diff --git "a/parse/train/H1cKvl-Rb/H1cKvl-Rb_middle.json" "b/parse/train/H1cKvl-Rb/H1cKvl-Rb_middle.json" new file mode 100644--- /dev/null +++ "b/parse/train/H1cKvl-Rb/H1cKvl-Rb_middle.json" @@ -0,0 +1,38464 @@ +{ + "pdf_info": [ + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 107, + 78, + 403, + 97 + ], + "lines": [ + { + "bbox": [ + 105, + 78, + 405, + 99 + ], + "spans": [ + { + "bbox": [ + 105, + 78, + 290, + 99 + ], + "score": 1.0, + "content": "UCB EXPLORATION VIA", + "type": "text" + }, + { + "bbox": [ + 290, + 79, + 307, + 98 + ], + "score": 0.73, + "content": "Q", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 78, + 405, + 99 + ], + "score": 1.0, + "content": "-ENSEMBLES", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 112, + 119, + 244, + 141 + ], + "lines": [ + { + "bbox": [ + 111, + 117, + 202, + 132 + ], + "spans": [ + { + "bbox": [ + 111, + 117, + 202, + 132 + ], + "score": 1.0, + "content": "Anonymous authors", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 111, + 129, + 245, + 141 + ], + "spans": [ + { + "bbox": [ + 111, + 129, + 245, + 141 + ], + "score": 1.0, + "content": "Paper under double-blind review", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1.5 + }, + { + "type": "title", + "bbox": [ + 278, + 169, + 333, + 182 + ], + "lines": [ + { + "bbox": [ + 277, + 169, + 335, + 183 + ], + "spans": [ + { + "bbox": [ + 277, + 169, + 335, + 183 + ], + "score": 1.0, + "content": "ABSTRACT", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 142, + 194, + 468, + 250 + ], + "lines": [ + { + "bbox": [ + 142, + 194, + 469, + 207 + ], + "spans": [ + { + "bbox": [ + 142, + 194, + 266, + 207 + ], + "score": 1.0, + "content": "We show how an ensemble of", + "type": "text" + }, + { + "bbox": [ + 267, + 195, + 280, + 207 + ], + "score": 0.9, + "content": "Q ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 194, + 469, + 207 + ], + "score": 1.0, + "content": "-functions can be leveraged for more effective", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 141, + 205, + 469, + 218 + ], + "spans": [ + { + "bbox": [ + 141, + 205, + 469, + 218 + ], + "score": 1.0, + "content": "exploration in deep reinforcement learning. 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We demonstrate that our algorithms significantly improve performance on the Atari", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 582, + 156, + 593 + ], + "spans": [ + { + "bbox": [ + 106, + 582, + 156, + 593 + ], + "score": 1.0, + "content": "benchmark.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 31.5 + }, + { + "type": "title", + "bbox": [ + 108, + 609, + 200, + 622 + ], + "lines": [ + { + "bbox": [ + 104, + 608, + 201, + 624 + ], + "spans": [ + { + "bbox": [ + 104, + 608, + 201, + 624 + ], + "score": 1.0, + "content": "2 BACKGROUND", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36 + }, + { + "type": "title", + "bbox": [ + 107, + 633, + 176, + 645 + ], + "lines": [ + { + "bbox": [ + 105, + 633, + 178, + 647 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 178, + 647 + ], + "score": 1.0, + "content": "2.1 NOTATION", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 106, + 654, + 505, + 734 + ], + "lines": [ + { + "bbox": [ + 105, + 654, + 506, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 506, + 667 + ], + "score": 1.0, + "content": "We model reinforcement learning as a Markov decision process (MDP). We define an MDP as", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 665, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 180, + 677 + ], + "score": 0.9, + "content": "( S , A , T , R , p _ { 0 } , \\gamma )", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 665, + 298, + 678 + ], + "score": 1.0, + "content": ", in which both the state space", + "type": "text" + }, + { + "bbox": [ + 298, + 666, + 306, + 676 + ], + "score": 0.83, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 665, + 373, + 678 + ], + "score": 1.0, + "content": "and action space", + "type": "text" + }, + { + "bbox": [ + 374, + 667, + 382, + 676 + ], + "score": 0.83, + "content": "\\mathcal { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 665, + 433, + 678 + ], + "score": 1.0, + "content": "are discrete,", + "type": "text" + }, + { + "bbox": [ + 433, + 666, + 505, + 676 + ], + "score": 0.89, + "content": "T : S \\times \\mathcal { A } \\times \\mathcal { S } \\mapsto", + "type": "inline_equation" + } + ], + "index": 39 + }, + { + "bbox": [ + 107, + 676, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 107, + 678, + 122, + 689 + ], + "score": 0.89, + "content": "\\mathbb { R } _ { + }", + "type": "inline_equation" + }, + { + "bbox": [ + 122, + 676, + 235, + 689 + ], + "score": 1.0, + "content": "is the transition distribution,", + "type": "text" + }, + { + "bbox": [ + 235, + 677, + 301, + 687 + ], + "score": 0.9, + "content": "R : S \\times \\mathcal { A } \\mapsto \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 301, + 676, + 505, + 689 + ], + "score": 1.0, + "content": "is the reward function, assumed deterministic given", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 208, + 699 + ], + "score": 1.0, + "content": "the state and action, and", + "type": "text" + }, + { + "bbox": [ + 208, + 688, + 249, + 700 + ], + "score": 0.92, + "content": "\\gamma \\in ( 0 , 1 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 688, + 351, + 699 + ], + "score": 1.0, + "content": "is a discount factor, and", + "type": "text" + }, + { + "bbox": [ + 351, + 690, + 362, + 699 + ], + "score": 0.8, + "content": "p _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 688, + 505, + 699 + ], + "score": 1.0, + "content": "is the initial state distribution. We", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 697, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 697, + 241, + 712 + ], + "score": 1.0, + "content": "denote a transition experience as", + "type": "text" + }, + { + "bbox": [ + 242, + 699, + 304, + 711 + ], + "score": 0.92, + "content": "\\tau = ( s , a , r , s ^ { \\prime } )", + "type": "inline_equation" + }, + { + "bbox": [ + 305, + 697, + 334, + 712 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 334, + 699, + 395, + 711 + ], + "score": 0.92, + "content": "s ^ { \\prime } \\sim T ( s ^ { \\prime } | s , a )", + "type": "inline_equation" + }, + { + "bbox": [ + 395, + 697, + 414, + 712 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 414, + 699, + 463, + 711 + ], + "score": 0.92, + "content": "r = R ( s , a )", + "type": "inline_equation" + }, + { + "bbox": [ + 463, + 697, + 505, + 712 + ], + "score": 1.0, + "content": ". 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We build on well established algorithms", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 141, + 216, + 469, + 230 + ], + "spans": [ + { + "bbox": [ + 141, + 216, + 327, + 230 + ], + "score": 1.0, + "content": "from the bandit setting, and adapt them to the", + "type": "text" + }, + { + "bbox": [ + 328, + 217, + 337, + 228 + ], + "score": 0.86, + "content": "Q", + "type": "inline_equation" + }, + { + "bbox": [ + 337, + 216, + 469, + 230 + ], + "score": 1.0, + "content": "-learning setting. We propose an", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 141, + 227, + 470, + 241 + ], + "spans": [ + { + "bbox": [ + 141, + 227, + 470, + 241 + ], + "score": 1.0, + "content": "exploration strategy based on upper-confidence bounds (UCB). Our experiments", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 141, + 239, + 333, + 250 + ], + "spans": [ + { + "bbox": [ + 141, + 239, + 333, + 250 + ], + "score": 1.0, + "content": "show significant gains on the Atari benchmark.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 6, + "bbox_fs": [ + 141, + 194, + 470, + 250 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 271, + 206, + 283 + ], + "lines": [ + { + "bbox": [ + 105, + 270, + 208, + 286 + ], + "spans": [ + { + "bbox": [ + 105, + 270, + 208, + 286 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 107, + 295, + 505, + 372 + ], + "lines": [ + { + "bbox": [ + 105, + 295, + 507, + 309 + ], + "spans": [ + { + "bbox": [ + 105, + 295, + 507, + 309 + ], + "score": 1.0, + "content": "Deep reinforcement learning seeks to learn mappings from high-dimensional observations to actions.", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 307, + 505, + 320 + ], + "spans": [ + { + "bbox": [ + 105, + 307, + 129, + 320 + ], + "score": 1.0, + "content": "Deep", + "type": "text" + }, + { + "bbox": [ + 129, + 307, + 138, + 318 + ], + "score": 0.82, + "content": "Q", + "type": "inline_equation" + }, + { + "bbox": [ + 139, + 307, + 505, + 320 + ], + "score": 1.0, + "content": "-learning (Mnih et al. (2015)) is a leading technique that has been used successfully, especially", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 317, + 505, + 332 + ], + "spans": [ + { + "bbox": [ + 105, + 317, + 505, + 332 + ], + "score": 1.0, + "content": "for video game benchmarks. However, fundamental challenges remain, for example, improving", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 329, + 505, + 342 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 505, + 342 + ], + "score": 1.0, + "content": "sample efficiency and ensuring convergence to high quality solutions. 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However these provably optimal exploration techniques do not extend to deep RL in a", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 361, + 191, + 375 + ], + "spans": [ + { + "bbox": [ + 105, + 361, + 191, + 375 + ], + "score": 1.0, + "content": "straightforward way.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 13, + "bbox_fs": [ + 105, + 295, + 507, + 375 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 379, + 505, + 500 + ], + "lines": [ + { + "bbox": [ + 105, + 379, + 505, + 391 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 505, + 391 + ], + "score": 1.0, + "content": "Bootstrapped DQN (Osband et al. (2016)) is a previous attempt at adapting a theoretically verified", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 389, + 506, + 403 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 506, + 403 + ], + "score": 1.0, + "content": "approach to deep RL. In particular, it draws inspiration from posterior sampling for reinforcement", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 399, + 506, + 414 + ], + "spans": [ + { + "bbox": [ + 105, + 399, + 506, + 414 + ], + "score": 1.0, + "content": "learning (PSRL, Osband et al. (2013); Osband and Van Roy (2016)), which has near-optimal regret", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 411, + 506, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 411, + 442, + 425 + ], + "score": 1.0, + "content": "bounds. PSRL samples an MDP from its posterior each episode and exactly solves", + "type": "text" + }, + { + "bbox": [ + 442, + 412, + 456, + 423 + ], + "score": 0.89, + "content": "Q ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 456, + 411, + 506, + 425 + ], + "score": 1.0, + "content": ", its optimal", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 107, + 423, + 505, + 435 + ], + "spans": [ + { + "bbox": [ + 107, + 423, + 115, + 434 + ], + "score": 0.82, + "content": "Q", + "type": "inline_equation" + }, + { + "bbox": [ + 116, + 423, + 505, + 435 + ], + "score": 1.0, + "content": "-function. However, in high-dimensional settings, both approximating the posterior over MDPs", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 433, + 506, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 506, + 447 + ], + "score": 1.0, + "content": "and solving the sampled MDP are intractable. 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In addition,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 455, + 505, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 404, + 469 + ], + "score": 1.0, + "content": "bootstrapped DQN uses a multi-headed neural network to represent the", + "type": "text" + }, + { + "bbox": [ + 405, + 456, + 414, + 467 + ], + "score": 0.85, + "content": "Q", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 455, + 505, + 469 + ], + "score": 1.0, + "content": "-ensemble. While the", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 466, + 505, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 505, + 480 + ], + "score": 1.0, + "content": "authors proposed bootstrapping to estimate the posterior distribution, their empirical findings show", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 477, + 505, + 489 + ], + "spans": [ + { + "bbox": [ + 106, + 477, + 505, + 489 + ], + "score": 1.0, + "content": "best performance is attained by simply relying on different initializations for the different heads, not", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 487, + 447, + 503 + ], + "spans": [ + { + "bbox": [ + 105, + 487, + 447, + 503 + ], + "score": 1.0, + "content": "requiring the sampling-with-replacement process that is prescribed by bootstrapping.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 22, + "bbox_fs": [ + 105, + 379, + 506, + 503 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 505, + 505, + 593 + ], + "lines": [ + { + "bbox": [ + 106, + 505, + 506, + 517 + ], + "spans": [ + { + "bbox": [ + 106, + 505, + 338, + 517 + ], + "score": 1.0, + "content": "In this paper, we design new algorithms that build on the", + "type": "text" + }, + { + "bbox": [ + 338, + 506, + 347, + 516 + ], + "score": 0.85, + "content": "Q", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 505, + 506, + 517 + ], + "score": 1.0, + "content": "-ensemble approach from Osband et al.", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 515, + 505, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 515, + 505, + 529 + ], + "score": 1.0, + "content": "(2016). However, instead of using posterior sampling for exploration, we construct uncertainty", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 527, + 505, + 540 + ], + "spans": [ + { + "bbox": [ + 106, + 527, + 182, + 540 + ], + "score": 1.0, + "content": "estimates from the", + "type": "text" + }, + { + "bbox": [ + 182, + 527, + 191, + 538 + ], + "score": 0.86, + "content": "Q", + "type": "inline_equation" + }, + { + "bbox": [ + 191, + 527, + 505, + 540 + ], + "score": 1.0, + "content": "-ensemble. Specifically, we first propose the Ensemble Voting algorithm where", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 538, + 504, + 550 + ], + "spans": [ + { + "bbox": [ + 106, + 538, + 299, + 550 + ], + "score": 1.0, + "content": "the agent takes action by a majority vote from the", + "type": "text" + }, + { + "bbox": [ + 299, + 538, + 308, + 549 + ], + "score": 0.86, + "content": "Q", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 538, + 504, + 550 + ], + "score": 1.0, + "content": "-ensemble. Next, we propose the UCB exploration", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 549, + 505, + 562 + ], + "spans": [ + { + "bbox": [ + 105, + 549, + 505, + 562 + ], + "score": 1.0, + "content": "strategy. This strategy is inspired by established UCB algorithms in the bandit setting and constructs", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 560, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 560, + 217, + 573 + ], + "score": 1.0, + "content": "uncertainty estimates of the", + "type": "text" + }, + { + "bbox": [ + 217, + 560, + 226, + 571 + ], + "score": 0.85, + "content": "Q", + "type": "inline_equation" + }, + { + "bbox": [ + 227, + 560, + 505, + 573 + ], + "score": 1.0, + "content": "-values. In this strategy, agents are optimistic and take actions with the", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 104, + 570, + 506, + 584 + ], + "spans": [ + { + "bbox": [ + 104, + 570, + 506, + 584 + ], + "score": 1.0, + "content": "highest UCB. We demonstrate that our algorithms significantly improve performance on the Atari", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 582, + 156, + 593 + ], + "spans": [ + { + "bbox": [ + 106, + 582, + 156, + 593 + ], + "score": 1.0, + "content": "benchmark.