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Due to its nature of without the access to interact with the MDP model (which causes", "type": "text" } ], "index": 38 }, { "bbox": [ 106, 692, 504, 703 ], "spans": [ { "bbox": [ 106, 692, 504, 703 ], "score": 1.0, "content": "the distributional mismatches), most of the literature that study the sample complexity / provable", "type": "text" } ], "index": 39 }, { "bbox": [ 106, 703, 506, 714 ], "spans": [ { "bbox": [ 106, 703, 506, 714 ], "score": 1.0, "content": "efficiency of offline RL (e.g. Le et al. [2019], Chen and Jiang [2019], Xie and Jiang [2020, 2021],", "type": "text" } ], "index": 40 } ], "index": 36.5, "bbox_fs": [ 105, 625, 506, 714 ] } ] }, { "preproc_blocks": [ { "type": "text", "bbox": [ 107, 72, 505, 128 ], "lines": [ { "bbox": [ 106, 71, 505, 86 ], "spans": [ { "bbox": [ 106, 71, 505, 86 ], "score": 1.0, "content": "Yin et al. [2021a,b], Ren et al. [2021], Rashidinejad et al. [2021], Xie et al. [2021b]) rely on making", "type": "text" } ], "index": 0 }, { "bbox": [ 106, 83, 504, 95 ], "spans": [ { "bbox": [ 106, 83, 504, 95 ], "score": 1.0, "content": "different data-coverage assumptions for making the problem learnable and provide the near-optimal", "type": "text" } ], "index": 1 }, { "bbox": [ 105, 94, 505, 107 ], "spans": [ { "bbox": [ 105, 94, 505, 107 ], "score": 1.0, "content": "worst-case performance bounds that depend on their data-coverage coefficients. Those results are", "type": "text" } ], "index": 2 }, { "bbox": [ 106, 106, 505, 118 ], "spans": [ { "bbox": [ 106, 106, 505, 118 ], "score": 1.0, "content": "valuable in general as they do not depend on the structure of the particular problem, therefore, remain", "type": "text" } ], "index": 3 }, { "bbox": [ 106, 117, 347, 128 ], "spans": [ { "bbox": [ 106, 117, 347, 128 ], "score": 1.0, "content": "valid even for pathological MDPs. But is this good enough?", "type": "text" } ], "index": 4 } ], "index": 2 }, { "type": "text", "bbox": [ 107, 132, 505, 253 ], "lines": [ { "bbox": [ 106, 132, 505, 145 ], "spans": [ { "bbox": [ 106, 132, 505, 145 ], "score": 1.0, "content": "In practice, the empirical performances of offline reinforcement learning (e.g. Gulcehre et al. [2020],", "type": "text" } ], "index": 5 }, { "bbox": [ 105, 143, 506, 156 ], "spans": [ { "bbox": [ 105, 143, 506, 156 ], "score": 1.0, "content": "Fu et al. [2020, 2021], Janner et al. [2021]) are often far better than what those non-adaptive /", "type": "text" } ], "index": 6 }, { "bbox": [ 105, 155, 505, 167 ], "spans": [ { "bbox": [ 105, 155, 505, 167 ], "score": 1.0, "content": "problem-independent bounds would indicate. Although empirical evidence can help explain why", "type": "text" } ], "index": 7 }, { "bbox": [ 105, 165, 506, 178 ], "spans": [ { "bbox": [ 105, 165, 506, 178 ], "score": 1.0, "content": "we may observe better or worse performances on different MDPs, a systematic understanding of", "type": "text" } ], "index": 8 }, { "bbox": [ 105, 176, 505, 189 ], "spans": [ { "bbox": [ 105, 176, 505, 189 ], "score": 1.0, "content": "what types of decision processes and what kinds of behavior policies are inherently easier or more", "type": "text" } ], "index": 9 }, { "bbox": [ 106, 187, 505, 199 ], "spans": [ { "bbox": [ 106, 187, 505, 199 ], "score": 1.0, "content": "challenging for offline RL is lacking. Besides, despite the fact that a non-adaptive bound can learn", "type": "text" } ], "index": 10 }, { "bbox": [ 106, 199, 504, 210 ], "spans": [ { "bbox": [ 106, 199, 504, 210 ], "score": 1.0, "content": "even the pathological examples within the assumption family, there is no guarantee for the instances", "type": "text" } ], "index": 11 }, { "bbox": [ 106, 210, 505, 221 ], "spans": [ { "bbox": [ 106, 210, 505, 221 ], "score": 1.0, "content": "outside the family. However, practical offline reinforcement learning problems are usually beyond", "type": "text" } ], "index": 12 }, { "bbox": [ 106, 221, 504, 232 ], "spans": [ { "bbox": [ 106, 221, 504, 232 ], "score": 1.0, "content": "the scope of certain data-coverage assumptions, which limits the applicability of those results. Can", "type": "text" } ], "index": 13 }, { "bbox": [ 105, 231, 505, 243 ], "spans": [ { "bbox": [ 105, 231, 505, 243 ], "score": 1.0, "content": "we make as few assumptions as possible? Or even more, what can we guarantee when no assumption", "type": "text" } ], "index": 14 }, { "bbox": [ 105, 241, 231, 254 ], "spans": [ { "bbox": [ 105, 241, 231, 254 ], "score": 1.0, "content": "is made about offline learning?", "type": "text" } ], "index": 15 } ], "index": 10 }, { "type": "text", "bbox": [ 107, 258, 505, 346 ], "lines": [ { "bbox": [ 105, 257, 505, 271 ], "spans": [ { "bbox": [ 105, 257, 505, 271 ], "score": 1.0, "content": "Those motivate us to derive the provably efficient bounds that are adaptive to the individual instances", "type": "text" } ], "index": 16 }, { "bbox": [ 105, 268, 505, 281 ], "spans": [ { "bbox": [ 105, 268, 505, 281 ], "score": 1.0, "content": "but only require minimal assumptions so they can be widely applied in most cases. Ideally, such", "type": "text" } ], "index": 17 }, { "bbox": [ 105, 280, 505, 293 ], "spans": [ { "bbox": [ 105, 280, 505, 293 ], "score": 1.0, "content": "bounds should characterize the system structures of the specific problems, hold even for peculiar", "type": "text" } ], "index": 18 }, { "bbox": [ 105, 291, 505, 303 ], "spans": [ { "bbox": [ 105, 291, 505, 303 ], "score": 1.0, "content": "instances that do not satisfy the standard data-coverage assumptions, and recover the worst-case", "type": "text" } ], "index": 19 }, { "bbox": [ 105, 302, 505, 314 ], "spans": [ { "bbox": [ 105, 302, 505, 314 ], "score": 1.0, "content": "guarantees when the assumptions are satisfied. As mentioned in Zanette and Brunskill [2019], a fully", "type": "text" } ], "index": 20 }, { "bbox": [ 105, 313, 505, 325 ], "spans": [ { "bbox": [ 105, 313, 505, 325 ], "score": 1.0, "content": "adaptive characterization in RL is important as it might bring considerable saving in the time spent", "type": "text" } ], "index": 21 }, { "bbox": [ 105, 324, 505, 336 ], "spans": [ { "bbox": [ 105, 324, 505, 336 ], "score": 1.0, "content": "designing domain-specific RL solutions and in training a human expert to judge and recognize the", "type": "text" } ], "index": 22 }, { "bbox": [ 105, 335, 243, 347 ], "spans": [ { "bbox": [ 105, 335, 243, 347 ], "score": 1.0, "content": "complexity of different problems.", "type": "text" } ], "index": 23 } ], "index": 19.5 }, { "type": "title", "bbox": [ 108, 367, 203, 379 ], "lines": [ { "bbox": [ 105, 365, 205, 381 ], "spans": [ { "bbox": [ 105, 365, 205, 381 ], "score": 1.0, "content": "1.1 Our contribution", "type": "text" } ], "index": 24 } ], "index": 24 }, { "type": "text", "bbox": [ 107, 390, 506, 480 ], "lines": [ { "bbox": [ 105, 391, 506, 403 ], "spans": [ { "bbox": [ 105, 391, 506, 403 ], "score": 1.0, "content": "In this work, we provide the analysis for the adaptive pessimistic value iteration (APVI) (Algorithm 1)", "type": "text" } ], "index": 25 }, { "bbox": [ 106, 402, 506, 414 ], "spans": [ { "bbox": [ 106, 402, 506, 414 ], "score": 1.0, "content": "with finite horizon time-inhomogeneous (non-stationary) MDPs and derive a strong adaptive bound", "type": "text" } ], "index": 26 }, { "bbox": [ 105, 412, 506, 427 ], "spans": [ { "bbox": [ 105, 414, 302, 427 ], "score": 1.0, "content": "that is near-optimal under the weak assumption", "type": "text" }, { "bbox": [ 302, 413, 365, 426 ], "score": 0.92, "content": "d _ { h } ^ { \\mu } ( s _ { h } , a _ { h } ) > 0", "type": "inline_equation" }, { "bbox": [ 365, 414, 375, 427 ], "score": 1.0, "content": "if", "type": "text" }, { "bbox": [ 375, 412, 442, 426 ], "score": 0.91, "content": "d _ { h } ^ { \\pi ^ { \\star } } ( s _ { h } , a _ { h } ) > \\bar { 0 }", "type": "inline_equation" }, { "bbox": [ 442, 414, 506, 427 ], "score": 1.0, "content": "(Theorem 4.1).", "type": "text" } ], "index": 27 }, { "bbox": [ 106, 425, 506, 437 ], "spans": [ { "bbox": [ 106, 425, 506, 437 ], "score": 1.0, "content": "Specifically, our bound (quantity (1)) explicitly depends on the marginal importance ratios (between", "type": "text" } ], "index": 28 }, { "bbox": [ 106, 436, 506, 448 ], "spans": [ { "bbox": [ 106, 436, 181, 448 ], "score": 1.0, "content": "the optimal policy", "type": "text" }, { "bbox": [ 181, 436, 193, 446 ], "score": 0.86, "content": "\\pi ^ { \\star }", "type": "inline_equation" }, { "bbox": [ 194, 436, 290, 448 ], "score": 1.0, "content": "and the behavior policy", "type": "text" }, { "bbox": [ 290, 437, 298, 447 ], "score": 0.73, "content": "\\mu _ { . }", "type": "inline_equation" }, { "bbox": [ 298, 436, 506, 448 ], "score": 1.0, "content": ") and the per-step conditional variances. In addition,", "type": "text" } ], "index": 29 }, { "bbox": [ 105, 447, 505, 459 ], "spans": [ { "bbox": [ 105, 447, 505, 459 ], "score": 1.0, "content": "we provide an instance-dependent (local minimax) lower bound (Theorem 4.3) to certify (1) is nearly", "type": "text" } ], "index": 30 }, { "bbox": [ 105, 457, 506, 471 ], "spans": [ { "bbox": [ 105, 457, 506, 471 ], "score": 1.0, "content": "optimal at the instance level for offline learning and call it the intrinsic offline learning bound. The", "type": "text" } ], "index": 31 }, { "bbox": [ 105, 468, 300, 481 ], "spans": [ { "bbox": [ 105, 468, 300, 481 ], "score": 1.0, "content": "intrinsic bound has the following consequences.", "type": "text" } ], "index": 32 } ], "index": 28.5 }, { "type": "text", "bbox": [ 133, 496, 505, 556 ], "lines": [ { "bbox": [ 132, 495, 505, 510 ], "spans": [ { "bbox": [ 132, 496, 448, 510 ], "score": 1.0, "content": "• In the non-adaptive / worst-case regime (4.1-4.3), the intrinsic bound implies", "type": "text" }, { "bbox": [ 449, 495, 505, 510 ], "score": 0.95, "content": "\\widetilde { O } ( H ^ { 3 } / d _ { m } \\epsilon ^ { 2 } )", "type": "inline_equation" } ], "index": 33 }, { "bbox": [ 141, 509, 506, 523 ], "spans": [ { "bbox": [ 141, 510, 349, 523 ], "score": 1.0, "content": "complexity under the uniform data-coverage 2.1,", "type": "text" }, { "bbox": [ 349, 509, 412, 522 ], "score": 0.91, "content": "\\tilde { O } ( H ^ { 3 } S C ^ { \\star } / \\epsilon ^ { 2 } )", "type": "inline_equation" }, { "bbox": [ 412, 510, 506, 523 ], "score": 1.0, "content": "complexity under the", "type": "text" } ], "index": 34 }, { "bbox": [ 141, 522, 506, 536 ], "spans": [ { "bbox": [ 141, 523, 337, 536 ], "score": 1.0, "content": "single policy concentrability assumption 2.3 and", "type": "text" }, { "bbox": [ 337, 522, 388, 536 ], "score": 0.95, "content": "\\widetilde { O } ( H / d _ { m } \\epsilon ^ { 2 } )", "type": "inline_equation" }, { "bbox": [ 388, 523, 506, 536 ], "score": 1.0, "content": "complexity when the sum of", "type": "text" } ], "index": 35 }, { "bbox": [ 141, 534, 505, 546 ], "spans": [ { "bbox": [ 141, 534, 505, 546 ], "score": 1.0, "content": "rewards is bounded by 1. All of those are optimal in their respectively regimes [Yin et al.,", "type": "text" } ], "index": 36 }, { "bbox": [ 141, 543, 418, 558 ], "spans": [ { "bbox": [ 141, 543, 418, 558 ], "score": 1.0, "content": "2021a, Rashidinejad et al., 2021, Xie et al., 2021b, Ren et al., 2021];", "type": "text" } ], "index": 37 } ], "index": 35 }, { "type": "text", "bbox": [ 132, 567, 504, 624 ], "lines": [ { "bbox": [ 132, 567, 505, 580 ], "spans": [ { "bbox": [ 132, 567, 505, 580 ], "score": 1.0, "content": "• In the adaptive domain (4.4), the intrinsic bound implies the tight problem-dependent", "type": "text" } ], "index": 38 }, { "bbox": [ 142, 578, 505, 592 ], "spans": [ { "bbox": [ 142, 579, 353, 592 ], "score": 1.0, "content": "counterpart of Zanette and Brunskill [2019], yields", "type": "text" }, { "bbox": [ 353, 578, 407, 592 ], "score": 0.93, "content": "{ \\tilde { O } } ( H ^ { 3 } / n d _ { m } )", "type": "inline_equation" }, { "bbox": [ 407, 579, 505, 592 ], "score": 1.0, "content": "fast convergence in the", "type": "text" } ], "index": 39 }, { "bbox": [ 142, 591, 505, 603 ], "spans": [ { "bbox": [ 142, 591, 505, 603 ], "score": 1.0, "content": "deterministic systems, has improved complexity in the partially deterministic systems and", "type": "text" } ], "index": 40 }, { "bbox": [ 142, 602, 505, 613 ], "spans": [ { "bbox": [ 142, 602, 505, 613 ], "score": 1.0, "content": "a family of highly mixing problems, and remains optimal when reducing to the tabular", "type": "text" } ], "index": 41 }, { "bbox": [ 141, 613, 219, 624 ], "spans": [ { "bbox": [ 141, 613, 219, 624 ], "score": 1.0, "content": "contextual bandits.", "type": "text" } ], "index": 42 } ], "index": 40 }, { "type": "text", "bbox": [ 108, 640, 505, 673 ], "lines": [ { "bbox": [ 105, 638, 505, 654 ], "spans": [ { "bbox": [ 105, 638, 505, 654 ], "score": 1.0, "content": "Beyond the above, due to the generic form of the intrinsic bound, we could come up with as many", "type": "text" } ], "index": 43 }, { "bbox": [ 106, 651, 505, 663 ], "spans": [ { "bbox": [ 106, 651, 505, 663 ], "score": 1.0, "content": "problem instances (that are of our interests) as possible and study their properties. In this sense, the", "type": "text" } ], "index": 44 }, { "bbox": [ 106, 662, 387, 673 ], "spans": [ { "bbox": [ 106, 662, 387, 673 ], "score": 1.0, "content": "intrinsic bound helps illuminate the fundamental nature of offline RL.", "type": "text" } ], "index": 45 } ], "index": 44 }, { "type": "text", "bbox": [ 108, 678, 504, 722 ], "lines": [ { "bbox": [ 106, 678, 505, 690 ], "spans": [ { "bbox": [ 106, 678, 505, 690 ], "score": 1.0, "content": "Furthermore, as a step towards assumption-free offline reinforcement learning, we build a modified", "type": "text" } ], "index": 46 }, { "bbox": [ 105, 688, 505, 701 ], "spans": [ { "bbox": [ 105, 688, 505, 701 ], "score": 1.0, "content": "AVPI and obtain an adaptive bound that could characterize the suboptimality gap in the state-action", "type": "text" } ], "index": 47 }, { "bbox": [ 105, 700, 506, 712 ], "spans": [ { "bbox": [ 105, 700, 506, 712 ], "score": 1.0, "content": "space that is agnostic to the behavior policy (Theorem 5.1). To the best of our knowledge, all of these", "type": "text" } ], "index": 48 }, { "bbox": [ 106, 711, 231, 722 ], "spans": [ { "bbox": [ 106, 711, 231, 722 ], "score": 1.0, "content": "results are the first of its kinds.", "type": "text" } ], "index": 49 } ], "index": 47.5 } ], "page_idx": 1, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 302, 741, 309, 750 ], "lines": [ { "bbox": [ 301, 740, 310, 753 ], "spans": [ { "bbox": [ 301, 740, 310, 753 ], "score": 1.0, "content": "2", "type": "text" } ] } ] } ], "para_blocks": [ { "type": "text", "bbox": [ 107, 72, 505, 128 ], "lines": [ { "bbox": [ 106, 71, 505, 86 ], "spans": [ { "bbox": [ 106, 71, 505, 86 ], "score": 1.0, "content": "Yin et al. [2021a,b], Ren et al. [2021], Rashidinejad et al. [2021], Xie et al. [2021b]) rely on making", "type": "text" } ], "index": 0 }, { "bbox": [ 106, 83, 504, 95 ], "spans": [ { "bbox": [ 106, 83, 504, 95 ], "score": 1.0, "content": "different data-coverage assumptions for making the problem learnable and provide the near-optimal", "type": "text" } ], "index": 1 }, { "bbox": [ 105, 94, 505, 107 ], "spans": [ { "bbox": [ 105, 94, 505, 107 ], "score": 1.0, "content": "worst-case performance bounds that depend on their data-coverage coefficients. Those results are", "type": "text" } ], "index": 2 }, { "bbox": [ 106, 106, 505, 118 ], "spans": [ { "bbox": [ 106, 106, 505, 118 ], "score": 1.0, "content": "valuable in general as they do not depend on the structure of the particular problem, therefore, remain", "type": "text" } ], "index": 3 }, { "bbox": [ 106, 117, 347, 128 ], "spans": [ { "bbox": [ 106, 117, 347, 128 ], "score": 1.0, "content": "valid even for pathological MDPs. But is this good enough?", "type": "text" } ], "index": 4 } ], "index": 2, "bbox_fs": [ 105, 71, 505, 128 ] }, { "type": "text", "bbox": [ 107, 132, 505, 253 ], "lines": [ { "bbox": [ 106, 132, 505, 145 ], "spans": [ { "bbox": [ 106, 132, 505, 145 ], "score": 1.0, "content": "In practice, the empirical performances of offline reinforcement learning (e.g. Gulcehre et al. [2020],", "type": "text" } ], "index": 5 }, { "bbox": [ 105, 143, 506, 156 ], "spans": [ { "bbox": [ 105, 143, 506, 156 ], "score": 1.0, "content": "Fu et al. [2020, 2021], Janner et al. [2021]) are often far better than what those non-adaptive /", "type": "text" } ], "index": 6 }, { "bbox": [ 105, 155, 505, 167 ], "spans": [ { "bbox": [ 105, 155, 505, 167 ], "score": 1.0, "content": "problem-independent bounds would indicate. Although empirical evidence can help explain why", "type": "text" } ], "index": 7 }, { "bbox": [ 105, 165, 506, 178 ], "spans": [ { "bbox": [ 105, 165, 506, 178 ], "score": 1.0, "content": "we may observe better or worse performances on different MDPs, a systematic understanding of", "type": "text" } ], "index": 8 }, { "bbox": [ 105, 176, 505, 189 ], "spans": [ { "bbox": [ 105, 176, 505, 189 ], "score": 1.0, "content": "what types of decision processes and what kinds of behavior policies are inherently easier or more", "type": "text" } ], "index": 9 }, { "bbox": [ 106, 187, 505, 199 ], "spans": [ { "bbox": [ 106, 187, 505, 199 ], "score": 1.0, "content": "challenging for offline RL is lacking. Besides, despite the fact that a non-adaptive bound can learn", "type": "text" } ], "index": 10 }, { "bbox": [ 106, 199, 504, 210 ], "spans": [ { "bbox": [ 106, 199, 504, 210 ], "score": 1.0, "content": "even the pathological examples within the assumption family, there is no guarantee for the instances", "type": "text" } ], "index": 11 }, { "bbox": [ 106, 210, 505, 221 ], "spans": [ { "bbox": [ 106, 210, 505, 221 ], "score": 1.0, "content": "outside the family. However, practical offline reinforcement learning problems are usually beyond", "type": "text" } ], "index": 12 }, { "bbox": [ 106, 221, 504, 232 ], "spans": [ { "bbox": [ 106, 221, 504, 232 ], "score": 1.0, "content": "the scope of certain data-coverage assumptions, which limits the applicability of those results. Can", "type": "text" } ], "index": 13 }, { "bbox": [ 105, 231, 505, 243 ], "spans": [ { "bbox": [ 105, 231, 505, 243 ], "score": 1.0, "content": "we make as few assumptions as possible? Or even more, what can we guarantee when no assumption", "type": "text" } ], "index": 14 }, { "bbox": [ 105, 241, 231, 254 ], "spans": [ { "bbox": [ 105, 241, 231, 254 ], "score": 1.0, "content": "is made about offline learning?", "type": "text" } ], "index": 15 } ], "index": 10, "bbox_fs": [ 105, 132, 506, 254 ] }, { "type": "text", "bbox": [ 107, 258, 505, 346 ], "lines": [ { "bbox": [ 105, 257, 505, 271 ], "spans": [ { "bbox": [ 105, 257, 505, 271 ], "score": 1.0, "content": "Those motivate us to derive the provably efficient bounds that are adaptive to the individual instances", "type": "text" } ], "index": 16 }, { "bbox": [ 105, 268, 505, 281 ], "spans": [ { "bbox": [ 105, 268, 505, 281 ], "score": 1.0, "content": "but only require minimal assumptions so they can be widely applied in most cases. Ideally, such", "type": "text" } ], "index": 17 }, { "bbox": [ 105, 280, 505, 293 ], "spans": [ { "bbox": [ 105, 280, 505, 293 ], "score": 1.0, "content": "bounds should characterize the system structures of the specific problems, hold even for peculiar", "type": "text" } ], "index": 18 }, { "bbox": [ 105, 291, 505, 303 ], "spans": [ { "bbox": [ 105, 291, 505, 303 ], "score": 1.0, "content": "instances that do not satisfy the standard data-coverage assumptions, and recover the worst-case", "type": "text" } ], "index": 19 }, { "bbox": [ 105, 302, 505, 314 ], "spans": [ { "bbox": [ 105, 302, 505, 314 ], "score": 1.0, "content": "guarantees when the assumptions are satisfied. As mentioned in Zanette and Brunskill [2019], a fully", "type": "text" } ], "index": 20 }, { "bbox": [ 105, 313, 505, 325 ], "spans": [ { "bbox": [ 105, 313, 505, 325 ], "score": 1.0, "content": "adaptive characterization in RL is important as it might bring considerable saving in the time spent", "type": "text" } ], "index": 21 }, { "bbox": [ 105, 324, 505, 336 ], "spans": [ { "bbox": [ 105, 324, 505, 336 ], "score": 1.0, "content": "designing domain-specific RL solutions and in training a human expert to judge and recognize the", "type": "text" } ], "index": 22 }, { "bbox": [ 105, 335, 243, 347 ], "spans": [ { "bbox": [ 105, 335, 243, 347 ], "score": 1.0, "content": "complexity of different problems.", "type": "text" } ], "index": 23 } ], "index": 19.5, "bbox_fs": [ 105, 257, 505, 347 ] }, { "type": "title", "bbox": [ 108, 367, 203, 379 ], "lines": [ { "bbox": [ 105, 365, 205, 381 ], "spans": [ { "bbox": [ 105, 365, 205, 381 ], "score": 1.0, "content": "1.1 Our contribution", "type": "text" } ], "index": 24 } ], "index": 24 }, { "type": "text", "bbox": [ 107, 390, 506, 480 ], "lines": [ { "bbox": [ 105, 391, 506, 403 ], "spans": [ { "bbox": [ 105, 391, 506, 403 ], "score": 1.0, "content": "In this work, we provide the analysis for the adaptive pessimistic value iteration (APVI) (Algorithm 1)", "type": "text" } ], "index": 25 }, { "bbox": [ 106, 402, 506, 414 ], "spans": [ { "bbox": [ 106, 402, 506, 414 ], "score": 1.0, "content": "with finite horizon time-inhomogeneous (non-stationary) MDPs and derive a strong adaptive bound", "type": "text" } ], "index": 26 }, { "bbox": [ 105, 412, 506, 427 ], "spans": [ { "bbox": [ 105, 414, 302, 427 ], "score": 1.0, "content": "that is near-optimal under the weak assumption", "type": "text" }, { "bbox": [ 302, 413, 365, 426 ], "score": 0.92, "content": "d _ { h } ^ { \\mu } ( s _ { h } , a _ { h } ) > 0", "type": "inline_equation" }, { "bbox": [ 365, 414, 375, 427 ], "score": 1.0, "content": "if", "type": "text" }, { "bbox": [ 375, 412, 442, 426 ], "score": 0.91, "content": "d _ { h } ^ { \\pi ^ { \\star } } ( s _ { h } , a _ { h } ) > \\bar { 0 }", "type": "inline_equation" }, { "bbox": [ 442, 414, 506, 427 ], "score": 1.0, "content": "(Theorem 4.1).", "type": "text" } ], "index": 27 }, { "bbox": [ 106, 425, 506, 437 ], "spans": [ { "bbox": [ 106, 425, 506, 437 ], "score": 1.0, "content": "Specifically, our bound (quantity (1)) explicitly depends on the marginal importance ratios (between", "type": "text" } ], "index": 28 }, { "bbox": [ 106, 436, 506, 448 ], "spans": [ { "bbox": [ 106, 436, 181, 448 ], "score": 1.0, "content": "the optimal policy", "type": "text" }, { "bbox": [ 181, 436, 193, 446 ], "score": 0.86, "content": "\\pi ^ { \\star }", "type": "inline_equation" }, { "bbox": [ 194, 436, 290, 448 ], "score": 1.0, "content": "and the behavior policy", "type": "text" }, { "bbox": [ 290, 437, 298, 447 ], "score": 0.73, "content": "\\mu _ { . }", "type": "inline_equation" }, { "bbox": [ 298, 436, 506, 448 ], "score": 1.0, "content": ") and the per-step conditional variances. In addition,", "type": "text" } ], "index": 29 }, { "bbox": [ 105, 447, 505, 459 ], "spans": [ { "bbox": [ 105, 447, 505, 459 ], "score": 1.0, "content": "we provide an instance-dependent (local minimax) lower bound (Theorem 4.3) to certify (1) is nearly", "type": "text" } ], "index": 30 }, { "bbox": [ 105, 457, 506, 471 ], "spans": [ { "bbox": [ 105, 457, 506, 471 ], "score": 1.0, "content": "optimal at the instance level for offline learning and call it the intrinsic offline learning bound. The", "type": "text" } ], "index": 31 }, { "bbox": [ 105, 468, 300, 481 ], "spans": [ { "bbox": [ 105, 468, 300, 481 ], "score": 1.0, "content": "intrinsic bound has the following consequences.", "type": "text" } ], "index": 32 } ], "index": 28.5, "bbox_fs": [ 105, 391, 506, 481 ] }, { "type": "text", "bbox": [ 133, 496, 505, 556 ], "lines": [ { "bbox": [ 132, 495, 505, 510 ], "spans": [ { "bbox": [ 132, 496, 448, 510 ], "score": 1.0, "content": "• In the non-adaptive / worst-case regime (4.1-4.3), the intrinsic bound implies", "type": "text" }, { "bbox": [ 449, 495, 505, 510 ], "score": 0.95, "content": "\\widetilde { O } ( H ^ { 3 } / d _ { m } \\epsilon ^ { 2 } )", "type": "inline_equation" } ], "index": 33 }, { "bbox": [ 141, 509, 506, 523 ], "spans": [ { "bbox": [ 141, 510, 349, 523 ], "score": 1.0, "content": "complexity under the uniform data-coverage 2.1,", "type": "text" }, { "bbox": [ 349, 509, 412, 522 ], "score": 0.91, "content": "\\tilde { O } ( H ^ { 3 } S C ^ { \\star } / \\epsilon ^ { 2 } )", "type": "inline_equation" }, { "bbox": [ 412, 510, 506, 523 ], "score": 1.0, "content": "complexity under the", "type": "text" } ], "index": 34 }, { "bbox": [ 141, 522, 506, 536 ], "spans": [ { "bbox": [ 141, 523, 337, 536 ], "score": 1.0, "content": "single policy concentrability assumption 2.3 and", "type": "text" }, { "bbox": [ 337, 522, 388, 536 ], "score": 0.95, "content": "\\widetilde { O } ( H / d _ { m } \\epsilon ^ { 2 } )", "type": "inline_equation" }, { "bbox": [ 388, 523, 506, 536 ], "score": 1.0, "content": "complexity when the sum of", "type": "text" } ], "index": 35 }, { "bbox": [ 141, 534, 505, 546 ], "spans": [ { "bbox": [ 141, 534, 505, 546 ], "score": 1.0, "content": "rewards is bounded by 1. All of those are optimal in their respectively regimes [Yin et al.,", "type": "text" } ], "index": 36 }, { "bbox": [ 141, 543, 418, 558 ], "spans": [ { "bbox": [ 141, 543, 418, 558 ], "score": 1.0, "content": "2021a, Rashidinejad et al., 2021, Xie et al., 2021b, Ren et al., 2021];", "type": "text" } ], "index": 37 } ], "index": 35, "bbox_fs": [ 132, 495, 506, 558 ] }, { "type": "text", "bbox": [ 132, 567, 504, 624 ], "lines": [ { "bbox": [ 132, 567, 505, 580 ], "spans": [ { "bbox": [ 132, 567, 505, 580 ], "score": 1.0, "content": "• In the adaptive domain (4.4), the intrinsic bound implies the tight problem-dependent", "type": "text" } ], "index": 38 }, { "bbox": [ 142, 578, 505, 592 ], "spans": [ { "bbox": [ 142, 579, 353, 592 ], "score": 1.0, "content": "counterpart of Zanette and Brunskill [2019], yields", "type": "text" }, { "bbox": [ 353, 578, 407, 592 ], "score": 0.93, "content": "{ \\tilde { O } } ( H ^ { 3 } / n d _ { m } )", "type": "inline_equation" }, { "bbox": [ 407, 579, 505, 592 ], "score": 1.0, "content": "fast convergence in the", "type": "text" } ], "index": 39 }, { "bbox": [ 142, 591, 505, 603 ], "spans": [ { "bbox": [ 142, 591, 505, 603 ], "score": 1.0, "content": "deterministic systems, has improved complexity in the partially deterministic systems and", "type": "text" } ], "index": 40 }, { "bbox": [ 142, 602, 505, 613 ], "spans": [ { "bbox": [ 142, 602, 505, 613 ], "score": 1.0, "content": "a family of highly mixing problems, and remains optimal when reducing to the tabular", "type": "text" } ], "index": 41 }, { "bbox": [ 141, 613, 219, 624 ], "spans": [ { "bbox": [ 141, 613, 219, 624 ], "score": 1.0, "content": "contextual bandits.", "type": "text" } ], "index": 42 } ], "index": 40, "bbox_fs": [ 132, 567, 505, 624 ] }, { "type": "text", "bbox": [ 108, 640, 505, 673 ], "lines": [ { "bbox": [ 105, 638, 505, 654 ], "spans": [ { "bbox": [ 105, 638, 505, 654 ], "score": 1.0, "content": "Beyond the above, due to the generic form of the intrinsic bound, we could come up with as many", "type": "text" } ], "index": 43 }, { "bbox": [ 106, 651, 505, 663 ], "spans": [ { "bbox": [ 106, 651, 505, 663 ], "score": 1.0, "content": "problem instances (that are of our interests) as possible and study their properties. In this sense, the", "type": "text" } ], "index": 44 }, { "bbox": [ 106, 662, 387, 673 ], "spans": [ { "bbox": [ 106, 662, 387, 673 ], "score": 1.0, "content": "intrinsic bound helps illuminate the fundamental nature of offline RL.", "type": "text" } ], "index": 45 } ], "index": 44, "bbox_fs": [ 105, 638, 505, 673 ] }, { "type": "text", "bbox": [ 108, 678, 504, 722 ], "lines": [ { "bbox": [ 106, 678, 505, 690 ], "spans": [ { "bbox": [ 106, 678, 505, 690 ], "score": 1.0, "content": "Furthermore, as a step towards assumption-free offline reinforcement learning, we build a modified", "type": "text" } ], "index": 46 }, { "bbox": [ 105, 688, 505, 701 ], "spans": [ { "bbox": [ 105, 688, 505, 701 ], "score": 1.0, "content": "AVPI and obtain an adaptive bound that could characterize the suboptimality gap in the state-action", "type": "text" } ], "index": 47 }, { "bbox": [ 105, 700, 506, 712 ], "spans": [ { "bbox": [ 105, 700, 506, 712 ], "score": 1.0, "content": "space that is agnostic to the behavior policy (Theorem 5.1). To the best of our knowledge, all of these", "type": "text" } ], "index": 48 }, { "bbox": [ 106, 711, 231, 722 ], "spans": [ { "bbox": [ 106, 711, 231, 722 ], "score": 1.0, "content": "results are the first of its kinds.", "type": "text" } ], "index": 49 } ], "index": 47.5, "bbox_fs": [ 105, 678, 506, 722 ] } ] }, { "preproc_blocks": [ { "type": "title", "bbox": [ 107, 72, 187, 84 ], "lines": [ { "bbox": [ 105, 71, 189, 86 ], "spans": [ { "bbox": [ 105, 71, 189, 86 ], "score": 1.0, "content": "1.2 Related work", "type": "text" } ], "index": 0 } ], "index": 0 }, { "type": "text", "bbox": [ 107, 94, 506, 258 ], "lines": [ { "bbox": [ 106, 93, 505, 106 ], "spans": [ { "bbox": [ 106, 93, 505, 106 ], "score": 1.0, "content": "Finite sample analysis for offline reinforcement learning can be traced back to Szepesvári and Munos", "type": "text" } ], "index": 1 }, { "bbox": [ 106, 104, 505, 117 ], "spans": [ { "bbox": [ 106, 104, 505, 117 ], "score": 1.0, "content": "[2005], Antos et al. [2008a,b] for the infinite horizon discounted setting via Fitted Q-Iteration (FQI)", "type": "text" } ], "index": 2 }, { "bbox": [ 105, 114, 506, 130 ], "spans": [ { "bbox": [ 105, 114, 506, 130 ], "score": 1.0, "content": "type function approximation algorithms. [Chen and Jiang, 2019, Le et al., 2019, Xie and Jiang,", "type": "text" } ], "index": 3 }, { "bbox": [ 106, 127, 505, 139 ], "spans": [ { "bbox": [ 106, 127, 505, 139 ], "score": 1.0, "content": "2021, 2020] follow this line of research and derive the information-theoretical bounds. Recently, Xie", "type": "text" } ], "index": 4 }, { "bbox": [ 106, 137, 505, 150 ], "spans": [ { "bbox": [ 106, 137, 505, 150 ], "score": 1.0, "content": "and Jiang [2021] considers the offline RL with only the realizability assumption, Liu et al. [2020],", "type": "text" } ], "index": 5 }, { "bbox": [ 106, 148, 505, 161 ], "spans": [ { "bbox": [ 106, 148, 505, 161 ], "score": 1.0, "content": "Chang et al. [2021] considers the offline RL without sufficient coverage and Kidambi et al. [2020],", "type": "text" } ], "index": 6 }, { "bbox": [ 105, 158, 506, 171 ], "spans": [ { "bbox": [ 105, 158, 506, 171 ], "score": 1.0, "content": "Uehara and Sun [2021] uses the model-based approach for addressing offline RL. Under those weak", "type": "text" } ], "index": 7 }, { "bbox": [ 106, 171, 505, 182 ], "spans": [ { "bbox": [ 106, 171, 505, 182 ], "score": 1.0, "content": "coverage assumption, their finite sample analysis are suboptimal (e.g. in terms of the effective", "type": "text" } ], "index": 8 }, { "bbox": [ 104, 180, 506, 194 ], "spans": [ { "bbox": [ 104, 180, 139, 194 ], "score": 1.0, "content": "horizon", "type": "text" }, { "bbox": [ 140, 181, 182, 193 ], "score": 0.9, "content": "( 1 - \\gamma ) ^ { \\bar { - } 1 } .", "type": "inline_equation" }, { "bbox": [ 182, 180, 506, 194 ], "score": 1.0, "content": "). Recently, Yin et al. [2021a,b], Ren et al. [2021] study the finite horizon case. In", "type": "text" } ], "index": 9 }, { "bbox": [ 105, 191, 505, 205 ], "spans": [ { "bbox": [ 105, 191, 505, 205 ], "score": 1.0, "content": "the linear MDP case, Jin et al. [2020] studies the pessimistic algorithm for offline policy learning", "type": "text" } ], "index": 10 }, { "bbox": [ 106, 203, 505, 215 ], "spans": [ { "bbox": [ 106, 203, 505, 215 ], "score": 1.0, "content": "under only the compliance assumption, and, concurrently, Xie et al. [2021a] proposes the general", "type": "text" } ], "index": 11 }, { "bbox": [ 105, 214, 506, 226 ], "spans": [ { "bbox": [ 105, 214, 506, 226 ], "score": 1.0, "content": "pessimistic function approximation framework with instantiation in linear MDP and Zanette et al.", "type": "text" } ], "index": 12 }, { "bbox": [ 106, 224, 505, 237 ], "spans": [ { "bbox": [ 106, 224, 505, 237 ], "score": 1.0, "content": "[2021] shows actor-critic style algorithm is near-optimal for linear Bellman complete model. In", "type": "text" } ], "index": 13 }, { "bbox": [ 105, 235, 506, 249 ], "spans": [ { "bbox": [ 105, 235, 506, 249 ], "score": 1.0, "content": "addition, Wang et al. [2021], Zanette [2021] prove some exponential lower bounds under their linear", "type": "text" } ], "index": 14 }, { "bbox": [ 105, 246, 257, 259 ], "spans": [ { "bbox": [ 105, 246, 257, 259 ], "score": 1.0, "content": "function approximation assumptions.", "type": "text" } ], "index": 15 } ], "index": 8 }, { "type": "text", "bbox": [ 106, 263, 506, 381 ], "lines": [ { "bbox": [ 106, 263, 505, 275 ], "spans": [ { "bbox": [ 106, 263, 505, 275 ], "score": 1.0, "content": "Among them, there are a few works that achieve the sample optimality under their respective", "type": "text" } ], "index": 16 }, { "bbox": [ 104, 274, 507, 286 ], "spans": [ { "bbox": [ 104, 274, 428, 286 ], "score": 1.0, "content": "assumptions. Under the uniform data coverage (minimal state-action probability", "type": "text" }, { "bbox": [ 429, 275, 461, 286 ], "score": 0.9, "content": "d _ { m } > 0", "type": "inline_equation" }, { "bbox": [ 461, 274, 507, 286 ], "score": 1.0, "content": "), Yin et al.", "type": "text" } ], "index": 17 }, { "bbox": [ 106, 285, 506, 299 ], "spans": [ { "bbox": [ 106, 285, 231, 299 ], "score": 1.0, "content": "[2021a] first proves the optimal", "type": "text" }, { "bbox": [ 231, 285, 287, 299 ], "score": 0.94, "content": "{ \\cal \\tilde { O } } ( H ^ { 3 } / d _ { m } \\epsilon ^ { 2 } )", "type": "inline_equation" }, { "bbox": [ 287, 285, 506, 299 ], "score": 1.0, "content": "complexity in the time-inhomogeneous MDP. Recently,", "type": "text" } ], "index": 18 }, { "bbox": [ 106, 298, 505, 312 ], "spans": [ { "bbox": [ 106, 298, 448, 312 ], "score": 1.0, "content": "Yin et al. [2021b] designs the offline variance reduction algorithm to achieve the optimal", "type": "text" }, { "bbox": [ 449, 298, 505, 311 ], "score": 0.94, "content": "\\tilde { O } ( H ^ { 2 } / d _ { m } \\epsilon ^ { 2 } )", "type": "inline_equation" } ], "index": 19 }, { "bbox": [ 105, 310, 505, 322 ], "spans": [ { "bbox": [ 105, 310, 505, 322 ], "score": 1.0, "content": "rate for the time-homogeneous case. Under the setting where the total cumulative reward is bounded", "type": "text" } ], "index": 20 }, { "bbox": [ 105, 321, 506, 335 ], "spans": [ { "bbox": [ 105, 322, 344, 335 ], "score": 1.0, "content": "by 1, Ren et al. [2021] obtains the horizon-free result with", "type": "text" }, { "bbox": [ 344, 321, 383, 334 ], "score": 0.92, "content": "\\tilde { O } ( 1 / d _ { m } )", "type": "inline_equation" }, { "bbox": [ 384, 322, 506, 335 ], "score": 1.0, "content": ". More recently, Rashidinejad", "type": "text" } ], "index": 21 }, { "bbox": [ 105, 334, 506, 348 ], "spans": [ { "bbox": [ 105, 334, 351, 348 ], "score": 1.0, "content": "et al. [2021] considers the single concentrability coefficient", "type": "text" }, { "bbox": [ 351, 334, 486, 347 ], "score": 0.92, "content": "\\begin{array} { r } { C ^ { \\star } : = \\operatorname* { m a x } _ { s , a } d ^ { \\pi ^ { \\star } } ( s , a ) / d ^ { \\mu } ( s , a ) } \\end{array}", "type": "inline_equation" }, { "bbox": [ 487, 334, 506, 348 ], "score": 1.0, "content": "and", "type": "text" } ], "index": 22 }, { "bbox": [ 105, 346, 505, 360 ], "spans": [ { "bbox": [ 105, 347, 212, 360 ], "score": 1.0, "content": "derives the upper bound", "type": "text" }, { "bbox": [ 212, 346, 302, 360 ], "score": 0.93, "content": "\\tilde { O } [ ( 1 - \\gamma ) ^ { - 5 } S C ^ { \\star } / \\epsilon ^ { 2 } ]", "type": "inline_equation" }, { "bbox": [ 303, 347, 505, 360 ], "score": 1.0, "content": "in the infinite horizon setting which is recently", "type": "text" } ], "index": 23 }, { "bbox": [ 106, 359, 506, 371 ], "spans": [ { "bbox": [ 106, 359, 506, 371 ], "score": 1.0, "content": "improved by the concurrent work Xie et al. [2021b]. While those worst-case guarantees are desirable,", "type": "text" } ], "index": 24 }, { "bbox": [ 105, 369, 374, 381 ], "spans": [ { "bbox": [ 105, 369, 374, 381 ], "score": 1.0, "content": "none of them can explain the hardness of the individual problems.1", "type": "text" } ], "index": 25 } ], "index": 20.5 }, { "type": "title", "bbox": [ 107, 400, 195, 414 ], "lines": [ { "bbox": [ 104, 399, 196, 416 ], "spans": [ { "bbox": [ 104, 399, 196, 416 ], "score": 1.0, "content": "2 Preliminaries", "type": "text" } ], "index": 26 } ], "index": 26 }, { "type": "text", "bbox": [ 106, 427, 506, 537 ], "lines": [ { "bbox": [ 106, 427, 507, 440 ], "spans": [ { "bbox": [ 106, 427, 507, 440 ], "score": 1.0, "content": "Episodic non-stationary (time-varying) reinforcement learning. A finite-horizon Markov Deci-", "type": "text" } ], "index": 27 }, { "bbox": [ 105, 438, 505, 450 ], "spans": [ { "bbox": [ 105, 438, 274, 450 ], "score": 1.0, "content": "sion Process (MDP) is denoted by a tuple", "type": "text" }, { "bbox": [ 275, 438, 373, 450 ], "score": 0.92, "content": "M = ( S , A , P , r , H , d _ { 1 } )", "type": "inline_equation" }, { "bbox": [ 374, 438, 505, 450 ], "score": 1.0, "content": "[Sutton and Barto, 2018], where", "type": "text" } ], "index": 28 }, { "bbox": [ 107, 448, 506, 461 ], "spans": [ { "bbox": [ 107, 450, 114, 459 ], "score": 0.79, "content": "s", "type": "inline_equation" }, { "bbox": [ 115, 448, 222, 461 ], "score": 1.0, "content": "is the finite state space and", "type": "text" }, { "bbox": [ 222, 450, 231, 459 ], "score": 0.8, "content": "\\mathcal { A }", "type": "inline_equation" }, { "bbox": [ 232, 448, 348, 461 ], "score": 1.0, "content": "is the finite action space with", "type": "text" }, { "bbox": [ 349, 450, 473, 461 ], "score": 0.9, "content": "S : = | S | < \\infty , A : = | A | < \\infty", "type": "inline_equation" }, { "bbox": [ 473, 448, 506, 461 ], "score": 1.0, "content": ". A non-", "type": "text" } ], "index": 29 }, { "bbox": [ 105, 460, 506, 474 ], "spans": [ { "bbox": [ 105, 460, 215, 474 ], "score": 1.0, "content": "stationary transition kernel", "type": "text" }, { "bbox": [ 215, 461, 317, 472 ], "score": 0.92, "content": "P _ { h } : S \\times \\mathcal { A } \\times \\mathcal { S } \\mapsto [ 0 , 1 ]", "type": "inline_equation" }, { "bbox": [ 317, 460, 408, 474 ], "score": 1.0, "content": "maps each state action", "type": "text" }, { "bbox": [ 408, 461, 441, 472 ], "score": 0.85, "content": "\\left( s _ { h } , a _ { h } \\right)", "type": "inline_equation" }, { "bbox": [ 441, 460, 506, 474 ], "score": 1.0, "content": "to a probability", "type": "text" } ], "index": 30 }, { "bbox": [ 105, 470, 507, 484 ], "spans": [ { "bbox": [ 105, 470, 154, 484 ], "score": 1.0, "content": "distribution", "type": "text" }, { "bbox": [ 155, 471, 205, 483 ], "score": 0.93, "content": "P _ { h } ( \\cdot | s _ { h } , a _ { h } )", "type": "inline_equation" }, { "bbox": [ 205, 470, 223, 484 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 223, 472, 236, 482 ], "score": 0.88, "content": "P _ { h }", "type": "inline_equation" }, { "bbox": [ 236, 470, 403, 484 ], "score": 1.0, "content": "can be different across the time. Besides,", "type": "text" }, { "bbox": [ 403, 472, 467, 482 ], "score": 0.91, "content": "r : S \\times A \\mapsto \\mathbb { R }", "type": "inline_equation" }, { "bbox": [ 467, 470, 507, 484 ], "score": 1.0, "content": "is the ex-", "type": "text" } ], "index": 31 }, { "bbox": [ 105, 482, 506, 494 ], "spans": [ { "bbox": [ 105, 482, 293, 494 ], "score": 1.0, "content": "pected instantaneous reward function satisfying", "type": "text" }, { "bbox": [ 294, 482, 336, 493 ], "score": 0.8, "content": "0 \\leq r \\leq 1", "type": "inline_equation" }, { "bbox": [ 336, 482, 340, 494 ], "score": 1.0, "content": ".", "type": "text" }, { "bbox": [ 340, 482, 351, 493 ], "score": 0.8, "content": "d _ { 1 }", "type": "inline_equation" }, { "bbox": [ 351, 482, 470, 494 ], "score": 1.0, "content": "is the initial state distribution.", "type": "text" }, { "bbox": [ 471, 483, 481, 492 ], "score": 0.79, "content": "H", "type": "inline_equation" }, { "bbox": [ 481, 482, 506, 494 ], "score": 1.0, "content": "is the", "type": "text" } ], "index": 32 }, { "bbox": [ 105, 493, 505, 506 ], "spans": [ { "bbox": [ 105, 493, 178, 506 ], "score": 1.0, "content": "horizon. 