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Most remarkably, AlphaGo (Silver et al., 2016) and AlphaZero (Silver et al., 2017b;a)", "type": "text" } ], "index": 29 }, { "bbox": [ 105, 517, 505, 530 ], "spans": [ { "bbox": [ 105, 517, 505, 530 ], "score": 1.0, "content": "couple MCTS with neural networks trained using Reinforcement Learning (RL) (Sutton & Barto,", "type": "text" } ], "index": 30 }, { "bbox": [ 105, 527, 505, 541 ], "spans": [ { "bbox": [ 105, 527, 213, 541 ], "score": 1.0, "content": "1998) methods, e.g., Deep", "type": "text" }, { "bbox": [ 214, 529, 223, 540 ], "score": 0.83, "content": "Q", "type": "inline_equation" }, { "bbox": [ 223, 527, 505, 541 ], "score": 1.0, "content": "-Learning (Mnih et al., 2015), to speed up learning of large scale prob-", "type": "text" } ], "index": 31 }, { "bbox": [ 105, 540, 504, 552 ], "spans": [ { "bbox": [ 105, 540, 504, 552 ], "score": 1.0, "content": "lems with continuous state space. 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This problem, combined with the high computational time", "type": "text" } ], "index": 39 }, { "bbox": [ 105, 627, 498, 640 ], "spans": [ { "bbox": [ 105, 627, 498, 640 ], "score": 1.0, "content": "to evaluate the deep neural networks, significantly hinder the applicability of both methodologies.", "type": "text" } ], "index": 40 } ], "index": 32.5 }, { "type": "text", "bbox": [ 107, 643, 504, 732 ], "lines": [ { "bbox": [ 105, 644, 505, 656 ], "spans": [ { "bbox": [ 105, 644, 505, 656 ], "score": 1.0, "content": "In this paper, we provide a unified theory of the use of convex regularization in MCTS, which proved", "type": "text" } ], "index": 41 }, { "bbox": [ 105, 655, 505, 667 ], "spans": [ { "bbox": [ 105, 655, 505, 667 ], "score": 1.0, "content": "to be an efficient solution for driving exploration and stabilizing learning in RL (Schulman et al.,", "type": "text" } ], "index": 42 }, { "bbox": [ 105, 666, 506, 678 ], "spans": [ { "bbox": [ 105, 666, 506, 678 ], "score": 1.0, "content": "2015; 2017a; Haarnoja et al., 2018; Buesing et al., 2020). In particular, we show how a regularized", "type": "text" } ], "index": 43 }, { "bbox": [ 106, 677, 505, 689 ], "spans": [ { "bbox": [ 106, 677, 505, 689 ], "score": 1.0, "content": "objective function in MCTS can be seen as an instance of the Legendre-Fenchel transform, similar to", "type": "text" } ], "index": 44 }, { "bbox": [ 105, 687, 506, 700 ], "spans": [ { "bbox": [ 105, 687, 506, 700 ], "score": 1.0, "content": "previous findings on the use of duality in RL (Mensch & Blondel, 2018; Geist et al., 2019; Nachum", "type": "text" } ], "index": 45 }, { "bbox": [ 105, 698, 506, 711 ], "spans": [ { "bbox": [ 105, 698, 506, 711 ], "score": 1.0, "content": "& Dai, 2020) and game theory (Shalev-Shwartz & Singer, 2006; Pavel, 2007). 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The recent AlphaGo and AlphaZero algorithms have shown", "type": "text" } ], "index": 6 }, { "bbox": [ 141, 232, 470, 245 ], "spans": [ { "bbox": [ 141, 232, 470, 245 ], "score": 1.0, "content": "how to successfully combine these two paradigms to solve large scale sequential", "type": "text" } ], "index": 7 }, { "bbox": [ 141, 243, 470, 257 ], "spans": [ { "bbox": [ 141, 243, 470, 257 ], "score": 1.0, "content": "decision problems. 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Second, we exploit our theoretical", "type": "text" } ], "index": 16 }, { "bbox": [ 141, 341, 470, 355 ], "spans": [ { "bbox": [ 141, 341, 470, 355 ], "score": 1.0, "content": "framework to introduce novel regularized backup operators for MCTS, based on", "type": "text" } ], "index": 17 }, { "bbox": [ 141, 352, 469, 366 ], "spans": [ { "bbox": [ 141, 352, 469, 366 ], "score": 1.0, "content": "the relative entropy of the policy update and on the Tsallis entropy of the policy.", "type": "text" } ], "index": 18 }, { "bbox": [ 141, 363, 469, 376 ], "spans": [ { "bbox": [ 141, 363, 469, 376 ], "score": 1.0, "content": "We provide an intuitive demonstration of the effect of each regularizer empirically", "type": "text" } ], "index": 19 }, { "bbox": [ 142, 375, 469, 387 ], "spans": [ { "bbox": [ 142, 375, 469, 387 ], "score": 1.0, "content": "verifying the consequence of our theoretical results on a toy problem. Finally, we", "type": "text" } ], "index": 20 }, { "bbox": [ 141, 385, 469, 398 ], "spans": [ { "bbox": [ 141, 385, 469, 398 ], "score": 1.0, "content": "show how our framework can easily be incorporated in AlphaGo and AlphaZero,", "type": "text" } ], "index": 21 }, { "bbox": [ 141, 396, 469, 409 ], "spans": [ { "bbox": [ 141, 396, 469, 409 ], "score": 1.0, "content": "and we empirically show the superiority of convex regularization w.r.t. represen-", "type": "text" } ], "index": 22 }, { "bbox": [ 141, 407, 436, 420 ], "spans": [ { "bbox": [ 141, 407, 436, 420 ], "score": 1.0, "content": "tative baselines, on well-known RL problems across several Atari games.", "type": "text" } ], "index": 23 } ], "index": 14, "bbox_fs": [ 141, 210, 470, 420 ] }, { "type": "title", "bbox": [ 108, 438, 205, 450 ], "lines": [ { "bbox": [ 105, 437, 208, 453 ], "spans": [ { "bbox": [ 105, 437, 208, 453 ], "score": 1.0, "content": "1 INTRODUCTION", "type": "text" } ], "index": 24 } ], "index": 24 }, { "type": "text", "bbox": [ 107, 462, 505, 638 ], "lines": [ { "bbox": [ 105, 462, 505, 475 ], "spans": [ { "bbox": [ 105, 462, 505, 475 ], "score": 1.0, "content": "Monte-Carlo Tree Search (MCTS) is a well-known algorithm to solve decision-making problems", "type": "text" } ], "index": 25 }, { "bbox": [ 105, 473, 505, 486 ], "spans": [ { "bbox": [ 105, 473, 505, 486 ], "score": 1.0, "content": "through the combination of Monte-Carlo planning with an incremental tree structure (Coulom,", "type": "text" } ], "index": 26 }, { "bbox": [ 106, 484, 505, 498 ], "spans": [ { "bbox": [ 106, 484, 505, 498 ], "score": 1.0, "content": "2006). Although standard MCTS is only suitable for problems with discrete state and action spaces,", "type": "text" } ], "index": 27 }, { "bbox": [ 105, 496, 505, 508 ], "spans": [ { "bbox": [ 105, 496, 505, 508 ], "score": 1.0, "content": "recent advances have shown how to enable MCTS in continuous problems (Silver et al., 2016; Yee", "type": "text" } ], "index": 28 }, { "bbox": [ 104, 505, 505, 519 ], "spans": [ { "bbox": [ 104, 505, 505, 519 ], "score": 1.0, "content": "et al., 2016). Most remarkably, AlphaGo (Silver et al., 2016) and AlphaZero (Silver et al., 2017b;a)", "type": "text" } ], "index": 29 }, { "bbox": [ 105, 517, 505, 530 ], "spans": [ { "bbox": [ 105, 517, 505, 530 ], "score": 1.0, "content": "couple MCTS with neural networks trained using Reinforcement Learning (RL) (Sutton & Barto,", "type": "text" } ], "index": 30 }, { "bbox": [ 105, 527, 505, 541 ], "spans": [ { "bbox": [ 105, 527, 213, 541 ], "score": 1.0, "content": "1998) methods, e.g., Deep", "type": "text" }, { "bbox": [ 214, 529, 223, 540 ], "score": 0.83, "content": "Q", "type": "inline_equation" }, { "bbox": [ 223, 527, 505, 541 ], "score": 1.0, "content": "-Learning (Mnih et al., 2015), to speed up learning of large scale prob-", "type": "text" } ], "index": 31 }, { "bbox": [ 105, 540, 504, 552 ], "spans": [ { "bbox": [ 105, 540, 504, 552 ], "score": 1.0, "content": "lems with continuous state space. In particular, a neural network is used to compute value function", "type": "text" } ], "index": 32 }, { "bbox": [ 105, 550, 506, 562 ], "spans": [ { "bbox": [ 105, 550, 506, 562 ], "score": 1.0, "content": "estimates of states as a replacement of time-consuming Monte-Carlo rollouts, and another neural", "type": "text" } ], "index": 33 }, { "bbox": [ 105, 561, 505, 574 ], "spans": [ { "bbox": [ 105, 561, 505, 574 ], "score": 1.0, "content": "network is used to estimate policies as a probability prior for the therein introduced PUCT action", "type": "text" } ], "index": 34 }, { "bbox": [ 105, 572, 506, 585 ], "spans": [ { "bbox": [ 105, 572, 506, 585 ], "score": 1.0, "content": "selection method, a variant of well-known UCT sampling strategy commonly used in MCTS for", "type": "text" } ], "index": 35 }, { "bbox": [ 105, 583, 505, 596 ], "spans": [ { "bbox": [ 105, 583, 505, 596 ], "score": 1.0, "content": "exploration (Kocsis et al., 2006). Despite AlphaGo and AlphaZero achieving state-of-the-art per-", "type": "text" } ], "index": 36 }, { "bbox": [ 105, 594, 504, 606 ], "spans": [ { "bbox": [ 105, 594, 504, 606 ], "score": 1.0, "content": "formance in games with high branching factor like Go (Silver et al., 2016) and Chess (Silver et al.,", "type": "text" } ], "index": 37 }, { "bbox": [ 105, 604, 505, 618 ], "spans": [ { "bbox": [ 105, 604, 505, 618 ], "score": 1.0, "content": "2017a), both methods suffer from poor sample-efficiency, mostly due to the polynomial conver-", "type": "text" } ], "index": 38 }, { "bbox": [ 105, 616, 505, 629 ], "spans": [ { "bbox": [ 105, 616, 505, 629 ], "score": 1.0, "content": "gence rate of PUCT (Xiao et al., 2019). This problem, combined with the high computational time", "type": "text" } ], "index": 39 }, { "bbox": [ 105, 627, 498, 640 ], "spans": [ { "bbox": [ 105, 627, 498, 640 ], "score": 1.0, "content": "to evaluate the deep neural networks, significantly hinder the applicability of both methodologies.", "type": "text" } ], "index": 40 } ], "index": 32.5, "bbox_fs": [ 104, 462, 506, 640 ] }, { "type": "text", "bbox": [ 107, 643, 504, 732 ], "lines": [ { "bbox": [ 105, 644, 505, 656 ], "spans": [ { "bbox": [ 105, 644, 505, 656 ], "score": 1.0, "content": "In this paper, we provide a unified theory of the use of convex regularization in MCTS, which proved", "type": "text" } ], "index": 41 }, { "bbox": [ 105, 655, 505, 667 ], "spans": [ { "bbox": [ 105, 655, 505, 667 ], "score": 1.0, "content": "to be an efficient solution for driving exploration and stabilizing learning in RL (Schulman et al.,", "type": "text" } ], "index": 42 }, { "bbox": [ 105, 666, 506, 678 ], "spans": [ { "bbox": [ 105, 666, 506, 678 ], "score": 1.0, "content": "2015; 2017a; Haarnoja et al., 2018; Buesing et al., 2020). In particular, we show how a regularized", "type": "text" } ], "index": 43 }, { "bbox": [ 106, 677, 505, 689 ], "spans": [ { "bbox": [ 106, 677, 505, 689 ], "score": 1.0, "content": "objective function in MCTS can be seen as an instance of the Legendre-Fenchel transform, similar to", "type": "text" } ], "index": 44 }, { "bbox": [ 105, 687, 506, 700 ], "spans": [ { "bbox": [ 105, 687, 506, 700 ], "score": 1.0, "content": "previous findings on the use of duality in RL (Mensch & Blondel, 2018; Geist et al., 2019; Nachum", "type": "text" } ], "index": 45 }, { "bbox": [ 105, 698, 506, 711 ], "spans": [ { "bbox": [ 105, 698, 506, 711 ], "score": 1.0, "content": "& Dai, 2020) and game theory (Shalev-Shwartz & Singer, 2006; Pavel, 2007). Establishing our", "type": "text" } ], "index": 46 }, { "bbox": [ 106, 710, 506, 722 ], "spans": [ { "bbox": [ 106, 710, 506, 722 ], "score": 1.0, "content": "theoretical framework, we can derive the first regret analysis of regularized MCTS, and prove that", "type": "text" } ], "index": 47 }, { "bbox": [ 105, 720, 506, 735 ], "spans": [ { "bbox": [ 105, 720, 506, 735 ], "score": 1.0, "content": "a generic convex regularizer guarantees an exponential convergence rate to the solution of the reg-", "type": "text" } ], "index": 48 }, { "bbox": [ 105, 82, 505, 95 ], "spans": [ { "bbox": [ 105, 82, 505, 95 ], "score": 1.0, "content": "ularized objective function, which improves on the polynomial rate of PUCT. These results provide", "type": "text", "cross_page": true } ], "index": 0 }, { "bbox": [ 105, 93, 506, 106 ], "spans": [ { "bbox": [ 105, 93, 506, 106 ], "score": 1.0, "content": "a theoretical ground for the use of arbitrary entropy-based regularizers in MCTS until now limited", "type": "text", "cross_page": true } ], "index": 1 }, { "bbox": [ 105, 105, 506, 117 ], "spans": [ { "bbox": [ 105, 105, 506, 117 ], "score": 1.0, "content": "to maximum entropy (Xiao et al., 2019), among which we specifically study the relative entropy of", "type": "text", "cross_page": true } ], "index": 2 }, { "bbox": [ 104, 115, 506, 128 ], "spans": [ { "bbox": [ 104, 115, 506, 128 ], "score": 1.0, "content": "policy updates, drawing on similarities with trust-region and proximal methods in RL (Schulman", "type": "text", "cross_page": true } ], "index": 3 }, { "bbox": [ 105, 126, 505, 139 ], "spans": [ { "bbox": [ 105, 126, 505, 139 ], "score": 1.0, "content": "et al., 2015; 2017b), and the Tsallis entropy, used for enforcing the learning of sparse policies (Lee", "type": "text", "cross_page": true } ], "index": 4 }, { "bbox": [ 105, 136, 505, 150 ], "spans": [ { "bbox": [ 105, 136, 505, 150 ], "score": 1.0, "content": "et al., 2018). Moreover, we provide an empirical analysis of the toy problem introduced in Xiao et al.", "type": "text", "cross_page": true } ], "index": 5 }, { "bbox": [ 106, 148, 505, 161 ], "spans": [ { "bbox": [ 106, 148, 505, 161 ], "score": 1.0, "content": "(2019) to intuitively evince the practical consequences of our theoretical results for each regularizer.", "type": "text", "cross_page": true } ], "index": 6 }, { "bbox": [ 106, 159, 506, 172 ], "spans": [ { "bbox": [ 106, 159, 506, 172 ], "score": 1.0, "content": "Finally, we empirically evaluate the proposed operators in AlphaGo and AlphaZero on problems of", "type": "text", "cross_page": true } ], "index": 7 }, { "bbox": [ 105, 171, 505, 183 ], "spans": [ { "bbox": [ 105, 171, 505, 183 ], "score": 1.0, "content": "increasing complexity, from classic RL problems to an extensive analysis of Atari games, confirm-", "type": "text", "cross_page": true } ], "index": 8 }, { "bbox": [ 106, 182, 505, 194 ], "spans": [ { "bbox": [ 106, 182, 505, 194 ], "score": 1.0, "content": "ing the benefit of our novel operators compared to maximum entropy and, in general, the superiority", "type": "text", "cross_page": true } ], "index": 9 }, { "bbox": [ 106, 192, 336, 204 ], "spans": [ { "bbox": [ 106, 192, 336, 204 ], "score": 1.0, "content": "of convex regularization in MCTS w.r.t. classic methods.", "type": "text", "cross_page": true } ], "index": 10 } ], "index": 44.5, "bbox_fs": [ 105, 644, 506, 735 ] } ] }, { "preproc_blocks": [ { "type": "text", "bbox": [ 106, 82, 505, 203 ], "lines": [ { "bbox": [ 105, 82, 505, 95 ], "spans": [ { "bbox": [ 105, 82, 505, 95 ], "score": 1.0, "content": "ularized objective function, which improves on the polynomial rate of PUCT. These results provide", "type": "text" } ], "index": 0 }, { "bbox": [ 105, 93, 506, 106 ], "spans": [ { "bbox": [ 105, 93, 506, 106 ], "score": 1.0, "content": "a theoretical ground for the use of arbitrary entropy-based regularizers in MCTS until now limited", "type": "text" } ], "index": 1 }, { "bbox": [ 105, 105, 506, 117 ], "spans": [ { "bbox": [ 105, 105, 506, 117 ], "score": 1.0, "content": "to maximum entropy (Xiao et al., 2019), among which we specifically study the relative entropy of", "type": "text" } ], "index": 2 }, { "bbox": [ 104, 115, 506, 128 ], "spans": [ { "bbox": [ 104, 115, 506, 128 ], "score": 1.0, "content": "policy updates, drawing on similarities with trust-region and proximal methods in RL (Schulman", "type": "text" } ], "index": 3 }, { "bbox": [ 105, 126, 505, 139 ], "spans": [ { "bbox": [ 105, 126, 505, 139 ], "score": 1.0, "content": "et al., 2015; 2017b), and the Tsallis entropy, used for enforcing the learning of sparse policies (Lee", "type": "text" } ], "index": 4 }, { "bbox": [ 105, 136, 505, 150 ], "spans": [ { "bbox": [ 105, 136, 505, 150 ], "score": 1.0, "content": "et al., 2018). Moreover, we provide an empirical analysis of the toy problem introduced in Xiao et al.", "type": "text" } ], "index": 5 }, { "bbox": [ 106, 148, 505, 161 ], "spans": [ { "bbox": [ 106, 148, 505, 161 ], "score": 1.0, "content": "(2019) to intuitively evince the practical consequences of our theoretical results for each regularizer.", "type": "text" } ], "index": 6 }, { "bbox": [ 106, 159, 506, 172 ], "spans": [ { "bbox": [ 106, 159, 506, 172 ], "score": 1.0, "content": "Finally, we empirically evaluate the proposed operators in AlphaGo and AlphaZero on problems of", "type": "text" } ], "index": 7 }, { "bbox": [ 105, 171, 505, 183 ], "spans": [ { "bbox": [ 105, 171, 505, 183 ], "score": 1.0, "content": "increasing complexity, from classic RL problems to an extensive analysis of Atari games, confirm-", "type": "text" } ], "index": 8 }, { "bbox": [ 106, 182, 505, 194 ], "spans": [ { "bbox": [ 106, 182, 505, 194 ], "score": 1.0, "content": "ing the benefit of our novel operators compared to maximum entropy and, in general, the superiority", "type": "text" } ], "index": 9 }, { "bbox": [ 106, 192, 336, 204 ], "spans": [ { "bbox": [ 106, 192, 336, 204 ], "score": 1.0, "content": "of convex regularization in MCTS w.r.t. classic methods.", "type": "text" } ], "index": 10 } ], "index": 5 }, { "type": "title", "bbox": [ 108, 219, 208, 232 ], "lines": [ { "bbox": [ 104, 218, 210, 235 ], "spans": [ { "bbox": [ 104, 218, 210, 235 ], "score": 1.0, "content": "2 PRELIMINARIES", "type": "text" } ], "index": 11 } ], "index": 11 }, { "type": "title", "bbox": [ 108, 244, 267, 256 ], "lines": [ { "bbox": [ 106, 244, 269, 257 ], "spans": [ { "bbox": [ 106, 244, 269, 257 ], "score": 1.0, "content": "2.1 MARKOV DECISION PROCESSES", "type": "text" } ], "index": 12 } ], "index": 12 }, { "type": "text", "bbox": [ 106, 265, 505, 443 ], "lines": [ { "bbox": [ 106, 265, 505, 277 ], "spans": [ { "bbox": [ 106, 265, 505, 277 ], "score": 1.0, "content": "We consider the classical definition of a finite-horizon Markov Decision Process (MDP) as a 5-", "type": "text" } ], "index": 13 }, { "bbox": [ 105, 275, 506, 290 ], "spans": [ { "bbox": [ 105, 275, 131, 290 ], "score": 1.0, "content": "tuple", "type": "text" }, { "bbox": [ 131, 276, 225, 288 ], "score": 0.9, "content": "\\mathcal { M } = \\langle \\mathcal { S } , \\mathcal { A } , \\mathcal { R } , \\mathcal { P } , \\gamma \\rangle", "type": "inline_equation" }, { "bbox": [ 225, 275, 259, 290 ], "score": 1.0, "content": ", where", "type": "text" }, { "bbox": [ 260, 277, 268, 286 ], "score": 0.8, "content": "s", "type": "inline_equation" }, { "bbox": [ 269, 275, 350, 290 ], "score": 1.0, "content": "is the state space,", "type": "text" }, { "bbox": [ 350, 277, 360, 286 ], "score": 0.73, "content": "\\mathcal { A }", "type": "inline_equation" }, { "bbox": [ 360, 275, 506, 290 ], "score": 1.0, "content": "is the finite discrete action space,", "type": "text" } ], "index": 14 }, { "bbox": [ 106, 287, 505, 299 ], "spans": [ { "bbox": [ 106, 288, 207, 298 ], "score": 0.9, "content": "\\mathcal { R } : \\mathcal { S } \\times \\mathcal { A } \\times \\mathcal { S } \\to \\mathbb { R }", "type": "inline_equation" }, { "bbox": [ 207, 287, 307, 299 ], "score": 1.0, "content": "is the reward function,", "type": "text" }, { "bbox": [ 307, 288, 387, 298 ], "score": 0.9, "content": "\\mathcal { P } : \\mathcal { S } \\times \\mathcal { A } \\mathcal { S }", "type": "inline_equation" }, { "bbox": [ 387, 287, 505, 299 ], "score": 1.0, "content": "is the transition kernel, and", "type": "text" } ], "index": 15 }, { "bbox": [ 106, 298, 506, 311 ], "spans": [ { "bbox": [ 106, 299, 149, 311 ], "score": 0.9, "content": "\\gamma \\in [ 0 , 1 )", "type": "inline_equation" }, { "bbox": [ 150, 298, 283, 311 ], "score": 1.0, "content": "is the discount factor. A policy", "type": "text" }, { "bbox": [ 284, 299, 378, 309 ], "score": 0.88, "content": "\\pi \\in \\Pi : \\mathcal { S } \\times \\mathcal { A } \\mathbb { R }", "type": "inline_equation" }, { "bbox": [ 379, 298, 506, 311 ], "score": 1.0, "content": "is a probability distribution of", "type": "text" } ], "index": 16 }, { "bbox": [ 105, 308, 506, 323 ], "spans": [ { "bbox": [ 105, 308, 238, 323 ], "score": 1.0, "content": "the event of executing an action", "type": "text" }, { "bbox": [ 238, 311, 245, 319 ], "score": 0.71, "content": "a", "type": "inline_equation" }, { "bbox": [ 245, 308, 285, 323 ], "score": 1.0, "content": "in a state", "type": "text" }, { "bbox": [ 286, 312, 291, 319 ], "score": 0.64, "content": "s", "type": "inline_equation" }, { "bbox": [ 292, 308, 335, 323 ], "score": 1.0, "content": ". A policy", "type": "text" }, { "bbox": [ 335, 311, 343, 319 ], "score": 0.74, "content": "\\pi", "type": "inline_equation" }, { "bbox": [ 343, 308, 506, 323 ], "score": 1.0, "content": "induces a value function corresponding", "type": "text" } ], "index": 17 }, { "bbox": [ 104, 319, 506, 334 ], "spans": [ { "bbox": [ 104, 319, 486, 334 ], "score": 1.0, "content": "to the expected cumulative discounted reward collected by the agent when executing action", "type": "text" }, { "bbox": [ 486, 323, 493, 330 ], "score": 0.66, "content": "a", "type": "inline_equation" }, { "bbox": [ 493, 319, 506, 334 ], "score": 1.0, "content": "in", "type": "text" } ], "index": 18 }, { "bbox": [ 101, 326, 509, 352 ], "spans": [ { "bbox": [ 101, 326, 509, 352 ], "score": 1.0, "content": "state s, and following the policy π thereafter: Qπ(s, a) , E -P∞k=0 γkri+k+1|si = s, ai = a, π\u0003,", "type": "text" } ], "index": 19 }, { "bbox": [ 105, 353, 505, 367 ], "spans": [ { "bbox": [ 105, 353, 161, 367 ], "score": 1.0, "content": "timal policy", "type": "text" }, { "bbox": [ 162, 355, 173, 365 ], "score": 0.83, "content": "\\pi ^ { * }", "type": "inline_equation" }, { "bbox": [ 174, 353, 505, 367 ], "score": 1.0, "content": ", which is the policy that maximizes the expected cumulative discounted re-", "type": "text" } ], "index": 20 }, { "bbox": [ 106, 365, 505, 378 ], "spans": [ { "bbox": [ 106, 365, 505, 378 ], "score": 1.0, "content": "ward. The optimal policy corresponds to the one satisfying the optimal Bellman equation (Bell-", "type": "text" } ], "index": 21 }, { "bbox": [ 105, 377, 506, 391 ], "spans": [ { "bbox": [ 105, 377, 155, 391 ], "score": 1.0, "content": "man, 1954)", "type": "text" }, { "bbox": [ 156, 377, 410, 390 ], "score": 0.91, "content": "\\begin{array} { r } { Q ^ { * } ( s , a ) \\triangleq \\int _ { S } \\mathcal { P } ( s ^ { \\prime } | s , a ) \\left[ \\mathcal { R } ( s , a , s ^ { \\prime } ) + \\gamma \\operatorname* { m a x } _ { a ^ { \\prime } } Q ^ { * } ( s ^ { \\prime } , a ^ { \\prime } ) \\right] d s ^ { \\prime } } \\end{array}", "type": "inline_equation" }, { "bbox": [ 411, 377, 506, 391 ], "score": 1.0, "content": ", and is the fixed point", "type": "text" } ], "index": 22 }, { "bbox": [ 105, 390, 505, 405 ], "spans": [ { "bbox": [ 105, 390, 241, 404 ], "score": 1.0, "content": "of the optimal Bellman operator", "type": "text" }, { "bbox": [ 241, 391, 501, 405 ], "score": 0.84, "content": "\\begin{array} { r } { \\mathcal { T } ^ { * } Q ( s , a ) \\triangleq \\int _ { S } \\mathcal { P } ( s ^ { \\prime } | s , a ) \\left[ \\mathcal { R } ( s , a , s ^ { \\prime } ) + \\gamma \\operatorname* { m a x } _ { a ^ { \\prime } } Q ( s ^ { \\prime } , a ^ { \\prime } ) \\right] d s ^ { \\prime } } \\end{array}", "type": "inline_equation" }, { "bbox": [ 501, 390, 505, 404 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 23 }, { "bbox": [ 105, 404, 504, 417 ], "spans": [ { "bbox": [ 105, 404, 409, 417 ], "score": 1.0, "content": "Additionally, we define the Bellman operator under the policy", "type": "text" }, { "bbox": [ 409, 407, 417, 415 ], "score": 0.72, "content": "\\pi", "type": "inline_equation" }, { "bbox": [ 418, 404, 440, 417 ], "score": 1.0, "content": "as", "type": "text" }, { "bbox": [ 440, 405, 482, 417 ], "score": 0.89, "content": "{ \\mathcal { T } } _ { \\pi } Q ( s , a )", "type": "inline_equation" }, { "bbox": [ 493, 404, 504, 415 ], "score": 1.0, "content": ",", "type": "text" } ], "index": 24 }, { "bbox": [ 107, 416, 506, 431 ], "spans": [ { "bbox": [ 107, 416, 341, 431 ], "score": 0.88, "content": "\\begin{array} { r } { \\int _ { \\mathcal { S } } \\mathcal { P } ( s ^ { \\prime } | s , a ) \\left[ \\mathcal { R } ( s , a , s ^ { \\prime } ) + \\gamma \\int _ { \\mathcal { A } } \\pi ( a ^ { \\prime } | s ^ { \\prime } ) Q ( s ^ { \\prime } , a ^ { \\prime } ) d a ^ { \\prime } \\right] d s ^ { \\prime } } \\end{array}", "type": "inline_equation" }, { "bbox": [ 341, 416, 462, 430 ], "score": 1.0, "content": ", the optimal value function", "type": "text" }, { "bbox": [ 463, 417, 489, 430 ], "score": 0.77, "content": "V ^ { \\ast } ( s )", "type": "inline_equation" }, { "bbox": [ 489, 416, 506, 430 ], "score": 1.0, "content": ",", "type": "text" } ], "index": 25 }, { "bbox": [ 107, 430, 476, 444 ], "spans": [ { "bbox": [ 107, 432, 179, 443 ], "score": 0.92, "content": "\\operatorname* { m a x } _ { a \\in \\mathcal { A } } Q ^ { * } ( s , a )", "type": "inline_equation" }, { "bbox": [ 179, 430, 340, 444 ], "score": 1.0, "content": ", and the value function under the policy", "type": "text" }, { "bbox": [ 341, 433, 349, 442 ], "score": 0.74, "content": "\\pi", "type": "inline_equation" }, { "bbox": [ 349, 430, 360, 444 ], "score": 1.0, "content": "as", "type": "text" }, { "bbox": [ 361, 430, 473, 443 ], "score": 0.91, "content": "V ^ { \\pi } ( s ) \\triangleq \\operatorname* { m a x } _ { a \\in \\mathcal { A } } Q ^ { \\pi } ( s , a )", "type": "inline_equation" }, { "bbox": [ 473, 430, 476, 444 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 26 } ], "index": 19.5 }, { "type": "title", "bbox": [ 107, 456, 462, 468 ], "lines": [ { "bbox": [ 105, 456, 464, 469 ], "spans": [ { "bbox": [ 105, 456, 464, 469 ], "score": 1.0, "content": "2.2 MONTE-CARLO TREE SEARCH AND UPPER CONFIDENCE BOUNDS FOR TREES", "type": "text" } ], "index": 27 } ], "index": 27 }, { "type": "text", "bbox": [ 107, 477, 505, 522 ], "lines": [ { "bbox": [ 105, 477, 505, 490 ], "spans": [ { "bbox": [ 105, 477, 505, 490 ], "score": 1.0, "content": "Monte-Carlo Tree Search (MCTS) is a planning strategy based on a combination of Monte-Carlo", "type": "text" } ], "index": 28 }, { "bbox": [ 105, 488, 505, 500 ], "spans": [ { "bbox": [ 105, 488, 505, 500 ], "score": 1.0, "content": "sampling and tree search to solve MDPs. MCTS builds a tree where the nodes are the visited states", "type": "text" } ], "index": 29 }, { "bbox": [ 105, 498, 505, 512 ], "spans": [ { "bbox": [ 105, 498, 505, 512 ], "score": 1.0, "content": "of the MDP, and the edges are the actions executed in each state. MCTS converges to the optimal", "type": "text" } ], "index": 30 }, { "bbox": [ 105, 510, 471, 524 ], "spans": [ { "bbox": [ 105, 510, 471, 524 ], "score": 1.0, "content": "policy (Kocsis et al., 2006; Xiao et al., 2019), iterating over a loop composed of four steps:", "type": "text" } ], "index": 31 } ], "index": 29.5 }, { "type": "text", "bbox": [ 129, 531, 505, 621 ], "lines": [ { "bbox": [ 129, 531, 505, 543 ], "spans": [ { "bbox": [ 129, 531, 505, 543 ], "score": 1.0, "content": "1. Selection: starting from the root node, a tree-policy is executed to navigate the tree until a", "type": "text" } ], "index": 32 }, { "bbox": [ 141, 541, 394, 554 ], "spans": [ { "bbox": [ 141, 541, 394, 554 ], "score": 1.0, "content": "node with unvisited children, i.e. expandable node, is reached;", "type": "text" } ], "index": 33 }, { "bbox": [ 128, 556, 428, 571 ], "spans": [ { "bbox": [ 128, 556, 428, 571 ], "score": 1.0, "content": "2. Expansion: the reached node is expanded according to the tree policy;", "type": "text" } ], "index": 34 }, { "bbox": [ 128, 571, 505, 585 ], "spans": [ { "bbox": [ 128, 571, 505, 585 ], "score": 1.0, "content": "3. Simulation: run a rollout, e.g. Monte-Carlo simulation, from the visited child of the cur-", "type": "text" } ], "index": 35 }, { "bbox": [ 141, 583, 285, 596 ], "spans": [ { "bbox": [ 141, 583, 285, 596 ], "score": 1.0, "content": "rent node to the end of the episode;", "type": "text" } ], "index": 36 }, { "bbox": [ 128, 597, 505, 611 ], "spans": [ { "bbox": [ 128, 597, 392, 611 ], "score": 1.0, "content": "4. Backup: use the collected reward to update the action-values", "type": "text" }, { "bbox": [ 392, 598, 412, 610 ], "score": 0.91, "content": "Q ( \\cdot )", "type": "inline_equation" }, { "bbox": [ 412, 597, 505, 611 ], "score": 1.0, "content": "of the nodes visited in", "type": "text" } ], "index": 37 }, { "bbox": [ 142, 609, 365, 622 ], "spans": [ { "bbox": [ 142, 609, 365, 622 ], "score": 1.0, "content": "the trajectory from the root node to the expanded node.", "type": "text" } ], "index": 38 } ], "index": 35 }, { "type": "text", "bbox": [ 107, 630, 505, 696 ], "lines": [ { "bbox": [ 106, 630, 504, 641 ], "spans": [ { "bbox": [ 106, 630, 504, 641 ], "score": 1.0, "content": "The tree-policy used to select the action to execute in each node needs to balance the use of al-", "type": "text" } ], "index": 39 }, { "bbox": [ 106, 641, 505, 653 ], "spans": [ { "bbox": [ 106, 641, 505, 653 ], "score": 1.0, "content": "ready known good actions, and the visitation of unknown states. The Upper Confidence bounds", "type": "text" } ], "index": 40 }, { "bbox": [ 105, 651, 505, 665 ], "spans": [ { "bbox": [ 105, 651, 505, 665 ], "score": 1.0, "content": "for Trees (UCT) sampling strategy (Kocsis et al., 2006) extends the use of the well-known UCB1", "type": "text" } ], "index": 41 }, { "bbox": [ 105, 663, 505, 675 ], "spans": [ { "bbox": [ 105, 663, 505, 675 ], "score": 1.0, "content": "sampling strategy for multi-armed bandits (Auer et al., 2002), to MCTS. 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The Upper Confidence bounds", "type": "text" } ], "index": 40 }, { "bbox": [ 105, 651, 505, 665 ], "spans": [ { "bbox": [ 105, 651, 505, 665 ], "score": 1.0, "content": "for Trees (UCT) sampling strategy (Kocsis et al., 2006) extends the use of the well-known UCB1", "type": "text" } ], "index": 41 }, { "bbox": [ 105, 663, 505, 675 ], "spans": [ { "bbox": [ 105, 663, 505, 675 ], "score": 1.0, "content": "sampling strategy for multi-armed bandits (Auer et al., 2002), to MCTS. 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However, the use of entropy regularization is MCTS is still mostly unexplored, although its", "type": "text" } ], "index": 9 }, { "bbox": [ 106, 223, 505, 235 ], "spans": [ { "bbox": [ 106, 223, 505, 235 ], "score": 1.0, "content": "advantageous exploration and value function estimation would be desirable to reduce the detrimen-", "type": "text" } ], "index": 10 }, { "bbox": [ 106, 234, 505, 245 ], "spans": [ { "bbox": [ 106, 234, 505, 245 ], "score": 1.0, "content": "tal effect of high-branching factor in AlphaGo and AlphaZero. To the best of our knowledge, the", "type": "text" } ], "index": 11 }, { "bbox": [ 104, 243, 506, 258 ], "spans": [ { "bbox": [ 104, 243, 506, 258 ], "score": 1.0, "content": "MENTS algorithm (Xiao et al., 2019) is the first and only method to combine MCTS and entropy", "type": "text" } ], "index": 12 }, { "bbox": [ 104, 254, 506, 269 ], "spans": [ { "bbox": [ 104, 254, 506, 269 ], "score": 1.0, "content": "regularization. In particular, MENTS uses a maximum entropy regularizer in AlphaGo, proving", "type": "text" } ], "index": 13 }, { "bbox": [ 105, 266, 506, 279 ], "spans": [ { "bbox": [ 105, 266, 506, 279 ], "score": 1.0, "content": "an exponential convergence rate to the solution of the respective softmax objective function and", "type": "text" } ], "index": 14 }, { "bbox": [ 105, 276, 505, 289 ], "spans": [ { "bbox": [ 105, 276, 505, 289 ], "score": 1.0, "content": "achieving state-of-the-art performance in some Atari games (Bellemare et al., 2013). In the fol-", "type": "text" } ], "index": 15 }, { "bbox": [ 104, 287, 506, 302 ], "spans": [ { "bbox": [ 104, 287, 506, 302 ], "score": 1.0, "content": "lowing, motivated by the success in RL and the promising results of MENTS, we derive a unified", "type": "text" } ], "index": 16 }, { "bbox": [ 106, 299, 506, 312 ], "spans": [ { "bbox": [ 106, 299, 506, 312 ], "score": 1.0, "content": "theory of regularization in MCTS based on the Legendre-Fenchel transform (Geist et al., 2019), that", "type": "text" } ], "index": 17 }, { "bbox": [ 105, 310, 506, 323 ], "spans": [ { "bbox": [ 105, 310, 506, 323 ], "score": 1.0, "content": "generalizes the use of maximum entropy of MENTS to an arbitrary convex regularizer. Notably, our", "type": "text" } ], "index": 18 }, { "bbox": [ 105, 320, 505, 334 ], "spans": [ { "bbox": [ 105, 320, 505, 334 ], "score": 1.0, "content": "theoretical framework enables to rigorously motivate the advantages of using maximum entropy and", "type": "text" } ], "index": 19 }, { "bbox": [ 106, 333, 505, 344 ], "spans": [ { "bbox": [ 106, 333, 505, 344 ], "score": 1.0, "content": "other entropy-based regularizers, such as relative entropy or Tsallis entropy, drawing connections", "type": "text" } ], "index": 20 }, { "bbox": [ 105, 342, 505, 356 ], "spans": [ { "bbox": [ 105, 342, 505, 356 ], "score": 1.0, "content": "with their RL counterparts TRPO (Schulman et al., 2015) and Sparse DQN (Lee et al., 2018), as", "type": "text" } ], "index": 21 }, { "bbox": [ 105, 353, 377, 366 ], "spans": [ { "bbox": [ 105, 353, 377, 366 ], "score": 1.0, "content": "MENTS does with Soft Actor-Critic (SAC) (Haarnoja et al., 2018).", "type": "text" } ], "index": 22 } ], "index": 14 }, { "type": "title", "bbox": [ 108, 378, 279, 389 ], "lines": [ { "bbox": [ 106, 378, 280, 391 ], "spans": [ { "bbox": [ 106, 378, 280, 391 ], "score": 1.0, "content": "3.1 LEGENDRE-FENCHEL TRANSFORM", "type": "text" } ], "index": 23 } ], "index": 23 }, { "type": "text", "bbox": [ 108, 398, 504, 432 ], "lines": [ { "bbox": [ 105, 396, 505, 412 ], "spans": [ { "bbox": [ 105, 396, 184, 412 ], "score": 1.0, "content": "Consider an MDP", "type": "text" }, { "bbox": [ 185, 399, 276, 411 ], "score": 0.91, "content": "\\mathcal { M } = \\langle \\mathcal { S } , \\mathcal { A } , \\mathcal { R } , \\mathcal { P } , \\gamma \\rangle", "type": "inline_equation" }, { "bbox": [ 276, 396, 392, 412 ], "score": 1.0, "content": ", as previously defined. Let", "type": "text" }, { "bbox": [ 393, 399, 447, 409 ], "score": 0.92, "content": "\\Omega : \\Pi \\mathbb { R }", "type": "inline_equation" }, { "bbox": [ 447, 396, 505, 412 ], "score": 1.0, "content": "be a strongly", "type": "text" } ], "index": 24 }, { "bbox": [ 105, 410, 506, 422 ], "spans": [ { "bbox": [ 105, 410, 225, 422 ], "score": 1.0, "content": "convex function. For a policy", "type": "text" }, { "bbox": [ 225, 411, 273, 421 ], "score": 0.86, "content": "\\pi _ { s } = \\pi ( \\cdot | s )", "type": "inline_equation" }, { "bbox": [ 273, 410, 291, 422 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 291, 410, 371, 422 ], "score": 0.93, "content": "\\bar { Q } _ { s } = Q ( \\bar { s } , \\cdot ) \\in \\mathbb { R } ^ { 4 }", "type": "inline_equation" }, { "bbox": [ 371, 410, 506, 422 ], "score": 1.0, "content": ", the Legendre-Fenchel transform", "type": "text" } ], "index": 25 }, { "bbox": [ 106, 420, 334, 434 ], "spans": [ { "bbox": [ 106, 420, 207, 434 ], "score": 1.0, "content": "(or convex conjugate) of", "type": "text" }, { "bbox": [ 207, 421, 215, 431 ], "score": 0.85, "content": "\\Omega", "type": "inline_equation" }, { "bbox": [ 215, 420, 225, 434 ], "score": 1.0, "content": "is", "type": "text" }, { "bbox": [ 226, 421, 283, 431 ], "score": 0.86, "content": "\\Omega ^ { * } : \\mathbb { R } ^ { A } \\xrightarrow [ ] { } \\mathbb { R }", "type": "inline_equation" }, { "bbox": [ 284, 420, 334, 434 ], "score": 1.0, "content": ", defined as:", "type": "text" } ], "index": 26 } ], "index": 25 }, { "type": "interline_equation", "bbox": [ 233, 436, 377, 457 ], "lines": [ { "bbox": [ 233, 436, 377, 457 ], "spans": [ { "bbox": [ 233, 436, 377, 457 ], "score": 0.93, "content": "\\Omega ^ { \\ast } ( Q _ { s } ) \\triangleq \\operatorname* { m a x } _ { \\pi _ { s } \\in \\Pi _ { s } } \\mathcal { T } _ { \\pi _ { s } } Q _ { s } - \\tau \\Omega ( \\pi _ { s } ) ,", "type": "interline_equation", "image_path": "705f296b30051d751481e31a1da74be16804249ff14598f8d3eee4cf17a742e9.jpg" } ] } ], "index": 27, "virtual_lines": [ { "bbox": [ 233, 436, 377, 457 ], "spans": [], "index": 27 } ] }, { "type": "text", "bbox": [ 107, 460, 505, 483 ], "lines": [ { "bbox": [ 106, 460, 505, 473 ], "spans": [ { "bbox": [ 106, 460, 197, 473 ], "score": 1.0, "content": "where the temperature", "type": "text" }, { "bbox": [ 197, 463, 204, 470 ], "score": 0.76, "content": "\\tau", "type": "inline_equation" }, { "bbox": [ 204, 460, 505, 473 ], "score": 1.0, "content": "specifies the strength of regularization. Among the several properties of the", "type": "text" } ], "index": 28 }, { "bbox": [ 105, 471, 496, 484 ], "spans": [ { "bbox": [ 105, 471, 496, 484 ], "score": 1.0, "content": "Legendre-Fenchel transform, we use the following (Mensch & Blondel, 2018; Geist et al., 2019).", "type": "text" } ], "index": 29 } ], "index": 28.5 }, { "type": "text", "bbox": [ 108, 491, 270, 503 ], "lines": [ { "bbox": [ 105, 490, 271, 506 ], "spans": [ { "bbox": [ 105, 490, 183, 506 ], "score": 1.0, "content": "Proposition 1 Let", "type": "text" }, { "bbox": [ 183, 492, 192, 502 ], "score": 0.57, "content": "\\Omega", "type": "inline_equation" }, { "bbox": [ 192, 490, 271, 506 ], "score": 1.0, "content": "be strongly convex.", "type": "text" } ], "index": 30 } ], "index": 30 }, { "type": "text", "bbox": [ 132, 511, 388, 524 ], "lines": [ { "bbox": [ 131, 510, 389, 526 ], "spans": [ { "bbox": [ 131, 510, 267, 526 ], "score": 1.0, "content": "• Unique maximizing argument:", "type": "text" }, { "bbox": [ 267, 512, 288, 522 ], "score": 0.86, "content": "\\nabla \\Omega ^ { * }", "type": "inline_equation" }, { "bbox": [ 288, 510, 389, 526 ], "score": 1.0, "content": "is Lipschitz and satisfies", "type": "text" } ], "index": 31 } ], "index": 31 }, { "type": "interline_equation", "bbox": [ 241, 527, 405, 548 ], "lines": [ { "bbox": [ 241, 527, 405, 548 ], "spans": [ { "bbox": [ 241, 527, 405, 548 ], "score": 0.93, "content": "\\nabla \\Omega ^ { * } ( Q _ { s } ) = \\arg \\operatorname* { m a x } _ { \\pi _ { s } \\in \\Pi _ { s } } \\mathcal { T } _ { \\pi _ { s } } Q _ { s } - \\tau \\Omega ( \\pi _ { s } ) .", "type": "interline_equation", "image_path": "e674e976ef76a93b807aba4a03db10634a2953047e64276826cc7fd8b94da3f0.jpg" } ] } ], "index": 32, "virtual_lines": [ { "bbox": [ 241, 527, 405, 548 ], "spans": [], "index": 32 } ] }, { "type": "text", "bbox": [ 130, 558, 504, 581 ], "lines": [ { "bbox": [ 131, 555, 505, 573 ], "spans": [ { "bbox": [ 131, 555, 288, 573 ], "score": 1.0, "content": "• Boundedness: if there are constants", "type": "text" }, { "bbox": [ 289, 559, 303, 569 ], "score": 0.91, "content": "L _ { \\Omega }", "type": "inline_equation" }, { "bbox": [ 303, 555, 321, 573 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 322, 559, 336, 569 ], "score": 0.9, "content": "U _ { \\Omega }", "type": "inline_equation" }, { "bbox": [ 336, 555, 403, 573 ], "score": 1.0, "content": "such that for all", "type": "text" }, { "bbox": [ 404, 558, 439, 569 ], "score": 0.92, "content": "\\pi _ { s } \\in \\Pi _ { s }", "type": "inline_equation" }, { "bbox": [ 439, 555, 479, 573 ], "score": 1.0, "content": ", we have", "type": "text" }, { "bbox": [ 479, 558, 505, 570 ], "score": 0.91, "content": "L _ { \\Omega } \\leq", "type": "inline_equation" } ], "index": 33 }, { "bbox": [ 142, 568, 218, 581 ], "spans": [ { "bbox": [ 142, 569, 194, 581 ], "score": 0.92, "content": "\\Omega ( \\pi _ { s } ) \\leq U _ { \\Omega }", "type": "inline_equation" }, { "bbox": [ 195, 568, 218, 581 ], "score": 1.0, "content": ", then", "type": "text" } ], "index": 34 } ], "index": 33.5 }, { "type": "interline_equation", "bbox": [ 215, 584, 431, 603 ], "lines": [ { "bbox": [ 215, 584, 431, 603 ], "spans": [ { "bbox": [ 215, 584, 431, 603 ], "score": 0.91, "content": "\\operatorname* { m a x } _ { a \\in \\mathcal { A } } Q _ { s } ( a ) - \\tau U _ { \\Omega } \\le \\Omega ^ { * } ( Q _ { s } ) \\le \\operatorname* { m a x } _ { a \\in \\mathcal { A } } Q _ { s } ( a ) - \\tau L _ { \\Omega } .", "type": "interline_equation", "image_path": "ef1948ef426ed74c7d1ad8271c006b4514d26b031d1e4810a3f96c9d9bd12c9d.jpg" } ] } ], "index": 35, "virtual_lines": [ { "bbox": [ 215, 584, 431, 603 ], "spans": [], "index": 35 } ] }, { "type": "text", "bbox": [ 133, 613, 297, 627 ], "lines": [ { "bbox": [ 131, 612, 296, 628 ], "spans": [ { "bbox": [ 131, 612, 228, 628 ], "score": 1.0, "content": "• Contraction: for any", "type": "text" }, { "bbox": [ 228, 613, 296, 626 ], "score": 0.89, "content": "Q _ { 1 } , Q _ { 2 } \\in \\mathbb { R } ^ { S \\times A }", "type": "inline_equation" } ], "index": 36 } ], "index": 36 }, { "type": "interline_equation", "bbox": [ 229, 630, 418, 644 ], "lines": [ { "bbox": [ 229, 630, 418, 644 ], "spans": [ { "bbox": [ 229, 630, 418, 644 ], "score": 0.86, "content": "\\parallel \\Omega ^ { * } ( Q _ { 1 } ) - \\Omega ^ { * } ( Q _ { 2 } ) \\parallel _ { \\infty } \\leq \\gamma \\parallel Q _ { 1 } - Q _ { 2 } \\parallel _ { \\infty } .", "type": "interline_equation", "image_path": "720371abba2eaebb2ef541f2248d9fef577e95eecaa3aff12d4e5f772ec37530.jpg" } ] } ], "index": 37, "virtual_lines": [ { "bbox": [ 229, 630, 418, 644 ], "spans": [], "index": 37 } ] }, { "type": "text", "bbox": [ 107, 653, 506, 676 ], "lines": [ { "bbox": [ 106, 653, 505, 666 ], "spans": [ { "bbox": [ 106, 653, 282, 666 ], "score": 1.0, "content": "Although the Legendre-Fenchel transform", "type": "text" }, { "bbox": [ 282, 654, 295, 664 ], "score": 0.87, "content": "\\Omega ^ { * }", "type": "inline_equation" }, { "bbox": [ 296, 653, 505, 666 ], "score": 1.0, "content": "applies to every strongly convex function, for the", "type": "text" } ], "index": 38 }, { "bbox": [ 105, 664, 438, 677 ], "spans": [ { "bbox": [ 105, 664, 438, 677 ], "score": 1.0, "content": "purpose of this work we only consider a representative set of entropic regularizers.", "type": "text" } ], "index": 39 } ], "index": 38.5 }, { "type": "title", "bbox": [ 107, 689, 312, 700 ], "lines": [ { "bbox": [ 106, 689, 312, 702 ], "spans": [ { "bbox": [ 106, 689, 312, 702 ], "score": 1.0, "content": "3.2 REGULARIZED BACKUP AND TREE POLICY", "type": "text" } ], "index": 40 } ], "index": 40 }, { "type": "text", "bbox": [ 108, 709, 505, 732 ], "lines": [ { "bbox": [ 106, 709, 504, 722 ], "spans": [ { "bbox": [ 106, 709, 312, 722 ], "score": 1.0, "content": "In MCTS, each node of the tree represents a state", "type": "text" }, { "bbox": [ 312, 710, 340, 720 ], "score": 0.91, "content": "s \\in S", "type": "inline_equation" }, { "bbox": [ 340, 709, 469, 722 ], "score": 1.0, "content": "and contains a visitation count", "type": "text" }, { "bbox": [ 469, 709, 501, 722 ], "score": 0.92, "content": "N ( s , a )", "type": "inline_equation" }, { "bbox": [ 501, 709, 504, 722 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 41 }, { "bbox": [ 107, 721, 505, 732 ], "spans": [ { "bbox": [ 107, 721, 228, 732 ], "score": 1.0, "content": "Given a trajectory, we define", "type": "text" }, { "bbox": [ 229, 721, 254, 732 ], "score": 0.95, "content": "n \\big ( s _ { T } \\big )", "type": "inline_equation" }, { "bbox": [ 255, 721, 469, 732 ], "score": 1.0, "content": "as the leaf node corresponding to the reached state", "type": "text" }, { "bbox": [ 470, 723, 482, 732 ], "score": 0.87, "content": "s _ { T }", "type": "inline_equation" }, { "bbox": [ 482, 721, 505, 732 ], "score": 1.0, "content": ". 