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The goal is to plan an intervention schedule that maximizes the expected", + "type": "text" + } + ], + "index": 16, + "is_list_start_line": true + }, + { + "bbox": [ + 90, + 387, + 469, + 399 + ], + "spans": [ + { + "bbox": [ + 90, + 389, + 99, + 398 + ], + "score": 1.0, + "content": "11", + "type": "text" + }, + { + "bbox": [ + 141, + 387, + 469, + 399 + ], + "score": 1.0, + "content": "reward while satisfying budget constraints on each worker as well as fairness in", + "type": "text" + } + ], + "index": 17, + "is_list_start_line": true + }, + { + "bbox": [ + 90, + 398, + 469, + 410 + ], + "spans": [ + { + "bbox": [ + 90, + 400, + 99, + 409 + ], + "score": 1.0, + "content": "12", + "type": "text" + }, + { + "bbox": [ + 142, + 398, + 469, + 410 + ], + "score": 1.0, + "content": "terms of the load assigned to each worker. Our contributions are two-fold: (1) we", + "type": "text" + } + ], + "index": 18, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 408, + 470, + 421 + ], + "spans": [ + { + "bbox": [ + 88, + 410, + 100, + 421 + ], + "score": 1.0, + "content": "13", + "type": "text" + }, + { + "bbox": [ + 141, + 408, + 470, + 421 + ], + "score": 1.0, + "content": "provide a multi-worker extension of the Whittle index to tackle heterogeneous", + "type": "text" + } + ], + "index": 19, + "is_list_start_line": true + }, + { + "bbox": [ + 90, + 419, + 470, + 433 + ], + "spans": [ + { + "bbox": [ + 90, + 422, + 99, + 431 + ], + "score": 1.0, + "content": "14", + "type": "text" + }, + { + "bbox": [ + 141, + 419, + 470, + 433 + ], + "score": 1.0, + "content": "costs and per-worker budget and (2) we develop an index-based scheduling policy", + "type": "text" + } + ], + "index": 20, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 430, + 469, + 443 + ], + "spans": [ + { + "bbox": [ + 88, + 432, + 100, + 443 + ], + "score": 1.0, + "content": "15", + "type": "text" + }, + { + "bbox": [ + 142, + 430, + 469, + 442 + ], + "score": 1.0, + "content": "to achieve fairness. Further, we evaluate our method on various cost structures and", + "type": "text" + } + ], + "index": 21, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 441, + 470, + 453 + ], + "spans": [ + { + "bbox": [ + 88, + 443, + 100, + 453 + ], + "score": 1.0, + "content": "16", + "type": "text" + }, + { + "bbox": [ + 142, + 441, + 470, + 453 + ], + "score": 1.0, + "content": "show that our method significantly outperforms other baselines in terms of fairness", + "type": "text" + } + ], + "index": 22, + "is_list_start_line": true + }, + { + "bbox": [ + 90, + 452, + 339, + 465 + ], + "spans": [ + { + "bbox": [ + 90, + 455, + 99, + 464 + ], + "score": 1.0, + "content": "17", + "type": "text" + }, + { + "bbox": [ + 142, + 452, + 339, + 465 + ], + "score": 1.0, + "content": "without sacrificing much in reward accumulated.", + "type": "text" + } + ], + "index": 23, + "is_list_start_line": true + } + ], + "index": 15, + "bbox_fs": [ + 88, + 277, + 470, + 465 + ] + }, + { + "type": "title", + "bbox": [ + 91, + 485, + 191, + 499 + ], + "lines": [ + { + "bbox": [ + 87, + 484, + 192, + 501 + ], + "spans": [ + { + "bbox": [ + 87, + 484, + 192, + 501 + ], + "score": 1.0, + "content": "18 1 Introduction", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "index", + "bbox": [ + 89, + 510, + 505, + 664 + ], + "lines": [ + { + "bbox": [ + 90, + 510, + 505, + 523 + ], + "spans": [ + { + "bbox": [ + 90, + 512, + 100, + 522 + ], + "score": 1.0, + "content": "19", + "type": "text" + }, + { + "bbox": [ + 105, + 510, + 505, + 523 + ], + "score": 1.0, + "content": "Restless multi-armed bandits (RMABs) Whittle [1988] have been used for sequential planning, where", + "type": "text" + } + ], + "index": 25, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 521, + 506, + 534 + ], + "spans": [ + { + "bbox": [ + 89, + 524, + 99, + 533 + ], + "score": 1.0, + "content": "20", + "type": "text" + }, + { + "bbox": [ + 105, + 521, + 247, + 534 + ], + "score": 1.0, + "content": "a planner allocates a limited set of", + "type": "text" + }, + { + "bbox": [ + 247, + 522, + 259, + 532 + ], + "score": 0.36, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 521, + 380, + 534 + ], + "score": 1.0, + "content": "intervention resources across", + "type": "text" + }, + { + "bbox": [ + 381, + 522, + 391, + 532 + ], + "score": 0.63, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 391, + 521, + 506, + 534 + ], + "score": 1.0, + "content": "independent heterogeneous", + "type": "text" + } + ], + "index": 26, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 533, + 505, + 544 + ], + "spans": [ + { + "bbox": [ + 88, + 534, + 99, + 543 + ], + "score": 1.0, + "content": "21", + "type": "text" + }, + { + "bbox": [ + 105, + 533, + 505, + 544 + ], + "score": 1.0, + "content": "arms (Markov Decision processes) at each time step in order to maximize the long-term expected", + "type": "text" + } + ], + "index": 27, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 543, + 505, + 555 + ], + "spans": [ + { + "bbox": [ + 88, + 545, + 99, + 555 + ], + "score": 1.0, + "content": "22", + "type": "text" + }, + { + "bbox": [ + 106, + 543, + 505, + 555 + ], + "score": 1.0, + "content": "reward. The term restless denotes that the arms undergo state-transitions even when they are not", + "type": "text" + } + ], + "index": 28, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 554, + 505, + 567 + ], + "spans": [ + { + "bbox": [ + 88, + 556, + 99, + 566 + ], + "score": 1.0, + "content": "23", + "type": "text" + }, + { + "bbox": [ + 105, + 554, + 505, + 567 + ], + "score": 1.0, + "content": "acted upon (with a different probability than when they are acted upon). RMABs have been receiving", + "type": "text" + } + ], + "index": 29, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 565, + 505, + 577 + ], + "spans": [ + { + "bbox": [ + 88, + 567, + 100, + 577 + ], + "score": 1.0, + "content": "24", + "type": "text" + }, + { + "bbox": [ + 105, + 565, + 425, + 577 + ], + "score": 1.0, + "content": "increasing attention across a wide range of applications such as maintenance", + "type": "text" + }, + { + "bbox": [ + 426, + 565, + 505, + 577 + ], + "score": 1.0, + "content": "[Abbou and Makis,", + "type": "text" + } + ], + "index": 30, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 575, + 504, + 589 + ], + "spans": [ + { + "bbox": [ + 88, + 577, + 100, + 588 + ], + "score": 1.0, + "content": "25", + "type": "text" + }, + { + "bbox": [ + 105, + 575, + 504, + 589 + ], + "score": 1.0, + "content": "2019], recommendation systems Meshram et al. 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Although, rangers", + "type": "text" + } + ], + "index": 32, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 598, + 506, + 611 + ], + "spans": [ + { + "bbox": [ + 88, + 600, + 99, + 609 + ], + "score": 1.0, + "content": "27", + "type": "text" + }, + { + "bbox": [ + 105, + 598, + 506, + 611 + ], + "score": 1.0, + "content": "in anti-poaching, healthcare workers in health intervention planning, and supervisors in machine", + "type": "text" + } + ], + "index": 33, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 609, + 505, + 621 + ], + "spans": [ + { + "bbox": [ + 88, + 610, + 100, + 621 + ], + "score": 1.0, + "content": "28", + "type": "text" + }, + { + "bbox": [ + 106, + 609, + 505, + 621 + ], + "score": 1.0, + "content": "maintenance are all commonly cited examples of human workforce used as intervention resources, the", + "type": "text" + } + ], + "index": 34, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 620, + 505, + 631 + ], + "spans": [ + { + "bbox": [ + 88, + 621, + 99, + 631 + ], + "score": 1.0, + "content": "29", + "type": "text" + }, + { + "bbox": [ + 106, + 620, + 505, + 631 + ], + "score": 1.0, + "content": "literature has so far ignored one key reality that the human workforce is heterogeneous—each worker", + "type": "text" + } + ], + "index": 35, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 630, + 506, + 643 + ], + "spans": [ + { + "bbox": [ + 88, + 633, + 99, + 642 + ], + "score": 1.0, + "content": "30", + "type": "text" + }, + { + "bbox": [ + 105, + 630, + 506, + 643 + ], + "score": 1.0, + "content": "has their own workload constraints and needs to commit a dedicated time duration for intervening on", + "type": "text" + } + ], + "index": 36, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 642, + 505, + 653 + ], + "spans": [ + { + "bbox": [ + 88, + 644, + 99, + 653 + ], + "score": 1.0, + "content": "31", + "type": "text" + }, + { + "bbox": [ + 105, + 642, + 505, + 653 + ], + "score": 1.0, + "content": "an arm. Thus, it is critical to restrict intervention workload for each worker and balance the workload", + "type": "text" + } + ], + "index": 37, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 652, + 441, + 665 + ], + "spans": [ + { + "bbox": [ + 88, + 655, + 100, + 664 + ], + "score": 1.0, + "content": "32", + "type": "text" + }, + { + "bbox": [ + 104, + 652, + 441, + 665 + ], + "score": 1.0, + "content": "across them, while also ensuring high effectiveness (reward) of the planning policy.", + "type": "text" + } + ], + "index": 38, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 668, + 506, + 682 + ], + "spans": [ + { + "bbox": [ + 89, + 671, + 100, + 681 + ], + "score": 1.0, + "content": "33", + "type": "text" + }, + { + "bbox": [ + 105, + 668, + 506, + 682 + ], + "score": 1.0, + "content": "RMAB literature does not consider this heterogeneity and mostly focuses on selecting best arms", + "type": "text" + } + ], + "index": 39, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 680, + 505, + 693 + ], + "spans": [ + { + "bbox": [ + 89, + 682, + 100, + 691 + ], + "score": 1.0, + "content": "34", + "type": "text" + }, + { + "bbox": [ + 106, + 680, + 505, + 693 + ], + "score": 1.0, + "content": "assuming that all intervention resources (workers) are interchangeable, i.e., as from a single pool", + "type": "text" + } + ], + "index": 40, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 690, + 505, + 703 + ], + "spans": [ + { + "bbox": [ + 89, + 693, + 100, + 703 + ], + "score": 1.0, + "content": "35", + "type": "text" + }, + { + "bbox": [ + 106, + 690, + 505, + 703 + ], + "score": 1.0, + "content": "(homogeneous). 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One concrete example is anti-poaching intervention planning Qian et al.", + "type": "text", + "cross_page": true + } + ], + "index": 0, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 83, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 88, + 86, + 100, + 95 + ], + "score": 1.0, + "content": "38", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 107, + 83, + 138, + 96 + ], + "score": 0.57, + "content": "\\pm \\pm { \\sqrt { 2 0 1 6 \\mathrm { a } } } ]", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 139, + 83, + 159, + 96 + ], + "score": 1.0, + "content": "with", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 160, + 84, + 170, + 93 + ], + "score": 0.79, + "content": "N", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 170, + 83, + 506, + 96 + ], + "score": 1.0, + "content": "areas in a national park where timely interventions (patrols) are required to detect as", + "type": "text", + "cross_page": true + } + ], + "index": 1, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 95, + 504, + 106 + ], + "spans": [ + { + "bbox": [ + 88, + 97, + 100, + 106 + ], + "score": 1.0, + "content": "39", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 95, + 492, + 106 + ], + "score": 1.0, + "content": "many snares as possible across all the areas. 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However, each ranger may incur heterogeneous cost (e.g., distance travelled,", + "type": "text", + "cross_page": true + } + ], + "index": 4, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 127, + 506, + 140 + ], + "spans": [ + { + "bbox": [ + 88, + 128, + 100, + 139 + ], + "score": 1.0, + "content": "42", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 127, + 506, + 140 + ], + "score": 1.0, + "content": "when assigned to intervene on a particular area) and the total cost incurred by any ranger (e.g., total", + "type": "text", + "cross_page": true + } + ], + "index": 5, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 137, + 505, + 151 + ], + "spans": [ + { + "bbox": [ + 88, + 140, + 100, + 150 + ], + "score": 1.0, + "content": "43", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 106, + 137, + 505, + 151 + ], + "score": 1.0, + "content": "distance traveled) must not exceed a given budget. Additionally, it is important to ensure that tasks", + "type": "text", + "cross_page": true + } + ], + "index": 6, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 148, + 506, + 162 + ], + "spans": [ + { + "bbox": [ + 88, + 151, + 100, + 161 + ], + "score": 1.0, + "content": "44", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 148, + 506, + 162 + ], + "score": 1.0, + "content": "are allocated fairly across rangers so that, for e.g., some rangers are not required to walk far greater", + "type": "text", + "cross_page": true + } + ], + "index": 7, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 159, + 501, + 173 + ], + "spans": [ + { + "bbox": [ + 88, + 162, + 100, + 171 + ], + "score": 1.0, + "content": "45", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 159, + 501, + 173 + ], + "score": 1.0, + "content": "distances than others. Adding this level of expressiveness to existing RMAB models is non-trivial.", + "type": "text", + "cross_page": true + } + ], + "index": 8, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 176, + 506, + 188 + ], + "spans": [ + { + "bbox": [ + 89, + 178, + 100, + 187 + ], + "score": 1.0, + "content": "46", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 106, + 176, + 506, + 188 + ], + "score": 1.0, + "content": "To address this, we introduce the multi-worker restless multi-armed bandits (MWRMAB) problem.", + "type": "text", + "cross_page": true + } + ], + "index": 9, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 186, + 506, + 200 + ], + "spans": [ + { + "bbox": [ + 89, + 189, + 100, + 199 + ], + "score": 1.0, + "content": "47", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 186, + 506, + 200 + ], + "score": 1.0, + "content": "Since MWRMABs are more general than the classical RMABs, they are at least PSPACE hard to", + "type": "text", + "cross_page": true + } + ], + "index": 10, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 197, + 506, + 210 + ], + "spans": [ + { + "bbox": [ + 88, + 200, + 99, + 210 + ], + "score": 1.0, + "content": "48", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 106, + 197, + 381, + 210 + ], + "score": 1.0, + "content": "solve optimally [Papadimitriou and Tsitsiklis, 1994]. 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A typical approach is to compute Whittle indices [Whittle, 1988] for", + "type": "text", + "cross_page": true + } + ], + "index": 13, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 230, + 506, + 243 + ], + "spans": [ + { + "bbox": [ + 88, + 232, + 99, + 243 + ], + "score": 1.0, + "content": "51", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 230, + 192, + 243 + ], + "score": 1.0, + "content": "each arm and choose", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 192, + 231, + 204, + 241 + ], + "score": 0.72, + "content": "M", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 205, + 230, + 506, + 243 + ], + "score": 1.0, + "content": "arms with highest index values—an asymptotically optimal solution under", + "type": "text", + "cross_page": true + } + ], + "index": 14, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 241, + 506, + 254 + ], + "spans": [ + { + "bbox": [ + 88, + 243, + 100, + 254 + ], + "score": 1.0, + "content": "52", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 106, + 241, + 506, + 254 + ], + "score": 1.0, + "content": "the technical condition indexability [Weber and Weiss, 1990]. 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A few papers", + "type": "text", + "cross_page": true + } + ], + "index": 16, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 262, + 505, + 277 + ], + "spans": [ + { + "bbox": [ + 88, + 266, + 100, + 275 + ], + "score": 1.0, + "content": "54", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 262, + 343, + 277 + ], + "score": 1.0, + "content": "on RMABs [Glazebrook et al., 2011; Meshram and Kaza,", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 343, + 263, + 370, + 276 + ], + "score": 0.46, + "content": "\\boxed { 2 0 2 0 }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 370, + 262, + 505, + 277 + ], + "score": 1.0, + "content": "study multiple interventions and", + "type": "text", + "cross_page": true + } + ], + "index": 17, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 275, + 505, + 287 + ], + "spans": [ + { + "bbox": [ + 88, + 276, + 99, + 286 + ], + "score": 1.0, + "content": "55", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 275, + 505, + 287 + ], + "score": 1.0, + "content": "non-unitary costs but assumes one global budget (instead of per-worker budget). Existing solutions", + "type": "text", + "cross_page": true + } + ], + "index": 18, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 286, + 505, + 298 + ], + "spans": [ + { + "bbox": [ + 88, + 287, + 100, + 298 + ], + "score": 1.0, + "content": "56", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 106, + 286, + 505, + 298 + ], + "score": 1.0, + "content": "aim at maximizing reward by selecting arms with highest index values that may not guarantee fairness", + "type": "text", + "cross_page": true + } + ], + "index": 19, + "is_list_start_line": true + }, + { + "bbox": [ + 88, + 297, + 371, + 308 + ], + "spans": [ + { + "bbox": [ + 88, + 298, + 100, + 308 + ], + "score": 1.0, + "content": "57", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 297, + 371, + 308 + ], + "score": 1.0, + "content": "towards the workers who are in charge of providing interventions.", + "type": "text", + "cross_page": true + } + ], + "index": 20, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 312, + 505, + 325 + ], + "spans": [ + { + "bbox": [ + 89, + 315, + 100, + 324 + ], + "score": 1.0, + "content": "58", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 106, + 312, + 505, + 325 + ], + "score": 1.0, + "content": "To the best of our knowledge, we are the first to introduce and formalize the multi-worker restless", + "type": "text", + "cross_page": true + } + ], + "index": 21, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 322, + 505, + 335 + ], + "spans": [ + { + "bbox": [ + 89, + 326, + 100, + 335 + ], + "score": 1.0, + "content": "59", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 106, + 322, + 505, + 335 + ], + "score": 1.0, + "content": "multi-armed bandit (MWRMAB) problem and a related worker-centric fairness constraint. We", + "type": "text", + "cross_page": true + } + ], + "index": 22, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 334, + 505, + 347 + ], + "spans": [ + { + "bbox": [ + 89, + 336, + 100, + 346 + ], + "score": 1.0, + "content": "60", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 106, + 334, + 505, + 347 + ], + "score": 1.0, + "content": "develop a novel framework for solving the MWRMAB problem. 