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The offline RL requires the agent to find a policy", "type": "text" }, { "bbox": [ 441, 653, 448, 660 ], "score": 0.76, "content": "\\pi", "type": "inline_equation" }, { "bbox": [ 448, 649, 506, 663 ], "score": 1.0, "content": "such that the", "type": "text" } ], "index": 45 }, { "bbox": [ 104, 658, 506, 680 ], "spans": [ { "bbox": [ 104, 658, 159, 680 ], "score": 1.0, "content": "performance", "type": "text" }, { "bbox": [ 160, 663, 171, 673 ], "score": 0.85, "content": "v ^ { \\pi }", "type": "inline_equation" }, { "bbox": [ 172, 658, 346, 680 ], "score": 1.0, "content": "is maximized, given only the episodic data", "type": "text" }, { "bbox": [ 347, 661, 462, 676 ], "score": 0.91, "content": "\\mathcal { D } = \\{ ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } , r _ { h } ^ { \\tau } , s _ { h + 1 } ^ { \\tau } ) \\} _ { \\tau \\in [ n ] } ^ { h \\in [ H ] }", "type": "inline_equation" }, { "bbox": [ 462, 658, 506, 680 ], "score": 1.0, "content": "rolled out", "type": "text" } ], "index": 46 }, { "bbox": [ 106, 675, 505, 687 ], "spans": [ { "bbox": [ 106, 675, 217, 687 ], "score": 1.0, "content": "from some behavior policy", "type": "text" }, { "bbox": [ 218, 677, 225, 686 ], "score": 0.78, "content": "\\mu", "type": "inline_equation" }, { "bbox": [ 225, 675, 413, 687 ], "score": 1.0, "content": ". The offline nature requires we cannot change", "type": "text" }, { "bbox": [ 414, 677, 421, 686 ], "score": 0.81, "content": "\\mu", "type": "inline_equation" }, { "bbox": [ 421, 675, 505, 687 ], "score": 1.0, "content": "and in particular we", "type": "text" } ], "index": 47 } ], "index": 46 } ], "page_idx": 2, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 108, 701, 505, 722 ], "lines": [ { "bbox": [ 119, 699, 505, 713 ], "spans": [ { "bbox": [ 119, 699, 505, 713 ], "score": 1.0, "content": "1We do mention Zanette et al. [2021] is near-optimal in their setting, but it is unclear whether it remains", "type": "text" } ] }, { "bbox": [ 105, 710, 472, 724 ], "spans": [ { "bbox": [ 105, 710, 241, 724 ], "score": 1.0, "content": "optimal in the standard setting where", "type": "text" }, { "bbox": [ 242, 712, 288, 722 ], "score": 0.92, "content": "Q ^ { \\pi } \\in [ 0 , H ]", "type": "inline_equation" }, { "bbox": [ 289, 710, 390, 724 ], "score": 1.0, "content": ", since there is an additional", "type": "text" }, { "bbox": [ 390, 713, 399, 720 ], "score": 0.82, "content": "H", "type": "inline_equation" }, { "bbox": [ 399, 710, 472, 724 ], "score": 1.0, "content": "factor by rescaling.", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 302, 741, 309, 750 ], "lines": [ { "bbox": [ 301, 740, 310, 752 ], "spans": [ { "bbox": [ 301, 740, 310, 752 ], "score": 1.0, "content": "3", "type": "text" } ] } ] } ], "para_blocks": [ { "type": "title", "bbox": [ 107, 72, 187, 84 ], "lines": [ { "bbox": [ 105, 71, 189, 86 ], "spans": [ { "bbox": [ 105, 71, 189, 86 ], "score": 1.0, "content": "1.2 Related work", "type": "text" } ], "index": 0 } ], "index": 0 }, { "type": "text", "bbox": [ 107, 94, 506, 258 ], "lines": [ { "bbox": [ 106, 93, 505, 106 ], "spans": [ { "bbox": [ 106, 93, 505, 106 ], "score": 1.0, "content": "Finite sample analysis for offline reinforcement learning can be traced back to Szepesvári and Munos", "type": "text" } ], "index": 1 }, { "bbox": [ 106, 104, 505, 117 ], "spans": [ { "bbox": [ 106, 104, 505, 117 ], "score": 1.0, "content": "[2005], Antos et al. [2008a,b] for the infinite horizon discounted setting via Fitted Q-Iteration (FQI)", "type": "text" } ], "index": 2 }, { "bbox": [ 105, 114, 506, 130 ], "spans": [ { "bbox": [ 105, 114, 506, 130 ], "score": 1.0, "content": "type function approximation algorithms. [Chen and Jiang, 2019, Le et al., 2019, Xie and Jiang,", "type": "text" } ], "index": 3 }, { "bbox": [ 106, 127, 505, 139 ], "spans": [ { "bbox": [ 106, 127, 505, 139 ], "score": 1.0, "content": "2021, 2020] follow this line of research and derive the information-theoretical bounds. Recently, Xie", "type": "text" } ], "index": 4 }, { "bbox": [ 106, 137, 505, 150 ], "spans": [ { "bbox": [ 106, 137, 505, 150 ], "score": 1.0, "content": "and Jiang [2021] considers the offline RL with only the realizability assumption, Liu et al. [2020],", "type": "text" } ], "index": 5 }, { "bbox": [ 106, 148, 505, 161 ], "spans": [ { "bbox": [ 106, 148, 505, 161 ], "score": 1.0, "content": "Chang et al. [2021] considers the offline RL without sufficient coverage and Kidambi et al. [2020],", "type": "text" } ], "index": 6 }, { "bbox": [ 105, 158, 506, 171 ], "spans": [ { "bbox": [ 105, 158, 506, 171 ], "score": 1.0, "content": "Uehara and Sun [2021] uses the model-based approach for addressing offline RL. Under those weak", "type": "text" } ], "index": 7 }, { "bbox": [ 106, 171, 505, 182 ], "spans": [ { "bbox": [ 106, 171, 505, 182 ], "score": 1.0, "content": "coverage assumption, their finite sample analysis are suboptimal (e.g. in terms of the effective", "type": "text" } ], "index": 8 }, { "bbox": [ 104, 180, 506, 194 ], "spans": [ { "bbox": [ 104, 180, 139, 194 ], "score": 1.0, "content": "horizon", "type": "text" }, { "bbox": [ 140, 181, 182, 193 ], "score": 0.9, "content": "( 1 - \\gamma ) ^ { \\bar { - } 1 } .", "type": "inline_equation" }, { "bbox": [ 182, 180, 506, 194 ], "score": 1.0, "content": "). Recently, Yin et al. [2021a,b], Ren et al. [2021] study the finite horizon case. In", "type": "text" } ], "index": 9 }, { "bbox": [ 105, 191, 505, 205 ], "spans": [ { "bbox": [ 105, 191, 505, 205 ], "score": 1.0, "content": "the linear MDP case, Jin et al. [2020] studies the pessimistic algorithm for offline policy learning", "type": "text" } ], "index": 10 }, { "bbox": [ 106, 203, 505, 215 ], "spans": [ { "bbox": [ 106, 203, 505, 215 ], "score": 1.0, "content": "under only the compliance assumption, and, concurrently, Xie et al. [2021a] proposes the general", "type": "text" } ], "index": 11 }, { "bbox": [ 105, 214, 506, 226 ], "spans": [ { "bbox": [ 105, 214, 506, 226 ], "score": 1.0, "content": "pessimistic function approximation framework with instantiation in linear MDP and Zanette et al.", "type": "text" } ], "index": 12 }, { "bbox": [ 106, 224, 505, 237 ], "spans": [ { "bbox": [ 106, 224, 505, 237 ], "score": 1.0, "content": "[2021] shows actor-critic style algorithm is near-optimal for linear Bellman complete model. In", "type": "text" } ], "index": 13 }, { "bbox": [ 105, 235, 506, 249 ], "spans": [ { "bbox": [ 105, 235, 506, 249 ], "score": 1.0, "content": "addition, Wang et al. [2021], Zanette [2021] prove some exponential lower bounds under their linear", "type": "text" } ], "index": 14 }, { "bbox": [ 105, 246, 257, 259 ], "spans": [ { "bbox": [ 105, 246, 257, 259 ], "score": 1.0, "content": "function approximation assumptions.", "type": "text" } ], "index": 15 } ], "index": 8, "bbox_fs": [ 104, 93, 506, 259 ] }, { "type": "text", "bbox": [ 106, 263, 506, 381 ], "lines": [ { "bbox": [ 106, 263, 505, 275 ], "spans": [ { "bbox": [ 106, 263, 505, 275 ], "score": 1.0, "content": "Among them, there are a few works that achieve the sample optimality under their respective", "type": "text" } ], "index": 16 }, { "bbox": [ 104, 274, 507, 286 ], "spans": [ { "bbox": [ 104, 274, 428, 286 ], "score": 1.0, "content": "assumptions. Under the uniform data coverage (minimal state-action probability", "type": "text" }, { "bbox": [ 429, 275, 461, 286 ], "score": 0.9, "content": "d _ { m } > 0", "type": "inline_equation" }, { "bbox": [ 461, 274, 507, 286 ], "score": 1.0, "content": "), Yin et al.", "type": "text" } ], "index": 17 }, { "bbox": [ 106, 285, 506, 299 ], "spans": [ { "bbox": [ 106, 285, 231, 299 ], "score": 1.0, "content": "[2021a] first proves the optimal", "type": "text" }, { "bbox": [ 231, 285, 287, 299 ], "score": 0.94, "content": "{ \\cal \\tilde { O } } ( H ^ { 3 } / d _ { m } \\epsilon ^ { 2 } )", "type": "inline_equation" }, { "bbox": [ 287, 285, 506, 299 ], "score": 1.0, "content": "complexity in the time-inhomogeneous MDP. Recently,", "type": "text" } ], "index": 18 }, { "bbox": [ 106, 298, 505, 312 ], "spans": [ { "bbox": [ 106, 298, 448, 312 ], "score": 1.0, "content": "Yin et al. [2021b] designs the offline variance reduction algorithm to achieve the optimal", "type": "text" }, { "bbox": [ 449, 298, 505, 311 ], "score": 0.94, "content": "\\tilde { O } ( H ^ { 2 } / d _ { m } \\epsilon ^ { 2 } )", "type": "inline_equation" } ], "index": 19 }, { "bbox": [ 105, 310, 505, 322 ], "spans": [ { "bbox": [ 105, 310, 505, 322 ], "score": 1.0, "content": "rate for the time-homogeneous case. Under the setting where the total cumulative reward is bounded", "type": "text" } ], "index": 20 }, { "bbox": [ 105, 321, 506, 335 ], "spans": [ { "bbox": [ 105, 322, 344, 335 ], "score": 1.0, "content": "by 1, Ren et al. [2021] obtains the horizon-free result with", "type": "text" }, { "bbox": [ 344, 321, 383, 334 ], "score": 0.92, "content": "\\tilde { O } ( 1 / d _ { m } )", "type": "inline_equation" }, { "bbox": [ 384, 322, 506, 335 ], "score": 1.0, "content": ". More recently, Rashidinejad", "type": "text" } ], "index": 21 }, { "bbox": [ 105, 334, 506, 348 ], "spans": [ { "bbox": [ 105, 334, 351, 348 ], "score": 1.0, "content": "et al. [2021] considers the single concentrability coefficient", "type": "text" }, { "bbox": [ 351, 334, 486, 347 ], "score": 0.92, "content": "\\begin{array} { r } { C ^ { \\star } : = \\operatorname* { m a x } _ { s , a } d ^ { \\pi ^ { \\star } } ( s , a ) / d ^ { \\mu } ( s , a ) } \\end{array}", "type": "inline_equation" }, { "bbox": [ 487, 334, 506, 348 ], "score": 1.0, "content": "and", "type": "text" } ], "index": 22 }, { "bbox": [ 105, 346, 505, 360 ], "spans": [ { "bbox": [ 105, 347, 212, 360 ], "score": 1.0, "content": "derives the upper bound", "type": "text" }, { "bbox": [ 212, 346, 302, 360 ], "score": 0.93, "content": "\\tilde { O } [ ( 1 - \\gamma ) ^ { - 5 } S C ^ { \\star } / \\epsilon ^ { 2 } ]", "type": "inline_equation" }, { "bbox": [ 303, 347, 505, 360 ], "score": 1.0, "content": "in the infinite horizon setting which is recently", "type": "text" } ], "index": 23 }, { "bbox": [ 106, 359, 506, 371 ], "spans": [ { "bbox": [ 106, 359, 506, 371 ], "score": 1.0, "content": "improved by the concurrent work Xie et al. [2021b]. While those worst-case guarantees are desirable,", "type": "text" } ], "index": 24 }, { "bbox": [ 105, 369, 374, 381 ], "spans": [ { "bbox": [ 105, 369, 374, 381 ], "score": 1.0, "content": "none of them can explain the hardness of the individual problems.1", "type": "text" } ], "index": 25 } ], "index": 20.5, "bbox_fs": [ 104, 263, 507, 381 ] }, { "type": "title", "bbox": [ 107, 400, 195, 414 ], "lines": [ { "bbox": [ 104, 399, 196, 416 ], "spans": [ { "bbox": [ 104, 399, 196, 416 ], "score": 1.0, "content": "2 Preliminaries", "type": "text" } ], "index": 26 } ], "index": 26 }, { "type": "text", "bbox": [ 106, 427, 506, 537 ], "lines": [ { "bbox": [ 106, 427, 507, 440 ], "spans": [ { "bbox": [ 106, 427, 507, 440 ], "score": 1.0, "content": "Episodic non-stationary (time-varying) reinforcement learning. A finite-horizon Markov Deci-", "type": "text" } ], "index": 27 }, { "bbox": [ 105, 438, 505, 450 ], "spans": [ { "bbox": [ 105, 438, 274, 450 ], "score": 1.0, "content": "sion Process (MDP) is denoted by a tuple", "type": "text" }, { "bbox": [ 275, 438, 373, 450 ], "score": 0.92, "content": "M = ( S , A , P , r , H , d _ { 1 } )", "type": "inline_equation" }, { "bbox": [ 374, 438, 505, 450 ], "score": 1.0, "content": "[Sutton and Barto, 2018], where", "type": "text" } ], "index": 28 }, { "bbox": [ 107, 448, 506, 461 ], "spans": [ { "bbox": [ 107, 450, 114, 459 ], "score": 0.79, "content": "s", "type": "inline_equation" }, { "bbox": [ 115, 448, 222, 461 ], "score": 1.0, "content": "is the finite state space and", "type": "text" }, { "bbox": [ 222, 450, 231, 459 ], "score": 0.8, "content": "\\mathcal { A }", "type": "inline_equation" }, { "bbox": [ 232, 448, 348, 461 ], "score": 1.0, "content": "is the finite action space with", "type": "text" }, { "bbox": [ 349, 450, 473, 461 ], "score": 0.9, "content": "S : = | S | < \\infty , A : = | A | < \\infty", "type": "inline_equation" }, { "bbox": [ 473, 448, 506, 461 ], "score": 1.0, "content": ". A non-", "type": "text" } ], "index": 29 }, { "bbox": [ 105, 460, 506, 474 ], "spans": [ { "bbox": [ 105, 460, 215, 474 ], "score": 1.0, "content": "stationary transition kernel", "type": "text" }, { "bbox": [ 215, 461, 317, 472 ], "score": 0.92, "content": "P _ { h } : S \\times \\mathcal { A } \\times \\mathcal { S } \\mapsto [ 0 , 1 ]", "type": "inline_equation" }, { "bbox": [ 317, 460, 408, 474 ], "score": 1.0, "content": "maps each state action", "type": "text" }, { "bbox": [ 408, 461, 441, 472 ], "score": 0.85, "content": "\\left( s _ { h } , a _ { h } \\right)", "type": "inline_equation" }, { "bbox": [ 441, 460, 506, 474 ], "score": 1.0, "content": "to a probability", "type": "text" } ], "index": 30 }, { "bbox": [ 105, 470, 507, 484 ], "spans": [ { "bbox": [ 105, 470, 154, 484 ], "score": 1.0, "content": "distribution", "type": "text" }, { "bbox": [ 155, 471, 205, 483 ], "score": 0.93, "content": "P _ { h } ( \\cdot | s _ { h } , a _ { h } )", "type": "inline_equation" }, { "bbox": [ 205, 470, 223, 484 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 223, 472, 236, 482 ], "score": 0.88, "content": "P _ { h }", "type": "inline_equation" }, { "bbox": [ 236, 470, 403, 484 ], "score": 1.0, "content": "can be different across the time. 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The offline RL requires the agent to find a policy", "type": "text" }, { "bbox": [ 441, 653, 448, 660 ], "score": 0.76, "content": "\\pi", "type": "inline_equation" }, { "bbox": [ 448, 649, 506, 663 ], "score": 1.0, "content": "such that the", "type": "text" } ], "index": 45 }, { "bbox": [ 104, 658, 506, 680 ], "spans": [ { "bbox": [ 104, 658, 159, 680 ], "score": 1.0, "content": "performance", "type": "text" }, { "bbox": [ 160, 663, 171, 673 ], "score": 0.85, "content": "v ^ { \\pi }", "type": "inline_equation" }, { "bbox": [ 172, 658, 346, 680 ], "score": 1.0, "content": "is maximized, given only the episodic data", "type": "text" }, { "bbox": [ 347, 661, 462, 676 ], "score": 0.91, "content": "\\mathcal { D } = \\{ ( s _ { h } ^ { \\tau } , a _ { h } ^ { \\tau } , r _ { h } ^ { \\tau } , s _ { h + 1 } ^ { \\tau } ) \\} _ { \\tau \\in [ n ] } ^ { h \\in [ H ] }", "type": "inline_equation" }, { "bbox": [ 462, 658, 506, 680 ], "score": 1.0, "content": "rolled out", "type": "text" } ], "index": 46 }, { "bbox": [ 106, 675, 505, 687 ], "spans": [ { "bbox": [ 106, 675, 217, 687 ], "score": 1.0, "content": "from some behavior policy", "type": "text" }, { "bbox": [ 218, 677, 225, 686 ], "score": 0.78, "content": "\\mu", "type": "inline_equation" }, { "bbox": [ 225, 675, 413, 687 ], "score": 1.0, "content": ". The offline nature requires we cannot change", "type": "text" }, { "bbox": [ 414, 677, 421, 686 ], "score": 0.81, "content": "\\mu", "type": "inline_equation" }, { "bbox": [ 421, 675, 505, 687 ], "score": 1.0, "content": "and in particular we", "type": "text" } ], "index": 47 }, { "bbox": [ 106, 72, 505, 85 ], "spans": [ { "bbox": [ 106, 72, 282, 85 ], "score": 1.0, "content": "do not assume the functional knowledge of", "type": "text", "cross_page": true }, { "bbox": [ 282, 75, 290, 84 ], "score": 0.79, "content": "\\mu", "type": "inline_equation", "cross_page": true }, { "bbox": [ 290, 72, 435, 85 ], "score": 1.0, "content": ". That is to say, given the batch data", "type": "text", "cross_page": true }, { "bbox": [ 436, 73, 445, 82 ], "score": 0.83, "content": "\\mathcal { D }", "type": "inline_equation", "cross_page": true }, { "bbox": [ 445, 72, 505, 85 ], "score": 1.0, "content": "and a targeted", "type": "text", "cross_page": true } ], "index": 0 }, { "bbox": [ 105, 83, 434, 97 ], "spans": [ { "bbox": [ 105, 83, 144, 97 ], "score": 1.0, "content": "accuracy", "type": "text", "cross_page": true }, { "bbox": [ 144, 84, 168, 94 ], "score": 0.89, "content": "\\epsilon > 0", "type": "inline_equation", "cross_page": true }, { "bbox": [ 168, 83, 316, 97 ], "score": 1.0, "content": ", the offline RL seeks to find a policy", "type": "text", "cross_page": true }, { "bbox": [ 316, 85, 332, 96 ], "score": 0.87, "content": "\\pi _ { \\mathrm { a l g } }", "type": "inline_equation", "cross_page": true }, { "bbox": [ 333, 83, 371, 97 ], "score": 1.0, "content": "such that", "type": "text", "cross_page": true }, { "bbox": [ 372, 84, 429, 95 ], "score": 0.91, "content": "v ^ { \\star } - v ^ { \\pi _ { \\mathrm { a l g } } } \\leq \\epsilon", "type": "inline_equation", "cross_page": true }, { "bbox": [ 429, 83, 434, 97 ], "score": 1.0, "content": ".", "type": "text", "cross_page": true } ], "index": 1 } ], "index": 46, "bbox_fs": [ 104, 649, 506, 687 ] } ] }, { "preproc_blocks": [ { "type": "text", "bbox": [ 105, 72, 505, 96 ], "lines": [ { "bbox": [ 106, 72, 505, 85 ], "spans": [ { "bbox": [ 106, 72, 282, 85 ], "score": 1.0, "content": "do not assume the functional knowledge of", "type": "text" }, { "bbox": [ 282, 75, 290, 84 ], "score": 0.79, "content": "\\mu", "type": "inline_equation" }, { "bbox": [ 290, 72, 435, 85 ], "score": 1.0, "content": ". That is to say, given the batch data", "type": "text" }, { "bbox": [ 436, 73, 445, 82 ], "score": 0.83, "content": "\\mathcal { D }", "type": "inline_equation" }, { "bbox": [ 445, 72, 505, 85 ], "score": 1.0, "content": "and a targeted", "type": "text" } ], "index": 0 }, { "bbox": [ 105, 83, 434, 97 ], "spans": [ { "bbox": [ 105, 83, 144, 97 ], "score": 1.0, "content": "accuracy", "type": "text" }, { "bbox": [ 144, 84, 168, 94 ], "score": 0.89, "content": "\\epsilon > 0", "type": "inline_equation" }, { "bbox": [ 168, 83, 316, 97 ], "score": 1.0, "content": ", the offline RL seeks to find a policy", "type": "text" }, { "bbox": [ 316, 85, 332, 96 ], "score": 0.87, "content": "\\pi _ { \\mathrm { a l g } }", "type": "inline_equation" }, { "bbox": [ 333, 83, 371, 97 ], "score": 1.0, "content": "such that", "type": "text" }, { "bbox": [ 372, 84, 429, 95 ], "score": 0.91, "content": "v ^ { \\star } - v ^ { \\pi _ { \\mathrm { a l g } } } \\leq \\epsilon", "type": "inline_equation" }, { "bbox": [ 429, 83, 434, 97 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 1 } ], "index": 0.5 }, { "type": "title", "bbox": [ 107, 109, 241, 120 ], "lines": [ { "bbox": [ 105, 108, 243, 122 ], "spans": [ { "bbox": [ 105, 108, 243, 122 ], "score": 1.0, "content": "2.1 Assumptions in offline RL", "type": "text" } ], "index": 2 } ], "index": 2 }, { "type": "text", "bbox": [ 107, 129, 504, 152 ], "lines": [ { "bbox": [ 106, 129, 505, 142 ], "spans": [ { "bbox": [ 106, 129, 505, 142 ], "score": 1.0, "content": "We revise several types of assumptions proposed by existing studies that can yield provably efficient", "type": "text" } ], "index": 3 }, { "bbox": [ 105, 140, 483, 154 ], "spans": [ { "bbox": [ 105, 140, 166, 154 ], "score": 1.0, "content": "results. Recall", "type": "text" }, { "bbox": [ 166, 140, 210, 153 ], "score": 0.93, "content": "d _ { h } ^ { \\mu } ( s _ { h } , a _ { h } )", "type": "inline_equation" }, { "bbox": [ 210, 140, 384, 154 ], "score": 1.0, "content": "is the marginal state-action probability and", "type": "text" }, { "bbox": [ 384, 142, 392, 152 ], "score": 0.82, "content": "\\mu", "type": "inline_equation" }, { "bbox": [ 392, 140, 483, 154 ], "score": 1.0, "content": "is the behavior policy.", "type": "text" } ], "index": 4 } ], "index": 3.5 }, { "type": "text", "bbox": [ 107, 155, 505, 189 ], "lines": [ { "bbox": [ 106, 154, 505, 169 ], "spans": [ { "bbox": [ 106, 154, 476, 169 ], "score": 1.0, "content": "Assumption 2.1 (Uniform data coverage [Yin et al., 2021a]). The behavior policy obeys that", "type": "text" }, { "bbox": [ 477, 156, 505, 167 ], "score": 0.87, "content": "d _ { m } : =", "type": "inline_equation" } ], "index": 5 }, { "bbox": [ 106, 165, 506, 182 ], "spans": [ { "bbox": [ 106, 167, 217, 179 ], "score": 0.9, "content": "\\begin{array} { r } { \\operatorname* { m i n } _ { h , s _ { h } , a _ { h } } d _ { h } ^ { \\mu } ( s _ { h } , a _ { h } ) > 0 . } \\end{array}", "type": "inline_equation" }, { "bbox": [ 217, 165, 506, 182 ], "score": 1.0, "content": ". Here the infimum is over all the states satisfying there exists certain", "type": "text" } ], "index": 6 }, { "bbox": [ 105, 177, 411, 191 ], "spans": [ { "bbox": [ 105, 177, 411, 191 ], "score": 1.0, "content": "policy so that this state can be reached by the current MDP with this policy.", "type": "text" } ], "index": 7 } ], "index": 6 }, { "type": "text", "bbox": [ 106, 198, 505, 265 ], "lines": [ { "bbox": [ 106, 199, 505, 210 ], "spans": [ { "bbox": [ 106, 199, 356, 210 ], "score": 1.0, "content": "This is the strongest assumption in offline RL as it requires", "type": "text" }, { "bbox": [ 357, 201, 364, 210 ], "score": 0.81, "content": "\\mu", "type": "inline_equation" }, { "bbox": [ 365, 199, 505, 210 ], "score": 1.0, "content": "to explore each state-action pairs", "type": "text" } ], "index": 8 }, { "bbox": [ 105, 208, 506, 222 ], "spans": [ { "bbox": [ 105, 208, 321, 222 ], "score": 1.0, "content": "with positive probability. Under 2.1, it mostly holds", "type": "text" }, { "bbox": [ 321, 209, 374, 221 ], "score": 0.93, "content": "1 / d _ { m } \\ge S A", "type": "inline_equation" }, { "bbox": [ 374, 208, 506, 222 ], "score": 1.0, "content": ". This reveals offline learning is", "type": "text" } ], "index": 9 }, { "bbox": [ 105, 219, 507, 234 ], "spans": [ { "bbox": [ 105, 219, 507, 234 ], "score": 1.0, "content": "generically harder than the generative model setting [Agarwal et al., 2020] in the statistical sense.", "type": "text" } ], "index": 10 }, { "bbox": [ 106, 231, 505, 243 ], "spans": [ { "bbox": [ 106, 231, 318, 243 ], "score": 1.0, "content": "On the other hand, this is required for the uniform", "type": "text" }, { "bbox": [ 319, 232, 340, 241 ], "score": 0.54, "content": "O P E", "type": "inline_equation" }, { "bbox": [ 341, 231, 505, 243 ], "score": 1.0, "content": "task in Yin et al. [2021a] as it seeks to", "type": "text" } ], "index": 11 }, { "bbox": [ 105, 242, 506, 255 ], "spans": [ { "bbox": [ 105, 242, 506, 255 ], "score": 1.0, "content": "simultaneously evaluate all the policies within the policy class and it is in general a harder task than", "type": "text" } ], "index": 12 }, { "bbox": [ 105, 253, 195, 266 ], "spans": [ { "bbox": [ 105, 253, 195, 266 ], "score": 1.0, "content": "offline learning itself.", "type": "text" } ], "index": 13 } ], "index": 10.5 }, { "type": "text", "bbox": [ 106, 268, 504, 293 ], "lines": [ { "bbox": [ 106, 267, 505, 281 ], "spans": [ { "bbox": [ 106, 267, 505, 281 ], "score": 1.0, "content": "Assumption 2.2 (Uniform concentrability Szepesvári and Munos [2005], Chen and Jiang [2019]).", "type": "text" } ], "index": 14 }, { "bbox": [ 105, 279, 353, 293 ], "spans": [ { "bbox": [ 105, 279, 186, 293 ], "score": 1.0, "content": "For all the policies,", "type": "text" }, { "bbox": [ 186, 279, 349, 293 ], "score": 0.92, "content": "\\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}", "type": "inline_equation" }, { "bbox": [ 350, 279, 353, 293 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 15 } ], "index": 14.5 }, { "type": "text", "bbox": [ 106, 299, 505, 344 ], "lines": [ { "bbox": [ 105, 299, 505, 313 ], "spans": [ { "bbox": [ 105, 299, 505, 313 ], "score": 1.0, "content": "This is a classical offline RL condition that is commonly assumed in the function approximation", "type": "text" } ], "index": 16 }, { "bbox": [ 106, 311, 505, 323 ], "spans": [ { "bbox": [ 106, 311, 505, 323 ], "score": 1.0, "content": "scheme (e.g. Fitted Q-Iteration). Qualitatively, this is a uniform data-coverage assumption that is", "type": "text" } ], "index": 17 }, { "bbox": [ 105, 322, 504, 335 ], "spans": [ { "bbox": [ 105, 322, 343, 335 ], "score": 1.0, "content": "similar to Assumption 2.1, but quantitatively, the coefficient", "type": "text" }, { "bbox": [ 343, 322, 357, 334 ], "score": 0.9, "content": "C _ { \\mu }", "type": "inline_equation" }, { "bbox": [ 357, 322, 435, 335 ], "score": 1.0, "content": "can be smaller than", "type": "text" }, { "bbox": [ 436, 322, 460, 334 ], "score": 0.91, "content": "1 / \\bar { d _ { m } } \\bar { }", "type": "inline_equation" }, { "bbox": [ 460, 322, 492, 335 ], "score": 1.0, "content": "due the", "type": "text" }, { "bbox": [ 492, 322, 504, 334 ], "score": 0.87, "content": "d _ { h } ^ { \\pi }", "type": "inline_equation" } ], "index": 18 }, { "bbox": [ 106, 333, 197, 344 ], "spans": [ { "bbox": [ 106, 333, 197, 344 ], "score": 1.0, "content": "term in the numerator.", "type": "text" } ], "index": 19 } ], "index": 17.5 }, { "type": "text", "bbox": [ 107, 348, 506, 383 ], "lines": [ { "bbox": [ 105, 346, 506, 361 ], "spans": [ { "bbox": [ 105, 346, 398, 361 ], "score": 1.0, "content": "Assumption 2.3 (Liu et al. [2019]). There exists one optimal policy", "type": "text" }, { "bbox": [ 398, 349, 410, 358 ], "score": 0.82, "content": "\\pi ^ { \\star }", "type": "inline_equation" }, { "bbox": [ 410, 346, 434, 361 ], "score": 1.0, "content": ", s.t.", "type": "text" }, { "bbox": [ 434, 348, 504, 360 ], "score": 0.88, "content": "\\forall s _ { h } , a _ { h } \\ \\in \\ S , A ,", "type": "inline_equation" }, { "bbox": [ 504, 346, 506, 361 ], "score": 1.0, "content": ",", "type": "text" } ], "index": 20 }, { "bbox": [ 107, 358, 506, 374 ], "spans": [ { "bbox": [ 107, 360, 174, 371 ], "score": 0.81, "content": "d _ { h } ^ { \\mu } ( s _ { h } , a _ { h } ) \\ > \\ 0", "type": "inline_equation" }, { "bbox": [ 174, 358, 185, 374 ], "score": 1.0, "content": "if", "type": "text" }, { "bbox": [ 185, 359, 256, 372 ], "score": 0.88, "content": "d _ { h } ^ { \\pi ^ { \\star } } ( s _ { h } , a _ { h } ) > 0", "type": "inline_equation" }, { "bbox": [ 256, 358, 428, 374 ], "score": 1.0, "content": ". We further denote the trackable set as", "type": "text" }, { "bbox": [ 428, 360, 502, 372 ], "score": 0.89, "content": "\\mathcal C _ { h } : = \\{ ( s _ { h } , a _ { h } )", "type": "inline_equation" }, { "bbox": [ 503, 358, 506, 374 ], "score": 1.0, "content": ":", "type": "text" } ], "index": 21 }, { "bbox": [ 107, 370, 178, 385 ], "spans": [ { "bbox": [ 107, 371, 173, 384 ], "score": 0.88, "content": "d _ { h } ^ { \\tilde { \\mu } } ( s _ { h } , a _ { h } ) > 0 \\}", "type": "inline_equation" }, { "bbox": [ 174, 370, 178, 385 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 22 } ], "index": 21 }, { "type": "text", "bbox": [ 106, 391, 506, 483 ], "lines": [ { "bbox": [ 106, 391, 505, 403 ], "spans": [ { "bbox": [ 106, 391, 505, 403 ], "score": 1.0, "content": "Assumption 2.3 is (arguably) the weakest assumption needed for accurately learning the optimal", "type": "text" } ], "index": 23 }, { "bbox": [ 105, 402, 506, 415 ], "spans": [ { "bbox": [ 105, 402, 130, 415 ], "score": 1.0, "content": "value", "type": "text" }, { "bbox": [ 130, 403, 141, 413 ], "score": 0.86, "content": "v ^ { \\star }", "type": "inline_equation" }, { "bbox": [ 142, 402, 400, 415 ], "score": 1.0, "content": "and we will use 2.3 for most parts of this paper. It only requires", "type": "text" }, { "bbox": [ 400, 405, 408, 414 ], "score": 0.81, "content": "\\mu", "type": "inline_equation" }, { "bbox": [ 408, 402, 506, 415 ], "score": 1.0, "content": "to trace the state-action", "type": "text" } ], "index": 24 }, { "bbox": [ 104, 414, 506, 426 ], "spans": [ { "bbox": [ 104, 414, 506, 426 ], "score": 1.0, "content": "space of one optimal policy and can be agnostic at other locations. Rashidinejad et al. [2021], Xie", "type": "text" } ], "index": 25 }, { "bbox": [ 105, 424, 505, 437 ], "spans": [ { "bbox": [ 105, 424, 505, 437 ], "score": 1.0, "content": "et al. [2021b] considers this assumption and provide analysis is based on the single concentrability", "type": "text" } ], "index": 26 }, { "bbox": [ 106, 435, 505, 449 ], "spans": [ { "bbox": [ 106, 437, 150, 449 ], "score": 1.0, "content": "coefficient", "type": "text" }, { "bbox": [ 150, 435, 285, 449 ], "score": 0.92, "content": "\\begin{array} { r } { C ^ { \\star } : = \\operatorname* { m a x } _ { s , a } { d ^ { \\pi ^ { \\star } } ( s , a ) } / { d ^ { \\mu } ( s , a ) } } \\end{array}", "type": "inline_equation" }, { "bbox": [ 286, 437, 367, 449 ], "score": 1.0, "content": ". The dependence on", "type": "text" }, { "bbox": [ 367, 437, 380, 447 ], "score": 0.87, "content": "C ^ { \\star }", "type": "inline_equation" }, { "bbox": [ 381, 437, 505, 449 ], "score": 1.0, "content": "makes their result less adaptive", "type": "text" } ], "index": 27 }, { "bbox": [ 105, 448, 505, 462 ], "spans": [ { "bbox": [ 105, 448, 332, 461 ], "score": 1.0, "content": "since there can be lots of locations that have the ratio", "type": "text" }, { "bbox": [ 332, 448, 408, 462 ], "score": 0.92, "content": "{ d ^ { \\pi } } ^ { \\star } ( s , a ) / { d ^ { \\mu } } ( s , a )", "type": "inline_equation" }, { "bbox": [ 408, 448, 489, 461 ], "score": 1.0, "content": "much smaller than", "type": "text" }, { "bbox": [ 489, 450, 502, 460 ], "score": 0.87, "content": "C ^ { \\star }", "type": "inline_equation" }, { "bbox": [ 503, 448, 505, 461 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 28 }, { "bbox": [ 105, 460, 505, 473 ], "spans": [ { "bbox": [ 105, 460, 505, 473 ], "score": 1.0, "content": "Furthermore, what could we end up with when 2.3 is not met? We will provide our answers in the", "type": "text" } ], "index": 29 }, { "bbox": [ 106, 471, 191, 484 ], "spans": [ { "bbox": [ 106, 471, 191, 484 ], "score": 1.0, "content": "subsequent sections.", "type": "text" } ], "index": 30 } ], "index": 26.5 }, { "type": "title", "bbox": [ 107, 498, 423, 513 ], "lines": [ { "bbox": [ 104, 498, 423, 515 ], "spans": [ { "bbox": [ 104, 498, 423, 515 ], "score": 1.0, "content": "3 A warm-up case study: Vanilla Pessimistic Value Iteration", "type": "text" } ], "index": 31 } ], "index": 31 }, { "type": "text", "bbox": [ 107, 524, 505, 568 ], "lines": [ { "bbox": [ 105, 524, 505, 537 ], "spans": [ { "bbox": [ 105, 524, 505, 537 ], "score": 1.0, "content": "As a step towards the optimal and strong adaptive offline RL bound, we analyze the vanilla pessimistic", "type": "text" } ], "index": 32 }, { "bbox": [ 106, 535, 506, 547 ], "spans": [ { "bbox": [ 106, 535, 506, 547 ], "score": 1.0, "content": "value iteration (VPVI), a tabular counterpart of pessimistic value iteration (PEVI initiated in Jin et al.", "type": "text" } ], "index": 33 }, { "bbox": [ 106, 546, 505, 559 ], "spans": [ { "bbox": [ 106, 546, 505, 559 ], "score": 1.0, "content": "[2020]), to understand what is missing for achieving the fully adaptivity. In particular, VPVI relies on", "type": "text" } ], "index": 34 }, { "bbox": [ 106, 558, 229, 569 ], "spans": [ { "bbox": [ 106, 558, 229, 569 ], "score": 1.0, "content": "the model-based construction.", "type": "text" } ], "index": 35 } ], "index": 33.5 }, { "type": "text", "bbox": [ 107, 572, 505, 608 ], "lines": [ { "bbox": [ 100, 564, 505, 594 ], "spans": [ { "bbox": [ 100, 564, 281, 594 ], "score": 1.0, "content": "Model-based Components. Given data n", "type": "text" }, { "bbox": [ 281, 572, 405, 586 ], "score": 0.92, "content": "\\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}", "type": "inline_equation" }, { "bbox": [ 405, 564, 459, 594 ], "score": 1.0, "content": ", we denote", "type": "text" }, { "bbox": [ 459, 574, 505, 585 ], "score": 0.84, "content": "n _ { s _ { h } , a _ { h } } : =", "type": "inline_equation" } ], "index": 36 }, { "bbox": [ 106, 584, 506, 609 ], "spans": [ { "bbox": [ 106, 585, 213, 599 ], "score": 0.92, "content": "\\begin{array} { r } { \\sum _ { \\tau = 1 } ^ { n } \\mathbf { 1 } [ s _ { h } ^ { \\tau } , { a } _ { h } ^ { \\tau } \\ = \\ s _ { h } , { a } _ { h } ] } \\end{array}", "type": "inline_equation" }, { "bbox": [ 213, 584, 327, 609 ], "score": 1.0, "content": "be the total counts that vis construct the estimators for", "type": "text" }, { "bbox": [ 337, 587, 370, 598 ], "score": 0.89, "content": "\\left( s _ { h } , a _ { h } \\right)", "type": "inline_equation" }, { "bbox": [ 370, 584, 424, 609 ], "score": 1.0, "content": "pair at time as:", "type": "text" }, { "bbox": [ 432, 584, 506, 609 ], "score": 1.0, "content": ", then we use the", "type": "text" } ], "index": 37 }, { "bbox": [ 327, 597, 369, 607 ], "spans": [ { "bbox": [ 327, 597, 340, 607 ], "score": 0.89, "content": "P _ { h }", "type": "inline_equation" }, { "bbox": [ 358, 599, 369, 607 ], "score": 0.86, "content": "r _ { h }", "type": "inline_equation" } ], "index": 38 } ], "index": 37 }, { "type": "interline_equation", "bbox": [ 111, 612, 504, 638 ], "lines": [ { "bbox": [ 111, 612, 504, 638 ], "spans": [ { "bbox": [ 111, 612, 504, 638 ], "score": 0.9, "content": "\\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 } } } ,", "type": "interline_equation", "image_path": "d06ad75b7bc6c9f05d1f248afa41dda6a85409c8e50be048ab90173d8ae25aa9.jpg" } ] } ], "index": 39, "virtual_lines": [ { "bbox": [ 111, 612, 504, 638 ], "spans": [], "index": 39 } ] }, { "type": "text", "bbox": [ 106, 646, 505, 686 ], "lines": [ { "bbox": [ 104, 644, 506, 663 ], "spans": [ { "bbox": [ 104, 644, 115, 663 ], "score": 1.0, "content": "if", "type": "text" }, { "bbox": [ 115, 647, 161, 659 ], "score": 0.91, "content": "n _ { s _ { h } , a _ { h } } > 0", "type": "inline_equation" }, { "bbox": [ 161, 644, 178, 663 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 178, 645, 328, 659 ], "score": 0.93, "content": "\\widehat { P } _ { h } ( s ^ { \\prime } | s _ { h } , a _ { h } ) = 1 / S , \\widehat { r } _ { h } ( s _ { h } , a _ { h } ) = 0", "type": "inline_equation" }, { "bbox": [ 328, 644, 337, 663 ], "score": 1.0, "content": "if", "type": "text" }, { "bbox": [ 337, 648, 383, 659 ], "score": 0.91, "content": "n _ { s _ { h } , a _ { h } } = 0", "type": "inline_equation" }, { "bbox": [ 383, 644, 506, 663 ], "score": 1.0, "content": ". In particular, we use the word", "type": "text" } ], "index": 40 }, { "bbox": [ 102, 656, 508, 675 ], "spans": [ { "bbox": [ 102, 656, 432, 675 ], "score": 1.0, "content": "b“vanilla” as it directly mirrors Jin et al. [2020] with a pessimistic penalty of order", "type": "text" }, { "bbox": [ 433, 659, 497, 673 ], "score": 0.93, "content": "O ( H / \\sqrt { n _ { s _ { h } , a _ { h } } } )", "type": "inline_equation" }, { "bbox": [ 497, 656, 508, 675 ], "score": 1.0, "content": ". 