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UCT asymptotically converges to the optimal action-value", "type": "text" } ], "index": 1 }, { "bbox": [ 106, 104, 505, 117 ], "spans": [ { "bbox": [ 106, 104, 142, 117 ], "score": 1.0, "content": "function", "type": "text" }, { "bbox": [ 142, 105, 156, 116 ], "score": 0.89, "content": "Q ^ { * }", "type": "inline_equation" }, { "bbox": [ 156, 104, 505, 117 ], "score": 1.0, "content": ", for all states and actions, with the probability of executing a suboptimal action at the", "type": "text" } ], "index": 2 }, { "bbox": [ 105, 114, 506, 129 ], "spans": [ { "bbox": [ 105, 114, 297, 129 ], "score": 1.0, "content": "root node approaching 0 with a polynomial rate", "type": "text" }, { "bbox": [ 297, 114, 320, 128 ], "score": 0.92, "content": "\\textstyle { \\dot { O } } ( { \\frac { 1 } { t } } )", "type": "inline_equation" }, { "bbox": [ 320, 114, 417, 129 ], "score": 1.0, "content": ", for a simulation budget", "type": "text" }, { "bbox": [ 418, 117, 423, 125 ], "score": 0.68, "content": "t", "type": "inline_equation" }, { "bbox": [ 423, 114, 506, 129 ], "score": 1.0, "content": "(Kocsis et al., 2006;", "type": "text" } ], "index": 3 }, { "bbox": [ 106, 126, 180, 138 ], "spans": [ { "bbox": [ 106, 126, 180, 138 ], "score": 1.0, "content": "Xiao et al., 2019).", "type": "text" } ], "index": 4 } ], "index": 2, "bbox_fs": [ 105, 82, 506, 138 ] }, { "type": "title", "bbox": [ 108, 153, 361, 166 ], "lines": [ { "bbox": [ 105, 152, 363, 168 ], "spans": [ { "bbox": [ 105, 152, 363, 168 ], "score": 1.0, "content": "3 REGULARIZED MONTE-CARLO TREE SEARCH", "type": "text" } ], "index": 5 } ], "index": 5 }, { "type": "text", "bbox": [ 106, 178, 505, 365 ], "lines": [ { "bbox": [ 105, 177, 505, 191 ], "spans": [ { "bbox": [ 105, 177, 505, 191 ], "score": 1.0, "content": "The success of RL methods based on entropy regularization comes from their ability to achieve", "type": "text" } ], "index": 6 }, { "bbox": [ 106, 190, 505, 201 ], "spans": [ { "bbox": [ 106, 190, 505, 201 ], "score": 1.0, "content": "state-of-the-art performance in decision making and control problems, while enjoying theoretical", "type": "text" } ], "index": 7 }, { "bbox": [ 104, 200, 506, 214 ], "spans": [ { "bbox": [ 104, 200, 506, 214 ], "score": 1.0, "content": "guarantees and ease of implementation (Haarnoja et al., 2018; Schulman et al., 2015; Lee et al.,", "type": "text" } ], "index": 8 }, { "bbox": [ 106, 211, 505, 225 ], "spans": [ { "bbox": [ 106, 211, 505, 225 ], "score": 1.0, "content": "2018). However, the use of entropy regularization is MCTS is still mostly unexplored, although its", "type": "text" } ], "index": 9 }, { "bbox": [ 106, 223, 505, 235 ], "spans": [ { "bbox": [ 106, 223, 505, 235 ], "score": 1.0, "content": "advantageous exploration and value function estimation would be desirable to reduce the detrimen-", "type": "text" } ], "index": 10 }, { "bbox": [ 106, 234, 505, 245 ], "spans": [ { "bbox": [ 106, 234, 505, 245 ], "score": 1.0, "content": "tal effect of high-branching factor in AlphaGo and AlphaZero. To the best of our knowledge, the", "type": "text" } ], "index": 11 }, { "bbox": [ 104, 243, 506, 258 ], "spans": [ { "bbox": [ 104, 243, 506, 258 ], "score": 1.0, "content": "MENTS algorithm (Xiao et al., 2019) is the first and only method to combine MCTS and entropy", "type": "text" } ], "index": 12 }, { "bbox": [ 104, 254, 506, 269 ], "spans": [ { "bbox": [ 104, 254, 506, 269 ], "score": 1.0, "content": "regularization. In particular, MENTS uses a maximum entropy regularizer in AlphaGo, proving", "type": "text" } ], "index": 13 }, { "bbox": [ 105, 266, 506, 279 ], "spans": [ { "bbox": [ 105, 266, 506, 279 ], "score": 1.0, "content": "an exponential convergence rate to the solution of the respective softmax objective function and", "type": "text" } ], "index": 14 }, { "bbox": [ 105, 276, 505, 289 ], "spans": [ { "bbox": [ 105, 276, 505, 289 ], "score": 1.0, "content": "achieving state-of-the-art performance in some Atari games (Bellemare et al., 2013). In the fol-", "type": "text" } ], "index": 15 }, { "bbox": [ 104, 287, 506, 302 ], "spans": [ { "bbox": [ 104, 287, 506, 302 ], "score": 1.0, "content": "lowing, motivated by the success in RL and the promising results of MENTS, we derive a unified", "type": "text" } ], "index": 16 }, { "bbox": [ 106, 299, 506, 312 ], "spans": [ { "bbox": [ 106, 299, 506, 312 ], "score": 1.0, "content": "theory of regularization in MCTS based on the Legendre-Fenchel transform (Geist et al., 2019), that", "type": "text" } ], "index": 17 }, { "bbox": [ 105, 310, 506, 323 ], "spans": [ { "bbox": [ 105, 310, 506, 323 ], "score": 1.0, "content": "generalizes the use of maximum entropy of MENTS to an arbitrary convex regularizer. Notably, our", "type": "text" } ], "index": 18 }, { "bbox": [ 105, 320, 505, 334 ], "spans": [ { "bbox": [ 105, 320, 505, 334 ], "score": 1.0, "content": "theoretical framework enables to rigorously motivate the advantages of using maximum entropy and", "type": "text" } ], "index": 19 }, { "bbox": [ 106, 333, 505, 344 ], "spans": [ { "bbox": [ 106, 333, 505, 344 ], "score": 1.0, "content": "other entropy-based regularizers, such as relative entropy or Tsallis entropy, drawing connections", "type": "text" } ], "index": 20 }, { "bbox": [ 105, 342, 505, 356 ], "spans": [ { "bbox": [ 105, 342, 505, 356 ], "score": 1.0, "content": "with their RL counterparts TRPO (Schulman et al., 2015) and Sparse DQN (Lee et al., 2018), as", "type": "text" } ], "index": 21 }, { "bbox": [ 105, 353, 377, 366 ], "spans": [ { "bbox": [ 105, 353, 377, 366 ], "score": 1.0, "content": "MENTS does with Soft Actor-Critic (SAC) (Haarnoja et al., 2018).", "type": "text" } ], "index": 22 } ], "index": 14, "bbox_fs": [ 104, 177, 506, 366 ] }, { "type": "title", "bbox": [ 108, 378, 279, 389 ], "lines": [ { "bbox": [ 106, 378, 280, 391 ], "spans": [ { "bbox": [ 106, 378, 280, 391 ], "score": 1.0, "content": "3.1 LEGENDRE-FENCHEL TRANSFORM", "type": "text" } ], "index": 23 } ], "index": 23 }, { "type": "text", "bbox": [ 108, 398, 504, 432 ], "lines": [ { "bbox": [ 105, 396, 505, 412 ], "spans": [ { "bbox": [ 105, 396, 184, 412 ], "score": 1.0, "content": "Consider an MDP", "type": "text" }, { "bbox": [ 185, 399, 276, 411 ], "score": 0.91, "content": "\\mathcal { M } = \\langle \\mathcal { S } , \\mathcal { A } , \\mathcal { R } , \\mathcal { P } , \\gamma \\rangle", "type": "inline_equation" }, { "bbox": [ 276, 396, 392, 412 ], "score": 1.0, "content": ", as previously defined. Let", "type": "text" }, { "bbox": [ 393, 399, 447, 409 ], "score": 0.92, "content": "\\Omega : \\Pi \\mathbb { R }", "type": "inline_equation" }, { "bbox": [ 447, 396, 505, 412 ], "score": 1.0, "content": "be a strongly", "type": "text" } ], "index": 24 }, { "bbox": [ 105, 410, 506, 422 ], "spans": [ { "bbox": [ 105, 410, 225, 422 ], "score": 1.0, "content": "convex function. For a policy", "type": "text" }, { "bbox": [ 225, 411, 273, 421 ], "score": 0.86, "content": "\\pi _ { s } = \\pi ( \\cdot | s )", "type": "inline_equation" }, { "bbox": [ 273, 410, 291, 422 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 291, 410, 371, 422 ], "score": 0.93, "content": "\\bar { Q } _ { s } = Q ( \\bar { s } , \\cdot ) \\in \\mathbb { R } ^ { 4 }", "type": "inline_equation" }, { "bbox": [ 371, 410, 506, 422 ], "score": 1.0, "content": ", the Legendre-Fenchel transform", "type": "text" } ], "index": 25 }, { "bbox": [ 106, 420, 334, 434 ], "spans": [ { "bbox": [ 106, 420, 207, 434 ], "score": 1.0, "content": "(or convex conjugate) of", "type": "text" }, { "bbox": [ 207, 421, 215, 431 ], "score": 0.85, "content": "\\Omega", "type": "inline_equation" }, { "bbox": [ 215, 420, 225, 434 ], "score": 1.0, "content": "is", "type": "text" }, { "bbox": [ 226, 421, 283, 431 ], "score": 0.86, "content": "\\Omega ^ { * } : \\mathbb { R } ^ { A } \\xrightarrow [ ] { } \\mathbb { R }", "type": "inline_equation" }, { "bbox": [ 284, 420, 334, 434 ], "score": 1.0, "content": ", defined as:", "type": "text" } ], "index": 26 } ], "index": 25, "bbox_fs": [ 105, 396, 506, 434 ] }, { "type": "interline_equation", "bbox": [ 233, 436, 377, 457 ], "lines": [ { "bbox": [ 233, 436, 377, 457 ], "spans": [ { "bbox": [ 233, 436, 377, 457 ], "score": 0.93, "content": "\\Omega ^ { \\ast } ( Q _ { s } ) \\triangleq \\operatorname* { m a x } _ { \\pi _ { s } \\in \\Pi _ { s } } \\mathcal { T } _ { \\pi _ { s } } Q _ { s } - \\tau \\Omega ( \\pi _ { s } ) ,", "type": "interline_equation", "image_path": "705f296b30051d751481e31a1da74be16804249ff14598f8d3eee4cf17a742e9.jpg" } ] } ], "index": 27, "virtual_lines": [ { "bbox": [ 233, 436, 377, 457 ], "spans": [], "index": 27 } ] }, { "type": "text", "bbox": [ 107, 460, 505, 483 ], "lines": [ { "bbox": [ 106, 460, 505, 473 ], "spans": [ { "bbox": [ 106, 460, 197, 473 ], "score": 1.0, "content": "where the temperature", "type": "text" }, { "bbox": [ 197, 463, 204, 470 ], "score": 0.76, "content": "\\tau", "type": "inline_equation" }, { "bbox": [ 204, 460, 505, 473 ], "score": 1.0, "content": "specifies the strength of regularization. Among the several properties of the", "type": "text" } ], "index": 28 }, { "bbox": [ 105, 471, 496, 484 ], "spans": [ { "bbox": [ 105, 471, 496, 484 ], "score": 1.0, "content": "Legendre-Fenchel transform, we use the following (Mensch & Blondel, 2018; Geist et al., 2019).", "type": "text" } ], "index": 29 } ], "index": 28.5, "bbox_fs": [ 105, 460, 505, 484 ] }, { "type": "text", "bbox": [ 108, 491, 270, 503 ], "lines": [ { "bbox": [ 105, 490, 271, 506 ], "spans": [ { "bbox": [ 105, 490, 183, 506 ], "score": 1.0, "content": "Proposition 1 Let", "type": "text" }, { "bbox": [ 183, 492, 192, 502 ], "score": 0.57, "content": "\\Omega", "type": "inline_equation" }, { "bbox": [ 192, 490, 271, 506 ], "score": 1.0, "content": "be strongly convex.", "type": "text" } ], "index": 30 } ], "index": 30, "bbox_fs": [ 105, 490, 271, 506 ] }, { "type": "text", "bbox": [ 132, 511, 388, 524 ], "lines": [ { "bbox": [ 131, 510, 389, 526 ], "spans": [ { "bbox": [ 131, 510, 267, 526 ], "score": 1.0, "content": "• Unique maximizing argument:", "type": "text" }, { "bbox": [ 267, 512, 288, 522 ], "score": 0.86, "content": "\\nabla \\Omega ^ { * }", "type": "inline_equation" }, { "bbox": [ 288, 510, 389, 526 ], "score": 1.0, "content": "is Lipschitz and satisfies", "type": "text" } ], "index": 31 } ], "index": 31, "bbox_fs": [ 131, 510, 389, 526 ] }, { "type": "interline_equation", "bbox": [ 241, 527, 405, 548 ], "lines": [ { "bbox": [ 241, 527, 405, 548 ], "spans": [ { "bbox": [ 241, 527, 405, 548 ], "score": 0.93, "content": "\\nabla \\Omega ^ { * } ( Q _ { s } ) = \\arg \\operatorname* { m a x } _ { \\pi _ { s } \\in \\Pi _ { s } } \\mathcal { T } _ { \\pi _ { s } } Q _ { s } - \\tau \\Omega ( \\pi _ { s } ) .", "type": "interline_equation", "image_path": "e674e976ef76a93b807aba4a03db10634a2953047e64276826cc7fd8b94da3f0.jpg" } ] } ], "index": 32, "virtual_lines": [ { "bbox": [ 241, 527, 405, 548 ], "spans": [], "index": 32 } ] }, { "type": "text", "bbox": [ 130, 558, 504, 581 ], "lines": [ { "bbox": [ 131, 555, 505, 573 ], "spans": [ { "bbox": [ 131, 555, 288, 573 ], "score": 1.0, "content": "• Boundedness: if there are constants", "type": "text" }, { "bbox": [ 289, 559, 303, 569 ], "score": 0.91, "content": "L _ { \\Omega }", "type": "inline_equation" }, { "bbox": [ 303, 555, 321, 573 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 322, 559, 336, 569 ], "score": 0.9, "content": "U _ { \\Omega }", "type": "inline_equation" }, { "bbox": [ 336, 555, 403, 573 ], "score": 1.0, "content": "such that for all", "type": "text" }, { "bbox": [ 404, 558, 439, 569 ], "score": 0.92, "content": "\\pi _ { s } \\in \\Pi _ { s }", "type": "inline_equation" }, { "bbox": [ 439, 555, 479, 573 ], "score": 1.0, "content": ", we have", "type": "text" }, { "bbox": [ 479, 558, 505, 570 ], "score": 0.91, "content": "L _ { \\Omega } \\leq", "type": "inline_equation" } ], "index": 33 }, { "bbox": [ 142, 568, 218, 581 ], "spans": [ { "bbox": [ 142, 569, 194, 581 ], "score": 0.92, "content": "\\Omega ( \\pi _ { s } ) \\leq U _ { \\Omega }", "type": "inline_equation" }, { "bbox": [ 195, 568, 218, 581 ], "score": 1.0, "content": ", then", "type": "text" } ], "index": 34 } ], "index": 33.5, "bbox_fs": [ 131, 555, 505, 581 ] }, { "type": "interline_equation", "bbox": [ 215, 584, 431, 603 ], "lines": [ { "bbox": [ 215, 584, 431, 603 ], "spans": [ { "bbox": [ 215, 584, 431, 603 ], "score": 0.91, "content": "\\operatorname* { m a x } _ { a \\in \\mathcal { A } } Q _ { s } ( a ) - \\tau U _ { \\Omega } \\le \\Omega ^ { * } ( Q _ { s } ) \\le \\operatorname* { m a x } _ { a \\in \\mathcal { A } } Q _ { s } ( a ) - \\tau L _ { \\Omega } .", "type": "interline_equation", "image_path": "ef1948ef426ed74c7d1ad8271c006b4514d26b031d1e4810a3f96c9d9bd12c9d.jpg" } ] } ], "index": 35, "virtual_lines": [ { "bbox": [ 215, 584, 431, 603 ], "spans": [], "index": 35 } ] }, { "type": "text", "bbox": [ 133, 613, 297, 627 ], "lines": [ { "bbox": [ 131, 612, 296, 628 ], "spans": [ { "bbox": [ 131, 612, 228, 628 ], "score": 1.0, "content": "• Contraction: for any", "type": "text" }, { "bbox": [ 228, 613, 296, 626 ], "score": 0.89, "content": "Q _ { 1 } , Q _ { 2 } \\in \\mathbb { R } ^ { S \\times A }", "type": "inline_equation" } ], "index": 36 } ], "index": 36, "bbox_fs": [ 131, 612, 296, 628 ] }, { "type": "interline_equation", "bbox": [ 229, 630, 418, 644 ], "lines": [ { "bbox": [ 229, 630, 418, 644 ], "spans": [ { "bbox": [ 229, 630, 418, 644 ], "score": 0.86, "content": "\\parallel \\Omega ^ { * } ( Q _ { 1 } ) - \\Omega ^ { * } ( Q _ { 2 } ) \\parallel _ { \\infty } \\leq \\gamma \\parallel Q _ { 1 } - Q _ { 2 } \\parallel _ { \\infty } .", "type": "interline_equation", "image_path": "720371abba2eaebb2ef541f2248d9fef577e95eecaa3aff12d4e5f772ec37530.jpg" } ] } ], "index": 37, "virtual_lines": [ { "bbox": [ 229, 630, 418, 644 ], "spans": [], "index": 37 } ] }, { "type": "text", "bbox": [ 107, 653, 506, 676 ], "lines": [ { "bbox": [ 106, 653, 505, 666 ], "spans": [ { "bbox": [ 106, 653, 282, 666 ], "score": 1.0, "content": "Although the Legendre-Fenchel transform", "type": "text" }, { "bbox": [ 282, 654, 295, 664 ], "score": 0.87, "content": "\\Omega ^ { * }", "type": "inline_equation" }, { "bbox": [ 296, 653, 505, 666 ], "score": 1.0, "content": "applies to every strongly convex function, for the", "type": "text" } ], "index": 38 }, { "bbox": [ 105, 664, 438, 677 ], "spans": [ { "bbox": [ 105, 664, 438, 677 ], "score": 1.0, "content": "purpose of this work we only consider a representative set of entropic regularizers.", "type": "text" } ], "index": 39 } ], "index": 38.5, "bbox_fs": [ 105, 653, 505, 677 ] }, { "type": "title", "bbox": [ 107, 689, 312, 700 ], "lines": [ { "bbox": [ 106, 689, 312, 702 ], "spans": [ { "bbox": [ 106, 689, 312, 702 ], "score": 1.0, "content": "3.2 REGULARIZED BACKUP AND TREE POLICY", "type": "text" } ], "index": 40 } ], "index": 40 }, { "type": "text", "bbox": [ 108, 709, 505, 732 ], "lines": [ { "bbox": [ 106, 709, 504, 722 ], "spans": [ { "bbox": [ 106, 709, 312, 722 ], "score": 1.0, "content": "In MCTS, each node of the tree represents a state", "type": "text" }, { "bbox": [ 312, 710, 340, 720 ], "score": 0.91, "content": "s \\in S", "type": "inline_equation" }, { "bbox": [ 340, 709, 469, 722 ], "score": 1.0, "content": "and contains a visitation count", "type": "text" }, { "bbox": [ 469, 709, 501, 722 ], "score": 0.92, "content": "N ( s , a )", "type": "inline_equation" }, { "bbox": [ 501, 709, 504, 722 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 41 }, { "bbox": [ 107, 721, 505, 732 ], "spans": [ { "bbox": [ 107, 721, 228, 732 ], "score": 1.0, "content": "Given a trajectory, we define", "type": "text" }, { "bbox": [ 229, 721, 254, 732 ], "score": 0.95, "content": "n \\big ( s _ { T } \\big )", "type": "inline_equation" }, { "bbox": [ 255, 721, 469, 732 ], "score": 1.0, "content": "as the leaf node corresponding to the reached state", "type": "text" }, { "bbox": [ 470, 723, 482, 732 ], "score": 0.87, "content": "s _ { T }", "type": "inline_equation" }, { "bbox": [ 482, 721, 505, 732 ], "score": 1.0, "content": ". Let", "type": "text" } ], "index": 42 }, { "bbox": [ 107, 82, 505, 95 ], "spans": [ { "bbox": [ 107, 85, 182, 94 ], "score": 0.86, "content": "s _ { 0 } , a _ { 0 } , s _ { 1 } , a _ { 1 } . . . , s _ { T }", "type": "inline_equation", "cross_page": true }, { "bbox": [ 182, 82, 395, 94 ], "score": 1.0, "content": "be the state action trajectory in a simulation, where", "type": "text", "cross_page": true }, { "bbox": [ 396, 82, 421, 95 ], "score": 0.93, "content": "n { \\left( { { s _ { T } } } \\right) }", "type": "inline_equation", "cross_page": true }, { "bbox": [ 421, 82, 492, 94 ], "score": 1.0, "content": "is a leaf node of", "type": "text", "cross_page": true }, { "bbox": [ 492, 83, 501, 93 ], "score": 0.79, "content": "\\tau", "type": "inline_equation", "cross_page": true }, { "bbox": [ 501, 82, 505, 94 ], "score": 1.0, "content": ".", "type": "text", "cross_page": true } ], "index": 0 }, { "bbox": [ 105, 92, 506, 106 ], "spans": [ { "bbox": [ 105, 92, 181, 106 ], "score": 1.0, "content": "Whenever a node", "type": "text", "cross_page": true }, { "bbox": [ 182, 94, 207, 105 ], "score": 0.91, "content": "n { \\left( { { s _ { T } } } \\right) }", "type": "inline_equation", "cross_page": true }, { "bbox": [ 208, 92, 506, 106 ], "score": 1.0, "content": "is expanded, the respective action values (Equation 6) are initialized as", "type": "text", "cross_page": true } ], "index": 1 }, { "bbox": [ 107, 104, 506, 117 ], "spans": [ { "bbox": [ 107, 104, 168, 116 ], "score": 0.91, "content": "Q _ { \\Omega } ( s _ { T } , a ) = 0", "type": "inline_equation", "cross_page": true }, { "bbox": [ 169, 104, 189, 117 ], "score": 1.0, "content": ", and", "type": "text", "cross_page": true }, { "bbox": [ 189, 105, 246, 116 ], "score": 0.89, "content": "\\dot { N } ( \\dot { s } _ { T } , a ) = 0", "type": "inline_equation", "cross_page": true }, { "bbox": [ 246, 104, 273, 117 ], "score": 1.0, "content": "for all", "type": "text", "cross_page": true }, { "bbox": [ 274, 105, 300, 115 ], "score": 0.89, "content": "a \\in { \\mathcal { A } }", "type": "inline_equation", "cross_page": true }, { "bbox": [ 300, 104, 506, 117 ], "score": 1.0, "content": ". For all nodes in the trajectory, the visitation count", "type": "text", "cross_page": true } ], "index": 2 }, { "bbox": [ 105, 115, 373, 128 ], "spans": [ { "bbox": [ 105, 115, 162, 128 ], "score": 1.0, "content": "is updated by", "type": "text", "cross_page": true }, { "bbox": [ 163, 116, 270, 127 ], "score": 0.9, "content": "N ( s _ { t } , a _ { t } ) = N ( s _ { t } , a _ { t } ) + 1", "type": "inline_equation", "cross_page": true }, { "bbox": [ 271, 115, 373, 128 ], "score": 1.0, "content": ", and the action-values by", "type": "text", "cross_page": true } ], "index": 3 } ], "index": 41.5, "bbox_fs": [ 106, 709, 505, 732 ] } ] }, { "preproc_blocks": [ { "type": "text", "bbox": [ 107, 82, 505, 127 ], "lines": [ { "bbox": [ 107, 82, 505, 95 ], "spans": [ { "bbox": [ 107, 85, 182, 94 ], "score": 0.86, "content": "s _ { 0 } , a _ { 0 } , s _ { 1 } , a _ { 1 } . . . , s _ { T }", "type": "inline_equation" }, { "bbox": [ 182, 82, 395, 94 ], "score": 1.0, "content": "be the state action trajectory in a simulation, where", "type": "text" }, { "bbox": [ 396, 82, 421, 95 ], "score": 0.93, "content": "n { \\left( { { s _ { T } } } \\right) }", "type": "inline_equation" }, { "bbox": [ 421, 82, 492, 94 ], "score": 1.0, "content": "is a leaf node of", "type": "text" }, { "bbox": [ 492, 83, 501, 93 ], "score": 0.79, "content": "\\tau", "type": "inline_equation" }, { "bbox": [ 501, 82, 505, 94 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 0 }, { "bbox": [ 105, 92, 506, 106 ], "spans": [ { "bbox": [ 105, 92, 181, 106 ], "score": 1.0, "content": "Whenever a node", "type": "text" }, { "bbox": [ 182, 94, 207, 105 ], "score": 0.91, "content": "n { \\left( { { s _ { T } } } \\right) }", "type": "inline_equation" }, { "bbox": [ 208, 92, 506, 106 ], "score": 1.0, "content": "is expanded, the respective action values (Equation 6) are initialized as", "type": "text" } ], "index": 1 }, { "bbox": [ 107, 104, 506, 117 ], "spans": [ { "bbox": [ 107, 104, 168, 116 ], "score": 0.91, "content": "Q _ { \\Omega } ( s _ { T } , a ) = 0", "type": "inline_equation" }, { "bbox": [ 169, 104, 189, 117 ], "score": 1.0, "content": ", and", "type": "text" }, { "bbox": [ 189, 105, 246, 116 ], "score": 0.89, "content": "\\dot { N } ( \\dot { s } _ { T } , a ) = 0", "type": "inline_equation" }, { "bbox": [ 246, 104, 273, 117 ], "score": 1.0, "content": "for all", "type": "text" }, { "bbox": [ 274, 105, 300, 115 ], "score": 0.89, "content": "a \\in { \\mathcal { A } }", "type": "inline_equation" }, { "bbox": [ 300, 104, 506, 117 ], "score": 1.0, "content": ". For all nodes in the trajectory, the visitation count", "type": "text" } ], "index": 2 }, { "bbox": [ 105, 115, 373, 128 ], "spans": [ { "bbox": [ 105, 115, 162, 128 ], "score": 1.0, "content": "is updated by", "type": "text" }, { "bbox": [ 163, 116, 270, 127 ], "score": 0.9, "content": "N ( s _ { t } , a _ { t } ) = N ( s _ { t } , a _ { t } ) + 1", "type": "inline_equation" }, { "bbox": [ 271, 115, 373, 128 ], "score": 1.0, "content": ", and the action-values by", "type": "text" } ], "index": 3 } ], "index": 1.5 }, { "type": "interline_equation", "bbox": [ 187, 134, 426, 163 ], "lines": [ { "bbox": [ 187, 134, 426, 163 ], "spans": [ { "bbox": [ 187, 134, 426, 163 ], "score": 0.91, "content": "Q _ { \\Omega } ( s _ { t } , a _ { t } ) = \\left\\{ \\begin{array} { l l } { r ( s _ { t } , a _ { t } ) + \\gamma \\rho } & { \\mathrm { i f ~ } t = T } \\\\ { r ( s _ { t } , a _ { t } ) + \\gamma \\Omega ^ { * } ( Q _ { \\Omega } ( s _ { t + 1 } ) / \\tau ) ) } & { \\mathrm { i f ~ } t < T } \\end{array} \\right.", "type": "interline_equation", "image_path": "1cbd7c1233c65cafaeab9cbbffdaf8a4f31c4ba57480c841757e7f90910ba381.jpg" } ] } ], "index": 4.5, "virtual_lines": [ { "bbox": [ 187, 134, 426, 148.5 ], "spans": [], "index": 4 }, { "bbox": [ 187, 148.5, 426, 163.0 ], "spans": [], "index": 5 } ] }, { "type": "text", "bbox": [ 107, 169, 505, 249 ], "lines": [ { "bbox": [ 105, 169, 506, 184 ], "spans": [ { "bbox": [ 105, 169, 133, 184 ], "score": 1.0, "content": "where", "type": "text" }, { "bbox": [ 134, 170, 201, 183 ], "score": 0.93, "content": "Q _ { \\Omega } ( s _ { t + 1 } ) \\in \\mathbb { R } ^ { A }", "type": "inline_equation" }, { "bbox": [ 202, 169, 274, 184 ], "score": 1.0, "content": "with components", "type": "text" }, { "bbox": [ 275, 171, 361, 183 ], "score": 0.92, "content": "Q _ { \\Omega } ( s _ { t + 1 } , a ) , \\forall a \\in \\mathcal { A }", "type": "inline_equation" }, { "bbox": [ 361, 169, 382, 184 ], "score": 1.0, "content": ", and", "type": "text" }, { "bbox": [ 382, 173, 389, 182 ], "score": 0.82, "content": "\\rho", "type": "inline_equation" }, { "bbox": [ 389, 169, 506, 184 ], "score": 1.0, "content": "is an estimate returned from", "type": "text" } ], "index": 6 }, { "bbox": [ 105, 181, 505, 194 ], "spans": [ { "bbox": [ 105, 181, 249, 194 ], "score": 1.0, "content": "an evaluation function computed in", "type": "text" }, { "bbox": [ 250, 183, 261, 193 ], "score": 0.84, "content": "s _ { T }", "type": "inline_equation" }, { "bbox": [ 262, 181, 505, 194 ], "score": 1.0, "content": ", e.g. a discounted cumulative reward averaged over multiple", "type": "text" } ], "index": 7 }, { "bbox": [ 105, 191, 507, 207 ], "spans": [ { "bbox": [ 105, 191, 265, 207 ], "score": 1.0, "content": "rollouts, or the value-function of node", "type": "text" }, { "bbox": [ 265, 193, 300, 205 ], "score": 0.92, "content": "n ( s _ { T + 1 } )", "type": "inline_equation" }, { "bbox": [ 301, 191, 507, 207 ], "score": 1.0, "content": "returned by a value-function approximator, e.g. a", "type": "text" } ], "index": 8 }, { "bbox": [ 106, 204, 505, 216 ], "spans": [ { "bbox": [ 106, 204, 254, 216 ], "score": 1.0, "content": "neural network pretrained with deep", "type": "text" }, { "bbox": [ 254, 204, 263, 215 ], "score": 0.86, "content": "Q", "type": "inline_equation" }, { "bbox": [ 264, 204, 505, 216 ], "score": 1.0, "content": "-learning (Mnih et al., 2015), as done in (Silver et al., 2016;", "type": "text" } ], "index": 9 }, { "bbox": [ 105, 214, 505, 227 ], "spans": [ { "bbox": [ 105, 214, 505, 227 ], "score": 1.0, "content": "Xiao et al., 2019). We revisit the E2W sampling strategy limited to maximum entropy regulariza-", "type": "text" } ], "index": 10 }, { "bbox": [ 105, 225, 506, 238 ], "spans": [ { "bbox": [ 105, 225, 506, 238 ], "score": 1.0, "content": "tion (Xiao et al., 2019) and, through the use of the convex conjugate in Equation (6), we derive a", "type": "text" } ], "index": 11 }, { "bbox": [ 106, 237, 371, 249 ], "spans": [ { "bbox": [ 106, 237, 371, 249 ], "score": 1.0, "content": "novel sampling strategy that generalizes to any convex regularizer", "type": "text" } ], "index": 12 } ], "index": 9 }, { "type": "interline_equation", "bbox": [ 202, 254, 408, 281 ], "lines": [ { "bbox": [ 202, 254, 408, 281 ], "spans": [ { "bbox": [ 202, 254, 408, 281 ], "score": 0.91, "content": "\\pi _ { t } ( a _ { t } | s _ { t } ) = ( 1 - \\lambda _ { s _ { t } } ) \\nabla \\Omega ^ { \\ast } ( Q _ { \\Omega } ( s _ { t } ) / \\tau ) ( a _ { t } ) + \\frac { \\lambda _ { s _ { t } } } { | \\mathcal { A } | } ,", "type": "interline_equation", "image_path": "78b532c26e70504ac790be8609e82938f1f4acbee16249ab81986e08036542ce.jpg" } ] } ], "index": 13.5, "virtual_lines": [ { "bbox": [ 202, 254, 408, 267.5 ], "spans": [], "index": 13 }, { "bbox": [ 202, 267.5, 408, 281.0 ], "spans": [], "index": 14 } ] }, { "type": "text", "bbox": [ 107, 287, 505, 332 ], "lines": [ { "bbox": [ 105, 286, 505, 300 ], "spans": [ { "bbox": [ 105, 286, 133, 300 ], "score": 1.0, "content": "where", "type": "text" }, { "bbox": [ 134, 288, 249, 299 ], "score": 0.92, "content": "\\lambda _ { s _ { t } } = \\epsilon | \\boldsymbol { \\mathcal { A } } | \\big / \\mathrm { l o g } ( \\sum _ { a } N ( s _ { t } , a ) + 1 )", "type": "inline_equation" }, { "bbox": [ 249, 286, 271, 300 ], "score": 1.0, "content": "with", "type": "text" }, { "bbox": [ 272, 288, 298, 298 ], "score": 0.91, "content": "\\epsilon > 0", "type": "inline_equation" }, { "bbox": [ 298, 286, 433, 300 ], "score": 1.0, "content": "as an exploration parameter, and", "type": "text" }, { "bbox": [ 434, 288, 455, 298 ], "score": 0.89, "content": "\\nabla \\Omega ^ { * }", "type": "inline_equation" }, { "bbox": [ 455, 286, 505, 300 ], "score": 1.0, "content": "depends on", "type": "text" } ], "index": 15 }, { "bbox": [ 105, 297, 505, 311 ], "spans": [ { "bbox": [ 105, 297, 505, 311 ], "score": 1.0, "content": "the measure in use (see Table 1 for maximum, relative, and Tsallis entropy). We call this sampling", "type": "text" } ], "index": 16 }, { "bbox": [ 105, 309, 505, 321 ], "spans": [ { "bbox": [ 105, 309, 505, 321 ], "score": 1.0, "content": "strategy Extended Empirical Exponential Weight (E3W) to highlight the extension of E2W from", "type": "text" } ], "index": 17 }, { "bbox": [ 105, 321, 309, 333 ], "spans": [ { "bbox": [ 105, 321, 309, 333 ], "score": 1.0, "content": "maximum entropy to a generic convex regularizer.", "type": "text" } ], "index": 18 } ], "index": 16.5 }, { "type": "title", "bbox": [ 107, 347, 350, 358 ], "lines": [ { "bbox": [ 105, 346, 351, 359 ], "spans": [ { "bbox": [ 105, 346, 351, 359 ], "score": 1.0, "content": "3.3 CONVERGENCE RATE TO REGULARIZED OBJECTIVE", "type": "text" } ], "index": 19 } ], "index": 19 }, { "type": "text", "bbox": [ 107, 367, 505, 401 ], "lines": [ { "bbox": [ 106, 366, 505, 380 ], "spans": [ { "bbox": [ 106, 366, 253, 380 ], "score": 1.0, "content": "We show that the regularized value", "type": "text" }, { "bbox": [ 253, 367, 267, 378 ], "score": 0.88, "content": "V _ { \\Omega }", "type": "inline_equation" }, { "bbox": [ 267, 366, 452, 380 ], "score": 1.0, "content": "can be effectively estimated at the root state", "type": "text" }, { "bbox": [ 452, 368, 480, 378 ], "score": 0.9, "content": "s \\in { \\mathcal { S } }", "type": "inline_equation" }, { "bbox": [ 480, 366, 505, 380 ], "score": 1.0, "content": ", with", "type": "text" } ], "index": 20 }, { "bbox": [ 106, 378, 505, 390 ], "spans": [ { "bbox": [ 106, 378, 295, 390 ], "score": 1.0, "content": "the assumption that each node in the tree has a", "type": "text" }, { "bbox": [ 296, 378, 307, 388 ], "score": 0.88, "content": "\\sigma ^ { 2 }", "type": "inline_equation" }, { "bbox": [ 308, 378, 505, 390 ], "score": 1.0, "content": "-subgaussian distribution. This result extends the", "type": "text" } ], "index": 21 }, { "bbox": [ 105, 389, 463, 403 ], "spans": [ { "bbox": [ 105, 389, 463, 403 ], "score": 1.0, "content": "analysis provided in (Xiao et al., 2019), which is limited to the use of maximum entropy.", "type": "text" } ], "index": 22 } ], "index": 21 }, { "type": "text", "bbox": [ 105, 410, 504, 435 ], "lines": [ { "bbox": [ 106, 410, 505, 423 ], "spans": [ { "bbox": [ 106, 410, 261, 423 ], "score": 1.0, "content": "Theorem 1 At the root node s where", "type": "text" }, { "bbox": [ 262, 411, 284, 423 ], "score": 0.9, "content": "N ( s )", "type": "inline_equation" }, { "bbox": [ 285, 410, 422, 423 ], "score": 1.0, "content": "is the number of visitations, with", "type": "text" }, { "bbox": [ 423, 411, 448, 421 ], "score": 0.85, "content": "\\epsilon > 0", "type": "inline_equation" }, { "bbox": [ 448, 410, 452, 423 ], "score": 1.0, "content": ",", "type": "text" }, { "bbox": [ 453, 411, 479, 423 ], "score": 0.88, "content": "V _ { \\Omega } ( s )", "type": "inline_equation" }, { "bbox": [ 479, 410, 505, 423 ], "score": 1.0, "content": "is the", "type": "text" } ], "index": 23 }, { "bbox": [ 106, 422, 305, 435 ], "spans": [ { "bbox": [ 106, 422, 230, 435 ], "score": 1.0, "content": "estimated value, with constant", "type": "text" }, { "bbox": [ 230, 424, 239, 433 ], "score": 0.8, "content": "C", "type": "inline_equation" }, { "bbox": [ 239, 422, 257, 435 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 258, 422, 266, 433 ], "score": 0.83, "content": "\\hat { C }", "type": "inline_equation" }, { "bbox": [ 266, 422, 305, 435 ], "score": 1.0, "content": ", we have", "type": "text" } ], "index": 24 } ], "index": 23.5 }, { "type": "interline_equation", "bbox": [ 181, 440, 429, 469 ], "lines": [ { "bbox": [ 181, 440, 429, 469 ], "spans": [ { "bbox": [ 181, 440, 429, 469 ], "score": 0.92, "content": "\\mathbb { P } ( | V _ { \\Omega } ( s ) - V _ { \\Omega } ^ { * } ( s ) | > \\epsilon ) \\le C \\exp \\{ - \\frac { N ( s ) \\epsilon } { \\hat { C } \\sigma ( \\log ( 2 + N ( s ) ) ) ^ { 2 } } \\} ,", "type": "interline_equation", "image_path": "ee73133dae2b3344936d0e895b0a982eb9ba57303ec807e16ea59258dfc7e4d4.jpg" } ] } ], "index": 26, "virtual_lines": [ { "bbox": [ 181, 440, 429, 449.6666666666667 ], "spans": [], "index": 25 }, { "bbox": [ 181, 449.6666666666667, 429, 459.33333333333337 ], "spans": [], "index": 26 }, { "bbox": [ 181, 459.33333333333337, 429, 469.00000000000006 ], "spans": [], "index": 27 } ] }, { "type": "text", "bbox": [ 106, 480, 504, 504 ], "lines": [ { "bbox": [ 106, 480, 506, 495 ], "spans": [ { "bbox": [ 106, 480, 133, 495 ], "score": 1.0, "content": "where", "type": "text" }, { "bbox": [ 134, 481, 204, 493 ], "score": 0.93, "content": "V _ { \\Omega } ( s ) = \\Omega ^ { * } ( Q _ { s } )", "type": "inline_equation" }, { "bbox": [ 205, 480, 223, 495 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 223, 481, 295, 493 ], "score": 0.93, "content": "V _ { \\Omega } ^ { * } ( s ) = \\Omega ^ { * } ( Q _ { s } ^ { * } )", "type": "inline_equation" }, { "bbox": [ 295, 480, 506, 495 ], "score": 1.0, "content": ". From this theorem, we obtain that the convergence", "type": "text" } ], "index": 28 }, { "bbox": [ 106, 492, 494, 505 ], "spans": [ { "bbox": [ 106, 492, 233, 505 ], "score": 1.0, "content": "rate of choosing the best action", "type": "text" }, { "bbox": [ 234, 494, 244, 502 ], "score": 0.86, "content": "a ^ { * }", "type": "inline_equation" }, { "bbox": [ 244, 492, 494, 505 ], "score": 1.0, "content": "at the root node, when using the E3W strategy, is exponential.", "type": "text" } ], "index": 29 } ], "index": 28.5 }, { "type": "text", "bbox": [ 107, 514, 501, 527 ], "lines": [ { "bbox": [ 105, 514, 502, 529 ], "spans": [ { "bbox": [ 105, 514, 172, 529 ], "score": 1.0, "content": "Theorem 2 Let", "type": "text" }, { "bbox": [ 173, 517, 182, 527 ], "score": 0.84, "content": "a _ { t }", "type": "inline_equation" }, { "bbox": [ 183, 514, 481, 529 ], "score": 1.0, "content": "be the action returned by E3W at step t. For large enough t and constants", "type": "text" }, { "bbox": [ 482, 514, 502, 527 ], "score": 0.88, "content": "C , { \\hat { C } }", "type": "inline_equation" } ], "index": 30 } ], "index": 30 }, { "type": "interline_equation", "bbox": [ 223, 533, 387, 560 ], "lines": [ { "bbox": [ 223, 533, 387, 560 ], "spans": [ { "bbox": [ 223, 533, 387, 560 ], "score": 0.94, "content": "\\mathbb { P } ( a _ { t } \\neq a ^ { * } ) \\leq C t \\exp \\{ - \\frac { t } { \\hat { C } \\sigma ( \\log ( t ) ) ^ { 3 } } \\} .", "type": "interline_equation", "image_path": "3ed970d53d860a4083973252de01a2ee3d2547d2f0af0e393dcd634d39e17148.jpg" } ] } ], "index": 31.5, "virtual_lines": [ { "bbox": [ 223, 533, 387, 546.5 ], "spans": [], "index": 31 }, { "bbox": [ 223, 546.5, 387, 560.0 ], "spans": [], "index": 32 } ] }, { "type": "title", "bbox": [ 106, 574, 378, 587 ], "lines": [ { "bbox": [ 105, 574, 379, 589 ], "spans": [ { "bbox": [ 105, 574, 379, 589 ], "score": 1.0, "content": "4 ENTROPY-REGULARIZATION BACKUP OPERATORS", "type": "text" } ], "index": 33 } ], "index": 33 }, { "type": "text", "bbox": [ 106, 599, 505, 732 ], "lines": [ { "bbox": [ 105, 600, 505, 613 ], "spans": [ { "bbox": [ 105, 600, 505, 613 ], "score": 1.0, "content": "From the introduction of a unified view of generic strongly convex regularizers as backup operators", "type": "text" } ], "index": 34 }, { "bbox": [ 105, 610, 506, 624 ], "spans": [ { "bbox": [ 105, 610, 506, 624 ], "score": 1.0, "content": "in MCTS, we narrow the analysis to entropy-based regularizers. For each entropy function, Table 1", "type": "text" } ], "index": 35 }, { "bbox": [ 105, 622, 505, 634 ], "spans": [ { "bbox": [ 105, 622, 505, 634 ], "score": 1.0, "content": "shows the Legendre-Fenchel transform and the maximizing argument, which can be respectively", "type": "text" } ], "index": 36 }, { "bbox": [ 104, 632, 506, 648 ], "spans": [ { "bbox": [ 104, 632, 506, 648 ], "score": 1.0, "content": "replaced in our backup operation (Equation 6) and sampling strategy E3W (Equation 7). Using", "type": "text" } ], "index": 37 }, { "bbox": [ 105, 643, 505, 657 ], "spans": [ { "bbox": [ 105, 643, 505, 657 ], "score": 1.0, "content": "maximum entropy retrieves the maximum entropy MCTS problem introduced in the MENTS algo-", "type": "text" } ], "index": 38 }, { "bbox": [ 105, 654, 506, 668 ], "spans": [ { "bbox": [ 105, 654, 506, 668 ], "score": 1.0, "content": "rithm (Xiao et al., 2019). This approach closely resembles the maximum entropy RL framework", "type": "text" } ], "index": 39 }, { "bbox": [ 106, 666, 505, 678 ], "spans": [ { "bbox": [ 106, 666, 505, 678 ], "score": 1.0, "content": "used to encourage exploration (Haarnoja et al., 2018; Schulman et al., 2017a). We introduce two", "type": "text" } ], "index": 40 }, { "bbox": [ 105, 677, 506, 689 ], "spans": [ { "bbox": [ 105, 677, 506, 689 ], "score": 1.0, "content": "novel MCTS algorithms based on the minimization of relative entropy of the policy update, inspired", "type": "text" } ], "index": 41 }, { "bbox": [ 105, 687, 505, 700 ], "spans": [ { "bbox": [ 105, 687, 505, 700 ], "score": 1.0, "content": "by trust-region (Schulman et al., 2015) and proximal optimization methods (Schulman et al., 2017b)", "type": "text" } ], "index": 42 }, { "bbox": [ 105, 699, 505, 711 ], "spans": [ { "bbox": [ 105, 699, 505, 711 ], "score": 1.0, "content": "in RL, and on the maximization of Tsallis entropy, which has been more recently introduced in RL", "type": "text" } ], "index": 43 }, { "bbox": [ 105, 709, 506, 723 ], "spans": [ { "bbox": [ 105, 709, 506, 723 ], "score": 1.0, "content": "as an effective solution to enforce the learning of sparse policies (Lee et al., 2018). We call these", "type": "text" } ], "index": 44 }, { "bbox": [ 106, 721, 505, 733 ], "spans": [ { "bbox": [ 106, 721, 505, 733 ], "score": 1.0, "content": "algorithms RENTS and TENTS. Contrary to maximum and relative entropy, the definition of the", "type": "text" } ], "index": 45 } ], "index": 39.5 } ], "page_idx": 3, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 302, 751, 308, 759 ], "lines": [ { "bbox": [ 302, 750, 310, 762 ], "spans": [ { "bbox": [ 302, 750, 310, 762 ], "score": 1.0, "content": "", "type": "text", "height": 12, "width": 8 } ] } ] }, { "type": "discarded", "bbox": [ 106, 26, 307, 38 ], "lines": [ { "bbox": [ 106, 25, 308, 39 ], "spans": [ { "bbox": [ 106, 25, 308, 39 ], "score": 1.0, "content": "Under review as a conference paper at ICLR 2021", "type": "text" } ] } ] } ], "para_blocks": [ { "type": "text", "bbox": [ 107, 82, 505, 127 ], "lines": [], "index": 1.5, "bbox_fs": [ 105, 82, 506, 128 ], "lines_deleted": true }, { "type": "interline_equation", "bbox": [ 187, 134, 426, 163 ], "lines": [ { "bbox": [ 187, 134, 426, 163 ], "spans": [ { "bbox": [ 187, 134, 426, 163 ], "score": 0.91, "content": "Q _ { \\Omega } ( s _ { t } , a _ { t } ) = \\left\\{ \\begin{array} { l l } { r ( s _ { t } , a _ { t } ) + \\gamma \\rho } & { \\mathrm { i f ~ } t = T } \\\\ { r ( s _ { t } , a _ { t } ) + \\gamma \\Omega ^ { * } ( Q _ { \\Omega } ( s _ { t + 1 } ) / \\tau ) ) } & { \\mathrm { i f ~ } t < T } \\end{array} \\right.", "type": "interline_equation", "image_path": "1cbd7c1233c65cafaeab9cbbffdaf8a4f31c4ba57480c841757e7f90910ba381.jpg" } ] } ], "index": 4.5, "virtual_lines": [ { "bbox": [ 187, 134, 426, 148.5 ], "spans": [], "index": 4 }, { "bbox": [ 187, 148.5, 426, 163.0 ], "spans": [], "index": 5 } ] }, { "type": "text", "bbox": [ 107, 169, 505, 249 ], "lines": [ { "bbox": [ 105, 169, 506, 184 ], "spans": [ { "bbox": [ 105, 169, 133, 184 ], "score": 1.0, "content": "where", "type": "text" }, { "bbox": [ 134, 170, 201, 183 ], "score": 0.93, "content": "Q _ { \\Omega } ( s _ { t + 1 } ) \\in \\mathbb { R } ^ { A }", "type": "inline_equation" }, { "bbox": [ 202, 169, 274, 184 ], "score": 1.0, "content": "with components", "type": "text" }, { "bbox": [ 275, 171, 361, 183 ], "score": 0.92, "content": "Q _ { \\Omega } ( s _ { t + 1 } , a ) , \\forall a \\in \\mathcal { A }", "type": "inline_equation" }, { "bbox": [ 361, 169, 382, 184 ], "score": 1.0, "content": ", and", "type": "text" }, { "bbox": [ 382, 173, 389, 182 ], "score": 0.82, "content": "\\rho", "type": "inline_equation" }, { "bbox": [ 389, 169, 506, 184 ], "score": 1.0, "content": "is an estimate returned from", "type": "text" } ], "index": 6 }, { "bbox": [ 105, 181, 505, 194 ], "spans": [ { "bbox": [ 105, 181, 249, 194 ], "score": 1.0, "content": "an evaluation function computed in", "type": "text" }, { "bbox": [ 250, 183, 261, 193 ], "score": 0.84, "content": "s _ { T }", "type": "inline_equation" }, { "bbox": [ 262, 181, 505, 194 ], "score": 1.0, "content": ", e.g. a discounted cumulative reward averaged over multiple", "type": "text" } ], "index": 7 }, { "bbox": [ 105, 191, 507, 207 ], "spans": [ { "bbox": [ 105, 191, 265, 207 ], "score": 1.0, "content": "rollouts, or the value-function of node", "type": "text" }, { "bbox": [ 265, 193, 300, 205 ], "score": 0.92, "content": "n ( s _ { T + 1 } )", "type": "inline_equation" }, { "bbox": [ 301, 191, 507, 207 ], "score": 1.0, "content": "returned by a value-function approximator, e.g. a", "type": "text" } ], "index": 8 }, { "bbox": [ 106, 204, 505, 216 ], "spans": [ { "bbox": [ 106, 204, 254, 216 ], "score": 1.0, "content": "neural network pretrained with deep", "type": "text" }, { "bbox": [ 254, 204, 263, 215 ], "score": 0.86, "content": "Q", "type": "inline_equation" }, { "bbox": [ 264, 204, 505, 216 ], "score": 1.0, "content": "-learning (Mnih et al., 2015), as done in (Silver et al., 2016;", "type": "text" } ], "index": 9 }, { "bbox": [ 105, 214, 505, 227 ], "spans": [ { "bbox": [ 105, 214, 505, 227 ], "score": 1.0, "content": "Xiao et al., 2019). We revisit the E2W sampling strategy limited to maximum entropy regulariza-", "type": "text" } ], "index": 10 }, { "bbox": [ 105, 225, 506, 238 ], "spans": [ { "bbox": [ 105, 225, 506, 238 ], "score": 1.0, "content": "tion (Xiao et al., 2019) and, through the use of the convex conjugate in Equation (6), we derive a", "type": "text" } ], "index": 11 }, { "bbox": [ 106, 237, 371, 249 ], "spans": [ { "bbox": [ 106, 237, 371, 249 ], "score": 1.0, "content": "novel sampling strategy that generalizes to any convex regularizer", "type": "text" } ], "index": 12 } ], "index": 9, "bbox_fs": [ 105, 169, 507, 249 ] }, { "type": "interline_equation", "bbox": [ 202, 254, 408, 281 ], "lines": [ { "bbox": [ 202, 254, 408, 281 ], "spans": [ { "bbox": [ 202, 254, 408, 281 ], "score": 0.91, "content": "\\pi _ { t } ( a _ { t } | s _ { t } ) = ( 1 - \\lambda _ { s _ { t } } ) \\nabla \\Omega ^ { \\ast } ( Q _ { \\Omega } ( s _ { t } ) / \\tau ) ( a _ { t } ) + \\frac { \\lambda _ { s _ { t } } } { | \\mathcal { A } | } ,", "type": "interline_equation", "image_path": "78b532c26e70504ac790be8609e82938f1f4acbee16249ab81986e08036542ce.jpg" } ] } ], "index": 13.5, "virtual_lines": [ { "bbox": [ 202, 254, 408, 267.5 ], "spans": [], "index": 13 }, { "bbox": [ 202, 267.5, 408, 281.0 ], "spans": [], "index": 14 } ] }, { "type": "text", "bbox": [ 107, 287, 505, 332 ], "lines": [ { "bbox": [ 105, 286, 505, 300 ], "spans": [ { "bbox": [ 105, 286, 133, 300 ], "score": 1.0, "content": "where", "type": "text" }, { "bbox": [ 134, 288, 249, 299 ], "score": 0.92, "content": "\\lambda _ { s _ { t } } = \\epsilon | \\boldsymbol { \\mathcal { A } } | \\big / \\mathrm { l o g } ( \\sum _ { a } N ( s _ { t } , a ) + 1 )", "type": "inline_equation" }, { "bbox": [ 249, 286, 271, 300 ], "score": 1.0, "content": "with", "type": "text" }, { "bbox": [ 272, 288, 298, 298 ], "score": 0.91, "content": "\\epsilon > 0", "type": "inline_equation" }, { "bbox": [ 298, 286, 433, 300 ], "score": 1.0, "content": "as an exploration parameter, and", "type": "text" }, { "bbox": [ 434, 288, 455, 298 ], "score": 0.89, "content": "\\nabla \\Omega ^ { * }", "type": "inline_equation" }, { "bbox": [ 455, 286, 505, 300 ], "score": 1.0, "content": "depends on", "type": "text" } ], "index": 15 }, { "bbox": [ 105, 297, 505, 311 ], "spans": [ { "bbox": [ 105, 297, 505, 311 ], "score": 1.0, "content": "the measure in use (see Table 1 for maximum, relative, and Tsallis entropy). We call this sampling", "type": "text" } ], "index": 16 }, { "bbox": [ 105, 309, 505, 321 ], "spans": [ { "bbox": [ 105, 309, 505, 321 ], "score": 1.0, "content": "strategy Extended Empirical Exponential Weight (E3W) to highlight the extension of E2W from", "type": "text" } ], "index": 17 }, { "bbox": [ 105, 321, 309, 333 ], "spans": [ { "bbox": [ 105, 321, 309, 333 ], "score": 1.0, "content": "maximum entropy to a generic convex regularizer.", "type": "text" } ], "index": 18 } ], "index": 16.5, "bbox_fs": [ 105, 286, 505, 333 ] }, { "type": "title", "bbox": [ 107, 347, 350, 358 ], "lines": [ { "bbox": [ 105, 346, 351, 359 ], "spans": [ { "bbox": [ 105, 346, 351, 359 ], "score": 1.0, "content": "3.3 CONVERGENCE RATE TO REGULARIZED OBJECTIVE", "type": "text" } ], "index": 19 } ], "index": 19 }, { "type": "text", "bbox": [ 107, 367, 505, 401 ], "lines": [ { "bbox": [ 106, 366, 505, 380 ], "spans": [ { "bbox": [ 106, 366, 253, 380 ], "score": 1.0, "content": "We show that the regularized value", "type": "text" }, { "bbox": [ 253, 367, 267, 378 ], "score": 0.88, "content": "V _ { \\Omega }", "type": "inline_equation" }, { "bbox": [ 267, 366, 452, 380 ], "score": 1.0, "content": "can be effectively estimated at the root state", "type": "text" }, { "bbox": [ 452, 368, 480, 378 ], "score": 0.9, "content": "s \\in { \\mathcal { S } }", "type": "inline_equation" }, { "bbox": [ 480, 366, 505, 380 ], "score": 1.0, "content": ", with", "type": "text" } ], "index": 20 }, { "bbox": [ 106, 378, 505, 390 ], "spans": [ { "bbox": [ 106, 378, 295, 390 ], "score": 1.0, "content": "the assumption that each node in the tree has a", "type": "text" }, { "bbox": [ 296, 378, 307, 388 ], "score": 0.88, "content": "\\sigma ^ { 2 }", "type": "inline_equation" }, { "bbox": [ 308, 378, 505, 390 ], "score": 1.0, "content": "-subgaussian distribution. This result extends the", "type": "text" } ], "index": 21 }, { "bbox": [ 105, 389, 463, 403 ], "spans": [ { "bbox": [ 105, 389, 463, 403 ], "score": 1.0, "content": "analysis provided in (Xiao et al., 2019), which is limited to the use of maximum entropy.", "type": "text" } ], "index": 22 } ], "index": 21, "bbox_fs": [ 105, 366, 505, 403 ] }, { "type": "text", "bbox": [ 105, 410, 504, 435 ], "lines": [ { "bbox": [ 106, 410, 505, 423 ], "spans": [ { "bbox": [ 106, 410, 261, 423 ], "score": 1.0, "content": "Theorem 1 At the root node s where", "type": "text" }, { "bbox": [ 262, 411, 284, 423 ], "score": 0.9, "content": "N ( s )", "type": "inline_equation" }, { "bbox": [ 285, 410, 422, 423 ], "score": 1.0, "content": "is the number of visitations, with", "type": "text" }, { "bbox": [ 423, 411, 448, 421 ], "score": 0.85, "content": "\\epsilon > 0", "type": "inline_equation" }, { "bbox": [ 448, 410, 452, 423 ], "score": 1.0, "content": ",", "type": "text" }, { "bbox": [ 453, 411, 479, 423 ], "score": 0.88, "content": "V _ { \\Omega } ( s )", "type": "inline_equation" }, { "bbox": [ 479, 410, 505, 423 ], "score": 1.0, "content": "is the", "type": "text" } ], "index": 23 }, { "bbox": [ 106, 422, 305, 435 ], "spans": [ { "bbox": [ 106, 422, 230, 435 ], "score": 1.0, "content": "estimated value, with constant", "type": "text" }, { "bbox": [ 230, 424, 239, 433 ], "score": 0.8, "content": "C", "type": "inline_equation" }, { "bbox": [ 239, 422, 257, 435 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 258, 422, 266, 433 ], "score": 0.83, "content": "\\hat { C }", "type": "inline_equation" }, { "bbox": [ 266, 422, 305, 435 ], "score": 1.0, "content": ", we have", "type": "text" } ], "index": 24 } ], "index": 23.5, "bbox_fs": [ 106, 410, 505, 435 ] }, { "type": "interline_equation", "bbox": [ 181, 440, 429, 469 ], "lines": [ { "bbox": [ 181, 440, 429, 469 ], "spans": [ { "bbox": [ 181, 440, 429, 469 ], "score": 0.92, "content": "\\mathbb { P } ( | V _ { \\Omega } ( s ) - V _ { \\Omega } ^ { * } ( s ) | > \\epsilon ) \\le C \\exp \\{ - \\frac { N ( s ) \\epsilon } { \\hat { C } \\sigma ( \\log ( 2 + N ( s ) ) ) ^ { 2 } } \\} ,", "type": "interline_equation", "image_path": "ee73133dae2b3344936d0e895b0a982eb9ba57303ec807e16ea59258dfc7e4d4.jpg" } ] } ], "index": 26, "virtual_lines": [ { "bbox": [ 181, 440, 429, 449.6666666666667 ], "spans": [], "index": 25 }, { "bbox": [ 181, 449.6666666666667, 429, 459.33333333333337 ], "spans": [], "index": 26 }, { "bbox": [ 181, 459.33333333333337, 429, 469.00000000000006 ], "spans": [], "index": 27 } ] }, { "type": "text", "bbox": [ 106, 480, 504, 504 ], "lines": [ { "bbox": [ 106, 480, 506, 495 ], "spans": [ { "bbox": [ 106, 480, 133, 495 ], "score": 1.0, "content": "where", "type": "text" }, { "bbox": [ 134, 481, 204, 493 ], "score": 0.93, "content": "V _ { \\Omega } ( s ) = \\Omega ^ { * } ( Q _ { s } )", "type": "inline_equation" }, { "bbox": [ 205, 480, 223, 495 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 223, 481, 295, 493 ], "score": 0.93, "content": "V _ { \\Omega } ^ { * } ( s ) = \\Omega ^ { * } ( Q _ { s } ^ { * } )", "type": "inline_equation" }, { "bbox": [ 295, 480, 506, 495 ], "score": 1.0, "content": ". From this theorem, we obtain that the convergence", "type": "text" } ], "index": 28 }, { "bbox": [ 106, 492, 494, 505 ], "spans": [ { "bbox": [ 106, 492, 233, 505 ], "score": 1.0, "content": "rate of choosing the best action", "type": "text" }, { "bbox": [ 234, 494, 244, 502 ], "score": 0.86, "content": "a ^ { * }", "type": "inline_equation" }, { "bbox": [ 244, 492, 494, 505 ], "score": 1.0, "content": "at the root node, when using the E3W strategy, is exponential.", "type": "text" } ], "index": 29 } ], "index": 28.5, "bbox_fs": [ 106, 480, 506, 505 ] }, { "type": "text", "bbox": [ 107, 514, 501, 527 ], "lines": [ { "bbox": [ 105, 514, 502, 529 ], "spans": [ { "bbox": [ 105, 514, 172, 529 ], "score": 1.0, "content": "Theorem 2 Let", "type": "text" }, { "bbox": [ 173, 517, 182, 527 ], "score": 0.84, "content": "a _ { t }", "type": "inline_equation" }, { "bbox": [ 183, 514, 481, 529 ], "score": 1.0, "content": "be the action returned by E3W at step t. For large enough t and constants", "type": "text" }, { "bbox": [ 482, 514, 502, 527 ], "score": 0.88, "content": "C , { \\hat { C } }", "type": "inline_equation" } ], "index": 30 } ], "index": 30, "bbox_fs": [ 105, 514, 502, 529 ] }, { "type": "interline_equation", "bbox": [ 223, 533, 387, 560 ], "lines": [ { "bbox": [ 223, 533, 387, 560 ], "spans": [ { "bbox": [ 223, 533, 387, 560 ], "score": 0.94, "content": "\\mathbb { P } ( a _ { t } \\neq a ^ { * } ) \\leq C t \\exp \\{ - \\frac { t } { \\hat { C } \\sigma ( \\log ( t ) ) ^ { 3 } } \\} .", "type": "interline_equation", "image_path": "3ed970d53d860a4083973252de01a2ee3d2547d2f0af0e393dcd634d39e17148.jpg" } ] } ], "index": 31.5, "virtual_lines": [ { "bbox": [ 223, 533, 387, 546.5 ], "spans": [], "index": 31 }, { "bbox": [ 223, 546.5, 387, 560.0 ], "spans": [], "index": 32 } ] }, { "type": "title", "bbox": [ 106, 574, 378, 587 ], "lines": [ { "bbox": [ 105, 574, 379, 589 ], "spans": [ { "bbox": [ 105, 574, 379, 589 ], "score": 1.0, "content": "4 ENTROPY-REGULARIZATION BACKUP OPERATORS", "type": "text" } ], "index": 33 } ], "index": 33 }, { "type": "text", "bbox": [ 106, 599, 505, 732 ], "lines": [ { "bbox": [ 105, 600, 505, 613 ], "spans": [ { "bbox": [ 105, 600, 505, 613 ], "score": 1.0, "content": "From the introduction of a unified view of generic strongly convex regularizers as backup operators", "type": "text" } ], "index": 34 }, { "bbox": [ 105, 610, 506, 624 ], "spans": [ { "bbox": [ 105, 610, 506, 624 ], "score": 1.0, "content": "in MCTS, we narrow the analysis to entropy-based regularizers. For each entropy function, Table 1", "type": "text" } ], "index": 35 }, { "bbox": [ 105, 622, 505, 634 ], "spans": [ { "bbox": [ 105, 622, 505, 634 ], "score": 1.0, "content": "shows the Legendre-Fenchel transform and the maximizing argument, which can be respectively", "type": "text" } ], "index": 36 }, { "bbox": [ 104, 632, 506, 648 ], "spans": [ { "bbox": [ 104, 632, 506, 648 ], "score": 1.0, "content": "replaced in our backup operation (Equation 6) and sampling strategy E3W (Equation 7). Using", "type": "text" } ], "index": 37 }, { "bbox": [ 105, 643, 505, 657 ], "spans": [ { "bbox": [ 105, 643, 505, 657 ], "score": 1.0, "content": "maximum entropy retrieves the maximum entropy MCTS problem introduced in the MENTS algo-", "type": "text" } ], "index": 38 }, { "bbox": [ 105, 654, 506, 668 ], "spans": [ { "bbox": [ 105, 654, 506, 668 ], "score": 1.0, "content": "rithm (Xiao et al., 2019). This approach closely resembles the maximum entropy RL framework", "type": "text" } ], "index": 39 }, { "bbox": [ 106, 666, 505, 678 ], "spans": [ { "bbox": [ 106, 666, 505, 678 ], "score": 1.0, "content": "used to encourage exploration (Haarnoja et al., 2018; Schulman et al., 2017a). We introduce two", "type": "text" } ], "index": 40 }, { "bbox": [ 105, 677, 506, 689 ], "spans": [ { "bbox": [ 105, 677, 506, 689 ], "score": 1.0, "content": "novel MCTS algorithms based on the minimization of relative entropy of the policy update, inspired", "type": "text" } ], "index": 41 }, { "bbox": [ 105, 687, 505, 700 ], "spans": [ { "bbox": [ 105, 687, 505, 700 ], "score": 1.0, "content": "by trust-region (Schulman et al., 2015) and proximal optimization methods (Schulman et al., 2017b)", "type": "text" } ], "index": 42 }, { "bbox": [ 105, 699, 505, 711 ], "spans": [ { "bbox": [ 105, 699, 505, 711 ], "score": 1.0, "content": "in RL, and on the maximization of Tsallis entropy, which has been more recently introduced in RL", "type": "text" } ], "index": 43 }, { "bbox": [ 105, 709, 506, 723 ], "spans": [ { "bbox": [ 105, 709, 506, 723 ], "score": 1.0, "content": "as an effective solution to enforce the learning of sparse policies (Lee et al., 2018). We call these", "type": "text" } ], "index": 44 }, { "bbox": [ 106, 721, 505, 733 ], "spans": [ { "bbox": [ 106, 721, 505, 733 ], "score": 1.0, "content": "algorithms RENTS and TENTS. Contrary to maximum and relative entropy, the definition of the", "type": "text" } ], "index": 45 } ], "index": 39.5, "bbox_fs": [ 104, 600, 506, 733 ] } ] }, { "preproc_blocks": [ { "type": "text", "bbox": [ 106, 82, 442, 95 ], "lines": [ { "bbox": [ 104, 80, 443, 98 ], "spans": [ { "bbox": [ 104, 80, 443, 98 ], "score": 1.0, "content": "Legendre-Fenchel and maximizing argument of Tsallis entropy is non-trivial, being", "type": "text" } ], "index": 0 } ], "index": 0 }, { "type": "interline_equation", "bbox": [ 185, 95, 426, 144 ], "lines": [ { "bbox": [ 185, 95, 426, 144 ], "spans": [ { "bbox": [ 185, 95, 426, 144 ], "score": 0.81, "content": "\\begin{array} { r l } & { ~ \\Omega ^ { * } ( Q _ { t } ) = \\tau \\cdot \\mathrm { s p m a x } ( Q _ { t } ( s , \\cdot ) / \\tau ) , } \\\\ & { ~ \\nabla \\Omega ^ { * } ( Q _ { t } ) = \\operatorname* { m a x } \\Bigg ( \\displaystyle \\frac { Q _ { t } ( s , a ) } { \\tau } - \\displaystyle \\frac { \\sum _ { a \\in { \\mathcal { K } } } Q _ { t } ( s , a ) / \\tau - 1 } { | { \\mathcal { K } } | } , 0 \\Bigg ) , } \\end{array}", "type": "interline_equation", "image_path": "c2bb83cafb01aadf64b3d47489e3e858bcae3f15f4903e114f3e981f0e50699b.jpg" } ] } ], "index": 2, "virtual_lines": [ { "bbox": [ 185, 95, 426, 111.33333333333333 ], "spans": [], "index": 1 }, { "bbox": [ 185, 111.33333333333333, 426, 127.66666666666666 ], "spans": [], "index": 2 }, { "bbox": [ 185, 127.66666666666666, 426, 144.0 ], "spans": [], "index": 3 } ] }, { "type": "text", "bbox": [ 107, 145, 347, 157 ], "lines": [ { "bbox": [ 106, 144, 347, 158 ], "spans": [ { "bbox": [ 106, 144, 269, 158 ], "score": 1.0, "content": "where spmax is defined for any function", "type": "text" }, { "bbox": [ 270, 146, 334, 157 ], "score": 0.9, "content": "f : \\mathcal { S } \\times \\mathcal { A } \\mathbb { R }", "type": "inline_equation" }, { "bbox": [ 335, 144, 347, 158 ], "score": 1.0, "content": "as", "type": "text" } ], "index": 4 } ], "index": 4 }, { "type": "interline_equation", "bbox": [ 172, 158, 438, 192 ], "lines": [ { "bbox": [ 172, 158, 438, 192 ], "spans": [ { "bbox": [ 172, 158, 438, 192 ], "score": 0.93, "content": "\\operatorname { s p m a x } ( f ( s , \\cdot ) ) \\triangleq \\sum _ { a \\in { \\mathcal { K } } } \\left( { \\frac { f ( s , a ) ^ { 2 } } { 2 } } - { \\frac { ( \\sum _ { a \\in { \\mathcal { K } } } f ( s , a ) - 1 ) ^ { 2 } } { 2 | K | ^ { 2 } } } \\right) + { \\frac { 1 } { 2 } } ,", "type": "interline_equation", "image_path": "1010175a551d5603043280b2589ec1948d19cb81c5e03e721432b08ede550e6c.jpg" } ] } ], "index": 6, "virtual_lines": [ { "bbox": [ 172, 158, 438, 169.33333333333334 ], "spans": [], "index": 5 }, { "bbox": [ 172, 169.33333333333334, 438, 180.66666666666669 ], "spans": [], "index": 6 }, { "bbox": [ 172, 180.66666666666669, 438, 192.00000000000003 ], "spans": [], "index": 7 } ] }, { "type": "text", "bbox": [ 107, 195, 504, 222 ], "lines": [ { "bbox": [ 103, 191, 508, 214 ], "spans": [ { "bbox": [ 103, 191, 123, 214 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 123, 197, 133, 207 ], "score": 0.82, "content": "\\kappa", "type": "inline_equation" }, { "bbox": [ 133, 191, 259, 214 ], "score": 1.0, "content": "is the set of actions that satisfy", "type": "text" }, { "bbox": [ 260, 195, 385, 211 ], "score": 0.94, "content": "\\begin{array} { r } { 1 + i f ( s , a _ { i } ) > \\sum _ { j = 1 } ^ { i } f ( s , a _ { j } ) } \\end{array}", "type": "inline_equation" }, { "bbox": [ 386, 191, 409, 214 ], "score": 1.0, "content": ", with", "type": "text" }, { "bbox": [ 410, 199, 419, 208 ], "score": 0.84, "content": "a _ { i }", "type": "inline_equation" }, { "bbox": [ 420, 191, 508, 214 ], "score": 1.0, "content": "indicating the action", "type": "text" } ], "index": 8 }, { "bbox": [ 106, 209, 325, 222 ], "spans": [ { "bbox": [ 106, 209, 141, 222 ], "score": 1.0, "content": "with the", "type": "text" }, { "bbox": [ 141, 210, 146, 219 ], "score": 0.73, "content": "i", "type": "inline_equation" }, { "bbox": [ 146, 209, 222, 222 ], "score": 1.0, "content": "-th largest value of", "type": "text" }, { "bbox": [ 222, 209, 251, 222 ], "score": 0.93, "content": "f ( s , a )", "type": "inline_equation" }, { "bbox": [ 251, 209, 325, 222 ], "score": 1.0, "content": "(Lee et al., 2018).", "type": "text" } ], "index": 9 } ], "index": 8.5 }, { "type": "table", "bbox": [ 117, 252, 495, 363 ], "blocks": [ { "type": "table_caption", "bbox": [ 106, 239, 503, 251 ], "group_id": 0, "lines": [ { "bbox": [ 106, 237, 505, 253 ], "spans": [ { "bbox": [ 106, 237, 505, 253 ], "score": 1.0, "content": "Table 1: List of entropy regularizers with Legendre-Fenchel transforms and maximizing arguments.", "type": "text" } ], "index": 10 } ], "index": 10 }, { "type": "table_body", "bbox": [ 117, 252, 495, 363 ], "group_id": 0, "lines": [ { "bbox": [ 117, 252, 495, 363 ], "spans": [ { "bbox": [ 117, 252, 495, 363 ], "score": 0.979, "html": "
