Datasets:
File size: 100,352 Bytes
034f953 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600 601 602 603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650 651 652 653 654 655 656 657 658 659 660 661 662 663 664 665 666 667 668 669 670 671 672 673 674 675 676 677 678 679 680 681 682 683 684 685 686 687 688 689 690 691 692 693 694 695 696 697 698 699 700 701 702 703 704 705 706 707 708 709 710 711 712 713 714 715 716 717 718 719 720 721 722 723 724 725 726 727 728 729 730 731 732 733 734 735 736 737 738 739 740 741 742 743 744 745 746 747 748 749 750 751 752 753 754 755 756 757 758 759 760 761 762 763 764 765 766 767 768 769 770 771 772 773 774 775 776 777 778 779 780 781 782 783 784 785 786 787 788 789 790 791 792 793 794 795 796 797 798 799 800 801 802 803 804 805 806 807 808 809 810 811 812 813 814 815 816 817 818 819 820 821 822 823 824 825 826 827 828 829 830 831 832 833 834 835 836 837 838 839 840 841 842 843 844 845 846 847 848 849 850 851 852 853 854 855 856 857 858 859 860 861 862 863 864 865 866 867 868 869 870 871 872 873 874 875 876 877 878 879 880 881 882 883 884 885 886 887 888 889 890 891 892 893 894 895 896 897 898 899 900 901 902 903 904 905 906 907 908 909 910 911 912 913 914 915 916 917 918 919 920 921 922 923 924 925 926 927 928 929 930 931 932 933 934 935 936 937 938 939 940 941 942 943 944 945 946 947 948 949 950 951 952 953 954 955 956 957 958 959 960 961 962 963 964 965 966 967 968 969 970 971 972 973 974 975 976 977 978 979 980 981 982 983 984 985 986 987 988 989 990 991 992 993 994 995 996 997 998 999 1000 1001 1002 1003 1004 1005 1006 1007 1008 1009 1010 1011 1012 1013 1014 1015 1016 1017 1018 1019 1020 1021 1022 1023 1024 1025 1026 1027 1028 1029 1030 1031 1032 1033 1034 1035 1036 1037 1038 1039 1040 1041 1042 1043 1044 1045 1046 1047 1048 1049 1050 1051 1052 1053 1054 1055 1056 1057 1058 1059 1060 1061 1062 1063 1064 1065 1066 1067 1068 1069 1070 1071 1072 1073 1074 1075 1076 1077 1078 1079 1080 1081 1082 1083 1084 1085 1086 1087 1088 1089 1090 1091 1092 1093 1094 1095 1096 1097 1098 1099 1100 1101 1102 1103 1104 1105 1106 1107 1108 1109 1110 1111 1112 1113 1114 1115 1116 1117 1118 1119 1120 1121 1122 1123 1124 1125 1126 1127 1128 1129 1130 1131 1132 1133 1134 1135 1136 1137 1138 1139 1140 1141 1142 1143 1144 1145 1146 1147 1148 1149 1150 1151 1152 1153 1154 1155 1156 1157 1158 1159 1160 1161 1162 1163 1164 1165 1166 1167 1168 1169 1170 1171 1172 1173 1174 1175 1176 1177 1178 1179 1180 1181 1182 1183 1184 1185 1186 1187 1188 1189 1190 1191 1192 1193 1194 1195 1196 1197 1198 1199 1200 1201 1202 1203 1204 1205 1206 1207 1208 1209 1210 1211 1212 1213 1214 1215 1216 1217 1218 1219 1220 1221 1222 1223 1224 1225 1226 1227 1228 1229 1230 1231 1232 1233 1234 1235 1236 1237 1238 1239 1240 1241 1242 1243 1244 1245 1246 1247 1248 1249 1250 1251 1252 1253 1254 1255 1256 1257 1258 1259 1260 1261 1262 1263 1264 1265 1266 1267 1268 1269 1270 1271 1272 1273 1274 1275 1276 1277 1278 1279 1280 1281 1282 1283 1284 1285 1286 1287 1288 1289 1290 1291 1292 1293 1294 1295 1296 1297 1298 1299 1300 1301 1302 1303 1304 1305 1306 1307 1308 1309 1310 1311 1312 1313 1314 1315 1316 1317 1318 1319 1320 1321 1322 1323 1324 1325 1326 1327 1328 1329 1330 1331 1332 1333 1334 1335 1336 1337 1338 1339 1340 1341 1342 1343 1344 1345 1346 1347 1348 1349 1350 1351 1352 1353 1354 1355 1356 1357 1358 1359 1360 1361 1362 1363 1364 1365 1366 1367 1368 1369 1370 1371 1372 1373 1374 1375 1376 1377 1378 1379 1380 1381 1382 1383 1384 1385 1386 1387 1388 1389 1390 1391 1392 1393 1394 1395 1396 1397 1398 1399 1400 1401 1402 1403 1404 1405 1406 1407 1408 1409 1410 1411 1412 1413 1414 1415 1416 1417 1418 1419 1420 1421 1422 1423 1424 1425 1426 1427 1428 1429 1430 1431 1432 1433 1434 1435 1436 1437 1438 1439 1440 1441 1442 1443 1444 1445 1446 1447 1448 1449 1450 1451 1452 1453 1454 1455 1456 1457 1458 1459 1460 1461 1462 1463 1464 1465 1466 1467 1468 1469 1470 1471 1472 1473 1474 1475 1476 1477 1478 1479 1480 1481 1482 1483 1484 1485 1486 1487 1488 1489 1490 1491 1492 1493 1494 1495 1496 1497 1498 1499 1500 1501 1502 1503 1504 1505 1506 1507 1508 1509 1510 1511 1512 1513 1514 1515 1516 1517 1518 1519 1520 1521 1522 1523 1524 1525 1526 1527 1528 1529 1530 1531 1532 1533 1534 1535 1536 1537 1538 1539 1540 1541 1542 1543 1544 1545 1546 1547 1548 1549 1550 1551 1552 1553 1554 1555 1556 1557 1558 1559 1560 1561 1562 1563 1564 1565 1566 1567 1568 1569 1570 1571 1572 1573 1574 1575 1576 1577 1578 1579 1580 1581 1582 1583 1584 1585 1586 1587 1588 1589 1590 1591 1592 1593 1594 1595 1596 1597 1598 1599 1600 1601 1602 1603 1604 1605 1606 1607 1608 1609 1610 1611 1612 1613 1614 1615 1616 1617 1618 1619 1620 1621 1622 1623 1624 1625 1626 1627 1628 1629 1630 1631 1632 1633 1634 1635 1636 1637 1638 1639 1640 1641 1642 1643 1644 1645 1646 1647 1648 1649 1650 1651 1652 1653 1654 1655 1656 1657 1658 1659 1660 1661 1662 1663 1664 1665 1666 1667 1668 1669 1670 1671 1672 1673 1674 1675 1676 1677 1678 1679 1680 1681 1682 1683 1684 1685 1686 1687 1688 1689 1690 1691 1692 1693 1694 1695 1696 1697 1698 1699 1700 1701 1702 1703 1704 1705 1706 1707 1708 1709 1710 1711 1712 1713 1714 1715 1716 1717 1718 1719 1720 1721 1722 1723 1724 1725 1726 1727 1728 1729 1730 1731 1732 1733 1734 1735 1736 1737 1738 1739 1740 1741 1742 1743 1744 1745 1746 1747 1748 1749 1750 1751 1752 1753 1754 1755 1756 1757 1758 1759 1760 1761 1762 1763 1764 1765 1766 1767 1768 1769 1770 1771 1772 1773 1774 1775 1776 1777 1778 1779 1780 1781 1782 1783 1784 1785 1786 1787 1788 1789 1790 1791 1792 1793 1794 1795 1796 1797 1798 1799 1800 1801 1802 1803 1804 1805 1806 1807 1808 1809 1810 1811 1812 1813 1814 1815 1816 1817 1818 1819 1820 1821 1822 1823 1824 1825 1826 1827 1828 1829 1830 1831 1832 1833 1834 1835 1836 1837 1838 1839 1840 1841 1842 1843 1844 1845 1846 1847 1848 1849 1850 1851 1852 1853 1854 1855 1856 1857 1858 1859 1860 1861 1862 1863 1864 1865 1866 1867 1868 1869 1870 1871 1872 1873 1874 1875 | [
{
"type": "text",
"text": "Towards Instance-Optimal Offline Reinforcement Learning with Pessimism ",
"text_level": 1,
"bbox": [
200,
122,
799,
172
],
"page_idx": 0
},
{
"type": "text",
"text": "Ming Yin 1,2 and Yu-Xiang Wang1 ",
"bbox": [
382,
222,
614,
238
],
"page_idx": 0
},
{
"type": "text",
"text": "1Department of Computer Science, UC Santa Barbara 2Department of Statistics and Applied Probability, UC Santa Barbara ming_yin@ucsb.edu yuxiangw@cs.ucsb.edu ",
"bbox": [
272,
250,
723,
292
],
"page_idx": 0
},
{
"type": "text",
"text": "Abstract ",
"text_level": 1,
"bbox": [
462,
333,
535,
349
],
"page_idx": 0
},
{
"type": "text",
"text": "We study the offline reinforcement learning (offline RL) problem, where the goal is to learn a reward-maximizing policy in an unknown Markov Decision Process (MDP) using the data coming from a policy $\\mu$ . In particular, we consider the sample complexity problems of offline RL for finite-horizon MDPs. Prior works study this problem based on different data-coverage assumptions, and their learning guarantees are expressed by the covering coefficients which lack the explicit characterization of system quantities. In this work, we analyze the Adaptive Pessimistic Value Iteration (APVI) algorithm and derive the suboptimality upper bound that nearly matches ",
"bbox": [
232,
364,
766,
489
],
"page_idx": 0
},
{
"type": "equation",
"img_path": "images/0cf94f59bc6ce48ff9b78391bb16bfa1365c7b8827a1bf2cdf0fb1aa10bd419c.jpg",
"text": "$$\nO \\left( \\sum _ { h = 1 } ^ { H } \\sum _ { s _ { h } , a _ { h } } { d _ { h } ^ { \\pi ^ { \\star } } ( s _ { h } , a _ { h } ) \\sqrt { \\frac { \\mathrm { V a r } _ { P _ { s _ { h } , a _ { h } } } ( V _ { h + 1 } ^ { \\star } + r _ { h } ) } { d _ { h } ^ { \\mu } ( s _ { h } , a _ { h } ) } } \\sqrt { \\frac { 1 } { n } } } \\right) .\n$$",
"text_format": "latex",
"bbox": [
302,
496,
696,
546
],
"page_idx": 0
},
{
"type": "text",
"text": "In complementary, we also prove a per-instance information-theoretical lower bound under the weak assumption that $d _ { h } ^ { \\mu } ( s _ { h } , a _ { h } ) > 0$ if $d _ { h } ^ { \\pi ^ { \\star } } ( s _ { h } , a _ { h } ) > 0$ . Different from the previous minimax lower bounds, the per-instance lower bound (via local minimaxity) is a much stronger criterion as it applies to individual instances separately. Here $\\pi ^ { \\star }$ is a optimal policy, $\\mu$ is the behavior policy and $d _ { h } ^ { \\mu }$ is the marginal state-action probability. We call (1) the intrinsic offline reinforcement learning bound since it directly implies all the existing optimal results: minimax rate under uniform data-coverage assumption, horizon-free setting, single policy concentrability, and the tight problem-dependent results. Later, we extend the result to the assumption-free regime (where we make no assumption on $\\mu$ ) and obtain the assumption-free intrinsic bound. Due to its generic form, we believe the intrinsic bound could help illuminate what makes a specific problem hard and reveal the fundamental challenges in offline RL. ",
"bbox": [
232,
551,
766,
733
],
"page_idx": 0
},
{
"type": "text",
"text": "1 Introduction ",
"text_level": 1,
"bbox": [
174,
758,
310,
776
],
"page_idx": 0
},
{
"type": "text",
"text": "In offline reinforcement learning (offline RL Levine et al. [2020], Lange et al. [2012]), the goal is to learn a reward-maximizing policy in an unknown environment (Markov Decision Process or MDP) using the historical data coming from a (fixed) behavior policy $\\mu$ . Unlike online RL, where the agent can keep interacting with the environment and gain new feedback by exploring unvisited state-action space, offline RL usually populates when such online interplays are expensive or even unethical. Due to its nature of without the access to interact with the MDP model (which causes the distributional mismatches), most of the literature that study the sample complexity / provable efficiency of offline RL (e.g. Le et al. [2019], Chen and Jiang [2019], Xie and Jiang [2020, 2021], ",
"bbox": [
174,
790,
825,
901
],
"page_idx": 0
},
{
"type": "text",
"text": "Yin et al. [2021a,b], Ren et al. [2021], Rashidinejad et al. [2021], Xie et al. [2021b]) rely on making different data-coverage assumptions for making the problem learnable and provide the near-optimal worst-case performance bounds that depend on their data-coverage coefficients. Those results are valuable in general as they do not depend on the structure of the particular problem, therefore, remain valid even for pathological MDPs. But is this good enough? ",
"bbox": [
174,
90,
825,
161
],
"page_idx": 1
},
{
"type": "text",
"text": "In practice, the empirical performances of offline reinforcement learning (e.g. Gulcehre et al. [2020], Fu et al. [2020, 2021], Janner et al. [2021]) are often far better than what those non-adaptive / problem-independent bounds would indicate. Although empirical evidence can help explain why we may observe better or worse performances on different MDPs, a systematic understanding of what types of decision processes and what kinds of behavior policies are inherently easier or more challenging for offline RL is lacking. Besides, despite the fact that a non-adaptive bound can learn even the pathological examples within the assumption family, there is no guarantee for the instances outside the family. However, practical offline reinforcement learning problems are usually beyond the scope of certain data-coverage assumptions, which limits the applicability of those results. Can we make as few assumptions as possible? Or even more, what can we guarantee when no assumption is made about offline learning? ",