2", "type": "text" } ], "index": 41 }, { "bbox": [ 105, 671, 473, 688 ], "spans": [ { "bbox": [ 105, 672, 128, 688 ], "score": 1.0, "content": "With", "type": "text" }, { "bbox": [ 128, 671, 155, 686 ], "score": 0.93, "content": "\\widehat { P } _ { h } , \\widehat { r } _ { h }", "type": "inline_equation" }, { "bbox": [ 155, 672, 473, 688 ], "score": 1.0, "content": "in Algorithm 2 (which we defer to Appendix), VPVI guarantees the following:", "type": "text" } ], "index": 42 } ], "index": 41 } ], "page_idx": 3, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 108, 697, 505, 722 ], "lines": [ { "bbox": [ 117, 694, 504, 714 ], "spans": [ { "bbox": [ 117, 694, 175, 714 ], "score": 1.0, "content": "2This is due to", "type": "text" }, { "bbox": [ 176, 695, 284, 713 ], "score": 0.89, "content": "\\sqrt { \\phi \\left( s _ { h } , a _ { h } \\right) ^ { \\top } \\Lambda _ { h } ^ { - 1 } \\phi \\left( s _ { h } , a _ { h } \\right) }", "type": "inline_equation" }, { "bbox": [ 284, 694, 325, 714 ], "score": 1.0, "content": "reduces to", "type": "text" }, { "bbox": [ 326, 699, 369, 711 ], "score": 0.91, "content": "\\sqrt { 1 / n _ { s _ { h } , a _ { h } } }", "type": "inline_equation" }, { "bbox": [ 369, 694, 420, 714 ], "score": 1.0, "content": "when setting", "type": "text" }, { "bbox": [ 420, 699, 504, 711 ], "score": 0.93, "content": "\\phi ( s _ { h } , a _ { h } ) = \\mathbf { 1 } ( s _ { h } , a _ { h } )", "type": "inline_equation" } ] }, { "bbox": [ 105, 709, 150, 723 ], "spans": [ { "bbox": [ 105, 709, 122, 723 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 122, 712, 145, 721 ], "score": 0.85, "content": "\\lambda = 0", "type": "inline_equation" }, { "bbox": [ 145, 709, 150, 723 ], "score": 1.0, "content": ".", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 302, 741, 309, 750 ], "lines": [ { "bbox": [ 301, 740, 310, 752 ], "spans": [ { "bbox": [ 301, 740, 310, 752 ], "score": 1.0, "content": "4", "type": "text" } ] } ] } ], "para_blocks": [ { "type": "text", "bbox": [ 105, 72, 505, 96 ], "lines": [], "index": 0.5, "bbox_fs": [ 105, 72, 505, 97 ], "lines_deleted": true }, { "type": "title", "bbox": [ 107, 109, 241, 120 ], "lines": [ { "bbox": [ 105, 108, 243, 122 ], "spans": [ { "bbox": [ 105, 108, 243, 122 ], "score": 1.0, "content": "2.1 Assumptions in offline RL", "type": "text" } ], "index": 2 } ], "index": 2 }, { "type": "text", "bbox": [ 107, 129, 504, 152 ], "lines": [ { "bbox": [ 106, 129, 505, 142 ], "spans": [ { "bbox": [ 106, 129, 505, 142 ], "score": 1.0, "content": "We revise several types of assumptions proposed by existing studies that can yield provably efficient", "type": "text" } ], "index": 3 }, { "bbox": [ 105, 140, 483, 154 ], "spans": [ { "bbox": [ 105, 140, 166, 154 ], "score": 1.0, "content": "results. Recall", "type": "text" }, { "bbox": [ 166, 140, 210, 153 ], "score": 0.93, "content": "d _ { h } ^ { \\mu } ( s _ { h } , a _ { h } )", "type": "inline_equation" }, { "bbox": [ 210, 140, 384, 154 ], "score": 1.0, "content": "is the marginal state-action probability and", "type": "text" }, { "bbox": [ 384, 142, 392, 152 ], "score": 0.82, "content": "\\mu", "type": "inline_equation" }, { "bbox": [ 392, 140, 483, 154 ], "score": 1.0, "content": "is the behavior policy.", "type": "text" } ], "index": 4 } ], "index": 3.5, "bbox_fs": [ 105, 129, 505, 154 ] }, { "type": "text", "bbox": [ 107, 155, 505, 189 ], "lines": [ { "bbox": [ 106, 154, 505, 169 ], "spans": [ { "bbox": [ 106, 154, 476, 169 ], "score": 1.0, "content": "Assumption 2.1 (Uniform data coverage [Yin et al., 2021a]). The behavior policy obeys that", "type": "text" }, { "bbox": [ 477, 156, 505, 167 ], "score": 0.87, "content": "d _ { m } : =", "type": "inline_equation" } ], "index": 5 }, { "bbox": [ 106, 165, 506, 182 ], "spans": [ { "bbox": [ 106, 167, 217, 179 ], "score": 0.9, "content": "\\begin{array} { r } { \\operatorname* { m i n } _ { h , s _ { h } , a _ { h } } d _ { h } ^ { \\mu } ( s _ { h } , a _ { h } ) > 0 . } \\end{array}", "type": "inline_equation" }, { "bbox": [ 217, 165, 506, 182 ], "score": 1.0, "content": ". Here the infimum is over all the states satisfying there exists certain", "type": "text" } ], "index": 6 }, { "bbox": [ 105, 177, 411, 191 ], "spans": [ { "bbox": [ 105, 177, 411, 191 ], "score": 1.0, "content": "policy so that this state can be reached by the current MDP with this policy.", "type": "text" } ], "index": 7 } ], "index": 6, "bbox_fs": [ 105, 154, 506, 191 ] }, { "type": "text", "bbox": [ 106, 198, 505, 265 ], "lines": [ { "bbox": [ 106, 199, 505, 210 ], "spans": [ { "bbox": [ 106, 199, 356, 210 ], "score": 1.0, "content": "This is the strongest assumption in offline RL as it requires", "type": "text" }, { "bbox": [ 357, 201, 364, 210 ], "score": 0.81, "content": "\\mu", "type": "inline_equation" }, { "bbox": [ 365, 199, 505, 210 ], "score": 1.0, "content": "to explore each state-action pairs", "type": "text" } ], "index": 8 }, { "bbox": [ 105, 208, 506, 222 ], "spans": [ { "bbox": [ 105, 208, 321, 222 ], "score": 1.0, "content": "with positive probability. Under 2.1, it mostly holds", "type": "text" }, { "bbox": [ 321, 209, 374, 221 ], "score": 0.93, "content": "1 / d _ { m } \\ge S A", "type": "inline_equation" }, { "bbox": [ 374, 208, 506, 222 ], "score": 1.0, "content": ". This reveals offline learning is", "type": "text" } ], "index": 9 }, { "bbox": [ 105, 219, 507, 234 ], "spans": [ { "bbox": [ 105, 219, 507, 234 ], "score": 1.0, "content": "generically harder than the generative model setting [Agarwal et al., 2020] in the statistical sense.", "type": "text" } ], "index": 10 }, { "bbox": [ 106, 231, 505, 243 ], "spans": [ { "bbox": [ 106, 231, 318, 243 ], "score": 1.0, "content": "On the other hand, this is required for the uniform", "type": "text" }, { "bbox": [ 319, 232, 340, 241 ], "score": 0.54, "content": "O P E", "type": "inline_equation" }, { "bbox": [ 341, 231, 505, 243 ], "score": 1.0, "content": "task in Yin et al. [2021a] as it seeks to", "type": "text" } ], "index": 11 }, { "bbox": [ 105, 242, 506, 255 ], "spans": [ { "bbox": [ 105, 242, 506, 255 ], "score": 1.0, "content": "simultaneously evaluate all the policies within the policy class and it is in general a harder task than", "type": "text" } ], "index": 12 }, { "bbox": [ 105, 253, 195, 266 ], "spans": [ { "bbox": [ 105, 253, 195, 266 ], "score": 1.0, "content": "offline learning itself.", "type": "text" } ], "index": 13 } ], "index": 10.5, "bbox_fs": [ 105, 199, 507, 266 ] }, { "type": "list", "bbox": [ 106, 268, 504, 293 ], "lines": [ { "bbox": [ 106, 267, 505, 281 ], "spans": [ { "bbox": [ 106, 267, 505, 281 ], "score": 1.0, "content": "Assumption 2.2 (Uniform concentrability Szepesvári and Munos [2005], Chen and Jiang [2019]).", "type": "text" } ], "index": 14, "is_list_end_line": true }, { "bbox": [ 105, 279, 353, 293 ], "spans": [ { "bbox": [ 105, 279, 186, 293 ], "score": 1.0, "content": "For all the policies,", "type": "text" }, { "bbox": [ 186, 279, 349, 293 ], "score": 0.92, "content": "\\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}", "type": "inline_equation" }, { "bbox": [ 350, 279, 353, 293 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 15, "is_list_start_line": true, "is_list_end_line": true } ], "index": 14.5, "bbox_fs": [ 105, 267, 505, 293 ] }, { "type": "text", "bbox": [ 106, 299, 505, 344 ], "lines": [ { "bbox": [ 105, 299, 505, 313 ], "spans": [ { "bbox": [ 105, 299, 505, 313 ], "score": 1.0, "content": "This is a classical offline RL condition that is commonly assumed in the function approximation", "type": "text" } ], "index": 16 }, { "bbox": [ 106, 311, 505, 323 ], "spans": [ { "bbox": [ 106, 311, 505, 323 ], "score": 1.0, "content": "scheme (e.g. Fitted Q-Iteration). Qualitatively, this is a uniform data-coverage assumption that is", "type": "text" } ], "index": 17 }, { "bbox": [ 105, 322, 504, 335 ], "spans": [ { "bbox": [ 105, 322, 343, 335 ], "score": 1.0, "content": "similar to Assumption 2.1, but quantitatively, the coefficient", "type": "text" }, { "bbox": [ 343, 322, 357, 334 ], "score": 0.9, "content": "C _ { \\mu }", "type": "inline_equation" }, { "bbox": [ 357, 322, 435, 335 ], "score": 1.0, "content": "can be smaller than", "type": "text" }, { "bbox": [ 436, 322, 460, 334 ], "score": 0.91, "content": "1 / \\bar { d _ { m } } \\bar { }", "type": "inline_equation" }, { "bbox": [ 460, 322, 492, 335 ], "score": 1.0, "content": "due the", "type": "text" }, { "bbox": [ 492, 322, 504, 334 ], "score": 0.87, "content": "d _ { h } ^ { \\pi }", "type": "inline_equation" } ], "index": 18 }, { "bbox": [ 106, 333, 197, 344 ], "spans": [ { "bbox": [ 106, 333, 197, 344 ], "score": 1.0, "content": "term in the numerator.", "type": "text" } ], "index": 19 } ], "index": 17.5, "bbox_fs": [ 105, 299, 505, 344 ] }, { "type": "text", "bbox": [ 107, 348, 506, 383 ], "lines": [ { "bbox": [ 105, 346, 506, 361 ], "spans": [ { "bbox": [ 105, 346, 398, 361 ], "score": 1.0, "content": "Assumption 2.3 (Liu et al. [2019]). There exists one optimal policy", "type": "text" }, { "bbox": [ 398, 349, 410, 358 ], "score": 0.82, "content": "\\pi ^ { \\star }", "type": "inline_equation" }, { "bbox": [ 410, 346, 434, 361 ], "score": 1.0, "content": ", s.t.", "type": "text" }, { "bbox": [ 434, 348, 504, 360 ], "score": 0.88, "content": "\\forall s _ { h } , a _ { h } \\ \\in \\ S , A ,", "type": "inline_equation" }, { "bbox": [ 504, 346, 506, 361 ], "score": 1.0, "content": ",", "type": "text" } ], "index": 20 }, { "bbox": [ 107, 358, 506, 374 ], "spans": [ { "bbox": [ 107, 360, 174, 371 ], "score": 0.81, "content": "d _ { h } ^ { \\mu } ( s _ { h } , a _ { h } ) \\ > \\ 0", "type": "inline_equation" }, { "bbox": [ 174, 358, 185, 374 ], "score": 1.0, "content": "if", "type": "text" }, { "bbox": [ 185, 359, 256, 372 ], "score": 0.88, "content": "d _ { h } ^ { \\pi ^ { \\star } } ( s _ { h } , a _ { h } ) > 0", "type": "inline_equation" }, { "bbox": [ 256, 358, 428, 374 ], "score": 1.0, "content": ". We further denote the trackable set as", "type": "text" }, { "bbox": [ 428, 360, 502, 372 ], "score": 0.89, "content": "\\mathcal C _ { h } : = \\{ ( s _ { h } , a _ { h } )", "type": "inline_equation" }, { "bbox": [ 503, 358, 506, 374 ], "score": 1.0, "content": ":", "type": "text" } ], "index": 21 }, { "bbox": [ 107, 370, 178, 385 ], "spans": [ { "bbox": [ 107, 371, 173, 384 ], "score": 0.88, "content": "d _ { h } ^ { \\tilde { \\mu } } ( s _ { h } , a _ { h } ) > 0 \\}", "type": "inline_equation" }, { "bbox": [ 174, 370, 178, 385 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 22 } ], "index": 21, "bbox_fs": [ 105, 346, 506, 385 ] }, { "type": "text", "bbox": [ 106, 391, 506, 483 ], "lines": [ { "bbox": [ 106, 391, 505, 403 ], "spans": [ { "bbox": [ 106, 391, 505, 403 ], "score": 1.0, "content": "Assumption 2.3 is (arguably) the weakest assumption needed for accurately learning the optimal", "type": "text" } ], "index": 23 }, { "bbox": [ 105, 402, 506, 415 ], "spans": [ { "bbox": [ 105, 402, 130, 415 ], "score": 1.0, "content": "value", "type": "text" }, { "bbox": [ 130, 403, 141, 413 ], "score": 0.86, "content": "v ^ { \\star }", "type": "inline_equation" }, { "bbox": [ 142, 402, 400, 415 ], "score": 1.0, "content": "and we will use 2.3 for most parts of this paper. It only requires", "type": "text" }, { "bbox": [ 400, 405, 408, 414 ], "score": 0.81, "content": "\\mu", "type": "inline_equation" }, { "bbox": [ 408, 402, 506, 415 ], "score": 1.0, "content": "to trace the state-action", "type": "text" } ], "index": 24 }, { "bbox": [ 104, 414, 506, 426 ], "spans": [ { "bbox": [ 104, 414, 506, 426 ], "score": 1.0, "content": "space of one optimal policy and can be agnostic at other locations. Rashidinejad et al. [2021], Xie", "type": "text" } ], "index": 25 }, { "bbox": [ 105, 424, 505, 437 ], "spans": [ { "bbox": [ 105, 424, 505, 437 ], "score": 1.0, "content": "et al. [2021b] considers this assumption and provide analysis is based on the single concentrability", "type": "text" } ], "index": 26 }, { "bbox": [ 106, 435, 505, 449 ], "spans": [ { "bbox": [ 106, 437, 150, 449 ], "score": 1.0, "content": "coefficient", "type": "text" }, { "bbox": [ 150, 435, 285, 449 ], "score": 0.92, "content": "\\begin{array} { r } { C ^ { \\star } : = \\operatorname* { m a x } _ { s , a } { d ^ { \\pi ^ { \\star } } ( s , a ) } / { d ^ { \\mu } ( s , a ) } } \\end{array}", "type": "inline_equation" }, { "bbox": [ 286, 437, 367, 449 ], "score": 1.0, "content": ". The dependence on", "type": "text" }, { "bbox": [ 367, 437, 380, 447 ], "score": 0.87, "content": "C ^ { \\star }", "type": "inline_equation" }, { "bbox": [ 381, 437, 505, 449 ], "score": 1.0, "content": "makes their result less adaptive", "type": "text" } ], "index": 27 }, { "bbox": [ 105, 448, 505, 462 ], "spans": [ { "bbox": [ 105, 448, 332, 461 ], "score": 1.0, "content": "since there can be lots of locations that have the ratio", "type": "text" }, { "bbox": [ 332, 448, 408, 462 ], "score": 0.92, "content": "{ d ^ { \\pi } } ^ { \\star } ( s , a ) / { d ^ { \\mu } } ( s , a )", "type": "inline_equation" }, { "bbox": [ 408, 448, 489, 461 ], "score": 1.0, "content": "much smaller than", "type": "text" }, { "bbox": [ 489, 450, 502, 460 ], "score": 0.87, "content": "C ^ { \\star }", "type": "inline_equation" }, { "bbox": [ 503, 448, 505, 461 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 28 }, { "bbox": [ 105, 460, 505, 473 ], "spans": [ { "bbox": [ 105, 460, 505, 473 ], "score": 1.0, "content": "Furthermore, what could we end up with when 2.3 is not met? We will provide our answers in the", "type": "text" } ], "index": 29 }, { "bbox": [ 106, 471, 191, 484 ], "spans": [ { "bbox": [ 106, 471, 191, 484 ], "score": 1.0, "content": "subsequent sections.", "type": "text" } ], "index": 30 } ], "index": 26.5, "bbox_fs": [ 104, 391, 506, 484 ] }, { "type": "title", "bbox": [ 107, 498, 423, 513 ], "lines": [ { "bbox": [ 104, 498, 423, 515 ], "spans": [ { "bbox": [ 104, 498, 423, 515 ], "score": 1.0, "content": "3 A warm-up case study: Vanilla Pessimistic Value Iteration", "type": "text" } ], "index": 31 } ], "index": 31 }, { "type": "text", "bbox": [ 107, 524, 505, 568 ], "lines": [ { "bbox": [ 105, 524, 505, 537 ], "spans": [ { "bbox": [ 105, 524, 505, 537 ], "score": 1.0, "content": "As a step towards the optimal and strong adaptive offline RL bound, we analyze the vanilla pessimistic", "type": "text" } ], "index": 32 }, { "bbox": [ 106, 535, 506, 547 ], "spans": [ { "bbox": [ 106, 535, 506, 547 ], "score": 1.0, "content": "value iteration (VPVI), a tabular counterpart of pessimistic value iteration (PEVI initiated in Jin et al.", "type": "text" } ], "index": 33 }, { "bbox": [ 106, 546, 505, 559 ], "spans": [ { "bbox": [ 106, 546, 505, 559 ], "score": 1.0, "content": "[2020]), to understand what is missing for achieving the fully adaptivity. In particular, VPVI relies on", "type": "text" } ], "index": 34 }, { "bbox": [ 106, 558, 229, 569 ], "spans": [ { "bbox": [ 106, 558, 229, 569 ], "score": 1.0, "content": "the model-based construction.", "type": "text" } ], "index": 35 } ], "index": 33.5, "bbox_fs": [ 105, 524, 506, 569 ] }, { "type": "text", "bbox": [ 107, 572, 505, 608 ], "lines": [ { "bbox": [ 100, 564, 505, 594 ], "spans": [ { "bbox": [ 100, 564, 281, 594 ], "score": 1.0, "content": "Model-based Components. Given data n", "type": "text" }, { "bbox": [ 281, 572, 405, 586 ], "score": 0.92, "content": "\\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}", "type": "inline_equation" }, { "bbox": [ 405, 564, 459, 594 ], "score": 1.0, "content": ", we denote", "type": "text" }, { "bbox": [ 459, 574, 505, 585 ], "score": 0.84, "content": "n _ { s _ { h } , a _ { h } } : =", "type": "inline_equation" } ], "index": 36 }, { "bbox": [ 106, 584, 506, 609 ], "spans": [ { "bbox": [ 106, 585, 213, 599 ], "score": 0.92, "content": "\\begin{array} { r } { \\sum _ { \\tau = 1 } ^ { n } \\mathbf { 1 } [ s _ { h } ^ { \\tau } , { a } _ { h } ^ { \\tau } \\ = \\ s _ { h } , { a } _ { h } ] } \\end{array}", "type": "inline_equation" }, { "bbox": [ 213, 584, 327, 609 ], "score": 1.0, "content": "be the total counts that vis construct the estimators for", "type": "text" }, { "bbox": [ 337, 587, 370, 598 ], "score": 0.89, "content": "\\left( s _ { h } , a _ { h } \\right)", "type": "inline_equation" }, { "bbox": [ 370, 584, 424, 609 ], "score": 1.0, "content": "pair at time as:", "type": "text" }, { "bbox": [ 432, 584, 506, 609 ], "score": 1.0, "content": ", then we use the", "type": "text" } ], "index": 37 }, { "bbox": [ 327, 597, 369, 607 ], "spans": [ { "bbox": [ 327, 597, 340, 607 ], "score": 0.89, "content": "P _ { h }", "type": "inline_equation" }, { "bbox": [ 358, 599, 369, 607 ], "score": 0.86, "content": "r _ { h }", "type": "inline_equation" } ], "index": 38 } ], "index": 37, "bbox_fs": [ 100, 564, 506, 609 ] }, { "type": "interline_equation", "bbox": [ 111, 612, 504, 638 ], "lines": [ { "bbox": [ 111, 612, 504, 638 ], "spans": [ { "bbox": [ 111, 612, 504, 638 ], "score": 0.9, "content": "\\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 } } } ,", "type": "interline_equation", "image_path": "d06ad75b7bc6c9f05d1f248afa41dda6a85409c8e50be048ab90173d8ae25aa9.jpg" } ] } ], "index": 39, "virtual_lines": [ { "bbox": [ 111, 612, 504, 638 ], "spans": [], "index": 39 } ] }, { "type": "text", "bbox": [ 106, 646, 505, 686 ], "lines": [ { "bbox": [ 104, 644, 506, 663 ], "spans": [ { "bbox": [ 104, 644, 115, 663 ], "score": 1.0, "content": "if", "type": "text" }, { "bbox": [ 115, 647, 161, 659 ], "score": 0.91, "content": "n _ { s _ { h } , a _ { h } } > 0", "type": "inline_equation" }, { "bbox": [ 161, 644, 178, 663 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 178, 645, 328, 659 ], "score": 0.93, "content": "\\widehat { P } _ { h } ( s ^ { \\prime } | s _ { h } , a _ { h } ) = 1 / S , \\widehat { r } _ { h } ( s _ { h } , a _ { h } ) = 0", "type": "inline_equation" }, { "bbox": [ 328, 644, 337, 663 ], "score": 1.0, "content": "if", "type": "text" }, { "bbox": [ 337, 648, 383, 659 ], "score": 0.91, "content": "n _ { s _ { h } , a _ { h } } = 0", "type": "inline_equation" }, { "bbox": [ 383, 644, 506, 663 ], "score": 1.0, "content": ". In particular, we use the word", "type": "text" } ], "index": 40 }, { "bbox": [ 102, 656, 508, 675 ], "spans": [ { "bbox": [ 102, 656, 432, 675 ], "score": 1.0, "content": "b“vanilla” as it directly mirrors Jin et al. [2020] with a pessimistic penalty of order", "type": "text" }, { "bbox": [ 433, 659, 497, 673 ], "score": 0.93, "content": "O ( H / \\sqrt { n _ { s _ { h } , a _ { h } } } )", "type": "inline_equation" }, { "bbox": [ 497, 656, 508, 675 ], "score": 1.0, "content": ". 2", "type": "text" } ], "index": 41 }, { "bbox": [ 105, 671, 473, 688 ], "spans": [ { "bbox": [ 105, 672, 128, 688 ], "score": 1.0, "content": "With", "type": "text" }, { "bbox": [ 128, 671, 155, 686 ], "score": 0.93, "content": "\\widehat { P } _ { h } , \\widehat { r } _ { h }", "type": "inline_equation" }, { "bbox": [ 155, 672, 473, 688 ], "score": 1.0, "content": "in Algorithm 2 (which we defer to Appendix), VPVI guarantees the following:", "type": "text" } ], "index": 42 } ], "index": 41, "bbox_fs": [ 102, 644, 508, 688 ] } ] }, { "preproc_blocks": [ { "type": "text", "bbox": [ 106, 71, 506, 109 ], "lines": [ { "bbox": [ 105, 71, 506, 87 ], "spans": [ { "bbox": [ 105, 71, 309, 87 ], "score": 1.0, "content": "Theorem 3.1. Under the Assumption 2.3, denote", "type": "text" }, { "bbox": [ 309, 72, 502, 86 ], "score": 0.91, "content": "\\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}", "type": "inline_equation" }, { "bbox": [ 502, 71, 506, 87 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 0 }, { "bbox": [ 104, 83, 504, 99 ], "spans": [ { "bbox": [ 104, 83, 140, 99 ], "score": 1.0, "content": "For any", "type": "text" }, { "bbox": [ 141, 85, 186, 96 ], "score": 0.9, "content": "0 < \\delta < 1", "type": "inline_equation" }, { "bbox": [ 186, 83, 317, 99 ], "score": 1.0, "content": ", there exists absolute constants", "type": "text" }, { "bbox": [ 318, 86, 362, 97 ], "score": 0.91, "content": "c _ { 0 } , C ^ { \\prime } > 0", "type": "inline_equation" }, { "bbox": [ 363, 83, 431, 99 ], "score": 1.0, "content": ", such that when", "type": "text" }, { "bbox": [ 432, 85, 504, 97 ], "score": 0.93, "content": "n > c _ { 0 } \\cdot 1 / \\bar { d } _ { m } \\cdot \\iota", "type": "inline_equation" } ], "index": 1 }, { "bbox": [ 108, 96, 428, 109 ], "spans": [ { "bbox": [ 108, 97, 183, 109 ], "score": 0.89, "content": "( \\iota = \\log ( H S A / \\delta ) )", "type": "inline_equation" }, { "bbox": [ 184, 96, 254, 109 ], "score": 1.0, "content": ", with probability", "type": "text" }, { "bbox": [ 254, 97, 277, 107 ], "score": 0.84, "content": "1 - \\delta", "type": "inline_equation" }, { "bbox": [ 277, 96, 350, 109 ], "score": 1.0, "content": ", the output policy", "type": "text" }, { "bbox": [ 350, 97, 358, 107 ], "score": 0.68, "content": "\\widehat { \\pi }", "type": "inline_equation" }, { "bbox": [ 358, 96, 428, 109 ], "score": 1.0, "content": "of VPVI satisfies", "type": "text" } ], "index": 2 } ], "index": 1 }, { "type": "interline_equation", "bbox": [ 169, 113, 442, 150 ], "lines": [ { "bbox": [ 169, 113, 442, 150 ], "spans": [ { "bbox": [ 169, 113, 442, 150 ], "score": 0.94, "content": "0 \\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 } ) } } .", "type": "interline_equation", "image_path": "d5aaa8742b8765850fe826ef7f579d0b1ca064fe3b3e881df803bc52cf418ca8.jpg" } ] } ], "index": 4, "virtual_lines": [ { "bbox": [ 169, 113, 442, 125.33333333333333 ], "spans": [], "index": 3 }, { "bbox": [ 169, 125.33333333333333, 442, 137.66666666666666 ], "spans": [], "index": 4 }, { "bbox": [ 169, 137.66666666666666, 442, 150.0 ], "spans": [], "index": 5 } ] }, { "type": "text", "bbox": [ 106, 160, 505, 228 ], "lines": [ { "bbox": [ 105, 159, 505, 173 ], "spans": [ { "bbox": [ 105, 159, 505, 173 ], "score": 1.0, "content": "The full proof can be found in Appendix C. Theorem 3.1 makes some improvements over the existing", "type": "text" } ], "index": 6 }, { "bbox": [ 106, 172, 505, 183 ], "spans": [ { "bbox": [ 106, 172, 505, 183 ], "score": 1.0, "content": "works. First, it is more adaptive than the results with uniform data-coverage Assumption 2.1 (Yin", "type": "text" } ], "index": 7 }, { "bbox": [ 104, 181, 505, 195 ], "spans": [ { "bbox": [ 104, 181, 505, 195 ], "score": 1.0, "content": "et al. [2021a], Ren et al. [2021]). In addition, by straightforward calculation (3) can be bounded by", "type": "text" } ], "index": 8 }, { "bbox": [ 107, 193, 507, 208 ], "spans": [ { "bbox": [ 107, 194, 177, 208 ], "score": 0.92, "content": "\\tilde { O } ( \\sqrt { H ^ { 4 } S C ^ { \\star } / n } )", "type": "inline_equation" }, { "bbox": [ 177, 193, 450, 208 ], "score": 1.0, "content": "which improves VI-LCB [Rashidinejad et al., 2021] by a factor of", "type": "text" }, { "bbox": [ 451, 195, 461, 205 ], "score": 0.75, "content": "H", "type": "inline_equation" }, { "bbox": [ 461, 193, 507, 208 ], "score": 1.0, "content": ".3 Besides,", "type": "text" } ], "index": 9 }, { "bbox": [ 105, 205, 506, 218 ], "spans": [ { "bbox": [ 105, 205, 506, 218 ], "score": 1.0, "content": "the analysis of VPVI also improves the direct reduction of PEVI [Jin et al., 2020] in the tabular case", "type": "text" } ], "index": 10 }, { "bbox": [ 106, 217, 318, 228 ], "spans": [ { "bbox": [ 106, 217, 151, 228 ], "score": 1.0, "content": "by a factor", "type": "text" }, { "bbox": [ 152, 217, 167, 227 ], "score": 0.87, "content": "_ { S A }", "type": "inline_equation" }, { "bbox": [ 167, 217, 212, 228 ], "score": 1.0, "content": "since their", "type": "text" }, { "bbox": [ 212, 217, 256, 228 ], "score": 0.92, "content": "\\beta = S A H", "type": "inline_equation" }, { "bbox": [ 256, 217, 281, 228 ], "score": 1.0, "content": "when", "type": "text" }, { "bbox": [ 281, 217, 315, 227 ], "score": 0.89, "content": "d = S A", "type": "inline_equation" }, { "bbox": [ 315, 217, 318, 228 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 11 } ], "index": 8.5 }, { "type": "text", "bbox": [ 106, 233, 505, 278 ], "lines": [ { "bbox": [ 106, 232, 505, 245 ], "spans": [ { "bbox": [ 106, 232, 354, 245 ], "score": 1.0, "content": "However, VPVI is not optimal as the dependence on horizon is", "type": "text" }, { "bbox": [ 354, 233, 369, 243 ], "score": 0.89, "content": "H ^ { 4 }", "type": "inline_equation" }, { "bbox": [ 370, 232, 505, 245 ], "score": 1.0, "content": "which does not match the optimal", "type": "text" } ], "index": 12 }, { "bbox": [ 105, 243, 505, 256 ], "spans": [ { "bbox": [ 105, 243, 189, 256 ], "score": 1.0, "content": "worst case guarantee", "type": "text" }, { "bbox": [ 189, 244, 204, 254 ], "score": 0.88, "content": "H ^ { 3 }", "type": "inline_equation" }, { "bbox": [ 204, 243, 505, 256 ], "score": 1.0, "content": "[Yin et al., 2021a] in the nonstationary setting. Also, the explicit dependence", "type": "text" } ], "index": 13 }, { "bbox": [ 106, 255, 505, 267 ], "spans": [ { "bbox": [ 106, 255, 119, 267 ], "score": 1.0, "content": "on", "type": "text" }, { "bbox": [ 119, 255, 129, 265 ], "score": 0.8, "content": "H", "type": "inline_equation" }, { "bbox": [ 130, 255, 505, 267 ], "score": 1.0, "content": "in (3) possibly hides some key features of the specific offline RL instances. For example, no", "type": "text" } ], "index": 14 }, { "bbox": [ 106, 266, 395, 278 ], "spans": [ { "bbox": [ 106, 266, 395, 278 ], "score": 1.0, "content": "improvement can be made if the system has the deterministic transition.", "type": "text" } ], "index": 15 } ], "index": 13.5 }, { "type": "title", "bbox": [ 106, 290, 466, 303 ], "lines": [ { "bbox": [ 105, 290, 466, 304 ], "spans": [ { "bbox": [ 105, 290, 466, 304 ], "score": 1.0, "content": "Algorithm 1 Adaptive (assumption-free) Pessimistic Value Iteration or LCBVI-Bernstein", "type": "text" } ], "index": 16 } ], "index": 16 }, { "type": "text", "bbox": [ 108, 305, 506, 440 ], "lines": [ { "bbox": [ 105, 298, 486, 325 ], "spans": [ { "bbox": [ 105, 298, 203, 325 ], "score": 1.0, "content": "1: Input: Offline dataset", "type": "text" }, { "bbox": [ 204, 306, 319, 319 ], "score": 0.9, "content": "\\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 }", "type": "inline_equation" }, { "bbox": [ 320, 298, 337, 325 ], "score": 1.0, "content": ". Set", "type": "text" }, { "bbox": [ 337, 306, 403, 317 ], "score": 0.88, "content": "C _ { 1 } = 2 , C _ { 2 } = 1 4", "type": "inline_equation" }, { "bbox": [ 403, 298, 473, 325 ], "score": 1.0, "content": ", failure probability", "type": "text" }, { "bbox": [ 473, 308, 478, 316 ], "score": 0.75, "content": "\\delta", "type": "inline_equation" }, { "bbox": [ 478, 298, 486, 325 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 17 }, { "bbox": [ 108, 318, 506, 332 ], "spans": [ { "bbox": [ 108, 318, 190, 332 ], "score": 1.0, "content": "2: Initialization: Set", "type": "text" }, { "bbox": [ 190, 318, 242, 331 ], "score": 0.89, "content": "\\widehat { V } _ { H + 1 } ( \\cdot ) 0", "type": "inline_equation" }, { "bbox": [ 242, 318, 259, 332 ], "score": 1.0, "content": ". Set", "type": "text" }, { "bbox": [ 259, 320, 325, 331 ], "score": 0.89, "content": "\\iota = \\log ( H S A / \\delta )", "type": "inline_equation" }, { "bbox": [ 326, 318, 412, 332 ], "score": 1.0, "content": ". (if assumption-free, set", "type": "text" }, { "bbox": [ 413, 318, 445, 331 ], "score": 0.71, "content": "M ^ { \\dagger } , \\widehat { M } ^ { \\dagger }", "type": "inline_equation" }, { "bbox": [ 446, 318, 506, 332 ], "score": 1.0, "content": "as in Section 5.)", "type": "text" } ], "index": 18 }, { "bbox": [ 109, 329, 247, 341 ], "spans": [ { "bbox": [ 109, 329, 153, 341 ], "score": 1.0, "content": "3: for time", "type": "text" }, { "bbox": [ 154, 331, 237, 340 ], "score": 0.81, "content": "h = H , H - 1 , \\ldots , 1 \\bullet", "type": "inline_equation" }, { "bbox": [ 237, 329, 247, 341 ], "score": 1.0, "content": "do", "type": "text" } ], "index": 19 }, { "bbox": [ 108, 339, 445, 354 ], "spans": [ { "bbox": [ 108, 339, 145, 354 ], "score": 1.0, "content": "4: Set", "type": "text" }, { "bbox": [ 146, 340, 372, 353 ], "score": 0.79, "content": "\\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 } )", "type": "inline_equation" }, { "bbox": [ 372, 339, 445, 354 ], "score": 1.0, "content": "if assumption-free)", "type": "text" } ], "index": 20 }, { "bbox": [ 109, 353, 481, 378 ], "spans": [ { "bbox": [ 109, 361, 120, 372 ], "score": 1.0, "content": "5:", "type": "text" }, { "bbox": [ 130, 362, 160, 372 ], "score": 0.79, "content": "\\forall s _ { h } , a _ { h }", "type": "inline_equation" }, { "bbox": [ 160, 360, 175, 373 ], "score": 1.0, "content": ", set", "type": "text" }, { "bbox": [ 175, 353, 366, 376 ], "score": 0.92, "content": "\\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}", "type": "inline_equation" }, { "bbox": [ 366, 353, 375, 378 ], "score": 1.0, "content": "if", "type": "text" }, { "bbox": [ 376, 361, 419, 372 ], "score": 0.86, "content": "n _ { s _ { h } , a _ { h } } \\ge 1", "type": "inline_equation" }, { "bbox": [ 419, 353, 461, 378 ], "score": 1.0, "content": ", o.w. set to", "type": "text" }, { "bbox": [ 462, 360, 479, 373 ], "score": 0.91, "content": "\\frac { C H \\iota } { 1 }", "type": "inline_equation" }, { "bbox": [ 480, 353, 481, 378 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 21 }, { "bbox": [ 109, 370, 503, 397 ], "spans": [ { "bbox": [ 109, 378, 120, 389 ], "score": 1.0, "content": "6:", "type": "text" }, { "bbox": [ 126, 370, 284, 397 ], "score": 1.0, "content": "(If assumption-free, use C1qVarPb†sh,ah (", "type": "text" }, { "bbox": [ 219, 377, 402, 393 ], "score": 0.92, "content": "\\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}", "type": "inline_equation" }, { "bbox": [ 373, 374, 503, 392 ], "score": 1.0, "content": "C2H·ιn if nsh,ah ≥ 1, o.w. use 0.)", "type": "text" } ], "index": 22 }, { "bbox": [ 109, 392, 506, 407 ], "spans": [ { "bbox": [ 109, 393, 120, 405 ], "score": 1.0, "content": "7:", "type": "text" }, { "bbox": [ 129, 392, 145, 407 ], "score": 1.0, "content": "Set", "type": "text" }, { "bbox": [ 145, 392, 255, 406 ], "score": 0.88, "content": "\\widehat { Q } _ { h } ^ { p } ( \\cdot , \\cdot ) \\gets \\widehat { Q } _ { h } ( \\cdot , \\cdot ) - \\dot { \\Gamma _ { h } } ( \\cdot , \\cdot )", "type": "inline_equation" }, { "bbox": [ 255, 392, 272, 407 ], "score": 1.0, "content": ". Set", "type": "text" }, { "bbox": [ 273, 393, 421, 406 ], "score": 0.89, "content": "\\overline { { { Q } } } _ { h } ( \\cdot , \\cdot ) \\operatorname* { m i n } \\{ \\widehat { Q } _ { h } ^ { p } ( \\cdot , \\cdot ) , H - h + 1 \\} ^ { + }", "type": "inline_equation" }, { "bbox": [ 421, 392, 506, 407 ], "score": 1.0, "content": ". // Pessmistic update", "type": "text" } ], "index": 23 }, { "bbox": [ 109, 405, 479, 419 ], "spans": [ { "bbox": [ 109, 406, 120, 417 ], "score": 1.0, "content": "8:", "type": "text" }, { "bbox": [ 130, 407, 146, 417 ], "score": 0.87, "content": "\\forall s _ { h }", "type": "inline_equation" }, { "bbox": [ 146, 405, 173, 419 ], "score": 1.0, "content": ", Select", "type": "text" }, { "bbox": [ 173, 406, 338, 419 ], "score": 0.89, "content": "\\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}", "type": "inline_equation" }, { "bbox": [ 339, 405, 356, 419 ], "score": 1.0, "content": ". Set", "type": "text" }, { "bbox": [ 356, 406, 475, 418 ], "score": 0.91, "content": "\\widehat { V } _ { h } ( s _ { h } ) \\gets \\langle \\overline { { Q } } _ { h } ( s _ { h } , \\cdot ) , \\widehat { \\pi } _ { h } ( \\cdot | s _ { h } ) \\rangle", "type": "inline_equation" }, { "bbox": [ 475, 405, 479, 419 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 24 }, { "bbox": [ 108, 415, 153, 428 ], "spans": [ { "bbox": [ 108, 415, 153, 428 ], "score": 1.0, "content": "9: end for", "type": "text" } ], "index": 25 }, { "bbox": [ 106, 425, 181, 439 ], "spans": [ { "bbox": [ 106, 425, 156, 439 ], "score": 1.0, "content": "10: Output:", "type": "text" }, { "bbox": [ 156, 426, 177, 438 ], "score": 0.87, "content": "\\left\\{ \\widehat { \\pi } _ { h } \\right\\}", "type": "inline_equation" }, { "bbox": [ 177, 425, 181, 439 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 26 } ], "index": 21.5 }, { "type": "title", "bbox": [ 106, 461, 372, 477 ], "lines": [ { "bbox": [ 104, 461, 373, 479 ], "spans": [ { "bbox": [ 104, 461, 373, 479 ], "score": 1.0, "content": "4 Intrinsic Offline Reinforcement Learning bound", "type": "text" } ], "index": 27 } ], "index": 27 }, { "type": "text", "bbox": [ 106, 485, 506, 546 ], "lines": [ { "bbox": [ 106, 486, 505, 498 ], "spans": [ { "bbox": [ 106, 486, 505, 498 ], "score": 1.0, "content": "Now we go deeper to understand what is the more intrinsic characterization for offline reinforcement", "type": "text" } ], "index": 28 }, { "bbox": [ 104, 497, 507, 513 ], "spans": [ { "bbox": [ 104, 498, 381, 513 ], "score": 1.0, "content": "learning. From the study of VPVI, penalizing the Q-function by", "type": "text" }, { "bbox": [ 382, 497, 446, 513 ], "score": 0.94, "content": "\\widetilde { O } ( H / \\sqrt { n _ { s _ { h } , a _ { h } } } )", "type": "inline_equation" }, { "bbox": [ 447, 498, 507, 513 ], "score": 1.0, "content": "is crude as it", "type": "text" } ], "index": 29 }, { "bbox": [ 105, 511, 505, 526 ], "spans": [ { "bbox": [ 105, 513, 241, 526 ], "score": 1.0, "content": "estimates the confidence width of", "type": "text" }, { "bbox": [ 242, 511, 255, 525 ], "score": 0.91, "content": "{ \\widehat { Q } } _ { h }", "type": "inline_equation" }, { "bbox": [ 256, 513, 505, 526 ], "score": 1.0, "content": "in Algorithm 2 too conservatively therefore loses the accuracy", "type": "text" } ], "index": 30 }, { "bbox": [ 106, 524, 506, 535 ], "spans": [ { "bbox": [ 106, 524, 506, 535 ], "score": 1.0, "content": "(the bound is suboptimal). This motivates us to use empirical standard deviation instead to create a", "type": "text" } ], "index": 31 }, { "bbox": [ 105, 533, 506, 546 ], "spans": [ { "bbox": [ 105, 533, 506, 546 ], "score": 1.0, "content": "more adaptive (and also less conservative) Bernstein-type confidence width as the pessimistic penalty:", "type": "text" } ], "index": 32 } ], "index": 30 }, { "type": "interline_equation", "bbox": [ 115, 557, 485, 593 ], "lines": [ { "bbox": [ 115, 557, 485, 593 ], "spans": [ { "bbox": [ 115, 557, 485, 593 ], "score": 0.93, "content": "\\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 ) .", "type": "interline_equation", "image_path": "1f9c995771f5021da4adef8ab8e4d1d821f9bc111b8c8d03bde8d1348b83f6f5.jpg" } ] } ], "index": 34, "virtual_lines": [ { "bbox": [ 115, 557, 485, 569.0 ], "spans": [], "index": 33 }, { "bbox": [ 115, 569.0, 485, 581.0 ], "spans": [], "index": 34 }, { "bbox": [ 115, 581.0, 485, 593.0 ], "spans": [], "index": 35 } ] }, { "type": "text", "bbox": [ 106, 598, 506, 694 ], "lines": [ { "bbox": [ 105, 598, 507, 619 ], "spans": [ { "bbox": [ 105, 599, 153, 617 ], "score": 1.0, "content": "and update", "type": "text" }, { "bbox": [ 154, 599, 220, 614 ], "score": 0.92, "content": "\\widehat { Q } _ { h } \\gets \\widehat { Q } _ { h } - \\Gamma _ { h }", "type": "inline_equation" }, { "bbox": [ 221, 599, 282, 617 ], "score": 1.0, "content": ". On one hand,", "type": "text" }, { "bbox": [ 282, 598, 415, 619 ], "score": 0.93, "content": "\\sqrt { \\mathrm { V a r } _ { \\widehat { P } _ { s _ { h } , a _ { h } } } ( \\widehat { r } _ { h } + \\widehat { V } _ { h + 1 } ) / n _ { s _ { h } , a _ { h } } }", "type": "inline_equation" }, { "bbox": [ 415, 600, 507, 616 ], "score": 1.0, "content": "is a “less pessimistic”", "type": "text" } ], "index": 36 }, { "bbox": [ 104, 618, 506, 638 ], "spans": [ { "bbox": [ 104, 621, 210, 636 ], "score": 1.0, "content": "penalty than VPVI due to", "type": "text" }, { "bbox": [ 211, 618, 317, 638 ], "score": 0.93, "content": "\\sqrt { \\mathrm { V a r } _ { \\widehat { P } } ( \\widehat { r } _ { h } + \\widehat { V } _ { h + 1 } ) } \\le H", "type": "inline_equation" }, { "bbox": [ 317, 621, 506, 636 ], "score": 1.0, "content": "and critically this design is more data-adaptive", "type": "text" } ], "index": 37 }, { "bbox": [ 106, 635, 505, 648 ], "spans": [ { "bbox": [ 106, 635, 505, 648 ], "score": 1.0, "content": "since it holds negative view towards the locations with high uncertainties and recommends the", "type": "text" } ], "index": 38 }, { "bbox": [ 106, 646, 505, 659 ], "spans": [ { "bbox": [ 106, 646, 505, 659 ], "score": 1.0, "content": "locations that we are confident about, as opposed to the online RL (which encourages exploration in", "type": "text" } ], "index": 39 }, { "bbox": [ 106, 657, 506, 670 ], "spans": [ { "bbox": [ 106, 657, 506, 670 ], "score": 1.0, "content": "the uncertain locations). Such principles are not reflected by the isotropic design in VPVI. On the", "type": "text" } ], "index": 40 }, { "bbox": [ 106, 668, 505, 682 ], "spans": [ { "bbox": [ 106, 669, 440, 682 ], "score": 1.0, "content": "other hand, it carries the extremely negative view towards fully agnostic locations", "type": "text" }, { "bbox": [ 440, 668, 466, 682 ], "score": 0.93, "content": "{ \\widetilde { O } } ( H )", "type": "inline_equation" }, { "bbox": [ 466, 669, 505, 682 ], "score": 1.0, "content": "which in", "type": "text" } ], "index": 41 }, { "bbox": [ 105, 681, 505, 694 ], "spans": [ { "bbox": [ 105, 681, 505, 694 ], "score": 1.0, "content": "turn causes the agent unlikely to choose them. We summarized the this adaptive pessimistic value", "type": "text" } ], "index": 42 } ], "index": 39 } ], "page_idx": 4, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 106, 701, 505, 723 ], "lines": [ { "bbox": [ 118, 699, 506, 714 ], "spans": [ { "bbox": [ 118, 699, 506, 714 ], "score": 1.0, "content": "3To be rigorous, translating the result from the infinite horizon setting to the finite horizon setting requires", "type": "text" } ] }, { "bbox": [ 105, 711, 295, 723 ], "spans": [ { "bbox": [ 105, 711, 295, 723 ], "score": 1.0, "content": "explanation. We add this discussion in Appendix D.", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 302, 741, 309, 750 ], "lines": [ { "bbox": [ 302, 741, 309, 752 ], "spans": [ { "bbox": [ 302, 741, 309, 752 ], "score": 1.0, "content": "5", "type": "text" } ] } ] } ], "para_blocks": [ { "type": "text", "bbox": [ 106, 71, 506, 109 ], "lines": [ { "bbox": [ 105, 71, 506, 87 ], "spans": [ { "bbox": [ 105, 71, 309, 87 ], "score": 1.0, "content": "Theorem 3.1. Under the Assumption 2.3, denote", "type": "text" }, { "bbox": [ 309, 72, 502, 86 ], "score": 0.91, "content": "\\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}", "type": "inline_equation" }, { "bbox": [ 502, 71, 506, 87 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 0 }, { "bbox": [ 104, 83, 504, 99 ], "spans": [ { "bbox": [ 104, 83, 140, 99 ], "score": 1.0, "content": "For any", "type": "text" }, { "bbox": [ 141, 85, 186, 96 ], "score": 0.9, "content": "0 < \\delta < 1", "type": "inline_equation" }, { "bbox": [ 186, 83, 317, 99 ], "score": 1.0, "content": ", there exists absolute constants", "type": "text" }, { "bbox": [ 318, 86, 362, 97 ], "score": 0.91, "content": "c _ { 0 } , C ^ { \\prime } > 0", "type": "inline_equation" }, { "bbox": [ 363, 83, 431, 99 ], "score": 1.0, "content": ", such that when", "type": "text" }, { "bbox": [ 432, 85, 504, 97 ], "score": 0.93, "content": "n > c _ { 0 } \\cdot 1 / \\bar { d } _ { m } \\cdot \\iota", "type": "inline_equation" } ], "index": 1 }, { "bbox": [ 108, 96, 428, 109 ], "spans": [ { "bbox": [ 108, 97, 183, 109 ], "score": 0.89, "content": "( \\iota = \\log ( H S A / \\delta ) )", "type": "inline_equation" }, { "bbox": [ 184, 96, 254, 109 ], "score": 1.0, "content": ", with probability", "type": "text" }, { "bbox": [ 254, 97, 277, 107 ], "score": 0.84, "content": "1 - \\delta", "type": "inline_equation" }, { "bbox": [ 277, 96, 350, 109 ], "score": 1.0, "content": ", the output policy", "type": "text" }, { "bbox": [ 350, 97, 358, 107 ], "score": 0.68, "content": "\\widehat { \\pi }", "type": "inline_equation" }, { "bbox": [ 358, 96, 428, 109 ], "score": 1.0, "content": "of VPVI satisfies", "type": "text" } ], "index": 2 } ], "index": 1, "bbox_fs": [ 104, 71, 506, 109 ] }, { "type": "interline_equation", "bbox": [ 169, 