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 31.5, + "bbox_fs": [ + 104, + 505, + 506, + 593 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 609, + 200, + 622 + ], + "lines": [ + { + "bbox": [ + 104, + 608, + 201, + 624 + ], + "spans": [ + { + "bbox": [ + 104, + 608, + 201, + 624 + ], + "score": 1.0, + "content": "2 BACKGROUND", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36 + }, + { + "type": "title", + "bbox": [ + 107, + 633, + 176, + 645 + ], + "lines": [ + { + "bbox": [ + 105, + 633, + 178, + 647 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 178, + 647 + ], + "score": 1.0, + "content": "2.1 NOTATION", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 106, + 654, + 505, + 734 + ], + "lines": [ + { + "bbox": [ + 105, + 654, + 506, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 506, + 667 + ], + "score": 1.0, + "content": "We model reinforcement learning as a Markov decision process (MDP). We define an MDP as", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 665, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 666, + 180, + 677 + ], + "score": 0.9, + "content": "( S , A , T , R , p _ { 0 } , \\gamma )", + "type": "inline_equation" + }, + { + "bbox": [ + 181, + 665, + 298, + 678 + ], + "score": 1.0, + "content": ", in which both the state space", + "type": "text" + }, + { + "bbox": [ + 298, + 666, + 306, + 676 + ], + "score": 0.83, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 306, + 665, + 373, + 678 + ], + "score": 1.0, + "content": "and action space", + "type": "text" + }, + { + "bbox": [ + 374, + 667, + 382, + 676 + ], + "score": 0.83, + "content": "\\mathcal { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 665, + 433, + 678 + ], + "score": 1.0, + "content": "are discrete,", + "type": "text" + }, + { + "bbox": [ + 433, + 666, + 505, + 676 + ], + "score": 0.89, + "content": "T : S \\times \\mathcal { A } \\times \\mathcal { S } \\mapsto", + "type": "inline_equation" + } + ], + "index": 39 + }, + { + "bbox": [ + 107, + 676, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 107, + 678, + 122, + 689 + ], + "score": 0.89, + "content": "\\mathbb { R } _ { + }", + "type": "inline_equation" + }, + { + "bbox": [ + 122, + 676, + 235, + 689 + ], + "score": 1.0, + "content": "is the transition distribution,", + "type": "text" + }, + { + "bbox": [ + 235, + 677, + 301, + 687 + ], + "score": 0.9, + "content": "R : S \\times \\mathcal { A } \\mapsto \\mathbb { R }", + "type": "inline_equation" + }, + { + "bbox": [ + 301, + 676, + 505, + 689 + ], + "score": 1.0, + "content": "is the reward function, assumed deterministic given", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 208, + 699 + ], + "score": 1.0, + "content": "the state and action, and", + "type": "text" + }, + { + "bbox": [ + 208, + 688, + 249, + 700 + ], + "score": 0.92, + "content": "\\gamma \\in ( 0 , 1 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 688, + 351, + 699 + ], + "score": 1.0, + "content": "is a discount factor, and", + "type": "text" + }, + { + "bbox": [ + 351, + 690, + 362, + 699 + ], + "score": 0.8, + "content": "p _ { 0 }", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 688, + 505, + 699 + ], + "score": 1.0, + "content": "is the initial state distribution. 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Later work developed reinforcement learning", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 236, + 506, + 248 + ], + "spans": [ + { + "bbox": [ + 106, + 236, + 506, + 248 + ], + "score": 1.0, + "content": "algorithms with provably fast (polynomial-time) convergence (Kearns and Singh (2002); Brafman", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 246, + 505, + 259 + ], + "spans": [ + { + "bbox": [ + 106, + 246, + 505, + 259 + ], + "score": 1.0, + "content": "and Tennenholtz (2002); Strehl et al. (2006)). At the core of these provably-optimal learning methods", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 258, + 506, + 270 + ], + "spans": [ + { + "bbox": [ + 106, + 258, + 506, + 270 + ], + "score": 1.0, + "content": "is some exploration strategy, which actively encourages the agent to visit novel state-action pairs. For", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 268, + 506, + 281 + ], + "spans": [ + { + "bbox": [ + 105, + 268, + 506, + 281 + ], + "score": 1.0, + "content": "example, R-MAX optimistically assumes that infrequently-visited states provide maximal reward,", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 280, + 506, + 292 + ], + "spans": [ + { + "bbox": [ + 106, + 280, + 156, + 292 + ], + "score": 1.0, + "content": "and delayed", + "type": "text" + }, + { + "bbox": [ + 156, + 280, + 165, + 291 + ], + "score": 0.84, + "content": "Q", + "type": "inline_equation" + }, + { + "bbox": [ + 166, + 280, + 258, + 292 + ], + "score": 1.0, + "content": "-learning initializes the", + "type": "text" + }, + { + "bbox": [ + 258, + 280, + 267, + 291 + ], + "score": 0.85, + "content": "Q", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 280, + 506, + 292 + ], + "score": 1.0, + "content": "-function with high values to ensure that each state-action is", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 291, + 290, + 302 + ], + "spans": [ + { + "bbox": [ + 106, + 291, + 290, + 302 + ], + "score": 1.0, + "content": "chosen enough times to drive the value down.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 107, + 307, + 505, + 427 + ], + "lines": [ + { + "bbox": [ + 105, + 306, + 507, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 306, + 507, + 321 + ], + "score": 1.0, + "content": "Since the theoretically sound RL algorithms are not computationally practical in the deep RL setting,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 318, + 505, + 331 + ], + "spans": [ + { + "bbox": [ + 106, + 318, + 402, + 331 + ], + "score": 1.0, + "content": "deep RL implementations often use simple exploration methods such as", + "type": "text" + }, + { + "bbox": [ + 403, + 320, + 408, + 328 + ], + "score": 0.54, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 318, + 505, + 331 + ], + "score": 1.0, + "content": "-greedy and Boltzmann", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 329, + 505, + 342 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 505, + 342 + ], + "score": 1.0, + "content": "exploration, which are often sample-inefficient and fail to find good policies. One common approach", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 339, + 506, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 339, + 506, + 354 + ], + "score": 1.0, + "content": "of exploration in deep RL is to construct an exploration bonus, which adds a reward for visiting", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 351, + 506, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 506, + 363 + ], + "score": 1.0, + "content": "state-action pairs that are deemed to be novel or informative. In particular, several prior methods", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 362, + 506, + 374 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 506, + 374 + ], + "score": 1.0, + "content": "define an exploration bonus based on a density model or dynamics model. Examples include VIME", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 373, + 505, + 385 + ], + "spans": [ + { + "bbox": [ + 105, + 373, + 505, + 385 + ], + "score": 1.0, + "content": "by Houthooft et al. (2016), which uses variational inference on the forward-dynamics model, and", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 384, + 506, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 506, + 397 + ], + "score": 1.0, + "content": "Tang et al. (2016), Bellemare et al. (2016), Ostrovski et al. (2017), Fu et al. (2017). While these", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 395, + 505, + 407 + ], + "spans": [ + { + "bbox": [ + 105, + 395, + 505, + 407 + ], + "score": 1.0, + "content": "methods yield successful exploration in some problems, a major drawback is that this exploration", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 406, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 505, + 419 + ], + "score": 1.0, + "content": "bonus does not depend on the rewards, so the exploration may focus on irrelevant aspects of the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 417, + 285, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 285, + 428 + ], + "score": 1.0, + "content": "environment, which are unrelated to reward.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 21 + }, + { + "type": "title", + "bbox": [ + 108, + 441, + 299, + 452 + ], + "lines": [ + { + "bbox": [ + 105, + 441, + 300, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 300, + 453 + ], + "score": 1.0, + "content": "2.3 BAYESIAN REINFORCEMENT LEARNING", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 107, + 461, + 505, + 604 + ], + "lines": [ + { + "bbox": [ + 106, + 462, + 505, + 474 + ], + "spans": [ + { + "bbox": [ + 106, + 462, + 505, + 474 + ], + "score": 1.0, + "content": "Earlier works on Bayesian reinforcement learning include Dearden et al. (1998; 1999). Dearden", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 472, + 506, + 485 + ], + "spans": [ + { + "bbox": [ + 106, + 472, + 225, + 485 + ], + "score": 1.0, + "content": "et al. (1998) studied Bayesian", + "type": "text" + }, + { + "bbox": [ + 225, + 473, + 234, + 484 + ], + "score": 0.85, + "content": "Q", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 472, + 488, + 485 + ], + "score": 1.0, + "content": "-learning in the model-free setting and learned the distribution of", + "type": "text" + }, + { + "bbox": [ + 489, + 473, + 502, + 484 + ], + "score": 0.88, + "content": "Q ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 472, + 506, + 485 + ], + "score": 1.0, + "content": "-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 482, + 506, + 498 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 506, + 498 + ], + "score": 1.0, + "content": "values through Bayesian updates. 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The algorithm samples", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 516, + 505, + 529 + ], + "spans": [ + { + "bbox": [ + 106, + 516, + 334, + 529 + ], + "score": 1.0, + "content": "from the MDP posterior multiple times and solving the", + "type": "text" + }, + { + "bbox": [ + 334, + 516, + 348, + 528 + ], + "score": 0.89, + "content": "Q ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 516, + 505, + 529 + ], + "score": 1.0, + "content": "values at every step. This approach is", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 527, + 506, + 540 + ], + "spans": [ + { + "bbox": [ + 106, + 527, + 506, + 540 + ], + "score": 1.0, + "content": "only feasible for RL problems with very small state space and action space. 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Later work developed reinforcement learning", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 236, + 506, + 248 + ], + "spans": [ + { + "bbox": [ + 106, + 236, + 506, + 248 + ], + "score": 1.0, + "content": "algorithms with provably fast (polynomial-time) convergence (Kearns and Singh (2002); Brafman", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 246, + 505, + 259 + ], + "spans": [ + { + "bbox": [ + 106, + 246, + 505, + 259 + ], + "score": 1.0, + "content": "and Tennenholtz (2002); Strehl et al. (2006)). At the core of these provably-optimal learning methods", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 258, + 506, + 270 + ], + "spans": [ + { + "bbox": [ + 106, + 258, + 506, + 270 + ], + "score": 1.0, + "content": "is some exploration strategy, which actively encourages the agent to visit novel state-action pairs. For", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 268, + 506, + 281 + ], + "spans": [ + { + "bbox": [ + 105, + 268, + 506, + 281 + ], + "score": 1.0, + "content": "example, R-MAX optimistically assumes that infrequently-visited states provide maximal reward,", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 280, + 506, + 292 + ], + "spans": [ + { + "bbox": [ + 106, + 280, + 156, + 292 + ], + "score": 1.0, + "content": "and delayed", + "type": "text" + }, + { + "bbox": [ + 156, + 280, + 165, + 291 + ], + "score": 0.84, + "content": "Q", + "type": "inline_equation" + }, + { + "bbox": [ + 166, + 280, + 258, + 292 + ], + "score": 1.0, + "content": "-learning initializes the", + "type": "text" + }, + { + "bbox": [ + 258, + 280, + 267, + 291 + ], + "score": 0.85, + "content": "Q", + "type": "inline_equation" + }, + { + "bbox": [ + 267, + 280, + 506, + 292 + ], + "score": 1.0, + "content": "-function with high values to ensure that each state-action is", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 291, + 290, + 302 + ], + "spans": [ + { + "bbox": [ + 106, + 291, + 290, + 302 + ], + "score": 1.0, + "content": "chosen enough times to drive the value down.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 10, + "bbox_fs": [ + 105, + 180, + 507, + 302 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 307, + 505, + 427 + ], + "lines": [ + { + "bbox": [ + 105, + 306, + 507, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 306, + 507, + 321 + ], + "score": 1.0, + "content": "Since the theoretically sound RL algorithms are not computationally practical in the deep RL setting,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 318, + 505, + 331 + ], + "spans": [ + { + "bbox": [ + 106, + 318, + 402, + 331 + ], + "score": 1.0, + "content": "deep RL implementations often use simple exploration methods such as", + "type": "text" + }, + { + "bbox": [ + 403, + 320, + 408, + 328 + ], + "score": 0.54, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 318, + 505, + 331 + ], + "score": 1.0, + "content": "-greedy and Boltzmann", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 329, + 505, + 342 + ], + "spans": [ + { + "bbox": [ + 105, + 329, + 505, + 342 + ], + "score": 1.0, + "content": "exploration, which are often sample-inefficient and fail to find good policies. One common approach", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 339, + 506, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 339, + 506, + 354 + ], + "score": 1.0, + "content": "of exploration in deep RL is to construct an exploration bonus, which adds a reward for visiting", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 351, + 506, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 506, + 363 + ], + "score": 1.0, + "content": "state-action pairs that are deemed to be novel or informative. In particular, several prior methods", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 362, + 506, + 374 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 506, + 374 + ], + "score": 1.0, + "content": "define an exploration bonus based on a density model or dynamics model. Examples include VIME", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 373, + 505, + 385 + ], + "spans": [ + { + "bbox": [ + 105, + 373, + 505, + 385 + ], + "score": 1.0, + "content": "by Houthooft et al. (2016), which uses variational inference on the forward-dynamics model, and", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 384, + 506, + 397 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 506, + 397 + ], + "score": 1.0, + "content": "Tang et al. (2016), Bellemare et al. (2016), Ostrovski et al. (2017), Fu et al. (2017). While these", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 395, + 505, + 407 + ], + "spans": [ + { + "bbox": [ + 105, + 395, + 505, + 407 + ], + "score": 1.0, + "content": "methods yield successful exploration in some problems, a major drawback is that this exploration", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 406, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 505, + 419 + ], + "score": 1.0, + "content": "bonus does not depend on the rewards, so the exploration may focus on irrelevant aspects of the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 417, + 285, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 285, + 428 + ], + "score": 1.0, + "content": "environment, which are unrelated to reward.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 21, + "bbox_fs": [ + 105, + 306, + 507, + 428 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 441, + 299, + 452 + ], + "lines": [ + { + "bbox": [ + 105, + 441, + 300, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 300, + 453 + ], + "score": 1.0, + "content": "2.3 BAYESIAN REINFORCEMENT LEARNING", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 107, + 461, + 505, + 604 + ], + "lines": [ + { + "bbox": [ + 106, + 462, + 505, + 474 + ], + "spans": [ + { + "bbox": [ + 106, + 462, + 505, + 474 + ], + "score": 1.0, + "content": "Earlier works on Bayesian reinforcement learning include Dearden et al. 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This", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 594, + 505, + 607 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 505, + 607 + ], + "score": 1.0, + "content": "phenomenon that that bagging deteriorates the performance of deep ensembles is also observed in", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 604, + 505, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 505, + 619 + ], + "score": 1.0, + "content": "supervised learning settings. Lee et al. (2015) observed that supervised learning trained with deep", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 616, + 506, + 629 + ], + "spans": [ + { + "bbox": [ + 106, + 616, + 506, + 629 + ], + "score": 1.0, + "content": "ensembles with random initializations perform better than bagging for deep ensembles. Lakshmi-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 104, + 626, + 506, + 641 + ], + "spans": [ + { + "bbox": [ + 104, + 626, + 506, + 641 + ], + "score": 1.0, + "content": "narayanan et al. (2016) used deep ensembles for uncertainty estimates and also observed that bagging", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 638, + 294, + 651 + ], + "spans": [ + { + "bbox": [ + 105, + 638, + 294, + 651 + ], + "score": 1.0, + "content": "deteriorated performance in their experiments.