"bbox": [
174,
166,
825,
319
],
"page_idx": 1
},
{
"type": "text",
"text": "Those motivate us to derive the provably efficient bounds that are adaptive to the individual instances but only require minimal assumptions so they can be widely applied in most cases. Ideally, such bounds should characterize the system structures of the specific problems, hold even for peculiar instances that do not satisfy the standard data-coverage assumptions, and recover the worst-case guarantees when the assumptions are satisfied. As mentioned in Zanette and Brunskill [2019], a fully adaptive characterization in RL is important as it might bring considerable saving in the time spent designing domain-specific RL solutions and in training a human expert to judge and recognize the complexity of different problems. ",
"bbox": [
174,
325,
825,
436
],
"page_idx": 1
},
{
"type": "text",
"text": "1.1 Our contribution ",
"text_level": 1,
"bbox": [
176,
463,
331,
478
],
"page_idx": 1
},
{
"type": "text",
"text": "In this work, we provide the analysis for the adaptive pessimistic value iteration (APVI) (Algorithm 1) with finite horizon time-inhomogeneous (non-stationary) MDPs and derive a strong adaptive bound that is near-optimal under the weak assumption $d _ { h } ^ { \\mu } ( s _ { h } , a _ { h } ) > 0$ if $d _ { h } ^ { \\pi ^ { \\star } } ( s _ { h } , a _ { h } ) > \\bar { 0 }$ (Theorem 4.1). Specifically, our bound (quantity (1)) explicitly depends on the marginal importance ratios (between the optimal policy $\\pi ^ { \\star }$ and the behavior policy $\\mu _ { . }$ ) and the per-step conditional variances. In addition, we provide an instance-dependent (local minimax) lower bound (Theorem 4.3) to certify (1) is nearly optimal at the instance level for offline learning and call it the intrinsic offline learning bound. The intrinsic bound has the following consequences. ",
"bbox": [
174,
492,
826,
606
],
"page_idx": 1
},
{
"type": "text",
"text": "• In the non-adaptive / worst-case regime (4.1-4.3), the intrinsic bound implies $\\widetilde { O } ( H ^ { 3 } / d _ { m } \\epsilon ^ { 2 } )$ complexity under the uniform data-coverage 2.1, $\\tilde { O } ( H ^ { 3 } S C ^ { \\star } / \\epsilon ^ { 2 } )$ complexity under the single policy concentrability assumption 2.3 and $\\widetilde { O } ( H / d _ { m } \\epsilon ^ { 2 } )$ complexity when the sum of rewards is bounded by 1. All of those are optimal in their respectively regimes [Yin et al., 2021a, Rashidinejad et al., 2021, Xie et al., 2021b, Ren et al., 2021]; ",
"bbox": [
217,
626,
825,
702
],
"page_idx": 1
},
{
"type": "text",
"text": "• In the adaptive domain (4.4), the intrinsic bound implies the tight problem-dependent counterpart of Zanette and Brunskill [2019], yields ${ \\tilde { O } } ( H ^ { 3 } / n d _ { m } )$ fast convergence in the deterministic systems, has improved complexity in the partially deterministic systems and a family of highly mixing problems, and remains optimal when reducing to the tabular contextual bandits. ",
"bbox": [
215,
715,
823,
787
],
"page_idx": 1
},
{
"type": "text",
"text": "Beyond the above, due to the generic form of the intrinsic bound, we could come up with as many problem instances (that are of our interests) as possible and study their properties. In this sense, the intrinsic bound helps illuminate the fundamental nature of offline RL. ",
"bbox": [
176,
808,
825,
849
],
"page_idx": 1
},
{
"type": "text",
"text": "Furthermore, as a step towards assumption-free offline reinforcement learning, we build a modified AVPI and obtain an adaptive bound that could characterize the suboptimality gap in the state-action space that is agnostic to the behavior policy (Theorem 5.1). To the best of our knowledge, all of these results are the first of its kinds. ",
"bbox": [
176,
856,
823,
911
],
"page_idx": 1
},
{
"type": "text",
"text": "1.2 Related work ",
"text_level": 1,
"bbox": [
174,
90,
305,
106
],
"page_idx": 2
},
{
"type": "text",
"text": "Finite sample analysis for offline reinforcement learning can be traced back to Szepesvári and Munos [2005], Antos et al. [2008a,b] for the infinite horizon discounted setting via Fitted Q-Iteration (FQI) type function approximation algorithms. [Chen and Jiang, 2019, Le et al., 2019, Xie and Jiang, 2021, 2020] follow this line of research and derive the information-theoretical bounds. Recently, Xie and Jiang [2021] considers the offline RL with only the realizability assumption, Liu et al. [2020], Chang et al. [2021] considers the offline RL without sufficient coverage and Kidambi et al. [2020], Uehara and Sun [2021] uses the model-based approach for addressing offline RL. Under those weak coverage assumption, their finite sample analysis are suboptimal (e.g. in terms of the effective horizon $( 1 - \\gamma ) ^ { \\bar { - } 1 } .$ ). Recently, Yin et al. [2021a,b], Ren et al. [2021] study the finite horizon case. In the linear MDP case, Jin et al. [2020] studies the pessimistic algorithm for offline policy learning under only the compliance assumption, and, concurrently, Xie et al. [2021a] proposes the general pessimistic function approximation framework with instantiation in linear MDP and Zanette et al. [2021] shows actor-critic style algorithm is near-optimal for linear Bellman complete model. In addition, Wang et al. [2021], Zanette [2021] prove some exponential lower bounds under their linear function approximation assumptions. ",
"bbox": [
174,
118,
826,
325
],
"page_idx": 2
},
{
"type": "text",
"text": "Among them, there are a few works that achieve the sample optimality under their respective assumptions. Under the uniform data coverage (minimal state-action probability $d _ { m } > 0$ ), Yin et al. [2021a] first proves the optimal ${ \\cal \\tilde { O } } ( H ^ { 3 } / d _ { m } \\epsilon ^ { 2 } )$ complexity in the time-inhomogeneous MDP. Recently, Yin et al. [2021b] designs the offline variance reduction algorithm to achieve the optimal $\\tilde { O } ( H ^ { 2 } / d _ { m } \\epsilon ^ { 2 } )$ rate for the time-homogeneous case. Under the setting where the total cumulative reward is bounded by 1, Ren et al. [2021] obtains the horizon-free result with $\\tilde { O } ( 1 / d _ { m } )$ . More recently, Rashidinejad et al. [2021] considers the single concentrability coefficient $\\begin{array} { r } { C ^ { \\star } : = \\operatorname* { m a x } _ { s , a } d ^ { \\pi ^ { \\star } } ( s , a ) / d ^ { \\mu } ( s , a ) } \\end{array}$ and derives the upper bound $\\tilde { O } [ ( 1 - \\gamma ) ^ { - 5 } S C ^ { \\star } / \\epsilon ^ { 2 } ]$ in the infinite horizon setting which is recently improved by the concurrent work Xie et al. [2021b]. While those worst-case guarantees are desirable, none of them can explain the hardness of the individual problems.1 ",
"bbox": [
173,
332,
826,
481
],
"page_idx": 2
},
{
"type": "text",
"text": "2 Preliminaries ",
"text_level": 1,
"bbox": [
174,
505,
318,
522
],
"page_idx": 2
},
{
"type": "text",
"text": "Episodic non-stationary (time-varying) reinforcement learning. A finite-horizon Markov Decision Process (MDP) is denoted by a tuple $M = ( S , A , P , r , H , d _ { 1 } )$ [Sutton and Barto, 2018], where $s$ is the finite state space and $\\mathcal { A }$ is the finite action space with $S : = | S | < \\infty , A : = | A | < \\infty$ . A nonstationary transition kernel $P _ { h } : S \\times \\mathcal { A } \\times \\mathcal { S } \\mapsto [ 0 , 1 ]$ maps each state action $\\left( s _ { h } , a _ { h } \\right)$ to a probability distribution $P _ { h } ( \\cdot | s _ { h } , a _ { h } )$ and $P _ { h }$ can be different across the time. Besides, $r : S \\times A \\mapsto \\mathbb { R }$ is the expected instantaneous reward function satisfying $0 \\leq r \\leq 1$ . $d _ { 1 }$ is the initial state distribution. $H$ is the horizon. A policy $\\pi = ( \\pi _ { 1 } , \\ldots , \\pi _ { H } )$ assigns each state $s _ { h } \\in S$ a probability distribution over actions according to the map $s _ { h } \\mapsto \\pi _ { h } ( \\cdot | s _ { h } ) \\forall h \\in [ H ]$ . An MDP together with a policy $\\pi$ induce a random trajectory $s _ { 1 } , a _ { 1 } , r _ { 1 } , \\ldots , s _ { H } , a _ { H } , r _ { H } , s _ { H + 1 }$ with $s _ { 1 } \\sim d _ { 1 } , a _ { h } \\sim \\pi ( \\cdot | s _ { h } ) , s _ { h + 1 } \\sim P _ { h } ( \\cdot | s _ { h } , a ) , \\forall h \\in [ H ]$ and $r _ { h }$ is a random realization given the observed $s _ { h } , a _ { h }$ . ",
"bbox": [
173,
539,
826,
678
],
"page_idx": 2
},
{
"type": "text",
"text": "$Q$ -values, Bellman (optimality) equations. The value function $V _ { h } ^ { \\pi } ( \\cdot ) \\in \\mathbb { R } ^ { S }$ and $\\mathbf { Q }$ -value function $Q _ { h } ^ { \\pi } ( \\cdot , \\cdot ) \\in \\mathbb { R } ^ { S \\times A }$ for any policy $\\pi$ is defined as: $V _ { h } ^ { \\pi } ( s ) ~ = ~ \\mathbb { E } _ { \\pi } [ \\sum _ { t = h } ^ { H } r _ { t } | s _ { h } = s ]$ , $Q _ { h } ^ { \\pi } ( s , a ) =$ $\\begin{array} { r } { \\mathbb { E } _ { \\pi } [ \\sum _ { t = h } ^ { H } r _ { t } | s _ { h } , a _ { h } \\ = \\ s , a ] } \\end{array}$ , $\\forall s , a \\in \\mathcal { S } , \\mathcal { A } , h \\in [ H ]$ . The performance is defined as $v ^ { \\pi } : =$ $\\begin{array} { r } { \\mathbb { E } _ { d _ { 1 } } \\left[ V _ { 1 } ^ { \\pi } \\right] = \\mathbb { E } _ { \\pi , d _ { 1 } } \\left[ \\sum _ { t = 1 } ^ { H } r _ { t } \\right] } \\end{array}$ , where we denote $V _ { h } ^ { \\pi } , Q _ { h } ^ { \\pi }$ as column vectors and $P _ { h } ~ \\in ~ \\mathbb { R } ^ { S A \\times S }$ the transition matrix, then the vector form Bellman (optimality) equations follow $\\forall h \\in [ H ]$ : $\\begin{array} { r } { Q _ { h } ^ { \\pi } = r _ { h } + P _ { h } V _ { h + 1 } ^ { \\pi } } \\end{array}$ , $V _ { h } ^ { \\pi } = \\mathbb { E } _ { a \\sim \\pi _ { h } } [ Q _ { h } ^ { \\pi } ]$ , $Q _ { h } ^ { \\star } = r _ { h } + P _ { h } V _ { h + 1 } ^ { \\star }$ , $V _ { h } ^ { \\star } = \\operatorname* { m a x } _ { a } Q _ { h } ^ { \\star } ( \\cdot , a )$ .In addition, we denote the per-step marginal state-action occupancy $d _ { h } ^ { \\pi } ( s , a )$ as: $d _ { h } ^ { \\pi } ( s , a ) : = \\mathbb { P } [ s _ { h } =$ $s | s _ { 1 } \\sim d _ { 1 } , \\pi ] \\cdot \\pi _ { h } ( a | s )$ , which is the marginal state-action probability at time $h$ . ",
"bbox": [
173,
683,
825,
815
],
"page_idx": 2
},
{
"type": "text",