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Adding this level of expressiveness to existing RMAB models is non-trivial.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 89, + 176, + 505, + 308 + ], + "lines": [ + { + "bbox": [ + 89, + 176, + 506, + 188 + ], + "spans": [ + { + "bbox": [ + 89, + 178, + 100, + 187 + ], + "score": 1.0, + "content": "46", + "type": "text" + }, + { + "bbox": [ + 106, + 176, + 506, + 188 + ], + "score": 1.0, + "content": "To address this, we introduce the multi-worker restless multi-armed bandits (MWRMAB) problem.", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 89, + 186, + 506, + 200 + ], + "spans": [ + { + "bbox": [ + 89, + 189, + 100, + 199 + ], + "score": 1.0, + "content": "47", + "type": "text" + }, + { + "bbox": [ + 105, + 186, + 506, + 200 + ], + "score": 1.0, + "content": "Since MWRMABs are more general than the classical RMABs, they are at least PSPACE hard to", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 88, + 197, + 506, + 210 + ], + "spans": [ + { + "bbox": [ + 88, + 200, + 99, + 210 + ], + "score": 1.0, + "content": "48", + "type": "text" + }, + { + "bbox": [ + 106, + 197, + 381, + 210 + ], + "score": 1.0, + "content": "solve optimally [Papadimitriou and Tsitsiklis, 1994]. 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Existing solutions", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 88, + 286, + 505, + 298 + ], + "spans": [ + { + "bbox": [ + 88, + 287, + 100, + 298 + ], + "score": 1.0, + "content": "56", + "type": "text" + }, + { + "bbox": [ + 106, + 286, + 505, + 298 + ], + "score": 1.0, + "content": "aim at maximizing reward by selecting arms with highest index values that may not guarantee fairness", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 88, + 297, + 371, + 308 + ], + "spans": [ + { + "bbox": [ + 88, + 298, + 100, + 308 + ], + "score": 1.0, + "content": "57", + "type": "text" + }, + { + "bbox": [ + 105, + 297, + 371, + 308 + ], + "score": 1.0, + "content": "towards the workers who are in charge of providing interventions.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 14.5 + }, + { + "type": "text", + "bbox": [ + 90, + 312, + 505, + 357 + ], + "lines": [ + { + "bbox": [ + 89, + 312, + 505, + 325 + ], + "spans": [ + { + "bbox": [ + 89, + 315, + 100, + 324 + ], + "score": 1.0, + "content": "58", + "type": "text" + }, + { + "bbox": [ + 106, + 312, + 505, + 325 + ], + "score": 1.0, + "content": "To the best of our knowledge, we are the first to introduce and formalize the multi-worker restless", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 89, + 322, + 505, + 335 + ], + "spans": [ + { + "bbox": [ + 89, + 326, + 100, + 335 + ], + "score": 1.0, + "content": "59", + "type": "text" + }, + { + "bbox": [ + 106, + 322, + 505, + 335 + ], + "score": 1.0, + "content": "multi-armed bandit (MWRMAB) problem and a related worker-centric fairness constraint. We", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 89, + 334, + 505, + 347 + ], + "spans": [ + { + "bbox": [ + 89, + 336, + 100, + 346 + ], + "score": 1.0, + "content": "60", + "type": "text" + }, + { + "bbox": [ + 106, + 334, + 505, + 347 + ], + "score": 1.0, + "content": "develop a novel framework for solving the MWRMAB problem. Further, we empirically evaluate our", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 89, + 344, + 442, + 359 + ], + "spans": [ + { + "bbox": [ + 89, + 347, + 100, + 357 + ], + "score": 1.0, + "content": "61", + "type": "text" + }, + { + "bbox": [ + 105, + 344, + 442, + 359 + ], + "score": 1.0, + "content": "algorithm to show that it is fair and scalable across a range of experimental settings.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 22.5 + }, + { + "type": "title", + "bbox": [ + 91, + 372, + 197, + 385 + ], + "lines": [ + { + "bbox": [ + 87, + 371, + 199, + 387 + ], + "spans": [ + { + "bbox": [ + 87, + 371, + 199, + 387 + ], + "score": 1.0, + "content": "62 2 Related Work", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 90, + 397, + 505, + 496 + ], + "lines": [ + { + "bbox": [ + 89, + 397, + 505, + 408 + ], + "spans": [ + { + "bbox": [ + 89, + 399, + 100, + 408 + ], + "score": 1.0, + "content": "63", + "type": "text" + }, + { + "bbox": [ + 106, + 397, + 505, + 408 + ], + "score": 1.0, + "content": "Multi-Action RMABs and Weakly Coupled MDPs Glazebrook et al. 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These papers do not", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 89, + 483, + 457, + 497 + ], + "spans": [ + { + "bbox": [ + 89, + 486, + 100, + 495 + ], + "score": 1.0, + "content": "71", + "type": "text" + }, + { + "bbox": [ + 105, + 483, + 457, + 497 + ], + "score": 1.0, + "content": "consider fairness, a crucial component of MWRMABs, which our algorithm addresses.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 90, + 501, + 505, + 566 + ], + "lines": [ + { + "bbox": [ + 89, + 500, + 505, + 514 + ], + "spans": [ + { + "bbox": [ + 89, + 503, + 100, + 512 + ], + "score": 1.0, + "content": "72", + "type": "text" + }, + { + "bbox": [ + 105, + 500, + 505, + 514 + ], + "score": 1.0, + "content": "Fairness in stochastic and contextual multi-armed bandits (MABs) [Patil et al., 2020; Joseph et al.,", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 89, + 511, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 89, + 514, + 100, + 523 + ], + "score": 1.0, + "content": "73", + "type": "text" + }, + { + "bbox": [ + 105, + 511, + 506, + 524 + ], + "score": 1.0, + "content": "2016; Chen et al., 2020] has been receiving significant attention. 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These papers do not", + "type": "text" + } + ], + "index": 33, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 483, + 457, + 497 + ], + "spans": [ + { + "bbox": [ + 89, + 486, + 100, + 495 + ], + "score": 1.0, + "content": "71", + "type": "text" + }, + { + "bbox": [ + 105, + 483, + 457, + 497 + ], + "score": 1.0, + "content": "consider fairness, a crucial component of MWRMABs, which our algorithm addresses.", + "type": "text" + } + ], + "index": 34, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 500, + 505, + 514 + ], + "spans": [ + { + "bbox": [ + 89, + 503, + 100, + 512 + ], + "score": 1.0, + "content": "72", + "type": "text" + }, + { + "bbox": [ + 105, + 500, + 505, + 514 + ], + "score": 1.0, + "content": "Fairness in stochastic and contextual multi-armed bandits (MABs) [Patil et al., 2020; Joseph et al.,", + "type": "text" + } + ], + "index": 35, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 511, + 506, + 524 + ], + "spans": [ + { + "bbox": [ + 89, + 514, + 100, + 523 + ], + "score": 1.0, + "content": "73", + "type": "text" + }, + { + "bbox": [ + 105, + 511, + 506, + 524 + ], + "score": 1.0, + "content": "2016; Chen et al., 2020] has been receiving significant attention. However, fairness in RMABs has", + "type": "text" + } + ], + "index": 36, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 522, + 505, + 535 + ], + "spans": [ + { + "bbox": [ + 89, + 525, + 100, + 534 + ], + "score": 1.0, + "content": "74", + "type": "text" + }, + { + "bbox": [ + 105, + 522, + 505, + 535 + ], + "score": 1.0, + "content": "been less explored. Recent work by Herlihy et al. [2021] considered quota-based fairness of RMAB", + "type": "text" + } + ], + "index": 37, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 533, + 506, + 546 + ], + "spans": [ + { + "bbox": [ + 89, + 536, + 100, + 545 + ], + "score": 1.0, + "content": "75", + "type": "text" + }, + { + "bbox": [ + 105, + 533, + 506, + 546 + ], + "score": 1.0, + "content": "arms assuming that arms correspond to human beneficiaries (for example, patients). 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We tackle this in Sec. 4.1.This step assumes that, for each", + "type": "text" + } + ], + "index": 23, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 486, + 506, + 498 + ], + "spans": [ + { + "bbox": [ + 86, + 488, + 100, + 497 + ], + "score": 1.0, + "content": "110", + "type": "text" + }, + { + "bbox": [ + 106, + 486, + 124, + 498 + ], + "score": 1.0, + "content": "arm", + "type": "text" + }, + { + "bbox": [ + 124, + 487, + 129, + 496 + ], + "score": 0.57, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 129, + 486, + 506, + 498 + ], + "score": 1.0, + "content": ", MDPs corresponding to any pair of workers are mutually independent. 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\\lambda _ { i j } c _ { i j } \\displaystyle + \\sum _ { s ^ { \\prime } \\in S _ { i } } P _ { s s ^ { \\prime } } ^ { i j } V _ { i , j } ^ { t + 1 } ( s ^ { \\prime } , \\lambda _ { i j } ) \\mathrm { , ~ i f ~ } x _ { i j } ( t ) = 1 } } \\\\ { { \\displaystyle R _ { i } ( s ) + \\sum _ { s ^ { \\prime } \\in S _ { i } } P _ { s s ^ { \\prime } } ^ { i 0 } V _ { i , j } ^ { t + 1 } ( s ^ { \\prime } , \\lambda _ { i j } ) \\mathrm { , ~ i f ~ } x _ { i j } ( t ) = 0 } } \\end{array} \\right.", + "type": "interline_equation", + "image_path": "ed50e4e6595c3c2e425105940af1f0372f43d67fed540df86e22a8627192b93e.jpg" + } + ] + } + ], + "index": 13, + "virtual_lines": [ + { + "bbox": [ + 154, + 226, + 466, + 243.66666666666666 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 154, + 243.66666666666666, + 466, + 261.3333333333333 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 154, + 261.3333333333333, + 466, + 279.0 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "text", + "bbox": [ + 87, + 275, + 99, + 283 + ], + "lines": [ + { + "bbox": [ + 86, + 275, + 99, + 284 + ], + "spans": [ + { + "bbox": [ + 86, + 275, + 99, + 284 + ], + "score": 1.0, + "content": "129", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "interline_equation", + "bbox": [ + 199, + 282, + 412, + 297 + ], + "lines": [ + { + "bbox": [ + 199, + 282, + 412, + 297 + ], + "spans": [ + { + "bbox": [ + 199, + 282, + 412, + 297 + ], + "score": 0.86, + "content": "\\lambda _ { i j } ^ { \\star } ( s ) = \\arg \\operatorname* { m i n } \\{ \\lambda : V _ { i , j } ^ { t } ( s , \\lambda , j ) = = V _ { i , j } ^ { t } ( s , \\lambda , 0 ) \\}", + "type": "interline_equation", + "image_path": "e9e74e1f83be86b7f8e80627e786e0a10d87c632b7b069310ed639a221d9d6e5.jpg" + } + ] + } + ], + "index": 16, + "virtual_lines": [ + { + "bbox": [ + 199, + 282, + 412, + 297 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "text", + "bbox": [ + 93, + 300, + 495, + 316 + ], + "lines": [ + { + "bbox": [ + 87, + 297, + 494, + 319 + ], + "spans": [ + { + "bbox": [ + 87, + 297, + 236, + 319 + ], + "score": 1.0, + "content": "We compute the Whittle indices 130", + "type": "text" + }, + { + "bbox": [ + 236, + 301, + 250, + 315 + ], + "score": 0.9, + "content": "{ \\lambda } _ { i j } ^ { \\star }", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 297, + 481, + 319 + ], + "score": 1.0, + "content": "(Eq. 5) [Qian et al., 2016b] (the algorithm is in Appendix", + "type": "text" + }, + { + "bbox": [ + 482, + 300, + 494, + 316 + ], + "score": 0.5, + "content": "\\mathbf { A } )", + "type": "inline_equation" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 90, + 317, + 505, + 351 + ], + "lines": [ + { + "bbox": [ + 89, + 317, + 505, + 330 + ], + "spans": [ + { + "bbox": [ + 89, + 320, + 99, + 329 + ], + "score": 1.0, + "content": "131", + "type": "text" + }, + { + "bbox": [ + 104, + 317, + 505, + 330 + ], + "score": 1.0, + "content": "Additionally, we establish that the Whittle indices of multiple workers are related when the costs", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 89, + 329, + 505, + 340 + ], + "spans": [ + { + "bbox": [ + 89, + 331, + 100, + 339 + ], + "score": 1.0, + "content": "132", + "type": "text" + }, + { + "bbox": [ + 104, + 329, + 505, + 340 + ], + "score": 1.0, + "content": "and transition probabilities possess certain characteristics, enabling simplification of Whittle Index", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 89, + 340, + 487, + 352 + ], + "spans": [ + { + "bbox": [ + 89, + 343, + 100, + 351 + ], + "score": 1.0, + "content": "133", + "type": "text" + }, + { + "bbox": [ + 104, + 340, + 487, + 352 + ], + "score": 1.0, + "content": "computation for multiple workers when there are certain structures in the MWRMAB problem.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 19 + }, + { + "type": "text", + "bbox": [ + 93, + 354, + 504, + 378 + ], + "lines": [ + { + "bbox": [ + 91, + 352, + 509, + 370 + ], + "spans": [ + { + "bbox": [ + 91, + 352, + 208, + 370 + ], + "score": 1.0, + "content": "Theorem 1. For an arm 34", + "type": "text" + }, + { + "bbox": [ + 208, + 356, + 212, + 366 + ], + "score": 0.61, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 352, + 307, + 370 + ], + "score": 1.0, + "content": ", and a pair of workers", + "type": "text" + }, + { + "bbox": [ + 307, + 356, + 313, + 367 + ], + "score": 0.75, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 352, + 331, + 370 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 332, + 356, + 340, + 367 + ], + "score": 0.86, + "content": "j ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 341, + 352, + 381, + 370 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 381, + 355, + 421, + 368 + ], + "score": 0.91, + "content": "c _ { i j } \\neq c _ { i j ^ { \\prime } }", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 352, + 440, + 370 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 441, + 352, + 488, + 368 + ], + "score": 0.92, + "content": "P _ { s s ^ { \\prime } } ^ { i j } = P _ { s s ^ { \\prime } } ^ { i j ^ { \\prime } }", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 353, + 509, + 370 + ], + "score": 1.0, + "content": "P ij0ss0 for", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 91, + 366, + 439, + 379 + ], + "spans": [ + { + "bbox": [ + 91, + 366, + 131, + 379 + ], + "score": 1.0, + "content": "every 35", + "type": "text" + }, + { + "bbox": [ + 131, + 366, + 169, + 378 + ], + "score": 0.9, + "content": "s , s ^ { \\prime } \\in S _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 170, + 366, + 439, + 379 + ], + "score": 1.0, + "content": ", then their Whittle Indices are inversely proportional to their costs.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21.5 + }, + { + "type": "interline_equation", + "bbox": [ + 231, + 379, + 379, + 409 + ], + "lines": [ + { + "bbox": [ + 231, + 379, + 379, + 409 + ], + "spans": [ + { + "bbox": [ + 231, + 379, + 379, + 409 + ], + "score": 0.92, + "content": "\\frac { { \\lambda } _ { i j } ^ { \\star } ( s ) } { { \\lambda } _ { i j ^ { \\prime } } ^ { \\star } ( s ) } = \\frac { c _ { i j ^ { \\prime } } } { c _ { i j } } f o r e a c h s t a t e s \\in \\mathcal { S } _ { i }", + "type": "interline_equation", + "image_path": "4b5eab5e9ad2e34d7ea223c9f13853f643d990bcde6fb1368e53c72402ba2df0.jpg" + } + ] + } + ], + "index": 23.5, + "virtual_lines": [ + { + "bbox": [ + 231, + 379, + 379, + 394.0 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 231, + 394.0, + 379, + 409.0 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "text", + "bbox": [ + 84, + 421, + 504, + 445 + ], + "lines": [ + { + "bbox": [ + 86, + 419, + 506, + 435 + ], + "spans": [ + { + "bbox": [ + 86, + 424, + 100, + 433 + ], + "score": 1.0, + "content": "136", + "type": "text" + }, + { + "bbox": [ + 102, + 421, + 230, + 435 + ], + "score": 1.0, + "content": "Proof. 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\\lambda _ { i j } c _ { i j } \\displaystyle + \\sum _ { s ^ { \\prime } \\in S _ { i } } P _ { s s ^ { \\prime } } ^ { i j } V _ { i , j } ^ { t + 1 } ( s ^ { \\prime } , \\lambda _ { i j } ) \\mathrm { , ~ i f ~ } x _ { i j } ( t ) = 1 } } \\\\ { { \\displaystyle R _ { i } ( s ) + \\sum _ { s ^ { \\prime } \\in S _ { i } } P _ { s s ^ { \\prime } } ^ { i 0 } V _ { i , j } ^ { t + 1 } ( s ^ { \\prime } , \\lambda _ { i j } ) \\mathrm { , ~ i f ~ } x _ { i j } ( t ) = 0 } } \\end{array} \\right.", + "type": "interline_equation", + "image_path": "ed50e4e6595c3c2e425105940af1f0372f43d67fed540df86e22a8627192b93e.jpg" + } + ] + } + ], + "index": 13, + "virtual_lines": [ + { + "bbox": [ + 154, + 226, + 466, + 243.66666666666666 + ], + "spans": [], + "index": 12 + }, + { + "bbox": [ + 154, + 243.66666666666666, + 466, + 261.3333333333333 + ], + "spans": [], + "index": 13 + }, + { + "bbox": [ + 154, + 261.3333333333333, + 466, + 279.0 + ], + "spans": [], + "index": 15 + } + ] + }, + { + "type": "text", + "bbox": [ + 87, + 275, + 99, + 283 + ], + "lines": [ + { + "bbox": [ + 86, + 275, + 99, + 284 + ], + "spans": [ + { + "bbox": [ + 86, + 275, + 99, + 284 + ], + "score": 1.0, + "content": "129", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14, + "bbox_fs": [ + 86, + 275, + 99, + 284 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 199, + 282, + 412, + 297 + ], + "lines": [ + { + "bbox": [ + 199, + 282, + 412, + 297 + ], + "spans": [ + { + "bbox": [ + 199, + 282, + 412, + 297 + ], + "score": 0.86, + "content": "\\lambda _ { i j } ^ { \\star } ( s ) = \\arg \\operatorname* { m i n } \\{ \\lambda : V _ { i , j } ^ { t } ( s , \\lambda , j ) = = V _ { i , j } ^ { t } ( s , \\lambda , 0 ) \\}", + "type": "interline_equation", + "image_path": "e9e74e1f83be86b7f8e80627e786e0a10d87c632b7b069310ed639a221d9d6e5.jpg" + } + ] + } + ], + "index": 16, + "virtual_lines": [ + { + "bbox": [ + 199, + 282, + 412, + 297 + ], + "spans": [], + "index": 16 + } + ] + }, + { + "type": "text", + "bbox": [ + 93, + 300, + 495, + 316 + ], + "lines": [ + { + "bbox": [ + 87, + 297, + 494, + 319 + ], + "spans": [ + { + "bbox": [ + 87, + 297, + 236, + 319 + ], + "score": 1.0, + "content": "We compute the Whittle indices 130", + "type": "text" + }, + { + "bbox": [ + 236, + 301, + 250, + 315 + ], + "score": 0.9, + "content": "{ \\lambda } _ { i j } ^ { \\star }", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 297, + 481, + 319 + ], + "score": 1.0, + "content": "(Eq. 5) [Qian et al., 2016b] (the algorithm is in Appendix", + "type": "text" + }, + { + "bbox": [ + 482, + 300, + 494, + 316 + ], + "score": 0.5, + "content": "\\mathbf { A } )", + "type": "inline_equation" + } + ], + "index": 17 + } + ], + "index": 17, + "bbox_fs": [ + 87, + 297, + 494, + 319 + ] + }, + { + "type": "index", + "bbox": [ + 90, + 317, + 505, + 351 + ], + "lines": [ + { + "bbox": [ + 89, + 317, + 505, + 330 + ], + "spans": [ + { + "bbox": [ + 89, + 320, + 99, + 329 + ], + "score": 1.0, + "content": "131", + "type": "text" + }, + { + "bbox": [ + 104, + 317, + 505, + 330 + ], + "score": 1.0, + "content": "Additionally, we establish that the Whittle indices of multiple workers are related when the costs", + "type": "text" + } + ], + "index": 18, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 329, + 505, + 340 + ], + "spans": [ + { + "bbox": [ + 89, + 331, + 100, + 339 + ], + "score": 1.0, + "content": "132", + "type": "text" + }, + { + "bbox": [ + 104, + 329, + 505, + 340 + ], + "score": 1.0, + "content": "and transition probabilities possess certain characteristics, enabling simplification of Whittle Index", + "type": "text" + } + ], + "index": 19, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 340, + 487, + 352 + ], + "spans": [ + { + "bbox": [ + 89, + 343, + 100, + 351 + ], + "score": 1.0, + "content": "133", + "type": "text" + }, + { + "bbox": [ + 104, + 340, + 487, + 352 + ], + "score": 1.0, + "content": "computation for multiple workers when there are certain structures in the MWRMAB problem.", + "type": "text" + } + ], + "index": 20, + "is_list_start_line": true + } + ], + "index": 19, + "bbox_fs": [ + 89, + 317, + 505, + 352 + ] + }, + { + "type": "text", + "bbox": [ + 93, + 354, + 504, + 378 + ], + "lines": [ + { + "bbox": [ + 91, + 352, + 509, + 370 + ], + "spans": [ + { + "bbox": [ + 91, + 352, + 208, + 370 + ], + "score": 1.0, + "content": "Theorem 1. For an arm 34", + "type": "text" + }, + { + "bbox": [ + 208, + 356, + 212, + 366 + ], + "score": 0.61, + "content": "i", + "type": "inline_equation" + }, + { + "bbox": [ + 213, + 352, + 307, + 370 + ], + "score": 1.0, + "content": ", and a pair of workers", + "type": "text" + }, + { + "bbox": [ + 307, + 356, + 313, + 367 + ], + "score": 0.75, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 352, + 331, + 370 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 332, + 356, + 340, + 367 + ], + "score": 0.86, + "content": "j ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 341, + 352, + 381, + 370 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 381, + 355, + 421, + 368 + ], + "score": 0.91, + "content": "c _ { i j } \\neq c _ { i j ^ { \\prime } }", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 352, + 440, + 370 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 441, + 352, + 488, + 368 + ], + "score": 0.92, + "content": "P _ { s s ^ { \\prime } } ^ { i j } = P _ { s s ^ { \\prime } } ^ { i j ^ { \\prime } }", + "type": "inline_equation" + }, + { + "bbox": [ + 470, + 353, + 509, + 370 + ], + "score": 1.0, + "content": "P ij0ss0 for", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 91, + 366, + 439, + 379 + ], + "spans": [ + { + "bbox": [ + 91, + 366, + 131, + 379 + ], + "score": 1.0, + "content": "every 35", + "type": "text" + }, + { + "bbox": [ + 131, + 366, + 169, + 378 + ], + "score": 0.9, + "content": "s , s ^ { \\prime } \\in S _ { i }", + "type": "inline_equation" + }, + { + "bbox": [ + 170, + 366, + 439, + 379 + ], + "score": 1.0, + "content": ", then their Whittle Indices are inversely proportional to their costs.