EntropyRegularizer Ω2(πs)Legendre-Fenchel Ω*(Q s) Max argument VΩ*(Qs)
Maximum∑π(a|s)logπ(a|s)log∑e Q(s,a) TQ(s,a） e T
Qt(s,a)Ωe Q(s,b) T Qt(s,a）
RelativeDkL(πt(a|s)llπt-1(a|s))log∑aTt-1(als)eTTt-1(a|s)e T
∑Tt-1(b|s)e Qt(s，b） T
Tsallis( π(a|s) -1)Equation (10)Equation (11)
", "type": "table", "image_path": "092676270489bb91efbe0f3744338184ab1c03ae1d497ac3d946fddb41b71820.jpg" } ] } ], "index": 12, "virtual_lines": [ { "bbox": [ 117, 252, 495, 289.0 ], "spans": [], "index": 11 }, { "bbox": [ 117, 289.0, 495, 326.0 ], "spans": [], "index": 12 }, { "bbox": [ 117, 326.0, 495, 363.0 ], "spans": [], "index": 13 } ] } ], "index": 11.0 }, { "type": "title", "bbox": [ 107, 379, 212, 390 ], "lines": [ { "bbox": [ 106, 379, 213, 391 ], "spans": [ { "bbox": [ 106, 379, 213, 391 ], "score": 1.0, "content": "4.1 REGRET ANALYSIS", "type": "text" } ], "index": 14 } ], "index": 14 }, { "type": "text", "bbox": [ 106, 399, 506, 455 ], "lines": [ { "bbox": [ 106, 401, 505, 411 ], "spans": [ { "bbox": [ 106, 401, 268, 411 ], "score": 1.0, "content": "At the root node, let each children node", "type": "text" }, { "bbox": [ 269, 401, 273, 410 ], "score": 0.75, "content": "i", "type": "inline_equation" }, { "bbox": [ 274, 401, 419, 411 ], "score": 1.0, "content": "be assigned with a random variable", "type": "text" }, { "bbox": [ 419, 401, 432, 411 ], "score": 0.89, "content": "X _ { i }", "type": "inline_equation" }, { "bbox": [ 432, 401, 505, 411 ], "score": 1.0, "content": ", with mean value", "type": "text" } ], "index": 15 }, { "bbox": [ 107, 410, 505, 423 ], "spans": [ { "bbox": [ 107, 411, 117, 422 ], "score": 0.86, "content": "V _ { i }", "type": "inline_equation" }, { "bbox": [ 117, 410, 378, 423 ], "score": 1.0, "content": ", while the quantities related to the optimal branch are denoted by", "type": "text" }, { "bbox": [ 378, 413, 385, 421 ], "score": 0.69, "content": "^ *", "type": "inline_equation" }, { "bbox": [ 385, 410, 453, 423 ], "score": 1.0, "content": ", e.g. mean value", "type": "text" }, { "bbox": [ 454, 411, 467, 421 ], "score": 0.86, "content": "V ^ { * }", "type": "inline_equation" }, { "bbox": [ 468, 410, 505, 423 ], "score": 1.0, "content": ". At each", "type": "text" } ], "index": 16 }, { "bbox": [ 102, 421, 508, 450 ], "spans": [ { "bbox": [ 102, 427, 508, 450 ], "score": 1.0, "content": "nthe root node, at timestep n, is defined as RUCTn = nV ∗ − Pnt=1 Vit . Similarly, we define the regret", "type": "text" }, { "bbox": [ 105, 421, 143, 434 ], "score": 1.0, "content": "timestep", "type": "text" }, { "bbox": [ 143, 424, 150, 432 ], "score": 0.73, "content": "n", "type": "inline_equation" }, { "bbox": [ 150, 421, 263, 434 ], "score": 1.0, "content": ", the mean value of variable", "type": "text" }, { "bbox": [ 263, 423, 276, 433 ], "score": 0.88, "content": "X _ { i }", "type": "inline_equation" }, { "bbox": [ 276, 421, 287, 434 ], "score": 1.0, "content": "is", "type": "text" }, { "bbox": [ 287, 422, 302, 433 ], "score": 0.86, "content": "V _ { i _ { n } }", "type": "inline_equation" }, { "bbox": [ 302, 421, 425, 434 ], "score": 1.0, "content": ". The pseudo-regret (Coquelin", "type": "text" }, { "bbox": [ 426, 423, 435, 432 ], "score": 0.45, "content": "\\&", "type": "inline_equation" }, { "bbox": [ 435, 421, 506, 434 ], "score": 1.0, "content": "Munos, 2007) at", "type": "text" } ], "index": 17 }, { "bbox": [ 105, 443, 259, 456 ], "spans": [ { "bbox": [ 105, 443, 117, 456 ], "score": 1.0, "content": "of", "type": "text" }, { "bbox": [ 117, 444, 139, 454 ], "score": 0.32, "content": "\\mathrm { E } 3 \\mathrm { W }", "type": "inline_equation" }, { "bbox": [ 140, 443, 259, 456 ], "score": 1.0, "content": "at the root node of the tree as", "type": "text" } ], "index": 18 } ], "index": 16.5 }, { "type": "interline_equation", "bbox": [ 146, 454, 465, 487 ], "lines": [ { "bbox": [ 146, 454, 465, 487 ], "spans": [ { "bbox": [ 146, 454, 465, 487 ], "score": 0.92, "content": "R _ { n } = n V ^ { * } - \\sum _ { t = 1 } ^ { n } V _ { i _ { t } } = n V ^ { * } - \\sum _ { t = 1 } ^ { n } \\mathbb { I } ( i _ { t } = i ) V _ { i _ { t } } = n V ^ { * } - \\sum _ { i } V _ { i } \\sum _ { t = 1 } ^ { n } { \\hat { \\pi } } _ { t } ( a _ { i } | s ) ,", "type": "interline_equation", "image_path": "4a34256777c532ef1f0207a9d83fd1cddb9ec1beb0ef725356833b2339bfb0c4.jpg" } ] } ], "index": 20, "virtual_lines": [ { "bbox": [ 146, 454, 465, 465.0 ], "spans": [], "index": 19 }, { "bbox": [ 146, 465.0, 465, 476.0 ], "spans": [], "index": 20 }, { "bbox": [ 146, 476.0, 465, 487.0 ], "spans": [], "index": 21 } ] }, { "type": "text", "bbox": [ 108, 489, 397, 502 ], "lines": [ { "bbox": [ 106, 488, 397, 503 ], "spans": [ { "bbox": [ 106, 488, 133, 503 ], "score": 1.0, "content": "where", "type": "text" }, { "bbox": [ 133, 489, 154, 501 ], "score": 0.92, "content": "\\hat { \\pi } _ { t } ( \\cdot )", "type": "inline_equation" }, { "bbox": [ 154, 488, 255, 503 ], "score": 1.0, "content": "is the policy at time step", "type": "text" }, { "bbox": [ 255, 491, 260, 500 ], "score": 0.71, "content": "t", "type": "inline_equation" }, { "bbox": [ 260, 488, 280, 503 ], "score": 1.0, "content": ", and", "type": "text" }, { "bbox": [ 281, 489, 296, 501 ], "score": 0.88, "content": "\\mathbb { I } ( \\cdot )", "type": "inline_equation" }, { "bbox": [ 296, 488, 397, 503 ], "score": 1.0, "content": "is the indicator function.", "type": "text" } ], "index": 22 } ], "index": 22 }, { "type": "text", "bbox": [ 107, 509, 505, 553 ], "lines": [ { "bbox": [ 104, 509, 503, 531 ], "spans": [ { "bbox": [ 104, 510, 173, 531 ], "score": 1.0, "content": "Theorem 3 Let", "type": "text" }, { "bbox": [ 174, 509, 325, 530 ], "score": 0.9, "content": "\\begin{array} { r } { \\kappa _ { i } = \\nabla \\Omega ^ { * } ( a _ { i } | s ) + \\frac { L } { p } \\sqrt { \\dot { C } \\sigma ^ { 2 } \\log \\frac { C } { \\delta } } / 2 n _ { i } } \\end{array}", "type": "inline_equation" }, { "bbox": [ 326, 510, 349, 531 ], "score": 1.0, "content": ", and", "type": "text" }, { "bbox": [ 349, 509, 503, 529 ], "score": 0.88, "content": "\\begin{array} { r } { \\chi _ { i } = \\nabla \\Omega ^ { * } ( a _ { i } | s ) - \\frac { L } { p } \\sqrt { \\hat { C } \\sigma ^ { 2 } \\log \\frac { C } { \\delta } / 2 n } , } \\end{array}", "type": "inline_equation" } ], "index": 23 }, { "bbox": [ 105, 526, 506, 542 ], "spans": [ { "bbox": [ 105, 526, 133, 542 ], "score": 1.0, "content": "where", "type": "text" }, { "bbox": [ 133, 528, 172, 540 ], "score": 0.92, "content": "\\nabla \\Omega ^ { * } ( . | s )", "type": "inline_equation" }, { "bbox": [ 172, 526, 375, 542 ], "score": 1.0, "content": "is the policy with respect to the mean value vector", "type": "text" }, { "bbox": [ 375, 528, 395, 540 ], "score": 0.9, "content": "V ( \\cdot )", "type": "inline_equation" }, { "bbox": [ 395, 526, 506, 542 ], "score": 1.0, "content": "at the root node s. For any", "type": "text" } ], "index": 24 }, { "bbox": [ 107, 540, 496, 554 ], "spans": [ { "bbox": [ 107, 541, 131, 552 ], "score": 0.88, "content": "\\delta > 0", "type": "inline_equation" }, { "bbox": [ 131, 540, 232, 554 ], "score": 1.0, "content": ", with probability at least", "type": "text" }, { "bbox": [ 232, 541, 255, 552 ], "score": 0.64, "content": "1 - \\delta", "type": "inline_equation" }, { "bbox": [ 255, 540, 303, 554 ], "score": 1.0, "content": ", ∃ constant", "type": "text" }, { "bbox": [ 304, 540, 345, 553 ], "score": 0.92, "content": "L , p , C , { \\hat { C } }", "type": "inline_equation" }, { "bbox": [ 345, 540, 446, 554 ], "score": 1.0, "content": "so that the pseudo regret", "type": "text" }, { "bbox": [ 447, 542, 460, 552 ], "score": 0.89, "content": "R _ { n }", "type": "inline_equation" }, { "bbox": [ 461, 540, 496, 554 ], "score": 1.0, "content": "satisfies", "type": "text" } ], "index": 25 } ], "index": 24 }, { "type": "interline_equation", "bbox": [ 113, 554, 495, 585 ], "lines": [ { "bbox": [ 113, 554, 495, 585 ], "spans": [ { "bbox": [ 113, 554, 495, 585 ], "score": 0.92, "content": "n V ^ { * } - n \\sum _ { i } V _ { i } \\Big ( \\kappa _ { i } + \\frac { L } { p } \\big ( \\frac { \\tau ( U _ { \\Omega } - L _ { \\Omega } ) } { 1 - \\gamma } \\big ) \\Big ) \\leq R _ { n } \\leq n V ^ { * } - n \\sum _ { i } V _ { i } \\Big ( \\chi _ { i } - \\frac { L } { p } \\big ( \\frac { \\tau ( U _ { \\Omega } - L _ { \\Omega } ) } { 1 - \\gamma } \\big ) \\Big ) .", "type": "interline_equation", "image_path": "791ab69721cfe57610b5a0e976125a0ffe0f20e1085911c0e680c7fcffa7a30b.jpg" } ] } ], "index": 26, "virtual_lines": [ { "bbox": [ 113, 554, 495, 585 ], "spans": [], "index": 26 } ] }, { "type": "text", "bbox": [ 107, 591, 504, 615 ], "lines": [ { "bbox": [ 105, 591, 505, 605 ], "spans": [ { "bbox": [ 105, 591, 457, 605 ], "score": 1.0, "content": "This theorem provides bounds for the regret of E3W using a generic convex regularizer", "type": "text" }, { "bbox": [ 457, 592, 465, 602 ], "score": 0.83, "content": "\\Omega", "type": "inline_equation" }, { "bbox": [ 465, 591, 505, 605 ], "score": 1.0, "content": "; thus, we", "type": "text" } ], "index": 27 }, { "bbox": [ 105, 601, 502, 617 ], "spans": [ { "bbox": [ 105, 601, 410, 617 ], "score": 1.0, "content": "can easily retrieve from it the regret bound for each entropy regularizer. Let", "type": "text" }, { "bbox": [ 410, 603, 497, 615 ], "score": 0.92, "content": "\\begin{array} { r } { m = \\operatorname* { m i n } _ { a } \\nabla \\Omega ^ { * } ( a | s ) } \\end{array}", "type": "inline_equation" }, { "bbox": [ 497, 601, 502, 617 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 28 } ], "index": 27.5 }, { "type": "text", "bbox": [ 105, 623, 437, 654 ], "lines": [ { "bbox": [ 106, 620, 238, 636 ], "spans": [ { "bbox": [ 106, 620, 238, 636 ], "score": 1.0, "content": "Corollary 1 Maximum entropy:", "type": "text" } ], "index": 29 }, { "bbox": [ 105, 632, 437, 654 ], "spans": [ { "bbox": [ 105, 632, 437, 654 ], "score": 0.87, "content": "\\begin{array} { r } { n V ^ { * } - \\tilde { n } \\sum _ { i } V _ { i } \\Big ( \\kappa _ { i } + L \\big ( \\frac { \\tau \\log | A | } { 1 - \\gamma } \\big ) \\Big ) \\leq R _ { n } \\leq n V ^ { * } - n \\sum _ { i } V _ { i } \\Big ( \\chi _ { i } - L \\big ( \\frac { \\tau \\log | A | } { 1 - \\gamma } \\big ) \\Big ) . } \\end{array}", "type": "inline_equation", "image_path": "cfe36a606698d4fb15ced64dd60a5f3b3a2a4be5ddc1a284438d631529a0eb0c.jpg" } ], "index": 30 } ], "index": 29.5 }, { "type": "text", "bbox": [ 105, 660, 475, 692 ], "lines": [ { "bbox": [ 105, 670, 475, 692 ], "spans": [ { "bbox": [ 105, 670, 475, 692 ], "score": 0.88, "content": "\\begin{array} { r } { n V ^ { * } - \\tilde { n } \\sum _ { i } V _ { i } \\Big ( \\kappa _ { i } + L \\big ( \\frac { \\tau ^ { \\langle \\log | A | - \\frac { 1 } { m } \\rangle } } { 1 - \\gamma } \\big ) \\Big ) \\leq R _ { n } \\leq n V ^ { * } - n \\sum _ { i } V _ { i } \\Big ( \\chi _ { i } - L \\big ( \\frac { \\tau ( \\log | A | - \\frac { 1 } { m } ) } { 1 - \\gamma } \\big ) \\Big ) . } \\end{array}", "type": "inline_equation", "image_path": "63f14ce96ec421e4d12f33665e44949d0ae360342d71641c8163cc8fcb3c5067.jpg" } ], "index": 31 } ], "index": 31 }, { "type": "title", "bbox": [ 107, 698, 224, 710 ], "lines": [ { "bbox": [ 106, 697, 225, 714 ], "spans": [ { "bbox": [ 106, 697, 225, 714 ], "score": 1.0, "content": "Corollary 3 Tsallis entropy:", "type": "text" } ], "index": 32 } ], "index": 32 }, { "type": "interline_equation", "bbox": [ 106, 709, 493, 735 ], "lines": [ { "bbox": [ 106, 709, 493, 735 ], "spans": [ { "bbox": [ 106, 709, 493, 735 ], "score": 0.9, "content": "\\begin{array} { r } { n V ^ { * } - n \\sum _ { i } V _ { i } \\Big ( \\kappa _ { i } + \\frac { L } { 2 } \\big ( \\frac { | A | - 1 } { 2 | A | } \\frac { \\tau } { 1 - \\gamma } \\big ) \\Big ) \\leq R _ { n } \\leq n V ^ { * } - n \\sum _ { i } V _ { i } \\Big ( \\chi _ { i } - \\frac { L } { 2 } \\big ( \\frac { | A | - 1 } { 2 | A | } \\frac { \\tau } { 1 - \\gamma } \\big ) \\Big ) . } \\end{array}", "type": "interline_equation", "image_path": "563e2c8a8f05ef4f64e0d998a92d1d2fee2116ca514366aa6d6487b767bdaa0c.jpg" } ] } ], "index": 33, "virtual_lines": [ { "bbox": [ 106, 709, 493, 735 ], "spans": [], "index": 33 } ] } ], "page_idx": 4, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 106, 26, 307, 38 ], "lines": [ { "bbox": [ 106, 25, 308, 38 ], "spans": [ { "bbox": [ 106, 25, 308, 38 ], "score": 1.0, "content": "Under review as a conference paper at ICLR 2021", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 302, 751, 309, 760 ], "lines": [ { "bbox": [ 302, 750, 309, 762 ], "spans": [ { "bbox": [ 302, 750, 309, 762 ], "score": 1.0, "content": "5", "type": "text" } ] } ] } ], "para_blocks": [ { "type": "text", "bbox": [ 106, 82, 442, 95 ], "lines": [ { "bbox": [ 104, 80, 443, 98 ], "spans": [ { "bbox": [ 104, 80, 443, 98 ], "score": 1.0, "content": "Legendre-Fenchel and maximizing argument of Tsallis entropy is non-trivial, being", "type": "text" } ], "index": 0 } ], "index": 0, "bbox_fs": [ 104, 80, 443, 98 ] }, { "type": "interline_equation", "bbox": [ 185, 95, 426, 144 ], "lines": [ { "bbox": [ 185, 95, 426, 144 ], "spans": [ { "bbox": [ 185, 95, 426, 144 ], "score": 0.81, "content": "\\begin{array} { r l } & { ~ \\Omega ^ { * } ( Q _ { t } ) = \\tau \\cdot \\mathrm { s p m a x } ( Q _ { t } ( s , \\cdot ) / \\tau ) , } \\\\ & { ~ \\nabla \\Omega ^ { * } ( Q _ { t } ) = \\operatorname* { m a x } \\Bigg ( \\displaystyle \\frac { Q _ { t } ( s , a ) } { \\tau } - \\displaystyle \\frac { \\sum _ { a \\in { \\mathcal { K } } } Q _ { t } ( s , a ) / \\tau - 1 } { | { \\mathcal { K } } | } , 0 \\Bigg ) , } \\end{array}", "type": "interline_equation", "image_path": "c2bb83cafb01aadf64b3d47489e3e858bcae3f15f4903e114f3e981f0e50699b.jpg" } ] } ], "index": 2, "virtual_lines": [ { "bbox": [ 185, 95, 426, 111.33333333333333 ], "spans": [], "index": 1 }, { "bbox": [ 185, 111.33333333333333, 426, 127.66666666666666 ], "spans": [], "index": 2 }, { "bbox": [ 185, 127.66666666666666, 426, 144.0 ], "spans": [], "index": 3 } ] }, { "type": "text", "bbox": [ 107, 145, 347, 157 ], "lines": [ { "bbox": [ 106, 144, 347, 158 ], "spans": [ { "bbox": [ 106, 144, 269, 158 ], "score": 1.0, "content": "where spmax is defined for any function", "type": "text" }, { "bbox": [ 270, 146, 334, 157 ], "score": 0.9, "content": "f : \\mathcal { S } \\times \\mathcal { A } \\mathbb { R }", "type": "inline_equation" }, { "bbox": [ 335, 144, 347, 158 ], "score": 1.0, "content": "as", "type": "text" } ], "index": 4 } ], "index": 4, "bbox_fs": [ 106, 144, 347, 158 ] }, { "type": "interline_equation", "bbox": [ 172, 158, 438, 192 ], "lines": [ { "bbox": [ 172, 158, 438, 192 ], "spans": [ { "bbox": [ 172, 158, 438, 192 ], "score": 0.93, "content": "\\operatorname { s p m a x } ( f ( s , \\cdot ) ) \\triangleq \\sum _ { a \\in { \\mathcal { K } } } \\left( { \\frac { f ( s , a ) ^ { 2 } } { 2 } } - { \\frac { ( \\sum _ { a \\in { \\mathcal { K } } } f ( s , a ) - 1 ) ^ { 2 } } { 2 | K | ^ { 2 } } } \\right) + { \\frac { 1 } { 2 } } ,", "type": "interline_equation", "image_path": "1010175a551d5603043280b2589ec1948d19cb81c5e03e721432b08ede550e6c.jpg" } ] } ], "index": 6, "virtual_lines": [ { "bbox": [ 172, 158, 438, 169.33333333333334 ], "spans": [], "index": 5 }, { "bbox": [ 172, 169.33333333333334, 438, 180.66666666666669 ], "spans": [], "index": 6 }, { "bbox": [ 172, 180.66666666666669, 438, 192.00000000000003 ], "spans": [], "index": 7 } ] }, { "type": "text", "bbox": [ 107, 195, 504, 222 ], "lines": [ { "bbox": [ 103, 191, 508, 214 ], "spans": [ { "bbox": [ 103, 191, 123, 214 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 123, 197, 133, 207 ], "score": 0.82, "content": "\\kappa", "type": "inline_equation" }, { "bbox": [ 133, 191, 259, 214 ], "score": 1.0, "content": "is the set of actions that satisfy", "type": "text" }, { "bbox": [ 260, 195, 385, 211 ], "score": 0.94, "content": "\\begin{array} { r } { 1 + i f ( s , a _ { i } ) > \\sum _ { j = 1 } ^ { i } f ( s , a _ { j } ) } \\end{array}", "type": "inline_equation" }, { "bbox": [ 386, 191, 409, 214 ], "score": 1.0, "content": ", with", "type": "text" }, { "bbox": [ 410, 199, 419, 208 ], "score": 0.84, "content": "a _ { i }", "type": "inline_equation" }, { "bbox": [ 420, 191, 508, 214 ], "score": 1.0, "content": "indicating the action", "type": "text" } ], "index": 8 }, { "bbox": [ 106, 209, 325, 222 ], "spans": [ { "bbox": [ 106, 209, 141, 222 ], "score": 1.0, "content": "with the", "type": "text" }, { "bbox": [ 141, 210, 146, 219 ], "score": 0.73, "content": "i", "type": "inline_equation" }, { "bbox": [ 146, 209, 222, 222 ], "score": 1.0, "content": "-th largest value of", "type": "text" }, { "bbox": [ 222, 209, 251, 222 ], "score": 0.93, "content": "f ( s , a )", "type": "inline_equation" }, { "bbox": [ 251, 209, 325, 222 ], "score": 1.0, "content": "(Lee et al., 2018).", "type": "text" } ], "index": 9 } ], "index": 8.5, "bbox_fs": [ 103, 191, 508, 222 ] }, { "type": "table", "bbox": [ 117, 252, 495, 363 ], "blocks": [ { "type": "table_caption", "bbox": [ 106, 239, 503, 251 ], "group_id": 0, "lines": [ { "bbox": [ 106, 237, 505, 253 ], "spans": [ { "bbox": [ 106, 237, 505, 253 ], "score": 1.0, "content": "Table 1: List of entropy regularizers with Legendre-Fenchel transforms and maximizing arguments.", "type": "text" } ], "index": 10 } ], "index": 10 }, { "type": "table_body", "bbox": [ 117, 252, 495, 363 ], "group_id": 0, "lines": [ { "bbox": [ 117, 252, 495, 363 ], "spans": [ { "bbox": [ 117, 252, 495, 363 ], "score": 0.979, "html": "
EntropyRegularizer Ω2(πs)Legendre-Fenchel Ω*(Q s) Max argument VΩ*(Qs)
Maximum∑π(a|s)logπ(a|s)log∑e Q(s,a) TQ(s,a） e T
Qt(s,a)Ωe Q(s,b) T Qt(s,a）
RelativeDkL(πt(a|s)llπt-1(a|s))log∑aTt-1(als)eTTt-1(a|s)e T
∑Tt-1(b|s)e Qt(s，b） T
Tsallis( π(a|s) -1)Equation (10)Equation (11)
", "type": "table", "image_path": "092676270489bb91efbe0f3744338184ab1c03ae1d497ac3d946fddb41b71820.jpg" } ] } ], "index": 12, "virtual_lines": [ { "bbox": [ 117, 252, 495, 289.0 ], "spans": [], "index": 11 }, { "bbox": [ 117, 289.0, 495, 326.0 ], "spans": [], "index": 12 }, { "bbox": [ 117, 326.0, 495, 363.0 ], "spans": [], "index": 13 } ] } ], "index": 11.0 }, { "type": "title", "bbox": [ 107, 379, 212, 390 ], "lines": [ { "bbox": [ 106, 379, 213, 391 ], "spans": [ { "bbox": [ 106, 379, 213, 391 ], "score": 1.0, "content": "4.1 REGRET ANALYSIS", "type": "text" } ], "index": 14 } ], "index": 14 }, { "type": "text", "bbox": [ 106, 399, 506, 455 ], "lines": [ { "bbox": [ 106, 401, 505, 411 ], "spans": [ { "bbox": [ 106, 401, 268, 411 ], "score": 1.0, "content": "At the root node, let each children node", "type": "text" }, { "bbox": [ 269, 401, 273, 410 ], "score": 0.75, "content": "i", "type": "inline_equation" }, { "bbox": [ 274, 401, 419, 411 ], "score": 1.0, "content": "be assigned with a random variable", "type": "text" }, { "bbox": [ 419, 401, 432, 411 ], "score": 0.89, "content": "X _ { i }", "type": "inline_equation" }, { "bbox": [ 432, 401, 505, 411 ], "score": 1.0, "content": ", with mean value", "type": "text" } ], "index": 15 }, { "bbox": [ 107, 410, 505, 423 ], "spans": [ { "bbox": [ 107, 411, 117, 422 ], "score": 0.86, "content": "V _ { i }", "type": "inline_equation" }, { "bbox": [ 117, 410, 378, 423 ], "score": 1.0, "content": ", while the quantities related to the optimal branch are denoted by", "type": "text" }, { "bbox": [ 378, 413, 385, 421 ], "score": 0.69, "content": "^ *", "type": "inline_equation" }, { "bbox": [ 385, 410, 453, 423 ], "score": 1.0, "content": ", e.g. mean value", "type": "text" }, { "bbox": [ 454, 411, 467, 421 ], "score": 0.86, "content": "V ^ { * }", "type": "inline_equation" }, { "bbox": [ 468, 410, 505, 423 ], "score": 1.0, "content": ". At each", "type": "text" } ], "index": 16 }, { "bbox": [ 102, 421, 508, 450 ], "spans": [ { "bbox": [ 102, 427, 508, 450 ], "score": 1.0, "content": "nthe root node, at timestep n, is defined as RUCTn = nV ∗ − Pnt=1 Vit . Similarly, we define the regret", "type": "text" }, { "bbox": [ 105, 421, 143, 434 ], "score": 1.0, "content": "timestep", "type": "text" }, { "bbox": [ 143, 424, 150, 432 ], "score": 0.73, "content": "n", "type": "inline_equation" }, { "bbox": [ 150, 421, 263, 434 ], "score": 1.0, "content": ", the mean value of variable", "type": "text" }, { "bbox": [ 263, 423, 276, 433 ], "score": 0.88, "content": "X _ { i }", "type": "inline_equation" }, { "bbox": [ 276, 421, 287, 434 ], "score": 1.0, "content": "is", "type": "text" }, { "bbox": [ 287, 422, 302, 433 ], "score": 0.86, "content": "V _ { i _ { n } }", "type": "inline_equation" }, { "bbox": [ 302, 421, 425, 434 ], "score": 1.0, "content": ". The pseudo-regret (Coquelin", "type": "text" }, { "bbox": [ 426, 423, 435, 432 ], "score": 0.45, "content": "\\&", "type": "inline_equation" }, { "bbox": [ 435, 421, 506, 434 ], "score": 1.0, "content": "Munos, 2007) at", "type": "text" } ], "index": 17 }, { "bbox": [ 105, 443, 259, 456 ], "spans": [ { "bbox": [ 105, 443, 117, 456 ], "score": 1.0, "content": "of", "type": "text" }, { "bbox": [ 117, 444, 139, 454 ], "score": 0.32, "content": "\\mathrm { E } 3 \\mathrm { W }", "type": "inline_equation" }, { "bbox": [ 140, 443, 259, 456 ], "score": 1.0, "content": "at the root node of the tree as", "type": "text" } ], "index": 18 } ], "index": 16.5, "bbox_fs": [ 102, 401, 508, 456 ] }, { "type": "interline_equation", "bbox": [ 146, 454, 465, 487 ], "lines": [ { "bbox": [ 146, 454, 465, 487 ], "spans": [ { "bbox": [ 146, 454, 465, 487 ], "score": 0.92, "content": "R _ { n } = n V ^ { * } - \\sum _ { t = 1 } ^ { n } V _ { i _ { t } } = n V ^ { * } - \\sum _ { t = 1 } ^ { n } \\mathbb { I } ( i _ { t } = i ) V _ { i _ { t } } = n V ^ { * } - \\sum _ { i } V _ { i } \\sum _ { t = 1 } ^ { n } { \\hat { \\pi } } _ { t } ( a _ { i } | s ) ,", "type": "interline_equation", "image_path": "4a34256777c532ef1f0207a9d83fd1cddb9ec1beb0ef725356833b2339bfb0c4.jpg" } ] } ], "index": 20, "virtual_lines": [ { "bbox": [ 146, 454, 465, 465.0 ], "spans": [], "index": 19 }, { "bbox": [ 146, 465.0, 465, 476.0 ], "spans": [], "index": 20 }, { "bbox": [ 146, 476.0, 465, 487.0 ], "spans": [], "index": 21 } ] }, { "type": "text", "bbox": [ 108, 489, 397, 502 ], "lines": [ { "bbox": [ 106, 488, 397, 503 ], "spans": [ { "bbox": [ 106, 488, 133, 503 ], "score": 1.0, "content": "where", "type": "text" }, { "bbox": [ 133, 489, 154, 501 ], "score": 0.92, "content": "\\hat { \\pi } _ { t } ( \\cdot )", "type": "inline_equation" }, { "bbox": [ 154, 488, 255, 503 ], "score": 1.0, "content": "is the policy at time step", "type": "text" }, { "bbox": [ 255, 491, 260, 500 ], "score": 0.71, "content": "t", "type": "inline_equation" }, { "bbox": [ 260, 488, 280, 503 ], "score": 1.0, "content": ", and", "type": "text" }, { "bbox": [ 281, 489, 296, 501 ], "score": 0.88, "content": "\\mathbb { I } ( \\cdot )", "type": "inline_equation" }, { "bbox": [ 296, 488, 397, 503 ], "score": 1.0, "content": "is the indicator function.", "type": "text" } ], "index": 22 } ], "index": 22, "bbox_fs": [ 106, 488, 397, 503 ] }, { "type": "text", "bbox": [ 107, 509, 505, 553 ], "lines": [ { "bbox": [ 104, 509, 503, 531 ], "spans": [ { "bbox": [ 104, 510, 173, 531 ], "score": 1.0, "content": "Theorem 3 Let", "type": "text" }, { "bbox": [ 174, 509, 325, 530 ], "score": 0.9, "content": "\\begin{array} { r } { \\kappa _ { i } = \\nabla \\Omega ^ { * } ( a _ { i } | s ) + \\frac { L } { p } \\sqrt { \\dot { C } \\sigma ^ { 2 } \\log \\frac { C } { \\delta } } / 2 n _ { i } } \\end{array}", "type": "inline_equation" }, { "bbox": [ 326, 510, 349, 531 ], "score": 1.0, "content": ", and", "type": "text" }, { "bbox": [ 349, 509, 503, 529 ], "score": 0.88, "content": "\\begin{array} { r } { \\chi _ { i } = \\nabla \\Omega ^ { * } ( a _ { i } | s ) - \\frac { L } { p } \\sqrt { \\hat { C } \\sigma ^ { 2 } \\log \\frac { C } { \\delta } / 2 n } , } \\end{array}", "type": "inline_equation" } ], "index": 23 }, { "bbox": [ 105, 526, 506, 542 ], "spans": [ { "bbox": [ 105, 526, 133, 542 ], "score": 1.0, "content": "where", "type": "text" }, { "bbox": [ 133, 528, 172, 540 ], "score": 0.92, "content": "\\nabla \\Omega ^ { * } ( . | s )", "type": "inline_equation" }, { "bbox": [ 172, 526, 375, 542 ], "score": 1.0, "content": "is the policy with respect to the mean value vector", "type": "text" }, { "bbox": [ 375, 528, 395, 540 ], "score": 0.9, "content": "V ( \\cdot )", "type": "inline_equation" }, { "bbox": [ 395, 526, 506, 542 ], "score": 1.0, "content": "at the root node s. For any", "type": "text" } ], "index": 24 }, { "bbox": [ 107, 540, 496, 554 ], "spans": [ { "bbox": [ 107, 541, 131, 552 ], "score": 0.88, "content": "\\delta > 0", "type": "inline_equation" }, { "bbox": [ 131, 540, 232, 554 ], "score": 1.0, "content": ", with probability at least", "type": "text" }, { "bbox": [ 232, 541, 255, 552 ], "score": 0.64, "content": "1 - \\delta", "type": "inline_equation" }, { "bbox": [ 255, 540, 303, 554 ], "score": 1.0, "content": ", ∃ constant", "type": "text" }, { "bbox": [ 304, 540, 345, 553 ], "score": 0.92, "content": "L , p , C , { \\hat { C } }", "type": "inline_equation" }, { "bbox": [ 345, 540, 446, 554 ], "score": 1.0, "content": "so that the pseudo regret", "type": "text" }, { "bbox": [ 447, 542, 460, 552 ], "score": 0.89, "content": "R _ { n }", "type": "inline_equation" }, { "bbox": [ 461, 540, 496, 554 ], "score": 1.0, "content": "satisfies", "type": "text" } ], "index": 25 } ], "index": 24, "bbox_fs": [ 104, 509, 506, 554 ] }, { "type": "interline_equation", "bbox": [ 113, 554, 495, 585 ], "lines": [ { "bbox": [ 113, 554, 495, 585 ], "spans": [ { "bbox": [ 113, 554, 495, 585 ], "score": 0.92, "content": "n V ^ { * } - n \\sum _ { i } V _ { i } \\Big ( \\kappa _ { i } + \\frac { L } { p } \\big ( \\frac { \\tau ( U _ { \\Omega } - L _ { \\Omega } ) } { 1 - \\gamma } \\big ) \\Big ) \\leq R _ { n } \\leq n V ^ { * } - n \\sum _ { i } V _ { i } \\Big ( \\chi _ { i } - \\frac { L } { p } \\big ( \\frac { \\tau ( U _ { \\Omega } - L _ { \\Omega } ) } { 1 - \\gamma } \\big ) \\Big ) .", "type": "interline_equation", "image_path": "791ab69721cfe57610b5a0e976125a0ffe0f20e1085911c0e680c7fcffa7a30b.jpg" } ] } ], "index": 26, "virtual_lines": [ { "bbox": [ 113, 554, 495, 585 ], "spans": [], "index": 26 } ] }, { "type": "text", "bbox": [ 107, 591, 504, 615 ], "lines": [ { "bbox": [ 105, 591, 505, 605 ], "spans": [ { "bbox": [ 105, 591, 457, 605 ], "score": 1.0, "content": "This theorem provides bounds for the regret of E3W using a generic convex regularizer", "type": "text" }, { "bbox": [ 457, 592, 465, 602 ], "score": 0.83, "content": "\\Omega", "type": "inline_equation" }, { "bbox": [ 465, 591, 505, 605 ], "score": 1.0, "content": "; thus, we", "type": "text" } ], "index": 27 }, { "bbox": [ 105, 601, 502, 617 ], "spans": [ { "bbox": [ 105, 601, 410, 617 ], "score": 1.0, "content": "can easily retrieve from it the regret bound for each entropy regularizer. Let", "type": "text" }, { "bbox": [ 410, 603, 497, 615 ], "score": 0.92, "content": "\\begin{array} { r } { m = \\operatorname* { m i n } _ { a } \\nabla \\Omega ^ { * } ( a | s ) } \\end{array}", "type": "inline_equation" }, { "bbox": [ 497, 601, 502, 617 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 28 } ], "index": 27.5, "bbox_fs": [ 105, 591, 505, 617 ] }, { "type": "text", "bbox": [ 105, 623, 437, 654 ], "lines": [ { "bbox": [ 106, 620, 238, 636 ], "spans": [ { "bbox": [ 106, 620, 238, 636 ], "score": 1.0, "content": "Corollary 1 Maximum entropy:", "type": "text" } ], "index": 29 }, { "bbox": [ 105, 632, 437, 654 ], "spans": [ { "bbox": [ 105, 632, 437, 654 ], "score": 0.87, "content": "\\begin{array} { r } { n V ^ { * } - \\tilde { n } \\sum _ { i } V _ { i } \\Big ( \\kappa _ { i } + L \\big ( \\frac { \\tau \\log | A | } { 1 - \\gamma } \\big ) \\Big ) \\leq R _ { n } \\leq n V ^ { * } - n \\sum _ { i } V _ { i } \\Big ( \\chi _ { i } - L \\big ( \\frac { \\tau \\log | A | } { 1 - \\gamma } \\big ) \\Big ) . } \\end{array}", "type": "inline_equation", "image_path": "cfe36a606698d4fb15ced64dd60a5f3b3a2a4be5ddc1a284438d631529a0eb0c.jpg" } ], "index": 30 } ], "index": 29.5, "bbox_fs": [ 105, 620, 437, 654 ] }, { "type": "text", "bbox": [ 105, 660, 475, 692 ], "lines": [ { "bbox": [ 105, 670, 475, 692 ], "spans": [ { "bbox": [ 105, 670, 475, 692 ], "score": 0.88, "content": "\\begin{array} { r } { n V ^ { * } - \\tilde { n } \\sum _ { i } V _ { i } \\Big ( \\kappa _ { i } + L \\big ( \\frac { \\tau ^ { \\langle \\log | A | - \\frac { 1 } { m } \\rangle } } { 1 - \\gamma } \\big ) \\Big ) \\leq R _ { n } \\leq n V ^ { * } - n \\sum _ { i } V _ { i } \\Big ( \\chi _ { i } - L \\big ( \\frac { \\tau ( \\log | A | - \\frac { 1 } { m } ) } { 1 - \\gamma } \\big ) \\Big ) . } \\end{array}", "type": "inline_equation", "image_path": "63f14ce96ec421e4d12f33665e44949d0ae360342d71641c8163cc8fcb3c5067.jpg" } ], "index": 31 } ], "index": 31, "bbox_fs": [ 105, 670, 475, 692 ] }, { "type": "title", "bbox": [ 107, 698, 224, 710 ], "lines": [ { "bbox": [ 106, 697, 225, 714 ], "spans": [ { "bbox": [ 106, 697, 225, 714 ], "score": 1.0, "content": "Corollary 3 Tsallis entropy:", "type": "text" } ], "index": 32 } ], "index": 32 }, { "type": "interline_equation", "bbox": [ 106, 709, 493, 735 ], "lines": [ { "bbox": [ 106, 709, 493, 735 ], "spans": [ { "bbox": [ 106, 709, 493, 735 ], "score": 0.9, "content": "\\begin{array} { r } { n V ^ { * } - n \\sum _ { i } V _ { i } \\Big ( \\kappa _ { i } + \\frac { L } { 2 } \\big ( \\frac { | A | - 1 } { 2 | A | } \\frac { \\tau } { 1 - \\gamma } \\big ) \\Big ) \\leq R _ { n } \\leq n V ^ { * } - n \\sum _ { i } V _ { i } \\Big ( \\chi _ { i } - \\frac { L } { 2 } \\big ( \\frac { | A | - 1 } { 2 | A | } \\frac { \\tau } { 1 - \\gamma } \\big ) \\Big ) . } \\end{array}", "type": "interline_equation", "image_path": "563e2c8a8f05ef4f64e0d998a92d1d2fee2116ca514366aa6d6487b767bdaa0c.jpg" } ] } ], "index": 33, "virtual_lines": [ { "bbox": [ 106, 709, 493, 735 ], "spans": [], "index": 33 } ] } ] }, { "preproc_blocks": [ { "type": "text", "bbox": [ 107, 82, 505, 182 ], "lines": [ { "bbox": [ 106, 82, 505, 95 ], "spans": [ { "bbox": [ 106, 82, 505, 95 ], "score": 1.0, "content": "Remarks. The regret bound of UCT and its variance have already been analyzed for non-", "type": "text" } ], "index": 0 }, { "bbox": [ 105, 93, 505, 106 ], "spans": [ { "bbox": [ 105, 93, 505, 106 ], "score": 1.0, "content": "regularized MCTS with binary tree (Coquelin & Munos, 2007). On the contrary, our regret bound", "type": "text" } ], "index": 1 }, { "bbox": [ 106, 105, 505, 117 ], "spans": [ { "bbox": [ 106, 105, 505, 117 ], "score": 1.0, "content": "analysis in Theorem 3 applies to generic regularized MCTS. From the specialized bounds in the", "type": "text" } ], "index": 2 }, { "bbox": [ 105, 115, 505, 128 ], "spans": [ { "bbox": [ 105, 115, 505, 128 ], "score": 1.0, "content": "corollaries, we observe that the maximum and relative entropy share similar results, although the", "type": "text" } ], "index": 3 }, { "bbox": [ 104, 125, 507, 143 ], "spans": [ { "bbox": [ 104, 125, 328, 143 ], "score": 1.0, "content": "bounds for relative entropy are slightly smaller due to", "type": "text" }, { "bbox": [ 328, 126, 338, 139 ], "score": 0.89, "content": "\\textstyle { \\frac { 1 } { m } }", "type": "inline_equation" }, { "bbox": [ 339, 125, 507, 143 ], "score": 1.0, "content": ". Remarkably, the bounds for Tsallis en-", "type": "text" } ], "index": 4 }, { "bbox": [ 105, 137, 505, 150 ], "spans": [ { "bbox": [ 105, 137, 505, 150 ], "score": 1.0, "content": "tropy become tighter for increasing number of actions, which translates in limited regret in problems", "type": "text" } ], "index": 5 }, { "bbox": [ 106, 149, 505, 160 ], "spans": [ { "bbox": [ 106, 149, 505, 160 ], "score": 1.0, "content": "with high branching factor. This result establishes the advantage of Tsallis entropy in complex prob-", "type": "text" } ], "index": 6 }, { "bbox": [ 105, 159, 505, 172 ], "spans": [ { "bbox": [ 105, 159, 505, 172 ], "score": 1.0, "content": "lems w.r.t. to other entropy regularizers, as empirically confirmed by the positive results in several", "type": "text" } ], "index": 7 }, { "bbox": [ 106, 171, 250, 182 ], "spans": [ { "bbox": [ 106, 171, 250, 182 ], "score": 1.0, "content": "Atari games described in Section 5.", "type": "text" } ], "index": 8 } ], "index": 4 }, { "type": "title", "bbox": [ 107, 194, 207, 205 ], "lines": [ { "bbox": [ 105, 193, 207, 207 ], "spans": [ { "bbox": [ 105, 193, 207, 207 ], "score": 1.0, "content": "4.2 ERROR ANALYSIS", "type": "text" } ], "index": 9 } ], "index": 9 }, { "type": "text", "bbox": [ 107, 214, 504, 238 ], "lines": [ { "bbox": [ 106, 214, 505, 228 ], "spans": [ { "bbox": [ 106, 214, 387, 228 ], "score": 1.0, "content": "We analyse the error of the regularized value estimate at the root node", "type": "text" }, { "bbox": [ 388, 215, 407, 227 ], "score": 0.91, "content": "n ( s )", "type": "inline_equation" }, { "bbox": [ 407, 214, 505, 228 ], "score": 1.0, "content": "w.r.t. the optimal value:", "type": "text" } ], "index": 10 }, { "bbox": [ 106, 225, 198, 240 ], "spans": [ { "bbox": [ 106, 226, 193, 238 ], "score": 0.92, "content": "\\varepsilon _ { \\Omega } = \\dot { V _ { \\Omega } } ( s ) - \\dot { V ^ { \\ast } } ( s )", "type": "inline_equation" }, { "bbox": [ 194, 225, 198, 240 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 11 } ], "index": 10.5 }, { "type": "text", "bbox": [ 106, 247, 504, 270 ], "lines": [ { "bbox": [ 105, 245, 505, 261 ], "spans": [ { "bbox": [ 105, 245, 191, 261 ], "score": 1.0, "content": "Theorem 4 For any", "type": "text" }, { "bbox": [ 191, 248, 217, 258 ], "score": 0.89, "content": "\\delta > 0", "type": "inline_equation" }, { "bbox": [ 217, 245, 344, 261 ], "score": 1.0, "content": "and generic convex regularizer", "type": "text" }, { "bbox": [ 344, 248, 353, 258 ], "score": 0.77, "content": "\\Omega", "type": "inline_equation" }, { "bbox": [ 353, 245, 436, 261 ], "score": 1.0, "content": ", with some constant", "type": "text" }, { "bbox": [ 437, 246, 457, 259 ], "score": 0.9, "content": "C , { \\hat { C } }", "type": "inline_equation" }, { "bbox": [ 457, 245, 505, 261 ], "score": 1.0, "content": ", with prob-", "type": "text" } ], "index": 12 }, { "bbox": [ 105, 257, 241, 271 ], "spans": [ { "bbox": [ 105, 257, 166, 271 ], "score": 1.0, "content": "ability at least", "type": "text" }, { "bbox": [ 166, 259, 189, 269 ], "score": 0.57, "content": "1 - \\delta", "type": "inline_equation" }, { "bbox": [ 189, 257, 192, 271 ], "score": 1.0, "content": ",", "type": "text" }, { "bbox": [ 193, 260, 205, 270 ], "score": 0.66, "content": "\\varepsilon _ { \\Omega }", "type": "inline_equation" }, { "bbox": [ 205, 257, 241, 271 ], "score": 1.0, "content": "satisfies", "type": "text" } ], "index": 13 } ], "index": 12.5 }, { "type": "interline_equation", "bbox": [ 189, 272, 422, 306 ], "lines": [ { "bbox": [ 189, 272, 422, 306 ], "spans": [ { "bbox": [ 189, 272, 422, 306 ], "score": 0.94, "content": "- \\sqrt { \\frac { \\hat { C } \\sigma ^ { 2 } \\log \\frac { C } { \\delta } } { 2 N ( s ) } } - \\frac { \\tau ( U _ { \\Omega } - L _ { \\Omega } ) } { 1 - \\gamma } \\leq \\varepsilon _ { \\Omega } \\leq \\sqrt { \\frac { \\hat { C } \\sigma ^ { 2 } \\log \\frac { C } { \\delta } } { 2 N ( s ) } } .", "type": "interline_equation", "image_path": "ae4cbab37a8eb4b4691183517f81846a12faf589033d60727a578b553c5954d2.jpg" } ] } ], "index": 14.5, "virtual_lines": [ { "bbox": [ 189, 272, 422, 289.0 ], "spans": [], "index": 14 }, { "bbox": [ 189, 289.0, 422, 306.0 ], "spans": [], "index": 15 } ] }, { "type": "text", "bbox": [ 106, 312, 505, 374 ], "lines": [ { "bbox": [ 106, 313, 505, 325 ], "spans": [ { "bbox": [ 106, 313, 505, 325 ], "score": 1.0, "content": "To give a better understanding of the effect of each entropy regularizer in Table 1, we specialize", "type": "text" } ], "index": 16 }, { "bbox": [ 106, 324, 504, 335 ], "spans": [ { "bbox": [ 106, 324, 504, 335 ], "score": 1.0, "content": "the bound in Equation 14 to each of them. From (Lee et al., 2018), we know that for maximum", "type": "text" } ], "index": 17 }, { "bbox": [ 105, 335, 505, 348 ], "spans": [ { "bbox": [ 105, 335, 140, 348 ], "score": 1.0, "content": "entropy", "type": "text" }, { "bbox": [ 141, 335, 234, 348 ], "score": 0.92, "content": "\\begin{array} { r } { \\Omega ( { \\boldsymbol \\pi } _ { t } ) \\stackrel { - } { = } \\sum _ { a } \\pi _ { t } \\log \\pi _ { t } } \\end{array}", "type": "inline_equation" }, { "bbox": [ 235, 335, 276, 348 ], "score": 1.0, "content": ", we have", "type": "text" }, { "bbox": [ 277, 335, 379, 347 ], "score": 0.9, "content": "- \\log | \\mathcal { A } | \\ \\leq \\ \\Omega ( \\pi _ { t } ) \\ \\leq \\ 0", "type": "inline_equation" }, { "bbox": [ 379, 335, 466, 348 ], "score": 1.0, "content": "; for relative entropy", "type": "text" }, { "bbox": [ 466, 335, 505, 347 ], "score": 0.89, "content": "\\Omega ( \\pi _ { t } ) =", "type": "inline_equation" } ], "index": 18 }, { "bbox": [ 107, 345, 504, 359 ], "spans": [ { "bbox": [ 107, 346, 163, 358 ], "score": 0.91, "content": "\\mathrm { K L } ( \\pi _ { t } | | \\pi _ { t - 1 } )", "type": "inline_equation" }, { "bbox": [ 163, 345, 219, 359 ], "score": 1.0, "content": ", if we define", "type": "text" }, { "bbox": [ 219, 347, 308, 358 ], "score": 0.89, "content": "m = \\mathrm { m i n } _ { a } \\pi _ { t - 1 } ( a | s )", "type": "inline_equation" }, { "bbox": [ 308, 345, 393, 359 ], "score": 1.0, "content": ", then we can derive", "type": "text" }, { "bbox": [ 393, 346, 504, 358 ], "score": 0.88, "content": "0 \\leq \\Omega ( \\pi _ { t } ) \\leq - \\log | \\mathcal { A } | +", "type": "inline_equation" } ], "index": 19 }, { "bbox": [ 106, 357, 485, 376 ], "spans": [ { "bbox": [ 106, 359, 131, 372 ], "score": 0.88, "content": "\\log { \\frac { 1 } { m } }", "type": "inline_equation" }, { "bbox": [ 132, 357, 228, 376 ], "score": 1.0, "content": "; and for Tsallis entropy", "type": "text" }, { "bbox": [ 229, 359, 326, 373 ], "score": 0.93, "content": "\\Omega ( \\pi _ { t } ) = { \\textstyle { \\frac { 1 } { 2 } } } ( \\parallel \\pi _ { t } \\parallel _ { 2 } ^ { 2 } - 1 )", "type": "inline_equation" }, { "bbox": [ 326, 357, 365, 376 ], "score": 1.0, "content": ", we have", "type": "text" }, { "bbox": [ 365, 357, 453, 374 ], "score": 0.94, "content": "- \\frac { | \\ r _ { A } | - 1 } { 2 | \\ r _ { A } | } \\le \\Omega ( \\pi _ { t } ) \\le 0", "type": "inline_equation" }, { "bbox": [ 454, 357, 485, 375 ], "score": 1.0, "content": ". Then,", "type": "text" } ], "index": 20 } ], "index": 18 }, { "type": "interline_equation", "bbox": [ 260, 381, 470, 415 ], "lines": [], "index": 21.5, "virtual_lines": [ { "bbox": [ 260, 381, 470, 398.0 ], "spans": [], "index": 21 }, { "bbox": [ 260, 398.0, 470, 415.0 ], "spans": [], "index": 22 } ] }, { "type": "text", "bbox": [ 106, 422, 505, 455 ], "lines": [ { "bbox": [ 104, 422, 504, 456 ], "spans": [ { "bbox": [ 104, 429, 269, 451 ], "score": 1.0, "content": "Corollary 5 relative entropy error: −", "type": "text" }, { "bbox": [ 252, 422, 504, 456 ], "score": 0.82, "content": "- \\sqrt { \\frac { \\hat { C } \\sigma ^ { 2 } \\log { \\frac { C } { \\delta } } } { 2 N ( s ) } } - \\frac { \\tau ( \\log | A | - \\log \\frac { 1 } { m } ) } { 1 - \\gamma } \\leq \\varepsilon _ { \\Omega } \\leq \\sqrt { \\frac { \\hat { C } \\sigma ^ { 2 } \\log \\frac { C } { \\delta } } { 2 N ( s ) } } .", "type": "inline_equation" } ], "index": 23 } ], "index": 23 }, { "type": "text", "bbox": [ 107, 462, 480, 496 ], "lines": [ { "bbox": [ 106, 462, 479, 496 ], "spans": [ { "bbox": [ 106, 472, 248, 489 ], "score": 1.0, "content": "Corollary 6 Tsallis entropy error:", "type": "text" }, { "bbox": [ 248, 462, 479, 496 ], "score": 0.7, "content": "- \\sqrt { \\frac { \\hat { C } \\sigma ^ { 2 } \\log \\frac { C } { \\delta } } { 2 N ( s ) } } - \\frac { | A | - 1 } { 2 | A | } \\frac { \\tau } { 1 - \\gamma } \\leq \\varepsilon _ { \\Omega } \\leq \\sqrt { \\frac { \\hat { C } \\sigma ^ { 2 } \\log \\frac { C } { \\delta } } { 2 N ( s ) } } .", "type": "inline_equation" } ], "index": 24 } ], "index": 24 }, { "type": "text", "bbox": [ 107, 503, 505, 526 ], "lines": [ { "bbox": [ 105, 503, 505, 516 ], "spans": [ { "bbox": [ 105, 503, 317, 516 ], "score": 1.0, "content": "These results show that when the number of actions", "type": "text" }, { "bbox": [ 317, 504, 332, 516 ], "score": 0.92, "content": "| { \\cal { A } } |", "type": "inline_equation" }, { "bbox": [ 333, 503, 505, 516 ], "score": 1.0, "content": "is large, TENTS enjoys the smallest error;", "type": "text" } ], "index": 25 }, { "bbox": [ 105, 514, 455, 527 ], "spans": [ { "bbox": [ 105, 514, 455, 527 ], "score": 1.0, "content": "moreover, we also see that lower bound of RENTS is always smaller than for MENTS.", "type": "text" } ], "index": 26 } ], "index": 25.5 }, { "type": "title", "bbox": [ 108, 542, 253, 554 ], "lines": [ { "bbox": [ 105, 541, 254, 556 ], "spans": [ { "bbox": [ 105, 541, 254, 556 ], "score": 1.0, "content": "5 EMPIRICAL EVALUATION", "type": "text" } ], "index": 27 } ], "index": 27 }, { "type": "text", "bbox": [ 107, 565, 505, 676 ], "lines": [ { "bbox": [ 106, 567, 504, 578 ], "spans": [ { "bbox": [ 106, 567, 504, 578 ], "score": 1.0, "content": "In this section, we empirically evaluate the benefit of the proposed entropy-based MCTS regular-", "type": "text" } ], "index": 28 }, { "bbox": [ 