"text": "Offline setting and the goal. The offline RL requires the agent to find a policy $\\pi$ such that the performance $v ^ { \\pi }$ is maximized, given only the episodic data $\\mathcal { D } = \\{ ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } , r _ { h } ^ { \\tau } , s _ { h + 1 } ^ { \\tau } ) \\} _ { \\tau \\in [ n ] } ^ { h \\in [ H ] }$ rolled out from some behavior policy $\\mu$ . The offline nature requires we cannot change $\\mu$ and in particular we do not assume the functional knowledge of $\\mu$ . That is to say, given the batch data $\\mathcal { D }$ and a targeted accuracy $\\epsilon > 0$ , the offline RL seeks to find a policy $\\pi _ { \\mathrm { a l g } }$ such that $v ^ { \\star } - v ^ { \\pi _ { \\mathrm { a l g } } } \\leq \\epsilon$ . ",
"bbox": [
174,
820,
825,
866
],
"page_idx": 2
},
{
"type": "text",
"text": "",
"bbox": [
171,
90,
825,
121
],
"page_idx": 3
},
{
"type": "text",
"text": "2.1 Assumptions in offline RL ",
"text_level": 1,
"bbox": [
174,
137,
393,
151
],
"page_idx": 3
},
{
"type": "text",
"text": "We revise several types of assumptions proposed by existing studies that can yield provably efficient results. Recall $d _ { h } ^ { \\mu } ( s _ { h } , a _ { h } )$ is the marginal state-action probability and $\\mu$ is the behavior policy. ",
"bbox": [
174,
162,
823,
191
],
"page_idx": 3
},
{
"type": "text",
"text": "Assumption 2.1 (Uniform data coverage [Yin et al., 2021a]). The behavior policy obeys that $d _ { m } : =$ $\\begin{array} { r } { \\operatorname* { m i n } _ { h , s _ { h } , a _ { h } } d _ { h } ^ { \\mu } ( s _ { h } , a _ { h } ) > 0 . } \\end{array}$ . Here the infimum is over all the states satisfying there exists certain policy so that this state can be reached by the current MDP with this policy. ",
"bbox": [
174,
195,
825,
238
],
"page_idx": 3
},
{
"type": "text",
"text": "This is the strongest assumption in offline RL as it requires $\\mu$ to explore each state-action pairs with positive probability. Under 2.1, it mostly holds $1 / d _ { m } \\ge S A$ . This reveals offline learning is generically harder than the generative model setting [Agarwal et al., 2020] in the statistical sense. On the other hand, this is required for the uniform $O P E$ task in Yin et al. [2021a] as it seeks to simultaneously evaluate all the policies within the policy class and it is in general a harder task than offline learning itself. ",
"bbox": [
173,
250,
825,
334
],
"page_idx": 3
},
{
"type": "text",
"text": "Assumption 2.2 (Uniform concentrability Szepesvári and Munos [2005], Chen and Jiang [2019]). \nFor all the policies, $\\begin{array} { r } { C _ { \\mu } : = \\operatorname* { s u p } _ { \\pi , h } | | d _ { h } ^ { \\pi } ( \\cdot , \\cdot ) / d _ { h } ^ { \\mu } ( \\cdot , \\cdot ) | | _ { \\infty } < \\infty . } \\end{array}$ . ",
"bbox": [
173,
338,
823,
369
],
"page_idx": 3
},
{
"type": "text",
"text": "This is a classical offline RL condition that is commonly assumed in the function approximation scheme (e.g. Fitted Q-Iteration). Qualitatively, this is a uniform data-coverage assumption that is similar to Assumption 2.1, but quantitatively, the coefficient $C _ { \\mu }$ can be smaller than $1 / \\bar { d _ { m } } \\bar { }$ due the $d _ { h } ^ { \\pi }$ term in the numerator. ",
"bbox": [
173,
377,
825,
434
],
"page_idx": 3
},
{
"type": "text",
"text": "Assumption 2.3 (Liu et al. [2019]). There exists one optimal policy $\\pi ^ { \\star }$ , s.t. $\\forall s _ { h } , a _ { h } \\ \\in \\ S , A ,$ , $d _ { h } ^ { \\mu } ( s _ { h } , a _ { h } ) \\ > \\ 0$ if $d _ { h } ^ { \\pi ^ { \\star } } ( s _ { h } , a _ { h } ) > 0$ . We further denote the trackable set as $\\mathcal C _ { h } : = \\{ ( s _ { h } , a _ { h } )$ : $d _ { h } ^ { \\tilde { \\mu } } ( s _ { h } , a _ { h } ) > 0 \\}$ . ",
"bbox": [
174,
439,
826,
483
],
"page_idx": 3
},
{
"type": "text",
"text": "Assumption 2.3 is (arguably) the weakest assumption needed for accurately learning the optimal value $v ^ { \\star }$ and we will use 2.3 for most parts of this paper. It only requires $\\mu$ to trace the state-action space of one optimal policy and can be agnostic at other locations. Rashidinejad et al. [2021], Xie et al. [2021b] considers this assumption and provide analysis is based on the single concentrability coefficient $\\begin{array} { r } { C ^ { \\star } : = \\operatorname* { m a x } _ { s , a } { d ^ { \\pi ^ { \\star } } ( s , a ) } / { d ^ { \\mu } ( s , a ) } } \\end{array}$ . The dependence on $C ^ { \\star }$ makes their result less adaptive since there can be lots of locations that have the ratio ${ d ^ { \\pi } } ^ { \\star } ( s , a ) / { d ^ { \\mu } } ( s , a )$ much smaller than $C ^ { \\star }$ . Furthermore, what could we end up with when 2.3 is not met? We will provide our answers in the subsequent sections. ",
"bbox": [
173,
493,
826,
609
],
"page_idx": 3
},
{
"type": "text",
"text": "3 A warm-up case study: Vanilla Pessimistic Value Iteration ",
"text_level": 1,
"bbox": [
174,
628,
691,
647
],
"page_idx": 3
},
{
"type": "text",
"text": "As a step towards the optimal and strong adaptive offline RL bound, we analyze the vanilla pessimistic value iteration (VPVI), a tabular counterpart of pessimistic value iteration (PEVI initiated in Jin et al. [2020]), to understand what is missing for achieving the fully adaptivity. In particular, VPVI relies on the model-based construction. ",
"bbox": [
174,
661,
825,
717
],
"page_idx": 3
},
{
"type": "text",
"text": "Model-based Components. Given data n $\\begin{array} { r } { { \\mathcal { D } } \\ = \\ \\left\\{ \\left( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } , r _ { h } ^ { \\tau } , s _ { h + 1 } ^ { \\tau } \\right) \\right\\} _ { \\tau \\in [ n ] } ^ { h \\in [ H ] } } \\end{array}$ , we denote $n _ { s _ { h } , a _ { h } } : =$ $\\begin{array} { r } { \\sum _ { \\tau = 1 } ^ { n } \\mathbf { 1 } [ s _ { h } ^ { \\tau } , { a } _ { h } ^ { \\tau } \\ = \\ s _ { h } , { a } _ { h } ] } \\end{array}$ be the total counts that vis construct the estimators for $\\left( s _ { h } , a _ { h } \\right)$ pair at time as: , then we use the $P _ { h }$ $r _ { h }$ ",
"bbox": [
174,
722,
825,
767
],
"page_idx": 3
},
{
"type": "equation",
"img_path": "images/d06ad75b7bc6c9f05d1f248afa41dda6a85409c8e50be048ab90173d8ae25aa9.jpg",
"text": "$$\n\\widehat { P } _ { h } ( s ^ { \\prime } | s _ { h } , a _ { h } ) = \\frac { \\sum _ { \\tau = 1 } ^ { n } \\mathbf { 1 } [ ( s _ { h + 1 } ^ { \\tau } , a _ { h } ^ { \\tau } , s _ { h } ^ { \\tau } ) = ( s ^ { \\prime } , s _ { h } , a _ { h } ) ] } { n _ { s _ { h } , a _ { h } } } , \\widehat { r } _ { h } ( s _ { h } , a _ { h } ) = \\frac { \\sum _ { \\tau = 1 } ^ { n } \\mathbf { 1 } [ ( a _ { h } ^ { \\tau } , s _ { h } ^ { \\tau } ) = ( s _ { h } , a _ { h } ) ] \\cdot r _ { h } ^ { \\tau } } { n _ { s _ { h } , a _ { h } } } ,\n$$",
"text_format": "latex",
"bbox": [
181,
772,
823,
805
],
"page_idx": 3
},
{
"type": "text",
"text": "if $n _ { s _ { h } , a _ { h } } > 0$ and $\\widehat { P } _ { h } ( s ^ { \\prime } | s _ { h } , a _ { h } ) = 1 / S , \\widehat { r } _ { h } ( s _ { h } , a _ { h } ) = 0$ if $n _ { s _ { h } , a _ { h } } = 0$ . In particular, we use the word b“vanilla” as it directly mirrors Jin et al. [2020] with a pessimistic penalty of order $O ( H / \\sqrt { n _ { s _ { h } , a _ { h } } } )$ . 2 With $\\widehat { P } _ { h } , \\widehat { r } _ { h }$ in Algorithm 2 (which we defer to Appendix), VPVI guarantees the following: ",
"bbox": [
173,
815,
825,
866
],
"page_idx": 3
},
{
"type": "text",
"text": "Theorem 3.1. Under the Assumption 2.3, denote $\\begin{array} { r } { \\bar { d } _ { m } : = \\operatorname* { m i n } _ { h \\in [ H ] } \\{ d _ { h } ^ { \\mu } ( s _ { h } , a _ { h } ) : d _ { h } ^ { \\mu } ( s _ { h } , a _ { h } ) > 0 \\} } \\end{array}$ . For any $0 < \\delta < 1$ , there exists absolute constants $c _ { 0 } , C ^ { \\prime } > 0$ , such that when $n > c _ { 0 } \\cdot 1 / \\bar { d } _ { m } \\cdot \\iota$ $( \\iota = \\log ( H S A / \\delta ) )$ , with probability $1 - \\delta$ , the output policy $\\widehat { \\pi }$ of VPVI satisfies ",
"bbox": [
173,
89,
826,
137
],
"page_idx": 4
},
{
"type": "equation",
"img_path": "images/d5aaa8742b8765850fe826ef7f579d0b1ca064fe3b3e881df803bc52cf418ca8.jpg",
"text": "$$\n0 \\leq v ^ { \\star } - v ^ { \\widehat { \\pi } } \\leq C ^ { \\prime } H \\sum _ { h = 1 } ^ { H } \\sum _ { ( s _ { h } , a _ { h } ) \\in \\mathcal { C } _ { h } } d _ { h } ^ { \\pi ^ { \\star } } ( s _ { h } , a _ { h } ) \\cdot \\sqrt { \\frac { \\iota } { n \\cdot d _ { h } ^ { \\mu } ( s _ { h } , a _ { h } ) } } .\n$$",
"text_format": "latex",
"bbox": [
276,
142,
722,
189
],
"page_idx": 4
},
{
"type": "text",
"text": "The full proof can be found in Appendix C. Theorem 3.1 makes some improvements over the existing works. First, it is more adaptive than the results with uniform data-coverage Assumption 2.1 (Yin et al. [2021a], Ren et al. [2021]). In addition, by straightforward calculation (3) can be bounded by $\\tilde { O } ( \\sqrt { H ^ { 4 } S C ^ { \\star } / n } )$ which improves VI-LCB [Rashidinejad et al., 2021] by a factor of $H$ .3 Besides, the analysis of VPVI also improves the direct reduction of PEVI [Jin et al., 2020] in the tabular case by a factor $_ { S A }$ since their $\\beta = S A H$ when $d = S A$ . ",
"bbox": [
173,
202,
825,
287
],
"page_idx": 4
},
{
"type": "text",
"text": "However, VPVI is not optimal as the dependence on horizon is $H ^ { 4 }$ which does not match the optimal worst case guarantee $H ^ { 3 }$ [Yin et al., 2021a] in the nonstationary setting. Also, the explicit dependence on $H$ in (3) possibly hides some key features of the specific offline RL instances. For example, no improvement can be made if the system has the deterministic transition. ",
"bbox": [
173,
294,
825,
351
],
"page_idx": 4
},
{
"type": "text",
"text": "Algorithm 1 Adaptive (assumption-free) Pessimistic Value Iteration or LCBVI-Bernstein ",
"text_level": 1,
"bbox": [
173,
366,
761,
382
],
"page_idx": 4
},
{
"type": "text",
"text": "1: Input: Offline dataset $\\boldsymbol { \\mathcal { D } } = \\{ ( \\boldsymbol { s } _ { h } ^ { \\tau } , \\boldsymbol { a } _ { h } ^ { \\tau } , \\boldsymbol { r } _ { h } ^ { \\tau } , \\boldsymbol { s } _ { h + 1 } ^ { \\tau } ) \\} _ { \\tau , h = 1 } ^ { n , H }$ . Set $C _ { 1 } = 2 , C _ { 2 } = 1 4$ , failure probability $\\delta$ . \n2: Initialization: Set $\\widehat { V } _ { H + 1 } ( \\cdot ) 0$ . Set $\\iota = \\log ( H S A / \\delta )$ . (if assumption-free, set $M ^ { \\dagger } , \\widehat { M } ^ { \\dagger }$ as in Section 5.) \n3: for time $h = H , H - 1 , \\ldots , 1 \\bullet$ do \n4: Set $\\widehat { Q } _ { h } ( \\cdot , \\cdot ) \\gets \\widehat { r } _ { h } ( \\cdot , \\cdot ) + ( \\widehat { P } _ { h } \\cdot \\widehat { V } _ { h + 1 } ) ( \\cdot , \\cdot ) \\mathrm { ~ } ( \\mathrm { u s e } \\widehat { r } _ { h } ^ { \\dag } + ( \\widehat { P } _ { h } ^ { \\dag } \\cdot \\widehat { V } _ { h + 1 } )$ if assumption-free) \n5: $\\forall s _ { h } , a _ { h }$ , set $\\begin{array} { r } { \\Gamma _ { h } ( s _ { h } , a _ { h } ) = C _ { 1 } \\sqrt { \\frac { \\operatorname { V a r } _ { \\widehat { P } _ { s _ { h } , a _ { h } } } ( \\widehat { r } _ { h } + \\widehat { V } _ { h + 1 } ) \\cdot \\iota } { { n _ { s _ { h } , a _ { h } } } } } + \\frac { C _ { 2 } H \\cdot \\iota } { n _ { s _ { h } , a _ { h } } } } \\end{array}$ if $n _ { s _ { h } , a _ { h } } \\ge 1$ , o.w. set to $\\frac { C H \\iota } { 1 }$ . \n6: (If assumption-free, use C1qVarPb†sh,ah ( $\\begin{array} { r } { C _ { 1 } \\sqrt { \\mathrm { V a r } _ { \\widehat { P } _ { s _ { h } , a _ { h } } ^ { \\dagger } } ( \\widehat { r } _ { h } ^ { \\dagger } + \\widehat { V } _ { h + 1 } ) \\cdot \\iota / n _ { s _ { h } , a _ { h } } } + \\frac { C _ { 2 } H \\cdot \\iota } { n _ { s _ { h } , a _ { h } } } } \\end{array}$ C2H·ιn if nsh,ah ≥ 1, o.w. use 0.) \n7: Set $\\widehat { Q } _ { h } ^ { p } ( \\cdot , \\cdot ) \\gets \\widehat { Q } _ { h } ( \\cdot , \\cdot ) - \\dot { \\Gamma _ { h } } ( \\cdot , \\cdot )$ . Set $\\overline { { { Q } } } _ { h } ( \\cdot , \\cdot ) \\operatorname* { m i n } \\{ \\widehat { Q } _ { h } ^ { p } ( \\cdot , \\cdot ) , H - h + 1 \\} ^ { + }$ . // Pessmistic update \n8: $\\forall s _ { h }$ , Select $\\begin{array} { r } { \\widehat \\pi _ { h } ( \\cdot | s _ { h } ) \\gets \\operatorname * { a r g m a x } _ { \\pi _ { h } } \\langle \\overline { Q } _ { h } ( s _ { h } , \\cdot ) , \\pi _ { h } ( \\cdot | s _ { h } ) \\rangle } \\end{array}$ . Set $\\widehat { V } _ { h } ( s _ { h } ) \\gets \\langle \\overline { { Q } } _ { h } ( s _ { h } , \\cdot ) , \\widehat { \\pi } _ { h } ( \\cdot | s _ { h } ) \\rangle$ . \n9: end for \n10: Output: $\\left\\{ \\widehat { \\pi } _ { h } \\right\\}$ . ",