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21.5, + "bbox_fs": [ + 91, + 352, + 509, + 379 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 231, + 379, + 379, + 409 + ], + "lines": [ + { + "bbox": [ + 231, + 379, + 379, + 409 + ], + "spans": [ + { + "bbox": [ + 231, + 379, + 379, + 409 + ], + "score": 0.92, + "content": "\\frac { { \\lambda } _ { i j } ^ { \\star } ( s ) } { { \\lambda } _ { i j ^ { \\prime } } ^ { \\star } ( s ) } = \\frac { c _ { i j ^ { \\prime } } } { c _ { i j } } f o r e a c h s t a t e s \\in \\mathcal { S } _ { i }", + "type": "interline_equation", + "image_path": "4b5eab5e9ad2e34d7ea223c9f13853f643d990bcde6fb1368e53c72402ba2df0.jpg" + } + ] + } + ], + "index": 23.5, + "virtual_lines": [ + { + "bbox": [ + 231, + 379, + 379, + 394.0 + ], + "spans": [], + "index": 23 + }, + { + "bbox": [ + 231, + 394.0, + 379, + 409.0 + ], + "spans": [], + "index": 24 + } + ] + }, + { + "type": "index", + "bbox": [ + 84, + 421, + 504, + 445 + ], + "lines": [ + { + "bbox": [ + 86, + 419, + 506, + 435 + ], + "spans": [ + { + "bbox": [ + 86, + 424, + 100, + 433 + ], + "score": 1.0, + "content": "136", + "type": "text" + }, + { + "bbox": [ + 102, + 421, + 230, + 435 + ], + "score": 1.0, + "content": "Proof. 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This", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 86, + 378, + 505, + 390 + ], + "spans": [ + { + "bbox": [ + 86, + 380, + 100, + 389 + ], + "score": 1.0, + "content": "175", + "type": "text" + }, + { + "bbox": [ + 105, + 378, + 505, + 390 + ], + "score": 1.0, + "content": "would correspond to an MDP that always gives 0 reward, and thus would artificially force the index", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 86, + 389, + 395, + 401 + ], + "spans": [ + { + "bbox": [ + 86, + 392, + 100, + 401 + ], + "score": 1.0, + "content": "176", + "type": "text" + }, + { + "bbox": [ + 105, + 389, + 367, + 401 + ], + "score": 1.0, + "content": "for ranger 1 to be 0, despite ranger 1 being the optimal action for", + "type": "text" + }, + { + "bbox": [ + 368, + 389, + 392, + 399 + ], + "score": 0.9, + "content": "s = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 389, + 395, + 401 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 86, + 405, + 505, + 493 + ], + "lines": [ + { + "bbox": [ + 87, + 405, + 505, + 417 + ], + "spans": [ + { + "bbox": [ + 87, + 407, + 99, + 416 + ], + "score": 1.0, + "content": "177", + "type": "text" + }, + { + "bbox": [ + 105, + 405, + 505, + 417 + ], + "score": 1.0, + "content": "To address this, we define a new index notion that accounts for such inter-action effects. The key idea", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 86, + 416, + 505, + 428 + ], + "spans": [ + { + "bbox": [ + 86, + 417, + 100, + 428 + ], + "score": 1.0, + "content": "178", + "type": "text" + }, + { + "bbox": [ + 105, + 416, + 505, + 428 + ], + "score": 1.0, + "content": "is that, when computing the index for a given worker, we will consider actions of all other workers", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 86, + 426, + 506, + 439 + ], + "spans": [ + { + "bbox": [ + 86, + 429, + 100, + 439 + ], + "score": 1.0, + "content": "179", + "type": "text" + }, + { + "bbox": [ + 105, + 426, + 459, + 439 + ], + "score": 1.0, + "content": "in future time steps. So in our poaching example, the new index value for ranger 1 in", + "type": "text" + }, + { + "bbox": [ + 460, + 428, + 485, + 437 + ], + "score": 0.88, + "content": "s = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 426, + 506, + 439 + ], + "score": 1.0, + "content": "will", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 86, + 438, + 505, + 450 + ], + "spans": [ + { + "bbox": [ + 86, + 440, + 99, + 449 + ], + "score": 1.0, + "content": "180", + "type": "text" + }, + { + "bbox": [ + 105, + 438, + 505, + 450 + ], + "score": 1.0, + "content": "increase compared to its decoupled index value, because the new index will take into account the", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 86, + 448, + 506, + 462 + ], + "spans": [ + { + "bbox": [ + 86, + 450, + 99, + 461 + ], + "score": 1.0, + "content": "181", + "type": "text" + }, + { + "bbox": [ + 105, + 448, + 336, + 462 + ], + "score": 1.0, + "content": "value of ranger 2’s actions when the system progresses to", + "type": "text" + }, + { + "bbox": [ + 336, + 449, + 360, + 459 + ], + "score": 0.9, + "content": "s = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 448, + 506, + 462 + ], + "score": 1.0, + "content": "in the future. Note that the methods", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 86, + 460, + 505, + 472 + ], + "spans": [ + { + "bbox": [ + 86, + 462, + 99, + 471 + ], + "score": 1.0, + "content": "182", + "type": "text" + }, + { + "bbox": [ + 105, + 460, + 289, + 472 + ], + "score": 1.0, + "content": "we build generalize to any number of workers", + "type": "text" + }, + { + "bbox": [ + 290, + 460, + 302, + 470 + ], + "score": 0.75, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 460, + 505, + 472 + ], + "score": 1.0, + "content": ". However, the manner in which we incorporate the", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 86, + 470, + 506, + 484 + ], + "spans": [ + { + "bbox": [ + 86, + 472, + 100, + 483 + ], + "score": 1.0, + "content": "183", + "type": "text" + }, + { + "bbox": [ + 105, + 470, + 506, + 484 + ], + "score": 1.0, + "content": "actions of other workers must be done carefully, We propose an approach and provide theoretical", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 86, + 482, + 465, + 494 + ], + "spans": [ + { + "bbox": [ + 86, + 484, + 99, + 493 + ], + "score": 1.0, + "content": "184", + "type": "text" + }, + { + "bbox": [ + 105, + 482, + 465, + 494 + ], + "score": 1.0, + "content": "results explaining why. Finally, we give the full algorithm for computing the new indices.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 40.5 + }, + { + "type": "text", + "bbox": [ + 89, + 497, + 505, + 595 + ], + "lines": [ + { + "bbox": [ + 86, + 498, + 505, + 510 + ], + "spans": [ + { + "bbox": [ + 86, + 499, + 100, + 510 + ], + "score": 1.0, + "content": "185", + "type": "text" + }, + { + "bbox": [ + 106, + 498, + 505, + 510 + ], + "score": 1.0, + "content": "New index notion: For a given arm, to account for the inter-worker action effects, we define the", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 86, + 509, + 506, + 520 + ], + "spans": [ + { + "bbox": [ + 86, + 510, + 100, + 520 + ], + "score": 1.0, + "content": "186", + "type": "text" + }, + { + "bbox": [ + 105, + 509, + 209, + 520 + ], + "score": 1.0, + "content": "new index for an action", + "type": "text" + }, + { + "bbox": [ + 209, + 510, + 216, + 520 + ], + "score": 0.83, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 509, + 446, + 520 + ], + "score": 1.0, + "content": "as the minimum charge that makes an intervention by", + "type": "text" + }, + { + "bbox": [ + 446, + 510, + 452, + 520 + ], + "score": 0.81, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 509, + 506, + 520 + ], + "score": 1.0, + "content": "on that arm", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 86, + 519, + 506, + 532 + ], + "spans": [ + { + "bbox": [ + 86, + 521, + 100, + 531 + ], + "score": 1.0, + "content": "187", + "type": "text" + }, + { + "bbox": [ + 105, + 519, + 240, + 532 + ], + "score": 1.0, + "content": "as valuable as any other worker", + "type": "text" + }, + { + "bbox": [ + 240, + 520, + 249, + 531 + ], + "score": 0.86, + "content": "j ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 519, + 367, + 532 + ], + "score": 1.0, + "content": "in the combined MDP, with", + "type": "text" + }, + { + "bbox": [ + 368, + 520, + 398, + 530 + ], + "score": 0.9, + "content": "M + 1", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 519, + 506, + 532 + ], + "score": 1.0, + "content": "actions. That is, we seek", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 86, + 529, + 506, + 544 + ], + "spans": [ + { + "bbox": [ + 86, + 532, + 100, + 542 + ], + "score": 1.0, + "content": "188", + "type": "text" + }, + { + "bbox": [ + 104, + 529, + 235, + 544 + ], + "score": 1.0, + "content": "the minimum charge for action", + "type": "text" + }, + { + "bbox": [ + 235, + 532, + 241, + 542 + ], + "score": 0.84, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 241, + 529, + 437, + 544 + ], + "score": 1.0, + "content": "that makes us indifferent between taking action", + "type": "text" + }, + { + "bbox": [ + 437, + 532, + 443, + 542 + ], + "score": 0.83, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 443, + 529, + 506, + 544 + ], + "score": 1.0, + "content": "and not taking", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 86, + 542, + 505, + 554 + ], + "spans": [ + { + "bbox": [ + 86, + 543, + 100, + 554 + ], + "score": 1.0, + "content": "189", + "type": "text" + }, + { + "bbox": [ + 106, + 542, + 133, + 554 + ], + "score": 1.0, + "content": "action", + "type": "text" + }, + { + "bbox": [ + 133, + 542, + 139, + 553 + ], + "score": 0.77, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 542, + 505, + 554 + ], + "score": 1.0, + "content": ", a multi-worker extension Whittle’s index notion. To capture this, we define an augmented", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 86, + 552, + 506, + 569 + ], + "spans": [ + { + "bbox": [ + 86, + 556, + 100, + 567 + ], + "score": 1.0, + "content": "190", + "type": "text" + }, + { + "bbox": [ + 104, + 552, + 172, + 569 + ], + "score": 1.0, + "content": "reward function", + "type": "text" + }, + { + "bbox": [ + 173, + 552, + 273, + 567 + ], + "score": 0.94, + "content": "R _ { \\lambda } ^ { \\dagger } ( s , j ) = R ( s ) - \\lambda _ { j } c _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 552, + 293, + 569 + ], + "score": 1.0, + "content": ". Let", + "type": "text" + }, + { + "bbox": [ + 293, + 555, + 301, + 565 + ], + "score": 0.79, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 552, + 365, + 569 + ], + "score": 1.0, + "content": "is the vector of", + "type": "text" + }, + { + "bbox": [ + 365, + 554, + 408, + 567 + ], + "score": 0.92, + "content": "\\{ \\lambda _ { j } \\} _ { j \\in [ M ] }", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 552, + 506, + 569 + ], + "score": 1.0, + "content": "charges. We define this", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 86, + 567, + 506, + 582 + ], + "spans": [ + { + "bbox": [ + 86, + 570, + 100, + 581 + ], + "score": 1.0, + "content": "191", + "type": "text" + }, + { + "bbox": [ + 105, + 567, + 186, + 582 + ], + "score": 1.0, + "content": "expanded MDP as", + "type": "text" + }, + { + "bbox": [ + 187, + 567, + 205, + 581 + ], + "score": 0.92, + "content": "\\mathcal { M } _ { \\lambda } ^ { \\dagger }", + "type": "inline_equation" + }, + { + "bbox": [ + 206, + 567, + 371, + 582 + ], + "score": 1.0, + "content": "and the corresponding value function as", + "type": "text" + }, + { + "bbox": [ + 372, + 567, + 385, + 581 + ], + "score": 0.89, + "content": "V _ { \\lambda } ^ { \\dagger }", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 567, + 506, + 582 + ], + "score": 1.0, + "content": ". We now find adjusted index", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 86, + 575, + 262, + 600 + ], + "spans": [ + { + "bbox": [ + 86, + 583, + 100, + 594 + ], + "score": 1.0, + "content": "192", + "type": "text" + }, + { + "bbox": [ + 106, + 580, + 133, + 597 + ], + "score": 0.92, + "content": "\\lambda _ { j , \\lambda _ { - j } } ^ { a d j , \\ast }", + "type": "inline_equation" + }, + { + "bbox": [ + 133, + 575, + 262, + 600 + ], + "score": 1.0, + "content": "using the following expression:", + "type": "text" + } + ], + "index": 52 + } + ], + "index": 48.5 + }, + { + "type": "interline_equation", + "bbox": [ + 189, + 599, + 421, + 621 + ], + "lines": [ + { + "bbox": [ + 189, + 599, + 421, + 621 + ], + "spans": [ + { + "bbox": [ + 189, + 599, + 421, + 621 + ], + "score": 0.93, + "content": "\\operatorname* { m i n } _ { j ^ { \\prime } \\in [ M ] \\setminus \\{ j \\} } \\arg \\operatorname* { m i n } _ { \\lambda _ { j } } \\{ \\lambda _ { j } \\colon V _ { \\lambda _ { - j } } ^ { \\dagger } ( s , \\lambda _ { j } , j ) = V _ { \\lambda _ { - j } } ^ { \\dagger } ( s , \\lambda _ { j } , j ^ { \\prime } ) \\}", + "type": "interline_equation", + "image_path": "7a04ca23e26454455bdcf6f9c4ed390aa588aa8f54ced8a42371a8bbf2b65e42.jpg" + } + ] + } + ], + "index": 53, + "virtual_lines": [ + { + "bbox": [ + 189, + 599, + 421, + 621 + ], + "spans": [], + "index": 53 + } + ] + }, + { + "type": "text", + "bbox": [ + 87, + 624, + 505, + 681 + ], + "lines": [ + { + "bbox": [ + 86, + 624, + 505, + 638 + ], + "spans": [ + { + "bbox": [ + 86, + 627, + 100, + 636 + ], + "score": 1.0, + "content": "193", + "type": "text" + }, + { + "bbox": [ + 104, + 624, + 133, + 638 + ], + "score": 1.0, + "content": "where", + "type": "text" + }, + { + "bbox": [ + 133, + 626, + 151, + 637 + ], + "score": 0.91, + "content": "\\lambda _ { - j }", + "type": "inline_equation" + }, + { + "bbox": [ + 152, + 624, + 289, + 638 + ], + "score": 1.0, + "content": "is a vector of fixed charges for all", + "type": "text" + }, + { + "bbox": [ + 290, + 625, + 316, + 637 + ], + "score": 0.91, + "content": "j ^ { \\prime } \\ne j", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 624, + 414, + 638 + ], + "score": 1.0, + "content": ", and the outer min over", + "type": "text" + }, + { + "bbox": [ + 414, + 625, + 423, + 636 + ], + "score": 0.87, + "content": "j ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 423, + 624, + 505, + 638 + ], + "score": 1.0, + "content": "simply captures the", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 86, + 636, + 506, + 648 + ], + "spans": [ + { + "bbox": [ + 86, + 638, + 100, + 647 + ], + "score": 1.0, + "content": "194", + "type": "text" + }, + { + "bbox": [ + 106, + 636, + 165, + 648 + ], + "score": 1.0, + "content": "specific action", + "type": "text" + }, + { + "bbox": [ + 165, + 636, + 174, + 648 + ], + "score": 0.88, + "content": "j ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 636, + 407, + 648 + ], + "score": 1.0, + "content": "that the optimal planner is indifferent to taking over action", + "type": "text" + }, + { + "bbox": [ + 407, + 637, + 413, + 647 + ], + "score": 0.83, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 413, + 636, + 506, + 648 + ], + "score": 1.0, + "content": "at the new index value.", + "type": "text" + } + ], + "index": 55 + }, + { + "bbox": [ + 86, + 646, + 506, + 660 + ], + "spans": [ + { + "bbox": [ + 86, + 649, + 100, + 658 + ], + "score": 1.0, + "content": "195", + "type": "text" + }, + { + "bbox": [ + 105, + 646, + 432, + 660 + ], + "score": 1.0, + "content": "Note, this is the natural extension of the decoupled two-action index definition, Eq.", + "type": "text" + }, + { + "bbox": [ + 433, + 647, + 445, + 659 + ], + "score": 0.74, + "content": "( 5 )", + "type": "inline_equation" + }, + { + "bbox": [ + 445, + 646, + 506, + 660 + ], + "score": 1.0, + "content": ", which defines", + "type": "text" + } + ], + "index": 56 + }, + { + "bbox": [ + 86, + 657, + 506, + 671 + ], + "spans": [ + { + "bbox": [ + 86, + 659, + 100, + 669 + ], + "score": 1.0, + "content": "196", + "type": "text" + }, + { + "bbox": [ + 105, + 657, + 215, + 671 + ], + "score": 1.0, + "content": "the index as the charge on", + "type": "text" + }, + { + "bbox": [ + 215, + 659, + 221, + 669 + ], + "score": 0.83, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 657, + 506, + 671 + ], + "score": 1.0, + "content": "that makes the planner indifferent between acting and, the only other", + "type": "text" + } + ], + "index": 57 + }, + { + "bbox": [ + 86, + 668, + 407, + 682 + ], + "spans": [ + { + "bbox": [ + 86, + 670, + 100, + 681 + ], + "score": 1.0, + "content": "197", + "type": "text" + }, + { + "bbox": [ + 105, + 668, + 407, + 682 + ], + "score": 1.0, + "content": "option, being passive. Our new adjusted index algorithm is given in Alg. 1.", + "type": "text" + } + ], + "index": 58 + } + ], + "index": 56 + }, + { + "type": "text", + "bbox": [ + 86, + 686, + 505, + 723 + ], + "lines": [ + { + "bbox": [ + 86, + 685, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 86, + 689, + 100, + 699 + ], + "score": 1.0, + "content": "198", + "type": "text" + }, + { + "bbox": [ + 104, + 686, + 396, + 701 + ], + "score": 1.0, + "content": "We use a binary search procedure to compute the adjusted indices since", + "type": "text" + }, + { + "bbox": [ + 396, + 685, + 453, + 702 + ], + "score": 0.93, + "content": "V _ { \\lambda _ { - j } } ^ { \\dagger } ( s , \\lambda _ { j } , j )", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 686, + 506, + 701 + ], + "score": 1.0, + "content": "is convex in", + "type": "text" + } + ], + "index": 59 + }, + { + "bbox": [ + 87, + 700, + 505, + 713 + ], + "spans": [ + { + "bbox": [ + 87, + 702, + 100, + 711 + ], + "score": 1.0, + "content": "199", + "type": "text" + }, + { + "bbox": [ + 106, + 700, + 118, + 712 + ], + "score": 0.84, + "content": "\\lambda _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 118, + 700, + 505, + 713 + ], + "score": 1.0, + "content": ". 