106, 578, 504, 589 ], "spans": [ { "bbox": [ 106, 578, 504, 589 ], "score": 1.0, "content": "izers. First, we complement our theoretical analysis with an empirical study of the synthetic tree", "type": "text" } ], "index": 29 }, { "bbox": [ 105, 588, 505, 601 ], "spans": [ { "bbox": [ 105, 588, 505, 601 ], "score": 1.0, "content": "toy problem introduced in Xiao et al. (2019), which serves as a simple scenario to give an inter-", "type": "text" } ], "index": 30 }, { "bbox": [ 105, 598, 506, 612 ], "spans": [ { "bbox": [ 105, 598, 506, 612 ], "score": 1.0, "content": "pretable demonstration of the effects of our theoretical results in practice. Second, we compare to", "type": "text" } ], "index": 31 }, { "bbox": [ 106, 610, 505, 622 ], "spans": [ { "bbox": [ 106, 610, 505, 622 ], "score": 1.0, "content": "AlphaGo and AlphaZero (Silver et al., 2016; 2017a), recently introduced to enable MCTS to solve", "type": "text" } ], "index": 32 }, { "bbox": [ 106, 622, 505, 633 ], "spans": [ { "bbox": [ 106, 622, 505, 633 ], "score": 1.0, "content": "large scale problems with high branching factor. Our implementation is a simplified version of the", "type": "text" } ], "index": 33 }, { "bbox": [ 105, 632, 506, 645 ], "spans": [ { "bbox": [ 105, 632, 506, 645 ], "score": 1.0, "content": "original algorithms, where we remove various tricks in favor of better interpretability. For the same", "type": "text" } ], "index": 34 }, { "bbox": [ 105, 641, 506, 656 ], "spans": [ { "bbox": [ 105, 641, 506, 656 ], "score": 1.0, "content": "reason, we do not compare with the most recent and state-of-the-art variant of AlphaZero known as", "type": "text" } ], "index": 35 }, { "bbox": [ 105, 654, 505, 666 ], "spans": [ { "bbox": [ 105, 654, 505, 666 ], "score": 1.0, "content": "MuZero (Schrittwieser et al., 2019), as this is a slightly different solution highly tuned to maximize", "type": "text" } ], "index": 36 }, { "bbox": [ 105, 664, 419, 677 ], "spans": [ { "bbox": [ 105, 664, 419, 677 ], "score": 1.0, "content": "performance, and a detailed description of its implementation is not available.", "type": "text" } ], "index": 37 } ], "index": 32.5 }, { "type": "title", "bbox": [ 107, 689, 206, 700 ], "lines": [ { "bbox": [ 106, 688, 207, 701 ], "spans": [ { "bbox": [ 106, 688, 207, 701 ], "score": 1.0, "content": "5.1 SYNTHETIC TREE", "type": "text" } ], "index": 38 } ], "index": 38 }, { "type": "text", "bbox": [ 107, 709, 505, 732 ], "lines": [ { "bbox": [ 106, 709, 505, 722 ], "spans": [ { "bbox": [ 106, 709, 505, 722 ], "score": 1.0, "content": "This toy problem is introduced in Xiao et al. (2019) to highlight the improvement of MENTS over", "type": "text" } ], "index": 39 }, { "bbox": [ 106, 720, 505, 733 ], "spans": [ { "bbox": [ 106, 720, 303, 733 ], "score": 1.0, "content": "UCT. It consists of a tree with branching factor", "type": "text" }, { "bbox": [ 304, 722, 310, 730 ], "score": 0.82, "content": "k", "type": "inline_equation" }, { "bbox": [ 310, 720, 355, 733 ], "score": 1.0, "content": "and depth", "type": "text" }, { "bbox": [ 356, 722, 361, 730 ], "score": 0.79, "content": "d", "type": "inline_equation" }, { "bbox": [ 362, 720, 505, 733 ], "score": 1.0, "content": ". Each edge of the tree is assigned", "type": "text" } ], "index": 40 } ], "index": 39.5 } ], "page_idx": 5, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 302, 752, 308, 760 ], "lines": [ { "bbox": [ 302, 751, 309, 762 ], "spans": [ { "bbox": [ 302, 751, 309, 762 ], "score": 1.0, "content": "6", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 107, 27, 307, 37 ], "lines": [ { "bbox": [ 106, 26, 308, 38 ], "spans": [ { "bbox": [ 106, 26, 308, 38 ], "score": 1.0, "content": "Under review as a conference paper at ICLR 2021", "type": "text" } ] } ] } ], "para_blocks": [ { "type": "text", "bbox": [ 107, 82, 505, 182 ], "lines": [ { "bbox": [ 106, 82, 505, 95 ], "spans": [ { "bbox": [ 106, 82, 505, 95 ], "score": 1.0, "content": "Remarks. The regret bound of UCT and its variance have already been analyzed for non-", "type": "text" } ], "index": 0 }, { "bbox": [ 105, 93, 505, 106 ], "spans": [ { "bbox": [ 105, 93, 505, 106 ], "score": 1.0, "content": "regularized MCTS with binary tree (Coquelin & Munos, 2007). On the contrary, our regret bound", "type": "text" } ], "index": 1 }, { "bbox": [ 106, 105, 505, 117 ], "spans": [ { "bbox": [ 106, 105, 505, 117 ], "score": 1.0, "content": "analysis in Theorem 3 applies to generic regularized MCTS. From the specialized bounds in the", "type": "text" } ], "index": 2 }, { "bbox": [ 105, 115, 505, 128 ], "spans": [ { "bbox": [ 105, 115, 505, 128 ], "score": 1.0, "content": "corollaries, we observe that the maximum and relative entropy share similar results, although the", "type": "text" } ], "index": 3 }, { "bbox": [ 104, 125, 507, 143 ], "spans": [ { "bbox": [ 104, 125, 328, 143 ], "score": 1.0, "content": "bounds for relative entropy are slightly smaller due to", "type": "text" }, { "bbox": [ 328, 126, 338, 139 ], "score": 0.89, "content": "\\textstyle { \\frac { 1 } { m } }", "type": "inline_equation" }, { "bbox": [ 339, 125, 507, 143 ], "score": 1.0, "content": ". Remarkably, the bounds for Tsallis en-", "type": "text" } ], "index": 4 }, { "bbox": [ 105, 137, 505, 150 ], "spans": [ { "bbox": [ 105, 137, 505, 150 ], "score": 1.0, "content": "tropy become tighter for increasing number of actions, which translates in limited regret in problems", "type": "text" } ], "index": 5 }, { "bbox": [ 106, 149, 505, 160 ], "spans": [ { "bbox": [ 106, 149, 505, 160 ], "score": 1.0, "content": "with high branching factor. This result establishes the advantage of Tsallis entropy in complex prob-", "type": "text" } ], "index": 6 }, { "bbox": [ 105, 159, 505, 172 ], "spans": [ { "bbox": [ 105, 159, 505, 172 ], "score": 1.0, "content": "lems w.r.t. to other entropy regularizers, as empirically confirmed by the positive results in several", "type": "text" } ], "index": 7 }, { "bbox": [ 106, 171, 250, 182 ], "spans": [ { "bbox": [ 106, 171, 250, 182 ], "score": 1.0, "content": "Atari games described in Section 5.", "type": "text" } ], "index": 8 } ], "index": 4, "bbox_fs": [ 104, 82, 507, 182 ] }, { "type": "title", "bbox": [ 107, 194, 207, 205 ], "lines": [ { "bbox": [ 105, 193, 207, 207 ], "spans": [ { "bbox": [ 105, 193, 207, 207 ], "score": 1.0, "content": "4.2 ERROR ANALYSIS", "type": "text" } ], "index": 9 } ], "index": 9 }, { "type": "text", "bbox": [ 107, 214, 504, 238 ], "lines": [ { "bbox": [ 106, 214, 505, 228 ], "spans": [ { "bbox": [ 106, 214, 387, 228 ], "score": 1.0, "content": "We analyse the error of the regularized value estimate at the root node", "type": "text" }, { "bbox": [ 388, 215, 407, 227 ], "score": 0.91, "content": "n ( s )", "type": "inline_equation" }, { "bbox": [ 407, 214, 505, 228 ], "score": 1.0, "content": "w.r.t. the optimal value:", "type": "text" } ], "index": 10 }, { "bbox": [ 106, 225, 198, 240 ], "spans": [ { "bbox": [ 106, 226, 193, 238 ], "score": 0.92, "content": "\\varepsilon _ { \\Omega } = \\dot { V _ { \\Omega } } ( s ) - \\dot { V ^ { \\ast } } ( s )", "type": "inline_equation" }, { "bbox": [ 194, 225, 198, 240 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 11 } ], "index": 10.5, "bbox_fs": [ 106, 214, 505, 240 ] }, { "type": "text", "bbox": [ 106, 247, 504, 270 ], "lines": [ { "bbox": [ 105, 245, 505, 261 ], "spans": [ { "bbox": [ 105, 245, 191, 261 ], "score": 1.0, "content": "Theorem 4 For any", "type": "text" }, { "bbox": [ 191, 248, 217, 258 ], "score": 0.89, "content": "\\delta > 0", "type": "inline_equation" }, { "bbox": [ 217, 245, 344, 261 ], "score": 1.0, "content": "and generic convex regularizer", "type": "text" }, { "bbox": [ 344, 248, 353, 258 ], "score": 0.77, "content": "\\Omega", "type": "inline_equation" }, { "bbox": [ 353, 245, 436, 261 ], "score": 1.0, "content": ", with some constant", "type": "text" }, { "bbox": [ 437, 246, 457, 259 ], "score": 0.9, "content": "C , { \\hat { C } }", "type": "inline_equation" }, { "bbox": [ 457, 245, 505, 261 ], "score": 1.0, "content": ", with prob-", "type": "text" } ], "index": 12 }, { "bbox": [ 105, 257, 241, 271 ], "spans": [ { "bbox": [ 105, 257, 166, 271 ], "score": 1.0, "content": "ability at least", "type": "text" }, { "bbox": [ 166, 259, 189, 269 ], "score": 0.57, "content": "1 - \\delta", "type": "inline_equation" }, { "bbox": [ 189, 257, 192, 271 ], "score": 1.0, "content": ",", "type": "text" }, { "bbox": [ 193, 260, 205, 270 ], "score": 0.66, "content": "\\varepsilon _ { \\Omega }", "type": "inline_equation" }, { "bbox": [ 205, 257, 241, 271 ], "score": 1.0, "content": "satisfies", "type": "text" } ], "index": 13 } ], "index": 12.5, "bbox_fs": [ 105, 245, 505, 271 ] }, { "type": "interline_equation", "bbox": [ 189, 272, 422, 306 ], "lines": [ { "bbox": [ 189, 272, 422, 306 ], "spans": [ { "bbox": [ 189, 272, 422, 306 ], "score": 0.94, "content": "- \\sqrt { \\frac { \\hat { C } \\sigma ^ { 2 } \\log \\frac { C } { \\delta } } { 2 N ( s ) } } - \\frac { \\tau ( U _ { \\Omega } - L _ { \\Omega } ) } { 1 - \\gamma } \\leq \\varepsilon _ { \\Omega } \\leq \\sqrt { \\frac { \\hat { C } \\sigma ^ { 2 } \\log \\frac { C } { \\delta } } { 2 N ( s ) } } .", "type": "interline_equation", "image_path": "ae4cbab37a8eb4b4691183517f81846a12faf589033d60727a578b553c5954d2.jpg" } ] } ], "index": 14.5, "virtual_lines": [ { "bbox": [ 189, 272, 422, 289.0 ], "spans": [], "index": 14 }, { "bbox": [ 189, 289.0, 422, 306.0 ], "spans": [], "index": 15 } ] }, { "type": "text", "bbox": [ 106, 312, 505, 374 ], "lines": [ { "bbox": [ 106, 313, 505, 325 ], "spans": [ { "bbox": [ 106, 313, 505, 325 ], "score": 1.0, "content": "To give a better understanding of the effect of each entropy regularizer in Table 1, we specialize", "type": "text" } ], "index": 16 }, { "bbox": [ 106, 324, 504, 335 ], "spans": [ { "bbox": [ 106, 324, 504, 335 ], "score": 1.0, "content": "the bound in Equation 14 to each of them. From (Lee et al., 2018), we know that for maximum", "type": "text" } ], "index": 17 }, { "bbox": [ 105, 335, 505, 348 ], "spans": [ { "bbox": [ 105, 335, 140, 348 ], "score": 1.0, "content": "entropy", "type": "text" }, { "bbox": [ 141, 335, 234, 348 ], "score": 0.92, "content": "\\begin{array} { r } { \\Omega ( { \\boldsymbol \\pi } _ { t } ) \\stackrel { - } { = } \\sum _ { a } \\pi _ { t } \\log \\pi _ { t } } \\end{array}", "type": "inline_equation" }, { "bbox": [ 235, 335, 276, 348 ], "score": 1.0, "content": ", we have", "type": "text" }, { "bbox": [ 277, 335, 379, 347 ], "score": 0.9, "content": "- \\log | \\mathcal { A } | \\ \\leq \\ \\Omega ( \\pi _ { t } ) \\ \\leq \\ 0", "type": "inline_equation" }, { "bbox": [ 379, 335, 466, 348 ], "score": 1.0, "content": "; for relative entropy", "type": "text" }, { "bbox": [ 466, 335, 505, 347 ], "score": 0.89, "content": "\\Omega ( \\pi _ { t } ) =", "type": "inline_equation" } ], "index": 18 }, { "bbox": [ 107, 345, 504, 359 ], "spans": [ { "bbox": [ 107, 346, 163, 358 ], "score": 0.91, "content": "\\mathrm { K L } ( \\pi _ { t } | | \\pi _ { t - 1 } )", "type": "inline_equation" }, { "bbox": [ 163, 345, 219, 359 ], "score": 1.0, "content": ", if we define", "type": "text" }, { "bbox": [ 219, 347, 308, 358 ], "score": 0.89, "content": "m = \\mathrm { m i n } _ { a } \\pi _ { t - 1 } ( a | s )", "type": "inline_equation" }, { "bbox": [ 308, 345, 393, 359 ], "score": 1.0, "content": ", then we can derive", "type": "text" }, { "bbox": [ 393, 346, 504, 358 ], "score": 0.88, "content": "0 \\leq \\Omega ( \\pi _ { t } ) \\leq - \\log | \\mathcal { A } | +", "type": "inline_equation" } ], "index": 19 }, { "bbox": [ 106, 357, 485, 376 ], "spans": [ { "bbox": [ 106, 359, 131, 372 ], "score": 0.88, "content": "\\log { \\frac { 1 } { m } }", "type": "inline_equation" }, { "bbox": [ 132, 357, 228, 376 ], "score": 1.0, "content": "; and for Tsallis entropy", "type": "text" }, { "bbox": [ 229, 359, 326, 373 ], "score": 0.93, "content": "\\Omega ( \\pi _ { t } ) = { \\textstyle { \\frac { 1 } { 2 } } } ( \\parallel \\pi _ { t } \\parallel _ { 2 } ^ { 2 } - 1 )", "type": "inline_equation" }, { "bbox": [ 326, 357, 365, 376 ], "score": 1.0, "content": ", we have", "type": "text" }, { "bbox": [ 365, 357, 453, 374 ], "score": 0.94, "content": "- \\frac { | \\ r _ { A } | - 1 } { 2 | \\ r _ { A } | } \\le \\Omega ( \\pi _ { t } ) \\le 0", "type": "inline_equation" }, { "bbox": [ 454, 357, 485, 375 ], "score": 1.0, "content": ". Then,", "type": "text" } ], "index": 20 } ], "index": 18, "bbox_fs": [ 105, 313, 505, 376 ] }, { "type": "interline_equation", "bbox": [ 260, 381, 470, 415 ], "lines": [], "index": 21.5, "virtual_lines": [ { "bbox": [ 260, 381, 470, 398.0 ], "spans": [], "index": 21 }, { "bbox": [ 260, 398.0, 470, 415.0 ], "spans": [], "index": 22 } ] }, { "type": "text", "bbox": [ 106, 422, 505, 455 ], "lines": [ { "bbox": [ 104, 422, 504, 456 ], "spans": [ { "bbox": [ 104, 429, 269, 451 ], "score": 1.0, "content": "Corollary 5 relative entropy error: −", "type": "text" }, { "bbox": [ 252, 422, 504, 456 ], "score": 0.82, "content": "- \\sqrt { \\frac { \\hat { C } \\sigma ^ { 2 } \\log { \\frac { C } { \\delta } } } { 2 N ( s ) } } - \\frac { \\tau ( \\log | A | - \\log \\frac { 1 } { m } ) } { 1 - \\gamma } \\leq \\varepsilon _ { \\Omega } \\leq \\sqrt { \\frac { \\hat { C } \\sigma ^ { 2 } \\log \\frac { C } { \\delta } } { 2 N ( s ) } } .", "type": "inline_equation" } ], "index": 23 } ], "index": 23, "bbox_fs": [ 104, 422, 504, 456 ] }, { "type": "text", "bbox": [ 107, 462, 480, 496 ], "lines": [ { "bbox": [ 106, 462, 479, 496 ], "spans": [ { "bbox": [ 106, 472, 248, 489 ], "score": 1.0, "content": "Corollary 6 Tsallis entropy error:", "type": "text" }, { "bbox": [ 248, 462, 479, 496 ], "score": 0.7, "content": "- \\sqrt { \\frac { \\hat { C } \\sigma ^ { 2 } \\log \\frac { C } { \\delta } } { 2 N ( s ) } } - \\frac { | A | - 1 } { 2 | A | } \\frac { \\tau } { 1 - \\gamma } \\leq \\varepsilon _ { \\Omega } \\leq \\sqrt { \\frac { \\hat { C } \\sigma ^ { 2 } \\log \\frac { C } { \\delta } } { 2 N ( s ) } } .", "type": "inline_equation" } ], "index": 24 } ], "index": 24, "bbox_fs": [ 106, 462, 479, 496 ] }, { "type": "list", "bbox": [ 107, 503, 505, 526 ], "lines": [ { "bbox": [ 105, 503, 505, 516 ], "spans": [ { "bbox": [ 105, 503, 317, 516 ], "score": 1.0, "content": "These results show that when the number of actions", "type": "text" }, { "bbox": [ 317, 504, 332, 516 ], "score": 0.92, "content": "| { \\cal { A } } |", "type": "inline_equation" }, { "bbox": [ 333, 503, 505, 516 ], "score": 1.0, "content": "is large, TENTS enjoys the smallest error;", "type": "text" } ], "index": 25, "is_list_end_line": true }, { "bbox": [ 105, 514, 455, 527 ], "spans": [ { "bbox": [ 105, 514, 455, 527 ], "score": 1.0, "content": "moreover, we also see that lower bound of RENTS is always smaller than for MENTS.", "type": "text" } ], "index": 26, "is_list_start_line": true, "is_list_end_line": true } ], "index": 25.5, "bbox_fs": [ 105, 503, 505, 527 ] }, { "type": "title", "bbox": [ 108, 542, 253, 554 ], "lines": [ { "bbox": [ 105, 541, 254, 556 ], "spans": [ { "bbox": [ 105, 541, 254, 556 ], "score": 1.0, "content": "5 EMPIRICAL EVALUATION", "type": "text" } ], "index": 27 } ], "index": 27 }, { "type": "text", "bbox": [ 107, 565, 505, 676 ], "lines": [ { "bbox": [ 106, 567, 504, 578 ], "spans": [ { "bbox": [ 106, 567, 504, 578 ], "score": 1.0, "content": "In this section, we empirically evaluate the benefit of the proposed entropy-based MCTS regular-", "type": "text" } ], "index": 28 }, { "bbox": [ 106, 578, 504, 589 ], "spans": [ { "bbox": [ 106, 578, 504, 589 ], "score": 1.0, "content": "izers. First, we complement our theoretical analysis with an empirical study of the synthetic tree", "type": "text" } ], "index": 29 }, { "bbox": [ 105, 588, 505, 601 ], "spans": [ { "bbox": [ 105, 588, 505, 601 ], "score": 1.0, "content": "toy problem introduced in Xiao et al. (2019), which serves as a simple scenario to give an inter-", "type": "text" } ], "index": 30 }, { "bbox": [ 105, 598, 506, 612 ], "spans": [ { "bbox": [ 105, 598, 506, 612 ], "score": 1.0, "content": "pretable demonstration of the effects of our theoretical results in practice. Second, we compare to", "type": "text" } ], "index": 31 }, { "bbox": [ 106, 610, 505, 622 ], "spans": [ { "bbox": [ 106, 610, 505, 622 ], "score": 1.0, "content": "AlphaGo and AlphaZero (Silver et al., 2016; 2017a), recently introduced to enable MCTS to solve", "type": "text" } ], "index": 32 }, { "bbox": [ 106, 622, 505, 633 ], "spans": [ { "bbox": [ 106, 622, 505, 633 ], "score": 1.0, "content": "large scale problems with high branching factor. Our implementation is a simplified version of the", "type": "text" } ], "index": 33 }, { "bbox": [ 105, 632, 506, 645 ], "spans": [ { "bbox": [ 105, 632, 506, 645 ], "score": 1.0, "content": "original algorithms, where we remove various tricks in favor of better interpretability. For the same", "type": "text" } ], "index": 34 }, { "bbox": [ 105, 641, 506, 656 ], "spans": [ { "bbox": [ 105, 641, 506, 656 ], "score": 1.0, "content": "reason, we do not compare with the most recent and state-of-the-art variant of AlphaZero known as", "type": "text" } ], "index": 35 }, { "bbox": [ 105, 654, 505, 666 ], "spans": [ { "bbox": [ 105, 654, 505, 666 ], "score": 1.0, "content": "MuZero (Schrittwieser et al., 2019), as this is a slightly different solution highly tuned to maximize", "type": "text" } ], "index": 36 }, { "bbox": [ 105, 664, 419, 677 ], "spans": [ { "bbox": [ 105, 664, 419, 677 ], "score": 1.0, "content": "performance, and a detailed description of its implementation is not available.", "type": "text" } ], "index": 37 } ], "index": 32.5, "bbox_fs": [ 105, 567, 506, 677 ] }, { "type": "title", "bbox": [ 107, 689, 206, 700 ], "lines": [ { "bbox": [ 106, 688, 207, 701 ], "spans": [ { "bbox": [ 106, 688, 207, 701 ], "score": 1.0, "content": "5.1 SYNTHETIC TREE", "type": "text" } ], "index": 38 } ], "index": 38 }, { "type": "text", "bbox": [ 107, 709, 505, 732 ], "lines": [ { "bbox": [ 106, 709, 505, 722 ], "spans": [ { "bbox": [ 106, 709, 505, 722 ], "score": 1.0, "content": "This toy problem is introduced in Xiao et al. (2019) to highlight the improvement of MENTS over", "type": "text" } ], "index": 39 }, { "bbox": [ 106, 720, 505, 733 ], "spans": [ { "bbox": [ 106, 720, 303, 733 ], "score": 1.0, "content": "UCT. It consists of a tree with branching factor", "type": "text" }, { "bbox": [ 304, 722, 310, 730 ], "score": 0.82, "content": "k", "type": "inline_equation" }, { "bbox": [ 310, 720, 355, 733 ], "score": 1.0, "content": "and depth", "type": "text" }, { "bbox": [ 356, 722, 361, 730 ], "score": 0.79, "content": "d", "type": "inline_equation" }, { "bbox": [ 362, 720, 505, 733 ], "score": 1.0, "content": ". Each edge of the tree is assigned", "type": "text" } ], "index": 40 }, { "bbox": [ 104, 549, 506, 561 ], "spans": [ { "bbox": [ 104, 549, 506, 561 ], "score": 1.0, "content": "a random value between 0 and 1. At each leaf, a Gaussian distribution is used as an evaluation", "type": "text", "cross_page": true } ], "index": 11 }, { "bbox": [ 106, 560, 505, 572 ], "spans": [ { "bbox": [ 106, 560, 505, 572 ], "score": 1.0, "content": "function resembling the return of random rollouts. The mean of the Gaussian distribution is the", "type": "text", "cross_page": true } ], "index": 12 }, { "bbox": [ 105, 571, 506, 583 ], "spans": [ { "bbox": [ 105, 571, 506, 583 ], "score": 1.0, "content": "sum of the values assigned to the edges connecting the root node to the considered leaf, while the", "type": "text", "cross_page": true } ], "index": 13 }, { "bbox": [ 106, 581, 505, 594 ], "spans": [ { "bbox": [ 106, 581, 192, 594 ], "score": 1.0, "content": "standard deviation is", "type": "text", "cross_page": true }, { "bbox": [ 193, 581, 237, 592 ], "score": 0.91, "content": "\\bar { \\sigma } = 0 . 0 5 ^ { 1 }", "type": "inline_equation", "cross_page": true }, { "bbox": [ 237, 581, 505, 594 ], "score": 1.0, "content": ". For stability, all the means are normalized between 0 and 1. As", "type": "text", "cross_page": true } ], "index": 14 }, { "bbox": [ 105, 593, 505, 605 ], "spans": [ { "bbox": [ 105, 593, 505, 605 ], "score": 1.0, "content": "in Xiao et al. (2019), we create 5 trees on which we perform 5 different runs in each, resulting in 25", "type": "text", "cross_page": true } ], "index": 15 }, { "bbox": [ 106, 604, 505, 617 ], "spans": [ { "bbox": [ 106, 604, 339, 617 ], "score": 1.0, "content": "experiments, for all the combinations of branching factor", "type": "text", "cross_page": true }, { "bbox": [ 339, 604, 461, 616 ], "score": 0.91, "content": "k = \\{ 2 , 4 , 6 , 8 , 1 0 , 1 2 , 1 4 , 1 6 \\}", "type": "inline_equation", "cross_page": true }, { "bbox": [ 462, 604, 505, 617 ], "score": 1.0, "content": "and depth", "type": "text", "cross_page": true } ], "index": 16 }, { "bbox": [ 107, 615, 506, 627 ], "spans": [ { "bbox": [ 107, 615, 180, 627 ], "score": 0.91, "content": "d = \\{ 1 , 2 , 3 , 4 , 5 \\}", "type": "inline_equation", "cross_page": true }, { "bbox": [ 181, 615, 506, 627 ], "score": 1.0, "content": ", computing: (i) the value estimation error at the root node w.r.t. the regularized", "type": "text", "cross_page": true } ], "index": 17 }, { "bbox": [ 105, 626, 506, 638 ], "spans": [ { "bbox": [ 105, 626, 166, 638 ], "score": 1.0, "content": "optimal value:", "type": "text", "cross_page": true }, { "bbox": [ 167, 626, 227, 637 ], "score": 0.9, "content": "\\begin{array} { r } { \\varepsilon _ { \\Omega } = V _ { \\Omega } - V * ; } \\end{array}", "type": "inline_equation", "cross_page": true }, { "bbox": [ 228, 626, 506, 638 ], "score": 1.0, "content": "(ii) the value estimation error at the root node w.r.t. the unregularized", "type": "text", "cross_page": true } ], "index": 18 }, { "bbox": [ 106, 637, 505, 650 ], "spans": [ { "bbox": [ 106, 637, 167, 650 ], "score": 1.0, "content": "optimal value:", "type": "text", "cross_page": true }, { "bbox": [ 168, 637, 256, 648 ], "score": 0.88, "content": "\\varepsilon _ { \\mathrm { U C T } } = V _ { \\Omega } - V * _ { \\mathrm { U C T } } ", "type": "inline_equation", "cross_page": true }, { "bbox": [ 256, 637, 318, 650 ], "score": 1.0, "content": "; (iii) the regret", "type": "text", "cross_page": true }, { "bbox": [ 318, 637, 328, 647 ], "score": 0.76, "content": "R", "type": "inline_equation", "cross_page": true }, { "bbox": [ 328, 637, 505, 650 ], "score": 1.0, "content": "as in Equation (13). For a fair comparison,", "type": "text", "cross_page": true } ], "index": 19 }, { "bbox": [ 106, 648, 505, 660 ], "spans": [ { "bbox": [ 106, 648, 159, 660 ], "score": 1.0, "content": "we use fixed", "type": "text", "cross_page": true }, { "bbox": [ 160, 649, 195, 658 ], "score": 0.88, "content": "\\tau = 0 . 1", "type": "inline_equation", "cross_page": true }, { "bbox": [ 195, 648, 213, 660 ], "score": 1.0, "content": "and", "type": "text", "cross_page": true }, { "bbox": [ 213, 649, 246, 658 ], "score": 0.88, "content": "\\epsilon = 0 . 1", "type": "inline_equation", "cross_page": true }, { "bbox": [ 247, 648, 505, 660 ], "score": 1.0, "content": "across all algorithms. Figure 1 and 2 show how UCT and each", "type": "text", "cross_page": true } ], "index": 20 }, { "bbox": [ 104, 658, 506, 671 ], "spans": [ { "bbox": [ 104, 658, 506, 671 ], "score": 1.0, "content": "regularizer behave for different configurations of the tree. We observe that, while RENTS and", "type": "text", "cross_page": true } ], "index": 21 }, { "bbox": [ 105, 669, 506, 682 ], "spans": [ { "bbox": [ 105, 669, 506, 682 ], "score": 1.0, "content": "MENTS converge slower for increasing tree sizes, TENTS is robust w.r.t. the size of the tree and", "type": "text", "cross_page": true } ], "index": 22 }, { "bbox": [ 106, 680, 505, 693 ], "spans": [ { "bbox": [ 106, 680, 505, 693 ], "score": 1.0, "content": "almost always converges faster than all other methods to the respective optimal value. Notably, the", "type": "text", "cross_page": true } ], "index": 23 }, { "bbox": [ 106, 692, 505, 704 ], "spans": [ { "bbox": [ 106, 692, 505, 704 ], "score": 1.0, "content": "optimal value of TENTS seems to be very close to the one of UCT, i.e. the optimal value of the", "type": "text", "cross_page": true } ], "index": 24 }, { "bbox": [ 105, 271, 505, 283 ], "spans": [ { "bbox": [ 105, 271, 505, 283 ], "score": 1.0, "content": "unregularized objective, and also converges faster than the one estimated by UCT, while MENTS", "type": "text", "cross_page": true } ], "index": 5 }, { "bbox": [ 105, 282, 505, 294 ], "spans": [ { "bbox": [ 105, 282, 505, 294 ], "score": 1.0, "content": "and RENTS are considerably further from this value. In terms of regret, UCT explores less than", "type": "text", "cross_page": true } ], "index": 6 }, { "bbox": [ 105, 293, 505, 306 ], "spans": [ { "bbox": [ 105, 293, 505, 306 ], "score": 1.0, "content": "the regularized methods and it is less prone to high regret, at the cost of slower convergence time.", "type": "text", "cross_page": true } ], "index": 7 }, { "bbox": [ 105, 304, 505, 316 ], "spans": [ { "bbox": [ 105, 304, 505, 316 ], "score": 1.0, "content": "Nevertheless, the regret of TENTS is the smallest between the ones of the other regularizers, which", "type": "text", "cross_page": true } ], "index": 8 }, { "bbox": [ 105, 314, 505, 328 ], "spans": [ { "bbox": [ 105, 314, 505, 328 ], "score": 1.0, "content": "seem to explore too much. These results show a general superiority of TENTS in this toy problem,", "type": "text", "cross_page": true } ], "index": 9 }, { "bbox": [ 105, 326, 505, 339 ], "spans": [ { "bbox": [ 105, 326, 505, 339 ], "score": 1.0, "content": "also confirming our theoretical findings about the advantage of TENTS in terms of approximation", "type": "text", "cross_page": true } ], "index": 10 }, { "bbox": [ 106, 338, 410, 349 ], "spans": [ { "bbox": [ 106, 338, 410, 349 ], "score": 1.0, "content": "error (Corollary 6) and regret (Corollary 3), in problems with many actions.", "type": "text", "cross_page": true } ], "index": 11 } ], "index": 39.5, "bbox_fs": [ 106, 709, 505, 733 ] } ] }, { "preproc_blocks": [ { "type": "image", "bbox": [ 126, 78, 485, 293 ], "blocks": [ { "type": "image_body", "bbox": [ 126, 78, 485, 293 ], "group_id": 0, "lines": [ { "bbox": [ 126, 78, 485, 293 ], "spans": [ { "bbox": [ 126, 78, 485, 293 ], "score": 0.97, "type": "image", "image_path": "9960ecdcaae3fb83b1b2bbe1e74f8fa6d0c0402e298a7b6c52eda1abbfac2d7e.jpg" } ] } ], "index": 1, "virtual_lines": [ { "bbox": [ 126, 78, 485, 149.66666666666669 ], "spans": [], "index": 0 }, { "bbox": [ 126, 149.66666666666669, 485, 221.33333333333337 ], "spans": [], "index": 1 }, { "bbox": [ 126, 221.33333333333337, 485, 293.00000000000006 ], "spans": [], "index": 2 } ] }, { "type": "image_caption", "bbox": [ 106, 302, 505, 325 ], "group_id": 0, "lines": [ { "bbox": [ 106, 303, 505, 314 ], "spans": [ { "bbox": [ 106, 303, 505, 314 ], "score": 1.0, "content": "Figure 1: For each algorithm, we show the convergence of the value estimate at the root node to the", "type": "text" } ], "index": 3 }, { "bbox": [ 105, 313, 470, 326 ], "spans": [ { "bbox": [ 105, 313, 470, 326 ], "score": 1.0, "content": "respective optimal value (top), to the UCT optimal value (middle), and the regret (bottom).", "type": "text" } ], "index": 4 } ], "index": 3.5 } ], "index": 2.25 }, { "type": "image", "bbox": [ 142, 338, 470, 482 ], "blocks": [ { "type": "image_body", "bbox": [ 142, 338, 470, 482 ], "group_id": 1, "lines": [ { "bbox": [ 142, 338, 470, 482 ], "spans": [ { "bbox": [ 142, 338, 470, 482 ], "score": 0.969, "type": "image", "image_path": "6252c86da6b7ad105e4698af7c9520c9024408f3ef1ad07638c795237f09b88b.jpg" } ] } ], "index": 6, "virtual_lines": [ { "bbox": [ 142, 338, 470, 386.0 ], "spans": [], "index": 5 }, { "bbox": [ 142, 386.0, 470, 434.0 ], "spans": [], "index": 6 }, { "bbox": [ 142, 434.0, 470, 482.0 ], "spans": [], "index": 7 } ] }, { "type": "image_caption", "bbox": [ 106, 495, 504, 529 ], "group_id": 1, "lines": [ { "bbox": [ 106, 495, 506, 507 ], "spans": [ { "bbox": [ 106, 495, 271, 507 ], "score": 1.0, "content": "Figure 2: For different branching factor", "type": "text" }, { "bbox": [ 271, 496, 278, 505 ], "score": 0.8, "content": "k", "type": "inline_equation" }, { "bbox": [ 279, 495, 351, 507 ], "score": 1.0, "content": "(rows) and depth", "type": "text" }, { "bbox": [ 352, 496, 358, 505 ], "score": 0.73, "content": "d", "type": "inline_equation" }, { "bbox": [ 359, 495, 506, 507 ], "score": 1.0, "content": "(columns), the heatmaps show: the", "type": "text" } ], "index": 8 }, { "bbox": [ 105, 505, 505, 519 ], "spans": [ { "bbox": [ 105, 505, 505, 519 ], "score": 1.0, "content": "absolute error of the value estimate at the root node after the last simulation of each algorithm w.r.t.", "type": "text" } ], "index": 9 }, { "bbox": [ 106, 517, 504, 529 ], "spans": [ { "bbox": [ 106, 517, 504, 529 ], "score": 1.0, "content": "the respective optimal value (a), and w.r.t. the optimal value of UCT (b); regret at the root node (c).", "type": "text" } ], "index": 10 } ], "index": 9 } ], "index": 7.5 }, { "type": "text", "bbox": [ 106, 549, 505, 703 ], "lines": [ { "bbox": [ 104, 549, 506, 561 ], "spans": [ { "bbox": [ 104, 549, 506, 561 ], "score": 1.0, "content": "a random value between 0 and 1. At each leaf, a Gaussian distribution is used as an evaluation", "type": "text" } ], "index": 11 }, { "bbox": [ 106, 560, 505, 572 ], "spans": [ { "bbox": [ 106, 560, 505, 572 ], "score": 1.0, "content": "function resembling the return of random rollouts. The mean of the Gaussian distribution is the", "type": "text" } ], "index": 12 }, { "bbox": [ 105, 571, 506, 583 ], "spans": [ { "bbox": [ 105, 571, 506, 583 ], "score": 1.0, "content": "sum of the values assigned to the edges connecting the root node to the considered leaf, while the", "type": "text" } ], "index": 13 }, { "bbox": [ 106, 581, 505, 594 ], "spans": [ { "bbox": [ 106, 581, 192, 594 ], "score": 1.0, "content": "standard deviation is", "type": "text" }, { "bbox": [ 193, 581, 237, 592 ], "score": 0.91, "content": "\\bar { \\sigma } = 0 . 0 5 ^ { 1 }", "type": "inline_equation" }, { "bbox": [ 237, 581, 505, 594 ], "score": 1.0, "content": ". For stability, all the means are normalized between 0 and 1. As", "type": "text" } ], "index": 14 }, { "bbox": [ 105, 593, 505, 605 ], "spans": [ { "bbox": [ 105, 593, 505, 605 ], "score": 1.0, "content": "in Xiao et al. (2019), we create 5 trees on which we perform 5 different runs in each, resulting in 25", "type": "text" } ], "index": 15 }, { "bbox": [ 106, 604, 505, 617 ], "spans": [ { "bbox": [ 106, 604, 339, 617 ], "score": 1.0, "content": "experiments, for all the combinations of branching factor", "type": "text" }, { "bbox": [ 339, 604, 461, 616 ], "score": 0.91, "content": "k = \\{ 2 , 4 , 6 , 8 , 1 0 , 1 2 , 1 4 , 1 6 \\}", "type": "inline_equation" }, { "bbox": [ 462, 604, 505, 617 ], "score": 1.0, "content": "and depth", "type": "text" } ], "index": 16 }, { "bbox": [ 107, 615, 506, 627 ], "spans": [ { "bbox": [ 107, 615, 180, 627 ], "score": 0.91, "content": "d = \\{ 1 , 2 , 3 , 4 , 5 \\}", "type": "inline_equation" }, { "bbox": [ 181, 615, 506, 627 ], "score": 1.0, "content": ", computing: (i) the value estimation error at the root node w.r.t. the regularized", "type": "text" } ], "index": 17 }, { "bbox": [ 105, 626, 506, 638 ], "spans": [ { "bbox": [ 105, 626, 166, 638 ], "score": 1.0, "content": "optimal value:", "type": "text" }, { "bbox": [ 167, 626, 227, 637 ], "score": 0.9, "content": "\\begin{array} { r } { \\varepsilon _ { \\Omega } = V _ { \\Omega } - V * ; } \\end{array}", "type": "inline_equation" }, { "bbox": [ 228, 626, 506, 638 ], "score": 1.0, "content": "(ii) the value estimation error at the root node w.r.t. the unregularized", "type": "text" } ], "index": 18 }, { "bbox": [ 106, 637, 505, 650 ], "spans": [ { "bbox": [ 106, 637, 167, 650 ], "score": 1.0, "content": "optimal value:", "type": "text" }, { "bbox": [ 168, 637, 256, 648 ], "score": 0.88, "content": "\\varepsilon _ { \\mathrm { U C T } } = V _ { \\Omega } - V * _ { \\mathrm { U C T } } ", "type": "inline_equation" }, { "bbox": [ 256, 637, 318, 650 ], "score": 1.0, "content": "; (iii) the regret", "type": "text" }, { "bbox": [ 318, 637, 328, 647 ], "score": 0.76, "content": "R", "type": "inline_equation" }, { "bbox": [ 328, 637, 505, 650 ], "score": 1.0, "content": "as in Equation (13). For a fair comparison,", "type": "text" } ], "index": 19 }, { "bbox": [ 106, 648, 505, 660 ], "spans": [ { "bbox": [ 106, 648, 159, 660 ], "score": 1.0, "content": "we use fixed", "type": "text" }, { "bbox": [ 160, 649, 195, 658 ], "score": 0.88, "content": "\\tau = 0 . 1", "type": "inline_equation" }, { "bbox": [ 195, 648, 213, 660 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 213, 649, 246, 658 ], "score": 0.88, "content": "\\epsilon = 0 . 1", "type": "inline_equation" }, { "bbox": [ 247, 648, 505, 660 ], "score": 1.0, "content": "across all algorithms. Figure 1 and 2 show how UCT and each", "type": "text" } ], "index": 20 }, { "bbox": [ 104, 658, 506, 671 ], "spans": [ { "bbox": [ 104, 658, 506, 671 ], "score": 1.0, "content": "regularizer behave for different configurations of the tree. We observe that, while RENTS and", "type": "text" } ], "index": 21 }, { "bbox": [ 105, 669, 506, 682 ], "spans": [ { "bbox": [ 105, 669, 506, 682 ], "score": 1.0, "content": "MENTS converge slower for increasing tree sizes, TENTS is robust w.r.t. the size of the tree and", "type": "text" } ], "index": 22 }, { "bbox": [ 106, 680, 505, 693 ], "spans": [ { "bbox": [ 106, 680, 505, 693 ], "score": 1.0, "content": "almost always converges faster than all other methods to the respective optimal value. Notably, the", "type": "text" } ], "index": 23 }, { "bbox": [ 106, 692, 505, 704 ], "spans": [ { "bbox": [ 106, 692, 505, 704 ], "score": 1.0, "content": "optimal value of TENTS seems to be very close to the one of UCT, i.e. the optimal value of the", "type": "text" } ], "index": 24 } ], "index": 17.5 } ], "page_idx": 6, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 106, 712, 506, 732 ], "lines": [ { "bbox": [ 119, 709, 505, 723 ], "spans": [ { "bbox": [ 119, 709, 505, 723 ], "score": 1.0, "content": "1The value of the standard deviation is not provided in Xiao et al. (2019). After trying different values, we", "type": "text" } ] }, { "bbox": [ 106, 720, 406, 732 ], "spans": [ { "bbox": [ 106, 720, 367, 732 ], "score": 1.0, "content": "observed that our results match the one in Xiao et al. 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Results are averaged over 5 and 10 seeds and show", "type": "text" }, { "bbox": [ 381, 240, 401, 251 ], "score": 0.88, "content": "9 5 \\%", "type": "inline_equation" }, { "bbox": [ 401, 240, 487, 252 ], "score": 1.0, "content": "confidence intervals.", "type": "text" } ], "index": 4 } ], "index": 3.5 } ], "index": 2.25 }, { "type": "text", "bbox": [ 107, 271, 505, 349 ], "lines": [ { "bbox": [ 105, 271, 505, 283 ], "spans": [ { "bbox": [ 105, 271, 505, 283 ], "score": 1.0, "content": "unregularized objective, and also converges faster than the one estimated by UCT, while MENTS", "type": "text" } ], "index": 5 }, { "bbox": [ 105, 282, 505, 294 ], "spans": [ { "bbox": [ 105, 282, 505, 294 ], "score": 1.0, "content": "and RENTS are considerably further from this value. In terms of regret, UCT explores less than", "type": "text" } ], "index": 6 }, { "bbox": [ 105, 293, 505, 306 ], "spans": [ { "bbox": [ 105, 293, 505, 306 ], "score": 1.0, "content": "the regularized methods and it is less prone to high regret, at the cost of slower convergence time.", "type": "text" } ], "index": 7 }, { "bbox": [ 105, 304, 505, 316 ], "spans": [ { "bbox": [ 105, 304, 505, 316 ], "score": 1.0, "content": "Nevertheless, the regret of TENTS is the smallest between the ones of the other regularizers, which", "type": "text" } ], "index": 8 }, { "bbox": [ 105, 314, 505, 328 ], "spans": [ { "bbox": [ 105, 314, 505, 328 ], "score": 1.0, "content": "seem to explore too much. These results show a general superiority of TENTS in this toy problem,", "type": "text" } ], "index": 9 }, { "bbox": [ 105, 326, 505, 339 ], "spans": [ { "bbox": [ 105, 326, 505, 339 ], "score": 1.0, "content": "also confirming our theoretical findings about the advantage of TENTS in terms of approximation", "type": "text" } ], "index": 10 }, { "bbox": [ 106, 338, 410, 349 ], "spans": [ { "bbox": [ 106, 338, 410, 349 ], "score": 1.0, "content": "error (Corollary 6) and regret (Corollary 3), in problems with many actions.", "type": "text" } ], "index": 11 } ], "index": 8 }, { "type": "title", "bbox": [ 108, 361, 294, 372 ], "lines": [ { "bbox": [ 106, 361, 294, 374 ], "spans": [ { "bbox": [ 106, 361, 294, 374 ], "score": 1.0, "content": "5.2 ENTROPY-REGULARIZED ALPHAZERO", "type": "text" } ], "index": 12 } ], "index": 12 }, { "type": "text", "bbox": [ 107, 381, 504, 405 ], "lines": [ { "bbox": [ 105, 381, 506, 395 ], "spans": [ { "bbox": [ 105, 381, 506, 395 ], "score": 1.0, "content": "In its standard form, AlphaZero (Silver et al., 2017a) uses the PUCT sampling strategy, a variant of", "type": "text" } ], "index": 13 }, { "bbox": [ 106, 391, 388, 407 ], "spans": [ { "bbox": [ 106, 391, 388, 407 ], "score": 1.0, "content": "UCT (Kocsis et al., 2006) that samples actions according to the policy", "type": "text" } ], "index": 14 } ], "index": 13.5 }, { "type": "interline_equation", "bbox": [ 206, 408, 404, 437 ], "lines": [ { "bbox": [ 206, 408, 404, 437 ], "spans": [ { "bbox": [ 206, 408, 404, 437 ], "score": 0.94, "content": "\\mathit { P U C T } ( s , a ) = { Q } ( s , a ) + \\epsilon { P } ( s , a ) { \\frac { \\sqrt { N ( s ) } } { 1 + N ( s , a ) } } ,", "type": "interline_equation", "image_path": "2de6e72e368e3c0714b9799d8fa9ff371232cdf343dbf42596c7aa63e4bf95bc.jpg" } ] } ], "index": 15.5, "virtual_lines": [ { "bbox": [ 206, 408, 404, 422.5 ], "spans": [], "index": 15 }, { "bbox": [ 206, 422.5, 404, 437.0 ], "spans": [], "index": 16 } ] }, { "type": "text", "bbox": [ 107, 440, 505, 583 ], "lines": [ { "bbox": [ 106, 441, 505, 453 ], "spans": [ { "bbox": [ 106, 441, 133, 453 ], "score": 1.0, "content": "where", "type": "text" }, { "bbox": [ 133, 441, 142, 451 ], "score": 0.84, "content": "P", "type": "inline_equation" }, { "bbox": [ 142, 441, 323, 453 ], "score": 1.0, "content": "is a prior probability on action selection, and", "type": "text" }, { "bbox": [ 323, 443, 329, 451 ], "score": 0.75, "content": "\\epsilon", "type": "inline_equation" }, { "bbox": [ 329, 441, 505, 453 ], "score": 1.0, "content": "is an exploration constant. A value network", "type": "text" } ], "index": 17 }, { "bbox": [ 106, 452, 505, 464 ], "spans": [ { "bbox": [ 106, 452, 439, 464 ], "score": 1.0, "content": "and a policy network are used to compute, respectively, the action-value function", "type": "text" }, { "bbox": [ 439, 452, 449, 463 ], "score": 0.85, "content": "Q", "type": "inline_equation" }, { "bbox": [ 449, 452, 505, 464 ], "score": 1.0, "content": "and the prior", "type": "text" } ], "index": 18 }, { "bbox": [ 104, 462, 506, 476 ], "spans": [ { "bbox": [ 104, 462, 134, 476 ], "score": 1.0, "content": "policy", "type": "text" }, { "bbox": [ 135, 463, 143, 473 ], "score": 0.8, "content": "P", "type": "inline_equation" }, { "bbox": [ 144, 462, 506, 476 ], "score": 1.0, "content": ". We use a single neural network, with 2 hidden layers composed of 128 ELU units, and", "type": "text" } ], "index": 19 }, { "bbox": [ 106, 474, 505, 486 ], "spans": [ { "bbox": [ 106, 474, 505, 486 ], "score": 1.0, "content": "two output layer respectively for the action-value function and the policy. We run 500 AlphaZero", "type": "text" } ], "index": 20 }, { "bbox": [ 105, 484, 506, 497 ], "spans": [ { "bbox": [ 105, 484, 506, 497 ], "score": 1.0, "content": "episodes, where each episode is composed of 300 steps. A step consists of running 32 MCTS", "type": "text" } ], "index": 21 }, { "bbox": [ 105, 496, 506, 509 ], "spans": [ { "bbox": [ 105, 496, 506, 509 ], "score": 1.0, "content": "simulations from the root node, as defined in Section 2, using the action-value function computed", "type": "text" } ], "index": 22 }, { "bbox": [ 105, 505, 506, 520 ], "spans": [ { "bbox": [ 105, 505, 506, 520 ], "score": 1.0, "content": "by the value network instead of using Monte-Carlo rollouts. 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Last,", "type": "text" } ], "index": 38 }, { "bbox": [ 104, 685, 506, 702 ], "spans": [ { "bbox": [ 104, 685, 506, 702 ], "score": 1.0, "content": "MENTS solves the problems slightly slower than RENTS, but reaches the same final performance.", "type": "text" } ], "index": 39 }, { "bbox": [ 105, 698, 505, 712 ], "spans": [ { "bbox": [ 105, 698, 505, 712 ], "score": 1.0, "content": "Although the results on these simple problems are not conclusive to assert the superiority of one", "type": "text" } ], "index": 40 }, { "bbox": [ 105, 709, 505, 722 ], "spans": [ { "bbox": [ 105, 709, 505, 722 ], "score": 1.0, "content": "method over the other, they definitely confirm the advantage of regularization in MCTS, and hint", "type": "text" } ], "index": 41 }, { "bbox": [ 105, 720, 505, 734 ], "spans": [ { "bbox": [ 105, 720, 505, 734 ], "score": 1.0, "content": "at the benefit of the use of relative entropy in control problems. Further analysis on more complex", "type": "text" } ], "index": 42 } ], "index": 36 } ], "page_idx": 7, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 108, 27, 306, 37 ], "lines": [ { "bbox": [ 106, 26, 308, 38 ], "spans": [ { "bbox": [ 106, 26, 308, 38 ], "score": 1.0, "content": "Under review as a conference paper at ICLR 2021", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 302, 752, 308, 760 ], "lines": [ { "bbox": [ 301, 750, 310, 762 ], "spans": [ { "bbox": [ 301, 750, 310, 762 ], "score": 1.0, "content": "", "type": "text", "height": 12, "width": 9 } ] } ] } ], "para_blocks": [ { "type": "image", "bbox": [ 133, 82, 482, 216 ], "blocks": [ { "type": "image_body", "bbox": [ 133, 82, 482, 216 ], "group_id": 0, "lines": [ { "bbox": [ 133, 82, 482, 216 ], "spans": [ { "bbox": [ 133, 82, 482, 216 ], "score": 0.972, "type": "image", "image_path": "42c28a1c16eea36bdb09be07654145c615b49198a9b8b0ff85b6ff1e59afac49.jpg" } ] } ], "index": 1, "virtual_lines": [ { "bbox": [ 133, 82, 482, 126.66666666666666 ], "spans": [], "index": 0 }, { "bbox": [ 133, 126.66666666666666, 482, 171.33333333333331 ], "spans": [], "index": 1 }, { "bbox": [ 133, 171.33333333333331, 482, 215.99999999999997 ], "spans": [], "index": 2 } ] }, { "type": "image_caption", "bbox": [ 106, 228, 504, 252 ], "group_id": 0, "lines": [ { "bbox": [ 105, 228, 506, 243 ], "spans": [ { "bbox": [ 105, 228, 506, 243 ], "score": 1.0, "content": "Figure 3: Cumulative rewards of AlphaZero with UCT and entropy-based operators, in CartPole (a)", "type": "text" } ], "index": 3 }, { "bbox": [ 105, 240, 487, 252 ], "spans": [ { "bbox": [ 105, 240, 381, 252 ], "score": 1.0, "content": "and Acrobot (b). Results are averaged over 5 and 10 seeds and show", "type": "text" }, { "bbox": [ 381, 240, 401, 251 ], "score": 0.88, "content": "9 5 \\%", "type": "inline_equation" }, { "bbox": [ 401, 240, 487, 252 ], "score": 1.0, "content": "confidence intervals.", "type": "text" } ], "index": 4 } ], "index": 3.5 } ], "index": 2.25 }, { "type": "text", "bbox": [ 107, 271, 505, 349 ], "lines": [], "index": 8, "bbox_fs": [ 105, 271, 505, 349 ], "lines_deleted": true }, { "type": "title", "bbox": [ 108, 361, 294, 372 ], "lines": [ { "bbox": [ 106, 361, 294, 374 ], "spans": [ { "bbox": [ 106, 361, 294, 374 ], "score": 1.0, "content": "5.2 ENTROPY-REGULARIZED ALPHAZERO", "type": "text" } ], "index": 12 } ], "index": 12 }, { "type": "text", "bbox": [ 107, 381, 504, 405 ], "lines": [ { "bbox": [ 105, 381, 506, 395 ], "spans": [ { "bbox": [ 105, 381, 506, 395 ], "score": 1.0, "content": "In its standard form, AlphaZero (Silver et al., 2017a) uses the PUCT sampling strategy, a variant of", "type": "text" } ], "index": 13 }, { "bbox": [ 106, 391, 388, 407 ], "spans": [ { "bbox": [ 106, 391, 388, 407 ], "score": 1.0, "content": "UCT (Kocsis et al., 2006) that samples actions according to the policy", "type": "text" } ], "index": 14 } ], "index": 13.5, "bbox_fs": [ 105, 381, 506, 407 ] }, { "type": "interline_equation", "bbox": [ 206, 408, 404, 437 ], "lines": [ { "bbox": [ 206, 408, 404, 437 ], "spans": [ { "bbox": [ 206, 408, 404, 437 ], "score": 0.94, "content": "\\mathit { P U C T } ( s , a ) = { Q } ( s , a ) + \\epsilon { P } ( s , a ) { \\frac { \\sqrt { N ( s ) } } { 1 + N ( s , a ) } } ,", "type": "interline_equation", "image_path": "2de6e72e368e3c0714b9799d8fa9ff371232cdf343dbf42596c7aa63e4bf95bc.jpg" } ] } ], "index": 15.5, "virtual_lines": [ { "bbox": [ 206, 408, 404, 422.5 ], "spans": [], "index": 15 }, { "bbox": [ 206, 422.5, 404, 437.0 ], "spans": [], "index": 16 } ] }, { "type": "text", "bbox": [ 107, 440, 505, 583 ], "lines": [ { "bbox": [ 106, 441, 505, 453 ], "spans": [ { "bbox": [ 106, 441, 133, 453 ], "score": 1.0, "content": "where", "type": "text" }, { "bbox": [ 133, 441, 142, 451 ], "score": 0.84, "content": "P", "type": "inline_equation" }, { "bbox": [ 142, 441, 323, 453 ], "score": 1.0, "content": "is a prior probability on action selection, and", "type": "text" }, { "bbox": [ 323, 443, 329, 451 ], "score": 0.75, "content": "\\epsilon", "type": "inline_equation" }, { "bbox": [ 329, 441, 505, 453 ], "score": 1.0, "content": "is an exploration constant. A value network", "type": "text" } ], "index": 17 }, { "bbox": [ 106, 452, 505, 464 ], "spans": [ { "bbox": [ 106, 452, 439, 464 ], "score": 1.0, "content": "and a policy network are used to compute, respectively, the action-value function", "type": "text" }, { "bbox": [ 439, 452, 449, 463 ], "score": 0.85, "content": "Q", "type": "inline_equation" }, { "bbox": [ 449, 452, 505, 464 ], "score": 1.0, "content": "and the prior", "type": "text" } ], "index": 18 }, { "bbox": [ 104, 462, 506, 476 ], "spans": [ { "bbox": [ 104, 462, 134, 476 ], "score": 1.0, "content": "policy", "type": "text" }, { "bbox": [ 135, 463, 143, 473 ], "score": 0.8, "content": "P", "type": "inline_equation" }, { "bbox": [ 144, 462, 506, 476 ], "score": 1.0, "content": ". 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The entropy-regularized variants of", "type": "text" } ], "index": 27 }, { "bbox": [ 105, 559, 505, 576 ], "spans": [ { "bbox": [ 105, 559, 505, 576 ], "score": 1.0, "content": "AlphaZero can be simply derived replacing the average backup operator, with the desired entropy", "type": "text" } ], "index": 28 }, { "bbox": [ 105, 572, 496, 585 ], "spans": [ { "bbox": [ 105, 572, 460, 585 ], "score": 1.0, "content": "function, and replacing PUCT with E3W using the respective maximizing argument and", "type": "text" }, { "bbox": [ 460, 573, 492, 583 ], "score": 0.88, "content": "\\epsilon = 0 . 1", "type": "inline_equation" }, { "bbox": [ 492, 572, 496, 585 ], "score": 1.0, "content": ".", "type": "text" } ], "index": 29 } ], "index": 23, "bbox_fs": [ 104, 441, 506, 585 ] }, { "type": "text", "bbox": [ 107, 589, 505, 732 ], "lines": [ { "bbox": [ 106, 588, 505, 601 ], "spans": [ { "bbox": [ 106, 588, 505, 601 ], "score": 1.0, "content": "Cartpole and Acrobot. 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First, although not", "type": "text" } ], "index": 33 }, { "bbox": [ 105, 633, 505, 646 ], "spans": [ { "bbox": [ 105, 633, 505, 646 ], "score": 1.0, "content": "significantly superior, RENTS exhibits the most stable learning and faster convergence, confirm-", "type": "text" } ], "index": 34 }, { "bbox": [ 106, 644, 505, 656 ], "spans": [ { "bbox": [ 106, 644, 505, 656 ], "score": 1.0, "content": "ing the benefit of relative entropy in control problems as already known for trust-region methods", "type": "text" } ], "index": 35 }, { "bbox": [ 105, 655, 506, 668 ], "spans": [ { "bbox": [ 105, 655, 506, 668 ], "score": 1.0, "content": "in RL (Schulman et al., 2015). Second, considering the small number of discrete actions in the", "type": "text" } ], "index": 36 }, { "bbox": [ 105, 665, 506, 678 ], "spans": [ { "bbox": [ 105, 665, 506, 678 ], "score": 1.0, "content": "problems, TENTS cannot benefit from the learning of sparse policies and shows slightly unstable", "type": "text" } ], "index": 37 }, { "bbox": [ 105, 677, 505, 690 ], "spans": [ { "bbox": [ 105, 677, 505, 690 ], "score": 1.0, "content": "learning in Cartpole, even though the overall performance is satisfying in both problems. Last,", "type": "text" } ], "index": 38 }, { "bbox": [ 104, 685, 506, 702 ], "spans": [ { "bbox": [ 104, 685, 506, 702 ], "score": 1.0, "content": "MENTS solves the problems slightly slower than RENTS, but reaches the same final performance.", "type": "text" } ], "index": 39 }, { "bbox": [ 105, 698, 505, 712 ], "spans": [ { "bbox": [ 105, 698, 505, 712 ], "score": 1.0, "content": "Although the results on these simple problems are not conclusive to assert the superiority of one", "type": "text" } ], "index": 40 }, { "bbox": [ 105, 709, 505, 722 ], "spans": [ { "bbox": [ 105, 709, 505, 722 ], "score": 1.0, "content": "method over the other, they definitely confirm the advantage of regularization in MCTS, and hint", "type": "text" } ], "index": 41 }, { "bbox": [ 105, 720, 505, 734 ], "spans": [ { "bbox": [ 105, 720, 505, 734 ], "score": 1.0, "content": "at the benefit of the use of relative entropy in control problems. Further analysis on more complex", "type": "text" } ], "index": 42 }, { "bbox": [ 107, 342, 505, 355 ], "spans": [ { "bbox": [ 107, 342, 505, 355 ], "score": 1.0, "content": "control problems will be desirable (e.g. MuJoCo (Todorov et al., 2012)), but the need to account for", "type": "text", "cross_page": true } ], "index": 18 }, { "bbox": [ 105, 352, 469, 367 ], "spans": [ { "bbox": [ 105, 352, 469, 367 ], "score": 1.0, "content": "continuous actions, a non-trivial setting for MCTS, makes it out of the scope of this paper.", "type": "text", "cross_page": true } ], "index": 19 } ], "index": 36, "bbox_fs": [ 104, 588, 506, 734 ] } ] }, { "preproc_blocks": [ { "type": "table", "bbox": [ 154, 113, 456, 321 ], "blocks": [ { "type": "table_caption", "bbox": [ 107, 89, 503, 111 ], "group_id": 0, "lines": [ { "bbox": [ 106, 89, 505, 102 ], "spans": [ { "bbox": [ 106, 89, 505, 102 ], "score": 1.0, "content": "Table 2: Average score in Atari over 100 seeds per game. Bold denotes no statistically significant", "type": "text" } ], "index": 0 }, { "bbox": [ 105, 99, 505, 114 ], "spans": [ { "bbox": [ 105, 99, 255, 114 ], "score": 1.0, "content": "difference to the highest mean (t-test,", "type": "text" }, { "bbox": [ 255, 101, 294, 112 ], "score": 0.88, "content": "p < 0 . 0 5", "type": "inline_equation" }, { "bbox": [ 294, 99, 505, 114 ], "score": 1.0, "content": "). Bottom row shows # no difference to highest mean.", "type": "text" } ], "index": 1 } ], "index": 0.5 }, { "type": "table_body", "bbox": [ 154, 113, 456, 321 ], "group_id": 0, "lines": [ { "bbox": [ 154, 113, 456, 321 ], "spans": [ { "bbox": [ 154, 113, 456, 321 ], "score": 0.984, "html": "