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 106, + 654, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 655, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 505, + 668 + ], + "score": 1.0, + "content": "Lu and Van Roy (2017) develop ensemble sampling for bandit problems with deep neural network", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 666, + 504, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 490, + 678 + ], + "score": 1.0, + "content": "parametrized policies and the theoretical justification. 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This", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 594, + 505, + 607 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 505, + 607 + ], + "score": 1.0, + "content": "phenomenon that that bagging deteriorates the performance of deep ensembles is also observed in", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 604, + 505, + 619 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 505, + 619 + ], + "score": 1.0, + "content": "supervised learning settings. Lee et al. (2015) observed that supervised learning trained with deep", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 616, + 506, + 629 + ], + "spans": [ + { + "bbox": [ + 106, + 616, + 506, + 629 + ], + "score": 1.0, + "content": "ensembles with random initializations perform better than bagging for deep ensembles. Lakshmi-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 104, + 626, + 506, + 641 + ], + "spans": [ + { + "bbox": [ + 104, + 626, + 506, + 641 + ], + "score": 1.0, + "content": "narayanan et al. 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(2016)?", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 107, + 172, + 505, + 250 + ], + "lines": [ + { + "bbox": [ + 106, + 173, + 506, + 184 + ], + "spans": [ + { + "bbox": [ + 106, + 173, + 506, + 184 + ], + "score": 1.0, + "content": "We evaluate the algorithms on each Atari game of the Arcade Learning Environment (Bellemare et al.", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 182, + 506, + 196 + ], + "spans": [ + { + "bbox": [ + 105, + 182, + 506, + 196 + ], + "score": 1.0, + "content": "(2013)). We use the multi-head neural net architecture of Osband et al. (2016). We fix the common", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 194, + 505, + 206 + ], + "spans": [ + { + "bbox": [ + 105, + 194, + 505, + 206 + ], + "score": 1.0, + "content": "hyperparameters of all algorithms based on a well-tuned double DQN implementation, which uses the", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 205, + 505, + 218 + ], + "spans": [ + { + "bbox": [ + 106, + 205, + 505, + 218 + ], + "score": 1.0, + "content": "Adam optimizer (Kingma and Ba (2014)), different learning rate and exploration schedules compared", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 216, + 505, + 229 + ], + "spans": [ + { + "bbox": [ + 106, + 217, + 431, + 229 + ], + "score": 1.0, + "content": "to Mnih et al. (2015). Appendix A tabulates the hyperparameters. The number of", + "type": "text" + }, + { + "bbox": [ + 432, + 216, + 455, + 228 + ], + "score": 0.93, + "content": "\\left\\{ Q _ { k } \\right\\}", + "type": "inline_equation" + }, + { + "bbox": [ + 455, + 217, + 505, + 229 + ], + "score": 1.0, + "content": "functions is", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 107, + 226, + 506, + 240 + ], + "spans": [ + { + "bbox": [ + 107, + 227, + 141, + 237 + ], + "score": 0.89, + "content": "K = 1 0", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 226, + 506, + 240 + ], + "score": 1.0, + "content": ". Experiments are conducted on the OpenAI Gym platform (Brockman et al. (2016)) and", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 237, + 336, + 251 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 336, + 251 + ], + "score": 1.0, + "content": "trained with 40 million frames and 2 trials on each game.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 106, + 255, + 424, + 267 + ], + "lines": [ + { + "bbox": [ + 106, + 254, + 425, + 268 + ], + "spans": [ + { + "bbox": [ + 106, + 254, + 425, + 268 + ], + "score": 1.0, + "content": "We take the following directions to evaluate the performance of our algorithms:", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 129, + 276, + 505, + 374 + ], + "lines": [ + { + "bbox": [ + 129, + 277, + 429, + 290 + ], + "spans": [ + { + "bbox": [ + 129, + 277, + 429, + 290 + ], + "score": 1.0, + "content": "1. we compare Algorithm 1 against Double DQN and bootstrapped DQN,", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 128, + 294, + 506, + 309 + ], + "spans": [ + { + "bbox": [ + 128, + 294, + 434, + 309 + ], + "score": 1.0, + "content": "2. we isolate the impact of UCB exploration by comparing Algorithm 2 with", + "type": "text" + }, + { + "bbox": [ + 435, + 295, + 468, + 306 + ], + "score": 0.89, + "content": "\\lambda = 0 . 1", + "type": "inline_equation" + }, + { + "bbox": [ + 468, + 294, + 506, + 309 + ], + "score": 1.0, + "content": ", denoted", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 307, + 487, + 318 + ], + "spans": [ + { + "bbox": [ + 141, + 307, + 487, + 318 + ], + "score": 1.0, + "content": "as ucb exploration, against Algorithm 1, Double DQN, and bootstrapped DQN.", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 128, + 323, + 506, + 337 + ], + "spans": [ + { + "bbox": [ + 128, + 323, + 506, + 337 + ], + "score": 1.0, + "content": "3. we compare Algorithm 1 and Algorithm 2 with the count-based exploration method of", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 335, + 239, + 347 + ], + "spans": [ + { + "bbox": [ + 141, + 335, + 239, + 347 + ], + "score": 1.0, + "content": "Bellemare et al. (2016).", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 128, + 352, + 505, + 365 + ], + "spans": [ + { + "bbox": [ + 128, + 352, + 505, + 365 + ], + "score": 1.0, + "content": "4. we aggregate the comparison according to different categories of games, to understand when", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 141, + 363, + 241, + 376 + ], + "spans": [ + { + "bbox": [ + 141, + 363, + 241, + 376 + ], + "score": 1.0, + "content": "our methods are suprior.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 107, + 385, + 505, + 418 + ], + "lines": [ + { + "bbox": [ + 106, + 384, + 506, + 398 + ], + "spans": [ + { + "bbox": [ + 106, + 384, + 506, + 398 + ], + "score": 1.0, + "content": "Figure 1 compares the normalized learning curves of all algorithms across Atari games. Over-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 396, + 505, + 408 + ], + "spans": [ + { + "bbox": [ + 106, + 396, + 505, + 408 + ], + "score": 1.0, + "content": "all, Ensemble Voting, Algorithm 1, outperforms both Double DQN and bootstrapped DQN. With", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 406, + 463, + 421 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 463, + 421 + ], + "score": 1.0, + "content": "exploration, ucb exploration improves further by outperforming Ensemble Voting.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 107, + 424, + 505, + 556 + ], + "lines": [ + { + "bbox": [ + 105, + 424, + 505, + 436 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 505, + 436 + ], + "score": 1.0, + "content": "In Appendix B, we tabulate detailed results that compare our algorithms, Ensemble Voting and", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 435, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 505, + 447 + ], + "score": 1.0, + "content": "ucb exploration, against prior methods. In Table 2, we tabulate the maximal mean reward in", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 104, + 444, + 507, + 459 + ], + "spans": [ + { + "bbox": [ + 104, + 444, + 507, + 459 + ], + "score": 1.0, + "content": "100 consecutive episodes for Ensemble Voting, ucb exploration, bootstrapped DQN and Dou-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 457, + 505, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 505, + 469 + ], + "score": 1.0, + "content": "ble DQN. Without exploration, Ensemble Voting already achieves higher maximal mean reward than", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 467, + 506, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 467, + 506, + 480 + ], + "score": 1.0, + "content": "both Double DQN and bootstrapped DQN in a majority of Atari games. Ensemble Voting performs", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 478, + 506, + 492 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 506, + 492 + ], + "score": 1.0, + "content": "better than Double DQN in 37 games out of the total 49 games evaluated, better than bootstrapped", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 488, + 505, + 503 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 505, + 503 + ], + "score": 1.0, + "content": "DQN in 41 games. ucb exploration achieves the highest maximal mean reward among these", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 501, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 505, + 513 + ], + "score": 1.0, + "content": "four algorithms in 30 games out of the total 49 games evaluated. Specifically, ucb exploration", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 510, + 506, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 506, + 525 + ], + "score": 1.0, + "content": "performs better than Double DQN in 38 out of 49 games evaluated, better than bootstrapped DQN in", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 522, + 506, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 506, + 536 + ], + "score": 1.0, + "content": "45 games, and better than Ensemble Voting in 35 games. Figure 2 displays the learning curves of", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 533, + 506, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 506, + 547 + ], + "score": 1.0, + "content": "these five algorithms on a set of six Atari games. Ensemble Voting outperforms Double DQN and", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 544, + 403, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 403, + 558 + ], + "score": 1.0, + "content": "bootstrapped DQN. ucb exploration outperforms Ensemble Voting.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 29.5 + }, + { + "type": "text", + "bbox": [ + 107, + 561, + 505, + 616 + ], + "lines": [ + { + "bbox": [ + 105, + 561, + 506, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 466, + 574 + ], + "score": 1.0, + "content": "In Table 3, we compare our proposed methods with the count-based exploration method", + "type": "text" + }, + { + "bbox": [ + 466, + 561, + 493, + 572 + ], + "score": 0.85, + "content": "{ \\bf A } 3 { \\bf C } +", + "type": "inline_equation" + }, + { + "bbox": [ + 494, + 561, + 506, + 574 + ], + "score": 1.0, + "content": "of", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 571, + 506, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 571, + 345, + 585 + ], + "score": 1.0, + "content": "Bellemare et al. (2016) based on their published results of", + "type": "text" + }, + { + "bbox": [ + 345, + 572, + 372, + 583 + ], + "score": 0.87, + "content": "{ \\bf A } 3 { \\bf C } +", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 571, + 506, + 585 + ], + "score": 1.0, + "content": "trained with 200 million frames.", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 582, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 106, + 582, + 505, + 596 + ], + "score": 1.0, + "content": "We point out that even though our methods were trained with only 40 million frames, much less", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 593, + 507, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 126, + 607 + ], + "score": 1.0, + "content": "than", + "type": "text" + }, + { + "bbox": [ + 126, + 594, + 152, + 605 + ], + "score": 0.81, + "content": "{ \\bf A } 3 { \\bf C } +", + "type": "inline_equation" + }, + { + "bbox": [ + 152, + 593, + 507, + 607 + ], + "score": 1.0, + "content": "’s 200 million frames, UCB exploration achieves the highest average reward in 28 games,", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 604, + 465, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 247, + 618 + ], + "score": 1.0, + "content": "Ensemble Voting in 10 games, and", + "type": "text" + }, + { + "bbox": [ + 247, + 605, + 273, + 615 + ], + "score": 0.84, + "content": "{ \\bf A } 3 { \\bf C } +", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 604, + 436, + 618 + ], + "score": 1.0, + "content": "in 10 games. Our approach outperforms", + "type": "text" + }, + { + "bbox": [ + 436, + 605, + 461, + 615 + ], + "score": 0.78, + "content": "{ \\bf A } 3 { \\bf C } +", + "type": "inline_equation" + }, + { + "bbox": [ + 462, + 604, + 465, + 618 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 38 + }, + { + "type": "text", + "bbox": [ + 107, + 622, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 621, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 505, + 635 + ], + "score": 1.0, + "content": "Finally to understand why and when the proposed methods are superior, we aggregate the comparison", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 633, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 505, + 646 + ], + "score": 1.0, + "content": "results according to four categories: Human Optimal, Score Explicit, Dense Reward, and Sparse", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 643, + 505, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 505, + 656 + ], + "score": 1.0, + "content": "Reward. These categories follow the taxonomy in Table 1 of Ostrovski et al. (2017). Out of all", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 654, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 506, + 668 + ], + "score": 1.0, + "content": "games evaluated, 23 games are Human Optimal, 8 are Score Explicit, 8 are Dense Reward, and 5 are", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 666, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 506, + 678 + ], + "score": 1.0, + "content": "Sparse Reward. The comparison results are tabulated in Table 4, where we see ucb exploration", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "score": 1.0, + "content": "achieves top performance in more games than Ensemble Voting, Double DQN, and Bootstrapped", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 688, + 505, + 699 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 505, + 699 + ], + "score": 1.0, + "content": "DQN in the categories of Human Optimal, Score Explicit, and Dense Reward. In Sparse Reward, both", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 699, + 507, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 507, + 711 + ], + "score": 1.0, + "content": "ucb exploration and Ensemble Voting achieve best performance in 2 games out of total of 5.", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "Thus, we conclude that ucb exploration improves prior methods consistently across different", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 721, + 342, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 721, + 342, + 733 + ], + "score": 1.0, + "content": "game categories within the Arcade Learning Environment.", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 45.5 + } + ], + "page_idx": 4, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 107, + 26, + 309, + 38 + ], + "spans": [ + { + "bbox": [ + 107, + 26, + 309, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 751, + 308, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 750, + 309, + 763 + ], + "spans": [ + { + "bbox": [ + 302, + 750, + 309, + 763 + ], + "score": 1.0, + "content": "5", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "list", + "bbox": [ + 130, + 82, + 505, + 162 + ], + "lines": [ + { + "bbox": [ + 129, + 81, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 129, + 81, + 505, + 96 + ], + "score": 1.0, + "content": "1. does Ensemble Voting, Algorithm 1, improve upon existing algorithms including Double", + "type": "text" + } + ], + "index": 0, + "is_list_start_line": true + }, + { + "bbox": [ + 141, + 93, + 266, + 107 + ], + "spans": [ + { + "bbox": [ + 141, + 93, + 266, + 107 + ], + "score": 1.0, + "content": "DQN and bootstrapped DQN?", + "type": "text" + } + ], + "index": 1, + "is_list_end_line": true + }, + { + "bbox": [ + 128, + 109, + 505, + 125 + ], + "spans": [ + { + "bbox": [ + 128, + 109, + 505, + 125 + ], + "score": 1.0, + "content": "2. is the proposed UCB exploration strategy of Algorithm 2 effective in improving learning", + "type": "text" + } + ], + "index": 2, + "is_list_start_line": true + }, + { + "bbox": [ + 141, + 122, + 403, + 135 + ], + "spans": [ + { + "bbox": [ + 141, + 122, + 403, + 135 + ], + "score": 1.0, + "content": "compared to Algorithm 1, Double DQN and bootstrapped DQN?", + "type": "text" + } + ], + "index": 3, + "is_list_end_line": true + }, + { + "bbox": [ + 128, + 138, + 505, + 153 + ], + "spans": [ + { + "bbox": [ + 128, + 138, + 505, + 153 + ], + "score": 1.0, + "content": "3. how does UCB exploration compare with prior exploration methods such as the count-based", + "type": "text" + } + ], + "index": 4, + "is_list_start_line": true + }, + { + "bbox": [ + 141, + 151, + 332, + 163 + ], + "spans": [ + { + "bbox": [ + 141, + 151, + 332, + 163 + ], + "score": 1.0, + "content": "exploration method of Bellemare et al. (2016)?", + "type": "text" + } + ], + "index": 5, + "is_list_end_line": true + } + ], + "index": 2.5, + "bbox_fs": [ + 128, + 81, + 505, + 163 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 172, + 505, + 250 + ], + "lines": [ + { + "bbox": [ + 106, + 173, + 506, + 184 + ], + "spans": [ + { + "bbox": [ + 106, + 173, + 506, + 184 + ], + "score": 1.0, + "content": "We evaluate the algorithms on each Atari game of the Arcade Learning Environment (Bellemare et al.", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 182, + 506, + 196 + ], + "spans": [ + { + "bbox": [ + 105, + 182, + 506, + 196 + ], + "score": 1.0, + "content": "(2013)). We use the multi-head neural net architecture of Osband et al. (2016). We fix the common", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 194, + 505, + 206 + ], + "spans": [ + { + "bbox": [ + 105, + 194, + 505, + 206 + ], + "score": 1.0, + "content": "hyperparameters of all algorithms based on a well-tuned double DQN implementation, which uses the", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 205, + 505, + 218 + ], + "spans": [ + { + "bbox": [ + 106, + 205, + 505, + 218 + ], + "score": 1.0, + "content": "Adam optimizer (Kingma and Ba (2014)), different learning rate and exploration schedules compared", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 216, + 505, + 229 + ], + "spans": [ + { + "bbox": [ + 106, + 217, + 431, + 229 + ], + "score": 1.0, + "content": "to Mnih et al. (2015). Appendix A tabulates the hyperparameters. The number of", + "type": "text" + }, + { + "bbox": [ + 432, + 216, + 455, + 228 + ], + "score": 0.93, + "content": "\\left\\{ Q _ { k } \\right\\}", + "type": "inline_equation" + }, + { + "bbox": [ + 455, + 217, + 505, + 229 + ], + "score": 1.0, + "content": "functions is", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 107, + 226, + 506, + 240 + ], + "spans": [ + { + "bbox": [ + 107, + 227, + 141, + 237 + ], + "score": 0.89, + "content": "K = 1 0", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 226, + 506, + 240 + ], + "score": 1.0, + "content": ". Experiments are conducted on the OpenAI Gym platform (Brockman et al. (2016)) and", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 237, + 336, + 251 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 336, + 251 + ], + "score": 1.0, + "content": "trained with 40 million frames and 2 trials on each game.