{},[59,93999,13377],{},[133,94001,94002],{"mathvariant":145},"≠",[539,94004,541],{},[63,94006,94007],{"encoding":65},"W_\\varepsilon(\\mu,\\mu)=\\varepsilon\\sum_i a_i\\log a_i\\ne0",[39,94009,94011,94084,94281],{"className":94010,"ariaHidden":70},[69],[39,94012,94014,94017,94057,94060,94063,94066,94069,94072,94075,94078,94081],{"className":94013},[74],[39,94015],{"className":94016,"style":245},[78],[39,94018,94020,94023],{"className":94019},[83],[39,94021,23703],{"className":94022,"style":178},[83,84],[39,94024,94026],{"className":94025},[192],[39,94027,94029,94049],{"className":94028},[196,649],[39,94030,94032,94046],{"className":94031},[200],[39,94033,94035],{"className":94034,"style":1728},[204],[39,94036,94037,94040],{"style":73958},[39,94038],{"className":94039,"style":213},[212],[39,94041,94043],{"className":94042},[217,218,219,220],[39,94044,58009],{"className":94045},[83,84,220],[39,94047,673],{"className":94048},[672],[39,94050,94052],{"className":94051},[200],[39,94053,94055],{"className":94054,"style":680},[204],[39,94056],{},[39,94058,136],{"className":94059},[182],[39,94061,73771],{"className":94062},[83,84],[39,94064,155],{"className":94065},[252],[39,94067],{"className":94068,"style":256},[230],[39,94070,73771],{"className":94071},[83,84],[39,94073,161],{"className":94074},[263],[39,94076],{"className":94077,"style":231},[230],[39,94079,793],{"className":94080},[235],[39,94082],{"className":94083,"style":231},[230],[39,94085,94087,94091,94094,94097,94138,94141,94181,94184,94189,94192,94232,94235,94278],{"className":94086},[74],[39,94088],{"className":94089,"style":94090},[78],"height:1.0497em;vertical-align:-0.2997em;",[39,94092,58009],{"className":94093},[83,84],[39,94095],{"className":94096,"style":256},[230],[39,94098,94100,94103],{"className":94099},[1744],[39,94101,3191],{"className":94102,"style":3953},[1744,3339,3952],[39,94104,94106],{"className":94105},[192],[39,94107,94109,94130],{"className":94108},[196,649],[39,94110,94112,94127],{"className":94111},[200],[39,94113,94116],{"className":94114,"style":94115},[204],"height:0.162em;",[39,94117,94118,94121],{"style":3969},[39,94119],{"className":94120,"style":213},[212],[39,94122,94124],{"className":94123},[217,218,219,220],[39,94125,13377],{"className":94126},[83,84,220],[39,94128,673],{"className":94129},[672],[39,94131,94133],{"className":94132},[200],[39,94134,94136],{"className":94135,"style":4025},[204],[39,94137],{},[39,94139],{"className":94140,"style":256},[230],[39,94142,94144,94147],{"className":94143},[83],[39,94145,158],{"className":94146},[83,84],[39,94148,94150],{"className":94149},[192],[39,94151,94153,94173],{"className":94152},[196,649],[39,94154,94156,94170],{"className":94155},[200],[39,94157,94159],{"className":94158,"style":13604},[204],[39,94160,94161,94164],{"style":966},[39,94162],{"className":94163,"style":213},[212],[39,94165,94167],{"className":94166},[217,218,219,220],[39,94168,13377],{"className":94169},[83,84,220],[39,94171,673],{"className":94172},[672],[39,94174,94176],{"className":94175},[200],[39,94177,94179],{"className":94178,"style":680},[204],[39,94180],{},[39,94182],{"className":94183,"style":256},[230],[39,94185,92914,94187],{"className":94186},[1744],[39,94188,39209],{"style":35078},[39,94190],{"className":94191,"style":256},[230],[39,94193,94195,94198],{"className":94194},[83],[39,94196,158],{"className":94197},[83,84],[39,94199,94201],{"className":94200},[192],[39,94202,94204,94224],{"className":94203},[196,649],[39,94205,94207,94221],{"className":94206},[200],[39,94208,94210],{"className":94209,"style":13604},[204],[39,94211,94212,94215],{"style":966},[39,94213],{"className":94214,"style":213},[212],[39,94216,94218],{"className":94217},[217,218,219,220],[39,94219,13377],{"className":94220},[83,84,220],[39,94222,673],{"className":94223},[672],[39,94225,94227],{"className":94226},[200],[39,94228,94230],{"className":94229,"style":680},[204],[39,94231],{},[39,94233],{"className":94234,"style":231},[230],[39,94236,94238,94271,94275],{"className":94237},[235],[39,94239,94241],{"className":94240},[235],[39,94242,94245],{"className":94243},[83,94244],"vbox",[39,94246,94249],{"className":94247},[94248],"thinbox",[39,94250,94253,94256,94267],{"className":94251},[94252],"rlap",[39,94254],{"className":94255,"style":51280},[78],[39,94257,94260],{"className":94258},[94259],"inner",[39,94261,94263],{"className":94262},[83],[39,94264,94266],{"className":94265},[235],"",[39,94268],{"className":94269},[94270],"fix",[39,94272],{"className":94273},[230,94274],"nobreak",[39,94276,793],{"className":94277},[235],[39,94279],{"className":94280,"style":231},[230],[39,94282,94284,94287],{"className":94283},[74],[39,94285],{"className":94286,"style":2473},[78],[39,94288,541],{"className":94289},[83],"。所以工程上常用的 Sinkhorn 散度要先去偏：",[39,94292,94294],{"className":94293,"translate":43},[746],[39,94295,94297,94409],{"className":94296,"translate":43},[42],[39,94298,94300],{"className":94299},[47],[49,94301,94302],{"xmlns":51,"display":755},[53,94303,94304,94406],{},[56,94305,94306,94312,94314,94316,94318,94320,94322,94324,94334,94336,94338,94340,94342,94344,94346,94354,94364,94366,94368,94370,94372,94374,94376,94384,94394,94396,94398,94400,94402,94404],{},[613,94307,94308,94310],{},[59,94309,61],{},[59,94311,58009],{},[133,94313,136],{"stretchy":135},[59,94315,73771],{},[133,94317,155],{"separator":70},[59,94319,73802],{},[133,94321,161],{"stretchy":135},[133,94323,793],{},[613,94325,94326,94332],{},[56,94327,94328,94330],{},[59,94329,92471],{"mathvariant":145},[59,94331,11772],{"mathvariant":145},[59,94333,58009],{},[133,94335,136],{"stretchy":135},[59,94337,73771],{},[133,94339,155],{"separator":70},[59,94341,73802],{},[133,94343,161],{"stretchy":135},[133,94345,5378],{},[770,94347,94348],{"scriptlevel":541,"displaystyle":135},[5362,94349,94350,94352],{},[539,94351,546],{},[539,94353,17359],{},[613,94355,94356,94362],{},[56,94357,94358,94360],{},[59,94359,92471],{"mathvariant":145},[59,94361,11772],{"mathvariant":145},[59,94363,58009],{},[133,94365,136],{"stretchy":135},[59,94367,73771],{},[133,94369,155],{"separator":70},[59,94371,73771],{},[133,94373,161],{"stretchy":135},[133,94375,5378],{},[770,94377,94378],{"scriptlevel":541,"displaystyle":135},[5362,94379,94380,94382],{},[539,94381,546],{},[539,94383,17359],{},[613,94385,94386,94392],{},[56,94387,94388,94390],{},[59,94389,92471],{"mathvariant":145},[59,94391,11772],{"mathvariant":145},[59,94393,58009],{},[133,94395,136],{"stretchy":135},[59,94397,73802],{},[133,94399,155],{"separator":70},[59,94401,73802],{},[133,94403,161],{"stretchy":135},[59,94405,2903],{"mathvariant":145},[63,94407,94408],{"encoding":65},"S_\\varepsilon(\\mu,\\nu)=\\mathrm{OT}_\\varepsilon(\\mu,\\nu)-\\tfrac12\\mathrm{OT}_\\varepsilon(\\mu,\\mu)-\\tfrac12\\mathrm{OT}_\\varepsilon(\\nu,\\nu).",[39,94410,94412,94486,94563,94707],{"className":94411,"ariaHidden":70},[69],[39,94413,94415,94418,94459,94462,94465,94468,94471,94474,94477,94480,94483],{"className":94414},[74],[39,94416],{"className":94417,"style":245},[78],[39,94419,94421,94424],{"className":94420},[83],[39,94422,61],{"className":94423,"style":85},[83,84],[39,94425,94427],{"className":94426},[192],[39,94428,94430,94451],{"className":94429},[196,649],[39,94431,94433,94448],{"className":94432},[200],[39,94434,94436],{"className":94435,"style":1728},[204],[39,94437,94439,94442],{"style":94438},"top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;",[39,94440],{"className":94441,"style":213},[212],[39,94443,94445],{"className":94444},[217,218,219,220],[39,94446,58009],{"className":94447},[83,84,220],[39,94449,673],{"className":94450},[672],[39,94452,94454],{"className":94453},[200],[39,94455,94457],{"className":94456,"style":680},[204],[39,94458],{},[39,94460,136],{"className":94461},[182],[39,94463,73771],{"className":94464},[83,84],[39,94466,155],{"className":94467},[252],[39,94469],{"className":94470,"style":256},[230],[39,94472,73802],{"className":94473,"style":73818},[83,84],[39,94475,161],{"className":94476},[263],[39,94478],{"className":94479,"style":231},[230],[39,94481,793],{"className":94482},[235],[39,94484],{"className":94485,"style":231},[230],[39,94487,94489,94492,94536,94539,94542,94545,94548,94551,94554,94557,94560],{"className":94488},[74],[39,94490],{"className":94491,"style":245},[78],[39,94493,94495,94502],{"className":94494},[83],[39,94496,94498],{"className":94497},[83],[39,94499,94501],{"className":94500},[83,35077],"OT",[39,94503,94505],{"className":94504},[192],[39,94506,94508,94528],{"className":94507},[196,649],[39,94509,94511,94525],{"className":94510},[200],[39,94512,94514],{"className":94513,"style":1728},[204],[39,94515,94516,94519],{"style":44021},[39,94517],{"className":94518,"style":213},[212],[39,94520,94522],{"className":94521},[217,218,219,220],[39,94523,58009],{"className":94524},[83,84,220],[39,94526,673],{"className":94527},[672],[39,94529,94531],{"className":94530},[200],[39,94532,94534],{"className":94533,"style":680},[204],[39,94535],{},[39,94537,136],{"className":94538},[182],[39,94540,73771],{"className":94541},[83,84],[39,94543,155],{"className":94544},[252],[39,94546],{"className":94547,"style":256},[230],[39,94549,73802],{"className":94550,"style":73818},[83,84],[39,94552,161],{"className":94553},[263],[39,94555],{"className":94556,"style":461},[230],[39,94558,5378],{"className":94559},[465],[39,94561],{"className":94562,"style":461},[230],[39,94564,94566,94569,94637,94680,94683,94686,94689,94692,94695,94698,94701,94704],{"className":94565},[74],[39,94567],{"className":94568,"style":74987},[78],[39,94570,94572,94575,94634],{"className":94571},[83],[39,94573],{"className":94574},[182,5463],[39,94576,94578],{"className":94577},[5362],[39,94579,94581,94626],{"className":94580},[196,649],[39,94582,94584,94623],{"className":94583},[200],[39,94585,94587,94601,94609],{"className":94586,"style":75006},[204],[39,94588,94589,94592],{"style":75009},[39,94590],{"className":94591,"style":935},[212],[39,94593,94595],{"className":94594},[217,218,219,220],[39,94596,94598],{"className":94597},[83,220],[39,94599,17359],{"className":94600},[83,220],[39,94602,94603,94606],{"style":5503},[39,94604],{"className":94605,"style":935},[212],[39,94607],{"className":94608,"style":5511},[5510],[39,94610,94611,94614],{"style":75032},[39,94612],{"className":94613,"style":935},[212],[39,94615,94617],{"className":94616},[217,218,219,220],[39,94618,94620],{"className":94619},[83,220],[39,94621,546],{"className":94622},[83,220],[39,94624,673],{"className":94625},[672],[39,94627,94629],{"className":94628},[200],[39,94630,94632],{"className":94631,"style":75054},[204],[39,94633],{},[39,94635],{"className":94636},[263,5463],[39,94638,94640,94646],{"className":94639},[83],[39,94641,94643],{"className":94642},[83],[39,94644,94501],{"className":94645},[83,35077],[39,94647,94649],{"className":94648},[192],[39,94650,94652,94672],{"className":94651},[196,649],[39,94653,94655,94669],{"className":94654},[200],[39,94656,94658],{"className":94657,"style":1728},[204],[39,94659,94660,94663],{"style":44021},[39,94661],{"className":94662,"style":213},[212],[39,94664,94666],{"className":94665},[217,218,219,220],[39,94667,58009],{"className":94668},[83,84,220],[39,94670,673],{"className":94671},[672],[39,94673,94675],{"className":94674},[200],[39,94676,94678],{"className":94677,"style":680},[204],[39,94679],{},[39,94681,136],{"className":94682},[182],[39,94684,73771],{"className":94685},[83,84],[39,94687,155],{"className":94688},[252],[39,94690],{"className":94691,"style":256},[230],[39,94693,73771],{"className":94694},[83,84],[39,94696,161],{"className":94697},[263],[39,94699],{"className":94700,"style":461},[230],[39,94702,5378],{"className":94703},[465],[39,94705],{"className":94706,"style":461},[230],[39,94708,94710,94713,94781,94824,94827,94830,94833,94836,94839,94842],{"className":94709},[74],[39,94711],{"className":94712,"style":74987},[78],[39,94714,94716,94719,94778],{"className":94715},[83],[39,94717],{"className":94718},[182,5463],[39,94720,94722],{"className":94721},[5362],[39,94723,94725,94770],{"className":94724},[196,649],[39,94726,94728,94767],{"className":94727},[200],[39,94729,94731,94745,94753],{"className":94730,"style":75006},[204],[39,94732,94733,94736],{"style":75009},[39,94734],{"className":94735,"style":935},[212],[39,94737,94739],{"className":94738},[217,218,219,220],[39,94740,94742],{"className":94741},[83,220],[39,94743,17359],{"className":94744},[83,220],[39,94746,94747,94750],{"style":5503},[39,94748],{"className":94749,"style":935},[212],[39,94751],{"className":94752,"style":5511},[5510],[39,94754,94755,94758],{"style":75032},[39,94756],{"className":94757,"style":935},[212],[39,94759,94761],{"className":94760},[217,218,219,220],[39,94762,94764],{"className":94763},[83,220],[39,94765,546],{"className":94766},[83,220],[39,94768,673],{"className":94769},[672],[39,94771,94773],{"className":94772},[200],[39,94774,94776],{"className":94775,"style":75054},[204],[39,94777],{},[39,94779],{"className":94780},[263,5463],[39,94782,94784,94790],{"className":94783},[83],[39,94785,94787],{"className":94786},[83],[39,94788,94501],{"className":94789},[83,35077],[39,94791,94793],{"className":94792},[192],[39,94794,94796,94816],{"className":94795},[196,649],[39,94797,94799,94813],{"className":94798},[200],[39,94800,94802],{"className":94801,"style":1728},[204],[39,94803,94804,94807],{"style":44021},[39,94805],{"className":94806,"style":213},[212],[39,94808,94810],{"className":94809},[217,218,219,220],[39,94811,58009],{"className":94812},[83,84,220],[39,94814,673],{"className":94815},[672],[39,94817,94819],{"className":94818},[200],[39,94820,94822],{"className":94821,"style":680},[204],[39,94823],{},[39,94825,136],{"className":94826},[182],[39,94828,73802],{"className":94829,"style":73818},[83,84],[39,94831,155],{"className":94832},[252],[39,94834],{"className":94835,"style":256},[230],[39,94837,73802],{"className":94838,"style":73818},[83,84],[39,94840,161],{"className":94841},[263],[39,94843,2903],{"className":94844},[83],[11,94846,94847,94848,94899],{},"去偏之后才非负、才在 ",[39,94849,94851,94869],{"className":94850,"translate":43},[42],[39,94852,94854],{"className":94853},[47],[49,94855,94856],{"xmlns":51},[53,94857,94858,94866],{},[56,94859,94860,94862,94864],{},[59,94861,73771],{},[133,94863,793],{},[59,94865,73802],{},[63,94867,94868],{"encoding":65},"\\mu=\\nu",[39,94870,94872,94890],{"className":94871,"ariaHidden":70},[69],[39,94873,94875,94878,94881,94884,94887],{"className":94874},[74],[39,94876],{"className":94877,"style":2654},[78],[39,94879,73771],{"className":94880},[83,84],[39,94882],{"className":94883,"style":231},[230],[39,94885,793],{"className":94886},[235],[39,94888],{"className":94889,"style":231},[230],[39,94891,94893,94896],{"className":94892},[74],[39,94894],{"className":94895,"style":290},[78],[39,94897,73802],{"className":94898,"style":73818},[83,84]," 处为零。",[11,94901,94902,94903,94973,94974,95002,95003,95031],{},"还有一个更根本的代价，来自样本。经验分布 ",[39,94904,94906,94924],{"className":94905,"translate":43},[42],[39,94907,94909],{"className":94908},[47],[49,94910,94911],{"xmlns":51},[53,94912,94913,94921],{},[56,94914,94915],{},[613,94916,94917,94919],{},[59,94918,73771],{},[59,94920,12962],{},[63,94922,94923],{"encoding":65},"\\mu_n",[39,94925,94927],{"className":94926,"ariaHidden":70},[69],[39,94928,94930,94933],{"className":94929},[74],[39,94931],{"className":94932,"style":2654},[78],[39,94934,94936,94939],{"className":94935},[83],[39,94937,73771],{"className":94938},[83,84],[39,94940,94942],{"className":94941},[192],[39,94943,94945,94965],{"className":94944},[196,649],[39,94946,94948,94962],{"className":94947},[200],[39,94949,94951],{"className":94950,"style":1728},[204],[39,94952,94953,94956],{"style":966},[39,94954],{"className":94955,"style":213},[212],[39,94957,94959],{"className":94958},[217,218,219,220],[39,94960,12962],{"className":94961},[83,84,220],[39,94963,673],{"className":94964},[672],[39,94966,94968],{"className":94967},[200],[39,94969,94971],{"className":94970,"style":680},[204],[39,94972],{}," 到真分布 ",[39,94975,94977,94990],{"className":94976,"translate":43},[42],[39,94978,94980],{"className":94979},[47],[49,94981,94982],{"xmlns":51},[53,94983,94984,94988],{},[56,94985,94986],{},[59,94987,73771],{},[63,94989,73774],{"encoding":65},[39,94991,94993],{"className":94992,"ariaHidden":70},[69],[39,94994,94996,94999],{"className":94995},[74],[39,94997],{"className":94998,"style":2654},[78],[39,95000,73771],{"className":95001},[83,84]," 的收敛速度，在 ",[39,95004,95006,95019],{"className":95005,"translate":43},[42],[39,95007,95009],{"className":95008},[47],[49,95010,95011],{"xmlns":51},[53,95012,95013,95017],{},[56,95014,95015],{},[59,95016,29425],{},[63,95018,29425],{"encoding":65},[39,95020,95022],{"className":95021,"ariaHidden":70},[69],[39,95023,95025,95028],{"className":95024},[74],[39,95026],{"className":95027,"style":28824},[78],[39,95029,29425],{"className":95030},[83,84]," 维空间里大致是",[39,95033,95035],{"className":95034,"translate":43},[746],[39,95036,95038,95111],{"className":95037,"translate":43},[42],[39,95039,95041],{"className":95040},[47],[49,95042,95043],{"xmlns":51,"display":755},[53,95044,95045,95108],{},[56,95046,95047,95049,95051,95057,95059,95065,95067,95069,95071,95073,95076,95078,95092,95094,95096,95098,95100,95102,95104,95106],{},[59,95048,4253],{"mathvariant":421},[35841,95050,35843],{},[613,95052,95053,95055],{},[59,95054,23703],{},[59,95056,11],{},[133,95058,136],{"stretchy":135},[613,95060,95061,95063],{},[59,95062,73771],{},[59,95064,12962],{},[133,95066,155],{"separator":70},[59,95068,73771],{},[133,95070,161],{"stretchy":135},[35841,95072,74416],{},[133,95074,95075],{},"≲",[35841,95077,74416],{},[138,95079,95080,95082],{},[59,95081,12962],{},[56,95083,95084,95086,95088,95090],{},[133,95085,5378],{},[539,95087,546],{},[59,95089,45691],{"mathvariant":145},[59,95091,29425],{},[230,95093],{"width":10251},[133,95095,136],{"stretchy":135},[59,95097,29425],{},[133,95099,31514],{},[539,95101,17359],{},[59,95103,11],{},[133,95105,161],{"stretchy":135},[59,95107,2903],{"mathvariant":145},[63,95109,95110],{"encoding":65},"\\mathbb E\\,W_p(\\mu_n,\\mu)\\ \\lesssim\\ n^{-1\u002Fd}\\qquad(d>2p).",[39,95112,95114,95237,95301],{"className":95113,"ariaHidden":70},[69],[39,95115,95117,95120,95123,95126,95166,95169,95209,95212,95215,95218,95221,95224,95227,95231,95234],{"className":95116},[74],[39,95118],{"className":95119,"style":41560},[78],[39,95121,4253],{"className":95122},[83,515],[39,95124],{"className":95125,"style":256},[230],[39,95127,95129,95132],{"className":95128},[83],[39,95130,23703],{"className":95131,"style":178},[83,84],[39,95133,95135],{"className":95134},[192],[39,95136,95138,95158],{"className":95137},[196,649],[39,95139,95141,95155],{"className":95140},[200],[39,95142,95144],{"className":95143,"style":1728},[204],[39,95145,95146,95149],{"style":73958},[39,95147],{"className":95148,"style":213},[212],[39,95150,95152],{"className":95151},[217,218,219,220],[39,95153,11],{"className":95154},[83,84,220],[39,95156,673],{"className":95157},[672],[39,95159,95161],{"className":95160},[200],[39,95162,95164],{"className":95163,"style":35396},[204],[39,95165],{},[39,95167,136],{"className":95168},[182],[39,95170,95172,95175],{"className":95171},[83],[39,95173,73771],{"className":95174},[83,84],[39,95176,95178],{"className":95177},[192],[39,95179,95181,95201],{"className":95180},[196,649],[39,95182,95184,95198],{"className":95183},[200],[39,95185,95187],{"className":95186,"style":1728},[204],[39,95188,95189,95192],{"style":966},[39,95190],{"className":95191,"style":213},[212],[39,95193,95195],{"className":95194},[217,218,219,220],[39,95196,12962],{"className":95197},[83,84,220],[39,95199,673],{"className":95200},[672],[39,95202,95204],{"className":95203},[200],[39,95205,95207],{"className":95206,"style":680},[204],[39,95208],{},[39,95210,155],{"className":95211},[252],[39,95213],{"className":95214,"style":256},[230],[39,95216,73771],{"className":95217},[83,84],[39,95219,161],{"className":95220},[263],[39,95222,74416],{"className":95223},[230],[39,95225],{"className":95226,"style":231},[230],[39,95228,95075],{"className":95229},[235,95230],"amsrm",[39,95232,74416],{"className":95233},[230],[39,95235],{"className":95236,"style":231},[230],[39,95238,95240,95244,95283,95286,95289,95292,95295,95298],{"className":95239},[74],[39,95241],{"className":95242,"style":95243},[78],"height:1.188em;vertical-align:-0.25em;",[39,95245,95247,95250],{"className":95246},[83],[39,95248,12962],{"className":95249},[83,84],[39,95251,95253],{"className":95252},[192],[39,95254,95256],{"className":95255},[196],[39,95257,95259],{"className":95258},[200],[39,95260,95263],{"className":95261,"style":95262},[204],"height:0.938em;",[39,95264,95265,95268],{"style":1027},[39,95266],{"className":95267,"style":213},[212],[39,95269,95271],{"className":95270},[217,218,219,220],[39,95272,95274,95277,95280],{"className":95273},[83,220],[39,95275,5378],{"className":95276},[83,220],[39,95278,76865],{"className":95279},[83,220],[39,95281,29425],{"className":95282},[83,84,220],[39,95284],{"className":95285,"style":10787},[230],[39,95287,136],{"className":95288},[182],[39,95290,29425],{"className":95291},[83,84],[39,95293],{"className":95294,"style":231},[230],[39,95296,31514],{"className":95297},[235],[39,95299],{"className":95300,"style":231},[230],[39,95302,95304,95307,95310,95313,95316],{"className":95303},[74],[39,95305],{"className":95306,"style":245},[78],[39,95308,17359],{"className":95309},[83],[39,95311,11],{"className":95312},[83,84],[39,95314,161],{"className":95315},[263],[39,95317,2903],{"className":95318},[83],[11,95320,95321,95322,95373,95374,95444,95445,95507],{},"指数太糟糕了。",[39,95323,95325,95343],{"className":95324,"translate":43},[42],[39,95326,95328],{"className":95327},[47],[49,95329,95330],{"xmlns":51},[53,95331,95332,95340],{},[56,95333,95334,95336,95338],{},[59,95335,29425],{},[133,95337,793],{},[539,95339,28724],{},[63,95341,95342],{"encoding":65},"d=100",[39,95344,95346,95364],{"className":95345,"ariaHidden":70},[69],[39,95347,95349,95352,95355,95358,95361],{"className":95348},[74],[39,95350],{"className":95351,"style":28824},[78],[39,95353,29425],{"className":95354},[83,84],[39,95356],{"className":95357,"style":231},[230],[39,95359,793],{"className":95360},[235],[39,95362],{"className":95363,"style":231},[230],[39,95365,95367,95370],{"className":95366},[74],[39,95368],{"className":95369,"style":2473},[78],[39,95371,28724],{"className":95372},[83]," 时误差按 ",[39,95375,95377,95400],{"className":95376,"translate":43},[42],[39,95378,95380],{"className":95379},[47],[49,95381,95382],{"xmlns":51},[53,95383,95384,95397],{},[56,95385,95386],{},[138,95387,95388,95390],{},[59,95389,12962],{},[56,95391,95392,95394],{},[133,95393,5378],{},[539,95395,95396],{},"0.01",[63,95398,95399],{"encoding":65},"n^{-0.01}",[39,95401,95403],{"className":95402,"ariaHidden":70},[69],[39,95404,95406,95409],{"className":95405},[74],[39,95407],{"className":95408,"style":19710},[78],[39,95410,95412,95415],{"className":95411},[83],[39,95413,12962],{"className":95414},[83,84],[39,95416,95418],{"className":95417},[192],[39,95419,95421],{"className":95420},[196],[39,95422,95424],{"className":95423},[200],[39,95425,95427],{"className":95426,"style":19710},[204],[39,95428,95429,95432],{"style":208},[39,95430],{"className":95431,"style":213},[212],[39,95433,95435],{"className":95434},[217,218,219,220],[39,95436,95438,95441],{"className":95437},[83,220],[39,95439,5378],{"className":95440},[83,220],[39,95442,95396],{"className":95443},[83,220]," 衰减，想把误差减半，样本量要乘 ",[39,95446,95448,95466],{"className":95447,"translate":43},[42],[39,95449,95451],{"className":95450},[47],[49,95452,95453],{"xmlns":51},[53,95454,95455,95463],{},[56,95456,95457],{},[138,95458,95459,95461],{},[539,95460,17359],{},[539,95462,28724],{},[63,95464,95465],{"encoding":65},"2^{100}",[39,95467,95469],{"className":95468,"ariaHidden":70},[69],[39,95470,95472,95475],{"className":95471},[74],[39,95473],{"className":95474,"style":19710},[78],[39,95476,95478,95481],{"className":95477},[83],[39,95479,17359],{"className":95480},[83],[39,95482,95484],{"className":95483},[192],[39,95485,95487],{"className":95486},[196],[39,95488,95490],{"className":95489},[200],[39,95491,95493],{"className":95492,"style":19710},[204],[39,95494,95495,95498],{"style":208},[39,95496],{"className":95497,"style":213},[212],[39,95499,95501],{"className":95500},[217,218,219,220],[39,95502,95504],{"className":95503},[83,220],[39,95505,28724],{"className":95506},[83,220],"。所以“直接在高维空间算 Wasserstein”在绝大多数真实问题上不可行。",[11,95509,95510,95511,95539,95540,95614],{},"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 ",[39,95512,95514,95527],{"className":95513,"translate":43},[42],[39,95515,95517],{"className":95516},[47],[49,95518,95519],{"xmlns":51},[53,95520,95521,95525],{},[56,95522,95523],{},[59,95524,23703],{},[63,95526,23703],{"encoding":65},[39,95528,95530],{"className":95529,"ariaHidden":70},[69],[39,95531,95533,95536],{"className":95532},[74],[39,95534],{"className":95535,"style":79},[78],[39,95537,23703],{"className":95538,"style":178},[83,84],"，再对方向平均（一维的收敛是 ",[39,95541,95543,95569],{"className":95542,"translate":43},[42],[39,95544,95546],{"className":95545},[47],[49,95547,95548],{"xmlns":51},[53,95549,95550,95566],{},[56,95551,95552],{},[138,95553,95554,95556],{},[59,95555,12962],{},[56,95557,95558,95560,95562,95564],{},[133,95559,5378],{},[539,95561,546],{},[59,95563,45691],{"mathvariant":145},[539,95565,17359],{},[63,95567,95568],{"encoding":65},"n^{-1\u002F2}",[39,95570,95572],{"className":95571,"ariaHidden":70},[69],[39,95573,95575,95579],{"className":95574},[74],[39,95576],{"className":95577,"style":95578},[78],"height:0.888em;",[39,95580,95582,95585],{"className":95581},[83],[39,95583,12962],{"className":95584},[83,84],[39,95586,95588],{"className":95587},[192],[39,95589,95591],{"className":95590},[196],[39,95592,95594],{"className":95593},[200],[39,95595,95597],{"className":95596,"style":95578},[204],[39,95598,95599,95602],{"style":208},[39,95600],{"className":95601,"style":213},[212],[39,95603,95605],{"className":95604},[217,218,219,220],[39,95606,95608,95611],{"className":95607},[83,220],[39,95609,5378],{"className":95610},[83,220],[39,95612,45696],{"className":95613},[83,220],"，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。",[21,95616,95617],{"id":95617},"单纯形上的尺子",[11,95619,95620,95621,75296,95675,95728,95729,75296,95782,1607],{},"我拿两个模型对同一张图跑一遍，输出分别是 ",[39,95622,95624,95648],{"className":95623,"translate":43},[42],[39,95625,95627],{"className":95626},[47],[49,95628,95629],{"xmlns":51},[53,95630,95631,95645],{},[56,95632,95633,95635,95638,95640,95643],{},[133,95634,136],{"stretchy":135},[539,95636,95637],{},"0.90",[133,95639,155],{"separator":70},[539,95641,95642],{},"0.10",[133,95644,161],{"stretchy":135},[63,95646,95647],{"encoding":65},"(0.90,0.10)",[39,95649,95651],{"className":95650,"ariaHidden":70},[69],[39,95652,95654,95657,95660,95663,95666,95669,95672],{"className":95653},[74],[39,95655],{"className":95656,"style":245},[78],[39,95658,136],{"className":95659},[182],[39,95661,95637],{"className":95662},[83],[39,95664,155],{"className":95665},[252],[39,95667],{"className":95668,"style":256},[230],[39,95670,95642],{"className":95671},[83],[39,95673,161],{"className":95674},[263],[39,95676,95678,95701],{"className":95677,"translate":43},[42],[39,95679,95681],{"className":95680},[47],[49,95682,95683],{"xmlns":51},[53,95684,95685,95698],{},[56,95686,95687,95689,95692,95694,95696],{},[133,95688,136],{"stretchy":135},[539,95690,95691],{},"0.99",[133,95693,155],{"separator":70},[539,95695,95396],{},[133,95697,161],{"stretchy":135},[63,95699,95700],{"encoding":65},"(0.99,0.01)",[39,95702,95704],{"className":95703,"ariaHidden":70},[69],[39,95705,95707,95710,95713,95716,95719,95722,95725],{"className":95706},[74],[39,95708],{"className":95709,"style":245},[78],[39,95711,136],{"className":95712},[182],[39,95714,95691],{"className":95715},[83],[39,95717,155],{"className":95718},[252],[39,95720],{"className":95721,"style":256},[230],[39,95723,95396],{"className":95724},[83],[39,95726,161],{"className":95727},[263],"。另一张图上，两个模型的输出是 ",[39,95730,95732,95755],{"className":95731,"translate":43},[42],[39,95733,95735],{"className":95734},[47],[49,95736,95737],{"xmlns":51},[53,95738,95739,95752],{},[56,95740,95741,95743,95746,95748,95750],{},[133,95742,136],{"stretchy":135},[539,95744,95745],{},"0.50",[133,95747,155],{"separator":70},[539,95749,95745],{},[133,95751,161],{"stretchy":135},[63,95753,95754],{"encoding":65},"(0.50,0.50)",[39,95756,95758],{"className":95757,"ariaHidden":70},[69],[39,95759,95761,95764,95767,95770,95773,95776,95779],{"className":95760},[74],[39,95762],{"className":95763,"style":245},[78],[39,95765,136],{"className":95766},[182],[39,95768,95745],{"className":95769},[83],[39,95771,155],{"className":95772},[252],[39,95774],{"className":95775,"style":256},[230],[39,95777,95745],{"className":95778},[83],[39,95780,161],{"className":95781},[263],[39,95783,95785,95809],{"className":95784,"translate":43},[42],[39,95786,95788],{"className":95787},[47],[49,95789,95790],{"xmlns":51},[53,95791,95792,95806],{},[56,95793,95794,95796,95799,95801,95804],{},[133,95795,136],{"stretchy":135},[539,95797,95798],{},"0.59",[133,95800,155],{"separator":70},[539,95802,95803],{},"0.41",[133,95805,161],{"stretchy":135},[63,95807,95808],{"encoding":65},"(0.59,0.41)",[39,95810,95812],{"className":95811,"ariaHidden":70},[69],[39,95813,95815,95818,95821,95824,95827,95830,95833],{"className":95814},[74],[39,95816],{"className":95817,"style":245},[78],[39,95819,136],{"className":95820},[182],[39,95822,95798],{"className":95823},[83],[39,95825,155],{"className":95826},[252],[39,95828],{"className":95829,"style":256},[230],[39,95831,95803],{"className":95832},[83],[39,95834,161],{"className":95835},[263],[11,95837,95838,95839,95868],{},"两组变化的欧氏距离都是 ",[39,95840,95842,95856],{"className":95841,"translate":43},[42],[39,95843,95845],{"className":95844},[47],[49,95846,95847],{"xmlns":51},[53,95848,95849,95854],{},[56,95850,95851],{},[539,95852,95853],{},"0.09",[63,95855,95853],{"encoding":65},[39,95857,95859],{"className":95858,"ariaHidden":70},[69],[39,95860,95862,95865],{"className":95861},[74],[39,95863],{"className":95864,"style":2473},[78],[39,95866,95853],{"className":95867},[83],"。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。",[11,95870,95871,95872,38249,95925,742,95954,38249,96006,96035,96036,34856,96064,96092],{},"这种直觉能不能写成数？能。把 Bernoulli 分布按 Fisher–Rao 距离量一下：",[39,95873,95875,95898],{"className":95874,"translate":43},[42],[39,95876,95878],{"className":95877},[47],[49,95879,95880],{"xmlns":51},[53,95881,95882,95895],{},[56,95883,95884,95886,95889,95891,95893],{},[133,95885,136],{"stretchy":135},[539,95887,95888],{},"0.5",[133,95890,155],{"separator":70},[539,95892,95798],{},[133,95894,161],{"stretchy":135},[63,95896,95897],{"encoding":65},"(0.5,0.59)",[39,95899,95901],{"className":95900,"ariaHidden":70},[69],[39,95902,95904,95907,95910,95913,95916,95919,95922],{"className":95903},[74],[39,95905],{"className":95906,"style":245},[78],[39,95908,136],{"className":95909},[182],[39,95911,95888],{"className":95912},[83],[39,95914,155],{"className":95915},[252],[39,95917],{"className":95918,"style":256},[230],[39,95920,95798],{"className":95921},[83],[39,95923,161],{"className":95924},[263],[39,95926,95928,95942],{"className":95927,"translate":43},[42],[39,95929,95931],{"className":95930},[47],[49,95932,95933],{"xmlns":51},[53,95934,95935,95940],{},[56,95936,95937],{},[539,95938,95939],{},"0.18",[63,95941,95939],{"encoding":65},[39,95943,95945],{"className":95944,"ariaHidden":70},[69],[39,95946,95948,95951],{"className":95947},[74],[39,95949],{"className":95950,"style":2473},[78],[39,95952,95939],{"className":95953},[83],[39,95955,95957,95979],{"className":95956,"translate":43},[42],[39,95958,95960],{"className":95959},[47],[49,95961,95962],{"xmlns":51},[53,95963,95964,95976],{},[56,95965,95966,95968,95970,95972,95974],{},[133,95967,136],{"stretchy":135},[539,95969,17739],{},[133,95971,155],{"separator":70},[539,95973,95691],{},[133,95975,161],{"stretchy":135},[63,95977,95978],{"encoding":65},"(0.9,0.99)",[39,95980,95982],{"className":95981,"ariaHidden":70},[69],[39,95983,95985,95988,95991,95994,95997,96000,96003],{"className":95984},[74],[39,95986],{"className":95987,"style":245},[78],[39,95989,136],{"className":95990},[182],[39,95992,17739],{"className":95993},[83],[39,95995,155],{"className":95996},[252],[39,95998],{"className":95999,"style":256},[230],[39,96001,95691],{"className":96002},[83],[39,96004,161],{"className":96005},[263],[39,96007,96009,96023],{"className":96008,"translate":43},[42],[39,96010,96012],{"className":96011},[47],[49,96013,96014],{"xmlns":51},[53,96015,96016,96021],{},[56,96017,96018],{},[539,96019,96020],{},"0.44",[63,96022,96020],{"encoding":65},[39,96024,96026],{"className":96025,"ariaHidden":70},[69],[39,96027,96029,96032],{"className":96028},[74],[39,96030],{"className":96031,"style":2473},[78],[39,96033,96020],{"className":96034},[83],"，后者是前者的两倍半。同样的坐标变化，在概率接近 ",[39,96037,96039,96052],{"className":96038,"translate":43},[42],[39,96040,96042],{"className":96041},[47],[49,96043,96044],{"xmlns":51},[53,96045,96046,96050],{},[56,96047,96048],{},[539,96049,541],{},[63,96051,541],{"encoding":65},[39,96053,96055],{"className":96054,"ariaHidden":70},[69],[39,96056,96058,96061],{"className":96057},[74],[39,96059],{"className":96060,"style":2473},[78],[39,96062,541],{"className":96063},[83],[39,96065,96067,96080],{"className":96066,"translate":43},[42],[39,96068,96070],{"className":96069},[47],[49,96071,96072],{"xmlns":51},[53,96073,96074,96078],{},[56,96075,96076],{},[539,96077,546],{},[63,96079,546],{"encoding":65},[39,96081,96083],{"className":96082,"ariaHidden":70},[69],[39,96084,96086,96089],{"className":96085},[74],[39,96087],{"className":96088,"style":2473},[78],[39,96090,546],{"className":96091},[83]," 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。",[11,96094,96095],{},"那么这把尺子是怎么来的？",[11,96097,96098],{},"模型的输出住在单纯形",[39,96100,96102],{"className":96101,"translate":43},[746],[39,96103,96105,96179],{"className":96104,"translate":43},[42],[39,96106,96108],{"className":96107},[47],[49,96109,96110],{"xmlns":51,"display":755},[53,96111,96112,96176],{},[56,96113,96114,96126,96128,96130,96132,96134,96140,96142,96144,96150,96152,96154,96156,96158,96164,96170,96172,96174],{},[138,96115,96116,96118],{},[59,96117,2134],{"mathvariant":145},[56,96119,96120,96122,96124],{},[59,96121,77380],{},[133,96123,5378],{},[539,96125,546],{},[133,96127,793],{},[133,96129,12951],{"fence":70,"stretchy":70,"minsize":65017,"maxsize":65017},[59,96131,11],{},[133,96133,534],{},[138,96135,96136,96138],{},[59,96137,422],{"mathvariant":421},[59,96139,77380],{},[133,96141,404],{},[35841,96143,74416],{},[613,96145,96146,96148],{},[59,96147,11],{},[59,96149,13377],{},[133,96151,25173],{},[539,96153,541],{},[133,96155,155],{"separator":70},[35841,96157,74416],{},[6556,96159,96160,96162],{},[133,96161,3191],{},[59,96163,13377],{},[613,96165,96166,96168],{},[59,96167,11],{},[59,96169,13377],{},[133,96171,793],{},[539,96173,546],{},[133,96175,12965],{"fence":70,"stretchy":70,"minsize":65017,"maxsize":65017},[63,96177,96178],{"encoding":65},"\\Delta^{K-1}=\\Bigl\\{p\\in\\mathbb R^K:\\ p_i\\ge0,\\ \\sum_i p_i=1\\Bigr\\}",[39,96180,96182,96236,96260,96307,96363,96477],{"className":96181,"ariaHidden":70},[69],[39,96183,96185,96189,96227,96230,96233],{"className":96184},[74],[39,96186],{"className":96187,"style":96188},[78],"height:0.8913em;",[39,96190,96192,96195],{"className":96191},[83],[39,96193,2134],{"className":96194},[83],[39,96196,96198],{"className":96197},[192],[39,96199,96201],{"className":96200},[196],[39,96202,96204],{"className":96203},[200],[39,96205,96207],{"className":96206,"style":96188},[204],[39,96208,96209,96212],{"style":1027},[39,96210],{"className":96211,"style":213},[212],[39,96213,96215],{"className":96214},[217,218,219,220],[39,96216,96218,96221,96224],{"className":96217},[83,220],[39,96219,77380],{"className":96220,"style":48487},[83,84,220],[39,96222,5378],{"className":96223},[465,220],[39,96225,546],{"className":96226},[83,220],[39,96228],{"className":96229,"style":231},[230],[39,96231,793],{"className":96232},[235],[39,96234],{"className":96235,"style":231},[230],[39,96237,96239,96242,96248,96251,96254,96257],{"className":96238},[74],[39,96240],{"className":96241,"style":75495},[78],[39,96243,96245],{"className":96244},[182],[39,96246,12951],{"className":96247},[18520,33204],[39,96249,11],{"className":96250},[83,84],[39,96252],{"className":96253,"style":231},[230],[39,96255,534],{"className":96256},[235],[39,96258],{"className":96259,"style":231},[230],[39,96261,96263,96266,96295,96298,96301,96304],{"className":96262},[74],[39,96264],{"className":96265,"style":96188},[78],[39,96267,96269,96272],{"className":96268},[83],[39,96270,422],{"className":96271},[83,515],[39,96273,96275],{"className":96274},[192],[39,96276,96278],{"className":96277},[196],[39,96279,96281],{"className":96280},[200],[39,96282,96284],{"className":96283,"style":96188},[204],[39,96285,96286,96289],{"style":1027},[39,96287],{"className":96288,"style":213},[212],[39,96290,96292],{"className":96291},[217,218,219,220],[39,96293,77380],{"className":96294,"style":48487},[83,84,220],[39,96296],{"className":96297,"style":231},[230],[39,96299,404],{"className":96300},[235],[39,96302,74416],{"className":96303},[230],[39,96305],{"className":96306,"style":231},[230],[39,96308,96310,96314,96354,96357,96360],{"className":96309},[74],[39,96311],{"className":96312,"style":96313},[78],"height:0.8304em;vertical-align:-0.1944em;",[39,96315,96317,96320],{"className":96316},[83],[39,96318,11],{"className":96319},[83,84],[39,96321,96323],{"className":96322},[192],[39,96324,96326,96346],{"className":96325},[196,649],[39,96327,96329,96343],{"className":96328},[200],[39,96330,96332],{"className":96331,"style":13604},[204],[39,96333,96334,96337],{"style":966},[39,96335],{"className":96336,"style":213},[212],[39,96338,96340],{"className":96339},[217,218,219,220],[39,96341,13377],{"className":96342},[83,84,220],[39,96344,673],{"className":96345},[672],[39,96347,96349],{"className":96348},[200],[39,96350,96352],{"className":96351,"style":680},[204],[39,96353],{},[39,96355],{"className":96356,"style":231},[230],[39,96358,25173],{"className":96359},[235],[39,96361],{"className":96362,"style":231},[230],[39,96364,96366,96369,96372,96375,96378,96381,96425,96428,96468,96471,96474],{"className":96365},[74],[39,96367],{"className":96368,"style":83027},[78],[39,96370,541],{"className":96371},[83],[39,96373,155],{"className":96374},[252],[39,96376,74416],{"className":96377},[230],[39,96379],{"className":96380,"style":256},[230],[39,96382,96384],{"className":96383},[1744,3294],[39,96385,96387,96417],{"className":96386},[196,649],[39,96388,96390,96414],{"className":96389},[200],[39,96391,96393,96404],{"className":96392,"style":6670},[204],[39,96394,96395,96398],{"style":13926},[39,96396],{"className":96397,"style":3311},[212],[39,96399,96401],{"className":96400},[217,218,219,220],[39,96402,13377],{"className":96403},[83,84,220],[39,96405,96406,96409],{"style":3330},[39,96407],{"className":96408,"style":3311},[212],[39,96410,96411],{},[39,96412,3191],{"className":96413},[1744,3339,3340],[39,96415,673],{"className":96416},[672],[39,96418,96420],{"className":96419},[200],[39,96421,96423],{"className":96422,"style":79128},[204],[39,96424],{},[39,96426],{"className":96427,"style":256},[230],[39,96429,96431,96434],{"className":96430},[83],[39,96432,11],{"className":96433},[83,84],[39,96435,96437],{"className":96436},[192],[39,96438,96440,96460],{"className":96439},[196,649],[39,96441,96443,96457],{"className":96442},[200],[39,96444,96446],{"className":96445,"style":13604},[204],[39,96447,96448,96451],{"style":966},[39,96449],{"className":96450,"style":213},[212],[39,96452,96454],{"className":96453},[217,218,219,220],[39,96455,13377],{"className":96456},[83,84,220],[39,96458,673],{"className":96459},[672],[39,96461,96463],{"className":96462},[200],[39,96464,96466],{"className":96465,"style":680},[204],[39,96467],{},[39,96469],{"className":96470,"style":231},[230],[39,96472,793],{"className":96473},[235],[39,96475],{"className":96476,"style":231},[230],[39,96478,96480,96483,96486],{"className":96479},[74],[39,96481],{"className":96482,"style":75495},[78],[39,96484,546],{"className":96485},[83],[39,96487,96489],{"className":96488},[263],[39,96490,12965],{"className":96491},[18520,33204],[11,96493,96494],{},"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。",[11,96496,96497,96498,96579],{},"Rao 在 1945 年的想法是：把参数化的分布族 ",[39,96499,96501,96524],{"className":96500,"translate":43},[42],[39,96502,96504],{"className":96503},[47],[49,96505,96506],{"xmlns":51},[53,96507,96508,96521],{},[56,96509,96510,96512,96519],{},[133,96511,12951],{"stretchy":135},[613,96513,96514,96516],{},[59,96515,11],{},[59,96517,96518],{},"θ",[133,96520,12965],{"stretchy":135},[63,96522,96523],{"encoding":65},"\\{p_\\theta\\}",[39,96525,96527],{"className":96526,"ariaHidden":70},[69],[39,96528,96530,96533,96536,96576],{"className":96529},[74],[39,96531],{"className":96532,"style":245},[78],[39,96534,12951],{"className":96535},[182],[39,96537,96539,96542],{"className":96538},[83],[39,96540,11],{"className":96541},[83,84],[39,96543,96545],{"className":96544},[192],[39,96546,96548,96568],{"className":96547},[196,649],[39,96549,96551,96565],{"className":96550},[200],[39,96552,96554],{"className":96553,"style":3419},[204],[39,96555,96556,96559],{"style":966},[39,96557],{"className":96558,"style":213},[212],[39,96560,96562],{"className":96561},[217,218,219,220],[39,96563,96518],{"className":96564,"style":438},[83,84,220],[39,96566,673],{"className":96567},[672],[39,96569,96571],{"className":96570},[200],[39,96572,96574],{"className":96573,"style":680},[204],[39,96575],{},[39,96577,12965],{"className":96578},[263]," 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",[39,96581,96583],{"className":96582,"translate":43},[746],[39,96584,96586,96707],{"className":96585,"translate":43},[42],[39,96587,96589],{"className":96588},[47],[49,96590,96591],{"xmlns":51,"display":755},[53,96592,96593,96704],{},[56,96594,96595,96605,96607,96609,96611,96613,96623,96625,96632,96634,96636,96642,96644,96650,96652,96654,96660,96662,96664,96666,96676,96678,96684,96690,96692,96694,96700,96702],{},[613,96596,96597,96599],{},[59,96598,39209],{},[56,96600,96601,96603],{},[59,96602,13377],{},[59,96604,13446],{},[133,96606,136],{"stretchy":135},[59,96608,96518],{},[133,96610,161],{"stretchy":135},[133,96612,793],{},[613,96614,96615,96617],{},[59,96616,4253],{"mathvariant":421},[613,96618,96619,96621],{},[59,96620,11],{},[59,96622,96518],{},[133,96624,537],{"fence":70,"stretchy":70,"minsize":74068,"maxsize":74068},[613,96626,96627,96630],{},[59,96628,96629],{"mathvariant":145},"∂",[59,96631,13377],{},[59,96633,92676],{},[133,96635,1690],{},[613,96637,96638,96640],{},[59,96639,11],{},[59,96641,96518],{},[133,96643,8885],{},[613,96645,96646,96648],{},[59,96647,96629],{"mathvariant":145},[59,96649,13446],{},[59,96651,92676],{},[133,96653,1690],{},[613,96655,96656,96658],{},[59,96657,11],{},[59,96659,96518],{},[133,96661,4280],{"fence":70,"stretchy":70,"minsize":74068,"maxsize":74068},[133,96663,793],{},[133,96665,5378],{},[613,96667,96668,96670],{},[59,96669,4253],{"mathvariant":421},[613,96671,96672,96674],{},[59,96673,11],{},[59,96675,96518],{},[133,96677,537],{"fence":70,"stretchy":70,"minsize":74068,"maxsize":74068},[613,96679,96680,96682],{},[59,96681,96629],{"mathvariant":145},[59,96683,13377],{},[613,96685,96686,96688],{},[59,96687,96629],{"mathvariant":145},[59,96689,13446],{},[59,96691,92676],{},[133,96693,1690],{},[613,96695,96696,96698],{},[59,96697,11],{},[59,96699,96518],{},[133,96701,4280],{"fence":70,"stretchy":70,"minsize":74068,"maxsize":74068},[133,96703,155],{"separator":70},[63,96705,96706],{"encoding":65},"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],",[39,96708,96710,96777,96972,97084],{"className":96709,"ariaHidden":70},[69],[39,96711,96713,96716,96759,96762,96765,96768,96771,96774],{"className":96712},[74],[39,96714],{"className":96715,"style":41560},[78],[39,96717,96719,96722],{"className":96718},[83],[39,96720,39209],{"className":96721,"style":2114},[83,84],[39,96723,96725],{"className":96724},[192],[39,96726,96728,96751],{"className":96727},[196,649],[39,96729,96731,96748],{"className":96730},[200],[39,96732,96734],{"className":96733,"style":13604},[204],[39,96735,96736,96739],{"style":41316},[39,96737],{"className":96738,"style":213},[212],[39,96740,96742],{"className":96741},[217,218,219,220],[39,96743,96745],{"className":96744},[83,220],[39,96746,13732],{"className":96747,"style":13731},[83,84,220],[39,96749,673],{"className":96750},[672],[39,96752,96754],{"className":96753},[200],[39,96755,96757],{"className":96756,"style":35396},[204],[39,96758],{},[39,96760,136],{"className":96761},[182],[39,96763,96518],{"className":96764,"style":438},[83,84],[39,96766,161],{"className":96767},[263],[39,96769],{"className":96770,"style":231},[230],[39,96772,793],{"className":96773},[235],[39,96775],{"className":96776,"style":231},[230],[39,96778,96780,96783,96865,96871,96912,96915,96920,96923,96963,96966,96969],{"className":96779},[74],[39,96781],{"className":96782,"style":908},[78],[39,96784,96786,96789],{"className":96785},[83],[39,96787,4253],{"className":96788},[83,515],[39,96790,96792],{"className":96791},[192],[39,96793,96795,96857],{"className":96794},[196,649],[39,96796,96798,96854],{"className":96797},[200],[39,96799,96801],{"className":96800,"style":1728},[204],[39,96802,96803,96806],{"style":966},[39,96804],{"className":96805,"style":213},[212],[39,96807,96809],{"className":96808},[217,218,219,220],[39,96810,96812],{"className":96811},[83,220],[39,96813,96815,96818],{"className":96814},[83,220],[39,96816,11],{"className":96817},[83,84,220],[39,96819,96821],{"className":96820},[192],[39,96822,96824,96845],{"className":96823},[196,649],[39,96825,96827,96842],{"className":96826},[200],[39,96828,96830],{"className":96829,"style":84537},[204],[39,96831,96833,96836],{"style":96832},"top:-2.3488em;margin-left:0em;margin-right:0.0714em;",[39,96834],{"className":96835,"style":4004},[212],[39,96837,96839],{"className":96838},[217,4008,4009,220],[39,96840,96518],{"className":96841,"style":438},[83,84,220],[39,96843,673],{"className":96844},[672],[39,96846,96848],{"className":96847},[200],[39,96849,96852],{"className":96850,"style":96851},[204],"height:0.1512em;",[39,96853],{},[39,96855,673],{"className":96856},[672],[39,96858,96860],{"className":96859},[200],[39,96861,96863],{"className":96862,"style":35396},[204],[39,96864],{},[39,96866,96868],{"className":96867},[182],[39,96869,537],{"className":96870},[18520,4009],[39,96872,96874,96877],{"className":96873},[83],[39,96875,96629],{"className":96876,"style":565},[83],[39,96878,96880],{"className":96879},[192],[39,96881,96883,96904],{"className":96882},[196,649],[39,96884,96886,96901],{"className":96885},[200],[39,96887,96889],{"className":96888,"style":13604},[204],[39,96890,96892,96895],{"style":96891},"top:-2.55em;margin-left:-0.0556em;margin-right:0.05em;",[39,96893],{"className":96894,"style":213},[212],[39,96896,96898],{"className":96897},[217,218,219,220],[39,96899,13377],{"className":96900},[83,84,220],[39,96902,673],{"className":96903},[672],[39,96905,96907],{"className":96906},[200],[39,96908,96910],{"className":96909,"style":680},[204],[39,96911],{},[39,96913],{"className":96914,"style":256},[230],[39,96916,92914,96918],{"className":96917},[1744],[39,96919,39209],{"style":35078},[39,96921],{"className":96922,"style":256},[230],[39,96924,96926,96929],{"className":96925},[83],[39,96927,11],{"className":96928},[83,84],[39,96930,96932],{"className":96931},[192],[39,96933,96935,96955],{"className":96934},[196,649],[39,96936,96938,96952],{"className":96937},[200],[39,96939,96941],{"className":96940,"style":3419},[204],[39,96942,96943,96946],{"style":966},[39,96944],{"className":96945,"style":213},[212],[39,96947,96949],{"className":96948},[217,218,219,220],[39,96950,96518],{"className":96951,"style":438},[83,84,220],[39,96953,673],{"className":96954},[672],[39,96956,96958],{"className":96957},[200],[39,96959,96961],{"className":96960,"style":680},[204],[39,96962],{},[39,96964],{"className":96965,"style":461},[230],[39,96967,8885],{"className":96968},[465],[39,96970],{"className":96971,"style":461},[230],[39,96973,96975,96978,97018,97021,97026,97029,97069,97075,97078,97081],{"className":96974},[74],[39,96976],{"className":96977,"style":908},[78],[39,96979,96981,96984],{"className":96980},[83],[39,96982,96629],{"className":96983,"style":565},[83],[39,96985,96987],{"className":96986},[192],[39,96988,96990,97010],{"className":96989},[196,649],[39,96991,96993,97007],{"className":96992},[200],[39,96994,96996],{"className":96995,"style":13604},[204],[39,96997,96998,97001],{"style":96891},[39,9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",[39,97323,97325,97359],{"className":97324,"translate":43},[42],[39,97326,97328],{"className":97327},[47],[49,97329,97330],{"xmlns":51},[53,97331,97332,97356],{},[56,97333,97334,97336,97338,97344,97346,97348,97350,97352,97354],{},[59,97335,4253],{"mathvariant":421},[133,97337,537],{"stretchy":135},[613,97339,97340,97342],{},[59,97341,96629],{"mathvariant":145},[59,97343,13377],{},[59,97345,92676],{},[133,97347,1690],{},[59,97349,11],{},[133,97351,4280],{"stretchy":135},[133,97353,793],{},[539,97355,541],{},[63,97357,97358],{"encoding":65},"\\mathbb