"bbox": [
176,
385,
826,
555
],
"page_idx": 4
},
{
"type": "text",
"text": "4 Intrinsic Offline Reinforcement Learning bound ",
"text_level": 1,
"bbox": [
173,
582,
607,
602
],
"page_idx": 4
},
{
"type": "text",
"text": "Now we go deeper to understand what is the more intrinsic characterization for offline reinforcement learning. From the study of VPVI, penalizing the Q-function by $\\widetilde { O } ( H / \\sqrt { n _ { s _ { h } , a _ { h } } } )$ is crude as it estimates the confidence width of ${ \\widehat { Q } } _ { h }$ in Algorithm 2 too conservatively therefore loses the accuracy (the bound is suboptimal). This motivates us to use empirical standard deviation instead to create a more adaptive (and also less conservative) Bernstein-type confidence width as the pessimistic penalty: ",
"bbox": [
173,
612,
826,
689
],
"page_idx": 4
},
{
"type": "equation",
"img_path": "images/1f9c995771f5021da4adef8ab8e4d1d821f9bc111b8c8d03bde8d1348b83f6f5.jpg",
"text": "$$\n\\Gamma _ { h } ( s _ { h } , a _ { h } ) = \\widetilde { \\mathcal { O } } \\bigg [ \\sqrt { \\frac { \\mathrm { V a r } _ { \\widetilde { P } _ { s _ { h } , a _ { h } } } ( \\widehat { r } _ { h } + \\widehat { V } _ { h + 1 } ) } { { n } _ { s _ { h } , a _ { h } } } } + \\frac { H } { { n } _ { s _ { h } , a _ { h } } } \\bigg ] \\ ( \\mathrm { i f } \\ n _ { s _ { h } , a _ { h } } > 0 ) ; \\ = \\widetilde { \\mathcal { O } } ( H ) \\ ( \\mathrm { i f } \\ n _ { s _ { h } , a _ { h } } = 0 ) .\n$$",
"text_format": "latex",
"bbox": [
187,
703,
792,
748
],
"page_idx": 4
},
{
"type": "text",
"text": "and update $\\widehat { Q } _ { h } \\gets \\widehat { Q } _ { h } - \\Gamma _ { h }$ . On one hand, $\\sqrt { \\mathrm { V a r } _ { \\widehat { P } _ { s _ { h } , a _ { h } } } ( \\widehat { r } _ { h } + \\widehat { V } _ { h + 1 } ) / n _ { s _ { h } , a _ { h } } }$ is a “less pessimistic” penalty than VPVI due to $\\sqrt { \\mathrm { V a r } _ { \\widehat { P } } ( \\widehat { r } _ { h } + \\widehat { V } _ { h + 1 } ) } \\le H$ and critically this design is more data-adaptive since it holds negative view towards the locations with high uncertainties and recommends the locations that we are confident about, as opposed to the online RL (which encourages exploration in the uncertain locations). Such principles are not reflected by the isotropic design in VPVI. On the other hand, it carries the extremely negative view towards fully agnostic locations ${ \\widetilde { O } } ( H )$ which in turn causes the agent unlikely to choose them. We summarized the this adaptive pessimistic value ",
"bbox": [
173,
755,
826,
876
],
"page_idx": 4
},
{
"type": "text",
"text": "iteration (APVI) into the Algorithm 1, with $\\widehat { P } _ { h } , \\widehat { r } _ { h }$ defined in (2). APVI has the following guarantee. \nbA sketch of the analysis is presented in Section B and Appendix F includes the full proof. ",
"bbox": [
171,
89,
825,
119
],
"page_idx": 5
},
{
"type": "text",
"text": "Theorem 4.1 (Intrinsic offline RL bound). Under the Assumption 2.3, denote $\\bar { d } _ { m } : =$ $\\mathrm { m i n } _ { h \\in [ H ] } \\{ d _ { h } ^ { \\mu } ( s _ { h } , a _ { h } ) \\ : \\ d _ { h } ^ { \\mu } ( s _ { h } , a _ { h } ) \\ > \\ 0 \\}$ . For any $0 ~ < ~ \\delta ~ < ~ 1$ , there exists absolute constants $c _ { 0 } , C ^ { \\prime } > 0$ , such that when $n > c _ { 0 } \\cdot 1 / \\bar { d } _ { m } \\cdot \\iota ( \\iota = \\log ( H S A / \\delta ) )$ , with probability $1 - \\delta$ , the output policy $\\widehat { \\pi }$ of APVI (Algorithm $I$ ) satisfies $\\widetilde O$ hides log factor and higher order terms) ",
"bbox": [
174,
122,
825,
186
],
"page_idx": 5
},
{
"type": "equation",
"img_path": "images/f8faee4d11f76593d45f80f3acf4eb263d1dc36bed744ce74f364b2ebd8d24c8.jpg",
"text": "$$\n0 \\leq v ^ { \\star } - v ^ { \\widehat { \\pi } } \\leq C ^ { \\prime } \\sum _ { h = 1 } ^ { H } \\sum _ { ( s _ { h } , a _ { h } ) \\in \\mathcal { C } _ { h } } d _ { h } ^ { \\pi ^ { \\star } } \\left( s _ { h } , a _ { h } \\right) \\cdot \\sqrt { \\frac { \\mathrm { V a r } _ { P _ { s _ { h } , a _ { h } } } \\left( r _ { h } + V _ { h + 1 } ^ { \\star } \\right) \\cdot \\iota } { n \\cdot d _ { h } ^ { \\mu } \\left( s _ { h } , a _ { h } \\right) } } + \\widetilde O \\left( \\frac { H ^ { 3 } } { n \\cdot \\bar { d } _ { m } } \\right)\n$$",
"text_format": "latex",
"bbox": [
184,
191,
792,
239
],
"page_idx": 5
},
{
"type": "text",
"text": "Remark 4.2. APVI (Algorithm $I$ ) can also be called LCBVI-Bernstein as it creates the offline counterpart of UCBVI in Azar et al. [2017]. However, to highlight that the resulting bound fully adapts to the specific system structure, we use the word “adaptive” instead. ",
"bbox": [
176,
244,
823,
287
],
"page_idx": 5
},
{
"type": "text",
"text": "APVI makes significant improvements in a lot of aspects. First and foremost, the dominate term is fully expressed by the system quantities that admits no explicit dependence on $H , S , A$ . To the best of our knowledge, this is the first offline RL bound that concretely depicts the interrelations within the problem when the problem instance is a tuple $( M , \\pi ^ { \\star } , \\mu )$ : an MDP $M$ (coupled with the optimal policy $\\pi ^ { \\star }$ ) with the data rolling from an offline logging policy $\\mu$ . As we will discuss later, this result indicates (nearly) all the optimal worst-case non-adaptive bounds (and clearly also the VPVI) under their respective regimes / assumptions. Thus, (5) is generic. More interestingly, Theorem 4.1 caters to the specific MDP structures and adaptively yields improved sample complexities (e.g. faster convergence in deterministic systems) that existing works cannot imply. Such features are crucial as it helps us to understand what type of problems are harder / easier than others, and even more, in a quantitative way. Last but not least, to illustrate this bound exhibits the intrinsic nature of offline RL, we prove a per-instance dependent information-theoretical lower bound that shares a similar formulation. The proof of Theorem 4.3 can be found in Appendix G. ",
"bbox": [
173,
297,
826,
478
],
"page_idx": 5
},
{
"type": "text",
"text": "Theorem 4.3 (Instance-dependent information theoretical offline lower bound). Denote $\\begin{array} { c c c c c c } { { \\mathcal G } } & { { : = } } & { { \\{ ( \\mu , M ) } } & { { : } } & { { \\exists \\pi ^ { \\star } } } & { { s } } \\end{array}$ .t. $d _ { h } ^ { \\mu } ( s , a ) \\quad > \\quad 0 \\quad i f \\quad d _ { h } ^ { \\pi ^ { \\star } } ( s , a ) \\quad > \\quad 0 \\}$ . Fix an instance $\\begin{array} { r l r l r } { { \\mathcal { P } } } & { { } = } & { ( \\mu , M ) } & { { } \\in { } } & { { \\mathcal { G } } } \\end{array}$ . Let $\\mathcal { D }$ consists of $n$ episodes and define $\\begin{array} { r l } { \\xi } & { { } = } \\end{array}$ $\\begin{array} { r } { \\operatorname* { s u p } _ { h , s _ { h } , a _ { h } , s _ { h + 1 } , d _ { h } ^ { \\mu } ( s _ { h } , a _ { h } ) \\cdot \\operatorname { V a r } _ { P _ { s _ { h } , a _ { h } } } ( V _ { h + 1 } ^ { \\star } ) > 0 } \\frac { P _ { h } ( s _ { h + 1 } | s _ { h } , a _ { h } ) \\big ( V _ { h + 1 } ^ { \\star } ( s _ { h + 1 } ) - \\mathbb { E } _ { P _ { s _ { h } , a _ { h } } } ( V _ { h + 1 } ^ { \\star } | ) \\big ) } { \\sqrt { 2 \\cdot d _ { h } ^ { \\mu } ( s _ { h } , a _ { h } ) \\cdot \\operatorname { V a r } _ { P _ { s _ { h } , a _ { h } } } ( V _ { h + 1 } ^ { \\star } ) } } } \\end{array}$ Ph(sh+1|sh,ah)(V ?h+1(sh+1)−EPsh,ah [V ?h+1]) . Let π to be the output of any algorithm. Define the local non-asymptotic minimax risk as ",
"bbox": [
173,
482,
826,
571
],
"page_idx": 5
},
{
"type": "equation",
"img_path": "images/bfd5ed1aee31bf18d0b649e656af2ebf48e938e4c9c6005c4991905767ec6df8.jpg",
"text": "$$\n\\mathfrak { R } _ { n } ( \\mathcal { P } ) : = \\operatorname* { s u p } _ { \\mathcal { P } ^ { \\prime } \\in \\mathcal { G } } \\operatorname* { i n f } _ { \\widehat { \\pi } } \\operatorname* { m a x } _ { \\mathcal { Q } \\in \\{ \\mathcal { P } , \\mathcal { P } ^ { \\prime } \\} } \\sqrt { n } \\cdot \\mathbb { E } _ { \\mathcal { Q } } \\left[ v ^ { \\star } ( \\mathcal { Q } ) - v ^ { \\widehat { \\pi } } \\right]\n$$",
"text_format": "latex",
"bbox": [
321,
578,
674,
609
],
"page_idx": 5
},
{
"type": "text",
"text": "where $v ^ { \\star } ( \\mathcal { Q } )$ denotes the optimal value under the instance $\\mathcal { Q }$ . Then there exists universal constants $c _ { 0 } , p , C > 0$ , such that if $\\begin{array} { r } { { \\bf \\ddot { \\sigma } } n \\geq c _ { 0 } H ^ { 6 } \\xi ^ { 4 } / ( \\sum _ { h = 1 } ^ { H } \\sum _ { s _ { h } , a _ { h } } { d _ { h } ^ { \\pi } } ^ { \\star } ( s _ { h } , a _ { h } ) \\sqrt { \\frac { \\mathrm { V a r } { P _ { s _ { h } , a _ { h } } } ( V _ { h + 1 } ^ { \\star } ) } { \\zeta \\cdot d _ { h } ^ { \\mu } ( s _ { h } , a _ { h } ) } } ) ^ { 2 } } \\end{array}$ , with constant probability $p > 0$ , Then we have (here $\\zeta = H / \\bar { d } _ { m , }$ ): ",
"bbox": [
174,
616,
826,
679
],
"page_idx": 5
},
{
"type": "equation",
"img_path": "images/cfc471f5a55860600a234cf4fe3c31689b97a7ff564bc439b67603e14bceb99e.jpg",
"text": "$$\n\\mathfrak { R } _ { n } ( \\mathcal { P } ) \\geq C \\cdot \\sum _ { h = 1 } ^ { H } \\sum _ { ( s _ { h } , a _ { h } ) \\in \\mathcal { C } _ { h } } d _ { h } ^ { \\pi ^ { \\star } } ( s _ { h } , a _ { h } ) \\cdot \\sqrt { \\frac { \\mathrm { V a r } _ { P _ { s _ { h } , a _ { h } } } ( r _ { h } + V _ { h + 1 } ^ { \\star } ) } { \\zeta \\cdot d _ { h } ^ { \\mu } ( s _ { h } , a _ { h } ) } } ,\n$$",
"text_format": "latex",
"bbox": [
269,
684,
725,
732
],
"page_idx": 5
},
{
"type": "text",
"text": "where $\\mathcal { P } = ( \\mu , M )$ and $M = \\left( S , A , P , r , H , d _ { 1 } \\right)$ . ",