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 9, + "bbox_fs": [ + 105, + 173, + 506, + 251 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 255, + 424, + 267 + ], + "lines": [ + { + "bbox": [ + 106, + 254, + 425, + 268 + ], + "spans": [ + { + "bbox": [ + 106, + 254, + 425, + 268 + ], + "score": 1.0, + "content": "We take the following directions to evaluate the performance of our algorithms:", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13, + "bbox_fs": [ + 106, + 254, + 425, + 268 + ] + }, + { + "type": "list", + "bbox": [ + 129, + 276, + 505, + 374 + ], + "lines": [ + { + "bbox": [ + 129, + 277, + 429, + 290 + ], + "spans": [ + { + "bbox": [ + 129, + 277, + 429, + 290 + ], + "score": 1.0, + "content": "1. we compare Algorithm 1 against Double DQN and bootstrapped DQN,", + "type": "text" + } + ], + "index": 14, + "is_list_start_line": true, + "is_list_end_line": true + }, + { + "bbox": [ + 128, + 294, + 506, + 309 + ], + "spans": [ + { + "bbox": [ + 128, + 294, + 434, + 309 + ], + "score": 1.0, + "content": "2. we isolate the impact of UCB exploration by comparing Algorithm 2 with", + "type": "text" + }, + { + "bbox": [ + 435, + 295, + 468, + 306 + ], + "score": 0.89, + "content": "\\lambda = 0 . 1", + "type": "inline_equation" + }, + { + "bbox": [ + 468, + 294, + 506, + 309 + ], + "score": 1.0, + "content": ", denoted", + "type": "text" + } + ], + "index": 15, + "is_list_start_line": true + }, + { + "bbox": [ + 141, + 307, + 487, + 318 + ], + "spans": [ + { + "bbox": [ + 141, + 307, + 487, + 318 + ], + "score": 1.0, + "content": "as ucb exploration, against Algorithm 1, Double DQN, and bootstrapped DQN.", + "type": "text" + } + ], + "index": 16, + "is_list_end_line": true + }, + { + "bbox": [ + 128, + 323, + 506, + 337 + ], + "spans": [ + { + "bbox": [ + 128, + 323, + 506, + 337 + ], + "score": 1.0, + "content": "3. we compare Algorithm 1 and Algorithm 2 with the count-based exploration method of", + "type": "text" + } + ], + "index": 17, + "is_list_start_line": true + }, + { + "bbox": [ + 141, + 335, + 239, + 347 + ], + "spans": [ + { + "bbox": [ + 141, + 335, + 239, + 347 + ], + "score": 1.0, + "content": "Bellemare et al. (2016).", + "type": "text" + } + ], + "index": 18, + "is_list_end_line": true + }, + { + "bbox": [ + 128, + 352, + 505, + 365 + ], + "spans": [ + { + "bbox": [ + 128, + 352, + 505, + 365 + ], + "score": 1.0, + "content": "4. we aggregate the comparison according to different categories of games, to understand when", + "type": "text" + } + ], + "index": 19, + "is_list_start_line": true + }, + { + "bbox": [ + 141, + 363, + 241, + 376 + ], + "spans": [ + { + "bbox": [ + 141, + 363, + 241, + 376 + ], + "score": 1.0, + "content": "our methods are suprior.", + "type": "text" + } + ], + "index": 20, + "is_list_end_line": true + } + ], + "index": 17, + "bbox_fs": [ + 128, + 277, + 506, + 376 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 385, + 505, + 418 + ], + "lines": [ + { + "bbox": [ + 106, + 384, + 506, + 398 + ], + "spans": [ + { + "bbox": [ + 106, + 384, + 506, + 398 + ], + "score": 1.0, + "content": "Figure 1 compares the normalized learning curves of all algorithms across Atari games. Over-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 396, + 505, + 408 + ], + "spans": [ + { + "bbox": [ + 106, + 396, + 505, + 408 + ], + "score": 1.0, + "content": "all, Ensemble Voting, Algorithm 1, outperforms both Double DQN and bootstrapped DQN. With", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 406, + 463, + 421 + ], + "spans": [ + { + "bbox": [ + 105, + 406, + 463, + 421 + ], + "score": 1.0, + "content": "exploration, ucb exploration improves further by outperforming Ensemble Voting.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22, + "bbox_fs": [ + 105, + 384, + 506, + 421 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 424, + 505, + 556 + ], + "lines": [ + { + "bbox": [ + 105, + 424, + 505, + 436 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 505, + 436 + ], + "score": 1.0, + "content": "In Appendix B, we tabulate detailed results that compare our algorithms, Ensemble Voting and", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 435, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 505, + 447 + ], + "score": 1.0, + "content": "ucb exploration, against prior methods. In Table 2, we tabulate the maximal mean reward in", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 104, + 444, + 507, + 459 + ], + "spans": [ + { + "bbox": [ + 104, + 444, + 507, + 459 + ], + "score": 1.0, + "content": "100 consecutive episodes for Ensemble Voting, ucb exploration, bootstrapped DQN and Dou-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 457, + 505, + 469 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 505, + 469 + ], + "score": 1.0, + "content": "ble DQN. Without exploration, Ensemble Voting already achieves higher maximal mean reward than", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 467, + 506, + 480 + ], + "spans": [ + { + "bbox": [ + 105, + 467, + 506, + 480 + ], + "score": 1.0, + "content": "both Double DQN and bootstrapped DQN in a majority of Atari games. Ensemble Voting performs", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 478, + 506, + 492 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 506, + 492 + ], + "score": 1.0, + "content": "better than Double DQN in 37 games out of the total 49 games evaluated, better than bootstrapped", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 488, + 505, + 503 + ], + "spans": [ + { + "bbox": [ + 105, + 488, + 505, + 503 + ], + "score": 1.0, + "content": "DQN in 41 games. ucb exploration achieves the highest maximal mean reward among these", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 501, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 505, + 513 + ], + "score": 1.0, + "content": "four algorithms in 30 games out of the total 49 games evaluated. Specifically, ucb exploration", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 510, + 506, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 506, + 525 + ], + "score": 1.0, + "content": "performs better than Double DQN in 38 out of 49 games evaluated, better than bootstrapped DQN in", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 522, + 506, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 522, + 506, + 536 + ], + "score": 1.0, + "content": "45 games, and better than Ensemble Voting in 35 games. Figure 2 displays the learning curves of", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 533, + 506, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 506, + 547 + ], + "score": 1.0, + "content": "these five algorithms on a set of six Atari games. Ensemble Voting outperforms Double DQN and", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 544, + 403, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 403, + 558 + ], + "score": 1.0, + "content": "bootstrapped DQN. ucb exploration outperforms Ensemble Voting.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 29.5, + "bbox_fs": [ + 104, + 424, + 507, + 558 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 561, + 505, + 616 + ], + "lines": [ + { + "bbox": [ + 105, + 561, + 506, + 574 + ], + "spans": [ + { + "bbox": [ + 105, + 561, + 466, + 574 + ], + "score": 1.0, + "content": "In Table 3, we compare our proposed methods with the count-based exploration method", + "type": "text" + }, + { + "bbox": [ + 466, + 561, + 493, + 572 + ], + "score": 0.85, + "content": "{ \\bf A } 3 { \\bf C } +", + "type": "inline_equation" + }, + { + "bbox": [ + 494, + 561, + 506, + 574 + ], + "score": 1.0, + "content": "of", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 571, + 506, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 571, + 345, + 585 + ], + "score": 1.0, + "content": "Bellemare et al. (2016) based on their published results of", + "type": "text" + }, + { + "bbox": [ + 345, + 572, + 372, + 583 + ], + "score": 0.87, + "content": "{ \\bf A } 3 { \\bf C } +", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 571, + 506, + 585 + ], + "score": 1.0, + "content": "trained with 200 million frames.", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 582, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 106, + 582, + 505, + 596 + ], + "score": 1.0, + "content": "We point out that even though our methods were trained with only 40 million frames, much less", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 593, + 507, + 607 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 126, + 607 + ], + "score": 1.0, + "content": "than", + "type": "text" + }, + { + "bbox": [ + 126, + 594, + 152, + 605 + ], + "score": 0.81, + "content": "{ \\bf A } 3 { \\bf C } +", + "type": "inline_equation" + }, + { + "bbox": [ + 152, + 593, + 507, + 607 + ], + "score": 1.0, + "content": "’s 200 million frames, UCB exploration achieves the highest average reward in 28 games,", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 604, + 465, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 247, + 618 + ], + "score": 1.0, + "content": "Ensemble Voting in 10 games, and", + "type": "text" + }, + { + "bbox": [ + 247, + 605, + 273, + 615 + ], + "score": 0.84, + "content": "{ \\bf A } 3 { \\bf C } +", + "type": "inline_equation" + }, + { + "bbox": [ + 274, + 604, + 436, + 618 + ], + "score": 1.0, + "content": "in 10 games. Our approach outperforms", + "type": "text" + }, + { + "bbox": [ + 436, + 605, + 461, + 615 + ], + "score": 0.78, + "content": "{ \\bf A } 3 { \\bf C } +", + "type": "inline_equation" + }, + { + "bbox": [ + 462, + 604, + 465, + 618 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 38, + "bbox_fs": [ + 105, + 561, + 507, + 618 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 622, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 621, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 505, + 635 + ], + "score": 1.0, + "content": "Finally to understand why and when the proposed methods are superior, we aggregate the comparison", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 633, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 505, + 646 + ], + "score": 1.0, + "content": "results according to four categories: Human Optimal, Score Explicit, Dense Reward, and Sparse", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 643, + 505, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 505, + 656 + ], + "score": 1.0, + "content": "Reward. These categories follow the taxonomy in Table 1 of Ostrovski et al. (2017). Out of all", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 654, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 506, + 668 + ], + "score": 1.0, + "content": "games evaluated, 23 games are Human Optimal, 8 are Score Explicit, 8 are Dense Reward, and 5 are", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 666, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 506, + 678 + ], + "score": 1.0, + "content": "Sparse Reward. The comparison results are tabulated in Table 4, where we see ucb exploration", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 506, + 690 + ], + "score": 1.0, + "content": "achieves top performance in more games than Ensemble Voting, Double DQN, and Bootstrapped", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 688, + 505, + 699 + ], + "spans": [ + { + "bbox": [ + 106, + 688, + 505, + 699 + ], + "score": 1.0, + "content": "DQN in the categories of Human Optimal, Score Explicit, and Dense Reward. 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hyperparameter total training frames40 milliondescriptionsLength of training for each game.
minibatch size32
replay buffer size1000000parameter update.The number of most recent frames
agent history length4
length.concatenated as input to the Q net- work. Total number of iterations = total training frames /agent history
target network update10000 frequency
discount factor0.99
action repeat4
update frequency 4
optimizerAdamOptimizer for parameter updates.
β1 0.9Adam optimizer parameter.
β0.99Adam optimizer parameter.
E 10-4Adam optimizer parameter.
learning rate schedule 210-4 Interp(10-4,5 * 10-5) 5*10-5t≤106 otherwise t>5*106Learning rate for Adam optimizer, as a function of iteration t.
exploration scheduleInterp(1,0.1) Interp(0.1,0.01) 0.01t<106 otherwise Probability of random action in e- t>5*106
greedy exploration, as a function of the iteration t . Number of uniform random ac- tions taken before learning starts.
replay start size50000
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value
hyperparameter total training frames40 milliondescriptionsLength of training for each game.
minibatch size32
replay buffer size1000000parameter update.The number of most recent frames
agent history length4
length.concatenated as input to the Q net- work. Total number of iterations = total training frames /agent history
target network update10000 frequency
discount factor0.99
action repeat4
update frequency 4
optimizerAdamOptimizer for parameter updates.
β1 0.9Adam optimizer parameter.
β0.99Adam optimizer parameter.
E 10-4Adam optimizer parameter.
learning rate schedule 210-4 Interp(10-4,5 * 10-5) 5*10-5t≤106 otherwise t>5*106Learning rate for Adam optimizer, as a function of iteration t.
exploration scheduleInterp(1,0.1) Interp(0.1,0.01) 0.01t<106 otherwise Probability of random action in e- t>5*106
greedy exploration, as a function of the iteration t . Number of uniform random ac- tions taken before learning starts.
replay start size50000
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UCB-Exploration
Bootstrapped DQNDoubleDQNEnsemble Voting 2282.8
Alien1445.12059.72817.6
Amidar430.58667.5683.72663.8
Assault2519.062820.613213.583702.76
Asterix3829.07639.58740.08732.0
Asteroids1009.51002.31149.31007.8
Atlantis1314058.01982677.01786305.02016145.0
Bank Heist795.1789.9869.4906.9
Battle Zone26230.024880.027430.026770.0
Beam Rider8006.587743.747991.99188.26
Bowling28.6230.9232.9238.06
Boxing85.9194.0794.4798.08
Breakout400.22467.45426.78411.31
Centipede5328.775177.516153.286237.18
Chopper Command2153.03260.03544.03677.0
Crazy Climber110926.0124456.0126677.0127754.0
Demon Attack9811.4523562.5530004.459861.9
Double Dunk-10.82-14.58-11.94-4.08
Enduro1314.311439.591999.882752.55
Fishing Derby21.8923.6930.0229.71
Freeway33.5732.9333.9233.96
Frostbite1284.8529.21196.01903.0
Gopher7652.212030.010993.212910.8
Gravitar227.5279.5371.5318.0
Ice Hockey-4.62-4.63-1.73-4.71
Jamesbond594.5594.0602.0710.0
Kangaroo8186.07787.08174.014196.0
Krull8537.528517.918669.179171.61
Kung Fu Master24153.032896.030988.031291.0
Montezuma Revenge2.04.01.04.0
Ms Pacman2508.72498.13039.73425.4
Name This Game8212.49806.99255.19570.5