UCTMaxMCTSMENTSRENTSTENTS
Alien1, 486.801,461.101, 508.601,547.801, 568.60
Amidar115.62124.92123.30125.58121.84
Asterix4,855.005,484.505,576.005,743.505,647.00
Asteroids873.40899.601,414.701,486.401,642.10
Atlantis35,182.0035,720.0036,277.0035,314.0035,756.00
BankHeist475.50458.60622.30636.70631.40
BeamRider2,616.722,661.302,822.182,558.942,804.88
Breakout303.04296.14309.03300.35316.68
Centipede1, 782.181,728.692,012.862,253.422,258.89
DemonAttack579.90640.801,044.501,124.701,113.30
Enduro129.28124.20128.79134.88132.05
Frostbite1,244.001,332.102,388.202,369.802,260.60
Gopher3,348.403,303.003,536.403,372.803,447.80
Hero3,009.953,010.553,044.553,077.203,074.00
MsPacman1,940.201,907.102,018.302,190.302,094.40
Phoenix2,747.302,626.603,098.302,582.303,975.30
Qbert7,987.258,033.508,051.258,254.008,437.75
Robotank11.4311.0011.5911.5111.47
Seaquest3,276.403,217.203,312.403,345.203,324.40
Solaris895.00923.201, 118.201,115.001,127.60
SpaceInvaders778.45835.90832.55867.35822.95
WizardOfWor685.00666.001,211.001,241.001,231.00
#Highest mean6/227/2217/2216/2222/22
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Each experimental run", "type": "text" } ], "index": 31 }, { "bbox": [ 105, 524, 505, 539 ], "spans": [ { "bbox": [ 105, 524, 324, 539 ], "score": 1.0, "content": "consists of 512 MCTS simulations. The temperature", "type": "text" }, { "bbox": [ 324, 528, 331, 536 ], "score": 0.76, "content": "\\tau", "type": "inline_equation" }, { "bbox": [ 331, 524, 505, 539 ], "score": 1.0, "content": "is optimized for each algorithm and game", "type": "text" } ], "index": 32 }, { "bbox": [ 106, 537, 505, 549 ], "spans": [ { "bbox": [ 106, 537, 341, 549 ], "score": 1.0, "content": "via grid-search between 0.01 and 1. 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Moreover, we test also", "type": "text" } ], "index": 35 }, { "bbox": [ 106, 569, 505, 581 ], "spans": [ { "bbox": [ 106, 569, 505, 581 ], "score": 1.0, "content": "AlphaGo using the MaxMCTS backup (Khandelwal et al., 2016) for further comparison with clas-", "type": "text" } ], "index": 36 }, { "bbox": [ 106, 581, 504, 592 ], "spans": [ { "bbox": [ 106, 581, 504, 592 ], "score": 1.0, "content": "sic baselines. We observe that regularized MCTS dominates other baselines, in particular TENTS", "type": "text" } ], "index": 37 }, { "bbox": [ 105, 591, 506, 604 ], "spans": [ { "bbox": [ 105, 591, 506, 604 ], "score": 1.0, "content": "achieves the highest scores in all the 22 games, showing that sparse policies are more effective in", "type": "text" } ], "index": 38 }, { "bbox": [ 106, 603, 506, 615 ], "spans": [ { "bbox": [ 106, 603, 506, 615 ], "score": 1.0, "content": "Atari. This can be explained by Corollary 6 which shows that Tsallis entropy can lead to a lower", "type": "text" } ], "index": 39 }, { "bbox": [ 105, 612, 505, 627 ], "spans": [ { "bbox": [ 105, 612, 505, 627 ], "score": 1.0, "content": "error at the root node even with a high number of actions compared to relative or maximum entropy.", "type": "text" } ], "index": 40 } ], "index": 32.5 }, { "type": "title", "bbox": [ 107, 640, 195, 653 ], "lines": [ { "bbox": [ 105, 639, 197, 656 ], "spans": [ { "bbox": [ 105, 639, 197, 656 ], "score": 1.0, "content": "6 CONCLUSION", "type": "text" } ], "index": 41 } ], "index": 41 }, { "type": "text", "bbox": [ 107, 666, 505, 732 ], "lines": [ { "bbox": [ 106, 666, 505, 677 ], "spans": [ { "bbox": [ 106, 666, 505, 677 ], "score": 1.0, "content": "We introduced a theory of convex regularization in Monte-Carlo Tree Search (MCTS) based on", "type": "text" } ], "index": 42 }, { "bbox": [ 106, 677, 505, 689 ], "spans": [ { "bbox": [ 106, 677, 505, 689 ], "score": 1.0, "content": "the Legendre-Fenchel transform. 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Bold denotes no statistically significant", "type": "text" } ], "index": 0 }, { "bbox": [ 105, 99, 505, 114 ], "spans": [ { "bbox": [ 105, 99, 255, 114 ], "score": 1.0, "content": "difference to the highest mean (t-test,", "type": "text" }, { "bbox": [ 255, 101, 294, 112 ], "score": 0.88, "content": "p < 0 . 0 5", "type": "inline_equation" }, { "bbox": [ 294, 99, 505, 114 ], "score": 1.0, "content": "). Bottom row shows # no difference to highest mean.", "type": "text" } ], "index": 1 } ], "index": 0.5 }, { "type": "table_body", "bbox": [ 154, 113, 456, 321 ], "group_id": 0, "lines": [ { "bbox": [ 154, 113, 456, 321 ], "spans": [ { "bbox": [ 154, 113, 456, 321 ], "score": 0.984, "html": "
UCTMaxMCTSMENTSRENTSTENTS
Alien1, 486.801,461.101, 508.601,547.801, 568.60
Amidar115.62124.92123.30125.58121.84
Asterix4,855.005,484.505,576.005,743.505,647.00
Asteroids873.40899.601,414.701,486.401,642.10
Atlantis35,182.0035,720.0036,277.0035,314.0035,756.00
BankHeist475.50458.60622.30636.70631.40
BeamRider2,616.722,661.302,822.182,558.942,804.88
Breakout303.04296.14309.03300.35316.68
Centipede1, 782.181,728.692,012.862,253.422,258.89
DemonAttack579.90640.801,044.501,124.701,113.30
Enduro129.28124.20128.79134.88132.05
Frostbite1,244.001,332.102,388.202,369.802,260.60
Gopher3,348.403,303.003,536.403,372.803,447.80
Hero3,009.953,010.553,044.553,077.203,074.00
MsPacman1,940.201,907.102,018.302,190.302,094.40
Phoenix2,747.302,626.603,098.302,582.303,975.30
Qbert7,987.258,033.508,051.258,254.008,437.75
Robotank11.4311.0011.5911.5111.47
Seaquest3,276.403,217.203,312.403,345.203,324.40
Solaris895.00923.201, 118.201,115.001,127.60
SpaceInvaders778.45835.90832.55867.35822.95
WizardOfWor685.00666.001,211.001,241.001,231.00
#Highest mean6/227/2217/2216/2222/22
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This can be explained by Corollary 6 which shows that Tsallis entropy can lead to a lower", "type": "text" } ], "index": 39 }, { "bbox": [ 105, 612, 505, 627 ], "spans": [ { "bbox": [ 105, 612, 505, 627 ], "score": 1.0, "content": "error at the root node even with a high number of actions compared to relative or maximum entropy.", "type": "text" } ], "index": 40 } ], "index": 32.5, "bbox_fs": [ 104, 448, 506, 627 ] }, { "type": "title", "bbox": [ 107, 640, 195, 653 ], "lines": [ { "bbox": [ 105, 639, 197, 656 ], "spans": [ { "bbox": [ 105, 639, 197, 656 ], "score": 1.0, "content": "6 CONCLUSION", "type": "text" } ], "index": 41 } ], "index": 41 }, { "type": "text", "bbox": [ 107, 666, 505, 732 ], "lines": [ { "bbox": [ 106, 666, 505, 677 ], "spans": [ { "bbox": [ 106, 666, 505, 677 ], "score": 1.0, "content": "We introduced a theory of convex regularization in Monte-Carlo Tree Search (MCTS) based on", "type": "text" } ], "index": 42 }, { "bbox": [ 106, 677, 505, 689 ], "spans": [ { "bbox": [ 106, 677, 505, 689 ], "score": 1.0, "content": "the Legendre-Fenchel transform. 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Finally, we test regularized MCTS", "type": "text" } ], "index": 46 }, { "bbox": [ 105, 721, 505, 733 ], "spans": [ { "bbox": [ 105, 721, 505, 733 ], "score": 1.0, "content": "algorithms in discrete control problems and Atari games, showing its advantages over other methods.", "type": "text" } ], "index": 47 } ], "index": 44.5, "bbox_fs": [ 105, 666, 506, 733 ] } ] }, { "preproc_blocks": [ { "type": "title", "bbox": [ 107, 81, 176, 94 ], "lines": [ { "bbox": [ 106, 82, 176, 94 ], "spans": [ { "bbox": [ 106, 82, 176, 94 ], "score": 1.0, "content": "REFERENCES", "type": "text" } ], "index": 0 } ], "index": 0 }, { "type": "text", "bbox": [ 106, 100, 504, 123 ], "lines": [ { "bbox": [ 106, 100, 505, 113 ], "spans": [ { "bbox": [ 106, 100, 505, 113 ], "score": 1.0, "content": "Peter Auer, Nicolo Cesa-Bianchi, and Paul Fischer. 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Recently (Geist et al., 2019) introduced regularized Markov Decision", "type": "text" } ], "index": 34 }, { "bbox": [ 105, 583, 505, 596 ], "spans": [ { "bbox": [ 105, 583, 505, 596 ], "score": 1.0, "content": "Processes, formalizing the RL objective with a generalized form of convex regularization, based", "type": "text" } ], "index": 35 }, { "bbox": [ 105, 594, 505, 607 ], "spans": [ { "bbox": [ 105, 594, 505, 607 ], "score": 1.0, "content": "on the Legendre-Fenchel transform. In this paper, we provide a novel study of convex regulariza-", "type": "text" } ], "index": 36 }, { "bbox": [ 105, 604, 505, 618 ], "spans": [ { "bbox": [ 105, 604, 505, 618 ], "score": 1.0, "content": "tion in MCTS, and derive relative entropy (KL-divergence) and Tsallis entropy regularized MCTS", "type": "text" } ], "index": 37 }, { "bbox": [ 105, 615, 506, 630 ], "spans": [ { "bbox": [ 105, 615, 506, 630 ], "score": 1.0, "content": "algorithms, i.e. RENTS and TENTS respectively. 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Typically specific forms of entropy are utilized such as maxi-", "type": "text" } ], "index": 27 }, { "bbox": [ 105, 506, 505, 519 ], "spans": [ { "bbox": [ 105, 506, 505, 519 ], "score": 1.0, "content": "mum entropy (Haarnoja et al., 2018) or relative entropy (Schulman et al., 2015). This approach", "type": "text" } ], "index": 28 }, { "bbox": [ 105, 518, 505, 530 ], "spans": [ { "bbox": [ 105, 518, 505, 530 ], "score": 1.0, "content": "is an instance of the more generic duality framework, commonly used in convex optimization the-", "type": "text" } ], "index": 29 }, { "bbox": [ 105, 528, 505, 541 ], "spans": [ { "bbox": [ 105, 528, 505, 541 ], "score": 1.0, "content": "ory. 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By choosing λs = l", "type": "text" }, { "bbox": [ 435, 329, 492, 345 ], "score": 0.94, "content": "\\begin{array} { r } { \\lambda _ { s } = \\frac { \\left. A \\right. } { \\log \\left( 1 + s \\right) } } \\end{array}", "type": "inline_equation" }, { "bbox": [ 456, 330, 509, 349 ], "score": 1.0, "content": "og(1+s) , it", "type": "text" } ], "index": 15 }, { "bbox": [ 105, 343, 235, 356 ], "spans": [ { "bbox": [ 105, 343, 183, 356 ], "score": 1.0, "content": "follows that for all", "type": "text" }, { "bbox": [ 183, 346, 189, 353 ], "score": 0.31, "content": "a", "type": "inline_equation" }, { "bbox": [ 190, 343, 208, 356 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 208, 344, 231, 354 ], "score": 0.89, "content": "t \\geq 4", "type": "inline_equation" }, { "bbox": [ 231, 343, 235, 356 ], "score": 1.0, "content": ",", "type": "text" } ], "index": 16 } ], "index": 15 }, { "type": "interline_equation", "bbox": [ 131, 362, 479, 428 ], "lines": [ { "bbox": [ 131, 362, 479, 428 ], "spans": [ { "bbox": [ 131, 362, 479, 428 ], "score": 0.93, "content": "\\begin{array} { l } { \\displaystyle \\hat { N } _ { t } ( a ) = \\sum _ { s = 1 } ^ { t } \\pi _ { s } ( a ) \\geq \\sum _ { s = 1 } ^ { t } \\frac { 1 } { \\log ( 1 + s ) } \\geq \\sum _ { s = 1 } ^ { t } \\frac { 1 } { \\log ( 1 + s ) } - \\frac { s / ( s + 1 ) } { ( \\log ( 1 + s ) ) ^ { 2 } } } \\\\ { \\displaystyle \\geq \\int _ { 1 } ^ { 1 + t } \\frac { 1 } { \\log ( 1 + s ) } - \\frac { s / ( s + 1 ) } { ( \\log ( 1 + s ) ) ^ { 2 } } d s = \\frac { 1 + t } { \\log ( 2 + t ) } - \\frac { 1 } { \\log 2 } \\geq \\frac { t } { 2 \\log ( 2 + t ) } . } \\end{array}", "type": "interline_equation", "image_path": "e30897b89c370c0fd5901413a22da49c6af5e76454ab29652a41ccb658a5c2c9.jpg" } ] } ], "index": 18, "virtual_lines": [ { "bbox": [ 131, 362, 479, 384.0 ], "spans": [], "index": 17 }, { "bbox": [ 131, 384.0, 479, 406.0 ], "spans": [], "index": 18 }, { "bbox": [ 131, 406.0, 479, 428.0 ], "spans": [], "index": 19 } ] }, { "type": "text", "bbox": [ 104, 432, 470, 445 ], "lines": [ { "bbox": [ 105, 430, 471, 447 ], "spans": [ { "bbox": [ 105, 430, 167, 447 ], "score": 1.0, "content": "From Theorem", "type": "text" }, { "bbox": [ 167, 433, 187, 443 ], "score": 0.44, "content": "2 . I 9", "type": "inline_equation" }, { "bbox": [ 187, 430, 471, 447 ], "score": 1.0, "content": "in Wainwright (2019), we have the following concentration inequality:", "type": "text" } ], "index": 20 } ], "index": 20 }, { "type": "interline_equation", "bbox": [ 169, 452, 440, 482 ], "lines": [ { "bbox": [ 169, 452, 440, 482 ], "spans": [ { "bbox": [ 169, 452, 440, 482 ], "score": 0.93, "content": "\\mathbb { P } ( | N _ { t } ( a ) - \\hat { N } _ { t } ( a ) | > \\epsilon ) \\le 2 \\exp \\{ - \\frac { \\epsilon ^ { 2 } } { 2 \\sum _ { s = 1 } ^ { t } \\sigma _ { s } ^ { 2 } } \\} \\le 2 \\exp \\{ - \\frac { 2 \\epsilon ^ { 2 } } { t } \\} ,", "type": "interline_equation", "image_path": "8bc46b1d1480f89b8e561cc3aa15c46e4ba31e934acfe1592a8fd31e8850873e.jpg" } ] } ], "index": 21, "virtual_lines": [ { "bbox": [ 169, 452, 440, 482 ], "spans": [], "index": 21 } ] }, { "type": "text", "bbox": [ 105, 489, 504, 513 ], "lines": [ { "bbox": [ 104, 487, 506, 504 ], "spans": [ { "bbox": [ 104, 487, 133, 504 ], "score": 1.0, "content": "where", "type": "text" }, { "bbox": [ 133, 489, 173, 502 ], "score": 0.92, "content": "\\sigma _ { s } ^ { 2 } \\le 1 / 4", "type": "inline_equation" }, { "bbox": [ 174, 487, 362, 504 ], "score": 1.0, "content": "is the variance of a Bernoulli distribution with", "type": "text" }, { "bbox": [ 362, 490, 405, 502 ], "score": 0.93, "content": "p = \\pi _ { s } ( k )", "type": "inline_equation" }, { "bbox": [ 405, 487, 455, 504 ], "score": 1.0, "content": "at time step", "type": "text" }, { "bbox": [ 455, 493, 460, 500 ], "score": 0.33, "content": "s", "type": "inline_equation" }, { "bbox": [ 460, 487, 506, 504 ], "score": 1.0, "content": ". We define", "type": "text" } ], "index": 22 }, { "bbox": [ 105, 500, 146, 514 ], "spans": [ { "bbox": [ 105, 500, 146, 514 ], "score": 1.0, "content": "the event", "type": "text" } ], "index": 23 } ], "index": 22.5 }, { "type": "interline_equation", "bbox": [ 226, 519, 384, 534 ], "lines": [ { "bbox": [ 226, 519, 384, 534 ], "spans": [ { "bbox": [ 226, 519, 384, 534 ], "score": 0.92, "content": "E _ { \\epsilon } = \\{ \\forall a \\in \\mathcal { A } , | \\hat { N } _ { t } ( a ) - N _ { t } ( a ) | \\leq \\epsilon \\} ,", "type": "interline_equation", "image_path": "f8f656896ba063383b85792a8ffb7403baa2d346c581722e11861667c65a20a0.jpg" } ] } ], "index": 24, "virtual_lines": [ { "bbox": [ 226, 519, 384, 534 ], "spans": [], "index": 24 } ] }, { "type": "text", "bbox": [ 106, 542, 178, 554 ], "lines": [ { "bbox": [ 105, 541, 179, 556 ], "spans": [ { "bbox": [ 105, 541, 179, 556 ], "score": 1.0, "content": "and consequently", "type": "text" } ], "index": 25 } ], "index": 25 }, { "type": "interline_equation", "bbox": [ 213, 561, 397, 587 ], "lines": [ { "bbox": [ 213, 561, 397, 587 ], "spans": [ { "bbox": [ 213, 561, 397, 587 ], "score": 0.91, "content": "\\mathbb { P } ( | \\hat { N } _ { t } ( a ) - N _ { t } ( a ) | \\geq \\epsilon ) \\leq 2 | A | \\exp ( - \\frac { 2 \\epsilon ^ { 2 } } { t } ) .", "type": "interline_equation", "image_path": "396fd84dc98b391cb1723025a62610b2abc46aa21ffb80ea2e70406dcbaf62c1.jpg" } ] } ], "index": 26, "virtual_lines": [ { "bbox": [ 213, 561, 397, 587 ], "spans": [], "index": 26 } ] }, { "type": "text", "bbox": [ 107, 593, 505, 620 ], "lines": [ { "bbox": [ 103, 590, 508, 614 ], "spans": [ { "bbox": [ 103, 590, 210, 614 ], "score": 1.0, "content": "Conditioned on the event", "type": "text" }, { "bbox": [ 211, 596, 223, 606 ], "score": 0.84, "content": "E _ { \\epsilon }", "type": "inline_equation" }, { "bbox": [ 223, 590, 241, 614 ], "score": 1.0, "content": ", for", "type": "text" }, { "bbox": [ 242, 594, 299, 610 ], "score": 0.94, "content": "\\begin{array} { r } { \\epsilon = \\frac { t } { 4 \\log ( 2 + t ) } } \\end{array}", "type": "inline_equation" }, { "bbox": [ 299, 590, 339, 614 ], "score": 1.0, "content": ", we have", "type": "text" }, { "bbox": [ 340, 594, 417, 610 ], "score": 0.93, "content": "\\begin{array} { r } { N _ { t } ( a ) \\geq \\frac { t } { 4 \\log ( 2 + t ) } } \\end{array}", "type": "inline_equation" }, { "bbox": [ 417, 590, 508, 614 ], "score": 1.0, "content": ". For any action a by", "type": "text" } ], "index": 27 }, { "bbox": [ 105, 606, 229, 622 ], "spans": [ { "bbox": [ 105, 606, 229, 622 ], "score": 1.0, "content": "the definition of sub-gaussian,", "type": "text" } ], "index": 28 } ], "index": 27.5 }, { "type": "interline_equation", "bbox": [ 118, 627, 493, 664 ], "lines": [ { "bbox": [ 118, 627, 493, 664 ], "spans": [ { "bbox": [ 118, 627, 493, 664 ], "score": 0.94, "content": "\\mathbb { P } \\Bigg ( \\vert r ( a ) - \\hat { r } _ { t } ( a ) \\vert > \\sqrt { \\frac { 8 \\sigma ^ { 2 } \\log ( \\frac { 2 } { \\delta } ) \\log ( 2 + t ) } { t } } \\Bigg ) \\leq \\mathbb { P } \\Bigg ( \\vert r ( a ) - \\hat { r } _ { t } ( a ) \\vert > \\sqrt { \\frac { 2 \\sigma ^ { 2 } \\log ( \\frac { 2 } { \\delta } ) } { N _ { t } ( a ) } } \\Bigg ) \\leq \\delta", "type": "interline_equation", "image_path": "4fec72fd985514bad06956af17dc60841afeba6da38726b0f6a98011da6d3131.jpg" } ] } ], "index": 30, "virtual_lines": [ { "bbox": [ 118, 627, 493, 639.3333333333334 ], "spans": [], "index": 29 }, { "bbox": [ 118, 639.3333333333334, 493, 651.6666666666667 ], "spans": [], "index": 30 }, { "bbox": [ 118, 651.6666666666667, 493, 664.0000000000001 ], "spans": [], "index": 31 } ] }, { "type": "text", "bbox": [ 106, 671, 334, 687 ], "lines": [ { "bbox": [ 102, 668, 339, 692 ], "spans": [ { "bbox": [ 102, 668, 164, 692 ], "score": 1.0, "content": "by choosing a", "type": "text" }, { "bbox": [ 164, 673, 170, 683 ], "score": 0.61, "content": "\\delta", "type": "inline_equation" }, { "bbox": [ 171, 668, 212, 692 ], "score": 1.0, "content": "satisfying", "type": "text" }, { "bbox": [ 212, 672, 295, 688 ], "score": 0.93, "content": "\\begin{array} { r } { \\log ( \\frac { 2 } { \\delta } ) = \\frac { 1 } { ( \\log ( 2 + t ) ) ^ { 3 } } } \\end{array}", "type": "inline_equation" }, { "bbox": [ 295, 668, 339, 692 ], "score": 1.0, "content": ", we have", "type": "text" } ], "index": 32 } ], "index": 32 }, { "type": "interline_equation", "bbox": [ 169, 695, 442, 731 ], "lines": [ { "bbox": [ 169, 695, 442, 731 ], "spans": [ { "bbox": [ 169, 695, 442, 731 ], "score": 0.94, "content": "\\mathbb { P } \\Bigg ( | r ( a ) - \\hat { r } _ { t } ( a ) | > \\sqrt { \\frac { 2 \\sigma ^ { 2 } \\log \\left( \\frac { 2 } { \\delta } \\right) } { N _ { t } ( a ) } } \\Bigg ) \\leq 2 \\exp \\Bigg ( - \\frac { 1 } { ( \\log ( 2 + t ) ) ^ { 3 } } \\Bigg ) .", "type": "interline_equation", "image_path": "c62a04b2490ecfa44b50e714cd6c7a1f7d05e733d5c9d3a12fc85457de44538d.jpg" } ] } ], "index": 34, "virtual_lines": [ { "bbox": [ 169, 695, 442, 707.0 ], "spans": [], "index": 33 }, { "bbox": [ 169, 707.0, 442, 719.0 ], "spans": [], "index": 34 }, { "bbox": [ 169, 719.0, 442, 731.0 ], "spans": [], "index": 35 } ] } ], "page_idx": 12, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 107, 27, 307, 37 ], "lines": [ { "bbox": [ 106, 26, 308, 38 ], "spans": [ { "bbox": [ 106, 26, 308, 38 ], "score": 1.0, "content": "Under review as a conference paper at ICLR 2021", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 300, 751, 311, 761 ], "lines": [ { "bbox": [ 299, 750, 313, 764 ], "spans": [ { "bbox": [ 299, 750, 313, 764 ], "score": 1.0, "content": "13", "type": "text" } ] } ] } ], "para_blocks": [ { "type": "text", "bbox": [ 106, 82, 505, 182 ], "lines": [], "index": 4, "bbox_fs": [ 105, 82, 506, 184 ], "lines_deleted": true }, { "type": "title", "bbox": [ 107, 200, 169, 213 ], "lines": [ { "bbox": [ 104, 198, 171, 216 ], "spans": [ { "bbox": [ 104, 198, 171, 216 ], "score": 1.0, "content": "B PROOFS", "type": "text" } ], "index": 9 } ], "index": 9 }, { "type": "text", "bbox": [ 106, 227, 504, 250 ], "lines": [ { "bbox": [ 106, 227, 505, 240 ], "spans": [ { "bbox": [ 106, 227, 122, 240 ], "score": 1.0, "content": "Let", "type": "text" }, { "bbox": [ 122, 228, 128, 237 ], "score": 0.79, "content": "\\hat { r }", "type": "inline_equation" }, { "bbox": [ 129, 227, 146, 240 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 146, 229, 152, 237 ], "score": 0.74, "content": "r", "type": "inline_equation" }, { "bbox": [ 153, 227, 505, 240 ], "score": 1.0, "content": "be respectively the average and the the expected reward at the leaf node, and the reward", "type": "text" } ], "index": 10 }, { "bbox": [ 105, 237, 303, 251 ], "spans": [ { "bbox": [ 105, 237, 230, 251 ], "score": 1.0, "content": "distribution at the leaf node be", "type": "text" }, { "bbox": [ 230, 238, 242, 248 ], "score": 0.88, "content": "\\sigma ^ { 2 }", "type": "inline_equation" }, { "bbox": [ 242, 237, 303, 251 ], "score": 1.0, "content": "-sub-Gaussian.", "type": "text" } ], "index": 11 } ], "index": 10.5, "bbox_fs": [ 105, 227, 505, 251 ] }, { "type": "text", "bbox": [ 105, 259, 417, 273 ], "lines": [ { "bbox": [ 105, 259, 418, 274 ], "spans": [ { "bbox": [ 105, 259, 391, 274 ], "score": 1.0, "content": "Lemma 1 For the stochastic bandit problem E3W guarantees that, for", "type": "text" }, { "bbox": [ 391, 261, 414, 272 ], "score": 0.87, "content": "t \\geq 4", "type": "inline_equation" }, { "bbox": [ 414, 259, 418, 274 ], "score": 1.0, "content": ",", "type": "text" } ], "index": 12 } ], "index": 12, "bbox_fs": [ 105, 259, 418, 274 ] }, { "type": "interline_equation", "bbox": [ 177, 279, 434, 306 ], "lines": [ { "bbox": [ 177, 279, 434, 306 ], "spans": [ { "bbox": [ 177, 279, 434, 306 ], "score": 0.92, "content": "\\mathbb { P } \\big ( \\mathrm { \\normalfont ~ } r - \\hat { r } _ { t } \\mathrm { \\normalfont ~ } _ { \\infty } \\ge \\frac { 2 \\sigma } { \\log ( 2 + t ) } \\big ) \\le 4 | A | \\exp \\Big ( - \\frac { t } { ( \\log ( 2 + t ) ) ^ { 3 } } \\Big ) .", "type": "interline_equation", "image_path": "5100e0eb2e62685c3aa1386bb73fe8284671304332ae38da84648106da88ea56.jpg" } ] } ], "index": 13, "virtual_lines": [ { "bbox": [ 177, 279, 434, 306 ], "spans": [], "index": 13 } ] }, { "type": "text", "bbox": [ 106, 316, 506, 356 ], "lines": [ { "bbox": [ 105, 316, 506, 329 ], "spans": [ { "bbox": [ 105, 316, 198, 329 ], "score": 1.0, "content": "Proof 1 Let us define", "type": "text" }, { "bbox": [ 198, 317, 224, 329 ], "score": 0.93, "content": "N _ { t } ( a )", "type": "inline_equation" }, { "bbox": [ 224, 316, 478, 329 ], "score": 1.0, "content": "as the number of times action a have been chosen until time", "type": "text" }, { "bbox": [ 478, 318, 483, 327 ], "score": 0.62, "content": "t", "type": "inline_equation" }, { "bbox": [ 483, 316, 506, 329 ], "score": 1.0, "content": ", and", "type": "text" } ], "index": 14 }, { "bbox": [ 103, 325, 509, 349 ], "spans": [ { "bbox": [ 103, 325, 463, 349 ], "score": 1.0, "content": "Nˆt(a) = Pts=1 πs(a), where πs(a) is the E3W policy at time step s. By choosing λs = l", "type": "text" }, { "bbox": [ 435, 329, 492, 345 ], "score": 0.94, "content": "\\begin{array} { r } { \\lambda _ { s } = \\frac { \\left. A \\right. } { \\log \\left( 1 + s \\right) } } \\end{array}", "type": "inline_equation" }, { "bbox": [ 456, 330, 509, 349 ], "score": 1.0, "content": "og(1+s) , it", "type": "text" } ], "index": 15 }, { "bbox": [ 105, 343, 235, 356 ], "spans": [ { "bbox": [ 105, 343, 183, 356 ], "score": 1.0, "content": "follows that for all", "type": "text" }, { "bbox": [ 183, 346, 189, 353 ], "score": 0.31, "content": "a", "type": "inline_equation" }, { "bbox": [ 190, 343, 208, 356 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 208, 344, 231, 354 ], "score": 0.89, "content": "t \\geq 4", "type": "inline_equation" }, { "bbox": [ 231, 343, 235, 356 ], "score": 1.0, "content": ",", "type": "text" } ], "index": 16 } ], "index": 15, "bbox_fs": [ 103, 316, 509, 356 ] }, { "type": "interline_equation", "bbox": [ 131, 362, 479, 428 ], "lines": [ { "bbox": [ 131, 362, 479, 428 ], "spans": [ { "bbox": [ 131, 362, 479, 428 ], "score": 0.93, "content": "\\begin{array} { l } { \\displaystyle \\hat { N } _ { t } ( a ) = \\sum _ { s = 1 } ^ { t } \\pi _ { s } ( a ) \\geq \\sum _ { s = 1 } ^ { t } \\frac { 1 } { \\log ( 1 + s ) } \\geq \\sum _ { s = 1 } ^ { t } \\frac { 1 } { \\log ( 1 + s ) } - \\frac { s / ( s + 1 ) } { ( \\log ( 1 + s ) ) ^ { 2 } } } \\\\ { \\displaystyle \\geq \\int _ { 1 } ^ { 1 + t } \\frac { 1 } { \\log ( 1 + s ) } - \\frac { s / ( s + 1 ) } { ( \\log ( 1 + s ) ) ^ { 2 } } d s = \\frac { 1 + t } { \\log ( 2 + t ) } - \\frac { 1 } { \\log 2 } \\geq \\frac { t } { 2 \\log ( 2 + t ) } . } \\end{array}", "type": "interline_equation", "image_path": "e30897b89c370c0fd5901413a22da49c6af5e76454ab29652a41ccb658a5c2c9.jpg" } ] } ], "index": 18, "virtual_lines": [ { "bbox": [ 131, 362, 479, 384.0 ], "spans": [], "index": 17 }, { "bbox": [ 131, 384.0, 479, 406.0 ], "spans": [], "index": 18 }, { "bbox": [ 131, 406.0, 479, 428.0 ], "spans": [], "index": 19 } ] }, { "type": "text", "bbox": [ 104, 432, 470, 445 ], "lines": [ { "bbox": [ 105, 430, 471, 447 ], "spans": [ { "bbox": [ 105, 430, 167, 447 ], "score": 1.0, "content": "From Theorem", "type": "text" }, { "bbox": [ 167, 433, 187, 443 ], "score": 0.44, "content": "2 . I 9", "type": "inline_equation" }, { "bbox": [ 187, 430, 471, 447 ], "score": 1.0, "content": "in Wainwright (2019), we have the following concentration inequality:", "type": "text" } ], "index": 20 } ], "index": 20, "bbox_fs": [ 105, 430, 471, 447 ] }, { "type": "interline_equation", "bbox": [ 169, 452, 440, 482 ], "lines": [ { "bbox": [ 169, 452, 440, 482 ], "spans": [ { "bbox": [ 169, 452, 440, 482 ], "score": 0.93, "content": "\\mathbb { P } ( | N _ { t } ( a ) - \\hat { N } _ { t } ( a ) | > \\epsilon ) \\le 2 \\exp \\{ - \\frac { \\epsilon ^ { 2 } } { 2 \\sum _ { s = 1 } ^ { t } \\sigma _ { s } ^ { 2 } } \\} \\le 2 \\exp \\{ - \\frac { 2 \\epsilon ^ { 2 } } { t } \\} ,", "type": "interline_equation", "image_path": "8bc46b1d1480f89b8e561cc3aa15c46e4ba31e934acfe1592a8fd31e8850873e.jpg" } ] } ], "index": 21, "virtual_lines": [ { "bbox": [ 169, 452, 440, 482 ], "spans": [], "index": 21 } ] }, { "type": "text", "bbox": [ 105, 489, 504, 513 ], "lines": [ { "bbox": [ 104, 487, 506, 504 ], "spans": [ { "bbox": [ 104, 487, 133, 504 ], "score": 1.0, "content": "where", "type": "text" }, { "bbox": [ 133, 489, 173, 502 ], "score": 0.92, "content": "\\sigma _ { s } ^ { 2 } \\le 1 / 4", "type": "inline_equation" }, { "bbox": [ 174, 487, 362, 504 ], "score": 1.0, "content": "is the variance of a Bernoulli distribution with", "type": "text" }, { "bbox": [ 362, 490, 405, 502 ], "score": 0.93, "content": "p = \\pi _ { s } ( k )", "type": "inline_equation" }, { "bbox": [ 405, 487, 455, 504 ], "score": 1.0, "content": "at time step", "type": "text" }, { "bbox": [ 455, 493, 460, 500 ], "score": 0.33, "content": "s", "type": "inline_equation" }, { "bbox": [ 460, 487, 506, 504 ], "score": 1.0, "content": ". We define", "type": "text" } ], "index": 22 }, { "bbox": [ 105, 500, 146, 514 ], "spans": [ { "bbox": [ 105, 500, 146, 514 ], "score": 1.0, "content": "the event", "type": "text" } ], "index": 23 } ], "index": 22.5, "bbox_fs": [ 104, 487, 506, 514 ] }, { "type": "interline_equation", "bbox": [ 226, 519, 384, 534 ], "lines": [ { "bbox": [ 226, 519, 384, 534 ], "spans": [ { "bbox": [ 226, 519, 384, 534 ], "score": 0.92, "content": "E _ { \\epsilon } = \\{ \\forall a \\in \\mathcal { A } , | \\hat { N } _ { t } ( a ) - N _ { t } ( a ) | \\leq \\epsilon \\} ,", "type": "interline_equation", "image_path": "f8f656896ba063383b85792a8ffb7403baa2d346c581722e11861667c65a20a0.jpg" } ] } ], "index": 24, "virtual_lines": [ { "bbox": [ 226, 519, 384, 534 ], "spans": [], "index": 24 } ] }, { "type": "text", "bbox": [ 106, 542, 178, 554 ], "lines": [ { "bbox": [ 105, 541, 179, 556 ], "spans": [ { "bbox": [ 105, 541, 179, 556 ], "score": 1.0, "content": "and consequently", "type": "text" } ], "index": 25 } ], "index": 25, "bbox_fs": [ 105, 541, 179, 556 ] }, { "type": "interline_equation", "bbox": [ 213, 561, 397, 587 ], "lines": [ { "bbox": [ 213, 561, 397, 587 ], "spans": [ { "bbox": [ 213, 561, 397, 587 ], "score": 0.91, "content": "\\mathbb { P } ( | \\hat { N } _ { t } ( a ) - N _ { t } ( a ) | \\geq \\epsilon ) \\leq 2 | A | \\exp ( - \\frac { 2 \\epsilon ^ { 2 } } { t } ) .", "type": "interline_equation", "image_path": "396fd84dc98b391cb1723025a62610b2abc46aa21ffb80ea2e70406dcbaf62c1.jpg" } ] } ], "index": 26, "virtual_lines": [ { "bbox": [ 213, 561, 397, 587 ], "spans": [], "index": 26 } ] }, { "type": "text", "bbox": [ 107, 593, 505, 620 ], "lines": [ { "bbox": [ 103, 590, 508, 614 ], "spans": [ { "bbox": [ 103, 590, 210, 614 ], "score": 1.0, "content": "Conditioned on the event", "type": "text" }, { "bbox": [ 211, 596, 223, 606 ], "score": 0.84, "content": "E _ { \\epsilon }", "type": "inline_equation" }, { "bbox": [ 223, 590, 241, 614 ], "score": 1.0, "content": ", for", "type": "text" }, { "bbox": [ 242, 594, 299, 610 ], "score": 0.94, "content": "\\begin{array} { r } { \\epsilon = \\frac { t } { 4 \\log ( 2 + t ) } } \\end{array}", "type": "inline_equation" }, { "bbox": [ 299, 590, 339, 614 ], "score": 1.0, "content": ", we have", "type": "text" }, { "bbox": [ 340, 594, 417, 610 ], "score": 0.93, "content": "\\begin{array} { r } { N _ { t } ( a ) \\geq \\frac { t } { 4 \\log ( 2 + t ) } } \\end{array}", "type": "inline_equation" }, { "bbox": [ 417, 590, 508, 614 ], "score": 1.0, "content": ". For any action a by", "type": "text" } ], "index": 27 }, { "bbox": [ 105, 606, 229, 622 ], "spans": [ { "bbox": [ 105, 606, 229, 622 ], "score": 1.0, "content": "the definition of sub-gaussian,", "type": "text" } ], "index": 28 } ], "index": 27.5, "bbox_fs": [ 103, 590, 508, 622 ] }, { "type": "interline_equation", "bbox": [ 118, 627, 493, 664 ], "lines": [ { "bbox": [ 118, 627, 493, 664 ], "spans": [ { "bbox": [ 118, 627, 493, 664 ], "score": 0.94, "content": "\\mathbb { P } \\Bigg ( \\vert r ( a ) - \\hat { r } _ { t } ( a ) \\vert > \\sqrt { \\frac { 8 \\sigma ^ { 2 } \\log ( \\frac { 2 } { \\delta } ) \\log ( 2 + t ) } { t } } \\Bigg ) \\leq \\mathbb { P } \\Bigg ( \\vert r ( a ) - \\hat { r } _ { t } ( a ) \\vert > \\sqrt { \\frac { 2 \\sigma ^ { 2 } \\log ( \\frac { 2 } { \\delta } ) } { N _ { t } ( a ) } } \\Bigg ) \\leq \\delta", "type": "interline_equation", "image_path": "4fec72fd985514bad06956af17dc60841afeba6da38726b0f6a98011da6d3131.jpg" } ] } ], "index": 30, "virtual_lines": [ { "bbox": [ 118, 627, 493, 639.3333333333334 ], "spans": [], "index": 29 }, { "bbox": [ 118, 639.3333333333334, 493, 651.6666666666667 ], "spans": [], "index": 30 }, { "bbox": [ 118, 651.6666666666667, 493, 664.0000000000001 ], "spans": [], "index": 31 } ] }, { "type": "text", "bbox": [ 106, 671, 334, 687 ], "lines": [ { "bbox": [ 102, 668, 339, 692 ], "spans": [ { "bbox": [ 102, 668, 164, 692 ], "score": 1.0, "content": "by choosing a", "type": "text" }, { "bbox": [ 164, 673, 170, 683 ], "score": 0.61, "content": "\\delta", "type": "inline_equation" }, { "bbox": [ 171, 668, 212, 692 ], "score": 1.0, "content": "satisfying", "type": "text" }, { "bbox": [ 212, 672, 295, 688 ], "score": 0.93, "content": "\\begin{array} { r } { \\log ( \\frac { 2 } { \\delta } ) = \\frac { 1 } { ( \\log ( 2 + t ) ) ^ { 3 } } } \\end{array}", "type": "inline_equation" }, { "bbox": [ 295, 668, 339, 692 ], "score": 1.0, "content": ", we have", "type": "text" } ], "index": 32 } ], "index": 32, "bbox_fs": [ 102, 668, 339, 692 ] }, { "type": "interline_equation", "bbox": [ 169, 695, 442, 731 ], "lines": [ { "bbox": [ 169, 695, 442, 731 ], "spans": [ { "bbox": [ 169, 695, 442, 731 ], "score": 0.94, "content": "\\mathbb { P } \\Bigg ( | r ( a ) - \\hat { r } _ { t } ( a ) | > \\sqrt { \\frac { 2 \\sigma ^ { 2 } \\log \\left( \\frac { 2 } { \\delta } \\right) } { N _ { t } ( a ) } } \\Bigg ) \\leq 2 \\exp \\Bigg ( - \\frac { 1 } { ( \\log ( 2 + t ) ) ^ { 3 } } \\Bigg ) .", "type": "interline_equation", "image_path": "c62a04b2490ecfa44b50e714cd6c7a1f7d05e733d5c9d3a12fc85457de44538d.jpg" } ] } ], "index": 34, "virtual_lines": [ { "bbox": [ 169, 695, 442, 707.0 ], "spans": [], "index": 33 }, { "bbox": [ 169, 707.0, 442, 719.0 ], "spans": [], "index": 34 }, { "bbox": [ 169, 719.0, 442, 731.0 ], "spans": [], "index": 35 } ] } ] }, { "preproc_blocks": [ { "type": "text", "bbox": [ 107, 82, 189, 94 ], "lines": [ { "bbox": [ 106, 81, 187, 96 ], "spans": [ { "bbox": [ 106, 81, 164, 96 ], "score": 1.0, "content": "Therefore, for", "type": "text" }, { "bbox": [ 164, 83, 187, 94 ], "score": 0.86, "content": "t \\geq 2", "type": "inline_equation" } ], "index": 0 } ], "index": 0 }, { "type": "interline_equation", "bbox": [ 126, 96, 485, 200 ], "lines": [ { "bbox": [ 126, 96, 485, 200 ], "spans": [ { "bbox": [ 126, 96, 485, 200 ], "score": 0.95, "content": "\\begin{array} { r l } & { \\mathbb { P } \\Bigg ( \\| r - \\hat { r } _ { t } \\| _ { \\infty } > \\frac { 2 \\sigma } { \\log ( 2 + t ) } \\Bigg ) \\leq \\mathbb { P } \\Bigg ( \\| r - \\hat { r } _ { t } \\| _ { \\infty } > \\frac { 2 \\sigma } { \\log ( 2 + t ) } \\Bigg | E _ { \\epsilon } \\Bigg ) + \\mathbb { P } ( E _ { \\epsilon } ^ { C } ) } \\\\ & { \\leq \\displaystyle \\sum _ { k } \\Bigg ( \\mathbb { P } \\Bigg ( | r ( a ) - \\hat { r } _ { t } ( a ) | > \\frac { 2 \\sigma } { \\log ( 2 + t ) } \\Bigg ) + \\mathbb { P } ( E _ { \\epsilon } ^ { C } ) \\leq 2 | A | \\exp \\Bigg ( - \\frac { 1 } { ( \\log ( 2 + t ) ) ^ { 3 } } \\Bigg ) \\Bigg ) } \\\\ & { + 2 | A | \\exp \\Bigg ( - \\frac { t } { ( \\log ( 2 + t ) ) ^ { 3 } } \\Bigg ) = 4 | A | \\exp \\Bigg ( - \\frac { t } { ( \\log ( 2 + t ) ) ^ { 3 } } \\Bigg ) . } \\end{array}", "type": "interline_equation", "image_path": "19381931196bce41bf0a09bd78dbdfc63c5bd280344f4b725e5d5c4a8c8a0cc2.jpg" } ] } ], "index": 2, "virtual_lines": [ { "bbox": [ 126, 96, 485, 130.66666666666666 ], "spans": [], "index": 1 }, { "bbox": [ 126, 130.66666666666666, 485, 165.33333333333331 ], "spans": [], "index": 2 }, { "bbox": [ 126, 165.33333333333331, 485, 199.99999999999997 ], "spans": [], "index": 3 } ] }, { "type": "text", "bbox": [ 104, 205, 455, 219 ], "lines": [ { "bbox": [ 104, 204, 457, 222 ], "spans": [ { "bbox": [ 104, 204, 228, 222 ], "score": 1.0, "content": "Lemma 2 Given two policies", "type": "text" }, { "bbox": [ 228, 206, 303, 220 ], "score": 0.9, "content": "\\pi ^ { ( 1 ) } = \\nabla \\Omega ^ { * } ( r ^ { ( 1 ) } )", "type": "inline_equation" }, { "bbox": [ 303, 204, 321, 222 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 322, 207, 413, 220 ], "score": 0.91, "content": "\\pi ^ { ( 2 ) } = \\nabla \\Omega ^ { * } ( r ^ { ( 2 ) } ) , \\exists L ,", "type": "inline_equation" }, { "bbox": [ 414, 204, 457, 222 ], "score": 1.0, "content": ", such that", "type": "text" } ], "index": 4 } ], "index": 4 }, { "type": "interline_equation", "bbox": [ 227, 223, 383, 238 ], "lines": [ { "bbox": [ 227, 223, 383, 238 ], "spans": [ { "bbox": [ 227, 223, 383, 238 ], "score": 0.84, "content": "\\parallel \\pi ^ { ( 1 ) } - \\pi ^ { ( 2 ) } \\parallel _ { p } \\leq L \\parallel r ^ { ( 1 ) } - r ^ { ( 2 ) } \\parallel _ { p } .", "type": "interline_equation", "image_path": "df954fa813734136547d1196525350a848ac2c04c73738243b785c7aa8c26a33.jpg" } ] } ], "index": 5, "virtual_lines": [ { "bbox": [ 227, 223, 383, 238 ], "spans": [], "index": 5 } ] }, { "type": "text", "bbox": [ 106, 245, 505, 291 ], "lines": [ { "bbox": [ 105, 245, 505, 259 ], "spans": [ { "bbox": [ 105, 245, 296, 259 ], "score": 1.0, "content": "Proof 2 This comes directly from the fact that", "type": "text" }, { "bbox": [ 296, 246, 350, 258 ], "score": 0.93, "content": "\\pi = \\nabla \\Omega ^ { * } ( r )", "type": "inline_equation" }, { "bbox": [ 351, 245, 467, 259 ], "score": 1.0, "content": "is Lipschitz continuous with", "type": "text" }, { "bbox": [ 467, 246, 477, 256 ], "score": 0.85, "content": "\\ell ^ { p \\ }", "type": "inline_equation" }, { "bbox": [ 477, 245, 505, 259 ], "score": 1.0, "content": "-norm.", "type": "text" } ], "index": 6 }, { "bbox": [ 105, 257, 505, 269 ], "spans": [ { "bbox": [ 105, 257, 146, 269 ], "score": 1.0, "content": "Note that", "type": "text" }, { "bbox": [ 146, 259, 153, 268 ], "score": 0.3, "content": "p", "type": "inline_equation" }, { "bbox": [ 153, 257, 505, 269 ], "score": 1.0, "content": "has different values according to the choice of regularizer. Refer to Niculae & Blondel", "type": "text" } ], "index": 7 }, { "bbox": [ 105, 267, 505, 281 ], "spans": [ { "bbox": [ 105, 267, 505, 281 ], "score": 1.0, "content": "(2017) for a discussion of each norm using Shannon entropy and Tsallis entropy regularizer. Relative", "type": "text" } ], "index": 8 }, { "bbox": [ 105, 279, 342, 293 ], "spans": [ { "bbox": [ 105, 279, 342, 293 ], "score": 1.0, "content": "entropy shares the same Properties with Shannon Entropy.", "type": "text" } ], "index": 9 } ], "index": 7.5 }, { "type": "text", "bbox": [ 106, 298, 506, 336 ], "lines": [ { "bbox": [ 105, 298, 505, 312 ], "spans": [ { "bbox": [ 105, 298, 479, 312 ], "score": 1.0, "content": "Lemma 3 Consider the E3W policy applied to a tree. At any node s of the tree with depth", "type": "text" }, { "bbox": [ 479, 300, 486, 309 ], "score": 0.28, "content": "d _ { \\mathrm { { z } } }", "type": "inline_equation" }, { "bbox": [ 486, 298, 505, 312 ], "score": 1.0, "content": ", Let", "type": "text" } ], "index": 10 }, { "bbox": [ 105, 309, 506, 327 ], "spans": [ { "bbox": [ 105, 309, 144, 327 ], "score": 1.0, "content": "us define", "type": "text" }, { "bbox": [ 145, 311, 232, 324 ], "score": 0.93, "content": "N _ { t } ^ { * } ( s , a ) = \\pi ^ { * } ( a | s ) . t", "type": "inline_equation" }, { "bbox": [ 233, 309, 254, 327 ], "score": 1.0, "content": ", and", "type": "text" }, { "bbox": [ 254, 310, 359, 324 ], "score": 0.92, "content": "\\begin{array} { r } { \\hat { N } _ { t } ( s , a ) = \\sum _ { s = 1 } ^ { t } \\pi _ { s } ( a | s ) } \\end{array}", "type": "inline_equation" }, { "bbox": [ 360, 309, 390, 327 ], "score": 1.0, "content": ", where", "type": "text" }, { "bbox": [ 390, 311, 422, 324 ], "score": 0.93, "content": "\\pi _ { k } ( a | s )", "type": "inline_equation" }, { "bbox": [ 423, 309, 506, 327 ], "score": 1.0, "content": "is the policy at time", "type": "text" } ], "index": 11 }, { "bbox": [ 105, 324, 286, 337 ], "spans": [ { "bbox": [ 105, 324, 125, 337 ], "score": 1.0, "content": "step", "type": "text" }, { "bbox": [ 125, 326, 132, 335 ], "score": 0.58, "content": "k", "type": "inline_equation" }, { "bbox": [ 132, 324, 209, 337 ], "score": 1.0, "content": ". There exists some", "type": "text" }, { "bbox": [ 209, 326, 218, 335 ], "score": 0.77, "content": "C", "type": "inline_equation" }, { "bbox": [ 218, 324, 237, 337 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 237, 324, 245, 335 ], "score": 0.85, "content": "\\hat { C }", "type": "inline_equation" }, { "bbox": [ 246, 324, 286, 337 ], "score": 1.0, "content": "such that", "type": "text" } ], "index": 12 } ], "index": 11 }, { "type": "interline_equation", "bbox": [ 182, 339, 428, 365 ], "lines": [ { "bbox": [ 182, 339, 428, 365 ], "spans": [ { "bbox": [ 182, 339, 428, 365 ], "score": 0.92, "content": "\\mathbb { P } \\big ( | \\hat { N } _ { t } ( s , a ) - N _ { t } ^ { * } ( s , a ) | > \\frac { C t } { \\log t } \\big ) \\leq \\hat { C } | A | t \\exp \\{ - \\frac { t } { ( \\log t ) ^ { 3 } } \\} .", "type": "interline_equation", "image_path": "702c42b5c4031194ac6b59f0387755d57d8d0de6aeab1fd01e1d41c0cfc59369.jpg" } ] } ], "index": 13, "virtual_lines": [ { "bbox": [ 182, 339, 428, 365 ], "spans": [], "index": 13 } ] }, { "type": "text", "bbox": [ 106, 373, 265, 385 ], "lines": [ { "bbox": [ 105, 371, 266, 387 ], "spans": [ { "bbox": [ 105, 371, 266, 387 ], "score": 1.0, "content": "Proof 3 We denote the following event,", "type": "text" } ], "index": 14 } ], "index": 14 }, { "type": "interline_equation", "bbox": [ 207, 387, 402, 413 ], "lines": [ { "bbox": [ 207, 387, 402, 413 ], "spans": [ { "bbox": [ 207, 387, 402, 413 ], "score": 0.9, "content": "E _ { r _ { k } } = \\{ \\| r ( s ^ { \\prime } , . ) - { \\hat { r } } _ { k } ( s ^ { \\prime } , . ) \\| _ { \\infty } < { \\frac { 2 \\sigma } { \\log ( 2 + k ) } } \\} .", "type": "interline_equation", "image_path": "6c1032db9de009a17624a8d89f97ca4709487ac2f090404635439e76359c88be.jpg" } ] } ], "index": 15, "virtual_lines": [ { "bbox": [ 207, 387, 402, 413 ], "spans": [], "index": 15 } ] }, { "type": "text", "bbox": [ 107, 416, 471, 431 ], "lines": [ { "bbox": [ 105, 416, 473, 432 ], "spans": [ { "bbox": [ 105, 416, 231, 432 ], "score": 1.0, "content": "Thus, conditioned on the event", "type": "text" }, { "bbox": [ 231, 416, 270, 431 ], "score": 0.93, "content": "\\textstyle \\bigcap _ { i = 1 } ^ { t } E _ { r _ { t } }", "type": "inline_equation" }, { "bbox": [ 270, 416, 303, 432 ], "score": 1.0, "content": "and for", "type": "text" }, { "bbox": [ 303, 418, 326, 429 ], "score": 0.88, "content": "t \\geq 4 ,", "type": "inline_equation" }, { "bbox": [ 327, 416, 371, 432 ], "score": 1.0, "content": ", we bound", "type": "text" }, { "bbox": [ 371, 416, 460, 430 ], "score": 0.89, "content": "\\vert \\hat { N } _ { t } ( s , a ) - N _ { t } ^ { \\ast } ( s , a ) \\vert", "type": "inline_equation" }, { "bbox": [ 460, 416, 473, 432 ], "score": 1.0, "content": "as", "type": "text" } ], "index": 16 } ], "index": 16 }, { "type": "interline_equation", "bbox": [ 117, 434, 497, 728 ], "lines": [ { "bbox": [ 117, 434, 497, 728 ], "spans": [ { "bbox": [ 117, 434, 497, 728 ], "score": 0.94, "content": "\\begin{array} { r l } { \\sum _ { k \\in \\partial _ { s } } \\sum _ { i = 1 } ^ { N } \\sum _ { j = 1 } ^ { N } \\sum _ { k = 0 } ^ { N } \\exp _ { i } ^ { - \\beta } \\sum _ { k = 0 } ^ { N } ( k ) \\sum _ { k = 1 } ^ { N } \\exp _ { i } ^ { - \\beta } ( k ) \\sum _ { k = 1 } ^ { N } \\exp _ { i } ^ { - \\beta } ( k ) \\sum _ { k = 1 } ^ { N } } & { } \\\\ & { \\leq \\sum _ { k = 1 } ^ { N } \\exp _ { i } ^ { - \\beta } \\exp _ { i } ^ { - \\beta } ( k ) \\exp _ { i } ^ { - \\beta } \\sum _ { k = 1 } ^ { N } } \\\\ & { \\leq \\sum _ { k = 1 } ^ { N } \\sum _ { i = 0 } ^ { N } \\exp _ { i } ^ { - \\beta } \\exp _ { i } ^ { - \\beta } \\sum _ { k = 1 } ^ { N } } \\\\ & { \\leq \\sum _ { k = 1 } ^ { N } \\sum _ { i = 0 } ^ { N } \\exp _ { i } ^ { - \\beta } \\exp _ { i } ^ { - \\beta } \\sum _ { k = 1 } ^ { N } \\sum _ { i = 0 } ^ { N } \\exp _ { i } ^ { - \\beta } } \\\\ & \\leq \\sum _ { k = 1 } ^ { N } \\sum _ { i = 0 } ^ { N } \\exp _ { i } ^ { - \\beta } \\exp _ { i } ^ { - \\beta } \\sum _ { k = 1 } ^ { N } \\exp _ { i } ^ { - \\beta } \\exp _ { i } ^ { - \\beta } \\exp _ { i } ^ { - \\beta } \\sum _ { k = 1 } ^ { N } \\exp _ { i } ^ { - \\beta } \\exp _ { i } ^ { - \\beta } \\exp _ { i } ^ { - \\beta } \\exp _ { i } ^ { - \\beta } \\sum _ { k = 1 } ^ { N } \\exp _ { i } ^ { - \\beta } \\exp _ { i } ^ { - \\beta } \\exp _ { i } ^ { - \\beta } \\exp _ { i } ^ { - \\beta } \\exp _ { i } ^ { - \\beta } \\exp _ { i } ^ { - \\beta } \\exp _ { i } ^ { - \\beta } \\exp _ { i } ^ { - \\beta } \\exp _ { i } ^ { - \\beta } \\exp _ { i } ^ { - \\beta } \\exp _ { i } ^ { - \\beta } \\exp _ { i } ^ { - \\beta } \\exp _ { i } ^ { - \\beta } \\exp _ { i } ^ { - \\beta } \\exp _ { i } ^ { - \\beta } \\exp _ { i } ^ { - \\beta } \\exp _ { i } ^ { - \\beta } \\exp _ { i } ^ - \\beta \\end{array}", "type": "interline_equation", "image_path": "9f4eea7254f9aa8e026747d1cc9fb431dfc18fed106114c5ed287ebd31cd1f4d.jpg" } ] } ], "index": 18, "virtual_lines": [ { "bbox": [ 117, 434, 497, 532.0 ], "spans": [], "index": 17 }, { "bbox": [ 117, 532.0, 497, 630.0 ], "spans": [], "index": 18 }, { "bbox": [ 117, 630.0, 497, 728.0 ], "spans": [], "index": 19 } ] } ], "page_idx": 13, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 106, 26, 307, 38 ], "lines": [ { "bbox": [ 106, 25, 307, 38 ], "spans": [ { "bbox": [ 106, 25, 307, 38 ], "score": 1.0, "content": "Under review as a conference paper at ICLR 2021", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 300, 751, 311, 760 ], "lines": [ { "bbox": [ 299, 750, 313, 764 ], "spans": [ { "bbox": [ 299, 750, 313, 764 ], "score": 1.0, "content": "14", "type": "text" } ] } ] } ], "para_blocks": [ { "type": "text", "bbox": [ 107, 82, 189, 94 ], "lines": [ { "bbox": [ 106, 81, 187, 96 ], "spans": [ { "bbox": [ 106, 81, 