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We show that a reasonable", + "type": "text" + } + ], + "index": 61 + } + ], + "index": 60 + } + ], + "page_idx": 4, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 92, + 73, + 266, + 84 + ], + "lines": [ + { + "bbox": [ + 87, + 72, + 268, + 86 + ], + "spans": [ + { + "bbox": [ + 87, + 72, + 268, + 86 + ], + "score": 1.0, + "content": "149 4.2 Adjusting for interaction effects", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 741, + 309, + 750 + ], + "lines": [ + { + "bbox": [ + 301, + 740, + 310, + 753 + ], + "spans": [ + { + "bbox": [ + 301, + 740, + 310, + 753 + ], + "score": 1.0, + "content": "5", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 101, + 92, + 505, + 126 + ], + "lines": [ + { + "bbox": [ + 105, + 92, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 92, + 234, + 106 + ], + "score": 1.0, + "content": "The indices obtained using Alg.", + "type": "text" + }, + { + "bbox": [ + 234, + 92, + 243, + 106 + ], + "score": 0.77, + "content": "\\begin{array} { l } { 3 } \\\\ { . } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 92, + 506, + 106 + ], + "score": 1.0, + "content": "are not indicative of the true long-term value of taking that action", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 103, + 504, + 116 + ], + "spans": [ + { + "bbox": [ + 105, + 103, + 497, + 116 + ], + "score": 1.0, + "content": "in the MWRMAB problem. This is because, for a given arm, the value of an intervention by worker", + "type": "text" + }, + { + "bbox": [ + 498, + 104, + 504, + 115 + ], + "score": 0.74, + "content": "j", + "type": "inline_equation" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 114, + 419, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 320, + 128 + ], + "score": 1.0, + "content": "in general depends on interventions by other workers", + "type": "text" + }, + { + "bbox": [ + 320, + 115, + 329, + 126 + ], + "score": 0.87, + "content": "j ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 329, + 114, + 419, + 128 + ], + "score": 1.0, + "content": "at different timesteps.", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1, + "bbox_fs": [ + 105, + 92, + 506, + 128 + ] + }, + { + "type": "index", + "bbox": [ + 93, + 131, + 505, + 154 + ], + "lines": [ + { + "bbox": [ + 92, + 130, + 505, + 143 + ], + "spans": [ + { + "bbox": [ + 92, + 130, + 505, + 143 + ], + "score": 1.0, + "content": "53 Consider a 2-worker MWRMAB corresponding to an anti-poaching patrol planning problem, where", + "type": "text" + } + ], + "index": 3, + "is_list_start_line": true + }, + { + "bbox": [ + 89, + 140, + 432, + 156 + ], + "spans": [ + { + "bbox": [ + 89, + 140, + 432, + 156 + ], + "score": 1.0, + "content": "154 each worker is a type of “specialist” with different equipment (detailed in Fig. 1).", + "type": "text" + } + ], + "index": 4, + "is_list_start_line": true + } + ], + "index": 3.5, + "bbox_fs": [ + 89, + 130, + 505, + 156 + ] + }, + { + "type": "text", + "bbox": [ + 99, + 158, + 297, + 324 + ], + "lines": [ + { + "bbox": [ + 106, + 156, + 298, + 171 + ], + "spans": [ + { + "bbox": [ + 106, + 156, + 210, + 171 + ], + "score": 1.0, + "content": "The first ranger (worker),", + "type": "text" + }, + { + "bbox": [ + 210, + 160, + 221, + 169 + ], + "score": 0.85, + "content": "a _ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 221, + 156, + 298, + 171 + ], + "score": 1.0, + "content": ", has special equip-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 169, + 298, + 181 + ], + "spans": [ + { + "bbox": [ + 106, + 169, + 298, + 181 + ], + "score": 1.0, + "content": "ment for clearing overgrown brush, and the sec-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 180, + 298, + 192 + ], + "spans": [ + { + "bbox": [ + 106, + 180, + 156, + 192 + ], + "score": 1.0, + "content": "ond ranger,", + "type": "text" + }, + { + "bbox": [ + 156, + 182, + 167, + 191 + ], + "score": 0.83, + "content": "a _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 167, + 180, + 298, + 192 + ], + "score": 1.0, + "content": ", has specialized equipment for", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 191, + 298, + 203 + ], + "spans": [ + { + "bbox": [ + 106, + 191, + 298, + 203 + ], + "score": 1.0, + "content": "detecting snares, e.g., a metal detector. 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Decoupled indices lead to sub-optimal", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 303, + 295, + 498, + 308 + ], + "spans": [ + { + "bbox": [ + 303, + 295, + 498, + 308 + ], + "score": 1.0, + "content": "policies, whereas adjusted indices perform well.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 26.5 + } + ], + "index": 24.0 + }, + { + "type": "index", + "bbox": [ + 86, + 324, + 505, + 401 + ], + "lines": [ + { + "bbox": [ + 86, + 324, + 505, + 336 + ], + "spans": [ + { + "bbox": [ + 86, + 325, + 99, + 335 + ], + "score": 1.0, + "content": "170", + "type": "text" + }, + { + "bbox": [ + 106, + 324, + 505, + 336 + ], + "score": 1.0, + "content": "mal policy is for ranger 1 to act on the arm in state “overgrown and snared” and ranger 2 to act on the", + "type": "text" + } + ], + "index": 30, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 332, + 504, + 349 + ], + "spans": [ + { + "bbox": [ + 86, + 336, + 99, + 345 + ], + "score": 1.0, + "content": "171", + "type": "text" + }, + { + "bbox": [ + 104, + 332, + 498, + 349 + ], + "score": 1.0, + "content": "arm in state “clear and snared”. However, the fully decoupled index computation for each ranger", + "type": "text" + }, + { + "bbox": [ + 498, + 335, + 504, + 346 + ], + "score": 0.78, + "content": "j", + "type": "inline_equation" + } + ], + "index": 31, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 345, + 505, + 358 + ], + "spans": [ + { + "bbox": [ + 86, + 347, + 100, + 357 + ], + "score": 1.0, + "content": "172", + "type": "text" + }, + { + "bbox": [ + 105, + 345, + 442, + 358 + ], + "score": 1.0, + "content": "would reason about restricted MDPs that only have passive action and ranger type", + "type": "text" + }, + { + "bbox": [ + 442, + 346, + 448, + 357 + ], + "score": 0.85, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 449, + 345, + 505, + 358 + ], + "score": 1.0, + "content": "available. So", + "type": "text" + } + ], + "index": 32, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 356, + 506, + 370 + ], + "spans": [ + { + "bbox": [ + 86, + 358, + 99, + 368 + ], + "score": 1.0, + "content": "173", + "type": "text" + }, + { + "bbox": [ + 105, + 356, + 297, + 370 + ], + "score": 1.0, + "content": "when computing, e.g., the index for ranger 1 in", + "type": "text" + }, + { + "bbox": [ + 297, + 357, + 321, + 366 + ], + "score": 0.89, + "content": "s = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 321, + 356, + 506, + 370 + ], + "score": 1.0, + "content": ", the restricted MDP would have 0 probability", + "type": "text" + } + ], + "index": 33, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 367, + 505, + 379 + ], + "spans": [ + { + "bbox": [ + 86, + 369, + 100, + 378 + ], + "score": 1.0, + "content": "174", + "type": "text" + }, + { + "bbox": [ + 105, + 367, + 505, + 379 + ], + "score": 1.0, + "content": "of reaching state “clear and not snared”, since it does not include ranger 2 in its restricted MDP. This", + "type": "text" + } + ], + "index": 34, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 378, + 505, + 390 + ], + "spans": [ + { + "bbox": [ + 86, + 380, + 100, + 389 + ], + "score": 1.0, + "content": "175", + "type": "text" + }, + { + "bbox": [ + 105, + 378, + 505, + 390 + ], + "score": 1.0, + "content": "would correspond to an MDP that always gives 0 reward, and thus would artificially force the index", + "type": "text" + } + ], + "index": 35, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 389, + 395, + 401 + ], + "spans": [ + { + "bbox": [ + 86, + 392, + 100, + 401 + ], + "score": 1.0, + "content": "176", + "type": "text" + }, + { + "bbox": [ + 105, + 389, + 367, + 401 + ], + "score": 1.0, + "content": "for ranger 1 to be 0, despite ranger 1 being the optimal action for", + "type": "text" + }, + { + "bbox": [ + 368, + 389, + 392, + 399 + ], + "score": 0.9, + "content": "s = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 389, + 395, + 401 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 36, + "is_list_start_line": true + }, + { + "bbox": [ + 87, + 405, + 505, + 417 + ], + "spans": [ + { + "bbox": [ + 87, + 407, + 99, + 416 + ], + "score": 1.0, + "content": "177", + "type": "text" + }, + { + "bbox": [ + 105, + 405, + 505, + 417 + ], + "score": 1.0, + "content": "To address this, we define a new index notion that accounts for such inter-action effects. The key idea", + "type": "text" + } + ], + "index": 37, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 416, + 505, + 428 + ], + "spans": [ + { + "bbox": [ + 86, + 417, + 100, + 428 + ], + "score": 1.0, + "content": "178", + "type": "text" + }, + { + "bbox": [ + 105, + 416, + 505, + 428 + ], + "score": 1.0, + "content": "is that, when computing the index for a given worker, we will consider actions of all other workers", + "type": "text" + } + ], + "index": 38, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 426, + 506, + 439 + ], + "spans": [ + { + "bbox": [ + 86, + 429, + 100, + 439 + ], + "score": 1.0, + "content": "179", + "type": "text" + }, + { + "bbox": [ + 105, + 426, + 459, + 439 + ], + "score": 1.0, + "content": "in future time steps. So in our poaching example, the new index value for ranger 1 in", + "type": "text" + }, + { + "bbox": [ + 460, + 428, + 485, + 437 + ], + "score": 0.88, + "content": "s = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 426, + 506, + 439 + ], + "score": 1.0, + "content": "will", + "type": "text" + } + ], + "index": 39, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 438, + 505, + 450 + ], + "spans": [ + { + "bbox": [ + 86, + 440, + 99, + 449 + ], + "score": 1.0, + "content": "180", + "type": "text" + }, + { + "bbox": [ + 105, + 438, + 505, + 450 + ], + "score": 1.0, + "content": "increase compared to its decoupled index value, because the new index will take into account the", + "type": "text" + } + ], + "index": 40, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 448, + 506, + 462 + ], + "spans": [ + { + "bbox": [ + 86, + 450, + 99, + 461 + ], + "score": 1.0, + "content": "181", + "type": "text" + }, + { + "bbox": [ + 105, + 448, + 336, + 462 + ], + "score": 1.0, + "content": "value of ranger 2’s actions when the system progresses to", + "type": "text" + }, + { + "bbox": [ + 336, + 449, + 360, + 459 + ], + "score": 0.9, + "content": "s = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 361, + 448, + 506, + 462 + ], + "score": 1.0, + "content": "in the future. Note that the methods", + "type": "text" + } + ], + "index": 41, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 460, + 505, + 472 + ], + "spans": [ + { + "bbox": [ + 86, + 462, + 99, + 471 + ], + "score": 1.0, + "content": "182", + "type": "text" + }, + { + "bbox": [ + 105, + 460, + 289, + 472 + ], + "score": 1.0, + "content": "we build generalize to any number of workers", + "type": "text" + }, + { + "bbox": [ + 290, + 460, + 302, + 470 + ], + "score": 0.75, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 460, + 505, + 472 + ], + "score": 1.0, + "content": ". However, the manner in which we incorporate the", + "type": "text" + } + ], + "index": 42, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 470, + 506, + 484 + ], + "spans": [ + { + "bbox": [ + 86, + 472, + 100, + 483 + ], + "score": 1.0, + "content": "183", + "type": "text" + }, + { + "bbox": [ + 105, + 470, + 506, + 484 + ], + "score": 1.0, + "content": "actions of other workers must be done carefully, We propose an approach and provide theoretical", + "type": "text" + } + ], + "index": 43, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 482, + 465, + 494 + ], + "spans": [ + { + "bbox": [ + 86, + 484, + 99, + 493 + ], + "score": 1.0, + "content": "184", + "type": "text" + }, + { + "bbox": [ + 105, + 482, + 465, + 494 + ], + "score": 1.0, + "content": "results explaining why. Finally, we give the full algorithm for computing the new indices.", + "type": "text" + } + ], + "index": 44, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 498, + 505, + 510 + ], + "spans": [ + { + "bbox": [ + 86, + 499, + 100, + 510 + ], + "score": 1.0, + "content": "185", + "type": "text" + }, + { + "bbox": [ + 106, + 498, + 505, + 510 + ], + "score": 1.0, + "content": "New index notion: For a given arm, to account for the inter-worker action effects, we define the", + "type": "text" + } + ], + "index": 45, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 509, + 506, + 520 + ], + "spans": [ + { + "bbox": [ + 86, + 510, + 100, + 520 + ], + "score": 1.0, + "content": "186", + "type": "text" + }, + { + "bbox": [ + 105, + 509, + 209, + 520 + ], + "score": 1.0, + "content": "new index for an action", + "type": "text" + }, + { + "bbox": [ + 209, + 510, + 216, + 520 + ], + "score": 0.83, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 509, + 446, + 520 + ], + "score": 1.0, + "content": "as the minimum charge that makes an intervention by", + "type": "text" + }, + { + "bbox": [ + 446, + 510, + 452, + 520 + ], + "score": 0.81, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 453, + 509, + 506, + 520 + ], + "score": 1.0, + "content": "on that arm", + "type": "text" + } + ], + "index": 46, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 519, + 506, + 532 + ], + "spans": [ + { + "bbox": [ + 86, + 521, + 100, + 531 + ], + "score": 1.0, + "content": "187", + "type": "text" + }, + { + "bbox": [ + 105, + 519, + 240, + 532 + ], + "score": 1.0, + "content": "as valuable as any other worker", + "type": "text" + }, + { + "bbox": [ + 240, + 520, + 249, + 531 + ], + "score": 0.86, + "content": "j ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 249, + 519, + 367, + 532 + ], + "score": 1.0, + "content": "in the combined MDP, with", + "type": "text" + }, + { + "bbox": [ + 368, + 520, + 398, + 530 + ], + "score": 0.9, + "content": "M + 1", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 519, + 506, + 532 + ], + "score": 1.0, + "content": "actions. 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To capture this, we define an augmented", + "type": "text" + } + ], + "index": 49, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 552, + 506, + 569 + ], + "spans": [ + { + "bbox": [ + 86, + 556, + 100, + 567 + ], + "score": 1.0, + "content": "190", + "type": "text" + }, + { + "bbox": [ + 104, + 552, + 172, + 569 + ], + "score": 1.0, + "content": "reward function", + "type": "text" + }, + { + "bbox": [ + 173, + 552, + 273, + 567 + ], + "score": 0.94, + "content": "R _ { \\lambda } ^ { \\dagger } ( s , j ) = R ( s ) - \\lambda _ { j } c _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 273, + 552, + 293, + 569 + ], + "score": 1.0, + "content": ". 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Though it may seem a natural heuristic, this corresponds to planning without considering", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 360, + 506, + 373 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 506, + 373 + ], + "score": 1.0, + "content": "the costs of other actions, which we show below can lead to arbitrarily low values of the indices,", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 372, + 344, + 384 + ], + "spans": [ + { + "bbox": [ + 106, + 372, + 344, + 384 + ], + "score": 1.0, + "content": "which subsequently can lead to poorly performing policies.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 18.5 + }, + { + "type": "text", + "bbox": [ + 106, + 385, + 505, + 441 + ], + "lines": [ + { + "bbox": [ + 104, + 383, + 505, + 403 + ], + "spans": [ + { + "bbox": [ + 104, + 383, + 175, + 403 + ], + "score": 1.0, + "content": "Theorem 2. As", + "type": "text" + }, + { + "bbox": [ + 176, + 387, + 254, + 400 + ], + "score": 0.78, + "content": "\\lambda _ { j ^ { \\prime } } 0 \\forall j ^ { \\prime } \\neq j", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 383, + 259, + 403 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 259, + 385, + 284, + 401 + ], + "score": 0.77, + "content": "\\lambda _ { j } ^ { a d j , * }", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 383, + 409, + 403 + ], + "score": 1.0, + "content": "will monotonically decrease,", + "type": "text" + }, + { + "bbox": [ + 410, + 385, + 505, + 403 + ], + "score": 0.89, + "content": "i f ( l ) \\ V _ { \\lambda _ { j ^ { \\prime } } } ^ { \\dagger } ( s , \\lambda _ { j } , j ^ { \\prime } ) \\ \\geq", + "type": "inline_equation" + } + ], + "index": 21 + }, + { + "bbox": [ + 107, + 401, + 506, + 420 + ], + "spans": [ + { + "bbox": [ + 107, + 402, + 161, + 420 + ], + "score": 0.9, + "content": "V _ { \\lambda _ { j ^ { \\prime } } } ^ { \\dagger } ( s , \\lambda _ { j } , 0 )", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 401, + 178, + 419 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 178, + 404, + 233, + 416 + ], + "score": 0.9, + "content": "O \\le \\lambda _ { j ^ { \\prime } } \\le \\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 401, + 390, + 419 + ], + "score": 1.0, + "content": "and (2) if the average cost of worker", + "type": "text" + }, + { + "bbox": [ + 390, + 404, + 399, + 416 + ], + "score": 0.87, + "content": "j ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 401, + 506, + 419 + ], + "score": 1.0, + "content": "under the optimal policy", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 416, + 505, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 416, + 186, + 432 + ], + "score": 1.0, + "content": "starting with action", + "type": "text" + }, + { + "bbox": [ + 186, + 419, + 195, + 430 + ], + "score": 0.85, + "content": "j ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 416, + 363, + 432 + ], + "score": 1.0, + "content": "is greater than the average cost of worker", + "type": "text" + }, + { + "bbox": [ + 363, + 419, + 371, + 430 + ], + "score": 0.87, + "content": "j ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 416, + 505, + 432 + ], + "score": 1.0, + "content": "under the optimal policy starting", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 428, + 164, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 154, + 443 + ], + "score": 1.0, + "content": "with action", + "type": "text" + }, + { + "bbox": [ + 154, + 430, + 159, + 441 + ], + "score": 0.71, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 428, + 164, + 443 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 22.5 + }, + { + "type": "text", + "bbox": [ + 105, + 448, + 506, + 560 + ], + "lines": [ + { + "bbox": [ + 105, + 448, + 507, + 463 