Pitfall-5.99-7.57-3.37-1.47
Pong21.020.6721.020.95
Private Eye1815.19788.631845.281252.01
Qbert10557.256529.512036.514198.25
Riverraid11528.011834.712785.815622.2
Road Runner52489.049039.054768.053596.0
Robotank21.0329.831.8341.04
Seaquest9320.718056.420458.624001.6
Space Invaders1549.91917.51890.82626.55
Star Gunner20115.052283.041684.047367.0
Tennis-15.11-14.04-11.63-7.8
Time Pilot5088.05548.06153.06490.0
Tutankham167.47223.43208.61200.76
Up N Down9049.111815.319528.319827.3
Venture115.096.078.067.0
Video Pinball364600.85374686.89343380.29372564.11
Wizard Of Wor2860.03877.05451.05873.0
Zaxxon592.08903.03901.03695.0
Times best17930
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UCB-Exploration
Bootstrapped DQNDoubleDQNEnsemble Voting 2282.8
Alien1445.12059.72817.6
Amidar430.58667.5683.72663.8
Assault2519.062820.613213.583702.76
Asterix3829.07639.58740.08732.0
Asteroids1009.51002.31149.31007.8
Atlantis1314058.01982677.01786305.02016145.0
Bank Heist795.1789.9869.4906.9
Battle Zone26230.024880.027430.026770.0
Beam Rider8006.587743.747991.99188.26
Bowling28.6230.9232.9238.06
Boxing85.9194.0794.4798.08
Breakout400.22467.45426.78411.31
Centipede5328.775177.516153.286237.18
Chopper Command2153.03260.03544.03677.0
Crazy Climber110926.0124456.0126677.0127754.0
Demon Attack9811.4523562.5530004.459861.9
Double Dunk-10.82-14.58-11.94-4.08
Enduro1314.311439.591999.882752.55
Fishing Derby21.8923.6930.0229.71
Freeway33.5732.9333.9233.96
Frostbite1284.8529.21196.01903.0
Gopher7652.212030.010993.212910.8
Gravitar227.5279.5371.5318.0
Ice Hockey-4.62-4.63-1.73-4.71
Jamesbond594.5594.0602.0710.0
Kangaroo8186.07787.08174.014196.0
Krull8537.528517.918669.179171.61
Kung Fu Master24153.032896.030988.031291.0
Montezuma Revenge2.04.01.04.0
Ms Pacman2508.72498.13039.73425.4
Name This Game8212.49806.99255.19570.5
Pitfall-5.99-7.57-3.37-1.47
Pong21.020.6721.020.95
Private Eye1815.19788.631845.281252.01
Qbert10557.256529.512036.514198.25
Riverraid11528.011834.712785.815622.2
Road Runner52489.049039.054768.053596.0
Robotank21.0329.831.8341.04
Seaquest9320.718056.420458.624001.6
Space Invaders1549.91917.51890.82626.55
Star Gunner20115.052283.041684.047367.0
Tennis-15.11-14.04-11.63-7.8
Time Pilot5088.05548.06153.06490.0
Tutankham167.47223.43208.61200.76
Up N Down9049.111815.319528.319827.3
Venture115.096.078.067.0
Video Pinball364600.85374686.89343380.29372564.11
Wizard Of Wor2860.03877.05451.05873.0
Zaxxon592.08903.03901.03695.0
Times best17930
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Ensemble VotingUCB-ExplorationA3C+
Alien2282.82817.61848.33
Amidar683.72663.8964.77
Assault3213.583702.762607.28
Asterix8740.08732.07262.77
Asteroids1149.31007.82257.92
Atlantis1786305.02016145.01733528.71
Bank Heist869.4906.9991.96
Battle Zone27430.026770.07428.99
Beam Rider7991.99188.265992.08
Bowling32.9238.0668.72
Boxing94.4798.0813.82
Breakout426.78411.31323.21
Centipede6153.286237.185338.24
Chopper Command3544.03677.05388.22
Crazy Climber126677.0127754.0104083.51
Demon Attack30004.459861.919589.95
Double Dunk-11.94-4.08-8.88
Enduro1999.882752.55749.11
Fishing Derby30.0229.7129.46
Freeway33.9233.9627.33
Frostbite1196.01903.0506.61
Gopher10993.212910.85948.40
Gravitar371.5318.0246.02
Ice Hockey-1.73-4.71-7.05
Jamesbond602.0710.01024.16
Kangaroo8174.014196.05475.73
Krull8669.179171.617587.58
Kung Fu Master30988.031291.026593.67
Montezuma Revenge1.04.0142.50
Ms Pacman3039.73425.42380.58
Name This Game9255.19570.56427.51
Pitfall-3.37-1.47-155.97
Pong21.020.9517.33
Private Eye1845.281252.01100.0
Qbert12036.514198.2515804.72
Riverraid12785.815622.210331.56
Road Runner54768.053596.049029.74
Robotank31.8341.046.68
Seaquest20458.624001.62274.06
Space Invaders1890.82626.551466.01
Star Gunner41684.047367.052466.84
Tennis-11.63-7.8-20.49
Time Pilot6153.06490.03816.38
Tutankham208.61200.76132.67
Up N Down19528.319827.38705.64
Venture78.067.00.00
Video Pinball343380.29372564.1135515.92
Wizard Of Wor5451.05873.03657.65
Zaxxon3901.03695.07956.05
Times Best102810
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Ensemble VotingUCB-ExplorationA3C+
Alien2282.82817.61848.33
Amidar683.72663.8964.77
Assault3213.583702.762607.28
Asterix8740.08732.07262.77
Asteroids1149.31007.82257.92
Atlantis1786305.02016145.01733528.71
Bank Heist869.4906.9991.96
Battle Zone27430.026770.07428.99
Beam Rider7991.99188.265992.08
Bowling32.9238.0668.72
Boxing94.4798.0813.82
Breakout426.78411.31323.21
Centipede6153.286237.185338.24
Chopper Command3544.03677.05388.22
Crazy Climber126677.0127754.0104083.51
Demon Attack30004.459861.919589.95
Double Dunk-11.94-4.08-8.88
Enduro1999.882752.55749.11
Fishing Derby30.0229.7129.46
Freeway33.9233.9627.33
Frostbite1196.01903.0506.61
Gopher10993.212910.85948.40
Gravitar371.5318.0246.02
Ice Hockey-1.73-4.71-7.05
Jamesbond602.0710.01024.16
Kangaroo8174.014196.05475.73
Krull8669.179171.617587.58
Kung Fu Master30988.031291.026593.67
Montezuma Revenge1.04.0142.50
Ms Pacman3039.73425.42380.58
Name This Game9255.19570.56427.51
Pitfall-3.37-1.47-155.97
Pong21.020.9517.33
Private Eye1845.281252.01100.0
Qbert12036.514198.2515804.72
Riverraid12785.815622.210331.56
Road Runner54768.053596.049029.74
Robotank31.8341.046.68
Seaquest20458.624001.62274.06
Space Invaders1890.82626.551466.01
Star Gunner41684.047367.052466.84
Tennis-11.63-7.8-20.49
Time Pilot6153.06490.03816.38
Tutankham208.61200.76132.67
Up N Down19528.319827.38705.64
Venture78.067.00.00
Video Pinball343380.29372564.1135515.92
Wizard Of Wor5451.05873.03657.65
Zaxxon3901.03695.07956.05
Times Best102810
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CategoryTotalBootstrapped DQNDouble DQNEnsemble VotingUCB-Exploration
Human Optimal2303515
Score Explicit80215
Dense Reward80116
Sparse Reward51022
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The Atari games are separated", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 160, + 506, + 175 + ], + "spans": [ + { + "bbox": [ + 105, + 160, + 506, + 175 + ], + "score": 1.0, + "content": "into four categories: human optimal, score explicit, dense reward, and sparse reward. In each row,", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 172, + 506, + 185 + ], + "spans": [ + { + "bbox": [ + 105, + 172, + 506, + 185 + ], + "score": 1.0, + "content": "we present the number of games in this category, the total number of games where each algorithm", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 182, + 506, + 197 + ], + "spans": [ + { + "bbox": [ + 105, + 182, + 506, + 197 + ], + "score": 1.0, + "content": "achieves the optimal performance according to Table 2. The game categories follow the taxonomy in", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 194, + 240, + 207 + ], + "spans": [ + { + "bbox": [ + 106, + 194, + 240, + 207 + ], + "score": 1.0, + "content": "Table 1 of Ostrovski et al. 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Next, we approximate", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 272, + 506, + 285 + ], + "spans": [ + { + "bbox": [ + 105, + 272, + 207, + 285 + ], + "score": 1.0, + "content": "the posterior update with", + "type": "text" + }, + { + "bbox": [ + 208, + 273, + 217, + 284 + ], + "score": 0.86, + "content": "Q", + "type": "inline_equation" + }, + { + "bbox": [ + 217, + 272, + 262, + 285 + ], + "score": 1.0, + "content": "-ensembles", + "type": "text" + }, + { + "bbox": [ + 263, + 272, + 286, + 284 + ], + "score": 0.92, + "content": "\\{ { \\bar { Q } } _ { k } \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 286, + 272, + 506, + 285 + ], + "score": 1.0, + "content": "and demonstrate that the Bellman equation emerges as", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 282, + 273, + 296 + ], + "spans": [ + { + "bbox": [ + 105, + 282, + 255, + 296 + ], + "score": 1.0, + "content": "the approximate update rule for each", + "type": "text" + }, + { + "bbox": [ + 255, + 284, + 269, + 294 + ], + "score": 0.89, + "content": "Q _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 269, + 282, + 273, + 296 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 10.5 + }, + { + "type": "title", + "bbox": [ + 107, + 307, + 322, + 319 + ], + "lines": [ + { + "bbox": [ + 105, + 306, + 323, + 322 + ], + "spans": [ + { + "bbox": [ + 105, + 306, + 259, + 322 + ], + "score": 1.0, + "content": "C.1 POSTERIOR UPDATE FOR THE", + "type": "text" + }, + { + "bbox": [ + 260, + 308, + 273, + 319 + ], + "score": 0.88, + "content": "Q ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 306, + 323, + 322 + ], + "score": 1.0, + "content": "-FUNCTION", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 106, + 327, + 505, + 405 + ], + "lines": [ + { + "bbox": [ + 105, + 328, + 505, + 340 + ], + "spans": [ + { + "bbox": [ + 105, + 328, + 306, + 340 + ], + "score": 1.0, + "content": "An MDP is specified by the transition probability", + "type": "text" + }, + { + "bbox": [ + 306, + 329, + 315, + 338 + ], + "score": 0.82, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 328, + 413, + 340 + ], + "score": 1.0, + "content": "and the reward function", + "type": "text" + }, + { + "bbox": [ + 414, + 329, + 422, + 338 + ], + "score": 0.79, + "content": "R", + "type": "inline_equation" + }, + { + "bbox": [ + 423, + 328, + 505, + 340 + ], + "score": 1.0, + "content": ". Unlike prior works", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 339, + 506, + 352 + ], + "spans": [ + { + "bbox": [ + 105, + 339, + 506, + 352 + ], + "score": 1.0, + "content": "outlined in Section 2.3 which learned the posterior of the MDP, we will consider the joint distribution", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 104, + 349, + 506, + 363 + ], + "spans": [ + { + "bbox": [ + 104, + 349, + 128, + 363 + ], + "score": 1.0, + "content": "over", + "type": "text" + }, + { + "bbox": [ + 128, + 350, + 160, + 362 + ], + "score": 0.92, + "content": "( Q ^ { * } , T )", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 349, + 208, + 363 + ], + "score": 1.0, + "content": ". 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So", + "type": "text" + }, + { + "bbox": [ + 385, + 350, + 417, + 362 + ], + "score": 0.92, + "content": "( Q ^ { * } , T )", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 349, + 506, + 363 + ], + "score": 1.0, + "content": "determines a unique", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 360, + 506, + 374 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 330, + 374 + ], + "score": 1.0, + "content": "MDP. In this section, we assume that the agent samples", + "type": "text" + }, + { + "bbox": [ + 330, + 362, + 353, + 373 + ], + "score": 0.91, + "content": "( s , a )", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 360, + 506, + 374 + ], + "score": 1.0, + "content": "according to a fixed distribution. The", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 104, + 372, + 506, + 385 + ], + "spans": [ + { + "bbox": [ + 104, + 372, + 198, + 385 + ], + "score": 1.0, + "content": "corresponding reward", + "type": "text" + }, + { + "bbox": [ + 198, + 374, + 205, + 382 + ], + "score": 0.75, + "content": "r", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 372, + 266, + 385 + ], + "score": 1.0, + "content": "and next state", + "type": "text" + }, + { + "bbox": [ + 266, + 372, + 274, + 382 + ], + "score": 0.84, + "content": "s ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 275, + 372, + 398, + 385 + ], + "score": 1.0, + "content": "given by the MDP append to", + "type": "text" + }, + { + "bbox": [ + 399, + 372, + 421, + 384 + ], + "score": 0.91, + "content": "( s , a )", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 372, + 506, + 385 + ], + "score": 1.0, + "content": "to form a transition", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 107, + 383, + 506, + 396 + ], + "spans": [ + { + "bbox": [ + 107, + 383, + 172, + 395 + ], + "score": 0.92, + "content": "{ \\boldsymbol \\tau } = ( s , a , r , s ^ { \\prime } )", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 383, + 299, + 396 + ], + "score": 1.0, + "content": ", for updating the posterior of", + "type": "text" + }, + { + "bbox": [ + 299, + 383, + 331, + 395 + ], + "score": 0.92, + "content": "( Q ^ { * } , T )", + "type": "inline_equation" + }, + { + "bbox": [ + 331, + 383, + 401, + 396 + ], + "score": 1.0, + "content": ". Recall that the", + "type": "text" + }, + { + "bbox": [ + 402, + 384, + 415, + 394 + ], + "score": 0.89, + "content": "Q ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 383, + 506, + 396 + ], + "score": 1.0, + "content": "-function satisfies the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 393, + 180, + 407 + ], + "spans": [ + { + "bbox": [ + 105, + 393, + 180, + 407 + ], + "score": 1.0, + "content": "Bellman equation", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 17 + }, + { + "type": "interline_equation", + "bbox": [ + 211, + 404, + 399, + 425 + ], + "lines": [ + { + "bbox": [ + 211, + 404, + 399, + 425 + ], + "spans": [ + { + "bbox": [ + 211, + 404, + 399, + 425 + ], + "score": 0.92, + "content": "Q ( s , a ) = r + \\mathbb { E } _ { s ^ { \\prime } \\sim T ( \\cdot | s , a ) } \\left[ \\gamma \\operatorname* { m a x } _ { a ^ { \\prime } } Q ( s ^ { \\prime } , a ^ { \\prime } ) \\right] .", + "type": "interline_equation", + "image_path": "a6677133076ccab2959a65d94d7e4fb0872abfdf9ae434ebc50645522db14b11.jpg" + } + ] + } + ], + "index": 21, + "virtual_lines": [ + { + "bbox": [ + 211, + 404, + 399, + 425 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 427, + 504, + 449 + ], + "lines": [ + { + "bbox": [ + 105, + 425, + 506, + 440 + ], + "spans": [ + { + "bbox": [ + 105, + 425, + 257, + 440 + ], + "score": 1.0, + "content": "Denote the joint prior distribution as", + "type": "text" + }, + { + "bbox": [ + 257, + 427, + 294, + 439 + ], + "score": 0.93, + "content": "p ( Q ^ { * } , T )", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 425, + 378, + 440 + ], + "score": 1.0, + "content": "and the posterior as", + "type": "text" + }, + { + "bbox": [ + 378, + 428, + 384, + 438 + ], + "score": 0.79, + "content": "\\tilde { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 425, + 506, + 440 + ], + "score": 1.0, + "content": ". We apply Bayes’ formula to", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 438, + 193, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 193, + 451 + ], + "score": 1.0, + "content": "expand the posterior:", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22.5 + }, + { + "type": "interline_equation", + "bbox": [ + 151, + 451, + 458, + 508 + ], + "lines": [ + { + "bbox": [ + 151, + 451, + 458, + 508 + ], + "spans": [ + { + "bbox": [ + 151, + 451, + 458, + 508 + ], + "score": 0.94, + "content": "\\begin{array} { r l } & { \\tilde { p } ( Q ^ { * } , T | \\tau ) = \\frac { p ( \\tau | Q ^ { * } , T ) \\cdot p ( Q ^ { * } , T ) } { Z ( \\tau ) } } \\\\ & { \\qquad = \\frac { p ( Q ^ { * } , T ) \\cdot p ( s ^ { \\prime } | Q ^ { * } , T , ( s , a ) ) \\cdot p ( r | Q ^ { * } , T , ( s , a , s ^ { \\prime } ) ) \\cdot p ( s , a ) } { Z ( \\tau ) } , } \\end{array}", + "type": "interline_equation", + "image_path": "49c85f5489553a7e3afb98aca65933136c3196eb0e7aba19aecddb350c6c2167.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 151, + 451, + 458, + 470.0 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 151, + 470.0, + 458, + 489.0 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 151, + 489.0, + 458, + 508.0 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 510, + 504, + 533 + ], + "lines": [ + { + "bbox": [ + 106, + 510, + 505, + 523 + ], + "spans": [ + { + "bbox": [ + 106, + 510, + 135, + 523 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 135, + 510, + 156, + 522 + ], + "score": 0.92, + "content": "Z ( \\tau )", + "type": "inline_equation" + }, + { + "bbox": [ + 157, + 510, + 416, + 523 + ], + "score": 1.0, + "content": "is a normalizing constant and the second equality is because", + "type": "text" + }, + { + "bbox": [ + 417, + 513, + 423, + 520 + ], + "score": 0.76, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 510, + 443, + 523 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 444, + 513, + 450, + 520 + ], + "score": 0.76, + "content": "a", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 510, + 505, + 523 + ], + "score": 1.0, + "content": "are sampled", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 521, + 468, + 534 + ], + "spans": [ + { + "bbox": [ + 106, + 521, + 169, + 534 + ], + "score": 1.0, + "content": "randomly from", + "type": "text" + }, + { + "bbox": [ + 169, + 522, + 177, + 531 + ], + "score": 0.84, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 521, + 195, + 534 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 195, + 522, + 205, + 531 + ], + "score": 0.79, + "content": "\\mathcal { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 521, + 468, + 534 + ], + "score": 1.0, + "content": ". Next, we calculate the two conditional probabilities in (1). 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Human Optimal2303515
Score Explicit80215