164, 96 ], "score": 1.0, "content": "Therefore, for", "type": "text" }, { "bbox": [ 164, 83, 187, 94 ], "score": 0.86, "content": "t \\geq 2", "type": "inline_equation" } ], "index": 0 } ], "index": 0, "bbox_fs": [ 106, 81, 187, 96 ] }, { "type": "interline_equation", "bbox": [ 126, 96, 485, 200 ], "lines": [ { "bbox": [ 126, 96, 485, 200 ], "spans": [ { "bbox": [ 126, 96, 485, 200 ], "score": 0.95, "content": "\\begin{array} { r l } & { \\mathbb { P } \\Bigg ( \\| r - \\hat { r } _ { t } \\| _ { \\infty } > \\frac { 2 \\sigma } { \\log ( 2 + t ) } \\Bigg ) \\leq \\mathbb { P } \\Bigg ( \\| r - \\hat { r } _ { t } \\| _ { \\infty } > \\frac { 2 \\sigma } { \\log ( 2 + t ) } \\Bigg | E _ { \\epsilon } \\Bigg ) + \\mathbb { P } ( E _ { \\epsilon } ^ { C } ) } \\\\ & { \\leq \\displaystyle \\sum _ { k } \\Bigg ( \\mathbb { P } \\Bigg ( | r ( a ) - \\hat { r } _ { t } ( a ) | > \\frac { 2 \\sigma } { \\log ( 2 + t ) } \\Bigg ) + \\mathbb { P } ( E _ { \\epsilon } ^ { C } ) \\leq 2 | A | \\exp \\Bigg ( - \\frac { 1 } { ( \\log ( 2 + t ) ) ^ { 3 } } \\Bigg ) \\Bigg ) } \\\\ & { + 2 | A | \\exp \\Bigg ( - \\frac { t } { ( \\log ( 2 + t ) ) ^ { 3 } } \\Bigg ) = 4 | A | \\exp \\Bigg ( - \\frac { t } { ( \\log ( 2 + t ) ) ^ { 3 } } \\Bigg ) . } \\end{array}", "type": "interline_equation", "image_path": "19381931196bce41bf0a09bd78dbdfc63c5bd280344f4b725e5d5c4a8c8a0cc2.jpg" } ] } ], "index": 2, "virtual_lines": [ { "bbox": [ 126, 96, 485, 130.66666666666666 ], "spans": [], "index": 1 }, { "bbox": [ 126, 130.66666666666666, 485, 165.33333333333331 ], "spans": [], "index": 2 }, { "bbox": [ 126, 165.33333333333331, 485, 199.99999999999997 ], "spans": [], "index": 3 } ] }, { "type": "text", "bbox": [ 104, 205, 455, 219 ], "lines": [ { "bbox": [ 104, 204, 457, 222 ], "spans": [ { "bbox": [ 104, 204, 228, 222 ], "score": 1.0, "content": "Lemma 2 Given two policies", "type": "text" }, { "bbox": [ 228, 206, 303, 220 ], "score": 0.9, "content": "\\pi ^ { ( 1 ) } = \\nabla \\Omega ^ { * } ( r ^ { ( 1 ) } )", "type": "inline_equation" }, { "bbox": [ 303, 204, 321, 222 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 322, 207, 413, 220 ], "score": 0.91, "content": "\\pi ^ { ( 2 ) } = \\nabla \\Omega ^ { * } ( r ^ { ( 2 ) } ) , \\exists L ,", "type": "inline_equation" }, { "bbox": [ 414, 204, 457, 222 ], "score": 1.0, "content": ", such that", "type": "text" } ], "index": 4 } ], "index": 4, "bbox_fs": [ 104, 204, 457, 222 ] }, { "type": "interline_equation", "bbox": [ 227, 223, 383, 238 ], "lines": [ { "bbox": [ 227, 223, 383, 238 ], "spans": [ { "bbox": [ 227, 223, 383, 238 ], "score": 0.84, "content": "\\parallel \\pi ^ { ( 1 ) } - \\pi ^ { ( 2 ) } \\parallel _ { p } \\leq L \\parallel r ^ { ( 1 ) } - r ^ { ( 2 ) } \\parallel _ { p } .", "type": "interline_equation", "image_path": "df954fa813734136547d1196525350a848ac2c04c73738243b785c7aa8c26a33.jpg" } ] } ], "index": 5, "virtual_lines": [ { "bbox": [ 227, 223, 383, 238 ], "spans": [], "index": 5 } ] }, { "type": "text", "bbox": [ 106, 245, 505, 291 ], "lines": [ { "bbox": [ 105, 245, 505, 259 ], "spans": [ { "bbox": [ 105, 245, 296, 259 ], "score": 1.0, "content": "Proof 2 This comes directly from the fact that", "type": "text" }, { "bbox": [ 296, 246, 350, 258 ], "score": 0.93, "content": "\\pi = \\nabla \\Omega ^ { * } ( r )", "type": "inline_equation" }, { "bbox": [ 351, 245, 467, 259 ], "score": 1.0, "content": "is Lipschitz continuous with", "type": "text" }, { "bbox": [ 467, 246, 477, 256 ], "score": 0.85, "content": "\\ell ^ { p \\ }", "type": "inline_equation" }, { "bbox": [ 477, 245, 505, 259 ], "score": 1.0, "content": "-norm.", "type": "text" } ], "index": 6 }, { "bbox": [ 105, 257, 505, 269 ], "spans": [ { "bbox": [ 105, 257, 146, 269 ], "score": 1.0, "content": "Note that", "type": "text" }, { "bbox": [ 146, 259, 153, 268 ], "score": 0.3, "content": "p", "type": "inline_equation" }, { "bbox": [ 153, 257, 505, 269 ], "score": 1.0, "content": "has different values according to the choice of regularizer. Refer to Niculae & Blondel", "type": "text" } ], "index": 7 }, { "bbox": [ 105, 267, 505, 281 ], "spans": [ { "bbox": [ 105, 267, 505, 281 ], "score": 1.0, "content": "(2017) for a discussion of each norm using Shannon entropy and Tsallis entropy regularizer. Relative", "type": "text" } ], "index": 8 }, { "bbox": [ 105, 279, 342, 293 ], "spans": [ { "bbox": [ 105, 279, 342, 293 ], "score": 1.0, "content": "entropy shares the same Properties with Shannon Entropy.", "type": "text" } ], "index": 9 } ], "index": 7.5, "bbox_fs": [ 105, 245, 505, 293 ] }, { "type": "text", "bbox": [ 106, 298, 506, 336 ], "lines": [ { "bbox": [ 105, 298, 505, 312 ], "spans": [ { "bbox": [ 105, 298, 479, 312 ], "score": 1.0, "content": "Lemma 3 Consider the E3W policy applied to a tree. At any node s of the tree with depth", "type": "text" }, { "bbox": [ 479, 300, 486, 309 ], "score": 0.28, "content": "d _ { \\mathrm { { z } } }", "type": "inline_equation" }, { "bbox": [ 486, 298, 505, 312 ], "score": 1.0, "content": ", Let", "type": "text" } ], "index": 10 }, { "bbox": [ 105, 309, 506, 327 ], "spans": [ { "bbox": [ 105, 309, 144, 327 ], "score": 1.0, "content": "us define", "type": "text" }, { "bbox": [ 145, 311, 232, 324 ], "score": 0.93, "content": "N _ { t } ^ { * } ( s , a ) = \\pi ^ { * } ( a | s ) . t", "type": "inline_equation" }, { "bbox": [ 233, 309, 254, 327 ], "score": 1.0, "content": ", and", "type": "text" }, { "bbox": [ 254, 310, 359, 324 ], "score": 0.92, "content": "\\begin{array} { r } { \\hat { N } _ { t } ( s , a ) = \\sum _ { s = 1 } ^ { t } \\pi _ { s } ( a | s ) } \\end{array}", "type": "inline_equation" }, { "bbox": [ 360, 309, 390, 327 ], "score": 1.0, "content": ", where", "type": "text" }, { "bbox": [ 390, 311, 422, 324 ], "score": 0.93, "content": "\\pi _ { k } ( a | s )", "type": "inline_equation" }, { "bbox": [ 423, 309, 506, 327 ], "score": 1.0, "content": "is the policy at time", "type": "text" } ], "index": 11 }, { "bbox": [ 105, 324, 286, 337 ], "spans": [ { "bbox": [ 105, 324, 125, 337 ], "score": 1.0, "content": "step", "type": "text" }, { "bbox": [ 125, 326, 132, 335 ], "score": 0.58, "content": "k", "type": "inline_equation" }, { "bbox": [ 132, 324, 209, 337 ], "score": 1.0, "content": ". There exists some", "type": "text" }, { "bbox": [ 209, 326, 218, 335 ], "score": 0.77, "content": "C", "type": "inline_equation" }, { "bbox": [ 218, 324, 237, 337 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 237, 324, 245, 335 ], "score": 0.85, "content": "\\hat { C }", "type": "inline_equation" }, { "bbox": [ 246, 324, 286, 337 ], "score": 1.0, "content": "such that", "type": "text" } ], "index": 12 } ], "index": 11, "bbox_fs": [ 105, 298, 506, 337 ] }, { "type": "interline_equation", "bbox": [ 182, 339, 428, 365 ], "lines": [ { "bbox": [ 182, 339, 428, 365 ], "spans": [ { "bbox": [ 182, 339, 428, 365 ], "score": 0.92, "content": "\\mathbb { P } \\big ( | \\hat { N } _ { t } ( s , a ) - N _ { t } ^ { * } ( s , a ) | > \\frac { C t } { \\log t } \\big ) \\leq \\hat { C } | A | t \\exp \\{ - \\frac { t } { ( \\log t ) ^ { 3 } } \\} .", "type": "interline_equation", "image_path": "702c42b5c4031194ac6b59f0387755d57d8d0de6aeab1fd01e1d41c0cfc59369.jpg" } ] } ], "index": 13, "virtual_lines": [ { "bbox": [ 182, 339, 428, 365 ], "spans": [], "index": 13 } ] }, { "type": "text", "bbox": [ 106, 373, 265, 385 ], "lines": [ { "bbox": [ 105, 371, 266, 387 ], "spans": [ { "bbox": [ 105, 371, 266, 387 ], "score": 1.0, "content": "Proof 3 We denote the following event,", "type": "text" } ], "index": 14 } ], "index": 14, "bbox_fs": [ 105, 371, 266, 387 ] }, { "type": "interline_equation", "bbox": [ 207, 387, 402, 413 ], "lines": [ { "bbox": [ 207, 387, 402, 413 ], "spans": [ { "bbox": [ 207, 387, 402, 413 ], "score": 0.9, "content": "E _ { r _ { k } } = \\{ \\| r ( s ^ { \\prime } , . ) - { \\hat { r } } _ { k } ( s ^ { \\prime } , . ) \\| _ { \\infty } < { \\frac { 2 \\sigma } { \\log ( 2 + k ) } } \\} .", "type": "interline_equation", "image_path": "6c1032db9de009a17624a8d89f97ca4709487ac2f090404635439e76359c88be.jpg" } ] } ], "index": 15, "virtual_lines": [ { "bbox": [ 207, 387, 402, 413 ], "spans": [], "index": 15 } ] }, { "type": "text", "bbox": [ 107, 416, 471, 431 ], "lines": [ { "bbox": [ 105, 416, 473, 432 ], "spans": [ { "bbox": [ 105, 416, 231, 432 ], "score": 1.0, "content": "Thus, conditioned on the event", "type": "text" }, { "bbox": [ 231, 416, 270, 431 ], "score": 0.93, "content": "\\textstyle \\bigcap _ { i = 1 } ^ { t } E _ { r _ { t } }", "type": "inline_equation" }, { "bbox": [ 270, 416, 303, 432 ], "score": 1.0, "content": "and for", "type": "text" }, { "bbox": [ 303, 418, 326, 429 ], "score": 0.88, "content": "t \\geq 4 ,", "type": "inline_equation" }, { "bbox": [ 327, 416, 371, 432 ], "score": 1.0, "content": ", we bound", "type": "text" }, { "bbox": [ 371, 416, 460, 430 ], "score": 0.89, "content": "\\vert \\hat { N } _ { t } ( s , a ) - N _ { t } ^ { \\ast } ( s , a ) \\vert", "type": "inline_equation" }, { "bbox": [ 460, 416, 473, 432 ], "score": 1.0, "content": "as", "type": "text" } ], "index": 16 } ], "index": 16, "bbox_fs": [ 105, 416, 473, 432 ] }, { "type": "interline_equation", "bbox": [ 117, 434, 497, 728 ], "lines": [ { "bbox": [ 117, 434, 497, 728 ], "spans": [ { "bbox": [ 117, 434, 497, 728 ], "score": 0.94, "content": "\\begin{array} { r l } { \\sum _ { k \\in \\partial _ { s } } \\sum _ { i = 1 } ^ { N } \\sum _ { j = 1 } ^ { N } \\sum _ { k = 0 } ^ { N } \\exp _ { i } ^ { - \\beta } \\sum _ { k = 0 } ^ { N } ( k ) \\sum _ { k = 1 } ^ { N } \\exp _ { i } ^ { - \\beta } ( k ) \\sum _ { k = 1 } ^ { N } \\exp _ { i } ^ { - \\beta } ( k ) \\sum _ { k = 1 } ^ { N } } & { } \\\\ & { \\leq \\sum _ { k = 1 } ^ { N } \\exp _ { i } ^ { - \\beta } \\exp _ { i } ^ { - \\beta } ( k ) \\exp _ { i } ^ { - \\beta } \\sum _ { k = 1 } ^ { N } } \\\\ & { \\leq \\sum _ { k = 1 } ^ { N } \\sum _ { i = 0 } ^ { N } \\exp _ { i } ^ { - \\beta } \\exp _ { i } ^ { - \\beta } \\sum _ { k = 1 } ^ { N } } \\\\ & { \\leq \\sum _ { k = 1 } ^ { N } \\sum _ { i = 0 } ^ { N } \\exp _ { i } ^ { - \\beta } \\exp _ { i } ^ { - \\beta } \\sum _ { k = 1 } ^ { N } \\sum _ { i = 0 } ^ { N } \\exp _ { i } ^ { - \\beta } } \\\\ & \\leq \\sum _ { k = 1 } ^ { N } \\sum _ { i = 0 } ^ { N } \\exp _ { i } ^ { - \\beta } \\exp _ { i } ^ { - \\beta } \\sum _ { k = 1 } ^ { N } \\exp _ { i } ^ { - \\beta } \\exp _ { i } ^ { - \\beta } \\exp _ { i } ^ { - \\beta } \\sum _ { k = 1 } ^ { N } \\exp _ { i } ^ { - \\beta } \\exp _ { i } ^ { - \\beta } \\exp _ { i } ^ { - \\beta } \\exp _ { i } ^ { - \\beta } \\sum _ { k = 1 } ^ { N } \\exp _ { i } ^ { - \\beta } \\exp _ { i } ^ { - \\beta } \\exp _ { i } ^ { - \\beta } \\exp _ { i } ^ { - \\beta } \\exp _ { i } ^ { - \\beta } \\exp _ { i } ^ { - \\beta } \\exp _ { i } ^ { - \\beta } \\exp _ { i } ^ { - \\beta } \\exp _ { i } ^ { - \\beta } \\exp _ { i } ^ { - \\beta } \\exp _ { i } ^ { - \\beta } \\exp _ { i } ^ { - \\beta } \\exp _ { i } ^ { - \\beta } \\exp _ { i } ^ { - \\beta } \\exp _ { i } ^ { - \\beta } \\exp _ { i } ^ { - \\beta } \\exp _ { i } ^ { - \\beta } \\exp _ { i } ^ - \\beta \\end{array}", "type": "interline_equation", "image_path": "9f4eea7254f9aa8e026747d1cc9fb431dfc18fed106114c5ed287ebd31cd1f4d.jpg" } ] } ], "index": 18, "virtual_lines": [ { "bbox": [ 117, 434, 497, 532.0 ], "spans": [], "index": 17 }, { "bbox": [ 117, 532.0, 497, 630.0 ], "spans": [], "index": 18 }, { "bbox": [ 117, 630.0, 497, 728.0 ], "spans": [], "index": 19 } ] } ] }, { "preproc_blocks": [ { "type": "text", "bbox": [ 104, 81, 368, 95 ], "lines": [ { "bbox": [ 104, 81, 368, 97 ], "spans": [ { "bbox": [ 104, 81, 180, 97 ], "score": 1.0, "content": "for some constant", "type": "text" }, { "bbox": [ 180, 83, 189, 92 ], "score": 0.76, "content": "C", "type": "inline_equation" }, { "bbox": [ 189, 81, 247, 97 ], "score": 1.0, "content": "depending on", "type": "text" }, { "bbox": [ 247, 82, 301, 95 ], "score": 0.9, "content": "| A | , p , d , \\sigma , L", "type": "inline_equation" }, { "bbox": [ 302, 81, 322, 97 ], "score": 1.0, "content": ", and", "type": "text" }, { "bbox": [ 322, 84, 330, 94 ], "score": 0.8, "content": "\\gamma", "type": "inline_equation" }, { "bbox": [ 330, 81, 368, 97 ], "score": 1.0, "content": ". Finally,", "type": "text" } ], "index": 0 } ], "index": 0 }, { "type": "interline_equation", "bbox": [ 139, 99, 471, 187 ], "lines": [ { "bbox": [ 139, 99, 471, 187 ], "spans": [ { "bbox": [ 139, 99, 471, 187 ], "score": 0.95, "content": "\\begin{array} { r l } { { \\mathbb { P } ( | \\hat { N _ { t } } ( s , a ) - N _ { t } ^ { * } ( s , a ) | \\geq \\frac { C t } { \\log t } ) \\leq \\sum _ { i = 1 } ^ { t } \\mathbb { P } ( E _ { r _ { t } } ^ { c } ) = \\displaystyle \\sum _ { i = 1 } ^ { t } 4 | A | \\exp ( - \\frac { t } { ( \\log ( 2 + t ) ) ^ { 3 } } ) } } \\\\ & { \\leq 4 | A | t \\exp ( - \\frac { t } { ( \\log ( 2 + t ) ) ^ { 3 } } ) } \\\\ & { = O ( t \\exp ( - \\frac { t } { ( \\log ( t ) ) ^ { 3 } } ) ) . } \\end{array}", "type": "interline_equation", "image_path": "8ff3c9f025e43ce9f87638de08fcf8f343e8e7e280cfdda75296ac27b0a76dae.jpg" } ] } ], "index": 2, "virtual_lines": [ { "bbox": [ 139, 99, 471, 128.33333333333334 ], "spans": [], "index": 1 }, { "bbox": [ 139, 128.33333333333334, 471, 157.66666666666669 ], "spans": [], "index": 2 }, { "bbox": [ 139, 157.66666666666669, 471, 187.00000000000003 ], "spans": [], "index": 3 } ] }, { "type": "text", "bbox": [ 107, 194, 505, 229 ], "lines": [ { "bbox": [ 105, 192, 506, 207 ], "spans": [ { "bbox": [ 105, 192, 207, 207 ], "score": 1.0, "content": "Lemma 4 Consider the", "type": "text" }, { "bbox": [ 208, 194, 230, 204 ], "score": 0.3, "content": "E 3 W", "type": "inline_equation" }, { "bbox": [ 230, 192, 336, 207 ], "score": 1.0, "content": "policy applied to a tree.", "type": "text" }, { "bbox": [ 337, 195, 348, 204 ], "score": 0.3, "content": "A t", "type": "inline_equation" }, { "bbox": [ 349, 192, 506, 207 ], "score": 1.0, "content": "any node s of the tree, Let us define", "type": "text" } ], "index": 4 }, { "bbox": [ 107, 204, 506, 218 ], "spans": [ { "bbox": [ 107, 205, 196, 217 ], "score": 0.91, "content": "N _ { t } ^ { * } ( s , a ) = \\pi ^ { * } ( a | s ) . t", "type": "inline_equation" }, { "bbox": [ 197, 204, 219, 218 ], "score": 1.0, "content": ", and", "type": "text" }, { "bbox": [ 220, 205, 254, 217 ], "score": 0.91, "content": "N _ { t } ( s , a )", "type": "inline_equation" }, { "bbox": [ 254, 204, 381, 218 ], "score": 1.0, "content": "as the number of times action", "type": "text" }, { "bbox": [ 381, 208, 387, 215 ], "score": 0.28, "content": "a", "type": "inline_equation" }, { "bbox": [ 388, 204, 506, 218 ], "score": 1.0, "content": "have been chosen until time", "type": "text" } ], "index": 5 }, { "bbox": [ 105, 217, 284, 230 ], "spans": [ { "bbox": [ 105, 218, 125, 230 ], "score": 1.0, "content": "step", "type": "text" }, { "bbox": [ 125, 220, 130, 228 ], "score": 0.46, "content": "t", "type": "inline_equation" }, { "bbox": [ 130, 218, 207, 230 ], "score": 1.0, "content": ". There exists some", "type": "text" }, { "bbox": [ 207, 218, 216, 228 ], "score": 0.82, "content": "C", "type": "inline_equation" }, { "bbox": [ 216, 218, 235, 230 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 235, 217, 244, 228 ], "score": 0.86, "content": "\\hat { C }", "type": "inline_equation" }, { "bbox": [ 244, 218, 284, 230 ], "score": 1.0, "content": "such that", "type": "text" } ], "index": 6 } ], "index": 5 }, { "type": "interline_equation", "bbox": [ 189, 234, 422, 260 ], "lines": [ { "bbox": [ 189, 234, 422, 260 ], "spans": [ { "bbox": [ 189, 234, 422, 260 ], "score": 0.93, "content": "\\mathbb { P } \\big ( | N _ { t } ( s , a ) - N _ { t } ^ { * } ( s , a ) | > \\frac { C t } { \\log t } \\big ) \\le \\hat { C } t \\exp \\{ - \\frac { t } { ( \\log t ) ^ { 3 } } \\} .", "type": "interline_equation", "image_path": "2aabca78cd25431ac6f23fcdd6bac4a6fe0a325e715324fe2dc243f108a9ad87.jpg" } ] } ], "index": 7, "virtual_lines": [ { "bbox": [ 189, 234, 422, 260 ], "spans": [], "index": 7 } ] }, { "type": "text", "bbox": [ 106, 268, 318, 281 ], "lines": [ { "bbox": [ 105, 267, 318, 282 ], "spans": [ { "bbox": [ 105, 267, 274, 282 ], "score": 1.0, "content": "Proof 4 Based on the result from Lemma", "type": "text" }, { "bbox": [ 274, 271, 280, 279 ], "score": 0.58, "content": "^ 3", "type": "inline_equation" }, { "bbox": [ 280, 267, 318, 282 ], "score": 1.0, "content": ", we have", "type": "text" } ], "index": 8 } ], "index": 8 }, { "type": "interline_equation", "bbox": [ 140, 285, 450, 390 ], "lines": [ { "bbox": [ 140, 285, 450, 390 ], "spans": [ { "bbox": [ 140, 285, 450, 390 ], "score": 0.94, "content": "\\begin{array} { r l } & { \\mathbb { P } \\big ( | N _ { t } ( s , a ) - N _ { t } ^ { * } ( s , a ) | > ( 1 + C ) \\displaystyle \\frac { t } { \\log t } \\big ) \\leq C t \\exp \\{ - \\displaystyle \\frac { t } { ( \\log t ) ^ { 3 } } \\} } \\\\ & { \\leq \\mathbb { P } \\big ( | \\hat { N } _ { t } ( s , a ) - N _ { t } ^ { * } ( s , a ) | > \\displaystyle \\frac { C t } { \\log t } \\big ) + \\mathbb { P } \\big ( | N _ { t } ( s , a ) - \\hat { N } _ { t } ( s , a ) | > \\displaystyle \\frac { t } { \\log t } \\big ) } \\\\ & { \\leq 4 | A | t \\exp \\{ - \\displaystyle \\frac { t } { ( \\log ( 2 + t ) ) ^ { 3 } } \\} + 2 | A | \\exp \\{ - \\displaystyle \\frac { t } { ( \\log ( 2 + t ) ) ^ { 2 } } \\} ( L e m m a 3 a n d ( \\log ( 2 + t ) ) ) } \\\\ & { \\leq O ( t \\exp ( - \\displaystyle \\frac { t } { ( \\log t ) ^ { 3 } } ) ) . } \\end{array}", "type": "interline_equation", "image_path": "1dcac355f10d8103cc1a2339cee649ac1b796639d183bc7c3f3795d4d2466ef6.jpg" } ] } ], "index": 10, "virtual_lines": [ { "bbox": [ 140, 285, 450, 320.0 ], "spans": [], "index": 9 }, { "bbox": [ 140, 320.0, 450, 355.0 ], "spans": [], "index": 10 }, { "bbox": [ 140, 355.0, 450, 390.0 ], "spans": [], "index": 11 } ] }, { "type": "text", "bbox": [ 106, 398, 505, 421 ], "lines": [ { "bbox": [ 105, 396, 505, 412 ], "spans": [ { "bbox": [ 105, 396, 309, 412 ], "score": 1.0, "content": "Theorem 1 At the root node s of the tree, defining", "type": "text" }, { "bbox": [ 310, 399, 332, 411 ], "score": 0.91, "content": "N ( s )", "type": "inline_equation" }, { "bbox": [ 333, 396, 462, 412 ], "score": 1.0, "content": "as the number of visitations and", "type": "text" }, { "bbox": [ 463, 398, 492, 411 ], "score": 0.91, "content": "V _ { \\Omega ^ { * } } ( s )", "type": "inline_equation" }, { "bbox": [ 493, 396, 505, 412 ], "score": 1.0, "content": "as", "type": "text" } ], "index": 12 }, { "bbox": [ 106, 409, 303, 421 ], "spans": [ { "bbox": [ 106, 409, 218, 421 ], "score": 1.0, "content": "the estimated value at node", "type": "text" }, { "bbox": [ 218, 412, 223, 419 ], "score": 0.42, "content": "s", "type": "inline_equation" }, { "bbox": [ 223, 409, 241, 421 ], "score": 1.0, "content": ", for", "type": "text" }, { "bbox": [ 241, 410, 264, 420 ], "score": 0.85, "content": "\\epsilon > 0", "type": "inline_equation" }, { "bbox": [ 265, 409, 303, 421 ], "score": 1.0, "content": ", we have", "type": "text" } ], "index": 13 } ], "index": 12.5 }, { "type": "interline_equation", "bbox": [ 185, 425, 426, 454 ], "lines": [ { "bbox": [ 185, 425, 426, 454 ], "spans": [ { "bbox": [ 185, 425, 426, 454 ], "score": 0.91, "content": "\\mathbb { P } ( | V _ { \\Omega } ( s ) - V _ { \\Omega } ^ { * } ( s ) | > \\epsilon ) \\le C \\exp \\{ - \\frac { N ( s ) \\epsilon } { \\hat { C } ( \\log ( 2 + N ( s ) ) ) ^ { 2 } } \\} .", "type": "interline_equation", "image_path": "f6e3a8443a668905551dd24efcf33683733e38bf4a91c4a60893a76afd39fcb2.jpg" } ] } ], "index": 14, "virtual_lines": [ { "bbox": [ 185, 425, 426, 454 ], "spans": [], "index": 14 } ] }, { "type": "text", "bbox": [ 105, 462, 504, 485 ], "lines": [ { "bbox": [ 105, 461, 505, 476 ], "spans": [ { "bbox": [ 105, 461, 473, 476 ], "score": 1.0, "content": "Proof 5 We prove this concentration inequality by induction. When the depth of the tree is", "type": "text" }, { "bbox": [ 473, 463, 501, 473 ], "score": 0.88, "content": "D = 1", "type": "inline_equation" }, { "bbox": [ 501, 461, 505, 476 ], "score": 1.0, "content": ",", "type": "text" } ], "index": 15 }, { "bbox": [ 104, 474, 216, 487 ], "spans": [ { "bbox": [ 104, 474, 177, 487 ], "score": 1.0, "content": "from Proposition", "type": "text" }, { "bbox": [ 177, 474, 182, 483 ], "score": 0.64, "content": "^ { l }", "type": "inline_equation" }, { "bbox": [ 182, 474, 216, 487 ], "score": 1.0, "content": ", we get", "type": "text" } ], "index": 16 } ], "index": 15.5 }, { "type": "interline_equation", "bbox": [ 130, 489, 481, 504 ], "lines": [ { "bbox": [ 130, 489, 481, 504 ], "spans": [ { "bbox": [ 130, 489, 481, 504 ], "score": 0.89, "content": "\\left| V _ { \\Omega } ( s ) - V _ { \\Omega } ^ { * } ( s ) \\right| = \\mid \\Omega ^ { * } ( Q _ { \\Omega } ( s , . ) ) - \\Omega ^ { * } ( Q _ { \\Omega } ^ { * } ( s , . ) ) \\mid \\mid _ { \\infty } \\le \\gamma \\mid \\mid \\hat { r } - r ^ { * } \\mid _ { \\infty } ( C o n t r a c t i o n )", "type": "interline_equation", "image_path": "04f271923bc197f7b3a71d4c67e6c4b977c9013e20b7c123c82c2f06b975deb0.jpg" } ] } ], "index": 17, "virtual_lines": [ { "bbox": [ 130, 489, 481, 504 ], "spans": [], "index": 17 } ] }, { "type": "text", "bbox": [ 112, 508, 388, 520 ], "lines": [ { "bbox": [ 110, 507, 374, 521 ], "spans": [ { "bbox": [ 110, 507, 133, 521 ], "score": 1.0, "content": "where", "type": "text" }, { "bbox": [ 133, 509, 139, 518 ], "score": 0.8, "content": "\\hat { r }", "type": "inline_equation" }, { "bbox": [ 140, 507, 250, 521 ], "score": 1.0, "content": "is the average rewards and", "type": "text" }, { "bbox": [ 250, 509, 261, 518 ], "score": 0.85, "content": "r ^ { * }", "type": "inline_equation" }, { "bbox": [ 261, 507, 374, 521 ], "score": 1.0, "content": "is the mean reward. So that", "type": "text" } ], "index": 18 } ], "index": 18 }, { "type": "interline_equation", "bbox": [ 203, 525, 406, 539 ], "lines": [ { "bbox": [ 203, 525, 406, 539 ], "spans": [ { "bbox": [ 203, 525, 406, 539 ], "score": 0.88, "content": "\\mathbb { P } ( | V _ { \\Omega } ( s ) - V _ { \\Omega } ^ { * } ( s ) | > \\epsilon ) \\le \\mathbb { P } ( \\gamma \\parallel \\hat { r } - r ^ { * } \\parallel _ { \\infty } > \\epsilon ) .", "type": "interline_equation", "image_path": "5a0dd09fb53a4d835ab652659f61fca36bd713cf16856dfce7198a72c1f3459f.jpg" } ] } ], "index": 19, "virtual_lines": [ { "bbox": [ 203, 525, 406, 539 ], "spans": [], "index": 19 } ] }, { "type": "text", "bbox": [ 106, 544, 295, 560 ], "lines": [ { "bbox": [ 102, 541, 299, 564 ], "spans": [ { "bbox": [ 102, 541, 162, 564 ], "score": 1.0, "content": "From Lemma", "type": "text" }, { "bbox": [ 163, 547, 168, 556 ], "score": 0.69, "content": "^ { l }", "type": "inline_equation" }, { "bbox": [ 168, 541, 192, 564 ], "score": 1.0, "content": ", with", "type": "text" }, { "bbox": [ 192, 544, 255, 560 ], "score": 0.94, "content": "\\begin{array} { r } { \\epsilon = \\frac { 2 \\sigma \\gamma } { \\log ( 2 + N ( s ) ) } } \\end{array}", "type": "inline_equation" }, { "bbox": [ 256, 541, 299, 564 ], "score": 1.0, "content": ", we have", "type": "text" } ], "index": 20 } ], "index": 20 }, { "type": "interline_equation", "bbox": [ 126, 565, 484, 622 ], "lines": [ { "bbox": [ 126, 565, 484, 622 ], "spans": [ { "bbox": [ 126, 565, 484, 622 ], "score": 0.93, "content": "\\begin{array} { r l } & { \\mathbb { P } ( | V _ { \\Omega } ( s ) - V _ { \\Omega } ^ { * } ( s ) | > \\epsilon ) \\le \\mathbb { P } ( \\gamma \\parallel \\hat { r } - r ^ { * } \\parallel _ { \\infty } > \\epsilon ) \\le 4 | A | \\exp \\{ - \\frac { N ( s ) \\epsilon } { 2 \\sigma \\gamma ( \\log ( 2 + N ( s ) ) ) ^ { 2 } } \\} } \\\\ & { \\quad \\quad \\quad = C \\exp \\{ - \\frac { N ( s ) \\epsilon } { \\hat { C } ( \\log ( 2 + N ( s ) ) ) ^ { 2 } } \\} . } \\end{array}", "type": "interline_equation", "image_path": "0892983841e4725b636e9902bd510d5fab9a6d3f595ac187e20f0892c0ea672f.jpg" } ] } ], "index": 22, "virtual_lines": [ { "bbox": [ 126, 565, 484, 584.0 ], "spans": [], "index": 21 }, { "bbox": [ 126, 584.0, 484, 603.0 ], "spans": [], "index": 22 }, { "bbox": [ 126, 603.0, 484, 622.0 ], "spans": [], "index": 23 } ] }, { "type": "text", "bbox": [ 105, 626, 504, 649 ], "lines": [ { "bbox": [ 105, 625, 505, 640 ], "spans": [ { "bbox": [ 105, 625, 338, 640 ], "score": 1.0, "content": "Let assume we have the concentration bound at the depth", "type": "text" }, { "bbox": [ 338, 627, 364, 637 ], "score": 0.88, "content": "D - 1", "type": "inline_equation" }, { "bbox": [ 365, 625, 421, 640 ], "score": 1.0, "content": ", Let us define", "type": "text" }, { "bbox": [ 422, 627, 501, 638 ], "score": 0.92, "content": "V _ { \\Omega } ( s _ { a } ) = Q _ { \\Omega } ( s , a )", "type": "inline_equation" }, { "bbox": [ 501, 625, 505, 640 ], "score": 1.0, "content": ",", "type": "text" } ], "index": 24 }, { "bbox": [ 105, 636, 421, 649 ], "spans": [ { "bbox": [ 105, 636, 133, 649 ], "score": 1.0, "content": "where", "type": "text" }, { "bbox": [ 133, 639, 144, 648 ], "score": 0.83, "content": "s _ { a }", "type": "inline_equation" }, { "bbox": [ 144, 636, 279, 649 ], "score": 1.0, "content": "is the state reached taking action", "type": "text" }, { "bbox": [ 279, 639, 285, 647 ], "score": 0.47, "content": "a", "type": "inline_equation" }, { "bbox": [ 286, 636, 393, 649 ], "score": 1.0, "content": "from state s. then at depth", "type": "text" }, { "bbox": [ 394, 638, 421, 648 ], "score": 0.88, "content": "D - 1", "type": "inline_equation" } ], "index": 25 } ], "index": 24.5 }, { "type": "interline_equation", "bbox": [ 177, 653, 433, 682 ], "lines": [ { "bbox": [ 177, 653, 433, 682 ], "spans": [ { "bbox": [ 177, 653, 433, 682 ], "score": 0.92, "content": "\\mathbb { P } ( | V _ { \\Omega } ( s _ { a } ) - V _ { \\Omega } ^ { * } ( s _ { a } ) | > \\epsilon ) \\le C \\exp \\{ - \\frac { N ( s _ { a } ) \\epsilon } { \\hat { C } ( \\log ( 2 + N ( s _ { a } ) ) ) ^ { 2 } } \\} .", "type": "interline_equation", "image_path": "c98cdea666f1dbe10bfdf596654a65d6083e32af6dd4b1a8b1d4e7331793f9a8.jpg" } ] } ], "index": 26, "virtual_lines": [ { "bbox": [ 177, 653, 433, 682 ], "spans": [], "index": 26 } ] }, { "type": "text", "bbox": [ 105, 685, 374, 698 ], "lines": [ { "bbox": [ 105, 684, 375, 700 ], "spans": [ { "bbox": [ 105, 684, 177, 700 ], "score": 1.0, "content": "Now at the depth", "type": "text" }, { "bbox": [ 177, 687, 186, 696 ], "score": 0.77, "content": "D", "type": "inline_equation" }, { "bbox": [ 186, 684, 375, 700 ], "score": 1.0, "content": ", because of the Contraction Property, we have", "type": "text" } ], "index": 27 } ], "index": 27 }, { "type": "interline_equation", "bbox": [ 208, 701, 401, 732 ], "lines": [ { "bbox": [ 208, 701, 401, 732 ], "spans": [ { "bbox": [ 208, 701, 401, 732 ], "score": 0.9, "content": "\\begin{array} { r l } & { | V _ { \\Omega } ( s ) - V _ { \\Omega } ^ { * } ( s ) | \\leq \\gamma \\parallel Q _ { \\Omega } ( s , . ) - Q _ { \\Omega } ^ { * } ( s , . ) \\parallel _ { \\infty } } \\\\ & { \\qquad = \\gamma | Q _ { \\Omega } ( s , a ) - Q _ { \\Omega } ^ { * } ( s , a ) | . } \\end{array}", "type": "interline_equation", "image_path": "5049be6d96fabe4e364692ce31cbd9be2d6bd65e81ffbdef49525ca6a9c676ba.jpg" } ] } ], "index": 28.5, "virtual_lines": [ { "bbox": [ 208, 701, 401, 716.5 ], "spans": [], "index": 28 }, { "bbox": [ 208, 716.5, 401, 732.0 ], "spans": [], "index": 29 } ] } ], "page_idx": 14, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 106, 26, 307, 38 ], "lines": [ { "bbox": [ 106, 25, 308, 38 ], "spans": [ { "bbox": [ 106, 25, 308, 38 ], "score": 1.0, "content": "Under review as a conference paper at ICLR 2021", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 300, 751, 311, 760 ], "lines": [ { "bbox": [ 299, 750, 313, 764 ], "spans": [ { "bbox": [ 299, 750, 313, 764 ], "score": 1.0, "content": "15", "type": "text" } ] } ] } ], "para_blocks": [ { "type": "text", "bbox": [ 104, 81, 368, 95 ], "lines": [ { "bbox": [ 104, 81, 368, 97 ], "spans": [ { "bbox": [ 104, 81, 180, 97 ], "score": 1.0, "content": "for some constant", "type": "text" }, { "bbox": [ 180, 83, 189, 92 ], "score": 0.76, "content": "C", "type": "inline_equation" }, { "bbox": [ 189, 81, 247, 97 ], "score": 1.0, "content": "depending on", "type": "text" }, { "bbox": [ 247, 82, 301, 95 ], "score": 0.9, "content": "| A | , p , d , \\sigma , L", "type": "inline_equation" }, { "bbox": [ 302, 81, 322, 97 ], "score": 1.0, "content": ", and", "type": "text" }, { "bbox": [ 322, 84, 330, 94 ], "score": 0.8, "content": "\\gamma", "type": "inline_equation" }, { "bbox": [ 330, 81, 368, 97 ], "score": 1.0, "content": ". Finally,", "type": "text" } ], "index": 0 } ], "index": 0, "bbox_fs": [ 104, 81, 368, 97 ] }, { "type": "interline_equation", "bbox": [ 139, 99, 471, 187 ], "lines": [ { "bbox": [ 139, 99, 471, 187 ], "spans": [ { "bbox": [ 139, 99, 471, 187 ], "score": 0.95, "content": "\\begin{array} { r l } { { \\mathbb { P } ( | \\hat { N _ { t } } ( s , a ) - N _ { t } ^ { * } ( s , a ) | \\geq \\frac { C t } { \\log t } ) \\leq \\sum _ { i = 1 } ^ { t } \\mathbb { P } ( E _ { r _ { t } } ^ { c } ) = \\displaystyle \\sum _ { i = 1 } ^ { t } 4 | A | \\exp ( - \\frac { t } { ( \\log ( 2 + t ) ) ^ { 3 } } ) } } \\\\ & { \\leq 4 | A | t \\exp ( - \\frac { t } { ( \\log ( 2 + t ) ) ^ { 3 } } ) } \\\\ & { = O ( t \\exp ( - \\frac { t } { ( \\log ( t ) ) ^ { 3 } } ) ) . } \\end{array}", "type": "interline_equation", "image_path": "8ff3c9f025e43ce9f87638de08fcf8f343e8e7e280cfdda75296ac27b0a76dae.jpg" } ] } ], "index": 2, "virtual_lines": [ { "bbox": [ 139, 99, 471, 128.33333333333334 ], "spans": [], "index": 1 }, { "bbox": [ 139, 128.33333333333334, 471, 157.66666666666669 ], "spans": [], "index": 2 }, { "bbox": [ 139, 157.66666666666669, 471, 187.00000000000003 ], "spans": [], "index": 3 } ] }, { "type": "text", "bbox": [ 107, 194, 505, 229 ], "lines": [ { "bbox": [ 105, 192, 506, 207 ], "spans": [ { "bbox": [ 105, 192, 207, 207 ], "score": 1.0, "content": "Lemma 4 Consider the", "type": "text" }, { "bbox": [ 208, 194, 230, 204 ], "score": 0.3, "content": "E 3 W", "type": "inline_equation" }, { "bbox": [ 230, 192, 336, 207 ], "score": 1.0, "content": "policy applied to a tree.", "type": "text" }, { "bbox": [ 337, 195, 348, 204 ], "score": 0.3, "content": "A t", "type": "inline_equation" }, { "bbox": [ 349, 192, 506, 207 ], "score": 1.0, "content": "any node s of the tree, Let us define", "type": "text" } ], "index": 4 }, { "bbox": [ 107, 204, 506, 218 ], "spans": [ { "bbox": [ 107, 205, 196, 217 ], "score": 0.91, "content": "N _ { t } ^ { * } ( s , a ) = \\pi ^ { * } ( a | s ) . t", "type": "inline_equation" }, { "bbox": [ 197, 204, 219, 218 ], "score": 1.0, "content": ", and", "type": "text" }, { "bbox": [ 220, 205, 254, 217 ], "score": 0.91, "content": "N _ { t } ( s , a )", "type": "inline_equation" }, { "bbox": [ 254, 204, 381, 218 ], "score": 1.0, "content": "as the number of times action", "type": "text" }, { "bbox": [ 381, 208, 387, 215 ], "score": 0.28, "content": "a", "type": "inline_equation" }, { "bbox": [ 388, 204, 506, 218 ], "score": 1.0, "content": "have been chosen until time", "type": "text" } ], "index": 5 }, { "bbox": [ 105, 217, 284, 230 ], "spans": [ { "bbox": [ 105, 218, 125, 230 ], "score": 1.0, "content": "step", "type": "text" }, { "bbox": [ 125, 220, 130, 228 ], "score": 0.46, "content": "t", "type": "inline_equation" }, { "bbox": [ 130, 218, 207, 230 ], "score": 1.0, "content": ". There exists some", "type": "text" }, { "bbox": [ 207, 218, 216, 228 ], "score": 0.82, "content": "C", "type": "inline_equation" }, { "bbox": [ 216, 218, 235, 230 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 235, 217, 244, 228 ], "score": 0.86, "content": "\\hat { C }", "type": "inline_equation" }, { "bbox": [ 244, 218, 284, 230 ], "score": 1.0, "content": "such that", "type": "text" } ], "index": 6 } ], "index": 5, "bbox_fs": [ 105, 192, 506, 230 ] }, { "type": "interline_equation", "bbox": [ 189, 234, 422, 260 ], "lines": [ { "bbox": [ 189, 234, 422, 260 ], "spans": [ { "bbox": [ 189, 234, 422, 260 ], "score": 0.93, "content": "\\mathbb { P } \\big ( | N _ { t } ( s , a ) - N _ { t } ^ { * } ( s , a ) | > \\frac { C t } { \\log t } \\big ) \\le \\hat { C } t \\exp \\{ - \\frac { t } { ( \\log t ) ^ { 3 } } \\} .", "type": "interline_equation", "image_path": "2aabca78cd25431ac6f23fcdd6bac4a6fe0a325e715324fe2dc243f108a9ad87.jpg" } ] } ], "index": 7, "virtual_lines": [ { "bbox": [ 189, 234, 422, 260 ], "spans": [], "index": 7 } ] }, { "type": "text", "bbox": [ 106, 268, 318, 281 ], "lines": [ { "bbox": [ 105, 267, 318, 282 ], "spans": [ { "bbox": [ 105, 267, 274, 282 ], "score": 1.0, "content": "Proof 4 Based on the result from Lemma", "type": "text" }, { "bbox": [ 274, 271, 280, 279 ], "score": 0.58, "content": "^ 3", "type": "inline_equation" }, { "bbox": [ 280, 267, 318, 282 ], "score": 1.0, "content": ", we have", "type": "text" } ], "index": 8 } ], "index": 8, "bbox_fs": [ 105, 267, 318, 282 ] }, { "type": "interline_equation", "bbox": [ 140, 285, 450, 390 ], "lines": [ { "bbox": [ 140, 285, 450, 390 ], "spans": [ { "bbox": [ 140, 285, 450, 390 ], "score": 0.94, "content": "\\begin{array} { r l } & { \\mathbb { P } \\big ( | N _ { t } ( s , a ) - N _ { t } ^ { * } ( s , a ) | > ( 1 + C ) \\displaystyle \\frac { t } { \\log t } \\big ) \\leq C t \\exp \\{ - \\displaystyle \\frac { t } { ( \\log t ) ^ { 3 } } \\} } \\\\ & { \\leq \\mathbb { P } \\big ( | \\hat { N } _ { t } ( s , a ) - N _ { t } ^ { * } ( s , a ) | > \\displaystyle \\frac { C t } { \\log t } \\big ) + \\mathbb { P } \\big ( | N _ { t } ( s , a ) - \\hat { N } _ { t } ( s , a ) | > \\displaystyle \\frac { t } { \\log t } \\big ) } \\\\ & { \\leq 4 | A | t \\exp \\{ - \\displaystyle \\frac { t } { ( \\log ( 2 + t ) ) ^ { 3 } } \\} + 2 | A | \\exp \\{ - \\displaystyle \\frac { t } { ( \\log ( 2 + t ) ) ^ { 2 } } \\} ( L e m m a 3 a n d ( \\log ( 2 + t ) ) ) } \\\\ & { \\leq O ( t \\exp ( - \\displaystyle \\frac { t } { ( \\log t ) ^ { 3 } } ) ) . } \\end{array}", "type": "interline_equation", "image_path": "1dcac355f10d8103cc1a2339cee649ac1b796639d183bc7c3f3795d4d2466ef6.jpg" } ] } ], "index": 10, "virtual_lines": [ { "bbox": [ 140, 285, 450, 320.0 ], "spans": [], "index": 9 }, { "bbox": [ 140, 320.0, 450, 355.0 ], "spans": [], "index": 10 }, { "bbox": [ 140, 355.0, 450, 390.0 ], "spans": [], "index": 11 } ] }, { "type": "text", "bbox": [ 106, 398, 505, 421 ], "lines": [ { "bbox": [ 105, 396, 505, 412 ], "spans": [ { "bbox": [ 105, 396, 309, 412 ], "score": 1.0, "content": "Theorem 1 At the root node s of the tree, defining", "type": "text" }, { "bbox": [ 310, 399, 332, 411 ], "score": 0.91, "content": "N ( s )", "type": "inline_equation" }, { "bbox": [ 333, 396, 462, 412 ], "score": 1.0, "content": "as the number of visitations and", "type": "text" }, { "bbox": [ 463, 398, 492, 411 ], "score": 0.91, "content": "V _ { \\Omega ^ { * } } ( s )", "type": "inline_equation" }, { "bbox": [ 493, 396, 505, 412 ], "score": 1.0, "content": "as", "type": "text" } ], "index": 12 }, { "bbox": [ 106, 409, 303, 421 ], "spans": [ { "bbox": [ 106, 409, 218, 421 ], "score": 1.0, "content": "the estimated value at node", "type": "text" }, { "bbox": [ 218, 412, 223, 419 ], "score": 0.42, "content": "s", "type": "inline_equation" }, { "bbox": [ 223, 409, 241, 421 ], "score": 1.0, "content": ", for", "type": "text" }, { "bbox": [ 241, 410, 264, 420 ], "score": 0.85, "content": "\\epsilon > 0", "type": "inline_equation" }, { "bbox": [ 265, 409, 303, 421 ], "score": 1.0, "content": ", we have", "type": "text" } ], "index": 13 } ], "index": 12.5, "bbox_fs": [ 105, 396, 505, 421 ] }, { "type": "interline_equation", "bbox": [ 185, 425, 426, 454 ], "lines": [ { "bbox": [ 185, 425, 426, 454 ], "spans": [ { "bbox": [ 185, 425, 426, 454 ], "score": 0.91, "content": "\\mathbb { P } ( | V _ { \\Omega } ( s ) - V _ { \\Omega } ^ { * } ( s ) | > \\epsilon ) \\le C \\exp \\{ - \\frac { N ( s ) \\epsilon } { \\hat { C } ( \\log ( 2 + N ( s ) ) ) ^ { 2 } } \\} .", "type": "interline_equation", "image_path": "f6e3a8443a668905551dd24efcf33683733e38bf4a91c4a60893a76afd39fcb2.jpg" } ] } ], "index": 14, "virtual_lines": [ { "bbox": [ 185, 425, 426, 454 ], "spans": [], "index": 14 } ] }, { "type": "text", "bbox": [ 105, 462, 504, 485 ], "lines": [ { "bbox": [ 105, 461, 505, 476 ], "spans": [ { "bbox": [ 105, 461, 473, 476 ], "score": 1.0, "content": "Proof 5 We prove this concentration inequality by induction. When the depth of the tree is", "type": "text" }, { "bbox": [ 473, 463, 501, 473 ], "score": 0.88, "content": "D = 1", "type": "inline_equation" }, { "bbox": [ 501, 461, 505, 476 ], "score": 1.0, "content": ",", "type": "text" } ], "index": 15 }, { "bbox": [ 104, 474, 216, 487 ], "spans": [ { "bbox": [ 104, 474, 177, 487 ], "score": 1.0, "content": "from Proposition", "type": "text" }, { "bbox": [ 177, 474, 182, 483 ], "score": 0.64, "content": "^ { l }", "type": "inline_equation" }, { "bbox": [ 182, 474, 216, 487 ], "score": 1.0, "content": ", we get", "type": "text" } ], "index": 16 } ], "index": 15.5, "bbox_fs": [ 104, 461, 505, 487 ] }, { "type": "interline_equation", "bbox": [ 130, 489, 481, 504 ], "lines": [ { "bbox": [ 130, 489, 481, 504 ], "spans": [ { "bbox": [ 130, 489, 481, 504 ], "score": 0.89, "content": "\\left| V _ { \\Omega } ( s ) - V _ { \\Omega } ^ { * } ( s ) \\right| = \\mid \\Omega ^ { * } ( Q _ { \\Omega } ( s , . ) ) - \\Omega ^ { * } ( Q _ { \\Omega } ^ { * } ( s , . ) ) \\mid \\mid _ { \\infty } \\le \\gamma \\mid \\mid \\hat { r } - r ^ { * } \\mid _ { \\infty } ( C o n t r a c t i o n )", "type": "interline_equation", "image_path": "04f271923bc197f7b3a71d4c67e6c4b977c9013e20b7c123c82c2f06b975deb0.jpg" } ] } ], "index": 17, "virtual_lines": [ { "bbox": [ 130, 489, 481, 504 ], "spans": [], "index": 17 } ] }, { "type": "text", "bbox": [ 112, 508, 388, 520 ], "lines": [ { "bbox": [ 110, 507, 374, 521 ], "spans": [ { "bbox": [ 110, 507, 133, 521 ], "score": 1.0, "content": "where", "type": "text" }, { "bbox": [ 133, 509, 139, 518 ], "score": 0.8, "content": "\\hat { r }", "type": "inline_equation" }, { "bbox": [ 140, 507, 250, 521 ], "score": 1.0, "content": "is the average rewards and", "type": "text" }, { "bbox": [ 250, 509, 261, 518 ], "score": 0.85, "content": "r ^ { * }", "type": "inline_equation" }, { "bbox": [ 261, 507, 374, 521 ], "score": 1.0, "content": "is the mean reward. So that", "type": "text" } ], "index": 18 } ], "index": 18, "bbox_fs": [ 110, 507, 374, 521 ] }, { "type": "interline_equation", "bbox": [ 203, 525, 406, 539 ], "lines": [ { "bbox": [ 203, 525, 406, 539 ], "spans": [ { "bbox": [ 203, 525, 406, 539 ], "score": 0.88, "content": "\\mathbb { P } ( | V _ { \\Omega } ( s ) - V _ { \\Omega } ^ { * } ( s ) | > \\epsilon ) \\le \\mathbb { P } ( \\gamma \\parallel \\hat { r } - r ^ { * } \\parallel _ { \\infty } > \\epsilon ) .", "type": "interline_equation", "image_path": "5a0dd09fb53a4d835ab652659f61fca36bd713cf16856dfce7198a72c1f3459f.jpg" } ] } ], "index": 19, "virtual_lines": [ { "bbox": [ 203, 525, 406, 539 ], "spans": [], "index": 19 } ] }, { "type": "text", "bbox": [ 106, 544, 295, 560 ], "lines": [ { "bbox": [ 102, 541, 299, 564 ], "spans": [ { "bbox": [ 102, 541, 162, 564 ], "score": 1.0, "content": "From Lemma", "type": "text" }, { "bbox": [ 163, 547, 168, 556 ], "score": 0.69, "content": "^ { l }", "type": "inline_equation" }, { "bbox": [ 168, 541, 192, 564 ], "score": 1.0, "content": ", with", "type": "text" }, { "bbox": [ 192, 544, 255, 560 ], "score": 0.94, "content": "\\begin{array} { r } { \\epsilon = \\frac { 2 \\sigma \\gamma } { \\log ( 2 + N ( s ) ) } } \\end{array}", "type": "inline_equation" }, { "bbox": [ 256, 541, 299, 564 ], "score": 1.0, "content": ", we have", "type": "text" } ], "index": 20 } ], "index": 20, "bbox_fs": [ 102, 541, 299, 564 ] }, { "type": "interline_equation", "bbox": [ 126, 565, 484, 622 ], "lines": [ { "bbox": [ 126, 565, 484, 622 ], "spans": [ { "bbox": [ 126, 565, 484, 622 ], "score": 0.93, "content": "\\begin{array} { r l } & { \\mathbb { P } ( | V _ { \\Omega } ( s ) - V _ { \\Omega } ^ { * } ( s ) | > \\epsilon ) \\le \\mathbb { P } ( \\gamma \\parallel \\hat { r } - r ^ { * } \\parallel _ { \\infty } > \\epsilon ) \\le 4 | A | \\exp \\{ - \\frac { N ( s ) \\epsilon } { 2 \\sigma \\gamma ( \\log ( 2 + N ( s ) ) ) ^ { 2 } } \\} } \\\\ & { \\quad \\quad \\quad = C \\exp \\{ - \\frac { N ( s ) \\epsilon } { \\hat { C } ( \\log ( 2 + N ( s ) ) ) ^ { 2 } } \\} . } \\end{array}", "type": "interline_equation", "image_path": "0892983841e4725b636e9902bd510d5fab9a6d3f595ac187e20f0892c0ea672f.jpg" } ] } ], "index": 22, "virtual_lines": [ { "bbox": [ 126, 565, 484, 584.0 ], "spans": [], "index": 21 }, { "bbox": [ 126, 584.0, 484, 603.0 ], "spans": [], "index": 22 }, { "bbox": [ 126, 603.0, 484, 622.0 ], "spans": [], "index": 23 } ] }, { "type": "text", "bbox": [ 105, 626, 504, 649 ], "lines": [ { "bbox": [ 105, 625, 505, 640 ], "spans": [ { "bbox": [ 105, 625, 338, 640 ], "score": 1.0, "content": "Let assume we have the concentration bound at the depth", "type": "text" }, { "bbox": [ 338, 627, 364, 637 ], "score": 0.88, "content": "D - 1", "type": "inline_equation" }, { "bbox": [ 365, 625, 421, 640 ], "score": 1.0, "content": ", Let us define", "type": "text" }, { "bbox": [ 422, 627, 501, 638 ], "score": 0.92, "content": "V _ { \\Omega } ( s _ { a } ) = Q _ { \\Omega } ( s , a )", "type": "inline_equation" }, { "bbox": [ 501, 625, 505, 640 ], "score": 1.0, "content": ",", "type": "text" } ], "index": 24 }, { "bbox": [ 105, 636, 421, 649 ], "spans": [ { "bbox": [ 105, 636, 133, 649 ], "score": 1.0, "content": "where", "type": "text" }, { "bbox": [ 133, 639, 144, 648 ], "score": 0.83, "content": "s _ { a }", "type": "inline_equation" }, { "bbox": [ 144, 636, 279, 649 ], "score": 1.0, "content": "is the state reached taking action", "type": "text" }, { "bbox": [ 279, 639, 285, 647 ], "score": 0.47, "content": "a", "type": "inline_equation" }, { "bbox": [ 286, 636, 393, 649 ], "score": 1.0, "content": "from state s. then at depth", "type": "text" }, { "bbox": [ 394, 638, 421, 648 ], "score": 0.88, "content": "D - 1", "type": "inline_equation" } ], "index": 25 } ], "index": 24.5, "bbox_fs": [ 105, 625, 505, 649 ] }, { "type": "interline_equation", "bbox": [ 177, 653, 433, 682 ], "lines": [ { "bbox": [ 177, 653, 433, 682 ], "spans": [ { "bbox": [ 177, 653, 433, 682 ], "score": 0.92, "content": "\\mathbb { P } ( | V _ { \\Omega } ( s _ { a } ) - V _ { \\Omega } ^ { * } ( s _ { a } ) | > \\epsilon ) \\le C \\exp \\{ - \\frac { N ( s _ { a } ) \\epsilon } { \\hat { C } ( \\log ( 2 + N ( s _ { a } ) ) ) ^ { 2 } } \\} .", "type": "interline_equation", "image_path": "c98cdea666f1dbe10bfdf596654a65d6083e32af6dd4b1a8b1d4e7331793f9a8.jpg" } ] } ], "index": 26, "virtual_lines": [ { "bbox": [ 177, 653, 433, 682 ], "spans": [], "index": 26 } ] }, { "type": "text", "bbox": [ 105, 685, 374, 698 ], "lines": [ { "bbox": [ 105, 684, 375, 700 ], "spans": [ { "bbox": [ 105, 684, 177, 700 ], "score": 1.0, "content": "Now at the depth", "type": "text" }, { "bbox": [ 177, 687, 186, 696 ], "score": 0.77, "content": "D", "type": "inline_equation" }, { "bbox": [ 186, 684, 375, 700 ], "score": 1.0, "content": ", because of the Contraction Property, we have", "type": "text" } ], "index": 27 } ], "index": 27, "bbox_fs": [ 105, 684, 375, 700 ] }, { "type": "interline_equation", "bbox": [ 208, 701, 401, 732 ], "lines": [ { "bbox": [ 208, 701, 401, 732 ], "spans": [ { "bbox": [ 208, 701, 401, 732 ], "score": 0.9, "content": "\\begin{array} { r l } & { | V _ { \\Omega } ( s ) - V _ { \\Omega } ^ { * } ( s ) | \\leq \\gamma \\parallel Q _ { \\Omega } ( s , . ) - Q _ { \\Omega } ^ { * } ( s , . ) \\parallel _ { \\infty } } \\\\ & { \\qquad = \\gamma | Q _ { \\Omega } ( s , a ) - Q _ { \\Omega } ^ { * } ( s , a ) | . } \\end{array}", "type": "interline_equation", "image_path": "5049be6d96fabe4e364692ce31cbd9be2d6bd65e81ffbdef49525ca6a9c676ba.jpg" } ] } ], "index": 28.5, "virtual_lines": [ { "bbox": [ 208, 701, 401, 716.5 ], "spans": [], "index": 28 }, { "bbox": [ 208, 716.5, 401, 732.0 ], "spans": [], "index": 29 } ] } ] }, { "preproc_blocks": [ { "type": "text", "bbox": [ 106, 82, 137, 94 ], "lines": [ { "bbox": [ 105, 81, 138, 95 ], "spans": [ { "bbox": [ 105, 81, 138, 95 ], "score": 1.0, "content": "So that", "type": "text" } ], "index": 0 } ], "index": 0 }, { "type": "interline_equation", "bbox": [ 179, 96, 432, 171 ], "lines": [ { "bbox": [ 179, 96, 432, 171 ], "spans": [ { "bbox": [ 179, 96, 432, 171 ], "score": 0.92, "content": "\\begin{array} { r l r } & { } & { { \\mathbb { P } } ( | V _ { \\Omega } ( s ) - V _ { \\Omega } ^ { * } ( s ) | > \\epsilon ) \\leq { \\mathbb { P } } ( \\gamma \\parallel Q _ { \\Omega } ( s , a ) - Q _ { \\Omega } ^ { * } ( s , a ) \\parallel > \\epsilon ) } \\\\ & { } & { \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad } \\\\ & { } & { \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad } \\\\ & { } & { \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad } \\\\ & { } & { \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\hat { C } _ { a } ( \\log ( 2 + N ( s ) ) ) ^ { 2 } } \\end{array}", "type": "interline_equation", "image_path": "862b189fb68ac340e865133ca8f70643304b546d003187815f9ea693fb009acb.jpg" } ] } ], "index": 2, "virtual_lines": [ { "bbox": [ 179, 96, 432, 121.0 ], "spans": [], "index": 1 }, { "bbox": [ 179, 121.0, 432, 146.0 ], "spans": [], "index": 2 }, { "bbox": [ 179, 146.0, 432, 171.0 ], "spans": [], "index": 3 } ] }, { "type": "text", "bbox": [ 108, 173, 503, 210 ], "lines": [ { "bbox": [ 106, 173, 504, 187 ], "spans": [ { "bbox": [ 106, 173, 131, 187 ], "score": 1.0, "content": "From", "type": "text" }, { "bbox": [ 132, 174, 149, 185 ], "score": 0.26, "content": "( I 7 )", "type": "inline_equation" }, { "bbox": [ 149, 173, 210, 187 ], "score": 1.0, "content": ", we can have", "type": "text" }, { "bbox": [ 210, 174, 301, 186 ], "score": 0.91, "content": "\\begin{array} { r } { \\operatorname* { l i m } _ { t \\to \\infty } N ( s _ { a } ) = \\infty } \\end{array}", "type": "inline_equation" }, { "bbox": [ 301, 173, 347, 187 ], "score": 1.0, "content": "because if", "type": "text" }, { "bbox": [ 348, 174, 418, 186 ], "score": 0.82, "content": "\\exists L , N ( s _ { a } ) \\ < \\ L", "type": "inline_equation" }, { "bbox": [ 418, 173, 475, 187 ], "score": 1.0, "content": ", we can find", "type": "text" }, { "bbox": [ 476, 174, 504, 184 ], "score": 0.84, "content": "\\epsilon > 0", "type": "inline_equation" } ], "index": 4 }, { "bbox": [ 106, 185, 505, 198 ], "spans": [ { "bbox": [ 106, 185, 330, 198 ], "score": 1.0, "content": "for which (17) is not satisfied. From Lemma 4, when", "type": "text" }, { "bbox": [ 330, 186, 352, 197 ], "score": 0.93, "content": "N ( s )", "type": "inline_equation" }, { "bbox": [ 353, 185, 461, 198 ], "score": 1.0, "content": "is large enough, we have", "type": "text" }, { "bbox": [ 462, 185, 505, 197 ], "score": 0.89, "content": "N ( s _ { a } ) ", "type": "inline_equation" } ], "index": 5 }, { "bbox": [ 107, 196, 500, 211 ], "spans": [ { "bbox": [ 107, 197, 160, 210 ], "score": 0.92, "content": "\\pi ^ { * } ( a | s ) N ( s )", "type": "inline_equation" }, { "bbox": [ 160, 196, 214, 211 ], "score": 1.0, "content": "(for example", "type": "text" }, { "bbox": [ 214, 196, 317, 211 ], "score": 0.93, "content": "\\begin{array} { r } { N ( s _ { a } ) > \\frac { 1 } { 2 } \\pi ^ { * } ( a | s ) N ( s ) ) } \\end{array}", "type": "inline_equation" }, { "bbox": [ 317, 196, 415, 211 ], "score": 1.0, "content": ", that means we can find", "type": "text" }, { "bbox": [ 415, 198, 424, 208 ], "score": 0.78, "content": "C", "type": "inline_equation" }, { "bbox": [ 425, 196, 443, 211 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 443, 196, 452, 208 ], "score": 0.86, "content": "\\hat { C }", "type": "inline_equation" }, { "bbox": [ 452, 196, 500, 211 ], "score": 1.0, "content": "that satisfy", "type": "text" } ], "index": 6 } ], "index": 5 }, { "type": "interline_equation", "bbox": [ 184, 214, 425, 243 ], "lines": [ { "bbox": [ 184, 214, 425, 243 ], "spans": [ { "bbox": [ 184, 214, 425, 243 ], "score": 0.91, "content": "\\mathbb { P } ( | V _ { \\Omega } ( s ) - V _ { \\Omega } ^ { * } ( s ) | > \\epsilon ) \\le C \\exp \\{ - \\frac { N ( s ) \\epsilon } { \\hat { C } ( \\log ( 2 + N ( s ) ) ) ^ { 2 } } \\} .", "type": "interline_equation", "image_path": "67bb4d4a844dd34e21d74312cc95cb564cfc1d0d3017caf87528586975e1352a.jpg" } ] } ], "index": 7.5, "virtual_lines": [ { "bbox": [ 184, 214, 425, 228.5 ], "spans": [], "index": 7 }, { "bbox": [ 184, 228.5, 425, 243.0 ], "spans": [], "index": 8 } ] }, { "type": "text", "bbox": [ 102, 250, 470, 263 ], "lines": [ { "bbox": [ 105, 249, 471, 264 ], "spans": [ { "bbox": [ 105, 249, 254, 264 ], "score": 1.0, "content": "Lemma 5 At any node s of the tree,", "type": "text" }, { "bbox": [ 254, 251, 276, 263 ], "score": 0.92, "content": "N ( s )", "type": "inline_equation" }, { "bbox": [ 277, 249, 471, 264 ], "score": 1.0, "content": "is the number of visitations. We define the event", "type": "text" } ], "index": 9 } ], "index": 9 }, { "type": "interline_equation", "bbox": [ 132, 267, 477, 291 ], "lines": [ { "bbox": [ 132, 267, 477, 291 ], "spans": [ { "bbox": [ 132, 267, 477, 291 ], "score": 0.9, "content": "E _ { s } = \\{ \\forall a i n \\boldsymbol { A } , | N ( s , a ) - N ^ { * } ( s , a ) | < \\frac { N ^ { * } ( s , a ) } { 2 } \\} w h e r e \\ N ^ { * } ( s , a ) = \\pi ^ { * } ( a | s ) N ( s ) ,", "type": "interline_equation", "image_path": "7b06e07c093361ffc5e8d58045792e29e574ae3c0b793df7b0c45085d63fcad3.jpg" } ] } ], "index": 10, "virtual_lines": [ { "bbox": [ 132, 267, 477, 291 ], "spans": [], "index": 10 } ] }, { "type": "text", "bbox": [ 107, 294, 372, 307 ], "lines": [ { "bbox": [ 106, 294, 372, 307 ], "spans": [ { "bbox": [ 106, 294, 133, 307 ], "score": 1.0, "content": "where", "type": "text" }, { "bbox": [ 133, 295, 156, 305 ], "score": 0.89, "content": "\\epsilon > 0", "type": "inline_equation" }, { "bbox": [ 157, 294, 175, 307 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 176, 295, 205, 307 ], "score": 0.92, "content": "V _ { \\Omega ^ { * } } ( s )", "type": "inline_equation" }, { "bbox": [ 205, 294, 326, 307 ], "score": 1.0, "content": "is the estimated value at node", "type": "text" }, { "bbox": [ 327, 298, 332, 304 ], "score": 0.42, "content": "s", "type": "inline_equation" }, { "bbox": [ 332, 294, 372, 307 ], "score": 1.0, "content": ". We have", "type": "text" } ], "index": 11 } ], "index": 11 }, { "type": "interline_equation", "bbox": [ 177, 310, 433, 339 ], "lines": [ { "bbox": [ 177, 310, 433, 339 ], "spans": [ { "bbox": [ 177, 310, 433, 339 ], "score": 0.91, "content": "\\mathbb { P } ( | V _ { \\Omega } ( s ) - V _ { \\Omega } ^ { * } ( s ) | > \\epsilon | E _ { s } ) \\le C \\exp \\{ - \\frac { N ( s ) \\epsilon } { \\hat { C } ( \\log ( 2 + N ( s ) ) ) ^ { 2 } } \\} .", "type": "interline_equation", "image_path": "2ed63c31de63326dd6e596ef8d553436fd927bd46f7290c28971da9e333ebaf5.jpg" } ] } ], "index": 13, "virtual_lines": [ { "bbox": [ 177, 310, 433, 319.6666666666667 ], "spans": [], "index": 12 }, { "bbox": [ 177, 319.6666666666667, 433, 329.33333333333337 ], "spans": [], "index": 13 }, { "bbox": [ 177, 329.33333333333337, 433, 339.00000000000006 ], "spans": [], "index": 14 } ] }, { "type": "text", "bbox": [ 106, 346, 502, 369 ], "lines": [ { "bbox": [ 105, 345, 504, 360 ], "spans": [ { "bbox": [ 105, 345, 504, 360 ], "score": 1.0, "content": "Proof 6 The proof is the same as in Theorem 2. We prove the concentration inequality by induction.", "type": "text" } ], "index": 15 }, { "bbox": [ 106, 356, 363, 371 ], "spans": [ { "bbox": [ 106, 356, 222, 371 ], "score": 1.0, "content": "When the depth of the tree is", "type": "text" }, { "bbox": [ 223, 358, 250, 368 ], "score": 0.89, "content": "D = 1", "type": "inline_equation" }, { "bbox": [ 251, 356, 324, 371 ], "score": 1.0, "content": ", from Proposition", "type": "text" }, { "bbox": [ 325, 358, 330, 367 ], "score": 0.57, "content": "^ { l }", "type": "inline_equation" }, { "bbox": [ 330, 356, 363, 371 ], "score": 1.0, "content": ", we get", "type": "text" } ], "index": 16 } ], "index": 15.5 }, { "type": "interline_equation", "bbox": [ 116, 373, 486, 387 ], "lines": [ { "bbox": [ 116, 373, 486, 387 ], "spans": [ { "bbox": [ 116, 373, 486, 387 ], "score": 0.78, "content": "\\left| V _ { \\Omega } ( s ) - V _ { \\Omega } ^ { * } ( s ) \\right| = \\left\\| \\ \\Omega ^ { * } ( Q _ { \\Omega } ( s , . ) ) - \\Omega ^ { * } ( Q _ { \\Omega } ^ { * } ( s , . ) ) \\ \\right\\| \\leq \\gamma \\ \\left\\| \\ \\hat { r } - r ^ { * } \\ \\right\\| _ { \\infty } \\left( C o n t r a c t i o n \\ P r o p e r t i o n s \\right) ,", "type": "interline_equation", "image_path": "dbb8c2d86ed50af634924d6947bfbc00c6a1be458cc7d266821ed42c33f5257a.jpg" } ] } ], "index": 17, "virtual_lines": [ { "bbox": [ 116, 373, 486, 387 ], "spans": [], "index": 17 } ] }, { "type": "text", "bbox": [ 108, 390, 387, 401 ], "lines": [ { "bbox": [ 106, 389, 378, 403 ], "spans": [ { "bbox": [ 106, 389, 133, 403 ], "score": 1.0, "content": "where", "type": "text" }, { "bbox": [ 133, 391, 139, 399 ], "score": 0.79, "content": "\\hat { r }", "type": "inline_equation" }, { "bbox": [ 139, 389, 250, 403 ], "score": 1.0, "content": "is the average rewards and", "type": "text" }, { "bbox": [ 250, 390, 260, 400 ], "score": 0.86, "content": "r ^ { * }", "type": "inline_equation" }, { "bbox": [ 261, 389, 378, 403 ], "score": 1.0, "content": "is the mean rewards. So that", "type": "text" } ], "index": 18 } ], "index": 18 }, { "type": "interline_equation", "bbox": [ 203, 405, 407, 419 ], "lines": [ { "bbox": [ 203, 405, 407, 419 ], "spans": [ { "bbox": [ 203, 405, 407, 419 ], "score": 0.87, "content": "\\mathbb { P } ( | V _ { \\Omega } ( s ) - V _ { \\Omega } ^ { * } ( s ) | > \\epsilon ) \\le \\mathbb { P } ( \\gamma \\parallel \\hat { r } - r ^ { * } \\parallel _ { \\infty } > \\epsilon ) .", "type": "interline_equation", "image_path": "5e0ac07aa2e7aced068c92a20f40cf55901cadc8b67aa2e74f81a7de8501fcaa.jpg" } ] } ], "index": 19, "virtual_lines": [ { "bbox": [ 203, 405, 407, 419 ], "spans": [], "index": 19 } ] }, { "type": "text", "bbox": [ 106, 423, 351, 440 ], "lines": [ { "bbox": [ 106, 421, 351, 441 ], "spans": [ { "bbox": [ 106, 421, 162, 441 ], "score": 1.0, "content": "From Lemma", "type": "text" }, { "bbox": [ 163, 426, 168, 435 ], "score": 0.68, "content": "^ { l }", "type": "inline_equation" }, { "bbox": [ 168, 421, 191, 441 ], "score": 1.0, "content": ", with", "type": "text" }, { "bbox": [ 192, 423, 256, 440 ], "score": 0.94, "content": "\\begin{array} { r } { \\epsilon = \\frac { 2 \\sigma \\gamma } { \\log ( 2 + N ( s ) ) } } \\end{array}", "type": "inline_equation" }, { "bbox": [ 256, 421, 299, 441 ], "score": 1.0, "content": "and given", "type": "text" }, { "bbox": [ 299, 425, 311, 436 ], "score": 0.88, "content": "E _ { s }", "type": "inline_equation" }, { "bbox": [ 312, 421, 351, 441 ], "score": 1.0, "content": ", we have", "type": "text" } ], "index": 20 } ], "index": 20 }, { "type": "interline_equation", "bbox": [ 126, 443, 484, 500 ], "lines": [ { "bbox": [ 126, 443, 484, 500 ], "spans": [ { "bbox": [ 126, 443, 484, 500 ], "score": 0.92, "content": "\\begin{array} { r l } & { \\mathbb { P } ( | V _ { \\Omega } ( s ) - V _ { \\Omega } ^ { * } ( s ) | > \\epsilon ) \\le \\mathbb { P } ( \\gamma \\parallel \\hat { r } - r ^ { * } \\parallel _ { \\infty } > \\epsilon ) \\le 4 | A | \\exp \\{ - \\frac { N ( s ) \\epsilon } { 2 \\sigma \\gamma ( \\log ( 2 + N ( s ) ) ) ^ { 2 } } \\} } \\\\ & { \\quad \\quad \\quad = C \\exp \\{ - \\frac { N ( s ) \\epsilon } { \\hat { C } ( \\log ( 2 + N ( s ) ) ) ^ { 2 } } \\} . } \\end{array}", "type": "interline_equation", "image_path": "617316a36ba69474015f9b8eb2f3f494365ed02641e1cc862897868d0caefdc4.jpg" } ] } ], "index": 22, "virtual_lines": [ { "bbox": [ 126, 443, 484, 462.0 ], "spans": [], "index": 21 }, { "bbox": [ 126, 462.0, 484, 481.0 ], "spans": [], "index": 22 }, { "bbox": [ 126, 481.0, 484, 500.0 ], "spans": [], "index": 23 } ] }, { "type": "text", "bbox": [ 105, 502, 506, 525 ], "lines": [ { "bbox": [ 105, 502, 505, 516 ], "spans": [ { "bbox": [ 105, 502, 338, 516 ], "score": 1.0, "content": "Let assume we have the concentration bound at the depth", "type": "text" }, { "bbox": [ 338, 503, 364, 513 ], "score": 0.82, "content": "D - 1", "type": "inline_equation" }, { "bbox": [ 365, 502, 421, 516 ], "score": 1.0, "content": ", Let us define", "type": "text" }, { "bbox": [ 422, 503, 501, 515 ], "score": 0.92, "content": "V _ { \\Omega } ( s _ { a } ) = Q _ { \\Omega } ( s , a )", "type": "inline_equation" }, { "bbox": [ 501, 502, 505, 516 ], "score": 1.0, "content": ",", "type": "text" } ], "index": 24 }, { "bbox": [ 105, 513, 420, 526 ], "spans": [ { "bbox": [ 105, 513, 133, 526 ], "score": 1.0, "content": "where", "type": "text" }, { "bbox": [ 133, 515, 144, 525 ], "score": 0.84, "content": "s _ { a }", "type": "inline_equation" }, { "bbox": [ 144, 513, 329, 526 ], "score": 1.0, "content": "is the state reached taking action a from state", "type": "text" }, { "bbox": [ 329, 516, 334, 523 ], "score": 0.57, "content": "s", "type": "inline_equation" }, { "bbox": [ 335, 513, 393, 526 ], "score": 1.0, "content": ", then at depth", "type": "text" }, { "bbox": [ 393, 514, 420, 524 ], "score": 0.88, "content": "D - 1", "type": "inline_equation" } ], "index": 25 } ], "index": 24.5 }, { "type": "interline_equation", "bbox": [ 177, 529, 433, 557 ], "lines": [ { "bbox": [ 177, 529, 433, 557 ], "spans": [ { "bbox": [ 177, 529, 433, 557 ], "score": 0.92, "content": "\\mathbb { P } ( | V _ { \\Omega } ( s _ { a } ) - V _ { \\Omega } ^ { * } ( s _ { a } ) | > \\epsilon ) \\le C \\exp \\{ - \\frac { N ( s _ { a } ) \\epsilon } { \\hat { C } ( \\log ( 2 + N ( s _ { a } ) ) ) ^ { 2 } } \\} .", "type": "interline_equation", "image_path": "1c85f24984a86a52582dcde14bacd5b5691ab32ad060ce088ba9d833cdb63415.jpg" } ] } ], "index": 26, "virtual_lines": [ { "bbox": [ 177, 529, 433, 557 ], "spans": [], "index": 26 } ] }, { "type": "text", "bbox": [ 107, 560, 416, 572 ], "lines": [ { "bbox": [ 106, 559, 416, 573 ], "spans": [ { "bbox": [ 106, 559, 162, 573 ], "score": 1.0, "content": "Now at depth", "type": "text" }, { "bbox": [ 162, 561, 172, 570 ], "score": 0.78, "content": "D", "type": "inline_equation" }, { "bbox": [ 172, 559, 365, 573 ], "score": 1.0, "content": ", because of the Contraction Property and given", "type": "text" }, { "bbox": [ 365, 561, 377, 571 ], "score": 0.88, "content": "E _ { s }", "type": "inline_equation" }, { "bbox": [ 377, 559, 416, 573 ], "score": 1.0, "content": ", we have", "type": "text" } ], "index": 27 } ], "index": 27 }, { "type": "interline_equation", "bbox": [ 184, 574, 425, 605 ], "lines": [ { "bbox": [ 184, 574, 425, 605 ], "spans": [ { "bbox": [ 184, 574, 425, 605 ], "score": 0.89, "content": "\\begin{array} { r l } & { | V _ { \\Omega } ( s ) - V _ { \\Omega } ^ { * } ( s ) | \\leq \\gamma \\parallel Q _ { \\Omega } ( s , . ) - Q _ { \\Omega } ^ { * } ( s , . ) \\parallel _ { \\infty } } \\\\ & { \\qquad = \\gamma | Q _ { \\Omega } ( s , a ) - Q _ { \\Omega } ^ { * } ( s , a ) | ( \\exists a , s a t i s f t e d ) . } \\end{array}", "type": "interline_equation", "image_path": "e0cb290fced55598b4b904c2a0f9354e46c6f8048b888fbace8168238bd2174a.jpg" } ] } ], "index": 28.5, "virtual_lines": [ { "bbox": [ 184, 574, 425, 589.5 ], "spans": [], "index": 28 }, { "bbox": [ 184, 589.5, 425, 605.0 ], "spans": [], "index": 29 } ] }, { "type": "text", "bbox": [ 106, 606, 137, 618 ], "lines": [ { "bbox": [ 105, 605, 138, 619 ], "spans": [ { "bbox": [ 105, 605, 138, 619 ], "score": 1.0, "content": "So that", "type": "text" } ], "index": 30 } ], "index": 30 }, { "type": "interline_equation", "bbox": [ 154, 621, 457, 725 ], "lines": [ { "bbox": [ 154, 621, 457, 725 ], "spans": [ { "bbox": [ 154, 621, 457, 725 ], "score": 0.94, "content": "\\begin{array} { r l } & { \\mathbb { P } ( | V _ { \\Omega } ( s ) - V _ { \\Omega } ^ { * } ( s ) | > \\epsilon ) \\leq \\mathbb { P } ( \\gamma \\parallel Q _ { \\Omega } ( s , a ) - Q _ { \\Omega } ^ { * } ( s , a ) \\parallel > \\epsilon ) } \\\\ & { \\qquad \\leq C _ { a } \\exp \\{ - \\frac { N ( s _ { a } ) \\epsilon } { \\hat { C } _ { a } ( \\log ( 2 + N ( s _ { a } ) ) ) ^ { 2 } } \\} } \\\\ & { \\qquad \\leq C _ { a } \\exp \\{ - \\frac { N ( s _ { a } ) \\epsilon } { \\hat { C } _ { a } ( \\log ( 2 + N ( s ) ) ) ^ { 2 } } \\} } \\\\ & { \\qquad \\leq C \\exp \\{ - \\frac { N ( s ) \\epsilon } { \\hat { C } ( \\log ( 2 + N ( s ) ) ) ^ { 2 } } \\} ( b e c a u s e o f E _ { s } ) } \\end{array}", "type": "interline_equation", "image_path": "ff386d89855cf3515a65d5b4ce2c91c90fed1dd0e25073e45066ebfcf59bbb21.jpg" } ] } ], "index": 32, "virtual_lines": [ { "bbox": [ 154, 621, 457, 655.6666666666666 ], "spans": [], "index": 31 }, { "bbox": [ 154, 655.6666666666666, 457, 690.3333333333333 ], "spans": [], "index": 32 }, { "bbox": [ 154, 690.3333333333333, 457, 724.9999999999999 ], "spans": [], "index": 33 } ] } ], "page_idx": 15, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 106, 26, 307, 38 ], "lines": [ { "bbox": [ 106, 25, 308, 39 ], "spans": [ { "bbox": [ 106, 25, 308, 39 ], "score": 1.0, "content": "Under review as a conference paper at ICLR 2021", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 300, 751, 311, 761 ], "lines": [ { "bbox": [ 299, 750, 313, 764 ], "spans": [ { "bbox": [ 299, 750, 313, 764 ], "score": 1.0, "content": "16", "type": "text" } ] } ] } ], "para_blocks": [ { "type": "text", "bbox": [ 106, 82, 137, 94 ], "lines": [ { "bbox": [ 105, 81, 138, 95 ], "spans": [ { "bbox": [ 105, 81, 138, 95 ], "score": 1.0, "content": "So that", "type": "text" } ], "index": 0 } ], "index": 0, "bbox_fs": [ 105, 81, 138, 95 ] }, { "type": "interline_equation", "bbox": [ 179, 96, 432, 171 ], "lines": [ { "bbox": [ 179, 96, 432, 171 ], "spans": [ { "bbox": [ 179, 96, 432, 171 ], "score": 0.92, "content": "\\begin{array} { r l r } & { } & { { \\mathbb { P } } ( | V _ { \\Omega } ( s ) - V _ { \\Omega } ^ { * } ( s ) | > \\epsilon ) \\leq { \\mathbb { P } } ( \\gamma \\parallel Q _ { \\Omega } ( s , a ) - Q _ { \\Omega } ^ { * } ( s , a ) \\parallel > \\epsilon ) } \\\\ & { } & { \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad } \\\\ & { } & { \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad } \\\\ & { } & { \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad } \\\\ & { } & { \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\hat { C } _ { a } ( \\log ( 2 + N ( s ) ) ) ^ { 2 } } \\end{array}", "type": "interline_equation", "image_path": "862b189fb68ac340e865133ca8f70643304b546d003187815f9ea693fb009acb.jpg" } ] } ], "index": 2, "virtual_lines": [ { "bbox": [ 179, 96, 432, 121.0 ], "spans": [], "index": 1 }, { "bbox": [ 179, 121.0, 432, 146.0 ], "spans": [], "index": 2 }, { "bbox": [ 179, 146.0, 432, 171.0 ], "spans": [], "index": 3 } ] }, { "type": "text", "bbox": [ 108, 173, 503, 210 ], "lines": [ { "bbox": [ 106, 173, 504, 187 ], "spans": [ { "bbox": [ 106, 173, 131, 187 ], "score": 1.0, "content": "From", "type": "text" }, { "bbox": [ 132, 174, 149, 185 ], "score": 0.26, "content": "( I 7 )", "type": "inline_equation" }, { "bbox": [ 149, 173, 210, 187 ], "score": 1.0, "content": ", we can have", "type": "text" }, { "bbox": [ 210, 174, 301, 186 ], "score": 0.91, "content": "\\begin{array} { r } { \\operatorname* { l i m } _ { t \\to \\infty } N ( s _ { a } ) = \\infty } \\end{array}", "type": "inline_equation" }, { "bbox": [ 301, 173, 347, 187 ], "score": 1.0, "content": "because if", "type": "text" }, { "bbox": [ 348, 174, 418, 186 ], "score": 0.82, "content": "\\exists L , N ( s _ { a } ) \\ < \\ L", "type": "inline_equation" }, { "bbox": [ 418, 173, 475, 187 ], "score": 1.0, "content": ", we can find", "type": "text" }, { "bbox": [ 476, 174, 504, 184 ], "score": 0.84, "content": "\\epsilon > 0", "type": "inline_equation" } ], "index": 4 }, { "bbox": [ 106, 185, 505, 198 ], "spans": [ { "bbox": [ 106, 185, 330, 198 ], "score": 1.0, "content": "for which (17) is not satisfied. From Lemma 4, when", "type": "text" }, { "bbox": [ 330, 186, 352, 197 ], "score": 0.93, "content": "N ( s )", "type": "inline_equation" }, { "bbox": [ 353, 185, 461, 198 ], "score": 1.0, "content": "is large enough, we have", "type": "text" }, { "bbox": [ 462, 185, 505, 197 ], "score": 0.89, "content": "N ( s _ { a } ) ", "type": "inline_equation" } ], "index": 5 }, { "bbox": [ 107, 196, 500, 211 ], "spans": [ { "bbox": [ 107, 197, 160, 210 ], "score": 0.92, "content": "\\pi ^ { * } ( a | s ) N ( s )", "type": "inline_equation" }, { "bbox": [ 160, 196, 214, 211 ], "score": 1.0, "content": "(for example", "type": "text" }, { "bbox": [ 214, 196, 317, 211 ], "score": 0.93, "content": "\\begin{array} { r } { N ( s _ { a } ) > \\frac { 1 } { 2 } \\pi ^ { * } ( a | s ) N ( s ) ) } \\end{array}", "type": "inline_equation" }, { "bbox": [ 317, 196, 415, 211 ], "score": 1.0, "content": ", that means we can find", "type": "text" }, { "bbox": [ 415, 198, 424, 208 ], "score": 0.78, "content": "C", "type": "inline_equation" }, { "bbox": [ 425, 196, 443, 211 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 443, 196, 452, 208 ], "score": 0.86, "content": "\\hat { C }", "type": "inline_equation" }, { "bbox": [ 452, 196, 500, 211 ], "score": 1.0, "content": "that satisfy", "type": "text" } ], "index": 6 } ], "index": 5, "bbox_fs": [ 106, 173, 505, 211 ] }, { "type": "interline_equation", "bbox": [ 184, 214, 425, 243 ], "lines": [ { "bbox": [ 184, 214, 425, 243 ], "spans": [ { "bbox": [ 184, 214, 425, 243 ], "score": 0.91, "content": "\\mathbb { P } ( | V _ { \\Omega } ( s ) - V _ { \\Omega } ^ { * } ( s ) | > \\epsilon ) \\le C \\exp \\{ - \\frac { N ( s ) \\epsilon } { \\hat { C } ( \\log ( 2 + N ( s ) ) ) ^ { 2 } } \\} .", "type": "interline_equation", "image_path": "67bb4d4a844dd34e21d74312cc95cb564cfc1d0d3017caf87528586975e1352a.jpg" } ] } ], "index": 7.5, "virtual_lines": [ { "bbox": [ 184, 214, 425, 228.5 ], "spans": [], "index": 7 }, { "bbox": [ 184, 228.5, 425, 243.0 ], "spans": [], "index": 8 } ] }, { "type": "text", "bbox": [ 102, 250, 470, 263 ], "lines": [ { "bbox": [ 105, 249, 471, 264 ], "spans": [ { "bbox": [ 105, 249, 254, 264 ], "score": 1.0, "content": "Lemma 5 At any node s of the tree,", "type": "text" }, { "bbox": [ 254, 251, 276, 263 ], "score": 0.92, "content": "N ( s )", "type": "inline_equation" }, { "bbox": [ 277, 249, 471, 264 ], "score": 1.0, "content": "is the number of visitations. We define the event", "type": "text" } ], "index": 9 } ], "index": 9, "bbox_fs": [ 105, 249, 471, 264 ] }, { "type": "interline_equation", "bbox": [ 132, 267, 477, 291 ], "lines": [ { "bbox": [ 132, 267, 477, 291 ], "spans": [ { "bbox": [ 132, 267, 477, 291 ], "score": 0.9, "content": "E _ { s } = \\{ \\forall a i n \\boldsymbol { A } , | N ( s , a ) - N ^ { * } ( s , a ) | < \\frac { N ^ { * } ( s , a ) } { 2 } \\} w h e r e \\ N ^ { * } ( s , a ) = \\pi ^ { * } ( a | s ) N ( s ) ,", "type": "interline_equation", "image_path": "7b06e07c093361ffc5e8d58045792e29e574ae3c0b793df7b0c45085d63fcad3.jpg" } ] } ], "index": 10, "virtual_lines": [ { "bbox": [ 132, 267, 477, 291 ], "spans": [], "index": 10 } ] }, { "type": "text", "bbox": [ 107, 294, 372, 307 ], "lines": [ { "bbox": [ 106, 294, 372, 307 ], "spans": [ { "bbox": [ 106, 294, 133, 307 ], "score": 1.0, "content": "where", "type": "text" }, { "bbox": [ 133, 295, 156, 305 ], "score": 0.89, "content": "\\epsilon > 0", "type": "inline_equation" }, { "bbox": [ 157, 294, 175, 307 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 176, 295, 205, 307 ], "score": 0.92, "content": "V _ { \\Omega ^ { * } } ( s )", "type": "inline_equation" }, { "bbox": [ 205, 294, 326, 307 ], "score": 1.0, "content": "is the estimated value at node", "type": "text" }, { "bbox": [ 327, 298, 332, 304 ], "score": 0.42, "content": "s", "type": "inline_equation" }, { "bbox": [ 332, 294, 372, 307 ], "score": 1.0, "content": ". We have", "type": "text" } ], "index": 11 } ], "index": 11, "bbox_fs": [ 106, 294, 372, 307 ] }, { "type": "interline_equation", "bbox": [ 177, 310, 433, 339 ], "lines": [ { "bbox": [ 177, 310, 433, 339 ], "spans": [ { "bbox": [ 177, 310, 433, 339 ], "score": 0.91, "content": "\\mathbb { P } ( | V _ { \\Omega } ( s ) - V _ { \\Omega } ^ { * } ( s ) | > \\epsilon | E _ { s } ) \\le C \\exp \\{ - \\frac { N ( s ) \\epsilon } { \\hat { C } ( \\log ( 2 + N ( s ) ) ) ^ { 2 } } \\} .", "type": "interline_equation", "image_path": "2ed63c31de63326dd6e596ef8d553436fd927bd46f7290c28971da9e333ebaf5.jpg" } ] } ], "index": 13, "virtual_lines": [ { "bbox": [ 177, 310, 433, 319.6666666666667 ], "spans": [], "index": 12 }, { "bbox": [ 177, 319.6666666666667, 433, 329.33333333333337 ], "spans": [], "index": 13 }, { "bbox": [ 177, 329.33333333333337, 433, 339.00000000000006 ], "spans": [], "index": 14 } ] }, { "type": "text", "bbox": [ 106, 346, 502, 369 ], "lines": [ { "bbox": [ 105, 345, 504, 360 ], "spans": [ { "bbox": [ 105, 345, 504, 360 ], "score": 1.0, "content": "Proof 6 The proof is the same as in Theorem 2. We prove the concentration inequality by induction.", "type": "text" } ], "index": 15 }, { "bbox": [ 106, 356, 363, 371 ], "spans": [ { "bbox": [ 106, 356, 222, 371 ], "score": 1.0, "content": "When the depth of the tree is", "type": "text" }, { "bbox": [ 223, 358, 250, 368 ], "score": 0.89, "content": "D = 1", "type": "inline_equation" }, { "bbox": [ 251, 356, 324, 371 ], "score": 1.0, "content": ", from Proposition", "type": "text" }, { "bbox": [ 325, 358, 330, 367 ], "score": 0.57, "content": "^ { l }", "type": "inline_equation" }, { "bbox": [ 330, 356, 363, 371 ], "score": 1.0, "content": ", we get", "type": "text" } ], "index": 16 } ], "index": 15.5, "bbox_fs": [ 105, 345, 504, 371 ] }, { "type": "interline_equation", "bbox": [ 116, 373, 486, 387 ], "lines": [ { "bbox": [ 116, 373, 486, 387 ], "spans": [ { "bbox": [ 116, 373, 486, 387 ], "score": 0.78, "content": "\\left| V _ { \\Omega } ( s ) - V _ { \\Omega } ^ { * } ( s ) \\right| = \\left\\| \\ \\Omega ^ { * } ( Q _ { \\Omega } ( s , . ) ) - \\Omega ^ { * } ( Q _ { \\Omega } ^ { * } ( s , . ) ) \\ \\right\\| \\leq \\gamma \\ \\left\\| \\ \\hat { r } - r ^ { * } \\ \\right\\| _ { \\infty } \\left( C o n t r a c t i o n \\ P r o p e r t i o n s \\right) ,", "type": "interline_equation", "image_path": "dbb8c2d86ed50af634924d6947bfbc00c6a1be458cc7d266821ed42c33f5257a.jpg" } ] } ], "index": 17, "virtual_lines": [ { "bbox": [ 116, 373, 486, 387 ], "spans": [], "index": 17 } ] }, { "type": "text", "bbox": [ 108, 390, 387, 401 ], "lines": [ { "bbox": [ 106, 389, 378, 403 ], "spans": [ { "bbox": [ 106, 389, 133, 403 ], "score": 1.0, "content": "where", "type": "text" }, { "bbox": [ 133, 391, 139, 399 ], "score": 0.79, "content": "\\hat { r }", "type": "inline_equation" }, { "bbox": [ 139, 389, 250, 403 ], "score": 1.0, "content": "is the average rewards and", "type": "text" }, { "bbox": [ 250, 390, 260, 400 ], "score": 0.86, "content": "r ^ { * }", "type": "inline_equation" }, { "bbox": [ 261, 389, 378, 403 ], "score": 1.0, "content": "is the mean rewards. So that", "type": "text" } ], "index": 18 } ], "index": 18, "bbox_fs": [ 106, 389, 378, 403 ] }, { "type": "interline_equation", "bbox": [ 203, 405, 407, 419 ], "lines": [ { "bbox": [ 203, 405, 407, 419 ], "spans": [ { "bbox": [ 203, 405, 407, 419 ], "score": 0.87, "content": "\\mathbb { P } ( | V _ { \\Omega } ( s ) - V _ { \\Omega } ^ { * } ( s ) | > \\epsilon ) \\le \\mathbb { P } ( \\gamma \\parallel \\hat { r } - r ^ { * } \\parallel _ { \\infty } > \\epsilon ) .", "type": "interline_equation", "image_path": "5e0ac07aa2e7aced068c92a20f40cf55901cadc8b67aa2e74f81a7de8501fcaa.jpg" } ] } ], "index": 19, "virtual_lines": [ { "bbox": [ 203, 405, 407, 419 ], "spans": [], "index": 19 } ] }, { "type": "text", "bbox": [ 106, 423, 351, 440 ], "lines": [ { "bbox": [ 106, 421, 351, 441 ], "spans": [ { "bbox": [ 106, 421, 162, 441 ], "score": 1.0, "content": "From Lemma", "type": "text" }, { "bbox": [ 163, 426, 168, 435 ], "score": 0.68, "content": "^ { l }", "type": "inline_equation" }, { "bbox": [ 168, 421, 191, 441 ], "score": 1.0, "content": ", with", "type": "text" }, { "bbox": [ 192, 423, 256, 440 ], "score": 0.94, "content": "\\begin{array} { r } { \\epsilon = \\frac { 2 \\sigma \\gamma } { \\log ( 2 + N ( s ) ) } } \\end{array}", "type": "inline_equation" }, { "bbox": [ 256, 421, 299, 441 ], "score": 1.0, "content": "and given", "type": "text" }, { "bbox": [ 299, 425, 311, 436 ], "score": 0.88, "content": "E _ { s }", "type": "inline_equation" }, { "bbox": [ 312, 421, 351, 441 ], "score": 1.0, "content": ", we have", "type": "text" } ], "index": 20 } ], "index": 20, "bbox_fs": [ 106, 421, 351, 441 ] }, { "type": "interline_equation", "bbox": [ 126, 443, 484, 500 ], "lines": [ { "bbox": [ 126, 443, 484, 500 ], "spans": [ { "bbox": [ 126, 443, 484, 500 ], "score": 0.92, "content": "\\begin{array} { r l } & { \\mathbb { P } ( | V _ { \\Omega } ( s ) - V _ { \\Omega } ^ { * } ( s ) | > \\epsilon ) \\le \\mathbb { P } ( \\gamma \\parallel \\hat { r } - r ^ { * } \\parallel _ { \\infty } > \\epsilon ) \\le 4 | A | \\exp \\{ - \\frac { N ( s ) \\epsilon } { 2 \\sigma \\gamma ( \\log ( 2 + N ( s ) ) ) ^ { 2 } } \\} } \\\\ & { \\quad \\quad \\quad = C \\exp \\{ - \\frac { N ( s ) \\epsilon } { \\hat { C } ( \\log ( 2 + N ( s ) ) ) ^ { 2 } } \\} . } \\end{array}", "type": "interline_equation", "image_path": "617316a36ba69474015f9b8eb2f3f494365ed02641e1cc862897868d0caefdc4.jpg" } ] } ], "index": 22, "virtual_lines": [ { "bbox": [ 126, 443, 484, 462.0 ], "spans": [], "index": 21 }, { "bbox": [ 126, 462.0, 484, 481.0 ], "spans": [], "index": 22 }, { "bbox": [ 126, 481.0, 484, 500.0 ], "spans": [], "index": 23 } ] }, { "type": "text", "bbox": [ 105, 502, 506, 525 ], "lines": [ { "bbox": [ 105, 502, 505, 516 ], "spans": [ { "bbox": [ 105, 502, 338, 516 ], "score": 1.0, "content": "Let assume we have the concentration bound at the depth", "type": "text" }, { "bbox": [ 338, 503, 364, 513 ], "score": 0.82, "content": "D - 1", "type": "inline_equation" }, { "bbox": [ 365, 502, 421, 516 ], "score": 1.0, "content": ", Let us define", "type": "text" }, { "bbox": [ 422, 503, 501, 515 ], "score": 0.92, "content": "V _ { \\Omega } ( s _ { a } ) = Q _ { \\Omega } ( s , a )", "type": "inline_equation" }, { "bbox": [ 501, 502, 505, 516 ], "score": 1.0, "content": ",", "type": "text" } ], "index": 24 }, { "bbox": [ 105, 513, 420, 526 ], "spans": [ { "bbox": [ 105, 513, 133, 526 ], "score": 1.0, "content": "where", "type": "text" }, { "bbox": [ 133, 515, 144, 525 ], "score": 0.84, "content": "s _ { a }", "type": "inline_equation" }, { "bbox": [ 144, 513, 329, 526 ], "score": 1.0, "content": "is the state reached taking action a from state", "type": "text" }, { "bbox": [ 329, 516, 334, 523 ], "score": 0.57, "content": "s", "type": "inline_equation" }, { "bbox": [ 335, 513, 393, 526 ], "score": 1.0, "content": ", then at depth", "type": "text" }, { "bbox": [ 393, 514, 420, 524 ], "score": 0.88, "content": "D - 1", "type": "inline_equation" } ], "index": 25 } ], "index": 24.5, "bbox_fs": [ 105, 502, 505, 526 ] }, { "type": "interline_equation", "bbox": [ 177, 529, 433, 557 ], "lines": [ { "bbox": [ 177, 529, 433, 557 ], "spans": [ { "bbox": [ 177, 529, 433, 557 ], "score": 0.92, "content": "\\mathbb { P } ( | V _ { \\Omega } ( s _ { a } ) - V _ { \\Omega } ^ { * } ( s _ { a } ) | > \\epsilon ) \\le C \\exp \\{ - \\frac { N ( s _ { a } ) \\epsilon } { \\hat { C } ( \\log ( 2 + N ( s _ { a } ) ) ) ^ { 2 } } \\} .", "type": "interline_equation", "image_path": "1c85f24984a86a52582dcde14bacd5b5691ab32ad060ce088ba9d833cdb63415.jpg" } ] } ], "index": 26, "virtual_lines": [ { "bbox": [ 177, 529, 433, 557 ], "spans": [], "index": 26 } ] }, { "type": "text", "bbox": [ 107, 560, 416, 572 ], "lines": [ { "bbox": [ 106, 559, 416, 573 ], "spans": [ { "bbox": [ 106, 559, 162, 573 ], "score": 1.0, "content": "Now at depth", "type": "text" }, { "bbox": [ 162, 561, 172, 570 ], "score": 0.78, "content": "D", "type": "inline_equation" }, { "bbox": [ 172, 559, 365, 573 ], "score": 1.0, "content": ", because of the Contraction Property and given", "type": "text" }, { "bbox": [ 365, 561, 377, 571 ], "score": 0.88, "content": "E _ { s }", "type": "inline_equation" }, { "bbox": [ 377, 559, 416, 573 ], "score": 1.0, "content": ", we have", "type": "text" } ], "index": 27 } ], "index": 27, "bbox_fs": [ 106, 559, 416, 573 ] }, { "type": "interline_equation", "bbox": [ 184, 574, 425, 605 ], "lines": [ { "bbox": [ 184, 574, 425, 605 ], "spans": [ { "bbox": [ 184, 574, 425, 605 ], "score": 0.89, "content": "\\begin{array} { r l } & { | V _ { \\Omega } ( s ) - V _ { \\Omega } ^ { * } ( s ) | \\leq \\gamma \\parallel Q _ { \\Omega } ( s , . ) - Q _ { \\Omega } ^ { * } ( s , . ) \\parallel _ { \\infty } } \\\\ & { \\qquad = \\gamma | Q _ { \\Omega } ( s , a ) - Q _ { \\Omega } ^ { * } ( s , a ) | ( \\exists a , s a t i s f t e d ) . } \\end{array}", "type": "interline_equation", "image_path": "e0cb290fced55598b4b904c2a0f9354e46c6f8048b888fbace8168238bd2174a.jpg" } ] } ], "index": 28.5, "virtual_lines": [ { "bbox": [ 184, 574, 425, 589.5 ], "spans": [], "index": 28 }, { "bbox": [ 184, 589.5, 425, 605.0 ], "spans": [], "index": 29 } ] }, { "type": "text", "bbox": [ 106, 606, 137, 618 ], "lines": [ { "bbox": [ 105, 605, 138, 619 ], "spans": [ { "bbox": [ 105, 605, 138, 619 ], "score": 1.0, "content": "So that", "type": "text" } ], "index": 30 } ], "index": 30, "bbox_fs": [ 105, 605, 138, 619 ] }, { "type": "interline_equation", "bbox": [ 154, 621, 457, 725 ], "lines": [ { "bbox": [ 154, 621, 457, 725 ], "spans": [ { "bbox": [ 154, 621, 457, 725 ], "score": 0.94, "content": "\\begin{array} { r l } & { \\mathbb { P } ( | V _ { \\Omega } ( s ) - V _ { \\Omega } ^ { * } ( s ) | > \\epsilon ) \\leq \\mathbb { P } ( \\gamma \\parallel Q _ { \\Omega } ( s , a ) - Q _ { \\Omega } ^ { * } ( s , a ) \\parallel > \\epsilon ) } \\\\ & { \\qquad \\leq C _ { a } \\exp \\{ - \\frac { N ( s _ { a } ) \\epsilon } { \\hat { C } _ { a } ( \\log ( 2 + N ( s _ { a } ) ) ) ^ { 2 } } \\} } \\\\ & { \\qquad \\leq C _ { a } \\exp \\{ - \\frac { N ( s _ { a } ) \\epsilon } { \\hat { C } _ { a } ( \\log ( 2 + N ( s ) ) ) ^ { 2 } } \\} } \\\\ & { \\qquad \\leq C \\exp \\{ - \\frac { N ( s ) \\epsilon } { \\hat { C } ( \\log ( 2 + N ( s ) ) ) ^ { 2 } } \\} ( b e c a u s e o f E _ { s } ) } \\end{array}", "type": "interline_equation", "image_path": "ff386d89855cf3515a65d5b4ce2c91c90fed1dd0e25073e45066ebfcf59bbb21.jpg" } ] } ], "index": 32, "virtual_lines": [ { "bbox": [ 154, 621, 457, 655.6666666666666 ], "spans": [], "index": 31 }, { "bbox": [ 154, 655.6666666666666, 457, 690.3333333333333 ], "spans": [], "index": 32 }, { "bbox": [ 154, 690.3333333333333, 457, 724.9999999999999 ], "spans": [], "index": 33 } ] } ] }, { "preproc_blocks": [ { "type": "text", "bbox": [ 104, 82, 504, 107 ], "lines": [ { "bbox": [ 105, 80, 504, 96 ], "spans": [ { "bbox": [ 105, 80, 172, 96 ], "score": 1.0, "content": "Theorem 2 Let", "type": "text" }, { "bbox": [ 173, 84, 182, 93 ], "score": 0.82, "content": "a _ { t }", "type": "inline_equation" }, { "bbox": [ 183, 80, 504, 96 ], "score": 1.0, "content": "be the action returned by algorithm E3W at iteration t. Then for t large enough,", "type": "text" } ], "index": 0 }, { "bbox": [ 105, 93, 213, 106 ], "spans": [ { "bbox": [ 105, 94, 189, 106 ], "score": 1.0, "content": "with some constants", "type": "text" }, { "bbox": [ 190, 93, 209, 106 ], "score": 0.9, "content": "C , { \\hat { C } }", "type": "inline_equation" }, { "bbox": [ 210, 94, 213, 106 ], "score": 1.0, "content": ",", "type": "text" } ], "index": 1 } ], "index": 0.5 }, { "type": "interline_equation", "bbox": [ 223, 110, 386, 138 ], "lines": [ { "bbox": [ 223, 110, 386, 138 ], "spans": [ { "bbox": [ 223, 110, 386, 138 ], "score": 0.94, "content": "\\mathbb { P } ( a _ { t } \\neq a ^ { * } ) \\leq C t \\exp \\{ - \\frac { t } { \\hat { C } \\sigma ( \\log ( t ) ) ^ { 3 } } \\} .", "type": "interline_equation", "image_path": "187609c71ee7364adab0a934a99f78ee2a6b0ee27164a257769da89212cd87b0.jpg" } ] } ], "index": 2, "virtual_lines": [ { "bbox": [ 223, 110, 386, 138 ], "spans": [], "index": 2 } ] }, { "type": "text", "bbox": [ 104, 145, 503, 169 ], "lines": [ { "bbox": [ 105, 145, 505, 159 ], "spans": [ { "bbox": [ 105, 145, 220, 159 ], "score": 1.0, "content": "Proof 7 Let us define event", "type": "text" }, { "bbox": [ 221, 146, 233, 157 ], "score": 0.89, "content": "E _ { s }", "type": "inline_equation" }, { "bbox": [ 233, 145, 316, 159 ], "score": 1.0, "content": "as in Lemma 5. Let", "type": "text" }, { "bbox": [ 317, 146, 327, 156 ], "score": 0.85, "content": "a ^ { * }", "type": "inline_equation" }, { "bbox": [ 327, 145, 505, 159 ], "score": 1.0, "content": "be the action with largest value estimate at", "type": "text" } ], "index": 3 }, { "bbox": [ 106, 156, 429, 170 ], "spans": [ { "bbox": [ 106, 156, 183, 170 ], "score": 1.0, "content": "the root node state", "type": "text" }, { "bbox": [ 183, 160, 189, 167 ], "score": 0.39, "content": "s", "type": "inline_equation" }, { "bbox": [ 189, 156, 275, 170 ], "score": 1.0, "content": ". The probability that", "type": "text" }, { "bbox": [ 275, 157, 297, 167 ], "score": 0.44, "content": "E 3 W", "type": "inline_equation" }, { "bbox": [ 297, 156, 429, 170 ], "score": 1.0, "content": "selects a sub-optimal arm at s is", "type": "text" } ], "index": 4 } ], "index": 3.5 }, { "type": "interline_equation", "bbox": [ 119, 173, 490, 225 ], "lines": [ { "bbox": [ 119, 173, 490, 225 ], "spans": [ { "bbox": [ 119, 173, 490, 225 ], "score": 0.91, "content": "\\begin{array} { r l } & { \\displaystyle \\mathbb { P } ( a _ { t } \\neq a ^ { * } ) \\leq \\sum _ { a } \\mathbb { P } ( V _ { \\Omega } ( s _ { a } ) ) > V _ { \\Omega } ( s _ { a ^ { * } } ) | E _ { s } ) + \\mathbb { P } ( E _ { s } ^ { c } ) } \\\\ & { \\displaystyle = \\sum _ { a } \\mathbb { P } ( ( V _ { \\Omega } ( s _ { a } ) - V _ { \\Omega } ^ { * } ( s _ { a } ) ) - ( V _ { \\Omega } ( s _ { a ^ { * } } ) - V _ { \\Omega } ^ { * } ( s _ { a ^ { * } } ) ) \\geq V _ { \\Omega } ^ { * } ( s _ { a ^ { * } } ) - V _ { \\Omega } ^ { * } ( s _ { a } ) | E _ { s } ) + \\mathbb { P } ( E _ { s } ^ { c } ) . } \\end{array}", "type": "interline_equation", "image_path": "2a36bc55ccbfbfc01643b8110338bfb7df7f42b6779c80f569c16274fee964a8.jpg" } ] } ], "index": 6, "virtual_lines": [ { "bbox": [ 119, 173, 490, 190.33333333333334 ], "spans": [], "index": 5 }, { "bbox": [ 119, 190.33333333333334, 490, 207.66666666666669 ], "spans": [], "index": 6 }, { "bbox": [ 119, 207.66666666666669, 490, 225.00000000000003 ], "spans": [], "index": 7 } ] }, { "type": "text", "bbox": [ 105, 229, 381, 243 ], "lines": [ { "bbox": [ 106, 229, 381, 244 ], "spans": [ { "bbox": [ 106, 229, 159, 244 ], "score": 1.0, "content": "Let us define", "type": "text" }, { "bbox": [ 160, 230, 258, 243 ], "score": 0.93, "content": "\\Delta = V _ { \\Omega } ^ { * } ( s _ { a ^ { * } } ) - V _ { \\Omega } ^ { * } ( s _ { a } )", "type": "inline_equation" }, { "bbox": [ 259, 229, 314, 244 ], "score": 1.0, "content": ", therefore for", "type": "text" }, { "bbox": [ 315, 231, 342, 241 ], "score": 0.89, "content": "\\Delta > 0", "type": "inline_equation" }, { "bbox": [ 342, 229, 381, 244 ], "score": 1.0, "content": ", we have", "type": "text" } ], "index": 8 } ], "index": 8 }, { "type": "interline_equation", "bbox": [ 126, 246, 484, 331 ], "lines": [ { "bbox": [ 126, 246, 484, 331 ], "spans": [ { "bbox": [ 126, 246, 484, 331 ], "score": 0.94, "content": "\\begin{array} { r l } & { \\displaystyle \\mathbb { P } ( a _ { t } \\neq a ^ { * } ) \\leq \\sum _ { a } \\mathbb { P } ( ( V _ { \\Omega } ( s _ { a } ) - V _ { \\Omega } ^ { * } ( s _ { a } ) ) - ( V _ { \\Omega } ( s _ { a ^ { * } } ) - V _ { \\Omega } ^ { * } ( s _ { a ^ { * } } ) ) \\geq \\Delta | E _ { s } ) + \\mathbb { P } ( E _ { s } ^ { c } ) } \\\\ & { \\leq \\displaystyle \\sum _ { a } \\mathbb { P } ( | V _ { \\Omega } ( s _ { a } ) - V _ { \\Omega } ^ { * } ( s _ { a } ) | \\geq \\alpha \\Delta | E _ { s } ) + \\mathbb { P } ( | V _ { \\Omega } ( s _ { a ^ { * } } ) - V _ { \\Omega } ^ { * } ( s _ { a ^ { * } } ) | \\geq \\beta \\Delta | E _ { s } ) + \\mathbb { P } ( E _ { s } ^ { c } ) } \\\\ & { \\leq \\displaystyle \\sum _ { a } C _ { a } \\exp \\{ - \\frac { N ( s ) ( \\alpha \\Delta ) } { \\hat { C } _ { a } ( \\log ( 2 + N ( s ) ) ) ^ { 2 } } \\} + C _ { a ^ { * } } \\exp \\{ - \\frac { N ( s ) ( \\beta \\Delta ) } { \\hat { C } _ { a ^ { * } } ( \\log ( 2 + N ( s ) ) ) ^ { 2 } } \\} + \\mathbb { P } ( E _ { s } ^ { c } ) , } \\end{array}", "type": "interline_equation", "image_path": "5a59f3ed97f3b3f900a43471079c10069b115391cbec54db671260e41e8d903e.jpg" } ] } ], "index": 10, "virtual_lines": [ { "bbox": [ 126, 246, 484, 274.3333333333333 ], "spans": [], "index": 9 }, { "bbox": [ 126, 274.3333333333333, 484, 302.66666666666663 ], "spans": [], "index": 10 }, { "bbox": [ 126, 302.66666666666663, 484, 330.99999999999994 ], "spans": [], "index": 11 } ] }, { "type": "text", "bbox": [ 106, 340, 504, 369 ], "lines": [ { "bbox": [ 106, 339, 505, 354 ], "spans": [ { "bbox": [ 106, 339, 133, 354 ], "score": 1.0, "content": "where", "type": "text" }, { "bbox": [ 133, 341, 235, 352 ], "score": 0.27, "content": "\\alpha + \\beta = 1 , \\alpha > 0 , \\beta > 0 ,", "type": "inline_equation" }, { "bbox": [ 235, 339, 255, 354 ], "score": 1.0, "content": ", and", "type": "text" }, { "bbox": [ 255, 341, 277, 353 ], "score": 0.91, "content": "N ( s )", "type": "inline_equation" }, { "bbox": [ 278, 339, 505, 354 ], "score": 1.0, "content": "is the number of visitations the root node s. Let us define", "type": "text" } ], "index": 12 }, { "bbox": [ 108, 350, 358, 372 ], "spans": [ { "bbox": [ 108, 352, 200, 369 ], "score": 0.91, "content": "\\begin{array} { r } { \\frac { 1 } { \\hat { C } } = \\operatorname* { m i n } \\{ \\frac { ( \\alpha \\Delta ) } { { C } _ { a } } , \\frac { ( \\beta \\Delta ) } { { C } _ { a ^ { * } } } \\} } \\end{array}", "type": "inline_equation" }, { "bbox": [ 201, 350, 222, 372 ], "score": 1.0, "content": ", and", "type": "text" }, { "bbox": [ 222, 354, 320, 369 ], "score": 0.92, "content": "\\begin{array} { r } { C = \\frac { 1 } { | A | } \\operatorname* { m a x } \\{ C _ { a } , C _ { a ^ { * } } \\} } \\end{array}", "type": "inline_equation" }, { "bbox": [ 321, 350, 358, 372 ], "score": 1.0, "content": "we have", "type": "text" } ], "index": 13 } ], "index": 12.5 }, { "type": "interline_equation", "bbox": [ 199, 374, 411, 401 ], "lines": [ { "bbox": [ 199, 374, 411, 401 ], "spans": [ { "bbox": [ 199, 374, 411, 401 ], "score": 0.91, "content": "\\mathbb { P } ( a \\neq a ^ { * } ) \\leq C \\exp \\{ - \\frac { t } { \\hat { C } \\sigma ( \\log ( 2 + t ) ) ^ { 2 } } \\} + \\mathbb { P } ( E _ { s } ^ { c } ) .", "type": "interline_equation", "image_path": "8ca849193e088f824b108a4d4b9d0c507d6f6e8848e7b0bd5adb83d7b3fe1562.jpg" } ] } ], "index": 14, "virtual_lines": [ { "bbox": [ 199, 374, 411, 401 ], "spans": [], "index": 14 } ] }, { "type": "text", "bbox": [ 107, 406, 247, 420 ], "lines": [ { "bbox": [ 105, 405, 247, 421 ], "spans": [ { "bbox": [ 105, 405, 171, 421 ], "score": 1.0, "content": "From Lemma 4,", "type": "text" }, { "bbox": [ 172, 405, 204, 420 ], "score": 0.88, "content": "\\exists C ^ { ' } , \\hat { C } ^ { \\prime }", "type": "inline_equation" }, { "bbox": [ 205, 405, 247, 421 ], "score": 1.0, "content": "for which", "type": "text" } ], "index": 15 } ], "index": 15 }, { "type": "interline_equation", "bbox": [ 233, 423, 376, 451 ], "lines": [ { "bbox": [ 233, 423, 376, 451 ], "spans": [ { "bbox": [ 233, 423, 376, 451 ], "score": 0.93, "content": "\\mathbb { P } ( E _ { s } ^ { c } ) \\le C ^ { ' } t \\exp \\{ - \\frac { t } { \\hat { C } ^ { ' } ( \\log ( t ) ) ^ { 3 } } \\} ,", "type": "interline_equation", "image_path": "2417eb7b4409e22142eb41c26b43311830dbd75e08c20b84c93e2f727e6ac097.jpg" } ] } ], "index": 16, "virtual_lines": [ { "bbox": [ 233, 423, 376, 451 ], "spans": [], "index": 16 } ] }, { "type": "text", "bbox": [ 106, 455, 136, 466 ], "lines": [ { "bbox": [ 104, 453, 137, 468 ], "spans": [ { "bbox": [ 104, 453, 137, 468 ], "score": 1.0, "content": "so that", "type": "text" } ], "index": 17 } ], "index": 17 }, { "type": "interline_equation", "bbox": [ 228, 469, 381, 495 ], "lines": [ { "bbox": [ 228, 469, 381, 495 ], "spans": [ { "bbox": [ 228, 469, 381, 495 ], "score": 0.93, "content": "\\mathbb { P } ( a \\neq a ^ { * } ) \\leq O ( t \\exp \\{ - \\frac { t } { ( \\log ( t ) ) ^ { 3 } } \\} ) .", "type": "interline_equation", "image_path": "d1f1bceb534ec1b0dff367e996af63fbf9bb17ea93c607fc579645fe922e0c43.jpg" } ] } ], "index": 18, "virtual_lines": [ { "bbox": [ 228, 469, 381, 495 ], "spans": [], "index": 18 } ] }, { "type": "text", "bbox": [ 106, 505, 506, 567 ], "lines": [ { "bbox": [ 104, 504, 502, 527 ], "spans": [ { "bbox": [ 104, 504, 209, 527 ], "score": 1.0, "content": "Theorem 3 Consider an", "type": "text" }, { "bbox": [ 209, 510, 231, 520 ], "score": 0.45, "content": "E 3 W", "type": "inline_equation" }, { "bbox": [ 231, 504, 354, 527 ], "score": 1.0, "content": "policy applied to the tree. Let", "type": "text" }, { "bbox": [ 354, 504, 502, 524 ], "score": 0.92, "content": "\\begin{array} { r } { \\kappa _ { i } = \\nabla \\Omega ^ { * } ( a _ { i } | s ) + \\frac { L } { p } \\sqrt { { \\hat { C } } \\sigma ^ { 2 } \\log \\frac { C } { \\delta } / 2 n } , } \\end{array}", "type": "inline_equation" } ], "index": 19 }, { "bbox": [ 106, 523, 506, 543 ], "spans": [ { "bbox": [ 106, 523, 255, 543 ], "score": 0.81, "content": "\\begin{array} { r } { \\chi _ { i } = \\nabla \\Omega ^ { * } ( a _ { i } | s ) - \\frac { L } { p } \\sqrt { \\hat { C } \\sigma ^ { 2 } \\log \\frac { C } { \\delta } / 2 n } , } \\end{array}", "type": "inline_equation" }, { "bbox": [ 255, 525, 286, 542 ], "score": 1.0, "content": ", where", "type": "text" }, { "bbox": [ 286, 528, 325, 540 ], "score": 0.92, "content": "\\nabla \\Omega ^ { * } ( . | s )", "type": "inline_equation" }, { "bbox": [ 326, 525, 506, 542 ], "score": 1.0, "content": "is the policy with respect to the mean value", "type": "text" } ], "index": 20 }, { "bbox": [ 105, 542, 504, 557 ], "spans": [ { "bbox": [ 105, 543, 134, 557 ], "score": 1.0, "content": "vector", "type": "text" }, { "bbox": [ 134, 545, 154, 556 ], "score": 0.89, "content": "V ( \\cdot )", "type": "inline_equation" }, { "bbox": [ 154, 543, 264, 557 ], "score": 1.0, "content": "at the root node s. For any", "type": "text" }, { "bbox": [ 265, 545, 289, 555 ], "score": 0.89, "content": "\\delta > 0", "type": "inline_equation" }, { "bbox": [ 289, 543, 390, 557 ], "score": 1.0, "content": ", with probability at least", "type": "text" }, { "bbox": [ 391, 545, 414, 555 ], "score": 0.62, "content": "1 - \\delta", "type": "inline_equation" }, { "bbox": [ 414, 543, 462, 557 ], "score": 1.0, "content": ", ∃ constant", "type": "text" }, { "bbox": [ 463, 542, 504, 556 ], "score": 0.92, "content": "L , p , C , { \\hat { C } }", "type": "inline_equation" } ], "index": 21 }, { "bbox": [ 106, 555, 257, 568 ], "spans": [ { "bbox": [ 106, 555, 207, 568 ], "score": 1.0, "content": "so that the pseudo regret", "type": "text" }, { "bbox": [ 207, 556, 221, 566 ], "score": 0.89, "content": "R _ { n }", "type": "inline_equation" }, { "bbox": [ 221, 555, 257, 568 ], "score": 1.0, "content": "satisfies", "type": "text" } ], "index": 22 } ], "index": 20.5 }, { "type": "interline_equation", "bbox": [ 114, 570, 495, 601 ], "lines": [ { "bbox": [ 114, 570, 495, 601 ], "spans": [ { "bbox": [ 114, 570, 495, 601 ], "score": 0.94, "content": "n V ^ { * } - n \\sum _ { i } V _ { i } \\Big ( \\kappa _ { i } + \\frac { L } { p } \\big ( \\frac { \\tau ( U _ { \\Omega } - L _ { \\Omega } ) } { 1 - \\gamma } \\big ) \\Big ) \\leq R _ { n } \\leq n V ^ { * } - n \\sum _ { i } V _ { i } \\Big ( \\chi _ { i } - \\frac { L } { p } \\big ( \\frac { \\tau ( U _ { \\Omega } - L _ { \\Omega } ) } { 1 - \\gamma } \\big ) \\Big ) .", "type": "interline_equation", "image_path": "902a8a5ba4718099795b9648b4785fd6500fe05c79dc2e19703b0fddc7fa33d1.jpg" } ] } ], "index": 24, "virtual_lines": [ { "bbox": [ 114, 570, 495, 580.3333333333334 ], "spans": [], "index": 23 }, { "bbox": [ 114, 580.3333333333334, 495, 590.6666666666667 ], "spans": [], "index": 24 }, { "bbox": [ 114, 590.6666666666667, 495, 601.0000000000001 ], "spans": [], "index": 25 } ] }, { "type": "text", "bbox": [ 105, 609, 504, 624 ], "lines": [ { "bbox": [ 103, 607, 507, 627 ], "spans": [ { "bbox": [ 103, 607, 279, 627 ], "score": 1.0, "content": "Proof 8 From Lemma 2 given two policies", "type": "text" }, { "bbox": [ 279, 610, 353, 624 ], "score": 0.89, "content": "\\pi ^ { ( 1 ) } = \\nabla \\Omega ^ { * } ( r ^ { ( 1 ) } )", "type": "inline_equation" }, { "bbox": [ 353, 607, 372, 627 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 372, 610, 464, 624 ], "score": 0.85, "content": "\\pi ^ { ( 2 ) } = \\nabla \\Omega ^ { * } ( r ^ { ( 2 ) } ) , \\exists L ,", "type": "inline_equation" }, { "bbox": [ 464, 607, 507, 627 ], "score": 1.0, "content": ", such that", "type": "text" } ], "index": 26 } ], "index": 26 }, { "type": "interline_equation", "bbox": [ 179, 628, 431, 653 ], "lines": [ { "bbox": [ 179, 628, 431, 653 ], "spans": [ { "bbox": [ 179, 628, 431, 653 ], "score": 0.92, "content": "\\parallel \\pi ^ { ( 1 ) } - \\pi ^ { ( 2 ) } \\parallel _ { p } \\leq L \\parallel r ^ { ( 1 ) } - r ^ { ( 2 ) } \\parallel _ { p } \\leq L \\frac { 1 } { p } \\parallel r ^ { ( 1 ) } - r ^ { ( 2 ) } \\parallel _ { \\infty } .", "type": "interline_equation", "image_path": "7b87807fa9d3b83f0d0fad0d520e1e4ec914379c5f64ac13eb89499683a579e4.jpg" } ] } ], "index": 27, "virtual_lines": [ { "bbox": [ 179, 628, 431, 653 ], "spans": [], "index": 27 } ] }, { "type": "text", "bbox": [ 107, 657, 227, 668 ], "lines": [ { "bbox": [ 106, 655, 227, 670 ], "spans": [ { "bbox": [ 106, 655, 227, 670 ], "score": 1.0, "content": "From (13), we have the regret", "type": "text" } ], "index": 28 } ], "index": 28 }, { "type": "interline_equation", "bbox": [ 236, 672, 374, 705 ], "lines": [ { "bbox": [ 236, 672, 374, 705 ], "spans": [ { "bbox": [ 236, 672, 374, 705 ], "score": 0.94, "content": "R _ { n } = n V ^ { * } - \\sum _ { i } V _ { i } \\sum _ { t = 1 } ^ { n } { \\hat { \\pi } } _ { t } ( a _ { i } | s ) ,", "type": "interline_equation", "image_path": "ace819a2ff0b913e28d002dfbe179a2dc737e8032d71b5890c40cbd287d722a8.jpg" } ] } ], "index": 29.5, "virtual_lines": [ { "bbox": [ 236, 672, 374, 688.5 ], "spans": [], "index": 29 }, { "bbox": [ 236, 688.5, 374, 705.0 ], "spans": [], "index": 30 } ] }, { "type": "text", "bbox": [ 