"bbox": [
173,
738,
496,
756
],
"page_idx": 5
},
{
"type": "text",
"text": "The interpretation of Theorem 4.3 is: for any instance $\\mathcal { P }$ , learning requires (7) (divided by $1 / \\sqrt { n } )$ for any algorithm. Note this notion is significantly stronger than the previous minimax offline lower bounds [Yin et al., 2021a, Rashidinejad et al., 2021, Xie et al., 2021b, Jin et al., 2020] (where they only select a particular family of hard problems), therefore, their lower bounds in general do not hold for individual instances. ",
"bbox": [
173,
765,
825,
835
],
"page_idx": 5
},
{
"type": "text",
"text": "The quantity (1) nearly-matches the per-instance lower bound (7) (they deviate by a factor of $\\zeta = \\hat { H } / \\hat { d } _ { m }$ due to the technical reason) and, in addition, we provide a matching minimax lower bound in Appendix H. These results certify Theorem 4.1 is not only adaptive but also near-optimal. ",
"bbox": [
173,
840,
825,
883
],
"page_idx": 5
},
{
"type": "text",
"text": "Hence, we call the quantity $\\begin{array} { r l } & { \\sum _ { h = 1 } ^ { H } \\sum _ { ( s _ { h } , a _ { h } ) \\in \\mathcal { C } _ { h } } { d _ { h } ^ { \\pi ^ { \\star } } ( s _ { h } , a _ { h } ) \\cdot \\sqrt { \\frac { \\operatorname { V a r } _ { P _ { s _ { h } , a _ { h } } } ( r _ { h } + V _ { h + 1 } ^ { \\star } ) } { n \\cdot d _ { h } ^ { \\mu } ( s _ { h } , a _ { h } ) } } } } \\end{array}$ intrinsic offline reinforcement learning bound. In the sequel, we provide thorough discussions to explain the intrinsic bound embraces the fundamental challenges in offline RL and the strong adaptivity. The detailed technical derivations that are missing in Section 4.1-4.4 are deferred to Appendix I. ",
"bbox": [
171,
886,
823,
915
],
"page_idx": 5
},
{
"type": "text",
"text": "",
"bbox": [
173,
90,
826,
133
],
"page_idx": 6
},
{
"type": "image",
"img_path": "images/6b83607ca2951ec3cf5d8f65a3f909d38e7e57c936199b24d666e4d9988b9fb3.jpg",
"image_caption": [
"Figure 1: A visualization on how intrinsic learning bound subsumes existing best-known results: uniform visitation, single concentrability (partial coverage) and adaptive domain. "
],
"image_footnote": [],
"bbox": [
316,
161,
679,
311
],
"page_idx": 6
},
{
"type": "text",
"text": "4.1 Optimality under Uniform data-coverage assumption ",
"text_level": 1,
"bbox": [
173,
382,
583,
397
],
"page_idx": 6
},
{
"type": "text",
"text": "Under the uniform exploration Assumption 2.1 with parameter $d _ { m } : = \\operatorname* { m i n } _ { h , s _ { h , } a _ { h } } d _ { h } ^ { \\mu } ( s _ { h } , a _ { h } ) > 0$ Yin et al. [2021a] analyzes the model-based plug-in approach and obtains the optimal sample complexity $\\widetilde { O } ( H ^ { 3 } / d _ { m } \\epsilon ^ { 2 } )$ and shows $\\Omega ( H ^ { 3 } / d _ { m } \\epsilon ^ { 2 } )$ is also the lower bound. Indeed, this rate can ebe directly implied by the intrinsic RL bound via Cauchy inequality and the Sum of Total Variance (Lemma J.6):4 ",
"bbox": [
174,
406,
825,
477
],
"page_idx": 6
},
{
"type": "equation",
"img_path": "images/e5ef868c25e2072276b0a54844afacd6756ec1bd12c00a9851ae9b4cd4604e8c.jpg",
"text": "$$\n\\begin{array} { r l } & { \\displaystyle \\sum _ { h = 1 } ^ { H } \\langle d _ { h } ^ { \\pi ^ { \\star } } ( \\cdot ) , \\sqrt { \\frac { \\mathrm { V a r } _ { P ( \\cdot ) } ( r _ { h } + V _ { h + 1 } ^ { \\star } ) } { n \\cdot d _ { h } ^ { \\mu } ( \\cdot ) } } \\rangle = \\displaystyle \\sum _ { h = 1 } ^ { H } \\langle \\sqrt { d _ { h } ^ { \\pi ^ { \\star } } ( \\cdot ) } , \\sqrt { \\frac { d _ { h } ^ { \\pi ^ { \\star } } ( \\cdot ) \\odot \\mathrm { V a r } _ { P ( \\cdot ) } ( r _ { h } + V _ { h + 1 } ^ { \\star } ) } { n \\cdot d _ { m } } } \\rangle } \\\\ & { \\displaystyle \\leq \\displaystyle \\sum _ { h = 1 } ^ { H } \\left\\| \\sqrt { d _ { h } ^ { \\pi ^ { \\star } } ( \\cdot ) } \\right\\| _ { 2 } \\left\\| \\sqrt { \\frac { d _ { h } ^ { \\pi ^ { \\star } } ( \\cdot ) \\odot \\mathrm { V a r } _ { P ( \\cdot ) } ( r _ { h } + V _ { h + 1 } ^ { \\star } ) } { n \\cdot d _ { m } } } \\right\\| _ { 2 } \\leq \\sqrt { \\frac { H \\cdot \\mathrm { V a r } _ { \\pi ^ { \\star } } ( \\cdot \\sum _ { h = 1 } ^ { H } r _ { h } ) } { n \\cdot d _ { m } } } \\leq \\sqrt { \\frac { H ^ { 3 } } { n \\cdot d _ { m } } } } \\end{array}\n$$",
"text_format": "latex",
"bbox": [
192,
484,
792,
574
],
"page_idx": 6
},
{
"type": "text",
"text": "which translates to $\\widetilde { O } ( H ^ { 3 } / d _ { m } \\epsilon ^ { 2 } )$ complexity. Our result maintains the optimal worst-case guarantee when $\\mu$ has the uniform data-coverage: ",
"bbox": [
173,
580,
823,
611
],
"page_idx": 6
},
{
"type": "text",
"text": "Proposition 4.4. Under Assumption 2.1 and apply Theorem 4.1, APVI achieves the sample complexity of minimax-rate $\\widetilde { O } ( H ^ { 3 } / d _ { m } \\epsilon ^ { 2 } )$ (Theorem 4.1 and Theorem G.2 in Yin et al. [2021a]). ",
"bbox": [
173,
614,
823,
647
],
"page_idx": 6
},
{
"type": "text",
"text": "Remark 4.5. We believe if the MDP is time-invariant, then by a modified construction of $\\widehat { P }$ , $\\widehat { r }$ in (2) our result will imply the minimax-rate of $\\widetilde { O } ( H ^ { 2 } / d _ { m } \\epsilon ^ { 2 } )$ as achieved in Yin et al. [2021b]. We include this discussion in Appendix I. ",
"bbox": [
173,
654,
825,
699
],
"page_idx": 6
},
{
"type": "text",
"text": "4.2 Bounded sum of total rewards and the Horizon-Free case ",
"text_level": 1,
"bbox": [
173,
715,
611,
731
],
"page_idx": 6
},
{
"type": "text",
"text": "$\\begin{array} { r } { r _ { h } \\geq 0 , \\sum _ { h = 1 } ^ { H } r _ { h } \\in [ 0 , 1 ] } \\end{array}$ f studies that follow the bounded sum of total rewards assumption: i.e. [Krishnamurthy et al., 2016, Jiang et al., 2017, Zhang et al., 2021]. Such a Jiang and Agarwal [2018]. In offline RL, Ren et al. [2021] derives the nearly horizon-free worst case bound $\\widetilde { O } ( \\sqrt { 1 / n d _ { m } } )$ for the time-invariant MDPs, under the Assumption 2.1. As a comparison, our Theorem 4.1 achieves the following guarantee for the time-varying (non-stationary) MDPs. ",
"bbox": [
173,
742,
826,
832
],
"page_idx": 6
},
{
"type": "text",
"text": "Proposition 4.6. Assume $r _ { h } \\ge 0 ,$ , $\\textstyle \\sum _ { h = 1 } ^ { H } r _ { h } \\leq 1$ . Then in the time-varying case AVPI (Theorem 4.1) outputs a policy $\\widehat { \\pi }$ such that the suboptimality gap $\\boldsymbol { v } ^ { \\star } - \\boldsymbol { v } ^ { \\widehat { \\pi } }$ is bounded by $\\widetilde { O } ( \\sqrt { H / n d _ { m } } )$ with high bprobability under the Assumption 2.1. ",
"bbox": [
174,
835,
825,
885
],
"page_idx": 6
},
{
"type": "text",
"text": "The derivation is straightforward by using $\\begin{array} { r } { \\operatorname { V a r } _ { \\pi ^ { \\star } } ( \\sum _ { h = 1 } ^ { H } r _ { h } ) \\le 1 } \\end{array}$ in (8). This proposition is interesting since it indicates when the MDP is non-stationary, $\\widetilde { O } ( H / d _ { m } \\epsilon ^ { 2 } )$ is required in the worst case even under $\\textstyle \\sum _ { h = 1 } ^ { H } r _ { h } \\leq 1$ .5 The extra $H$ factor resembles the challenge that we have $H$ transitions $( P _ { 1 } , \\dots , P _ { H } )$ to learn, as opposed to the bandit-type $1 / d _ { m } \\epsilon ^ { 2 }$ result due to there is only one $P$ throughout (time-invariant). This reveals that one hardness in solving the MDP is in proportion to the number of different transition kernels within the MDP. Such a finding could help researchers understand the special settings like low switching cost in transitions [Bai et al., 2019] or nonstationarity [Cheung et al., 2020]. ",
"bbox": [
173,
88,
826,
210
],
"page_idx": 7
},
{
"type": "text",
"text": "4.3 Optimality with Single Concentrability ",
"text_level": 1,
"bbox": [
174,
227,
483,
243
],
"page_idx": 7
},
{
"type": "text",
"text": "In the finite horizon discounted setting, Rashidinejad et al. [2021] proposes the single policy concentrability assumption which is defined as C? := maxh,s,a dh (s,a)dµ(s,a) in the current episodic nonstationary MDP setting. As discussed in Appendix D, their lower bound translates to $\\Omega ( { \\sqrt { \\frac { H ^ { 3 } S C ^ { \\star } } { n } } } )$ and their VI-LCB algorithm yields q H5SC?n ) suboptimality gap in H-horizon case. Since single policy concentrability is strictly weaker than its uniform version (Assumption 2.2), we only discuss this set up. In particular, we have the following implication from our Theorem 4.1 (whose derivation can be found in Appendix I): ",
"bbox": [
173,
252,
826,
382
],
"page_idx": 7
},
{
"type": "text",
"text": "Proposition 4.7. Let π? be a deterministic policy such that C? := maxh,s,a dπ h (s,a)dµ(s,a) . Then by Theorem 4.1, with high probability the output policy of APVI satisfies the suboptimality gap $\\widetilde { O } ( \\sqrt { \\frac { H ^ { 3 } S C ^ { \\star } } { n } } )$ in the time-varying (non-stationary) MDPs. ",
"bbox": [
173,
387,
825,
446
],
"page_idx": 7
},
{
"type": "text",
"text": "This can computed similar to (8) except we use $\\begin{array} { r } { \\frac { \\boldsymbol { d } _ { h } ^ { \\pi ^ { \\star } } ( s , a ) } { \\boldsymbol { d } _ { h } ^ { \\mu } ( s , a ) } \\leq C ^ { \\star } } \\end{array}$ . Our implication improves the VI-LCB by the factor $H ^ { 2 }$ (in terms of sample complexity) and is optimal (recover the concurrent Xie et al. [2021b]). Qualitatively, single concentrability is the same as Assumption 2.3, but the use of $C ^ { \\star }$ makes the bound highly problem independent and limits the adaptivity. Problem dependent bound is a more interesting domain as it tailors to each MDP separately. We discuss it now. ",
"bbox": [
173,
460,
825,
541
],
"page_idx": 7
},
{
"type": "text",
"text": "4.4 Problem dependent domain ",
"text_level": 1,
"bbox": [
174,
558,
406,
574
],
"page_idx": 7
},
{
"type": "text",
"text": "We define the pre-step environmental norm (the finite horizon counterpart of Maillard et al. [2014]) as: $\\begin{array} { r } { \\mathbb { Q } _ { h } ^ { \\star } = \\operatorname* { m a x } _ { s _ { h } , a _ { h } } { \\operatorname { V a r } _ { P _ { s _ { h } , a _ { h } } } { \\left( r _ { h } + V _ { h + 1 } ^ { \\star } \\right) } } } \\end{array}$ for all $h \\in [ H ]$ , and relax the total sum of rewards to be bounded by any arbitrary value $\\boldsymbol { B }$ (i.e. $\\textstyle \\sum _ { h = 1 } ^ { H } r _ { h } \\leq B )$ , then Theorem 4.1 implies: ",
"bbox": [
173,
583,
825,
633
],
"page_idx": 7
},
{
"type": "text",
"text": "Proposition 4.8. Under Assumption 2.1, with high probability, subopmality of AVPI is bounded by ",
"bbox": [
173,
636,
820,
651
],
"page_idx": 7
},
{
"type": "equation",
"img_path": "images/1f6383d73d7e96784480eaaec32ca332fed385bc82436c24398afd8a492b8716.jpg",
"text": "$$\n\\operatorname* { m i n } \\bigg \\{ \\widetilde O \\big ( \\sum _ { h = 1 } ^ { H } \\sqrt { \\frac { \\mathbb { Q } _ { h } ^ { \\star } } { n \\bar { d } _ { m } } } \\big ) , \\widetilde O \\big ( \\sqrt { \\frac { H \\cdot \\mathcal { B } ^ { 2 } } { n \\bar { d } _ { m } } } \\big ) \\bigg \\} + \\widetilde O ( \\frac { H ^ { 3 } } { n \\bar { d } _ { m } } ) .\n$$",