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 129, + 462 + ], + "score": 1.0, + "content": "Thm.", + "type": "text" + }, + { + "bbox": [ + 129, + 448, + 139, + 462 + ], + "score": 0.74, + "content": "2", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 448, + 219, + 462 + ], + "score": 1.0, + "content": "(proof in Appendix", + "type": "text" + }, + { + "bbox": [ + 219, + 448, + 233, + 463 + ], + "score": 0.73, + "content": "\\boxed { \\mathbf { B } }", + "type": "inline_equation" + }, + { + "bbox": [ + 233, + 448, + 362, + 462 + ], + "score": 1.0, + "content": "confirms that, although setting", + "type": "text" + }, + { + "bbox": [ + 362, + 450, + 396, + 461 + ], + "score": 0.92, + "content": "\\lambda _ { j ^ { \\prime } } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 448, + 425, + 462 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 425, + 450, + 434, + 461 + ], + "score": 0.87, + "content": "j ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 448, + 507, + 462 + ], + "score": 1.0, + "content": "may seem like a", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 461, + 505, + 472 + ], + "spans": [ + { + "bbox": [ + 106, + 461, + 430, + 472 + ], + "score": 1.0, + "content": "natural option, in many cases it will artificially reduce the index value for action", + "type": "text" + }, + { + "bbox": [ + 431, + 461, + 437, + 471 + ], + "score": 0.8, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 461, + 505, + 472 + ], + "score": 1.0, + "content": ". This is because", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 470, + 505, + 484 + ], + "spans": [ + { + "bbox": [ + 106, + 471, + 139, + 484 + ], + "score": 0.91, + "content": "\\lambda _ { j ^ { \\prime } } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 139, + 470, + 287, + 484 + ], + "score": 1.0, + "content": "corresponds to planning as if action", + "type": "text" + }, + { + "bbox": [ + 287, + 471, + 296, + 483 + ], + "score": 0.87, + "content": "j ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 296, + 470, + 505, + 484 + ], + "score": 1.0, + "content": "comes with no charge. Naturally then, as we try to", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 482, + 505, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 227, + 495 + ], + "score": 1.0, + "content": "determine the non-zero charge", + "type": "text" + }, + { + "bbox": [ + 227, + 483, + 238, + 494 + ], + "score": 0.88, + "content": "\\lambda _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 482, + 363, + 495 + ], + "score": 1.0, + "content": "we are willing to pay for action", + "type": "text" + }, + { + "bbox": [ + 364, + 483, + 370, + 493 + ], + "score": 0.8, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 482, + 465, + 495 + ], + "score": 1.0, + "content": ", i.e., the index of action", + "type": "text" + }, + { + "bbox": [ + 465, + 483, + 471, + 494 + ], + "score": 0.78, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 471, + 482, + 505, + 495 + ], + "score": 1.0, + "content": ", we will", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 493, + 506, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 365, + 506 + ], + "score": 1.0, + "content": "be less willing to pay higher charges, since there are free actions", + "type": "text" + }, + { + "bbox": [ + 366, + 493, + 374, + 504 + ], + "score": 0.86, + "content": "j ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 493, + 506, + 506 + ], + "score": 1.0, + "content": ". Note that conditions (1) and (2)", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 504, + 505, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 504, + 505, + 516 + ], + "score": 1.0, + "content": "of the above proof are not restrictive. The first is a common epsilon-neighborhood condition, which", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 515, + 505, + 527 + ], + "spans": [ + { + "bbox": [ + 105, + 515, + 421, + 527 + ], + "score": 1.0, + "content": "requires that value functions do not change in arbitrarily non-smooth ways with", + "type": "text" + }, + { + "bbox": [ + 422, + 515, + 429, + 525 + ], + "score": 0.78, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 515, + 505, + 527 + ], + "score": 1.0, + "content": "values near 0. The", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 526, + 506, + 539 + ], + "spans": [ + { + "bbox": [ + 105, + 526, + 338, + 539 + ], + "score": 1.0, + "content": "second requires that a policy’s accumulated costs of action", + "type": "text" + }, + { + "bbox": [ + 338, + 526, + 347, + 537 + ], + "score": 0.87, + "content": "j ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 526, + 493, + 539 + ], + "score": 1.0, + "content": "are greater when starting with action", + "type": "text" + }, + { + "bbox": [ + 494, + 526, + 502, + 537 + ], + "score": 0.86, + "content": "j ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 526, + 506, + 539 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 536, + 505, + 549 + ], + "spans": [ + { + "bbox": [ + 105, + 536, + 505, + 549 + ], + "score": 1.0, + "content": "than starting from any other action— this is same as assuming that the MDPs do not have arbitrarily", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 547, + 506, + 560 + ], + "spans": [ + { + "bbox": [ + 106, + 549, + 278, + 560 + ], + "score": 1.0, + "content": "long mixing times. That is to say that Thm.", + "type": "text" + }, + { + "bbox": [ + 278, + 547, + 288, + 560 + ], + "score": 0.4, + "content": "2", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 549, + 506, + 560 + ], + "score": 1.0, + "content": "applies to a wide range of problems that we care about.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 29.5 + }, + { + "type": "text", + "bbox": [ + 86, + 563, + 506, + 723 + ], + "lines": [ + { + "bbox": [ + 105, + 564, + 506, + 577 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 456, + 577 + ], + "score": 1.0, + "content": "The key question then is: what are reasonable values of charges for other actions", + "type": "text" + }, + { + "bbox": [ + 456, + 564, + 474, + 576 + ], + "score": 0.86, + "content": "\\lambda _ { - j }", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 564, + 506, + 577 + ], + "score": 1.0, + "content": ", when", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 574, + 506, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 236, + 588 + ], + "score": 1.0, + "content": "computing the index for action", + "type": "text" + }, + { + "bbox": [ + 236, + 575, + 244, + 586 + ], + "score": 0.55, + "content": "j ^ { \\check { \\mathbf { \\ell } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 574, + 435, + 588 + ], + "score": 1.0, + "content": "? We propose that a good choice is to set each", + "type": "text" + }, + { + "bbox": [ + 435, + 575, + 480, + 587 + ], + "score": 0.92, + "content": "\\lambda _ { j ^ { \\prime } } \\in \\lambda _ { - j }", + "type": "inline_equation" + }, + { + "bbox": [ + 481, + 574, + 506, + 588 + ], + "score": 1.0, + "content": "to its", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 586, + 506, + 599 + ], + "spans": [ + { + "bbox": [ + 106, + 586, + 369, + 599 + ], + "score": 1.0, + "content": "corresponding decoupled index value for the current state, i.e.,", + "type": "text" + }, + { + "bbox": [ + 369, + 586, + 396, + 599 + ], + "score": 0.93, + "content": "\\lambda _ { j ^ { \\prime } } ^ { * } ( s )", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 586, + 506, + 599 + ], + "score": 1.0, + "content": ". The reason relies on the", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 597, + 506, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 597, + 277, + 613 + ], + "score": 1.0, + "content": "following key idea: we know that at charge", + "type": "text" + }, + { + "bbox": [ + 277, + 598, + 303, + 612 + ], + "score": 0.93, + "content": "\\lambda _ { j ^ { \\prime } } ^ { * } ( s )", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 597, + 506, + 613 + ], + "score": 1.0, + "content": ", the optimal policy is indifferent between choosing", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 610, + 506, + 624 + ], + "spans": [ + { + "bbox": [ + 106, + 610, + 150, + 624 + ], + "score": 1.0, + "content": "that action", + "type": "text" + }, + { + "bbox": [ + 150, + 611, + 159, + 623 + ], + "score": 0.86, + "content": "j ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 610, + 303, + 624 + ], + "score": 1.0, + "content": "and the passive action, at least when", + "type": "text" + }, + { + "bbox": [ + 303, + 611, + 312, + 622 + ], + "score": 0.87, + "content": "j ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 610, + 506, + 624 + ], + "score": 1.0, + "content": "is the only action available. Now, assume we are", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 622, + 506, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 284, + 634 + ], + "score": 1.0, + "content": "computing the new adjusted index for action", + "type": "text" + }, + { + "bbox": [ + 284, + 623, + 290, + 633 + ], + "score": 0.8, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 622, + 506, + 634 + ], + "score": 1.0, + "content": ", when combined in planning with the aforementioned", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 632, + 506, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 134, + 645 + ], + "score": 1.0, + "content": "action", + "type": "text" + }, + { + "bbox": [ + 134, + 633, + 143, + 644 + ], + "score": 0.87, + "content": "j ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 632, + 185, + 645 + ], + "score": 1.0, + "content": "at charge", + "type": "text" + }, + { + "bbox": [ + 185, + 633, + 211, + 646 + ], + "score": 0.91, + "content": "\\lambda _ { j ^ { \\prime } } ^ { * } ( s )", + "type": "inline_equation" + }, + { + "bbox": [ + 211, + 632, + 302, + 645 + ], + "score": 1.0, + "content": ". Since the charge for", + "type": "text" + }, + { + "bbox": [ + 302, + 633, + 311, + 644 + ], + "score": 0.88, + "content": "j ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 632, + 506, + 645 + ], + "score": 1.0, + "content": "is already set at a level that makes the planner", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 644, + 506, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 644, + 187, + 658 + ], + "score": 1.0, + "content": "indifferent between", + "type": "text" + }, + { + "bbox": [ + 187, + 646, + 196, + 657 + ], + "score": 0.86, + "content": "j ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 644, + 311, + 658 + ], + "score": 1.0, + "content": "and being passive, if adding", + "type": "text" + }, + { + "bbox": [ + 311, + 645, + 320, + 657 + ], + "score": 0.87, + "content": "j ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 320, + 644, + 429, + 658 + ], + "score": 1.0, + "content": "to the planning space with", + "type": "text" + }, + { + "bbox": [ + 429, + 646, + 435, + 657 + ], + "score": 0.8, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 435, + 644, + 506, + 658 + ], + "score": 1.0, + "content": "does not provide", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 655, + 506, + 669 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 428, + 669 + ], + "score": 1.0, + "content": "any additional benefit over the passive action, then the new adjusted index for", + "type": "text" + }, + { + "bbox": [ + 428, + 657, + 434, + 668 + ], + "score": 0.74, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 655, + 506, + 669 + ], + "score": 1.0, + "content": "will be the same", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 667, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 218, + 680 + ], + "score": 1.0, + "content": "as the decoupled index for", + "type": "text" + }, + { + "bbox": [ + 218, + 668, + 224, + 679 + ], + "score": 0.78, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 225, + 667, + 333, + 680 + ], + "score": 1.0, + "content": ", which only planned with", + "type": "text" + }, + { + "bbox": [ + 333, + 669, + 338, + 679 + ], + "score": 0.84, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 667, + 506, + 680 + ], + "score": 1.0, + "content": "and the passive action. This avoids the", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 677, + 506, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 477, + 691 + ], + "score": 1.0, + "content": "undesirable effect of getting artificially reduced indices due to under-charging for other actions", + "type": "text" + }, + { + "bbox": [ + 478, + 678, + 486, + 690 + ], + "score": 0.87, + "content": "j ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 677, + 506, + 691 + ], + "score": 1.0, + "content": ", i.e.,", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 688, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 106, + 689, + 128, + 701 + ], + "score": 1.0, + "content": "Thm.", + "type": "text" + }, + { + "bbox": [ + 128, + 688, + 139, + 702 + ], + "score": 0.4, + "content": "2 .", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 689, + 386, + 701 + ], + "score": 1.0, + "content": "The ideas follow similarly for whether the adjusted index for", + "type": "text" + }, + { + "bbox": [ + 387, + 691, + 391, + 700 + ], + "score": 0.86, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 689, + 506, + 701 + ], + "score": 1.0, + "content": "should increase or decrease", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 104, + 699, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 104, + 699, + 481, + 712 + ], + "score": 1.0, + "content": "relative to its decoupled index value. 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As the charge", + "type": "text" + }, + { + "bbox": [ + 197, + 300, + 236, + 312 + ], + "score": 0.9, + "content": "\\lambda _ { j ^ { \\prime } } \\to \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 298, + 240, + 314 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 241, + 299, + 317, + 312 + ], + "score": 0.85, + "content": "R _ { \\lambda } ^ { \\dagger } ( s , j ^ { \\prime } ) \\ - \\infty", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 298, + 469, + 314 + ], + "score": 1.0, + "content": ", making it undesirable to take action", + "type": "text" + }, + { + "bbox": [ + 470, + 300, + 478, + 312 + ], + "score": 0.87, + "content": "j ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 479, + 298, + 505, + 314 + ], + "score": 1.0, + "content": "in the", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 311, + 505, + 324 + ], + "spans": [ + { + "bbox": [ + 106, + 311, + 389, + 324 + ], + "score": 1.0, + "content": "optimal policy. Thus, the optimal policy would only consider actions", + "type": "text" + }, + { + "bbox": [ + 390, + 311, + 414, + 323 + ], + "score": 0.93, + "content": "\\{ 0 , j \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 311, + 505, + 324 + ], + "score": 1.0, + "content": ", which reduces to the", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 322, + 318, + 335 + ], + "spans": [ + { + "bbox": [ + 105, + 322, + 318, + 335 + ], + "score": 1.0, + "content": "restricted MDP of the decoupled index computation.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 14.5, + "bbox_fs": [ + 104, + 284, + 505, + 335 + ] + }, + { + "type": "text", + "bbox": [ + 104, + 338, + 505, + 383 + ], + "lines": [ + { + "bbox": [ + 105, + 338, + 506, + 351 + ], + "spans": [ + { + "bbox": [ + 105, + 338, + 289, + 351 + ], + "score": 1.0, + "content": "Next we analyze a potential naive choice for", + "type": "text" + }, + { + "bbox": [ + 290, + 339, + 308, + 351 + ], + "score": 0.9, + "content": "\\lambda _ { - j }", + "type": "inline_equation" + }, + { + "bbox": [ + 308, + 338, + 462, + 351 + ], + "score": 1.0, + "content": "when computing the indices for each", + "type": "text" + }, + { + "bbox": [ + 462, + 339, + 468, + 350 + ], + "score": 0.8, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 468, + 338, + 506, + 351 + ], + "score": 1.0, + "content": ", namely,", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 349, + 506, + 363 + ], + "spans": [ + { + "bbox": [ + 106, + 349, + 143, + 362 + ], + "score": 0.9, + "content": "\\lambda _ { - j } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 349, + 506, + 363 + ], + "score": 1.0, + "content": ". Though it may seem a natural heuristic, this corresponds to planning without considering", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 360, + 506, + 373 + ], + "spans": [ + { + "bbox": [ + 105, + 360, + 506, + 373 + ], + "score": 1.0, + "content": "the costs of other actions, which we show below can lead to arbitrarily low values of the indices,", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 372, + 344, + 384 + ], + "spans": [ + { + "bbox": [ + 106, + 372, + 344, + 384 + ], + "score": 1.0, + "content": "which subsequently can lead to poorly performing policies.