113, 442, 150 ], "lines": [ { "bbox": [ 169, 113, 442, 150 ], "spans": [ { "bbox": [ 169, 113, 442, 150 ], "score": 0.94, "content": "0 \\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 } ) } } .", "type": "interline_equation", "image_path": "d5aaa8742b8765850fe826ef7f579d0b1ca064fe3b3e881df803bc52cf418ca8.jpg" } ] } ], "index": 4, "virtual_lines": [ { "bbox": [ 169, 113, 442, 125.33333333333333 ], "spans": [], "index": 3 }, { "bbox": [ 169, 125.33333333333333, 442, 137.66666666666666 ], "spans": [], "index": 4 }, { "bbox": [ 169, 137.66666666666666, 442, 150.0 ], "spans": [], "index": 5 } ] }, { "type": "text", "bbox": [ 106, 160, 505, 228 ], "lines": [ { "bbox": [ 105, 159, 505, 173 ], "spans": [ { "bbox": [ 105, 159, 505, 173 ], "score": 1.0, "content": "The full proof can be found in Appendix C. Theorem 3.1 makes some improvements over the existing", "type": "text" } ], "index": 6 }, { "bbox": [ 106, 172, 505, 183 ], "spans": [ { "bbox": [ 106, 172, 505, 183 ], "score": 1.0, "content": "works. First, it is more adaptive than the results with uniform data-coverage Assumption 2.1 (Yin", "type": "text" } ], "index": 7 }, { "bbox": [ 104, 181, 505, 195 ], "spans": [ { "bbox": [ 104, 181, 505, 195 ], "score": 1.0, "content": "et al. [2021a], Ren et al. [2021]). In addition, by straightforward calculation (3) can be bounded by", "type": "text" } ], "index": 8 }, { "bbox": [ 107, 193, 507, 208 ], "spans": [ { "bbox": [ 107, 194, 177, 208 ], "score": 0.92, "content": "\\tilde { O } ( \\sqrt { H ^ { 4 } S C ^ { \\star } / n } )", "type": "inline_equation" }, { "bbox": [ 177, 193, 450, 208 ], "score": 1.0, "content": "which improves VI-LCB [Rashidinejad et al., 2021] by a factor of", "type": "text" }, { "bbox": [ 451, 195, 461, 205 ], "score": 0.75, "content": "H", "type": "inline_equation" }, { "bbox": [ 461, 193, 507, 208 ], "score": 1.0, "content": ".3 Besides,", "type": "text" } ], "index": 9 }, { "bbox": [ 105, 205, 506, 218 ], "spans": [ { "bbox": [ 105, 205, 506, 218 ], "score": 1.0, "content": "the analysis of VPVI also improves the direct reduction of PEVI [Jin et al., 2020] in the tabular case", "type": "text" } ], "index": 10 }, { "bbox": [ 106, 217, 318, 228 ], "spans": [ { "bbox": [ 106, 217, 151, 228 ], "score": 1.0, "content": "by a factor", "type": "text" }, { "bbox": [ 152, 217, 167, 227 ], "score": 0.87, "content": "_ { S A }", "type": "inline_equation" }, { "bbox": [ 167, 217, 212, 228 ], "score": 1.0, "content": "since their", "type": "text" }, { "bbox": [ 212, 217, 256, 228 ], "score": 0.92, "content": "\\beta = S A H", "type": "inline_equation" }, { "bbox": [ 256, 217, 281, 228 ], "score": 1.0, "content": "when", "type": "text" }, { "bbox": [ 281, 217, 315, 227 ], "score": 0.89, "content": "d = S A", "type": "inline_equation" }, { "bbox": [ 315, 217, 318, 228 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 11 } ], "index": 8.5, "bbox_fs": [ 104, 159, 507, 228 ] }, { "type": "text", "bbox": [ 106, 233, 505, 278 ], "lines": [ { "bbox": [ 106, 232, 505, 245 ], "spans": [ { "bbox": [ 106, 232, 354, 245 ], "score": 1.0, "content": "However, VPVI is not optimal as the dependence on horizon is", "type": "text" }, { "bbox": [ 354, 233, 369, 243 ], "score": 0.89, "content": "H ^ { 4 }", "type": "inline_equation" }, { "bbox": [ 370, 232, 505, 245 ], "score": 1.0, "content": "which does not match the optimal", "type": "text" } ], "index": 12 }, { "bbox": [ 105, 243, 505, 256 ], "spans": [ { "bbox": [ 105, 243, 189, 256 ], "score": 1.0, "content": "worst case guarantee", "type": "text" }, { "bbox": [ 189, 244, 204, 254 ], "score": 0.88, "content": "H ^ { 3 }", "type": "inline_equation" }, { "bbox": [ 204, 243, 505, 256 ], "score": 1.0, "content": "[Yin et al., 2021a] in the nonstationary setting. Also, the explicit dependence", "type": "text" } ], "index": 13 }, { "bbox": [ 106, 255, 505, 267 ], "spans": [ { "bbox": [ 106, 255, 119, 267 ], "score": 1.0, "content": "on", "type": "text" }, { "bbox": [ 119, 255, 129, 265 ], "score": 0.8, "content": "H", "type": "inline_equation" }, { "bbox": [ 130, 255, 505, 267 ], "score": 1.0, "content": "in (3) possibly hides some key features of the specific offline RL instances. For example, no", "type": "text" } ], "index": 14 }, { "bbox": [ 106, 266, 395, 278 ], "spans": [ { "bbox": [ 106, 266, 395, 278 ], "score": 1.0, "content": "improvement can be made if the system has the deterministic transition.", "type": "text" } ], "index": 15 } ], "index": 13.5, "bbox_fs": [ 105, 232, 505, 278 ] }, { "type": "title", "bbox": [ 106, 290, 466, 303 ], "lines": [ { "bbox": [ 105, 290, 466, 304 ], "spans": [ { "bbox": [ 105, 290, 466, 304 ], "score": 1.0, "content": "Algorithm 1 Adaptive (assumption-free) Pessimistic Value Iteration or LCBVI-Bernstein", "type": "text" } ], "index": 16 } ], "index": 16 }, { "type": "index", "bbox": [ 108, 305, 506, 440 ], "lines": [ { "bbox": [ 105, 298, 486, 325 ], "spans": [ { "bbox": [ 105, 298, 203, 325 ], "score": 1.0, "content": "1: Input: Offline dataset", "type": "text" }, { "bbox": [ 204, 306, 319, 319 ], "score": 0.9, "content": "\\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 }", "type": "inline_equation" }, { "bbox": [ 320, 298, 337, 325 ], "score": 1.0, "content": ". Set", "type": "text" }, { "bbox": [ 337, 306, 403, 317 ], "score": 0.88, "content": "C _ { 1 } = 2 , C _ { 2 } = 1 4", "type": "inline_equation" }, { "bbox": [ 403, 298, 473, 325 ], "score": 1.0, "content": ", failure probability", "type": "text" }, { "bbox": [ 473, 308, 478, 316 ], "score": 0.75, "content": "\\delta", "type": "inline_equation" }, { "bbox": [ 478, 298, 486, 325 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 17, "is_list_start_line": true }, { "bbox": [ 108, 318, 506, 332 ], "spans": [ { "bbox": [ 108, 318, 190, 332 ], "score": 1.0, "content": "2: Initialization: Set", "type": "text" }, { "bbox": [ 190, 318, 242, 331 ], "score": 0.89, "content": "\\widehat { V } _ { H + 1 } ( \\cdot ) 0", "type": "inline_equation" }, { "bbox": [ 242, 318, 259, 332 ], "score": 1.0, "content": ". Set", "type": "text" }, { "bbox": [ 259, 320, 325, 331 ], "score": 0.89, "content": "\\iota = \\log ( H S A / \\delta )", "type": "inline_equation" }, { "bbox": [ 326, 318, 412, 332 ], "score": 1.0, "content": ". (if assumption-free, set", "type": "text" }, { "bbox": [ 413, 318, 445, 331 ], "score": 0.71, "content": "M ^ { \\dagger } , \\widehat { M } ^ { \\dagger }", "type": "inline_equation" }, { "bbox": [ 446, 318, 506, 332 ], "score": 1.0, "content": "as in Section 5.)", "type": "text" } ], "index": 18, "is_list_start_line": true }, { "bbox": [ 109, 329, 247, 341 ], "spans": [ { "bbox": [ 109, 329, 153, 341 ], "score": 1.0, "content": "3: for time", "type": "text" }, { "bbox": [ 154, 331, 237, 340 ], "score": 0.81, "content": "h = H , H - 1 , \\ldots , 1 \\bullet", "type": "inline_equation" }, { "bbox": [ 237, 329, 247, 341 ], "score": 1.0, "content": "do", "type": "text" } ], "index": 19, "is_list_start_line": true }, { "bbox": [ 108, 339, 445, 354 ], "spans": [ { "bbox": [ 108, 339, 145, 354 ], "score": 1.0, "content": "4: Set", "type": "text" }, { "bbox": [ 146, 340, 372, 353 ], "score": 0.79, "content": "\\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 } )", "type": "inline_equation" }, { "bbox": [ 372, 339, 445, 354 ], "score": 1.0, "content": "if assumption-free)", "type": "text" } ], "index": 20, "is_list_start_line": true }, { "bbox": [ 109, 353, 481, 378 ], "spans": [ { "bbox": [ 109, 361, 120, 372 ], "score": 1.0, "content": "5:", "type": "text" }, { "bbox": [ 130, 362, 160, 372 ], "score": 0.79, "content": "\\forall s _ { h } , a _ { h }", "type": "inline_equation" }, { "bbox": [ 160, 360, 175, 373 ], "score": 1.0, "content": ", set", "type": "text" }, { "bbox": [ 175, 353, 366, 376 ], "score": 0.92, "content": "\\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}", "type": "inline_equation" }, { "bbox": [ 366, 353, 375, 378 ], "score": 1.0, "content": "if", "type": "text" }, { "bbox": [ 376, 361, 419, 372 ], "score": 0.86, "content": "n _ { s _ { h } , a _ { h } } \\ge 1", "type": "inline_equation" }, { "bbox": [ 419, 353, 461, 378 ], "score": 1.0, "content": ", o.w. set to", "type": "text" }, { "bbox": [ 462, 360, 479, 373 ], "score": 0.91, "content": "\\frac { C H \\iota } { 1 }", "type": "inline_equation" }, { "bbox": [ 480, 353, 481, 378 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 21, "is_list_start_line": true }, { "bbox": [ 109, 370, 503, 397 ], "spans": [ { "bbox": [ 109, 378, 120, 389 ], "score": 1.0, "content": "6:", "type": "text" }, { "bbox": [ 126, 370, 284, 397 ], "score": 1.0, "content": "(If assumption-free, use C1qVarPb†sh,ah (", "type": "text" }, { "bbox": [ 219, 377, 402, 393 ], "score": 0.92, "content": "\\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}", "type": "inline_equation" }, { "bbox": [ 373, 374, 503, 392 ], "score": 1.0, "content": "C2H·ιn if nsh,ah ≥ 1, o.w. use 0.)", "type": "text" } ], "index": 22, "is_list_start_line": true }, { "bbox": [ 109, 392, 506, 407 ], "spans": [ { "bbox": [ 109, 393, 120, 405 ], "score": 1.0, "content": "7:", "type": "text" }, { "bbox": [ 129, 392, 145, 407 ], "score": 1.0, "content": "Set", "type": "text" }, { "bbox": [ 145, 392, 255, 406 ], "score": 0.88, "content": "\\widehat { Q } _ { h } ^ { p } ( \\cdot , \\cdot ) \\gets \\widehat { Q } _ { h } ( \\cdot , \\cdot ) - \\dot { \\Gamma _ { h } } ( \\cdot , \\cdot )", "type": "inline_equation" }, { "bbox": [ 255, 392, 272, 407 ], "score": 1.0, "content": ". Set", "type": "text" }, { "bbox": [ 273, 393, 421, 406 ], "score": 0.89, "content": "\\overline { { { Q } } } _ { h } ( \\cdot , \\cdot ) \\operatorname* { m i n } \\{ \\widehat { Q } _ { h } ^ { p } ( \\cdot , \\cdot ) , H - h + 1 \\} ^ { + }", "type": "inline_equation" }, { "bbox": [ 421, 392, 506, 407 ], "score": 1.0, "content": ". // Pessmistic update", "type": "text" } ], "index": 23, "is_list_start_line": true }, { "bbox": [ 109, 405, 479, 419 ], "spans": [ { "bbox": [ 109, 406, 120, 417 ], "score": 1.0, "content": "8:", "type": "text" }, { "bbox": [ 130, 407, 146, 417 ], "score": 0.87, "content": "\\forall s _ { h }", "type": "inline_equation" }, { "bbox": [ 146, 405, 173, 419 ], "score": 1.0, "content": ", Select", "type": "text" }, { "bbox": [ 173, 406, 338, 419 ], "score": 0.89, "content": "\\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}", "type": "inline_equation" }, { "bbox": [ 339, 405, 356, 419 ], "score": 1.0, "content": ". Set", "type": "text" }, { "bbox": [ 356, 406, 475, 418 ], "score": 0.91, "content": "\\widehat { V } _ { h } ( s _ { h } ) \\gets \\langle \\overline { { Q } } _ { h } ( s _ { h } , \\cdot ) , \\widehat { \\pi } _ { h } ( \\cdot | s _ { h } ) \\rangle", "type": "inline_equation" }, { "bbox": [ 475, 405, 479, 419 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 24, "is_list_start_line": true }, { "bbox": [ 108, 415, 153, 428 ], "spans": [ { "bbox": [ 108, 415, 153, 428 ], "score": 1.0, "content": "9: end for", "type": "text" } ], "index": 25, "is_list_start_line": true }, { "bbox": [ 106, 425, 181, 439 ], "spans": [ { "bbox": [ 106, 425, 156, 439 ], "score": 1.0, "content": "10: Output:", "type": "text" }, { "bbox": [ 156, 426, 177, 438 ], "score": 0.87, "content": "\\left\\{ \\widehat { \\pi } _ { h } \\right\\}", "type": "inline_equation" }, { "bbox": [ 177, 425, 181, 439 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 26, "is_list_start_line": true } ], "index": 21.5, "bbox_fs": [ 105, 298, 506, 439 ] }, { "type": "title", "bbox": [ 106, 461, 372, 477 ], "lines": [ { "bbox": [ 104, 461, 373, 479 ], "spans": [ { "bbox": [ 104, 461, 373, 479 ], "score": 1.0, "content": "4 Intrinsic Offline Reinforcement Learning bound", "type": "text" } ], "index": 27 } ], "index": 27 }, { "type": "text", "bbox": [ 106, 485, 506, 546 ], "lines": [ { "bbox": [ 106, 486, 505, 498 ], "spans": [ { "bbox": [ 106, 486, 505, 498 ], "score": 1.0, "content": "Now we go deeper to understand what is the more intrinsic characterization for offline reinforcement", "type": "text" } ], "index": 28 }, { "bbox": [ 104, 497, 507, 513 ], "spans": [ { "bbox": [ 104, 498, 381, 513 ], "score": 1.0, "content": "learning. From the study of VPVI, penalizing the Q-function by", "type": "text" }, { "bbox": [ 382, 497, 446, 513 ], "score": 0.94, "content": "\\widetilde { O } ( H / \\sqrt { n _ { s _ { h } , a _ { h } } } )", "type": "inline_equation" }, { "bbox": [ 447, 498, 507, 513 ], "score": 1.0, "content": "is crude as it", "type": "text" } ], "index": 29 }, { "bbox": [ 105, 511, 505, 526 ], "spans": [ { "bbox": [ 105, 513, 241, 526 ], "score": 1.0, "content": "estimates the confidence width of", "type": "text" }, { "bbox": [ 242, 511, 255, 525 ], "score": 0.91, "content": "{ \\widehat { Q } } _ { h }", "type": "inline_equation" }, { "bbox": [ 256, 513, 505, 526 ], "score": 1.0, "content": "in Algorithm 2 too conservatively therefore loses the accuracy", "type": "text" } ], "index": 30 }, { "bbox": [ 106, 524, 506, 535 ], "spans": [ { "bbox": [ 106, 524, 506, 535 ], "score": 1.0, "content": "(the bound is suboptimal). This motivates us to use empirical standard deviation instead to create a", "type": "text" } ], "index": 31 }, { "bbox": [ 105, 533, 506, 546 ], "spans": [ { "bbox": [ 105, 533, 506, 546 ], "score": 1.0, "content": "more adaptive (and also less conservative) Bernstein-type confidence width as the pessimistic penalty:", "type": "text" } ], "index": 32 } ], "index": 30, "bbox_fs": [ 104, 486, 507, 546 ] }, { "type": "interline_equation", "bbox": [ 115, 557, 485, 593 ], "lines": [ { "bbox": [ 115, 557, 485, 593 ], "spans": [ { "bbox": [ 115, 557, 485, 593 ], "score": 0.93, "content": "\\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 ) .", "type": "interline_equation", "image_path": "1f9c995771f5021da4adef8ab8e4d1d821f9bc111b8c8d03bde8d1348b83f6f5.jpg" } ] } ], "index": 34, "virtual_lines": [ { "bbox": [ 115, 557, 485, 569.0 ], "spans": [], "index": 33 }, { "bbox": [ 115, 569.0, 485, 581.0 ], "spans": [], "index": 34 }, { "bbox": [ 115, 581.0, 485, 593.0 ], "spans": [], "index": 35 } ] }, { "type": "text", "bbox": [ 106, 598, 506, 694 ], "lines": [ { "bbox": [ 105, 598, 507, 619 ], "spans": [ { "bbox": [ 105, 599, 153, 617 ], "score": 1.0, "content": "and update", "type": "text" }, { "bbox": [ 154, 599, 220, 614 ], "score": 0.92, "content": "\\widehat { Q } _ { h } \\gets \\widehat { Q } _ { h } - \\Gamma _ { h }", "type": "inline_equation" }, { "bbox": [ 221, 599, 282, 617 ], "score": 1.0, "content": ". On one hand,", "type": "text" }, { "bbox": [ 282, 598, 415, 619 ], "score": 0.93, "content": "\\sqrt { \\mathrm { V a r } _ { \\widehat { P } _ { s _ { h } , a _ { h } } } ( \\widehat { r } _ { h } + \\widehat { V } _ { h + 1 } ) / n _ { s _ { h } , a _ { h } } }", "type": "inline_equation" }, { "bbox": [ 415, 600, 507, 616 ], "score": 1.0, "content": "is a “less pessimistic”", "type": "text" } ], "index": 36 }, { "bbox": [ 104, 618, 506, 638 ], "spans": [ { "bbox": [ 104, 621, 210, 636 ], "score": 1.0, "content": "penalty than VPVI due to", "type": "text" }, { "bbox": [ 211, 618, 317, 638 ], "score": 0.93, "content": "\\sqrt { \\mathrm { V a r } _ { \\widehat { P } } ( \\widehat { r } _ { h } + \\widehat { V } _ { h + 1 } ) } \\le H", "type": "inline_equation" }, { "bbox": [ 317, 621, 506, 636 ], "score": 1.0, "content": "and critically this design is more data-adaptive", "type": "text" } ], "index": 37 }, { "bbox": [ 106, 635, 505, 648 ], "spans": [ { "bbox": [ 106, 635, 505, 648 ], "score": 1.0, "content": "since it holds negative view towards the locations with high uncertainties and recommends the", "type": "text" } ], "index": 38 }, { "bbox": [ 106, 646, 505, 659 ], "spans": [ { "bbox": [ 106, 646, 505, 659 ], "score": 1.0, "content": "locations that we are confident about, as opposed to the online RL (which encourages exploration in", "type": "text" } ], "index": 39 }, { "bbox": [ 106, 657, 506, 670 ], "spans": [ { "bbox": [ 106, 657, 506, 670 ], "score": 1.0, "content": "the uncertain locations). Such principles are not reflected by the isotropic design in VPVI. On the", "type": "text" } ], "index": 40 }, { "bbox": [ 106, 668, 505, 682 ], "spans": [ { "bbox": [ 106, 669, 440, 682 ], "score": 1.0, "content": "other hand, it carries the extremely negative view towards fully agnostic locations", "type": "text" }, { "bbox": [ 440, 668, 466, 682 ], "score": 0.93, "content": "{ \\widetilde { O } } ( H )", "type": "inline_equation" }, { "bbox": [ 466, 669, 505, 682 ], "score": 1.0, "content": "which in", "type": "text" } ], "index": 41 }, { "bbox": [ 105, 681, 505, 694 ], "spans": [ { "bbox": [ 105, 681, 505, 694 ], "score": 1.0, "content": "turn causes the agent unlikely to choose them. We summarized the this adaptive pessimistic value", "type": "text" } ], "index": 42 } ], "index": 39, "bbox_fs": [ 104, 598, 507, 694 ] } ] }, { "preproc_blocks": [ { "type": "text", "bbox": [ 105, 71, 505, 95 ], "lines": [ { "bbox": [ 105, 70, 507, 86 ], "spans": [ { "bbox": [ 105, 70, 280, 86 ], "score": 1.0, "content": "iteration (APVI) into the Algorithm 1, with", "type": "text" }, { "bbox": [ 280, 70, 307, 84 ], "score": 0.93, "content": "\\widehat { P } _ { h } , \\widehat { r } _ { h }", "type": "inline_equation" }, { "bbox": [ 307, 70, 507, 86 ], "score": 1.0, "content": "defined in (2). APVI has the following guarantee.", "type": "text" } ], "index": 0 }, { "bbox": [ 105, 83, 465, 96 ], "spans": [ { "bbox": [ 105, 83, 465, 96 ], "score": 1.0, "content": "bA sketch of the analysis is presented in Section B and Appendix F includes the full proof.", "type": "text" } ], "index": 1 } ], "index": 0.5 }, { "type": "text", "bbox": [ 107, 97, 505, 148 ], "lines": [ { "bbox": [ 105, 95, 505, 112 ], "spans": [ { "bbox": [ 105, 95, 468, 112 ], "score": 1.0, "content": "Theorem 4.1 (Intrinsic offline RL bound). Under the Assumption 2.3, denote", "type": "text" }, { "bbox": [ 468, 97, 505, 110 ], "score": 0.86, "content": "\\bar { d } _ { m } : =", "type": "inline_equation" } ], "index": 2 }, { "bbox": [ 106, 109, 505, 122 ], "spans": [ { "bbox": [ 106, 109, 276, 122 ], "score": 0.91, "content": "\\mathrm { m i n } _ { h \\in [ H ] } \\{ d _ { h } ^ { \\mu } ( s _ { h } , a _ { h } ) \\ : \\ d _ { h } ^ { \\mu } ( s _ { h } , a _ { h } ) \\ > \\ 0 \\}", "type": "inline_equation" }, { "bbox": [ 276, 109, 320, 122 ], "score": 1.0, "content": ". For any", "type": "text" }, { "bbox": [ 321, 110, 371, 120 ], "score": 0.9, "content": "0 ~ < ~ \\delta ~ < ~ 1", "type": "inline_equation" }, { "bbox": [ 371, 109, 505, 122 ], "score": 1.0, "content": ", there exists absolute constants", "type": "text" } ], "index": 3 }, { "bbox": [ 107, 121, 505, 135 ], "spans": [ { "bbox": [ 107, 123, 150, 134 ], "score": 0.91, "content": "c _ { 0 } , C ^ { \\prime } > 0", "type": "inline_equation" }, { "bbox": [ 150, 121, 216, 135 ], "score": 1.0, "content": ", such that when", "type": "text" }, { "bbox": [ 216, 122, 365, 134 ], "score": 0.67, "content": "n > c _ { 0 } \\cdot 1 / \\bar { d } _ { m } \\cdot \\iota ( \\iota = \\log ( H S A / \\delta ) )", "type": "inline_equation" }, { "bbox": [ 365, 121, 435, 135 ], "score": 1.0, "content": ", with probability", "type": "text" }, { "bbox": [ 435, 122, 459, 133 ], "score": 0.76, "content": "1 - \\delta", "type": "inline_equation" }, { "bbox": [ 459, 121, 505, 135 ], "score": 1.0, "content": ", the output", "type": "text" } ], "index": 4 }, { "bbox": [ 105, 134, 444, 149 ], "spans": [ { "bbox": [ 105, 135, 133, 149 ], "score": 1.0, "content": "policy", "type": "text" }, { "bbox": [ 133, 136, 141, 145 ], "score": 0.73, "content": "\\widehat { \\pi }", "type": "inline_equation" }, { "bbox": [ 141, 135, 222, 149 ], "score": 1.0, "content": "of APVI (Algorithm", "type": "text" }, { "bbox": [ 223, 137, 228, 146 ], "score": 0.25, "content": "I", "type": "inline_equation" }, { "bbox": [ 228, 135, 270, 149 ], "score": 1.0, "content": ") satisfies", "type": "text" }, { "bbox": [ 271, 134, 280, 146 ], "score": 0.74, "content": "\\widetilde O", "type": "inline_equation" }, { "bbox": [ 280, 135, 444, 149 ], "score": 1.0, "content": "hides log factor and higher order terms)", "type": "text" } ], "index": 5 } ], "index": 3.5 }, { "type": "interline_equation", "bbox": [ 113, 152, 485, 190 ], "lines": [ { "bbox": [ 113, 152, 485, 190 ], "spans": [ { "bbox": [ 113, 152, 485, 190 ], "score": 0.94, "content": "0 \\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)", "type": "interline_equation", "image_path": "f8faee4d11f76593d45f80f3acf4eb263d1dc36bed744ce74f364b2ebd8d24c8.jpg" } ] } ], "index": 7, "virtual_lines": [ { "bbox": [ 113, 152, 485, 164.66666666666666 ], "spans": [], "index": 6 }, { "bbox": [ 113, 164.66666666666666, 485, 177.33333333333331 ], "spans": [], "index": 7 }, { "bbox": [ 113, 177.33333333333331, 485, 189.99999999999997 ], "spans": [], "index": 8 } ] }, { "type": "text", "bbox": [ 108, 194, 504, 228 ], "lines": [ { "bbox": [ 106, 194, 505, 207 ], "spans": [ { "bbox": [ 106, 194, 239, 207 ], "score": 1.0, "content": "Remark 4.2. APVI (Algorithm", "type": "text" }, { "bbox": [ 239, 196, 245, 205 ], "score": 0.26, "content": "I", "type": "inline_equation" }, { "bbox": [ 245, 194, 505, 207 ], "score": 1.0, "content": ") can also be called LCBVI-Bernstein as it creates the offline", "type": "text" } ], "index": 9 }, { "bbox": [ 105, 205, 505, 219 ], "spans": [ { "bbox": [ 105, 205, 505, 219 ], "score": 1.0, "content": "counterpart of UCBVI in Azar et al. [2017]. However, to highlight that the resulting bound fully", "type": "text" } ], "index": 10 }, { "bbox": [ 106, 216, 411, 229 ], "spans": [ { "bbox": [ 106, 216, 411, 229 ], "score": 1.0, "content": "adapts to the specific system structure, we use the word “adaptive” instead.", "type": "text" } ], "index": 11 } ], "index": 10 }, { "type": "text", "bbox": [ 106, 236, 506, 379 ], "lines": [ { "bbox": [ 105, 236, 506, 250 ], "spans": [ { "bbox": [ 105, 236, 506, 250 ], "score": 1.0, "content": "APVI makes significant improvements in a lot of aspects. First and foremost, the dominate term", "type": "text" } ], "index": 12 }, { "bbox": [ 105, 248, 506, 261 ], "spans": [ { "bbox": [ 105, 248, 439, 261 ], "score": 1.0, "content": "is fully expressed by the system quantities that admits no explicit dependence on", "type": "text" }, { "bbox": [ 439, 248, 471, 259 ], "score": 0.91, "content": "H , S , A", "type": "inline_equation" }, { "bbox": [ 472, 248, 506, 261 ], "score": 1.0, "content": ". To the", "type": "text" } ], "index": 13 }, { "bbox": [ 105, 258, 506, 272 ], "spans": [ { "bbox": [ 105, 258, 506, 272 ], "score": 1.0, "content": "best of our knowledge, this is the first offline RL bound that concretely depicts the interrelations", "type": "text" } ], "index": 14 }, { "bbox": [ 105, 270, 506, 282 ], "spans": [ { "bbox": [ 105, 270, 334, 282 ], "score": 1.0, "content": "within the problem when the problem instance is a tuple", "type": "text" }, { "bbox": [ 334, 270, 378, 282 ], "score": 0.93, "content": "( M , \\pi ^ { \\star } , \\mu )", "type": "inline_equation" }, { "bbox": [ 379, 270, 419, 282 ], "score": 1.0, "content": ": an MDP", "type": "text" }, { "bbox": [ 419, 270, 431, 280 ], "score": 0.67, "content": "M", "type": "inline_equation" }, { "bbox": [ 432, 270, 506, 282 ], "score": 1.0, "content": "(coupled with the", "type": "text" } ], "index": 15 }, { "bbox": [ 105, 281, 506, 294 ], "spans": [ { "bbox": [ 105, 281, 164, 294 ], "score": 1.0, "content": "optimal policy", "type": "text" }, { "bbox": [ 165, 281, 177, 291 ], "score": 0.81, "content": "\\pi ^ { \\star }", "type": "inline_equation" }, { "bbox": [ 177, 281, 380, 294 ], "score": 1.0, "content": ") with the data rolling from an offline logging policy", "type": "text" }, { "bbox": [ 381, 282, 388, 292 ], "score": 0.73, "content": "\\mu", "type": "inline_equation" }, { "bbox": [ 388, 281, 506, 294 ], "score": 1.0, "content": ". As we will discuss later, this", "type": "text" } ], "index": 16 }, { "bbox": [ 106, 291, 505, 303 ], "spans": [ { "bbox": [ 106, 291, 505, 303 ], "score": 1.0, "content": "result indicates (nearly) all the optimal worst-case non-adaptive bounds (and clearly also the VPVI)", "type": "text" } ], "index": 17 }, { "bbox": [ 105, 303, 505, 316 ], "spans": [ { "bbox": [ 105, 303, 505, 316 ], "score": 1.0, "content": "under their respective regimes / assumptions. Thus, (5) is generic. More interestingly, Theorem 4.1", "type": "text" } ], "index": 18 }, { "bbox": [ 105, 312, 506, 327 ], "spans": [ { "bbox": [ 105, 312, 506, 327 ], "score": 1.0, "content": "caters to the specific MDP structures and adaptively yields improved sample complexities (e.g. faster", "type": "text" } ], "index": 19 }, { "bbox": [ 105, 324, 506, 336 ], "spans": [ { "bbox": [ 105, 324, 506, 336 ], "score": 1.0, "content": "convergence in deterministic systems) that existing works cannot imply. Such features are crucial as", "type": "text" } ], "index": 20 }, { "bbox": [ 105, 336, 505, 347 ], "spans": [ { "bbox": [ 105, 336, 505, 347 ], "score": 1.0, "content": "it helps us to understand what type of problems are harder / easier than others, and even more, in", "type": "text" } ], "index": 21 }, { "bbox": [ 104, 347, 506, 358 ], "spans": [ { "bbox": [ 104, 347, 506, 358 ], "score": 1.0, "content": "a quantitative way. Last but not least, to illustrate this bound exhibits the intrinsic nature of offline", "type": "text" } ], "index": 22 }, { "bbox": [ 105, 357, 506, 370 ], "spans": [ { "bbox": [ 105, 357, 506, 370 ], "score": 1.0, "content": "RL, we prove a per-instance dependent information-theoretical lower bound that shares a similar", "type": "text" } ], "index": 23 }, { "bbox": [ 106, 369, 383, 380 ], "spans": [ { "bbox": [ 106, 369, 383, 380 ], "score": 1.0, "content": "formulation. The proof of Theorem 4.3 can be found in Appendix G.", "type": "text" } ], "index": 24 } ], "index": 18 }, { "type": "text", "bbox": [ 106, 382, 506, 453 ], "lines": [ { "bbox": [ 105, 380, 506, 395 ], "spans": [ { "bbox": [ 105, 380, 506, 395 ], "score": 1.0, "content": "Theorem 4.3 (Instance-dependent information theoretical offline lower bound). Denote", "type": "text" } ], "index": 25 }, { "bbox": [ 107, 392, 507, 408 ], "spans": [ { "bbox": [ 107, 394, 241, 406 ], "score": 0.39, "content": "\\begin{array} { c c c c c c } { { \\mathcal G } } & { { : = } } & { { \\{ ( \\mu , M ) } } & { { : } } & { { \\exists \\pi ^ { \\star } } } & { { s } } \\end{array}", "type": "inline_equation" }, { "bbox": [ 241, 392, 259, 408 ], "score": 1.0, "content": ".t.", "type": "text" }, { "bbox": [ 259, 394, 435, 407 ], "score": 0.46, "content": "d _ { h } ^ { \\mu } ( s , a ) \\quad > \\quad 0 \\quad i f \\quad d _ { h } ^ { \\pi ^ { \\star } } ( s , a ) \\quad > \\quad 0 \\}", "type": "inline_equation" }, { "bbox": [ 435, 392, 507, 408 ], "score": 1.0, "content": ". Fix an in-", "type": "text" } ], "index": 26 }, { "bbox": [ 105, 403, 505, 420 ], "spans": [ { "bbox": [ 105, 404, 139, 419 ], "score": 1.0, "content": "stance", "type": "text" }, { "bbox": [ 140, 407, 250, 418 ], "score": 0.85, "content": "\\begin{array} { r l r l r } { { \\mathcal { P } } } & { { } = } & { ( \\mu , M ) } & { { } \\in { } } & { { \\mathcal { G } } } \\end{array}", "type": "inline_equation" }, { "bbox": [ 251, 404, 259, 419 ], "score": 1.0, "content": ".", "type": "text" }, { "bbox": [ 263, 403, 292, 420 ], "score": 1.0, "content": "Let", "type": "text" }, { "bbox": [ 293, 407, 302, 416 ], "score": 0.68, "content": "\\mathcal { D }", "type": "inline_equation" }, { "bbox": [ 303, 403, 364, 420 ], "score": 1.0, "content": "consists of", "type": "text" }, { "bbox": [ 365, 408, 372, 416 ], "score": 0.54, "content": "n", "type": "inline_equation" }, { "bbox": [ 372, 403, 476, 420 ], "score": 1.0, "content": "episodes and define", "type": "text" }, { "bbox": [ 476, 406, 505, 418 ], "score": 0.86, "content": "\\begin{array} { r l } { \\xi } & { { } = } \\end{array}", "type": "inline_equation" } ], "index": 27 }, { "bbox": [ 106, 413, 507, 443 ], "spans": [ { "bbox": [ 106, 421, 444, 443 ], "score": 0.76, "content": "\\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}", "type": "inline_equation" }, { "bbox": [ 278, 413, 507, 434 ], "score": 1.0, "content": "Ph(sh+1|sh,ah)(V ?h+1(sh+1)−EPsh,ah [V ?h+1]) . Let π to be", "type": "text" } ], "index": 28 }, { "bbox": [ 106, 441, 417, 454 ], "spans": [ { "bbox": [ 106, 441, 417, 454 ], "score": 1.0, "content": "the output of any algorithm. Define the local non-asymptotic minimax risk as", "type": "text" } ], "index": 29 } ], "index": 27 }, { "type": "interline_equation", "bbox": [ 197, 458, 413, 483 ], "lines": [ { "bbox": [ 197, 458, 413, 483 ], "spans": [ { "bbox": [ 197, 458, 413, 483 ], "score": 0.89, "content": "\\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]", "type": "interline_equation", "image_path": "bfd5ed1aee31bf18d0b649e656af2ebf48e938e4c9c6005c4991905767ec6df8.jpg" } ] } ], "index": 30, "virtual_lines": [ { "bbox": [ 197, 458, 413, 483 ], "spans": [], "index": 30 } ] }, { "type": "text", "bbox": [ 107, 488, 506, 538 ], "lines": [ { "bbox": [ 106, 487, 506, 501 ], "spans": [ { "bbox": [ 106, 487, 134, 501 ], "score": 1.0, "content": "where", "type": "text" }, { "bbox": [ 135, 489, 161, 501 ], "score": 0.92, "content": "v ^ { \\star } ( \\mathcal { Q } )", "type": "inline_equation" }, { "bbox": [ 162, 487, 354, 501 ], "score": 1.0, "content": "denotes the optimal value under the instance", "type": "text" }, { "bbox": [ 355, 489, 364, 499 ], "score": 0.73, "content": "\\mathcal { Q }", "type": "inline_equation" }, { "bbox": [ 364, 487, 506, 501 ], "score": 1.0, "content": ". Then there exists universal con-", "type": "text" } ], "index": 31 }, { "bbox": [ 102, 500, 507, 527 ], "spans": [ { "bbox": [ 102, 500, 132, 527 ], "score": 1.0, "content": "stants", "type": "text" }, { "bbox": [ 132, 509, 182, 520 ], "score": 0.91, "content": "c _ { 0 } , p , C > 0", "type": "inline_equation" }, { "bbox": [ 182, 500, 231, 527 ], "score": 1.0, "content": ", such that if", "type": "text" }, { "bbox": [ 231, 501, 481, 526 ], "score": 0.93, "content": "\\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}", "type": "inline_equation" }, { "bbox": [ 481, 501, 507, 523 ], "score": 1.0, "content": ", with", "type": "text" } ], "index": 32 }, { "bbox": [ 106, 525, 351, 538 ], "spans": [ { "bbox": [ 106, 525, 189, 538 ], "score": 1.0, "content": "constant probability", "type": "text" }, { "bbox": [ 189, 526, 213, 537 ], "score": 0.91, "content": "p > 0", "type": "inline_equation" }, { "bbox": [ 214, 525, 298, 538 ], "score": 1.0, "content": ", Then we have (here", "type": "text" }, { "bbox": [ 298, 525, 344, 538 ], "score": 0.9, "content": "\\zeta = H / \\bar { d } _ { m , }", "type": "inline_equation" }, { "bbox": [ 345, 525, 351, 538 ], "score": 1.0, "content": "):", "type": "text" } ], "index": 33 } ], "index": 32 }, { "type": "interline_equation", "bbox": [ 165, 542, 444, 580 ], "lines": [ { "bbox": [ 165, 542, 444, 580 ], "spans": [ { "bbox": [ 165, 542, 444, 580 ], "score": 0.93, "content": "\\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 } ) } } ,", "type": "interline_equation", "image_path": "cfc471f5a55860600a234cf4fe3c31689b97a7ff564bc439b67603e14bceb99e.jpg" } ] } ], "index": 35, "virtual_lines": [ { "bbox": [ 165, 542, 444, 554.6666666666666 ], "spans": [], "index": 34 }, { "bbox": [ 165, 554.6666666666666, 444, 567.3333333333333 ], "spans": [], "index": 35 }, { "bbox": [ 165, 567.3333333333333, 444, 579.9999999999999 ], "spans": [], "index": 36 } ] }, { "type": "text", "bbox": [ 106, 585, 304, 599 ], "lines": [ { "bbox": [ 106, 585, 305, 600 ], "spans": [ { "bbox": [ 106, 585, 133, 600 ], "score": 1.0, "content": "where", "type": "text" }, { "bbox": [ 133, 586, 184, 598 ], "score": 0.93, "content": "\\mathcal { P } = ( \\mu , M )", "type": "inline_equation" }, { "bbox": [ 184, 585, 203, 600 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 203, 586, 301, 598 ], "score": 0.9, "content": "M = \\left( S , A , P , r , H , d _ { 1 } \\right)", "type": "inline_equation" }, { "bbox": [ 302, 585, 305, 600 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 37 } ], "index": 37 }, { "type": "text", "bbox": [ 106, 606, 505, 662 ], "lines": [ { "bbox": [ 105, 605, 504, 619 ], "spans": [ { "bbox": [ 105, 605, 328, 619 ], "score": 1.0, "content": "The interpretation of Theorem 4.3 is: for any instance", "type": "text" }, { "bbox": [ 328, 607, 337, 617 ], "score": 0.81, "content": "\\mathcal { P }", "type": "inline_equation" }, { "bbox": [ 337, 605, 475, 619 ], "score": 1.0, "content": ", learning requires (7) (divided by", "type": "text" }, { "bbox": [ 476, 606, 504, 618 ], "score": 0.9, "content": "1 / \\sqrt { n } )", "type": "inline_equation" } ], "index": 38 }, { "bbox": [ 106, 617, 506, 630 ], "spans": [ { "bbox": [ 106, 617, 506, 630 ], "score": 1.0, "content": "for any algorithm. Note this notion is significantly stronger than the previous minimax offline lower", "type": "text" } ], "index": 39 }, { "bbox": [ 105, 626, 506, 642 ], "spans": [ { "bbox": [ 105, 626, 506, 642 ], "score": 1.0, "content": "bounds [Yin et al., 2021a, Rashidinejad et al., 2021, Xie et al., 2021b, Jin et al., 2020] (where they", "type": "text" } ], "index": 40 }, { "bbox": [ 106, 639, 505, 651 ], "spans": [ { "bbox": [ 106, 639, 505, 651 ], "score": 1.0, "content": "only select a particular family of hard problems), therefore, their lower bounds in general do not hold", "type": "text" } ], "index": 41 }, { "bbox": [ 105, 649, 205, 662 ], "spans": [ { "bbox": [ 105, 649, 205, 662 ], "score": 1.0, "content": "for individual instances.", "type": "text" } ], "index": 42 } ], "index": 40 }, { "type": "text", "bbox": [ 106, 666, 505, 700 ], "lines": [ { "bbox": [ 106, 666, 506, 678 ], "spans": [ { "bbox": [ 106, 666, 506, 678 ], "score": 1.0, "content": "The quantity (1) nearly-matches the per-instance lower bound (7) (they deviate by a factor of", "type": "text" } ], "index": 43 }, { "bbox": [ 107, 676, 506, 690 ], "spans": [ { "bbox": [ 107, 677, 154, 689 ], "score": 0.93, "content": "\\zeta = \\hat { H } / \\hat { d } _ { m }", "type": "inline_equation" }, { "bbox": [ 155, 676, 506, 690 ], "score": 1.0, "content": "due to the technical reason) and, in addition, we provide a matching minimax lower", "type": "text" } ], "index": 44 }, { "bbox": [ 106, 689, 506, 701 ], "spans": [ { "bbox": [ 106, 689, 506, 701 ], "score": 1.0, "content": "bound in Appendix H. These results certify Theorem 4.1 is not only adaptive but also near-optimal.", "type": "text" } ], "index": 45 } ], "index": 44 }, { "type": "text", "bbox": [ 105, 702, 504, 725 ], "lines": [ { "bbox": [ 102, 698, 506, 727 ], "spans": [ { "bbox": [ 102, 698, 218, 727 ], "score": 1.0, "content": "Hence, we call the quantity", "type": "text" }, { "bbox": [ 218, 700, 441, 726 ], "score": 0.9, "content": "\\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}", "type": "inline_equation" }, { "bbox": [ 441, 698, 506, 726 ], "score": 1.0, "content": "intrinsic offline", "type": "text" } ], "index": 46 } ], "index": 46 } ], "page_idx": 5, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 302, 741, 309, 750 ], "lines": [ { "bbox": [ 302, 741, 310, 752 ], "spans": [ { "bbox": [ 302, 741, 310, 752 ], "score": 1.0, "content": "6", "type": "text" } ] } ] } ], "para_blocks": [ { "type": "list", "bbox": [ 105, 71, 505, 95 ], "lines": [ { "bbox": [ 105, 70, 507, 86 ], "spans": [ { "bbox": [ 105, 70, 280, 86 ], "score": 1.0, "content": "iteration (APVI) into the Algorithm 1, with", "type": "text" }, { "bbox": [ 280, 70, 307, 84 ], "score": 0.93, "content": "\\widehat { P } _ { h } , \\widehat { r } _ { h }", "type": "inline_equation" }, { "bbox": [ 307, 70, 507, 86 ], "score": 1.0, "content": "defined in (2). APVI has the following guarantee.", "type": "text" } ], "index": 0, "is_list_end_line": true }, { "bbox": [ 105, 83, 465, 96 ], "spans": [ { "bbox": [ 105, 83, 465, 96 ], "score": 1.0, "content": "bA sketch of the analysis is presented in Section B and Appendix F includes the full proof.", "type": "text" } ], "index": 1, "is_list_start_line": true, "is_list_end_line": true } ], "index": 0.5, "bbox_fs": [ 105, 70, 507, 96 ] }, { "type": "text", "bbox": [ 107, 97, 505, 148 ], "lines": [ { "bbox": [ 105, 95, 505, 112 ], "spans": [ { "bbox": [ 105, 95, 468, 112 ], "score": 1.0, "content": "Theorem 4.1 (Intrinsic offline RL bound). Under the Assumption 2.3, denote", "type": "text" }, { "bbox": [ 468, 97, 505, 110 ], "score": 0.86, "content": "\\bar { d } _ { m } : =", "type": "inline_equation" } ], "index": 2 }, { "bbox": [ 106, 109, 505, 122 ], "spans": [ { "bbox": [ 106, 109, 276, 122 ], "score": 0.91, "content": "\\mathrm { m i n } _ { h \\in [ H ] } \\{ d _ { h } ^ { \\mu } ( s _ { h } , a _ { h } ) \\ : \\ d _ { h } ^ { \\mu } ( s _ { h } , a _ { h } ) \\ > \\ 0 \\}", "type": "inline_equation" }, { "bbox": [ 276, 109, 320, 122 ], "score": 1.0, "content": ". For any", "type": "text" }, { "bbox": [ 321, 110, 371, 120 ], "score": 0.9, "content": "0 ~ < ~ \\delta ~ < ~ 1", "type": "inline_equation" }, { "bbox": [ 371, 109, 505, 122 ], "score": 1.0, "content": ", there exists absolute constants", "type": "text" } ], "index": 3 }, { "bbox": [ 107, 121, 505, 135 ], "spans": [ { "bbox": [ 107, 123, 150, 134 ], "score": 0.91, "content": "c _ { 0 } , C ^ { \\prime } > 0", "type": "inline_equation" }, { "bbox": [ 150, 121, 216, 135 ], "score": 1.0, "content": ", such that when", "type": "text" }, { "bbox": [ 216, 122, 365, 134 ], "score": 0.67, "content": "n > c _ { 0 } \\cdot 1 / \\bar { d } _ { m } \\cdot \\iota ( \\iota = \\log ( H S A / \\delta ) )", "type": "inline_equation" }, { "bbox": [ 365, 121, 435, 135 ], "score": 1.0, "content": ", with probability", "type": "text" }, { "bbox": [ 435, 122, 459, 133 ], "score": 0.76, "content": "1 - \\delta", "type": "inline_equation" }, { "bbox": [ 459, 121, 505, 135 ], "score": 1.0, "content": ", the output", "type": "text" } ], "index": 4 }, { "bbox": [ 105, 134, 444, 149 ], "spans": [ { "bbox": [ 105, 135, 133, 149 ], "score": 1.0, "content": "policy", "type": "text" }, { "bbox": [ 133, 136, 141, 145 ], "score": 0.73, "content": "\\widehat { \\pi }", "type": "inline_equation" }, { "bbox": [ 141, 135, 222, 149 ], "score": 1.0, "content": "of APVI (Algorithm", "type": "text" }, { "bbox": [ 223, 137, 228, 146 ], "score": 0.25, "content": "I", "type": "inline_equation" }, { "bbox": [ 