"text_format": "latex",
"bbox": [
320,
659,
674,
703
],
"page_idx": 7
},
{
"type": "text",
"text": "Such a result mirrors the online version of the tight problem-dependent bound Zanette and Brunskill [2019] but with a more general pre-step environmental norm for the non-stationary MDPs.6 For the problem instances with either small $\\boldsymbol { B }$ or small $\\mathbb { Q } _ { h } ^ { \\star }$ , our result yields much better performances, as discussed in the following. ",
"bbox": [
173,
717,
825,
773
],
"page_idx": 7
},
{
"type": "text",
"text": "Deterministic systems. For many practical applications of interest, the systems are equipped with low stochasticity, e.g. robotics, or even deterministic dynamics, e.g. the game of GO. In those scenarios, the agent needs less experience for each state-action therefore the learning procedure could be much faster. In particular, when the system is fully deterministic (in both transitions and rewards) then $\\mathbb { Q } _ { h } ^ { \\star } = 0$ for all $h$ . This enables a faster convergence rate of order $\\frac { H ^ { 3 } } { n \\bar { d } _ { m } }$ and significantly improves over the existing non-adaptive results that have order $\\scriptstyle { \\frac { 1 } { \\sqrt { n } } }$ . The convergence rate $\\textstyle { \\frac { 1 } { n } }$ matches Wen and ) regret into the PAC bound. ",
"bbox": [
173,
779,
825,
856
],
"page_idx": 7
},
{
"type": "text",
"text": "",
"bbox": [
171,
89,
825,
122
],
"page_idx": 8
},
{
"type": "text",
"text": "Partially deterministic systems. Practical worlds are complicated and we could sometimes have a mixture model which contains both deterministic and stochastic steps. In those scenarios, the main complexity is decided by the number of stochastic stages: suppose there are $t$ stochastic $P _ { h } , r _ { h }$ ’s and $H - t$ deterministic $P _ { h ^ { \\prime } } , r _ { h ^ { \\prime } }$ ’s, then completing the offline learning guarantees $t \\cdot \\sqrt { \\operatorname* { m a x } Q _ { h } ^ { \\star } / n \\bar { d } _ { m } }$ suboptimality gap, which could be much smaller than $H \\cdot \\sqrt { \\operatorname* { m a x } Q _ { h } ^ { \\star } / n \\bar { d } _ { m } }$ when $t \\ll H$ . ",
"bbox": [
173,
127,
825,
214
],
"page_idx": 8
},
{
"type": "text",
"text": "Fast mixing domains. Consider a class of highly mixing non-stationary MDPs (a variant of Zanette and Brunskill [2018]) that satisfies the transition $P _ { h } ( \\cdot | s _ { h } , a _ { h } ) : = \\nu _ { h } ( \\cdot ) \\overline { { } }$ depends on neither the state $s _ { h }$ nor the action $a _ { h }$ . Define $\\bar { s } _ { t } : = \\arg \\operatorname* { m a x } V _ { t } ^ { \\star } ( s )$ and $\\underline { { s } } _ { t } : = \\arg \\operatorname* { m a x } V _ { t } ^ { \\star } ( s$ . Also, denote $\\mathrm { r n g } V _ { h } ^ { \\star }$ to be the range of $V _ { h } ^ { \\star }$ . In such cases, Bellman optimality equations have the form ",
"bbox": [
173,
219,
825,
276
],
"page_idx": 8
},
{
"type": "equation",
"img_path": "images/04a0eeb9dbb60365f7f2f7a9257ad850888db79f47b72addae2c9eaa7d42f5a4.jpg",
"text": "$$\nV _ { h } ^ { \\star } \\left( \\bar { s } _ { h } \\right) = \\operatorname* { m a x } _ { a } \\left( r _ { h } \\left( \\bar { s } _ { h } , a \\right) + \\nu _ { h } ^ { \\top } V _ { h + 1 } ^ { \\star } \\right) , \\ V _ { h } ^ { \\star } \\left( { { s } _ { h } } \\right) = \\operatorname* { m a x } _ { a } \\left( r _ { h } \\left( { { s } _ { h } } , a \\right) + \\nu _ { h } ^ { \\top } V _ { h + 1 } ^ { \\star } \\right)\n$$",
"text_format": "latex",
"bbox": [
223,
280,
767,
303
],
"page_idx": 8
},
{
"type": "text",
"text": "which yield $\\begin{array} { r } { \\mathfrak { s } \\mathrm { \\ r n g } { V } _ { h } ^ { \\star } = V _ { h } ^ { \\star } \\left( \\bar { s } _ { h } \\right) - V _ { h } ^ { \\star } \\left( \\underline { { s } } _ { h } \\right) = \\operatorname* { m a x } _ { a } r _ { h } \\left( \\bar { s } _ { h } , a \\right) - \\operatorname* { m i n } _ { a } r _ { h } \\left( \\underline { { s } } _ { h } , a \\right) \\le 1 , } \\end{array}$ , and this in turn gives $\\mathbb { Q } _ { h } ^ { \\star } \\le 1 + ( \\mathrm { r n g } V _ { h } ^ { \\star } ) ^ { 2 } = 2$ . As a result, the suboptimality is bounded by $\\widetilde { O } ( \\sqrt { H ^ { 2 } / n d _ { m } } )$ in the worst case. This result reveals, although this is a family of stochastic non-stationary MDPs, but it is only as hard as the family of stationary MDPs in the minimax sense $\\left( \\Omega ( H ^ { 2 } / d _ { m } \\epsilon ^ { 2 } ) \\right)$ ). ",
"bbox": [
173,
305,
825,
367
],
"page_idx": 8
},
{
"type": "text",
"text": "Tabular contextual bandits. Our result also implies $\\begin{array} { r l } & { \\widetilde { \\cal O } ( \\sum _ { x _ { 1 } , a _ { 1 } } { d _ { 1 } ^ { \\pi ^ { \\star } } ( x _ { 1 } , a _ { 1 } ) \\sqrt { \\frac { \\mathrm { V a r } ( r _ { 1 } ) } { n \\cdot d _ { 1 } ^ { \\mu } ( x _ { 1 } , a _ { 1 } ) } } } ) } \\end{array}$ q Var(r1)n·dµ1 (x1,a1) ) gap for the offline tabular contextual bandit problem and improves to $\\widetilde { \\cal O } ( 1 / n d _ { m } )$ when the reward is deterministic. In either cases, the result is optimal and this is due to: when $r _ { 1 }$ is deterministic, the agent only needs one sample at every location (see Bubeck and Cesa-Bianchi [2012] for a survey). ",
"bbox": [
173,
373,
826,
443
],
"page_idx": 8
},
{
"type": "text",
"text": "5 Towards Assumption-Free Offline RL ",
"text_level": 1,
"bbox": [
173,
459,
521,
478
],
"page_idx": 8
},
{
"type": "text",
"text": "While assumption 2.3 is (arguably) the weakest assumption for correctly learning the optimal value, for the real-world applications even this might not be guaranteed. Can we still learn something meaningful? In this section, we consider this most general setting where the behavior policy $\\mu$ can be arbitrary. In this case, $\\mu$ might not cover any optimal policy $\\pi ^ { \\star }$ (i.e. there might be high reward location $( s , a )$ that $\\mu$ can never visit, e.g. in the extreme case where a clumsy doctor only uses one treatment all the time), and, irrelevant to the number of episode $n$ , a constant suboptimality gap needs to be suffered. To tackle this problem, we create a fictitious augmented MDP $M ^ { \\dagger }$ that can help characterize the discrepancy of the values between the original MDP $M$ and the estimated MDP $\\widehat { M } ^ { \\dag }$ . In particular, $M ^ { \\dagger }$ is negative towards agnostic state-actions $s _ { h } , a _ { h }$ by setting $r _ { h } ^ { \\dagger } = 0$ cand transitions to an absorbing state s†h+1. ",
"bbox": [
173,
489,
826,
640
],
"page_idx": 8
},
{
"type": "text",
"text": "Pessimistic augmented MDP. $M ^ { \\dagger }$ is defined with one extra state $s _ { h } ^ { \\dagger }$ for all $h \\in \\{ 2 , \\dots , H + 1 \\}$ with the augmented state space $S ^ { \\dagger } = S \\cup \\{ s _ { h } ^ { \\dagger } \\}$ . The transition and the reward are defined as follows: ",
"bbox": [
174,
646,
823,
681
],
"page_idx": 8
},
{
"type": "equation",
"img_path": "images/ac7425d3170fce5cc0e3f0d9a9d4917aedb573be48daab7516765aff0b37b324.jpg",
"text": "$$\nP _ { h } ^ { \\dagger } ( \\cdot \\mid s _ { h } , a _ { h } ) = \\left\\{ \\begin{array} { l l } { P _ { h } ( \\cdot \\mid s _ { h } , a _ { h } ) , n _ { s _ { h } , a _ { h } } > 0 , } \\\\ { \\delta _ { s _ { h + 1 } ^ { \\dagger } } , s _ { h } = s _ { h } ^ { \\dagger } \\mathrm { o r } n _ { s _ { h } , a _ { h } } = 0 . } \\end{array} \\right. \\quad r ^ { \\dagger } ( s _ { h } , a _ { h } ) = \\left\\{ \\begin{array} { l l } { r ( s _ { h } , a _ { h } ) , n _ { s _ { h } , a _ { h } } > 0 , } \\\\ { 0 , s _ { h } = s _ { h } ^ { \\dagger } \\mathrm { o r } n _ { s _ { h } , a _ { h } } = 0 . } \\end{array} \\right.\n$$",
"text_format": "latex",
"bbox": [
194,
684,
790,
722
],
"page_idx": 8
},
{
"type": "text",
"text": "here $\\delta _ { s }$ is the Dirac measure and we denote $V _ { h } ^ { \\dag \\pi }$ and $v ^ { \\dag \\pi }$ to be the values under $M ^ { \\dagger }$ . $\\widehat { M } ^ { \\dagger }$ is the empirical counterpart of $M ^ { \\dagger }$ with $\\widehat { P }$ , $\\widehat { r }$ (the same as (2)) replacing $P , r .$ By Algorithm 1, we have bTheorem 5.1 (Assumption-free offline reinforcement learning). Let us make no assumption for $\\mu$ and still denote $\\bar { d } _ { m } : = \\operatorname * { m i n } _ { h \\in [ H ] } \\{ d _ { h } ^ { \\mu } ( s _ { h } , a _ { h } ) : d _ { h } ^ { \\mu } ( s _ { h } , a _ { h } ) > 0 \\}$ . For any $0 < \\delta < 1$ , there exists absolute constants $c _ { 0 } , C ^ { \\prime } > 0$ , such that when $n > c _ { 0 } \\cdot 1 / \\bar { d } _ { m } \\cdot \\iota ( \\iota = \\log ( H S A / \\delta ) )$ , with probability $1 - \\delta$ , the output policy $\\widehat { \\pi }$ of APVI satisfies (recall $\\mathcal { C } _ { h } : = \\{ ( s _ { h } , a _ { h } ) : d _ { h } ^ { \\mu } ( s _ { h } , a _ { h } ) > 0 \\}$ ) ",
"bbox": [
173,
724,
826,
821
],
"page_idx": 8
},
{
"type": "equation",
"img_path": "images/2f801afdfa20cca687c2bb53ebbb35e04b08e7d49d57f9af1205563dc12e9b3f.jpg",
"text": "$$\nv ^ { \\star } - v ^ { \\widehat { \\pi } } \\leq \\sum _ { h = 2 } ^ { H + 1 } d _ { h } ^ { \\dagger \\pi ^ { \\star } } \\left( s _ { h } ^ { \\dagger } \\right) + C ^ { \\prime } \\sum _ { h = 1 } ^ { H } \\sum _ { ( s _ { h } , a _ { h } ) \\in { \\mathcal { C } } _ { h } } d _ { h } ^ { \\dagger \\pi ^ { \\star } } \\left( s _ { h } , a _ { h } \\right) \\cdot \\sqrt { \\frac { \\operatorname { V a r } _ { P _ { \\hat { s } _ { h } , a _ { h } } ^ { \\dagger } } \\left( r _ { h } ^ { \\dagger } + V _ { h + 1 } ^ { \\dagger \\pi ^ { \\star } } \\right) \\cdot \\iota } { n \\cdot d _ { h } ^ { \\mu } \\left( s _ { h } , a _ { h } \\right) } } + \\widetilde { O } \\left( \\frac { H ^ { 3 } } { n \\bar { d } _ { m } } \\right) ,\n$$",
"text_format": "latex",
"bbox": [
178,
825,
799,
872
],
"page_idx": 8
},
{
"type": "text",
"text": "where $d _ { h } ^ { \\dagger \\pi ^ { \\star } } ( s _ { h } , a _ { h } ) \\leq d _ { h } ^ { \\pi ^ { \\star } } ( s _ { h } , a _ { h } ) , V _ { h } ^ { \\dagger \\pi ^ { \\star } } ( s _ { h } ) \\leq V _ { h } ^ { \\star } ( s _ { h } ) .$ for all $s _ { h } , a _ { h } \\in S \\times \\mathcal { A } ,$ , and for all $h \\in [ H ]$ , $\\begin{array} { r } { d _ { h } ^ { \\dagger \\pi ^ { \\star } } ( s _ { h } ^ { \\dagger } ) = \\sum _ { t = 1 } ^ { h - 1 } \\sum _ { ( s _ { t } , a _ { t } ) \\in \\mathcal { S } \\times \\mathcal { A } \\backslash \\mathcal { C } _ { t } } d _ { t } ^ { \\dagger \\pi ^ { \\star } } \\big ( s _ { t } , a _ { t } \\big ) } \\end{array}$ . The proof is in Appendix $E$ . ",
"bbox": [
173,
876,
833,
915
],
"page_idx": 8
},
{
"type": "text",