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 18.5, + "bbox_fs": [ + 105, + 338, + 506, + 384 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 385, + 505, + 441 + ], + "lines": [ + { + "bbox": [ + 104, + 383, + 505, + 403 + ], + "spans": [ + { + "bbox": [ + 104, + 383, + 175, + 403 + ], + "score": 1.0, + "content": "Theorem 2. As", + "type": "text" + }, + { + "bbox": [ + 176, + 387, + 254, + 400 + ], + "score": 0.78, + "content": "\\lambda _ { j ^ { \\prime } } 0 \\forall j ^ { \\prime } \\neq j", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 383, + 259, + 403 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 259, + 385, + 284, + 401 + ], + "score": 0.77, + "content": "\\lambda _ { j } ^ { a d j , * }", + "type": "inline_equation" + }, + { + "bbox": [ + 284, + 383, + 409, + 403 + ], + "score": 1.0, + "content": "will monotonically decrease,", + "type": "text" + }, + { + "bbox": [ + 410, + 385, + 505, + 403 + ], + "score": 0.89, + "content": "i f ( l ) \\ V _ { \\lambda _ { j ^ { \\prime } } } ^ { \\dagger } ( s , \\lambda _ { j } , j ^ { \\prime } ) \\ \\geq", + "type": "inline_equation" + } + ], + "index": 21 + }, + { + "bbox": [ + 107, + 401, + 506, + 420 + ], + "spans": [ + { + "bbox": [ + 107, + 402, + 161, + 420 + ], + "score": 0.9, + "content": "V _ { \\lambda _ { j ^ { \\prime } } } ^ { \\dagger } ( s , \\lambda _ { j } , 0 )", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 401, + 178, + 419 + ], + "score": 1.0, + "content": "for", + "type": "text" + }, + { + "bbox": [ + 178, + 404, + 233, + 416 + ], + "score": 0.9, + "content": "O \\le \\lambda _ { j ^ { \\prime } } \\le \\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 401, + 390, + 419 + ], + "score": 1.0, + "content": "and (2) if the average cost of worker", + "type": "text" + }, + { + "bbox": [ + 390, + 404, + 399, + 416 + ], + "score": 0.87, + "content": "j ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 399, + 401, + 506, + 419 + ], + "score": 1.0, + "content": "under the optimal policy", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 416, + 505, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 416, + 186, + 432 + ], + "score": 1.0, + "content": "starting with action", + "type": "text" + }, + { + "bbox": [ + 186, + 419, + 195, + 430 + ], + "score": 0.85, + "content": "j ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 416, + 363, + 432 + ], + "score": 1.0, + "content": "is greater than the average cost of worker", + "type": "text" + }, + { + "bbox": [ + 363, + 419, + 371, + 430 + ], + "score": 0.87, + "content": "j ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 372, + 416, + 505, + 432 + ], + "score": 1.0, + "content": "under the optimal policy starting", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 428, + 164, + 443 + ], + "spans": [ + { + "bbox": [ + 105, + 428, + 154, + 443 + ], + "score": 1.0, + "content": "with action", + "type": "text" + }, + { + "bbox": [ + 154, + 430, + 159, + 441 + ], + "score": 0.71, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 160, + 428, + 164, + 443 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 22.5, + "bbox_fs": [ + 104, + 383, + 506, + 443 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 448, + 506, + 560 + ], + "lines": [ + { + "bbox": [ + 105, + 448, + 507, + 463 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 129, + 462 + ], + "score": 1.0, + "content": "Thm.", + "type": "text" + }, + { + "bbox": [ + 129, + 448, + 139, + 462 + ], + "score": 0.74, + "content": "2", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 448, + 219, + 462 + ], + "score": 1.0, + "content": "(proof in Appendix", + "type": "text" + }, + { + "bbox": [ + 219, + 448, + 233, + 463 + ], + "score": 0.73, + "content": "\\boxed { \\mathbf { B } }", + "type": "inline_equation" + }, + { + "bbox": [ + 233, + 448, + 362, + 462 + ], + "score": 1.0, + "content": "confirms that, although setting", + "type": "text" + }, + { + "bbox": [ + 362, + 450, + 396, + 461 + ], + "score": 0.92, + "content": "\\lambda _ { j ^ { \\prime } } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 448, + 425, + 462 + ], + "score": 1.0, + "content": "for all", + "type": "text" + }, + { + "bbox": [ + 425, + 450, + 434, + 461 + ], + "score": 0.87, + "content": "j ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 448, + 507, + 462 + ], + "score": 1.0, + "content": "may seem like a", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 461, + 505, + 472 + ], + "spans": [ + { + "bbox": [ + 106, + 461, + 430, + 472 + ], + "score": 1.0, + "content": "natural option, in many cases it will artificially reduce the index value for action", + "type": "text" + }, + { + "bbox": [ + 431, + 461, + 437, + 471 + ], + "score": 0.8, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 461, + 505, + 472 + ], + "score": 1.0, + "content": ". This is because", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 470, + 505, + 484 + ], + "spans": [ + { + "bbox": [ + 106, + 471, + 139, + 484 + ], + "score": 0.91, + "content": "\\lambda _ { j ^ { \\prime } } = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 139, + 470, + 287, + 484 + ], + "score": 1.0, + "content": "corresponds to planning as if action", + "type": "text" + }, + { + "bbox": [ + 287, + 471, + 296, + 483 + ], + "score": 0.87, + "content": "j ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 296, + 470, + 505, + 484 + ], + "score": 1.0, + "content": "comes with no charge. Naturally then, as we try to", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 482, + 505, + 495 + ], + "spans": [ + { + "bbox": [ + 105, + 482, + 227, + 495 + ], + "score": 1.0, + "content": "determine the non-zero charge", + "type": "text" + }, + { + "bbox": [ + 227, + 483, + 238, + 494 + ], + "score": 0.88, + "content": "\\lambda _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 482, + 363, + 495 + ], + "score": 1.0, + "content": "we are willing to pay for action", + "type": "text" + }, + { + "bbox": [ + 364, + 483, + 370, + 493 + ], + "score": 0.8, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 482, + 465, + 495 + ], + "score": 1.0, + "content": ", i.e., the index of action", + "type": "text" + }, + { + "bbox": [ + 465, + 483, + 471, + 494 + ], + "score": 0.78, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 471, + 482, + 505, + 495 + ], + "score": 1.0, + "content": ", we will", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 493, + 506, + 506 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 365, + 506 + ], + "score": 1.0, + "content": "be less willing to pay higher charges, since there are free actions", + "type": "text" + }, + { + "bbox": [ + 366, + 493, + 374, + 504 + ], + "score": 0.86, + "content": "j ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 374, + 493, + 506, + 506 + ], + "score": 1.0, + "content": ". Note that conditions (1) and (2)", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 504, + 505, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 504, + 505, + 516 + ], + "score": 1.0, + "content": "of the above proof are not restrictive. The first is a common epsilon-neighborhood condition, which", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 515, + 505, + 527 + ], + "spans": [ + { + "bbox": [ + 105, + 515, + 421, + 527 + ], + "score": 1.0, + "content": "requires that value functions do not change in arbitrarily non-smooth ways with", + "type": "text" + }, + { + "bbox": [ + 422, + 515, + 429, + 525 + ], + "score": 0.78, + "content": "\\lambda", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 515, + 505, + 527 + ], + "score": 1.0, + "content": "values near 0. The", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 526, + 506, + 539 + ], + "spans": [ + { + "bbox": [ + 105, + 526, + 338, + 539 + ], + "score": 1.0, + "content": "second requires that a policy’s accumulated costs of action", + "type": "text" + }, + { + "bbox": [ + 338, + 526, + 347, + 537 + ], + "score": 0.87, + "content": "j ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 526, + 493, + 539 + ], + "score": 1.0, + "content": "are greater when starting with action", + "type": "text" + }, + { + "bbox": [ + 494, + 526, + 502, + 537 + ], + "score": 0.86, + "content": "j ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 526, + 506, + 539 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 536, + 505, + 549 + ], + "spans": [ + { + "bbox": [ + 105, + 536, + 505, + 549 + ], + "score": 1.0, + "content": "than starting from any other action— this is same as assuming that the MDPs do not have arbitrarily", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 547, + 506, + 560 + ], + "spans": [ + { + "bbox": [ + 106, + 549, + 278, + 560 + ], + "score": 1.0, + "content": "long mixing times. That is to say that Thm.", + "type": "text" + }, + { + "bbox": [ + 278, + 547, + 288, + 560 + ], + "score": 0.4, + "content": "2", + "type": "inline_equation" + }, + { + "bbox": [ + 288, + 549, + 506, + 560 + ], + "score": 1.0, + "content": "applies to a wide range of problems that we care about.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 29.5, + "bbox_fs": [ + 105, + 448, + 507, + 560 + ] + }, + { + "type": "text", + "bbox": [ + 86, + 563, + 506, + 723 + ], + "lines": [ + { + "bbox": [ + 105, + 564, + 506, + 577 + ], + "spans": [ + { + "bbox": [ + 105, + 564, + 456, + 577 + ], + "score": 1.0, + "content": "The key question then is: what are reasonable values of charges for other actions", + "type": "text" + }, + { + "bbox": [ + 456, + 564, + 474, + 576 + ], + "score": 0.86, + "content": "\\lambda _ { - j }", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 564, + 506, + 577 + ], + "score": 1.0, + "content": ", when", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 574, + 506, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 574, + 236, + 588 + ], + "score": 1.0, + "content": "computing the index for action", + "type": "text" + }, + { + "bbox": [ + 236, + 575, + 244, + 586 + ], + "score": 0.55, + "content": "j ^ { \\check { \\mathbf { \\ell } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 244, + 574, + 435, + 588 + ], + "score": 1.0, + "content": "? We propose that a good choice is to set each", + "type": "text" + }, + { + "bbox": [ + 435, + 575, + 480, + 587 + ], + "score": 0.92, + "content": "\\lambda _ { j ^ { \\prime } } \\in \\lambda _ { - j }", + "type": "inline_equation" + }, + { + "bbox": [ + 481, + 574, + 506, + 588 + ], + "score": 1.0, + "content": "to its", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 106, + 586, + 506, + 599 + ], + "spans": [ + { + "bbox": [ + 106, + 586, + 369, + 599 + ], + "score": 1.0, + "content": "corresponding decoupled index value for the current state, i.e.,", + "type": "text" + }, + { + "bbox": [ + 369, + 586, + 396, + 599 + ], + "score": 0.93, + "content": "\\lambda _ { j ^ { \\prime } } ^ { * } ( s )", + "type": "inline_equation" + }, + { + "bbox": [ + 396, + 586, + 506, + 599 + ], + "score": 1.0, + "content": ". The reason relies on the", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 597, + 506, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 597, + 277, + 613 + ], + "score": 1.0, + "content": "following key idea: we know that at charge", + "type": "text" + }, + { + "bbox": [ + 277, + 598, + 303, + 612 + ], + "score": 0.93, + "content": "\\lambda _ { j ^ { \\prime } } ^ { * } ( s )", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 597, + 506, + 613 + ], + "score": 1.0, + "content": ", the optimal policy is indifferent between choosing", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 610, + 506, + 624 + ], + "spans": [ + { + "bbox": [ + 106, + 610, + 150, + 624 + ], + "score": 1.0, + "content": "that action", + "type": "text" + }, + { + "bbox": [ + 150, + 611, + 159, + 623 + ], + "score": 0.86, + "content": "j ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 610, + 303, + 624 + ], + "score": 1.0, + "content": "and the passive action, at least when", + "type": "text" + }, + { + "bbox": [ + 303, + 611, + 312, + 622 + ], + "score": 0.87, + "content": "j ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 312, + 610, + 506, + 624 + ], + "score": 1.0, + "content": "is the only action available. Now, assume we are", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 622, + 506, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 284, + 634 + ], + "score": 1.0, + "content": "computing the new adjusted index for action", + "type": "text" + }, + { + "bbox": [ + 284, + 623, + 290, + 633 + ], + "score": 0.8, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 290, + 622, + 506, + 634 + ], + "score": 1.0, + "content": ", when combined in planning with the aforementioned", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 632, + 506, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 134, + 645 + ], + "score": 1.0, + "content": "action", + "type": "text" + }, + { + "bbox": [ + 134, + 633, + 143, + 644 + ], + "score": 0.87, + "content": "j ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 144, + 632, + 185, + 645 + ], + "score": 1.0, + "content": "at charge", + "type": "text" + }, + { + "bbox": [ + 185, + 633, + 211, + 646 + ], + "score": 0.91, + "content": "\\lambda _ { j ^ { \\prime } } ^ { * } ( s )", + "type": "inline_equation" + }, + { + "bbox": [ + 211, + 632, + 302, + 645 + ], + "score": 1.0, + "content": ". 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This avoids the", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 677, + 506, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 477, + 691 + ], + "score": 1.0, + "content": "undesirable effect of getting artificially reduced indices due to under-charging for other actions", + "type": "text" + }, + { + "bbox": [ + 478, + 678, + 486, + 690 + ], + "score": 0.87, + "content": "j ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 677, + 506, + 691 + ], + "score": 1.0, + "content": ", i.e.,", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 688, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 106, + 689, + 128, + 701 + ], + "score": 1.0, + "content": "Thm.", + "type": "text" + }, + { + "bbox": [ + 128, + 688, + 139, + 702 + ], + "score": 0.4, + "content": "2 .", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 689, + 386, + 701 + ], + "score": 1.0, + "content": "The ideas follow similarly for whether the adjusted index for", + "type": "text" + }, + { + "bbox": [ + 387, + 691, + 391, + 700 + ], + "score": 0.86, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 392, + 689, + 506, + 701 + ], + "score": 1.0, + "content": "should increase or decrease", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 104, + 699, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 104, + 699, + 481, + 712 + ], + "score": 1.0, + "content": "relative to its decoupled index value. 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On the other hand, if", + "type": "text" + }, + { + "bbox": [ + 324, + 84, + 333, + 95 + ], + "score": 0.87, + "content": "j ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 84, + 376, + 96 + ], + "score": 1.0, + "content": "dominates", + "type": "text" + }, + { + "bbox": [ + 377, + 84, + 383, + 95 + ], + "score": 0.84, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 383, + 84, + 506, + 96 + ], + "score": 1.0, + "content": "in terms of intervention effect,", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 86, + 95, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 86, + 96, + 100, + 106 + ], + "score": 1.0, + "content": "246", + "type": "text" + }, + { + "bbox": [ + 106, + 95, + 250, + 106 + ], + "score": 1.0, + "content": "then even at a reasonable charge for", + "type": "text" + }, + { + "bbox": [ + 251, + 95, + 259, + 106 + ], + "score": 0.85, + "content": "j ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 95, + 422, + 106 + ], + "score": 1.0, + "content": ", we will be less willing to pay for action", + "type": "text" + }, + { + "bbox": [ + 423, + 95, + 428, + 106 + ], + "score": 0.82, + "content": "j", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 95, + 505, + 106 + ], + "score": 1.0, + "content": "when both options", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 85, + 105, + 505, + 118 + ], + "spans": [ + { + "bbox": [ + 85, + 107, + 100, + 118 + ], + "score": 1.0, + "content": "247", + "type": "text" + }, + { + "bbox": [ + 105, + 105, + 505, + 118 + ], + "score": 1.0, + "content": "are available, and so the adjusted index will decrease. We give our new adjusted index algorithm in", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 86, + 116, + 396, + 129 + ], + "spans": [ + { + "bbox": [ + 86, + 118, + 100, + 127 + ], + "score": 1.0, + "content": "248", + "type": "text" + }, + { + "bbox": [ + 104, + 116, + 396, + 129 + ], + "score": 1.0, + "content": "Alg. 1, and provide experimental results demonstrating its effectiveness.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 2 + }, + { + "type": "title", + "bbox": [ + 104, + 140, + 221, + 152 + ], + "lines": [ + { + "bbox": [ + 105, + 138, + 222, + 155 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 222, + 155 + ], + "score": 1.0, + "content": "4.3 Allocation Algorithm", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 105, + 159, + 505, + 309 + ], + "lines": [ + { + "bbox": [ + 105, + 159, + 505, + 173 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 318, + 173 + ], + "score": 1.0, + "content": "We provide a method called Balanced Allocation", + "type": "text" + }, + { + "bbox": [ + 319, + 160, + 353, + 173 + ], + "score": 0.39, + "content": "( \\operatorname { A l g } . 2 )", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 159, + 505, + 173 + ], + "score": 1.0, + "content": "to tackle the problem of allocating", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 171, + 506, + 185 + ], + "spans": [ + { + "bbox": [ + 105, + 171, + 506, + 185 + ], + "score": 1.0, + "content": "intervention tasks to each worker in a balanced way. At each time step, given the current states of all", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 181, + 505, + 196 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 144, + 196 + ], + "score": 1.0, + "content": "the arms", + "type": "text" + }, + { + "bbox": [ + 144, + 182, + 183, + 195 + ], + "score": 0.9, + "content": "\\{ s _ { i } ^ { t } \\} _ { i \\in [ N ] }", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 181, + 206, + 196 + ], + "score": 1.0, + "content": ", Alg.", + "type": "text" + }, + { + "bbox": [ + 206, + 181, + 216, + 196 + ], + "score": 0.69, + "content": "2", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 181, + 306, + 196 + ], + "score": 1.0, + "content": "creates an ordered list", + "type": "text" + }, + { + "bbox": [ + 307, + 184, + 314, + 192 + ], + "score": 0.77, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 181, + 505, + 196 + ], + "score": 1.0, + "content": "among workers based on their highest Whittle", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 195, + 506, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 195, + 137, + 208 + ], + "score": 1.0, + "content": "Indices", + "type": "text" + }, + { + "bbox": [ + 137, + 195, + 187, + 211 + ], + "score": 0.91, + "content": "\\operatorname* { m a x } _ { i } \\lambda _ { i j } ( s _ { i } ^ { t } )", + "type": "inline_equation" + }, + { + "bbox": [ + 187, + 195, + 506, + 208 + ], + "score": 1.0, + "content": ". It then allocates the best possible (in terms of Whittle Indices) available arm to", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 210, + 506, + 223 + ], + "spans": [ + { + "bbox": [ + 106, + 210, + 248, + 223 + ], + "score": 1.0, + "content": "each worker according to the order", + "type": "text" + }, + { + "bbox": [ + 249, + 213, + 256, + 220 + ], + "score": 0.77, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 210, + 506, + 223 + ], + "score": 1.0, + "content": "in a round-robin way (allocate one arm to a worker and move", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 222, + 505, + 233 + ], + "spans": [ + { + "bbox": [ + 106, + 222, + 505, + 233 + ], + "score": 1.0, + "content": "on to the next worker until the stopping criterion is met). Note that this satisfies the constraint that the", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 232, + 505, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 232, + 505, + 244 + ], + "score": 1.0, + "content": "same arm cannot be allocated to more than one worker. In situations where the best possible available", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 243, + 506, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 243, + 244, + 255 + ], + "score": 1.0, + "content": "arm leads to the budget violation", + "type": "text" + }, + { + "bbox": [ + 244, + 243, + 253, + 253 + ], + "score": 0.79, + "content": "B", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 243, + 506, + 255 + ], + "score": 1.0, + "content": ", an attempt is made to allocate the next best. This process is", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 254, + 506, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 254, + 506, + 266 + ], + "score": 1.0, + "content": "repeated until there are no more arms left to be allocated. 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When all workers are homogeneous (same costs and transition probabilities on arms", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 86, + 564, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 86, + 567, + 100, + 575 + ], + "score": 1.0, + "content": "264", + "type": "text" + }, + { + "bbox": [ + 104, + 564, + 505, + 578 + ], + "score": 1.0, + "content": "after intervention) and satisfy indexability, then our framework outputs the optimal policy while being", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 87, + 575, + 215, + 587 + ], + "spans": [ + { + "bbox": [ + 87, + 578, + 100, + 586 + ], + "score": 1.0, + "content": "265", + "type": "text" + }, + { + "bbox": [ + 105, + 575, + 215, + 587 + ], + "score": 1.0, + "content": "exactly fair to the workers.