E[\\partial_i\\log p]=0",[39,97360,97362,97440],{"className":97361,"ariaHidden":70},[69],[39,97363,97365,97368,97371,97374,97414,97417,97422,97425,97428,97431,97434,97437],{"className":97364},[74],[39,97366],{"className":97367,"style":245},[78],[39,97369,4253],{"className":97370},[83,515],[39,97372,537],{"className":97373},[182],[39,97375,97377,97380],{"className":97376},[83],[39,97378,96629],{"className":97379,"style":565},[83],[39,97381,97383],{"className":97382},[192],[39,97384,97386,97406],{"className":97385},[196,649],[39,97387,97389,97403],{"className":97388},[200],[39,97390,97392],{"className":97391,"style":13604},[204],[39,97393,97394,97397],{"style":96891},[39,97395],{"className":97396,"style":213},[212],[39,97398,97400],{"className":97399},[217,218,219,220],[39,97401,13377],{"className":97402},[83,84,220],[39,97404,673],{"className":97405},[672],[39,97407,97409],{"className":97408},[200],[39,97410,97412],{"className":97411,"style":680},[204],[39,97413],{},[39,97415],{"className":97416,"style":256},[230],[39,97418,92914,97420],{"className":97419},[1744],[39,97421,39209],{"style":35078},[39,97423],{"className":97424,"style":256},[230],[39,97426,11],{"className":97427},[83,84],[39,97429,4280],{"className":97430},[263],[39,97432],{"className":97433,"style":231},[230],[39,97435,793],{"className":97436},[235],[39,97438],{"className":97439,"style":231},[230],[39,97441,97443,97446],{"className":97442},[74],[39,97444],{"className":97445,"style":2473},[78],[39,97447,541],{"className":97448},[83],"，因为对 ",[39,97451,97453,97467],{"className":97452,"translate":43},[42],[39,97454,97456],{"className":97455},[47],[49,97457,97458],{"xmlns":51},[53,97459,97460,97464],{},[56,97461,97462],{},[59,97463,96518],{},[63,97465,97466],{"encoding":65},"\\theta",[39,97468,97470],{"className":97469,"ariaHidden":70},[69],[39,97471,97473,97476],{"className":97472},[74],[39,97474],{"className":97475,"style":28824},[78],[39,97477,96518],{"className":97478,"style":438},[83,84]," 求导再积分，",[39,97481,97483,97533],{"className":97482,"translate":43},[42],[39,97484,97486],{"className":97485},[47],[49,97487,97488],{"xmlns":51},[53,97489,97490,97530],{},[56,97491,97492,97494,97500,97502,97504,97506,97508,97510,97516,97518,97520,97522,97524,97526,97528],{},[133,97493,69087],{},[613,97495,97496,97498],{},[59,97497,96629],{"mathvariant":145},[59,97499,13377],{},[59,97501,11],{},[35841,97503,35843],{},[59,97505,29425],{},[59,97507,1755],{},[133,97509,793],{},[613,97511,97512,97514],{},[59,97513,96629],{"mathvariant":145},[59,97515,13377],{},[133,97517,69087],{},[59,97519,11],{},[35841,97521,35843],{},[59,97523,29425],{},[59,97525,1755],{},[133,97527,793],{},[539,97529,541],{},[63,97531,97532],{"encoding":65},"\\int\\partial_i p\\,dx=\\partial_i\\int 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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。",[11,98781,98782],{},"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。",[11,98784,98785],{},"第三，对指数族它恰好是势函数的 Hessian，下一节说。",[11,98787,98788,98789,742],{},"把 Bernoulli 族算一遍。",[39,98790,98792,98844],{"className":98791,"translate":43},[42],[39,98793,98795],{"className":98794},[47],[49,98796,98797],{"xmlns":51},[53,98798,98799,98841],{},[56,98800,98801,98803,98805,98807,98809,98811,98813,98815,98821,98823,98825,98827,98829],{},[59,98802,11],{},[133,98804,136],{"stretchy":135},[59,98806,1755],{},[133,98808,150],{},[59,98810,96518],{},[133,98812,161],{"stretchy":135},[133,98814,793],{},[138,98816,98817,98819],{},[59,98818,96518],{},[59,98820,1755],{},[133,98822,136],{"stretchy":135},[539,98824,546],{},[133,98826,5378],{},[59,98828,96518],{},[138,98830,98831,98833],{},[133,98832,161],{"stretchy":135},[56,98834,98835,98837,98839],{},[539,98836,546],{},[133,98838,5378],{},[59,98840,1755],{},[63,98842,98843],{"encoding":65},"p(x\\mid\\theta)=\\theta^x(1-\\theta)^{1-x}",[39,98845,98847,98871,98892,98942],{"className":98846,"ariaHidden":70},[69],[39,98848,98850,98853,98856,98859,98862,98865,98868],{"className":98849},[74],[39,98851],{"className":98852,"style":245},[78],[39,98854,11],{"className":98855},[83,84],[39,98857,136],{"className":98858},[182],[39,98860,1755],{"className":98861},[83,84],[39,98863],{"className":98864,"style":231},[230],[39,98866,150],{"className":98867},[235],[39,98869],{"className":98870,"style":231},[230],[39,98872,98874,98877,98880,98883,98886,98889],{"className":98873},[74],[39,98875],{"className":98876,"style":245},[78],[39,98878,96518],{"className":98879,"style":438},[83,84],[39,98881,161],{"className":98882},[263],[39,98884],{"className":98885,"style":231},[230],[39,98887,793],{"className":98888},[235],[39,98890],{"className":98891,"style":231},[230],[39,98893,98895,98898,98927,98930,98933,98936,98939],{"className":98894},[74],[39,98896],{"className":98897,"style":245},[78],[39,98899,98901,98904],{"className":98900},[83],[39,98902,96518],{"className":98903,"style":438},[83,84],[39,98905,98907],{"className":98906},[192],[39,98908,98910],{"className":98909},[196],[39,98911,98913],{"className":98912},[200],[39,98914,98916],{"className":98915,"style":6410},[204],[39,98917,98918,98921],{"style":208},[39,98919],{"className":98920,"style":213},[212],[39,98922,98924],{"className":98923},[217,218,219,220],[39,98925,1755],{"className":98926},[83,84,220],[39,98928,136],{"className":98929},[182],[39,98931,546],{"className":98932},[83],[39,98934],{"className":98935,"style":461},[230],[39,98937,5378],{"className":98938},[465],[39,98940],{"className":98941,"style":461},[230],[39,98943,98945,98948,98951],{"className":98944},[74],[39,98946],{"className":98947,"style":84808},[78],[39,98949,96518],{"className":98950,"style":438},[83,84],[39,98952,98954,98957],{"className":98953},[263],[39,98955,161],{"className":98956},[263],[39,98958,98960],{"className":98959},[192],[39,98961,98963],{"className":98962},[196],[39,98964,98966],{"className":98965},[200],[39,98967,98969],{"className":98968,"style":19710},[204],[39,98970,98971,98974],{"style":208},[39,98972],{"className":98973,"style":213},[212],[39,98975,98977],{"className":98976},[217,218,219,220],[39,98978,98980,98983,98986],{"className":98979},[83,220],[39,98981,546],{"className":98982},[83,220],[39,98984,5378],{"className":98985},[465,220],[39,98987,1755],{"className":98988},[83,84,220],[39,98990,98992],{"className":98991,"translate":43},[746],[39,98993,98995,99037],{"className":98994,"translate":43},[42],[39,98996,98998],{"className":98997},[47],[49,98999,99000],{"xmlns":51,"display":755},[53,99001,99002,99034],{},[56,99003,99004,99006,99008,99010,99012,99014,99032],{},[59,99005,39209],{},[133,99007,136],{"stretchy":135},[59,99009,96518],{},[133,99011,161],{"stretchy":135},[133,99013,793],{},[5362,99015,99016,99018],{},[539,99017,546],{},[56,99019,99020,99022,99024,99026,99028,99030],{},[59,99021,96518],{},[133,99023,136],{"stretchy":135},[539,99025,546],{},[133,99027,5378],{},[59,99029,96518],{},[133,99031,161],{"stretchy":135},[59,99033,2903],{"mathvariant":145},[63,99035,99036],{"encoding":65},"g(\\theta)=\\frac{1}{\\theta(1-\\theta)} 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的距离是 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0”走到“几乎确定是 1”，不管两端有多极端，路径长度都是 ",[39,100134,100136,100149],{"className":100135,"translate":43},[42],[39,100137,100139],{"className":100138},[47],[49,100140,100141],{"xmlns":51},[53,100142,100143,100147],{},[56,100144,100145],{},[59,100146,2098],{},[63,100148,2101],{"encoding":65},[39,100150,100152],{"className":100151,"ariaHidden":70},[69],[39,100153,100155,100158],{"className":100154},[74],[39,100156],{"className":100157,"style":290},[78],[39,100159,2098],{"className":100160,"style":2114},[83,84],"；而 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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。",[11,100245,100246,100247,100346],{},"那个 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不是巧合。对一般的离散分布，",[39,100348,100350],{"className":100349,"translate":43},[746],[39,100351,100353,100423],{"className":100352,"translate":43},[42],[39,100354,100356],{"className":100355},[47],[49,100357,100358],{"xmlns":51,"display":755},[53,100359,100360,100420],{},[56,100361,100362,100372,100374,100376,100378,100381,100383,100385,100387,100390,100392,100394,100400,100416,100418],{},[613,100363,100364,100366],{},[59,100365,29425],{},[56,100367,100368,100370],{},[59,100369,618],{"mathvariant":145},[59,100371,422],{"mathvariant":145},[133,100373,136],{"stretchy":135},[59,100375,11],{},[133,100377,155],{"separator":70},[59,100379,100380],{},"q",[133,100382,161],{"stretchy":135},[133,100384,793],{},[539,100386,17359],{},[59,100388,100389],{},"arccos",[133,100391,1690],{},[133,100393,136],{"fence":70,"stretchy":70,"minsize":65017,"maxsize":65017},[6556,100395,100396,100398],{},[133,100397,3191],{},[59,100399,13377],{},[99189,100401,100402],{},[56,100403,100404,100410],{},[613,100405,100406,100408],{},[59,100407,11],{},[59,100409,13377],{},[613,100411,100412,100414],{},[59,100413,100380],{},[59,100415,13377],{},[133,100417,161],{"fence":70,"stretchy":70,"minsize":65017,"maxsize":65017},[133,100419,155],{"separator":70},[63,100421,100422],{"encoding":65},"d_{\\mathrm{FR}}(p,q)=2\\arccos\\Bigl(\\sum_i\\sqrt{p_iq_i}\\Bigr),",[39,100424,100426,100506],{"className":100425,"ariaHidden":70},[69],[39,100427,100429,100432,100479,100482,100485,100488,100491,100494,100497,100500,100503],{"className":100428},[74],[39,100430],{"className":100431,"style":245},[78],[39,100433,100435,100438],{"className":100434},[83],[39,100436,29425],{"className":100437},[83,84],[39,100439,100441],{"className":100440},[192],[39,100442,100444,100471],{"className":100443},[196,649],[39,100445,100447,100468],{"className":100446},[200],[39,100448,100450],{"className":100449,"style":62921},[204],[39,100451,100452,100455],{"style":966},[39,100453],{"className":100454,"style":213},[212],[39,100456,100458],{"className":100457},[217,218,219,220],[39,100459,100461],{"className":100460},[83,220],[39,100462,100464],{"className":100463},[83,220],[39,100465,100467],{"className":100466},[83,35077,220],"FR",[39,100469,673],{"className":100470},[672],[39,100472,100474],{"className":100473},[200],[39,100475,100477],{"className":100476,"style":680},[204],[39,100478],{},[39,100480,136],{"className":100481},[182],[39,100483,11],{"className":100484},[83,84],[39,100486,155],{"className":100487},[252],[39,100489],{"className":100490,"style":256},[230],[39,100492,100380],{"className":100493,"style":2114},[83,84],[39,100495,161],{"className":100496},[263],[39,100498],{"className":100499,"style":231},[230],[39,100501,793],{"className":100502},[235],[39,100504],{"className":100505,"style":231},[230],[39,100507,100509,100513,100516,100519,100522,100528,100572,100575,100702,100708],{"className":100508},[74],[39,100510],{"className":100511,"style":100512},[78],"height:2.4277em;vertical-align:-1.2777em;",[39,100514,17359],{"className":100515},[83],[39,100517],{"className":100518,"style":256},[230],[39,100520,100389],{"className":100521},[1744],[39,100523,100525],{"className":100524},[182],[39,100526,136],{"className":100527},[18520,33204],[39,100529,100531],{"className":100530},[1744,3294],[39,100532,100534,100564],{"className":100533},[196,649],[39,100535,100537,100561],{"className":100536},[200],[39,100538,100540,100551],{"className":100539,"style":6670},[204],[39,100541,100542,100545],{"style":13926},[39,100543],{"className":100544,"style":3311},[212],[39,100546,100548],{"className":100547},[217,218,219,220],[39,100549,13377],{"className":100550},[83,84,220],[39,100552,100553,100556],{"style":3330},[39,100554],{"className":100555,"style":3311},[212],[39,100557,100558],{},[39,100559,3191],{"className":100560},[1744,3339,3340],[39,100562,673],{"className":100563},[672],[39,100565,100567],{"className":100566},[200],[39,100568,100570],{"className":100569,"style":79128},[204],[39,100571],{},[39,100573],{"className":100574,"style":256},[230],[39,100576,100578],{"className":100577},[83,99251],[39,100579,100581,100693],{"className":100580},[196,649],[39,100582,100584,100690],{"className":100583},[200],[39,100585,100587,100677],{"className":100586,"style":205},[204],[39,100588,100590,100593],{"className":100589,"style":21599},[99265],[39,100591],{"className":100592,"style":935},[212],[39,100594,100597,100637],{"className":100595,"style":100596},[83],"padding-left:0.833em;",[39,100598,100600,100603],{"className":100599},[83],[39,100601,11],{"className":100602},[83,84],[39,100604,100606],{"className":100605},[192],[39,100607,100609,100629],{"className":100608},[196,649],[39,100610,100612,100626],{"className":100611},[200],[39,100613,100615],{"className":100614,"style":13604},[204],[39,100616,100617,100620],{"style":966},[39,100618],{"className":100619,"style":213},[212],[39,100621,100623],{"className":100622},[217,218,219,220],[39,100624,13377],{"className":100625},[83,84,220],[39,1006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",[39,100715,100717,100743],{"className":100716,"translate":43},[42],[39,100718,100720],{"className":100719},[47],[49,100721,100722],{"xmlns":51},[53,100723,100724,100740],{},[56,100725,100726,100728,100730,100732,100734,100736],{},[59,100727,11],{},[133,100729,69813],{},[59,100731,35321],{},[133,100733,793],{},[539,100735,17359],{},[99189,100737,100738],{},[59,100739,11],{},[63,100741,100742],{"encoding":65},"p\\mapsto u=2\\sqrt p",[39,100744,100746,100765,100783],{"className":100745,"ariaHidden":70},[69],[39,100747,100749,100753,100756,100759,100762],{"className":100748},[74],[39,100750],{"className":100751,"style":100752},[78],"height:0.7054em;vertical-align:-0.1944em;",[39,100754,11],{"className":100755},[83,84],[39,100757],{"className":100758,"style":231},[230],[39,100760,69813],{"className":100761},[235],[39,100763],{"className":100764,"style":231},[230],[39,100766,100768,100771,100774,100777,100780],{"className":100767},[74],[39,100769],{"className":100770,"style":290},[78],[39,100772,35321],{"className":100773},[83,84],[39,100775],{"className":100776,"style":231},[230],[39,100778,793],{"className":100779},[235],[39,100781],{"className":100782,"style":231},[230],[39,100784,100786,100790,100793],{"className":100785},[74],[39,100787],{"className":100788,"style":100789},[78],"height:1.04em;vertical-align:-0.3369em;",[39,100791,17359],{"className":100792},[83],[39,100794,100796],{"className":100795},[83,99251],[39,100797,100799,100831],{"className":100798},[196,649],[39,100800,100802,100828],{"className":100801},[200],[39,100803,100806,100815],{"className":100804,"style":100805},[204],"height:0.7031em;",[39,100807,100809,100812],{"className":100808,"style":21599},[99265],[39,100810],{"className":100811,"style":935},[212],[39,100813,11],{"className":100814,"style":100596},[83,84],[39,100816,100818,100821],{"style":100817},"top:-2.6631em;",[39,100819],{"className":100820,"style":935},[212],[39,100822,100824],{"className":100823,"style":100326},[99307],[86106,100825,100826],{"xmlns":86108,"width":99311,"height":100329,"viewBox":100330,"preserveAspectRatio":99314},[86112,100827],{"d":100333},[39,100829,673],{"className":100830},[672],[39,100832,100834],{"className":100833},[200],[39,100835,100838],{"className":100836,"style":100837},[204],"height:0.3369em;",[39,100839],{}," 把单纯形嵌入到 ",[39,100842,100844,100862],{"className":100843,"translate":43},[42],[39,100845,100847],{"className":100846},[47],[49,100848,100849],{"xmlns":51},[53,100850,100851,100859],{},[56,100852,100853],{},[138,100854,100855,100857],{},[59,100856,422],{"mathvariant":421},[59,100858,77380],{},[63,100860,100861],{"encoding":65},"\\mathbb R^K",[39,100863,100865],{"className":100864,"ariaHidden":70},[69],[39,100866,100868,100872],{"className":100867},[74],[39,100869],{"className":100870,"style":100871},[78],"height:0.8413em;",[39,100873,100875,100878],{"className":100874},[83],[39,100876,422],{"className":100877},[83,515],[39,100879,100881],{"className":100880},[192],[39,100882,100884],{"className":100883},[196],[39,100885,100887],{"className":100886},[200],[39,100888,100890],{"className":100889,"style":100871},[204],[39,100891,100892,100895],{"style":208},[39,100893],{"className":100894,"style":213},[212],[39,100896,100898],{"className":100897},[217,218,219,220],[39,100899,77380],{"className":100900,"style":48487},[83,84,220]," 中半径为 ",[39,100903,100905,100918],{"className":100904,"translate":43},[42],[39,100906,100908],{"className":100907},[47],[49,100909,100910],{"xmlns":51},[53,100911,100912,100916],{},[56,100913,100914],{},[539,100915,17359],{},[63,100917,17359],{"encoding":65},[39,100919,100921],{"className":100920,"ariaHidden":70},[69],[39,100922,100924,100927],{"className":100923},[74],[39,100925],{"className":100926,"style":2473},[78],[39,100928,17359],{"className":100929},[83]," 的球面上（因为 ",[39,100932,100934,100964],{"className":100933,"translate":43},[42],[39,100935,100937],{"className":100936},[47],[49,100938,100939],{"xmlns":51},[53,100940,100941,100961],{},[56,100942,100943,100949,100951,100957,100959],{},[613,100944,100945,100947],{},[133,100946,3191],{},[59,100948,13377],{},[539,100950,59223],{},[613,100952,100953,100955],{},[59,100954,11],{},[59,100956,13377],{},[133,100958,793],{},[539,100960,59223],{},[63,100962,100963],{"encoding":65},"\\sum_i4p_i=4",[39,100965,100967,101068],{"className":100966,"ariaHidden":70},[69],[39,100968,100970,100973,101013,101016,101019,101059,101062,101065],{"className":100969},[74],[39,100971],{"className":100972,"style":94090},[78],[39,100974,100976,100979],{"className":100975},[1744],[39,100977,3191],{"className":100978,"style":3953},[1744,3339,3952],[39,100980,100982],{"className":100981},[192],[39,100983,100985,101005],{"className":100984},[196,649],[39,100986,100988,101002],{"className":100987},[200],[39,100989,100991],{"className":100990,"style":94115},[204],[39,100992,100993,100996],{"style":3969},[39,100994],{"className":100995,"style":213},[212],[39,100997,100999],{"className":100998},[217,218,219,220],[39,101000,13377],{"className":101001},[83,84,220],[39,101003,673],{"className":101004},[672],[39,101006,101008],{"className":101007},[200],[39,101009,101011],{"className":101010,"style":4025},[204],[39,101012],{},[39,101014],{"className":101015,"style":256},[230],[39,101017,59223],{"className":101018},[83],[39,101020,101022,101025],{"className":101021},[83],[39,101023,11],{"className":101024},[83,84],[39,101026,101028],{"className":101027},[192],[39,101029,101031,101051],{"className":101030},[196,649],[39,101032,101034,101048],{"className":101033},[200],[39,101035,101037],{"className":101036,"style":13604},[204],[39,101038,101039,101042],{"style":966},[39,101040],{"className":101041,"style":213},[212],[39,101043,101045],{"className":101044},[217,218,219,220],[39,101046,13377],{"className":101047},[83,84,220],[39,101049,673],{"className":101050},[672],[39,101052,101054],{"className":101053},[200],[39,101055,101057],{"className":101056,"style":680},[204],[39,101058],{},[39,101060],{"className":101061,"style":231},[230],[39,101063,793],{"className":101064},[235],[39,101066],{"className":101067,"style":231},[230],[39,101069,101071,101074],{"className":101070},[74],[39,101072],{"className":101073,"style":2473},[78],[39,101075,59223],{"className":101076},[83],"），Fisher–Rao 距离正好是球面上的大圆距离。这个嵌入叫 Hellinger 嵌入，",[39,101079,101081,101115],{"className":101080,"translate":43},[42],[39,101082,101084],{"className":101083},[47],[49,101085,101086],{"xmlns":51},[53,101087,101088,101112],{},[56,101089,101090,101096],{},[613,101091,101092,101094],{},[133,101093,3191],{},[59,101095,13377],{},[99189,101097,101098],{},[56,101099,101100,101106],{},[613,101101,101102,101104],{},[59,101103,11],{},[59,101105,13377],{},[613,101107,101108,101110],{},[59,101109,100380],{},[59,101111,13377],{},[63,101113,101114],{"encoding":65},"\\sum_i\\sqrt{p_iq_i}",[39,101116,101118],{"className":101117,"ariaHidden":70},[69],[39,101119,101121,101125,101165,101168],{"className":101120},[74],[39,101122],{"className":101123,"style":101124},[78],"height:1.0869em;vertical-align:-0.3369em;",[39,101126,101128,101131],{"className":101127},[1744],[39,101129,3191],{"className":101130,"style":3953},[1744,3339,3952],[39,101132,101134],{"className":101133},[192],[39,101135,101137,101157],{"className":101136},[196,649],[39,101138,101140,101154],{"className":101139},[200],[39,101141,101143],{"className":101142,"style":94115},[204],[39,101144,101145,101148],{"style":3969},[39,101146],{"className":101147,"style":213},[212],[39,101149,101151],{"className":101150},[217,218,219,220],[39,101152,13377],{"className":101153},[83,84,220],[39,101155,673],{"className":101156},[672],[39,101158,101160],{"className":101159},[200],[39,101161,101163],{"className":101162,"style":4025},[204],[39,101164],{},[39,101166],{"className":101167,"style":256},[230],[39,101169,101171],{"className":101170},[83,99251],[39,101172,101174,101284],{"className":101173},[196,649],[39,101175,101177,101281],{"className":101176},[200],[39,101178,101180,101269],{"className":101179,"style":100805},[204],[39,101181,101183,101186],{"className":101182,"style":21599},[99265],[39,101184],{"className":101185,"style":935},[212],[39,101187,101189,101229],{"className":101188,"style":100596},[83],[39,101190,101192,101195],{"className":101191},[83],[39,101193,11],{"className":101194},[83,84],[39,101196,101198],{"className":101197},[192],[39,101199,101201,101221],{"className":101200},[196,649],[39,101202,101204,101218],{"className":101203},[200],[39,101205,101207],{"className":101206,"style":13604},[204],[39,101208,101209,101212],{"style":966},[39,101210],{"className":101211,"style":213},[212],[39,101213,101215],{"className":101214},[217,218,219,220],[39,101216,13377],{"className":101217},[83,84,220],[39,101219,673],{"className":101220},[672],[39,101222,101224],{"className":101223},[200],[39,101225,101227],{"className":101226,"style":680},[204],[39,101228],{},[39,101230,101232,101235],{"className":101231},[83],[39,101233,100380],{"className":101234,"style":2114},[83,84],[39,101236,101238],{"className":101237},[192],[39,101239,101241,101261],{"className":101240},[196,649],[39,101242,101244,101258],{"className":101243},[200],[39,101245,101247],{"className":101246,"style":13604},[204],[39,101248,101249,101252],{"style":41316},[39,101250],{"className":101251,"style":213},[212],[39,101253,101255],{"className":101254},[217,218,219,220],[39,101256,13377],{"className":101257},[83,84,220],[39,101259,673],{"className":101260},[672],[39,101262,101264],{"className":101263},[200],[39,101265,101267],{"className":101266,"style":680},[204],[39,101268],{},[39,101270,101271,101274],{"style":100817},[39,101272],{"className":101273,"style":935},[212],[39,101275,101277],{"className":101276,"style":100326},[99307],[86106,101278,101279],{"xmlns":86108,"width":99311,"height":100329,"viewBox":100330,"preserveAspectRatio":99314},[86112,101280],{"d":100333},[39,101282,673],{"className":101283},[672],[39,101285,101287],{"className":101286},[200],[39,101288,101290],{"className":101289,"style":100837},[204],[39,101291],{}," 叫 Bhattacharyya 系数。",[11,101294,101295],{},"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",[39,101297,101299],{"className":101298,"translate":43},[746],[39,101300,101302,101421],{"className":101301,"translate":43},[42],[39,101303,101305],{"className":101304},[47],[49,101306,101307],{"xmlns":51,"display":755},[53,101308,101309,101418],{},[56,101310,101311,101317,101319,101360,101362,101364,101370,101372,101378,101380,101382,101384,101386,101388,101390,101398,101400,101402,101410,101416],{},[613,101312,101313,101315],{},[59,101314,11],{},[59,101316,621],{},[133,101318,793],{},[5362,101320,101321,101353],{},[56,101322,101323,101325,101327,101329,101331,101333,101335,101339,101341,101343,101347],{},[133,101324,136],{"fence":70,"stretchy":70,"minsize":74068,"maxsize":74068},[133,101326,136],{"stretchy":135},[539,101328,546],{},[133,101330,5378],{},[59,101332,621],{},[133,101334,161],{"stretchy":135},[99189,101336,101337],{},[59,101338,11],{},[133,101340,788],{},[59,101342,621],{},[99189,101344,101345],{},[59,101346,100380],{},[138,101348,101349,101351],{},[133,101350,161],{"fence":70,"stretchy":70,"minsize":74068,"maxsize":74068},[539,101352,17359],{},[613,101354,101355,101358],{},[59,101356,101357],{},"Z",[59,101359,621],{},[133,101361,155],{"separator":70},[230,101363],{"width":10251},[613,101365,101366,101368],{},[59,101367,101357],{},[59,101369,621],{},[133,101371,793],{},[6556,101373,101374,101376],{},[133,101375,3191],{},[59,101377,13377],{},[133,101379,136],{"fence":70,"stretchy":70,"minsize":74068,"maxsize":74068},[133,101381,136],{"stretchy":135},[539,101383,546],{},[133,101385,5378],{},[59,101387,621],{},[133,101389,161],{"stretchy":135},[99189,101391,101392],{},[613,101393,101394,101396],{},[59,101395,11],{},[59,101397,13377],{},[133,101399,788],{},[59,101401,621],{},[99189,101403,101404],{},[613,101405,101406,101408],{},[59,101407,100380],{},[59,101409,13377],{},[138,101411,101412,101414],{},[133,101413,161],{"fence":70,"stretchy":70,"minsize":74068,"maxsize":74068},[539,101415,17359],{},[59,101417,2903],{"mathvariant":145},[63,101419,101420],{"encoding":65},"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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01969},[200],[39,101971,101973],{"className":101972,"style":100699},[204],[39,101974],{},[39,101976],{"className":101977,"style":461},[230],[39,101979,788],{"className":101980},[465],[39,101982],{"className":101983,"style":461},[230],[39,101985,101987,101991,101994,102078,102110],{"className":101986},[74],[39,101988],{"className":101989,"style":101990},[78],"height:1.404em;vertical-align:-0.35em;",[39,101992,621],{"className":101993},[83,84],[39,101995,101997],{"className":101996},[83,99251],[39,101998,102000,102070],{"className":101999},[196,649],[39,102001,102003,102067],{"className":102002},[200],[39,102004,102006,102055],{"className":102005,"style":205},[204],[39,102007,102009,102012],{"className":102008,"style":21599},[99265],[39,102010],{"className":102011,"style":935},[212],[39,102013,102015],{"className":102014,"style":100596},[83],[39,102016,102018,102021],{"className":102017},[83],[39,102019,100380],{"className":102020,"style":2114},[83,84],[39,102022,102024],{"className":102023},[192],[39,102025,102027,102047],{"className":102026},[196,649],[39,102028,102030,102044],{"className":102029},[200],[39,102031,102033],{"className":102032,"style":13604},[204],[39,102034,102035,102038],{"style":41316},[39,102036],{"className":102037,"style":213},[212],[39,102039,102041],{"className":102040},[217,218,219,220],[39,102042,13377],{"className":102043},[83,84,220],[39,102045,673],{"className":102046},[672],[39,102048,102050],{"className":102049},[200],[39,102051,102053],{"className":102052,"style":680},[204],[39,102054],{},[39,102056,102057,102060],{"style":100679},[39,102058],{"className":102059,"style":935},[212],[39,102061,102063],{"className":102062,"style":100326},[99307],[86106,102064,102065],{"xmlns":86108,"width":99311,"height":100329,"viewBox":100330,"preserveAspectRatio":99314},[86112,102066],{"d":100333},[39,102068,673],{"className":102069},[672],[39,102071,102073],{"className":102072},[200],[39,102074,102076],{"className":102075,"style":100699},[204],[39,102077],{},[39,102079,102081,102087],{"className":102080},[263],[39,102082,102084],{"className":102083},[263],[39,102085,161],{"className":102086},[18520,4009],[39,102088,102090],{"className":102089},[192],[39,102091,102093],{"className":102092},[196],[39,102094,102096],{"className":102095},[200],[39,102097,102099],{"className":102098,"style":101722},[204],[39,102100,102101,102104],{"style":86329},[39,102102],{"className":102103,"style":213},[212],[39,102105,102107],{"className":102106},[217,218,219,220],[39,102108,17359],{"className":102109},[83,220],[39,102111,2903],{"className":102112},[83],[11,102114,102115],{},"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。",[21,102117,102118],{"id":102118},"指数族",[11,102120,102121,102122,102150,102151,102179],{},"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 ",[39,102123,102125,102138],{"className":102124,"translate":43},[42],[39,102126,102128],{"className":102127},[47],[49,102129,102130],{"xmlns":51},[53,102131,102132,102136],{},[56,102133,102134],{},[59,102135,96518],{},[63,102137,97466],{"encoding":65},[39,102139,102141],{"className":102140,"ariaHidden":70},[69],[39,102142,102144,102147],{"className":102143},[74],[39,102145],{"className":102146,"style":28824},[78],[39,102148,96518],{"className":102149,"style":438},[83,84],"，正面次数服从二项分布，而样本对 ",[39,102152,102154,102167],{"className":102153,"translate":43},[42],[39,102155,102157],{"className":102156},[47],[49,102158,102159],{"xmlns":51},[53,102160,102161,102165],{},[56,102162,102163],{},[59,102164,96518],{},[63,102166,97466],{"encoding":65},[39,102168,102170],{"className":102169,"ariaHidden":70},[69],[39,102171,102173,102176],{"className":102172},[74],[39,102174],{"className":102175,"style":28824},[78],[39,102177,96518],{"className":102178,"style":438},[83,84]," 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。",[11,102181,102182],{},"指数族把这个结构写成一个模板：",[39,102184,102186],{"className":102185,"translate":43},[746],[39,102187,102189,102257],{"className":102188,"translate":43},[42],[39,102190,102192],{"className":102191},[47],[49,102193,102194],{"xmlns":51,"display":755},[53,102195,102196,102254],{},[56,102197,102198,102200,102202,102204,102206,102208,102210,102212,102214,102216,102218,102220,102222,102224,102226,102232,102234,102236,102238,102240,102242,102244,102246,102248,102250,102252],{},[59,102199,11],{},[133,102201,136],{"stretchy":135},[59,102203,1755],{},[133,102205,150],{},[59,102207,96518],{},[133,102209,161],{"stretchy":135},[133,102211,793],{},[59,102213,69998],{},[133,102215,136],{"stretchy":135},[59,102217,1755],{},[133,102219,161],{"stretchy":135},[59,102221,93036],{},[133,102223,1690],{},[133,102225,136],{"fence":70,"stretchy":70,"minsize":74068,"maxsize":74068},[138,102227,102228,102230],{},[59,102229,96518],{},[59,102231,93534],{"mathvariant":145},[59,102233,11772],{},[133,102235,136],{"stretchy":135},[59,102237,1755],{},[133,102239,161],{"stretchy":135},[133,102241,5378],{},[59,102243,80846],{},[133,102245,136],{"stretchy":135},[59,102247,96518],{},[133,102249,161],{"stretchy":135},[133,102251,161],{"fence":70,"stretchy":70,"minsize":74068,"maxsize":74068},[59,102253,2903],{"mathvariant":145},[63,102255,102256],{"encoding":65},"p(x\\mid\\theta)=h(x)\\exp\\bigl(\\theta^\\top T(x)-\\psi(\\theta)\\bigr).",[39,102258,102260,102284,102305,102386],{"className":102259,"ariaHidden":70},[69],[39,102261,102263,102266,102269,102272,102275,102278,102281],{"className":102262},[74],[39,102264],{"className":102265,"style":245},[78],[39,102267,11],{"className":102268},[83,84],[39,102270,136],{"className":102271},[182],[39,102273,1755],{"className":102274},[83,84],[39,102276],{"className":102277,"style":231},[230],[39,102279,150],{"className":102280},[235],[39,102282],{"className":102283,"style":231},[230],[39,102285,102287,102290,102293,102296,102299,102302],{"className":102286},[74],[39,102288],{"className":102289,"style":245},[78],[39,102291,96518],{"className":102292,"style":438},[83,84],[39,102294,161],{"className":102295},[263],[39,102297],{"className":102298,"style":231},[230],[39,102300,793],{"className":102301},[235],[39,102303],{"className":102304,"style":231},[230],[39,102306,102308,102312,102315,102318,102321,102324,102327,102330,102336,102365,102368,102371,102374,102377,102380,102383],{"className":102307},[74],[39,102309],{"className":102310,"style":102311},[78],"height:1.2491em;vertical-align:-0.35em;",[39,102313,69998],{"className":102314},[83,84],[39,102316,136],{"className":102317},[182],[39,102319,1755],{"className":102320},[83,84],[39,102322,161],{"className":102323},[263],[39,102325],{"className":102326,"style":256},[230],[39,102328,93036],{"className":102329},[1744],[39,102331,102333],{"className":102332},[182],[39,102334,136],{"className":102335},[18520,4009],[39,102337,102339,102342],{"className":102338},[83],[39,102340,96518],{"className":102341,"style":438},[83,84],[39,102343,102345],{"className":102344},[192],[39,102346,102348],{"className":102347},[196],[39,102349,102351],{"className":102350},[200],[39,102352,102354],{"className":102353,"style":3389},[204],[39,102355,102356,102359],{"style":1027},[39,102357],{"className":102358,"style":213},[212],[39,102360,102362],{"className":102361},[217,218,219,220],[39,102363,93534],{"className":102364},[83,220],[39,102366,11772],{"className":102367,"style":178},[83,84],[39,102369,136],{"className":102370},[182],[39,102372,1755],{"className":102373},[83,84],[39,102375,161],{"className":102376},[263],[39,102378],{"className":102379,"style":461},[230],[39,102381,5378],{"className":102382},[465],[39,102384],{"className":102385,"style":461},[230],[39,102387,102389,102392,102395,102398,102401,102404,102410],{"className":102388},[74],[39,102390],{"className":102391,"style":908},[78],[39,102393,80846],{"className":102394,"style":2114},[83,84],[39,102396,136],{"className":102397},[182],[39,102399,96518],{"className":102400,"style":438},[83,84],[39,102402,161],{"className":102403},[263],[39,102405,102407],{"className":102406},[263],[39,102408,161],{"className":102409},[18520,4009],[39,102411,2903],{"className":102412},[83],[11,102414,102415,102443,102444,102472,102473,102502,102503,88033,102569,88033,102717,1607],{},[39,102416,102418,102431],{"className":102417,"translate":43},[42],[39,102419,102421],{"className":102420},[47],[49,102422,102423],{"xmlns":51},[53,102424,102425,102429],{},[56,102426,102427],{},[59,102428,11772],{},[63,102430,11772],{"encoding":65},[39,102432,102434],{"className":102433,"ariaHidden":70},[69],[39,102435,102437,102440],{"className":102436},[74],[39,102438],{"className":102439,"style":79},[78],[39,102441,11772],{"className":102442,"style":178},[83,84]," 是充分统计量，",[39,102445,102447,102460],{"className":102446,"translate":43},[42],[39,102448,102450],{"className":102449},[47],[49,102451,102452],{"xmlns":51},[53,102453,102454,102458],{},[56,102455,102456],{},[59,102457,96518],{},[63,102459,97466],{"encoding":65},[39,102461,102463],{"className":102462,"ariaHidden":70},[69],[39,102464,102466,102469],{"className":102465},[74],[39,102467],{"className":102468,"style":28824},[78],[39,102470,96518],{"className":102471,"style":438},[83,84]," 是自然参数，",[39,102474,102476,102490],{"className":102475,"translate":43},[42],[39,102477,102479],{"className":102478},[47],[49,102480,102481],{"xmlns":51},[53,102482,102483,102487],{},[56,102484,102485],{},[59,102486,80846],{},[63,102488,102489],{"encoding":65},"\\psi",[39,102491,102493],{"className":102492,"ariaHidden":70},[69],[39,102494,102496,102499],{"className":102495},[74],[39,102497],{"className":102498,"style":51280},[78],[39,102500,80846],{"className":102501,"style":2114},[83,84]," 是对数配分函数，负责把总质量归一化成 1。抛硬币就是 ",[39,102504,102506,102530],{"className":102505,"translate":43},[42],[39,102507,102509],{"className":102508},[47],[49,102510,102511],{"xmlns":51},[53,102512,102513,102527],{},[56,102514,102515,102517,102519,102521,102523,102525],{},[59,102516,11772],{},[133,102518,136],{"stretchy":135},[59,102520,1755],{},[133,102522,161],{"stretchy":135},[133,102524,793],{},[59,102526,1755],{},[63,102528,102529],{"encoding":65},"T(x)=x",[39,102531,102533,102560],{"className":102532,"ariaHidden":70},[69],[39,102534,102536,102539,102542,102545,102548,102551,102554,102557],{"className":102535},[74],[39,102537],{"className":102538,"style":245},[78],[39,102540,11772],{"className":102541,"style":178},[83,84],[39,102543,136],{"className":102544},[182],[39,102546,1755],{"className":102547},[83,84],[39,102549,161],{"className":102550},[263],[39,102552],{"className":102553,"style":231},[230],[39,102555,793],{"className":102556},[235],[39,102558],{"className":102559,"style":231},[230],[39,102561,102563,102566],{"className":102562},[74],[39,102564],{"className":102565,"style":290},[78],[39,102567,1755],{"className":102568},[83,84],[39,102570,102572,102604],{"className":102571,"translate":43},[42],[39,102573,102575],{"className":102574},[47],[49,102576,102577],{"xmlns":51},[53,102578,102579,102601],{},[56,102580,102581,102583,102585,102587,102589],{},[59,102582,96518],{},[133,102584,793],{},[59,102586,92676],{},[133,102588,1690],{},[5362,102590,102591,102593],{},[59,102592,11],{},[56,102594,102595,102597,102599],{},[539,102596,546],{},[133,102598,5378],{},[59,102600,11],{},[63,102602,102603],{"encoding":65},"\\theta=\\log\\frac{p}{1-p}",[39,102605,102607,102625],{"className":102606,"ariaHidden":70},[69],[39,102608,102610,102613,102616,102619,102622],{"className":102609},[74],[39,102611],{"className":102612,"style":28824},[78],[39,102614,96518],{"className":102615,"style":438},[83,84],[39,102617],{"className":102618,"style":231},[230],[39,102620,793],{"className":102621},[235],[39,102623],{"className":102624,"style":231},[230],[39,102626,102628,102632,102637,102640],{"className":102627},[74],[39,102629],{"className":102630,"style":102631},[78],"height:1.2286em;vertical-align:-0.4811em;",[39,102633,92914,102635],{"className":102634},[1744],[39,102636,39209],{"style":35078},[39,102638],{"className":102639,"style":256},[230],[39,102641,102643,102646,102714],{"className":102642},[83],[39,102644],{"className":102645},[182,5463],[39,102647,102649],{"className":102648},[5362],[39,102650,102652,102705],{"className":102651},[196,649],[39,102653,102655,102702],{"className":102654},[200],[39,102656,102659,102679,102687],{"className":102657,"style":102658},[204],"height:0.7475em;",[39,102660,102661,102664],{"style":75009},[39,102662],{"className":102663,"style":935},[212],[39,102665,102667],{"className":102666},[217,218,219,220],[39,102668,102670,102673,102676],{"className":102669},[83,220],[39,102671,546],{"className":102672},[83,220],[39,102674,5378],{"className":102675},[465,220],[39,102677,11],{"className":102678},[83,84,220],[39,102680,102681,102684],{"style":5503},[39,102682],{"className":102683,"style":935},[212],[39,102685],{"className":102686,"style":5511},[5510],[39,102688,102690,102693],{"style":102689},"top:-3.4461em;",[39,102691],{"className":102692,"style":935},[212],[39,102694,102696],{"className":102695},[217,218,219,220],[39,102697,102699],{"className":102698},[83,220],[39,102700,11],{"className":102701},[83,84,220],[39,102703,673],{"className":102704},[672],[39,102706,102708],{"className":102707},[200],[39,102709,102712],{"className":102710,"style":102711},[204],"height:0.4811em;",[39,102713],{},[39,102715],{"className":102716},[263,5463],[39,102718,102720,102761],{"className":102719,"translate":43},[42],[39,102721,102723],{"className":102722},[47],[49,102724,102725],{"xmlns":51},[53,102726,102727,102758],{},[56,102728,102729,102731,102733,102735,102737,102739,102741,102743,102745,102747,102749,102756],{},[59,102730,80846],{},[133,102732,136],{"stretchy":135},[59,102734,96518],{},[133,102736,161],{"stretchy":135},[133,102738,793],{},[59,102740,92676],{},[133,102742,1690],{},[133,102744,136],{"stretchy":135},[539,102746,546],{},[133,102748,788],{},[138,102750,102751,102754],{},[59,102752,102753],{},"e",[59,102755,96518],{},[133,102757,161],{"stretchy":135},[63,102759,102760],{"encoding":65},"\\psi(\\theta)=\\log(1+e^\\theta)",[39,102762,102764,102791,102817],{"className":102763,"ariaHidden":70},[69],[39,102765,102767,102770,102773,102776,102779,102782,102785,102788],{"className":102766},[74],[39,102768],{"className":102769,"style":245},[78],[39,102771,80846],{"className":102772,"style":2114},[83,84],[39,102774,136],{"className":102775},[182],[39,102777,96518],{"className":102778,"style":438},[83,84],[39,102780,161],{"className":102781},[263],[39,102783],{"className":102784,"style":231},[230],[39,102786,793],{"className":102787},[235],[39,102789],{"className":102790,"style":231},[230],[39,102792,102794,102797,102802,102805,102808,102811,102814],{"className":102793},[74],[39,102795],{"className":102796,"style":245},[78],[39,102798,92914,102800],{"className":102799},[1744],[39,102801,39209],{"style":35078},[39,102803,136],{"className":102804},[182],[39,102806,546],{"className":102807},[83],[39,102809],{"className":102810,"style":461},[230],[39,102812,788],{"className":102813},[465],[39,102815],{"className":102816,"style":461},[230],[39,102818,102820,102823,102852],{"className":102819},[74],[39,102821],{"className":102822,"style":15555},[78],[39,102824,102826,102829],{"className":102825},[83],[39,102827,102753],{"className":102828},[83,84],[39,102830,102832],{"className":102831},[192],[39,102833,102835],{"className":102834},[196],[39,102836,102838],{"className":102837},[200],[39,102839,102841],{"className":102840,"style":15577},[204],[39,102842,102843,102846],{"style":208},[39,102844],{"className":102845,"style":213},[212],[39,102847,102849],{"className":102848},[217,218,219,220],[39,102850,96518],{"className":102851,"style":438},[83,84,220],[39,102853,161],{"className":102854},[263],[11,102856,102857,102885,102886,742,103026,103205,103206,103234],{},[39,102858,102860,102873],{"className":102859,"translate":43},[42],[39,102861,102863],{"className":102862},[47],[49,102864,102865],{"xmlns":51},[53,102866,102867,102871],{},[56,102868,102869],{},[59,102870,80846],{},[63,102872,102489],{"encoding":65},[39,102874,102876],{"className":102875,"ariaHidden":70},[69],[39,102877,102879,102882],{"className":102878},[74],[39,102880],{"className":102881,"style":51280},[78],[39,102883,80846],{"className":102884,"style":2114},[83,84]," 的性质很好：",[39,102887,102889,102931],{"className":102888,"translate":43},[42],[39,102890,102892],{"className":102891},[47],[49,102893,102894],{"xmlns":51},[53,102895,102896,102928],{},[56,102897,102898,102900,102902,102904,102906,102908,102910,102916,102918,102920,102922,102924,102926],{},[59,102899,84918],{"mathvariant":145},[59,102901,80846],{},[133,102903,136],{"stretchy":135},[59,102905,96518],{},[133,102907,161],{"stretchy":135},[133,102909,793],{},[613,102911,102912,102914],{},[59,102913,4253],{"mathvariant":421},[59,102915,96518],{},[133,102917,537],{"stretchy":135},[59,102919,11772],{},[133,102921,136],{"stretchy":135},[59,102923,29420],{},[133,102925,161],{"stretchy":135},[133,102927,4280],{"stretchy":135},[63,102929,102930],{"encoding":65},"\\nabla\\psi(\\theta)=\\mathbb E_\\theta[T(X)]",[39,102932,102934,102964],{"className":102933,"ariaHidden":70},[69],[39,102935,102937,102940,102943,102946,102949,102952,102955,102958,102961],{"className":102936},[74],[39,102938],{"className":102939,"style":245},[78],[39,102941,84918],{"className":102942},[83],[39,102944,80846],{"className":102945,"style":2114},[83,84],[39,102947,136],{"className":102948},[182],[39,102950,96518],{"className":102951,"style":438},[83,84],[39,102953,161],{"className":102954},[263],[39,102956],{"className":102957,"style":231},[230],[39,102959,793],{"className":102960},[235],[39,102962],{"className":102963,"style":231},[230],[39,102965,102967,102970,103010,103013,103016,103019,103022],{"className":102966},[74],[39,102968],{"className":102969,"style":245},[78],[39,102971,102973,102976],{"className":102972},[83],[39,102974,4253],{"className":102975},[83,515],[39,102977,102979],{"className":102978},[192],[39,102980,102982,103002],{"className":102981},[196,649],[39,102983,102985,102999],{"className":102984},[200],[39,102986,102988],{"className":102987,"style":3419},[204],[39,102989,102990,102993],{"style":966},[39,102991],{"className":102992,"style":213},[212],[39,102994,102996],{"className":102995},[217,218,219,220],[39,102997,96518],{"className":102998,"style":438},[83,84,220],[39,103000,673],{"className":103001},[672],[39,103003,103005],{"className":103004},[200],[39,103006,103008],{"className":103007,"style":680},[204],[39,103009],{},[39,103011,537],{"className":103012},[182],[39,103014,11772],{"className":103015,"style":178},[83,84],[39,103017,136],{"className":103018},[182],[39,103020,29420],{"className":103021,"style":14420},[83,84],[39,103023,103025],{"className":103024},[263],")]",[39,103027,103029,103081],{"className":103028,"translate":43},[42],[39,103030,103032],{"className":103031},[47],[49,103033,103034],{"xmlns":51},[53,103035,103036,103078],{},[56,103037,103038,103044,103046,103048,103050,103052,103054,103066,103068,103070,103072,103074,103076],{},[138,103039,103040,103042],{},[59,103041,84918],{"mathvariant":145},[539,103043,17359],{},[59,103045,80846],{},[133,103047,136],{"stretchy":135},[59,103049,96518],{},[133,103051,161],{"stretchy":135},[133,103053,793],{},[613,103055,103056,103064],{},[56,103057,103058,103060,103062],{},[59,103059,48444],{"mathvariant":145},[59,103061,35010],{"mathvariant":145},[59,103063,78164],{"mathvariant":145},[59,103065,96518],{},[133,103067,537],{"stretchy":135},[59,103069,11772],{},[133,103071,136],{"stretchy":135},[59,103073,29420],{},[133,103075,161],{"stretchy":135},[133,103077,4280],{"stretchy":135},[63,103079,103080],{"encoding":65},"\\nabla^2\\psi(\\theta)=\\mathrm{Cov}_\\theta[T(X)]",[39,103082,103084,103140],{"className":103083,"ariaHidden":70},[69],[39,103085,103087,103090,103119,103122,103125,103128,103131,103134,103137],{"className":103086},[74],[39,103088],{"className":103089,"style":84808},[78],[39,103091,103093,103096],{"className":103092},[83],[39,103094,84918],{"className":103095},[83],[39,103097,103099],{"className":103098},[192],[39,103100,103102],{"className":103101},[196],[39,103103,103105],{"className":103104},[200],[39,103106,103108],{"className":103107,"style":19710},[204],[39,103109,103110,103113],{"style":208},[39,103111],{"className":103112,"style":213},[212],[39,103114,103116],{"className":103115},[217,218,219,220],[39,103117,17359],{"className":103118},[83,220],[39,103120,80846],{"className":103121,"style":2114},[83,84],[39,103123,136],{"className":103124},[182],[39,103126,96518],{"className":103127,"style":438},[83,84],[39,103129,161],{"className":103130},[263],[39,103132],{"className":103133,"style":231},[230],[39,103135,793],{"className":103136},[235],[39,103138],{"className":103139,"style":231},[230],[39,103141,103143,103146,103190,103193,103196,103199,103202],{"className":103142},[74],[39,103144],{"className":103145,"style":245},[78],[39,103147,103149,103156],{"className":103148},[83],[39,103150,103152],{"className":103151},[83],[39,103153,103155],{"className":103154,"style":35078},[83,35077],"Cov",[39,103157,103159],{"className":103158},[192],[39,103160,103162,103182],{"className":103161},[196,649],[39,103163,103165,103179],{"className":103164},[200],[39,103166,103168],{"className":103167,"style":3419},[204],[39,103169,103170,103173],{"style":44021},[39,103171],{"className":103172,"style":213},[212],[39,103174,103176],{"className":103175},[217,218,219,220],[39,103177,96518],{"className":103178,"style":438},[83,84,220],[39,103180,673],{"className":103181},[672],[39,103183,103185],{"className":103184},[200],[39,103186,103188],{"className":103187,"style":680},[204],[39,103189],{},[39,103191,537],{"className":103192},[182],[39,103194,11772],{"className":103195,"style":178},[83,84],[39,103197,136],{"className":103198},[182],[39,103200,29420],{"className":103201,"style":14420},[83,84],[39,103203,103025],{"className":103204},[263],"。