"text": "Take-aways of Theorem 5.1. In $M ^ { \\dagger }$ , there is no agnostic location any more since the original unknown spaces now all have known deterministic transitions to $s ^ { \\dagger }$ in $M ^ { \\dagger }$ . At a price, the algorithm has to suffer the constant suboptimality PH+1h=2 $\\textstyle \\sum _ { h = 2 } ^ { H + 1 } d _ { h } ^ { \\dagger \\pi ^ { \\star } } ( s _ { h } ^ { \\dagger } )$ due to no data in the region. The quantity $\\textstyle \\sum _ { h = 2 } ^ { H + 1 } d _ { h } ^ { \\dag \\pi ^ { \\star } } ( s _ { h } ^ { \\dag } )$ helps characterize the hardness when nothing is assumed about $\\mu$ : it is always less than $H$ (cannot suffer more than suboptimality); under Assumption 2.1, it is 0 since with high probability (by Chernoff bound) and this causes $\\boldsymbol { \\mathcal { S } } \\times \\boldsymbol { \\mathcal { A } } \\backslash \\boldsymbol { \\bar { \\mathcal { C } } } _ { h } = \\boldsymbol { \\emptyset }$ ; under Assumption 2.3, it is also 0 and 5.1 reduces to Theorem 4.1 (see Appendix F). ",
"bbox": [
173,
90,
825,
198
],
"page_idx": 9
},
{
"type": "text",
"text": "5.1 Assumption Free vs Without Great Coverage (Partial Coverage) ",
"text_level": 1,
"bbox": [
176,
219,
658,
234
],
"page_idx": 9
},
{
"type": "text",
"text": "Recently there is a surge of studies that aim at weakening the assumptions of provable offline $/$ batch RL. Those learning bounds are derived (mostly) under the insufficient data coverage assumptions. One type of works consider the assumption without great coverage (or partial coverage): Chang et al. [2021], Uehara and Sun [2021] assume $\\begin{array} { r } { \\operatorname* { m a x } _ { s , a } d ^ { \\pi _ { e } } ( s , a ) / \\mu ( s , a ) < \\infty } \\end{array}$ where $\\pi _ { e }$ is either an expert policy or a policy of great quality and they further compete against with this policy $\\pi _ { e }$ . Those assumptions are similar to 2.3 and therefore are stronger than the assumption-free RL we considered in 5.1. ",
"bbox": [
173,
246,
826,
344
],
"page_idx": 9
},
{
"type": "text",
"text": "In addition, there are other studies that apply to the case where $\\mu$ can be arbitrary: Liu et al. [2020] considers the behavior policy with insufficient coverage probability $\\epsilon _ { \\zeta }$ (see their Definition 1), and they end up with the constant suboptimality gap Vmax\u000fζ1−γ (their Theorem 1), when the insufficient coverage probability $\\epsilon _ { \\zeta } > 0$ , this gap has order $( 1 - \\gamma ) ^ { - 2 }$ , which is larger in order than the biggest possible suboptimality gap $( 1 - \\gamma ) ^ { - 1 }$ therefore unable to characterize the essential statistical gap over the region that can never be visited by the behavior policy (and this happens similarly in Kidambi et al. [2020], see their Theorem 1); Jin et al. [2020] derive the nice assumption-free result via regularization and their bound can incur $O ( H ^ { 2 } )$ constant gap when there is at least one $\\left( s _ { h } , a _ { h } \\right)$ cannot be obtained by $\\mu$ for all $h \\in [ H ]$ (i.e. replacing $n d _ { h } ^ { \\bar { \\mu } } ( { \\bar { s } } _ { h } , a _ { h } )$ by 1 in (3)). The concurrent work Xie et al. [2021a] provides a better characterization (and they call it off-support error) with roughly $\\begin{array} { r } { \\frac { 1 } { 1 - \\gamma } \\sum _ { \\left( s , a \\right) \\in S \\times A } \\left( d _ { \\pi } \\backslash \\nu \\right) \\left( s , a \\right) \\left[ \\Delta f _ { \\pi } ( s , a ) - ( \\mathcal { T } ^ { \\pi } \\Delta f _ { \\pi } ) \\left( s , \\bar { a } \\right) \\right] } \\end{array}$ , however, in the worst case $\\Delta f _ { \\pi } ( s , a ) - ( T ^ { \\pi } \\Delta f _ { \\pi } ) ( s , a )$ might be large (which depends on the quality (assumption) of the function approximation class). ",
"bbox": [
173,
351,
825,
542
],
"page_idx": 9
},
{
"type": "text",
"text": "In contrast, our 1) describes the $\\textstyle \\sum _ { h = 2 } ^ { H + 1 } d _ { h } ^ { \\dag \\pi ^ { \\star } } ( s _ { h } ^ { \\dag } )$ quantity (with p in a more pr $\\begin{array} { r } { d _ { h } ^ { \\dagger \\pi ^ { \\star } } ( s _ { h } ^ { \\dagger } ) = \\sum _ { t = 1 } ^ { h - 1 } \\sum _ { ( s _ { t } , a _ { t } ) \\in \\mathcal { S } \\times \\mathcal { A } \\backslash \\mathcal { C } _ { t } } d _ { t } ^ { \\dagger \\pi ^ { \\star } } ( s _ { t } , a _ { t } ) \\leq } \\end{array}$ into $s ^ { \\dagger }$ and it is always bounded between 0 and $H$ . It reduces to 0 when $\\pi ^ { \\star }$ is covered. The gap is always of order $H$ (as opposed to $O ( H ^ { 2 } ) _ { * }$ ). ",
"bbox": [
173,
547,
825,
609
],
"page_idx": 9
},
{
"type": "text",
"text": "6 Discussion and Conclusion ",
"text_level": 1,
"bbox": [
176,
633,
428,
651
],
"page_idx": 9
},
{
"type": "text",
"text": "This work studies the offline reinforcement learning problem and contributes the intrinsic offline learning bound which is a near-optimal and strong adaptive bound that subsumes existing worst-case bounds under various assumptions. The adaptive characterization of the intrinsic bound abandons the explicit dependence on $H , S , A , C ^ { \\star } , d _ { m }$ and helps reveal the fundamental hardness of each individual instances. In this sense, it draws a clearer picture of what offline reinforcement learning looks like and serves as a step towards instance optimality in offline RL. ",
"bbox": [
173,
669,
825,
752
],
"page_idx": 9
},
{
"type": "text",
"text": "Nevertheless, it is still unclear whether (5) is optimal over all the instances. For example, for fully deterministic systems, our bound provides a faster convergence $H ^ { 3 } / n \\bar { d } _ { m }$ , however, $\\bar { H } ^ { 3 }$ might be very suboptimal comparing to algorithms that are designed specifically for deterministic MDPs, since the agent only need to experience each location $( s , a )$ once to fully acquire the dynamic $P ( \\cdot | s , a )$ and $r ( s , a )$ . Recently, Xiao et al. [2021] goes beyond the minimax (worst case) optimality and studies the instance optimality behavior for the simplified batch bandit setting. One of their findings is: for “easy enough” tasks, different type of algorithms can be equally good, provably. This seems to suggest instance optimality only matters for problems that are hard to learn. How to formally define the instance optimality metric for different problems remains an open problem and how to design a single algorithm that can achieve optimality for all instances could be challenging (or even infeasible). We leave those as the future works. ",
"bbox": [
173,
758,
825,
911
],
"page_idx": 9
},
{
"type": "text",
"text": "Acknowledgment ",
"text_level": 1,
"bbox": [
174,
92,
297,
106
],
"page_idx": 10
},
{
"type": "text",
"text": "The research is partially supported by NSF Awards #2007117 and #2003257. MY would like to thank Chenjun Xiao for bringing up a related literature [Xiao et al., 2021] and Masatoshi Uehara, Yu Bai for helpful suggestions. ",
"bbox": [
173,
116,
825,
159
],
"page_idx": 10
},
{
"type": "text",
"text": "References ",
"text_level": 1,
"bbox": [
174,
179,
267,
195
],
"page_idx": 10
},
{
"type": "text",
"text": "Alekh Agarwal, Sham Kakade, and Lin F Yang. Model-based reinforcement learning with a generative model is minimax optimal. In Conference on Learning Theory, pages 67–83, 2020. ",
"bbox": [
173,
202,
823,
232
],
"page_idx": 10
},
{
"type": "text",
"text": "Andras Antos, Remi Munos, and Csaba Szepesvari. Fitted q-iteration in continuous action-space mdps. In Advances in Neural Information Processing Systems, pages 9–16, 2008a. ",
"bbox": [
171,
239,
823,
270
],
"page_idx": 10
},
{
"type": "text",
"text": "András Antos, Csaba Szepesvári, and Rémi Munos. Learning near-optimal policies with bellmanresidual minimization based fitted policy iteration and a single sample path. Machine Learning, 71 (1):89–129, 2008b. ",
"bbox": [
173,
279,
823,
321
],
"page_idx": 10
},
{
"type": "text",
"text": "Mohammad Gheshlaghi Azar, Ian Osband, and Rémi Munos. Minimax regret bounds for reinforcement learning. In Proceedings of the 34th International Conference on Machine Learning-Volume 70, pages 263–272. JMLR. org, 2017. ",
"bbox": [
174,
330,
823,
373
],
"page_idx": 10
},
{
"type": "text",
"text": "Yu Bai, Tengyang Xie, Nan Jiang, and Yu-Xiang Wang. Provably efficient q-learning with low switching cost. In Advances in Neural Information Processing Systems, volume 32, 2019. ",
"bbox": [
174,
382,
821,
412
],
"page_idx": 10
},
{
"type": "text",
"text": "Ronen I Brafman and Moshe Tennenholtz. R-max-a general polynomial time algorithm for nearoptimal reinforcement learning. Journal of Machine Learning Research, 3(Oct):213–231, 2002. ",
"bbox": [
173,
420,
823,
450
],
"page_idx": 10
},
{
"type": "text",
"text": "Sébastien Bubeck and Nicolo Cesa-Bianchi. Regret analysis of stochastic and nonstochastic multiarmed bandit problems. Foundations and Trends in Machine Learning, 2012. ",
"bbox": [
174,
458,
821,
488
],
"page_idx": 10
},
{
"type": "text",
"text": "Jacob Buckman, Carles Gelada, and Marc G Bellemare. The importance of pessimism in fixed-dataset policy optimization. International Conference on Learning Representations, 2021. ",
"bbox": [
173,
496,
823,
526
],
"page_idx": 10
},
{
"type": "text",
"text": "Qi Cai, Zhuoran Yang, Chi Jin, and Zhaoran Wang. Provably efficient exploration in policy optimization. In International Conference on Machine Learning, pages 1283–1294. PMLR, 2020. ",
"bbox": [
173,
535,
823,
565
],
"page_idx": 10
},
{
"type": "text",
"text": "T Tony Cai and Mark G Low. An adaptation theory for nonparametric confidence intervals. The Annals of statistics, 32(5):1805–1840, 2004. ",
"bbox": [
174,
573,
823,
603
],
"page_idx": 10
},
{
"type": "text",
"text": "Jonathan D Chang, Masatoshi Uehara, Dhruv Sreenivas, Rahul Kidambi, and Wen Sun. Mitigating covariate shift in imitation learning via offline data without great coverage. Advances in Neural Information Processing Systems, 2021. ",
"bbox": [
173,
611,
825,
655
],
"page_idx": 10
},
{
"type": "text",
"text": "Jinglin Chen and Nan Jiang. Information-theoretic considerations in batch reinforcement learning. In International Conference on Machine Learning, pages 1042–1051, 2019. ",
"bbox": [
169,
664,
823,
694
],
"page_idx": 10
},
{
"type": "text",
"text": "Herman Chernoff et al. A measure of asymptotic efficiency for tests of a hypothesis based on the sum of observations. The Annals of Mathematical Statistics, 23(4):493–507, 1952. ",
"bbox": [
171,
702,
823,
732
],
"page_idx": 10
},
{
"type": "text",
"text": "Wang Chi Cheung, David Simchi-Levi, and Ruihao Zhu. Reinforcement learning for non-stationary markov decision processes: The blessing of (more) optimism. In International Conference on Machine Learning, pages 1843–1854. PMLR, 2020. ",
"bbox": [
176,
739,
823,
784
],
"page_idx": 10
},
{
"type": "text",
"text": "Yaqi Duan, Zeyu Jia, and Mengdi Wang. Minimax-optimal off-policy evaluation with linear function approximation. In International Conference on Machine Learning, pages 8334–8342, 2020. ",
"bbox": [
173,
791,
823,
821
],
"page_idx": 10
},
{
"type": "text",
"text": "Justin Fu, Aviral Kumar, Ofir Nachum, George Tucker, and Sergey Levine. D4rl: Datasets for deep data-driven reinforcement learning. arXiv preprint arXiv:2004.07219, 2020. ",
"bbox": [
171,
830,
825,
859
],
"page_idx": 10
},
{
"type": "text",