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 40 + }, + { + "type": "text", + "bbox": [ + 102, + 594, + 505, + 662 + ], + "lines": [ + { + "bbox": [ + 105, + 593, + 505, + 608 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 505, + 608 + ], + "score": 1.0, + "content": "Proof sketch. The proof consists of two components: (1) optimality, which can be proved using", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 605, + 506, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 506, + 618 + ], + "score": 1.0, + "content": "Corollary 1 (Whittle Indices for homogeneous workers are the same), and the fact that the same costs", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 616, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 505, + 628 + ], + "score": 1.0, + "content": "lead to considering all workers from the same pool of actions, and (2) perfect fairness, using the fact", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 627, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 505, + 639 + ], + "score": 1.0, + "content": "that, when costs are equal, Step 3 of our algorithm divides the arms among workers in a way such", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 637, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 505, + 650 + ], + "score": 1.0, + "content": "that the difference between the number of allocations between two workers differs by at most 1 (see", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 649, + 235, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 235, + 662 + ], + "score": 1.0, + "content": "complete proof in Appendix D).", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 44.5 + }, + { + "type": "title", + "bbox": [ + 101, + 675, + 236, + 689 + ], + "lines": [ + { + "bbox": [ + 104, + 675, + 237, + 693 + ], + "spans": [ + { + "bbox": [ + 104, + 675, + 237, + 693 + ], + "score": 1.0, + "content": "5 Empirical Evaluation", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 48 + }, + { + "type": "text", + "bbox": [ + 89, + 700, + 506, + 722 + ], + "lines": [ + { + "bbox": [ + 88, + 699, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 88, + 702, + 101, + 711 + ], + "score": 1.0, + "content": "273", + "type": "text" + }, + { + "bbox": [ + 103, + 699, + 506, + 713 + ], + "score": 1.0, + "content": "We evaluate our framework on three domains, namely constant unitary costs, ordered workers,", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 85, + 710, + 505, + 724 + ], + "spans": [ + { + "bbox": [ + 85, + 710, + 505, + 724 + ], + "score": 1.0, + "content": "274 and specialist domain, each highlighting various challenging dimensions of the MWRMAB problem", + "type": "text" + } + ], + "index": 50 + } + ], + "index": 49.5 + } + ], + "page_idx": 6, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 741, + 309, + 750 + ], + "lines": [ + { + "bbox": [ + 302, + 741, + 309, + 752 + ], + "spans": [ + { + "bbox": [ + 302, + 741, + 309, + 752 + ], + "score": 1.0, + "content": "7", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "index", + "bbox": [ + 86, + 73, + 505, + 128 + ], + "lines": [ + { + "bbox": [ + 86, + 72, + 506, + 86 + ], + "spans": [ + { + "bbox": [ + 86, + 75, + 100, + 84 + ], + "score": 1.0, + "content": "244", + "type": "text" + }, + { + "bbox": [ + 105, + 72, + 303, + 86 + ], + "score": 1.0, + "content": "then we will become more willing to pay a charge", + "type": "text" + }, + { + "bbox": [ + 303, + 73, + 314, + 85 + ], + "score": 0.88, + "content": "\\lambda _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 72, + 477, + 86 + ], + "score": 1.0, + "content": "now to help reach states where the action", + "type": "text" + }, + { + "bbox": [ + 478, + 73, + 486, + 84 + ], + "score": 0.87, + "content": "j ^ { \\prime }", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 72, + 506, + 86 + ], + "score": 1.0, + "content": "will", + "type": "text" + } + ], + "index": 0, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 84, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 86, + 86, + 100, + 95 + ], + "score": 1.0, + "content": "245", + "type": "text" + }, + { + "bbox": [ + 106, + 84, + 324, + 96 + ], + "score": 1.0, + "content": "let us achieve that higher reward. 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We give our new adjusted index algorithm in", + "type": "text" + } + ], + "index": 3, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 116, + 396, + 129 + ], + "spans": [ + { + "bbox": [ + 86, + 118, + 100, + 127 + ], + "score": 1.0, + "content": "248", + "type": "text" + }, + { + "bbox": [ + 104, + 116, + 396, + 129 + ], + "score": 1.0, + "content": "Alg. 1, and provide experimental results demonstrating its effectiveness.", + "type": "text" + } + ], + "index": 4, + "is_list_start_line": true + } + ], + "index": 2, + "bbox_fs": [ + 85, + 72, + 506, + 129 + ] + }, + { + "type": "title", + "bbox": [ + 104, + 140, + 221, + 152 + ], + "lines": [ + { + "bbox": [ + 105, + 138, + 222, + 155 + ], + "spans": [ + { + "bbox": [ + 105, + 138, + 222, + 155 + ], + "score": 1.0, + "content": "4.3 Allocation Algorithm", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 105, + 159, + 505, + 309 + ], + "lines": [ + { + "bbox": [ + 105, + 159, + 505, + 173 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 318, + 173 + ], + "score": 1.0, + "content": "We provide a method called Balanced Allocation", + "type": "text" + }, + { + "bbox": [ + 319, + 160, + 353, + 173 + ], + "score": 0.39, + "content": "( \\operatorname { A l g } . 2 )", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 159, + 505, + 173 + ], + "score": 1.0, + "content": "to tackle the problem of allocating", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 171, + 506, + 185 + ], + "spans": [ + { + "bbox": [ + 105, + 171, + 506, + 185 + ], + "score": 1.0, + "content": "intervention tasks to each worker in a balanced way. 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When all workers are homogeneous (same costs and transition probabilities on arms", + "type": "text" + } + ], + "index": 39, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 564, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 86, + 567, + 100, + 575 + ], + "score": 1.0, + "content": "264", + "type": "text" + }, + { + "bbox": [ + 104, + 564, + 505, + 578 + ], + "score": 1.0, + "content": "after intervention) and satisfy indexability, then our framework outputs the optimal policy while being", + "type": "text" + } + ], + "index": 40, + "is_list_start_line": true + }, + { + "bbox": [ + 87, + 575, + 215, + 587 + ], + "spans": [ + { + "bbox": [ + 87, + 578, + 100, + 586 + ], + "score": 1.0, + "content": "265", + "type": "text" + }, + { + "bbox": [ + 105, + 575, + 215, + 587 + ], + "score": 1.0, + "content": "exactly fair to the workers.", + "type": "text" + } + ], + "index": 41, + "is_list_start_line": true + } + ], + "index": 29, + "bbox_fs": [ + 105, + 333, + 502, + 542 + ] + }, + { + "type": "index", + "bbox": [ + 88, + 553, + 505, + 586 + ], + "lines": [], + "index": 40, + "bbox_fs": [ + 86, + 552, + 506, + 587 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 102, + 594, + 505, + 662 + ], + "lines": [ + { + "bbox": [ + 105, + 593, + 505, + 608 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 505, + 608 + ], + "score": 1.0, + "content": "Proof sketch. The proof consists of two components: (1) optimality, which can be proved using", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 605, + 506, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 506, + 618 + ], + "score": 1.0, + "content": "Corollary 1 (Whittle Indices for homogeneous workers are the same), and the fact that the same costs", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 616, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 105, + 616, + 505, + 628 + ], + "score": 1.0, + "content": "lead to considering all workers from the same pool of actions, and (2) perfect fairness, using the fact", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 627, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 505, + 639 + ], + "score": 1.0, + "content": "that, when costs are equal, Step 3 of our algorithm divides the arms among workers in a way such", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 637, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 505, + 650 + ], + "score": 1.0, + "content": "that the difference between the number of allocations between two workers differs by at most 1 (see", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 649, + 235, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 235, + 662 + ], + "score": 1.0, + "content": "complete proof in Appendix D).", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 44.5, + "bbox_fs": [ + 105, + 593, + 506, + 662 + ] + }, + { + "type": "title", + "bbox": [ + 101, + 675, + 236, + 689 + ], + "lines": [ + { + "bbox": [ + 104, + 675, + 237, + 693 + ], + "spans": [ + { + "bbox": [ + 104, + 675, + 237, + 693 + ], + "score": 1.0, + "content": "5 Empirical Evaluation", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 48 + }, + { + "type": "index", + "bbox": [ + 89, + 700, + 506, + 722 + ], + "lines": [ + { + "bbox": [ + 88, + 699, + 506, + 713 + ], + "spans": [ + { + "bbox": [ + 88, + 702, + 101, + 711 + ], + "score": 1.0, + "content": "273", + "type": "text" + }, + { + "bbox": [ + 103, + 699, + 506, + 713 + ], + "score": 1.0, + "content": "We evaluate our framework on three domains, namely constant unitary costs, ordered workers,", + "type": "text" + } + ], + "index": 49, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 710, + 505, + 724 + ], + "spans": [ + { + "bbox": [ + 85, + 710, + 505, + 724 + ], + "score": 1.0, + "content": "274 and specialist domain, each highlighting various challenging dimensions of the MWRMAB problem", + "type": "text" + } + ], + "index": 50, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 72, + 506, + 85 + ], + "spans": [ + { + "bbox": [ + 85, + 74, + 100, + 84 + ], + "score": 1.0, + "content": "275", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 73, + 196, + 85 + ], + "score": 1.0, + "content": "(detailed in Appendix", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 197, + 72, + 209, + 85 + ], + "score": 0.81, + "content": "\\mathbf { C } )", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 210, + 73, + 506, + 85 + ], + "score": 1.0, + "content": ". In the first domain, the cost associated with all worker-arm pairs is the", + "type": "text", + "cross_page": true + } + ], + "index": 0, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 83, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 86, + 86, + 99, + 95 + ], + "score": 1.0, + "content": "276", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 83, + 505, + 96 + ], + "score": 1.0, + "content": "same, but transition probabilities differ; the main challenge is in finding optimal assignments, though", + "type": "text", + "cross_page": true + } + ], + "index": 1, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 93, + 505, + 107 + ], + "spans": [ + { + "bbox": [ + 85, + 96, + 100, + 106 + ], + "score": 1.0, + "content": "277", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 93, + 505, + 107 + ], + "score": 1.0, + "content": "fairness is still considered. In the second domain, there exists an ordering among the workers such", + "type": "text", + "cross_page": true + } + ], + "index": 2, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 104, + 506, + 119 + ], + "spans": [ + { + "bbox": [ + 85, + 107, + 100, + 117 + ], + "score": 1.0, + "content": "278", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 104, + 506, + 119 + ], + "score": 1.0, + "content": "that the highest (or lowest) ranked worker has the highest (or lowest) probability of transitioning any", + "type": "text", + "cross_page": true + } + ], + "index": 3, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 114, + 507, + 131 + ], + "spans": [ + { + "bbox": [ + 85, + 118, + 100, + 128 + ], + "score": 1.0, + "content": "279", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 104, + 114, + 507, + 131 + ], + "score": 1.0, + "content": "arm to “good” state; which makes balancing optimal assignments with fair assignments challenging.", + "type": "text", + "cross_page": true + } + ], + "index": 4, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 126, + 424, + 140 + ], + "spans": [ + { + "bbox": [ + 86, + 129, + 99, + 138 + ], + "score": 1.0, + "content": "280", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 126, + 424, + 140 + ], + "score": 1.0, + "content": "The final domain highlights the need to consider inter-action effects via Step 2.", + "type": "text", + "cross_page": true + } + ], + "index": 5, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 143, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 86, + 146, + 99, + 155 + ], + "score": 1.0, + "content": "281", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 143, + 505, + 156 + ], + "score": 1.0, + "content": "We run experiments by varying the number of arms for each domain. For the first and third domains", + "type": "text", + "cross_page": true + } + ], + "index": 6, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 153, + 506, + 167 + ], + "spans": [ + { + "bbox": [ + 85, + 156, + 100, + 166 + ], + "score": 1.0, + "content": "282", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 153, + 231, + 167 + ], + "score": 1.0, + "content": "that consider unit costs, we use", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 231, + 155, + 258, + 165 + ], + "score": 0.9, + "content": "B = 4", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 259, + 153, + 506, + 167 + ], + "score": 1.0, + "content": "budget per worker, and for the second domain where costs are", + "type": "text", + "cross_page": true + } + ], + "index": 7, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 165, + 504, + 177 + ], + "spans": [ + { + "bbox": [ + 85, + 167, + 100, + 177 + ], + "score": 1.0, + "content": "283", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 165, + 245, + 177 + ], + "score": 1.0, + "content": "in the range [1, 10], we use budget", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 246, + 165, + 279, + 176 + ], + "score": 0.9, + "content": "B = 1 8", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 279, + 165, + 468, + 177 + ], + "score": 1.0, + "content": ". We ran all the experiments on Apple M1 with", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 469, + 165, + 504, + 176 + ], + "score": 0.56, + "content": "3 . 2 \\mathrm { G H z }", + "type": "inline_equation", + "cross_page": true + } + ], + "index": 8, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 176, + 506, + 190 + ], + "spans": [ + { + "bbox": [ + 85, + 178, + 100, + 188 + ], + "score": 1.0, + "content": "284", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 104, + 176, + 506, + 190 + ], + "score": 1.0, + "content": "Processor and 16 GB RAM. We evaluate the average reward per arm over a fixed time horizon of", + "type": "text", + "cross_page": true + } + ], + "index": 9, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 187, + 505, + 199 + ], + "spans": [ + { + "bbox": [ + 85, + 189, + 100, + 199 + ], + "score": 1.0, + "content": "285", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 106, + 187, + 505, + 199 + ], + "score": 1.0, + "content": "100 steps and averaged over 50 epochs with random or fixed transition probabilities that follow the", + "type": "text", + "cross_page": true + } + ], + "index": 10, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 198, + 232, + 210 + ], + "spans": [ + { + "bbox": [ + 85, + 200, + 100, + 210 + ], + "score": 1.0, + "content": "286", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 105, + 198, + 232, + 209 + ], + "score": 1.0, + "content": "characteristics of each domain.", + "type": "text", + "cross_page": true + } + ], + "index": 11, + "is_list_start_line": true + } + ], + "index": 49.5, + "bbox_fs": [ + 85, + 699, + 506, + 724 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 86, + 72, + 505, + 138 + ], + "lines": [ + { + "bbox": [ + 85, + 72, + 506, + 85 + ], + "spans": [ + { + "bbox": [ + 85, + 74, + 100, + 84 + ], + "score": 1.0, + "content": "275", + "type": "text" + }, + { + "bbox": [ + 105, + 73, + 196, + 85 + ], + "score": 1.0, + "content": "(detailed in Appendix", + "type": "text" + }, + { + "bbox": [ + 197, + 72, + 209, + 85 + ], + "score": 0.81, + "content": "\\mathbf { C } )", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 73, + 506, + 85 + ], + "score": 1.0, + "content": ". In the first domain, the cost associated with all worker-arm pairs is the", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 86, + 83, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 86, + 86, + 99, + 95 + ], + "score": 1.0, + "content": "276", + "type": "text" + }, + { + "bbox": [ + 105, + 83, + 505, + 96 + ], + "score": 1.0, + "content": "same, but transition probabilities differ; the main challenge is in finding optimal assignments, though", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 85, + 93, + 505, + 107 + ], + "spans": [ + { + "bbox": [ + 85, + 96, + 100, + 106 + ], + "score": 1.0, + "content": "277", + "type": "text" + }, + { + "bbox": [ + 105, + 93, + 505, + 107 + ], + "score": 1.0, + "content": "fairness is still considered. In the second domain, there exists an ordering among the workers such", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 85, + 104, + 506, + 119 + ], + "spans": [ + { + "bbox": [ + 85, + 107, + 100, + 117 + ], + "score": 1.0, + "content": "278", + "type": "text" + }, + { + "bbox": [ + 105, + 104, + 506, + 119 + ], + "score": 1.0, + "content": "that the highest (or lowest) ranked worker has the highest (or lowest) probability of transitioning any", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 85, + 114, + 507, + 131 + ], + "spans": [ + { + "bbox": [ + 85, + 118, + 100, + 128 + ], + "score": 1.0, + "content": "279", + "type": "text" + }, + { + "bbox": [ + 104, + 114, + 507, + 131 + ], + "score": 1.0, + "content": "arm to “good” state; which makes balancing optimal assignments with fair assignments challenging.", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 86, + 126, + 424, + 140 + ], + "spans": [ + { + "bbox": [ + 86, + 129, + 99, + 138 + ], + "score": 1.0, + "content": "280", + "type": "text" + }, + { + "bbox": [ + 105, + 126, + 424, + 140 + ], + "score": 1.0, + "content": "The final domain highlights the need to consider inter-action effects via Step 2.