第二个等式说明 Fisher 度量就是 ",[39,103207,103209,103222],{"className":103208,"translate":43},[42],[39,103210,103212],{"className":103211},[47],[49,103213,103214],{"xmlns":51},[53,103215,103216,103220],{},[56,103217,103218],{},[59,103219,80846],{},[63,103221,102489],{"encoding":65},[39,103223,103225],{"className":103224,"ariaHidden":70},[69],[39,103226,103228,103231],{"className":103227},[74],[39,103229],{"className":103230,"style":51280},[78],[39,103232,80846],{"className":103233,"style":2114},[83,84]," 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。",[11,103236,60385,103237,103309,103310,103338,103339,103565,103566,103667,103668,103696,103697,103726],{},[39,103238,103240,103267],{"className":103239,"translate":43},[42],[39,103241,103243],{"className":103242},[47],[49,103244,103245],{"xmlns":51},[53,103246,103247,103264],{},[56,103248,103249,103252,103254,103256,103258,103260,103262],{},[59,103250,103251],{},"η",[133,103253,793],{},[59,103255,84918],{"mathvariant":145},[59,103257,80846],{},[133,103259,136],{"stretchy":135},[59,103261,96518],{},[133,103263,161],{"stretchy":135},[63,103265,103266],{"encoding":65},"\\eta=\\nabla\\psi(\\theta)",[39,103268,103270,103288],{"className":103269,"ariaHidden":70},[69],[39,103271,103273,103276,103279,103282,103285],{"className":103272},[74],[39,103274],{"className":103275,"style":2654},[78],[39,103277,103251],{"className":103278,"style":2114},[83,84],[39,103280],{"className":103281,"style":231},[230],[39,103283,793],{"className":103284},[235],[39,103286],{"className":103287,"style":231},[230],[39,103289,103291,103294,103297,103300,103303,103306],{"className":103290},[74],[39,103292],{"className":103293,"style":245},[78],[39,103295,84918],{"className":103296},[83],[39,103298,80846],{"className":103299,"style":2114},[83,84],[39,103301,136],{"className":103302},[182],[39,103304,96518],{"className":103305,"style":438},[83,84],[39,103307,161],{"className":103308},[263]," 为期望参数，它与 ",[39,103311,103313,103326],{"className":103312,"translate":43},[42],[39,103314,103316],{"className":103315},[47],[49,103317,103318],{"xmlns":51},[53,103319,103320,103324],{},[56,103321,103322],{},[59,103323,96518],{},[63,103325,97466],{"encoding":65},[39,103327,103329],{"className":103328,"ariaHidden":70},[69],[39,103330,103332,103335],{"className":103331},[74],[39,103333],{"className":103334,"style":28824},[78],[39,103336,96518],{"className":103337,"style":438},[83,84]," 之间是 Legendre 变换的关系：",[39,103340,103342,103400],{"className":103341,"translate":43},[42],[39,103343,103345],{"className":103344},[47],[49,103346,103347],{"xmlns":51},[53,103348,103349,103397],{},[56,103350,103351,103357,103359,103361,103363,103365,103375,103377,103383,103385,103387,103389,103391,103393,103395],{},[138,103352,103353,103355],{},[59,103354,80846],{},[133,103356,21884],{},[133,103358,136],{"stretchy":135},[59,103360,103251],{},[133,103362,161],{"stretchy":135},[133,103364,793],{},[613,103366,103367,103373],{},[56,103368,103369,103371],{},[59,103370,21465],{},[133,103372,1690],{},[59,103374,96518],{},[133,103376,136],{"stretchy":135},[138,103378,103379,103381],{},[59,103380,96518],{},[59,103382,93534],{"mathvariant":145},[59,103384,103251],{},[133,103386,5378],{},[59,103388,80846],{},[133,103390,136],{"stretchy":135},[59,103392,96518],{},[133,103394,161],{"stretchy":135},[133,103396,161],{"stretchy":135},[63,103398,103399],{"encoding":65},"\\psi^*(\\eta)=\\sup_\\theta(\\theta^\\top\\eta-\\psi(\\theta))",[39,103401,103403,103456,103547],{"className":103402,"ariaHidden":70},[69],[39,103404,103406,103409,103438,103441,103444,103447,103450,103453],{"className":103405},[74],[39,103407],{"className":103408,"style":245},[78],[39,103410,103412,103415],{"className":103411},[83],[39,103413,80846],{"className":103414,"style":2114},[83,84],[39,103416,103418],{"className":103417},[192],[39,103419,103421],{"className":103420},[196],[39,103422,103424],{"className":103423},[200],[39,103425,103427],{"className":103426,"style":21897},[204],[39,103428,103429,103432],{"style":208},[39,103430],{"className":103431,"style":213},[212],[39,103433,103435],{"className":103434},[217,218,219,220],[39,103436,21884],{"className":103437},[465,220],[39,103439,136],{"className":103440},[182],[39,103442,103251],{"className":103443,"style":2114},[83,84],[39,103445,161],{"className":103446},[263],[39,103448],{"className":103449,"style":231},[230],[39,103451,793],{"className":103452},[235],[39,103454],{"className":103455,"style":231},[230],[39,103457,103459,103462,103503,103506,103535,103538,103541,103544],{"className":103458},[74],[39,103460],{"className":103461,"style":15555},[78],[39,103463,103465,103468],{"className":103464},[1744],[39,103466,21465],{"className":103467},[1744],[39,103469,103471],{"className":103470},[192],[39,103472,103474,103495],{"className":103473},[196,649],[39,103475,103477,103492],{"className":103476},[200],[39,103478,103481],{"className":103479,"style":103480},[204],"height:0.242em;",[39,103482,103483,103486],{"style":63928},[39,103484],{"className":103485,"style":213},[212],[39,103487,103489],{"className":103488},[217,218,219,220],[39,103490,96518],{"className":103491,"style":438},[83,84,220],[39,103493,673],{"className":103494},[672],[39,103496,103498],{"className":103497},[200],[39,103499,103501],{"className":103500,"style":63947},[204],[39,103502],{},[39,103504,136],{"className":103505},[182],[39,103507,103509,103512],{"className":103508},[83],[39,103510,96518],{"className":103511,"style":438},[83,84],[39,103513,103515],{"className":103514},[192],[39,103516,103518],{"className":103517},[196],[39,103519,103521],{"className":103520},[200],[39,103522,103524],{"className":103523,"style":15577},[204],[39,103525,103526,103529],{"style":208},[39,103527],{"className":103528,"style":213},[212],[39,103530,103532],{"className":103531},[217,218,219,220],[39,103533,93534],{"className":103534},[83,220],[39,103536,103251],{"className":103537,"style":2114},[83,84],[39,103539],{"className":103540,"style":461},[230],[39,103542,5378],{"className":103543},[465],[39,103545],{"className":103546,"style":461},[230],[39,103548,103550,103553,103556,103559,103562],{"className":103549},[74],[39,103551],{"className":103552,"style":245},[78],[39,103554,80846],{"className":103555,"style":2114},[83,84],[39,103557,136],{"className":103558},[182],[39,103560,96518],{"className":103561,"style":438},[83,84],[39,103563,74368],{"className":103564},[263],"，且 ",[39,103567,103569,103599],{"className":103568,"translate":43},[42],[39,103570,103572],{"className":103571},[47],[49,103573,103574],{"xmlns":51},[53,103575,103576,103596],{},[56,103577,103578,103580,103586,103588,103590,103592,103594],{},[59,103579,84918],{"mathvariant":145},[138,103581,103582,103584],{},[59,103583,80846],{},[133,103585,21884],{},[133,103587,136],{"stretchy":135},[59,103589,103251],{},[133,103591,161],{"stretchy":135},[133,103593,793],{},[59,103595,96518],{},[63,103597,103598],{"encoding":65},"\\nabla\\psi^*(\\eta)=\\theta",[39,103600,103602,103658],{"className":103601,"ariaHidden":70},[69],[39,103603,103605,103608,103611,103640,103643,103646,103649,103652,103655],{"className":103604},[74],[39,103606],{"className":103607,"style":245},[78],[39,103609,84918],{"className":103610},[83],[39,103612,103614,103617],{"className":103613},[83],[39,103615,80846],{"className":103616,"style":2114},[83,84],[39,103618,103620],{"className":103619},[192],[39,103621,103623],{"className":103622},[196],[39,103624,103626],{"className":103625},[200],[39,103627,103629],{"className":103628,"style":21897},[204],[39,103630,103631,103634],{"style":208},[39,103632],{"className":103633,"style":213},[212],[39,103635,103637],{"className":103636},[217,218,219,220],[39,103638,21884],{"className":103639},[465,220],[39,103641,136],{"className":103642},[182],[39,103644,103251],{"className":103645,"style":2114},[83,84],[39,103647,161],{"className":103648},[263],[39,103650],{"className":103651,"style":231},[230],[39,103653,793],{"className":103654},[235],[39,103656],{"className":103657,"style":231},[230],[39,103659,103661,103664],{"className":103660},[74],[39,103662],{"className":103663,"style":28824},[78],[39,103665,96518],{"className":103666,"style":438},[83,84],"。所以同一个分布有两套坐标，自然参数 ",[39,103669,103671,103684],{"className":103670,"translate":43},[42],[39,103672,103674],{"className":103673},[47],[49,103675,103676],{"xmlns":51},[53,103677,103678,103682],{},[56,103679,103680],{},[59,103681,96518],{},[63,103683,97466],{"encoding":65},[39,103685,103687],{"className":103686,"ariaHidden":70},[69],[39,103688,103690,103693],{"className":103689},[74],[39,103691],{"className":103692,"style":28824},[78],[39,103694,96518],{"className":103695,"style":438},[83,84]," 和期望参数 ",[39,103698,103700,103714],{"className":103699,"translate":43},[42],[39,103701,103703],{"className":103702},[47],[49,103704,103705],{"xmlns":51},[53,103706,103707,103711],{},[56,103708,103709],{},[59,103710,103251],{},[63,103712,103713],{"encoding":65},"\\eta",[39,103715,103717],{"className":103716,"ariaHidden":70},[69],[39,103718,103720,103723],{"className":103719},[74],[39,103721],{"className":103722,"style":2654},[78],[39,103724,103251],{"className":103725,"style":2114},[83,84],"，互为对偶。",[11,103728,103729],{},"现在算两个成员之间的 KL：",[39,103731,103733],{"className":103732,"translate":43},[746],[39,103734,103736,104048],{"className":103735,"translate":43},[42],[39,103737,103739],{"className":103738},[47],[49,103740,103741],{"xmlns":51,"display":755},[53,103742,103743,104045],{},[759,103744,103745,103827,103903,103973],{"rowspacing":761,"columnalign":762,"columnspacing":146},[764,103746,103747,103781],{},[767,103748,103749],{},[770,103750,103751],{"scriptlevel":541,"displaystyle":70},[56,103752,103753,103759,103761,103767,103769,103779],{},[56,103754,103755,103757],{},[59,103756,77380],{"mathvariant":145},[59,103758,77383],{"mathvariant":145},[133,103760,136],{"stretchy":135},[613,103762,103763,103765],{},[59,103764,11],{},[59,103766,96518],{},[59,103768,15389],{"mathvariant":145},[613,103770,103771,103773],{},[59,103772,11],{},[138,103774,103775,103777],{},[59,103776,96518],{},[133,103778,147],{"mathvariant":145,"lspace":146,"rspace":146},[133,103780,161],{"stretchy":135},[767,103782,103783],{},[770,103784,103785],{"scriptlevel":541,"displaystyle":70},[56,103786,103787,103789,103791,103797,103799,103801,103803,103809,103811,103813,103815,103825],{},[56,103788],{},[133,103790,793],{},[613,103792,103793,103795],{},[59,103794,4253],{"mathvariant":421},[59,103796,96518],{},[133,103798,537],{"fence":70,"stretchy":70,"minsize":74068,"maxsize":74068},[59,103800,92676],{},[133,103802,1690],{},[613,103804,103805,103807],{},[59,103806,11],{},[59,103808,96518],{},[133,103810,5378],{},[59,103812,92676],{},[133,103814,1690],{},[613,103816,103817,103819],{},[59,103818,11],{},[138,103820,103821,103823],{},[59,103822,96518],{},[133,103824,147],{"mathvariant":145,"lspace":146,"rspace":146},[133,103826,4280],{"fence":70,"stretchy":70,"minsize":74068,"maxsize":74068},[764,103828,103829,103835],{},[767,103830,103831],{},[770,103832,103833],{"scriptlevel":541,"displaystyle":70},[56,103834],{},[767,103836,103837],{},[770,103838,103839],{"scriptlevel":541,"displaystyle":70},[56,103840,103841,103843,103845,103851,103853,103855,103857,103859,103865,103871,103873,103875,103877,103879,103881,103883,103885,103887,103889,103891,103893,103899,103901],{},[56,103842],{},[133,103844,793],{},[613,103846,103847,103849],{},[59,103848,4253],{"mathvariant":421},[59,103850,96518],{},[133,103852,537],{"fence":70,"stretchy":70,"minsize":74068,"maxsize":74068},[133,103854,136],{"stretchy":135},[59,103856,96518],{},[133,103858,5378],{},[138,103860,103861,103863],{},[59,103862,96518],{},[133,103864,147],{"mathvariant":145,"lspace":146,"rspace":146},[138,103866,103867,103869],{},[133,103868,161],{"stretchy":135},[59,103870,93534],{"mathvariant":145},[59,103872,11772],{},[133,103874,4280],{"fence":70,"stretchy":70,"minsize":74068,"maxsize":74068},[133,103876,5378],{},[133,103878,136],{"fence":70,"stretchy":70,"minsize":74068,"maxsize":74068},[59,103880,80846],{},[133,103882,136],{"stretchy":135},[59,103884,96518],{},[133,103886,161],{"stretchy":135},[133,103888,5378],{},[59,103890,80846],{},[133,103892,136],{"stretchy":135},[138,103894,103895,103897],{},[59,103896,96518],{},[133,103898,147],{"mathvariant":145,"lspace":146,"rspace":146},[133,103900,161],{"stretchy":135},[133,103902,161],{"fence":70,"stretchy":70,"minsize":74068,"maxsize":74068},[764,103904,103905,103911],{},[767,103906,103907],{},[770,103908,103909],{"scriptlevel":541,"displaystyle":70},[56,103910],{},[767,103912,103913],{},[770,103914,103915],{"scriptlevel":541,"displaystyle":70},[56,103916,103917,103919,103921,103923,103925,103927,103933,103939,103941,103943,103945,103947,103949,103951,103953,103955,103957,103959,103961,103963,103965,103971],{},[56,103918],{},[133,103920,793],{},[133,103922,136],{"stretchy":135},[59,103924,96518],{},[133,103926,5378],{},[138,103928,103929,103931],{},[59,103930,96518],{},[133,103932,147],{"mathvariant":145,"lspace":146,"rspace":146},[138,103934,103935,103937],{},[133,103936,161],{"stretchy":135},[59,103938,93534],{"mathvariant":145},[59,103940,84918],{"mathvariant":145},[59,103942,80846],{},[133,103944,136],{"stretchy":135},[59,103946,96518],{},[133,103948,161],{"stretchy":135},[133,103950,5378],{},[59,103952,80846],{},[133,103954,136],{"stretchy":135},[59,103956,96518],{},[133,103958,161],{"stretchy":135},[133,103960,788],{},[59,103962,80846],{},[133,103964,136],{"stretchy":135},[138,103966,103967,103969],{},[59,103968,96518],{},[133,103970,147],{"mathvariant":145,"lspace":146,"rspace":146},[133,103972,161],{"stretchy":135},[764,103974,103975,103981],{},[767,103976,103977],{},[770,103978,103979],{"scriptlevel":541,"displaystyle":70},[56,103980],{},[767,103982,103983],{},[770,103984,103985],{"scriptlevel":541,"displaystyle":70},[56,103986,103987,103989,103991,103993,103995,104001,104003,104005,104007,104009,104011,104013,104015,104017,104019,104021,104023,104029,104031,104037,104039,104041,104043],{},[56,103988],{},[133,103990,793],{},[59,103992,80846],{},[133,103994,136],{"stretchy":135},[138,103996,103997,103999],{},[59,103998,96518],{},[133,104000,147],{"mathvariant":145,"ls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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}",[39,104049,104051],{"className":104050,"ariaHidden":70},[69],[39,104052,104054,104058],{"className":104053},[74],[39,104055],{"className":104056,"style":104057},[78],"height:5.8873em;vertical-align:-2.6937em;",[39,104059,104061],{"className":104060},[83],[39,104062,104064,104252],{"className":104063},[759],[39,104065,104067],{"className":104066},[918],[39,104068,104070,104243],{"className":104069},[196,649],[39,104071,104073,104240],{"className":104072},[200],[39,104074,104077,104213,104222,104231],{"className":104075,"style":104076},[204],"height:3.1937em;",[39,104078,104080,104083],{"style":104079},"top:-5.3437em;",[39,104081],{"className":104082,"style":935},[212],[39,104084,104086,104092,104095,104135,104138,104210],{"className":104085},[83],[39,104087,104089],{"className":104088},[83],[39,104090,77426],{"className":104091},[83,35077],[39,104093,136],{"className":104094},[182],[39,104096,104098,104101],{"className":104097},[83],[39,104099,11],{"className":104100},[83,84],[39,104102,104104],{"className":104103},[192],[39,104105,104107,104127],{"className":104106},[196,649],[39,104108,104110,104124],{"className":104109},[200],[39,104111,104113],{"className":104112,"style":3419},[204],[39,104114,104115,104118],{"style":966},[39,104116],{"className":104117,"style":213},[212],[39,104119,104121],{"className":104120},[217,218,219,220],[39,104122,96518],{"className":104123,"style":438},[83,84,220],[39,104125,673],{"className":104126},[672],[39,104128,104130],{"className":104129},[200],[39,104131,104133],{"className":104132,"style":680},[204],[39,104134],{},[39,104136,15389],{"className":104137},[83],[39,104139,104141,104144],{"className":104140},[83],[39,104142,11],{"className":104143},[83,84],[39,104145,104147],{"className":104146},[192],[39,104148,104150,104202],{"className":104149},[196,649],[39,104151,104153,104199],{"className":104152},[200],[39,104154,104156],{"className":104155,"style":3419},[204],[39,104157,104158,104161],{"style":966},[39,104159],{"className":104160,"style":213},[212],[39,104162,104164],{"className":104163},[217,218,219,220],[39,104165,104167],{"className":104166},[83,220],[39,104168,104170,104173],{"className":104169},[83,220],[39,104171,96518],{"className":104172,"style":438},[83,84,220],[39,104174,104176],{"className":104175},[192],[39,104177,104179],{"className":104178},[196],[39,104180,104182],{"className":104181},[200],[39,104183,104185],{"className":104184,"style":3997},[204],[39,104186,104187,104190],{"style":4000},[39,104188],{"className":104189,"style":4004},[212],[39,104191,104193],{"className":104192},[217,4008,4009,220],[39,104194,104196],{"className":104195},[83,220],[39,104197,147],{"className":104198},[83,220],[39,104200,673],{"className":104201},[672],[39,104203,104205],{"className":104204},[200],[39,104206,104208],{"className":104207,"style":680},[204],[39,104209],{},[39,104211,161],{"className":104212},[263],[39,104214,104216,104219],{"style":104215},"top:-3.7846em;",[39,104217],{"className":104218,"style":935},[212],[39,104220],{"className":104221},[83],[39,104223,104225,104228],{"style":104224},"top:-2.2254em;",[39,104226],{"className":104227,"style":935},[212],[39,104229],{"className":104230},[83],[39,104232,104234,104237],{"style":104233},"top:-0.6663em;",[39,104235],{"className":104236,"style":935},[212],[39,104238],{"className":104239},[83],[39,104241,673],{"className":104242},[672],[39,104244,104246],{"className":104245},[200],[39,104247,104250],{"className":104248,"style":104249},[204],"height:2.6937em;",[39,104251],{},[39,104253,104255],{"className":104254},[1223],[39,104256,104258,105077],{"className":104257},[196,649],[39,104259,104261,105074],{"className":104260},[200],[39,104262,104264,104473,104707,104889],{"className":104263,"style":104076},[204],[39,104265,104266,104269],{"style":104079},[39,104267],{"className":104268,"style":935},[212],[39,104270,104272,104275,104278,104281,104284,104324,104330,104335,104338,104378,104381,104384,104387,104392,104395,104467],{"className":104271},[83],[39,104273],{"className":104274},[83],[39,104276],{"className":104277,"style":231},[230],[39,104279,793],{"className":104280},[235],[39,104282],{"className":104283,"style":231},[230],[39,104285,104287,104290],{"className":104286},[83],[39,104288,4253],{"className":104289},[83,515],[39,104291,104293],{"className":104292},[192],[39,104294,104296,104316],{"className":104295},[196,649],[39,104297,104299,104313],{"className":104298},[200],[39,104300,104302],{"className":104301,"style":3419},[204],[39,104303,104304,104307],{"style":966},[39,104305],{"className":104306,"style":213},[212],[39,104308,104310],{"className":104309},[217,218,219,220],[39,104311,96518],{"className":104312,"style":438},[83,84,220],[39,104314,673],{"className":104315}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处高出的那部分。它有名字：Bregman 散度。",[11,105209,105210,105211,742,105262,105365],{},"拿方差固定为 1 的高斯族验一下。此时 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6447,106448,106451],{"style":5503},[39,106449],{"className":106450,"style":935},[212],[39,106452],{"className":106453,"style":5511},[5510],[39,106455,106456,106459],{"style":75032},[39,106457],{"className":106458,"style":935},[212],[39,106460,106462],{"className":106461},[217,218,219,220],[39,106463,106465],{"className":106464},[83,220],[39,106466,546],{"className":106467},[83,220],[39,106469,673],{"className":106470},[672],[39,106472,106474],{"className":106473},[200],[39,106475,106477],{"className":106476,"style":75054},[204],[39,106478],{},[39,106480],{"className":106481},[263,5463],[39,106483,136],{"className":106484},[182],[39,106486,106488,106491],{"className":106487},[83],[39,106489,96518],{"className":106490,"style":438},[83,84],[39,106492,106494],{"className":106493},[192],[39,106495,106497,106517],{"className":106496},[196,649],[39,106498,106500,106514],{"className":106499},[200],[39,106501,106503],{"className":106502,"style":963},[204],[39,106504,106505,106508],{"style":2937},[39,106506],{"className":106507,"style":213},[212],[39,106509,106511],{"className":106510},[217,218,219,220],[39,106512,17359],{"className":106513},[83,220],[39,106515,673],{"className":106516},[672],[39,106518,106520],{"className":106519},[200],[39,106521,106523],{"className":106522,"style":680},[204],[39,106524],{},[39,106526],{"className":106527,"style":461},[230],[39,106529,5378],{"className":106530},[465],[39,106532],{"className":106533,"style":461},[230],[39,106535,106537,106540,106580],{"className":106536},[74],[39,106538],{"className":106539,"style":84808},[78],[39,106541,106543,106546],{"className":106542},[83],[39,106544,96518],{"className":106545,"style":438},[83,84],[39,106547,106549],{"className":106548},[192],[39,106550,106552,106572],{"className":106551},[196,649],[39,106553,106555,106569],{"className":106554},[200],[39,106556,106558],{"className":106557,"style":963},[204],[39,106559,106560,106563],{"style":2937},[39,106561],{"className":106562,"style":213},[212],[39,106564,106566],{"className":106565},[217,218,219,220],[39,106567,546],{"className":106568},[83,220],[39,106570,673],{"className":106571},[672],[39,106573,106575],{"className":106574},[200],[39,106576,106578],{"className":106577,"style":680},[204],[39,106579],{},[39,106581,106583,106586],{"className":106582},[263],[39,106584,161],{"className":106585},[263],[39,106587,106589],{"className":106588},[192],[39,106590,106592],{"className":106591},[196],[39,106593,106595],{"className":106594},[200],[39,106596,106598],{"className":106597,"style":19710},[204],[39,106599,106600,106603],{"style":208},[39,106601],{"className":106602,"style":213},[212],[39,106604,106606],{"className":106605},[217,218,219,220],[39,106607,17359],{"className":106608},[83,220],"。同一个数。",[11,106611,106612],{},"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。",[21,106614,106616],{"id":106615},"bregman","Bregman",[11,106618,106619,106620,106694,106695,106769],{},"先看一个会让人不舒服的例子。",[39,106621,106623,106649],{"className":106622,"translate":43},[42],[39,106624,106626],{"className":106625},[47],[49,106627,106628],{"xmlns":51},[53,106629,106630,106646],{},[56,106631,106632,106634,106636,106638,106640,106642,106644],{},[59,106633,11],{},[133,106635,793],{},[133,106637,136],{"stretchy":135},[539,106639,546],{},[133,106641,155],{"separator":70},[539,106643,541],{},[133,106645,161],{"stretchy":135},[63,106647,106648],{"encoding":65},"p=(1,0)",[39,106650,106652,106670],{"className":106651,"ariaHidden":70},[69],[39,106653,106655,106658,106661,106664,106667],{"className":106654},[74],[39,106656],{"className":106657,"style":2654},[78],[39,106659,11],{"className":106660},[83,84],[39,106662],{"className":106663,"style":231},[230],[39,106665,793],{"className":106666},[235],[39,106668],{"className":106669,"style":231},[230],[39,106671,106673,106676,106679,106682,106685,106688,106691],{"className":106672},[74],[39,106674],{"className":106675,"style":245},[78],[39,106677,136],{"className":106678},[182],[39,106680,546],{"className":106681},[83],[39,106683,155],{"className":106684},[252],[39,106686],{"className":106687,"style":256},[230],[39,106689,541],{"className":106690},[83],[39,106692,161],{"className":106693},[263]," 是“真相”，",[39,106696,106698,106724],{"className":106697,"translate":43},[42],[39,106699,106701],{"className":106700},[47],[49,106702,106703],{"xmlns":51},[53,106704,106705,106721],{},[56,106706,106707,106709,106711,106713,106715,106717,106719],{},[59,106708,100380],{},[133,106710,793],{},[133,106712,136],{"stretchy":135},[539,106714,95888],{},[133,106716,155],{"separator":70},[539,106718,95888],{},[133,106720,161],{"stretchy":135},[63,106722,106723],{"encoding":65},"q=(0.5,0.5)",[39,106725,106727,106745],{"className":106726,"ariaHidden":70},[69],[39,106728,106730,106733,106736,106739,106742],{"className":106729},[74],[39,106731],{"className":106732,"style":2654},[78],[39,106734,100380],{"className":106735,"style":2114},[83,84],[39,106737],{"className":106738,"style":231},[230],[39,106740,793],{"className":106741},[235],[39,106743],{"className":106744,"style":231},[230],[39,106746,106748,106751,106754,106757,106760,106763,106766],{"className":106747},[74],[39,106749],{"className":106750,"style":245},[78],[39,106752,136],{"className":106753},[182],[39,106755,95888],{"className":106756},[83],[39,106758,155],{"className":106759},[252],[39,106761],{"className":106762,"style":256},[230],[39,106764,95888],{"className":106765},[83],[39,106767,161],{"className":106768},[263]," 是“模型的预测”。",[39,106771,106773],{"className":106772,"translate":43},[746],[39,106774,106776,106845],{"className":106775,"translate":43},[42],[39,106777,106779],{"className":106778},[47],[49,106780,106781],{"xmlns":51,"display":755},[53,106782,106783,106842],{},[56,106784,106785,106791,106793,106795,106797,106799,106801,106803,106805,106807,106809,106811,106814,106816,106818,106824,106826,106828,106830,106832,106834,106836,106838,106840],{},[56,106786,106787,106789],{},[59,106788,77380],{"mathvariant":145},[59,106790,77383],{"mathvariant":145},[133,106792,136],{"stretchy":135},[59,106794,11],{},[59,106796,15389],{"mathvariant":145},[59,106798,100380],{},[133,106800,161],{"stretchy":135},[133,106802,793],{},[59,106804,92676],{},[133,106806,1690],{},[539,106808,17359],{},[133,106810,99937],{},[539,106812,106813],{},"0.693",[133,106815,155],{"separator":70},[230,106817],{"width":10251},[56,106819,106820,106822],{},[59,106821,77380],{"mathvariant":145},[59,106823,77383],{"mathvariant":145},[133,106825,136],{"stretchy":135},[59,106827,100380],{},[59,106829,15389],{"mathvariant":145},[59,106831,11],{},[133,106833,161],{"stretchy":135},[133,106835,793],{},[133,106837,788],{},[59,106839,1696],{"mathvariant":145},[59,106841,2903],{"mathvariant":145},[63,106843,106844],{"encoding":65},"\\mathrm{KL}(p\\|q)=\\log2\\approx0.693,\n\\qquad\n\\mathrm{KL}(q\\|p)=+\\infty .",[39,106846,106848,106884,106910,106958],{"className":106847,"ariaHidden":70},[69],[39,106849,106851,106854,106860,106863,106866,106869,106872,106875,106878,106881],{"className":106850},[74],[39,106852],{"className":106853,"style":245},[78],[39,106855,106857],{"className":106856},[83],[39,106858,77426],{"className":106859},[83,35077],[39,106861,136],{"className":106862},[182],[39,106864,11],{"className":106865},[83,84],[39,106867,15389],{"className":106868},[83],[39,106870,100380],{"className":106871,"style":2114},[83,84],[39,106873,161],{"className":106874},[263],[39,106876],{"className":106877,"style":231},[230],[39,106879,793],{"className":106880},[235],[39,106882],{"className":106883,"style":231},[230],[39,106885,106887,106890,106895,106898,106901,106904,106907],{"className":106886},[74],[39,106888],{"className":106889,"style":51280},[78],[39,106891,92914,106893],{"className":106892},[1744],[39,106894,39209],{"style":35078},[39,106896],{"className":106897,"style":256},[230],[39,106899,17359],{"className":106900},[83],[39,106902],{"className":106903,"style":231},[230],[39,106905,99937],{"className":106906},[235],[39,106908],{"className":106909,"style":231},[230],[39,106911,106913,106916,106919,106922,106925,106928,106934,106937,106940,106943,106946,106949,106952,106955],{"className":106912},[74],[39,106914],{"className":106915,"style":245},[78],[39,106917,106813],{"className":106918},[83],[39,106920,155],{"className":106921},[252],[39,106923],{"className":106924,"style":10787},[230],[39,106926],{"className":106927,"style":256},[230],[39,106929,106931],{"className":106930},[83],[39,106932,77426],{"className":106933},[83,35077],[39,106935,136],{"className":106936},[182],[39,106938,100380],{"className":106939,"style":2114},[83,84],[39,106941,15389],{"className":106942},[83],[39,106944,11],{"className":106945},[83,84],[39,106947,161],{"className":106948},[263],[39,106950],{"className":106951,"style":231},[230],[39,106953,793],{"className":106954},[235],[39,106956],{"className":106957,"style":231},[230],[39,106959,106961,106964,106967],{"className":106960},[74],[39,106962],{"className":106963,"style":73236},[78],[39,106965,788],{"className":106966},[83],[39,106968,106970],{"className":106969},[83],"∞.",[11,106972,106973,106974,107002,107003,107031,107032,107060,107061,107089,107090,107060,107118,107146,107147,107175],{},"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：",[39,106975,106977,106990],{"className":106976,"translate":43},[42],[39,106978,106980],{"className":106979},[47],[49,106981,106982],{"xmlns":51},[53,106983,106984,106988],{},[56,106985,106986],{},[59,106987,11],{},[63,106989,11],{"encoding":65},[39,106991,106993],{"className":106992,"ariaHidden":70},[69],[39,106994,106996,106999],{"className":106995},[74],[39,106997],{"className":106998,"style":2654},[78],[39,107000,11],{"className":107001},[83,84]," 在第二位上放了 0，而 ",[39,107004,107006,107019],{"className":107005,"translate":43},[42],[39,107007,107009],{"className":107008},[47],[49,107010,107011],{"xmlns":51},[53,107012,107013,107017],{},[56,107014,107015],{},[59,107016,100380],{},[63,107018,100380],{"encoding":65},[39,107020,107022],{"className":107021,"ariaHidden":70},[69],[39,107023,107025,107028],{"className":107024},[74],[39,107026],{"className":107027,"style":2654},[78],[39,107029,100380],{"className":107030,"style":2114},[83,84]," 在那里放了 0.5，用 ",[39,107033,107035,107048],{"className":107034,"translate":43},[42],[39,107036,107038],{"className":107037},[47],[49,107039,107040],{"xmlns":51},[53,107041,107042,107046],{},[56,107043,107044],{},[59,107045,11],{},[63,107047,11],{"encoding":65},[39,107049,107051],{"className":107050,"ariaHidden":70},[69],[39,107052,107054,107057],{"className":107053},[74],[39,107055],{"className":107056,"style":2654},[78],[39,107058,11],{"className":107059},[83,84]," 去看 ",[39,107062,107064,107077],{"className":107063,"translate":43},[42],[39,107065,107067],{"className":107066},[47],[49,107068,107069],{"xmlns":51},[53,107070,107071,107075],{},[56,107072,107073],{},[59,107074,100380],{},[63,107076,100380],{"encoding":65},[39,107078,107080],{"className":107079,"ariaHidden":70},[69],[39,107081,107083,107086],{"className":107082},[74],[39,107084],{"className":107085,"style":2654},[78],[39,107087,100380],{"className":107088,"style":2114},[83,84],"，只是多花了 0.69 nat；用 ",[39,107091,107093,107106],{"className":107092,"translate":43},[42],[39,107094,107096],{"className":107095},[47],[49,107097,107098],{"xmlns":51},[53,107099,107100,107104],{},[56,107101,107102],{},[59,107103,100380],{},[63,107105,100380],{"encoding":65},[39,107107,107109],{"className":107108,"ariaHidden":70},[69],[39,107110,107112,107115],{"className":107111},[74],[39,107113],{"className":107114,"style":2654},[78],[39,107116,100380],{"className":107117,"style":2114},[83,84],[39,107119,107121,107134],{"className":107120,"translate":43},[42],[39,107122,107124],{"className":107123},[47],[49,107125,107126],{"xmlns":51},[53,107127,107128,107132],{},[56,107129,107130],{},[59,107131,11],{},[63,107133,11],{"encoding":65},[39,107135,107137],{"className":107136,"ariaHidden":70},[69],[39,107138,107140,107143],{"className":107139},[74],[39,107141],{"className":107142,"style":2654},[78],[39,107144,11],{"className":107145},[83,84],"，等于说“我给了一个 ",[39,107148,107150,107163],{"className":107149,"translate":43},[42],[39,107151,107153],{"className":107152},[47],[49,107154,107155],{"xmlns":51},[53,107156,107157,107161],{},[56,107158,107159],{},[59,107160,11],{},[63,107162,11],{"encoding":65},[39,107164,107166],{"className":107165,"ariaHidden":70},[69],[39,107167,107169,107172],{"className":107168},[74],[39,107170],{"className":107171,"style":2654},[78],[39,107173,11],{"className":107174},[83,84]," 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。",[11,107177,107178,107179,107240,107241,107269,107270,107298],{},"这正是交叉熵写成 ",[39,107180,107182,107210],{"className":107181,"translate":43},[42],[39,107183,107185],{"className":107184},[47],[49,107186,107187],{"xmlns":51},[53,107188,107189,107207],{},[56,107190,107191,107197,107199,107201,107203,107205],{},[56,107192,107193,107195],{},[59,107194,77380],{"mathvariant":145},[59,107196,77383],{"mathvariant":145},[133,107198,136],{"stretchy":135},[59,107200,29625],{},[59,107202,15389],{"mathvariant":145},[59,107204,100380],{},[133,107206,161],{"stretchy":135},[63,107208,107209],{"encoding":65},"\\mathrm{KL}(y\\|q)",[39,107211,107213],{"className":107212,"ariaHidden":70},[69],[39,107214,107216,107219,107225,107228,107231,107234,107237],{"className":107215},[74],[39,107217],{"className":107218,"style":245},[78],[39,107220,107222],{"className":107221},[83],[39,107223,77426],{"className":107224},[83,35077],[39,107226,136],{"className":107227},[182],[39,107229,29625],{"className":107230,"style":2114},[83,84],[39,107232,15389],{"className":107233},[83],[39,107235,100380],{"className":107236,"style":2114},[83,84],[39,107238,161],{"className":107239},[263]," 而不是反过来的原因：",[39,107242,107244,107257],{"className":107243,"translate":43},[42],[39,107245,107247],{"className":107246},[47],[49,107248,107249],{"xmlns":51},[53,107250,107251,107255],{},[56,107252,107253],{},[59,107254,29625],{},[63,107256,29625],{"encoding":65},[39,107258,107260],{"className":107259,"ariaHidden":70},[69],[39,107261,107263,107266],{"className":107262},[74],[39,107264],{"className":107265,"style":2654},[78],[39,107267,29625],{"className":107268,"style":2114},[83,84]," 是真相，模型 ",[39,107271,107273,107286],{"className":107272,"translate":43},[42],[39,107274,107276],{"className":107275},[47],[49,107277,107278],{"xmlns":51},[53,107279,107280,107284],{},[56,107281,107282],{},[59,107283,100380],{},[63,107285,100380],{"encoding":65},[39,107287,107289],{"className":107288,"ariaHidden":70},[69],[39,107290,107292,107295],{"className":107291},[74],[39,107293],{"className":107294,"style":2654},[78],[39,107296,100380],{"className":107297,"style":2114},[83,84]," 不该在真答案上给 0 概率。",[11,107300,107301,107302,107513,107514,107695,107696,107776,107777,107805,107806,75296,107835,107863],{},"再看一个尺度上的例子。Itakura–Saito 散度 ",[39,107303,107305,107365],{"className":107304,"translate":43},[42],[39,107306,107308],{"className":107307},[47],[49,107309,107310],{"xmlns":51},[53,107311,107312,107362],{},[56,107313,107314,107324,107326,107328,107330,107332,107334,107336,107338,107340,107342,107344,107346,107348,107350,107352,107354,107356,107358,107360],{},[613,107315,107316,107318],{},[59,107317,48449],{},[56,107319,107320,107322],{},[59,107321,14373],{"mathvariant":145},[59,107323,61],{"mathvariant":145},[133,107325,136],{"stretchy":135},[59,107327,1755],{},[59,107329,15389],{"mathvariant":145},[59,107331,29625],{},[133,107333,161],{"stretchy":135},[133,107335,793],{},[59,107337,1755],{},[59,107339,45691],{"mathvariant":145},[59,107341,29625],{},[133,107343,5378],{},[59,107345,92676],{},[133,107347,1690],{},[133,107349,136],{"stretchy":135},[59,107351,1755],{},[59,107353,45691],{"mathvariant":145},[59,107355,29625],{},[133,107357,161],{"stretchy":135},[133,107359,5378],{},[539,107361,546],{},[63,107363,107364],{"encoding":65},"D_{\\mathrm{IS}}(x\\|y)=x\u002Fy-\\log(x\u002Fy)-1",[39,107366,107368,107445,107469,107504],{"className":107367,"ariaHidden":70},[69],[39,107369,107371,107374,107421,107424,107427,107430,107433,107436,107439,107442],{"className":107370},[74],[39,107372],{"className":107373,"style":245},[78],[39,107375,107377,107380],{"className":107376},[83],[39,107378,48449],{"className":107379,"style":438},[83,84],[39,107381,107383],{"className":107382},[192],[39,107384,107386,107413],{"className":107385},[196,649],[39,107387,107389,107410],{"className":107388},[200],[39,107390,107392],{"className":107391,"style":62921},[204],[39,107393,107394,107397],{"style":2937},[39,107395],{"className":107396,"style":213},[212],[39,107398,107400],{"className":107399},[217,218,219,220],[39,107401,107403],{"className":107402},[83,220],[39,107404,107406],{"className":107405},[83,220],[39,107407,107409],{"className":107408},[83,35077,220],"IS",[39,107411,673],{"className":107412},[672],[39,107414,107416],{"className":107415},[200],[39,107417,107419],{"className":107418,"style":680},[204],[39,107420],{},[39,107422,136],{"className":107423},[182],[39,107425,1755],{"className":107426},[83,84],[39,107428,15389],{"className":107429},[83],[39,107431,29625],{"className":107432,"style":2114},[83,84],[39,107434,161],{"className":107435},[263],[39,107437],{"className":107438,"style":231},[230],[39,107440,793],{"className":107441},[235],[39,107443],{"className":107444,"style":231},[230],[39,107446,107448,107451,107454,107457,107460,107463,107466],{"className":107447},[74],[39,107449],{"className":107450,"style":245},[78],[39,107452,1755],{"className":107453},[83,84],[39,107455,45691],{"className":107456},[83],[39,107458,29625],{"className":107459,"style":2114},[83,84],[39,107461],{"className":107462,"style":461},[230],[39,107464,5378],{"className":107465},[465],[39,107467],{"className":107468,"style":461},[230],[39,107470,107472,107475,107480,107483,107486,107489,107492,107495,107498,107501],{"className":107471},[74],[39,107473],{"className":107474,"style":245},[78],[39,107476,92914,107478],{"className":107477},[1744],[39,107479,39209],{"style":35078},[39,107481,136],{"className":107482},[182],[39,107484,1755],{"className":107485},[83,84],[39,107487,45691],{"className":107488},[83],[39,107490,29625],{"className":107491,"style":2114},[83,84],[39,107493,161],{"className":107494},[263],[39,107496],{"className":107497,"style":461},[230],[39,107499,5378],{"className":107500},[465],[39,107502],{"className":107503,"style":461},[230],[39,107505,107507,107510],{"className":107506},[74],[39,107508],{"className":107509,"style":2473},[78],[39,107511,546],{"className":107512},[83],"：把音量从 1 减半到 0.5，",[39,107515,107517,107560],{"className":107516,"translate":43},[42],[39,107518,107520],{"className":107519},[47],[49,107521,107522],{"xmlns":51},[53,107523,107524,107557],{},[56,107525,107526,107536,107538,107540,107542,107544,107546,107548,107550,107552,107554],{},[613,107527,107528,107530],{},[59,107529,48449],{},[56,107531,107532,107534],{},[59,107533,14373],{"mathvariant":145},[59,107535,61],{"mathvariant":145},[133,107537,793],{},[539,107539,17359],{},[133,107541,5378],{},[59,107543,92676],{},[133,107545,1690],{},[539,107547,17359],{},[133,107549,5378],{},[539,107551,546],{},[133,107553,793],{},[539,107555,107556],{},"0.307",[63,107558,107559],{"encoding":65},"D_{\\mathrm{IS}}=2-\\log2-1=0.307",[39,107561,107563,107624,107642,107668,107686],{"className":107562,"ariaHidden":70},[69],[39,107564,107566,107569,107615,107618,107621],{"className":107565},[74],[39,107567],{"className":107568,"style":634},[78],[39,107570,107572,107575],{"className":107571},[83],[39,107573,48449],{"className":107574,"style":438},[83,84],[39,107576,107578],{"className":107577},[192],[39,107579,107581,107607],{"className":107580},[196,649],[39,107582,107584,107604],{"className":107583},[200],[39,107585,107587],{"className":107586,"style":62921},[204],[39,107588,107589,107592],{"style":2937},[39,107590],{"className":107591,"style":213},[212],[39,107593,107595],{"className":107594},[217,218,219,220],[39,107596,107598],{"className":107597},[83,220],[39,107599,107601],{"className":107600},[83,220],[39,107602,107409],{"className":107603},[83,35077,220],[39,107605,673],{"className":107606},[672],[39,107608,107610],{"className":107609},[200],[39,107611,107613],{"className":107612,"style":680},[204],[39,107614],{},[39,107616],{"className":107617,"style":231},[230],[39,107619,793],{"className":107620},[235],[39,107622],{"className":107623,"style":231},[230],[39,107625,107627,107630,107633,107636,107639],{"className":107626},[74],[39,107628],{"className":107629,"style":17882},[78],[39,107631,17359],{"className":107632},[83],[39,107634],{"className":107635,"style":461},[230],[39,107637,5378],{"className":107638},[465],[39,107640],{"className":107641,"style":461},[230],[39,107643,107645,107648,107653,107656,107659,107662,107665],{"className":107644},[74],[39,107646],{"className":107647,"style":51280},[78],[39,107649,92914,107651],{"className":107650},[1744],[39,107652,39209],{"style":35078},[39,107654],{"className":107655,"style":256},[230],[39,107657,17359],{"className":107658},[83],[39,107660],{"className":107661,"style":461},[230],[39,107663,5378],{"className":107664},[465],[39,107666],{"className":107667,"style":461},[230],[39,107669,107671,107674,107677,107680,107683],{"className":107670},[74],[39,107672],{"className":107673,"style":2473},[78],[39,107675,546],{"className":107676},[83],[39,107678],{"className":107679,"style":231},[230],[39,107681,793],{"className":107682},[235],[39,107684],{"className":107685,"style":231},[230],[39,107687,107689,107692],{"className":107688},[74],[39,107690],{"className":107691,"style":2473},[78],[39,107693,107556],{"className":107694},[83],"；从 2 减半到 1，",[39,107697,107699,107721],{"className":107698,"translate":43},[42],[39,107700,107702],{"className":107701},[47],[49,107703,107704],{"xmlns":51},[53,107705,107706,107718],{},[56,107707,107708],{},[613,107709,107710,107712],{},[59,107711,48449],{},[56,107713,107714,107716],{},[59,107715,14373],{"mathvariant":145},[59,107717,61],{"mathvariant":145},[63,107719,107720],{"encoding":65},"D_{\\mathrm{IS}}",[39,107722,107724],{"className":107723,"ariaHidden":70},[69],[39,107725,107727,107730],{"className":107726},[74],[39,107728],{"className":107729,"style":634},[78],[39,107731,107733,107736],{"className":107732},[83],[39,107734,48449],{"className":107735,"style":438},[83,84],[39,107737,107739],{"className":107738},[192],[39,107740,107742,107768],{"className":107741},[196,649],[39,107743,107745,107765],{"className":107744},[200],[39,107746,107748],{"className":107747,"style":62921},[204],[39,107749,107750,107753],{"style":2937},[39,107751],{"className":107752,"style":213},[212],[39,107754,107756],{"className":107755},[217,218,219,220],[39,107757,107759],{"className":107758},[83,220],[39,107760,107762],{"className":107761},[83,220],[39,107763,107409],{"className":107764},[83,35077,220],[39,107766,673],{"className":107767},[672],[39,107769,107771],{"className":107770},[200],[39,107772,107774],{"className":107773,"style":680},[204],[39,107775],{}," 还是 ",[39,107778,107780,107793],{"className":107779,"translate":43},[42],[39,107781,107783],{"className":107782},[47],[49,107784,107785],{"xmlns":51},[53,107786,107787,107791],{},[56,107788,107789],{},[539,107790,107556],{},[63,107792,107556],{"encoding":65},[39,107794,107796],{"className":107795,"ariaHidden":70},[69],[39,107797,107799,107802],{"className":107798},[74],[39,107800],{"className":107801,"style":2473},[78],[39,107803,107556],{"className":107804},[83],"。同样的比例变化，距离一样。而欧氏距离给的是 ",[39,107807,107809,107823],{"className":107808,"translate":43},[42],[39,107810,107812],{"className":107811},[47],[49,107813,107814],{"xmlns":51},[53,107815,107816,107821],{},[56,107817,107818],{},[539,107819,107820],{},"0.25",[63,107822,107820],{"encoding":65},[39,107824,107826],{"className":107825,"ariaHidden":70},[69],[39,107827,107829,107832],{"className":107828},[74],[39,107830],{"className":107831,"style":2473},[78],[39,107833,107820],{"className":107834},[83],[39,107836,107838,107851],{"className":107837,"translate":43},[42],[39,107839,107841],{"className":107840},[47],[49,107842,107843],{"xmlns":51},[53,107844,107845,107849],{},[56,107846,107847],{},[539,107848,546],{},[63,107850,546],{"encoding":65},[39,107852,107854],{"className":107853,"ariaHidden":70},[69],[39,107855,107857,107860],{"className":107856},[74],[39,107858],{"className":107859,"style":2473},[78],[39,107861,546],{"className":107862},[83],"，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。",[11,107865,107866],{},"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。",[11,107868,107869,107870,107899,107900,742],{},"一般地，给定凸函数 ",[39,107871,107873,107886],{"className":107872,"translate":43},[42],[39,107874,107876],{"className":107875},[47],[49,107877,107878],{"xmlns":51},[53,107879,107880,107884],{},[56,107881,107882],{},[59,107883,74528],{},[63,107885,74528],{"encoding":65},[39,107887,107889],{"className":107888,"ariaHidden":70},[69],[39,107890,107892,107895],{"className":107891},[74],[39,107893],{"className":107894,"style":51280},[78],[39,107896,74528],{"className":107897,"style":107898},[83,84],"margin-right:0.1076em;"," 和可微点 ",[39,107901,107903,107916],{"className":107902,"translate":43},[42],[39,107904,107906],{"className":107905},[47],[49,107907,107908],{"xmlns":51},[53,107909,107910,107914],{},[56,107911,107912],{},[59,107913,29625],{},[63,107915,29625],{"encoding":65},[39,107917,107919],{"className":107918,"ariaHidden":70},[69],[39,107920,107922,107925],{"className":107921},[74],[39,107923],{"className":107924,"style":2654},[78],[39,107926,29625],{"className":107927,"style":2114},[83,84],[39,107929,107931],{"className":107930,"translate":43},[746],[39,107932,107934,108010],{"className":107933,"translate":43},[42],[39,107935,107937],{"className":107936},[47],[49,107938,107939],{"xmlns":51,"display":755},[53,107940,107941,108007],{},[56,107942,107943,107949,107951,107953,107955,107957,107959,107961,107963,107965,107967,107969,107971,107973,107975,107977,107979,107981,107983,107985,107987,107989,107995,107997,107999,108001,108003,108005],{},[613,107944,107945,107947],{},[59,107946,48449],{},[59,107948,74528],{},[133,107950,136],{"stretchy":135},[59,107952,1755],{},[59,107954,15389],{"mathvariant":145},[59,107956,29625],{},[133,107958,161],{"stretchy":135},[133,107960,793],{},[59,107962,74528],{},[133,107964,136],{"stretchy":135},[59,107966,1755],{},[133,107968,161],{"stretchy":135},[133,107970,5378],{},[59,107972,74528],{},[133,107974,136],{"stretchy":135},[59,107976,29625],{},[133,107978,161],{"stretchy":135},[133,107980,5378],{},[59,107982,84918],{"mathvariant":145},[59,107984,74528],{},[133,107986,136],{"stretchy":135},[59,107988,29625],{},[138,107990,107991,107993],{},[133,107992,161],{"stretchy":135},[59,107994,93534],{"mathvariant":145},[133,107996,136],{"stretchy":135},[59,107998,1755],{},[133,108000,5378],{},[59,108002,29625],{},[133,108004,161],{"stretchy":135},[59,108006,2903],{"mathvariant":145},[63,108008,108009],{"encoding":65},"D_f(x\\|y)=f(x)-f(y)-\\nabla f(y)^\\top(x-y).",[39,108011,108013,108083,108110,108137,108199],{"className":108012,"ariaHidden":70},[69],[39,108014,108016,108019,108059,108062,108065,108068,108071,108074,108077,108080],{"className":108015},[74],[39,108017],{"className":108018,"style":41560},[78],[39,108020,108022,108025],{"className":108021},[83],[39,108023,48449],{"className":108024,"style":438},[83,84],[39,108026,108028],{"className":108027},[192],[39,108029,108031,108051],{"className":108030},[196,649],[39,108032,108034,108048],{"className":108033},[200],[39,108035,108037],{"className":108036,"style":3419},[204],[39,108038,108039,108042],{"style":2937},[39,108040],{"className":108041,"style":213},[212],[39,108043,108045],{"className":108044},[217,218,219,220],[39,108046,74528],{"className":108047,"style":107898},[83,84,220],[39,108049,673],{"className":108050},[672],[39,108052,108054],{"className":108053},[200],[39,108055,108057],{"className":108056,"style":35396},[204],[39,108058],{},[39,108060,136],{"className":108061},[182],[39,108063,1755],{"className":108064},[83,84],[39,108066,15389],{"className":108067},[83],[39,108069,29625],{"className":108070,"style":2114},[83,84],[39,108072,161],{"className":108073},[263],[39,108075],{"className":108076,"style":231},[230],[39,108078,793],{"className":108079},[235],[39,108081],{"className":108082,"style":231},[230],[39,108084,108086,108089,108092,108095,108098,108101,108104,108107],{"className":108085},[74],[39,108087],{"className":108088,"style":245},[78],[39,108090,74528],{"className":108091,"style":107898},[83,84],[39,108093,136],{"className":108094},[182],[39,108096,1755],{"className":108097},[83,84],[39,108099,161],{"className":108100},[263],[39,108102],{"className":108103,"style":461},[230],[39,108105,5378],{"className":108106},[465],[39,108108],{"className":108109,"style":461},[230],[39,108111,108113,108116,108119,108122,108125,108128,108131,108134],{"className":108112},[74],[39,108114],{"className":108115,"style":245},[78],[39,108117,74528],{"className":108118,"style":107898},[83,84],[39,108120,136],{"className":108121},[182],[39,108123,29625],{"className":108124,"style":2114},[83,84],[39,108126,161],{"className":108127},[263],[39,108129],{"className":108130,"style":461},[230],[39,108132,5378],{"className":108133},[465],[39,108135],{"className":108136,"style":461},[230],[39,108138,108140,108143,108146,108149,108152,108155,108184,108187,108190,108193,108196],{"className":108139},[74],[39,108141],{"className":108142,"style":30346},[78],[39,108144,84918],{"className":108145},[83],[39,108147,74528],{"className":108148,"style":107898},[83,84],[39,108150,136],{"className":108151},[182],[39,108153,29625],{"className":108154,"style":2114},[83,84],[39,108156,108158,108161],{"className":108157},[263],[39,108159,161],{"className":108160},[263],[39,108162,108164],{"className":108163},[192],[39,108165,108167],{"className":108166},[196],[39,108168,108170],{"className":108169},[200],[39,108171,108173],{"className":108172,"style":3389},[204],[39,108174,108175,108178],{"style":1027},[39,108176],{"className":108177,"style":213},[212],[39,108179,108181],{"className":108180},[217,218,219,220],[39,108182,93534],{"className":108183},[83,220],[39,108185,136],{"className":108186},[182],[39,108188,1755],{"className":108189},[83,84],[39,108191],{"className":108192,"style":461},[230],[39,108194,5378],{"className":108195},[465],[39,108197],{"className":108198,"style":461},[230],[39,108200,108202,108205,108208,108211],{"className":108201},[74],[39,108203],{"className":108204,"style":245},[78],[39,108206,29625],{"className":108207,"style":2114},[83,84],[39,108209,161],{"className":108210},[263],[39,108212,2903],{"className":108213},[83],[11,108215,108216,108217,108245,108246,108274,108275,108303,108304,108332,108333,108384],{},"几何上，这是 ",[39,108218,108220,108233],{"className":108219,"translate":43},[42],[39,108221,108223],{"className":108222},[47],[49,108224,108225],{"xmlns":51},[53,108226,108227,108231],{},[56,108228,108229],{},[59,108230,74528],{},[63,108232,74528],{"encoding":65},[39,108234,108236],{"className":108235,"ariaHidden":70},[69],[39,108237,108239,108242],{"className":108238},[74],[39,108240],{"className":108241,"style":51280},[78],[39,108243,74528],{"className":108244,"style":107898},[83,84]," 的图像在 ",[39,108247,108249,108262],{"className":108248,"translate":43},[42],[39,108250,108252],{"className":108251},[47],[49,108253,108254],{"xmlns":51},[53,108255,108256,108260],{},[56,108257,108258],{},[59,108259,1755],{},[63,108261,1755],{"encoding":65},[39,108263,108265],{"className":108264,"ariaHidden":70},[69],[39,108266,108268,108271],{"className":108267},[74],[39,108269],{"className":108270,"style":290},[78],[39,108272,1755],{"className":108273},[83,84]," 处比 ",[39,108276,108278,108291],{"className":108277,"translate":43},[42],[39,108279,108281],{"className":108280},[47],[49,108282,108283],{"xmlns":51},[53,108284,108285,108289],{},[56,108286,108287],{},[59,108288,29625],{},[63,108290,29625],{"encoding":65},[39,108292,108294],{"className":108293,"ariaHidden":70},[69],[39,108295,108297,108300],{"className":108296},[74],[39,108298],{"className":108299,"style":2654},[78],[39,108301,29625],{"className":108302,"style":2114},[83,84]," 处的切平面高出多少。",[39,108305,108307,108320],{"className":108306,"translate":43},[42],[39,108308,108310],{"className":108309},[47],[49,108311,108312],{"xmlns":51},[53,108313,108314,108318],{},[56,108315,108316],{},[59,108317,74528],{},[63,108319,74528],{"encoding":65},[39,108321,108323],{"className":108322,"ariaHidden":70},[69],[39,108324,108326,108329],{"className":108325},[74],[39,108327],{"className":108328,"style":51280},[78],[39,108330,74528],{"className":108331,"style":107898},[83,84]," 凸所以非负，严格凸所以只在 ",[39,108334,108336,108354],{"className":108335,"translate":43},[42],[39,108337,108339],{"className":108338},[47],[49,108340,108341],{"xmlns":51},[53,108342,108343,108351],{},[56,108344,108345,108347,108349],{},[59,108346,1755],{},[133,108348,793],{},[59,108350,29625],{},[63,108352,108353],{"encoding":65},"x=y",[39,108355,108357,108375],{"className":108356,"ariaHidden":70},[69],[39,108358,108360,108363,108366,108369,108372],{"className":108359},[74],[39,108361],{"className":108362,"style":290},[78],[39,108364,1755],{"className":108365},[83,84],[39,108367],{"className":108368,"style":231},[230],[39,108370,793],{"className":108371},[235],[39,108373],{"className":108374,"style":231},[230],[39,108376,108378,108381],{"className":108377},[74],[39,108379],{"className":108380,"style":2654},[78],[39,108382,29625],{"className":108383,"style":2114},[83,84]," 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。",[11,108386,108387,108388,108570,108571,108738,108739,108956,108957,108738,109018,109194,109195,109378],{},"几个我常用的：",[39,108389,108391,108431],{"className":108390,"translate":43},[42],[39,108392,108394],{"className":108393},[47],[49,108395,108396],{"xmlns":51},[53,108397,108398,108428],{},[56,108399,108400,108402,108404,108406,108408,108410,108418,108420,108422],{},[59,108401,74528],{},[133,108403,136],{"stretchy":135},[59,108405,1755],{},[133,108407,161],{"stretchy":135},[133,108409,793],{},[770,108411,108412],{"scriptlevel":541,"displaystyle":135},[5362,108413,108414,108416],{},[539,108415,546],{},[539,108417,17359],{},[59,108419,15389],{"mathvariant":145},[59,108421,1755],{},[138,108423,108424,108426],{},[59,108425,15389],{"mathvariant":145},[539,108427,17359],{},[63,108429,108430],{"encoding":65},"f(x)=\\tfrac12\\|x\\|^2",[39,108432,108434,108461],{"className":108433,"ariaHidden":70},[69],[39,108435,108437,108440,108443,108446,108449,108452,108455,108458],{"className":108436},[74],[39,108438],{"className":108439,"style":245},[78],[39,108441,74528],{"className":108442,"style":107898},[83,84],[39,108444,136],{"className":108445},[182],[39,108447,1755],{"className":108448},[83,84],[39,108450,161],{"className":108451},[263],[39,108453],{"className":108454,"style":231},[230],[39,108456,793],{"className":108457},[235],[39,108459],{"className":108460,"style":231},[230],[39,108462,108464,108467,108535,108538,108541],{"className":108463},[74],[39,108465],{"className":108466,"style":74987},[78],[39,108468,108470,108473,108532],{"className":108469},[83],[39,108471],{"className":108472},[182,5463],[39,108474,108476],{"className":108475},[5362],[39,108477,108479,108524],{"className":108478},[196,649],[39,108480,108482,108521],{"className":108481},[200],[39,108483,108485,108499,108507],{"className":108484,"style":75006},[204],[39,108486,108487,108490],{"style":75009},[39,108488],{"className":108489,"style":935},[212],[39,108491,108493],{"className":108492},[217,218,219,220],[39,108494,108496],{"className":108495},[83,220],[39,108497,17359],{"className":108498},[83,220],[39,108500,108501,108504],{"style":5503},[39,108502],{"className":108503,"style":935},[212],[39,108505],{"className":108506,"style":5511},[5510],[39,108508,108509,108512],{"style":75032},[39,108510],{"className":108511,"style":935},[212],[39,108513,108515],{"className":108514},[217,218,219,220],[39,108516,108518],{"className":108517},[83,220],[39,108519,546],{"className":108520},[83,220],[39,108522,673],{"className":108523},[672],[39,108525,108527],{"className":108526},[200],[39,108528,108530],{"className":108529,"style":75054},[204],[39,108531],{},[39,108533],{"className":108534},[263,5463],[39,108536,15389],{"className":108537},[83],[39,108539,1755],{"className":108540},[83,84],[39,108542,108544,108547],{"className":108543},[83],[39,108545,15389],{"className":108546},[83],[39,108548,108550],{"className":108549},[192],[39,108551,108553],{"className":108552},[196],[39,108554,108556],{"className":108555},[200],[39,108557,108559],{"className":108558,"style":19710},[204],[39,108560,108561,108564],{"style":208},[39,108562],{"className":108563,"style":213},[212],[39,108565,108567],{"className":108566},[217,218,219,220],[39,108568,17359],{"className":108569},[83,220]," 给 ",[39,108572,108574,108608],{"className":108573,"translate":43},[42],[39,108575,108577],{"className":108576},[47],[49,108578,108579],{"xmlns":51},[53,108580,108581,108605],{},[56,108582,108583,108591,108593,108595,108597,108599],{},[770,108584,108585],{"scriptlevel":541,"displaystyle":135},[5362,108586,108587,108589],{},[539,108588,546],{},[539,108590,17359],{},[59,108592,15389],{"mathvariant":145},[59,108594,1755],{},[133,108596,5378],{},[59,108598,29625],{},[138,108600,108601,108603],{},[59,108602,15389],{"mathvariant":145},[539,108604,17359],{},[63,108606,108607],{"encoding":65},"\\tfrac12\\|x-y\\|^2",[39,108609,108611,108700],{"className":108610,"ariaHidden":70},[69],[39,108612,108614,108617,108685,108688,108691,108694,108697],{"className":108613},[74],[39,108615],{"className":108616,"style":74987},[78],[39,108618,108620,108623,108682],{"className":108619},[83],[39,108621],{"className":108622},[182,5463],[39,108624,108626],{"className":108625},[5362],[39,108627,108629,108674],{"className":108628},[196,649],[39,108630,108632,108671],{"className":108631},[200],[39,108633,108635,108649,108657],{"className":108634,"style":75006},[204],[39,108636,108637,108640],{"style":75009},[39,108638],{"className":108639,"style":935},[212],[39,108641,108643],{"className":108642},[217,218,219,220],[39,108644,108646],{"className":108645},[83,220],[39,108647,17359],{"className":108648},[83,220],[39,108650,108651,108654],{"style":5503},[39,108652],{"className":108653,"style":935},[212],[39,108655],{"className":108656,"style":5511},[5510],[39,108658,108659,108662],{"style":75032},[39,108660],{"className":108661,"style":935},[212],[39,108663,108665],{"className":108664},[217,218,219,220],[39,108666,108668],{"className":108667},[83,220],[39,108669,546],{"className":108670},[83,220],[39,108672,673],{"className":108673},[672],[39,108675,108677],{"className":108676},[200],[39,108678,108680],{"className":108679,"style":75054},[204],[39,108681],{},[39,108683],{"className":108684},[263,5463],[39,108686,15389],{"className":108687},[83],[39,108689,1755],{"className":108690},[83,84],[39,108692],{"className":108693,"style":461},[230],[39,108695,5378],{"className":108696},[465],[39,108698],{"className":108699,"style":461},[230],[39,108701,108703,108706,108709],{"className":108702},[74],[39,108704],{"className":108705,"style":84808},[78],[39,108707,29625],{"className":108708,"style":2114},[83,84],[39,108710,108712,108715],{"className":108711},[83],[39,108713,15389],{"className":108714},[83],[39,108716,108718],{"className":108717},[192],[39,108719,108721],{"className":108720},[196],[39,108722,108724],{"className":108723},[200],[39,108725,108727],{"className":108726,"style":19710},[204],[39,108728,108729,108732],{"style":208},[39,108730],{"className":108731,"style":213},[212],[39,108733,108735],{"className":108734},[217,218,219,220],[39,108736,17359],{"className":108737},[83,220],"；",[39,108740,108742,108786],{"className":108741,"translate":43},[42],[39,108743,108745],{"className":108744},[47],[49,108746,108747],{"xmlns":51},[53,108748,108749,108783],{},[56,108750,108751,108753,108755,108757,108759,108761,108767,108773,108775,108777],{},[59,108752,74528],{},[133,108754,136],{"stretchy":135},[59,108756,11],{},[133,108758,161],{"stretchy":135},[133,108760,793],{},[613,108762,108763,108765],{},[133,108764,3191],{},[59,108766,13377],{},[613,108768,108769,108771],{},[59,108770,11],{},[59,108772,13377],{},[59,108774,92676],{},[133,108776,1690],{},[613,108778,108779,108781],{},[59,108780,11],{},[59,108782,13377],{},[63,108784,108785],{"encoding":65},"f(p)=\\sum_ip_i\\log p_i",[39,108787,108789,108816],{"className":108788,"ariaHidden":70},[69],[39,108790,108792,108795,108798,108801,108804,108807,108810,108813],{"className":108791},[74],[39,108793],{"className":108794,"style":245},[78],[39,108796,74528],{"className":108797,"style":107898},[83,84],[39,108799,136],{"className":108800},[182],[39,108802,11],{"className":108803},[83,84],[39,108805,161],{"className":108806},[263],[39,108808],{"className":108809,"style":231},[230],[39,108811,793],{"className":108812},[235],[39,108814],{"className":108815,"style":231},[230],[39,108817,108819,108822,108862,108865,108905,108908,108913,108916],{"className":108818},[74],[39,108820],{"className":108821,"style":94090},[78],[39,108823,108825,108828],{"className":108824},[1744],[39,108826,3191],{"className":108827,"style":3953},[1744,3339,3952],[39,108829,108831],{"className":108830},[192],[39,108832,108834,108854],{"className":108833},[196,649],[39,108835,108837,108851],{"className":108836},[200],[39,108838,108840],{"className":108839,"style":94115},[204],[39,108841,108842,108845],{"style":3969},[39,108843],{"className":108844,"style":213},[212],[39,108846,108848],{"className":108847},[217,218,219,220],[39,108849,13377],{"className":108850},[83,84,220],[39,108852,673],{"className":108853},[672],[39,108855,108857],{"className":108856},[200],[39,108858,108860],{"className":108859,"style":4025},[204],[39,108861],{},[39,108863],{"className":108864,"style":256},[230],[39,108866,108868,108871],{"className":108867},[83],[39,108869,11],{"className":108870},[83,84],[39,108872,108874],{"className":108873},[192],[39,108875,108877,108897],{"className":108876},[196,649],[39,108878,108880,108894],{"className":108879},[200],[39,108881,108883],{"className":108882,"style":13604},[204],[39,108884,108885,108888],{"style":966},[39,108886],{"className":108887,"style":213},[212],[39,108889,108891],{"className":108890},[217,218,219,220],[39,108892,13377],{"className":108893},[83,84,220],[39,108895,673],{"className":108896},[672],[39,108898,108900],{"className":108899},[200],[39,108901,108903],{"className":108902,"style":680},[204],[39,108904],{},[39,108906],{"className":108907,"style":256},[230],[39,108909,92914,108911],{"className":108910},[1744],[39,108912,39209],{"style":35078},[39,108914],{"className":108915,"style":256},[230],[39,108917,108919,108922],{"className":108918},[83],[39,108920,11],{"className":108921},[83,84],[39,108923,108925],{"className":108924},[192],[39,108926,108928,108948],{"className":108927},[196,649],[39,108929,108931,108945],{"className":108930},[200],[39,108932,108934],{"className":108933,"style":13604},[204],[39,108935,108936,108939],{"style":966},[39,108937],{"className":108938,"style":213},[212],[39,108940,108942],{"className":108941},[217,218,219,220],[39,108943,13377],{"className":108944},[83,84,220],[39,108946,673],{"className":108947},[672],[39,108949,108951],{"className":108950},[200],[39,108952,108954],{"className":108953,"style":680},[204],[39,108955],{},"（负熵）给 ",[39,108958,108960,108988],{"className":108959,"translate":43},[42],[39,108961,108963],{"className":108962},[47],[49,108964,108965],{"xmlns":51},[53,108966,108967,108985],{},[56,108968,108969,108975,108977,108979,108981,108983],{},[56,108970,108971,108973],{},[59,108972,77380],{"mathvariant":145},[59,108974,77383],{"mathvariant":145},[133,108976,136],{"stretchy":135},[59,108978,11],{},[59,108980,15389],{"mathvariant":145},[59,108982,100380],{},[133,108984,161],{"stretchy":135},[63,108986,108987],{"encoding":65},"\\mathrm{KL}(p\\|q)",[39,108989,108991],{"className":108990,"ariaHidden":70},[69],[39,108992,108994,108997,109003,109006,109009,109012,109015],{"className":108993},[74],[39,108995],{"className":108996,"style":245},[78],[39,108998,109000],{"className":108999},[83],[39,109001,77426],{"className":109002},[83,35077],[39,109004,136],{"className":109005},[182],[39,109007,11],{"className":109008},[83,84],[39,109010,15389],{"className":109011},[83],[39,109013,100380],{"className":109014,"style":2114},[83,84],[39,109016,161],{"className":109017},[263],[39,109019,109021,109061],{"className":109020,"translate":43},[42],[39,109022,109024],{"className":109023},[47],[49,109025,109026],{"xmlns":51},[53,109027,109028,109058],{},[56,109029,109030,109032,109034,109036,109038,109040,109042,109048,109050,109052],{},[59,109031,74528],{},[133,109033,136],{"stretchy":135},[59,109035,11],{},[133,109037,161],{"stretchy":135},[133,109039,793],{},[133,109041,5378],{},[613,109043,109044,109046],{},[133,109045,3191],{},[59,109047,13377],{},[59,109049,92676],{},[133,109051,1690],{},[613,109053,109054,109056],{},[59,109055,11],{},[59,109057,13377],{},[63,109059,109060],{"encoding":65},"f(p)=-\\sum_i\\log p_i",[39,109062,109064,109091],{"className":109063,"ariaHidden":70},[69],[39,109065,109067,109070,109073,109076,109079,109082,109085,109088],{"className":109066},[74],[39,109068],{"className":109069,"style":245},[78],[39,109071,74528],{"className":109072,"style":107898},[83,84],[39,109074,136],{"className":109075},[182],[39,109077,11],{"className":109078},[83,84],[39,109080,161],{"className":109081},[263],[39,109083],{"className":109084,"style":231},[230],[39,109086,793],{"className":109087},[235],[39,109089],{"className":109090,"style":231},[230],[39,109092,109094,109097,109100,109103,109143,109146,109151,109154],{"className":109093},[74],[39,109095],{"className":109096,"style":94090},[78],[39,109098,5378],{"className":109099},[83],[39,109101],{"className":109102,"style":256},[230],[39,109104,109106,109109],{"className":109105},[1744],[39,109107,3191],{"className":109108,"style":3953},[1744,3339,3952],[39,109110,109112],{"className":109111},[192],[39,109113,109115,109135],{"className":109114},[196,649],[39,109116,109118,109132],{"className":109117},[200],[39,109119,109121],{"className":109120,"style":94115},[204],[39,109122,109123,109126],{"style":3969},[39,109124],{"className":109125,"style":213},[212],[39,109127,109129],{"className":109128},[217,218,219,220],[39,109130,13377],{"className":109131},[83,84,220],[39,109133,673],{"className":109134},[672],[39,109136,109138],{"className":109137},[200],[39,109139,109141],{"className":109140,"style":4025},[204],[39,109142],{},[39,109144],{"className":109145,"style":256},[230],[39,109147,92914,109149],{"className":109148},[1744],[39,109150,39209],{"style":35078},[39,109152],{"className":109153,"style":256},[230],[39,109155,109157,109160],{"className":109156},[83],[39,109158,11],{"className":109159},[83,84],[39,109161,109163],{"className":109162},[192],[39,109164,109166,109186],{"className":109165},[196,649],[39,109167,109169,109183],{"className":109168},[200],[39,109170,109172],{"className":109171,"style":13604},[204],[39,109173,109174,109177],{"style":966},[39,109175],{"className":109176,"style":213},[212],[39,109178,109180],{"className":109179},[217,218,219,220],[39,109181,13377],{"className":109182},[83,84,220],[39,109184,673],{"className":109185},[672],[39,109187,109189],{"className":109188},[200],[39,109190,109192],{"className":109191,"style":680},[204],[39,109193],{}," 给 Itakura–Saito；",[39,109196,109198,109238],{"className":109197,"translate":43},[42],[39,109199,109201],{"className":109200},[47],[49,109202,109203],{"xmlns":51},[53,109204,109205,109235],{},[56,109206,109207,109209,109211,109213,109215,109217,109225,109231,109233],{},[59,109208,74528],{},[133,109210,136],{"stretchy":135},[59,109212,1755],{},[133,109214,161],{"stretchy":135},[133,109216,793],{},[770,109218,109219],{"scriptlevel":541,"displaystyle":135},[5362,109220,109221,109223],{},[539,109222,546],{},[539,109224,17359],{},[138,109226,109227,109229],{},[59,109228,1755],{},[59,109230,93534],{"mathvariant":145},[59,109232,101],{},[59,109234,1755],{},[63,109236,109237],{"encoding":65},"f(x)=\\tfrac12x^\\top Ax",[39,109239,109241,109268],{"className":109240,"ariaHidden":70},[69],[39,109242,109244,109247,109250,109253,109256,109259,109262,109265],{"className":109243},[74],[39,109245],{"className":109246,"style":245},[78],[39,109248,74528],{"className":109249,"style":107898},[83,84],[39,109251,136],{"className":109252},[182],[39,109254,1755],{"className":109255},[83,84],[39,109257,161],{"className":109258},[263],[39,109260],{"className":109261,"style":231},[230],[39,109263,793],{"className":109264},[235],[39,109266],{"className":109267,"style":231},[230],[39,109269,109271,109275,109343,109372,109375],{"className":109270},[74],[39,109272],{"className":109273,"style":109274},[78],"height:1.1941em;vertical-align:-0.345em;",[39,109276,109278,109281,109340],{"className":109277},[83],[39,109279],{"className":109280},[182,5463],[39,109282,109284],{"className":109283},[5362],[39,109285,109287,109332],{"className":109286},[196,649],[39,109288,109290,109329],{"className":109289},[200],[39,109291,109293,109307,109315],{"className":109292,"style":75006},[204],[39,109294,109295,109298],{"style":75009},[39,109296],{"className":109297,"style":935},[212],[39,109299,109301],{"className":109300},[217,218,219,220],[39,109302,109304],{"className":109303},[83,220],[39,109305,17359],{"className":109306},[83,220],[39,109308,109309,109312],{"style":5503},[39,109310],{"className":109311,"style":935},[212],[39,109313],{"className":109314,"style":5511},[5510],[39,109316,109317,109320],{"style":75032},[39,109318],{"className":109319,"style":935},[212],[39,109321,109323],{"className":109322},[217,218,219,220],[39,109324,109326],{"className":109325},[83,220],[39,109327,546],{"className":109328},[83,220],[39,109330,673],{"className":109331},[672],[39,109333,109335],{"className":109334},[200],[39,109336,109338],{"className":109337,"style":75054},[204],[39,109339],{},[39,109341],{"className":109342},[263,5463],[39,109344,109346,109349],{"className":109345},[83],[39,109347,1755],{"className":109348},[83,84],[39,109350,109352],{"className":109351},[192],[39,109353,109355],{"className":109354},[196],[39,109356,109358],{"className":109357},[200],[39,109359,109361],{"className":109360,"style":15577},[204],[39,109362,109363,109366],{"style":208},[39,109364],{"className":109365,"style":213},[212],[39,109367,109369],{"className":109368},[217,218,219,220],[39,109370,93534],{"className":109371},[83,220],[39,109373,101],{"className":109374},[83,84],[39,109376,1755],{"className":109377},[83,84]," 给 Mahalanobis 距离。",[11,109380,109381],{},"它在机器学习里出现的频率比多数人以为的高。交叉熵",[39,109383,109385],{"className":109384,"translate":43},[746],[39,109386,109388,109505],{"className":109387,"translate":43},[42],[39,109389,109391],{"className":109390},[47],[49,109392,109393],{"xmlns":51,"display":755},[53,109394,109395,109502],{},[56,109396,109397,109403,109405,109407,109409,109411,109413,109415,109417,109423,109429,109431,109433,109439,109441,109447,109449,109451,109453,109455,109457,109459,109462,109464,109466,109468,109470,109480,109482,109484,109486,109488,109490,109492,109494,109496,109498,109500],{},[56,109398,109399,109401],{},[59,109400,48444],{"mathvariant":145},[59,109402,4253],{"mathvariant":145},[133,109404,136],{"stretchy":135},[59,109406,29625],{},[133,109408,155],{"separator":70},[59,109410,100380],{},[133,109412,161],{"stretchy":135},[133,109414,793],{},[133,109416,5378],{},[6556,109418,109419,109421],{},[133,109420,3191],{},[59,109422,13377],{},[613,109424,109425,109427],{},[59,109426,29625],{},[59,109428,13377],{},[59,109430,92676],{},[133,109432,1690],{},[613,109434,109435,109437],{},[59,109436,100380],{},[59,109438,13377],{},[133,109440,793],{},[56,109442,109443,109445],{},[59,109444,77380],{"mathvariant":145},[59,109446,77383],{"mathvariant":145},[133,109448,136],{"stretchy":135},[59,109450,29625],{},[59,109452,15389],{"mathvariant":145},[59,109454,100380],{},[133,109456,161],{"stretchy":135},[133,109458,788],{},[59,109460,109461],{},"H",[133,109463,136],{"stretchy":135},[59,109465,29625],{},[133,109467,161],{"stretchy":135},[133,109469,793],{},[613,109471,109472,109474],{},[59,109473,48449],{},[56,109475,109476,109478],{},[133,109477,5378],{},[59,109479,109461],{},[133,109481,136],{"stretchy":135},[59,109483,29625],{},[59,109485,15389],{"mathvariant":145},[59,109487,100380],{},[133,109489,161],{"stretchy":135},[133,109491,788],{},[59,109493,109461],{},[133,109495,136],{"stretchy":135},[59,109497,29625],{},[133,109499,161],{"stretchy":135},[133,109501,155],{"separator":70},[63,109503,109504],{"encoding":65},"\\mathrm{CE}(y,q)=-\\sum_iy_i\\log q_i=\\mathrm{KL}(y\\|q)+H(y)=D_{-H}(y\\|q)+H(y),",[39,109506,109508,109548,109707,109743,109771,109847],{"className":109507,"ariaHidden":70},[69],[39,109509,109511,109514,109521,109524,109527,109530,109533,109536,109539,109542,109545],{"className":109510},[74],[39,109512],{"className":109513,"style":245},[78],[39,109515,109517],{"className":109516},[83],[39,109518,109520],{"className":109519},[83,35077],"CE",[39,109522,136],{"className":109523},[182],[39,109525,29625],{"className":109526,"style":2114},[83,84],[39,109528,155],{"className":109529},[252],[39,109531],{"className":109532,"style":256},[230],[39,109534,100380],{"className":109535,"style":2114},[83,84],[39,109537,161],{"className":109538},[263],[39,109540],{"className":109541,"style":231},[230],[39,109543,793],{"className":109544},[235],[39,109546],{"className":109547,"style":231},[230],[39,109549,109551,109554,109557,109560,109604,109607,109647,109650,109655,109658,109698,109701,109704],{"className":109550},[74],[39,109552],{"className":109553,"style":83027},[78],[39,109555,5378],{"className":109556},[83],[39,109558],{"className":109559,"style":256},[230],[39,109561,109563],{"className":109562},[1744,3294],[39,109564,109566,109596],{"className":109565},[196,649],[39,109567,109569,109593],{"className":109568},[200],[39,109570,109572,109583],{"className":109571,"style":6670},[204],[39,109573,109574,109577],{"style":13926},[39,109575],{"className":109576,"style":3311},[212],[39,109578,109580],{"className":109579},[217,218,219,220],[39,109581,13377],{"className":109582},[83,84,220],[39,109584,109585,109588],{"style":3330},[39,109586],{"className":109587,"style":3311},[212],[39,109589,109590],{},[39,109591,3191],{"className":109592},[1744,3339,3340],[39,109594,673],{"className":109595},[672],[39,109597,109599],{"className":109598},[200],[39,109600,109602],{"className":109601,"style":79128},[204],[39,109603],{},[39,109605],{"className":109606,"style":256},[230],[39,109608,109610,109613],{"className":109609},[83],[39,109611,29625],{"className":109612,"style":2114},[83,84],[39,109614,109616],{"className":109615},[192],[39,109617,109619,109639],{"className":109618},[196,649],[39,109620,109622,109636],{"className":109621},[200],[39,109623,109625],{"className":109624,"style":13604},[204],[39,109626,109627,109630],{"style":41316},[39,109628],{"className":109629,"style":213},[212],[39,109631,109633],{"className":109632},[217,218,219,220],[39,109634,13377],{"className":109635},[83,84,220],[39,109637,673],{"className":109638},[672],[39,109640,109642],{"className":109641},[200],[39,109643,109645],{"className":109644,"style":680},[204],[39,109646],{},[39,109648],{"className":109649,"style":256},[230],[39,109651,92914,109653],{"className":109652},[1744],[39,109654,39209],{"style":35078},[39,109656],{"className":109657,"style":256},[230],[39,109659,109661,109664],{"className":109660},[83],[39,109662,100380],{"className":109663,"style":2114},[83,84],[39,109665,109667],{"className":109666},[192],[39,109668,109670,109690],{"className":109669},[196,649],[39,109671,109673,109687],{"className":109672},[200],[39,109674,109676],{"className":109675,"style":13604},[204],[39,109677,109678,109681],{"style":41316},[39,109679],{"className":109680,"style":213},[212],[39,109682,109684],{"className":109683},[217,218,219,220],[39,109685,13377],{"className":109686},[83,84,220],[39,109688,673],{"className":109689},[672],[39,109691,109693],{"className":109692},[200],[39,109694,109696],{"className":109695,"style":680},[204],[39,109697],{},[39,109699],{"className":109700,"style":231},[230],[39,109702,793],{"className":109703},[235],[39,109705],{"className":109706,"style":231},[230],[39,109708,109710,109713,109719,109722,109725,109728,109731,109734,109737,109740],{"className":109709},[74],[39,109711],{"className":109712,"style":245},[78],[39,109714,109716],{"className":109715},[83],[39,109717,77426],{"className":109718},[83,35077],[39,109720,136],{"className":109721},[182],[39,109723,29625],{"className":109724,"style":2114},[83,84],[39,109726,15389],{"className":109727},[83],[39,109729,100380],{"className":109730,"style":2114},[83,84],[39,109732,161],{"className":109733},[263],[39,109735],{"className":109736,"style":461},[230],[39,109738,788],{"className":109739},[465],[39,109741],{"className":109742,"style":461},[230],[39,109744,109746,109749,109753,109756,109759,109762,109765,109768],{"className":109745},[74],[39,109747],{"className":109748,"style":245},[78],[39,109750,109461],{"className":109751,"style":109752},[83,84],"margin-right:0.0813em;",[39,109754,136],{"className":109755},[182],[39,109757,29625],{"className":109758,"style":2114},[83,84],[39,109760,161],{"className":109761},[263],[39,109763],{"className":109764,"style":231},[230],[39,109766,793],{"className":109767},[235],[39,109769],{"className":109770,"style":231},[230],[39,109772,109774,109777,109823,109826,109829,109832,109835,109838,109841,109844],{"className":109773},[74],[39,109775],{"className":109776,"style":245},[78],[3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散度，",[39,109948,109950],{"className":109949,"translate":43},[746],[39,109951,109953,110029],{"className":109952,"translate":43},[42],[39,109954,109956],{"className":109955},[47],[49,109957,109958],{"xmlns":51,"display":755},[53,109959,109960,110026],{},[56,109961,109962,109964,109966,109976,109978,109984,109990,109992,109998,110000,110002,110004,110006,110012,110018,110024],{},[59,109963,39047],{},[133,109965,1690],{},[6556,109967,109968,109974],{},[56,109969,109970,109972],{},[59,109971,92615],{},[133,109973,1690],{},[59,109975,24077],{},[35841,109977,74416],{},[6556,109979,109980,109982],{},[133,109981,3191],{},[59,109983,13377],{},[613,109985,109986,109988],{},[59,109987,48449],{},[59,109989,74528],{},[133,109991,136],{"stretchy":135},[613,109993,109994,109996],{},[59,109995,1755],{},[59,109997,13377],{},[59,109999,15389],{"mathvariant":145},[59,110001,24077],{},[133,110003,161],{"stretchy":135},[133,110005,793],{},[5362,110007,110008,110010],{},[539,110009,546],{},[59,110011,12962],{},[6556,110013,110014,110016],{},[133,110015,3191],{},[59,110017,13377],{},[613,110019,110020,110022],{},[59,110021,1755],{},[59,110023,13377],{},[59,110025,2903],{"mathvariant":145},[63,110027,110028],{"encoding":65},"\\arg\\min_c\\ \\sum_iD_f(x_i\\|c)=\\frac1n\\sum_ix_i 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ame":110312},[263,5463],[39,110314],{"className":110315,"style":256},[230],[39,110317,110319],{"className":110318},[1744,3294],[39,110320,110322,110352],{"className":110321},[196,649],[39,110323,110325,110349],{"className":110324},[200],[39,110326,110328,110339],{"className":110327,"style":6670},[204],[39,110329,110330,110333],{"style":13926},[39,110331],{"className":110332,"style":3311},[212],[39,110334,110336],{"className":110335},[217,218,219,220],[39,110337,13377],{"className":110338},[83,84,220],[39,110340,110341,110344],{"style":3330},[39,110342],{"className":110343,"style":3311},[212],[39,110345,110346],{},[39,110347,3191],{"className":110348},[1744,3339,3340],[39,110350,673],{"className":110351},[672],[39,110353,110355],{"className":110354},[200],[39,110356,110358],{"className":110357,"style":79128},[204],[39,110359],{},[39,110361],{"className":110362,"style":256},[230],[39,110364,110366,110369],{"className":110365},[83],[39,110367,1755],{"className":110368},[83,84],[39,110370,110372],{"className":110371},[192],[39,110373,110375,110395],{"className":110374},[196,649],[39,110376,110378,110392],{"className":110377},[200],[39,110379,110381],{"className":110380,"style":13604},[204],[39,110382,110383,110386],{"style":966},[39,110384],{"className":110385,"style":213},[212],[39,110387,110389],{"className":110388},[217,218,219,220],[39,110390,13377],{"className":110391},[83,84,220],[39,110393,673],{"className":110394},[672],[39,110396,110398],{"className":110397},[200],[39,110399,110401],{"className":110400,"style":680},[204],[39,110402],{},[39,110404,2903],{"className":110405},[83],[11,110407,110408,110409,110638],{},"求导得到 ",[39,110410,110412,110462],{"className":110411,"translate":43},[42],[39,110413,110415],{"className":110414},[47],[49,110416,110417],{"xmlns":51},[53,110418,110419,110459],{},[56,110420,110421,110427,110433,110435,110437,110439,110441,110443,110449,110451,110453,110455,110457],{},[613,110422,110423,110425],{},[133,110424,3191],{},[59,110426,13377],{},[138,110428,110429,110431],{},[59,110430,84918],{"mathvariant":145},[539,110432,17359],{},[59,110434,74528],{},[133,110436,136],{"stretchy":135},[59,110438,24077],{},[133,110440,161],{"stretchy":135},[133,110442,136],{"stretchy":135},[613,110444,110445,110447],{},[59,110446,1755],{},[59,110448,13377],{},[133,110450,5378],{},[59,110452,24077],{},[133,110454,161],{"stretchy":135},[133,110456,793],{},[539,110458,541],{},[63,110460,110461],{"encoding":65},"\\sum_i\\nabla^2f(c)(x_i-c)=0",[39,110463,110465,110608,110629],{"className":110464,"ariaHidden":70},[69],[39,110466,110468,110472,110512,110515,110544,110547,110550,110553,110556,110559,110599,110602,110605],{"className":110467},[74],[39,110469],{"className":110470,"style":110471},[78],"height:1.1138em;vertical-align:-0.2997em;",[39,110473,110475,110478],{"className":110474},[1744],[39,110476,3191],{"className":110477,"style":3953},[1744,3339,3952],[39,110479,110481],{"className":110480},[192],[39,110482,110484,110504],{"className":110483},[196,649],[39,110485,110487,110501],{"className":110486},[200],[39,110488,110490],{"className":110489,"style":94115},[204],[39,110491,110492,110495],{"style":3969},[39,110493],{"className":110494,"style":213},[212],[39,110496,110498],{"className":110497},[217,218,219,220],[39,110499,13377],{"className":110500},[83,84,220],[39,110502,673],{"className":110503},[672],[39,110505,110507],{"className":110506},[200],[39,110508,110510],{"className":110509,"style":4025},[204],[39,110511],{},[39,110513],{"className":110514,"style":256},[230],[39,110516,110518,110521],{"className":110517},[83],[39,110519,84918],{"className":110520},[83],[39,110522,110524],{"className":110523},[192],[39,110525,110527],{"className":110526},[196],[39,110528,110530],{"className":110529},[200],[39,110531,110533],{"className":110532,"style":19710},[204],[39,110534,110535,110538],{"style":208},[39,110536],{"className":110537,"style":213},[212],[39,110539,110541],{"className":110540},[217,218,219,220],[39,110542,17359],{"className":110543},[83,220],[39,110545,74528],{"className":110546,"style":107898},[83,84],[39,110548,136],{"className":110549},[182],[39,110551,24077],{"className":110552},[83,84],[39,110554,161],{"className":110555},[263],[39,110557,136],{"className":110558},[182],[39,110560,110562,110565],{"className":110561},[83],[39,110563,1755],{"className":110564},[83,84],[39,110566,110568],{"className":110567},[192],[39,110569,110571,110591],{"className":110570},[196,649],[39,110572,110574,110588],{"className":110573},[200],[39,110575,110577],{"className":110576,"style":13604},[204],[39,110578,110579,110582],{"style":966},[39,110580],{"className":110581,"style":213},[212],[39,110583,110585],{"className":110584},[217,218,219,220],[39,110586,13377],{"className":110587},[83,84,220],[39,110589,673],{"className":110590},[672],[39,110592,110594],{"className":110593},[200],[39,110595,110597],{"className":110596,"style":680},[204],[39,110598],{},[39,110600],{"className":110601,"style":461},[230],[39,110603,5378],{"className":110604},[465],[39,110606],{"className":110607,"style":461},[230],[39,110609,110611,110614,110617,110620,110623,110626],{"className":110610},[74],[39,110612],{"className":110613,"style":245},[78],[39,110615,24077],{"className":110616},[83,84],[39,110618,161],{"className":110619},[263],[39,110621],{"className":110622,"style":231},[230],[39,110624,793],{"className":110625},[235],[39,110627],{"className":110628,"style":231},[230],[39,110630,110632,110635],{"className":110631},[74],[39,110633],{"className":110634,"style":2473},[78],[39,110636,541],{"className":110637},[83],"，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。",[11,110640,110641,110642,110766,110767,1639,110903,111039],{},"这一点值得展开，因为它关系到分类边界。设两个类原型 ",[39,110643,110645,110671],{"className":110644,"translate":43},[42],[39,110646,110648],{"className":110647},[47],[49,110649,110650],{"xmlns":51},[53,110651,110652,110668],{},[56,110653,110654,110660,110662],{},[613,110655,110656,110658],{},[59,110657,24077],{},[539,110659,546],{},[133,110661,155],{"separator":70},[613,110663,110664,110666],{},[59,110665,24077],{},[539,110667,17359],{},[63,110669,110670],{"encoding":65},"c_1,c_2",[39,110672,110674],{"className":110673,"ariaHidden":70},[69],[39,110675,110677,110680,110720,110723,110726],{"className":110676},[74],[39,110678],{"className":110679,"style":2654},[78],[39,110681,110683,110686],{"className":110682},[83],[39,110684,24077],{"className":110685},[83,84],[39,110687,110689],{"className":110688},[192],[39,110690,110692,110712],{"className":110691},[196,649],[39,110693,110695,110709],{"className":110694},[200],[39,110696,110698],{"className":110697,"style":963},[204],[39,110699,110700,110703],{"style":966},[39,110701],{"className":110702,"style":213},[212],[39,110704,110706],{"className":110705},[217,218,219,220],[39,110707,546],{"className":110708},[83,220],[39,110710,673],{"className":110711},[672],[39,110713,110715],{"className":110714},[200],[39,110716,110718],{"className":110717,"style":680},[204],[39,110719],{},[39,110721,155],{"className":110722},[252],[39,110724],{"className":110725,"style":256},[230],[39,110727,110729,110732],{"className":110728},[83],[39,110730,24077],{"className":110731},[83,84],[39,110733,110735],{"className":110734},[192],[39,110736,110738,110758],{"className":110737},[196,649],[39,110739,110741,110755],{"className":110740},[200],[39,110742,110744],{"className":110743,"style":963},[204],[39,110745,110746,110749],{"style":966},[39,110747],{"className":110748,"style":213},[212],[39,110750,110752],{"className":110751},[217,218,219,220],[39,110753,17359],{"className":110754},[83,220],[39,110756,673],{"className":110757},[672],[39,110759,110761],{"className":110760},[200],[39,110762,110764],{"className":110763,"style":680},[204],[39,110765],{},"，最近原型判类就是比较 ",[39,110768,110770,110802],{"className":110769,"translate":43},[42],[39,110771,110773],{"className":110772},[47],[49,110774,110775],{"xmlns":51},[53,110776,110777,110799],{},[56,110778,110779,110785,110787,110789,110791,110797],{},[613,110780,110781,110783],{},[59,110782,48449],{},[59,110784,74528],{},[133,110786,136],{"stretchy":135},[59,110788,1755],{},[59,110790,15389],{"mathvariant":145},[613,110792,110793,110795],{},[59,110794,24077],{},[539,110796,546],{},[133,110798,161],{"stretchy":135},[63,110800,110801],{"encoding":65},"D_f(x\\|c_1)",[39,110803,110805],{"className":110804,"ariaHidden":70},[69],[39,110806,110808,110811,110851,110854,110857,110860,110900],{"className":110807},[74],[39,110809],{"className":110810,"style":41560},[78],[39,110812,110814,110817],{"className":110813},[83],[39,110815,48449],{"className":110816,"style":438},[83,84],[39,110818,110820],{"className":110819},[192],[39,110821,110823,110843],{"className":110822},[196,649],[39,110824,110826,110840],{"className":110825},[200],[39,110827,110829],{"className":110828,"style":3419},[204],[39,110830,110831,110834],{"style":2937},[39,110832],{"className":110833,"style":213},[212],[39,110835,110837],{"className":110836},[217,218,219,220],[39,110838,74528],{"className":110839,"style":107898},[83,84,220],[39,110841,673],{"className":110842},[672],[39,110844,110846],{"className":110845},[200],[39,110847,110849],{"className":110848,"style":35396},[204],[39,110850],{},[39,110852,136],{"className":110853},[182],[39,110855,1755],{"className":110856},[83,84],[39,110858,15389],{"className":110859},[83],[39,110861,110863,110866],{"className":110862},[83],[39,110864,24077],{"className":110865},[83,84],[39,110867,110869],{"className":110868},[192],[39,110870,110872,110892],{"className":110871},[196,649],[39,110873,110875,110889],{"className":110874},[200],[39,110876,110878],{"className":110877,"style":963},[204],[39,110879,110880,110883],{"style":966},[39,110881],{"className":110882,"style":213},[212],[39,110884,110886],{"className":110885},[217,218,219,220],[39,110887,546],{"className":110888},[83,220],[39,110890,673],{"className":110891},[672],[39,110893,110895],{"className":110894},[200],[39,110896,110898],{"className":110897,"style":680},[204],[39,110899],{},[39,110901,161],{"className":110902},[263],[39,110904,110906,110938],{"className":110905,"translate":43},[42],[39,110907,110909],{"className":110908},[47],[49,110910,110911],{"xmlns":51},[53,110912,110913,110935],{},[56,110914,110915,110921,110923,110925,110927,110933],{},[613,110916,110917,110919],{},[59,110918,48449],{},[59,110920,74528],{},[133,110922,136],{"stretchy":135},[59,110924,1755],{},[59,110926,15389],{"mathvariant":145},[613,110928,110929,110931],{},[59,110930,24077],{},[539,110932,17359],{},[133,110934,161],{"stretchy":135},[63,110936,110937],{"encoding":65},"D_f(x\\|c_2)",[39,110939,110941],{"className":110940,"ariaHidden":70},[69],[39,110942,110944,110947,110987,110990,110993,110996,111036],{"className":110943},[74],[39,110945],{"className":110946,"style":41560},[78],[39,110948,110950,110953],{"className":110949},[83],[39,110951,48449],{"className":110952,"style":438},[83,84],[39,110954,110956],{"className":110955},[192],[39,110957,110959,110979],{"className":110958},[196,649],[39,110960,110962,110976],{"className":110961},[200],[39,110963,110965],{"className":110964,"style":3419},[204],[39,110966,110967,110970],{"style":2937},[39,110968],{"className":110969,"style":213},[212],[39,110971,110973],{"className":110972},[217,218,219,220],[39,110974,74528],{"className":110975,"style":107898},[83,84,220],[39,110977,673],{"className":110978},[672],[39,110980,110982],{"className":110981},[200],[39,110983,110985],{"className":110984,"style":35396},[204],[39,110986],{},[39,110988,136],{"className":110989},[182],[39,110991,1755],{"className":110992},[83,84],[39,110994,15389],{"className":110995},[83],[39,110997,110999,111002],{"className":110998},[83],[39,111000,24077],{"className":111001},[83,84],[39,111003,111005],{"className":111004},[192],[39,111006,111008,111028],{"className":111007},[196,649],[39,111009,111011,111025],{"className":111010},[200],[39,111012,111014],{"className":111013,"style":963},[204],[39,111015,111016,111019],{"style":966},[39,111017],{"className":111018,"style":213},[212],[39,111020,111022],{"className":111021},[217,218,219,220],[39,111023,17359],{"className":111024},[83,220],[39,111026,673],{"className":111027},[672],[39,111029,111031],{"className":111030},[200],[39,111032,111034],{"className":111033,"style":680},[204],[39,111035],{},[39,111037,161],{"className":111038},[263],"。把定义代进去，",[39,111041,111043],{"className":111042,"translate":43},[746],[39,111044,111046,111410],{"className":111045,"translate":43},[42],[39,111047,111049],{"className":111048},[47],[49,111050,111051],{"xmlns":51,"display":755},[53,111052,111053,111407],{},[759,111054,111055,111113,111253],{"rowspacing":761,"columnalign":762,"columnspacing":146},[764,111056,111057,111063],{},[767,111058,111059],{},[770,111060,111061],{"scriptlevel":541,"displaystyle":70},[56,111062],{},[767,111064,111065],{},[770,111066,111067],{"scriptlevel":541,"displaystyle":70},[56,111068,111069,111071,111077,111079,111081,111083,111089,111091,111093,111099,111101,111103,111105,111111],{},[56,111070],{},[613,111072,111073,111075],{},[59,111074,48449],{},[59,111076,74528],{},[133,111078,136],{"stretchy":135},[59,111080,1755],{},[59,111082,15389],{"mathvariant":145},[613,111084,111085,111087],{},[59,111086,24077],{},[539,111088,546],{},[133,111090,161],{"stretchy":135},[133,111092,5378],{},[613,111094,111095,111097],{},[59,111096,48449],{},[59,111098,74528],{},[133,111100,136],{"stretchy":135},[59,111102,1755],{},[59,111104,15389],{"mathvariant":145},[613,111106,111107,111109],{},[59,111108,24077],{},[539,111110,17359],{},[133,111112,161],{"stretchy":135},[764,111114,111115,111121],{},[767,111116,111117],{},[770,111118,111119],{"scriptlevel":541,"displaystyle":70},[56,111120],{},[767,111122,111123],{},[770,111124,111125],{"scriptlevel":541,"displaystyle":70},[56,111126,111127,111129,111131,111133,111135,111137,111139,111141,111143,111145,111147,111153,111155,111157,111159,111161,111163,111169,111175,111177,111179,111181,111187,111189,111191,111193,111195,111197,111199,111201,111203,111205,111207,111209,111215,111217,111219,111221,111223,111225,111231,111237,111239,111241,111243,111249,111251],{},[56,111128],{},[133,111130,793],{},[133,111132,537],{"fence":70,"stretchy":70,"minsize":74068,"maxsize":74068},[59,111134,74528],{},[133,111136,136],{"stretchy":135},[59,111138,1755],{},[133,111140,161],{"stretchy":135},[133,111142,5378],{},[59,111144,74528],{},[133,111146,136],{"stretchy":135},[613,111148,111149,111151],{},[59,111150,24077],{},[539,111152,546],{},[133,111154,161],{"stretchy":135},[133,111156,5378],{},[59,111158,84918],{"mathvariant":145},[59,111160,74528],{},[133,111162,136],{"stretchy":135},[613,111164,111165,111167],{},[59,111166,24077],{},[539,111168,546],{},[138,111170,111171,111173],{},[133,111172,161],{"stretchy":135},[59,111174,93534],{"mathvariant":145},[133,111176,136],{"stretchy":135},[59,111178,1755],{},[133,111180,5378],{},[613,111182,111183,111185],{},[59,111184,24077],{},[539,111186,546],{},[133,111188,161],{"stretchy":135},[133,111190,4280],{"fence":70,"stretchy":70,"minsize":74068,"maxsize":74068},[133,111192,5378],{},[133,111194,537],{"fence":70,"stretchy":70,"minsize":74068,"maxsize":74068},[59,111196,74528],{},[133,111198,136],{"stretchy":135},[59,111200,1755],{},[133,111202,161],{"stretchy":135},[133,111204,5378],{},[59,111206,74528],{},[133,111208,136],{"stretchy":135},[613,111210,111211,111213],{},[59,111212,24077],{},[539,111214,17359],{},[133,111216,161],{"stretchy":135},[133,111218,5378],{},[59,111220,84918],{"mathvariant":145},[59,111222,74528],{},[133,111224,136],{"stretchy":135},[613,111226,111227,111229],{},[59,111228,24077],{},[539,111230,17359],{},[138,111232,111233,111235],{},[133,111234,161],{"stretchy":135},[59,111236,93534],{"mathvariant":145},[133,111238,136],{"stretchy":135},[59,111240,1755],{},[133,111242,5378],{},[613,111244,111245,111247],{},[59,111246,24077],{},[539,111248,17359],{},[133,111250,161],{"stretchy":135},[133,111252,4280],{"fence":70,"stretchy":70,"minsize":74068,"maxsize":74068},[764,111254,111255,111261],{},[767,111256,111257],{},[770,111258,111259],{"scriptlevel":541,"displaystyle":70},[56,111260],{},[767,111262,111263],{},[770,111264,111265],{"scriptlevel":541,"displaystyle":70},[56,111266,111267,111269,111271,111363,111365,111367,111369,111371,111373,111379,111381,111383,111385,111387,111389,111395,111397,111403,111405],{},[56,111268],{},[133,111270,793],{},[6556,111272,111273,111360],{},[6556,111274,111275,111357],{},[56,111276,111277,111279,111281,111283,111289,111291,111293,111295,111297,111303,111305,111307,111309,111311,111313,111319,111325,111331,111333,111335,111337,111339,111345,111351],{},[133,111278,5378],{},[59,111280,74528],{},[133,111282,136],{"stretchy":135},[613,111284,111285,111287],{},[59,111286,24077],{},[539,111288,546],{},[133,111290,161],{"stretchy":135},[133,111292,788],{},[59,111294,74528],{},[133,111296,136],{"stretchy":135},[613,111298,111299,111301],{},[59,111300,24077],{},[539,111302,17359],{},[133,111304,161],{"stretchy":135},[133,111306,788],{},[59,111308,84918],{"mathvariant":145},[59,111310,74528],{},[133,111312,136],{"stretchy":135},[613,111314,111315,111317],{},[59,111316,24077],{},[539,111318,546],{},[138,111320,111321,111323],{},[133,111322,161],{"stretchy":135},[59,111324,93534],{"mathvariant":145},[613,111326,111327,111329],{},[59,111328,24077],{},[539,111330,546],{},[133,111332,5378],{},[59,111334,84918],{"mathvariant":145},[59,111336,74528],{},[133,111338,136],{"stretchy":135},[613,111340,111341,111343],{},[59,111342,24077],{},[539,111344,17359],{},[138,111346,111347,111349],{},[133,111348,161],{"stretchy":135},[59,111350,93534],{"mathvariant":145},[613,111352,111353,111355],{},[59,111354,24077],{},[539,111356,17359],{},[133,111358,111359],{"stretchy":70},"⏟",[35841,111361,111362],{},"常数",[133,111364,788],{},[133,111366,136],{"fence":70,"stretchy":70,"minsize":74068,"maxsize":74068},[59,111368,84918],{"mathvariant":145},[59,111370,74528],{},[133,111372,136],{"stretchy":135},[613,111374,111375,111377],{},[59,111376,24077],{},[539,111378,17359],{},[133,111380,161],{"stretchy":135},[133,111382,5378],{},[59,111384,84918],{"mathvariant":145},[59,111386,74528],{},[133,111388,136],{"stretchy":135},[613,111390,111391,111393],{},[59,111392,24077],{},[539,111394,546],{},[133,111396,161],{"stretchy":135},[138,111398,111399,111401],{},[133,111400,161],{"fence":70,"stretchy":70,"minsize":74068,"maxsize":74068},[59,111402,93534],{"mathvariant":145},[59,111404,1755],{},[59,111406,2903],{"mathvariant":145},[63,111408,111409],{"encoding":65},"\\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}",[39,111411,111413],{"className":111412,"ariaHidden":70},[69],[39,111414,111416,111420],{"className":111415},[74],[39,111417],{"className":111418,"style":111419},[78],"height:5.7244em;vertical-align:-2.6122em;",[39,111421,111423],{"className":111422},[83],[39,111424,111426,111479],{"className":111425},[759],[39,111427,111429],{"className":111428},[918],[39,111430,111432,111470],{"className":111431},[196,649],[39,111433,111435,111467],{"className":111434},[200],[39,111436,111439,111449,111458],{"className":111437,"style":111438},[204],"height:3.1122em;",[39,111440,111442,111446],{"style":111441},"top:-5.3612em;",[39,111443],{"className":111444,"style":111445},[212],"height:3.089em;",[39,111447],{"className":111448},[83],[39,111450,111452,111455],{"style":111451},"top:-3.8021em;",[39,111453],{"className":111454,"style":111445},[212],[39,111456],{"className":111457},[83],[39,111459,111461,111464],{"style":111460},"top:-2.0531em;",[39,111462],{"className":111463,"style":111445},[212],[39,111465],{"className":111466},[83],[39,111468,673],{"className":111469},[672],[39,111471,111473],{"className":111472},[200],[39,111474,111477],{"className":111475,"style":111476},[204],"height:2.6122em;",[39,111478],{},[39,111480,111482],{"className":111481},[1223],[39,111483,111485,112855],{"className":111484},[196,649],[39,111486,111488,112852],{"className":111487},[200],[39,111489,111491,111695,112178],{"className":111490,"style":111438},[204],[39,111492,111493,111496],{"style":111441},[39,111494],{"className":111495,"style":111445},[212],[39,111497,111499,111502,111542,111545,111548,111551,111591,111594,111597,111600,111603,111643,111646,111649,111652,111692],{"className":111498},[83],[39,111500],{"className":111501},[83],[39,111503,111505,111508],{"className":111504},[83],[39,111506,48449],{"className":111507,"style":438},[83,84],[39,111509,111511],{"className":111510},[192],[39,111512,111514,111534],{"className":111513},[196,649],[39,111515,111517,111531],{"className":111516},[200],[39,111518,111520],{"className":111519,"style":3419},[204],[39,111521,111522,111525],{"style":2937},[39,111523],{"className":111524,"style":213},[212],[39,111526,111528],{"className":111527},[217,218,219,220],[39,111529,74528],{"className":111530,"style":107898},[83,84,220],[39,111532,673],{"className":111533},[672],[39,111535,111537],{"className":111536},[200],[39,111538,111540],{"className":111539,"style":35396},[204],[39,111541],{},[39,111543,136],{"className":111544},[182],[39,111546,1755],{"className":111547},[83,84],[39,111549,15389],{"className":111550},[83],[39,111552,111554,111557],{"className":111553},[83],[39,111555,24077],{"className":111556},[83,84],[39,111558,111560],{"className":111559},[192],[39,111561,111563,111583],{"className":111562},[196,649],[39,111564,111566,111580],{"className":111565},[200],[39,111567,111569],{"className":111568,"style":963},[204],[39,111570,111571,111574],{"style":966},[39,111572],{"className":111573,"style":213},[212],[39,111575,111577],{"className":111576},[217,218,219,220],[39,111578,546],{"className":111579},[83,220],[39,111581,673],{"className":111582},[672],[39,111584,111586],{"className":111585},[200],[39,111587,111589],{"className":111588,"style":680},[204],[39,111590],{},[39,111592,161