228, 135, 270, 149 ], "score": 1.0, "content": ") satisfies", "type": "text" }, { "bbox": [ 271, 134, 280, 146 ], "score": 0.74, "content": "\\widetilde O", "type": "inline_equation" }, { "bbox": [ 280, 135, 444, 149 ], "score": 1.0, "content": "hides log factor and higher order terms)", "type": "text" } ], "index": 5 } ], "index": 3.5, "bbox_fs": [ 105, 95, 505, 149 ] }, { "type": "interline_equation", "bbox": [ 113, 152, 485, 190 ], "lines": [ { "bbox": [ 113, 152, 485, 190 ], "spans": [ { "bbox": [ 113, 152, 485, 190 ], "score": 0.94, "content": "0 \\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)", "type": "interline_equation", "image_path": "f8faee4d11f76593d45f80f3acf4eb263d1dc36bed744ce74f364b2ebd8d24c8.jpg" } ] } ], "index": 7, "virtual_lines": [ { "bbox": [ 113, 152, 485, 164.66666666666666 ], "spans": [], "index": 6 }, { "bbox": [ 113, 164.66666666666666, 485, 177.33333333333331 ], "spans": [], "index": 7 }, { "bbox": [ 113, 177.33333333333331, 485, 189.99999999999997 ], "spans": [], "index": 8 } ] }, { "type": "text", "bbox": [ 108, 194, 504, 228 ], "lines": [ { "bbox": [ 106, 194, 505, 207 ], "spans": [ { "bbox": [ 106, 194, 239, 207 ], "score": 1.0, "content": "Remark 4.2. APVI (Algorithm", "type": "text" }, { "bbox": [ 239, 196, 245, 205 ], "score": 0.26, "content": "I", "type": "inline_equation" }, { "bbox": [ 245, 194, 505, 207 ], "score": 1.0, "content": ") can also be called LCBVI-Bernstein as it creates the offline", "type": "text" } ], "index": 9 }, { "bbox": [ 105, 205, 505, 219 ], "spans": [ { "bbox": [ 105, 205, 505, 219 ], "score": 1.0, "content": "counterpart of UCBVI in Azar et al. [2017]. However, to highlight that the resulting bound fully", "type": "text" } ], "index": 10 }, { "bbox": [ 106, 216, 411, 229 ], "spans": [ { "bbox": [ 106, 216, 411, 229 ], "score": 1.0, "content": "adapts to the specific system structure, we use the word “adaptive” instead.", "type": "text" } ], "index": 11 } ], "index": 10, "bbox_fs": [ 105, 194, 505, 229 ] }, { "type": "text", "bbox": [ 106, 236, 506, 379 ], "lines": [ { "bbox": [ 105, 236, 506, 250 ], "spans": [ { "bbox": [ 105, 236, 506, 250 ], "score": 1.0, "content": "APVI makes significant improvements in a lot of aspects. First and foremost, the dominate term", "type": "text" } ], "index": 12 }, { "bbox": [ 105, 248, 506, 261 ], "spans": [ { "bbox": [ 105, 248, 439, 261 ], "score": 1.0, "content": "is fully expressed by the system quantities that admits no explicit dependence on", "type": "text" }, { "bbox": [ 439, 248, 471, 259 ], "score": 0.91, "content": "H , S , A", "type": "inline_equation" }, { "bbox": [ 472, 248, 506, 261 ], "score": 1.0, "content": ". To the", "type": "text" } ], "index": 13 }, { "bbox": [ 105, 258, 506, 272 ], "spans": [ { "bbox": [ 105, 258, 506, 272 ], "score": 1.0, "content": "best of our knowledge, this is the first offline RL bound that concretely depicts the interrelations", "type": "text" } ], "index": 14 }, { "bbox": [ 105, 270, 506, 282 ], "spans": [ { "bbox": [ 105, 270, 334, 282 ], "score": 1.0, "content": "within the problem when the problem instance is a tuple", "type": "text" }, { "bbox": [ 334, 270, 378, 282 ], "score": 0.93, "content": "( M , \\pi ^ { \\star } , \\mu )", "type": "inline_equation" }, { "bbox": [ 379, 270, 419, 282 ], "score": 1.0, "content": ": an MDP", "type": "text" }, { "bbox": [ 419, 270, 431, 280 ], "score": 0.67, "content": "M", "type": "inline_equation" }, { "bbox": [ 432, 270, 506, 282 ], "score": 1.0, "content": "(coupled with the", "type": "text" } ], "index": 15 }, { "bbox": [ 105, 281, 506, 294 ], "spans": [ { "bbox": [ 105, 281, 164, 294 ], "score": 1.0, "content": "optimal policy", "type": "text" }, { "bbox": [ 165, 281, 177, 291 ], "score": 0.81, "content": "\\pi ^ { \\star }", "type": "inline_equation" }, { "bbox": [ 177, 281, 380, 294 ], "score": 1.0, "content": ") with the data rolling from an offline logging policy", "type": "text" }, { "bbox": [ 381, 282, 388, 292 ], "score": 0.73, "content": "\\mu", "type": "inline_equation" }, { "bbox": [ 388, 281, 506, 294 ], "score": 1.0, "content": ". As we will discuss later, this", "type": "text" } ], "index": 16 }, { "bbox": [ 106, 291, 505, 303 ], "spans": [ { "bbox": [ 106, 291, 505, 303 ], "score": 1.0, "content": "result indicates (nearly) all the optimal worst-case non-adaptive bounds (and clearly also the VPVI)", "type": "text" } ], "index": 17 }, { "bbox": [ 105, 303, 505, 316 ], "spans": [ { "bbox": [ 105, 303, 505, 316 ], "score": 1.0, "content": "under their respective regimes / assumptions. Thus, (5) is generic. More interestingly, Theorem 4.1", "type": "text" } ], "index": 18 }, { "bbox": [ 105, 312, 506, 327 ], "spans": [ { "bbox": [ 105, 312, 506, 327 ], "score": 1.0, "content": "caters to the specific MDP structures and adaptively yields improved sample complexities (e.g. faster", "type": "text" } ], "index": 19 }, { "bbox": [ 105, 324, 506, 336 ], "spans": [ { "bbox": [ 105, 324, 506, 336 ], "score": 1.0, "content": "convergence in deterministic systems) that existing works cannot imply. Such features are crucial as", "type": "text" } ], "index": 20 }, { "bbox": [ 105, 336, 505, 347 ], "spans": [ { "bbox": [ 105, 336, 505, 347 ], "score": 1.0, "content": "it helps us to understand what type of problems are harder / easier than others, and even more, in", "type": "text" } ], "index": 21 }, { "bbox": [ 104, 347, 506, 358 ], "spans": [ { "bbox": [ 104, 347, 506, 358 ], "score": 1.0, "content": "a quantitative way. Last but not least, to illustrate this bound exhibits the intrinsic nature of offline", "type": "text" } ], "index": 22 }, { "bbox": [ 105, 357, 506, 370 ], "spans": [ { "bbox": [ 105, 357, 506, 370 ], "score": 1.0, "content": "RL, we prove a per-instance dependent information-theoretical lower bound that shares a similar", "type": "text" } ], "index": 23 }, { "bbox": [ 106, 369, 383, 380 ], "spans": [ { "bbox": [ 106, 369, 383, 380 ], "score": 1.0, "content": "formulation. The proof of Theorem 4.3 can be found in Appendix G.", "type": "text" } ], "index": 24 } ], "index": 18, "bbox_fs": [ 104, 236, 506, 380 ] }, { "type": "text", "bbox": [ 106, 382, 506, 453 ], "lines": [ { "bbox": [ 105, 380, 506, 395 ], "spans": [ { "bbox": [ 105, 380, 506, 395 ], "score": 1.0, "content": "Theorem 4.3 (Instance-dependent information theoretical offline lower bound). Denote", "type": "text" } ], "index": 25 }, { "bbox": [ 107, 392, 507, 408 ], "spans": [ { "bbox": [ 107, 394, 241, 406 ], "score": 0.39, "content": "\\begin{array} { c c c c c c } { { \\mathcal G } } & { { : = } } & { { \\{ ( \\mu , M ) } } & { { : } } & { { \\exists \\pi ^ { \\star } } } & { { s } } \\end{array}", "type": "inline_equation" }, { "bbox": [ 241, 392, 259, 408 ], "score": 1.0, "content": ".t.", "type": "text" }, { "bbox": [ 259, 394, 435, 407 ], "score": 0.46, "content": "d _ { h } ^ { \\mu } ( s , a ) \\quad > \\quad 0 \\quad i f \\quad d _ { h } ^ { \\pi ^ { \\star } } ( s , a ) \\quad > \\quad 0 \\}", "type": "inline_equation" }, { "bbox": [ 435, 392, 507, 408 ], "score": 1.0, "content": ". Fix an in-", "type": "text" } ], "index": 26 }, { "bbox": [ 105, 403, 505, 420 ], "spans": [ { "bbox": [ 105, 404, 139, 419 ], "score": 1.0, "content": "stance", "type": "text" }, { "bbox": [ 140, 407, 250, 418 ], "score": 0.85, "content": "\\begin{array} { r l r l r } { { \\mathcal { P } } } & { { } = } & { ( \\mu , M ) } & { { } \\in { } } & { { \\mathcal { G } } } \\end{array}", "type": "inline_equation" }, { "bbox": [ 251, 404, 259, 419 ], "score": 1.0, "content": ".", "type": "text" }, { "bbox": [ 263, 403, 292, 420 ], "score": 1.0, "content": "Let", "type": "text" }, { "bbox": [ 293, 407, 302, 416 ], "score": 0.68, "content": "\\mathcal { D }", "type": "inline_equation" }, { "bbox": [ 303, 403, 364, 420 ], "score": 1.0, "content": "consists of", "type": "text" }, { "bbox": [ 365, 408, 372, 416 ], "score": 0.54, "content": "n", "type": "inline_equation" }, { "bbox": [ 372, 403, 476, 420 ], "score": 1.0, "content": "episodes and define", "type": "text" }, { "bbox": [ 476, 406, 505, 418 ], "score": 0.86, "content": "\\begin{array} { r l } { \\xi } & { { } = } \\end{array}", "type": "inline_equation" } ], "index": 27 }, { "bbox": [ 106, 413, 507, 443 ], "spans": [ { "bbox": [ 106, 421, 444, 443 ], "score": 0.76, "content": "\\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}", "type": "inline_equation" }, { "bbox": [ 278, 413, 507, 434 ], "score": 1.0, "content": "Ph(sh+1|sh,ah)(V ?h+1(sh+1)−EPsh,ah [V ?h+1]) . Let π to be", "type": "text" } ], "index": 28 }, { "bbox": [ 106, 441, 417, 454 ], "spans": [ { "bbox": [ 106, 441, 417, 454 ], "score": 1.0, "content": "the output of any algorithm. Define the local non-asymptotic minimax risk as", "type": "text" } ], "index": 29 } ], "index": 27, "bbox_fs": [ 105, 380, 507, 454 ] }, { "type": "interline_equation", "bbox": [ 197, 458, 413, 483 ], "lines": [ { "bbox": [ 197, 458, 413, 483 ], "spans": [ { "bbox": [ 197, 458, 413, 483 ], "score": 0.89, "content": "\\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]", "type": "interline_equation", "image_path": "bfd5ed1aee31bf18d0b649e656af2ebf48e938e4c9c6005c4991905767ec6df8.jpg" } ] } ], "index": 30, "virtual_lines": [ { "bbox": [ 197, 458, 413, 483 ], "spans": [], "index": 30 } ] }, { "type": "text", "bbox": [ 107, 488, 506, 538 ], "lines": [ { "bbox": [ 106, 487, 506, 501 ], "spans": [ { "bbox": [ 106, 487, 134, 501 ], "score": 1.0, "content": "where", "type": "text" }, { "bbox": [ 135, 489, 161, 501 ], "score": 0.92, "content": "v ^ { \\star } ( \\mathcal { Q } )", "type": "inline_equation" }, { "bbox": [ 162, 487, 354, 501 ], "score": 1.0, "content": "denotes the optimal value under the instance", "type": "text" }, { "bbox": [ 355, 489, 364, 499 ], "score": 0.73, "content": "\\mathcal { Q }", "type": "inline_equation" }, { "bbox": [ 364, 487, 506, 501 ], "score": 1.0, "content": ". Then there exists universal con-", "type": "text" } ], "index": 31 }, { "bbox": [ 102, 500, 507, 527 ], "spans": [ { "bbox": [ 102, 500, 132, 527 ], "score": 1.0, "content": "stants", "type": "text" }, { "bbox": [ 132, 509, 182, 520 ], "score": 0.91, "content": "c _ { 0 } , p , C > 0", "type": "inline_equation" }, { "bbox": [ 182, 500, 231, 527 ], "score": 1.0, "content": ", such that if", "type": "text" }, { "bbox": [ 231, 501, 481, 526 ], "score": 0.93, "content": "\\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}", "type": "inline_equation" }, { "bbox": [ 481, 501, 507, 523 ], "score": 1.0, "content": ", with", "type": "text" } ], "index": 32 }, { "bbox": [ 106, 525, 351, 538 ], "spans": [ { "bbox": [ 106, 525, 189, 538 ], "score": 1.0, "content": "constant probability", "type": "text" }, { "bbox": [ 189, 526, 213, 537 ], "score": 0.91, "content": "p > 0", "type": "inline_equation" }, { "bbox": [ 214, 525, 298, 538 ], "score": 1.0, "content": ", Then we have (here", "type": "text" }, { "bbox": [ 298, 525, 344, 538 ], "score": 0.9, "content": "\\zeta = H / \\bar { d } _ { m , }", "type": "inline_equation" }, { "bbox": [ 345, 525, 351, 538 ], "score": 1.0, "content": "):", "type": "text" } ], "index": 33 } ], "index": 32, "bbox_fs": [ 102, 487, 507, 538 ] }, { "type": "interline_equation", "bbox": [ 165, 542, 444, 580 ], "lines": [ { "bbox": [ 165, 542, 444, 580 ], "spans": [ { "bbox": [ 165, 542, 444, 580 ], "score": 0.93, "content": "\\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 } ) } } ,", "type": "interline_equation", "image_path": "cfc471f5a55860600a234cf4fe3c31689b97a7ff564bc439b67603e14bceb99e.jpg" } ] } ], "index": 35, "virtual_lines": [ { "bbox": [ 165, 542, 444, 554.6666666666666 ], "spans": [], "index": 34 }, { "bbox": [ 165, 554.6666666666666, 444, 567.3333333333333 ], "spans": [], "index": 35 }, { "bbox": [ 165, 567.3333333333333, 444, 579.9999999999999 ], "spans": [], "index": 36 } ] }, { "type": "text", "bbox": [ 106, 585, 304, 599 ], "lines": [ { "bbox": [ 106, 585, 305, 600 ], "spans": [ { "bbox": [ 106, 585, 133, 600 ], "score": 1.0, "content": "where", "type": "text" }, { "bbox": [ 133, 586, 184, 598 ], "score": 0.93, "content": "\\mathcal { P } = ( \\mu , M )", "type": "inline_equation" }, { "bbox": [ 184, 585, 203, 600 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 203, 586, 301, 598 ], "score": 0.9, "content": "M = \\left( S , A , P , r , H , d _ { 1 } \\right)", "type": "inline_equation" }, { "bbox": [ 302, 585, 305, 600 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 37 } ], "index": 37, "bbox_fs": [ 106, 585, 305, 600 ] }, { "type": "text", "bbox": [ 106, 606, 505, 662 ], "lines": [ { "bbox": [ 105, 605, 504, 619 ], "spans": [ { "bbox": [ 105, 605, 328, 619 ], "score": 1.0, "content": "The interpretation of Theorem 4.3 is: for any instance", "type": "text" }, { "bbox": [ 328, 607, 337, 617 ], "score": 0.81, "content": "\\mathcal { P }", "type": "inline_equation" }, { "bbox": [ 337, 605, 475, 619 ], "score": 1.0, "content": ", learning requires (7) (divided by", "type": "text" }, { "bbox": [ 476, 606, 504, 618 ], "score": 0.9, "content": "1 / \\sqrt { n } )", "type": "inline_equation" } ], "index": 38 }, { "bbox": [ 106, 617, 506, 630 ], "spans": [ { "bbox": [ 106, 617, 506, 630 ], "score": 1.0, "content": "for any algorithm. Note this notion is significantly stronger than the previous minimax offline lower", "type": "text" } ], "index": 39 }, { "bbox": [ 105, 626, 506, 642 ], "spans": [ { "bbox": [ 105, 626, 506, 642 ], "score": 1.0, "content": "bounds [Yin et al., 2021a, Rashidinejad et al., 2021, Xie et al., 2021b, Jin et al., 2020] (where they", "type": "text" } ], "index": 40 }, { "bbox": [ 106, 639, 505, 651 ], "spans": [ { "bbox": [ 106, 639, 505, 651 ], "score": 1.0, "content": "only select a particular family of hard problems), therefore, their lower bounds in general do not hold", "type": "text" } ], "index": 41 }, { "bbox": [ 105, 649, 205, 662 ], "spans": [ { "bbox": [ 105, 649, 205, 662 ], "score": 1.0, "content": "for individual instances.", "type": "text" } ], "index": 42 } ], "index": 40, "bbox_fs": [ 105, 605, 506, 662 ] }, { "type": "text", "bbox": [ 106, 666, 505, 700 ], "lines": [ { "bbox": [ 106, 666, 506, 678 ], "spans": [ { "bbox": [ 106, 666, 506, 678 ], "score": 1.0, "content": "The quantity (1) nearly-matches the per-instance lower bound (7) (they deviate by a factor of", "type": "text" } ], "index": 43 }, { "bbox": [ 107, 676, 506, 690 ], "spans": [ { "bbox": [ 107, 677, 154, 689 ], "score": 0.93, "content": "\\zeta = \\hat { H } / \\hat { d } _ { m }", "type": "inline_equation" }, { "bbox": [ 155, 676, 506, 690 ], "score": 1.0, "content": "due to the technical reason) and, in addition, we provide a matching minimax lower", "type": "text" } ], "index": 44 }, { "bbox": [ 106, 689, 506, 701 ], "spans": [ { "bbox": [ 106, 689, 506, 701 ], "score": 1.0, "content": "bound in Appendix H. These results certify Theorem 4.1 is not only adaptive but also near-optimal.", "type": "text" } ], "index": 45 } ], "index": 44, "bbox_fs": [ 106, 666, 506, 701 ] }, { "type": "text", "bbox": [ 105, 702, 504, 725 ], "lines": [ { "bbox": [ 102, 698, 506, 727 ], "spans": [ { "bbox": [ 102, 698, 218, 727 ], "score": 1.0, "content": "Hence, we call the quantity", "type": "text" }, { "bbox": [ 218, 700, 441, 726 ], "score": 0.9, "content": "\\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}", "type": "inline_equation" }, { "bbox": [ 441, 698, 506, 726 ], "score": 1.0, "content": "intrinsic offline", "type": "text" } ], "index": 46 }, { "bbox": [ 105, 72, 505, 86 ], "spans": [ { "bbox": [ 105, 72, 505, 86 ], "score": 1.0, "content": "reinforcement learning bound. In the sequel, we provide thorough discussions to explain the intrinsic", "type": "text", "cross_page": true } ], "index": 0 }, { "bbox": [ 105, 83, 505, 96 ], "spans": [ { "bbox": [ 105, 83, 505, 96 ], "score": 1.0, "content": "bound embraces the fundamental challenges in offline RL and the strong adaptivity. The detailed", "type": "text", "cross_page": true } ], "index": 1 }, { "bbox": [ 105, 95, 441, 107 ], "spans": [ { "bbox": [ 105, 95, 441, 107 ], "score": 1.0, "content": "technical derivations that are missing in Section 4.1-4.4 are deferred to Appendix I.", "type": "text", "cross_page": true } ], "index": 2 } ], "index": 46, "bbox_fs": [ 102, 698, 506, 727 ] } ] }, { "preproc_blocks": [ { "type": "text", "bbox": [ 106, 72, 506, 106 ], "lines": [ { "bbox": [ 105, 72, 505, 86 ], "spans": [ { "bbox": [ 105, 72, 505, 86 ], "score": 1.0, "content": "reinforcement learning bound. In the sequel, we provide thorough discussions to explain the intrinsic", "type": "text" } ], "index": 0 }, { "bbox": [ 105, 83, 505, 96 ], "spans": [ { "bbox": [ 105, 83, 505, 96 ], "score": 1.0, "content": "bound embraces the fundamental challenges in offline RL and the strong adaptivity. The detailed", "type": "text" } ], "index": 1 }, { "bbox": [ 105, 95, 441, 107 ], "spans": [ { "bbox": [ 105, 95, 441, 107 ], "score": 1.0, "content": "technical derivations that are missing in Section 4.1-4.4 are deferred to Appendix I.", "type": "text" } ], "index": 2 } ], "index": 1 }, { "type": "image", "bbox": [ 194, 128, 416, 247 ], "blocks": [ { "type": "image_body", "bbox": [ 194, 128, 416, 247 ], "group_id": 0, "lines": [ { "bbox": [ 194, 128, 416, 247 ], "spans": [ { "bbox": [ 194, 128, 416, 247 ], "score": 0.959, "type": "image", "image_path": "6b83607ca2951ec3cf5d8f65a3f909d38e7e57c936199b24d666e4d9988b9fb3.jpg" } ] } ], "index": 6.5, "virtual_lines": [ { "bbox": [ 194, 128, 416, 142.875 ], "spans": [], "index": 3 }, { "bbox": [ 194, 142.875, 416, 157.75 ], "spans": [], "index": 4 }, { "bbox": [ 194, 157.75, 416, 172.625 ], "spans": [], "index": 5 }, { "bbox": [ 194, 172.625, 416, 187.5 ], "spans": [], "index": 6 }, { "bbox": [ 194, 187.5, 416, 202.375 ], "spans": [], "index": 7 }, { "bbox": [ 194, 202.375, 416, 217.25 ], "spans": [], "index": 8 }, { "bbox": [ 194, 217.25, 416, 232.125 ], "spans": [], "index": 9 }, { "bbox": [ 194, 232.125, 416, 247.0 ], "spans": [], "index": 10 } ] }, { "type": "image_caption", "bbox": [ 105, 260, 505, 283 ], "group_id": 0, "lines": [ { "bbox": [ 106, 259, 506, 272 ], "spans": [ { "bbox": [ 106, 259, 506, 272 ], "score": 1.0, "content": "Figure 1: A visualization on how intrinsic learning bound subsumes existing best-known results:", "type": "text" } ], "index": 11 }, { "bbox": [ 106, 271, 432, 283 ], "spans": [ { "bbox": [ 106, 271, 432, 283 ], "score": 1.0, "content": "uniform visitation, single concentrability (partial coverage) and adaptive domain.", "type": "text" } ], "index": 12 } ], "index": 11.5 } ], "index": 9.0 }, { "type": "title", "bbox": [ 106, 303, 357, 315 ], "lines": [ { "bbox": [ 105, 302, 358, 318 ], "spans": [ { "bbox": [ 105, 302, 358, 318 ], "score": 1.0, "content": "4.1 Optimality under Uniform data-coverage assumption", "type": "text" } ], "index": 13 } ], "index": 13 }, { "type": "text", "bbox": [ 107, 322, 505, 378 ], "lines": [ { "bbox": [ 105, 320, 502, 337 ], "spans": [ { "bbox": [ 105, 320, 365, 337 ], "score": 1.0, "content": "Under the uniform exploration Assumption 2.1 with parameter", "type": "text" }, { "bbox": [ 365, 322, 502, 335 ], "score": 0.9, "content": "d _ { m } : = \\operatorname* { m i n } _ { h , s _ { h , } a _ { h } } d _ { h } ^ { \\mu } ( s _ { h } , a _ { h } ) > 0", "type": "inline_equation" } ], "index": 14 }, { "bbox": [ 106, 332, 505, 345 ], "spans": [ { "bbox": [ 106, 332, 505, 345 ], "score": 1.0, "content": "Yin et al. [2021a] analyzes the model-based plug-in approach and obtains the optimal sample", "type": "text" } ], "index": 15 }, { "bbox": [ 106, 344, 506, 358 ], "spans": [ { "bbox": [ 106, 344, 155, 358 ], "score": 1.0, "content": "complexity", "type": "text" }, { "bbox": [ 156, 344, 211, 358 ], "score": 0.92, "content": "\\widetilde { O } ( H ^ { 3 } / d _ { m } \\epsilon ^ { 2 } )", "type": "inline_equation" }, { "bbox": [ 212, 344, 259, 358 ], "score": 1.0, "content": "and shows", "type": "text" }, { "bbox": [ 259, 345, 315, 358 ], "score": 0.92, "content": "\\Omega ( H ^ { 3 } / d _ { m } \\epsilon ^ { 2 } )", "type": "inline_equation" }, { "bbox": [ 315, 344, 506, 358 ], "score": 1.0, "content": "is also the lower bound. Indeed, this rate can", "type": "text" } ], "index": 16 }, { "bbox": [ 105, 354, 505, 369 ], "spans": [ { "bbox": [ 105, 354, 505, 369 ], "score": 1.0, "content": "ebe directly implied by the intrinsic RL bound via Cauchy inequality and the Sum of Total Variance", "type": "text" } ], "index": 17 }, { "bbox": [ 104, 366, 167, 379 ], "spans": [ { "bbox": [ 104, 366, 167, 379 ], "score": 1.0, "content": "(Lemma J.6):4", "type": "text" } ], "index": 18 } ], "index": 16 }, { "type": "interline_equation", "bbox": [ 118, 384, 485, 455 ], "lines": [ { "bbox": [ 118, 384, 485, 455 ], "spans": [ { "bbox": [ 118, 384, 485, 455 ], "score": 0.94, "content": "\\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}", "type": "interline_equation", "image_path": "e5ef868c25e2072276b0a54844afacd6756ec1bd12c00a9851ae9b4cd4604e8c.jpg" } ] } ], "index": 20, "virtual_lines": [ { "bbox": [ 118, 384, 485, 407.6666666666667 ], "spans": [], "index": 19 }, { "bbox": [ 118, 407.6666666666667, 485, 431.33333333333337 ], "spans": [], "index": 20 }, { "bbox": [ 118, 431.33333333333337, 485, 455.00000000000006 ], "spans": [], "index": 21 } ] }, { "type": "text", "bbox": [ 106, 460, 504, 484 ], "lines": [ { "bbox": [ 105, 459, 506, 475 ], "spans": [ { "bbox": [ 105, 459, 183, 475 ], "score": 1.0, "content": "which translates to", "type": "text" }, { "bbox": [ 183, 459, 239, 473 ], "score": 0.93, "content": "\\widetilde { O } ( H ^ { 3 } / d _ { m } \\epsilon ^ { 2 } )", "type": "inline_equation" }, { "bbox": [ 239, 459, 506, 475 ], "score": 1.0, "content": "complexity. Our result maintains the optimal worst-case guarantee", "type": "text" } ], "index": 22 }, { "bbox": [ 106, 471, 264, 485 ], "spans": [ { "bbox": [ 106, 471, 130, 485 ], "score": 1.0, "content": "when", "type": "text" }, { "bbox": [ 130, 474, 138, 484 ], "score": 0.8, "content": "\\mu", "type": "inline_equation" }, { "bbox": [ 138, 471, 264, 485 ], "score": 1.0, "content": "has the uniform data-coverage:", "type": "text" } ], "index": 23 } ], "index": 22.5 }, { "type": "text", "bbox": [ 106, 487, 504, 513 ], "lines": [ { "bbox": [ 105, 485, 505, 501 ], "spans": [ { "bbox": [ 105, 485, 505, 501 ], "score": 1.0, "content": "Proposition 4.4. Under Assumption 2.1 and apply Theorem 4.1, APVI achieves the sample complexity", "type": "text" } ], "index": 24 }, { "bbox": [ 106, 498, 447, 514 ], "spans": [ { "bbox": [ 106, 499, 173, 514 ], "score": 1.0, "content": "of minimax-rate", "type": "text" }, { "bbox": [ 173, 498, 229, 513 ], "score": 0.92, "content": "\\widetilde { O } ( H ^ { 3 } / d _ { m } \\epsilon ^ { 2 } )", "type": "inline_equation" }, { "bbox": [ 229, 499, 447, 514 ], "score": 1.0, "content": "(Theorem 4.1 and Theorem G.2 in Yin et al. [2021a]).", "type": "text" } ], "index": 25 } ], "index": 24.5 }, { "type": "text", "bbox": [ 106, 518, 505, 554 ], "lines": [ { "bbox": [ 105, 516, 506, 532 ], "spans": [ { "bbox": [ 105, 516, 461, 532 ], "score": 1.0, "content": "Remark 4.5. We believe if the MDP is time-invariant, then by a modified construction of", "type": "text" }, { "bbox": [ 462, 516, 470, 528 ], "score": 0.75, "content": "\\widehat { P }", "type": "inline_equation" }, { "bbox": [ 470, 516, 474, 532 ], "score": 1.0, "content": ",", "type": "text" }, { "bbox": [ 474, 519, 481, 528 ], "score": 0.46, "content": "\\widehat { r }", "type": "inline_equation" }, { "bbox": [ 482, 516, 506, 532 ], "score": 1.0, "content": "in (2)", "type": "text" } ], "index": 26 }, { "bbox": [ 105, 529, 506, 544 ], "spans": [ { "bbox": [ 105, 530, 269, 544 ], "score": 1.0, "content": "our result will imply the minimax-rate of", "type": "text" }, { "bbox": [ 269, 529, 325, 543 ], "score": 0.93, "content": "\\widetilde { O } ( H ^ { 2 } / d _ { m } \\epsilon ^ { 2 } )", "type": "inline_equation" }, { "bbox": [ 325, 530, 506, 544 ], "score": 1.0, "content": "as achieved in Yin et al. [2021b]. We include", "type": "text" } ], "index": 27 }, { "bbox": [ 106, 542, 226, 554 ], "spans": [ { "bbox": [ 106, 542, 226, 554 ], "score": 1.0, "content": "this discussion in Appendix I.", "type": "text" } ], "index": 28 } ], "index": 27 }, { "type": "title", "bbox": [ 106, 567, 374, 579 ], "lines": [ { "bbox": [ 105, 567, 374, 581 ], "spans": [ { "bbox": [ 105, 567, 374, 581 ], "score": 1.0, "content": "4.2 Bounded sum of total rewards and the Horizon-Free case", "type": "text" } ], "index": 29 } ], "index": 29 }, { "type": "text", "bbox": [ 106, 588, 506, 659 ], "lines": [ { "bbox": [ 106, 587, 511, 625 ], "spans": [ { "bbox": [ 106, 599, 210, 614 ], "score": 0.8, "content": "\\begin{array} { r } { r _ { h } \\geq 0 , \\sum _ { h = 1 } ^ { H } r _ { h } \\in [ 0 , 1 ] } \\end{array}", "type": "inline_equation" }, { "bbox": [ 210, 587, 511, 625 ], "score": 1.0, "content": "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", "type": "text" } ], "index": 30 }, { "bbox": [ 105, 622, 505, 636 ], "spans": [ { "bbox": [ 105, 622, 505, 636 ], "score": 1.0, "content": "Jiang and Agarwal [2018]. In offline RL, Ren et al. [2021] derives the nearly horizon-free worst case", "type": "text" } ], "index": 31 }, { "bbox": [ 105, 634, 506, 649 ], "spans": [ { "bbox": [ 105, 635, 134, 649 ], "score": 1.0, "content": "bound", "type": "text" }, { "bbox": [ 135, 634, 189, 649 ], "score": 0.93, "content": "\\widetilde { O } ( \\sqrt { 1 / n d _ { m } } )", "type": "inline_equation" }, { "bbox": [ 189, 635, 506, 649 ], "score": 1.0, "content": "for the time-invariant MDPs, under the Assumption 2.1. As a comparison, our", "type": "text" } ], "index": 32 }, { "bbox": [ 105, 645, 472, 659 ], "spans": [ { "bbox": [ 105, 645, 472, 659 ], "score": 1.0, "content": "Theorem 4.1 achieves the following guarantee for the time-varying (non-stationary) MDPs.", "type": "text" } ], "index": 33 } ], "index": 31.5 }, { "type": "text", "bbox": [ 107, 662, 505, 701 ], "lines": [ { "bbox": [ 104, 662, 507, 680 ], "spans": [ { "bbox": [ 104, 663, 210, 680 ], "score": 1.0, "content": "Proposition 4.6. Assume", "type": "text" }, { "bbox": [ 211, 664, 240, 676 ], "score": 0.8, "content": "r _ { h } \\ge 0 ,", "type": "inline_equation" }, { "bbox": [ 240, 663, 243, 680 ], "score": 1.0, "content": ",", "type": "text" }, { "bbox": [ 244, 662, 300, 677 ], "score": 0.89, "content": "\\textstyle \\sum _ { h = 1 } ^ { H } r _ { h } \\leq 1", "type": "inline_equation" }, { "bbox": [ 300, 663, 507, 680 ], "score": 1.0, "content": ". Then in the time-varying case AVPI (Theorem 4.1)", "type": "text" } ], "index": 34 }, { "bbox": [ 105, 676, 505, 691 ], "spans": [ { "bbox": [ 105, 677, 173, 691 ], "score": 1.0, "content": "outputs a policy", "type": "text" }, { "bbox": [ 173, 678, 181, 688 ], "score": 0.71, "content": "\\widehat { \\pi }", "type": "inline_equation" }, { "bbox": [ 181, 677, 311, 691 ], "score": 1.0, "content": "such that the suboptimality gap", "type": "text" }, { "bbox": [ 311, 677, 344, 689 ], "score": 0.89, "content": "\\boldsymbol { v } ^ { \\star } - \\boldsymbol { v } ^ { \\widehat { \\pi } }", "type": "inline_equation" }, { "bbox": [ 345, 677, 405, 691 ], "score": 1.0, "content": "is bounded by", "type": "text" }, { "bbox": [ 405, 676, 463, 691 ], "score": 0.93, "content": "\\widetilde { O } ( \\sqrt { H / n d _ { m } } )", "type": "inline_equation" }, { "bbox": [ 463, 677, 505, 691 ], "score": 1.0, "content": "with high", "type": "text" } ], "index": 35 }, { "bbox": [ 105, 689, 259, 702 ], "spans": [ { "bbox": [ 105, 689, 259, 702 ], "score": 1.0, "content": "bprobability under the Assumption 2.1.", "type": "text" } ], "index": 36 } ], "index": 35 } ], "page_idx": 6, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 118, 711, 416, 723 ], "lines": [ { "bbox": [ 118, 709, 417, 724 ], "spans": [ { "bbox": [ 118, 709, 142, 724 ], "score": 1.0, "content": "4Here", "type": "text" }, { "bbox": [ 142, 712, 151, 721 ], "score": 0.76, "content": "\\odot", "type": "inline_equation" }, { "bbox": [ 151, 709, 376, 724 ], "score": 1.0, "content": "denotes element-wise multiplication. 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[2021a] analyzes the model-based plug-in approach and obtains the optimal sample", "type": "text" } ], "index": 15 }, { "bbox": [ 106, 344, 506, 358 ], "spans": [ { "bbox": [ 106, 344, 155, 358 ], "score": 1.0, "content": "complexity", "type": "text" }, { "bbox": [ 156, 344, 211, 358 ], "score": 0.92, "content": "\\widetilde { O } ( H ^ { 3 } / d _ { m } \\epsilon ^ { 2 } )", "type": "inline_equation" }, { "bbox": [ 212, 344, 259, 358 ], "score": 1.0, "content": "and shows", "type": "text" }, { "bbox": [ 259, 345, 315, 358 ], "score": 0.92, "content": "\\Omega ( H ^ { 3 } / d _ { m } \\epsilon ^ { 2 } )", "type": "inline_equation" }, { "bbox": [ 315, 344, 506, 358 ], "score": 1.0, "content": "is also the lower bound. Indeed, this rate can", "type": "text" } ], "index": 16 }, { "bbox": [ 105, 354, 505, 369 ], "spans": [ { "bbox": [ 105, 354, 505, 369 ], "score": 1.0, "content": "ebe directly implied by the intrinsic RL bound via Cauchy inequality and the Sum of Total Variance", "type": "text" } ], "index": 17 }, { "bbox": [ 104, 366, 167, 379 ], "spans": [ { "bbox": [ 104, 366, 167, 379 ], "score": 1.0, "content": "(Lemma J.6):4", "type": "text" } ], "index": 18 } ], "index": 16, "bbox_fs": [ 104, 320, 506, 379 ] }, { "type": "interline_equation", "bbox": [ 118, 384, 485, 455 ], "lines": [ { "bbox": [ 118, 384, 485, 455 ], "spans": [ { "bbox": [ 118, 384, 485, 455 ], "score": 0.94, "content": "\\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}", "type": "interline_equation", "image_path": "e5ef868c25e2072276b0a54844afacd6756ec1bd12c00a9851ae9b4cd4604e8c.jpg" } ] } ], "index": 20, "virtual_lines": [ { "bbox": [ 118, 384, 485, 407.6666666666667 ], "spans": [], "index": 19 }, { "bbox": [ 118, 407.6666666666667, 485, 431.33333333333337 ], "spans": [], "index": 20 }, { "bbox": [ 118, 431.33333333333337, 485, 455.00000000000006 ], "spans": [], "index": 21 } ] }, { "type": "text", "bbox": [ 106, 460, 504, 484 ], "lines": [ { "bbox": [ 105, 459, 506, 475 ], "spans": [ { "bbox": [ 105, 459, 183, 475 ], "score": 1.0, "content": "which translates to", "type": "text" }, { "bbox": [ 183, 459, 239, 473 ], "score": 0.93, "content": "\\widetilde { O } ( H ^ { 3 } / d _ { m } \\epsilon ^ { 2 } )", "type": "inline_equation" }, { "bbox": [ 239, 459, 506, 475 ], "score": 1.0, "content": "complexity. Our result maintains the optimal worst-case guarantee", "type": "text" } ], "index": 22 }, { "bbox": [ 106, 471, 264, 485 ], "spans": [ { "bbox": [ 106, 471, 130, 485 ], "score": 1.0, "content": "when", "type": "text" }, { "bbox": [ 130, 474, 138, 484 ], "score": 0.8, "content": "\\mu", "type": "inline_equation" }, { "bbox": [ 138, 471, 264, 485 ], "score": 1.0, "content": "has the uniform data-coverage:", "type": "text" } ], "index": 23 } ], "index": 22.5, "bbox_fs": [ 105, 459, 506, 485 ] }, { "type": "text", "bbox": [ 106, 487, 504, 513 ], "lines": [ { "bbox": [ 105, 485, 505, 501 ], "spans": [ { "bbox": [ 105, 485, 505, 501 ], "score": 1.0, "content": "Proposition 4.4. Under Assumption 2.1 and apply Theorem 4.1, APVI achieves the sample complexity", "type": "text" } ], "index": 24 }, { "bbox": [ 106, 498, 447, 514 ], "spans": [ { "bbox": [ 106, 499, 173, 514 ], "score": 1.0, "content": "of minimax-rate", "type": "text" }, { "bbox": [ 173, 498, 229, 513 ], "score": 0.92, "content": "\\widetilde { O } ( H ^ { 3 } / d _ { m } \\epsilon ^ { 2 } )", "type": "inline_equation" }, { "bbox": [ 229, 499, 447, 514 ], "score": 1.0, "content": "(Theorem 4.1 and Theorem G.2 in Yin et al. [2021a]).", "type": "text" } ], "index": 25 } ], "index": 24.5, "bbox_fs": [ 105, 485, 505, 514 ] }, { "type": "text", "bbox": [ 106, 518, 505, 554 ], "lines": [ { "bbox": [ 105, 516, 506, 532 ], "spans": [ { "bbox": [ 105, 516, 461, 532 ], "score": 1.0, "content": "Remark 4.5. We believe if the MDP is time-invariant, then by a modified construction of", "type": "text" }, { "bbox": [ 462, 516, 470, 528 ], "score": 0.75, "content": "\\widehat { P }", "type": "inline_equation" }, { "bbox": [ 470, 516, 474, 532 ], "score": 1.0, "content": ",", "type": "text" }, { "bbox": [ 474, 519, 481, 528 ], "score": 0.46, "content": "\\widehat { r }", "type": "inline_equation" }, { "bbox": [ 482, 516, 506, 532 ], "score": 1.0, "content": "in (2)", "type": "text" } ], "index": 26 }, { "bbox": [ 105, 529, 506, 544 ], "spans": [ { "bbox": [ 105, 530, 269, 544 ], "score": 1.0, "content": "our result will imply the minimax-rate of", "type": "text" }, { "bbox": [ 269, 529, 325, 543 ], "score": 0.93, "content": "\\widetilde { O } ( H ^ { 2 } / d _ { m } \\epsilon ^ { 2 } )", "type": "inline_equation" }, { "bbox": [ 325, 530, 506, 544 ], "score": 1.0, "content": "as achieved in Yin et al. [2021b]. We include", "type": "text" } ], "index": 27 }, { "bbox": [ 106, 542, 226, 554 ], "spans": [ { "bbox": [ 106, 542, 226, 554 ], "score": 1.0, "content": "this discussion in Appendix I.", "type": "text" } ], "index": 28 } ], "index": 27, "bbox_fs": [ 105, 516, 506, 554 ] }, { "type": "title", "bbox": [ 106, 567, 374, 579 ], "lines": [ { "bbox": [ 105, 567, 374, 581 ], "spans": [ { "bbox": [ 105, 567, 374, 581 ], "score": 1.0, "content": "4.2 Bounded sum of total rewards and the Horizon-Free case", "type": "text" } ], "index": 29 } ], "index": 29 }, { "type": "text", "bbox": [ 106, 588, 506, 659 ], "lines": [ { "bbox": [ 106, 587, 511, 625 ], "spans": [ { "bbox": [ 106, 599, 210, 614 ], "score": 0.8, "content": "\\begin{array} { r } { r _ { h } \\geq 0 , \\sum _ { h = 1 } ^ { H } r _ { h } \\in [ 0 , 1 ] } \\end{array}", "type": "inline_equation" }, { "bbox": [ 210, 587, 511, 625 ], "score": 1.0, "content": "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", "type": "text" } ], "index": 30 }, { "bbox": [ 105, 622, 505, 636 ], "spans": [ { "bbox": [ 105, 622, 505, 636 ], "score": 1.0, "content": "Jiang and Agarwal [2018]. In offline RL, Ren et al. [2021] derives the nearly horizon-free worst case", "type": "text" } ], "index": 31 }, { "bbox": [ 105, 634, 506, 649 ], "spans": [ { "bbox": [ 105, 635, 134, 649 ], "score": 1.0, "content": "bound", "type": "text" }, { "bbox": [ 135, 634, 189, 649 ], "score": 0.93, "content": "\\widetilde { O } ( \\sqrt { 1 / n d _ { m } } )", "type": "inline_equation" }, { "bbox": [ 189, 635, 506, 649 ], "score": 1.0, "content": "for the time-invariant MDPs, under the Assumption 2.1. As a comparison, our", "type": "text" } ], "index": 32 }, { "bbox": [ 105, 645, 472, 659 ], "spans": [ { "bbox": [ 105, 645, 472, 659 ], "score": 1.0, "content": "Theorem 4.1 achieves the following guarantee for the time-varying (non-stationary) MDPs.", "type": "text" } ], "index": 33 } ], "index": 31.5, "bbox_fs": [ 105, 587, 511, 659 ] }, { "type": "text", "bbox": [ 107, 662, 505, 701 ], "lines": [ { "bbox": [ 104, 662, 507, 680 ], "spans": [ { "bbox": [ 104, 663, 210, 680 ], "score": 1.0, "content": "Proposition 4.6. Assume", "type": "text" }, { "bbox": [ 211, 664, 240, 676 ], "score": 0.8, "content": "r _ { h } \\ge 0 ,", "type": "inline_equation" }, { "bbox": [ 240, 663, 243, 680 ], "score": 1.0, "content": ",", "type": "text" }, { "bbox": [ 244, 662, 300, 677 ], "score": 0.89, "content": "\\textstyle \\sum _ { h = 1 } ^ { H } r _ { h } \\leq 1", "type": "inline_equation" }, { "bbox": [ 300, 663, 507, 680 ], "score": 1.0, "content": ". Then in the time-varying case AVPI (Theorem 4.1)", "type": "text" } ], "index": 34 }, { "bbox": [ 105, 676, 505, 691 ], "spans": [ { "bbox": [ 105, 677, 173, 691 ], "score": 1.0, "content": "outputs a policy", "type": "text" }, { "bbox": [ 173, 678, 181, 688 ], "score": 0.71, "content": "\\widehat { \\pi }", "type": "inline_equation" }, { "bbox": [ 181, 677, 311, 691 ], "score": 1.0, "content": "such that the suboptimality gap", "type": "text" }, { "bbox": [ 311, 677, 344, 689 ], "score": 0.89, "content": "\\boldsymbol { v } ^ { \\star } - \\boldsymbol { v } ^ { \\widehat { \\pi } }", "type": "inline_equation" }, { "bbox": [ 345, 677, 405, 691 ], "score": 1.0, "content": "is bounded by", "type": "text" }, { "bbox": [ 405, 676, 463, 691 ], "score": 0.93, "content": "\\widetilde { O } ( \\sqrt { H / n d _ { m } } )", "type": "inline_equation" }, { "bbox": [ 463, 677, 505, 691 ], "score": 1.0, "content": "with high", "type": "text" } ], "index": 35 }, { "bbox": [ 105, 689, 259, 702 ], "spans": [ { "bbox": [ 105, 689, 259, 702 ], "score": 1.0, "content": "bprobability under the Assumption 2.1.", "type": "text" } ], "index": 36 } ], "index": 35, "bbox_fs": [ 104, 662, 507, 702 ] } ] }, { "preproc_blocks": [ { "type": "text", "bbox": [ 106, 70, 506, 167 ], "lines": [ { "bbox": [ 103, 70, 507, 89 ], "spans": [ { "bbox": [ 103, 70, 270, 89 ], "score": 1.0, "content": "The derivation is straightforward by using", "type": "text" }, { "bbox": [ 271, 70, 358, 86 ], "score": 0.92, "content": "\\begin{array} { r } { \\operatorname { V a r } _ { \\pi ^ { \\star } } ( \\sum _ { h = 1 } ^ { H } r _ { h } ) \\le 1 } \\end{array}", "type": "inline_equation" }, { "bbox": [ 359, 70, 507, 89 ], "score": 1.0, "content": "in (8). This proposition is interesting", "type": "text" } ], "index": 0 }, { "bbox": [ 105, 85, 505, 99 ], "spans": [ { "bbox": [ 105, 86, 314, 99 ], "score": 1.0, "content": "since it indicates when the MDP is non-stationary,", "type": "text" }, { "bbox": [ 315, 85, 365, 99 ], "score": 0.93, "content": "\\widetilde { O } ( H / d _ { m } \\epsilon ^ { 2 } )", "type": "inline_equation" }, { "bbox": [ 366, 86, 505, 99 ], "score": 1.0, "content": "is required in the worst case even", "type": "text" } ], "index": 1 }, { "bbox": [ 102, 97, 507, 114 ], "spans": [ { "bbox": [ 102, 97, 133, 114 ], "score": 1.0, "content": "under", "type": "text" }, { "bbox": [ 133, 97, 194, 112 ], "score": 0.93, "content": "\\textstyle \\sum _ { h = 1 } ^ { H } r _ { h } \\leq 1", "type": "inline_equation" }, { "bbox": [ 195, 97, 248, 114 ], "score": 1.0, "content": ".5 The extra", "type": "text" }, { "bbox": [ 249, 100, 259, 110 ], "score": 0.77, "content": "H", "type": "inline_equation" }, { "bbox": [ 259, 97, 447, 114 ], "score": 1.0, "content": "factor resembles the challenge that we have", "type": "text" }, { "bbox": [ 447, 100, 458, 110 ], "score": 0.78, "content": "H", "type": "inline_equation" }, { "bbox": [ 