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Recall that the", + "type": "text" + }, + { + "bbox": [ + 402, + 384, + 415, + 394 + ], + "score": 0.89, + "content": "Q ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 383, + 506, + 396 + ], + "score": 1.0, + "content": "-function satisfies the", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 393, + 180, + 407 + ], + "spans": [ + { + "bbox": [ + 105, + 393, + 180, + 407 + ], + "score": 1.0, + "content": "Bellman equation", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 17, + "bbox_fs": [ + 104, + 328, + 506, + 407 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 211, + 404, + 399, + 425 + ], + "lines": [ + { + "bbox": [ + 211, + 404, + 399, + 425 + ], + "spans": [ + { + "bbox": [ + 211, + 404, + 399, + 425 + ], + "score": 0.92, + "content": "Q ( s , a ) = r + \\mathbb { E } _ { s ^ { \\prime } \\sim T ( \\cdot | s , a ) } \\left[ \\gamma \\operatorname* { m a x } _ { a ^ { \\prime } } Q ( s ^ { \\prime } , a ^ { \\prime } ) \\right] .", + "type": "interline_equation", + "image_path": "a6677133076ccab2959a65d94d7e4fb0872abfdf9ae434ebc50645522db14b11.jpg" + } + ] + } + ], + "index": 21, + "virtual_lines": [ + { + "bbox": [ + 211, + 404, + 399, + 425 + ], + "spans": [], + "index": 21 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 427, + 504, + 449 + ], + "lines": [ + { + "bbox": [ + 105, + 425, + 506, + 440 + ], + "spans": [ + { + "bbox": [ + 105, + 425, + 257, + 440 + ], + "score": 1.0, + "content": "Denote the joint prior distribution as", + "type": "text" + }, + { + "bbox": [ + 257, + 427, + 294, + 439 + ], + "score": 0.93, + "content": "p ( Q ^ { * } , T )", + "type": "inline_equation" + }, + { + "bbox": [ + 295, + 425, + 378, + 440 + ], + "score": 1.0, + "content": "and the posterior as", + "type": "text" + }, + { + "bbox": [ + 378, + 428, + 384, + 438 + ], + "score": 0.79, + "content": "\\tilde { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 425, + 506, + 440 + ], + "score": 1.0, + "content": ". We apply Bayes’ formula to", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 438, + 193, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 193, + 451 + ], + "score": 1.0, + "content": "expand the posterior:", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22.5, + "bbox_fs": [ + 105, + 425, + 506, + 451 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 151, + 451, + 458, + 508 + ], + "lines": [ + { + "bbox": [ + 151, + 451, + 458, + 508 + ], + "spans": [ + { + "bbox": [ + 151, + 451, + 458, + 508 + ], + "score": 0.94, + "content": "\\begin{array} { r l } & { \\tilde { p } ( Q ^ { * } , T | \\tau ) = \\frac { p ( \\tau | Q ^ { * } , T ) \\cdot p ( Q ^ { * } , T ) } { Z ( \\tau ) } } \\\\ & { \\qquad = \\frac { p ( Q ^ { * } , T ) \\cdot p ( s ^ { \\prime } | Q ^ { * } , T , ( s , a ) ) \\cdot p ( r | Q ^ { * } , T , ( s , a , s ^ { \\prime } ) ) \\cdot p ( s , a ) } { Z ( \\tau ) } , } \\end{array}", + "type": "interline_equation", + "image_path": "49c85f5489553a7e3afb98aca65933136c3196eb0e7aba19aecddb350c6c2167.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 151, + 451, + 458, + 470.0 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 151, + 470.0, + 458, + 489.0 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 151, + 489.0, + 458, + 508.0 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 510, + 504, + 533 + ], + "lines": [ + { + "bbox": [ + 106, + 510, + 505, + 523 + ], + "spans": [ + { + "bbox": [ + 106, + 510, + 135, + 523 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 135, + 510, + 156, + 522 + ], + "score": 0.92, + "content": "Z ( \\tau )", + "type": "inline_equation" + }, + { + "bbox": [ + 157, + 510, + 416, + 523 + ], + "score": 1.0, + "content": "is a normalizing constant and the second equality is because", + "type": "text" + }, + { + "bbox": [ + 417, + 513, + 423, + 520 + ], + "score": 0.76, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 510, + 443, + 523 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 444, + 513, + 450, + 520 + ], + "score": 0.76, + "content": "a", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 510, + 505, + 523 + ], + "score": 1.0, + "content": "are sampled", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 521, + 468, + 534 + ], + "spans": [ + { + "bbox": [ + 106, + 521, + 169, + 534 + ], + "score": 1.0, + "content": "randomly from", + "type": "text" + }, + { + "bbox": [ + 169, + 522, + 177, + 531 + ], + "score": 0.84, + "content": "s", + "type": "inline_equation" + }, + { + "bbox": [ + 177, + 521, + 195, + 534 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 195, + 522, + 205, + 531 + ], + "score": 0.79, + "content": "\\mathcal { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 205, + 521, + 468, + 534 + ], + "score": 1.0, + "content": ". Next, we calculate the two conditional probabilities in (1). First,", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 27.5, + "bbox_fs": [ + 106, + 510, + 505, + 534 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 205, + 535, + 405, + 550 + ], + "lines": [ + { + "bbox": [ + 205, + 535, + 405, + 550 + ], + "spans": [ + { + "bbox": [ + 205, + 535, + 405, + 550 + ], + "score": 0.87, + "content": "p ( s ^ { \\prime } | Q ^ { * } , T , ( s , a ) ) = p ( s ^ { \\prime } | T , ( s , a ) ) = T ( s ^ { \\prime } | s , a ) ,", + "type": "interline_equation", + "image_path": "602cdad18535c501841b89ecb1826559681ee676df0c9a35d39a813d615bf0be.jpg" + } + ] + } + ], + "index": 29, + "virtual_lines": [ + { + "bbox": [ + 205, + 535, + 405, + 550 + ], + "spans": [], + "index": 29 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 552, + 460, + 564 + ], + "lines": [ + { + "bbox": [ + 105, + 551, + 462, + 566 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 267, + 566 + ], + "score": 1.0, + "content": "where the first equality is because given", + "type": "text" + }, + { + "bbox": [ + 268, + 553, + 276, + 563 + ], + "score": 0.5, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 551, + 280, + 566 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 280, + 553, + 293, + 564 + ], + "score": 0.62, + "content": "Q ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 551, + 462, + 566 + ], + "score": 1.0, + "content": "does not influence the transition. Second,", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30, + "bbox_fs": [ + 105, + 551, + 462, + 566 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 173, + 566, + 438, + 613 + ], + "lines": [ + { + "bbox": [ + 173, + 566, + 438, + 613 + ], + "spans": [ + { + "bbox": [ + 173, + 566, + 438, + 613 + ], + "score": 0.94, + "content": "\\begin{array} { r l } & { p ( r | Q ^ { * } , T , ( s , a , s ^ { \\prime } ) ) = p ( r | Q ^ { * } , T , ( s , a ) ) } \\\\ & { \\phantom { p s p a c e } = \\mathbb { 1 } _ { \\{ Q ^ { * } ( s , a ) = r + \\gamma \\cdot \\mathbb { E } _ { s ^ { \\prime \\prime } \\sim T ( \\cdot \\cdot \\vert s , a ) } \\operatorname* { m a x } _ { a ^ { \\prime } } Q ^ { * } ( s ^ { \\prime \\prime } , a ^ { \\prime } ) \\} } } \\\\ & { \\phantom { p s p a c e } : = \\mathbb { 1 } ( Q ^ { * } , T ) , } \\end{array}", + "type": "interline_equation", + "image_path": "fb0955c02f65e1e496f065faeae25ba9c05cbf13f12d277ebac49da28a4215c6.jpg" + } + ] + } + ], + "index": 32, + "virtual_lines": [ + { + "bbox": [ + 173, + 566, + 438, + 581.6666666666666 + ], + "spans": [], + "index": 31 + }, + { + "bbox": [ + 173, + 581.6666666666666, + 438, + 597.3333333333333 + ], + "spans": [], + "index": 32 + }, + { + "bbox": [ + 173, + 597.3333333333333, + 438, + 612.9999999999999 + ], + "spans": [], + "index": 33 + } + ] + }, + { + "type": "text", + "bbox": [ + 108, + 614, + 505, + 648 + ], + "lines": [ + { + "bbox": [ + 105, + 613, + 507, + 627 + ], + "spans": [ + { + "bbox": [ + 105, + 613, + 134, + 627 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 135, + 615, + 151, + 627 + ], + "score": 0.9, + "content": "\\mathbb { 1 } _ { \\{ \\cdot \\} }", + "type": "inline_equation" + }, + { + "bbox": [ + 152, + 613, + 431, + 627 + ], + "score": 1.0, + "content": "is the indicator function and in the last equation we abbreviate it as", + "type": "text" + }, + { + "bbox": [ + 432, + 614, + 470, + 626 + ], + "score": 0.92, + "content": "\\mathbb { 1 } ( Q ^ { * } , T )", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 613, + 507, + 627 + ], + "score": 1.0, + "content": ". Substi-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 106, + 626, + 505, + 638 + ], + "spans": [ + { + "bbox": [ + 106, + 626, + 343, + 638 + ], + "score": 1.0, + "content": "tuting (2) and (3) into (1), we obtain the joint posterior of", + "type": "text" + }, + { + "bbox": [ + 343, + 626, + 357, + 637 + ], + "score": 0.93, + "content": "Q ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 626, + 375, + 638 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 376, + 626, + 384, + 635 + ], + "score": 0.85, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 626, + 505, + 638 + ], + "score": 1.0, + "content": "after observing an additional", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 106, + 636, + 230, + 649 + ], + "spans": [ + { + "bbox": [ + 106, + 636, + 223, + 649 + ], + "score": 1.0, + "content": "randomly sampled transition", + "type": "text" + }, + { + "bbox": [ + 223, + 639, + 230, + 646 + ], + "score": 0.76, + "content": "\\tau", + "type": "inline_equation" + } + ], + "index": 36 + } + ], + "index": 35, + "bbox_fs": [ + 105, + 613, + 507, + 649 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 190, + 650, + 420, + 677 + ], + "lines": [ + { + "bbox": [ + 190, + 650, + 420, + 677 + ], + "spans": [ + { + "bbox": [ + 190, + 650, + 420, + 677 + ], + "score": 0.94, + "content": "\\tilde { p } ( Q ^ { * } , T | \\tau ) = \\frac { p ( Q ^ { * } , T ) \\cdot T ( s ^ { \\prime } | s , a ) \\cdot p ( s , a ) } { Z ( \\tau ) } \\cdot \\mathbb { 1 } ( Q ^ { * } , T ) .", + "type": "interline_equation", + "image_path": "9e1f8436cc01293f7e533e96b6775e8cf9531c631fae4a8f68650d6cb6ddb3a0.jpg" + } + ] + } + ], + "index": 37.5, + "virtual_lines": [ + { + "bbox": [ + 190, + 650, + 420, + 663.5 + ], + "spans": [], + "index": 37 + }, + { + "bbox": [ + 190, + 663.5, + 420, + 677.0 + ], + "spans": [], + "index": 38 + } + ] + }, + { + "type": "title", + "bbox": [ + 106, + 689, + 302, + 701 + ], + "lines": [ + { + "bbox": [ + 105, + 687, + 302, + 703 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 237, + 703 + ], + "score": 1.0, + "content": "C.2 APPROXIMATIONS WITH", + "type": "text" + }, + { + "bbox": [ + 237, + 690, + 246, + 701 + ], + "score": 0.86, + "content": "Q", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 687, + 302, + 703 + ], + "score": 1.0, + "content": "-ENSEMBLES", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 39 + }, + { + "type": "text", + "bbox": [ + 106, + 709, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 106, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 709, + 149, + 722 + ], + "score": 1.0, + "content": "The exact", + "type": "text" + }, + { + "bbox": [ + 149, + 710, + 162, + 721 + ], + "score": 0.9, + "content": "Q ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "-posterior update (4) is intractable in high-dimensional RL due to the large space of", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 720, + 504, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 138, + 732 + ], + "score": 0.92, + "content": "( Q ^ { * } , T )", + "type": "inline_equation" + }, + { + "bbox": [ + 138, + 720, + 328, + 733 + ], + "score": 1.0, + "content": ". 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The fourth", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 421, + 506, + 433 + ], + "spans": [ + { + "bbox": [ + 105, + 421, + 506, + 433 + ], + "score": 1.0, + "content": "equation is due to the bounded convergence theorem. The inequality is Jensen’s inequality. The last", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 432, + 337, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 337, + 444 + ], + "score": 1.0, + "content": "equation (9) replaces the limit with an indicator function.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18 + }, + { + "type": "text", + "bbox": [ + 105, + 448, + 504, + 471 + ], + "lines": [ + { + "bbox": [ + 105, + 447, + 505, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 505, + 462 + ], + "score": 1.0, + "content": "A sufficient condition for (8) is to maximize the lower-bound of the posterior distribution in (9) by", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 459, + 471, + 472 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 471, + 472 + ], + "score": 1.0, + "content": "ensuring the indicator function in (9) to hold. We can replace (8) with the following update", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20.5 + }, + { + "type": "interline_equation", + "bbox": [ + 112, + 476, + 498, + 501 + ], + "lines": [ + { + "bbox": [ + 112, + 476, + 498, + 501 + ], + "spans": [ + { + "bbox": [ + 112, + 476, + 498, + 501 + ], + "score": 0.92, + "content": "Q _ { k , i + 1 } \\underset { Q _ { k , i + 1 } } { \\mathrm { a r g m i n } } \\mathbb { E } _ { T \\sim \\tilde { p } ( T | Q _ { k , i } , \\tau ) } [ Q _ { k , i + 1 } ( s , a ) - ( r + \\gamma \\cdot \\mathbb { E } _ { s ^ { \\prime \\prime } \\sim T ( \\cdot | s , a ) } \\operatorname* { m a x } _ { a ^ { \\prime } } Q _ { k , i + 1 } ( s ^ { \\prime \\prime } , a ^ { \\prime } ) ) ] ^ { 2 } .", + "type": "interline_equation", + "image_path": "1cc85ed7d62f85ce8d16545a0958e89d101ba108cc4e25fe6629d2e2ecef62f7.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 112, + 476, + 498, + 501 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 509, + 505, + 555 + ], + "lines": [ + { + "bbox": [ + 105, + 509, + 505, + 522 + ], + "spans": [ + { + "bbox": [ + 105, + 509, + 505, + 522 + ], + "score": 1.0, + "content": "However, (10) is not tractable because the expectation in (10) is taken with respect to the posterior", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 107, + 520, + 506, + 533 + ], + "spans": [ + { + "bbox": [ + 107, + 520, + 158, + 533 + ], + "score": 0.92, + "content": "\\tilde { p } ( T | Q _ { k , i } , \\tau )", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 520, + 223, + 533 + ], + "score": 1.0, + "content": "of the transition", + "type": "text" + }, + { + "bbox": [ + 224, + 521, + 232, + 531 + ], + "score": 0.79, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 520, + 506, + 533 + ], + "score": 1.0, + "content": ". To overcome this challenge, we approximate the posterior update by", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 104, + 531, + 506, + 544 + ], + "spans": [ + { + "bbox": [ + 104, + 531, + 238, + 544 + ], + "score": 1.0, + "content": "reusing the one-sample next state", + "type": "text" + }, + { + "bbox": [ + 238, + 532, + 246, + 541 + ], + "score": 0.88, + "content": "s ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 247, + 531, + 269, + 544 + ], + "score": 1.0, + "content": "from", + "type": "text" + }, + { + "bbox": [ + 269, + 533, + 276, + 541 + ], + "score": 0.73, + "content": "\\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 531, + 418, + 544 + ], + "score": 1.0, + "content": ". 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Instead of updating", + "type": "text" + }, + { + "bbox": [ + 288, + 583, + 301, + 594 + ], + "score": 0.89, + "content": "Q _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 581, + 506, + 596 + ], + "score": 1.0, + "content": "after each transition, we use an experience replay", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 593, + 506, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 133, + 606 + ], + "score": 1.0, + "content": "buffer", + "type": "text" + }, + { + "bbox": [ + 134, + 595, + 143, + 604 + ], + "score": 0.83, + "content": "B", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 593, + 362, + 606 + ], + "score": 1.0, + "content": "to store observed transitions and sample a minibatch", + "type": "text" + }, + { + "bbox": [ + 363, + 594, + 388, + 605 + ], + "score": 0.9, + "content": "B _ { \\mathrm { m i n i } }", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 593, + 446, + 606 + ], + "score": 1.0, + "content": "of transitions", + "type": "text" + }, + { + "bbox": [ + 446, + 594, + 489, + 605 + ], + "score": 0.92, + "content": "( s , a , r , s ^ { \\prime } )", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 593, + 506, + 606 + ], + "score": 1.0, + "content": "for", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 604, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 324, + 617 + ], + "score": 1.0, + "content": "each update. 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We found it had some benefits on", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 709, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 723 + ], + "score": 1.0, + "content": "top of Ensemble Voting, but no uniform additional benefits once already using Q-ensembles on top of", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 719, + 394, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 719, + 394, + 734 + ], + "score": 1.0, + "content": "Double DQN. We describe the approach and our experimental findings.