],{"className":111593},[263],[39,111595],{"className":111596,"style":461},[230],[39,111598,5378],{"className":111599},[465],[39,111601],{"className":111602,"style":461},[230],[39,111604,111606,111609],{"className":111605},[83],[39,111607,48449],{"className":111608,"style":438},[83,84],[39,111610,111612],{"className":111611},[192],[39,111613,111615,111635],{"className":111614},[196,649],[39,111616,111618,111632],{"className":111617},[200],[39,111619,111621],{"className":111620,"style":3419},[204],[39,111622,111623,111626],{"style":2937},[39,111624],{"className":111625,"style":213},[212],[39,111627,111629],{"className":111628},[217,218,219,220],[39,111630,74528],{"className":111631,"style":107898},[83,84,220],[39,111633,673],{"className":111634},[672],[39,111636,111638],{"className":111637},[200],[39,111639,111641],{"className":111640,"style":35396},[204],[39,111642],{},[39,111644,136],{"className":111645},[182],[39,111647,1755],{"className":111648},[83,84],[39,111650,15389],{"className":111651},[83],[39,111653,111655,111658],{"className":111654},[83],[39,111656,24077],{"className":111657},[83,84],[39,111659,111661],{"className":111660},[192],[39,111662,111664,111684],{"className":111663},[196,649],[39,111665,111667,111681],{"className":111666},[200],[39,111668,111670],{"className":111669,"style":963},[204],[39,111671,111672,111675],{"style":966},[39,111673],{"className":111674,"style":213},[212],[39,111676,111678],{"className":111677},[217,218,219,220],[39,111679,17359],{"className":111680},[83,220],[39,111682,673],{"className":111683},[672],[39,111685,111687],{"className":111686},[200],[39,111688,111690],{"className":111689,"style":680},[204],[39,111691],{},[39,111693,161],{"className":111694},[2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20-1h17z",[39,112290,112291,112294],{"style":21599},[39,112292],{"className":112293,"style":935},[212],[39,112295,112297,112300,112303,112306,112346,112349,112352,112355,112358,112361,112364,112404,112407,112410,112413,112416,112419,112422,112425,112465,112494,112534,112537,112540,112543,112546,112549,112552,112592,112621],{"className":112296},[83],[39,112298,5378],{"className":112299},[83],[39,112301,74528],{"className":112302,"style":107898},[83,84],[39,112304,136],{"className":112305},[182],[39,112307,112309,112312],{"className":112308},[83],[39,112310,24077],{"className":112311},[83,84],[39,112313,112315],{"className":112314},[192],[39,112316,112318,112338],{"className":112317},[196,649],[39,112319,112321,112335],{"className":112320},[200],[39,112322,112324],{"className":112323,"style":963},[204],[39,112325,112326,112329],{"style":966},[39,112327],{"className":112328,"style":213},[212],[39,112330,112332],{"className":112331},[217,218,219,220],[39,112333,546],{"className":112334},[83,220],[39,112336,673],{"className":112337},[672],[39,112339,112341],{"className":112340},[200],[39,112342,112344],{"className":112343,"style":680},[204],[39,112345],{},[39,112347,161],{"className":112348},[263],[39,112350],{"className":112351,"style":461},[230],[39,112353,788],{"className":112354},[465],[39,112356],{"className":112357,"style":461},[230],[39,112359,74528],{"className":112360,"style":107898},[83,84],[39,112362,136],{"className":112363},[182],[39,112365,112367,112370],{"className":112366},[83],[39,112368,24077],{"className":112369},[83,84],[39,112371,112373],{"className":112372},[192],[39,112374,112376,112396],{"className":112375},[196,649],[39,112377,112379,112393],{"className":112378},[200],[39,112380,112382],{"className":112381,"style":963},[204],[39,112383,112384,112387],{"style":966},[39,112385],{"className":112386,"style":213},[212],[39,112388,112390],{"className":112389},[217,218,219,220],[39,112391,17359],{"className":112392},[83,220],[39,112394,673],{"className":112395},[672],[39,112397,112399],{"className":112398},[200],[39,112400,112402],{"className":112401,"style":680},[204],[39,112403],{},[39,112405,161],{"className":112406},[263],[39,112408],{"className":112409,"style":461},[230],[39,112411,788],{"className":112412},[465],[39,112414],{"className":112415,"style":461},[230],[39,112417,84918],{"className":112418},[83],[39,112420,74528],{"className":112421,"style":107898},[83,84],[39,112423,136],{"className":112424},[182],[39,112426,112428,112431],{"className":112427},[83],[39,112429,24077],{"className":112430},[83,84],[39,112432,112434],{"className":112433},[192],[39,112435,112437,112457],{"className":112436},[196,649],[39,112438,112440,112454],{"className":112439},[200],[39,112441,112443],{"className":112442,"style":963},[204],[39,112444,112445,112448],{"style":966},[39,112446],{"className":112447,"style":213},[212],[39,112449,112451],{"className":112450},[217,218,219,220],[39,112452,546],{"className":112453},[83,220],[39,112455,673],{"className":112456},[672],[39,112458,112460],{"className":112459},[200],[39,112461,112463],{"className":112462,"style":680},[204],[39,112464],{},[39,112466,112468,112471],{"className":112467},[263],[39,112469,161],{"className":112470},[263],[39,112472,112474],{"className":112473},[192],[39,112475,112477],{"className":112476},[196],[39,112478,112480],{"className":112479},[200],[39,112481,112483],{"className":112482,"style":3389},[204],[39,112484,112485,112488],{"style":1027},[39,112486],{"className":112487,"style":213},[212],[39,112489,112491],{"className":112490},[217,218,219,220],[39,112492,93534],{"className":112493},[83,220],[39,112495,112497,112500],{"className":112496},[83],[39,112498,24077],{"className":112499},[83,84],[39,112501,112503],{"className":112502},[192],[39,112504,112506,112526],{"className":112505},[196,649],[39,112507,112509,112523],{"className":112508},[200],[39,112510,112512],{"className":112511,"style":963},[204],[39,112513,112514,112517],{"style":966},[39,112515],{"className":112516,"style":213},[212],[39,112518,112520],{"className":112519},[217,218,219,220],[39,112521,546],{"className":112522},[83,220],[39,112524,673],{"className":112525},[672],[39,112527,112529],{"className":112528},[200],[39,112530,112532],{"className":112531,"style":680},[204],[39,112533],{},[39,112535],{"className":112536,"style":461},[230],[39,112538,5378],{"className":112539},[465],[39,112541],{"className":112542,"style":461},[230],[39,112544,84918],{"className":112545},[83],[39,112547,74528],{"className":112548,"style":107898},[83,84],[39,112550,136],{"className":112551},[182],[39,112553,112555,112558],{"className":112554},[83],[39,112556,24077],{"className":112557},[83,84],[39,112559,112561],{"className":112560},[192],[39,112562,112564,112584],{"className":112563},[196,649],[39,112565,112567,112581],{"className":112566},[200],[39,112568,112570],{"className":112569,"style":963},[204],[39,112571,112572,112575],{"style":966},[39,112573],{"className":112574,"style":213},[212],[39,112576,112578],{"className":112577},[217,218,219,220],[39,112579,17359],{"className":112580},[83,220],[39,112582,673],{"className":112583},[672],[39,112585,112587],{"className":112586},[200],[39,112588,112590],{"className":112589,"style":680},[204],[39,112591],{},[39,112593,112595,112598],{"className":112594},[263],[39,112596,161],{"className":112597},[263],[39,112599,112601],{"className":112600},[192],[39,112602,112604],{"className":112603},[196],[39,112605,112607],{"className":112606},[200],[39,112608,112610],{"className":112609,"style":3389},[204],[39,112611,112612,112615],{"style":1027},[39,112613],{"className":112614,"style":213},[212],[39,112616,112618],{"className":112617},[217,218,219,220],[39,112619,93534],{"className":112620},[83,220],[39,112622,112624,112627],{"className":112623},[83],[39,112625,24077],{"className":112626},[83,84],[39,112628,112630],{"className":112629},[192],[39,112631,112633,112653],{"className":112632},[196,649],[39,112634,112636,112650],{"className":112635},[200],[39,112637,112639],{"className":112638,"style":963},[204],[39,112640,112641,112644],{"style":966},[39,112642],{"className":112643,"style":213},[212],[39,112645,112647],{"className":112646},[217,218,219,220],[39,112648,17359],{"className":112649},[83,220],[39,112651,673],{"className":112652},[672],[39,112654,112656],{"className":112655},[200],[39,112657,112659],{"className":112658,"style":680},[204],[39,112660],{},[39,112662,673],{"className":112663},[672],[39,112665,112667],{"className":112666},[200],[39,112668,112671],{"className":112669,"style":112670},[204],"height:0.898em;",[39,112672],{},[39,112674,673],{"className":112675},[672],[39,112677,112679],{"className":112678},[200],[39,112680,112683],{"className":112681,"style":112682},[204],"height:1.5763em;",[39,112684],{},[39,112686],{"className":112687,"style":461},[230],[39,112689,788],{"className":112690},[465],[39,112692],{"className":112693,"style":461},[230],[39,112695,112697],{"className":112696},[182],[39,112698,136],{"className":112699},[18520,4009],[39,112701,84918],{"className":112702},[83],[39,112704,74528],{"className":112705,"style":107898},[83,84],[39,112707,136],{"className":112708},[182],[39,112710,112712,112715],{"className":112711},[83],[39,112713,24077],{"className":112714},[83,84],[39,112716,112718],{"className":112717},[192],[39,112719,112721,112741],{"className":112720},[196,649],[39,112722,112724,112738],{"className":112723},[200],[39,112725,112727],{"className":112726,"style":963},[204],[39,112728,112729,112732],{"style":966},[39,112730],{"className":112731,"style":213},[212],[39,112733,112735],{"className":112734},[217,218,219,220],[39,112736,17359],{"className":112737},[83,220],[39,112739,673],{"className":112740},[672],[39,112742,112744],{"className":112743},[200],[39,112745,112747],{"className":112746,"style":680},[204],[39,112748],{},[39,112750,161],{"className":112751},[263],[39,112753],{"className":112754,"style":461},[230],[39,112756,5378],{"className":112757},[465],[39,112759],{"className":112760,"style":461},[230],[39,112762,84918],{"className":112763},[83],[39,112765,74528],{"className":112766,"style":107898},[83,84],[39,112768,136],{"className":112769},[182],[39,112771,112773,112776],{"className":112772},[83],[39,112774,24077],{"className":112775},[83,84],[39,112777,112779],{"className":112778},[192],[39,112780,112782,112802],{"className":112781},[196,649],[39,112783,112785,112799],{"className":112784},[200],[39,112786,112788],{"className":112787,"style":963},[204],[39,112789,112790,112793],{"style":966},[39,112791],{"className":112792,"style":213},[212],[39,112794,112796],{"className":112795},[217,218,219,220],[39,112797,546],{"className":112798},[83,220],[39,112800,673],{"className":112801},[672],[39,112803,112805],{"className":112804},[200],[39,112806,112808],{"className":112807,"style":680},[204],[39,112809],{},[39,112811,161],{"className":112812},[263],[39,112814,112816,112822],{"className":112815},[263],[39,112817,112819],{"className":112818},[263],[39,112820,161],{"className":112821},[18520,4009],[39,112823,112825],{"className":112824},[192],[39,112826,112828],{"className":112827},[196],[39,112829,112831],{"className":112830},[200],[39,112832,112835],{"className":112833,"style":112834},[204],"height:1.089em;",[39,112836,112837,112840],{"style":86329},[39,112838],{"className":112839,"style":213},[212],[39,112841,112843],{"className":112842},[217,218,219,220],[39,112844,93534],{"className":112845},[83,220],[39,112847,1755],{"className":112848},[83,84],[39,112850,2903],{"className":112851},[83],[39,112853,673],{"className":112854},[672],[39,112856,112858],{"className":112857},[200],[39,112859,112861],{"className":112860,"style":111476},[204],[39,112862],{},[11,112864,112865,112909,112910,112938],{},[39,112866,112868,112888],{"className":112867,"translate":43},[42],[39,112869,112871],{"className":112870},[47],[49,112872,112873],{"xmlns":51},[53,112874,112875,112885],{},[56,112876,112877,112879,112881,112883],{},[59,112878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那一项消掉了，剩下的是 ",[39,112911,112913,112926],{"className":112912,"translate":43},[42],[39,112914,112916],{"className":112915},[47],[49,112917,112918],{"xmlns":51},[53,112919,112920,112924],{},[56,112921,112922],{},[59,112923,1755],{},[63,112925,1755],{"encoding":65},[39,112927,112929],{"className":112928,"ariaHidden":70},[69],[39,112930,112932,112935],{"className":112931},[74],[39,112933],{"className":112934,"style":290},[78],[39,112936,1755],{"className":112937},[83,84]," 的仿射函数。所以边界仍然是一个超平面。",[11,112940,112941],{},"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。",[11,112943,112944,112945,112996,112997,75296,113026,113055,113056,113109,113110,113139,113140,113169],{},"具体到二维单纯形（其实就是线段 ",[39,112946,112948,112969],{"className":112947,"translate":43},[42],[39,112949,112951],{"className":112950},[47],[49,112952,112953],{"xmlns":51},[53,112954,112955,112967],{},[56,112956,112957,112959,112961,112963,112965],{},[133,112958,537],{"stretchy":135},[539,112960,541],{},[133,112962,155],{"separator":70},[539,112964,546],{},[133,112966,4280],{"stretchy":135},[63,112968,87938],{"encoding":65},[39,112970,112972],{"className":112971,"ariaHidden":70},[69],[39,112973,112975,112978,112981,112984,112987,112990,112993],{"className":112974},[74],[39,112976],{"className":112977,"style":245},[78],[39,112979,537],{"className":112980},[182],[39,112982,541],{"className":112983},[83],[39,112985,155],{"className":112986},[252],[39,112988],{"className":112989,"style":256},[230],[39,112991,546],{"className":112992},[83],[39,112994,4280],{"className":112995},[263],"）上，两个类原型 ",[39,112998,113000,113014],{"className":112999,"translate":43},[42],[39,113001,113003],{"className":113002},[47],[49,113004,113005],{"xmlns":51},[53,113006,113007,113012],{},[56,113008,113009],{},[539,113010,113011],{},"0.2",[63,113013,113011],{"encoding":65},[39,113015,113017],{"className":113016,"ariaHidden":70},[69],[39,113018,113020,113023],{"className":113019},[74],[39,113021],{"className":113022,"style":2473},[78],[39,113024,113011],{"className":113025},[83],[39,113027,113029,113043],{"className":113028,"translate":43},[42],[39,113030,113032],{"className":113031},[47],[49,113033,113034],{"xmlns":51},[53,113035,113036,113041],{},[56,113037,113038],{},[539,113039,113040],{},"0.6",[63,113042,113040],{"encoding":65},[39,113044,113046],{"className":113045,"ariaHidden":70},[69],[39,113047,113049,113052],{"className":113048},[74],[39,113050],{"className":113051,"style":2473},[78],[39,113053,113040],{"className":113054},[83],"，用 KL 判类的分界点在 ",[39,113057,113059,113078],{"className":113058,"translate":43},[42],[39,113060,113062],{"className":113061},[47],[49,113063,113064],{"xmlns":51},[53,113065,113066,113075],{},[56,113067,113068,113070,113072],{},[59,113069,11],{},[133,113071,99937],{},[539,113073,113074],{},"0.387",[63,113076,113077],{"encoding":65},"p\\approx0.387",[39,113079,113081,113100],{"className":113080,"ariaHidden":70},[69],[39,113082,113084,113088,113091,113094,113097],{"className":113083},[74],[39,113085],{"className":113086,"style":113087},[78],"height:0.6776em;vertical-align:-0.1944em;",[39,113089,11],{"className":113090},[83,84],[39,113092],{"className":113093,"style":231},[230],[39,113095,99937],{"className":113096},[235],[39,113098],{"className":113099,"style":231},[230],[39,113101,113103,113106],{"className":113102},[74],[39,113104],{"className":113105,"style":2473},[78],[39,113107,113074],{"className":113108},[83],"；欧氏中点是 ",[39,113111,113113,113127],{"className":113112,"translate":43},[42],[39,113114,113116],{"className":113115},[47],[49,113117,113118],{"xmlns":51},[53,113119,113120,113125],{},[56,113121,113122],{},[539,113123,113124],{},"0.4",[63,113126,113124],{"encoding":65},[39,113128,113130],{"className":113129,"ariaHidden":70},[69],[39,113131,113133,113136],{"className":113132},[74],[39,113134],{"className":113135,"style":2473},[78],[39,113137,113124],{"className":113138},[83],"；而在 logit 坐标里取中点又得到 ",[39,113141,113143,113157],{"className":113142,"translate":43},[42],[39,113144,113146],{"className":113145},[47],[49,113147,113148],{"xmlns":51},[53,113149,113150,113155],{},[56,113151,113152],{},[539,113153,113154],{},"0.380",[63,113156,113154],{"encoding":65},[39,113158,113160],{"className":113159,"ariaHidden":70},[69],[39,113161,113163,113166],{"className":113162},[74],[39,113164],{"className":113165,"style":2473},[78],[39,113167,113154],{"className":113168},[83],"。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。",[11,113171,113172,113173,113277,113278,29832,113603,113631,113632,113660],{},"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 ",[39,113174,113176,113222],{"className":113175,"translate":43},[42],[39,113177,113179],{"className":113178},[47],[49,113180,113181],{"xmlns":51},[53,113182,113183,113219],{},[56,113184,113185,113187,113189,113191,113193,113195,113211,113213,113215,113217],{},[59,113186,100380],{},[133,113188,136],{"stretchy":135},[59,113190,1755],{},[133,113192,161],{"stretchy":135},[133,113194,793],{},[56,113196,113197,113199,113201,113203,113205,113207,113209],{},[59,113198,142],{"mathvariant":145},[59,113200,35010],{"mathvariant":145},[59,113202,74528],{"mathvariant":145},[59,113204,621],{"mathvariant":145},[59,113206,1748],{"mathvariant":145},[59,113208,158],{"mathvariant":145},[59,113210,1755],{"mathvariant":145},[133,113212,136],{"stretchy":135},[59,113214,23703],{},[59,113216,1755],{},[133,113218,161],{"stretchy":135},[63,113220,113221],{"encoding":65},"q(x)=\\mathrm{softmax}(Wx)",[39,113223,113225,113252],{"className":113224,"ariaHidden":70},[69],[39,113226,113228,113231,113234,113237,113240,113243,113246,113249],{"className":113227},[74],[39,113229],{"className":113230,"style":245},[78],[39,113232,100380],{"className":113233,"style":2114},[83,84],[39,113235,136],{"className":113236},[182],[39,113238,1755],{"className":113239},[83,84],[39,113241,161],{"className":113242},[263],[39,113244],{"className":113245,"style":231},[230],[39,113247,793],{"className":113248},[235],[39,113250],{"className":113251,"style":231},[230],[39,113253,113255,113258,113265,113268,113271,113274],{"className":113254},[74],[39,113256],{"className":113257,"style":245},[78],[39,113259,113261],{"className":113260},[83],[39,113262,113264],{"className":113263},[83,35077],"softmax",[39,113266,136],{"className":113267},[182],[39,113269,23703],{"className":113270,"style":178},[83,84],[39,113272,1755],{"className":113273},[83,84],[39,113275,161],{"className":113276},[263],"，那么 ",[39,113279,113281,113355],{"className":113280,"translate":43},[42],[39,113282,113284],{"className":113283},[47],[49,113285,113286],{"xmlns":51},[53,113287,113288,113352],{},[56,113289,113290,113292,113294,113296,113302,113304,113306,113308,113310,113312,113314,113320,113322,113324,113330,113332,113334,113336,113338,113340,113342,113348,113350],{},[133,113291,12951],{"stretchy":135},[59,113293,1755],{},[133,113295,404],{},[613,113297,113298,113300],{},[59,113299,48449],{},[59,113301,74528],{},[133,113303,136],{"stretchy":135},[59,113305,100380],{},[133,113307,136],{"stretchy":135},[59,113309,1755],{},[133,113311,161],{"stretchy":135},[59,113313,15389],{"mathvariant":145},[613,113315,113316,113318],{},[59,113317,24077],{},[539,113319,546],{},[133,113321,161],{"stretchy":135},[133,113323,793],{},[613,113325,113326,113328],{},[59,113327,48449],{},[59,113329,74528],{},[133,113331,136],{"stretchy":135},[59,113333,100380],{},[133,113335,136],{"stretchy":135},[59,113337,1755],{},[133,113339,161],{"stretchy":135},[59,113341,15389],{"mathvariant":145},[613,113343,113344,113346],{},[59,113345,24077],{},[539,113347,17359],{},[133,113349,161],{"stretchy":135},[133,113351,12965],{"stretchy":135},[63,113353,113354],{"encoding":65},"\\{x:D_f(q(x)\\|c_1)=D_f(q(x)\\|c_2)\\}",[39,113356,113358,113379,113495],{"className":113357,"ariaHidden":70},[69],[39,113359,113361,113364,113367,113370,113373,113376],{"className":113360},[74],[39,113362],{"className":113363,"style":245},[78],[39,113365,12951],{"className":113366},[182],[39,113368,1755],{"className":113369},[83,84],[39,113371],{"className":113372,"style":231},[230],[39,113374,404],{"className":113375},[235],[39,113377],{"className":113378,"style":231},[230],[39,113380,113382,113385,113425,113428,113431,113434,113437,113440,113443,113483,113486,113489,113492],{"className":113381},[74],[39,113383],{"className":113384,"style":41560},[78],[39,113386,113388,113391],{"className":113387},[83],[39,113389,48449],{"className":113390,"style":438},[83,84],[39,113392,113394],{"className":113393},[192],[39,113395,113397,113417],{"className":113396},[196,649],[39,113398,113400,113414],{"className":113399},[200],[39,113401,113403],{"className":113402,"style":3419},[204],[39,113404,113405,113408],{"style":2937},[39,113406],{"className":113407,"style":213},[212],[39,113409,113411],{"className":113410},[217,218,219,220],[39,113412,74528],{"className":113413,"style":107898},[83,84,220],[39,113415,673],{"className":113416},[672],[39,113418,113420],{"className":113419},[200],[39,113421,113423],{"className":113422,"style":35396},[204],[39,113424],{},[39,113426,136],{"className":113427},[182],[39,113429,100380],{"className":113430,"style":2114},[83,84],[39,113432,136],{"className":113433},[182],[39,113435,1755],{"className":113436},[83,84],[39,113438,161],{"className":113439},[263],[39,113441,15389],{"className":113442},[83],[39,113444,113446,113449],{"className":113445},[83],[39,113447,24077],{"className":113448},[83,84],[39,113450,113452],{"className":113451},[192],[39,113453,113455,113475],{"className":113454},[196,649],[39,113456,113458,113472],{"className":113457},[200],[39,113459,113461],{"className":113460,"style":963},[204],[39,113462,113463,113466],{"style":966},[39,113464],{"className":113465,"style":213},[212],[39,113467,113469],{"className":113468},[217,218,219,220],[39,113470,546],{"className":113471},[83,220],[39,113473,673],{"className":113474},[672],[39,113476,113478],{"className":113477},[200],[39,113479,113481],{"className":113480,"style":680},[204],[39,113482],{},[39,113484,161],{"className":113485},[263],[39,113487],{"className":113488,"style":231},[230],[39,113490,793],{"className":113491},[235],[39,113493],{"className":113494,"style":231},[230],[39,113496,113498,113501,113541,113544,113547,113550,113553,113556,113559,113599],{"className":113497},[74],[39,113499],{"className":113500,"style":41560},[78],[39,113502,113504,113507],{"className":113503},[83],[39,113505,48449],{"className":113506,"style":438},[83,84],[39,113508,113510],{"className":113509},[192],[39,113511,113513,113533],{"className":113512},[196,649],[39,113514,113516,113530],{"className":113515},[200],[39,113517,113519],{"className":113518,"style":3419},[204],[39,113520,113521,113524],{"style":2937},[39,113522],{"className":113523,"style":213},[212],[39,113525,113527],{"className":113526},[217,218,219,220],[39,113528,74528],{"className":113529,"style":107898},[83,84,220],[39,113531,673],{"className":113532},[672],[39,113534,113536],{"className":113535},[200],[39,113537,113539],{"className":113538,"style":35396},[204],[39,113540],{},[39,113542,136],{"className":113543},[182],[39,113545,100380],{"className":113546,"style":2114},[83,84],[39,113548,136],{"className":113549},[182],[39,113551,1755],{"className":113552},[83,84],[39,113554,161],{"className":113555},[263],[39,113557,15389],{"className":113558},[83],[39,113560,113562,113565],{"className":113561},[83],[39,113563,24077],{"className":113564},[83,84],[39,11356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空间是超平面，在 ",[39,113633,113635,113648],{"className":113634,"translate":43},[42],[39,113636,113638],{"className":113637},[47],[49,113639,113640],{"xmlns":51},[53,113641,113642,113646],{},[56,113643,113644],{},[59,113645,1755],{},[63,113647,1755],{"encoding":65},[39,113649,113651],{"className":113650,"ariaHidden":70},[69],[39,113652,113654,113657],{"className":113653},[74],[39,113655],{"className":113656,"style":290},[78],[39,113658,1755],{"className":113659},[83,84]," 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。",[11,113662,113663],{},"Bregman 还给出两个优化工具。一个是镜像下降：",[39,113665,113667],{"className":113666,"translate":43},[746],[39,113668,113670,113756],{"className":113669,"translate":43},[42],[39,113671,113673],{"className":113672},[47],[49,113674,113675],{"xmlns":51,"display":755},[53,113676,113677,113753],{},[56,113678,113679,113691,113693,113695,113697,113707,113709,113711,113717,113719,113721,113723,113725,113731,113737,113739,113741,113743,113749,113751],{},[613,113680,113681,113683],{},[59,113682,1755],{},[56,113684,113685,113687,113689],{},[59,113686,621],{},[133,113688,788],{},[539,113690,546],{},[133,113692,793],{},[59,113694,39047],{},[133,113696,1690],{},[6556,113698,113699,113705],{},[56,113700,113701,113703],{},[59,113702,92615],{},[133,113704,1690],{},[59,113706,1755],{},[35841,113708,74416],{},[133,113710,92640],{"stretchy":135},[613,113712,113713,113715],{},[59,113714,39209],{},[59,113716,621],{},[133,113718,155],{"separator":70},[59,113720,1755],{},[133,113722,92649],{"stretchy":135},[133,113724,788],{},[5362,113726,113727,113729],{},[539,113728,546],{},[59,113730,103251],{},[613,113732,113733,113735],{},[59,113734,48449],{},[59,113736,74528],{},[133,113738,136],{"stretchy":135},[59,113740,1755],{},[59,113742,15389],{"mathvariant":145},[613,113744,113745,113747],{},[59,113746,1755],{},[59,113748,621],{},[133,113750,161],{"stretchy":135},[133,113752,155],{"separator":70},[63,113754,113755],{"encoding":65},"x_{t+1}=\\arg\\min_x\\ \\langle g_t,x\\rangle+\\frac1\\eta D_f(x\\|x_t),",[39,113757,113759,113823,113949],{"className":113758,"ariaHidden":70},[69],[39,113760,113762,113765,113814,113817,113820],{"className":113761},[74],[39,113763],{"className":113764,"style":3635},[78],[39,113766,113768,113771],{"className":113767},[83],[39,113769,1755],{"className":113770},[83,84],[39,113772,113774],{"className":113773},[192],[39,113775,113777,113806],{"className":113776},[196,649],[39,113778,113780,113803],{"className":113779},[200],[39,113781,113783],{"className":113782,"style":963},[204],[39,113784,113785,113788],{"style":966},[39,113786],{"className":113787,"style":213},[212],[39,113789,113791],{"className":113790},[217,218,219,220],[39,113792,113794,113797,113800],{"className":113793},[83,220],[39,113795,621],{"className":113796},[83,84,220],[39,113798,788],{"className":113799},[465,220],[39,113801,546],{"className":113802},[83,220],[39,113804,673],{"className":113805},[672],[39,113807,113809],{"className":113808},[200],[39,113810,113812],{"className":113811,"style":994},[204],[39,113813],{},[39,113815],{"className":113816,"style":231},[230],[39,113818,793],{"className":113819},[235],[39,113821],{"className":113822,"style":231},[230],[39,113824,113826,113830,113835,113838,113882,113885,113888,113928,113931,113934,113937,113940,113943,113946],{"className":113825},[74],[39,113827],{"className":113828,"style":113829},[78],"height:1.45em;vertical-align:-0.7em;",[39,113831,39206,113833],{"className":113832},[1744],[39,113834,39209],{"style":35078},[39,113836],{"className":113837,"style":256},[230],[39,113839,113841],{"className":113840},[1744,3294],[39,113842,113844,113874],{"className":113843},[196,649],[39,113845,113847,113871],{"className":113846},[200],[39,113848,113850,113861],{"className":113849,"style":92716},[204],[39,113851,113852,113855],{"style":22299},[39,113853],{"className":113854,"style":935},[212],[39,113856,113858],{"className":113857},[217,218,219,220],[39,113859,1755],{"className":113860},[83,84,220],[39,113862,113863,113866],{"style":21599},[39,113864],{"className":113865,"style":935},[212],[39,113867,113868],{},[39,113869,92615],{"className":113870},[1744],[39,113872,673],{"className":113873},[672],[39,113875,113877],{"className":113876},[200],[39,113878,113880],{"className":113879,"style":22331},[204],[39,113881],{},[39,113883,74416],{"className":113884},[230],[39,113886,92640],{"className":113887},[182],[39,113889,113891,113894],{"className":113890},[83],[39,113892,39209],{"className":113893,"style":2114},[83,84],[39,113895,113897],{"className":113896},[192],[39,113898,113900,113920],{"className":113899},[196,649],[39,113901,113903,113917],{"className":113902},[200],[39,113904,113906],{"className":113905,"style":656},[204],[39,113907,113908,113911],{"style":41316},[39,113909],{"className":113910,"style":213},[212],[39,113912,113914],{"className":113913},[217,218,219,220],[39,113915,621],{"className":113916},[83,84,220],[39,113918,673],{"className":113919},[672],[39,113921,113923],{"className":113922},[200],[39,113924,113926],{"className":113925,"style":680},[204],[39,113927],{},[39,113929,155],{"className":113930},[252],[39,113932],{"className":113933,"style":256},[230],[39,113935,1755],{"className":113936},[83,84],[39,113938,92649],{"className":113939},[263],[39,113941],{"className":113942,"style":461},[230],[39,113944,788],{"className":113945},[465],[39,113947],{"className":113948,"style":461},[230],[39,113950,113952,113955,114017,114057,114060,114063,114066,114106,114109],{"className":113951},[74],[39,113953],{"className":113954,"style":72267},[78],[39,113956,113958,113961,114014],{"className":113957},[83],[39,113959],{"className":113960},[182,5463],[39,113962,113964],{"className":113963},[5362],[39,113965,113967,114006],{"className":113966},[196,649],[39,113968,113970,114003],{"className":113969},[200],[39,113971,113973,113984,113992],{"className":113972,"style":18240},[204],[39,113974,113975,113978],{"style":5479},[39,113976],{"className":113977,"style":935},[212],[39,113979,113981],{"className":113980},[83],[39,113982,103251],{"className":113983,"style":2114},[83,84],[39,113985,113986,113989],{"style":5503},[39,113987],{"className":113988,"style":935},[212],[39,113990],{"className":113991,"style":5511},[5510],[39,113993,113994,113997],{"style":5514},[39,113995],{"className":113996,"style":935},[212],[39,113998,114000],{"className":113999},[83],[39,114001,546],{"className":114002},[83],[39,114004,673],{"className":114005},[672],[39,114007,114009],{"className":114008},[200],[39,114010,114012],{"className":114011,"style":5582},[204],[39,114013],{},[39,114015],{"className":114016},[263,5463],[39,114018,114020,114023],{"className":114019},[83],[39,114021,48449],{"className":114022,"style":438},[83,84],[39,114024,114026],{"className":114025},[192],[39,114027,114029,114049],{"className":114028},[196,649],[39,114030,114032,114046],{"className":114031},[200],[39,114033,114035],{"className":114034,"style":3419},[204],[39,114036,114037,114040],{"style":2937},[39,114038],{"className":114039,"style":213},[212],[39,114041,114043],{"className":114042},[217,218,219,220],[39,114044,74528],{"className":114045,"style":107898},[83,84,220],[39,114047,673],{"className":114048},[672],[39,114050,114052],{"className":114051},[200],[39,114053,114055],{"className":114054,"style":35396},[204],[39,114056],{},[39,114058,136],{"className":114059},[182],[39,114061,1755],{"className":114062},[83,84],[39,114064,15389],{"className":114065},[83],[39,114067,114069,114072],{"className":114068},[83],[39,114070,1755],{"className":114071},[83,84],[39,114073,114075],{"className":114074},[192],[39,114076,114078,114098],{"className":114077},[196,649],[39,114079,114081,114095],{"className":114080},[200],[39,114082,114084],{"className":114083,"style":656},[204],[39,114085,114086,114089],{"style":966},[39,114087],{"className":114088,"style":213},[212],[39,114090,114092],{"className":114091},[217,218,219,220],[39,114093,621],{"className":114094},[83,84,220],[39,114096,673],{"className":114097},[672],[39,114099,114101],{"className":114100},[200],[39,114102,114104],{"className":114103,"style":680},[204],[39,114105],{},[39,114107,161],{"className":114108},[263],[39,114110,155],{"className":114111},[252],[11,114113,114114,114115,114143,114144,114311,114312,114340,114341,2183,114586,114614],{},"直觉是“走一步，但别离原来的位置太远”，而“太远”用 ",[39,114116,114118,114131],{"className":114117,"translate":43},[42],[39,114119,114121],{"className":114120},[47],[49,114122,114123],{"xmlns":51},[53,114124,114125,114129],{},[56,114126,114127],{},[59,114128,74528],{},[63,114130,74528],{"encoding":65},[39,114132,114134],{"className":114133,"ariaHidden":70},[69],[39,114135,114137,114140],{"className":114136},[74],[39,114138],{"className":114139,"style":51280},[78],[39,114141,74528],{"className":114142,"style":107898},[83,84]," 来量。取 ",[39,114145,114147,114181],{"className":114146,"translate":43},[42],[39,114148,114150],{"className":114149},[47],[49,114151,114152],{"xmlns":51},[53,114153,114154,114178],{},[56,114155,114156,114158,114160,114168,114170,114172],{},[59,114157,74528],{},[133,114159,793],{},[770,114161,114162],{"scriptlevel":541,"displaystyle":135},[5362,114163,114164,114166],{},[539,114165,546],{},[539,114167,17359],{},[59,114169,15389],{"mathvariant":145},[59,114171,1755],{},[138,114173,114174,114176],{},[59,114175,15389],{"mathvariant":145},[539,114177,17359],{},[63,114179,114180],{"encoding":65},"f=\\tfrac12\\|x\\|^2",[39,114182,114184,114202],{"className":114183,"ariaHidden":70},[69],[39,114185,114187,114190,114193,114196,114199],{"className":114186},[74],[39,114188],{"className":114189,"style":51280},[78],[39,114191,74528],{"className":114192,"style":107898},[83,84],[39,114194],{"className":114195,"style":231},[230],[39,114197,793],{"className":114198},[235],[39,114200],{"className":114201,"style":231},[230],[39,114203,114205,114208,114276,114279,114282],{"className":114204},[74],[39,114206],{"className":114207,"style":74987},[78],[39,114209,114211,114214,114273],{"className":114210},[83],[39,114212],{"className":114213},[182,5463],[39,114215,114217],{"className":114216},[5362],[39,114218,114220,114265],{"className":114219},[196,649],[39,114221,114223,114262],{"className":114222},[200],[39,114224,114226,114240,114248],{"className":114225,"style":75006},[204],[39,114227,114228,114231],{"style":75009},[39,114229],{"className":114230,"style":935},[212],[39,114232,114234],{"className":114233},[217,218,219,220],[39,114235,114237],{"className":114236},[83,220],[39,114238,17359],{"className":114239},[83,220],[39,114241,114242,114245],{"style":5503},[39,114243],{"className":114244,"style":935},[212],[39,114246],{"className":114247,"style":5511},[5510],[39,114249,114250,114253],{"style":75032},[39,114251],{"className":114252,"style":935},[212],[39,114254,114256],{"className":114255},[217,218,219,220],[39,114257,114259],{"className":114258},[83,220],[39,114260,546],{"className":114261},[83,220],[39,114263,673],{"className":114264},[672],[39,114266,114268],{"className":114267},[200],[39,114269,114271],{"className":114270,"style":75054},[204],[39,114272],{},[39,114274],{"className":114275},[263,5463],[39,114277,15389],{"className":114278},[83],[39,114280,1755],{"className":114281},[83,84],[39,114283,114285,114288],{"className":114284},[83],[39,114286,15389],{"className":114287},[83],[39,114289,114291],{"className":114290},[192],[39,114292,114294],{"className":114293},[196],[39,114295,114297],{"className":114296},[200],[39,114298,114300],{"className":114299,"style":19710},[204],[39,114301,114302,114305],{"style":208},[39,114303],{"className":114304,"style":213},[212],[39,114306,114308],{"className":114307},[217,218,219,220],[39,114309,17359],{"className":114310},[83,220]," 就是梯度下降；取 ",[39,114313,114315,114328],{"className":114314,"translate":43},[42],[39,114316,114318],{"className":114317},[47],[49,114319,114320],{"xmlns":51},[53,114321,114322,114326],{},[56,114323,114324],{},[59,114325,74528],{},[63,114327,74528],{"encoding":65},[39,114329,114331],{"className":114330,"ariaHidden":70},[69],[39,114332,114334,114337],{"className":114333},[74],[39,114335],{"className":114336,"style":51280},[78],[39,114338,74528],{"className":114339,"style":107898},[83,84]," 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：",[39,114342,114344,114402],{"className":114343,"translate":43},[42],[39,114345,114347],{"className":114346},[47],[49,114348,114349],{"xmlns":51},[53,114350,114351,114399],{},[56,114352,114353,114365,114367,114373,114375,114377,114379,114381,114383,114385,114395,114397],{},[613,114354,114355,114357],{},[59,114356,96518],{},[56,114358,114359,114361,114363],{},[59,114360,621],{},[133,114362,788],{},[539,114364,546],{},[133,114366,793],{},[613,114368,114369,114371],{},[59,114370,96518],{},[59,114372,621],{},[133,114374,5378],{},[59,114376,103251],{},[35841,114378,35843],{},[59,114380,3181],{},[133,114382,136],{"stretchy":135},[59,114384,96518],{},[138,114386,114387,114389],{},[133,114388,161],{"stretchy":135},[56,114390,114391,114393],{},[133,114392,5378],{},[539,114394,546],{},[59,114396,84918],{"mathvariant":145},[59,114398,77383],{},[63,114400,114401],{"encoding":65},"\\theta_{t+1}=\\theta_t-\\eta\\,G(\\theta)^{-1}\\nabla L",[39,114403,114405,114469,114524],{"className":114404,"ariaHidden":70},[69],[39,114406,114408,114411,114460,114463,114466],{"className":114407},[74],[39,114409],{"className":114410,"style":97742},[78],[39,114412,114414,114417],{"className":114413},[83],[39,114415,96518],{"className":114416,"style":438},[83,84],[39,114418,114420],{"className":114419},[192],[39,114421,114423,114452],{"className":114422},[196,649],[39,114424,114426,114449],{"className":114425},[200],[39,114427,114429],{"className":114428,"style":963},[204],[39,114430,114431,114434],{"style":2937},[39,114432],{"className":114433,"style":213},[212],[39,114435,114437],{"className":114436},[217,218,219,220],[39,114438,114440,114443,114446],{"className":114439},[83,220],[39,114441,621],{"className":114442},[83,84,220],[39,114444,788],{"className":114445},[465,220],[39,114447,546],{"className":114448},[83,220],[39,114450,673],{"className":114451},[672],[39,114453,114455],{"className":114454},[200],[39,114456,114458],{"className":114457,"style":994},[204],[39,114459],{},[39,114461],{"className":114462,"style":231},[230],[39,114464,793],{"className":114465},[235],[39,114467],{"className":114468,"style":231},[230],[39,114470,114472,114475,114515,114518,114521],{"className":114471},[74],[39,114473],{"className":114474,"style":74867},[78],[39,114476,114478,114481],{"className":114477},[83],[39,114479,96518],{"className":114480,"style":438},[83,84],[39,114482,114484],{"className":114483},[192],[39,114485,114487,114507],{"className":114486},[196,649],[39,114488,114490,114504],{"className":114489},[200],[39,114491,114493],{"className":114492,"style":656},[204],[39,114494,114495,114498],{"style":2937},[39,114496],{"className":114497,"style":213},[212],[39,114499,114501],{"className":114500},[217,218,219,220],[39,114502,621],{"className":114503},[83,84,220],[39,114505,673],{"className":114506},[672],[39,114508,114510],{"className":114509},[200],[39,114511,114513],{"className":114512,"style":680},[204],[39,114514],{},[39,114516],{"className":114517,"style":461},[230],[39,114519,5378],{"className":114520},[465],[39,114522],{"className":114523,"style":461},[230],[39,114525,114527,114530,114533,114536,114539,114542,114545,114580,114583],{"className":114526},[74],[39,114528],{"className":114529,"style":84808},[78],[39,114531,103251],{"className":114532,"style":2114},[83,84],[39,114534],{"className":114535,"style":256},[230],[39,114537,3181],{"className":114538},[83,84],[39,114540,136],{"className":114541},[182],[39,114543,96518],{"className":114544,"style":438},[83,84],[39,114546,114548,114551],{"className":114547},[263],[39,114549,161],{"className":114550},[263],[39,114552,114554],{"className":114553},[192],[39,114555,114557],{"className":114556},[196],[39,114558,114560],{"className":114559},[200],[39,114561,114563],{"className":114562,"style":19710},[204],[39,114564,114565,114568],{"style":208},[39,114566],{"className":114567,"style":213},[212],[39,114569,114571],{"className":114570},[217,218,219,220],[39,114572,114574,114577],{"className":114573},[83,220],[39,114575,5378],{"className":114576},[83,220],[39,114578,546],{"className":114579},[83,220],[39,114581,84918],{"className":114582},[83],[39,114584,77383],{"className":114585},[83,84],[39,114587,114589,114602],{"className":114588,"translate":43},[42],[39,114590,114592],{"className":114591},[47],[49,114593,114594],{"xmlns":51},[53,114595,114596,114600],{},[56,114597,114598],{},[59,114599,3181],{},[63,114601,3181],{"encoding":65},[39,114603,114605],{"className":114604,"ariaHidden":70},[69],[39,114606,114608,114611],{"className":114607},[74],[39,114609],{"className":114610,"style":79},[78],[39,114612,3181],{"className":114613},[83,84]," 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”。",[21,114616,114617],{"id":114617},"点态密度",[11,114619,114620,114621,114726,114727,114755],{},"手上有五个一维样本：",[39,114622,114624,114666],{"className":114623,"translate":43},[42],[39,114625,114627],{"className":114626},[47],[49,114628,114629],{"xmlns":51},[53,114630,114631,114663],{},[56,114632,114633,114635,114637,114639,114642,114644,114646,114649,114651,114653,114656,114658,114660],{},[539,114634,17739],{},[133,114636,155],{"separator":70},[35841,114638,74416],{},[539,114640,114641],{},"1.0",[133,114643,155],{"separator":70},[35841,114645,74416],{},[539,114647,114648],{},"1.1",[133,114650,155],{"separator":70},[35841,114652,74416],{},[539,114654,114655],{},"5.0",[133,114657,155],{"separator":70},[35841,114659,74416],{},[539,114661,114662],{},"5.1",[63,114664,114665],{"encoding":65},"0.9,\\ 1.0,\\ 1.1,\\ 5.0,\\ 5.1",[39,114667,114669],{"className":114668,"ariaHidden":70},[69],[39,114670,114672,114675,114678,114681,114684,114687,114690,114693,114696,114699,114702,114705,114708,114711,114714,114717,114720,114723],{"className":114671},[74],[39,114673],{"className":114674,"style":17578},[78],[39,114676,17739],{"className":114677},[83],[39,114679,155],{"className":114680},[252],[39,114682,74416],{"className":114683},[230],[39,114685],{"className":114686,"style":256},[230],[39,114688,114641],{"className":114689},[83],[39,114691,155],{"className":114692},[252],[39,114694,74416],{"className":114695},[230],[39,114697],{"className":114698,"style":256},[230],[39,114700,114648],{"className":114701},[83],[39,114703,155],{"className":114704},[252],[39,114706,74416],{"className":114707},[230],[39,114709],{"className":114710,"style":256},[230],[39,114712,114655],{"className":114713},[83],[39,114715,155],{"className":114716},[252],[39,114718,74416],{"className":114719},[230],[39,114721],{"className":114722,"style":256},[230],[39,114724,114662],{"className":114725},[83],"。肉眼看出两簇，中间在 ",[39,114728,114730,114743],{"className":114729,"translate":43},[42],[39,114731,114733],{"className":114732},[47],[49,114734,114735],{"xmlns":51},[53,114736,114737,114741],{},[56,114738,114739],{},[539,114740,59187],{},[63,114742,59187],{"encoding":65},[39,114744,114746],{"className":114745,"ariaHidden":70},[69],[39,114747,114749,114752],{"className":114748},[74],[39,114750],{"className":114751,"style":2473},[78],[39,114753,59187],{"className":114754},[83]," 附近是空的。",[11,114757,114758,114759,114788,114789,114817,114818,114869,114870,114899,114900,114951,114952,114981,114982,114985],{},"现在用核密度估计去拟合，只用高斯核、只改带宽。带宽取 ",[39,114760,114762,114776],{"className":114761,"translate":43},[42],[39,114763,114765],{"className":114764},[47],[49,114766,114767],{"xmlns":51},[53,114768,114769,114774],{},[56,114770,114771],{},[539,114772,114773],{},"0.1",[63,114775,114773],{"encoding":65},[39,114777,114779],{"className":114778,"ariaHidden":70},[69],[39,114780,114782,114785],{"className":114781},[74],[39,114783],{"className":114784,"style":2473},[78],[39,114786,114773],{"className":114787},[83],"，每个点都是一个尖峰，曲线在两个簇内部剧烈起伏；带宽取 ",[39,114790,114792,114805],{"className":114791,"translate":43},[42],[39,114793,114795],{"className":114794},[47],[49,114796,114797],{"xmlns":51},[53,114798,114799,114803],{},[56,114800,114801],{},[539,114802,59259],{},[63,114804,59259],{"encoding":65},[39,114806,114808],{"className":114807,"ariaHidden":70},[69],[39,114809,114811,114814],{"className":114810},[74],[39,114812],{"className":114813,"style":2473},[78],[39,114815,59259],{"className":114816},[83],"，曲线被抹平。抹平到什么程度？