459, 97, 507, 114 ], "score": 1.0, "content": "transitions", "type": "text" } ], "index": 2 }, { "bbox": [ 106, 110, 504, 125 ], "spans": [ { "bbox": [ 106, 112, 161, 123 ], "score": 0.9, "content": "( P _ { 1 } , \\dots , P _ { H } )", "type": "inline_equation" }, { "bbox": [ 162, 110, 329, 125 ], "score": 1.0, "content": "to learn, as opposed to the bandit-type", "type": "text" }, { "bbox": [ 329, 111, 362, 124 ], "score": 0.93, "content": "1 / d _ { m } \\epsilon ^ { 2 }", "type": "inline_equation" }, { "bbox": [ 362, 110, 495, 125 ], "score": 1.0, "content": "result due to there is only one", "type": "text" }, { "bbox": [ 495, 112, 504, 122 ], "score": 0.78, "content": "P", "type": "inline_equation" } ], "index": 3 }, { "bbox": [ 105, 121, 506, 135 ], "spans": [ { "bbox": [ 105, 121, 506, 135 ], "score": 1.0, "content": "throughout (time-invariant). This reveals that one hardness in solving the MDP is in proportion to", "type": "text" } ], "index": 4 }, { "bbox": [ 106, 133, 505, 145 ], "spans": [ { "bbox": [ 106, 133, 505, 145 ], "score": 1.0, "content": "the number of different transition kernels within the MDP. Such a finding could help researchers", "type": "text" } ], "index": 5 }, { "bbox": [ 106, 144, 507, 157 ], "spans": [ { "bbox": [ 106, 144, 507, 157 ], "score": 1.0, "content": "understand the special settings like low switching cost in transitions [Bai et al., 2019] or non-", "type": "text" } ], "index": 6 }, { "bbox": [ 105, 155, 243, 168 ], "spans": [ { "bbox": [ 105, 155, 243, 168 ], "score": 1.0, "content": "stationarity [Cheung et al., 2020].", "type": "text" } ], "index": 7 } ], "index": 3.5 }, { "type": "title", "bbox": [ 107, 180, 296, 193 ], "lines": [ { "bbox": [ 104, 178, 297, 196 ], "spans": [ { "bbox": [ 104, 178, 297, 196 ], "score": 1.0, "content": "4.3 Optimality with Single Concentrability", "type": "text" } ], "index": 8 } ], "index": 8 }, { "type": "text", "bbox": [ 106, 200, 506, 303 ], "lines": [ { "bbox": [ 105, 200, 506, 214 ], "spans": [ { "bbox": [ 105, 200, 506, 214 ], "score": 1.0, "content": "In the finite horizon discounted setting, Rashidinejad et al. [2021] proposes the single policy concen-", "type": "text" } ], "index": 9 }, { "bbox": [ 104, 211, 507, 233 ], "spans": [ { "bbox": [ 104, 214, 369, 233 ], "score": 1.0, "content": "trability assumption which is defined as C? := maxh,s,a dh (s,a)dµ(s,a)", "type": "text" }, { "bbox": [ 393, 211, 507, 231 ], "score": 1.0, "content": "in the current episodic non-", "type": "text" } ], "index": 10 }, { "bbox": [ 104, 230, 505, 252 ], "spans": [ { "bbox": [ 104, 230, 450, 251 ], "score": 1.0, "content": "stationary MDP setting. As discussed in Appendix D, their lower bound translates to", "type": "text" }, { "bbox": [ 450, 231, 505, 252 ], "score": 0.94, "content": "\\Omega ( { \\sqrt { \\frac { H ^ { 3 } S C ^ { \\star } } { n } } } )", "type": "inline_equation" } ], "index": 11 }, { "bbox": [ 105, 251, 508, 272 ], "spans": [ { "bbox": [ 105, 253, 246, 270 ], "score": 1.0, "content": "and their VI-LCB algorithm yields", "type": "text" }, { "bbox": [ 264, 251, 508, 272 ], "score": 1.0, "content": "q H5SC?n ) suboptimality gap in H-horizon case. Since single", "type": "text" } ], "index": 12 }, { "bbox": [ 106, 269, 506, 282 ], "spans": [ { "bbox": [ 106, 269, 506, 282 ], "score": 1.0, "content": "policy concentrability is strictly weaker than its uniform version (Assumption 2.2), we only discuss", "type": "text" } ], "index": 13 }, { "bbox": [ 106, 280, 505, 293 ], "spans": [ { "bbox": [ 106, 280, 505, 293 ], "score": 1.0, "content": "this set up. In particular, we have the following implication from our Theorem 4.1 (whose derivation", "type": "text" } ], "index": 14 }, { "bbox": [ 106, 291, 224, 303 ], "spans": [ { "bbox": [ 106, 291, 224, 303 ], "score": 1.0, "content": "can be found in Appendix I):", "type": "text" } ], "index": 15 } ], "index": 12 }, { "type": "text", "bbox": [ 106, 307, 505, 354 ], "lines": [ { "bbox": [ 102, 304, 506, 328 ], "spans": [ { "bbox": [ 102, 306, 454, 328 ], "score": 1.0, "content": "Proposition 4.7. Let π? be a deterministic policy such that C? := maxh,s,a dπ h (s,a)dµ(s,a)", "type": "text" }, { "bbox": [ 477, 304, 506, 325 ], "score": 1.0, "content": ". Then", "type": "text" } ], "index": 16 }, { "bbox": [ 105, 323, 505, 338 ], "spans": [ { "bbox": [ 105, 323, 505, 338 ], "score": 1.0, "content": "by Theorem 4.1, with high probability the output policy of APVI satisfies the suboptimality gap", "type": "text" } ], "index": 17 }, { "bbox": [ 107, 335, 340, 357 ], "spans": [ { "bbox": [ 107, 336, 162, 356 ], "score": 0.93, "content": "\\widetilde { O } ( \\sqrt { \\frac { H ^ { 3 } S C ^ { \\star } } { n } } )", "type": "inline_equation" }, { "bbox": [ 162, 335, 340, 357 ], "score": 1.0, "content": "in the time-varying (non-stationary) MDPs.", "type": "text" } ], "index": 18 } ], "index": 17 }, { "type": "text", "bbox": [ 106, 365, 505, 429 ], "lines": [ { "bbox": [ 106, 363, 505, 388 ], "spans": [ { "bbox": [ 106, 363, 291, 388 ], "score": 1.0, "content": "This can computed similar to (8) except we use", "type": "text" }, { "bbox": [ 292, 365, 349, 385 ], "score": 0.94, "content": "\\begin{array} { r } { \\frac { \\boldsymbol { d } _ { h } ^ { \\pi ^ { \\star } } ( s , a ) } { \\boldsymbol { d } _ { h } ^ { \\mu } ( s , a ) } \\leq C ^ { \\star } } \\end{array}", "type": "inline_equation" }, { "bbox": [ 350, 363, 505, 388 ], "score": 1.0, "content": ". Our implication improves the VI-LCB", "type": "text" } ], "index": 19 }, { "bbox": [ 106, 383, 506, 397 ], "spans": [ { "bbox": [ 106, 383, 160, 397 ], "score": 1.0, "content": "by the factor", "type": "text" }, { "bbox": [ 161, 384, 175, 394 ], "score": 0.86, "content": "H ^ { 2 }", "type": "inline_equation" }, { "bbox": [ 176, 383, 506, 397 ], "score": 1.0, "content": "(in terms of sample complexity) and is optimal (recover the concurrent Xie et al.", "type": "text" } ], "index": 20 }, { "bbox": [ 105, 394, 504, 407 ], "spans": [ { "bbox": [ 105, 394, 490, 407 ], "score": 1.0, "content": "[2021b]). Qualitatively, single concentrability is the same as Assumption 2.3, but the use of", "type": "text" }, { "bbox": [ 491, 396, 504, 405 ], "score": 0.79, "content": "C ^ { \\star }", "type": "inline_equation" } ], "index": 21 }, { "bbox": [ 106, 407, 505, 418 ], "spans": [ { "bbox": [ 106, 407, 505, 418 ], "score": 1.0, "content": "makes the bound highly problem independent and limits the adaptivity. Problem dependent bound is", "type": "text" } ], "index": 22 }, { "bbox": [ 105, 417, 436, 430 ], "spans": [ { "bbox": [ 105, 417, 436, 430 ], "score": 1.0, "content": "a more interesting domain as it tailors to each MDP separately. We discuss it now.", "type": "text" } ], "index": 23 } ], "index": 21 }, { "type": "title", "bbox": [ 107, 442, 249, 455 ], "lines": [ { "bbox": [ 106, 442, 249, 456 ], "spans": [ { "bbox": [ 106, 442, 249, 456 ], "score": 1.0, "content": "4.4 Problem dependent domain", "type": "text" } ], "index": 24 } ], "index": 24 }, { "type": "text", "bbox": [ 106, 462, 505, 502 ], "lines": [ { "bbox": [ 106, 462, 505, 475 ], "spans": [ { "bbox": [ 106, 462, 505, 475 ], "score": 1.0, "content": "We define the pre-step environmental norm (the finite horizon counterpart of Maillard et al. [2014])", "type": "text" } ], "index": 25 }, { "bbox": [ 103, 472, 507, 490 ], "spans": [ { "bbox": [ 103, 472, 120, 490 ], "score": 1.0, "content": "as:", "type": "text" }, { "bbox": [ 121, 474, 279, 488 ], "score": 0.91, "content": "\\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}", "type": "inline_equation" }, { "bbox": [ 280, 472, 307, 490 ], "score": 1.0, "content": "for all", "type": "text" }, { "bbox": [ 307, 474, 341, 486 ], "score": 0.92, "content": "h \\in [ H ]", "type": "inline_equation" }, { "bbox": [ 341, 472, 507, 490 ], "score": 1.0, "content": ", and relax the total sum of rewards to be", "type": "text" } ], "index": 26 }, { "bbox": [ 105, 486, 436, 505 ], "spans": [ { "bbox": [ 105, 486, 233, 505 ], "score": 1.0, "content": "bounded by any arbitrary value", "type": "text" }, { "bbox": [ 233, 489, 241, 499 ], "score": 0.76, "content": "\\boldsymbol { B }", "type": "inline_equation" }, { "bbox": [ 242, 486, 261, 505 ], "score": 1.0, "content": "(i.e.", "type": "text" }, { "bbox": [ 261, 487, 322, 502 ], "score": 0.9, "content": "\\textstyle \\sum _ { h = 1 } ^ { H } r _ { h } \\leq B )", "type": "inline_equation" }, { "bbox": [ 322, 486, 436, 505 ], "score": 1.0, "content": ", then Theorem 4.1 implies:", "type": "text" } ], "index": 27 } ], "index": 26 }, { "type": "text", "bbox": [ 106, 504, 502, 516 ], "lines": [ { "bbox": [ 105, 502, 503, 519 ], "spans": [ { "bbox": [ 105, 502, 503, 519 ], "score": 1.0, "content": "Proposition 4.8. Under Assumption 2.1, with high probability, subopmality of AVPI is bounded by", "type": "text" } ], "index": 28 } ], "index": 28 }, { "type": "interline_equation", "bbox": [ 196, 522, 413, 557 ], "lines": [ { "bbox": [ 196, 522, 413, 557 ], "spans": [ { "bbox": [ 196, 522, 413, 557 ], "score": 0.93, "content": "\\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 } } ) .", "type": "interline_equation", "image_path": "1f6383d73d7e96784480eaaec32ca332fed385bc82436c24398afd8a492b8716.jpg" } ] } ], "index": 29.5, "virtual_lines": [ { "bbox": [ 196, 522, 413, 539.5 ], "spans": [], "index": 29 }, { "bbox": [ 196, 539.5, 413, 557.0 ], "spans": [], "index": 30 } ] }, { "type": "text", "bbox": [ 106, 568, 505, 613 ], "lines": [ { "bbox": [ 105, 568, 506, 581 ], "spans": [ { "bbox": [ 105, 568, 506, 581 ], "score": 1.0, "content": "Such a result mirrors the online version of the tight problem-dependent bound Zanette and Brunskill", "type": "text" } ], "index": 31 }, { "bbox": [ 105, 578, 506, 591 ], "spans": [ { "bbox": [ 105, 578, 506, 591 ], "score": 1.0, "content": "[2019] but with a more general pre-step environmental norm for the non-stationary MDPs.6 For the", "type": "text" } ], "index": 32 }, { "bbox": [ 105, 590, 506, 603 ], "spans": [ { "bbox": [ 105, 590, 253, 603 ], "score": 1.0, "content": "problem instances with either small", "type": "text" }, { "bbox": [ 253, 591, 261, 600 ], "score": 0.82, "content": "\\boldsymbol { B }", "type": "inline_equation" }, { "bbox": [ 262, 590, 298, 603 ], "score": 1.0, "content": "or small", "type": "text" }, { "bbox": [ 298, 590, 312, 603 ], "score": 0.9, "content": "\\mathbb { Q } _ { h } ^ { \\star }", "type": "inline_equation" }, { "bbox": [ 312, 590, 506, 603 ], "score": 1.0, "content": ", our result yields much better performances, as", "type": "text" } ], "index": 33 }, { "bbox": [ 105, 599, 216, 614 ], "spans": [ { "bbox": [ 105, 599, 216, 614 ], "score": 1.0, "content": "discussed in the following.", "type": "text" } ], "index": 34 } ], "index": 32.5 }, { "type": "text", "bbox": [ 106, 617, 505, 678 ], "lines": [ { "bbox": [ 105, 616, 506, 630 ], "spans": [ { "bbox": [ 105, 616, 506, 630 ], "score": 1.0, "content": "Deterministic systems. For many practical applications of interest, the systems are equipped with", "type": "text" } ], "index": 35 }, { "bbox": [ 106, 628, 506, 641 ], "spans": [ { "bbox": [ 106, 628, 506, 641 ], "score": 1.0, "content": "low stochasticity, e.g. robotics, or even deterministic dynamics, e.g. the game of GO. In those", "type": "text" } ], "index": 36 }, { "bbox": [ 105, 639, 506, 652 ], "spans": [ { "bbox": [ 105, 639, 506, 652 ], "score": 1.0, "content": "scenarios, the agent needs less experience for each state-action therefore the learning procedure could", "type": "text" } ], "index": 37 }, { "bbox": [ 106, 651, 505, 662 ], "spans": [ { "bbox": [ 106, 651, 505, 662 ], "score": 1.0, "content": "be much faster. In particular, when the system is fully deterministic (in both transitions and rewards)", "type": "text" } ], "index": 38 }, { "bbox": [ 105, 661, 505, 680 ], "spans": [ { "bbox": [ 105, 663, 126, 680 ], "score": 1.0, "content": "then", "type": "text" }, { "bbox": [ 126, 663, 158, 676 ], "score": 0.92, "content": "\\mathbb { Q } _ { h } ^ { \\star } = 0", "type": "inline_equation" }, { "bbox": [ 158, 663, 185, 680 ], "score": 1.0, "content": "for all", "type": "text" }, { "bbox": [ 185, 664, 192, 673 ], "score": 0.77, "content": "h", "type": "inline_equation" }, { "bbox": [ 192, 663, 380, 680 ], "score": 1.0, "content": ". This enables a faster convergence rate of order", "type": "text" }, { "bbox": [ 380, 661, 398, 678 ], "score": 0.93, "content": "\\frac { H ^ { 3 } } { n \\bar { d } _ { m } }", "type": "inline_equation" }, { "bbox": [ 398, 663, 505, 680 ], "score": 1.0, "content": "and significantly improves", "type": "text" } ], "index": 39 } ], "index": 37 } ], "page_idx": 7, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 106, 687, 505, 722 ], "lines": [ { "bbox": [ 118, 686, 506, 701 ], "spans": [ { "bbox": [ 118, 686, 255, 701 ], "score": 1.0, "content": "5Suppose in this case we can achieve", "type": "text" }, { "bbox": [ 255, 687, 299, 700 ], "score": 0.92, "content": "\\widetilde { O } ( 1 / d _ { m } \\epsilon ^ { 2 } )", "type": "inline_equation" }, { "bbox": [ 299, 686, 506, 701 ], "score": 1.0, "content": "just like Ren et al. [2021], then by a rescaling we obtain", "type": "text" } ] }, { "bbox": [ 105, 698, 488, 712 ], "spans": [ { "bbox": [ 105, 698, 120, 712 ], "score": 1.0, "content": "the", "type": "text" }, { "bbox": [ 120, 699, 172, 712 ], "score": 0.91, "content": "\\widetilde { O } ( H ^ { 2 } / d _ { m } \\epsilon ^ { 2 } )", "type": "inline_equation" }, { "bbox": [ 172, 698, 230, 712 ], "score": 1.0, "content": "under the usual", "type": "text" }, { "bbox": [ 230, 701, 274, 711 ], "score": 0.9, "content": "0 \\leq r _ { h } \\leq 1", "type": "inline_equation" }, { "bbox": [ 275, 698, 386, 712 ], "score": 1.0, "content": "assumption which violates the", "type": "text" }, { "bbox": [ 387, 699, 437, 712 ], "score": 0.93, "content": "\\Omega ( H ^ { 3 } / d _ { m } \\epsilon ^ { 2 } )", "type": "inline_equation" }, { "bbox": [ 438, 698, 488, 712 ], "score": 1.0, "content": "lower bound.", "type": "text" } ] }, { "bbox": [ 118, 708, 407, 724 ], "spans": [ { "bbox": [ 118, 708, 396, 724 ], "score": 1.0, "content": "e6Zanette and Brunskill [2019] uses the maximal version by maximizing over", "type": "text" }, { "bbox": [ 396, 712, 402, 720 ], "score": 0.8, "content": "h", "type": "inline_equation" }, { "bbox": [ 403, 708, 407, 724 ], "score": 1.0, "content": ".", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 302, 741, 309, 750 ], "lines": [ { "bbox": [ 302, 740, 309, 752 ], "spans": [ { "bbox": [ 302, 740, 309, 752 ], "score": 1.0, "content": "8", "type": "text" } ] } ] } ], "para_blocks": [ { "type": "text", "bbox": [ 106, 70, 506, 167 ], "lines": [ { "bbox": [ 103, 70, 507, 89 ], "spans": [ { "bbox": [ 103, 70, 270, 89 ], "score": 1.0, "content": "The derivation is straightforward by using", "type": "text" }, { "bbox": [ 271, 70, 358, 86 ], "score": 0.92, "content": "\\begin{array} { r } { \\operatorname { V a r } _ { \\pi ^ { \\star } } ( \\sum _ { h = 1 } ^ { H } r _ { h } ) \\le 1 } \\end{array}", "type": "inline_equation" }, { "bbox": [ 359, 70, 507, 89 ], "score": 1.0, "content": "in (8). This proposition is interesting", "type": "text" } ], "index": 0 }, { "bbox": [ 105, 85, 505, 99 ], "spans": [ { "bbox": [ 105, 86, 314, 99 ], "score": 1.0, "content": "since it indicates when the MDP is non-stationary,", "type": "text" }, { "bbox": [ 315, 85, 365, 99 ], "score": 0.93, "content": "\\widetilde { O } ( H / d _ { m } \\epsilon ^ { 2 } )", "type": "inline_equation" }, { "bbox": [ 366, 86, 505, 99 ], "score": 1.0, "content": "is required in the worst case even", "type": "text" } ], "index": 1 }, { "bbox": [ 102, 97, 507, 114 ], "spans": [ { "bbox": [ 102, 97, 133, 114 ], "score": 1.0, "content": "under", "type": "text" }, { "bbox": [ 133, 97, 194, 112 ], "score": 0.93, "content": "\\textstyle \\sum _ { h = 1 } ^ { H } r _ { h } \\leq 1", "type": "inline_equation" }, { "bbox": [ 195, 97, 248, 114 ], "score": 1.0, "content": ".5 The extra", "type": "text" }, { "bbox": [ 249, 100, 259, 110 ], "score": 0.77, "content": "H", "type": "inline_equation" }, { "bbox": [ 259, 97, 447, 114 ], "score": 1.0, "content": "factor resembles the challenge that we have", "type": "text" }, { "bbox": [ 447, 100, 458, 110 ], "score": 0.78, "content": "H", "type": "inline_equation" }, { "bbox": [ 459, 97, 507, 114 ], "score": 1.0, "content": "transitions", "type": "text" } ], "index": 2 }, { "bbox": [ 106, 110, 504, 125 ], "spans": [ { "bbox": [ 106, 112, 161, 123 ], "score": 0.9, "content": "( P _ { 1 } , \\dots , P _ { H } )", "type": "inline_equation" }, { "bbox": [ 162, 110, 329, 125 ], "score": 1.0, "content": "to learn, as opposed to the bandit-type", "type": "text" }, { "bbox": [ 329, 111, 362, 124 ], "score": 0.93, "content": "1 / d _ { m } \\epsilon ^ { 2 }", "type": "inline_equation" }, { "bbox": [ 362, 110, 495, 125 ], "score": 1.0, "content": "result due to there is only one", "type": "text" }, { "bbox": [ 495, 112, 504, 122 ], "score": 0.78, "content": "P", "type": "inline_equation" } ], "index": 3 }, { "bbox": [ 105, 121, 506, 135 ], "spans": [ { "bbox": [ 105, 121, 506, 135 ], "score": 1.0, "content": "throughout (time-invariant). This reveals that one hardness in solving the MDP is in proportion to", "type": "text" } ], "index": 4 }, { "bbox": [ 106, 133, 505, 145 ], "spans": [ { "bbox": [ 106, 133, 505, 145 ], "score": 1.0, "content": "the number of different transition kernels within the MDP. Such a finding could help researchers", "type": "text" } ], "index": 5 }, { "bbox": [ 106, 144, 507, 157 ], "spans": [ { "bbox": [ 106, 144, 507, 157 ], "score": 1.0, "content": "understand the special settings like low switching cost in transitions [Bai et al., 2019] or non-", "type": "text" } ], "index": 6 }, { "bbox": [ 105, 155, 243, 168 ], "spans": [ { "bbox": [ 105, 155, 243, 168 ], "score": 1.0, "content": "stationarity [Cheung et al., 2020].", "type": "text" } ], "index": 7 } ], "index": 3.5, "bbox_fs": [ 102, 70, 507, 168 ] }, { "type": "title", "bbox": [ 107, 180, 296, 193 ], "lines": [ { "bbox": [ 104, 178, 297, 196 ], "spans": [ { "bbox": [ 104, 178, 297, 196 ], "score": 1.0, "content": "4.3 Optimality with Single Concentrability", "type": "text" } ], "index": 8 } ], "index": 8 }, { "type": "text", "bbox": [ 106, 200, 506, 303 ], "lines": [ { "bbox": [ 105, 200, 506, 214 ], "spans": [ { "bbox": [ 105, 200, 506, 214 ], "score": 1.0, "content": "In the finite horizon discounted setting, Rashidinejad et al. [2021] proposes the single policy concen-", "type": "text" } ], "index": 9 }, { "bbox": [ 104, 211, 507, 233 ], "spans": [ { "bbox": [ 104, 214, 369, 233 ], "score": 1.0, "content": "trability assumption which is defined as C? := maxh,s,a dh (s,a)dµ(s,a)", "type": "text" }, { "bbox": [ 393, 211, 507, 231 ], "score": 1.0, "content": "in the current episodic non-", "type": "text" } ], "index": 10 }, { "bbox": [ 104, 230, 505, 252 ], "spans": [ { "bbox": [ 104, 230, 450, 251 ], "score": 1.0, "content": "stationary MDP setting. As discussed in Appendix D, their lower bound translates to", "type": "text" }, { "bbox": [ 450, 231, 505, 252 ], "score": 0.94, "content": "\\Omega ( { \\sqrt { \\frac { H ^ { 3 } S C ^ { \\star } } { n } } } )", "type": "inline_equation" } ], "index": 11 }, { "bbox": [ 105, 251, 508, 272 ], "spans": [ { "bbox": [ 105, 253, 246, 270 ], "score": 1.0, "content": "and their VI-LCB algorithm yields", "type": "text" }, { "bbox": [ 264, 251, 508, 272 ], "score": 1.0, "content": "q H5SC?n ) suboptimality gap in H-horizon case. Since single", "type": "text" } ], "index": 12 }, { "bbox": [ 106, 269, 506, 282 ], "spans": [ { "bbox": [ 106, 269, 506, 282 ], "score": 1.0, "content": "policy concentrability is strictly weaker than its uniform version (Assumption 2.2), we only discuss", "type": "text" } ], "index": 13 }, { "bbox": [ 106, 280, 505, 293 ], "spans": [ { "bbox": [ 106, 280, 505, 293 ], "score": 1.0, "content": "this set up. In particular, we have the following implication from our Theorem 4.1 (whose derivation", "type": "text" } ], "index": 14 }, { "bbox": [ 106, 291, 224, 303 ], "spans": [ { "bbox": [ 106, 291, 224, 303 ], "score": 1.0, "content": "can be found in Appendix I):", "type": "text" } ], "index": 15 } ], "index": 12, "bbox_fs": [ 104, 200, 508, 303 ] }, { "type": "text", "bbox": [ 106, 307, 505, 354 ], "lines": [ { "bbox": [ 102, 304, 506, 328 ], "spans": [ { "bbox": [ 102, 306, 454, 328 ], "score": 1.0, "content": "Proposition 4.7. Let π? be a deterministic policy such that C? := maxh,s,a dπ h (s,a)dµ(s,a)", "type": "text" }, { "bbox": [ 477, 304, 506, 325 ], "score": 1.0, "content": ". Then", "type": "text" } ], "index": 16 }, { "bbox": [ 105, 323, 505, 338 ], "spans": [ { "bbox": [ 105, 323, 505, 338 ], "score": 1.0, "content": "by Theorem 4.1, with high probability the output policy of APVI satisfies the suboptimality gap", "type": "text" } ], "index": 17 }, { "bbox": [ 107, 335, 340, 357 ], "spans": [ { "bbox": [ 107, 336, 162, 356 ], "score": 0.93, "content": "\\widetilde { O } ( \\sqrt { \\frac { H ^ { 3 } S C ^ { \\star } } { n } } )", "type": "inline_equation" }, { "bbox": [ 162, 335, 340, 357 ], "score": 1.0, "content": "in the time-varying (non-stationary) MDPs.", "type": "text" } ], "index": 18 } ], "index": 17, "bbox_fs": [ 102, 304, 506, 357 ] }, { "type": "text", "bbox": [ 106, 365, 505, 429 ], "lines": [ { "bbox": [ 106, 363, 505, 388 ], "spans": [ { "bbox": [ 106, 363, 291, 388 ], "score": 1.0, "content": "This can computed similar to (8) except we use", "type": "text" }, { "bbox": [ 292, 365, 349, 385 ], "score": 0.94, "content": "\\begin{array} { r } { \\frac { \\boldsymbol { d } _ { h } ^ { \\pi ^ { \\star } } ( s , a ) } { \\boldsymbol { d } _ { h } ^ { \\mu } ( s , a ) } \\leq C ^ { \\star } } \\end{array}", "type": "inline_equation" }, { "bbox": [ 350, 363, 505, 388 ], "score": 1.0, "content": ". Our implication improves the VI-LCB", "type": "text" } ], "index": 19 }, { "bbox": [ 106, 383, 506, 397 ], "spans": [ { "bbox": [ 106, 383, 160, 397 ], "score": 1.0, "content": "by the factor", "type": "text" }, { "bbox": [ 161, 384, 175, 394 ], "score": 0.86, "content": "H ^ { 2 }", "type": "inline_equation" }, { "bbox": [ 176, 383, 506, 397 ], "score": 1.0, "content": "(in terms of sample complexity) and is optimal (recover the concurrent Xie et al.", "type": "text" } ], "index": 20 }, { "bbox": [ 105, 394, 504, 407 ], "spans": [ { "bbox": [ 105, 394, 490, 407 ], "score": 1.0, "content": "[2021b]). Qualitatively, single concentrability is the same as Assumption 2.3, but the use of", "type": "text" }, { "bbox": [ 491, 396, 504, 405 ], "score": 0.79, "content": "C ^ { \\star }", "type": "inline_equation" } ], "index": 21 }, { "bbox": [ 106, 407, 505, 418 ], "spans": [ { "bbox": [ 106, 407, 505, 418 ], "score": 1.0, "content": "makes the bound highly problem independent and limits the adaptivity. Problem dependent bound is", "type": "text" } ], "index": 22 }, { "bbox": [ 105, 417, 436, 430 ], "spans": [ { "bbox": [ 105, 417, 436, 430 ], "score": 1.0, "content": "a more interesting domain as it tailors to each MDP separately. We discuss it now.", "type": "text" } ], "index": 23 } ], "index": 21, "bbox_fs": [ 105, 363, 506, 430 ] }, { "type": "title", "bbox": [ 107, 442, 249, 455 ], "lines": [ { "bbox": [ 106, 442, 249, 456 ], "spans": [ { "bbox": [ 106, 442, 249, 456 ], "score": 1.0, "content": "4.4 Problem dependent domain", "type": "text" } ], "index": 24 } ], "index": 24 }, { "type": "text", "bbox": [ 106, 462, 505, 502 ], "lines": [ { "bbox": [ 106, 462, 505, 475 ], "spans": [ { "bbox": [ 106, 462, 505, 475 ], "score": 1.0, "content": "We define the pre-step environmental norm (the finite horizon counterpart of Maillard et al. [2014])", "type": "text" } ], "index": 25 }, { "bbox": [ 103, 472, 507, 490 ], "spans": [ { "bbox": [ 103, 472, 120, 490 ], "score": 1.0, "content": "as:", "type": "text" }, { "bbox": [ 121, 474, 279, 488 ], "score": 0.91, "content": "\\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}", "type": "inline_equation" }, { "bbox": [ 280, 472, 307, 490 ], "score": 1.0, "content": "for all", "type": "text" }, { "bbox": [ 307, 474, 341, 486 ], "score": 0.92, "content": "h \\in [ H ]", "type": "inline_equation" }, { "bbox": [ 341, 472, 507, 490 ], "score": 1.0, "content": ", and relax the total sum of rewards to be", "type": "text" } ], "index": 26 }, { "bbox": [ 105, 486, 436, 505 ], "spans": [ { "bbox": [ 105, 486, 233, 505 ], "score": 1.0, "content": "bounded by any arbitrary value", "type": "text" }, { "bbox": [ 233, 489, 241, 499 ], "score": 0.76, "content": "\\boldsymbol { B }", "type": "inline_equation" }, { "bbox": [ 242, 486, 261, 505 ], "score": 1.0, "content": "(i.e.", "type": "text" }, { "bbox": [ 261, 487, 322, 502 ], "score": 0.9, "content": "\\textstyle \\sum _ { h = 1 } ^ { H } r _ { h } \\leq B )", "type": "inline_equation" }, { "bbox": [ 322, 486, 436, 505 ], "score": 1.0, "content": ", then Theorem 4.1 implies:", "type": "text" } ], "index": 27 } ], "index": 26, "bbox_fs": [ 103, 462, 507, 505 ] }, { "type": "text", "bbox": [ 106, 504, 502, 516 ], "lines": [ { "bbox": [ 105, 502, 503, 519 ], "spans": [ { "bbox": [ 105, 502, 503, 519 ], "score": 1.0, "content": "Proposition 4.8. Under Assumption 2.1, with high probability, subopmality of AVPI is bounded by", "type": "text" } ], "index": 28 } ], "index": 28, "bbox_fs": [ 105, 502, 503, 519 ] }, { "type": "interline_equation", "bbox": [ 196, 522, 413, 557 ], "lines": [ { "bbox": [ 196, 522, 413, 557 ], "spans": [ { "bbox": [ 196, 522, 413, 557 ], "score": 0.93, "content": "\\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 } } ) .", "type": "interline_equation", "image_path": "1f6383d73d7e96784480eaaec32ca332fed385bc82436c24398afd8a492b8716.jpg" } ] } ], "index": 29.5, "virtual_lines": [ { "bbox": [ 196, 522, 413, 539.5 ], "spans": [], "index": 29 }, { "bbox": [ 196, 539.5, 413, 557.0 ], "spans": [], "index": 30 } ] }, { "type": "text", "bbox": [ 106, 568, 505, 613 ], "lines": [ { "bbox": [ 105, 568, 506, 581 ], "spans": [ { "bbox": [ 105, 568, 506, 581 ], "score": 1.0, "content": "Such a result mirrors the online version of the tight problem-dependent bound Zanette and Brunskill", "type": "text" } ], "index": 31 }, { "bbox": [ 105, 578, 506, 591 ], "spans": [ { "bbox": [ 105, 578, 506, 591 ], "score": 1.0, "content": "[2019] but with a more general pre-step environmental norm for the non-stationary MDPs.6 For the", "type": "text" } ], "index": 32 }, { "bbox": [ 105, 590, 506, 603 ], "spans": [ { "bbox": [ 105, 590, 253, 603 ], "score": 1.0, "content": "problem instances with either small", "type": "text" }, { "bbox": [ 253, 591, 261, 600 ], "score": 0.82, "content": "\\boldsymbol { B }", "type": "inline_equation" }, { "bbox": [ 262, 590, 298, 603 ], "score": 1.0, "content": "or small", "type": "text" }, { "bbox": [ 298, 590, 312, 603 ], "score": 0.9, "content": "\\mathbb { Q } _ { h } ^ { \\star }", "type": "inline_equation" }, { "bbox": [ 312, 590, 506, 603 ], "score": 1.0, "content": ", our result yields much better performances, as", "type": "text" } ], "index": 33 }, { "bbox": [ 105, 599, 216, 614 ], "spans": [ { "bbox": [ 105, 599, 216, 614 ], "score": 1.0, "content": "discussed in the following.", "type": "text" } ], "index": 34 } ], "index": 32.5, "bbox_fs": [ 105, 568, 506, 614 ] }, { "type": "text", "bbox": [ 106, 617, 505, 678 ], "lines": [ { "bbox": [ 105, 616, 506, 630 ], "spans": [ { "bbox": [ 105, 616, 506, 630 ], "score": 1.0, "content": "Deterministic systems. For many practical applications of interest, the systems are equipped with", "type": "text" } ], "index": 35 }, { "bbox": [ 106, 628, 506, 641 ], "spans": [ { "bbox": [ 106, 628, 506, 641 ], "score": 1.0, "content": "low stochasticity, e.g. robotics, or even deterministic dynamics, e.g. the game of GO. In those", "type": "text" } ], "index": 36 }, { "bbox": [ 105, 639, 506, 652 ], "spans": [ { "bbox": [ 105, 639, 506, 652 ], "score": 1.0, "content": "scenarios, the agent needs less experience for each state-action therefore the learning procedure could", "type": "text" } ], "index": 37 }, { "bbox": [ 106, 651, 505, 662 ], "spans": [ { "bbox": [ 106, 651, 505, 662 ], "score": 1.0, "content": "be much faster. In particular, when the system is fully deterministic (in both transitions and rewards)", "type": "text" } ], "index": 38 }, { "bbox": [ 105, 661, 505, 680 ], "spans": [ { "bbox": [ 105, 663, 126, 680 ], "score": 1.0, "content": "then", "type": "text" }, { "bbox": [ 126, 663, 158, 676 ], "score": 0.92, "content": "\\mathbb { Q } _ { h } ^ { \\star } = 0", "type": "inline_equation" }, { "bbox": [ 158, 663, 185, 680 ], "score": 1.0, "content": "for all", "type": "text" }, { "bbox": [ 185, 664, 192, 673 ], "score": 0.77, "content": "h", "type": "inline_equation" }, { "bbox": [ 192, 663, 380, 680 ], "score": 1.0, "content": ". This enables a faster convergence rate of order", "type": "text" }, { "bbox": [ 380, 661, 398, 678 ], "score": 0.93, "content": "\\frac { H ^ { 3 } } { n \\bar { d } _ { m } }", "type": "inline_equation" }, { "bbox": [ 398, 663, 505, 680 ], "score": 1.0, "content": "and significantly improves", "type": "text" } ], "index": 39 }, { "bbox": [ 102, 65, 509, 93 ], "spans": [ { "bbox": [ 102, 65, 317, 93 ], "score": 1.0, "content": "over the existing non-adaptive results that have order", "type": "text", "cross_page": true }, { "bbox": [ 318, 72, 332, 87 ], "score": 0.91, "content": "\\scriptstyle { \\frac { 1 } { \\sqrt { n } } }", "type": "inline_equation", "cross_page": true }, { "bbox": [ 333, 65, 423, 93 ], "score": 1.0, "content": ". The convergence rate", "type": "text", "cross_page": true }, { "bbox": [ 424, 72, 432, 86 ], "score": 0.86, "content": "\\textstyle { \\frac { 1 } { n } }", "type": "inline_equation", "cross_page": true }, { "bbox": [ 432, 65, 509, 93 ], "score": 1.0, "content": "matches Wen and", "type": "text", "cross_page": true } ], "index": 0 }, { "bbox": [ 308, 85, 420, 97 ], "spans": [ { "bbox": [ 308, 85, 420, 97 ], "score": 1.0, "content": ") regret into the PAC bound.", "type": "text", "cross_page": true } ], "index": 1 } ], "index": 37, "bbox_fs": [ 105, 616, 506, 680 ] } ] }, { "preproc_blocks": [ { "type": "text", "bbox": [ 105, 71, 505, 97 ], "lines": [ { "bbox": [ 102, 65, 509, 93 ], "spans": [ { "bbox": [ 102, 65, 317, 93 ], "score": 1.0, "content": "over the existing non-adaptive results that have order", "type": "text" }, { "bbox": [ 318, 72, 332, 87 ], "score": 0.91, "content": "\\scriptstyle { \\frac { 1 } { \\sqrt { n } } }", "type": "inline_equation" }, { "bbox": [ 333, 65, 423, 93 ], "score": 1.0, "content": ". The convergence rate", "type": "text" }, { "bbox": [ 424, 72, 432, 86 ], "score": 0.86, "content": "\\textstyle { \\frac { 1 } { n } }", "type": "inline_equation" }, { "bbox": [ 432, 65, 509, 93 ], "score": 1.0, "content": "matches Wen and", "type": "text" } ], "index": 0 }, { "bbox": [ 308, 85, 420, 97 ], "spans": [ { "bbox": [ 308, 85, 420, 97 ], "score": 1.0, "content": ") regret into the PAC bound.", "type": "text" } ], "index": 1 } ], "index": 0.5 }, { "type": "text", "bbox": [ 106, 101, 505, 170 ], "lines": [ { "bbox": [ 105, 101, 506, 114 ], "spans": [ { "bbox": [ 105, 101, 506, 114 ], "score": 1.0, "content": "Partially deterministic systems. Practical worlds are complicated and we could sometimes have a", "type": "text" } ], "index": 2 }, { "bbox": [ 106, 113, 505, 125 ], "spans": [ { "bbox": [ 106, 113, 505, 125 ], "score": 1.0, "content": "mixture model which contains both deterministic and stochastic steps. In those scenarios, the main", "type": "text" } ], "index": 3 }, { "bbox": [ 105, 123, 505, 137 ], "spans": [ { "bbox": [ 105, 123, 420, 137 ], "score": 1.0, "content": "complexity is decided by the number of stochastic stages: suppose there are", "type": "text" }, { "bbox": [ 421, 126, 426, 134 ], "score": 0.73, "content": "t", "type": "inline_equation" }, { "bbox": [ 426, 123, 470, 137 ], "score": 1.0, "content": "stochastic", "type": "text" }, { "bbox": [ 470, 124, 497, 135 ], "score": 0.88, "content": "P _ { h } , r _ { h }", "type": "inline_equation" }, { "bbox": [ 497, 123, 505, 137 ], "score": 1.0, "content": "’s", "type": "text" } ], "index": 4 }, { "bbox": [ 105, 135, 504, 153 ], "spans": [ { "bbox": [ 105, 138, 123, 153 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 123, 138, 149, 149 ], "score": 0.88, "content": "H - t", "type": "inline_equation" }, { "bbox": [ 149, 138, 203, 153 ], "score": 1.0, "content": "deterministic", "type": "text" }, { "bbox": [ 203, 138, 235, 150 ], "score": 0.91, "content": "P _ { h ^ { \\prime } } , r _ { h ^ { \\prime } }", "type": "inline_equation" }, { "bbox": [ 235, 138, 431, 153 ], "score": 1.0, "content": "’s, then completing the offline learning guarantees", "type": "text" }, { "bbox": [ 432, 135, 504, 153 ], "score": 0.93, "content": "t \\cdot \\sqrt { \\operatorname* { m a x } Q _ { h } ^ { \\star } / n \\bar { d } _ { m } }", "type": "inline_equation" } ], "index": 5 }, { "bbox": [ 105, 152, 460, 171 ], "spans": [ { "bbox": [ 105, 154, 322, 170 ], "score": 1.0, "content": "suboptimality gap, which could be much smaller than", "type": "text" }, { "bbox": [ 323, 152, 401, 171 ], "score": 0.93, "content": "H \\cdot \\sqrt { \\operatorname* { m a x } Q _ { h } ^ { \\star } / n \\bar { d } _ { m } }", "type": "inline_equation" }, { "bbox": [ 401, 154, 426, 170 ], "score": 1.0, "content": "when", "type": "text" }, { "bbox": [ 426, 156, 455, 167 ], "score": 0.88, "content": "t \\ll H", "type": "inline_equation" }, { "bbox": [ 456, 154, 460, 170 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 6 } ], "index": 4 }, { "type": "text", "bbox": [ 106, 174, 505, 219 ], "lines": [ { "bbox": [ 106, 174, 505, 186 ], "spans": [ { "bbox": [ 106, 174, 505, 186 ], "score": 1.0, "content": "Fast mixing domains. Consider a class of highly mixing non-stationary MDPs (a variant of Zanette", "type": "text" } ], "index": 7 }, { "bbox": [ 104, 184, 506, 198 ], "spans": [ { "bbox": [ 104, 184, 303, 198 ], "score": 1.0, "content": "and Brunskill [2018]) that satisfies the transition", "type": "text" }, { "bbox": [ 303, 185, 390, 198 ], "score": 0.92, "content": "P _ { h } ( \\cdot | s _ { h } , a _ { h } ) : = \\nu _ { h } ( \\cdot ) \\overline { { } }", "type": "inline_equation" }, { "bbox": [ 390, 184, 506, 198 ], "score": 1.0, "content": "depends on neither the state", "type": "text" } ], "index": 8 }, { "bbox": [ 107, 195, 506, 210 ], "spans": [ { "bbox": [ 107, 198, 117, 208 ], "score": 0.83, "content": "s _ { h }", "type": "inline_equation" }, { "bbox": [ 118, 195, 174, 210 ], "score": 1.0, "content": "nor the action", "type": "text" }, { "bbox": [ 175, 198, 186, 208 ], "score": 0.85, "content": "a _ { h }", "type": "inline_equation" }, { "bbox": [ 186, 195, 219, 210 ], "score": 1.0, "content": ". Define", "type": "text" }, { "bbox": [ 219, 196, 305, 208 ], "score": 0.81, "content": "\\bar { s } _ { t } : = \\arg \\operatorname* { m a x } V _ { t } ^ { \\star } ( s )", "type": "inline_equation" }, { "bbox": [ 306, 195, 322, 210 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 323, 198, 406, 208 ], "score": 0.69, "content": "\\underline { { s } } _ { t } : = \\arg \\operatorname* { m a x } V _ { t } ^ { \\star } ( s", "type": "inline_equation" }, { "bbox": [ 409, 195, 465, 210 ], "score": 1.0, "content": ". Also, denote", "type": "text" }, { "bbox": [ 465, 196, 493, 209 ], "score": 0.91, "content": "\\mathrm { r n g } V _ { h } ^ { \\star }", "type": "inline_equation" }, { "bbox": [ 494, 195, 506, 210 ], "score": 1.0, "content": "to", "type": "text" } ], "index": 9 }, { "bbox": [ 105, 207, 423, 220 ], "spans": [ { "bbox": [ 105, 207, 168, 220 ], "score": 1.0, "content": "be the range of", "type": "text" }, { "bbox": [ 169, 208, 182, 220 ], "score": 0.88, "content": "V _ { h } ^ { \\star }", "type": "inline_equation" }, { "bbox": [ 182, 207, 423, 220 ], "score": 1.0, "content": ". In such cases, Bellman optimality equations have the form", "type": "text" } ], "index": 10 } ], "index": 8.5 }, { "type": "interline_equation", "bbox": [ 137, 222, 470, 240 ], "lines": [ { "bbox": [ 137, 222, 470, 240 ], "spans": [ { "bbox": [ 137, 222, 470, 240 ], "score": 0.81, "content": "V _ { 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)", "type": "interline_equation", "image_path": "04a0eeb9dbb60365f7f2f7a9257ad850888db79f47b72addae2c9eaa7d42f5a4.jpg" } ] } ], "index": 11, "virtual_lines": [ { "bbox": [ 137, 222, 470, 240 ], "spans": [], "index": 11 } ] }, { "type": "text", "bbox": [ 106, 242, 505, 291 ], "lines": [ { "bbox": [ 106, 242, 506, 257 ], "spans": [ { "bbox": [ 106, 242, 156, 257 ], "score": 1.0, "content": "which yield", "type": "text" }, { "bbox": [ 156, 243, 456, 256 ], "score": 0.8, "content": "\\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}", "type": "inline_equation" }, { "bbox": [ 456, 242, 506, 257 ], "score": 1.0, "content": ", and this in", "type": "text" } ], "index": 12 }, { "bbox": [ 105, 255, 506, 270 ], "spans": [ { "bbox": [ 105, 255, 147, 270 ], "score": 1.0, "content": "turn gives", "type": "text" }, { "bbox": [ 148, 257, 250, 270 ], "score": 0.92, "content": "\\mathbb { Q } _ { h } ^ { \\star } \\le 1 + ( \\mathrm { r n g } V _ { h } ^ { \\star } ) ^ { 2 } = 2", "type": "inline_equation" }, { "bbox": [ 250, 255, 430, 270 ], "score": 1.0, "content": ". As a result, the suboptimality is bounded by", "type": "text" }, { "bbox": [ 430, 255, 493, 270 ], "score": 0.92, "content": "\\widetilde { O } ( \\sqrt { H ^ { 2 } / n d _ { m } } )", "type": "inline_equation" }, { "bbox": [ 493, 255, 506, 270 ], "score": 1.0, "content": "in", "type": "text" } ], "index": 13 }, { "bbox": [ 106, 268, 506, 280 ], "spans": [ { "bbox": [ 106, 268, 506, 280 ], "score": 1.0, "content": "the worst case. This result reveals, although this is a family of stochastic non-stationary MDPs, but it", "type": "text" } ], "index": 14 }, { "bbox": [ 105, 278, 454, 291 ], "spans": [ { "bbox": [ 105, 279, 389, 291 ], "score": 1.0, "content": "is only as hard as the family of stationary MDPs in the minimax sense", "type": "text" }, { "bbox": [ 389, 278, 449, 291 ], "score": 0.88, "content": "\\left( \\Omega ( H ^ { 2 } / d _ { m } \\epsilon ^ { 2 } ) \\right)", "type": "inline_equation" }, { "bbox": [ 449, 279, 454, 291 ], "score": 1.0, "content": ").", "type": "text" } ], "index": 