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 35.5 + } + ], + "page_idx": 12, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 26, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 309, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 309, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2018", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 310, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 14, + "width": 13 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 81, + 505, + 127 + ], + "lines": [], + "index": 1.5, + "bbox_fs": [ + 104, + 79, + 507, + 129 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 131, + 505, + 155 + ], + "lines": [ + { + "bbox": [ + 105, + 130, + 507, + 146 + ], + "spans": [ + { + "bbox": [ + 105, + 130, + 226, + 146 + ], + "score": 1.0, + "content": "We derive the update rule for", + "type": "text" + }, + { + "bbox": [ + 227, + 132, + 250, + 144 + ], + "score": 0.93, + "content": "\\left\\{ Q _ { k } \\right\\}", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 130, + 382, + 146 + ], + "score": 1.0, + "content": "after observing a new transition", + "type": "text" + }, + { + "bbox": [ + 382, + 132, + 444, + 144 + ], + "score": 0.92, + "content": "\\tau = ( s , a , r , s ^ { \\prime } )", + "type": "inline_equation" + }, + { + "bbox": [ + 444, + 130, + 498, + 146 + ], + "score": 1.0, + "content": ". At iteration", + "type": "text" + }, + { + "bbox": [ + 498, + 133, + 502, + 142 + ], + "score": 0.6, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 130, + 507, + 146 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 143, + 456, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 143, + 131, + 156 + ], + "score": 1.0, + "content": "given", + "type": "text" + }, + { + "bbox": [ + 131, + 143, + 200, + 156 + ], + "score": 0.94, + "content": "Q ^ { * } = Q _ { k , i } ( \\cdot ; \\theta _ { k } )", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 143, + 268, + 156 + ], + "score": 1.0, + "content": "parametrized by", + "type": "text" + }, + { + "bbox": [ + 268, + 144, + 279, + 154 + ], + "score": 0.88, + "content": "\\theta _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 280, + 143, + 373, + 156 + ], + "score": 1.0, + "content": "the joint probability of", + "type": "text" + }, + { + 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480, + 249 + ], + "score": 0.92, + "content": "\\begin{array} { r l r } { { \\tilde { p } ( Q _ { k , i + 1 } , T | \\tau ) = \\frac { p ( T | Q _ { k , i } ) \\cdot T ( s ^ { \\prime } | s , a ) \\cdot p ( s , a ) } { Z ( \\tau ) } \\cdot \\mathbb { 1 } ( Q _ { k , i + 1 } , T ) . } } \\\\ & { } & { \\tilde { p } ( Q _ { k , i + 1 } | \\tau ) = \\int _ { T } \\tilde { p } ( Q _ { k , i + 1 } , T | \\tau ) \\mathrm { d } T = p ( s , a ) \\cdot \\int _ { T } \\tilde { p } ( T | Q _ { k , i } , \\tau ) \\cdot \\mathbb { 1 } ( Q _ { k , i + 1 } , T ) \\mathrm { d } T . } \\end{array}", + "type": "interline_equation", + "image_path": "2161b1d580d9c2964aed6a05d3774e58926355fa4fa34b423c443d47e0cad512.jpg" + } + ] + } + ], + "index": 9, + "virtual_lines": [ + { + "bbox": [ + 129, + 194, + 480, + 212.33333333333334 + ], + "spans": [], + "index": 8 + }, + { + "bbox": [ + 129, + 212.33333333333334, + 480, + 230.66666666666669 + ], + "spans": [], + "index": 9 + }, + { + "bbox": [ + 129, + 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\\operatorname { m a x } } Q _ { k , i + 1 } ( s ^ { \\prime \\prime } , a ^ { \\prime } ) ] ^ { 2 } ) } \\\\ & { = p ( s , a ) \\cdot \\underset { c \\to + \\infty } { \\operatorname* { i m } } \\mathbb { E } _ { T \\sim \\tilde { p } ( T \\lvert Q _ { k , i } , \\tau ) } \\exp ( - c [ Q _ { k , i + 1 } ( s , a ) - r - \\gamma \\mathbb { E } _ { s ^ { \\prime \\prime } \\sim T ( \\cdot \\lvert s , a ) } \\underset { a ^ { \\prime } } { \\operatorname { m a x } } Q _ { k , i + 1 } ( s ^ { \\prime \\prime } , a ^ { \\prime } ) ] ^ { 2 } ) } \\\\ & { \\geq p ( s , a ) \\cdot \\underset { c \\to + \\infty } { \\operatorname* { i m } } \\exp ( - c \\mathbb { E } _ { T \\sim \\tilde { p } ( T \\lvert Q _ { k , i } , \\tau ) } [ Q _ { k , i + 1 } ( s , a ) - r - \\gamma \\mathbb { E } _ { s ^ { \\prime \\prime } \\sim T ( \\cdot \\lvert s , a ) } \\underset { a ^ { \\prime } } { \\operatorname { m a x } } Q _ { k , i + 1 } ( s ^ { \\prime \\prime } , a ^ { \\prime } ) ] ^ { 2 } ) } \\\\ & { = p ( s , a ) \\cdot \\underset { c \\to + \\infty } { \\operatorname* { l i m } } \\underset { c \\to \\tau ( \\tau \\lvert Q _ { k , i } , \\tau ) } { \\operatorname* { i m } } [ Q _ { k , i + 1 } ( s , a ) - r - \\gamma \\mathbb { E } _ { s ^ { \\prime \\prime } \\sim T ( \\cdot \\lvert s , a ) } \\underset { a ^ { \\prime } } { \\operatorname* { m a x } } Q _ { k , i + 1 } ( s ^ { \\prime \\prime } , a ^ { \\prime } ) ] ^ { 2 } } \\\\ & = p ( s , a ) \\cdot \\mathbb { E } _ { T \\sim \\tilde { p } ( T \\lvert Q _ { k , i } , \\tau ) } [ Q _ k , \\end{array}", + "type": "interline_equation", + "image_path": "aadb27b42c7df13b92fe96ca8b2bd2e3c4f0eb84124981ebf634d4b0c43dae83.jpg" + } + ] + } + ], + "index": 15, + "virtual_lines": [ + { + "bbox": [ + 111, + 311, + 513, + 342.6666666666667 + ], + "spans": [], + "index": 14 + }, + { + "bbox": [ + 111, + 342.6666666666667, + 513, + 374.33333333333337 + ], + "spans": [], + "index": 15 + }, + { + "bbox": [ + 111, + 374.33333333333337, + 513, 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The fourth", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 421, + 506, + 433 + ], + "spans": [ + { + "bbox": [ + 105, + 421, + 506, + 433 + ], + "score": 1.0, + "content": "equation is due to the bounded convergence theorem. The inequality is Jensen’s inequality. The last", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 432, + 337, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 337, + 444 + ], + "score": 1.0, + "content": "equation (9) replaces the limit with an indicator function.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18, + "bbox_fs": [ + 105, + 410, + 506, + 444 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 448, + 504, + 471 + ], + "lines": [ + { + "bbox": [ + 105, + 447, + 505, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 505, + 462 + ], + "score": 1.0, + "content": "A sufficient condition for (8) is to maximize the lower-bound of the posterior distribution in (9) by", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 459, + 471, + 472 + ], + "spans": [ + { + "bbox": [ + 105, + 459, + 471, + 472 + ], + "score": 1.0, + "content": "ensuring the indicator function in (9) to hold. We can replace (8) with the following update", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20.5, + "bbox_fs": [ + 105, + 447, + 505, + 472 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 112, + 476, + 498, + 501 + ], + "lines": [ + { + "bbox": [ + 112, + 476, + 498, + 501 + ], + "spans": [ + { + "bbox": [ + 112, + 476, + 498, + 501 + ], + "score": 0.92, + "content": "Q _ { k , i + 1 } \\underset { Q _ { k , i + 1 } } { \\mathrm { a r g m i n } } \\mathbb { E } _ { T \\sim \\tilde { p } ( T | Q _ { k , i } , \\tau ) } [ Q _ { k , i + 1 } ( s , a ) - ( r + \\gamma \\cdot \\mathbb { E } _ { s ^ { \\prime \\prime } \\sim T ( \\cdot | s , a ) } \\operatorname* { m a x } _ { a ^ { \\prime } } Q _ { k , i + 1 } ( s ^ { \\prime \\prime } , a ^ { \\prime } ) ) ] ^ { 2 } .", + "type": "interline_equation", + "image_path": "1cc85ed7d62f85ce8d16545a0958e89d101ba108cc4e25fe6629d2e2ecef62f7.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 112, + 476, + 498, + 501 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 509, + 505, + 555 + ], + "lines": [ + { + "bbox": [ + 105, + 509, + 505, + 522 + ], + "spans": [ + { + "bbox": [ + 105, + 509, + 505, + 522 + ], + "score": 1.0, + "content": "However, (10) is not tractable because the expectation in (10) is taken with respect to the posterior", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 107, + 520, + 506, + 533 + ], + "spans": [ + { + "bbox": [ + 107, + 520, + 158, + 533 + ], + "score": 0.92, + "content": "\\tilde { p } ( T | Q _ { k , i } , \\tau )", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 520, + 223, + 533 + ], + "score": 1.0, + "content": "of the transition", + "type": "text" + }, + { + "bbox": [ + 224, + 521, + 232, + 531 + ], + "score": 0.79, + "content": "T", + "type": "inline_equation" + }, + { + "bbox": [ + 232, + 520, + 506, + 533 + ], + "score": 1.0, + "content": ". To overcome this challenge, we approximate the posterior update by", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 104, + 531, + 506, + 544 + ], + "spans": [ + { + "bbox": [ + 104, + 531, + 238, + 544 + ], + "score": 1.0, + "content": "reusing the one-sample next state", + "type": "text" + }, + { + "bbox": [ + 238, + 532, + 246, + 541 + ], + "score": 0.88, + "content": "s ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 247, + 531, + 269, + 544 + ], + "score": 1.0, + "content": "from", + "type": "text" + }, + { + "bbox": [ + 269, + 533, + 276, + 541 + ], + "score": 0.73, + "content": "\\tau", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 531, + 418, + 544 + ], + "score": 1.0, + "content": ". Solving the exact minimal for each", + "type": "text" + }, + { + "bbox": [ + 419, + 532, + 447, + 543 + ], + "score": 0.92, + "content": "Q _ { k , i + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 448, + 531, + 506, + 544 + ], + "score": 1.0, + "content": "is impractical,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 542, + 406, + 556 + ], + "spans": [ + { + "bbox": [ + 105, + 542, + 232, + 556 + ], + "score": 1.0, + "content": "thus we take a gradient step on", + "type": "text" + }, + { + "bbox": [ + 232, + 543, + 261, + 555 + ], + "score": 0.92, + "content": "Q _ { k , i + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 542, + 406, + 556 + ], + "score": 1.0, + "content": "according to the following gradient", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 24.5, + "bbox_fs": [ + 104, + 509, + 506, + 556 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 149, + 559, + 461, + 578 + ], + "lines": [ + { + "bbox": [ + 149, + 559, + 461, + 578 + ], + "spans": [ + { + "bbox": [ + 149, + 559, + 461, + 578 + ], + "score": 0.91, + "content": "\\theta _ { k } \\theta _ { k } + \\eta \\cdot ( Q _ { k } ( s , a ; \\theta _ { k } ) - ( r + \\gamma \\cdot \\operatorname* { m a x } _ { a ^ { \\prime } } Q _ { k } ( s ^ { \\prime } , a ^ { \\prime } ; \\theta _ { k } ) ) ) \\nabla _ { \\theta _ { k } } Q _ { k } ( s , a ; \\theta _ { k } ) ,", + "type": "interline_equation", + "image_path": "4bea0dec3faa79e0caee8d86ff93eb1847812a86435a690e5db1076c1a7d9095.jpg" + } + ] + } + ], + "index": 27, + "virtual_lines": [ + { + "bbox": [ + 149, + 559, + 461, + 578 + ], + "spans": [], + "index": 27 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 582, + 505, + 627 + ], + "lines": [ + { + "bbox": [ + 104, + 581, + 506, + 596 + ], + "spans": [ + { + "bbox": [ + 104, + 581, + 133, + 596 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 585, + 140, + 594 + ], + "score": 0.8, + "content": "\\eta", + "type": "inline_equation" + }, + { + "bbox": [ + 141, + 581, + 287, + 596 + ], + "score": 1.0, + "content": "is the step size. Instead of updating", + "type": "text" + }, + { + "bbox": [ + 288, + 583, + 301, + 594 + ], + "score": 0.89, + "content": "Q _ { k }", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 581, + 506, + 596 + ], + "score": 1.0, + "content": "after each transition, we use an experience replay", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 593, + 506, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 133, + 606 + ], + "score": 1.0, + "content": "buffer", + "type": "text" + }, + { + "bbox": [ + 134, + 595, + 143, + 604 + ], + "score": 0.83, + "content": "B", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 593, + 362, + 606 + ], + "score": 1.0, + "content": "to store observed transitions and sample a minibatch", + "type": "text" + }, + { + "bbox": [ + 363, + 594, + 388, + 605 + ], + "score": 0.9, + "content": "B _ { \\mathrm { m i n i } }", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 593, + 446, + 606 + ], + "score": 1.0, + "content": "of transitions", + "type": "text" + }, + { + "bbox": [ + 446, + 594, + 489, + 605 + ], + "score": 0.92, + "content": "( s , a , r , s ^ { \\prime } )", + "type": "inline_equation" + }, + { + "bbox": [ + 489, + 593, + 506, + 606 + ], + "score": 1.0, + "content": "for", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 604, + 505, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 324, + 617 + ], + "score": 1.0, + "content": "each update. In this case, the batched update of each", + "type": "text" + }, + { + "bbox": [ + 324, + 605, + 343, + 617 + ], + "score": 0.91, + "content": "Q _ { k , i }", + "type": "inline_equation" + }, + { + "bbox": [ + 343, + 604, + 355, + 617 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 355, + 605, + 384, + 617 + ], + "score": 0.92, + "content": "Q _ { k , i + 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 604, + 505, + 617 + ], + "score": 1.0, + "content": "becomes a standard Bellman", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 615, + 137, + 628 + ], + "spans": [ + { + "bbox": [ + 104, + 615, + 137, + 628 + ], + "score": 1.0, + "content": "update", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 29.5, + "bbox_fs": [ + 104, + 581, + 506, + 628 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 112, + 630, + 497, + 649 + ], + "lines": [ + { + "bbox": [ + 112, + 630, + 497, + 649 + ], + "spans": [ + { + "bbox": [ + 112, + 630, + 497, + 649 + ], + "score": 0.91, + "content": "\\theta _ { k } \\gets \\theta _ { k } + \\eta \\cdot \\mathbb { E } _ { ( s , a , r , s ^ { \\prime } ) \\in B _ { \\operatorname* { m i n } } } \\big [ \\big ( Q _ { k } \\big ( s , a ; \\theta _ { k } \\big ) - \\big ( r + \\gamma \\cdot \\operatorname* { m a x } _ { a ^ { \\prime } } Q _ { k } \\big ( s ^ { \\prime } , a ^ { \\prime } ; \\theta _ { k } \\big ) \\big ) \\big ) \\nabla _ { \\theta _ { k } } Q _ { k } \\big ( s , a ; \\theta _ { k } \\big ) \\big ] .", + "type": "interline_equation", + "image_path": "d70b3f36248f6004e8bc44efa652a042d00c1404f0bb9b03800759a5a683cd41.jpg" + } + ] + } + ], + "index": 32, + "virtual_lines": [ + { + "bbox": [ + 112, + 630, + 497, + 649 + ], + "spans": [], + "index": 32 + } + ] + }, + { + "type": "title", + "bbox": [ + 108, + 663, + 258, + 676 + ], + "lines": [ + { + "bbox": [ + 105, + 662, + 259, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 662, + 259, + 678 + ], + "score": 1.0, + "content": "D INFOGAIN EXPLORATION", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 107, + 687, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 505, + 700 + ], + "score": 1.0, + "content": "In this section, we also studied an “InfoGain” exploration bonus, which encourages agents to gain", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 699, + 505, + 711 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 195, + 711 + ], + "score": 1.0, + "content": "information about the", + "type": "text" + }, + { + "bbox": [ + 195, + 699, + 209, + 710 + ], + "score": 0.9, + "content": "Q ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 699, + 505, + 711 + ], + "score": 1.0, + "content": "-function and examine its effectiveness. We found it had some benefits on", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 709, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 723 + ], + "score": 1.0, + "content": "top of Ensemble Voting, but no uniform additional benefits once already using Q-ensembles on top of", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 719, + 394, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 719, + 394, + 734 + ], + "score": 1.0, + "content": "Double DQN. We describe the approach and our experimental findings.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 35.5, + "bbox_fs": [ + 105, + 687, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 105, + 82, + 504, + 105 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "Similar to Sun et al. (2011), we define the information gain from observing an additional transition", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 107, + 95, + 131, + 106 + ], + "spans": [ + { + "bbox": [ + 107, + 96, + 117, + 105 + ], + "score": 0.76, + "content": "\\tau _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 118, + 95, + 131, + 106 + ], + "score": 1.0, + "content": "as", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "interline_equation", + "bbox": [ + 178, + 105, + 431, + 120 + ], + "lines": [ + { + "bbox": [ + 178, + 105, + 431, + 120 + ], + "spans": [ + { + "bbox": [ + 178, + 105, + 431, + 120 + ], + "score": 0.88, + "content": "H _ { \\tau _ { t } | \\tau _ { 1 } , \\dots , \\tau _ { n - 1 } } = D _ { K L } ( \\tilde { p } ( Q ^ { * } | \\tau _ { 1 } , \\dots , \\tau _ { n } ) | | \\tilde { p } ( Q ^ { * } | \\tau _ { 1 } , \\dots , \\tau _ { n - 1 } ) )", + "type": "interline_equation", + "image_path": "92d0a80953a40c9a25364420585521da683a2bbc55554267f030e06126cf824a.