"text": "Justin Fu, Mohammad Norouzi, Ofir Nachum, George Tucker, Ziyu Wang, Alexander Novikov, Mengjiao Yang, Michael R Zhang, Yutian Chen, Aviral Kumar, et al. Benchmarks for deep off-policy evaluation. International Conference on Learning Representations, 2021. ",
"bbox": [
173,
868,
826,
911
],
"page_idx": 10
},
{
"type": "text",
"text": "Caglar Gulcehre, Ziyu Wang, Alexander Novikov, Tom Le Paine, Sergio Gómez Colmenarejo, Konrad Zolna, Rishabh Agarwal, Josh Merel, Daniel Mankowitz, Cosmin Paduraru, et al. Rl unplugged: Benchmarks for offline reinforcement learning. Advances in neural information processing systems, 2020. ",
"bbox": [
173,
90,
826,
146
],
"page_idx": 11
},
{
"type": "text",
"text": "Michael Janner, Qiyang Li, and Sergey Levine. Reinforcement learning as one big sequence modeling problem. Advances in neural information processing systems, 2021. ",
"bbox": [
173,
157,
821,
188
],
"page_idx": 11
},
{
"type": "text",
"text": "Nan Jiang and Alekh Agarwal. Open problem: The dependence of sample complexity lower bounds on planning horizon. In Conference On Learning Theory, pages 3395–3398, 2018. ",
"bbox": [
173,
196,
821,
227
],
"page_idx": 11
},
{
"type": "text",
"text": "Nan Jiang and Lihong Li. Doubly robust off-policy value evaluation for reinforcement learning. In Proceedings of the 33rd International Conference on International Conference on Machine Learning-Volume 48, pages 652–661. JMLR. org, 2016. ",
"bbox": [
173,
236,
823,
280
],
"page_idx": 11
},
{
"type": "text",
"text": "Nan Jiang, Akshay Krishnamurthy, Alekh Agarwal, John Langford, and Robert E Schapire. Contextual decision processes with low bellman rank are pac-learnable. In International Conference on Machine Learning-Volume 70, pages 1704–1713, 2017. ",
"bbox": [
173,
289,
825,
333
],
"page_idx": 11
},
{
"type": "text",
"text": "Ying Jin, Zhuoran Yang, and Zhaoran Wang. Is pessimism provably efficient for offline rl? International Conference on Machine Learning, 2020. ",
"bbox": [
169,
342,
825,
372
],
"page_idx": 11
},
{
"type": "text",
"text": "Tobias Jung and Peter Stone. Gaussian processes for sample efficient reinforcement learning with rmax-like exploration. In Joint European Conference on Machine Learning and Knowledge Discovery in Databases, pages 601–616. Springer, 2010. ",
"bbox": [
176,
381,
823,
425
],
"page_idx": 11
},
{
"type": "text",
"text": "Koulik Khamaru, Ashwin Pananjady, Feng Ruan, Martin J Wainwright, and Michael I Jordan. Is temporal difference learning optimal? an instance-dependent analysis. SIAM Journal on Mathematics of Data Science, 2021a. ",
"bbox": [
173,
434,
826,
478
],
"page_idx": 11
},
{
"type": "text",
"text": "Koulik Khamaru, Eric Xia, Martin J Wainwright, and Michael I Jordan. Instance-optimality in optimal value estimation: Adaptivity via variance-reduced q-learning. arXiv preprint arXiv:2106.14352, 2021b. ",
"bbox": [
174,
487,
825,
530
],
"page_idx": 11
},
{
"type": "text",
"text": "Rahul Kidambi, Aravind Rajeswaran, Praneeth Netrapalli, and Thorsten Joachims. Morel: Modelbased offline reinforcement learning. Advances in neural information processing systems, 2020. ",
"bbox": [
171,
540,
825,
570
],
"page_idx": 11
},
{
"type": "text",
"text": "Akshay Krishnamurthy, Alekh Agarwal, and John Langford. PAC reinforcement learning with rich observations. In Advances in Neural Information Processing Systems, pages 1840–1848, 2016. ",
"bbox": [
171,
579,
823,
609
],
"page_idx": 11
},
{
"type": "text",
"text": "Sascha Lange, Thomas Gabel, and Martin Riedmiller. Batch reinforcement learning. In Reinforcement learning, pages 45–73. Springer, 2012. ",
"bbox": [
173,
619,
823,
648
],
"page_idx": 11
},
{
"type": "text",
"text": "Hoang Le, Cameron Voloshin, and Yisong Yue. Batch policy learning under constraints. In International Conference on Machine Learning, pages 3703–3712, 2019. ",
"bbox": [
173,
659,
823,
688
],
"page_idx": 11
},
{
"type": "text",
"text": "Sergey Levine, Aviral Kumar, George Tucker, and Justin Fu. Offline reinforcement learning: Tutorial, review, and perspectives on open problems. arXiv preprint arXiv:2005.01643, 2020. ",
"bbox": [
173,
696,
825,
728
],
"page_idx": 11
},
{
"type": "text",
"text": "Yao Liu, Adith Swaminathan, Alekh Agarwal, and Emma Brunskill. Off-policy policy gradient with state distribution correction. In Uncertainty in Artificial Intelligence, 2019. ",
"bbox": [
173,
737,
825,
767
],
"page_idx": 11
},
{
"type": "text",
"text": "Yao Liu, Adith Swaminathan, Alekh Agarwal, and Emma Brunskill. Provably good batch reinforcement learning without great exploration. Advances in neural information processing systems, 2020. ",
"bbox": [
174,
776,
825,
819
],
"page_idx": 11
},
{
"type": "text",
"text": "Odalric-Ambrym Maillard, Timothy A Mann, and Shie Mannor. How hard is my mdp?\" the distribution-norm to the rescue\". Advances in Neural Information Processing Systems, 27:1835– 1843, 2014. ",
"bbox": [
173,
829,
825,
872
],
"page_idx": 11
},
{
"type": "text",
"text": "Andreas Maurer and Massimiliano Pontil. Empirical bernstein bounds and sample variance penaliza tion. Conference on Learning Theory, 2009. ",
"bbox": [
173,
882,
821,
911
],
"page_idx": 11
},
{
"type": "text",
"text": "Paria Rashidinejad, Banghua Zhu, Cong Ma, Jiantao Jiao, and Stuart Russell. Bridging offline reinforcement learning and imitation learning: A tale of pessimism. arXiv preprint arXiv:2103.12021, 2021. ",
"bbox": [
176,
90,
825,
133
],
"page_idx": 12
},
{
"type": "text",
"text": "Tongzheng Ren, Jialian Li, Bo Dai, Simon S Du, and Sujay Sanghavi. Nearly horizon-free offline reinforcement learning. Advances in neural information processing systems, 2021. ",
"bbox": [
173,
143,
823,
172
],
"page_idx": 12
},
{
"type": "text",
"text": "Karthik Sridharan. A gentle introduction to concentration inequalities. Dept. Comput. Sci., Cornell Univ., Tech. Rep, 2002. ",
"bbox": [
174,
183,
823,
213
],
"page_idx": 12
},
{
"type": "text",
"text": "Richard S Sutton and Andrew G Barto. Reinforcement learning: An introduction. MIT press, 2018. ",
"bbox": [
173,
222,
823,
238
],
"page_idx": 12
},
{
"type": "text",
"text": "Csaba Szepesvári and Rémi Munos. Finite time bounds for sampling based fitted value iteration. In Proceedings of the 22nd international conference on Machine learning, pages 880–887, 2005. ",
"bbox": [
173,
247,
825,
279
],
"page_idx": 12
},
{
"type": "text",
"text": "Joel Tropp et al. Freedman’s inequality for matrix martingales. Electronic Communications in Probability, 16:262–270, 2011. ",
"bbox": [
173,
287,
825,
316
],
"page_idx": 12
},
{
"type": "text",
"text": "Masatoshi Uehara and Wen Sun. Pessimistic model-based offline rl: Pac bounds and posterior sampling under partial coverage. arXiv preprint arXiv:2107.06226, 2021. ",
"bbox": [
171,
327,
823,
357
],
"page_idx": 12
},
{
"type": "text",
"text": "Ruosong Wang, Dean P Foster, and Sham M Kakade. What are the statistical limits of offline rl with linear function approximation? International Conference on Machine Learning, 2021. ",
"bbox": [
173,
366,
825,
396
],
"page_idx": 12
},
{
"type": "text",
"text": "Zheng Wen and Benjamin Van Roy. Efficient exploration and value function generalization in deterministic systems. Advances in Neural Information Processing Systems, 26, 2013. ",
"bbox": [
171,
405,
825,
435
],
"page_idx": 12
},
{
"type": "text",
"text": "Chenjun Xiao, Yifan Wu, Jincheng Mei, Bo Dai, Tor Lattimore, Lihong Li, Csaba Szepesvari, and Dale Schuurmans. On the optimality of batch policy optimization algorithms. In International Conference on Machine Learning, pages 11362–11371. PMLR, 2021. ",
"bbox": [
173,
444,
825,
488
],
"page_idx": 12
},
{
"type": "text",
"text": "Tengyang Xie and Nan Jiang. $\\mathbf { Q } ^ { * }$ approximation schemes for batch reinforcement learning: A theoretical comparison. In Uncertainty in Artificial Intelligence, pages 550–559, 2020. ",
"bbox": [
173,
498,
823,
529
],
"page_idx": 12
},
{
"type": "text",
"text": "Tengyang Xie and Nan Jiang. Batch value-function approximation with only realizability. International Conference on Machine Learning, 2021. ",
"bbox": [
171,
537,
823,
568
],
"page_idx": 12
},
{
"type": "text",
"text": "Tengyang Xie, Ching-An Cheng, Nan Jiang, Paul Mineiro, and Alekh Agarwal. Bellman-consistent pessimism for offline reinforcement learning. Advances in neural information processing systems, 2021a. ",
"bbox": [
174,
577,
823,
619
],
"page_idx": 12
},
{
"type": "text",
"text": "Tengyang Xie, Nan Jiang, Huan Wang, Caiming Xiong, and Yu Bai. Policy finetuning: Bridging sample-efficient offline and online reinforcement learning. Advances in neural information processing systems, 2021b. ",
"bbox": [
174,
630,
825,
674
],
"page_idx": 12
},
{
"type": "text",
"text": "Ming Yin and Yu-Xiang Wang. Asymptotically efficient off-policy evaluation for tabular reinforcement learning. In International Conference on Artificial Intelligence and Statistics, pages 3948–3958. PMLR, 2020. ",
"bbox": [
173,
683,
826,
727
],
"page_idx": 12
},
{
"type": "text",
"text": "Ming Yin and Yu-Xiang Wang. Optimal uniform ope and model-based offline reinforcement learning in time-homogeneous, reward-free and task-agnostic settings. Advances in neural information processing systems, 2021. ",
"bbox": [
173,
737,
826,
780
],
"page_idx": 12
},
{
"type": "text",
"text": "Ming Yin, Yu Bai, and Yu-Xiang Wang. Near-optimal provable uniform convergence in offline policy evaluation for reinforcement learning. In International Conference on Artificial Intelligence and Statistics, pages 1567–1575. PMLR, 2021a. ",
"bbox": [
173,
790,
825,
833
],
"page_idx": 12
},
{
"type": "text",
"text": "Ming Yin, Yu Bai, and Yu-Xiang Wang. Near-optimal offline reinforcement learning via double variance reduction. Advances in neural information processing systems, 2021b. ",
"bbox": [
169,
843,
823,
872
],
"page_idx": 12
},
{
"type": "text",
"text": "Andrea Zanette. Exponential lower bounds for batch reinforcement learning: Batch rl can be exponentially harder than online rl. International Conference on Machine Learning, 2021. ",
"bbox": [
173,
882,
820,
911
],
"page_idx": 12
},
{
"type": "text",
"text": "Andrea Zanette and Emma Brunskill. Problem dependent reinforcement learning bounds which can identify bandit structure in mdps. In International Conference on Machine Learning, pages 5747–5755. PMLR, 2018. ",
"bbox": [
173,
90,
825,
133
],
"page_idx": 13
},
{
"type": "text",
"text": "Andrea Zanette and Emma Brunskill. Tighter problem-dependent regret bounds in reinforcement learning without domain knowledge using value function bounds. In International Conference on Machine Learning, pages 7304–7312. PMLR, 2019. ",
"bbox": [
173,
142,
823,
184
],
"page_idx": 13
},
{
"type": "text",
"text": "Andrea Zanette, Martin J Wainwright, and Emma Brunskill. Provable benefits of actor-critic methods for offline reinforcement learning. Advances in neural information processing systems, 2021. ",
"bbox": [
171,
194,
825,
222
],
"page_idx": 13
},
{
"type": "text",
"text": "Zihan Zhang, Xiangyang Ji, and Simon S Du. Is reinforcement learning more difficult than bandits? a near-optimal algorithm escaping the curse of horizon. Conference of Learning Theory, 2021. ",
"bbox": [
174,
231,
821,
260
],
"page_idx": 13
}
] |