算一下会发现，",[39,114819,114821,114839],{"className":114820,"translate":43},[42],[39,114822,114824],{"className":114823},[47],[49,114825,114826],{"xmlns":51},[53,114827,114828,114836],{},[56,114829,114830,114832,114834],{},[59,114831,1755],{},[133,114833,793],{},[539,114835,59187],{},[63,114837,114838],{"encoding":65},"x=3",[39,114840,114842,114860],{"className":114841,"ariaHidden":70},[69],[39,114843,114845,114848,114851,114854,114857],{"className":114844},[74],[39,114846],{"className":114847,"style":290},[78],[39,114849,1755],{"className":114850},[83,84],[39,114852],{"className":114853,"style":231},[230],[39,114855,793],{"className":114856},[235],[39,114858],{"className":114859,"style":231},[230],[39,114861,114863,114866],{"className":114862},[74],[39,114864],{"className":114865,"style":2473},[78],[39,114867,59187],{"className":114868},[83]," 那个空洞处的密度大约是 ",[39,114871,114873,114887],{"className":114872,"translate":43},[42],[39,114874,114876],{"className":114875},[47],[49,114877,114878],{"xmlns":51},[53,114879,114880,114885],{},[56,114881,114882],{},[539,114883,114884],{},"0.073",[63,114886,114884],{"encoding":65},[39,114888,114890],{"className":114889,"ariaHidden":70},[69],[39,114891,114893,114896],{"className":114892},[74],[39,114894],{"className":114895,"style":2473},[78],[39,114897,114884],{"className":114898},[83],"，而簇心 ",[39,114901,114903,114921],{"className":114902,"translate":43},[42],[39,114904,114906],{"className":114905},[47],[49,114907,114908],{"xmlns":51},[53,114909,114910,114918],{},[56,114911,114912,114914,114916],{},[59,114913,1755],{},[133,114915,793],{},[539,114917,546],{},[63,114919,114920],{"encoding":65},"x=1",[39,114922,114924,114942],{"className":114923,"ariaHidden":70},[69],[39,114925,114927,114930,114933,114936,114939],{"className":114926},[74],[39,114928],{"className":114929,"style":290},[78],[39,114931,1755],{"className":114932},[83,84],[39,114934],{"className":114935,"style":231},[230],[39,114937,793],{"className":114938},[235],[39,114940],{"className":114941,"style":231},[230],[39,114943,114945,114948],{"className":114944},[74],[39,114946],{"className":114947,"style":2473},[78],[39,114949,546],{"className":114950},[83]," 处是 ",[39,114953,114955,114969],{"className":114954,"translate":43},[42],[39,114956,114958],{"className":114957},[47],[49,114959,114960],{"xmlns":51},[53,114961,114962,114967],{},[56,114963,114964],{},[539,114965,114966],{},"0.071",[63,114968,114966],{"encoding":65},[39,114970,114972],{"className":114971,"ariaHidden":70},[69],[39,114973,114975,114978],{"className":114974},[74],[39,114976],{"className":114977,"style":2473},[78],[39,114979,114966],{"className":114980},[83],"——",[1489,114983,114984],{},"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。",[11,114987,114988,114989,115017,115018,115046,115047,115075],{},"这个例子里，带宽就是分辨率。",[39,114990,114992,115005],{"className":114991,"translate":43},[42],[39,114993,114995],{"className":114994},[47],[49,114996,114997],{"xmlns":51},[53,114998,114999,115003],{},[56,115000,115001],{},[59,115002,69998],{},[63,115004,69998],{"encoding":65},[39,115006,115008],{"className":115007,"ariaHidden":70},[69],[39,115009,115011,115014],{"className":115010},[74],[39,115012],{"className":115013,"style":28824},[78],[39,115015,69998],{"className":115016},[83,84]," 太小，方差爆炸；",[39,115019,115021,115034],{"className":115020,"translate":43},[42],[39,115022,115024],{"className":115023},[47],[49,115025,115026],{"xmlns":51},[53,115027,115028,115032],{},[56,115029,115030],{},[59,115031,69998],{},[63,115033,69998],{"encoding":65},[39,115035,115037],{"className":115036,"ariaHidden":70},[69],[39,115038,115040,115043],{"className":115039},[74],[39,115041],{"className":115042,"style":28824},[78],[39,115044,69998],{"className":115045},[83,84]," 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 ",[39,115048,115050,115063],{"className":115049,"translate":43},[42],[39,115051,115053],{"className":115052},[47],[49,115054,115055],{"xmlns":51},[53,115056,115057,115061],{},[56,115058,115059],{},[59,115060,69998],{},[63,115062,69998],{"encoding":65},[39,115064,115066],{"className":115065,"ariaHidden":70},[69],[39,115067,115069,115072],{"className":115068},[74],[39,115070],{"className":115071,"style":28824},[78],[39,115073,69998],{"className":115074},[83,84]," 其实是在调“我允许多大的类间间隙”。",[11,115077,115078],{},"形式上是",[39,115080,115082],{"className":115081,"translate":43},[746],[39,115083,115085,115198],{"className":115084,"translate":43},[42],[39,115086,115088],{"className":115087},[47],[49,115089,115090],{"xmlns":51,"display":755},[53,115091,115092,115195],{},[56,115093,115094,115105,115107,115109,115111,115113,115119,115133,115139,115141,115143,115145,115151,115153,115155,115157,115163,115165,115167,115169,115171,115181,115183,115185,115187,115189,115191,115193],{},[613,115095,115096,115103],{},[39689,115097,115098,115100],{"accent":70},[59,115099,11],{},[133,115101,115102],{},"^",[59,115104,69998],{},[133,115106,136],{"stretchy":135},[59,115108,1755],{},[133,115110,161],{"stretchy":135},[133,115112,793],{},[5362,115114,115115,115117],{},[539,115116,546],{},[59,115118,12962],{},[3187,115120,115121,115123,115131],{},[133,115122,3191],{},[56,115124,115125,115127,115129],{},[59,115126,13377],{},[133,115128,793],{},[539,115130,546],{},[59,115132,12962],{},[613,115134,115135,115137],{},[59,115136,77380],{},[59,115138,69998],{},[133,115140,136],{"stretchy":135},[59,115142,1755],{},[133,115144,5378],{},[613,115146,115147,115149],{},[59,115148,1755],{},[59,115150,13377],{},[133,115152,161],{"stretchy":135},[133,115154,155],{"separator":70},[230,115156],{"width":10251},[613,115158,115159,115161],{},[59,115160,77380],{},[59,115162,69998],{},[133,115164,136],{"stretchy":135},[59,115166,35321],{},[133,115168,161],{"stretchy":135},[133,115170,793],{},[138,115172,115173,115175],{},[59,115174,69998],{},[56,115176,115177,115179],{},[133,115178,5378],{},[59,115180,29425],{},[59,115182,77380],{},[133,115184,136],{"stretchy":135},[59,115186,35321],{},[59,115188,45691],{"mathvariant":145},[59,115190,69998],{},[133,115192,161],{"stretchy":135},[59,115194,2903],{"mathvariant":145},[63,115196,115197],{"encoding":65},"\\hat p_h(x)=\\frac1n\\sum_{i=1}^nK_h(x-x_i),\\qquad K_h(u)=h^{-d}K(u\u002Fh).",[39,115199,115201,115306,115500,115616],{"className":115200,"ariaHidden":70},[69],[39,115202,115204,115207,115288,115291,115294,115297,115300,115303],{"className":115203},[74],[39,115205],{"className":115206,"style":245},[78],[39,115208,115210,115254],{"className":115209},[83],[39,115211,115213],{"className":115212},[83,39712],[39,115214,115216,115245],{"className":115215},[196,649],[39,115217,115219,115242],{"className":115218},[200],[39,115220,115222,115230],{"className":115221,"style":28824},[204],[39,115223,115224,115227],{"style":21599},[39,115225],{"className":115226,"style":935},[212],[39,115228,11],{"className":115229},[83,84],[39,115231,115232,115235],{"style":21599},[39,115233],{"className":115234,"style":935},[212],[39,115236,115239],{"className":115237,"style":115238},[39739],"left:-0.1667em;",[39,115240,115102],{"className":115241},[83],[39,115243,673],{"className":115244},[672],[39,115246,115248],{"className":115247},[200],[39,115249,115252],{"className":115250,"style":115251},[204],"height:0.1944em;",[39,115253],{},[39,115255,115257],{"className":115256},[192],[39,115258,115260,115280],{"className":115259},[196,649],[39,115261,115263,115277],{"className":115262},[200],[39,115264,115266],{"className":115265,"style":3419},[204],[39,115267,115268,115271],{"style":966},[39,115269],{"className":115270,"style":213},[212],[39,115272,115274],{"className":115273},[217,218,219,220],[39,115275,69998],{"className":115276},[83,84,220],[39,115278,673],{"className":115279},[672],[39,115281,115283],{"className":115282},[200],[39,115284,115286],{"className":115285,"style":680},[204],[39,115287],{},[39,115289,136],{"className":115290},[182],[39,115292,1755],{"className":115293},[83,84],[39,115295,161],{"className":115296},[263],[39,115298],{"className":115299,"style":231},[230],[39,115301,793],{"className":115302},[235],[39,115304],{"className":115305,"style":231},[230],[39,115307,115309,115313,115375,115378,115442,115445,115485,115488,115491,115494,115497],{"className":115308},[74],[39,115310],{"className":115311,"style":115312},[78],"height:2.9291em;vertical-align:-1.2777em;",[39,115314,115316,115319,115372],{"className":115315},[83],[39,115317],{"className":115318},[182,5463],[39,115320,115322],{"className":115321},[5362],[39,115323,115325,115364],{"className":115324},[196,649],[39,115326,115328,115361],{"className":115327},[200],[39,115329,115331,115342,115350],{"className":115330,"style":18240},[204],[39,115332,115333,115336],{"style":5479},[39,115334],{"className":115335,"style":935},[212],[39,115337,115339],{"className":115338},[83],[39,115340,12962],{"className":115341},[83,84],[39,115343,115344,115347],{"style":5503},[39,115345],{"className":115346,"style":935},[212],[39,115348],{"className":115349,"style":5511},[5510],[39,115351,115352,115355],{"style":5514},[39,115353],{"className":115354,"style":935},[212],[39,115356,115358],{"className":115357},[83],[39,115359,546],{"className":115360},[83],[39,115362,673],{"className":115363},[672],[39,115365,115367],{"className":115366},[200],[39,115368,115370],{"className":115369,"style":19487},[204],[39,115371],{},[39,115373],{"className":115374},[263,5463],[39,115376],{"className":115377,"style":256},[230],[39,115379,115381],{"className":115380},[1744,3294],[39,115382,115384,115434],{"className":115383},[196,649],[39,115385,115387,115431],{"className":115386},[200],[39,115388,115390,115410,115420],{"className":115389,"style":3304},[204],[39,115391,115392,115395],{"style":13926},[39,115393],{"className":115394,"style":3311},[212],[39,115396,115398],{"className":115397},[217,218,219,220],[39,115399,115401,115404,115407],{"className":115400},[83,220],[39,115402,13377],{"className":115403},[83,84,220],[39,115405,793],{"className":115406},[235,220],[39,115408,546],{"className":115409},[83,220],[39,115411,115412,115415],{"style":3330},[39,115413],{"className":115414,"style":3311},[212],[39,115416,115417],{},[39,115418,3191],{"className":115419},[1744,3339,3340],[39,115421,115422,115425],{"style":3343},[39,115423],{"className":115424,"style":3311},[212],[39,115426,115428],{"className":115427},[217,218,219,220],[39,115429,12962],{"className":115430},[83,84,220],[39,115432,673],{"className":115433},[672],[39,115435,115437],{"className":115436},[200],[39,115438,115440],{"className":115439,"style":79128},[204],[39,115441],{},[39,115443],{"className":115444,"style":256},[230],[39,115446,115448,115451],{"className":115447},[83],[39,115449,77380],{"className":115450,"style":48487},[83,84],[39,115452,115454],{"className":115453},[192],[39,115455,115457,115477],{"className":115456},[196,649],[39,115458,115460,115474],{"className":115459},[200],[39,115461,115463],{"className":115462,"style":3419},[204],[39,115464,115465,115468],{"style":81320},[39,115466],{"className":115467,"style":213},[212],[39,115469,115471],{"className":115470},[217,218,219,220],[39,115472,69998],{"className":115473},[83,84,220],[39,115475,673],{"className":115476},[672],[39,115478,115480],{"className":115479},[200],[39,115481,115483],{"className":115482,"style":680},[204],[39,115484],{},[39,115486,136],{"className":115487},[182],[39,115489,1755],{"className":115490},[83,84],[39,115492],{"className":115493,"style":461},[230],[39,115495,5378],{"className":115496},[465],[39,115498],{"className":115499,"style":461},[230],[39,115501,115503,115506,115546,115549,115552,115555,115558,115598,115601,115604,115607,115610,115613],{"className":115502},[74],[39,115504],{"className":115505,"style":245},[78],[39,115507,115509,115512],{"className":115508},[83],[39,115510,1755],{"className":115511},[83,84],[39,115513,115515],{"className":115514},[192],[39,115516,115518,115538],{"className":115517},[196,649],[39,115519,115521,115535],{"className":115520},[200],[39,115522,115524],{"className":115523,"style":13604},[204],[39,115525,115526,115529],{"style":966},[39,115527],{"className":115528,"style":213},[212],[39,115530,115532],{"className":115531},[217,218,219,220],[39,115533,13377],{"className":115534},[83,84,220],[39,115536,673],{"className":115537},[672],[39,115539,115541],{"className":115540},[200],[39,115542,115544],{"className":115543,"style":680},[204],[39,115545],{},[39,115547,161],{"className":115548},[263],[39,115550,155],{"className":115551},[252],[39,115553],{"className":115554,"style":10787},[230],[39,115556],{"className":115557,"style":256},[230],[39,115559,115561,115564],{"className":115560},[83],[39,115562,77380],{"className":115563,"style":48487},[83,84],[39,115565,115567],{"className":115566},[192],[39,115568,115570,115590],{"className":115569},[196,649],[39,115571,115573,115587],{"className":115572},[200],[39,115574,115576],{"className":115575,"style":3419},[204],[39,115577,115578,115581],{"style":81320},[39,115579],{"className":115580,"style":213},[212],[39,115582,115584],{"className":115583},[217,218,219,220],[39,115585,69998],{"className":115586},[83,84,220],[39,115588,673],{"className":115589},[672],[39,115591,115593],{"className":115592},[200],[39,115594,115596],{"className":115595,"style":680},[204],[39,115597],{},[39,115599,136],{"className":115600},[182],[39,115602,35321],{"className":115603},[83,84],[39,115605,161],{"className":115606},[263],[39,115608],{"className":115609,"style":231},[230],[39,115611,793],{"className":115612},[235],[39,115614],{"className":115615,"style":231},[230],[39,115617,115619,115622,115657,115660,115663,115666,115669,115672,115675],{"className":115618},[74],[39,115620],{"className":115621,"style":30346},[78],[39,115623,115625,115628],{"className":115624},[83],[39,115626,69998],{"className":115627},[83,84],[39,115629,115631],{"className":115630},[192],[39,115632,115634],{"className":115633},[196],[39,115635,115637],{"className":115636},[200],[39,115638,115640],{"className":115639,"style":3389},[204],[39,115641,115642,115645],{"style":1027},[39,115643],{"className":115644,"style":213},[212],[39,115646,115648],{"className":115647},[217,218,219,220],[39,115649,115651,115654],{"className":115650},[83,220],[39,115652,5378],{"className":115653},[83,220],[39,115655,29425],{"className":115656},[83,84,220],[39,115658,77380],{"className":115659,"style":48487},[83,84],[39,115661,136],{"className":115662},[182],[39,115664,35321],{"className":115665},[83,84],[39,115667,45691],{"className":115668},[83],[39,115670,69998],{"className":115671},[83,84],[39,115673,161],{"className":115674},[263],[39,115676,2903],{"className":115677},[83],[11,115679,115680,115681,742,115740,742,115927,116081],{},"它的偏差和方差可以算出来（记 ",[39,115682,115684,115704],{"className":115683,"translate":43},[42],[39,115685,115687],{"className":115686},[47],[49,115688,115689],{"xmlns":51},[53,115690,115691,115701],{},[56,115692,115693,115695,115697,115699],{},[133,115694,69087],{},[59,115696,77380],{},[133,115698,793],{},[539,115700,546],{},[63,115702,115703],{"encoding":65},"\\int K=1",[39,115705,115707,115731],{"className":115706,"ariaHidden":70},[69],[39,115708,115710,115713,115716,115719,115722,115725,115728],{"className":115709},[74],[39,115711],{"className":115712,"style":74333},[78],[39,115714,69087],{"className":115715,"style":74337},[1744,3339,3952],[39,115717],{"className":115718,"style":256},[230],[39,115720,77380],{"className":115721,"style":48487},[83,84],[39,115723],{"className":115724,"style":231},[230],[39,115726,793],{"className":115727},[235],[39,115729],{"className":115730,"style":231},[230],[39,115732,115734,115737],{"className":115733},[74],[39,115735],{"className":115736,"style":2473},[78],[39,115738,546],{"className":115739},[83],[39,115741,115743,115789],{"className":115742,"translate":43},[42],[39,115744,115746],{"className":115745},[47],[49,115747,115748],{"xmlns":51},[53,115749,115750,115786],{},[56,115751,115752,115754,115756,115758,115764,115766,115768,115770,115772,115774,115776,115778],{},[133,115753,69087],{},[59,115755,15389],{"mathvariant":145},[59,115757,35321],{},[138,115759,115760,115762],{},[59,115761,15389],{"mathvariant":145},[539,115763,17359],{},[59,115765,77380],{},[133,115767,136],{"stretchy":135},[59,115769,35321],{},[133,115771,161],{"stretchy":135},[59,115773,29425],{},[59,115775,35321],{},[133,115777,793],{},[9932,115779,115780,115782,115784],{},[59,115781,90086],{},[59,115783,77380],{},[539,115785,17359],{},[63,115787,115788],{"encoding":65},"\\int\\|u\\|^2K(u)du=\\sigma_K^2",[39,115790,115792,115867],{"className":115791,"ariaHidden":70},[69],[39,115793,115795,115799,115802,115805,115808,115811,115840,115843,115846,115849,115852,115855,115858,115861,115864],{"className":115794},[74],[39,115796],{"className":115797,"style":115798},[78],"height:1.1202em;vertical-align:-0.3061em;",[39,115800,69087],{"className":115801,"style":74337},[1744,3339,3952],[39,115803],{"className":115804,"style":256},[230],[39,115806,15389],{"className":115807},[83],[39,115809,35321],{"className":115810},[83,84],[39,115812,115814,115817],{"className":115813},[83],[39,115815,15389],{"className":115816},[83],[39,115818,115820],{"className":115819},[192],[39,115821,115823],{"className":115822},[196],[39,115824,115826],{"className":115825},[200],[39,115827,115829],{"className":115828,"style":19710},[204],[39,115830,115831,115834],{"style":208},[39,115832],{"className":115833,"style":213},[212],[39,115835,115837],{"className":115836},[217,218,219,220],[39,115838,17359],{"className":115839},[83,220],[39,115841,77380],{"className":115842,"style":48487},[83,84],[39,115844,136],{"className":115845},[182],[39,115847,35321],{"className":115848},[83,84],[39,115850,161],{"className":115851},[263],[39,115853,29425],{"className":115854},[83,84],[39,115856,35321],{"className":115857},[83,84],[39,115859],{"className":115860,"style":231},[230],[39,115862,793],{"className":115863},[235],[39,115865],{"className":115866,"style":231},[230],[39,115868,115870,115874],{"className":115869},[74],[39,115871],{"className":115872,"style":115873},[78],"height:1.0894em;vertical-align:-0.2753em;",[39,115875,115877,115880],{"className":115876},[83],[39,115878,90086],{"className":115879,"style":2114},[83,84],[39,115881,115883],{"className":115882},[192],[39,115884,115886,115918],{"className":115885},[196,649],[39,115887,115889,115915],{"className":115888},[200],[39,115890,115892,115904],{"className":115891,"style":19710},[204],[39,115893,115895,115898],{"style":115894},"top:-2.4247em;margin-left:-0.0359em;margin-right:0.05em;",[39,115896],{"className":115897,"style":213},[212],[39,115899,115901],{"className":115900},[217,218,219,220],[39,115902,77380],{"className":115903,"style":48487},[83,84,220],[39,115905,115906,115909],{"style":208},[39,115907],{"className":115908,"style":213},[212],[39,115910,115912],{"className":115911},[217,218,219,220],[39,115913,17359],{"className":115914},[83,220],[39,115916,673],{"className":115917},[672],[39,115919,115921],{"className":115920},[200],[39,115922,115925],{"className":115923,"style":115924},[204],"height:0.2753em;",[39,115926],{},[39,115928,115930,115964],{"className":115929,"translate":43},[42],[39,115931,115933],{"className":115932},[47],[49,115934,115935],{"xmlns":51},[53,115936,115937,115961],{},[56,115938,115939,115941,115943,115951,115953,115955],{},[59,115940,15389],{"mathvariant":145},[59,115942,77380],{},[9932,115944,115945,115947,115949],{},[59,115946,15389],{"mathvariant":145},[539,115948,17359],{},[539,115950,17359],{},[133,115952,793],{},[133,115954,69087],{},[138,115956,115957,115959],{},[59,115958,77380],{},[539,115960,17359],{},[63,115962,115963],{"encoding":65},"\\|K\\|_2^2=\\int K^2",[39,115965,115967,116040],{"className":115966,"ariaHidden":70},[69],[39,115968,115970,115973,115976,115979,116031,116034,116037],{"className":115969},[74],[39,115971],{"className":115972,"style":84808},[78],[39,115974,15389],{"className":115975},[83],[39,115977,77380],{"className":115978,"style":48487},[83,84],[39,115980,115982,115985],{"className":115981},[83],[39,115983,15389],{"className":115984},[83],[39,115986,115988],{"className":115987},[192],[39,115989,115991,116023],{"className":115990},[196,649],[39,115992,115994,116020],{"className":115993},[200],[39,115995,115997,116009],{"className":115996,"style":19710},[204],[39,115998,116000,116003],{"style":115999},"top:-2.4519em;margin-left:0em;margin-right:0.05em;",[39,116001],{"className":116002,"style":213},[212],[39,116004,116006],{"className":116005},[217,218,219,220],[39,116007,17359],{"className":116008},[83,220],[39,116010,116011,116014],{"style":208},[39,116012],{"className":116013,"style":213},[212],[39,116015,116017],{"className":116016},[217,218,219,220],[39,116018,17359],{"className":116019},[83,220],[39,116021,673],{"className":116022},[672],[39,116024,116026],{"className":116025},[200],[39,116027,116029],{"className":116028,"style":90540},[204],[39,116030],{},[39,116032],{"className":116033,"style":231},[230],[39,116035,793],{"className":116036},[235],[39,116038],{"className":116039,"style":231},[230],[39,116041,116043,116046,116049,116052],{"className":116042},[74],[39,116044],{"className":116045,"style":115798},[78],[39,116047,69087],{"className":116048,"style":74337},[1744,3339,3952],[39,116050],{"className":116051,"style":256},[230],[39,116053,116055,116058],{"className":116054},[83],[39,116056,77380],{"className":116057,"style":48487},[83,84],[39,116059,116061],{"className":116060},[192],[39,116062,116064],{"className":116063},[196],[39,116065,116067],{"className":116066},[200],[39,116068,116070],{"className":116069,"style":19710},[204],[39,116071,116072,116075],{"style":208},[39,116073],{"className":116074,"style":213},[212],[39,116076,116078],{"className":116077},[217,218,219,220],[39,116079,17359],{"className":116080},[83,220],"）：",[39,116083,116085],{"className":116084,"translate":43},[746],[39,116086,116088,116230],{"className":116087,"translate":43},[42],[39,116089,116091],{"className":116090},[47],[49,116092,116093],{"xmlns":51,"display":755},[53,116094,116095,116227],{},[56,116096,116097,116099,116101,116111,116113,116115,116117,116119,116121,116123,116125,116127,116129,116149,116151,116153,116155,116157,116159,116161,116163,116171,116173,116183,116185,116187,116189,116191,116225],{},[59,116098,4253],{"mathvariant":421},[35841,116100,35843],{},[613,116102,116103,116109],{},[39689,116104,116105,116107],{"accent":70},[59,116106,11],{},[133,116108,115102],{},[59,116110,69998],{},[133,116112,136],{"stretchy":135},[59,116114,1755],{},[133,116116,161],{"stretchy":135},[133,116118,5378],{},[59,116120,11],{},[133,116122,136],{"stretchy":135},[59,116124,1755],{},[133,116126,161],{"stretchy":135},[133,116128,99937],{},[5362,116130,116131,116147],{},[56,116132,116133,116139],{},[138,116134,116135,116137],{},[59,116136,69998],{},[539,116138,17359],{},[9932,116140,116141,116143,116145],{},[59,116142,90086],{},[59,116144,77380],{},[539,116146,17359],{},[539,116148,17359],{},[59,116150,2134],{"mathvariant":145},[59,116152,11],{},[133,116154,136],{"stretchy":135},[59,116156,1755],{},[133,116158,161],{"stretchy":135},[133,116160,155],{"separator":70},[230,116162],{"width":10251},[56,116164,116165,116167,116169],{},[59,116166,4230],{"mathvariant":145},[59,116168,158],{"mathvariant":145},[59,116170,401],{"mathvariant":145},[35841,116172,35843],{},[613,116174,116175,116181],{},[39689,116176,116177,116179],{"accent":70},[59,116178,11],{},[133,116180,115102],{},[59,116182,69998],{},[133,116184,136],{"stretchy":135},[59,116186,1755],{},[133,116188,161],{"stretchy":135},[133,116190,99937],{},[5362,116192,116193,116215],{},[56,116194,116195,116197,116199,116201,116203,116205,116207],{},[59,116196,11],{},[133,116198,136],{"stretchy":135},[59,116200,1755],{},[133,116202,161],{"stretchy":135},[59,116204,15389],{"mathvariant":145},[59,116206,77380],{},[9932,116208,116209,116211,116213],{},[59,116210,15389],{"mathvariant":145},[539,116212,17359],{},[539,116214,17359],{},[56,116216,116217,116219],{},[59,116218,12962],{},[138,116220,116221,116223],{},[59,116222,69998],{},[59,116224,29425],{},[59,116226,2903],{"mathvariant":145},[63,116228,116229],{"encoding":65},"\\mathbb E\\,\\hat p_h(x)-p(x)\\approx\\frac{h^2\\sigma_K^2}{2}\\Delta p(x),\n\\qquad\n\\mathrm{Var}\\,\\hat p_h(x)\\approx\\frac{p(x)\\|K\\|_2^2}{nh^d}.",[39,116231,116233,116342,116369,116645],{"className":116232,"ariaHidden":70},[69],[39,116234,116236,116239,116242,116245,116324,116327,116330,116333,116336,116339],{"className":116235},[74],[39,116237],{"className":116238,"style":245},[78],[39,116240,4253],{"className":116241},[83,515],[39,116243],{"className":116244,"style":256},[230],[39,116246,116248,116290],{"className":116247},[83],[39,116249,116251],{"className":116250},[83,39712],[39,116252,116254,116282],{"className":116253},[196,649],[39,116255,116257,116279],{"className":116256},[200],[39,116258,116260,116268],{"className":116259,"style":28824},[204],[39,116261,116262,116265],{"style":21599},[39,116263],{"className":116264,"style":935},[212],[39,116266,11],{"className":116267},[83,84],[39,116269,116270,116273],{"style":21599},[39,116271],{"className":116272,"style":935},[212],[39,116274,116276],{"className":116275,"style":115238},[39739],[39,116277,115102],{"className":116278},[83],[39,116280,673],{"className":116281},[672],[39,116283,116285],{"className":116284},[200],[39,116286,116288],{"className":116287,"style":115251},[204],[39,116289],{},[39,116291,116293],{"className":116292},[192],[39,116294,116296,116316],{"className":116295},[196,649],[39,116297,116299,116313],{"className":116298},[200],[39,116300,116302],{"className":116301,"style":3419},[204],[39,116303,116304,116307],{"style":966},[39,116305],{"className":116306,"style":213},[212],[39,116308,116310],{"className":116309},[217,218,219,220],[39,116311,69998],{"className":116312},[83,84,220],[39,116314,673],{"className":116315},[672],[39,116317,116319],{"className":116318},[200],[39,116320,116322],{"className":116321,"style":680},[204],[39,116323],{},[39,116325,136],{"className":116326},[182],[39,116328,1755],{"className":116329},[83,84],[39,116331,161],{"className":116332},[263],[39,116334],{"className":116335,"style":461},[230],[39,116337,5378],{"className":116338},[465],[39,116340],{"className":116341,"style":461},[230],[39,116343,116345,116348,116351,116354,116357,116360,116363,116366],{"className":116344},[74],[39,116346],{"className":116347,"style":245},[78],[39,116349,11],{"className":116350},[83,84],[39,116352,136],{"className":116353},[182],[39,116355,1755],{"className":116356},[83,84],[39,116358,161],{"className":116359},[263],[39,116361],{"className":116362,"style":231},[230],[39,116364,99937],{"className":116365},[235],[39,116367],{"className":116368,"style":231},[230],[39,116370,116372,116375,116514,116517,116520,116523,116526,116529,116532,116535,116538,116545,116548,116627,116630,116633,116636,116639,116642],{"className":116371},[74],[39,116373],{"className":116374,"style":77884},[78],[39,116376,116378,116381,116511],{"className":116377},[83],[39,116379],{"className":116380},[182,5463],[39,116382,116384],{"className":116383},[5362],[39,116385,116387,116503],{"className":116386},[196,649],[39,116388,116390,116500],{"className":116389},[200],[39,116391,116393,116404,116412],{"className":116392,"style":77903},[204],[39,116394,116395,116398],{"style":5479},[39,116396],{"className":116397,"style":935},[212],[39,116399,116401],{"className":116400},[83],[39,116402,17359],{"className":116403},[83],[39,116405,116406,116409],{"style":5503},[39,116407],{"className":116408,"style":935},[212],[39,116410],{"className":116411,"style":5511},[5510],[39,116413,116414,116417],{"style":5514},[39,116415],{"className":116416,"style":935},[212],[39,116418,116420,116449],{"className":116419},[83],[39,116421,116423,116426],{"className":116422},[83],[39,116424,69998],{"className":116425},[83,84],[39,116427,116429],{"className":116428},[192],[39,116430,116432],{"className":116431},[196],[39,116433,116435],{"className":116434},[200],[39,116436,116438],{"className":116437,"style":19710},[204],[39,116439,116440,116443],{"style":208},[39,116441],{"className":116442,"style":213},[212],[39,116444,116446],{"className":116445},[217,218,219,220],[39,116447,17359],{"className":116448},[83,220],[39,116450,116452,116455],{"className":116451},[83],[39,116453,90086],{"className":116454,"style":2114},[83,84],[39,116456,116458],{"className":116457},[192],[39,116459,116461,116492],{"className":116460},[196,649],[39,116462,116464,116489],{"className":116463},[200],[39,116465,116467,116478],{"className":116466,"style":19710},[204],[39,116468,116469,116472],{"style":115894},[39,116470],{"className":116471,"style":213},[212],[39,116473,116475],{"className":116474},[217,218,219,220],[39,116476,77380],{"className":116477,"style":48487},[83,84,220],[39,116479,116480,116483],{"style":208},[39,116481],{"className":116482,"style":213},[212],[39,116484,116486],{"className":116485},[217,218,219,220],[39,116487,17359],{"className":116488},[83,220],[39,116490,673],{"className":116491},[672],[39,116493,116495],{"className":116494},[200],[39,116496,116498],{"className":116497,"style":115924},[204],[39,116499],{},[39,116501,673],{"className":116502},[672],[39,116504,116506],{"className":1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ame":116800},[200],[39,116802,116804],{"className":116803,"style":19487},[204],[39,116805],{},[39,116807],{"className":116808},[263,5463],[39,116810,2903],{"className":116811},[83],[11,116813,116814,116815,116843,116844,116872,116873,117022,117023,1607],{},"两项一个随 ",[39,116816,116818,116831],{"className":116817,"translate":43},[42],[39,116819,116821],{"className":116820},[47],[49,116822,116823],{"xmlns":51},[53,116824,116825,116829],{},[56,116826,116827],{},[59,116828,69998],{},[63,116830,69998],{"encoding":65},[39,116832,116834],{"className":116833,"ariaHidden":70},[69],[39,116835,116837,116840],{"className":116836},[74],[39,116838],{"className":116839,"style":28824},[78],[39,116841,69998],{"className":116842},[83,84]," 增、一个随 ",[39,116845,116847,116860],{"className":116846,"translate":43},[42],[39,116848,116850],{"className":116849},[47],[49,116851,116852],{"xmlns":51},[53,116853,116854,116858],{},[56,116855,116856],{},[59,116857,69998],{},[63,116859,69998],{"encoding":65},[39,116861,116863],{"className":116862,"ariaHidden":70},[69],[39,116864,116866,116869],{"className":116865},[74],[39,116867],{"className":116868,"style":28824},[78],[39,116870,69998],{"className":116871},[83,84]," 减，最优带宽 ",[39,116874,116876,116919],{"className":116875,"translate":43},[42],[39,116877,116879],{"className":116878},[47],[49,116880,116881],{"xmlns":51},[53,116882,116883,116916],{},[56,116884,116885,116891,116894],{},[138,116886,116887,116889],{},[59,116888,69998],{},[133,116890,21884],{},[133,116892,116893],{},"∝",[138,116895,116896,116898],{},[59,116897,12962],{},[56,116899,116900,116902,116904,116906,116908,116910,116912,116914],{},[133,116901,5378],{},[539,116903,546],{},[59,116905,45691],{"mathvariant":145},[133,116907,136],{"stretchy":135},[59,116909,29425],{},[133,116911,788],{},[539,116913,59223],{},[133,116915,161],{"stretchy":135},[63,116917,116918],{"encoding":65},"h^*\\propto n^{-1\u002F(d+4)}",[39,116920,116922,116966],{"className":116921,"ariaHidden":70},[69],[39,116923,116925,116928,116957,116960,116963],{"className":116924},[74],[39,116926],{"className":116927,"style":28824},[78],[39,116929,116931,116934],{"className":116930},[83],[39,116932,69998],{"className":116933},[83,84],[39,116935,116937],{"className":116936},[192],[39,116938,116940],{"className":116939},[196],[39,116941,116943],{"className":116942},[200],[39,116944,116946],{"className":116945,"style":21897},[204],[39,116947,116948,116951],{"style":208},[39,116949],{"className":116950,"style":213},[212],[39,116952,116954],{"className":116953},[217,218,219,220],[39,116955,21884],{"className":116956},[465,220],[39,116958],{"className":116959,"style":231},[230],[39,116961,116893],{"className":116962},[235],[39,116964],{"className":116965,"style":231},[230],[39,116967,116969,116972],{"className":116968},[74],[39,116970],{"className":116971,"style":95578},[78],[39,116973,116975,116978],{"className":116974},[83],[39,116976,12962],{"className":116977},[83,84],[39,116979,116981],{"className":116980},[192],[39,116982,116984],{"className":116983},[196],[39,116985,116987],{"className":116986},[200],[39,116988,116990],{"className":116989,"style":95578},[204],[39,116991,116992,116995],{"style":208},[39,116993],{"className":116994,"style":213},[212],[39,116996,116998],{"className":116997},[217,218,219,220],[39,116999,117001,117004,117007,117010,117013,117016,117019],{"className":117000},[83,220],[39,117002,5378],{"className":117003},[83,220],[39,117005,76865],{"className":117006},[83,220],[39,117008,136],{"className":117009},[182,220],[39,117011,29425],{"className":117012},[83,84,220],[39,117014,788],{"className":117015},[465,220],[39,117017,59223],{"className":117018},[83,220],[39,117020,161],{"className":117021},[263,220],"，此时 ",[39,117024,117026,117070],{"className":117025,"translate":43},[42],[39,117027,117029],{"className":117028},[47],[49,117030,117031],{"xmlns":51},[53,117032,117033,117067],{},[56,117034,117035,117043,117045],{},[56,117036,117037,117039,117041],{},[59,117038,1507],{"mathvariant":145},[59,117040,61],{"mathvariant":145},[59,117042,4253],{"mathvariant":145},[133,117044,116893],{},[138,117046,117047,117049],{},[59,117048,12962],{},[56,117050,117051,117053,117055,117057,117059,117061,117063,117065],{},[133,117052,5378],{},[539,117054,59223],{},[59,117056,45691],{"mathvariant":145},[133,117058,136],{"stretchy":135},[59,117060,29425],{},[133,117062,788],{},[539,117064,59223],{},[133,117066,161],{"stretchy":135},[63,117068,117069],{"encoding":65},"\\mathrm{MSE}\\propto n^{-4\u002F(d+4)}",[39,117071,117073,117095],{"className":117072,"ariaHidden":70},[69],[39,117074,117076,117079,117086,117089,117092],{"className":117075},[74],[39,117077],{"className":117078,"style":79},[78],[39,117080,117082],{"className":117081},[83],[39,117083,117085],{"className":117084},[83,35077],"MSE",[39,117087],{"className":117088,"style":231},[230],[39,117090,116893],{"className":117091},[235],[39,117093],{"className":117094,"style":231},[230],[39,117096,117098,117101],{"className":117097},[74],[39,117099],{"className":117100,"style":95578},[78],[39,117102,117104,117107],{"className":117103},[83],[39,117105,12962],{"className":117106},[83,84],[39,117108,117110],{"className":117109},[192],[39,117111,117113],{"className":117112},[196],[39,117114,117116],{"className":117115},[200],[39,117117,117119],{"className":117118,"style":95578},[204],[39,117120,117121,117124],{"style":208},[39,117122],{"className":117123,"style":213},[212],[39,117125,117127],{"className":117126},[217,218,219,220],[39,117128,117130,117133,117137,117140,117143,117146,117149],{"className":117129},[83,220],[39,117131,5378],{"className":117132},[83,220],[39,117134,117136],{"className":117135},[83,220],"4\u002F",[39,117138,136],{"className":117139},[182,220],[39,117141,29425],{"className":117142},[83,84,220],[39,117144,788],{"className":117145},[465,220],[39,117147,59223],{"className":117148},[83,220],[39,117150,161],{"className":117151},[263,220],[11,117153,117154,117205,117206,117280,117281,117205,117332,117407,117408,117493],{},[39,117155,117157,117175],{"className":117156,"translate":43},[42],[39,117158,117160],{"className":117159},[47],[49,117161,117162],{"xmlns":51},[53,117163,117164,117172],{},[56,117165,117166,117168,117170],{},[59,117167,29425],{},[133,117169,793],{},[539,117171,546],{},[63,117173,117174],{"encoding":65},"d=1",[39,117176,117178,117196],{"className":117177,"ariaHidden":70},[69],[39,117179,117181,117184,117187,117190,117193],{"className":117180},[74],[39,117182],{"className":117183,"style":28824},[78],[39,117185,29425],{"className":117186},[83,84],[39,117188],{"className":117189,"style":231},[230],[39,117191,793],{"className":117192},[235],[39,117194],{"className":117195,"style":231},[230],[39,117197,117199,117202],{"className":117198},[74],[39,117200],{"className":117201,"style":2473},[78],[39,117203,546],{"className":117204},[83]," 时是 ",[39,117207,117209,117235],{"className":117208,"translate":43},[42],[39,117210,117212],{"className":117211},[47],[49,117213,117214],{"xmlns":51},[53,117215,117216,117232],{},[56,117217,117218],{},[138,117219,117220,117222],{},[59,117221,12962],{},[56,117223,117224,117226,117228,117230],{},[133,117225,5378],{},[539,117227,59223],{},[59,117229,45691],{"mathvariant":145},[539,117231,59259],{},[63,117233,117234],{"encoding":65},"n^{-4\u002F5}",[39,117236,117238],{"className":117237,"ariaHidden":70},[69],[39,117239,117241,117244],{"className":117240},[74],[39,117242],{"className":117243,"style":95578},[78],[39,117245,117247,117250],{"className":117246},[83],[39,117248,12962],{"className":117249},[83,84],[39,117251,117253],{"className":117252},[192],[39,117254,117256],{"className":117255},[196],[39,117257,117259],{"className":117258},[200],[39,117260,117262],{"className":117261,"style":95578},[204],[39,117263,117264,117267],{"style":208},[39,117265],{"className":117266,"style":213},[212],[39,117268,117270],{"className":117269},[217,218,219,220],[39,117271,117273,117276],{"className":117272},[83,220],[39,117274,5378],{"className":117275},[83,220],[39,117277,117279],{"className":117278},[83,220],"4\u002F5","，很不错。",[39,117282,117284,117302],{"className":117283,"translate":43},[42],[39,117285,117287],{"className":117286},[47],[49,117288,117289],{"xmlns":51},[53,117290,117291,117299],{},[56,117292,117293,117295,117297],{},[59,117294,29425],{},[133,117296,793],{},[539,117298,48919],{},[63,117300,117301],{"encoding":65},"d=10",[39,117303,117305,117323],{"className":117304,"ariaHidden":70},[69],[39,117306,117308,117311,117314,117317,117320],{"className":117307},[74],[39,117309],{"className":117310,"style":28824},[78],[39,117312,29425],{"className":117313},[83,84],[39,117315],{"className":117316,"style":231},[230],[39,117318,793],{"className":117319},[235],[39,117321],{"className":117322,"style":231},[230],[39,117324,117326,117329],{"className":117325},[74],[39,117327],{"className":117328,"style":2473},[78],[39,117330,48919],{"className":117331},[83],[39,117333,117335,117362],{"className":117334,"translate":43},[42],[39,117336,117338],{"className":117337},[47],[49,117339,117340],{"xmlns":51},[53,117341,117342,117359],{},[56,117343,117344],{},[138,117345,117346,117348],{},[59,117347,12962],{},[56,117349,117350,117352,117354,117356],{},[133,117351,5378],{},[539,117353,17359],{},[59,117355,45691],{"mathvariant":145},[539,117357,117358],{},"7",[63,117360,117361],{"encoding":65},"n^{-2\u002F7}",[39,117363,117365],{"className":117364,"ariaHidden":70},[69],[39,117366,117368,117371],{"className":117367},[74],[39,117369],{"className":117370,"style":95578},[78],[39,117372,117374,117377],{"className":117373},[83],[39,117375,12962],{"className":117376},[83,84],[39,117378,117380],{"className":117379},[192],[39,117381,117383],{"className":117382},[196],[39,117384,117386],{"className":117385},[200],[39,117387,117389],{"className":117388,"style":95578},[204],[39,117390,117391,117394],{"style":208},[39,117392],{"className":117393,"style":213},[212],[39,117395,117397],{"className":117396},[217,218,219,220],[39,117398,117400,117403],{"className":117399},[83,220],[39,117401,5378],{"className":117402},[83,220],[39,117404,117406],{"className":117405},[83,220],"2\u002F7","，想把误差减半，样本量要乘 ",[39,117409,117411,117434],{"className":117410,"translate":43},[42],[39,117412,117414],{"className":117413},[47],[49,117415,117416],{"xmlns":51},[53,117417,117418,117431],{},[56,117419,117420,117427,117429],{},[138,117421,117422,117424],{},[539,117423,17359],{},[539,117425,117426],{},"3.5",[133,117428,99937],{},[539,117430,73406],{},[63,117432,117433],{"encoding":65},"2^{3.5}\\approx11",[39,117435,117437,117484],{"className":117436,"ariaHidden":70},[69],[39,117438,117440,117443,117475,117478,117481],{"className":117439},[74],[39,117441],{"className":117442,"style":19710},[78],[39,117444,117446,117449],{"className":117445},[83],[39,117447,17359],{"className":117448},[83],[39,117450,117452],{"className":117451},[192],[39,117453,117455],{"className":117454},[196],[39,117456,117458],{"className":117457},[200],[39,117459,117461],{"className":117460,"style":19710},[204],[39,117462,117463,117466],{"style":208},[39,117464],{"className":117465,"style":213},[212],[39,117467,117469],{"className":117468},[217,218,219,220],[39,117470,117472],{"className":117471},[83,220],[39,117473,117426],{"className":117474},[83,220],[39,117476],{"className":117477,"style":231},[230],[39,117479,99937],{"className":117480},[235],[39,117482],{"className":117483,"style":231},[230],[39,117485,117487,117490],{"className":117486},[74],[39,117488],{"className":117489,"style":2473},[78],[39,117491,73406],{"className":117492},[83],"。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”。",[11,117495,117496,117497,117596,117597,117689,117690,117762,117763,117864,117865,118008,118009,118080,118081,117205,118131,1607],{},"但 KDE 在高维还有一层跟带宽无关的麻烦。考虑标准正态 ",[39,117498,117500,117528],{"className":117499,"translate":43},[42],[39,117501,117503],{"className":117502},[47],[49,117504,117505],{"xmlns":51},[53,117506,117507,117525],{},[56,117508,117509,117511,117513,117515,117517,117523],{},[59,117510,77752],{"mathvariant":617},[133,117512,136],{"stretchy":135},[539,117514,541],{},[133,117516,155],{"separator":70},[613,117518,117519,117521],{},[59,117520,14373],{},[59,117522,29425],{},[133,117524,161],{"stretchy":135},[63,117526,117527],{"encoding":65},"\\mathcal N(0,I_d)",[39,117529,117531],{"className":117530,"ariaHidden":70},[69],[39,117532,117534,117537,117540,117543,117546,117549,117552,117593],{"className":117533},[74],[39,117535],{"className":117536,"style":245},[78],[39,117538,77752],{"className":117539,"style":77820},[83,641],[39,117541,136],{"className":117542},[182],[39,117544,541],{"className":117545},[83],[39,117547,155],{"className":117548},[252],[39,117550],{"className":117551,"style":256},[230],[39,117553,117555,117558],{"className":117554},[83],[39,117556,14373],{"className":117557,"style":14420},[83,84],[39,117559,117561],{"className":117560},[192],[39,117562,117564,117585],{"className":117563},[196,649],[39,117565,117567,117582],{"className":117566},[200],[39,117568,117570],{"className":117569,"style":3419},[204],[39,117571,117573,117576],{"style":117572},"top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;",[39,117574],{"className":117575,"style":213},[212],[39,117577,117579],{"className":117578},[217,218,219,220],[39,117580,29425],{"className":117581},[83,84,220],[39,117583,673],{"className":117584},[672],[39,117586,117588],{"className":117587},[200],[39,117589,117591],{"className":117590,"style":680},[204],[39,117592],{},[39,117594,161],{"className":117595},[263],"。众数在原点的密度是 ",[39,117598,117600,117632],{"className":117599,"translate":43},[42],[39,117601,117603],{"className":117602},[47],[49,117604,117605],{"xmlns":51},[53,117606,117607,117629],{},[56,117608,117609,117611,117613,117615],{},[133,117610,136],{"stretchy":135},[539,117612,17359],{},[59,117614,2098],{},[138,117616,117617,117619],{},[133,117618,161],{"stretchy":135},[56,117620,117621,117623,117625,117627],{},[133,117622,5378],{},[59,117624,29425],{},[59,117626,45691],{"mathvariant":145},[539,117628,17359],{},[63,117630,117631],{"encoding":65},"(2\\pi)^{-d\u002F2}",[39,117633,117635],{"className":117634,"ariaHidden":70},[69],[39,117636,117638,117642,117645,117648,117651],{"className":117637},[74],[39,117639],{"className":117640,"style":117641},[78],"height:1.138em;vertical-align:-0.25em;",[39,117643,136],{"className":117644},[182],[39,117646,17359],{"className":117647},[83],[39,117649,2098],{"className":117650,"style":2114},[83,84],[39,117652,117654,117657],{"className":117653},[263],[39,117655,161],{"className":117656},[263],[39,117658,117660],{"className":117659},[192],[39,117661,117663],{"className":117662},[196],[39,117664,117666],{"className":117665},[200],[39,117667,117669],{"className":117668,"style":95578},[204],[39,117670,117671,117674],{"style":208},[39,117672],{"className":117673,"style":213},[212],[39,117675,117677],{"className":117676},[217,218,219,220],[39,117678,117680,117683,117686],{"className":117679},[83,220],[39,117681,5378],{"className":117682},[83,220],[39,117684,29425],{"className":117685},[83,84,220],[39,117687,88107],{"className":117688},[83,220],"；而一个“典型”样本落在半径 ",[39,117691,117693,117709],{"className":117692,"translate":43},[42],[39,117694,117696],{"className":117695},[47],[49,117697,117698],{"xmlns":51},[53,117699,117700,117706],{},[56,117701,117702],{},[99189,117703,117704],{},[59,117705,29425],{},[63,117707,117708],{"encoding":65},"\\sqrt d",[39,117710,117712],{"className":117711,"ariaHidden":70},[69],[39,117713,117715,117718],{"className":117714},[74],[39,117716],{"className":117717,"style":100281},[78],[39,117719,117721],{"className":117720},[83,99251],[39,117722,117724,117754],{"className":117723},[196,649],[39,117725,117727,117751],{"className":117726},[200],[39,117728,117730,117739],{"className":117729,"style":100306},[204],[39,117731,117733,117736],{"className":117732,"style":21599},[99265],[39,117734],{"className":117735,"style":935},[212],[39,117737,29425],{"className":117738,"style":100596},[83,84],[39,117740,117741,117744],{"style":100319},[39,117742],{"className":117743,"style":935},[212],[39,117745,117747],{"className":117746,"style":100326},[99307],[86106,117748,117749],{"xmlns":86108,"width":99311,"height":100329,"viewBox":100330,"preserveAspectRatio":99314},[86112,117750],{"d":100333},[39,117752,673],{"className":117753},[672],[39,117755,117757],{"className":117756},[200],[39,117758,117760],{"className":117759,"style":100343},[204],[39,117761],{}," 的球壳上（因为 ",[39,117764,117766,117796],{"className":117765,"translate":43},[42],[39,117767,117769],{"className":117768},[47],[49,117770,117771],{"xmlns":51},[53,117772,117773,117793],{},[56,117774,117775,117777,117779,117785,117787,117789,117791],{},[59,117776,15389],{"mathvariant":145},[59,117778,29420],{},[138,117780,117781,117783],{},[59,117782,15389],{"mathvariant":145},[539,117784,17359],{},[59,117786,45691],{"mathvariant":145},[59,117788,29425],{},[133,117790,418],{},[539,117792,546],{},[63,117794,117795],{"encoding":65},"\\|X\\|^2\u002Fd\\to1",[39,117797,117799,117855],{"className":117798,"ariaHidden":70},[69],[39,117800,117802,117805,117808,117811,117840,117843,117846,117849,117852],{"className":117801},[74],[39,117803],{"className":117804,"style":84808},[78],[39,117806,15389],{"className":117807},[83],[39,117809,29420],{"className":117810,"style":14420},[83,84],[39,117812,117814,117817],{"className":117813},[83],[39,117815,15389],{"className":117816},[83],[39,117818,117820],{"className":117819},[192],[39,117821,117823],{"className":117822},[196],[39,117824,117826],{"className":117825},[200],[39,117827,117829],{"className":117828,"style":19710},[204],[39,117830,117831,117834],{"style":208},[39,117832],{"className":117833,"style":213},[212],[39,117835,117837],{"className":117836},[217,218,219,220],[39,117838,17359],{"className":117839},[83,220],[39,117841,45691],{"className":117842},[83],[39,117844,29425],{"className":117845},[83,84],[39,117847],{"className":117848,"style":231},[230],[39,117850,418],{"className":117851},[235],[39,117853],{"className":117854,"style":231},[230],[39,117856,117858,117861],{"className":117857},[74],[39,117859],{"className":117860,"style":2473},[78],[39,117862,546],{"className":117863},[83],"），那里的密度是 ",[39,117866,117868,117914],{"className":117867,"translate":43},[42],[39,117869,117871],{"className":117870},[47],[49,117872,117873],{"xmlns":51},[53,117874,117875,117911],{},[56,117876,117877,117879,117881,117883,117897],{},[133,117878,136],{"stretchy":135},[539,117880,17359],{},[59,117882,2098],{},[138,117884,117885,117887],{},[133,117886,161],{"stretchy":135},[56,117888,117889,117891,117893,117895],{},[133,117890,5378],{},[59,117892,29425],{},[59,117894,45691],{"mathvariant":145},[539,117896,17359],{},[138,117898,117899,117901],{},[59,117900,102753],{},[56,117902,117903,117905,117907,117909],{},[133,117904,5378],{},[59,117906,29425],{},[59,117908,45691],{"mathvariant":145},[539,117910,17359],{},[63,117912,117913],{"encoding":65},"(2\\pi)^{-d\u002F2}e^{-d\u002F2}",[39,117915,117917],{"className":117916,"ariaHidden":70},[69],[39,117918,117920,117923,117926,117929,117932,117970],{"className":117919},[74],[39,117921],{"className":117922,"style":117641},[78],[39,117924,136],{"className":117925},[182],[39,117927,17359],{"className":117928},[83],[39,117930,2098],{"className":117931,"style":2114},[83,84],[39,117933,117935,117938],{"className":117934},[263],[39,117936,161],{"className":117937},[263],[39,117939,117941],{"className":117940},[192],[39,117942,117944],{"className":117943},[196],[39,117945,117947],{"className":117946},[200],[39,117948,117950],{"className":117949,"style":95578},[204],[39,117951,117952,117955],{"style":208},[39,117953],{"className":117954,"style":213},[212],[39,117956,117958],{"className":117957},[217,218,219,220],[39,117959,117961,117964,117967],{"className":117960},[83,220],[39,117962,5378],{"className":117963},[83,220],[39,117965,29425],{"className":117966},[83,84,220],[39,117968,88107],{"className":117969},[83,220],[39,117971,117973,117976],{"className":117972},[83],[39,117974,102753],{"className":117975},[83,84],[39,117977,117979],{"className":117978},[192],[39,117980,117982],{"className":117981},[196],[39,117983,117985],{"className":117984},[200],[39,117986,117988],{"className":117987,"style":95578},[204],[39,117989,117990,117993],{"style":208},[39,117991],{"className":117992,"style":213},[212],[39,117994,117996],{"className":117995},[217,218,219,220],[39,117997,117999,118002,118005],{"className":117998},[83,220],[39,118000,5378],{"className":118001},[83,220],[39,118003,29425],{"className":118004},[83,84,220],[39,118006,88107],{"className":118007},[83,220],"。两者差 ",[39,118010,118012,118036],{"className":118011,"translate":43},[42],[39,118013,118015],{"className":118014},[47],[49,118016,118017],{"xmlns":51},[53,118018,118019,118033],{},[56,118020,118021],{},[138,118022,118023,118025],{},[59,118024,102753],{},[56,118026,118027,118029,118031],{},[59,118028,29425],{},[59,118030,45691],{"mathvariant":145},[539,118032,17359],{},[63,118034,118035],{"encoding":65},"e^{d\u002F2}",[39,118037,118039],{"className":118038,"ariaHidden":70},[69],[39,118040,118042,118045],{"className":118041},[74],[39,118043],{"className":118044,"style":95578},[78],[39,118046,118048,118051],{"className":118047},[83],[39,118049,102753],{"className":118050},[83,84],[39,118052,118054],{"className":118053},[192],[39,118055,118057],{"className":118056},[196],[39,118058,118060],{"className":118059},[200],[39,118061,118063],{"className":118062,"style":95578},[204],[39,118064,118065,118068],{"style":208},[39,118066],{"className":118067,"style":213},[212],[39,118069,118071],{"className":118070},[217,218,219,220],[39,118072,118074,118077],{"className":118073},[83,220],[39,118075,29425],{"className":118076},[83,84,220],[39,118078,88107],{"className":118079},[83,220]," 倍，",[39,118082,118084,118101],{"className":118083,"translate":43},[42],[39,118085,118087],{"className":118086},[47],[49,118088,118089],{"xmlns":51},[53,118090,118091,118099],{},[56,118092,118093,118095,118097],{},[59,118094,29425],{},[133,118096,793],{},[539,118098,28724],{},[63,118100,95342],{"encoding":65},[39,118102,118104,118122],{"className":118103,"ariaHidden":70},[69],[39,118105,118107,118110,118113,118116,118119],{"className":118106},[74],[39,118108],{"className":118109,"style":28824},[78],[39,118111,29425],{"className":118112},[83,84],[39,118114],{"className":118115,"style":231},[230],[39,118117,793],{"className":118118},[235],[39,118120],{"className":118121,"style":231},[230],[39,118123,118125,118128],{"className":118124},[74],[39,118126],{"className":118127,"style":2473},[78],[39,118129,28724],{"className":118130},[83],[39,118132,118134,118165],{"className":118133,"translate":43},[42],[39,118135,118137],{"className":118136},[47],[49,118138,118139],{"xmlns":51},[53,118140,118141,118162],{},[56,118142,118143,118149,118151,118153,118155],{},[138,118144,118145,118147],{},[59,118146,102753],{},[539,118148,61535],{},[133,118150,99937],{},[539,118152,59259],{},[133,118154,409],{},[138,118156,118157,118159],{},[539,118158,48919],{},[539,118160,118161],{},"21",[63,118163,118164],{"encoding":65},"e^{50}\\approx5\\times10^{21}",[39,118166,118168,118215,118233],{"className":118167,"ariaHidden":70},[69],[39,118169,118171,118174,118206,118209,118212],{"className":118170},[74],[39,118172],{"className":118173,"style":19710},[78],[39,118175,118177,118180],{"className":118176},[83],[39,118178,102753],{"className":118179},[83,84],[39,118181,118183],{"className":118182},[192],[39,118184,118186],{"className":118185},[196],[39,118187,118189],{"className":118188},[200],[39,118190,118192],{"className":118191,"style":19710},[204],[39,118193,118194,118197],{"style":208},[39,118195],{"className":118196,"style":213},[212],[39,118198,118200],{"className":118199},[217,218,219,220],[39,118201,118203],{"className":118202},[83,220],[39,118204,61535],{"className":118205},[83,220],[39,118207],{"className":118208,"style":231},[230],[39,118210,99937],{"className":118211},[235],[39,118213],{"className":118214,"style":231},[230],[39,118216,118218,118221,118224,118227,118230],{"className":118217},[74],[39,118219],{"className":118220,"style":17882},[78],[39,118222,59259],{"className":118223},[83],[39,118225],{"className":118226,"style":461},[230],[39,118228,409],{"className":118229},[465],[39,118231],{"className":118232,"style":461},[230],[39,118234,118236,118239,118242],{"className":118235},[74],[39,118237],{"className":118238,"style":19710},[78],[39,118240,546],{"className":118241},[83],[39,118243,118245,118248],{"className":118244},[83],[39,118246,541],{"className":118247},[83],[39,118249,118251],{"className":118250},[192],[39,118252,118254],{"className":118253},[196],[39,118255,118257],{"className":118256},[200],[39,118258,118260],{"className":118259,"style":19710},[204],[39,118261,118262,118265],{"style":208},[39,118263],{"className":118264,"style":213},[212],[39,118266,118268],{"className":118267},[217,218,219,220],[39,118269,118271],{"className":118270},[83,220],[39,118272,118161],{"className":118273},[83,220],[11,118275,118276],{},"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。",[11,118278,118279],{},"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。",[11,118281,118282],{},"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。",[21,118284,118285],{"id":118285},"可逆变换",[11,118287,118288,118289,118293,118294,118350],{},"写采样代码时我总会用到同一招：",[118290,118291,118292],"code",{},"random()"," 只能给 ",[39,118295,118297,118320],{"className":118296,"translate":43},[42],[39,118298,118300],{"className":118299},[47],[49,118301,118302],{"xmlns":51},[53,118303,118304,118318],{},[56,118305,118306,118308,118310,118312,118314,118316],{},[59,118307,85587],{},[133,118309,136],{"stretchy":135},[539,118311,541],{},[133,118313,155],{"separator":70},[539,118315,546],{},[133,118317,161],{"stretchy":135},[63,118319,85659],{"encoding":65},[39,118321,118323],{"className":118322,"ariaHidden":70},[69],[39,118324,118326,118329,118332,118335,118338,118341,118344,118347],{"className":118325},[74],[39,118327],{"className":118328,"style":245},[78],[39,118330,85587],{"className":118331,"style":85613},[83,84],[39,118333,136],{"className":118334},[182],[39,118336,541],{"className":118337},[83],[39,118339,155],{"className":118340},[252],[39,118342],{"className":118343,"style":256},[230],[39,118345,546],{"className":118346},[83],[39,118348,161],{"className":118349},[263],"，可我想要指数分布，于是取",[39,118352,118354],{"className":118353,"translate":43},[746],[39,118355,118357,118398],{"className":118356,"translate":43},[42],[39,118358,118360],{"className":118359},[47],[49,118361,118362],{"xmlns":51,"display":755},[53,118363,118364,118395],{},[56,118365,118366,118368,118370,118372,118393],{},[59,118367,1755],{},[133,118369,793],{},[133,118371,5378],{},[5362,118373,118374,118390],{},[56,118375,118376,118378,118380,118382,118384,118386,118388],{},[59,118377,92676],{},[133,118379,1690],{},[133,118381,136],{"stretchy":135},[539,118383,546],{},[133,118385,5378],{},[59,118387,35321],{},[133,118389,161],{"stretchy":135},[59,118391,118392],{},"λ",[59,118394,2903],{"mathvariant":145},[63,118396,118397],{"encoding":65},"x=-\\frac{\\log(1-u)}{\\lambda}.",[39,118399,118401,118419],{"className":118400,"ariaHidden":70},[69],[39,118402,118404,118407,118410,118413,118416],{"className":118403},[74],[39,118405],{"className":118406,"style":290},[78],[39,118408,1755],{"className":118409},[83,84],[39,118411],{"className":118412,"style":231},[230],[39,118414,793],{"className":118415},[235],[39,118417],{"className":118418,"style":231},[230],[39,118420,118422,118426,118429,118514],{"className":118421},[74],[39,118423],{"className":118424,"style":118425},[78],"height:2.113em;vertical-align:-0.686em;",[39,118427,5378],{"className":118428},[83],[39,118430,118432,118435,118511],{"className":118431},[83],[39,118433],{"className":118434},[182,5463],[39,118436,118438],{"className":118437},[5362],[39,118439,118441,118503],{"className":118440},[196,649],[39,118442,118444,118500],{"className":118443},[200],[39,118445,118447,118458,118466],{"className":118446,"style":58292},[204],[39,118448,118449,118452],{"style":5479},[39,118450],{"className":118451,"style":935},[212],[39,118453,118455],{"className":118454},[83],[39,118456,118392],{"className":118457},[83,84],[39,118459,118460,118463],{"style":5503},[39,118461],{"className":118462,"style":935},[212],[39,118464],{"className":118465,"style":5511},[5510],[39,118467,118468,118471],{"style":5514},[39,118469],{"className":118470,"style":935},[212],[39,118472,118474,118479,118482,118485,118488,118491,118494,118497],{"className":118473},[83],[39,118475,92914,118477],{"className":118476},[1744],[39,118478,39209],{"style":35078},[39,118480,136],{"className":118481},[182],[39,118483,546],{"className":118484},[83],[39,118486],{"className":118487,"style":461},[230],[39,118489,5378],{"className":118490},[465],[39,118492],{"className":118493,"style":461},[230],[39,118495,35321],{"className":118496},[83,84],[39,118498,161],{"className":118499},[263],[39,118501,673],{"className":118502},[672],[39,118504,118506],{"className":118505},[200],[39,118507,118509],{"className":118508,"style":19487},[204],[39,118510],{},[39,118512],{"className":118513},[263,5463],[39,118515,2903],{"className":118516},[83],[11,118518,118519,118520,38249,118638,118666],{},"这一个式子就是一个标准化流。