107, 709, 504, 733 ], "lines": [ { "bbox": [ 106, 709, 505, 722 ], "spans": [ { "bbox": [ 106, 709, 133, 722 ], "score": 1.0, "content": "where", "type": "text" }, { "bbox": [ 133, 710, 154, 722 ], "score": 0.9, "content": "\\hat { \\pi } _ { t } ( \\cdot )", "type": "inline_equation" }, { "bbox": [ 154, 709, 256, 722 ], "score": 1.0, "content": "is the policy at time step", "type": "text" }, { "bbox": [ 257, 713, 260, 719 ], "score": 0.7, "content": "t", "type": "inline_equation" }, { "bbox": [ 261, 709, 282, 722 ], "score": 1.0, "content": ", and", "type": "text" }, { "bbox": [ 283, 711, 298, 722 ], "score": 0.85, "content": "\\mathbb { I } ( \\cdot )", "type": "inline_equation" }, { "bbox": [ 298, 709, 402, 722 ], "score": 1.0, "content": "is the indicator function.", "type": "text" }, { "bbox": [ 402, 712, 415, 720 ], "score": 0.87, "content": "V ^ { * }", "type": "inline_equation" }, { "bbox": [ 415, 709, 505, 722 ], "score": 1.0, "content": "is the optimal branch", "type": "text" } ], "index": 31 }, { "bbox": [ 105, 720, 504, 733 ], "spans": [ { "bbox": [ 105, 720, 174, 733 ], "score": 1.0, "content": "at the root node,", "type": "text" }, { "bbox": [ 174, 721, 185, 732 ], "score": 0.86, "content": "V _ { i }", "type": "inline_equation" }, { "bbox": [ 185, 720, 437, 733 ], "score": 1.0, "content": "is the mean value function of the branch with respect to action", "type": "text" }, { "bbox": [ 437, 723, 441, 730 ], "score": 0.66, "content": "i", "type": "inline_equation" }, { "bbox": [ 441, 720, 445, 733 ], "score": 1.0, "content": ",", "type": "text" }, { "bbox": [ 446, 722, 464, 733 ], "score": 0.91, "content": "V ( \\cdot )", "type": "inline_equation" }, { "bbox": [ 464, 720, 490, 733 ], "score": 1.0, "content": "is the", "type": "text" }, { "bbox": [ 490, 722, 504, 732 ], "score": 0.91, "content": "| A |", "type": "inline_equation" } ], "index": 32 } ], "index": 31.5 } ], "page_idx": 16, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 106, 26, 307, 38 ], "lines": [ { "bbox": [ 106, 25, 308, 39 ], "spans": [ { "bbox": [ 106, 25, 308, 39 ], "score": 1.0, "content": "Under review as a conference paper at ICLR 2021", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 300, 750, 311, 760 ], "lines": [ { "bbox": [ 299, 750, 313, 764 ], "spans": [ { "bbox": [ 299, 750, 313, 764 ], "score": 1.0, "content": "17", "type": "text" } ] } ] } ], "para_blocks": [ { "type": "text", "bbox": [ 104, 82, 504, 107 ], "lines": [ { "bbox": [ 105, 80, 504, 96 ], "spans": [ { "bbox": [ 105, 80, 172, 96 ], "score": 1.0, "content": "Theorem 2 Let", "type": "text" }, { "bbox": [ 173, 84, 182, 93 ], "score": 0.82, "content": "a _ { t }", "type": "inline_equation" }, { "bbox": [ 183, 80, 504, 96 ], "score": 1.0, "content": "be the action returned by algorithm E3W at iteration t. Then for t large enough,", "type": "text" } ], "index": 0 }, { "bbox": [ 105, 93, 213, 106 ], "spans": [ { "bbox": [ 105, 94, 189, 106 ], "score": 1.0, "content": "with some constants", "type": "text" }, { "bbox": [ 190, 93, 209, 106 ], "score": 0.9, "content": "C , { \\hat { C } }", "type": "inline_equation" }, { "bbox": [ 210, 94, 213, 106 ], "score": 1.0, "content": ",", "type": "text" } ], "index": 1 } ], "index": 0.5, "bbox_fs": [ 105, 80, 504, 106 ] }, { "type": "interline_equation", "bbox": [ 223, 110, 386, 138 ], "lines": [ { "bbox": [ 223, 110, 386, 138 ], "spans": [ { "bbox": [ 223, 110, 386, 138 ], "score": 0.94, "content": "\\mathbb { P } ( a _ { t } \\neq a ^ { * } ) \\leq C t \\exp \\{ - \\frac { t } { \\hat { C } \\sigma ( \\log ( t ) ) ^ { 3 } } \\} .", "type": "interline_equation", "image_path": "187609c71ee7364adab0a934a99f78ee2a6b0ee27164a257769da89212cd87b0.jpg" } ] } ], "index": 2, "virtual_lines": [ { "bbox": [ 223, 110, 386, 138 ], "spans": [], "index": 2 } ] }, { "type": "text", "bbox": [ 104, 145, 503, 169 ], "lines": [ { "bbox": [ 105, 145, 505, 159 ], "spans": [ { "bbox": [ 105, 145, 220, 159 ], "score": 1.0, "content": "Proof 7 Let us define event", "type": "text" }, { "bbox": [ 221, 146, 233, 157 ], "score": 0.89, "content": "E _ { s }", "type": "inline_equation" }, { "bbox": [ 233, 145, 316, 159 ], "score": 1.0, "content": "as in Lemma 5. Let", "type": "text" }, { "bbox": [ 317, 146, 327, 156 ], "score": 0.85, "content": "a ^ { * }", "type": "inline_equation" }, { "bbox": [ 327, 145, 505, 159 ], "score": 1.0, "content": "be the action with largest value estimate at", "type": "text" } ], "index": 3 }, { "bbox": [ 106, 156, 429, 170 ], "spans": [ { "bbox": [ 106, 156, 183, 170 ], "score": 1.0, "content": "the root node state", "type": "text" }, { "bbox": [ 183, 160, 189, 167 ], "score": 0.39, "content": "s", "type": "inline_equation" }, { "bbox": [ 189, 156, 275, 170 ], "score": 1.0, "content": ". The probability that", "type": "text" }, { "bbox": [ 275, 157, 297, 167 ], "score": 0.44, "content": "E 3 W", "type": "inline_equation" }, { "bbox": [ 297, 156, 429, 170 ], "score": 1.0, "content": "selects a sub-optimal arm at s is", "type": "text" } ], "index": 4 } ], "index": 3.5, "bbox_fs": [ 105, 145, 505, 170 ] }, { "type": "interline_equation", "bbox": [ 119, 173, 490, 225 ], "lines": [ { "bbox": [ 119, 173, 490, 225 ], "spans": [ { "bbox": [ 119, 173, 490, 225 ], "score": 0.91, "content": "\\begin{array} { r l } & { \\displaystyle \\mathbb { P } ( a _ { t } \\neq a ^ { * } ) \\leq \\sum _ { a } \\mathbb { P } ( V _ { \\Omega } ( s _ { a } ) ) > V _ { \\Omega } ( s _ { a ^ { * } } ) | E _ { s } ) + \\mathbb { P } ( E _ { s } ^ { c } ) } \\\\ & { \\displaystyle = \\sum _ { a } \\mathbb { P } ( ( V _ { \\Omega } ( s _ { a } ) - V _ { \\Omega } ^ { * } ( s _ { a } ) ) - ( V _ { \\Omega } ( s _ { a ^ { * } } ) - V _ { \\Omega } ^ { * } ( s _ { a ^ { * } } ) ) \\geq V _ { \\Omega } ^ { * } ( s _ { a ^ { * } } ) - V _ { \\Omega } ^ { * } ( s _ { a } ) | E _ { s } ) + \\mathbb { P } ( E _ { s } ^ { c } ) . } \\end{array}", "type": "interline_equation", "image_path": "2a36bc55ccbfbfc01643b8110338bfb7df7f42b6779c80f569c16274fee964a8.jpg" } ] } ], "index": 6, "virtual_lines": [ { "bbox": [ 119, 173, 490, 190.33333333333334 ], "spans": [], "index": 5 }, { "bbox": [ 119, 190.33333333333334, 490, 207.66666666666669 ], "spans": [], "index": 6 }, { "bbox": [ 119, 207.66666666666669, 490, 225.00000000000003 ], "spans": [], "index": 7 } ] }, { "type": "text", "bbox": [ 105, 229, 381, 243 ], "lines": [ { "bbox": [ 106, 229, 381, 244 ], "spans": [ { "bbox": [ 106, 229, 159, 244 ], "score": 1.0, "content": "Let us define", "type": "text" }, { "bbox": [ 160, 230, 258, 243 ], "score": 0.93, "content": "\\Delta = V _ { \\Omega } ^ { * } ( s _ { a ^ { * } } ) - V _ { \\Omega } ^ { * } ( s _ { a } )", "type": "inline_equation" }, { "bbox": [ 259, 229, 314, 244 ], "score": 1.0, "content": ", therefore for", "type": "text" }, { "bbox": [ 315, 231, 342, 241 ], "score": 0.89, "content": "\\Delta > 0", "type": "inline_equation" }, { "bbox": [ 342, 229, 381, 244 ], "score": 1.0, "content": ", we have", "type": "text" } ], "index": 8 } ], "index": 8, "bbox_fs": [ 106, 229, 381, 244 ] }, { "type": "interline_equation", "bbox": [ 126, 246, 484, 331 ], "lines": [ { "bbox": [ 126, 246, 484, 331 ], "spans": [ { "bbox": [ 126, 246, 484, 331 ], "score": 0.94, "content": "\\begin{array} { r l } & { \\displaystyle \\mathbb { P } ( a _ { t } \\neq a ^ { * } ) \\leq \\sum _ { a } \\mathbb { P } ( ( V _ { \\Omega } ( s _ { a } ) - V _ { \\Omega } ^ { * } ( s _ { a } ) ) - ( V _ { \\Omega } ( s _ { a ^ { * } } ) - V _ { \\Omega } ^ { * } ( s _ { a ^ { * } } ) ) \\geq \\Delta | E _ { s } ) + \\mathbb { P } ( E _ { s } ^ { c } ) } \\\\ & { \\leq \\displaystyle \\sum _ { a } \\mathbb { P } ( | V _ { \\Omega } ( s _ { a } ) - V _ { \\Omega } ^ { * } ( s _ { a } ) | \\geq \\alpha \\Delta | E _ { s } ) + \\mathbb { P } ( | V _ { \\Omega } ( s _ { a ^ { * } } ) - V _ { \\Omega } ^ { * } ( s _ { a ^ { * } } ) | \\geq \\beta \\Delta | E _ { s } ) + \\mathbb { P } ( E _ { s } ^ { c } ) } \\\\ & { \\leq \\displaystyle \\sum _ { a } C _ { a } \\exp \\{ - \\frac { N ( s ) ( \\alpha \\Delta ) } { \\hat { C } _ { a } ( \\log ( 2 + N ( s ) ) ) ^ { 2 } } \\} + C _ { a ^ { * } } \\exp \\{ - \\frac { N ( s ) ( \\beta \\Delta ) } { \\hat { C } _ { a ^ { * } } ( \\log ( 2 + N ( s ) ) ) ^ { 2 } } \\} + \\mathbb { P } ( E _ { s } ^ { c } ) , } \\end{array}", "type": "interline_equation", "image_path": "5a59f3ed97f3b3f900a43471079c10069b115391cbec54db671260e41e8d903e.jpg" } ] } ], "index": 10, "virtual_lines": [ { "bbox": [ 126, 246, 484, 274.3333333333333 ], "spans": [], "index": 9 }, { "bbox": [ 126, 274.3333333333333, 484, 302.66666666666663 ], "spans": [], "index": 10 }, { "bbox": [ 126, 302.66666666666663, 484, 330.99999999999994 ], "spans": [], "index": 11 } ] }, { "type": "text", "bbox": [ 106, 340, 504, 369 ], "lines": [ { "bbox": [ 106, 339, 505, 354 ], "spans": [ { "bbox": [ 106, 339, 133, 354 ], "score": 1.0, "content": "where", "type": "text" }, { "bbox": [ 133, 341, 235, 352 ], "score": 0.27, "content": "\\alpha + \\beta = 1 , \\alpha > 0 , \\beta > 0 ,", "type": "inline_equation" }, { "bbox": [ 235, 339, 255, 354 ], "score": 1.0, "content": ", and", "type": "text" }, { "bbox": [ 255, 341, 277, 353 ], "score": 0.91, "content": "N ( s )", "type": "inline_equation" }, { "bbox": [ 278, 339, 505, 354 ], "score": 1.0, "content": "is the number of visitations the root node s. Let us define", "type": "text" } ], "index": 12 }, { "bbox": [ 108, 350, 358, 372 ], "spans": [ { "bbox": [ 108, 352, 200, 369 ], "score": 0.91, "content": "\\begin{array} { r } { \\frac { 1 } { \\hat { C } } = \\operatorname* { m i n } \\{ \\frac { ( \\alpha \\Delta ) } { { C } _ { a } } , \\frac { ( \\beta \\Delta ) } { { C } _ { a ^ { * } } } \\} } \\end{array}", "type": "inline_equation" }, { "bbox": [ 201, 350, 222, 372 ], "score": 1.0, "content": ", and", "type": "text" }, { "bbox": [ 222, 354, 320, 369 ], "score": 0.92, "content": "\\begin{array} { r } { C = \\frac { 1 } { | A | } \\operatorname* { m a x } \\{ C _ { a } , C _ { a ^ { * } } \\} } \\end{array}", "type": "inline_equation" }, { "bbox": [ 321, 350, 358, 372 ], "score": 1.0, "content": "we have", "type": "text" } ], "index": 13 } ], "index": 12.5, "bbox_fs": [ 106, 339, 505, 372 ] }, { "type": "interline_equation", "bbox": [ 199, 374, 411, 401 ], "lines": [ { "bbox": [ 199, 374, 411, 401 ], "spans": [ { "bbox": [ 199, 374, 411, 401 ], "score": 0.91, "content": "\\mathbb { P } ( a \\neq a ^ { * } ) \\leq C \\exp \\{ - \\frac { t } { \\hat { C } \\sigma ( \\log ( 2 + t ) ) ^ { 2 } } \\} + \\mathbb { P } ( E _ { s } ^ { c } ) .", "type": "interline_equation", "image_path": "8ca849193e088f824b108a4d4b9d0c507d6f6e8848e7b0bd5adb83d7b3fe1562.jpg" } ] } ], "index": 14, "virtual_lines": [ { "bbox": [ 199, 374, 411, 401 ], "spans": [], "index": 14 } ] }, { "type": "text", "bbox": [ 107, 406, 247, 420 ], "lines": [ { "bbox": [ 105, 405, 247, 421 ], "spans": [ { "bbox": [ 105, 405, 171, 421 ], "score": 1.0, "content": "From Lemma 4,", "type": "text" }, { "bbox": [ 172, 405, 204, 420 ], "score": 0.88, "content": "\\exists C ^ { ' } , \\hat { C } ^ { \\prime }", "type": "inline_equation" }, { "bbox": [ 205, 405, 247, 421 ], "score": 1.0, "content": "for which", "type": "text" } ], "index": 15 } ], "index": 15, "bbox_fs": [ 105, 405, 247, 421 ] }, { "type": "interline_equation", "bbox": [ 233, 423, 376, 451 ], "lines": [ { "bbox": [ 233, 423, 376, 451 ], "spans": [ { "bbox": [ 233, 423, 376, 451 ], "score": 0.93, "content": "\\mathbb { P } ( E _ { s } ^ { c } ) \\le C ^ { ' } t \\exp \\{ - \\frac { t } { \\hat { C } ^ { ' } ( \\log ( t ) ) ^ { 3 } } \\} ,", "type": "interline_equation", "image_path": "2417eb7b4409e22142eb41c26b43311830dbd75e08c20b84c93e2f727e6ac097.jpg" } ] } ], "index": 16, "virtual_lines": [ { "bbox": [ 233, 423, 376, 451 ], "spans": [], "index": 16 } ] }, { "type": "text", "bbox": [ 106, 455, 136, 466 ], "lines": [ { "bbox": [ 104, 453, 137, 468 ], "spans": [ { "bbox": [ 104, 453, 137, 468 ], "score": 1.0, "content": "so that", "type": "text" } ], "index": 17 } ], "index": 17, "bbox_fs": [ 104, 453, 137, 468 ] }, { "type": "interline_equation", "bbox": [ 228, 469, 381, 495 ], "lines": [ { "bbox": [ 228, 469, 381, 495 ], "spans": [ { "bbox": [ 228, 469, 381, 495 ], "score": 0.93, "content": "\\mathbb { P } ( a \\neq a ^ { * } ) \\leq O ( t \\exp \\{ - \\frac { t } { ( \\log ( t ) ) ^ { 3 } } \\} ) .", "type": "interline_equation", "image_path": "d1f1bceb534ec1b0dff367e996af63fbf9bb17ea93c607fc579645fe922e0c43.jpg" } ] } ], "index": 18, "virtual_lines": [ { "bbox": [ 228, 469, 381, 495 ], "spans": [], "index": 18 } ] }, { "type": "text", "bbox": [ 106, 505, 506, 567 ], "lines": [ { "bbox": [ 104, 504, 502, 527 ], "spans": [ { "bbox": [ 104, 504, 209, 527 ], "score": 1.0, "content": "Theorem 3 Consider an", "type": "text" }, { "bbox": [ 209, 510, 231, 520 ], "score": 0.45, "content": "E 3 W", "type": "inline_equation" }, { "bbox": [ 231, 504, 354, 527 ], "score": 1.0, "content": "policy applied to the tree. Let", "type": "text" }, { "bbox": [ 354, 504, 502, 524 ], "score": 0.92, "content": "\\begin{array} { r } { \\kappa _ { i } = \\nabla \\Omega ^ { * } ( a _ { i } | s ) + \\frac { L } { p } \\sqrt { { \\hat { C } } \\sigma ^ { 2 } \\log \\frac { C } { \\delta } / 2 n } , } \\end{array}", "type": "inline_equation" } ], "index": 19 }, { "bbox": [ 106, 523, 506, 543 ], "spans": [ { "bbox": [ 106, 523, 255, 543 ], "score": 0.81, "content": "\\begin{array} { r } { \\chi _ { i } = \\nabla \\Omega ^ { * } ( a _ { i } | s ) - \\frac { L } { p } \\sqrt { \\hat { C } \\sigma ^ { 2 } \\log \\frac { C } { \\delta } / 2 n } , } \\end{array}", "type": "inline_equation" }, { "bbox": [ 255, 525, 286, 542 ], "score": 1.0, "content": ", where", "type": "text" }, { "bbox": [ 286, 528, 325, 540 ], "score": 0.92, "content": "\\nabla \\Omega ^ { * } ( . | s )", "type": "inline_equation" }, { "bbox": [ 326, 525, 506, 542 ], "score": 1.0, "content": "is the policy with respect to the mean value", "type": "text" } ], "index": 20 }, { "bbox": [ 105, 542, 504, 557 ], "spans": [ { "bbox": [ 105, 543, 134, 557 ], "score": 1.0, "content": "vector", "type": "text" }, { "bbox": [ 134, 545, 154, 556 ], "score": 0.89, "content": "V ( \\cdot )", "type": "inline_equation" }, { "bbox": [ 154, 543, 264, 557 ], "score": 1.0, "content": "at the root node s. For any", "type": "text" }, { "bbox": [ 265, 545, 289, 555 ], "score": 0.89, "content": "\\delta > 0", "type": "inline_equation" }, { "bbox": [ 289, 543, 390, 557 ], "score": 1.0, "content": ", with probability at least", "type": "text" }, { "bbox": [ 391, 545, 414, 555 ], "score": 0.62, "content": "1 - \\delta", "type": "inline_equation" }, { "bbox": [ 414, 543, 462, 557 ], "score": 1.0, "content": ", ∃ constant", "type": "text" }, { "bbox": [ 463, 542, 504, 556 ], "score": 0.92, "content": "L , p , C , { \\hat { C } }", "type": "inline_equation" } ], "index": 21 }, { "bbox": [ 106, 555, 257, 568 ], "spans": [ { "bbox": [ 106, 555, 207, 568 ], "score": 1.0, "content": "so that the pseudo regret", "type": "text" }, { "bbox": [ 207, 556, 221, 566 ], "score": 0.89, "content": "R _ { n }", "type": "inline_equation" }, { "bbox": [ 221, 555, 257, 568 ], "score": 1.0, "content": "satisfies", "type": "text" } ], "index": 22 } ], "index": 20.5, "bbox_fs": [ 104, 504, 506, 568 ] }, { "type": "interline_equation", "bbox": [ 114, 570, 495, 601 ], "lines": [ { "bbox": [ 114, 570, 495, 601 ], "spans": [ { "bbox": [ 114, 570, 495, 601 ], "score": 0.94, "content": "n V ^ { * } - n \\sum _ { i } V _ { i } \\Big ( \\kappa _ { i } + \\frac { L } { p } \\big ( \\frac { \\tau ( U _ { \\Omega } - L _ { \\Omega } ) } { 1 - \\gamma } \\big ) \\Big ) \\leq R _ { n } \\leq n V ^ { * } - n \\sum _ { i } V _ { i } \\Big ( \\chi _ { i } - \\frac { L } { p } \\big ( \\frac { \\tau ( U _ { \\Omega } - L _ { \\Omega } ) } { 1 - \\gamma } \\big ) \\Big ) .", "type": "interline_equation", "image_path": "902a8a5ba4718099795b9648b4785fd6500fe05c79dc2e19703b0fddc7fa33d1.jpg" } ] } ], "index": 24, "virtual_lines": [ { "bbox": [ 114, 570, 495, 580.3333333333334 ], "spans": [], "index": 23 }, { "bbox": [ 114, 580.3333333333334, 495, 590.6666666666667 ], "spans": [], "index": 24 }, { "bbox": [ 114, 590.6666666666667, 495, 601.0000000000001 ], "spans": [], "index": 25 } ] }, { "type": "text", "bbox": [ 105, 609, 504, 624 ], "lines": [ { "bbox": [ 103, 607, 507, 627 ], "spans": [ { "bbox": [ 103, 607, 279, 627 ], "score": 1.0, "content": "Proof 8 From Lemma 2 given two policies", "type": "text" }, { "bbox": [ 279, 610, 353, 624 ], "score": 0.89, "content": "\\pi ^ { ( 1 ) } = \\nabla \\Omega ^ { * } ( r ^ { ( 1 ) } )", "type": "inline_equation" }, { "bbox": [ 353, 607, 372, 627 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 372, 610, 464, 624 ], "score": 0.85, "content": "\\pi ^ { ( 2 ) } = \\nabla \\Omega ^ { * } ( r ^ { ( 2 ) } ) , \\exists L ,", "type": "inline_equation" }, { "bbox": [ 464, 607, 507, 627 ], "score": 1.0, "content": ", such that", "type": "text" } ], "index": 26 } ], "index": 26, "bbox_fs": [ 103, 607, 507, 627 ] }, { "type": "interline_equation", "bbox": [ 179, 628, 431, 653 ], "lines": [ { "bbox": [ 179, 628, 431, 653 ], "spans": [ { "bbox": [ 179, 628, 431, 653 ], "score": 0.92, "content": "\\parallel \\pi ^ { ( 1 ) } - \\pi ^ { ( 2 ) } \\parallel _ { p } \\leq L \\parallel r ^ { ( 1 ) } - r ^ { ( 2 ) } \\parallel _ { p } \\leq L \\frac { 1 } { p } \\parallel r ^ { ( 1 ) } - r ^ { ( 2 ) } \\parallel _ { \\infty } .", "type": "interline_equation", "image_path": "7b87807fa9d3b83f0d0fad0d520e1e4ec914379c5f64ac13eb89499683a579e4.jpg" } ] } ], "index": 27, "virtual_lines": [ { "bbox": [ 179, 628, 431, 653 ], "spans": [], "index": 27 } ] }, { "type": "text", "bbox": [ 107, 657, 227, 668 ], "lines": [ { "bbox": [ 106, 655, 227, 670 ], "spans": [ { "bbox": [ 106, 655, 227, 670 ], "score": 1.0, "content": "From (13), we have the regret", "type": "text" } ], "index": 28 } ], "index": 28, "bbox_fs": [ 106, 655, 227, 670 ] }, { "type": "interline_equation", "bbox": [ 236, 672, 374, 705 ], "lines": [ { "bbox": [ 236, 672, 374, 705 ], "spans": [ { "bbox": [ 236, 672, 374, 705 ], "score": 0.94, "content": "R _ { n } = n V ^ { * } - \\sum _ { i } V _ { i } \\sum _ { t = 1 } ^ { n } { \\hat { \\pi } } _ { t } ( a _ { i } | s ) ,", "type": "interline_equation", "image_path": "ace819a2ff0b913e28d002dfbe179a2dc737e8032d71b5890c40cbd287d722a8.jpg" } ] } ], "index": 29.5, "virtual_lines": [ { "bbox": [ 236, 672, 374, 688.5 ], "spans": [], "index": 29 }, { "bbox": [ 236, 688.5, 374, 705.0 ], "spans": [], "index": 30 } ] }, { "type": "text", "bbox": [ 107, 709, 504, 733 ], "lines": [ { "bbox": [ 106, 709, 505, 722 ], "spans": [ { "bbox": [ 106, 709, 133, 722 ], "score": 1.0, "content": "where", "type": "text" }, { "bbox": [ 133, 710, 154, 722 ], "score": 0.9, "content": "\\hat { \\pi } _ { t } ( \\cdot )", "type": "inline_equation" }, { "bbox": [ 154, 709, 256, 722 ], "score": 1.0, "content": "is the policy at time step", "type": "text" }, { "bbox": [ 257, 713, 260, 719 ], "score": 0.7, "content": "t", "type": "inline_equation" }, { "bbox": [ 261, 709, 282, 722 ], "score": 1.0, "content": ", and", "type": "text" }, { "bbox": [ 283, 711, 298, 722 ], "score": 0.85, "content": "\\mathbb { I } ( \\cdot )", "type": "inline_equation" }, { "bbox": [ 298, 709, 402, 722 ], "score": 1.0, "content": "is the indicator function.", "type": "text" }, { "bbox": [ 402, 712, 415, 720 ], "score": 0.87, "content": "V ^ { * }", "type": "inline_equation" }, { "bbox": [ 415, 709, 505, 722 ], "score": 1.0, "content": "is the optimal branch", "type": "text" } ], "index": 31 }, { "bbox": [ 105, 720, 504, 733 ], "spans": [ { "bbox": [ 105, 720, 174, 733 ], "score": 1.0, "content": "at the root node,", "type": "text" }, { "bbox": [ 174, 721, 185, 732 ], "score": 0.86, "content": "V _ { i }", "type": "inline_equation" }, { "bbox": [ 185, 720, 437, 733 ], "score": 1.0, "content": "is the mean value function of the branch with respect to action", "type": "text" }, { "bbox": [ 437, 723, 441, 730 ], "score": 0.66, "content": "i", "type": "inline_equation" }, { "bbox": [ 441, 720, 445, 733 ], "score": 1.0, "content": ",", "type": "text" }, { "bbox": [ 446, 722, 464, 733 ], "score": 0.91, "content": "V ( \\cdot )", "type": 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1 - \\delta } + \\sqrt { \\frac { \\hat { C } \\sigma ^ { 2 } \\log \\frac { C } { \\delta } } { 2 N ( s ) } } \\biggr ) ( T h e o r e m 4 ) } \\end{array}", "type": "interline_equation", "image_path": "e95dcabfc16631fc6957467a8f17b537a0e85f91ca93803174671c4ff1fbfd38.jpg" } ] } ], "index": 4, "virtual_lines": [ { "bbox": [ 142, 124, 467, 144.66666666666666 ], "spans": [], "index": 3 }, { "bbox": [ 142, 144.66666666666666, 467, 165.33333333333331 ], "spans": [], "index": 4 }, { "bbox": [ 142, 165.33333333333331, 467, 185.99999999999997 ], "spans": [], "index": 5 } ] }, { "type": "text", "bbox": [ 106, 187, 137, 198 ], "lines": [ { "bbox": [ 105, 185, 138, 200 ], "spans": [ { "bbox": [ 105, 185, 138, 200 ], "score": 1.0, "content": "So that", "type": "text" } ], "index": 6 } ], "index": 6 }, { "type": "interline_equation", "bbox": [ 111, 200, 514, 233 ], "lines": [ { "bbox": [ 111, 200, 514, 233 ], "spans": [ { "bbox": [ 111, 200, 514, 233 ], "score": 0.89, "content": "\\tau ( a _ { i } | s ) - \\frac { L } { p } \\Bigg ( \\frac { \\tau ( U _ { \\Omega } - L _ { \\Omega } ) } { 1 - \\delta } + \\sqrt { \\frac { \\hat { C } \\sigma ^ { 2 } \\log \\frac { C } { \\delta } } { 2 N ( s ) } } \\Bigg ) \\leq \\hat { \\pi } _ { t } ( a _ { i } | s ) \\leq \\pi ( a _ { i } | s ) + \\frac { L } { p } \\Bigg ( \\frac { \\tau ( U _ { \\Omega } - L _ { \\Omega } ) } { 1 - \\delta } + \\sqrt { \\frac { \\hat { C } \\sigma ^ { 2 } \\log \\frac { C } { \\delta } } { 2 N ( s ) } } \\Bigg ) .", "type": "interline_equation", "image_path": "25decafdb5eb6fe0e0fb9542c801d8829b40af1be49e578dedf4a5485f23d763.jpg" } ] } ], "index": 8, "virtual_lines": [ { "bbox": [ 111, 200, 514, 211.0 ], "spans": [], "index": 7 }, { "bbox": [ 111, 211.0, 514, 222.0 ], "spans": [], "index": 8 }, { "bbox": [ 111, 222.0, 514, 233.0 ], "spans": [], "index": 9 } ] }, { "type": "text", "bbox": [ 106, 236, 136, 246 ], "lines": [ { "bbox": [ 105, 234, 137, 248 ], "spans": [ { "bbox": [ 105, 234, 137, 248 ], "score": 1.0, "content": "so that", "type": "text" } ], "index": 10 } ], "index": 10 }, { "type": "interline_equation", "bbox": [ 111, 247, 514, 364 ], "lines": [ { "bbox": [ 111, 247, 514, 364 ], "spans": [ { "bbox": [ 111, 247, 514, 364 ], "score": 0.91, "content": "\\begin{array} { l } { \\displaystyle \\mathfrak { l } _ { n } = n V ^ { * } - \\sum _ { i } V _ { i } \\sum _ { t = 1 } ^ { n } \\hat { \\pi } _ { t } ( a _ { i } | s ) \\le n V ^ { * } - \\sum _ { i } V _ { i } \\sum _ { t = 1 } ^ { n } \\left( \\pi ( a _ { i } | s ) - \\displaystyle \\frac { L } { p } \\big ( \\displaystyle \\frac { \\tau ( U _ { \\Omega } - L _ { \\Omega } ) } { 1 - \\delta } + \\sqrt { \\displaystyle \\frac { \\hat { C } \\sigma ^ { 2 } \\log \\frac { C } { \\delta } } { 2 n } } \\big ) \\right) } \\\\ { \\displaystyle \\mathfrak { l } _ { n } \\le n V ^ { * } - \\sum _ { i } V _ { i } \\sum _ { t = 1 } ^ { n } \\left( \\pi ( a _ { i } | s ) - \\displaystyle \\frac { L } { p } \\big ( \\displaystyle \\frac { \\tau ( U _ { \\Omega } - L _ { \\Omega } ) } { 1 - \\delta } + \\sqrt { \\displaystyle \\frac { \\hat { C } \\sigma ^ { 2 } \\log \\frac { C } { \\delta } } { 2 n } } \\big ) \\right) } \\\\ { \\displaystyle \\mathfrak { l } _ { n } \\le n V ^ { * } - n \\sum _ { i } V _ { i } \\left( \\pi ( a _ { i } | s ) - \\displaystyle \\frac { L } { p } \\big ( \\displaystyle \\frac { \\tau ( U _ { \\Omega } - L _ { \\Omega } ) } { 1 - \\delta } + \\sqrt { \\displaystyle \\frac { \\hat { C } \\sigma ^ { 2 } \\log \\frac { C } { \\delta } } { 2 n } } \\big ) \\right) } \\end{array}", "type": "interline_equation", "image_path": "45466167a5c6fe37df4f18a104b0329b7b71a1e4f2e84a5d4763a6262a89ba6b.jpg" } ] } ], "index": 12, "virtual_lines": [ { "bbox": [ 111, 247, 514, 286.0 ], "spans": [], "index": 11 }, { "bbox": [ 111, 286.0, 514, 325.0 ], "spans": [], "index": 12 }, { "bbox": [ 111, 325.0, 514, 364.0 ], "spans": [], "index": 13 } ] }, { "type": "text", "bbox": [ 106, 378, 126, 389 ], "lines": [ { "bbox": [ 105, 376, 127, 390 ], "spans": [ { "bbox": [ 105, 376, 127, 390 ], "score": 1.0, "content": "And", "type": "text" } ], "index": 14 } ], "index": 14 }, { "type": "interline_equation", "bbox": [ 157, 391, 453, 465 ], "lines": [ { "bbox": [ 157, 391, 453, 465 ], "spans": [ { "bbox": [ 157, 391, 453, 465 ], "score": 0.91, "content": "\\begin{array} { l } { { R _ { n } \\geq n V ^ { \\ast } - \\displaystyle \\sum _ { i } V _ { i } \\sum _ { t = 1 } ^ { n } \\left( \\pi ( a _ { i } | s ) + \\frac { L } { p } \\big ( \\frac { \\tau ( U _ { \\Omega } - L _ { \\Omega } ) } { 1 - \\delta } + \\sqrt { \\frac { \\hat { C } \\sigma ^ { 2 } \\log \\frac { C } { \\delta } } { 2 n } } \\big ) \\right) } } \\\\ { { R _ { n } \\geq n V ^ { \\ast } - n \\displaystyle \\sum _ { i } V _ { i } \\Big ( \\pi ( a _ { i } | s ) + \\frac { L } { p } \\big ( \\frac { \\tau ( U _ { \\Omega } - L _ { \\Omega } ) } { 1 - \\delta } + \\sqrt { \\frac { \\hat { C } \\sigma ^ { 2 } \\log \\frac { C } { \\delta } } { 2 n } } \\big ) \\Big ) } } \\end{array}", "type": "interline_equation", "image_path": "27f8ec5114898add53e98642164403e5fc0daf4eb313a161ff25528384b728a5.jpg" } ] } ], "index": 16, "virtual_lines": [ { "bbox": [ 157, 391, 453, 415.6666666666667 ], "spans": [], "index": 15 }, { "bbox": [ 157, 415.6666666666667, 453, 440.33333333333337 ], "spans": [], "index": 16 }, { "bbox": [ 157, 440.33333333333337, 453, 465.00000000000006 ], "spans": [], "index": 17 } ] }, { "type": "text", "bbox": [ 106, 468, 373, 479 ], "lines": [ { "bbox": [ 105, 466, 374, 481 ], "spans": [ { "bbox": [ 105, 466, 311, 481 ], "score": 1.0, "content": "In case of Maximum Entropy and Relative Entropy", "type": "text" }, { "bbox": [ 311, 469, 335, 479 ], "score": 0.9, "content": "p = 1", "type": "inline_equation" }, { "bbox": [ 335, 466, 374, 481 ], "score": 1.0, "content": ", because", "type": "text" } ], "index": 18 } ], "index": 18 }, { "type": "interline_equation", "bbox": [ 223, 482, 387, 497 ], "lines": [ { "bbox": [ 223, 482, 387, 497 ], "spans": [ { "bbox": [ 223, 482, 387, 497 ], "score": 0.9, "content": "\\parallel \\pi ^ { ( 1 ) } - \\pi ^ { ( 2 ) } \\parallel _ { \\infty } \\leq L \\parallel r ^ { ( 1 ) } - r ^ { ( 2 ) } \\parallel _ { \\infty } .", "type": "interline_equation", "image_path": "2efed660ac2d7e13b9af72230a0af68e84c0f7f3615ab215ccbfc754084cafb5.jpg" } ] } ], "index": 19, "virtual_lines": [ { "bbox": [ 223, 482, 387, 497 ], "spans": [], "index": 19 } ] }, { "type": "text", "bbox": [ 106, 500, 220, 511 ], "lines": [ { "bbox": [ 106, 500, 220, 513 ], "spans": [ { "bbox": [ 106, 500, 220, 513 ], "score": 1.0, "content": "So that we have for MENTS", "type": "text" } ], "index": 20 } ], "index": 20 }, { "type": "interline_equation", "bbox": [ 132, 514, 478, 545 ], "lines": [ { "bbox": [ 132, 514, 478, 545 ], "spans": [ { "bbox": [ 132, 514, 478, 545 ], "score": 0.92, "content": "n V ^ { * } - n \\sum _ { i } V _ { i } \\Big ( \\kappa _ { i } + L \\big ( \\frac { \\tau \\log | A | } { 1 - \\gamma } \\big ) \\Big ) \\leq R _ { n } \\leq n V ^ { * } - n \\sum _ { i } V _ { i } \\Big ( \\chi _ { i } - L \\big ( \\frac { \\tau \\log | A | } { 1 - \\gamma } \\big ) \\Big ) .", "type": "interline_equation", "image_path": "231adc8dfe6afc7cad125769c3ff5a118f083e161db2ee46ade231908f37ca4f.jpg" } ] } ], "index": 22, "virtual_lines": [ { "bbox": [ 132, 514, 478, 524.3333333333334 ], "spans": [], "index": 21 }, { "bbox": [ 132, 524.3333333333334, 478, 534.6666666666667 ], "spans": [], "index": 22 }, { "bbox": [ 132, 534.6666666666667, 478, 545.0000000000001 ], "spans": [], "index": 23 } ] }, { "type": "text", "bbox": [ 107, 552, 192, 564 ], "lines": [ { "bbox": [ 105, 551, 193, 565 ], "spans": [ { "bbox": [ 105, 551, 193, 565 ], "score": 1.0, "content": "For RENTS, we have", "type": "text" } ], "index": 24 } ], "index": 24 }, { "type": "interline_equation", "bbox": [ 111, 566, 505, 598 ], "lines": [ { "bbox": [ 111, 566, 505, 598 ], "spans": [ { "bbox": [ 111, 566, 505, 598 ], "score": 0.92, "content": "\\imath V ^ { * } - n \\sum _ { i } V _ { i } \\Big ( \\kappa _ { i } + L \\big ( \\frac { \\tau ( \\log | A | - \\frac { 1 } { m } ) } { 1 - \\gamma } \\big ) \\Big ) \\leq R _ { n } \\leq n V ^ { * } - n \\sum _ { i } V _ { i } \\Big ( \\chi _ { i } - L \\big ( \\frac { \\tau ( \\log | A | - \\frac { 1 } { m } ) } { 1 - \\gamma } \\big ) \\Big )", "type": "interline_equation", "image_path": "d4fd869a735dd27c880e7b0fa501e80127e2b752e3879689307993b1c44cf911.jpg" } ] } ], "index": 26, "virtual_lines": [ { "bbox": [ 111, 566, 505, 576.6666666666666 ], "spans": [], "index": 25 }, { "bbox": [ 111, 576.6666666666666, 505, 587.3333333333333 ], "spans": [], "index": 26 }, { "bbox": [ 111, 587.3333333333333, 505, 597.9999999999999 ], "spans": [], "index": 27 } ] }, { "type": "text", "bbox": [ 106, 601, 208, 612 ], "lines": [ { "bbox": [ 106, 599, 205, 614 ], "spans": [ { "bbox": [ 106, 599, 133, 614 ], "score": 1.0, "content": "where", "type": "text" }, { "bbox": [ 133, 601, 205, 613 ], "score": 0.92, "content": "\\begin{array} { r } { m = \\operatorname* { m i n } _ { a } \\pi ( a | s ) } \\end{array}", "type": "inline_equation" } ], "index": 28 } ], "index": 28 }, { "type": "text", "bbox": [ 108, 612, 384, 624 ], "lines": [ { "bbox": [ 106, 612, 384, 624 ], "spans": [ { "bbox": [ 106, 612, 210, 624 ], "score": 1.0, "content": "In case of Tsallis Entropy", "type": "text" }, { "bbox": [ 211, 613, 236, 624 ], "score": 0.89, "content": "p = 2", "type": "inline_equation" }, { "bbox": [ 236, 612, 384, 624 ], "score": 1.0, "content": "( Niculae & Blondel (2017)), so that", "type": "text" } ], "index": 29 } ], "index": 29 }, { "type": "interline_equation", "bbox": [ 111, 626, 500, 657 ], "lines": [ { "bbox": [ 111, 626, 500, 657 ], "spans": [ { "bbox": [ 111, 626, 500, 657 ], "score": 0.93, "content": "n V ^ { * } - n \\sum _ { i } V _ { i } \\Big ( \\kappa _ { i } + \\frac { L } { 2 } \\big ( \\frac { | A | - 1 } { 2 | A | } \\frac { \\tau } { 1 - \\gamma } \\big ) \\Big ) \\leq R _ { n } \\leq n V ^ { * } - n \\sum _ { i } V _ { i } \\Big ( \\chi _ { i } - \\frac { L } { 2 } \\big ( \\frac { | A | - 1 } { 2 | A | } \\frac { \\tau } { 1 - \\gamma } \\big ) \\Big )", "type": "interline_equation", "image_path": "b1190e1157284e901e0760831cabe7555d7f6cf86fc5ce242e261553f1bd9a0b.jpg" } ] } ], "index": 31, "virtual_lines": [ { "bbox": [ 111, 626, 500, 636.3333333333334 ], "spans": [], "index": 30 }, { "bbox": [ 111, 636.3333333333334, 500, 646.6666666666667 ], "spans": [], "index": 31 }, { "bbox": [ 111, 646.6666666666667, 500, 657.0000000000001 ], "spans": [], "index": 32 } ] }, { "type": "text", "bbox": [ 106, 664, 435, 677 ], "lines": [ { "bbox": [ 106, 664, 434, 677 ], "spans": [ { "bbox": [ 106, 664, 434, 677 ], "score": 1.0, "content": "Before derive the next theorem, we state here the Theorem 2 in Geist et al. (2019)", "type": "text" } ], "index": 33 } ], "index": 33 }, { "type": "text", "bbox": [ 128, 684, 504, 707 ], "lines": [ { "bbox": [ 131, 682, 505, 699 ], "spans": [ { "bbox": [ 131, 682, 271, 699 ], "score": 1.0, "content": "• Boundedness: for two constants", "type": "text" }, { "bbox": [ 271, 685, 285, 696 ], "score": 0.89, "content": "L _ { \\Omega }", "type": "inline_equation" }, { "bbox": [ 285, 682, 302, 699 ], "score": 1.0, "content": "and", "type": "text" }, { "bbox": [ 303, 685, 316, 696 ], "score": 0.89, "content": "U _ { \\Omega }", "type": "inline_equation" }, { "bbox": [ 317, 682, 380, 699 ], "score": 1.0, "content": "such that for all", "type": "text" }, { "bbox": [ 380, 685, 407, 695 ], "score": 0.91, "content": "\\pi \\in \\Pi", "type": "inline_equation" }, { "bbox": [ 408, 682, 444, 699 ], "score": 1.0, "content": ", we have", "type": "text" }, { "bbox": [ 445, 684, 505, 697 ], "score": 0.92, "content": "L _ { \\Omega } \\leq \\Omega ( \\pi ) \\leq", "type": "inline_equation" } ], "index": 34 }, { "bbox": [ 143, 694, 181, 708 ], "spans": [ { "bbox": [ 143, 696, 156, 707 ], "score": 0.87, "content": "U _ { \\Omega }", "type": "inline_equation" }, { "bbox": [ 156, 694, 181, 708 ], "score": 1.0, "content": ", then", "type": "text" } ], "index": 35 } ], "index": 34.5 }, { "type": "interline_equation", "bbox": [ 237, 709, 410, 735 ], "lines": [ { "bbox": [ 237, 709, 410, 735 ], "spans": [ { "bbox": [ 237, 709, 410, 735 ], "score": 0.95, "content": "V ^ { \\ast } ( s ) - \\frac { \\tau ( U _ { \\Omega } - L _ { \\Omega } ) } { 1 - \\gamma } \\leq V _ { \\Omega } ^ { \\ast } ( s ) \\leq V ^ { \\ast } ( s ) .", "type": "interline_equation", "image_path": "129b22fc0d1ba768cd119e4007acabac31d107163162779b0786e704a19378d2.jpg" } ] } ], "index": 36, "virtual_lines": [ { "bbox": [ 237, 709, 410, 735 ], "spans": [], "index": 36 } ] } ], "page_idx": 17, "page_size": [ 612, 792 ], "discarded_blocks": [ { "type": "discarded", "bbox": [ 300, 751, 311, 760 ], "lines": [ { "bbox": [ 299, 750, 313, 764 ], "spans": [ { "bbox": [ 299, 750, 313, 764 ], "score": 1.0, "content": "18", "type": "text" } ] } ] }, { "type": "discarded", "bbox": [ 106, 26, 307, 38 ], "lines": [ { "bbox": [ 106, 25, 308, 38 ], "spans": [ { "bbox": [ 106, 25, 308, 38 ], "score": 1.0, "content": "Under review as a conference paper at ICLR 2021", "type": "text" } ] } ] } ], "para_blocks": [ { "type": "text", "bbox": [ 106, 81, 505, 106 ], "lines": [ { "bbox": [ 106, 81, 505, 95 ], "spans": [ { "bbox": [ 106, 82, 275, 95 ], "score": 1.0, "content": "vector of value function at the root node.", "type": "text" }, { "bbox": [ 275, 81, 295, 95 ], "score": 0.9, "content": "\\hat { V } ( \\cdot )", "type": "inline_equation" }, { "bbox": [ 295, 82, 321, 95 ], "score": 1.0, "content": "is the", "type": "text" }, { "bbox": [ 322, 82, 335, 95 ], "score": 0.89, "content": "| A |", "type": "inline_equation" }, { "bbox": [ 336, 82, 505, 95 ], "score": 1.0, "content": "estimation vector of value function at the", "type": "text" } ], "index": 0 }, { "bbox": [ 106, 93, 478, 107 ], "spans": [ { "bbox": [ 106, 93, 150, 107 ], "score": 1.0, "content": "root node.", "type": "text" }, { "bbox": [ 150, 93, 235, 106 ], "score": 0.93, "content": "\\pi ( . | s ) = \\nabla \\Omega ^ { * } ( V ( \\cdot ) )", "type": "inline_equation" }, { "bbox": [ 235, 93, 362, 107 ], "score": 1.0, "content": "is the policy with respect to the", "type": "text" }, { "bbox": [ 362, 93, 382, 106 ], "score": 0.88, "content": "V ( \\cdot )", "type": "inline_equation" }, { "bbox": [ 382, 93, 478, 107 ], "score": 1.0, "content": "vector at the root node.", "type": "text" } ], "index": 1 } ], "index": 0.5, "bbox_fs": [ 106, 81, 505, 107 ] }, { "type": "text", "bbox": [ 106, 110, 348, 122 ], "lines": [ { "bbox": [ 106, 109, 348, 124 ], "spans": [ { "bbox": [ 106, 109, 160, 124 ], "score": 1.0, "content": "Then for any", "type": "text" }, { "bbox": [ 160, 111, 184, 121 ], "score": 0.88, "content": "\\delta > 0", "type": "inline_equation" }, { "bbox": [ 185, 109, 285, 124 ], "score": 1.0, "content": ", with probability at least", "type": "text" }, { "bbox": [ 286, 111, 308, 121 ], "score": 0.77, "content": "1 - \\delta", "type": "inline_equation" }, { "bbox": [ 309, 109, 348, 124 ], "score": 1.0, "content": ", we have", "type": "text" } ], "index": 2 } ], "index": 2, "bbox_fs": [ 106, 109, 348, 124 ] }, { "type": "interline_equation", "bbox": [ 142, 124, 467, 186 ], "lines": [ { "bbox": [ 142, 124, 467, 186 ], "spans": [ { "bbox": [ 142, 124, 467, 186 ], "score": 0.92, "content": "\\begin{array} { l } { \\displaystyle \\lvert \\pi ( a _ { i } \\vert s ) - \\hat { \\pi } _ { t } ( a _ { i } \\vert s ) \\rvert \\le \\parallel \\pi ( . \\vert s ) - \\hat { \\pi } _ { t } ( . \\vert s ) \\parallel _ { \\infty } \\le \\displaystyle \\frac { L } { p } \\parallel V ( \\cdot ) - \\hat { V } ( \\cdot ) \\parallel _ { \\infty } ( L e m m a 2 ) } \\\\ { \\displaystyle \\le \\displaystyle \\frac { L } { p } \\lvert V ( \\cdot ) - \\hat { V } ( \\cdot ) \\rvert \\le \\displaystyle \\frac { L } { p } \\biggl ( \\frac { \\tau ( U _ { \\Omega } - L _ { \\Omega } ) } { 1 - \\delta } + \\sqrt { \\frac { \\hat { C } \\sigma ^ { 2 } \\log \\frac { C } { \\delta } } { 2 N ( s ) } } \\biggr ) ( T h e o r e m 4 ) } \\end{array}", "type": "interline_equation", "image_path": "e95dcabfc16631fc6957467a8f17b537a0e85f91ca93803174671c4ff1fbfd38.jpg" } ] } ], "index": 4, "virtual_lines": [ { "bbox": [ 142, 124, 467, 144.66666666666666 ], "spans": [], "index": 3 }, { "bbox": [ 142, 144.66666666666666, 467, 165.33333333333331 ], "spans": [], "index": 4 }, { "bbox": [ 142, 165.33333333333331, 467, 185.99999999999997 ], "spans": [], "index": 5 } ] }, { "type": "text", "bbox": [ 106, 187, 137, 198 ], "lines": [ { "bbox": [ 105, 185, 138, 200 ], "spans": [ { "bbox": [ 105, 185, 138, 200 ], "score": 1.0, "content": "So that", "type": "text" } ], "index": 6 } ], "index": 6, "bbox_fs": [ 105, 185, 138, 200 ] }, { "type": "interline_equation", "bbox": [ 111, 200, 514, 233 ], "lines": [ { "bbox": [ 111, 200, 514, 233 ], "spans": [ { "bbox": [ 111, 200, 514, 233 ], "score": 0.89, "content": "\\tau ( a _ { i } | s ) - \\frac { L } { p } \\Bigg ( \\frac { \\tau ( U _ { \\Omega } - L _ { \\Omega } ) } { 1 - \\delta } + \\sqrt { \\frac { \\hat { C } \\sigma ^ { 2 } \\log \\frac { C } { \\delta } } { 2 N ( s ) } } \\Bigg ) \\leq \\hat { \\pi } _ { t } ( a _ { i } | s ) \\leq \\pi ( a _ { i } | s ) + \\frac { L } { p } \\Bigg ( \\frac { \\tau ( U _ { \\Omega } - L _ { \\Omega } ) } { 1 - \\delta } + \\sqrt { \\frac { \\hat { C } \\sigma ^ { 2 } \\log \\frac { C } { \\delta } } { 2 N ( s ) } } \\Bigg ) .", "type": "interline_equation", "image_path": "25decafdb5eb6fe0e0fb9542c801d8829b40af1be49e578dedf4a5485f23d763.jpg" } ] } ], "index": 8, "virtual_lines": [ { "bbox": [ 111, 200, 514, 211.0 ], "spans": [], "index": 7 }, { "bbox": [ 111, 211.0, 514, 222.0 ], "spans": [], "index": 8 }, { "bbox": [ 111, 222.0, 514, 233.0 ], "spans": [], "index": 9 } ] }, { "type": "text", "bbox": [ 106, 236, 136, 246 ], "lines": [ { "bbox": [ 105, 234, 137, 248 ], "spans": [ { "bbox": [ 105, 234, 137, 248 ], "score": 1.0, "content": "so that", "type": "text" } ], "index": 10 } ], "index": 10, "bbox_fs": [ 105, 234, 137, 248 ] }, { "type": "interline_equation", "bbox": [ 111, 247, 514, 364 ], "lines": [ { "bbox": [ 111, 247, 514, 364 ], "spans": [ { "bbox": [ 111, 247, 514, 364 ], "score": 0.91, "content": "\\begin{array} { l } { \\displaystyle \\mathfrak { l } _ { n } = n V ^ { * } - \\sum _ { i } V _ { i } \\sum _ { t = 1 } ^ { n } \\hat { \\pi } _ { t } ( a _ { i } | s ) \\le n V ^ { * } - \\sum _ { i } V _ { i } \\sum _ { t = 1 } ^ { n } \\left( \\pi ( a _ { i } | s ) - \\displaystyle \\frac { L } { p } \\big ( \\displaystyle \\frac { \\tau ( U _ { \\Omega } - L _ { \\Omega } ) } { 1 - \\delta } + \\sqrt { \\displaystyle \\frac { \\hat { C } \\sigma ^ { 2 } \\log \\frac { C } { \\delta } } { 2 n } } \\big ) \\right) } \\\\ { \\displaystyle \\mathfrak { l } _ { n } \\le n V ^ { * } - \\sum _ { i } V _ { i } \\sum _ { t = 1 } ^ { n } \\left( \\pi ( a _ { i } | s ) - \\displaystyle \\frac { L } { p } \\big ( \\displaystyle \\frac { \\tau ( U _ { \\Omega } - L _ { \\Omega } ) } { 1 - \\delta } + \\sqrt { \\displaystyle \\frac { \\hat { C } \\sigma ^ { 2 } \\log \\frac { C } { \\delta } } { 2 n } } \\big ) \\right) } \\\\ { \\displaystyle \\mathfrak { l } _ { n } \\le n V ^ { * } - n \\sum _ { i } V _ { i } \\left( \\pi ( a _ { i } | s ) - \\displaystyle \\frac { L } { p } \\big ( \\displaystyle \\frac { \\tau ( U _ { \\Omega } - L _ { \\Omega } ) } { 1 - \\delta } + \\sqrt { \\displaystyle \\frac { \\hat { C } \\sigma ^ { 2 } \\log \\frac { C } { \\delta } } { 2 n } } \\big ) \\right) } \\end{array}", "type": "interline_equation", "image_path": "45466167a5c6fe37df4f18a104b0329b7b71a1e4f2e84a5d4763a6262a89ba6b.jpg" } ] } ], "index": 12, "virtual_lines": [ { "bbox": [ 111, 247, 514, 286.0 ], "spans": [], "index": 11 }, { "bbox": [ 111, 286.0, 514, 325.0 ], "spans": [], "index": 12 }, { "bbox": [ 111, 325.0, 514, 364.0 ], "spans": [], "index": 13 } ] }, { "type": "text", "bbox": [ 106, 378, 126, 389 ], "lines": [ { "bbox": [ 105, 376, 127, 390 ], "spans": [ { "bbox": [ 105, 376, 127, 390 ], "score": 1.0, "content": "And", "type": "text" } ], "index": 14 } ], "index": 14, "bbox_fs": [ 105, 376, 127, 390 ] }, { "type": "interline_equation", "bbox": [ 157, 391, 453, 465 ], "lines": [ { "bbox": [ 157, 391, 453, 465 ], "spans": [ { "bbox": [ 157, 391, 453, 465 ], "score": 0.91, "content": "\\begin{array} { l } { { R _ { n } \\geq n V ^ { \\ast } - \\displaystyle \\sum _ { i } V _ { i } \\sum _ { t = 1 } ^ { n } \\left( \\pi ( a _ { i } | s ) + \\frac { L } { p } \\big ( \\frac { \\tau ( U _ { \\Omega } - L _ { \\Omega } ) } { 1 - \\delta } + \\sqrt { \\frac { \\hat { C } \\sigma ^ { 2 } \\log \\frac { C } { \\delta } } { 2 n } } \\big ) \\right) } } \\\\ { { R _ { n } \\geq n V ^ { \\ast } - n \\displaystyle \\sum _ { i } V _ { i } \\Big ( \\pi ( a _ { i } | s ) + \\frac { L } { p } \\big ( \\frac { \\tau ( U _ { \\Omega } - L _ { \\Omega } ) } { 1 - \\delta } + \\sqrt { \\frac { \\hat { C } \\sigma ^ { 2 } \\log \\frac { C } { \\delta } } { 2 n } } \\big ) \\Big ) } } \\end{array}", "type": "interline_equation", "image_path": "27f8ec5114898add53e98642164403e5fc0daf4eb313a161ff25528384b728a5.jpg" } ] } ], "index": 16, "virtual_lines": [ { "bbox": [ 157, 391, 453, 415.6666666666667 ], "spans": [], "index": 15 }, { "bbox": [ 157, 415.6666666666667, 453, 440.33333333333337 ], "spans": [], "index": 16 }, { "bbox": [ 157, 440.33333333333337, 453, 465.00000000000006 ], "spans": [], "index": 17 } ] }, { "type": "text", "bbox": [ 106, 468, 373, 479 ], "lines": [ { "bbox": [ 105, 466, 374, 481 ], "spans": [ { "bbox": [ 105, 466, 311, 481 ], "score": 1.0, "content": "In case of Maximum Entropy and Relative Entropy", "type": "text" }, { "bbox": [ 311, 469, 335, 479 ], "score": 0.9, "content": "p = 1", "type": "inline_equation" }, { "bbox": [ 335, 466, 374, 481 ], "score": 1.0, "content": ", because", "type": "text" } ], "index": 18 } ], "index": 18, "bbox_fs": [ 105, 466, 374, 481 ] }, { "type": "interline_equation", "bbox": [ 223, 482, 387, 497 ], "lines": [ { "bbox": [ 223, 482, 387, 497 ], "spans": [ { "bbox": [ 223, 482, 387, 497 ], "score": 0.9, "content": "\\parallel \\pi ^ { ( 1 ) } - \\pi ^ { ( 2 ) } \\parallel _ { \\infty } \\leq L \\parallel r ^ { ( 1 ) } - r ^ { ( 2 ) } \\parallel _ { \\infty } .", "type": "interline_equation", "image_path": "2efed660ac2d7e13b9af72230a0af68e84c0f7f3615ab215ccbfc754084cafb5.jpg" } ] } ], "index": 19, "virtual_lines": [ { "bbox": [ 223, 482, 387, 497 ], "spans": [], "index": 19 } ] }, { "type": "text", "bbox": [ 106, 500, 220, 511 ], "lines": [ { "bbox": [ 106, 500, 220, 513 ], "spans": [ { "bbox": [ 106, 500, 220, 513 ], "score": 1.0, "content": "So that we have for MENTS", "type": "text" } ], "index": 20 } ], "index": 20, "bbox_fs": [ 106, 500, 220, 513 ] }, { "type": "interline_equation", "bbox": [ 132, 514, 478, 545 ], "lines": [ { "bbox": [ 132, 514, 478, 545 ], "spans": [ { "bbox": [ 132, 514, 478, 545 ], "score": 0.92, "content": "n V ^ { * } - n \\sum _ { i } V _ { i } \\Big ( \\kappa _ { i } + L \\big ( \\frac { \\tau \\log | A | } { 1 - \\gamma } \\big ) \\Big ) \\leq R _ { n } \\leq n V ^ { * } - n \\sum _ { i } V _ { i } \\Big ( \\chi _ { i } - L \\big ( \\frac { \\tau \\log | A | } { 1 - \\gamma } \\big ) \\Big ) .", "type": "interline_equation", "image_path": "231adc8dfe6afc7cad125769c3ff5a118f083e161db2ee46ade231908f37ca4f.jpg" } ] } ], "index": 22, "virtual_lines": [ { "bbox": [ 132, 514, 478, 524.3333333333334 ], "spans": [], "index": 21 }, { "bbox": [ 132, 524.3333333333334, 478, 534.6666666666667 ], "spans": [], "index": 22 }, { "bbox": [ 132, 534.6666666666667, 478, 545.0000000000001 ], "spans": [], "index": 23 } ] }, { "type": "text", "bbox": [ 107, 552, 192, 564 ], "lines": [ { "bbox": [ 105, 551, 193, 565 ], "spans": [ { "bbox": [ 105, 551, 193, 565 ], "score": 1.0, "content": "For RENTS, we have", "type": "text" } ], "index": 24 } ], "index": 24, "bbox_fs": [ 105, 551, 193, 565 ] }, { "type": "interline_equation", "bbox": [ 111, 566, 505, 598 ], "lines": [ { "bbox": [ 111, 566, 505, 598 ], "spans": [ { "bbox": [ 111, 566, 505, 598 ], "score": 0.92, "content": "\\imath V ^ { * } - n \\sum _ { i } V _ { i } \\Big ( \\kappa _ { i } + L \\big ( \\frac { \\tau ( \\log | A | - \\frac { 1 } { m } ) } { 1 - \\gamma } \\big ) \\Big ) \\leq R _ { n } \\leq n V ^ { * } - n \\sum _ { i } V _ { i } \\Big ( \\chi _ { i } - L \\big ( \\frac { \\tau ( \\log | A | - \\frac { 1 } { m } ) } { 1 - \\gamma } \\big ) \\Big )", "type": "interline_equation", "image_path": "d4fd869a735dd27c880e7b0fa501e80127e2b752e3879689307993b1c44cf911.jpg" } ] } ], "index": 26, "virtual_lines": [ { "bbox": [ 111, 566, 505, 576.6666666666666 ], "spans": [], "index": 25 }, { "bbox": [ 111, 576.6666666666666, 505, 587.3333333333333 ], "spans": [], "index": 26 }, { "bbox": [ 111, 587.3333333333333, 505, 597.9999999999999 ], "spans": [], "index": 27 } ] }, { "type": "text", "bbox": [ 106, 601, 208, 612 ], "lines": [ { "bbox": [ 106, 599, 205, 614 ], "spans": [ { "bbox": [ 106, 599, 133, 614 ], "score": 1.0, "content": "where", "type": "text" }, { "bbox": [ 133, 601, 205, 613 ], "score": 0.92, "content": "\\begin{array} { r } { m = \\operatorname* { m i n } _ { a } \\pi ( a | s ) } \\end{array}", "type": "inline_equation" } ], "index": 28 } ], "index": 28, "bbox_fs": [ 106, 599, 205, 614 ] }, { "type": "text", "bbox": [ 108, 612, 384, 624 ], "lines": [ { "bbox": [ 106, 612, 384, 624 ], "spans": [ { "bbox": [ 106, 612, 210, 624 ], "score": 1.0, "content": "In case of Tsallis Entropy", "type": "text" }, { "bbox": [ 211, 613, 236, 624 ], "score": 0.89, "content": "p = 2", "type": "inline_equation" }, { "bbox": [ 236, 612, 384, 624 ], "score": 1.0, "content": "( Niculae & Blondel (2017)), so that", "type": "text" } ], "index": 29 } ], "index": 29, "bbox_fs": [ 106, 612, 384, 624 ] }, { "type": "interline_equation", "bbox": [ 111, 626, 500, 657 ], "lines": [ { "bbox": [ 111, 626, 500, 657 ], "spans": [ { "bbox": [ 111, 626, 500, 657 ], "score": 0.93, "content": "n V ^ { * } - n \\sum _ { i } V _ { i } \\Big ( \\kappa _ { i } + \\frac { L } { 2 } \\big ( \\frac { | A | - 1 } { 2 | A | } \\frac { \\tau } { 1 - \\gamma } \\big ) \\Big ) \\leq R _ { n } \\leq n V ^ { * } - n \\sum _ { i } V _ { i } \\Big ( \\chi _ { i } - \\frac { L } { 2 } \\big ( \\frac { | A | - 1 } { 2 | A | } \\frac { \\tau } { 1 - \\gamma } \\big ) \\Big )", "type": "interline_equation", "image_path": "b1190e1157284e901e0760831cabe7555d7f6cf86fc5ce242e261553f1bd9a0b.jpg" } ] } ], "index": 31, "virtual_lines": [ { "bbox": [ 111, 626, 500, 636.3333333333334 ], "spans": [], "index": 30 }, { "bbox": [ 111, 636.3333333333334, 500, 646.6666666666667 ], "spans": [], "index": 31 }, { "bbox": [ 111, 646.6666666666667, 500, 657.0000000000001 ], "spans": [], "index": 32 } ] }, { "type": "text", "bbox": [ 106, 664, 435, 677 ], "lines": [ { "bbox": [ 106, 664, 434, 677 ], "spans": [ { "bbox": [ 106, 664, 434, 677 ], "score": 1.0, "content": "Before derive the next theorem, we state here the Theorem 2 in Geist et al. 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