15 } ], "index": 13.5 }, { "type": "text", "bbox": [ 106, 296, 506, 351 ], "lines": [ { "bbox": [ 104, 295, 507, 317 ], "spans": [ { "bbox": [ 104, 295, 323, 316 ], "score": 1.0, "content": "Tabular contextual bandits. Our result also implies", "type": "text" }, { "bbox": [ 324, 296, 472, 316 ], "score": 0.93, "content": "\\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}", "type": "inline_equation" }, { "bbox": [ 419, 299, 507, 317 ], "score": 1.0, "content": "q Var(r1)n·dµ1 (x1,a1) ) gap for", "type": "text" } ], "index": 16 }, { "bbox": [ 106, 315, 507, 329 ], "spans": [ { "bbox": [ 106, 316, 356, 329 ], "score": 1.0, "content": "the offline tabular contextual bandit problem and improves to", "type": "text" }, { "bbox": [ 356, 315, 401, 329 ], "score": 0.91, "content": "\\widetilde { \\cal O } ( 1 / n d _ { m } )", "type": "inline_equation" }, { "bbox": [ 401, 316, 507, 329 ], "score": 1.0, "content": "when the reward is deter-", "type": "text" } ], "index": 17 }, { "bbox": [ 105, 327, 506, 340 ], "spans": [ { "bbox": [ 105, 327, 387, 340 ], "score": 1.0, "content": "ministic. In either cases, the result is optimal and this is due to: when", "type": "text" }, { "bbox": [ 388, 330, 398, 339 ], "score": 0.85, "content": "r _ { 1 }", "type": "inline_equation" }, { "bbox": [ 398, 327, 506, 340 ], "score": 1.0, "content": "is deterministic, the agent", "type": "text" } ], "index": 18 }, { "bbox": [ 105, 337, 477, 352 ], "spans": [ { "bbox": [ 105, 337, 477, 352 ], "score": 1.0, "content": "only needs one sample at every location (see Bubeck and Cesa-Bianchi [2012] for a survey).", "type": "text" } ], "index": 19 } ], "index": 17.5 }, { "type": "title", "bbox": [ 106, 364, 319, 379 ], "lines": [ { "bbox": [ 105, 364, 320, 380 ], "spans": [ { "bbox": [ 105, 364, 320, 380 ], "score": 1.0, "content": "5 Towards Assumption-Free Offline RL", "type": "text" } ], "index": 20 } ], "index": 20 }, { "type": "text", "bbox": [ 106, 388, 506, 507 ], "lines": [ { "bbox": [ 104, 388, 507, 402 ], "spans": [ { "bbox": [ 104, 388, 507, 402 ], "score": 1.0, "content": "While assumption 2.3 is (arguably) the weakest assumption for correctly learning the optimal value,", "type": "text" } ], "index": 21 }, { "bbox": [ 105, 399, 506, 414 ], "spans": [ { "bbox": [ 105, 399, 506, 414 ], "score": 1.0, "content": "for the real-world applications even this might not be guaranteed. Can we still learn something", "type": "text" } ], "index": 22 }, { "bbox": [ 105, 410, 505, 424 ], "spans": [ { "bbox": [ 105, 410, 480, 424 ], "score": 1.0, "content": "meaningful? In this section, we consider this most general setting where the behavior policy", "type": "text" }, { "bbox": [ 480, 412, 488, 423 ], "score": 0.79, "content": "\\mu", "type": "inline_equation" }, { "bbox": [ 488, 410, 505, 424 ], "score": 1.0, "content": "can", "type": "text" } ], "index": 23 }, { "bbox": [ 105, 421, 506, 435 ], "spans": [ { "bbox": [ 105, 421, 209, 435 ], "score": 1.0, "content": "be arbitrary. In this case,", "type": "text" }, { "bbox": [ 209, 424, 216, 434 ], "score": 0.78, "content": "\\mu", "type": "inline_equation" }, { "bbox": [ 217, 421, 361, 435 ], "score": 1.0, "content": "might not cover any optimal policy", "type": "text" }, { "bbox": [ 362, 423, 374, 433 ], "score": 0.84, "content": "\\pi ^ { \\star }", "type": "inline_equation" }, { "bbox": [ 374, 421, 506, 435 ], "score": 1.0, "content": "(i.e. there might be high reward", "type": "text" } ], "index": 24 }, { "bbox": [ 105, 432, 505, 446 ], "spans": [ { "bbox": [ 105, 432, 142, 446 ], "score": 1.0, "content": "location", "type": "text" }, { "bbox": [ 142, 433, 165, 444 ], "score": 0.91, "content": "( s , a )", "type": "inline_equation" }, { "bbox": [ 165, 432, 183, 446 ], "score": 1.0, "content": "that", "type": "text" }, { "bbox": [ 184, 434, 191, 444 ], "score": 0.82, "content": "\\mu", "type": "inline_equation" }, { "bbox": [ 192, 432, 505, 446 ], "score": 1.0, "content": "can never visit, e.g. in the extreme case where a clumsy doctor only uses one", "type": "text" } ], "index": 25 }, { "bbox": [ 105, 443, 505, 457 ], "spans": [ { "bbox": [ 105, 443, 372, 457 ], "score": 1.0, "content": "treatment all the time), and, irrelevant to the number of episode", "type": "text" }, { "bbox": [ 372, 446, 379, 453 ], "score": 0.72, "content": "n", "type": "inline_equation" }, { "bbox": [ 380, 443, 505, 457 ], "score": 1.0, "content": ", a constant suboptimality gap", "type": "text" } ], "index": 26 }, { "bbox": [ 105, 454, 505, 467 ], "spans": [ { "bbox": [ 105, 455, 435, 467 ], "score": 1.0, "content": "needs to be suffered. To tackle this problem, we create a fictitious augmented MDP", "type": "text" }, { "bbox": [ 435, 454, 451, 465 ], "score": 0.86, "content": "M ^ { \\dagger }", "type": "inline_equation" }, { "bbox": [ 451, 455, 505, 467 ], "score": 1.0, "content": "that can help", "type": "text" } ], "index": 27 }, { "bbox": [ 105, 465, 506, 480 ], "spans": [ { "bbox": [ 105, 466, 379, 480 ], "score": 1.0, "content": "characterize the discrepancy of the values between the original MDP", "type": "text" }, { "bbox": [ 379, 468, 391, 478 ], "score": 0.7, "content": "M", "type": "inline_equation" }, { "bbox": [ 391, 466, 487, 480 ], "score": 1.0, "content": "and the estimated MDP", "type": "text" }, { "bbox": [ 487, 465, 502, 478 ], "score": 0.89, "content": "\\widehat { M } ^ { \\dag }", "type": "inline_equation" }, { "bbox": [ 503, 466, 506, 480 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 28 }, { "bbox": [ 101, 472, 510, 498 ], "spans": [ { "bbox": [ 101, 472, 160, 498 ], "score": 1.0, "content": "In particular,", "type": "text" }, { "bbox": [ 160, 479, 176, 491 ], "score": 0.88, "content": "M ^ { \\dagger }", "type": "inline_equation" }, { "bbox": [ 177, 472, 344, 498 ], "score": 1.0, "content": "is negative towards agnostic state-actions", "type": "text" }, { "bbox": [ 345, 482, 370, 492 ], "score": 0.89, "content": "s _ { h } , a _ { h }", "type": "inline_equation" }, { "bbox": [ 371, 472, 413, 498 ], "score": 1.0, "content": "by setting", "type": "text" }, { "bbox": [ 414, 478, 443, 493 ], "score": 0.93, "content": "r _ { h } ^ { \\dagger } = 0", "type": "inline_equation" }, { "bbox": [ 443, 472, 510, 498 ], "score": 1.0, "content": "cand transitions", "type": "text" } ], "index": 29 }, { "bbox": [ 104, 493, 217, 509 ], "spans": [ { "bbox": [ 104, 493, 217, 509 ], "score": 1.0, "content": "to an absorbing state s†h+1.", "type": "text" } ], "index": 30 } ], "index": 25.5 }, { "type": "text", "bbox": [ 107, 512, 504, 540 ], "lines": [ { "bbox": [ 105, 511, 506, 528 ], "spans": [ { "bbox": [ 105, 511, 231, 528 ], "score": 1.0, "content": "Pessimistic augmented MDP.", "type": "text" }, { "bbox": [ 231, 513, 246, 524 ], "score": 0.89, "content": "M ^ { \\dagger }", "type": "inline_equation" }, { "bbox": [ 247, 511, 365, 528 ], "score": 1.0, "content": "is defined with one extra state", "type": "text" }, { "bbox": [ 366, 512, 376, 527 ], "score": 0.91, "content": "s _ { h } ^ { \\dagger }", "type": "inline_equation" }, { "bbox": [ 377, 511, 403, 528 ], "score": 1.0, "content": "for all", "type": "text" }, { "bbox": [ 403, 514, 484, 527 ], "score": 0.92, "content": "h \\in \\{ 2 , \\dots , H + 1 \\}", "type": "inline_equation" }, { "bbox": [ 484, 511, 506, 528 ], "score": 1.0, "content": "with", "type": "text" } ], "index": 31 }, { "bbox": [ 105, 525, 492, 541 ], "spans": [ { "bbox": [ 105, 526, 213, 541 ], "score": 1.0, "content": "the augmented state space", "type": "text" }, { "bbox": [ 213, 525, 276, 540 ], "score": 0.93, "content": "S ^ { \\dagger } = S \\cup \\{ s _ { h } ^ { \\dagger } \\}", "type": "inline_equation" }, { "bbox": [ 276, 526, 492, 541 ], "score": 1.0, "content": ". The transition and the reward are defined as follows:", "type": "text" } ], "index": 32 } ], "index": 31.5 }, { "type": "interline_equation", "bbox": [ 119, 542, 484, 572 ], "lines": [ { "bbox": [ 119, 542, 484, 572 ], "spans": [ { "bbox": [ 119, 542, 484, 572 ], "score": 0.92, "content": "P _ { 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.", "type": "interline_equation", "image_path": "ac7425d3170fce5cc0e3f0d9a9d4917aedb573be48daab7516765aff0b37b324.jpg" } ] } ], "index": 33, "virtual_lines": [ { "bbox": [ 119, 542, 484, 572 ], "spans": [], "index": 33 } ] }, { "type": "text", "bbox": [ 106, 574, 506, 651 ], "lines": [ { "bbox": [ 105, 574, 506, 590 ], "spans": [ { "bbox": [ 105, 574, 127, 590 ], "score": 1.0, "content": "here", "type": "text" }, { "bbox": [ 127, 577, 137, 588 ], "score": 0.87, "content": "\\delta _ { s }", "type": "inline_equation" }, { "bbox": [ 137, 574, 290, 590 ], "score": 1.0, "content": "is the Dirac measure and we denote", "type": "text" }, { "bbox": [ 290, 575, 308, 589 ], "score": 0.95, "content": "V _ { h } ^ { \\dag \\pi }", "type": "inline_equation" }, { "bbox": [ 308, 574, 327, 590 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 328, 576, 343, 587 ], "score": 0.89, "content": "v ^ { \\dag \\pi }", "type": "inline_equation" }, { "bbox": [ 344, 574, 439, 590 ], "score": 1.0, "content": "to be the values under", "type": "text" }, { "bbox": [ 440, 575, 455, 587 ], "score": 0.77, "content": "M ^ { \\dagger }", "type": "inline_equation" }, { "bbox": [ 456, 574, 462, 590 ], "score": 1.0, "content": ".", "type": "text" }, { "bbox": [ 462, 574, 478, 587 ], "score": 0.81, "content": "\\widehat { M } ^ { \\dagger }", "type": "inline_equation" }, { "bbox": [ 479, 574, 506, 590 ], "score": 1.0, "content": "is the", "type": "text" } ], "index": 34 }, { "bbox": [ 105, 588, 499, 604 ], "spans": [ { "bbox": [ 105, 589, 205, 604 ], "score": 1.0, "content": "empirical counterpart of", "type": "text" }, { "bbox": [ 205, 589, 221, 601 ], "score": 0.89, "content": "M ^ { \\dagger }", "type": "inline_equation" }, { "bbox": [ 222, 589, 243, 604 ], "score": 1.0, "content": "with", "type": "text" }, { "bbox": [ 243, 588, 252, 601 ], "score": 0.77, "content": "\\widehat { P }", "type": "inline_equation" }, { "bbox": [ 252, 589, 255, 604 ], "score": 1.0, "content": ",", "type": "text" }, { "bbox": [ 255, 590, 263, 600 ], "score": 0.72, "content": "\\widehat { r }", "type": "inline_equation" }, { "bbox": [ 263, 589, 371, 604 ], "score": 1.0, "content": "(the same as (2)) replacing", "type": "text" }, { "bbox": [ 372, 591, 391, 601 ], "score": 0.53, "content": "P , r .", "type": "inline_equation" }, { "bbox": [ 392, 589, 499, 604 ], "score": 1.0, "content": "By Algorithm 1, we have", "type": "text" } ], "index": 35 }, { "bbox": [ 105, 601, 504, 617 ], "spans": [ { "bbox": [ 105, 601, 496, 617 ], "score": 1.0, "content": "bTheorem 5.1 (Assumption-free offline reinforcement learning). Let us make no assumption for", "type": "text" }, { "bbox": [ 496, 605, 504, 615 ], "score": 0.78, "content": "\\mu", "type": "inline_equation" } ], "index": 36 }, { "bbox": [ 105, 613, 506, 628 ], "spans": [ { "bbox": [ 105, 613, 172, 628 ], "score": 1.0, "content": "and still denote", "type": "text" }, { "bbox": [ 173, 614, 367, 628 ], "score": 0.9, "content": "\\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 \\}", "type": "inline_equation" }, { "bbox": [ 368, 613, 407, 628 ], "score": 1.0, "content": ". For any", "type": "text" }, { "bbox": [ 408, 615, 452, 625 ], "score": 0.89, "content": "0 < \\delta < 1", "type": "inline_equation" }, { "bbox": [ 453, 613, 506, 628 ], "score": 1.0, "content": ", there exists", "type": "text" } ], "index": 37 }, { "bbox": [ 105, 627, 505, 640 ], "spans": [ { "bbox": [ 105, 627, 181, 640 ], "score": 1.0, "content": "absolute constants", "type": "text" }, { "bbox": [ 181, 628, 225, 639 ], "score": 0.91, "content": "c _ { 0 } , C ^ { \\prime } > 0", "type": "inline_equation" }, { "bbox": [ 225, 627, 289, 640 ], "score": 1.0, "content": ", such that when", "type": "text" }, { "bbox": [ 290, 627, 436, 639 ], "score": 0.74, "content": "n > c _ { 0 } \\cdot 1 / \\bar { d } _ { m } \\cdot \\iota ( \\iota = \\log ( H S A / \\delta ) )", "type": "inline_equation" }, { "bbox": [ 437, 627, 505, 640 ], "score": 1.0, "content": ", with probability", "type": "text" } ], "index": 38 }, { "bbox": [ 106, 638, 453, 652 ], "spans": [ { "bbox": [ 106, 639, 129, 649 ], "score": 0.85, "content": "1 - \\delta", "type": "inline_equation" }, { "bbox": [ 129, 638, 202, 652 ], "score": 1.0, "content": ", the output policy", "type": "text" }, { "bbox": [ 202, 639, 210, 649 ], "score": 0.73, "content": "\\widehat { \\pi }", "type": "inline_equation" }, { "bbox": [ 210, 638, 308, 652 ], "score": 1.0, "content": "of APVI satisfies (recall", "type": "text" }, { "bbox": [ 308, 639, 448, 651 ], "score": 0.9, "content": "\\mathcal { C } _ { h } : = \\{ ( s _ { h } , a _ { h } ) : d _ { h } ^ { \\mu } ( s _ { h } , a _ { h } ) > 0 \\}", "type": "inline_equation" }, { "bbox": [ 448, 638, 453, 652 ], "score": 1.0, "content": ")", "type": "text" } ], "index": 39 } ], "index": 36.5 }, { "type": "interline_equation", "bbox": [ 109, 654, 489, 691 ], "lines": [ { "bbox": [ 109, 654, 489, 691 ], "spans": [ { "bbox": [ 109, 654, 489, 691 ], "score": 0.92, "content": "v ^ { \\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) ,", "type": "interline_equation", "image_path": "2f801afdfa20cca687c2bb53ebbb35e04b08e7d49d57f9af1205563dc12e9b3f.jpg" } ] } ], "index": 41, "virtual_lines": [ { "bbox": [ 109, 654, 489, 666.3333333333334 ], "spans": [], "index": 40 }, { "bbox": [ 109, 666.3333333333334, 489, 678.6666666666667 ], "spans": [], "index": 41 }, { "bbox": [ 109, 678.6666666666667, 489, 691.0000000000001 ], "spans": [], "index": 42 } ] }, { "type": "text", "bbox": [ 106, 694, 510, 725 ], "lines": [ { "bbox": [ 103, 691, 508, 712 ], "spans": [ { "bbox": [ 103, 691, 131, 712 ], "score": 1.0, "content": "where", "type": "text" }, { "bbox": [ 131, 695, 334, 709 ], "score": 0.85, "content": "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 } ) .", "type": "inline_equation" }, { "bbox": [ 334, 691, 359, 712 ], "score": 1.0, "content": "for all", "type": "text" }, { "bbox": [ 359, 697, 423, 708 ], "score": 0.9, "content": "s _ { h } , a _ { h } \\in S \\times \\mathcal { A } ,", "type": "inline_equation" }, { "bbox": [ 423, 691, 469, 712 ], "score": 1.0, "content": ", and for all", "type": "text" }, { "bbox": [ 470, 696, 503, 709 ], "score": 0.92, "content": "h \\in [ H ]", "type": "inline_equation" }, { "bbox": [ 503, 691, 508, 712 ], "score": 1.0, "content": ",", "type": "text" } ], "index": 43 }, { "bbox": [ 108, 702, 424, 730 ], "spans": [ { "bbox": [ 108, 709, 303, 725 ], "score": 0.92, "content": "\\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}", "type": "inline_equation" }, { "bbox": [ 303, 702, 407, 730 ], "score": 1.0, "content": ". The proof is in Appendix", "type": "text" }, { "bbox": [ 408, 712, 416, 721 ], "score": 0.75, "content": "E", "type": "inline_equation" }, { "bbox": [ 416, 702, 424, 730 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 44 } ], "index": 43.5 } ], "page_idx": 8, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 302, 741, 309, 750 ], "lines": [ { "bbox": [ 302, 741, 309, 752 ], "spans": [ { "bbox": [ 302, 741, 309, 752 ], "score": 1.0, "content": "9", "type": "text" } ] } ] } ], "para_blocks": [ { "type": "text", "bbox": [ 105, 71, 505, 97 ], "lines": [], "index": 0.5, "bbox_fs": [ 102, 65, 509, 97 ], "lines_deleted": true }, { "type": "text", "bbox": [ 106, 101, 505, 170 ], "lines": [ { "bbox": [ 105, 101, 506, 114 ], "spans": [ { "bbox": [ 105, 101, 506, 114 ], "score": 1.0, "content": "Partially deterministic systems. Practical worlds are complicated and we could sometimes have a", "type": "text" } ], "index": 2 }, { "bbox": [ 106, 113, 505, 125 ], "spans": [ { "bbox": [ 106, 113, 505, 125 ], "score": 1.0, "content": "mixture model which contains both deterministic and stochastic steps. In those scenarios, the main", "type": "text" } ], "index": 3 }, { "bbox": [ 105, 123, 505, 137 ], "spans": [ { "bbox": [ 105, 123, 420, 137 ], "score": 1.0, "content": "complexity is decided by the number of stochastic stages: suppose there are", "type": "text" }, { "bbox": [ 421, 126, 426, 134 ], "score": 0.73, "content": "t", "type": "inline_equation" }, { "bbox": [ 426, 123, 470, 137 ], "score": 1.0, "content": "stochastic", "type": "text" }, { "bbox": [ 470, 124, 497, 135 ], "score": 0.88, "content": "P _ { h } , r _ { h }", "type": "inline_equation" }, { "bbox": [ 497, 123, 505, 137 ], "score": 1.0, "content": "’s", "type": "text" } ], "index": 4 }, { "bbox": [ 105, 135, 504, 153 ], "spans": [ { "bbox": [ 105, 138, 123, 153 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 123, 138, 149, 149 ], "score": 0.88, "content": "H - t", "type": "inline_equation" }, { "bbox": [ 149, 138, 203, 153 ], "score": 1.0, "content": "deterministic", "type": "text" }, { "bbox": [ 203, 138, 235, 150 ], "score": 0.91, "content": "P _ { h ^ { \\prime } } , r _ { h ^ { \\prime } }", "type": "inline_equation" }, { "bbox": [ 235, 138, 431, 153 ], "score": 1.0, "content": "’s, then completing the offline learning guarantees", "type": "text" }, { "bbox": [ 432, 135, 504, 153 ], "score": 0.93, "content": "t \\cdot \\sqrt { \\operatorname* { m a x } Q _ { h } ^ { \\star } / n \\bar { d } _ { m } }", "type": "inline_equation" } ], "index": 5 }, { "bbox": [ 105, 152, 460, 171 ], "spans": [ { "bbox": [ 105, 154, 322, 170 ], "score": 1.0, "content": "suboptimality gap, which could be much smaller than", "type": "text" }, { "bbox": [ 323, 152, 401, 171 ], "score": 0.93, "content": "H \\cdot \\sqrt { \\operatorname* { m a x } Q _ { h } ^ { \\star } / n \\bar { d } _ { m } }", "type": "inline_equation" }, { "bbox": [ 401, 154, 426, 170 ], "score": 1.0, "content": "when", "type": "text" }, { "bbox": [ 426, 156, 455, 167 ], "score": 0.88, "content": "t \\ll H", "type": "inline_equation" }, { "bbox": [ 456, 154, 460, 170 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 6 } ], "index": 4, "bbox_fs": [ 105, 101, 506, 171 ] }, { "type": "text", "bbox": [ 106, 174, 505, 219 ], "lines": [ { "bbox": [ 106, 174, 505, 186 ], "spans": [ { "bbox": [ 106, 174, 505, 186 ], "score": 1.0, "content": "Fast mixing domains. Consider a class of highly mixing non-stationary MDPs (a variant of Zanette", "type": "text" } ], "index": 7 }, { "bbox": [ 104, 184, 506, 198 ], "spans": [ { "bbox": [ 104, 184, 303, 198 ], "score": 1.0, "content": "and Brunskill [2018]) that satisfies the transition", "type": "text" }, { "bbox": [ 303, 185, 390, 198 ], "score": 0.92, "content": "P _ { h } ( \\cdot | s _ { h } , a _ { h } ) : = \\nu _ { h } ( \\cdot ) \\overline { { } }", "type": "inline_equation" }, { "bbox": [ 390, 184, 506, 198 ], "score": 1.0, "content": "depends on neither the state", "type": "text" } ], "index": 8 }, { "bbox": [ 107, 195, 506, 210 ], "spans": [ { "bbox": [ 107, 198, 117, 208 ], "score": 0.83, "content": "s _ { h }", "type": "inline_equation" }, { "bbox": [ 118, 195, 174, 210 ], "score": 1.0, "content": "nor the action", "type": "text" }, { "bbox": [ 175, 198, 186, 208 ], "score": 0.85, "content": "a _ { h }", "type": "inline_equation" }, { "bbox": [ 186, 195, 219, 210 ], "score": 1.0, "content": ". Define", "type": "text" }, { "bbox": [ 219, 196, 305, 208 ], "score": 0.81, "content": "\\bar { s } _ { t } : = \\arg \\operatorname* { m a x } V _ { t } ^ { \\star } ( s )", "type": "inline_equation" }, { "bbox": [ 306, 195, 322, 210 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 323, 198, 406, 208 ], "score": 0.69, "content": "\\underline { { s } } _ { t } : = \\arg \\operatorname* { m a x } V _ { t } ^ { \\star } ( s", "type": "inline_equation" }, { "bbox": [ 409, 195, 465, 210 ], "score": 1.0, "content": ". Also, denote", "type": "text" }, { "bbox": [ 465, 196, 493, 209 ], "score": 0.91, "content": "\\mathrm { r n g } V _ { h } ^ { \\star }", "type": "inline_equation" }, { "bbox": [ 494, 195, 506, 210 ], "score": 1.0, "content": "to", "type": "text" } ], "index": 9 }, { "bbox": [ 105, 207, 423, 220 ], "spans": [ { "bbox": [ 105, 207, 168, 220 ], "score": 1.0, "content": "be the range of", "type": "text" }, { "bbox": [ 169, 208, 182, 220 ], "score": 0.88, "content": "V _ { h } ^ { \\star }", "type": "inline_equation" }, { "bbox": [ 182, 207, 423, 220 ], "score": 1.0, "content": ". In such cases, Bellman optimality equations have the form", "type": "text" } ], "index": 10 } ], "index": 8.5, "bbox_fs": [ 104, 174, 506, 220 ] }, { "type": "interline_equation", "bbox": [ 137, 222, 470, 240 ], "lines": [ { "bbox": [ 137, 222, 470, 240 ], "spans": [ { "bbox": [ 137, 222, 470, 240 ], "score": 0.81, "content": "V _ { 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)", "type": "interline_equation", "image_path": "04a0eeb9dbb60365f7f2f7a9257ad850888db79f47b72addae2c9eaa7d42f5a4.jpg" } ] } ], "index": 11, "virtual_lines": [ { "bbox": [ 137, 222, 470, 240 ], "spans": [], "index": 11 } ] }, { "type": "text", "bbox": [ 106, 242, 505, 291 ], "lines": [ { "bbox": [ 106, 242, 506, 257 ], "spans": [ { "bbox": [ 106, 242, 156, 257 ], "score": 1.0, "content": "which yield", "type": "text" }, { "bbox": [ 156, 243, 456, 256 ], "score": 0.8, "content": "\\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}", "type": "inline_equation" }, { "bbox": [ 456, 242, 506, 257 ], "score": 1.0, "content": ", and this in", "type": "text" } ], "index": 12 }, { "bbox": [ 105, 255, 506, 270 ], "spans": [ { "bbox": [ 105, 255, 147, 270 ], "score": 1.0, "content": "turn gives", "type": "text" }, { "bbox": [ 148, 257, 250, 270 ], "score": 0.92, "content": "\\mathbb { Q } _ { h } ^ { \\star } \\le 1 + ( \\mathrm { r n g } V _ { h } ^ { \\star } ) ^ { 2 } = 2", "type": "inline_equation" }, { "bbox": [ 250, 255, 430, 270 ], "score": 1.0, "content": ". As a result, the suboptimality is bounded by", "type": "text" }, { "bbox": [ 430, 255, 493, 270 ], "score": 0.92, "content": "\\widetilde { O } ( \\sqrt { H ^ { 2 } / n d _ { m } } )", "type": "inline_equation" }, { "bbox": [ 493, 255, 506, 270 ], "score": 1.0, "content": "in", "type": "text" } ], "index": 13 }, { "bbox": [ 106, 268, 506, 280 ], "spans": [ { "bbox": [ 106, 268, 506, 280 ], "score": 1.0, "content": "the worst case. This result reveals, although this is a family of stochastic non-stationary MDPs, but it", "type": "text" } ], "index": 14 }, { "bbox": [ 105, 278, 454, 291 ], "spans": [ { "bbox": [ 105, 279, 389, 291 ], "score": 1.0, "content": "is only as hard as the family of stationary MDPs in the minimax sense", "type": "text" }, { "bbox": [ 389, 278, 449, 291 ], "score": 0.88, "content": "\\left( \\Omega ( H ^ { 2 } / d _ { m } \\epsilon ^ { 2 } ) \\right)", "type": "inline_equation" }, { "bbox": [ 449, 279, 454, 291 ], "score": 1.0, "content": ").", "type": "text" } ], "index": 15 } ], "index": 13.5, "bbox_fs": [ 105, 242, 506, 291 ] }, { "type": "text", "bbox": [ 106, 296, 506, 351 ], "lines": [ { "bbox": [ 104, 295, 507, 317 ], "spans": [ { "bbox": [ 104, 295, 323, 316 ], "score": 1.0, "content": "Tabular contextual bandits. Our result also implies", "type": "text" }, { "bbox": [ 324, 296, 472, 316 ], "score": 0.93, "content": "\\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}", "type": "inline_equation" }, { "bbox": [ 419, 299, 507, 317 ], "score": 1.0, "content": "q Var(r1)n·dµ1 (x1,a1) ) gap for", "type": "text" } ], "index": 16 }, { "bbox": [ 106, 315, 507, 329 ], "spans": [ { "bbox": [ 106, 316, 356, 329 ], "score": 1.0, "content": "the offline tabular contextual bandit problem and improves to", "type": "text" }, { "bbox": [ 356, 315, 401, 329 ], "score": 0.91, "content": "\\widetilde { \\cal O } ( 1 / n d _ { m } )", "type": "inline_equation" }, { "bbox": [ 401, 316, 507, 329 ], "score": 1.0, "content": "when the reward is deter-", "type": "text" } ], "index": 17 }, { "bbox": [ 105, 327, 506, 340 ], "spans": [ { "bbox": [ 105, 327, 387, 340 ], "score": 1.0, "content": "ministic. In either cases, the result is optimal and this is due to: when", "type": "text" }, { "bbox": [ 388, 330, 398, 339 ], "score": 0.85, "content": "r _ { 1 }", "type": "inline_equation" }, { "bbox": [ 398, 327, 506, 340 ], "score": 1.0, "content": "is deterministic, the agent", "type": "text" } ], "index": 18 }, { "bbox": [ 105, 337, 477, 352 ], "spans": [ { "bbox": [ 105, 337, 477, 352 ], "score": 1.0, "content": "only needs one sample at every location (see Bubeck and Cesa-Bianchi [2012] for a survey).", "type": "text" } ], "index": 19 } ], "index": 17.5, "bbox_fs": [ 104, 295, 507, 352 ] }, { "type": "title", "bbox": [ 106, 364, 319, 379 ], "lines": [ { "bbox": [ 105, 364, 320, 380 ], "spans": [ { "bbox": [ 105, 364, 320, 380 ], "score": 1.0, "content": "5 Towards Assumption-Free Offline RL", "type": "text" } ], "index": 20 } ], "index": 20 }, { "type": "text", "bbox": [ 106, 388, 506, 507 ], "lines": [ { "bbox": [ 104, 388, 507, 402 ], "spans": [ { "bbox": [ 104, 388, 507, 402 ], "score": 1.0, "content": "While assumption 2.3 is (arguably) the weakest assumption for correctly learning the optimal value,", "type": "text" } ], "index": 21 }, { "bbox": [ 105, 399, 506, 414 ], "spans": [ { "bbox": [ 105, 399, 506, 414 ], "score": 1.0, "content": "for the real-world applications even this might not be guaranteed. Can we still learn something", "type": "text" } ], "index": 22 }, { "bbox": [ 105, 410, 505, 424 ], "spans": [ { "bbox": [ 105, 410, 480, 424 ], "score": 1.0, "content": "meaningful? In this section, we consider this most general setting where the behavior policy", "type": "text" }, { "bbox": [ 480, 412, 488, 423 ], "score": 0.79, "content": "\\mu", "type": "inline_equation" }, { "bbox": [ 488, 410, 505, 424 ], "score": 1.0, "content": "can", "type": "text" } ], "index": 23 }, { "bbox": [ 105, 421, 506, 435 ], "spans": [ { "bbox": [ 105, 421, 209, 435 ], "score": 1.0, "content": "be arbitrary. In this case,", "type": "text" }, { "bbox": [ 209, 424, 216, 434 ], "score": 0.78, "content": "\\mu", "type": "inline_equation" }, { "bbox": [ 217, 421, 361, 435 ], "score": 1.0, "content": "might not cover any optimal policy", "type": "text" }, { "bbox": [ 362, 423, 374, 433 ], "score": 0.84, "content": "\\pi ^ { \\star }", "type": "inline_equation" }, { "bbox": [ 374, 421, 506, 435 ], "score": 1.0, "content": "(i.e. there might be high reward", "type": "text" } ], "index": 24 }, { "bbox": [ 105, 432, 505, 446 ], "spans": [ { "bbox": [ 105, 432, 142, 446 ], "score": 1.0, "content": "location", "type": "text" }, { "bbox": [ 142, 433, 165, 444 ], "score": 0.91, "content": "( s , a )", "type": "inline_equation" }, { "bbox": [ 165, 432, 183, 446 ], "score": 1.0, "content": "that", "type": "text" }, { "bbox": [ 184, 434, 191, 444 ], "score": 0.82, "content": "\\mu", "type": "inline_equation" }, { "bbox": [ 192, 432, 505, 446 ], "score": 1.0, "content": "can never visit, e.g. in the extreme case where a clumsy doctor only uses one", "type": "text" } ], "index": 25 }, { "bbox": [ 105, 443, 505, 457 ], "spans": [ { "bbox": [ 105, 443, 372, 457 ], "score": 1.0, "content": "treatment all the time), and, irrelevant to the number of episode", "type": "text" }, { "bbox": [ 372, 446, 379, 453 ], "score": 0.72, "content": "n", "type": "inline_equation" }, { "bbox": [ 380, 443, 505, 457 ], "score": 1.0, "content": ", a constant suboptimality gap", "type": "text" } ], "index": 26 }, { "bbox": [ 105, 454, 505, 467 ], "spans": [ { "bbox": [ 105, 455, 435, 467 ], "score": 1.0, "content": "needs to be suffered. To tackle this problem, we create a fictitious augmented MDP", "type": "text" }, { "bbox": [ 435, 454, 451, 465 ], "score": 0.86, "content": "M ^ { \\dagger }", "type": "inline_equation" }, { "bbox": [ 451, 455, 505, 467 ], "score": 1.0, "content": "that can help", "type": "text" } ], "index": 27 }, { "bbox": [ 105, 465, 506, 480 ], "spans": [ { "bbox": [ 105, 466, 379, 480 ], "score": 1.0, "content": "characterize the discrepancy of the values between the original MDP", "type": "text" }, { "bbox": [ 379, 468, 391, 478 ], "score": 0.7, "content": "M", "type": "inline_equation" }, { "bbox": [ 391, 466, 487, 480 ], "score": 1.0, "content": "and the estimated MDP", "type": "text" }, { "bbox": [ 487, 465, 502, 478 ], "score": 0.89, "content": "\\widehat { M } ^ { \\dag }", "type": "inline_equation" }, { "bbox": [ 503, 466, 506, 480 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 28 }, { "bbox": [ 101, 472, 510, 498 ], "spans": [ { "bbox": [ 101, 472, 160, 498 ], "score": 1.0, "content": "In particular,", "type": "text" }, { "bbox": [ 160, 479, 176, 491 ], "score": 0.88, "content": "M ^ { \\dagger }", "type": "inline_equation" }, { "bbox": [ 177, 472, 344, 498 ], "score": 1.0, "content": "is negative towards agnostic state-actions", "type": "text" }, { "bbox": [ 345, 482, 370, 492 ], "score": 0.89, "content": "s _ { h } , a _ { h }", "type": "inline_equation" }, { "bbox": [ 371, 472, 413, 498 ], "score": 1.0, "content": "by setting", "type": "text" }, { "bbox": [ 414, 478, 443, 493 ], "score": 0.93, "content": "r _ { h } ^ { \\dagger } = 0", "type": "inline_equation" }, { "bbox": [ 443, 472, 510, 498 ], "score": 1.0, "content": "cand transitions", "type": "text" } ], "index": 29 }, { "bbox": [ 104, 493, 217, 509 ], "spans": [ { "bbox": [ 104, 493, 217, 509 ], "score": 1.0, "content": "to an absorbing state s†h+1.", "type": "text" } ], "index": 30 } ], "index": 25.5, "bbox_fs": [ 101, 388, 510, 509 ] }, { "type": "text", "bbox": [ 107, 512, 504, 540 ], "lines": [ { "bbox": [ 105, 511, 506, 528 ], "spans": [ { "bbox": [ 105, 511, 231, 528 ], "score": 1.0, "content": "Pessimistic augmented MDP.", "type": "text" }, { "bbox": [ 231, 513, 246, 524 ], "score": 0.89, "content": "M ^ { \\dagger }", "type": "inline_equation" }, { "bbox": [ 247, 511, 365, 528 ], "score": 1.0, "content": "is defined with one extra state", "type": "text" }, { "bbox": [ 366, 512, 376, 527 ], "score": 0.91, "content": "s _ { h } ^ { \\dagger }", "type": "inline_equation" }, { "bbox": [ 377, 511, 403, 528 ], "score": 1.0, "content": "for all", "type": "text" }, { "bbox": [ 403, 514, 484, 527 ], "score": 0.92, "content": "h \\in \\{ 2 , \\dots , H + 1 \\}", "type": "inline_equation" }, { "bbox": [ 484, 511, 506, 528 ], "score": 1.0, "content": "with", "type": "text" } ], "index": 31 }, { "bbox": [ 105, 525, 492, 541 ], "spans": [ { "bbox": [ 105, 526, 213, 541 ], "score": 1.0, "content": "the augmented state space", "type": "text" }, { "bbox": [ 213, 525, 276, 540 ], "score": 0.93, "content": "S ^ { \\dagger } = S \\cup \\{ s _ { h } ^ { \\dagger } \\}", "type": "inline_equation" }, { "bbox": [ 276, 526, 492, 541 ], "score": 1.0, "content": ". The transition and the reward are defined as follows:", "type": "text" } ], "index": 32 } ], "index": 31.5, "bbox_fs": [ 105, 511, 506, 541 ] }, { "type": "interline_equation", "bbox": [ 119, 542, 484, 572 ], "lines": [ { "bbox": [ 119, 542, 484, 572 ], "spans": [ { "bbox": [ 119, 542, 484, 572 ], "score": 0.92, "content": "P _ { 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.", "type": "interline_equation", "image_path": "ac7425d3170fce5cc0e3f0d9a9d4917aedb573be48daab7516765aff0b37b324.jpg" } ] } ], "index": 33, "virtual_lines": [ { "bbox": [ 119, 542, 484, 572 ], "spans": [], "index": 33 } ] }, { "type": "text", "bbox": [ 106, 574, 506, 651 ], "lines": [ { "bbox": [ 105, 574, 506, 590 ], "spans": [ { "bbox": [ 105, 574, 127, 590 ], "score": 1.0, "content": "here", "type": "text" }, { "bbox": [ 127, 577, 137, 588 ], "score": 0.87, "content": "\\delta _ { s }", "type": "inline_equation" }, { "bbox": [ 137, 574, 290, 590 ], "score": 1.0, "content": "is the Dirac measure and we denote", "type": "text" }, { "bbox": [ 290, 575, 308, 589 ], "score": 0.95, "content": "V _ { h } ^ { \\dag \\pi }", "type": "inline_equation" }, { "bbox": [ 308, 574, 327, 590 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 328, 576, 343, 587 ], "score": 0.89, "content": "v ^ { \\dag \\pi }", "type": "inline_equation" }, { "bbox": [ 344, 574, 439, 590 ], "score": 1.0, "content": "to be the values under", "type": "text" }, { "bbox": [ 440, 575, 455, 587 ], "score": 0.77, "content": "M ^ { \\dagger }", "type": "inline_equation" }, { "bbox": [ 456, 574, 462, 590 ], "score": 1.0, "content": ".", "type": "text" }, { "bbox": [ 462, 574, 478, 587 ], "score": 0.81, "content": "\\widehat { M } ^ { \\dagger }", "type": "inline_equation" }, { "bbox": [ 479, 574, 506, 590 ], "score": 1.0, "content": "is the", "type": "text" } ], "index": 34 }, { "bbox": [ 105, 588, 499, 604 ], "spans": [ { "bbox": [ 105, 589, 205, 604 ], "score": 1.0, "content": "empirical counterpart of", "type": "text" }, { "bbox": [ 205, 589, 221, 601 ], "score": 0.89, "content": "M ^ { \\dagger }", "type": "inline_equation" }, { "bbox": [ 222, 589, 243, 604 ], "score": 1.0, "content": "with", "type": "text" }, { "bbox": [ 243, 588, 252, 601 ], "score": 0.77, "content": "\\widehat { P }", "type": "inline_equation" }, { "bbox": [ 252, 589, 255, 604 ], "score": 1.0, "content": ",", "type": "text" }, { "bbox": [ 255, 590, 263, 600 ], "score": 0.72, "content": "\\widehat { r }", "type": "inline_equation" }, { "bbox": [ 263, 589, 371, 604 ], "score": 1.0, "content": "(the same as (2)) replacing", "type": "text" }, { "bbox": [ 372, 591, 391, 601 ], "score": 0.53, "content": "P , r .", "type": "inline_equation" }, { "bbox": [ 392, 589, 499, 604 ], "score": 1.0, "content": "By Algorithm 1, we have", "type": "text" } ], "index": 35 }, { "bbox": [ 105, 601, 504, 617 ], "spans": [ { "bbox": [ 105, 601, 496, 617 ], "score": 1.0, "content": "bTheorem 5.1 (Assumption-free offline reinforcement learning). Let us make no assumption for", "type": "text" }, { "bbox": [ 496, 605, 504, 615 ], "score": 0.78, "content": "\\mu", "type": "inline_equation" } ], "index": 36 }, { "bbox": [ 105, 613, 506, 628 ], "spans": [ { "bbox": [ 105, 613, 172, 628 ], "score": 1.0, "content": "and still denote", "type": "text" }, { "bbox": [ 173, 614, 367, 628 ], "score": 0.9, "content": "\\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 \\}", "type": "inline_equation" }, { "bbox": [ 368, 613, 407, 628 ], "score": 1.0, "content": ". For any", "type": "text" }, { "bbox": [ 408, 615, 452, 625 ], "score": 0.89, "content": "0 < \\delta < 1", "type": "inline_equation" }, { "bbox": [ 453, 613, 506, 628 ], "score": 1.0, "content": ", there exists", "type": "text" } ], "index": 37 }, { "bbox": [ 105, 627, 505, 640 ], "spans": [ { "bbox": [ 105, 627, 181, 640 ], "score": 1.0, "content": "absolute constants", "type": "text" }, { "bbox": [ 181, 628, 225, 639 ], "score": 0.91, "content": "c _ { 0 } , C ^ { \\prime } > 0", "type": "inline_equation" }, { "bbox": [ 225, 627, 289, 640 ], "score": 1.0, "content": ", such that when", "type": "text" }, { "bbox": [ 290, 627, 436, 639 ], "score": 0.74, "content": "n > c _ { 0 } \\cdot 1 / \\bar { d } _ { m } \\cdot \\iota ( \\iota = \\log ( H S A / \\delta ) )", "type": "inline_equation" }, { "bbox": [ 437, 627, 505, 640 ], "score": 1.0, "content": ", with probability", "type": "text" } ], "index": 38 }, { "bbox": [ 106, 638, 453, 652 ], "spans": [ { "bbox": [ 106, 639, 129, 649 ], "score": 0.85, "content": "1 - \\delta", "type": "inline_equation" }, { "bbox": [ 129, 638, 202, 652 ], "score": 1.0, "content": ", the output policy", "type": "text" }, { "bbox": [ 202, 639, 210, 649 ], "score": 0.73, "content": "\\widehat { \\pi }", "type": "inline_equation" }, { "bbox": [ 210, 638, 308, 652 ], "score": 1.0, "content": "of APVI satisfies (recall", "type": "text" }, { "bbox": [ 308, 639, 448, 651 ], "score": 0.9, "content": "\\mathcal { C } _ { h } : = \\{ ( s _ { h } , a _ { h } ) : d _ { h } ^ { \\mu } ( s _ { h } , a _ { h } ) > 0 \\}", "type": "inline_equation" }, { "bbox": [ 448, 638, 453, 652 ], "score": 1.0, "content": ")", "type": "text" } ], "index": 39 } ], "index": 36.5, "bbox_fs": [ 105, 574, 506, 652 ] }, { "type": "interline_equation", "bbox": [ 109, 654, 489, 691 ], "lines": [ { "bbox": [ 109, 654, 489, 691 ], "spans": [ { "bbox": [ 109, 654, 489, 691 ], "score": 0.92, "content": "v ^ { \\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) ,", "type": "interline_equation", "image_path": "2f801afdfa20cca687c2bb53ebbb35e04b08e7d49d57f9af1205563dc12e9b3f.jpg" } ] } ], "index": 41, "virtual_lines": [ { "bbox": [ 109, 654, 489, 666.3333333333334 ], "spans": [], "index": 40 }, { "bbox": [ 109, 666.3333333333334, 489, 678.6666666666667 ], "spans": [], "index": 41 }, { "bbox": [ 109, 678.6666666666667, 489, 691.0000000000001 ], "spans": [], "index": 42 } ] }, { "type": "text", "bbox": [ 106, 694, 510, 725 ], "lines": [ { "bbox": [ 103, 691, 508, 712 ], "spans": [ { "bbox": [ 103, 691, 131, 712 ], "score": 1.0, "content": "where", "type": "text" }, { "bbox": [ 131, 695, 334, 709 ], "score": 0.85, "content": "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 } ) .", "type": "inline_equation" }, { "bbox": [ 334, 691, 359, 712 ], "score": 1.0, "content": "for all", "type": "text" }, { "bbox": [ 359, 697, 423, 708 ], "score": 0.9, "content": "s _ { h } , a _ { h } \\in S \\times \\mathcal { A } ,", "type": "inline_equation" }, { "bbox": [ 423, 691, 469, 712 ], "score": 1.0, "content": ", and for all", "type": "text" }, { "bbox": [ 470, 696, 503, 709 ], "score": 0.92, "content": "h \\in [ H ]", "type": "inline_equation" }, { "bbox": [ 503, 691, 508, 712 ], "score": 1.0, "content": ",", "type": "text" } ], "index": 43 }, { "bbox": [ 108, 702, 424, 730 ], "spans": [ { "bbox": [ 108, 709, 303, 725 ], "score": 0.92, "content": "\\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}", "type": "inline_equation" }, { "bbox": [ 303, 702, 407, 730 ], "score": 1.0, "content": ". The proof is in Appendix", "type": "text" }, { "bbox": [ 408, 712, 416, 721 ], "score": 0.75, "content": "E", "type": "inline_equation" }, { "bbox": [ 416, 702, 424, 730 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 44 } ], "index": 43.5, "bbox_fs": [ 103, 691, 508, 730 ] } ] }, { "preproc_blocks": [ { "type": "text", "bbox": [ 106, 72, 505, 157 ], "lines": [ { "bbox": [ 105, 71, 505, 85 ], "spans": [ { "bbox": [ 105, 71, 246, 85 ], "score": 1.0, "content": "Take-aways of Theorem 5.1. In", "type": "text" }, { "bbox": [ 246, 72, 262, 83 ], "score": 0.89, "content": "M ^ { \\dagger }", "type": "inline_equation" }, { "bbox": [ 262, 71, 505, 85 ], "score": 1.0, "content": ", there is no agnostic location any more since the original", "type": "text" } ], "index": 0 }, { "bbox": [ 105, 82, 506, 97 ], "spans": [ { "bbox": [ 105, 82, 363, 97 ], "score": 1.0, "content": "unknown spaces now all have known