反过来看，",[39,118521,118523,118555],{"className":118522,"translate":43},[42],[39,118524,118526],{"className":118525},[47],[49,118527,118528],{"xmlns":51},[53,118529,118530,118552],{},[56,118531,118532,118534,118536,118538,118540],{},[59,118533,35321],{},[133,118535,793],{},[539,118537,546],{},[133,118539,5378],{},[138,118541,118542,118544],{},[59,118543,102753],{},[56,118545,118546,118548,118550],{},[133,118547,5378],{},[59,118549,118392],{},[59,118551,1755],{},[63,118553,118554],{"encoding":65},"u=1-e^{-\\lambda x}",[39,118556,118558,118576,118594],{"className":118557,"ariaHidden":70},[69],[39,118559,118561,118564,118567,118570,118573],{"className":118560},[74],[39,118562],{"className":118563,"style":290},[78],[39,118565,35321],{"className":118566},[83,84],[39,118568],{"className":118569,"style":231},[230],[39,118571,793],{"className":118572},[235],[39,118574],{"className":118575,"style":231},[230],[39,118577,118579,118582,118585,118588,118591],{"className":118578},[74],[39,118580],{"className":118581,"style":17882},[78],[39,118583,546],{"className":118584},[83],[39,118586],{"className":118587,"style":461},[230],[39,118589,5378],{"className":118590},[465],[39,118592],{"className":118593,"style":461},[230],[39,118595,118597,118600],{"className":118596},[74],[39,118598],{"className":118599,"style":15577},[78],[39,118601,118603,118606],{"className":118602},[83],[39,118604,102753],{"className":118605},[83,84],[39,118607,118609],{"className":118608},[192],[39,118610,118612],{"className":118611},[196],[39,118613,118615],{"className":118614},[200],[39,118616,118618],{"className":118617,"style":15577},[204],[39,118619,118620,118623],{"style":208},[39,118621],{"className":118622,"style":213},[212],[39,118624,118626],{"className":118625},[217,218,219,220],[39,118627,118629,118632,118635],{"className":118628},[83,220],[39,118630,5378],{"className":118631},[83,220],[39,118633,118392],{"className":118634},[83,84,220],[39,118636,1755],{"className":118637},[83,84,220],[39,118639,118641,118654],{"className":118640,"translate":43},[42],[39,118642,118644],{"className":118643},[47],[49,118645,118646],{"xmlns":51},[53,118647,118648,118652],{},[56,118649,118650],{},[59,118651,1755],{},[63,118653,1755],{"encoding":65},[39,118655,118657],{"className":118656,"ariaHidden":70},[69],[39,118658,118660,118663],{"className":118659},[74],[39,118661],{"className":118662,"style":290},[78],[39,118664,1755],{"className":118665},[83,84]," 的 CDF，而密度按变量替换是",[39,118668,118670],{"className":118669,"translate":43},[746],[39,118671,118673,118753],{"className":118672,"translate":43},[42],[39,118674,118676],{"className":118675},[47],[49,118677,118678],{"xmlns":51,"display":755},[53,118679,118680,118750],{},[56,118681,118682,118688,118690,118692,118694,118696,118702,118704,118706,118708,118728,118730,118732,118734,118736,118748],{},[613,118683,118684,118686],{},[59,118685,11],{},[59,118687,1755],{},[133,118689,136],{"stretchy":135},[59,118691,1755],{},[133,118693,161],{"stretchy":135},[133,118695,793],{},[613,118697,118698,118700],{},[59,118699,11],{},[59,118701,35321],{},[133,118703,136],{"stretchy":135},[59,118705,35321],{},[133,118707,161],{"stretchy":135},[56,118709,118710,118712,118726],{},[133,118711,150],{"fence":70},[5362,118713,118714,118720],{},[56,118715,118716,118718],{},[59,118717,29425],{},[59,118719,35321],{},[56,118721,118722,118724],{},[59,118723,29425],{},[59,118725,1755],{},[133,118727,150],{"fence":70},[133,118729,793],{},[539,118731,546],{},[133,118733,8885],{},[59,118735,118392],{},[138,118737,118738,118740],{},[59,118739,102753],{},[56,118741,118742,118744,118746],{},[133,118743,5378],{},[59,118745,118392],{},[59,118747,1755],{},[133,118749,155],{"separator":70},[63,118751,118752],{"encoding":65},"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|.",[39,119815,119817,119889,119961],{"className":119816,"ariaHidden":70},[69],[39,119818,119820,119823,119828,119831,119871,119874,119877,119880,119883,119886],{"className":119819},[74],[39,119821],{"className":119822,"style":245},[78],[39,119824,92914,119826],{"className":119825},[1744],[39,119827,39209],{"style":35078},[39,119829],{"className":119830,"style":256},[230],[39,119832,119834,119837],{"className":119833},[83],[39,119835,11],{"className":119836},[83,84],[39,119838,119840],{"className":119839},[192],[39,119841,119843,119863],{"className":119842},[196,649],[39,119844,119846,119860],{"className":119845},[200],[39,119847,119849],{"className":119848,"style":1728},[204],[39,119850,119851,119854],{"style":966},[39,119852],{"className":119853,"style":213},[212],[39,119855,119857],{"className":119856},[217,218,219,220],[39,119858,1755],{"className":119859},[83,84,220],[39,119861,673],{"className":119862},[672],[39,119864,119866],{"className":119865},[200],[39,119867,119869],{"className":119868,"style":680},[204],[39,119870],{},[39,119872,136],{"className":119873},[182],[39,119875,1755],{"className":119876},[83,84],[39,119878,161],{"className":119879},[263],[39,119881],{"className":119882,"style":231},[230],[39,119884,793],{"className":119885},[235],[39,119887],{"className":119888,"style":231},[230],[39,119890,119892,119895,119900,119903,119943,119946,119949,119952,119955,119958],{"className":119891},[74],[39,119893],{"className":119894,"style":245},[78],[39,119896,92914,119898],{"className":119897},[1744],[39,119899,39209],{"style":35078},[39,119901],{"className":119902,"style":256},[230],[39,119904,119906,119909],{"className":119905},[83],[39,119907,11],{"className":119908},[83,84],[39,119910,119912],{"className":119911},[192],[39,119913,119915,119935],{"className":119914},[196,649],[39,119916,119918,119932],{"className":119917},[200],[39,119919,119921],{"className":119920,"style":1728},[204],[39,119922,119923,119926],{"style":966},[39,119924],{"className":119925,"style":213},[212],[39,119927,119929],{"className":119928},[217,218,219,220],[39,119930,119136],{"className":119931,"style":119178},[83,84,220],[39,119933,673],{"className":119934},[672],[39,119936,119938],{"className":119937},[200],[39,119939,119941],{"className":119940,"style":680},[204],[39,119942],{},[39,119944,136],{"className":119945},[182],[39,119947,119136],{"className":119948,"style":119178},[83,84],[39,119950,161],{"className":119951},[263],[39,119953],{"className":119954,"style":461},[230],[39,119956,5378],{"className":119957},[465],[39,119959],{"className":119960,"style":461},[230],[39,119962,119964,119967,119972,120009,120012,120015,120057,120060,120063,120066,120103],{"className":119963},[74],[39,119965],{"className":119966,"style":908},[78],[39,119968,92914,119970],{"className":119969},[1744],[39,119971,39209],{"style":35078},[39,119973,119975],{"className":119974},[182],[39,119976,119978],{"className":119977},[18520,86085],[39,119979,119981,120001],{"className":119980},[196,649],[39,119982,119984,119998],{"className":119983},[200],[39,119985,119987],{"className":119986,"style":928},[204],[39,119988,119989,119992],{"style":86097},[39,119990],{"className":119991,"style":86101},[212],[39,119993,119994],{"style":86104},[86106,119995,119996],{"xmlns":86108,"width":86109,"height":74068,"viewBox":86110},[86112,119997],{"d":86114},[39,119999,673],{"className":120000},[672],[39,120002,120004],{"className":120003},[200],[39,120005,120007],{"className":120006,"style":1217},[204],[39,120008],{},[39,120010,119336],{"className":120011},[1744],[39,120013],{"className":120014,"style":256},[230],[39,120016,120018,120022],{"className":120017},[83],[39,120019,119798],{"className":120020,"style":120021},[83,84],"margin-right:0.0962em;",[39,120023,120025],{"className":120024},[192],[39,120026,120028,120049],{"className":120027},[196,649],[39,120029,120031,120046],{"className":120030},[200],[39,120032,120034],{"className":120033,"style":3419},[204],[39,120035,120037,120040],{"style":120036},"top:-2.55em;margin-left:-0.0962em;margin-right:0.05em;",[39,120038],{"className":120039,"style":213},[212],[39,120041,120043],{"className":120042},[217,218,219,220],[39,120044,74528],{"className":120045,"style":107898},[83,84,220],[39,120047,673],{"className":120048},[672],[39,120050,120052],{"className":120051},[200],[39,120053,120055],{"className":120054,"style":35396},[204],[39,120056],{},[39,120058,136],{"className":120059},[182],[39,120061,119136],{"className":120062,"style":119178},[83,84],[39,120064,161],{"className":120065},[263],[39,120067,120069],{"className":120068},[263],[39,120070,120072],{"className":120071},[18520,86085],[39,120073,120075,120095],{"className":120074},[196,649],[39,120076,120078,120092],{"className":120077},[200],[39,120079,120081],{"className":120080,"style":928},[204],[39,120082,120083,120086],{"style":86097},[39,120084],{"className":120085,"style":86101},[212],[39,120087,120088],{"style":86104},[86106,120089,120090],{"xmlns":86108,"width":86109,"height":74068,"viewBox":86110},[86112,120091],{"d":86114},[39,120093,673],{"className":120094},[672],[39,120096,120098],{"className":120097},[200],[39,120099,120101],{"className":120100,"style":1217},[204],[39,120102],{},[39,120104,2903],{"className":120105},[83],[11,120107,100246,120108,120167],{},[39,120109,120111,120135],{"className":120110,"translate":43},[42],[39,120112,120114],{"className":120113},[47],[49,120115,120116],{"xmlns":51},[53,120117,120118,120132],{},[56,120119,120120,120122,120124,120126,120128,120130],{},[59,120121,92676],{},[133,120123,1690],{},[59,120125,150],{"mathvariant":145},[59,120127,119336],{},[133,120129,1690],{},[59,120131,150],{"mathvariant":145},[63,120133,120134],{"encoding":65},"\\log|\\det|",[39,120136,120138],{"className":120137,"ariaHidden":70},[69],[39,120139,120141,120144,120149,120152,120155,120158,120161,120164],{"className":120140},[74],[39,120142],{"className":120143,"style":245},[78],[39,120145,92914,120147],{"className":120146},[1744],[39,120148,39209],{"style":35078},[39,120150],{"className":120151,"style":256},[230],[39,120153,150],{"className":120154},[83],[39,120156],{"className":120157,"style":256},[230],[39,120159,119336],{"className":120160},[1744],[39,120162],{"className":120163,"style":256},[230],[39,120165,150],{"className":120166},[83]," 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）。",[11,120169,120170,120171,88033,120250,742,120341,120406,120407,120435,120436,742,120515,120629,120630,1607],{},"一维的情形可以推到极致：任意一维分布都能写成逆 CDF。取 ",[39,120172,120174,120202],{"className":120173,"translate":43},[42],[39,120175,120177],{"className":120176},[47],[49,120178,120179],{"xmlns":51},[53,120180,120181,120199],{},[56,120182,120183,120185,120187,120189,120191,120193,120195,120197],{},[59,120184,119136],{},[133,120186,119198],{},[59,120188,85587],{},[133,120190,136],{"stretchy":135},[539,120192,541],{},[133,120194,155],{"separator":70},[539,120196,546],{},[133,120198,161],{"stretchy":135},[63,120200,120201],{"encoding":65},"z\\sim U(0,1)",[39,120203,120205,120223],{"className":120204,"ariaHidden":70},[69],[39,120206,120208,120211,120214,120217,120220],{"className":120207},[74],[39,120209],{"className":120210,"style":290},[78],[39,120212,119136],{"className":120213,"style":119178},[83,84],[39,120215],{"className":120216,"style":231},[230],[39,120218,119198],{"className":120219},[235],[39,120221],{"className":120222,"style":231},[230],[39,120224,120226,120229,120232,120235,120238,120241,120244,120247],{"className":120225},[74],[39,120227],{"className":120228,"style":245},[78],[39,120230,85587],{"className":120231,"style":85613},[83,84],[39,120233,136],{"className":120234},[182],[39,120236,541],{"className":120237},[83],[39,120239,155],{"className":120240},[252],[39,120242],{"className":120243,"style":256},[230],[39,120245,546],{"className":120246},[83],[39,120248,161],{"className":120249},[263],[39,120251,120253,120279],{"className":120252,"translate":43},[42],[39,120254,120256],{"className":120255},[47],[49,120257,120258],{"xmlns":51},[53,120259,120260,120276],{},[56,120261,120262,120264,120266],{},[59,120263,74528],{},[133,120265,793],{},[138,120267,120268,120270],{},[59,120269,618],{},[56,120271,120272,120274],{},[133,120273,5378],{},[539,120275,546],{},[63,120277,120278],{"encoding":65},"f=F^{-1}",[39,120280,120282,120300],{"className":120281,"ariaHidden":70},[69],[39,120283,120285,120288,120291,120294,120297],{"className":120284},[74],[39,120286],{"className":120287,"style":51280},[78],[39,120289,74528],{"className":120290,"style":107898},[83,84],[39,120292],{"className":120293,"style":231},[230],[39,120295,793],{"className":120296},[235],[39,120298],{"className":120299,"style":231},[230],[39,120301,120303,120306],{"className":120302},[74],[39,120304],{"className":120305,"style":19710},[78],[39,120307,120309,120312],{"className":120308},[83],[39,120310,618],{"className":120311,"style":178},[83,84],[39,120313,120315],{"className":120314},[192],[39,120316,120318],{"className":120317},[196],[39,120319,120321],{"className":120320},[200],[39,120322,120324],{"className":120323,"style":19710},[204],[39,120325,120326,120329],{"style":208},[39,120327],{"className":120328,"style":213},[212],[39,120330,120332],{"className":120331},[217,218,219,220],[39,120333,120335,120338],{"className":120334},[83,220],[39,120336,5378],{"className":120337},[83,220],[39,120339,546],{"className":120340},[83,220],[39,120342,120344,120367],{"className":120343,"translate":43},[42],[39,120345,120347],{"className":120346},[47],[49,120348,120349],{"xmlns":51},[53,120350,120351,120365],{},[56,120352,120353,120355,120357,120359,120361,120363],{},[59,120354,1755],{},[133,120356,793],{},[59,120358,74528],{},[133,120360,136],{"stretchy":135},[59,120362,119136],{},[133,120364,161],{"stretchy":135},[63,120366,119141],{"encoding":65},[39,120368,120370,120388],{"className":120369,"ariaHidden":70},[69],[39,120371,120373,120376,120379,120382,120385],{"className":120372},[74],[39,120374],{"className":120375,"style":290},[78],[39,120377,1755],{"className":120378},[83,84],[39,120380],{"className":120381,"style":231},[230],[39,120383,793],{"className":120384},[235],[39,120386],{"className":120387,"style":231},[230],[39,120389,120391,120394,120397,120400,120403],{"className":120390},[74],[39,120392],{"className":120393,"style":245},[78],[39,120395,74528],{"className":120396,"style":107898},[83,84],[39,120398,136],{"className":120399},[182],[39,120401,119136],{"className":120402,"style":119178},[83,84],[39,120404,161],{"className":120405},[263]," 的分布就是 ",[39,120408,120410,120423],{"className":120409,"translate":43},[42],[39,120411,120413],{"className":120412},[47],[49,120414,120415],{"xmlns":51},[53,120416,120417,120421],{},[56,120418,120419],{},[59,120420,618],{},[63,120422,618],{"encoding":65},[39,120424,120426],{"className":120425,"ariaHidden":70},[69],[39,120427,120429,120432],{"className":120428},[74],[39,120430],{"className":120431,"style":79},[78],[39,120433,618],{"className":120434,"style":178},[83,84],"，这是精确的，不是近似。取 ",[39,120437,120439,120467],{"className":120438,"translate":43},[42],[39,120440,120442],{"className":120441},[47],[49,120443,120444],{"xmlns":51},[53,120445,120446,120464],{},[56,120447,120448,120450,120452,120454,120456,120458,120460,120462],{},[59,120449,119136],{},[133,120451,119198],{},[59,120453,77752],{"mathvariant":617},[133,120455,136],{"stretchy":135},[539,120457,541],{},[133,120459,155],{"separator":70},[539,120461,546],{},[133,120463,161],{"stretchy":135},[63,120465,120466],{"encoding":65},"z\\sim\\mathcal N(0,1)",[39,120468,120470,120488],{"className":120469,"ariaHidden":70},[69],[39,120471,120473,120476,120479,120482,120485],{"className":120472},[74],[39,120474],{"className":120475,"style":290},[78],[39,120477,119136],{"className":120478,"style":119178},[83,84],[39,120480],{"className":120481,"style":231},[230],[39,120483,119198],{"className":120484},[235],[39,120486],{"className":120487,"style":231},[230],[39,120489,120491,120494,120497,120500,120503,120506,120509,120512],{"className":120490},[74],[39,120492],{"className":120493,"style":245},[78],[39,120495,77752],{"className":120496,"style":77820},[83,641],[39,120498,136],{"className":120499},[182],[39,120501,541],{"className":120502},[83],[39,120504,155],{"className":120505},[252],[39,120507],{"className":120508,"style":256},[230],[39,120510,546],{"className":120511},[83],[39,120513,161],{"className":120514},[263],[39,120516,120518,120549],{"className":120517,"translate":43},[42],[39,120519,120521],{"className":120520},[47],[49,120522,120523],{"xmlns":51},[53,120524,120525,120546],{},[56,120526,120527,120529,120531,120541,120543],{},[59,120528,74528],{},[133,120530,793],{},[138,120532,120533,120535],{},[59,120534,618],{},[56,120536,120537,120539],{},[133,120538,5378],{},[539,120540,546],{},[133,120542,88956],{},[59,120544,120545],{"mathvariant":145},"Φ",[63,120547,120548],{"encoding":65},"f=F^{-1}\\circ\\Phi",[39,120550,120552,120570,120620],{"className":120551,"ariaHidden":70},[69],[39,120553,120555,120558,120561,120564,120567],{"className":120554},[74],[39,120556],{"className":120557,"style":51280},[78],[39,120559,74528],{"className":120560,"style":107898},[83,84],[39,120562],{"className":120563,"style":231},[230],[39,120565,793],{"className":120566},[235],[39,120568],{"className":120569,"style":231},[230],[39,120571,120573,120576,120611,120614,120617],{"className":120572},[74],[39,120574],{"className":120575,"style":19710},[78],[39,120577,120579,120582],{"className":120578},[83],[39,120580,618],{"className":120581,"style":178},[83,84],[39,120583,120585],{"className":120584},[192],[39,120586,120588],{"className":120587},[196],[39,120589,120591],{"className":120590},[200],[39,120592,120594],{"className":120593,"style":19710},[204],[39,120595,120596,120599],{"style":208},[39,120597],{"className":120598,"style":213},[212],[39,120600,120602],{"className":120601},[217,218,219,220],[39,120603,120605,120608],{"className":120604},[83,220],[39,120606,5378],{"className":120607},[83,220],[39,120609,546],{"className":120610},[83,220],[39,120612],{"className":120613,"style":461},[230],[39,120615,88956],{"className":120616},[465],[39,120618],{"className":120619,"style":461},[230],[39,120621,120623,120626],{"className":120622},[74],[39,120624],{"className":120625,"style":79},[78],[39,120627,120545],{"className":120628},[83]," 就得到任意一维分布。上面那个指数分布的例子就是 ",[39,120631,120633,120683],{"className":120632,"translate":43},[42],[39,120634,120636],{"className":120635},[47],[49,120637,120638],{"xmlns":51},[53,120639,120640,120680],{},[56,120641,120642,120652,120654,120656,120658,120660,120662,120664,120666,120668,120670,120672,120674,120676,120678],{},[138,120643,120644,120646],{},[59,120645,618],{},[56,120647,120648,120650],{},[133,120649,5378],{},[539,120651,546],{},[133,120653,136],{"stretchy":135},[59,120655,35321],{},[133,120657,161],{"stretchy":135},[133,120659,793],{},[133,120661,5378],{},[59,120663,92676],{},[133,120665,1690],{},[133,120667,136],{"stretchy":135},[539,120669,546],{},[133,120671,5378],{},[59,120673,35321],{},[133,120675,161],{"stretchy":135},[59,120677,45691],{"mathvariant":145},[59,120679,118392],{},[63,120681,120682],{"encoding":65},"F^{-1}(u)=-\\log(1-u)\u002F\\lambda",[39,120684,120686,120745,120777],{"className":120685,"ariaHidden":70},[69],[39,120687,120689,120692,120727,120730,120733,120736,120739,120742],{"className":120688},[74],[39,120690],{"className":120691,"style":84808},[78],[39,120693,120695,120698],{"className":120694},[83],[39,120696,618],{"className":120697,"style":178},[83,84],[39,120699,120701],{"className":120700},[192],[39,120702,120704],{"className":120703},[196],[39,120705,120707],{"className":120706},[200],[39,120708,120710],{"className":120709,"style":19710},[204],[39,120711,120712,120715],{"style":208},[39,120713],{"className":120714,"style":213},[212],[39,120716,120718],{"className":120717},[217,218,219,220],[39,120719,120721,120724],{"className":120720},[83,220],[39,120722,5378],{"className":120723},[83,220],[39,120725,546],{"className":120726},[83,220],[39,120728,136],{"className":120729},[182],[39,120731,35321],{"className":120732},[83,84],[39,120734,161],{"className":120735},[263],[39,120737],{"className":120738,"style":231},[230],[39,120740,793],{"className":120741},[235],[39,120743],{"className":120744,"style":231},[230],[39,120746,120748,120751,120754,120757,120762,120765,120768,120771,120774],{"className":120747},[74],[39,120749],{"className":120750,"style":245},[78],[39,120752,5378],{"className":120753},[83],[39,120755],{"className":120756,"style":256},[230],[39,120758,92914,120760],{"className":120759},[1744],[39,120761,39209],{"style":35078},[39,120763,136],{"className":120764},[182],[39,120766,546],{"className":120767},[83],[39,120769],{"className":120770,"style":461},[230],[39,120772,5378],{"className":120773},[465],[39,120775],{"className":120776,"style":461},[230],[39,120778,120780,120783,120786,120789,120792],{"className":120779},[74],[39,120781],{"className":120782,"style":245},[78],[39,120784,35321],{"className":120785},[83,84],[39,120787,161],{"className":120788},[263],[39,120790,45691],{"className":120791},[83],[39,120793,118392],{"className":120794},[83,84],[11,120796,120797,120798,121090,121091,121119,121120,121412,121413,121525,121526,121578,121579,121646,121647,121675,121676,121743],{},"举个能算的例子。",[39,120799,120801,120861],{"className":120800,"translate":43},[42],[39,120802,120804],{"className":120803},[47],[49,120805,120806],{"xmlns":51},[53,120807,120808,120858],{},[56,120809,120810,120812,120814,120816,120818,120820,120828,120830,120832,120834,120836,120838,120846,120848,120850,120852,120854,120856],{},[59,120811,11],{},[133,120813,136],{"stretchy":135},[59,120815,1755],{},[133,120817,161],{"stretchy":135},[133,120819,793],{},[770,120821,120822],{"scriptlevel":541,"displaystyle":135},[5362,120823,120824,120826],{},[539,120825,546],{},[539,120827,17359],{},[59,120829,80841],{},[133,120831,136],{"stretchy":135},[59,120833,1755],{},[133,120835,161],{"stretchy":135},[133,120837,788],{},[770,120839,120840],{"scriptlevel":541,"displaystyle":135},[5362,120841,120842,120844],{},[539,120843,546],{},[539,120845,17359],{},[59,120847,80841],{},[133,120849,136],{"stretchy":135},[59,120851,1755],{},[133,120853,5378],{},[539,120855,59187],{},[133,120857,161],{"stretchy":135},[63,120859,120860],{"encoding":65},"p(x)=\\tfrac12\\varphi(x)+\\tfrac12\\varphi(x-3)",[39,120862,120864,120891,120986,121078],{"className":120863,"ariaHidden":70},[69],[39,120865,120867,120870,120873,120876,120879,120882,120885,120888],{"className":120866},[74],[39,120868],{"className":120869,"style":245},[78],[39,120871,11],{"className":120872},[83,84],[39,120874,136],{"className":120875},[182],[39,120877,1755],{"className":120878},[83,84],[39,120880,161],{"className":120881},[263],[39,120883],{"className":120884,"style":231},[230],[39,120886,793],{"className":120887},[235],[39,120889],{"className":120890,"style":231},[230],[39,120892,120894,120897,120965,120968,120971,120974,120977,120980,120983],{"className":120893},[74],[39,120895],{"className":120896,"style":74987},[78],[39,120898,120900,120903,120962],{"className":120899},[83],[39,120901],{"className":120902},[182,5463],[39,120904,120906],{"className":120905},[5362],[39,120907,120909,120954],{"className":120908},[196,649],[39,120910,120912,120951],{"className":120911},[200],[39,120913,120915,120929,120937],{"className":120914,"style":75006},[204],[39,120916,120917,120920],{"style":75009},[39,120918],{"className":120919,"style":935},[212],[39,120921,120923],{"className":120922},[217,218,219,220],[39,120924,120926],{"className":120925},[83,220],[39,120927,17359],{"className":120928},[83,220],[39,120930,120931,120934],{"style":5503},[39,120932],{"className":120933,"style":935},[212],[39,120935],{"className":120936,"style":5511},[5510],[39,120938,120939,120942],{"style":75032},[39,120940],{"className":120941,"style":935},[212],[39,120943,120945],{"className":120944},[217,218,219,220],[39,120946,120948],{"className":120947},[83,220],[39,120949,546],{"className":120950},[83,220],[39,120952,673],{"className":120953},[672],[39,120955,120957],{"className":120956},[200],[39,120958,120960],{"className":120959,"style":75054},[204],[39,120961],{},[39,120963],{"className":120964},[263,5463],[39,120966,80841],{"className":120967},[83,84],[39,120969,136],{"className":120970},[182],[39,120972,1755],{"className":120973},[83,84],[39,120975,161],{"className":120976},[263],[39,120978],{"className":120979,"style":461},[230],[39,120981,788],{"className":120982},[465],[39,120984],{"className":120985,"style":461},[230],[39,120987,120989,120992,121060,121063,121066,121069,121072,121075],{"className":120988},[74],[39,120990],{"className":120991,"style":74987},[78],[39,120993,120995,120998,121057],{"className":120994},[83],[39,120996],{"className":120997},[182,5463],[39,120999,121001],{"className":121000},[5362],[39,121002,121004,121049],{"className":121003},[196,649],[39,121005,121007,121046],{"className":121006},[200],[39,121008,121010,121024,121032],{"className":121009,"style":75006},[204],[39,121011,121012,121015],{"style":75009},[39,121013],{"className":121014,"style":935},[212],[39,121016,121018],{"className":121017},[217,218,219,220],[39,121019,121021],{"className":121020},[83,220],[39,121022,17359],{"className":121023},[83,220],[39,121025,121026,121029],{"style":5503},[39,121027],{"className":121028,"style":935},[212],[39,121030],{"className":121031,"style":5511},[5510],[39,121033,121034,121037],{"style":75032},[39,121035],{"className":121036,"style":935},[212],[39,121038,121040],{"className":121039},[217,218,219,220],[39,121041,121043],{"className":121042},[83,220],[39,121044,546],{"className":121045},[83,220],[39,121047,673],{"className":121048},[672],[39,121050,121052],{"className":121051},[200],[39,121053,121055],{"className":121054,"style":75054},[204],[39,121056],{},[39,121058],{"className":121059},[263,5463],[39,121061,80841],{"className":121062},[83,84],[39,121064,136],{"className":121065},[182],[39,121067,1755],{"className":121068},[83,84],[39,121070],{"className":121071,"style":461},[230],[39,121073,5378],{"className":121074},[465],[39,121076],{"className":121077,"style":461},[230],[39,121079,121081,121084,121087],{"className":121080},[74],[39,121082],{"className":121083,"style":245},[78],[39,121085,59187],{"className":121086},[83],[39,121088,161],{"className":121089},[263],"，两个单位方差高斯，中心相距 ",[39,121092,121094,121107],{"className":121093,"translate":43},[42],[39,121095,121097],{"className":121096},[47],[49,121098,121099],{"xmlns":51},[53,121100,121101,121105],{},[56,121102,121103],{},[539,121104,59187],{},[63,121106,59187],{"encoding":65},[39,121108,121110],{"className":121109,"ariaHidden":70},[69],[39,121111,121113,121116],{"className":121112},[74],[39,121114],{"className":121115,"style":2473},[78],[39,121117,59187],{"className":121118},[83],"。它的 CDF 是 ",[39,121121,121123,121183],{"className":121122,"translate":43},[42],[39,121124,121126],{"className":121125},[47],[49,121127,121128],{"xmlns":51},[53,121129,121130,121180],{},[56,121131,121132,121134,121136,121138,121140,121142,121150,121152,121154,121156,121158,121160,121168,121170,121172,121174,121176,121178],{},[59,121133,618],{},[133,121135,136],{"stretchy":135},[59,121137,1755],{},[133,121139,161],{"stretchy":135},[133,121141,793],{},[770,121143,121144],{"scriptlevel":541,"displaystyle":135},[5362,121145,121146,121148],{},[539,121147,546],{},[539,121149,17359],{},[59,121151,120545],{"mathvariant":145},[133,121153,136],{"stretchy":135},[59,121155,1755],{},[133,121157,161],{"stretchy":135},[133,121159,788],{},[770,121161,121162],{"scriptlevel":541,"displaystyle":135},[5362,121163,121164,121166],{},[539,121165,546],{},[539,121167,17359],{},[59,121169,120545],{"mathvariant":145},[133,121171,136],{"stretchy":135},[59,121173,1755],{},[133,121175,5378],{},[539,121177,59187],{},[133,121179,161],{"stretchy":135},[63,121181,121182],{"encoding":65},"F(x)=\\tfrac12\\Phi(x)+\\tfrac12\\Phi(x-3)",[39,121184,121186,121213,121308,121400],{"className":121185,"ariaHidden":70},[69],[39,121187,121189,121192,121195,121198,121201,121204,121207,121210],{"className":121188},[74],[39,121190],{"className":121191,"style":245},[78],[39,121193,618],{"className":121194,"style":178},[83,84],[39,121196,136],{"className":121197},[182],[39,121199,1755],{"className":121200},[83,84],[39,121202,161],{"className":121203},[263],[39,121205],{"className":121206,"style":231},[230],[39,121208,793],{"className":121209},[235],[39,121211],{"className":121212,"style":231},[230],[39,121214,121216,121219,121287,121290,121293,121296,121299,121302,121305],{"className":121215},[74],[39,121217],{"className":121218,"style":74987},[78],[39,121220,121222,121225,121284],{"className":121221},[83],[39,121223],{"className":121224},[182,5463],[39,121226,121228],{"className":121227},[5362],[39,121229,121231,121276],{"className":121230},[196,649],[39,121232,121234,121273],{"className":121233},[200],[39,121235,121237,121251,121259],{"className":121236,"style":75006},[204],[39,121238,121239,121242],{"style":75009},[39,121240],{"className":121241,"style":935},[212],[39,121243,121245],{"className":121244},[217,218,219,220],[39,121246,121248],{"className":121247},[83,220],[39,121249,17359],{"className":121250},[83,220],[39,121252,121253,121256],{"style":5503},[39,121254],{"className":121255,"style":935},[212],[39,121257],{"className":121258,"style":5511},[5510],[39,121260,121261,121264],{"style":75032},[39,121262],{"className":121263,"style":935},[212],[39,121265,121267],{"className":121266},[217,218,219,220],[39,121268,121270],{"className":121269},[83,220],[39,121271,546],{"className":121272},[83,220],[39,121274,673],{"className":121275},[672],[39,121277,121279],{"className":121278},[200],[39,121280,121282],{"className":121281,"style":75054},[204],[39,121283],{},[39,121285],{"className":121286},[263,5463],[39,121288,120545],{"className":121289},[83],[39,121291,136],{"className":121292},[182],[39,121294,1755],{"className":121295},[83,84],[39,121297,161],{"className":121298},[263],[39,121300],{"className":121301,"style":461},[230],[39,121303,788],{"className":121304},[465],[39,121306],{"className":121307,"style":461},[230],[39,121309,121311,121314,121382,121385,121388,121391,121394,121397],{"className":121310},[74],[39,121312],{"className":121313,"style":74987},[78],[39,121315,121317,121320,121379],{"className":121316},[83],[39,121318],{"className":121319},[182,5463],[39,121321,121323],{"className":121322},[5362],[39,121324,121326,121371],{"className":121325},[196,649],[39,121327,121329,121368],{"className":121328},[200],[39,121330,121332,121346,121354],{"className":121331,"style":75006},[204],[39,121333,121334,121337],{"style":75009},[39,121335],{"className":121336,"style":935},[212],[39,121338,121340],{"className":121339},[217,218,219,220],[39,121341,121343],{"className":121342},[83,220],[39,121344,17359],{"className":121345},[83,220],[39,121347,121348,121351],{"style":5503},[39,121349],{"className":121350,"style":935},[212],[39,121352],{"className":121353,"style":5511},[5510],[39,121355,121356,121359],{"style":75032},[39,121357],{"className":121358,"style":935},[212],[39,121360,121362],{"className":121361},[217,218,219,220],[39,121363,121365],{"className":121364},[83,220],[39,121366,546],{"className":121367},[83,220],[39,121369,673],{"className":121370},[672],[39,121372,121374],{"className":121373},[200],[39,121375,121377],{"className":121376,"style":75054},[204],[39,121378],{},[39,121380],{"className":121381},[263,5463],[39,121383,120545],{"className":121384},[83],[39,121386,136],{"className":121387},[182],[39,121389,1755],{"className":121390},[83,84],[39,121392],{"className":121393,"style":461},[230],[39,121395,5378],{"className":121396},[465],[39,121398],{"className":121399,"style":461},[230],[39,121401,121403,121406,121409],{"className":121402},[74],[39,121404],{"className":121405,"style":245},[78],[39,121407,59187],{"className":121408},[83],[39,121410,161],{"className":121411},[263],"，流就是 ",[39,121414,121416,121445],{"className":121415,"translate":43},[42],[39,121417,121419],{"className":121418},[47],[49,121420,121421],{"xmlns":51},[53,121422,121423,121443],{},[56,121424,121425,121427,121429,121439,121441],{},[59,121426,74528],{},[133,121428,793],{},[138,121430,121431,121433],{},[59,121432,618],{},[56,121434,121435,121437],{},[133,121436,5378],{},[539,121438,546],{},[133,121440,88956],{},[59,121442,120545],{"mathvariant":145},[63,121444,120548],{"encoding":65},[39,121446,121448,121466,121516],{"className":121447,"ariaHidden":70},[69],[39,121449,121451,121454,121457,121460,121463],{"className":121450},[74],[39,121452],{"className":121453,"style":51280},[78],[39,121455,74528],{"className":121456,"style":107898},[83,84],[39,121458],{"className":121459,"style":231},[230],[39,121461,793],{"className":121462},[235],[39,121464],{"className":121465,"style":231},[230],[39,121467,121469,121472,121507,121510,121513],{"className":121468},[74],[39,121470],{"className":121471,"style":19710},[78],[39,121473,121475,121478],{"className":121474},[83],[39,121476,618],{"className":121477,"style":178},[83,84],[39,121479,121481],{"className":121480},[192],[39,121482,121484],{"className":121483},[196],[39,121485,121487],{"className":121486},[200],[39,121488,121490],{"className":121489,"style":19710},[204],[39,121491,121492,121495],{"style":208},[39,121493],{"className":121494,"style":213},[212],[39,121496,121498],{"className":121497},[217,218,219,220],[39,121499,121501,121504],{"className":121500},[83,220],[39,121502,5378],{"className":121503},[83,220],[39,121505,546],{"className":121506},[83,220],[39,121508],{"className":121509,"style":461},[230],[39,121511,88956],{"className":121512},[465],[39,121514],{"className":121515,"style":461},[230],[39,121517,121519,121522],{"className":121518},[74],[39,121520],{"className":121521,"style":79},[78],[39,121523,120545],{"className":121524},[83],"。两峰之间 ",[39,121527,121529,121548],{"className":121528,"translate":43},[42],[39,121530,121532],{"className":121531},[47],[49,121533,121534],{"xmlns":51},[53,121535,121536,121545],{},[56,121537,121538,121540,121542],{},[59,121539,1755],{},[133,121541,793],{},[539,121543,121544],{},"1.5",[63,121546,121547],{"encoding":65},"x=1.5",[39,121549,121551,121569],{"className":121550,"ariaHidden":70},[69],[39,121552,121554,121557,121560,121563,121566],{"className":121553},[74],[39,121555],{"className":121556,"style":290},[78],[39,121558,1755],{"className":121559},[83,84],[39,121561],{"className":121562,"style":231},[230],[39,121564,793],{"className":121565},[235],[39,121567],{"className":121568,"style":231},[230],[39,121570,121572,121575],{"className":121571},[74],[39,121573],{"className":121574,"style":2473},[78],[39,121576,121544],{"className":121577},[83]," 处的真实密度是 ",[39,121580,121582,121607],{"className":121581,"translate":43},[42],[39,121583,121585],{"className":121584},[47],[49,121586,121587],{"xmlns":51},[53,121588,121589,121604],{},[56,121590,121591,121593,121595,121597,121599,121601],{},[59,121592,80841],{},[133,121594,136],{"stretchy":135},[539,121596,121544],{},[133,121598,161],{"stretchy":135},[133,121600,99937],{},[539,121602,121603],{},"0.1295",[63,121605,121606],{"encoding":65},"\\varphi(1.5)\\approx0.1295",[39,121608,121610,121637],{"className":121609,"ariaHidden":70},[69],[39,121611,121613,121616,121619,121622,121625,121628,121631,121634],{"className":121612},[74],[39,121614],{"className":121615,"style":245},[78],[39,121617,80841],{"className":121618},[83,84],[39,121620,136],{"className":121621},[182],[39,121623,121544],{"className":121624},[83],[39,121626,161],{"className":121627},[263],[39,121629],{"className":121630,"style":231},[230],[39,121632,99937],{"className":121633},[235],[39,121635],{"className":121636,"style":231},[230],[39,121638,121640,121643],{"className":121639},[74],[39,121641],{"className":121642,"style":2473},[78],[39,121644,121603],{"className":121645},[83],"；如果错当成单个高斯（中心在 ",[39,121648,121650,121663],{"className":121649,"translate":43},[42],[39,121651,121653],{"className":121652},[47],[49,121654,121655],{"xmlns":51},[53,121656,121657,121661],{},[56,121658,121659],{},[539,121660,121544],{},[63,121662,121544],{"encoding":65},[39,121664,121666],{"className":121665,"ariaHidden":70},[69],[39,121667,121669,121672],{"className":121668},[74],[39,121670],{"className":121671,"style":2473},[78],[39,121673,121544],{"className":121674},[83],"、单峰），会给出 ",[39,121677,121679,121704],{"className":121678,"translate":43},[42],[39,121680,121682],{"className":121681},[47],[49,121683,121684],{"xmlns":51},[53,121685,121686,121701],{},[56,121687,121688,121690,121692,121694,121696,121698],{},[59,121689,80841],{},[133,121691,136],{"stretchy":135},[539,121693,541],{},[133,121695,161],{"stretchy":135},[133,121697,99937],{},[539,121699,121700],{},"0.3989",[63,121702,121703],{"encoding":65},"\\varphi(0)\\approx0.3989",[39,121705,121707,121734],{"className":121706,"ariaHidden":70},[69],[39,121708,121710,121713,121716,121719,121722,121725,121728,121731],{"className":121709},[74],[39,121711],{"className":121712,"style":245},[78],[39,121714,80841],{"className":121715},[83,84],[39,121717,136],{"className":121718},[182],[39,121720,541],{"className":121721},[83],[39,121723,161],{"className":121724},[263],[39,121726],{"className":121727,"style":231},[230],[39,121729,99937],{"className":121730},[235],[39,121732],{"className":121733,"style":231},[230],[39,121735,121737,121740],{"className":121736},[74],[39,121738],{"className":121739,"style":2473},[78],[39,121741,121700],{"className":121742},[83],"，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。",[11,121745,121746,121747,121846],{},"连续情形就是 Neural ODE：",[39,121748,121750,121786],{"className":121749,"translate":43},[42],[39,121751,121753],{"className":121752},[47],[49,121754,121755],{"xmlns":51},[53,121756,121757,121783],{},[56,121758,121759,121761,121763,121765,121767,121769,121771,121773,121775,121777,121779,121781],{},[59,121760,29425],{},[59,121762,119136],{},[59,121764,45691],{"mathvariant":145},[59,121766,29425],{},[59,121768,621],{},[133,121770,793],{},[59,121772,78164],{},[133,121774,136],{"stretchy":135},[59,121776,119136],{},[133,121778,155],{"separator":70},[59,121780,621],{},[133,121782,161],{"stretchy":135},[63,121784,121785],{"encoding":65},"dz\u002Fdt=v(z,t)",[39,121787,121789,121819],{"className":121788,"ariaHidden":70},[69],[39,121790,121792,121795,121798,121801,121804,121807,121810,121813,121816],{"className":121791},[74],[39,121793],{"className":121794,"style":245},[78],[39,121796,29425],{"className":121797},[83,84],[39,121799,119136],{"className":121800,"style":119178},[83,84],[39,121802,45691],{"className":121803},[83],[39,121805,29425],{"className":121806},[83,84],[39,121808,621],{"className":121809},[83,84],[39,121811],{"className":121812,"style":231},[230],[39,121814,793],{"className":121815},[235],[39,121817],{"className":121818,"style":231},[230],[39,121820,121822,121825,121828,121831,121834,121837,121840,121843],{"className":121821},[74],[39,121823],{"className":121824,"style":245},[78],[39,121826,78164],{"className":121827,"style":2114},[83,84],[39,121829,136],{"className":121830},[182],[39,121832,119136],{"className":121833,"style":119178},[83,84],[39,121835,155],{"className":121836},[252],[39,121838],{"className":121839,"style":256},[230],[39,121841,621],{"className":121842},[83,84],[39,121844,161],{"className":121845},[263],"，密度沿轨迹演化，",[39,121848,121850],{"className":121849,"translate":43},[746],[39,121851,121853,121925],{"className":121852,"translate":43},[42],[39,121854,121856],{"className":121855},[47],[49,121857,121858],{"xmlns":51,"display":755},[53,121859,121860,121922],{},[56,121861,121862,121872,121874,121876,121882,121884,121890,121892,121894,121896,121902,121904,121918,121920],{},[5362,121863,121864,121866],{},[59,121865,29425],{},[56,121867,121868,121870],{},[59,121869,29425],{},[59,121871,621],{},[59,121873,92676],{},[133,121875,1690],{},[613,121877,121878,121880],{},[59,121879,11],{},[59,121881,621],{},[133,121883,136],{"stretchy":135},[613,121885,121886,121888],{},[59,121887,119136],{},[59,121889,621],{},[133,121891,161],{"stretchy":135},[133,121893,793],{},[133,121895,5378],{},[56,121897,121898,121900],{},[59,121899,621],{"mathvariant":145},[59,121901,401],{"mathvariant":145},[133,121903,136],{"fence":70,"stretchy":70,"minsize":65017,"maxsize":65017},[5362,121905,121906,121912],{},[56,121907,121908,121910],{},[59,121909,96629],{"mathvariant":145},[59,121911,78164],{},[56,121913,121914,121916],{},[59,121915,96629],{"mathvariant":145},[59,121917,119136],{},[133,121919,161],{"fence":70,"stretchy":70,"minsize":65017,"maxsize":65017},[133,121921,155],{"separator":70},[63,121923,121924],{"encoding":65},"\\frac{d}{dt}\\log p_t(z_t)=-\\mathrm{tr}\\Bigl(\\frac{\\partial v}{\\partial z}\\Bigr),",[39,121926,121928,122107],{"className":121927,"ariaHidden":70},[69],[39,121929,121931,121935,122000,122003,122008,122011,122051,122054,122095,122098,122101,122104],{"className":121930},[74],[39,121932],{"className":121933,"style":121934},[78],"height:2.0574em;vertical-align:-0.686em;",[39,121936,121938,121941,121997],{"className":121937},[83],[39,121939],{"className":121940},[182,5463],[39,121942,121944],{"className":121943},[5362],[39,121945,121947,121989],{"className":121946},[196,649],[39,121948,121950,121986],{"className":121949},[200],[39,121951,121953,121967,121975],{"className":121952,"style":118942},[204],[39,121954,121955,121958],{"style":5479},[39,121956],{"className":121957,"style":935},[212],[39,121959,121961,121964],{"className":121960},[83],[39,121962,29425],{"className":121963},[83,84],[39,121965,621],{"className":121966},[83,84],[39,121968,121969,121972],{"style":5503},[39,121970],{"className":121971,"style":935},[212],[39,121973],{"className":121974,"style":5511},[5510],[39,121976,121977,121980],{"style":5514},[39,121978],{"className":121979,"style":935},[212],[39,121981,121983],{"className":121982},[83],[39,121984,29425],{"className":121985},[83,84],[39,121987,673],{"className":121988},[672],[39,121990,121992],{"className":121991},[200],[39,121993,121995],{"className":121994,"style":19487},[204],[39,121996],{},[39,121998],{"className":121999},[263,5463],[39,122001],{"className":122002,"style":256},[230],[39,122004,92914,122006],{"className":122005},[1744],[39,122007,39209],{"style":35078},[39,122009],{"className":122010,"style":256},[230],[39,122012,122014,122017],{"className":122013},[83],[39,122015,11],{"className":122016},[83,84],[39,122018,122020],{"className":122019},[192],[39,122021,122023,122043],{"className":122022},[196,649],[39,122024,122026,122040],{"className":122025},[200],[39,122027,122029],{"className":122028,"style":656},[204],[39,122030,122031,122034],{"style":966},[39,122032],{"className":122033,"style":213},[212],[39,122035,122037],{"className":122036},[217,218,219,220],[39,122038,621],{"className":122039},[83,84,220],[39,122041,673],{"className":122042},[672],[39,122044,122046],{"className":122045},[200],[39,122047,122049],{"className":122048,"style":680},[204],[39,122050],{},[39,122052,136],{"className":122053},[182],[39,122055,122057,122060],{"className":122056},[83],[39,122058,119136],{"className":122059,"style":119178},[83,84],[39,122061,122063],{"className":122062},[192],[39,122064,122066,122087],{"className":122065},[196,649],[39,122067,122069,122084],{"className":122068},[200],[39,122070,122072],{"className":122071,"style":656},[204],[39,122073,122075,122078],{"style":122074},"top:-2.55em;margin-left:-0.044em;margin-right:0.05em;",[39,122076],{"className":122077,"style":213},[212],[39,122079,122081],{"className":122080},[217,218,219,220],[39,122082,621],{"className":122083},[83,84,220],[39,122085,673],{"className":122086},[672],[39,122088,122090],{"className":122089},[200],[39,122091,122093],{"className":122092,"style":680},[204],[39,122094],{},[39,122096,161],{"className":122097},[263],[39,122099],{"className":122100,"style":231},[230],[39,122102,793],{"className":122103},[235],[39,122105],{"className":122106,"style":231},[230],[39,122108,122110,122113,122116,122122,122128,122196,122202],{"className":122109},[74],[39,122111],{"className":122112,"style":121934},[78],[39,122114,5378],{"className":122115},[83],[39,122117,122119],{"className":122118},[83],[39,122120,89718],{"className":122121},[83,35077],[39,122123,122125],{"className":122124},[182],[39,122126,136],{"className":122127},[18520,33204],[39,122129,122131,122134,122193],{"className":122130},[83],[39,122132],{"className":122133},[182,5463],[39,122135,122137],{"className":122136},[5362],[39,122138,122140,122185],{"className":122139},[196,649],[39,122141,122143,122182],{"className":122142},[200],[39,122144,122146,122160,122168],{"className":122145,"style":118942},[204],[39,122147,122148,122151],{"style":5479},[39,122149],{"className":122150,"style":935},[212],[39,122152,122154,122157],{"className":122153},[83],[39,122155,96629],{"className":122156,"style":565},[83],[39,122158,119136],{"className":122159,"style":119178},[83,84],[39,122161,122162,122165],{"style":5503},[39,122163],{"className":122164,"style":935},[212],[39,122166],{"className":122167,"style":5511},[5510],[39,122169,122170,122173],{"style":5514},[39,122171],{"className":122172,"style":935},[212],[39,122174,122176,122179],{"className":122175},[83],[39,122177,96629],{"className":122178,"style":565},[83],[39,122180,78164],{"className":122181,"style":2114},[83,84],[39,122183,673],{"className":122184},[672],[39,122186,122188],{"className":122187},[200],[39,122189,122191],{"className":122190,"style":19487},[204],[39,122192],{},[39,122194],{"className":122195},[263,5463],[39,122197,122199],{"className":122198},[263],[39,122200,161],{"className":122201},[18520,33204],[39,122203,155],{"className":122204},[252],[11,122206,122207,122208,742,122377,122405],{},"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 ",[39,122209,122211,122257],{"className":122210,"translate":43},[42],[39,122212,122214],{"className":122213},[47],[49,122215,122216],{"xmlns":51},[53,122217,122218,122254],{},[56,122219,122220,122226,122228,122230,122232,122234,122240,122242,122248,122250,122252],{},[56,122221,122222,122224],{},[59,122223,621],{"mathvariant":145},[59,122225,401],{"mathvariant":145},[133,122227,136],{"stretchy":135},[59,122229,101],{},[133,122231,161],{"stretchy":135},[133,122233,99937],{},[613,122235,122236,122238],{},[59,122237,4253],{"mathvariant":421},[59,122239,58009],{},[133,122241,537],{"stretchy":135},[138,122243,122244,122246],{},[59,122245,58009],{},[59,122247,93534],{"mathvariant":145},[59,122249,101],{},[59,122251,58009],{},[133,122253,4280],{"stretchy":135},[63,122255,122256],{"encoding":65},"\\mathrm{tr}(A)\\approx\\mathbb E_\\varepsilon[\\varepsilon^\\top A\\varepsilon]",[39,122258,122260,122290],{"className":122259,"ariaHidden":70},[69],[39,122261,122263,122266,122272,122275,122278,122281,122284,122287],{"className":122262},[74],[39,122264],{"className":122265,"style":245},[78],[39,122267,122269],{"className":122268},[83],[39,122270,89718],{"className":122271},[83,35077],[39,122273,136],{"className":122274},[182],[39,122276,101],{"className":122277},[83,84],[39,122279,161],{"className":122280},[263],[39,122282],{"className":122283,"style":231},[230],[39,122285,99937],{"className":122286},[235],[39,122288],{"className":122289,"style":231},[230],[39,122291,122293,122296,122336,122339,122368,122371,122374],{"className":122292},[74],[39,122294],{"className":122295,"style":15555},[78],[39,122297,122299,122302],{"className":122298},[83],[39,122300,4253],{"className":122301},[83,515],[39,122303,122305],{"className":122304},[192],[39,122306,122308,122328],{"className":122307},[196,649],[39,122309,122311,122325],{"className":122310},[200],[39,122312,122314],{"className":122313,"style":1728},[204],[39,122315,122316,122319],{"style":966},[39,122317],{"className":122318,"style":213},[212],[39,122320,122322],{"className":122321},[217,218,219,220],[39,122323,58009],{"className":122324},[83,84,220],[39,122326,673],{"className":122327},[672],[39,122329,122331],{"className":122330},[200],[39,122332,122334],{"className":122333,"style":680},[204],[39,122335],{},[39,122337,537],{"className":122338},[182],[39,122340,122342,122345],{"className":122341},[83],[39,122343,58009],{"className":122344},[83,84],[39,122346,122348],{"className":122347},[192],[39,122349,122351],{"className":122350},[196],[39,122352,122354],{"className":122353},[200],[39,122355,122357],{"className":122356,"style":15577},[204],[39,122358,122359,122362],{"style":208},[39,122360],{"className":122361,"style":213},[212],[39,122363,122365],{"className":122364},[217,218,219,220],[39,122366,93534],{"className":122367},[83,220],[39,122369,101],{"className":122370},[83,84],[39,122372,58009],{"className":122373},[83,84],[39,122375,4280],{"className":122376},[263],[39,122378,122380,122393],{"className":122379,"translate":43},[42],[39,122381,122383],{"className":122382},[47],[49,122384,122385],{"xmlns":51},[53,122386,122387,122391],{},[56,122388,122389],{},[59,122390,58009],{},[63,122392,77615],{"encoding":65},[39,122394,122396],{"className":122395,"ariaHidden":70},[69],[39,122397,122399,122402],{"className":122398},[74],[39,122400],{"className":122401,"style":290},[78],[39,122403,58009],{"className":122404},[83,84]," 取 Rademacher 向量，代价是方差。",[11,122407,122408,122409,122442,122443,122476],{},"流与最优传输有一个漂亮的接口。