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 178, + 105, + 431, + 120 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 124, + 502, + 147 + ], + "lines": [ + { + "bbox": [ + 105, + 123, + 505, + 138 + ], + "spans": [ + { + "bbox": [ + 105, + 123, + 133, + 138 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 125, + 203, + 137 + ], + "score": 0.91, + "content": "\\tilde { p } ( Q ^ { * } | \\tau _ { 1 } , \\dots , \\tau _ { n } )", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 123, + 326, + 138 + ], + "score": 1.0, + "content": "is the posterior distribution of", + "type": "text" + }, + { + "bbox": [ + 326, + 125, + 339, + 136 + ], + "score": 0.9, + "content": "Q ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 123, + 505, + 138 + ], + "score": 1.0, + "content": "after observing a sequence of transitions", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 135, + 277, + 149 + ], + "spans": [ + { + "bbox": [ + 106, + 136, + 155, + 148 + ], + "score": 0.9, + "content": "\\left( \\tau _ { 1 } , \\dots , \\tau _ { n } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 135, + 277, + 149 + ], + "score": 1.0, + "content": ". The total information gain is", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + }, + { + "type": "interline_equation", + "bbox": [ + 232, + 154, + 378, + 177 + ], + "lines": [ + { + "bbox": [ + 232, + 154, + 378, + 177 + ], + "spans": [ + { + "bbox": [ + 232, + 154, + 378, + 177 + ], + "score": 0.94, + "content": "H _ { \\tau _ { 1 } , \\dots , \\tau _ { N } } = \\sum _ { n = 1 } ^ { N } H _ { \\tau _ { n } | \\tau _ { 1 } , \\dots , \\tau _ { n - 1 } } .", + "type": "interline_equation", + "image_path": "f989e42d3a3df5cd3eacb6baeb2a0fdf4ed66ccb011a106b222986f026240b2d.jpg" + } + ] + } + ], + "index": 5, + "virtual_lines": [ + { + "bbox": [ + 232, + 154, + 378, + 177 + ], + "spans": [], + "index": 5 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 182, + 505, + 217 + ], + "lines": [ + { + "bbox": [ + 105, + 182, + 506, + 196 + ], + "spans": [ + { + "bbox": [ + 105, + 182, + 376, + 196 + ], + "score": 1.0, + "content": "Our Ensemble Voting, Algorithm 1, does not maintain the posterior", + "type": "text" + }, + { + "bbox": [ + 377, + 184, + 383, + 194 + ], + "score": 0.83, + "content": "\\tilde { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 182, + 506, + 196 + ], + "score": 1.0, + "content": ", thus we cannot calculate (11)", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 194, + 505, + 206 + ], + "spans": [ + { + "bbox": [ + 106, + 194, + 505, + 206 + ], + "score": 1.0, + "content": "explicitly. Instead, inspired by Lakshminarayanan et al. (2016), we define an InfoGain exploration", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 204, + 358, + 218 + ], + "spans": [ + { + "bbox": [ + 105, + 204, + 290, + 218 + ], + "score": 1.0, + "content": "bonus that measures the disagreement among", + "type": "text" + }, + { + "bbox": [ + 290, + 205, + 313, + 217 + ], + "score": 0.91, + "content": "\\left\\{ Q _ { k } \\right\\}", + "type": "inline_equation" + }, + { + "bbox": [ + 314, + 204, + 358, + 218 + ], + "score": 1.0, + "content": ". Note that", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7 + }, + { + "type": "interline_equation", + "bbox": [ + 208, + 223, + 402, + 237 + ], + "lines": [ + { + "bbox": [ + 208, + 223, + 402, + 237 + ], + "spans": [ + { + "bbox": [ + 208, + 223, + 402, + 237 + ], + "score": 0.9, + "content": "H _ { \\tau _ { 1 } , \\dots , \\tau _ { N } } + \\mathsf { H } ( \\tilde { p } ( Q ^ { * } | \\tau _ { 1 } , \\dots , \\tau _ { N } ) ) = \\mathsf { H } ( p ( Q ^ { * } ) ) ,", + "type": "interline_equation", + "image_path": "da85240ec8e2f545f5e402235f477ab4c68b55ff03768b899da2490a4bc78ae3.jpg" + } + ] + } + ], + "index": 9, + "virtual_lines": [ + { + "bbox": [ + 208, + 223, + 402, + 237 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 243, + 505, + 299 + ], + "lines": [ + { + "bbox": [ + 105, + 241, + 506, + 257 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 133, + 257 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 244, + 152, + 255 + ], + "score": 0.9, + "content": "\\mathsf { H } ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 152, + 241, + 221, + 257 + ], + "score": 1.0, + "content": "is the entropy. If", + "type": "text" + }, + { + "bbox": [ + 222, + 244, + 261, + 255 + ], + "score": 0.91, + "content": "H _ { \\tau _ { 1 } , \\dots , \\tau _ { N } }", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 243, + 506, + 256 + ], + "score": 1.0, + "content": "is small, then the posterior distribution has high entropy and", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 254, + 506, + 268 + ], + "spans": [ + { + "bbox": [ + 105, + 254, + 238, + 268 + ], + "score": 1.0, + "content": "high residual information. Since", + "type": "text" + }, + { + "bbox": [ + 239, + 255, + 262, + 267 + ], + "score": 0.9, + "content": "\\left\\{ Q _ { k } \\right\\}", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 254, + 506, + 268 + ], + "score": 1.0, + "content": "are approximate samples from the posterior, high entropy of", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 265, + 506, + 279 + ], + "spans": [ + { + "bbox": [ + 106, + 265, + 298, + 279 + ], + "score": 1.0, + "content": "the posterior leads to large discrepancy among", + "type": "text" + }, + { + "bbox": [ + 299, + 265, + 322, + 277 + ], + "score": 0.92, + "content": "\\left\\{ Q _ { k } \\right\\}", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 265, + 506, + 279 + ], + "score": 1.0, + "content": ". 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(2011), we define the information gain from observing an additional transition", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 107, + 95, + 131, + 106 + ], + "spans": [ + { + "bbox": [ + 107, + 96, + 117, + 105 + ], + "score": 0.76, + "content": "\\tau _ { n }", + "type": "inline_equation" + }, + { + "bbox": [ + 118, + 95, + 131, + 106 + ], + "score": 1.0, + "content": "as", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5, + "bbox_fs": [ + 106, + 82, + 505, + 106 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 178, + 105, + 431, + 120 + ], + "lines": [ + { + "bbox": [ + 178, + 105, + 431, + 120 + ], + "spans": [ + { + "bbox": [ + 178, + 105, + 431, + 120 + ], + "score": 0.88, + "content": "H _ { \\tau _ { t } | \\tau _ { 1 } , \\dots , \\tau _ { n - 1 } } = D _ { K L } ( \\tilde { p } ( Q ^ { * } | \\tau _ { 1 } , \\dots , \\tau _ { n } ) | | \\tilde { p } ( Q ^ { * } | \\tau _ { 1 } , \\dots , \\tau _ { n - 1 } ) )", + "type": "interline_equation", + "image_path": "92d0a80953a40c9a25364420585521da683a2bbc55554267f030e06126cf824a.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 178, + 105, + 431, + 120 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 124, + 502, + 147 + ], + "lines": [ + { + "bbox": [ + 105, + 123, + 505, + 138 + ], + "spans": [ + { + "bbox": [ + 105, + 123, + 133, + 138 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 125, + 203, + 137 + ], + "score": 0.91, + "content": "\\tilde { p } ( Q ^ { * } | \\tau _ { 1 } , \\dots , \\tau _ { n } )", + "type": "inline_equation" + }, + { + "bbox": [ + 203, + 123, + 326, + 138 + ], + "score": 1.0, + "content": "is the posterior distribution of", + "type": "text" + }, + { + "bbox": [ + 326, + 125, + 339, + 136 + ], + "score": 0.9, + "content": "Q ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 123, + 505, + 138 + ], + "score": 1.0, + "content": "after observing a sequence of transitions", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 135, + 277, + 149 + ], + "spans": [ + { + "bbox": [ + 106, + 136, + 155, + 148 + ], + "score": 0.9, + "content": "\\left( \\tau _ { 1 } , \\dots , \\tau _ { n } \\right)", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 135, + 277, + 149 + ], + "score": 1.0, + "content": ". The total information gain is", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5, + "bbox_fs": [ + 105, + 123, + 505, + 149 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 232, + 154, + 378, + 177 + ], + "lines": [ + { + "bbox": [ + 232, + 154, + 378, + 177 + ], + "spans": [ + { + "bbox": [ + 232, + 154, + 378, + 177 + ], + "score": 0.94, + "content": "H _ { \\tau _ { 1 } , \\dots , \\tau _ { N } } = \\sum _ { n = 1 } ^ { N } H _ { \\tau _ { n } | \\tau _ { 1 } , \\dots , \\tau _ { n - 1 } } .", + "type": "interline_equation", + "image_path": "f989e42d3a3df5cd3eacb6baeb2a0fdf4ed66ccb011a106b222986f026240b2d.jpg" + } + ] + } + ], + "index": 5, + "virtual_lines": [ + { + "bbox": [ + 232, + 154, + 378, + 177 + ], + "spans": [], + "index": 5 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 182, + 505, + 217 + ], + "lines": [ + { + "bbox": [ + 105, + 182, + 506, + 196 + ], + "spans": [ + { + "bbox": [ + 105, + 182, + 376, + 196 + ], + "score": 1.0, + "content": "Our Ensemble Voting, Algorithm 1, does not maintain the posterior", + "type": "text" + }, + { + "bbox": [ + 377, + 184, + 383, + 194 + ], + "score": 0.83, + "content": "\\tilde { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 182, + 506, + 196 + ], + "score": 1.0, + "content": ", thus we cannot calculate (11)", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 194, + 505, + 206 + ], + "spans": [ + { + "bbox": [ + 106, + 194, + 505, + 206 + ], + "score": 1.0, + "content": "explicitly. Instead, inspired by Lakshminarayanan et al. (2016), we define an InfoGain exploration", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 204, + 358, + 218 + ], + "spans": [ + { + "bbox": [ + 105, + 204, + 290, + 218 + ], + "score": 1.0, + "content": "bonus that measures the disagreement among", + "type": "text" + }, + { + "bbox": [ + 290, + 205, + 313, + 217 + ], + "score": 0.91, + "content": "\\left\\{ Q _ { k } \\right\\}", + "type": "inline_equation" + }, + { + "bbox": [ + 314, + 204, + 358, + 218 + ], + "score": 1.0, + "content": ". Note that", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7, + "bbox_fs": [ + 105, + 182, + 506, + 218 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 208, + 223, + 402, + 237 + ], + "lines": [ + { + "bbox": [ + 208, + 223, + 402, + 237 + ], + "spans": [ + { + "bbox": [ + 208, + 223, + 402, + 237 + ], + "score": 0.9, + "content": "H _ { \\tau _ { 1 } , \\dots , \\tau _ { N } } + \\mathsf { H } ( \\tilde { p } ( Q ^ { * } | \\tau _ { 1 } , \\dots , \\tau _ { N } ) ) = \\mathsf { H } ( p ( Q ^ { * } ) ) ,", + "type": "interline_equation", + "image_path": "da85240ec8e2f545f5e402235f477ab4c68b55ff03768b899da2490a4bc78ae3.jpg" + } + ] + } + ], + "index": 9, + "virtual_lines": [ + { + "bbox": [ + 208, + 223, + 402, + 237 + ], + "spans": [], + "index": 9 + } + ] + }, + { + "type": "text", + "bbox": [ + 106, + 243, + 505, + 299 + ], + "lines": [ + { + "bbox": [ + 105, + 241, + 506, + 257 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 133, + 257 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 244, + 152, + 255 + ], + "score": 0.9, + "content": "\\mathsf { H } ( \\cdot )", + "type": "inline_equation" + }, + { + "bbox": [ + 152, + 241, + 221, + 257 + ], + "score": 1.0, + "content": "is the entropy. If", + "type": "text" + }, + { + "bbox": [ + 222, + 244, + 261, + 255 + ], + "score": 0.91, + "content": "H _ { \\tau _ { 1 } , \\dots , \\tau _ { N } }", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 243, + 506, + 256 + ], + "score": 1.0, + "content": "is small, then the posterior distribution has high entropy and", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 254, + 506, + 268 + ], + "spans": [ + { + "bbox": [ + 105, + 254, + 238, + 268 + ], + "score": 1.0, + "content": "high residual information. Since", + "type": "text" + }, + { + "bbox": [ + 239, + 255, + 262, + 267 + ], + "score": 0.9, + "content": "\\left\\{ Q _ { k } \\right\\}", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 254, + 506, + 268 + ], + "score": 1.0, + "content": "are approximate samples from the posterior, high entropy of", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 265, + 506, + 279 + ], + "spans": [ + { + "bbox": [ + 106, + 265, + 298, + 279 + ], + "score": 1.0, + "content": "the posterior leads to large discrepancy among", + "type": "text" + }, + { + "bbox": [ + 299, + 265, + 322, + 277 + ], + "score": 0.92, + "content": "\\left\\{ Q _ { k } \\right\\}", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 265, + 506, + 279 + ], + "score": 1.0, + "content": ". 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Next, calculate the average Boltzmann distribution", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17, + "bbox_fs": [ + 105, + 341, + 476, + 355 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 238, + 360, + 372, + 384 + ], + "lines": [ + { + "bbox": [ + 238, + 360, + 372, + 384 + ], + "spans": [ + { + "bbox": [ + 238, + 360, + 372, + 384 + ], + "score": 0.94, + "content": "P _ { \\mathsf { T } , \\mathrm { a v g } } = \\frac { 1 } { K } \\cdot \\sum _ { k = 1 } ^ { K } P _ { \\mathsf { T } , k } ( a | s ) .", + "type": "interline_equation", + "image_path": "fd7f46f57b39c6d76b6ea0551a04fbcf7f0fff4f0f12971f16ea7d9371731d99.jpg" + } + ] + } + ], + "index": 18, + "virtual_lines": [ + { + "bbox": [ + 238, + 360, + 372, + 384 + ], + "spans": [], + "index": 18 + } + ] + }, + { + "type": "text", + "bbox": [ + 105, + 390, + 462, + 404 + ], + "lines": [ + { + "bbox": [ + 104, + 387, + 462, + 408 + ], + "spans": [ + { + "bbox": [ + 104, + 387, + 378, + 408 + ], + "score": 1.0, + "content": "The InfoGain exploration bonus is the average KL-divergence from", + "type": "text" + }, + { + "bbox": [ + 378, + 390, + 423, + 404 + ], + "score": 0.93, + "content": "\\{ P _ { \\mathsf { T } , k } \\} _ { k = 1 } ^ { K }", + "type": "inline_equation" + }, + { + "bbox": [ + 423, + 387, + 434, + 408 + ], + "score": 1.0, + "content": "to", + "type": "text" + }, + { + "bbox": [ + 434, + 392, + 462, + 404 + ], + "score": 0.83, + "content": "P _ { \\mathrm { { T , a v g } } }", + "type": "inline_equation" + } + ], + "index": 19 + } + ], + "index": 19, + "bbox_fs": [ + 104, + 387, + 462, + 408 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 221, + 410, + 390, + 434 + ], + "lines": [ + { + "bbox": [ + 221, + 410, + 390, + 434 + ], + "spans": [ + { + "bbox": [ + 221, + 410, + 390, + 434 + ], + "score": 0.93, + "content": "b _ { \\mathsf { T } } ( s ) = \\frac { 1 } { K } \\cdot \\sum _ { k = 1 } ^ { K } \\mathrm { D } _ { K L } [ P _ { \\mathsf { T } , k } | | P _ { \\mathsf { T } , \\mathrm { a v g } } ] .", + "type": "interline_equation", + "image_path": "0b2a7fedeb71301cadf9a63b4396515b553a6a5e5eaf3244c7354a20783153fb.jpg" + } + ] + } + ], + "index": 20, + "virtual_lines": [ + { + "bbox": [ + 221, + 410, + 390, + 434 + ], + "spans": [], + "index": 20 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 440, + 200, + 451 + ], + "lines": [ + { + "bbox": [ + 106, + 439, + 201, + 452 + ], + "spans": [ + { + "bbox": [ + 106, + 439, + 201, + 452 + ], + "score": 1.0, + "content": "The modified reward is", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21, + "bbox_fs": [ + 106, + 439, + 201, + 452 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 240, + 452, + 371, + 466 + ], + "lines": [ + { + "bbox": [ + 240, + 452, + 371, + 466 + ], + "spans": [ + { + "bbox": [ + 240, + 452, + 371, + 466 + ], + "score": 0.94, + "content": "\\hat { r } ( s , a , s ^ { \\prime } ) = r ( s , a ) + \\rho \\cdot b \\tau ( s ) ,", + "type": "interline_equation", + "image_path": "2271a374b1bd2f8522eea625ab54732f3037b9931d39f79a368d1c20741583c3.jpg" + } + ] + } + ], + "index": 22, + "virtual_lines": [ + { + "bbox": [ + 240, + 452, + 371, + 466 + ], + "spans": [], + "index": 22 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 470, + 403, + 482 + ], + "lines": [ + { + "bbox": [ + 105, + 469, + 404, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 133, + 484 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 471, + 165, + 482 + ], + "score": 0.93, + "content": "\\rho \\in \\mathbb { R } _ { + }", + "type": "inline_equation" + }, + { + "bbox": [ + 165, + 469, + 404, + 484 + ], + "score": 1.0, + "content": "is a hyperparameter that controls the degree of exploration.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23, + "bbox_fs": [ + 105, + 469, + 404, + 484 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 487, + 505, + 532 + ], + "lines": [ + { + "bbox": [ + 104, + 486, + 506, + 501 + ], + "spans": [ + { + "bbox": [ + 104, + 486, + 196, + 501 + ], + "score": 1.0, + "content": "The exploration bonus", + "type": "text" + }, + { + "bbox": [ + 196, + 487, + 223, + 499 + ], + "score": 0.92, + "content": "b _ { \\mathsf { T } } ( s _ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 223, + 486, + 375, + 501 + ], + "score": 1.0, + "content": "encourages the agent to explore where", + "type": "text" + }, + { + "bbox": [ + 376, + 487, + 399, + 499 + ], + "score": 0.91, + "content": "\\left\\{ Q _ { k } \\right\\}", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 486, + 506, + 501 + ], + "score": 1.0, + "content": "disagree. 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