deterministic transitions to", "type": "text" }, { "bbox": [ 363, 83, 373, 93 ], "score": 0.87, "content": "s ^ { \\dagger }", "type": "inline_equation" }, { "bbox": [ 374, 82, 385, 97 ], "score": 1.0, "content": "in", "type": "text" }, { "bbox": [ 385, 83, 401, 93 ], "score": 0.88, "content": "M ^ { \\dagger }", "type": "inline_equation" }, { "bbox": [ 401, 82, 506, 97 ], "score": 1.0, "content": ". At a price, the algorithm", "type": "text" } ], "index": 1 }, { "bbox": [ 104, 94, 506, 112 ], "spans": [ { "bbox": [ 104, 94, 300, 112 ], "score": 1.0, "content": "has to suffer the constant suboptimality PH+1h=2", "type": "text" }, { "bbox": [ 268, 94, 335, 110 ], "score": 0.94, "content": "\\textstyle \\sum _ { h = 2 } ^ { H + 1 } d _ { h } ^ { \\dagger \\pi ^ { \\star } } ( s _ { h } ^ { \\dagger } )", "type": "inline_equation" }, { "bbox": [ 335, 94, 506, 112 ], "score": 1.0, "content": "due to no data in the region. The quantity", "type": "text" } ], "index": 2 }, { "bbox": [ 107, 103, 506, 130 ], "spans": [ { "bbox": [ 107, 108, 173, 124 ], "score": 0.92, "content": "\\textstyle \\sum _ { h = 2 } ^ { H + 1 } d _ { h } ^ { \\dag \\pi ^ { \\star } } ( s _ { h } ^ { \\dag } )", "type": "inline_equation" }, { "bbox": [ 174, 103, 428, 130 ], "score": 1.0, "content": "helps characterize the hardness when nothing is assumed about", "type": "text" }, { "bbox": [ 429, 113, 436, 123 ], "score": 0.75, "content": "\\mu", "type": "inline_equation" }, { "bbox": [ 436, 103, 506, 130 ], "score": 1.0, "content": ": it is always less", "type": "text" } ], "index": 3 }, { "bbox": [ 105, 123, 462, 135 ], "spans": [ { "bbox": [ 105, 123, 127, 135 ], "score": 1.0, "content": "than", "type": "text" }, { "bbox": [ 127, 124, 137, 134 ], "score": 0.78, "content": "H", "type": "inline_equation" }, { "bbox": [ 137, 123, 241, 135 ], "score": 1.0, "content": "(cannot suffer more than", "type": "text" }, { "bbox": [ 252, 123, 462, 135 ], "score": 1.0, "content": "suboptimality); under Assumption 2.1, it is 0 since", "type": "text" } ], "index": 4 }, { "bbox": [ 105, 133, 506, 147 ], "spans": [ { "bbox": [ 105, 133, 338, 147 ], "score": 1.0, "content": "with high probability (by Chernoff bound) and this causes", "type": "text" }, { "bbox": [ 339, 134, 400, 146 ], "score": 0.93, "content": "\\boldsymbol { \\mathcal { S } } \\times \\boldsymbol { \\mathcal { A } } \\backslash \\boldsymbol { \\bar { \\mathcal { C } } } _ { h } = \\boldsymbol { \\emptyset }", "type": "inline_equation" }, { "bbox": [ 401, 133, 506, 147 ], "score": 1.0, "content": "; under Assumption 2.3, it", "type": "text" } ], "index": 5 }, { "bbox": [ 105, 145, 343, 157 ], "spans": [ { "bbox": [ 105, 145, 343, 157 ], "score": 1.0, "content": "is also 0 and 5.1 reduces to Theorem 4.1 (see Appendix F).", "type": "text" } ], "index": 6 } ], "index": 3 }, { "type": "title", "bbox": [ 108, 174, 403, 186 ], "lines": [ { "bbox": [ 105, 173, 404, 190 ], "spans": [ { "bbox": [ 105, 173, 404, 190 ], "score": 1.0, "content": "5.1 Assumption Free vs Without Great Coverage (Partial Coverage)", "type": "text" } ], "index": 7 } ], "index": 7 }, { "type": "text", "bbox": [ 106, 195, 506, 273 ], "lines": [ { "bbox": [ 106, 196, 506, 208 ], "spans": [ { "bbox": [ 106, 196, 475, 208 ], "score": 1.0, "content": "Recently there is a surge of studies that aim at weakening the assumptions of provable offline", "type": "text" }, { "bbox": [ 475, 197, 480, 207 ], "score": 0.34, "content": "/", "type": "inline_equation" }, { "bbox": [ 481, 196, 506, 208 ], "score": 1.0, "content": "batch", "type": "text" } ], "index": 8 }, { "bbox": [ 105, 207, 507, 220 ], "spans": [ { "bbox": [ 105, 207, 507, 220 ], "score": 1.0, "content": "RL. Those learning bounds are derived (mostly) under the insufficient data coverage assumptions.", "type": "text" } ], "index": 9 }, { "bbox": [ 105, 217, 506, 231 ], "spans": [ { "bbox": [ 105, 217, 506, 231 ], "score": 1.0, "content": "One type of works consider the assumption without great coverage (or partial coverage): Chang", "type": "text" } ], "index": 10 }, { "bbox": [ 105, 229, 506, 241 ], "spans": [ { "bbox": [ 105, 229, 291, 241 ], "score": 1.0, "content": "et al. [2021], Uehara and Sun [2021] assume", "type": "text" }, { "bbox": [ 291, 229, 416, 241 ], "score": 0.91, "content": "\\begin{array} { r } { \\operatorname* { m a x } _ { s , a } d ^ { \\pi _ { e } } ( s , a ) / \\mu ( s , a ) < \\infty } \\end{array}", "type": "inline_equation" }, { "bbox": [ 416, 229, 445, 241 ], "score": 1.0, "content": "where", "type": "text" }, { "bbox": [ 445, 230, 456, 240 ], "score": 0.86, "content": "\\pi _ { e }", "type": "inline_equation" }, { "bbox": [ 457, 229, 506, 241 ], "score": 1.0, "content": "is either an", "type": "text" } ], "index": 11 }, { "bbox": [ 105, 240, 505, 253 ], "spans": [ { "bbox": [ 105, 240, 462, 253 ], "score": 1.0, "content": "expert policy or a policy of great quality and they further compete against with this policy", "type": "text" }, { "bbox": [ 463, 241, 474, 252 ], "score": 0.84, "content": "\\pi _ { e }", "type": "inline_equation" }, { "bbox": [ 474, 240, 505, 253 ], "score": 1.0, "content": ". Those", "type": "text" } ], "index": 12 }, { "bbox": [ 106, 252, 505, 263 ], "spans": [ { "bbox": [ 106, 252, 505, 263 ], "score": 1.0, "content": "assumptions are similar to 2.3 and therefore are stronger than the assumption-free RL we considered", "type": "text" } ], "index": 13 }, { "bbox": [ 104, 261, 135, 274 ], "spans": [ { "bbox": [ 104, 261, 135, 274 ], "score": 1.0, "content": "in 5.1.", "type": "text" } ], "index": 14 } ], "index": 11 }, { "type": "text", "bbox": [ 106, 278, 505, 430 ], "lines": [ { "bbox": [ 106, 278, 505, 290 ], "spans": [ { "bbox": [ 106, 278, 360, 290 ], "score": 1.0, "content": "In addition, there are other studies that apply to the case where", "type": "text" }, { "bbox": [ 360, 281, 368, 290 ], "score": 0.82, "content": "\\mu", "type": "inline_equation" }, { "bbox": [ 368, 278, 505, 290 ], "score": 1.0, "content": "can be arbitrary: Liu et al. [2020]", "type": "text" } ], "index": 15 }, { "bbox": [ 105, 289, 505, 302 ], "spans": [ { "bbox": [ 105, 289, 380, 302 ], "score": 1.0, "content": "considers the behavior policy with insufficient coverage probability", "type": "text" }, { "bbox": [ 380, 291, 390, 302 ], "score": 0.85, "content": "\\epsilon _ { \\zeta }", "type": "inline_equation" }, { "bbox": [ 390, 289, 505, 302 ], "score": 1.0, "content": "(see their Definition 1), and", "type": "text" } ], "index": 16 }, { "bbox": [ 105, 299, 507, 318 ], "spans": [ { "bbox": [ 105, 299, 308, 318 ], "score": 1.0, "content": "they end up with the constant suboptimality gap", "type": "text" }, { "bbox": [ 305, 299, 507, 317 ], "score": 1.0, "content": "Vmax\u000fζ1−γ (their Theorem 1), when the insufficient", "type": "text" } ], "index": 17 }, { "bbox": [ 104, 314, 506, 331 ], "spans": [ { "bbox": [ 104, 314, 191, 331 ], "score": 1.0, "content": "coverage probability", "type": "text" }, { "bbox": [ 191, 317, 219, 329 ], "score": 0.9, "content": "\\epsilon _ { \\zeta } > 0", "type": "inline_equation" }, { "bbox": [ 220, 314, 297, 331 ], "score": 1.0, "content": ", this gap has order", "type": "text" }, { "bbox": [ 298, 317, 339, 329 ], "score": 0.92, "content": "( 1 - \\gamma ) ^ { - 2 }", "type": "inline_equation" }, { "bbox": [ 339, 314, 506, 331 ], "score": 1.0, "content": ", which is larger in order than the biggest", "type": "text" } ], "index": 18 }, { "bbox": [ 106, 328, 506, 341 ], "spans": [ { "bbox": [ 106, 329, 213, 341 ], "score": 1.0, "content": "possible suboptimality gap", "type": "text" }, { "bbox": [ 214, 328, 254, 340 ], "score": 0.92, "content": "( 1 - \\gamma ) ^ { - 1 }", "type": "inline_equation" }, { "bbox": [ 254, 329, 506, 341 ], "score": 1.0, "content": "therefore unable to characterize the essential statistical gap over", "type": "text" } ], "index": 19 }, { "bbox": [ 105, 339, 505, 351 ], "spans": [ { "bbox": [ 105, 339, 505, 351 ], "score": 1.0, "content": "the region that can never be visited by the behavior policy (and this happens similarly in Kidambi", "type": "text" } ], "index": 20 }, { "bbox": [ 106, 350, 506, 363 ], "spans": [ { "bbox": [ 106, 350, 506, 363 ], "score": 1.0, "content": "et al. [2020], see their Theorem 1); Jin et al. [2020] derive the nice assumption-free result via", "type": "text" } ], "index": 21 }, { "bbox": [ 105, 360, 505, 375 ], "spans": [ { "bbox": [ 105, 360, 275, 375 ], "score": 1.0, "content": "regularization and their bound can incur", "type": "text" }, { "bbox": [ 276, 361, 306, 373 ], "score": 0.92, "content": "O ( H ^ { 2 } )", "type": "inline_equation" }, { "bbox": [ 306, 360, 471, 375 ], "score": 1.0, "content": "constant gap when there is at least one", "type": "text" }, { "bbox": [ 472, 362, 505, 374 ], "score": 0.92, "content": "\\left( s _ { h } , a _ { h } \\right)", "type": "inline_equation" } ], "index": 22 }, { "bbox": [ 106, 372, 506, 385 ], "spans": [ { "bbox": [ 106, 372, 200, 385 ], "score": 1.0, "content": "cannot be obtained by", "type": "text" }, { "bbox": [ 200, 374, 208, 384 ], "score": 0.79, "content": "\\mu", "type": "inline_equation" }, { "bbox": [ 208, 372, 237, 385 ], "score": 1.0, "content": "for all", "type": "text" }, { "bbox": [ 237, 372, 272, 384 ], "score": 0.92, "content": "h \\in [ H ]", "type": "inline_equation" }, { "bbox": [ 272, 372, 335, 385 ], "score": 1.0, "content": "(i.e. replacing", "type": "text" }, { "bbox": [ 335, 372, 385, 385 ], "score": 0.93, "content": "n d _ { h } ^ { \\bar { \\mu } } ( { \\bar { s } } _ { h } , a _ { h } )", "type": "inline_equation" }, { "bbox": [ 385, 372, 506, 385 ], "score": 1.0, "content": "by 1 in (3)). The concurrent", "type": "text" } ], "index": 23 }, { "bbox": [ 105, 382, 505, 396 ], "spans": [ { "bbox": [ 105, 382, 505, 396 ], "score": 1.0, "content": "work Xie et al. [2021a] provides a better characterization (and they call it off-support error) with", "type": "text" } ], "index": 24 }, { "bbox": [ 104, 392, 507, 411 ], "spans": [ { "bbox": [ 104, 392, 141, 411 ], "score": 1.0, "content": "roughly", "type": "text" }, { "bbox": [ 142, 394, 390, 407 ], "score": 0.89, "content": "\\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}", "type": "inline_equation" }, { "bbox": [ 390, 392, 507, 411 ], "score": 1.0, "content": ", however, in the worst case", "type": "text" } ], "index": 25 }, { "bbox": [ 106, 406, 506, 420 ], "spans": [ { "bbox": [ 106, 408, 225, 420 ], "score": 0.9, "content": "\\Delta f _ { \\pi } ( s , a ) - ( T ^ { \\pi } \\Delta f _ { \\pi } ) ( s , a )", "type": "inline_equation" }, { "bbox": [ 225, 406, 506, 420 ], "score": 1.0, "content": "might be large (which depends on the quality (assumption) of the", "type": "text" } ], "index": 26 }, { "bbox": [ 106, 418, 231, 431 ], "spans": [ { "bbox": [ 106, 418, 231, 431 ], "score": 1.0, "content": "function approximation class).", "type": "text" } ], "index": 27 } ], "index": 21 }, { "type": "text", "bbox": [ 106, 434, 505, 483 ], "lines": [ { "bbox": [ 105, 426, 506, 463 ], "spans": [ { "bbox": [ 105, 426, 170, 463 ], "score": 1.0, "content": "In contrast, our 1) describes the", "type": "text" }, { "bbox": [ 170, 434, 236, 450 ], "score": 0.93, "content": "\\textstyle \\sum _ { h = 2 } ^ { H + 1 } d _ { h } ^ { \\dag \\pi ^ { \\star } } ( s _ { h } ^ { \\dag } )", "type": "inline_equation" }, { "bbox": [ 237, 426, 296, 463 ], "score": 1.0, "content": "quantity (with p in a more pr", "type": "text" }, { "bbox": [ 297, 434, 506, 451 ], "score": 0.91, "content": "\\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}", "type": "inline_equation" } ], "index": 28 }, { "bbox": [ 105, 459, 506, 473 ], "spans": [ { "bbox": [ 105, 459, 125, 473 ], "score": 1.0, "content": "into", "type": "text" }, { "bbox": [ 125, 460, 135, 470 ], "score": 0.87, "content": "s ^ { \\dagger }", "type": "inline_equation" }, { "bbox": [ 135, 459, 300, 473 ], "score": 1.0, "content": "and it is always bounded between 0 and", "type": "text" }, { "bbox": [ 300, 461, 311, 470 ], "score": 0.82, "content": "H", "type": "inline_equation" }, { "bbox": [ 311, 459, 400, 473 ], "score": 1.0, "content": ". It reduces to 0 when", "type": "text" }, { "bbox": [ 400, 461, 412, 470 ], "score": 0.87, "content": "\\pi ^ { \\star }", "type": "inline_equation" }, { "bbox": [ 412, 459, 506, 473 ], "score": 1.0, "content": "is covered. The gap is", "type": "text" } ], "index": 29 }, { "bbox": [ 106, 471, 279, 484 ], "spans": [ { "bbox": [ 106, 471, 171, 484 ], "score": 1.0, "content": "always of order", "type": "text" }, { "bbox": [ 171, 472, 181, 482 ], "score": 0.8, "content": "H", "type": "inline_equation" }, { "bbox": [ 182, 471, 242, 484 ], "score": 1.0, "content": "(as opposed to", "type": "text" }, { "bbox": [ 243, 471, 273, 483 ], "score": 0.91, "content": "O ( H ^ { 2 } ) _ { * }", "type": "inline_equation" }, { "bbox": [ 274, 471, 279, 484 ], "score": 1.0, "content": ").", "type": "text" } ], "index": 30 } ], "index": 29 }, { "type": "title", "bbox": [ 108, 502, 262, 516 ], "lines": [ { "bbox": [ 105, 502, 263, 518 ], "spans": [ { "bbox": [ 105, 502, 263, 518 ], "score": 1.0, "content": "6 Discussion and Conclusion", "type": "text" } ], "index": 31 } ], "index": 31 }, { "type": "text", "bbox": [ 106, 530, 505, 596 ], "lines": [ { "bbox": [ 106, 531, 505, 542 ], "spans": [ { "bbox": [ 106, 531, 505, 542 ], "score": 1.0, "content": "This work studies the offline reinforcement learning problem and contributes the intrinsic offline", "type": "text" } ], "index": 32 }, { "bbox": [ 105, 541, 505, 555 ], "spans": [ { "bbox": [ 105, 541, 505, 555 ], "score": 1.0, "content": "learning bound which is a near-optimal and strong adaptive bound that subsumes existing worst-case", "type": "text" } ], "index": 33 }, { "bbox": [ 106, 552, 505, 565 ], "spans": [ { "bbox": [ 106, 552, 505, 565 ], "score": 1.0, "content": "bounds under various assumptions. The adaptive characterization of the intrinsic bound abandons the", "type": "text" } ], "index": 34 }, { "bbox": [ 106, 563, 505, 576 ], "spans": [ { "bbox": [ 106, 563, 199, 576 ], "score": 1.0, "content": "explicit dependence on", "type": "text" }, { "bbox": [ 199, 564, 264, 575 ], "score": 0.92, "content": "H , S , A , C ^ { \\star } , d _ { m }", "type": "inline_equation" }, { "bbox": [ 265, 563, 505, 576 ], "score": 1.0, "content": "and helps reveal the fundamental hardness of each individual", "type": "text" } ], "index": 35 }, { "bbox": [ 106, 574, 505, 586 ], "spans": [ { "bbox": [ 106, 574, 505, 586 ], "score": 1.0, "content": "instances. In this sense, it draws a clearer picture of what offline reinforcement learning looks like", "type": "text" } ], "index": 36 }, { "bbox": [ 106, 586, 355, 597 ], "spans": [ { "bbox": [ 106, 586, 355, 597 ], "score": 1.0, "content": "and serves as a step towards instance optimality in offline RL.", "type": "text" } ], "index": 37 } ], "index": 34.5 }, { "type": "text", "bbox": [ 106, 601, 505, 722 ], "lines": [ { "bbox": [ 105, 601, 505, 614 ], "spans": [ { "bbox": [ 105, 601, 505, 614 ], "score": 1.0, "content": "Nevertheless, it is still unclear whether (5) is optimal over all the instances. For example, for fully", "type": "text" } ], "index": 38 }, { "bbox": [ 104, 610, 506, 626 ], "spans": [ { "bbox": [ 104, 610, 368, 626 ], "score": 1.0, "content": "deterministic systems, our bound provides a faster convergence", "type": "text" }, { "bbox": [ 368, 612, 406, 624 ], "score": 0.93, "content": "H ^ { 3 } / n \\bar { d } _ { m }", "type": "inline_equation" }, { "bbox": [ 406, 610, 450, 626 ], "score": 1.0, "content": ", however,", "type": "text" }, { "bbox": [ 450, 612, 465, 623 ], "score": 0.88, "content": "\\bar { H } ^ { 3 }", "type": "inline_equation" }, { "bbox": [ 465, 610, 506, 626 ], "score": 1.0, "content": "might be", "type": "text" } ], "index": 39 }, { "bbox": [ 105, 624, 505, 636 ], "spans": [ { "bbox": [ 105, 624, 505, 636 ], "score": 1.0, "content": "very suboptimal comparing to algorithms that are designed specifically for deterministic MDPs, since", "type": "text" } ], "index": 40 }, { "bbox": [ 105, 634, 506, 647 ], "spans": [ { "bbox": [ 105, 634, 294, 647 ], "score": 1.0, "content": "the agent only need to experience each location", "type": "text" }, { "bbox": [ 294, 635, 316, 646 ], "score": 0.92, "content": "( s , a )", "type": "inline_equation" }, { "bbox": [ 317, 634, 451, 647 ], "score": 1.0, "content": "once to fully acquire the dynamic", "type": "text" }, { "bbox": [ 451, 635, 487, 646 ], "score": 0.93, "content": "P ( \\cdot | s , a )", "type": "inline_equation" }, { "bbox": [ 487, 634, 506, 647 ], "score": 1.0, "content": "and", "type": "text" } ], "index": 41 }, { "bbox": [ 107, 645, 506, 658 ], "spans": [ { "bbox": [ 107, 645, 134, 657 ], "score": 0.92, "content": "r ( s , a )", "type": "inline_equation" }, { "bbox": [ 135, 645, 506, 658 ], "score": 1.0, "content": ". Recently, Xiao et al. [2021] goes beyond the minimax (worst case) optimality and studies the", "type": "text" } ], "index": 42 }, { "bbox": [ 105, 656, 506, 669 ], "spans": [ { "bbox": [ 105, 656, 506, 669 ], "score": 1.0, "content": "instance optimality behavior for the simplified batch bandit setting. One of their findings is: for “easy", "type": "text" } ], "index": 43 }, { "bbox": [ 106, 667, 505, 680 ], "spans": [ { "bbox": [ 106, 667, 505, 680 ], "score": 1.0, "content": "enough” tasks, different type of algorithms can be equally good, provably. This seems to suggest", "type": "text" } ], "index": 44 }, { "bbox": [ 105, 678, 505, 690 ], "spans": [ { "bbox": [ 105, 678, 505, 690 ], "score": 1.0, "content": "instance optimality only matters for problems that are hard to learn. How to formally define the", "type": "text" } ], "index": 45 }, { "bbox": [ 106, 689, 505, 701 ], "spans": [ { "bbox": [ 106, 689, 505, 701 ], "score": 1.0, "content": "instance optimality metric for different problems remains an open problem and how to design a single", "type": "text" } ], "index": 46 }, { "bbox": [ 105, 699, 505, 713 ], "spans": [ { "bbox": [ 105, 699, 505, 713 ], "score": 1.0, "content": "algorithm that can achieve optimality for all instances could be challenging (or even infeasible). We", "type": "text" } ], "index": 47 }, { "bbox": [ 106, 711, 234, 722 ], "spans": [ { "bbox": [ 106, 711, 234, 722 ], "score": 1.0, "content": "leave those as the future works.", "type": "text" } ], "index": 48 } ], "index": 43 } ], "page_idx": 9, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 300, 741, 311, 750 ], "lines": [ { "bbox": [ 299, 740, 313, 754 ], "spans": [ { "bbox": [ 299, 740, 313, 754 ], "score": 1.0, "content": "10", "type": "text" } ] } ] } ], "para_blocks": [ { "type": "text", "bbox": [ 106, 72, 505, 157 ], "lines": [ { "bbox": [ 105, 71, 505, 85 ], "spans": [ { "bbox": [ 105, 71, 246, 85 ], "score": 1.0, "content": "Take-aways of Theorem 5.1. In", "type": "text" }, { "bbox": [ 246, 72, 262, 83 ], "score": 0.89, "content": "M ^ { \\dagger }", "type": "inline_equation" }, { "bbox": [ 262, 71, 505, 85 ], "score": 1.0, "content": ", there is no agnostic location any more since the original", "type": "text" } ], "index": 0 }, { "bbox": [ 105, 82, 506, 97 ], "spans": [ { "bbox": [ 105, 82, 363, 97 ], "score": 1.0, "content": "unknown spaces now all have known deterministic transitions to", "type": "text" }, { "bbox": [ 363, 83, 373, 93 ], "score": 0.87, "content": "s ^ { \\dagger }", "type": "inline_equation" }, { "bbox": [ 374, 82, 385, 97 ], "score": 1.0, "content": "in", "type": "text" }, { "bbox": [ 385, 83, 401, 93 ], "score": 0.88, "content": "M ^ { \\dagger }", "type": "inline_equation" }, { "bbox": [ 401, 82, 506, 97 ], "score": 1.0, "content": ". At a price, the algorithm", "type": "text" } ], "index": 1 }, { "bbox": [ 104, 94, 506, 112 ], "spans": [ { "bbox": [ 104, 94, 300, 112 ], "score": 1.0, "content": "has to suffer the constant suboptimality PH+1h=2", "type": "text" }, { "bbox": [ 268, 94, 335, 110 ], "score": 0.94, "content": "\\textstyle \\sum _ { h = 2 } ^ { H + 1 } d _ { h } ^ { \\dagger \\pi ^ { \\star } } ( s _ { h } ^ { \\dagger } )", "type": "inline_equation" }, { "bbox": [ 335, 94, 506, 112 ], "score": 1.0, "content": "due to no data in the region. The quantity", "type": "text" } ], "index": 2 }, { "bbox": [ 107, 103, 506, 130 ], "spans": [ { "bbox": [ 107, 108, 173, 124 ], "score": 0.92, "content": "\\textstyle \\sum _ { h = 2 } ^ { H + 1 } d _ { h } ^ { \\dag \\pi ^ { \\star } } ( s _ { h } ^ { \\dag } )", "type": "inline_equation" }, { "bbox": [ 174, 103, 428, 130 ], "score": 1.0, "content": "helps characterize the hardness when nothing is assumed about", "type": "text" }, { "bbox": [ 429, 113, 436, 123 ], "score": 0.75, "content": "\\mu", "type": "inline_equation" }, { "bbox": [ 436, 103, 506, 130 ], "score": 1.0, "content": ": it is always less", "type": "text" } ], "index": 3 }, { "bbox": [ 105, 123, 462, 135 ], "spans": [ { "bbox": [ 105, 123, 127, 135 ], "score": 1.0, "content": "than", "type": "text" }, { "bbox": [ 127, 124, 137, 134 ], "score": 0.78, "content": "H", "type": "inline_equation" }, { "bbox": [ 137, 123, 241, 135 ], "score": 1.0, "content": "(cannot suffer more than", "type": "text" }, { "bbox": [ 252, 123, 462, 135 ], "score": 1.0, "content": "suboptimality); under Assumption 2.1, it is 0 since", "type": "text" } ], "index": 4 }, { "bbox": [ 105, 133, 506, 147 ], "spans": [ { "bbox": [ 105, 133, 338, 147 ], "score": 1.0, "content": "with high probability (by Chernoff bound) and this causes", "type": "text" }, { "bbox": [ 339, 134, 400, 146 ], "score": 0.93, "content": "\\boldsymbol { \\mathcal { S } } \\times \\boldsymbol { \\mathcal { A } } \\backslash \\boldsymbol { \\bar { \\mathcal { C } } } _ { h } = \\boldsymbol { \\emptyset }", "type": "inline_equation" }, { "bbox": [ 401, 133, 506, 147 ], "score": 1.0, "content": "; under Assumption 2.3, it", "type": "text" } ], "index": 5 }, { "bbox": [ 105, 145, 343, 157 ], "spans": [ { "bbox": [ 105, 145, 343, 157 ], "score": 1.0, "content": "is also 0 and 5.1 reduces to Theorem 4.1 (see Appendix F).", "type": "text" } ], "index": 6 } ], "index": 3, "bbox_fs": [ 104, 71, 506, 157 ] }, { "type": "title", "bbox": [ 108, 174, 403, 186 ], "lines": [ { "bbox": [ 105, 173, 404, 190 ], "spans": [ { "bbox": [ 105, 173, 404, 190 ], "score": 1.0, "content": "5.1 Assumption Free vs Without Great Coverage (Partial Coverage)", "type": "text" } ], "index": 7 } ], "index": 7 }, { "type": "text", "bbox": [ 106, 195, 506, 273 ], "lines": [ { "bbox": [ 106, 196, 506, 208 ], "spans": [ { "bbox": [ 106, 196, 475, 208 ], "score": 1.0, "content": "Recently there is a surge of studies that aim at weakening the assumptions of provable offline", "type": "text" }, { "bbox": [ 475, 197, 480, 207 ], "score": 0.34, "content": "/", "type": "inline_equation" }, { "bbox": [ 481, 196, 506, 208 ], "score": 1.0, "content": "batch", "type": "text" } ], "index": 8 }, { "bbox": [ 105, 207, 507, 220 ], "spans": [ { "bbox": [ 105, 207, 507, 220 ], "score": 1.0, "content": "RL. Those learning bounds are derived (mostly) under the insufficient data coverage assumptions.", "type": "text" } ], "index": 9 }, { "bbox": [ 105, 217, 506, 231 ], "spans": [ { "bbox": [ 105, 217, 506, 231 ], "score": 1.0, "content": "One type of works consider the assumption without great coverage (or partial coverage): Chang", "type": "text" } ], "index": 10 }, { "bbox": [ 105, 229, 506, 241 ], "spans": [ { "bbox": [ 105, 229, 291, 241 ], "score": 1.0, "content": "et al. [2021], Uehara and Sun [2021] assume", "type": "text" }, { "bbox": [ 291, 229, 416, 241 ], "score": 0.91, "content": "\\begin{array} { r } { \\operatorname* { m a x } _ { s , a } d ^ { \\pi _ { e } } ( s , a ) / \\mu ( s , a ) < \\infty } \\end{array}", "type": "inline_equation" }, { "bbox": [ 416, 229, 445, 241 ], "score": 1.0, "content": "where", "type": "text" }, { "bbox": [ 445, 230, 456, 240 ], "score": 0.86, "content": "\\pi _ { e }", "type": "inline_equation" }, { "bbox": [ 457, 229, 506, 241 ], "score": 1.0, "content": "is either an", "type": "text" } ], "index": 11 }, { "bbox": [ 105, 240, 505, 253 ], "spans": [ { "bbox": [ 105, 240, 462, 253 ], "score": 1.0, "content": "expert policy or a policy of great quality and they further compete against with this policy", "type": "text" }, { "bbox": [ 463, 241, 474, 252 ], "score": 0.84, "content": "\\pi _ { e }", "type": "inline_equation" }, { "bbox": [ 474, 240, 505, 253 ], "score": 1.0, "content": ". Those", "type": "text" } ], "index": 12 }, { "bbox": [ 106, 252, 505, 263 ], "spans": [ { "bbox": [ 106, 252, 505, 263 ], "score": 1.0, "content": "assumptions are similar to 2.3 and therefore are stronger than the assumption-free RL we considered", "type": "text" } ], "index": 13 }, { "bbox": [ 104, 261, 135, 274 ], "spans": [ { "bbox": [ 104, 261, 135, 274 ], "score": 1.0, "content": "in 5.1.", "type": "text" } ], "index": 14 } ], "index": 11, "bbox_fs": [ 104, 196, 507, 274 ] }, { "type": "text", "bbox": [ 106, 278, 505, 430 ], "lines": [ { "bbox": [ 106, 278, 505, 290 ], "spans": [ { "bbox": [ 106, 278, 360, 290 ], "score": 1.0, "content": "In addition, there are other studies that apply to the case where", "type": "text" }, { "bbox": [ 360, 281, 368, 290 ], "score": 0.82, "content": "\\mu", "type": "inline_equation" }, { "bbox": [ 368, 278, 505, 290 ], "score": 1.0, "content": "can be arbitrary: Liu et al. [2020]", "type": "text" } ], "index": 15 }, { "bbox": [ 105, 289, 505, 302 ], "spans": [ { "bbox": [ 105, 289, 380, 302 ], "score": 1.0, "content": "considers the behavior policy with insufficient coverage probability", "type": "text" }, { "bbox": [ 380, 291, 390, 302 ], "score": 0.85, "content": "\\epsilon _ { \\zeta }", "type": "inline_equation" }, { "bbox": [ 390, 289, 505, 302 ], "score": 1.0, "content": "(see their Definition 1), and", "type": "text" } ], "index": 16 }, { "bbox": [ 105, 299, 507, 318 ], "spans": [ { "bbox": [ 105, 299, 308, 318 ], "score": 1.0, "content": "they end up with the constant suboptimality gap", "type": "text" }, { "bbox": [ 305, 299, 507, 317 ], "score": 1.0, "content": "Vmax\u000fζ1−γ (their Theorem 1), when the insufficient", "type": "text" } ], "index": 17 }, { "bbox": [ 104, 314, 506, 331 ], "spans": [ { "bbox": [ 104, 314, 191, 331 ], "score": 1.0, "content": "coverage probability", "type": "text" }, { "bbox": [ 191, 317, 219, 329 ], "score": 0.9, "content": "\\epsilon _ { \\zeta } > 0", "type": "inline_equation" }, { "bbox": [ 220, 314, 297, 331 ], "score": 1.0, "content": ", this gap has order", "type": "text" }, { "bbox": [ 298, 317, 339, 329 ], "score": 0.92, "content": "( 1 - \\gamma ) ^ { - 2 }", "type": "inline_equation" }, { "bbox": [ 339, 314, 506, 331 ], "score": 1.0, "content": ", which is larger in order than the biggest", "type": "text" } ], "index": 18 }, { "bbox": [ 106, 328, 506, 341 ], "spans": [ { "bbox": [ 106, 329, 213, 341 ], "score": 1.0, "content": "possible suboptimality gap", "type": "text" }, { "bbox": [ 214, 328, 254, 340 ], "score": 0.92, "content": "( 1 - \\gamma ) ^ { - 1 }", "type": "inline_equation" }, { "bbox": [ 254, 329, 506, 341 ], "score": 1.0, "content": "therefore unable to characterize the essential statistical gap over", "type": "text" } ], "index": 19 }, { "bbox": [ 105, 339, 505, 351 ], "spans": [ { "bbox": [ 105, 339, 505, 351 ], "score": 1.0, "content": "the region that can never be visited by the behavior policy (and this happens similarly in Kidambi", "type": "text" } ], "index": 20 }, { "bbox": [ 106, 350, 506, 363 ], "spans": [ { "bbox": [ 106, 350, 506, 363 ], "score": 1.0, "content": "et al. [2020], see their Theorem 1); Jin et al. [2020] derive the nice assumption-free result via", "type": "text" } ], "index": 21 }, { "bbox": [ 105, 360, 505, 375 ], "spans": [ { "bbox": [ 105, 360, 275, 375 ], "score": 1.0, "content": "regularization and their bound can incur", "type": "text" }, { "bbox": [ 276, 361, 306, 373 ], "score": 0.92, "content": "O ( H ^ { 2 } )", "type": "inline_equation" }, { "bbox": [ 306, 360, 471, 375 ], "score": 1.0, "content": "constant gap when there is at least one", "type": "text" }, { "bbox": [ 472, 362, 505, 374 ], "score": 0.92, "content": "\\left( s _ { h } , a _ { h } \\right)", "type": "inline_equation" } ], "index": 22 }, { "bbox": [ 106, 372, 506, 385 ], "spans": [ { "bbox": [ 106, 372, 200, 385 ], "score": 1.0, "content": "cannot be obtained by", "type": "text" }, { "bbox": [ 200, 374, 208, 384 ], "score": 0.79, "content": "\\mu", "type": "inline_equation" }, { "bbox": [ 208, 372, 237, 385 ], "score": 1.0, "content": "for all", "type": "text" }, { "bbox": [ 237, 372, 272, 384 ], "score": 0.92, "content": "h \\in [ H ]", "type": "inline_equation" }, { "bbox": [ 272, 372, 335, 385 ], "score": 1.0, "content": "(i.e. replacing", "type": "text" }, { "bbox": [ 335, 372, 385, 385 ], "score": 0.93, "content": "n d _ { h } ^ { \\bar { \\mu } } ( { \\bar { s } } _ { h } , a _ { h } )", "type": "inline_equation" }, { "bbox": [ 385, 372, 506, 385 ], "score": 1.0, "content": "by 1 in (3)). The concurrent", "type": "text" } ], "index": 23 }, { "bbox": [ 105, 382, 505, 396 ], "spans": [ { "bbox": [ 105, 382, 505, 396 ], "score": 1.0, "content": "work Xie et al. [2021a] provides a better characterization (and they call it off-support error) with", "type": "text" } ], "index": 24 }, { "bbox": [ 104, 392, 507, 411 ], "spans": [ { "bbox": [ 104, 392, 141, 411 ], "score": 1.0, "content": "roughly", "type": "text" }, { "bbox": [ 142, 394, 390, 407 ], "score": 0.89, "content": "\\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}", "type": "inline_equation" }, { "bbox": [ 390, 392, 507, 411 ], "score": 1.0, "content": ", however, in the worst case", "type": "text" } ], "index": 25 }, { "bbox": [ 106, 406, 506, 420 ], "spans": [ { "bbox": [ 106, 408, 225, 420 ], "score": 0.9, "content": "\\Delta f _ { \\pi } ( s , a ) - ( T ^ { \\pi } \\Delta f _ { \\pi } ) ( s , a )", "type": "inline_equation" }, { "bbox": [ 225, 406, 506, 420 ], "score": 1.0, "content": "might be large (which depends on the quality (assumption) of the", "type": "text" } ], "index": 26 }, { "bbox": [ 106, 418, 231, 431 ], "spans": [ { "bbox": [ 106, 418, 231, 431 ], "score": 1.0, "content": "function approximation class).", "type": "text" } ], "index": 27 } ], "index": 21, "bbox_fs": [ 104, 278, 507, 431 ] }, { "type": "text", "bbox": [ 106, 434, 505, 483 ], "lines": [ { "bbox": [ 105, 426, 506, 463 ], "spans": [ { "bbox": [ 105, 426, 170, 463 ], "score": 1.0, "content": "In contrast, our 1) describes the", "type": "text" }, { "bbox": [ 170, 434, 236, 450 ], "score": 0.93, "content": "\\textstyle \\sum _ { h = 2 } ^ { H + 1 } d _ { h } ^ { \\dag \\pi ^ { \\star } } ( s _ { h } ^ { \\dag } )", "type": "inline_equation" }, { "bbox": [ 237, 426, 296, 463 ], "score": 1.0, "content": "quantity (with p in a more pr", "type": "text" }, { "bbox": [ 297, 434, 506, 451 ], "score": 0.91, "content": "\\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}", "type": "inline_equation" } ], "index": 28 }, { "bbox": [ 105, 459, 506, 473 ], "spans": [ { "bbox": [ 105, 459, 125, 473 ], "score": 1.0, "content": "into", "type": "text" }, { "bbox": [ 125, 460, 135, 470 ], "score": 0.87, "content": "s ^ { \\dagger }", "type": "inline_equation" }, { "bbox": [ 135, 459, 300, 473 ], "score": 1.0, "content": "and it is always bounded between 0 and", "type": "text" }, { "bbox": [ 300, 461, 311, 470 ], "score": 0.82, "content": "H", "type": "inline_equation" }, { "bbox": [ 311, 459, 400, 473 ], "score": 1.0, "content": ". It reduces to 0 when", "type": "text" }, { "bbox": [ 400, 461, 412, 470 ], "score": 0.87, "content": "\\pi ^ { \\star }", "type": "inline_equation" }, { "bbox": [ 412, 459, 506, 473 ], "score": 1.0, "content": "is covered. The gap is", "type": "text" } ], "index": 29 }, { "bbox": [ 106, 471, 279, 484 ], "spans": [ { "bbox": [ 106, 471, 171, 484 ], "score": 1.0, "content": "always of order", "type": "text" }, { "bbox": [ 171, 472, 181, 482 ], "score": 0.8, "content": "H", "type": "inline_equation" }, { "bbox": [ 182, 471, 242, 484 ], "score": 1.0, "content": "(as opposed to", "type": "text" }, { "bbox": [ 243, 471, 273, 483 ], "score": 0.91, "content": "O ( H ^ { 2 } ) _ { * }", "type": "inline_equation" }, { "bbox": [ 274, 471, 279, 484 ], "score": 1.0, "content": ").", "type": "text" } ], "index": 30 } ], "index": 29, "bbox_fs": [ 105, 426, 506, 484 ] }, { "type": "title", "bbox": [ 108, 502, 262, 516 ], "lines": [ { "bbox": [ 105, 502, 263, 518 ], "spans": [ { "bbox": [ 105, 502, 263, 518 ], "score": 1.0, "content": "6 Discussion and Conclusion", "type": "text" } ], "index": 31 } ], "index": 31 }, { "type": "text", "bbox": [ 106, 530, 505, 596 ], "lines": [ { "bbox": [ 106, 531, 505, 542 ], "spans": [ { "bbox": [ 106, 531, 505, 542 ], "score": 1.0, "content": "This work studies the offline reinforcement learning problem and contributes the intrinsic offline", "type": "text" } ], "index": 32 }, { "bbox": [ 105, 541, 505, 555 ], "spans": [ { "bbox": [ 105, 541, 505, 555 ], "score": 1.0, "content": "learning bound which is a near-optimal and strong adaptive bound that subsumes existing worst-case", "type": "text" } ], "index": 33 }, { "bbox": [ 106, 552, 505, 565 ], "spans": [ { "bbox": [ 106, 552, 505, 565 ], "score": 1.0, "content": "bounds under various assumptions. The adaptive characterization of the intrinsic bound abandons the", "type": "text" } ], "index": 34 }, { "bbox": [ 106, 563, 505, 576 ], "spans": [ { "bbox": [ 106, 563, 199, 576 ], "score": 1.0, "content": "explicit dependence on", "type": "text" }, { "bbox": [ 199, 564, 264, 575 ], "score": 0.92, "content": "H , S , A , C ^ { \\star } , d _ { m }", "type": "inline_equation" }, { "bbox": [ 265, 563, 505, 576 ], "score": 1.0, "content": "and helps reveal the fundamental hardness of each individual", "type": "text" } ], "index": 35 }, { "bbox": [ 106, 574, 505, 586 ], "spans": [ { "bbox": [ 106, 574, 505, 586 ], "score": 1.0, "content": "instances. In this sense, it draws a clearer picture of what offline reinforcement learning looks like", "type": "text" } ], "index": 36 }, { "bbox": [ 106, 586, 355, 597 ], "spans": [ { "bbox": [ 106, 586, 355, 597 ], "score": 1.0, "content": "and serves as a step towards instance optimality in offline RL.", "type": "text" } ], "index": 37 } ], "index": 34.5, "bbox_fs": [ 105, 531, 505, 597 ] }, { "type": "text", "bbox": [ 106, 601, 505, 722 ], "lines": [ { "bbox": [ 105, 601, 505, 614 ], "spans": [ { "bbox": [ 105, 601, 505, 614 ], "score": 1.0, "content": "Nevertheless, it is still unclear whether (5) is optimal over all the instances. For example, for fully", "type": "text" } ], "index": 38 }, { "bbox": [ 104, 610, 506, 626 ], "spans": [ { "bbox": [ 104, 610, 368, 626 ], "score": 1.0, "content": "deterministic systems, our bound provides a faster convergence", "type": "text" }, { "bbox": [ 368, 612, 406, 624 ], "score": 0.93, "content": "H ^ { 3 } / n \\bar { d } _ { m }", "type": "inline_equation" }, { "bbox": [ 406, 610, 450, 626 ], "score": 1.0, "content": ", however,", "type": "text" }, { "bbox": [ 450, 612, 465, 623 ], "score": 0.88, "content": "\\bar { H } ^ { 3 }", "type": "inline_equation" }, { "bbox": [ 465, 610, 506, 626 ], "score": 1.0, "content": "might be", "type": "text" } ], "index": 39 }, { "bbox": [ 105, 624, 505, 636 ], "spans": [ { "bbox": [ 105, 624, 505, 636 ], "score": 1.0, "content": "very suboptimal comparing to algorithms that are designed specifically for deterministic MDPs, since", "type": "text" } ], "index": 40 }, { "bbox": [ 105, 634, 506, 647 ], "spans": [ { "bbox": [ 105, 634, 294, 647 ], "score": 1.0, "content": "the agent only need to experience each location", "type": "text" }, { "bbox": [ 294, 635, 316, 646 ], "score": 0.92, "content": "( s , a )", "type": "inline_equation" }, { "bbox": [ 317, 634, 451, 647 ], "score": 1.0, "content": "once to fully acquire the dynamic", "type": "text" }, { "bbox": [ 451, 635, 487, 646 ], "score": 0.93, "content": "P ( \\cdot | s , a )", "type": "inline_equation" }, { "bbox": [ 487, 634, 506, 647 ], "score": 1.0, "content": "and", "type": "text" } ], "index": 41 }, { "bbox": [ 107, 645, 506, 658 ], "spans": [ { "bbox": [ 107, 645, 134, 657 ], "score": 0.92, "content": "r ( s , a )", "type": "inline_equation" }, { "bbox": [ 135, 645, 506, 658 ], "score": 1.0, "content": ". Recently, Xiao et al. [2021] goes beyond the minimax (worst case) optimality and studies the", "type": "text" } ], "index": 42 }, { "bbox": [ 105, 656, 506, 669 ], "spans": [ { "bbox": [ 105, 656, 506, 669 ], "score": 1.0, "content": "instance optimality behavior for the simplified batch bandit setting. One of their findings is: for “easy", "type": "text" } ], "index": 43 }, { "bbox": [ 106, 667, 505, 680 ], "spans": [ { "bbox": [ 106, 667, 505, 680 ], "score": 1.0, "content": "enough” tasks, different type of algorithms can be equally good, provably. This seems to suggest", "type": "text" } ], "index": 44 }, { "bbox": [ 105, 678, 505, 690 ], "spans": [ { "bbox": [ 105, 678, 505, 690 ], "score": 1.0, "content": "instance optimality only matters for problems that are hard to learn. How to formally define the", "type": "text" } ], "index": 45 }, { "bbox": [ 106, 689, 505, 701 ], "spans": [ { "bbox": [ 106, 689, 505, 701 ], "score": 1.0, "content": "instance optimality metric for different problems remains an open problem and how to design a single", "type": "text" } ], "index": 46 }, { "bbox": [ 105, 699, 505, 713 ], "spans": [ { "bbox": [ 105, 699, 505, 713 ], "score": 1.0, "content": "algorithm that can achieve optimality for all instances could be challenging (or even infeasible). We", "type": "text" } ], "index": 47 }, { "bbox": [ 106, 711, 234, 722 ], "spans": [ { "bbox": [ 106, 711, 234, 722 ], "score": 1.0, "content": "leave those as the future works.", "type": "text" } ], "index": 48 } ], "index": 43, "bbox_fs": [ 104, 601, 506, 722 ] } ] }, { "preproc_blocks": [ { "type": "title", "bbox": [ 107, 73, 182, 84 ], "lines": [ { "bbox": [ 106, 72, 183, 86 ], "spans": [ { "bbox": [ 106, 72, 183, 86 ], "score": 1.0, "content": "Acknowledgment", "type": "text" } ], "index": 0 } ], "index": 0 }, { "type": "text", "bbox": [ 106, 92, 505, 126 ], "lines": [ { "bbox": [ 105, 92, 506, 104 ], "spans": [ { "bbox": [ 105, 92, 506, 104 ], "score": 1.0, "content": "The research is partially supported by NSF Awards #2007117 and #2003257. 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