Brenier 说最优映射是 ",[39,122410,122412,122427],{"className":122411,"translate":43},[42],[39,122413,122415],{"className":122414},[47],[49,122416,122417],{"xmlns":51},[53,122418,122419,122425],{},[56,122420,122421,122423],{},[59,122422,84918],{"mathvariant":145},[59,122424,35321],{},[63,122426,85005],{"encoding":65},[39,122428,122430],{"className":122429,"ariaHidden":70},[69],[39,122431,122433,122436,122439],{"className":122432},[74],[39,122434],{"className":122435,"style":79},[78],[39,122437,84918],{"className":122438},[83],[39,122440,35321],{"className":122441},[83,84],"，流说数据由可逆映射得到；当映射取 ",[39,122444,122446,122461],{"className":122445,"translate":43},[42],[39,122447,122449],{"className":122448},[47],[49,122450,122451],{"xmlns":51},[53,122452,122453,122459],{},[56,122454,122455,122457],{},[59,122456,84918],{"mathvariant":145},[59,122458,35321],{},[63,122460,85005],{"encoding":65},[39,122462,122464],{"className":122463,"ariaHidden":70},[69],[39,122465,122467,122470,122473],{"className":122466},[74],[39,122468],{"className":122469,"style":79},[78],[39,122471,84918],{"className":122472},[83],[39,122474,35321],{"className":122475},[83,84]," 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",[39,122478,122480],{"className":122479,"translate":43},[746],[39,122481,122483,122611],{"className":122482,"translate":43},[42],[39,122484,122486],{"className":122485},[47],[49,122487,122488],{"xmlns":51,"display":755},[53,122489,122490,122608],{},[56,122491,122492,122494,122496,122520,122522,122528,122530,122536,122538,122540,122542,122544,122546,122552,122554,122560,122562,122568,122570,122572,122578,122580,122582,122584,122586,122588,122590,122596,122598,122600,122606],{},[59,122493,77383],{"mathvariant":617},[133,122495,793],{},[613,122497,122498,122500],{},[59,122499,4253],{"mathvariant":421},[56,122501,122502,122504,122506,122512,122514],{},[59,122503,621],{},[133,122505,155],{"separator":70},[613,122507,122508,122510],{},[59,122509,1755],{},[539,122511,541],{},[133,122513,155],{"separator":70},[613,122515,122516,122518],{},[59,122517,1755],{},[539,122519,546],{},[133,122521,15389],{"fence":70,"stretchy":70,"minsize":74068,"maxsize":74068},[613,122523,122524,122526],{},[59,122525,78164],{},[59,122527,96518],{},[133,122529,136],{"stretchy":135},[613,122531,122532,122534],{},[59,122533,1755],{},[59,122535,621],{},[133,122537,155],{"separator":70},[59,122539,621],{},[133,122541,161],{"stretchy":135},[133,122543,5378],{},[133,122545,136],{"stretchy":135},[613,122547,122548,122550],{},[59,122549,1755],{},[539,122551,546],{},[133,122553,5378],{},[613,122555,122556,122558],{},[59,122557,1755],{},[539,122559,541],{},[133,122561,161],{"stretchy":135},[138,122563,122564,122566],{},[133,122565,15389],{"fence":70,"stretchy":70,"minsize":74068,"maxsize":74068},[539,122567,17359],{},[133,122569,155],{"separator":70},[230,122571],{"width":10251},[613,122573,122574,122576],{},[59,122575,1755],{},[59,122577,621],{},[133,122579,793],{},[133,122581,136],{"stretchy":135},[539,122583,546],{},[133,122585,5378],{},[59,122587,621],{},[133,122589,161],{"stretchy":135},[613,122591,122592,122594],{},[59,122593,1755],{},[539,122595,541],{},[133,122597,788],{},[59,122599,621],{},[613,122601,122602,122604],{},[59,122603,1755],{},[539,122605,546],{},[133,122607,155],{"separator":70},[63,122609,122610],{"encoding":65},"\\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,",[39,122612,122614,122632,122910,122968,123138,123159,123220],{"className":122613,"ariaHidden":70},[69],[39,122615,122617,122620,122623,122626,122629],{"className":122616},[74],[39,122618],{"className":122619,"style":79},[78],[39,122621,77383],{"className":122622},[83,641],[39,122624],{"className":122625,"style":231},[230],[39,122627,793],{"className":122628},[235],[39,122630],{"className":122631,"style":231},[230],[39,122633,122635,122638,122769,122806,122846,122849,122889,122892,122895,122898,122901,122904,122907],{"className":122634},[74],[39,122636],{"className":122637,"style":908},[78],[39,122639,122641,122644],{"className":122640},[83],[39,122642,4253],{"className":122643},[83,515],[39,122645,122647],{"className":122646},[192],[39,122648,122650,122761],{"className":122649},[196,649],[39,122651,122653,122758],{"className":122652},[200],[39,122654,122656],{"className":122655,"style":656},[204],[39,122657,122658,122661],{"style":966},[39,122659],{"className":122660,"style":213},[212],[39,122662,122664],{"className":122663},[217,218,219,220],[39,122665,122667,122670,122673,122715,122718],{"className":122666},[83,220],[39,122668,621],{"className":122669},[83,84,220],[39,122671,155],{"className":122672},[252,220],[39,122674,122676,122679],{"className":122675},[83,220],[39,122677,1755],{"className":122678},[83,84,220],[39,122680,122682],{"className":122681},[192],[39,122683,122685,122707],{"className":122684},[196,649],[39,122686,122688,122704],{"className":122687},[200],[39,122689,122692],{"className":122690,"style":122691},[204],"height:0.3173em;",[39,122693,122695,122698],{"style":122694},"top:-2.357em;margin-left:0em;margin-right:0.0714em;",[39,122696],{"className":122697,"style":4004},[212],[39,122699,122701],{"className":122700},[217,4008,4009,220],[39,122702,541],{"className":122703},[83,220],[39,122705,673],{"className":122706},[672],[39,122708,122710],{"className":122709},[200],[39,122711,122713],{"className":122712,"style":64389},[204],[39,122714],{},[39,122716,155],{"className":122717},[252,220],[39,122719,122721,122724],{"className":122720},[83,220],[39,122722,1755],{"className":122723},[83,84,220],[39,122725,122727],{"className":122726},[192],[39,122728,122730,122750],{"className":122729},[196,649],[39,122731,122733,122747],{"className":122732},[200],[39,122734,122736],{"className":122735,"style":122691},[204],[39,122737,122738,122741],{"style":122694},[39,122739],{"className":122740,"style":4004},[212],[39,122742,122744],{"className":122743},[217,4008,4009,220],[39,122745,546],{"className":122746},[83,220],[39,122748,673],{"className":122749},[672],[39,122751,122753],{"className":122752},[200],[39,122754,122756],{"className":122755,"style":64389},[204],[39,122757],{},[39,122759,673],{"className":122760},[672],[39,122762,122764],{"className":122763},[200],[39,122765,122767],{"className":122766,"style":35396},[204],[39,122768],{},[39,122770,122772],{"className":122771},[182],[39,122773,122775],{"className":122774},[18520,86085],[39,122776,122778,122798],{"className":122777},[196,649],[39,122779,122781,122795],{"className":122780},[200],[39,122782,122784],{"className":122783,"style":928},[204],[39,122785,122786,122789],{"style":86097},[39,122787],{"className":122788,"style":86101},[212],[39,122790,122791],{"style":105561},[86106,122792,122793],{"xmlns":86108,"width":105564,"height":74068,"viewBox":105565},[86112,122794],{"d":105568},[39,122796,673],{"className":122797},[672],[39,122799,122801],{"className":122800},[200],[39,122802,122804],{"className":122803,"style":1217},[204],[39,122805],{},[39,122807,122809,122812],{"className":122808},[83],[39,122810,78164],{"className":122811,"style":2114},[83,84],[39,122813,122815],{"className":122814},[192],[39,122816,122818,122838],{"className":122817},[196,649],[39,122819,122821,122835],{"className":122820},[200],[39,122822,122824],{"className":122823,"style":3419},[204],[39,122825,122826,122829],{"style":41316},[39,122827],{"className":122828,"style":213},[212],[39,122830,122832],{"className":122831},[217,218,219,220],[39,122833,96518],{"className":122834,"style":438},[83,84,220],[39,122836,673],{"className":122837},[672],[39,122839,122841],{"className":122840},[200],[39,122842,122844],{"className":122843,"style":680},[204],[39,122845],{},[39,122847,136],{"className":122848},[182],[39,122850,122852,122855],{"className":122851},[83],[39,122853,1755],{"className":122854},[83,84],[39,122856,122858],{"className":122857},[192],[39,122859,122861,122881],{"className":122860},[196,649],[39,122862,122864,122878],{"className":122863},[200],[39,122865,122867],{"className":122866,"style":656},[204],[39,122868,122869,122872],{"style":966},[39,122870],{"className":122871,"style":213},[212],[39,122873,122875],{"className":122874},[217,218,219,220],[39,122876,621],{"className":122877},[83,84,220],[39,122879,673],{"className":122880},[672],[39,122882,122884],{"className":122883},[200],[39,122885,122887],{"className":122886,"style":680},[204],[39,122888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是噪声与数据的配对。如果配对来自 OT 计划，这条直线就是 ",[39,123411,123413,123430],{"className":123412,"translate":43},[42],[39,123414,123416],{"className":123415},[47],[49,123417,123418],{"xmlns":51},[53,123419,123420,123428],{},[56,123421,123422],{},[613,123423,123424,123426],{},[59,123425,23703],{},[539,123427,17359],{},[63,123429,91827],{"encoding":65},[39,123431,123433],{"className":123432,"ariaHidden":70},[69],[39,123434,123436,123439],{"className":123435},[74],[39,123437],{"className":123438,"style":634},[78],[39,123440,123442,123445],{"className":123441},[83],[39,123443,23703],{"className":123444,"style":178},[83,84],[39,123446,123448],{"className":123447},[192],[39,123449,123451,123471],{"className":123450},[196,649],[39,123452,123454,123468],{"className":123453},[200],[39,123455,123457],{"className":123456,"style":963},[204],[39,123458,123459,123462],{"style":73958},[39,123460],{"className":123461,"style":213},[212],[39,123463,123465],{"className":123464},[217,218,219,220],[39,123466,17359],{"className":123467},[83,220],[39,123469,673],{"className":123470},[672],[39,123472,123474],{"className":123473},[200],[39,123475,123477],{"className":123476,"style":680},[204],[39,123478],{}," 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。",[11,123481,123482],{},"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。",[21,123484,123485],{"id":123485},"落地",[11,123487,123488],{},"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。",[11,123490,123491,123492,123533,123534,123603],{},"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 ",[39,123493,123495,123513],{"className":123494,"translate":43},[42],[39,123496,123498],{"className":123497},[47],[49,123499,123500],{"xmlns":51},[53,123501,123502,123510],{},[56,123503,123504,123506,123508],{},[59,123505,92676],{},[133,123507,1690],{},[539,123509,17359],{},[63,123511,123512],{"encoding":65},"\\log2",[39,123514,123516],{"className":123515,"ariaHidden":70},[69],[39,123517,123519,123522,123527,123530],{"className":123518},[74],[39,123520],{"className":123521,"style":51280},[78],[39,123523,92914,123525],{"className":123524},[1744],[39,123526,39209],{"style":35078},[39,123528],{"className":123529,"style":256},[230],[39,123531,17359],{"className":123532},[83],"，梯度为零——这是早期 GAN 训练不稳定的理论解释之一。WGAN 换成 ",[39,123535,123537,123554],{"className":123536,"translate":43},[42],[39,123538,123540],{"className":123539},[47],[49,123541,123542],{"xmlns":51},[53,123543,123544,123552],{},[56,123545,123546],{},[613,123547,123548,123550],{},[59,123549,23703],{},[539,123551,546],{},[63,123553,78032],{"encoding":65},[39,123555,123557],{"className":123556,"ariaHidden":70},[69],[39,123558,123560,123563],{"className":123559},[74],[39,123561],{"className":123562,"style":634},[78],[39,123564,123566,123569],{"className":123565},[83],[39,123567,23703],{"className":123568,"style":178},[83,84],[39,123570,123572],{"className":123571},[192],[39,123573,123575,123595],{"className":123574},[196,649],[39,123576,123578,123592],{"className":123577},[200],[39,123579,123581],{"className":123580,"style":963},[204],[39,123582,123583,123586],{"style":73958},[39,123584],{"className":123585,"style":213},[212],[39,123587,123589],{"className":123588},[217,218,219,220],[39,123590,546],{"className":123591},[83,220],[39,123593,673],{"className":123594},[672],[39,123596,123598],{"className":123597},[200],[39,123599,123601],{"className":123600,"style":680},[204],[39,123602],{},"，用对偶式",[39,123605,123607],{"className":123606,"translate":43},[746],[39,123608,123610,123712],{"className":123609,"translate":43},[42],[39,123611,123613],{"className":123612},[47],[49,123614,123615],{"xmlns":51,"display":755},[53,123616,123617,123709],{},[56,123618,123619,123629,123659,123661,123671,123673,123675,123677,123679,123681,123683,123685,123695,123697,123699,123701,123703,123705,123707],{},[6556,123620,123621,123627],{},[56,123622,123623,123625],{},[59,123624,92615],{},[133,123626,1690],{},[59,123628,3181],{},[6556,123630,123631,123637],{},[56,123632,123633,123635],{},[59,123634,1687],{},[133,123636,1690],{},[56,123638,123639,123641,123643,123655,123657],{},[59,123640,15389],{"mathvariant":145},[59,123642,74528],{},[613,123644,123645,123647],{},[59,123646,15389],{"mathvariant":145},[56,123648,123649,123651,123653],{},[59,123650,77383],{"mathvariant":145},[59,123652,13377],{"mathvariant":145},[59,123654,11],{"mathvariant":145},[133,123656,1824],{},[539,123658,546],{},[35841,123660,74416],{},[613,123662,123663,123665],{},[59,123664,4253],{"mathvariant":421},[613,123666,123667,123669],{},[59,123668,131],{},[59,123670,401],{},[133,123672,537],{"stretchy":135},[59,123674,74528],{},[133,123676,136],{"stretchy":135},[59,123678,1755],{},[133,123680,161],{"stretchy":135},[133,123682,4280],{"stretchy":135},[133,123684,5378],{},[613,123686,123687,123689],{},[59,123688,4253],{"mathvariant":421},[613,123690,123691,123693],{},[59,123692,131],{},[59,123694,39209],{},[133,123696,537],{"stretchy":135},[59,123698,74528],{},[133,123700,136],{"stretchy":135},[59,123702,1755],{},[133,123704,161],{"stretchy":135},[133,123706,4280],{"stretchy":135},[133,123708,155],{"separator":70},[63,123710,123711],{"encoding":65},"\\min_G\\max_{\\|f\\|_{\\mathrm{Lip}}\\le1}\\ \\mathbb E_{P_r}[f(x)]-\\mathbb E_{P_g}[f(x)],",[39,123713,123715,123984],{"className":123714,"ariaHidden":70},[69],[39,123716,123718,123722,123766,123769,123872,123875,123878,123960,123963,123966,123969,123972,123975,123978,123981],{"className":123717},[74],[39,123719],{"className":123720,"style":123721},[78],"height:1.7385em;vertical-align:-0.9885em;",[39,123723,123725],{"className":123724},[1744,3294],[39,123726,123728,123758],{"className":123727},[196,649],[39,123729,123731,123755],{"className":123730},[200],[39,123732,123734,123745],{"className":123733,"style":92716},[204],[39,123735,123736,123739],{"style":74501},[39,123737],{"className":123738,"style":935},[212],[39,123740,123742],{"className":123741},[217,218,219,220],[39,123743,3181],{"className":123744},[83,84,220],[39,123746,123747,123750],{"style":21599},[39,123748],{"className":123749,"style":935},[212],[39,123751,123752],{},[39,123753,92615],{"className":123754},[1744],[39,123756,673],{"className":123757},[672],[39,123759,123761],{"className":123760},[200],[39,123762,123764],{"className":123763,"style":74538},[204],[39,123765],{},[39,123767],{"className":123768,"style":256},[230],[39,123770,123772],{"className":123771},[1744,3294],[39,123773,123775,123863],{"className":123774},[196,649],[39,123776,123778,123860],{"className":123777},[200],[39,123779,123781,123850],{"className":123780,"style":290},[204],[39,123782,123783,123786],{"style":76132},[39,123784],{"className":123785,"style":935},[212],[39,123787,123789],{"className":123788},[217,218,219,220],[39,123790,123792,123795,123798,123844,123847],{"className":123791},[83,220],[39,123793,15389],{"className":123794},[83,220],[39,123796,74528],{"className":123797,"style":107898},[83,84,220],[39,123799,123801,123804],{"className":123800},[83,220],[39,123802,15389],{"className":123803},[83,220],[39,123805,123807],{"className":123806},[192],[39,123808,123810,123836],{"className":123809},[196,649],[39,123811,123813,123833],{"className":123812},[200],[39,123814,123816],{"className":123815,"style":84537},[204],[39,123817,123818,123821],{"style":84540},[39,123819],{"className":123820,"style":4004},[212],[39,123822,123824],{"className":123823},[217,4008,4009,220],[39,123825,123827],{"className":123826},[83,220],[39,123828,123830],{"className":123829},[83,220],[39,123831,84556],{"className":123832},[83,35077,220],[39,123834,673],{"className":123835},[672],[39,123837,123839],{"className":123838},[200],[39,123840,123842],{"className":123841,"style":84566},[204],[39,123843],{},[39,123845,1824],{"className":123846},[235,220],[39,123848,546],{"className":123849},[83,220],[39,123851,123852,123855],{"style":21599},[39,123853],{"className":123854,"style":935},[212],[39,123856,123857],{},[39,123858,1687],{"className":123859},[1744],[39,123861,673],{"className":123862},[672],[39,123864,123866],{"className":123865},[200],[39,123867,123870],{"className":123868,"style":123869},[204],"height:0.9885em;",[39,123871],{},[39,123873,74416],{"className":123874},[230],[39,123876],{"className":123877,"style":256},[230],[39,123879,123881,123884],{"className":123880},[83],[39,123882,4253],{"className":123883},[83,515],[39,123885,123887],{"className":123886},[192],[39,123888,123890,123951],{"className":123889},[196,649],[39,123891,123893,123948],{"className":123892},[200],[39,123894,123896],{"className":123895,"style":62921},[204],[39,123897,123898,123901],{"style":966},[39,123899],{"className":123900,"style":213},[212],[39,123902,123904],{"className":123903},[217,218,219,220],[39,123905,123907],{"className":123906},[83,220],[39,123908,123910,123913],{"className":123909},[83,220],[39,123911,131],{"className":123912,"style":178},[83,84,220],[39,123914,123916],{"className":123915},[192],[39,123917,123919,123940],{"className":123918},[196,649],[39,123920,123922,123937],{"className":123921},[200],[39,123923,123925],{"className":123924,"style":42105},[204],[39,123926,123928,123931],{"style":123927},"top:-2.357em;margin-left:-0.1389em;margin-right:0.0714em;",[39,123929],{"className":123930,"style":4004},[212],[39,123932,123934],{"className":123933},[217,4008,4009,220],[39,123935,401],{"className":123936,"style":438},[83,84,220],[39,123938,673],{"className":123939},[672],[39,123941,123943],{"className":123942},[200],[39,123944,123946],{"className":123945,"style":64389},[204],[39,123947],{},[39,123949,673],{"className":123950},[672],[39,123952,123954],{"className":123953},[200],[39,123955,123958],{"className":123956,"style":123957},[204],"height:0.2501em;",[39,123959],{},[39,123961,537],{"className":123962},[182],[39,123964,74528],{"className":123965,"style":107898},[83,84],[39,123967,136],{"className":123968},[182],[39,123970,1755],{"className":123971},[83,84],[39,123973,103025],{"className":123974},[263],[39,123976],{"className":123977,"style":461},[230],[39,123979,5378],{"className":123980},[465],[39,123982],{"className":123983,"style":461},[230],[39,123985,123987,123991,124072,124075,124078,124081,124084,124087],{"className":123986},[74],[39,123988],{"className":123989,"style":123990},[78],"height:1.0973em;vertical-align:-0.3473em;",[39,123992,123994,123997],{"className":123993},[83],[39,123995,4253],{"className":123996},[83,515],[39,123998,124000],{"className":123999},[192],[39,124001,124003,124063],{"className":124002},[196,649],[39,124004,124006,124060],{"className":124005},[200],[39,124007,124009],{"className":124008,"style":62921},[204],[39,124010,124011,124014],{"style":966},[39,124012],{"className":124013,"style":213},[212],[39,124015,124017],{"className":124016},[217,218,219,220],[39,124018,124020],{"className":124019},[83,220],[39,124021,124023,124026],{"className":124022},[83,220],[39,124024,131],{"className":124025,"style":178},[83,84,220],[39,124027,124029],{"className":124028},[192],[39,124030,124032,124052],{"className":124031},[196,649],[39,124033,124035,124049],{"className":124034},[200],[39,124036,124038],{"className":124037,"style":42105},[204],[39,124039,124040,124043],{"style":123927},[39,124041],{"className":124042,"style":4004},[212],[39,124044,124046],{"className":124045},[217,4008,4009,220],[39,124047,39209],{"className":124048,"style":2114},[83,84,220],[39,124050,673],{"className":124051},[672],[39,124053,124055],{"className":124054},[200],[39,124056,124058],{"className":124057,"style":42127},[204],[39,124059],{},[39,124061,673],{"className":124062},[672],[39,124064,124066],{"className":124065},[200],[39,124067,124070],{"className":124068,"style":124069},[204],"height:0.3473em;",[39,124071],{},[39,124073,537],{"className":124074},[182],[39,124076,74528],{"className":124077,"style":107898},[83,84],[39,124079,136],{"className":124080},[182],[39,124082,1755],{"className":124083},[83,84],[39,124085,103025],{"className":124086},[263],[39,124088,155],{"className":124089},[252],[11,124091,124092,124093,124121],{},"critic ",[39,124094,124096,124109],{"className":124095,"translate":43},[42],[39,124097,124099],{"className":124098},[47],[49,124100,124101],{"xmlns":51},[53,124102,124103,124107],{},[56,124104,124105],{},[59,124106,74528],{},[63,124108,74528],{"encoding":65},[39,124110,124112],{"className":124111,"ariaHidden":70},[69],[39,124113,124115,124118],{"className":124114},[74],[39,124116],{"className":124117,"style":51280},[78],[39,124119,74528],{"className":124120,"style":107898},[83,84]," 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。",[11,124123,124124],{},"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。",[11,124126,124127,124128,124156,124157,124281,124282,124421,124422,124450,124451,124630,124631,124700],{},"域适应与分布偏移是 ",[39,124129,124131,124144],{"className":124130,"translate":43},[42],[39,124132,124134],{"className":124133},[47],[49,124135,124136],{"xmlns":51},[53,124137,124138,124142],{},[56,124139,124140],{},[59,124141,23703],{},[63,124143,23703],{"encoding":65},[39,124145,124147],{"className":124146,"ariaHidden":70},[69],[39,124148,124150,124153],{"className":124149},[74],[39,124151],{"className":124152,"style":79},[78],[39,124154,23703],{"className":124155,"style":178},[83,84]," 最能发挥的地方。给定源域和目标域的特征分布 ",[39,124158,124160,124186],{"className":124159,"translate":43},[42],[39,124161,124163],{"className":124162},[47],[49,124164,124165],{"xmlns":51},[53,124166,124167,124183],{},[56,124168,124169,124175,124177],{},[613,124170,124171,124173],{},[59,124172,73771],{},[59,124174,61],{},[133,124176,155],{"separator":70},[613,124178,124179,124181],{},[59,124180,73771],{},[59,124182,11772],{},[63,124184,124185],{"encoding":65},"\\mu_S,\\mu_T",[39,124187,124189],{"className":124188,"ariaHidden":70},[69],[39,124190,124192,124195,124235,124238,124241],{"className":124191},[74],[39,124193],{"className":124194,"style":2654},[78],[39,124196,124198,124201],{"className":124197},[83],[39,124199,73771],{"className":124200},[83,84],[39,124202,124204],{"className":124203},[192],[39,124205,124207,124227],{"className":124206},[196,649],[39,124208,124210,124224],{"className":124209},[200],[39,124211,124213],{"className":124212,"style":62921},[204],[39,124214,124215,124218],{"style":966},[39,124216],{"className":124217,"style":213},[212],[39,124219,124221],{"className":124220},[217,218,219,220],[39,124222,61],{"className":124223,"style":85},[83,84,220],[39,124225,673],{"className":124226},[672],[39,124228,124230],{"className":124229},[200],[39,124231,124233],{"className":124232,"style":680},[204],[39,124234],{},[39,124236,155],{"className":124237},[252],[39,124239],{"className":124240,"style":256},[230],[39,124242,124244,124247],{"className":124243},[83],[39,124245,73771],{"className":124246},[83,84],[39,124248,124250],{"className":124249},[192],[39,124251,124253,124273],{"className":124252},[196,649],[39,124254,124256,124270],{"className":124255},[200],[39,124257,124259],{"className":124258,"style":62921},[204],[39,124260,124261,124264],{"style":966},[39,124262],{"className":124263,"style":213},[212],[39,124265,124267],{"className":124266},[217,218,219,220],[39,124268,11772],{"className":124269,"style":178},[83,84,220],[39,124271,673],{"className":124272},[672],[39,124274,124276],{"className":124275},[200],[39,124277,124279],{"className":124278,"style":680},[204],[39,124280],{},"，直接最小化 ",[39,124283,124285,124317],{"className":124284,"translate":43},[42],[39,124286,124288],{"className":124287},[47],[49,124289,124290],{"xmlns":51},[53,124291,124292,124314],{},[56,124293,124294,124296,124298,124304,124306,124312],{},[59,124295,23703],{},[133,124297,136],{"stretchy":135},[613,124299,124300,124302],{},[59,124301,73771],{},[59,124303,61],{},[133,124305,155],{"separator":70},[613,124307,124308,124310],{},[59,124309,73771],{},[59,124311,11772],{},[133,124313,161],{"stretchy":135},[63,124315,124316],{"encoding":65},"W(\\mu_S,\\mu_T)",[39,124318,124320],{"className":124319,"ariaHidden":70},[69],[39,124321,124323,124326,124329,124332,124372,124375,124378,124418],{"className":124322},[74],[39,124324],{"className":124325,"style":245},[78],[39,124327,23703],{"className":124328,"style":178},[83,84],[39,124330,136],{"className":124331},[182],[39,124333,124335,124338],{"className":124334},[83],[39,124336,73771],{"className":124337},[83,84],[39,124339,124341],{"className":124340},[192],[39,124342,124344,124364],{"className":124343},[196,649],[39,124345,124347,124361],{"className":124346},[200],[39,124348,124350],{"className":124349,"style":62921},[204],[39,124351,124352,124355],{"style":966},[39,124353],{"className":124354,"style":213},[212],[39,124356,124358],{"className":124357},[217,218,219,220],[39,124359,61],{"className":124360,"style":85},[83,84,220],[39,124362,673],{"className":124363},[672],[39,124365,124367],{"className":124366},[200],[39,124368,124370],{"className":124369,"style":680},[204],[39,124371],{},[39,124373,155],{"className":124374},[252],[39,124376],{"className":124377,"style":256},[230],[39,124379,124381,124384],{"className":124380},[83],[39,124382,73771],{"className":124383},[83,84],[39,124385,124387],{"className":124386},[192],[39,124388,124390,124410],{"className":124389},[196,649],[39,124391,124393,124407],{"className":124392},[200],[39,124394,124396],{"className":124395,"style":62921},[204],[39,124397,124398,124401],{"style":966},[39,124399],{"className":124400,"style":213},[212],[39,124402,124404],{"className":124403},[217,218,219,220],[39,124405,11772],{"className":124406,"style":178},[83,84,220],[39,124408,673],{"className":124409},[672],[39,124411,124413],{"className":124412},[200],[39,124414,124416],{"className":124415,"style":680},[204],[39,124417],{},[39,124419,161],{"className":124420},[263]," 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 ",[39,124423,124425,124438],{"className":124424,"translate":43},[42],[39,124426,124428],{"className":124427},[47],[49,124429,124430],{"xmlns":51},[53,124431,124432,124436],{},[56,124433,124434],{},[59,124435,80841],{},[63,124437,84278],{"encoding":65},[39,124439,124441],{"className":124440,"ariaHidden":70},[69],[39,124442,124444,124447],{"className":124443},[74],[39,124445],{"className":124446,"style":2654},[78],[39,124448,80841],{"className":124449},[83,84]," 去最大化 ",[39,124452,124454,124496],{"className":124453,"translate":43},[42],[39,124455,124457],{"className":124456},[47],[49,124458,124459],{"xmlns":51},[53,124460,124461,124493],{},[56,124462,124463,124465,124467,124469,124471,124477,124479,124481,124483,124485,124487],{},[133,124464,69087],{},[59,124466,80841],{},[35841,124468,35843],{},[59,124470,29425],{},[613,124472,124473,124475],{},[59,124474,73771],{},[59,124476,61],{},[133,124478,5378],{},[133,124480,69087],{},[59,124482,80841],{},[35841,124484,35843],{},[59,124486,29425],{},[613,124488,124489,124491],{},[59,124490,73771],{},[59,124492,11772],{},[63,124494,124495],{"encoding":65},"\\int\\varphi\\,d\\mu_S-\\int\\varphi\\,d\\mu_T",[39,124497,124499,124569],{"className":124498,"ariaHidden":70},[69],[39,124500,124502,124505,124508,124511,124514,124517,124520,124560,124563,124566],{"className":124501},[74],[39,124503],{"className":124504,"style":74333},[78],[39,124506,69087],{"className":124507,"style":74337},[1744,3339,3952],[39,124509],{"className":124510,"style":256},[230],[39,124512,80841],{"className":124513},[83,84],[39,124515],{"className":124516,"style":256},[230],[39,124518,29425],{"className":124519},[83,84],[39,124521,124523,124526],{"className":124522},[83],[39,124524,73771],{"className":124525},[83,84],[39,124527,124529],{"className":124528},[192],[39,124530,124532,124552],{"className":124531},[196,649],[39,124533,124535,124549],{"className":124534},[200],[39,124536,124538],{"className":124537,"style":62921},[204],[39,124539,124540,124543],{"style":966},[39,124541],{"className":124542,"style":213},[212],[39,124544,124546],{"className":124545},[217,218,219,220],[39,124547,61],{"className":124548,"style":85},[83,84,220],[39,124550,673],{"className":124551},[672],[39,124553,124555],{"className":124554},[200],[39,124556,124558],{"className":124557,"style":680},[204],[39,124559],{},[39,124561],{"className":124562,"style":461},[230],[39,124564,5378],{"className":124565},[465],[39,124567],{"className":124568,"style":461},[230],[39,124570,124572,124575,124578,124581,124584,124587,124590],{"className":124571},[74],[39,124573],{"className":124574,"style":74333},[78],[39,124576,69087],{"className":124577,"style":74337},[1744,3339,3952],[39,124579],{"className":124580,"style":256},[230],[39,124582,80841],{"className":124583},[83,84],[39,124585],{"className":124586,"style":256},[230],[39,124588,29425],{"className":124589},[83,84],[39,124591,124593,124596],{"className":124592},[83],[39,124594,73771],{"className":124595},[83,84],[39,124597,124599],{"className":124598},[192],[39,124600,124602,124622],{"className":124601},[196,649],[39,124603,124605,124619],{"className":124604},[200],[39,124606,124608],{"className":124607,"style":62921},[204],[39,124609,124610,124613],{"style":966},[39,124611],{"className":124612,"style":213},[212],[39,124614,124616],{"className":124615},[217,218,219,220],[39,124617,11772],{"className":124618,"style":178},[83,84,220],[39,124620,673],{"className":124621},[672],[39,124623,124625],{"className":124624},[200],[39,124626,124628],{"className":124627,"style":680},[204],[39,124629],{},"，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 ",[39,124632,124634,124651],{"className":124633,"translate":43},[42],[39,124635,124637],{"className":124636},[47],[49,124638,124639],{"xmlns":51},[53,124640,124641,124649],{},[56,124642,124643],{},[613,124644,124645,124647],{},[59,124646,23703],{},[539,124648,17359],{},[63,124650,91827],{"encoding":65},[39,124652,124654],{"className":124653,"ariaHidden":70},[69],[39,124655,124657,124660],{"className":124656},[74],[39,124658],{"className":124659,"style":634},[78],[39,124661,124663,124666],{"className":124662},[83],[39,124664,23703],{"className":124665,"style":178},[83,84],[39,124667,124669],{"className":124668},[192],[39,124670,124672,124692],{"className":124671},[196,649],[39,124673,124675,124689],{"className":124674},[200],[39,124676,124678],{"className":124677,"style":963},[204],[39,124679,124680,124683],{"style":73958},[39,124681],{"className":124682,"style":213},[212],[39,124684,124686],{"className":124685},[217,218,219,220],[39,124687,17359],{"className":124688},[83,220],[39,124690,673],{"className":124691},[672],[39,124693,124695],{"className":124694},[200],[39,124696,124698],{"className":124697,"style":680},[204],[39,124699],{}," 空间里求“平均分布”，这正是 Fréchet 均值在分布空间里的实例。",[11,124702,124703],{},"OOD 检测，也就是我开头那个 0.99 的问题，我会分三层来做。",[11,124705,124706],{},"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。",[11,124708,124709,124710,124858],{},"点态一层比较样本密度：对每个类拟合 KDE 或流，取 ",[39,124711,124713,124747],{"className":124712,"translate":43},[42],[39,124714,124716],{"className":124715},[47],[49,124717,124718],{"xmlns":51},[53,124719,124720,124744],{},[56,124721,124722,124724,124726,124728,124738,124740,124742],{},[133,124723,5378],{},[59,124725,92676],{},[133,124727,1690],{},[613,124729,124730,124736],{},[39689,124731,124732,124734],{"accent":70},[59,124733,11],{},[133,124735,115102],{},[59,124737,3196],{},[133,124739,136],{"stretchy":135},[59,124741,1755],{},[133,124743,161],{"stretchy":135},[63,124745,124746],{"encoding":65},"-\\log\\hat p_k(x)",[39,124748,124750],{"className":124749,"ariaHidden":70},[69],[39,124751,124753,124756,124759,124762,124767,124770,124849,124852,124855],{"className":124752},[74],[39,124754],{"className":124755,"style":245},[78],[39,124757,5378],{"className":124758},[83],[39,124760],{"className":124761,"style":256},[230],[39,124763,92914,124765],{"className":124764},[1744],[39,124766,39209],{"style":35078},[39,124768],{"className":124769,"style":256},[230],[39,124771,124773,124815],{"className":124772},[83],[39,124774,124776],{"className":124775},[83,39712],[39,124777,124779,124807],{"className":124778},[196,649],[39,124780,124782,124804],{"className":124781},[200],[39,124783,124785,124793],{"className":124784,"style":28824},[204],[39,124786,124787,124790],{"style":21599},[39,124788],{"className":124789,"style":935},[212],[39,124791,11],{"className":124792},[83,84],[39,124794,124795,124798],{"style":21599},[39,124796],{"className":124797,"style":935},[212],[39,124799,124801],{"className":124800,"style":115238},[39739],[39,124802,115102],{"className":124803},[83],[39,124805,673],{"className":124806},[672],[39,124808,124810],{"className":124809},[200],[39,124811,124813],{"className":124812,"style":115251},[204],[39,124814],{},[39,124816,124818],{"className":124817},[192],[39,124819,124821,124841],{"className":124820},[196,649],[39,124822,124824,124838],{"className":124823},[200],[39,124825,124827],{"className":124826,"style":3419},[204],[39,124828,124829,124832],{"style":966},[39,124830],{"className":124831,"style":213},[212],[39,124833,124835],{"className":124834},[217,218,219,220],[39,124836,3196],{"className":124837,"style":3321},[83,84,220],[39,124839,673],{"className":124840},[672],[39,124842,124844],{"className":124843},[200],[39,124845,124847],{"className":124846,"style":680},[204],[39,124848],{},[39,124850,136],{"className":124851},[182],[39,124853,1755],{"className":124854},[83,84],[39,124856,161],{"className":124857},[263]," 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。",[11,124860,124861,124862,124931,124932,125008,125009,125037,125038,742,125066,125110],{},"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 ",[39,124863,124865,124882],{"className":124864,"translate":43},[42],[39,124866,124868],{"className":124867},[47],[49,124869,124870],{"xmlns":51},[53,124871,124872,124880],{},[56,124873,124874],{},[613,124875,124876,124878],{},[59,124877,23703],{},[539,124879,17359],{},[63,124881,91827],{"encoding":65},[39,124883,124885],{"className":124884,"ariaHidden":70},[69],[39,124886,124888,124891],{"className":124887},[74],[39,124889],{"className":124890,"style":634},[78],[39,124892,124894,124897],{"className":124893},[83],[39,124895,23703],{"className":124896,"style":178},[83,84],[39,124898,124900],{"className":124899},[192],[39,124901,124903,124923],{"className":124902},[196,649],[39,124904,124906,124920],{"className":124905},[200],[39,124907,124909],{"className":124908,"style":963},[204],[39,124910,124911,124914],{"style":73958},[39,124912],{"className":124913,"style":213},[212],[39,124915,124917],{"className":124916},[217,218,219,220],[39,124918,17359],{"className":124919},[83,220],[39,124921,673],{"className":124922},[672],[39,124924,124926],{"className":124925},[200],[39,124927,124929],{"className":124928,"style":680},[204],[39,124930],{},"，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 ",[39,124933,124935,124961],{"className":124934,"translate":43},[42],[39,124936,124938],{"className":124937},[47],[49,124939,124940],{"xmlns":51},[53,124941,124942,124958],{},[56,124943,124944],{},[138,124945,124946,124948],{},[59,124947,12962],{},[56,124949,124950,124952,124954,124956],{},[133,124951,5378],{},[539,124953,546],{},[59,124955,45691],{"mathvariant":145},[59,124957,29425],{},[63,124959,124960],{"encoding":65},"n^{-1\u002Fd}",[39,124962,124964],{"className":124963,"ariaHidden":70},[69],[39,124965,124967,124970],{"className":124966},[74],[39,124968],{"className":124969,"style":95578},[78],[39,124971,124973,124976],{"className":124972},[83],[39,124974,12962],{"className":124975},[83,84],[39,124977,124979],{"className":124978},[192],[39,124980,124982],{"className":124981},[196],[39,124983,124985],{"className":124984},[200],[39,124986,124988],{"className":124987,"style":95578},[204],[39,124989,124990,124993],{"style":208},[39,124991],{"className":124992,"style":213},[212],[39,124994,124996],{"className":124995},[217,218,219,220],[39,124997,124999,125002,125005],{"className":124998},[83,220],[39,125000,5378],{"className":125001},[83,220],[39,125003,76865],{"className":125004},[83,220],[39,125006,29425],{"className":125007},[83,84,220]," 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 ",[39,125010,125012,125025],{"className":125011,"translate":43},[42],[39,125013,125015],{"className":125014},[47],[49,125016,125017],{"xmlns":51},[53,125018,125019,125023],{},[56,125020,125021],{},[59,125022,80841],{},[63,125024,84278],{"encoding":65},[39,125026,125028],{"className":125027,"ariaHidden":70},[69],[39,125029,125031,125034],{"className":125030},[74],[39,125032],{"className":125033,"style":2654},[78],[39,125035,80841],{"className":125036},[83,84],"——一旦在一批数据上解出 ",[39,125039,125041,125054],{"className":125040,"translate":43},[42],[39,125042,125044],{"className":125043},[47],[49,125045,125046],{"xmlns":51},[53,125047,125048,125052],{},[56,125049,125050],{},[59,125051,80841],{},[63,125053,84278],{"encoding":65},[39,125055,125057],{"className":125056,"ariaHidden":70},[69],[39,125058,125060,125063],{"className":125059},[74],[39,125061],{"className":125062,"style":2654},[78],[39,125064,80841],{"className":125065},[83,84],[39,125067,125069,125089],{"className":125068,"translate":43},[42],[39,125070,125072],{"className":125071},[47],[49,125073,125074],{"xmlns":51},[53,125075,125076,125086],{},[56,125077,125078,125080,125082,125084],{},[59,125079,80841],{},[133,125081,136],{"stretchy":135},[59,125083,1755],{},[133,125085,161],{"stretchy":135},[63,125087,125088],{"encoding":65},"\\varphi(x)",[39,125090,125092],{"className":125091,"ariaHidden":70},[69],[39,125093,125095,125098,125101,125104,125107],{"className":125094},[74],[39,125096],{"className":125097,"style":245},[78],[39,125099,80841],{"className":125100},[83,84],[39,125102,136],{"className":125103},[182],[39,125105,1755],{"className":125106},[83,84],[39,125108,161],{"className":125109},[263]," 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。",[11,125112,125113],{},"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。",[11,125115,125116],{},"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。",[21,125118,125119],{"id":125119},"回到开头",[11,125121,125122],{},"回到那个 0.99。",[11,125124,125125,125126,125154],{},"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 ",[39,125127,125129,125142],{"className":125128,"translate":43},[42],[39,125130,125132],{"className":125131},[47],[49,125133,125134],{"xmlns":51},[53,125135,125136,125140],{},[56,125137,125138],{},[539,125139,546],{},[63,125141,546],{"encoding":65},[39,125143,125145],{"className":125144,"ariaHidden":70},[69],[39,125146,125148,125151],{"className":125147},[74],[39,125149],{"className":125150,"style":2473},[78],[39,125152,546],{"className":125153},[83],"。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。",[11,125156,125157],{},"如果只留一个式子，我会留 Kantorovich 对偶：",[39,125159,125161],{"className":125160,"translate":43},[746],[39,125162,125164,125244],{"className":125163,"translate":43},[42],[39,125165,125167],{"className":125166},[47],[49,125168,125169],{"xmlns":51,"display":755},[53,125170,125171,125241],{},[56,125172,125173,125179,125181,125183,125185,125187,125189,125191,125221,125223,125225,125227,125229,125231,125233,125235,125237,125239],{},[613,125174,125175,125177],{},[59,125176,23703],{},[539,125178,546],{},[133,125180,136],{"stretchy":135},[59,125182,73771],{},[133,125184,155],{"separator":70},[59,125186,73802],{},[133,125188,161],{"stretchy":135},[133,125190,793],{},[6556,125192,125193,125199],{},[56,125194,125195,125197],{},[59,125196,21465],{},[133,125198,1690],{},[56,125200,125201,125203,125205,125217,125219],{},[59,125202,15389],{"mathvariant":145},[59,125204,80841],{},[613,125206,125207,125209],{},[59,125208,15389],{"mathvariant":145},[56,125210,125211,125213,125215],{},[59,125212,77383],{"mathvariant":145},[59,125214,13377],{"mathvariant":145},[59,125216,11],{"mathvariant":145},[133,125218,1824],{},[539,125220,546],{},[133,125222,69087],{},[59,125224,80841],{},[35841,125226,35843],{},[59,125228,29425],{},[133,125230,136],{"stretchy":135},[59,125232,73771],{},[133,125234,5378],{},[59,125236,73802],{},[133,125238,161],{"stretchy":135},[59,125240,2903],{"mathvariant":145},[63,125242,125243],{"encoding":65},"W_1(\\mu,\\nu)=\\sup_{\\|\\varphi\\|_{\\mathrm{Lip}}\\le1}\\int\\varphi\\,d(\\mu-\\nu).",[39,125245,125247,125320,125461],{"className":125246,"ariaHidden":70},[69],[39,125248,125250,125253,125293,125296,125299,125302,125305,125308,125311,125314,125317],{"className":125249},[74],[39,125251],{"className":125252,"style":245},[78],[39,125254,125256,125259],{"className":125255},[83],[39,125257,23703],{"className":125258,"style":178},[83,84],[39,125260,125262],{"className":125261},[192],[39,125263,125265,125285],{"className":125264},[196,649],[39,125266,125268,125282],{"className":125267},[200],[39,125269,125271],{"className":125270,"style":963},[204],[39,125272,125273,125276],{"style":73958},[39,125274],{"className":125275,"style":213},[212],[39,125277,125279],{"className":125278},[217,218,219,220],[39,125280,546],{"className":125281},[83,220],[39,125283,673],{"className":125284},[672],[39,125286,125288],{"className":125287},[200],[39,125289,125291],{"className":125290,"style":680},[204],[39,125292],{},[39,125294,136],{"className":125295},[182],[39,125297,73771],{"className":125298},[83,84],[39,125300,155],{"className":125301},[252],[39,125303],{"className":125304,"style":256},[230],[39,125306,73802],{"className":125307,"style":73818},[83,84],[39,125309,161],{"className":125310},[263],[39,125312],{"className":125313,"style":231},[230],[39,125315,793],{"className":125316},[235],[39,125318],{"className":125319,"style":231},[230],[39,125321,125323,125326,125428,125431,125434,125437,125440,125443,125446,125449,125452,125455,125458],{"className":125322},[74],[39,125324],{"className":125325,"style":84488},[78],[39,125327,125329],{"className":125328},[1744,3294],[39,125330,125332,125420],{"className":125331},[196,649],[39,125333,125335,125417],{"className":125334},[200],[39,125336,125338,125407],{"className":125337,"style":290},[204],[39,125339,125340,125343],{"style":84503},[39,125341],{"className":125342,"style":935},[212],[39,125344,125346],{"className":125345},[217,218,219,220],[39,125347,125349,125352,125355,125401,125404],{"className":125348},[83,220],[39,125350,15389],{"className":125351},[83,220],[39,125353,80841],{"className":125354},[83,84,220],[39,125356,125358,125361],{"className":125357},[83,220],[39,125359,15389],{"className":125360},[83,220],[39,125362,125364],{"className":125363},[192],[39,125365,125367,125393],{"className":125366},[196,649],[39,125368,125370,125390],{"className":125369},[200],[39,125371,125373],{"className":125372,"style":84537},[204],[39,125374,125375,125378],{"style":84540},[39,125376],{"className":125377,"style":4004},[212],[39,125379,125381],{"className":125380},[217,4008,4009,220],[39,125382,125384],{"className":125383},[83,220],[39,125385,125387],{"className":125386},[83,220],[39,125388,84556],{"className":125389},[83,35077,220],[39,125391,673],{"className":125392},[672],[39,125394,125396],{"className":125395},[200],[39,125397,125399],{"className":125398,"style":84566},[204],[39,125400],{},[39,125402,1824],{"className":125403},[235,220],[39,125405,546],{"className":125406},[83,220],[39,125408,125409,125412],{"style":21599},[39,125410],{"className":125411,"style":935},[212],[39,125413,125414],{},[39,125415,21465],{"className":125416},[1744],[39,125418,673],{"className":125419},[672],[39,125421,125423],{"className":125422},[200],[39,125424,125426],{"className":125425,"style":84594},[204],[39,125427],{},[39,125429],{"className":125430,"style":256},[230],[39,125432,69087],{"className":125433,"style":69380},[1744,3339,3340],[39,125435],{"className":125436,"style":256},[230],[39,125438,80841],{"className":125439},[83,84],[39,125441],{"className":125442,"style":256},[230],[39,125444,29425],{"className":125445},[83,84],[39,125447,136],{"className":125448},[182],[39,125450,73771],{"className":125451},[83,84],[39,125453],{"className":125454,"style":461},[230],[39,125456,5378],{"className":125457},[465],[39,125459],{"className":125460,"style":461},[230],[39,125462,125464,125467,125470,125473],{"className":125463},[74],[39,125465],{"className":125466,"style":245},[78],[39,125468,73802],{"className":125469,"style":73818},[83,84],[39,125471,161],{"className":125472},[263],[39,125474,2903],{"className":125475},[83],[11,125477,125478],{},"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。",[11,125480,125481],{},"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":72720,"searchDepth":72721,"depth":72721,"links":125483},[125484,125485,125486,125487,125488,125489,125490,125491,125492,125493,125494],{"id":72987,"depth":72721,"text":72987},{"id":78114,"depth":72721,"text":78114},{"id":85023,"depth":72721,"text":85023},{"id":92179,"depth":72721,"text":92179},{"id":95617,"depth":72721,"text":95617},{"id":102118,"depth":72721,"text":102118},{"id":106615,"depth":72721,"text":106616},{"id":114617,"depth":72721,"text":114617},{"id":118285,"depth":72721,"text":118285},{"id":123485,"depth":72721,"text":123485},{"id":125119,"depth":72721,"text":125119},{},"\u002Fblog\u002F2026\u002F0014-zui-you-chuan-shu-ji-he-jian-lun",{"title":72976,"description":72981},"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。",[72778,72777,125501],"statistics","2026-10-31T20:17:23+08:00","IAT5rYeOtscnEe8Mhh1eCeIF4PJR1Ewyc86Z_G_0ncI",1794483690791]