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 86, + 144, + 505, + 209 + ], + "lines": [ + { + "bbox": [ + 86, + 143, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 86, + 146, + 99, + 155 + ], + "score": 1.0, + "content": "281", + "type": "text" + }, + { + "bbox": [ + 105, + 143, + 505, + 156 + ], + "score": 1.0, + "content": "We run experiments by varying the number of arms for each domain. For the first and third domains", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 85, + 153, + 506, + 167 + ], + "spans": [ + { + "bbox": [ + 85, + 156, + 100, + 166 + ], + "score": 1.0, + "content": "282", + "type": "text" + }, + { + "bbox": [ + 105, + 153, + 231, + 167 + ], + "score": 1.0, + "content": "that consider unit costs, we use", + "type": "text" + }, + { + "bbox": [ + 231, + 155, + 258, + 165 + ], + "score": 0.9, + "content": "B = 4", + "type": "inline_equation" + }, + { + "bbox": [ + 259, + 153, + 506, + 167 + ], + "score": 1.0, + "content": "budget per worker, and for the second domain where costs are", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 85, + 165, + 504, + 177 + ], + "spans": [ + { + "bbox": [ + 85, + 167, + 100, + 177 + ], + "score": 1.0, + "content": "283", + "type": "text" + }, + { + "bbox": [ + 105, + 165, + 245, + 177 + ], + "score": 1.0, + "content": "in the range [1, 10], we use budget", + "type": "text" + }, + { + "bbox": [ + 246, + 165, + 279, + 176 + ], + "score": 0.9, + "content": "B = 1 8", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 165, + 468, + 177 + ], + "score": 1.0, + "content": ". We ran all the experiments on Apple M1 with", + "type": "text" + }, + { + "bbox": [ + 469, + 165, + 504, + 176 + ], + "score": 0.56, + "content": "3 . 2 \\mathrm { G H z }", + "type": "inline_equation" + } + ], + "index": 8 + }, + { + "bbox": [ + 85, + 176, + 506, + 190 + ], + "spans": [ + { + "bbox": [ + 85, + 178, + 100, + 188 + ], + "score": 1.0, + "content": "284", + "type": "text" + }, + { + "bbox": [ + 104, + 176, + 506, + 190 + ], + "score": 1.0, + "content": "Processor and 16 GB RAM. We evaluate the average reward per arm over a fixed time horizon of", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 85, + 187, + 505, + 199 + ], + "spans": [ + { + "bbox": [ + 85, + 189, + 100, + 199 + ], + "score": 1.0, + "content": "285", + "type": "text" + }, + { + "bbox": [ + 106, + 187, + 505, + 199 + ], + "score": 1.0, + "content": "100 steps and averaged over 50 epochs with random or fixed transition probabilities that follow the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 85, + 198, + 232, + 210 + ], + "spans": [ + { + "bbox": [ + 85, + 200, + 100, + 210 + ], + "score": 1.0, + "content": "286", + "type": "text" + }, + { + "bbox": [ + 105, + 198, + 232, + 209 + ], + "score": 1.0, + "content": "characteristics of each domain.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 8.5 + }, + { + "type": "text", + "bbox": [ + 97, + 222, + 505, + 245 + ], + "lines": [ + { + "bbox": [ + 99, + 221, + 507, + 236 + ], + "spans": [ + { + 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P W I + B A }", + "type": "inline_equation" + }, + { + "bbox": [ + 156, + 249, + 505, + 263 + ], + "score": 1.0, + "content": "(Per arm-worker Whittle Index with Balanced Allocation) that combines Steps 1 and 3", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 114, + 261, + 356, + 274 + ], + "spans": [ + { + "bbox": [ + 114, + 261, + 356, + 274 + ], + "score": 1.0, + "content": "of our approach, skipping Step 2 (adjusted index algorithm)", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14.5 + }, + { + "type": "text", + "bbox": [ + 105, + 277, + 504, + 311 + ], + "lines": [ + { + "bbox": [ + 105, + 276, + 505, + 290 + ], + "spans": [ + { + "bbox": [ + 105, + 276, + 114, + 290 + ], + "score": 1.0, + "content": "•", + "type": "text" + }, + { + "bbox": [ + 115, + 277, + 158, + 288 + ], + "score": 0.66, + "content": "\\mathbf { C W I + G A }", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 276, + 505, + 290 + ], + "score": 1.0, + "content": "(Combined arm-worker Whittle Index with Greedy Allocation) that combines Steps", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 113, + 288, + 505, + 300 + ], + "spans": [ + { + "bbox": [ + 113, + 288, + 505, + 300 + ], + "score": 1.0, + "content": "1 and 2 and, instead of Step 3 (balanced allocation), the highest values of indices are used for", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 114, + 298, + 408, + 313 + ], + "spans": [ + { + "bbox": [ + 114, + 298, + 408, + 313 + ], + "score": 1.0, + "content": "allocating arms to workers while ensuring budget constraint per timestep", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 106, + 314, + 503, + 338 + ], + "lines": [ + { + "bbox": [ + 105, + 314, + 505, + 329 + ], + "spans": [ + { + "bbox": [ + 105, + 314, + 155, + 329 + ], + "score": 1.0, + "content": "• Hawkins", + "type": "text" + }, + { + "bbox": [ + 155, + 315, + 183, + 327 + ], + "score": 0.7, + "content": "\\underline { { \\| 2 0 0 3 \\| } }", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 314, + 322, + 329 + ], + "score": 1.0, + "content": "solves a discounted version of Eq.", + "type": "text" + }, + { + "bbox": [ + 322, + 315, + 332, + 328 + ], + "score": 0.53, + "content": "2", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 314, + 505, + 329 + ], + "score": 1.0, + "content": "without the fairness constraint, to compute", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 114, + 327, + 369, + 339 + ], + "spans": [ + { + "bbox": [ + 114, + 327, + 153, + 338 + ], + "score": 1.0, + "content": "values of", + "type": "text" + }, + { + "bbox": [ + 153, + 327, + 164, + 339 + ], + "score": 0.88, + "content": "\\lambda _ { j }", + "type": "inline_equation" + }, + { + "bbox": [ + 164, + 327, + 281, + 338 + ], + "score": 1.0, + "content": ", then solves a knapsack over", + "type": "text" + }, + { + "bbox": [ + 281, + 327, + 292, + 339 + ], + "score": 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fairness constraint.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21.5 + }, + { + "type": "text", + "bbox": [ + 102, + 370, + 504, + 392 + ], + "lines": [ + { + "bbox": [ + 107, + 369, + 505, + 382 + ], + "spans": [ + { + "bbox": [ + 107, + 369, + 505, + 382 + ], + "score": 1.0, + "content": "• OPT-fair follows OPT, but adds the fairness constraints. These optimal algorithms are exponential", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 113, + 380, + 499, + 393 + ], + "spans": [ + { + "bbox": [ + 113, + 380, + 499, + 393 + ], + "score": 1.0, + "content": "in the number of arms, states, and workers, and thus, could only be executed on small instances.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23.5 + }, + { + "type": "text", + "bbox": [ + 106, + 396, + 504, + 420 + ], + "lines": [ + { + "bbox": [ + 105, + 394, + 506, + 410 + ], + "spans": [ + { + "bbox": [ + 105, + 394, + 241, + 410 + ], + "score": 1.0, + "content": "• Random takes random actions", + "type": "text" + }, + { + "bbox": [ + 241, + 396, + 302, + 409 + ], + "score": 0.93, + "content": "j \\in [ M ] \\cup \\{ 0 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 302, + 394, + 506, + 410 + ], + "score": 1.0, + "content": "on every arm while maintaining budget feasibility", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 114, + 406, + 251, + 421 + ], + "spans": [ + { + "bbox": [ + 114, + 406, + 251, + 421 + ], + "score": 1.0, + "content": "for every worker at each timestep", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25.5 + }, + { + "type": "text", + "bbox": [ + 88, + 432, + 506, + 575 + ], + "lines": [ + { + "bbox": [ + 86, + 432, + 506, + 445 + ], + "spans": [ + { + "bbox": [ + 86, + 435, + 100, + 443 + ], + "score": 1.0, + "content": "302", + "type": "text" + }, + { + "bbox": [ + 105, + 432, + 174, + 445 + ], + "score": 1.0, + "content": "Results Figure", + "type": "text" + }, + { + "bbox": [ + 174, + 432, + 183, + 445 + ], + "score": 0.6, + "content": "2", + "type": "inline_equation" + }, + { + "bbox": [ + 183, + 432, + 376, + 445 + ], + "score": 1.0, + "content": "shows that reward obtained using our framework", + "type": "text" + }, + { + "bbox": [ + 376, + 433, + 418, + 443 + ], + "score": 0.66, + "content": "\\mathrm { ( C W I + B A ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 419, + 432, + 506, + 445 + ], + "score": 1.0, + "content": "is comparable to that", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 85, + 443, + 506, + 455 + ], + "spans": [ + { + "bbox": [ + 85, + 445, + 100, + 455 + ], + "score": 1.0, + "content": "303", + "type": "text" + }, + { + "bbox": [ + 106, + 443, + 506, + 455 + ], + "score": 1.0, + "content": "of the reward maximizing baselines (Hawkins and OPT) across all the domains. We observe at most", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 86, + 454, + 505, + 466 + ], + "spans": [ + { + "bbox": [ + 86, + 457, + 99, + 465 + ], + "score": 1.0, + "content": "304", + "type": "text" + }, + { + "bbox": [ + 106, + 454, + 138, + 465 + ], + "score": 0.87, + "content": "1 8 . 9 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 138, + 454, + 505, + 466 + ], + "score": 1.0, + "content": "reduction in reward compared to OPT, where the highest reduction occurs for ordered workers", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 85, + 464, + 506, + 478 + ], + "spans": [ + { + "bbox": [ + 85, + 467, + 100, + 477 + ], + "score": 1.0, + "content": "305", + "type": "text" + }, + { + "bbox": [ + 106, + 464, + 257, + 478 + ], + "score": 1.0, + "content": "in Fig. 2(b). In terms of fairness, Figs.", + "type": "text" + }, + { + "bbox": [ + 257, + 465, + 275, + 477 + ], + "score": 0.26, + "content": "2 ( \\mathbf { a } )", + "type": "inline_equation" + }, + { + "bbox": [ + 275, + 464, + 344, + 478 + ], + "score": 1.0, + "content": "and (c) show that", + "type": "text" + }, + { + "bbox": [ + 345, + 465, + 384, + 476 + ], + "score": 0.58, + "content": "\\mathrm { C W I + B A }", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 464, + 506, + 478 + ], + "score": 1.0, + "content": "achieves fair allocation among", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 86, + 475, + 506, + 489 + ], + "spans": [ + { + "bbox": [ + 86, + 478, + 100, + 487 + ], + "score": 1.0, + "content": "306", + "type": "text" + }, + { + "bbox": [ + 105, + 475, + 241, + 489 + ], + "score": 1.0, + "content": "workers at all timesteps. In Figure", + "type": "text" + }, + { + "bbox": [ + 241, + 476, + 260, + 488 + ], + "score": 0.49, + "content": "\\boxed { 2 } ( 6 )", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 476, + 301, + 487 + ], + "score": 0.61, + "content": "\\mathbf { C W I + B A }", + "type": "inline_equation" + }, + { + "bbox": [ + 301, + 475, + 506, + 489 + ], + "score": 1.0, + "content": "achieves fair allocation in almost all timesteps. The", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 85, + 487, + 505, + 499 + ], + "spans": [ + { + "bbox": [ + 85, + 489, + 100, + 499 + ], + "score": 1.0, + "content": "307", + "type": "text" + }, + { + "bbox": [ + 106, + 487, + 303, + 499 + ], + "score": 1.0, + "content": "fraction of timesteps where fairness is attained by", + "type": "text" + }, + { + "bbox": [ + 304, + 487, + 343, + 498 + ], + "score": 0.48, + "content": "\\mathrm { C W I + B A }", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 487, + 505, + 499 + ], + "score": 1.0, + "content": "is significantly higher than Hawkins and", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 86, + 497, + 506, + 510 + ], + "spans": [ + { + "bbox": [ + 86, + 500, + 100, + 509 + ], + "score": 1.0, + "content": "308", + "type": "text" + }, + { + "bbox": [ + 106, + 497, + 159, + 510 + ], + "score": 1.0, + "content": "OPT. In fact,", + "type": "text" + }, + { + "bbox": [ + 159, + 497, + 192, + 510 + ], + "score": 0.53, + "content": "\\mathrm { F i g } \\bigstar \\bigstar ( \\mathsf { b } )", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 497, + 506, + 510 + ], + "score": 1.0, + "content": "also shows that Hawkins obtains unfair solutions at every timesteps (0 fairness)", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 85, + 509, + 506, + 521 + ], + "spans": [ + { + "bbox": [ + 85, + 510, + 100, + 521 + ], + "score": 1.0, + "content": "309", + "type": "text" + }, + { + "bbox": [ + 106, + 509, + 130, + 520 + ], + "score": 1.0, + "content": "when", + "type": "text" + }, + { + "bbox": [ + 131, + 509, + 150, + 519 + ], + "score": 0.7, + "content": "{ \\Nu } { = } 5", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 509, + 168, + 520 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 169, + 510, + 192, + 519 + ], + "score": 0.85, + "content": "_ { \\mathrm { B = } 1 8 }", + "type": "inline_equation" + }, + { + "bbox": [ + 193, + 509, + 240, + 520 + ], + "score": 1.0, + "content": ", and, when", + "type": "text" + }, + { + "bbox": [ + 241, + 509, + 266, + 519 + ], + "score": 0.84, + "content": "{ \\Nu } { = } 1 0", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 509, + 284, + 520 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 284, + 509, + 309, + 519 + ], + "score": 0.82, + "content": "_ { \\mathrm { N = 1 5 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 509, + 506, + 520 + ], + "score": 1.0, + "content": ", Hawkins is fair only 0.41 and 0.67 fractions of", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 85, + 519, + 506, + 532 + ], + "spans": [ + { + "bbox": [ + 85, + 521, + 100, + 532 + ], + "score": 1.0, + "content": "310", + "type": "text" + }, + { + "bbox": [ + 105, + 519, + 506, + 532 + ], + "score": 1.0, + "content": "the time, respectively. 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In terms of fairness, Figs.", + "type": "text" + }, + { + "bbox": [ + 257, + 465, + 275, + 477 + ], + "score": 0.26, + "content": "2 ( \\mathbf { a } )", + "type": "inline_equation" + }, + { + "bbox": [ + 275, + 464, + 344, + 478 + ], + "score": 1.0, + "content": "and (c) show that", + "type": "text" + }, + { + "bbox": [ + 345, + 465, + 384, + 476 + ], + "score": 0.58, + "content": "\\mathrm { C W I + B A }", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 464, + 506, + 478 + ], + "score": 1.0, + "content": "achieves fair allocation among", + "type": "text" + } + ], + "index": 30, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 475, + 506, + 489 + ], + "spans": [ + { + "bbox": [ + 86, + 478, + 100, + 487 + ], + "score": 1.0, + "content": "306", + "type": "text" + }, + { + "bbox": [ + 105, + 475, + 241, + 489 + ], + "score": 1.0, + "content": "workers at all timesteps. In Figure", + "type": "text" + }, + { + "bbox": [ + 241, + 476, + 260, + 488 + ], + "score": 0.49, + "content": "\\boxed { 2 } ( 6 )", + "type": "inline_equation" + }, + { + "bbox": [ + 261, + 476, + 301, + 487 + ], + "score": 0.61, + "content": "\\mathbf { C W I + B A }", + "type": "inline_equation" + }, + { + "bbox": [ + 301, + 475, + 506, + 489 + ], + "score": 1.0, + "content": "achieves fair allocation in almost all timesteps. The", + "type": "text" + } + ], + "index": 31, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 487, + 505, + 499 + ], + "spans": [ + { + "bbox": [ + 85, + 489, + 100, + 499 + ], + "score": 1.0, + "content": "307", + "type": "text" + }, + { + "bbox": [ + 106, + 487, + 303, + 499 + ], + "score": 1.0, + "content": "fraction of timesteps where fairness is attained by", + "type": "text" + }, + { + "bbox": [ + 304, + 487, + 343, + 498 + ], + "score": 0.48, + "content": "\\mathrm { C W I + B A }", + "type": "inline_equation" + }, + { + "bbox": [ + 344, + 487, + 505, + 499 + ], + "score": 1.0, + "content": "is significantly higher than Hawkins and", + "type": "text" + } + ], + "index": 32, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 497, + 506, + 510 + ], + "spans": [ + { + "bbox": [ + 86, + 500, + 100, + 509 + ], + "score": 1.0, + "content": "308", + "type": "text" + }, + { + "bbox": [ + 106, + 497, + 159, + 510 + ], + "score": 1.0, + "content": "OPT. In fact,", + "type": "text" + }, + { + "bbox": [ + 159, + 497, + 192, + 510 + ], + "score": 0.53, + "content": "\\mathrm { F i g } \\bigstar \\bigstar ( \\mathsf { b } )", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 497, + 506, + 510 + ], + "score": 1.0, + "content": "also shows that Hawkins obtains unfair solutions at every timesteps (0 fairness)", + "type": "text" + } + ], + "index": 33, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 509, + 506, + 521 + ], + "spans": [ + { + "bbox": [ + 85, + 510, + 100, + 521 + ], + "score": 1.0, + "content": "309", + "type": "text" + }, + { + "bbox": [ + 106, + 509, + 130, + 520 + ], + "score": 1.0, + "content": "when", + "type": "text" + }, + { + "bbox": [ + 131, + 509, + 150, + 519 + ], + "score": 0.7, + "content": "{ \\Nu } { = } 5", + "type": "inline_equation" + }, + { + "bbox": [ + 151, + 509, + 168, + 520 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 169, + 510, + 192, + 519 + ], + "score": 0.85, + "content": "_ { \\mathrm { B = } 1 8 }", + "type": "inline_equation" + }, + { + "bbox": [ + 193, + 509, + 240, + 520 + ], + "score": 1.0, + "content": ", and, when", + "type": "text" + }, + { + "bbox": [ + 241, + 509, + 266, + 519 + ], + "score": 0.84, + "content": "{ \\Nu } { = } 1 0", + "type": "inline_equation" + }, + { + "bbox": [ + 266, + 509, + 284, + 520 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 284, + 509, + 309, + 519 + ], + "score": 0.82, + "content": "_ { \\mathrm { N = 1 5 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 309, + 509, + 506, + 520 + ], + "score": 1.0, + "content": ", Hawkins is fair only 0.41 and 0.67 fractions of", + "type": "text" + } + ], + "index": 34, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 519, + 506, + 532 + ], + "spans": [ + { + "bbox": [ + 85, + 521, + 100, + 532 + ], + "score": 1.0, + "content": "310", + "type": "text" + }, + { + "bbox": [ + 105, + 519, + 506, + 532 + ], + "score": 1.0, + "content": "the time, respectively. Thus, compared to reward maximizing baselines (Hawkins and OPT),", + "type": "text" + } + ], + "index": 35, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 529, + 506, + 543 + ], + "spans": [ + { + "bbox": [ + 85, + 532, + 100, + 542 + ], + "score": 1.0, + "content": "311", + "type": "text" + }, + { + "bbox": [ + 106, + 531, + 148, + 542 + ], + "score": 0.64, + "content": "\\mathbf { C W I + B A }", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 529, + 506, + 543 + ], + "score": 1.0, + "content": "achieves the highest fairness. We also compare against two versions of our solution", + "type": "text" + } + ], + "index": 36, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 541, + 506, + 555 + ], + "spans": [ + { + "bbox": [ + 85, + 543, + 100, + 553 + ], + "score": 1.0, + "content": "312", + "type": "text" + }, + { + "bbox": [ + 105, + 541, + 180, + 555 + ], + "score": 1.0, + "content": "approach, namely,", + "type": "text" + }, + { + "bbox": [ + 181, + 542, + 219, + 552 + ], + "score": 0.54, + "content": "\\mathrm { P W I + B A }", + "type": "inline_equation" + }, + { + "bbox": [ + 219, + 541, + 236, + 555 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 236, + 542, + 276, + 552 + ], + "score": 0.6, + "content": "\\mathrm { C W I + G A }", + "type": "inline_equation" + }, + { + "bbox": [ + 277, + 541, + 345, + 555 + ], + "score": 1.0, + "content": ". We observe that", + "type": "text" + }, + { + "bbox": [ + 346, + 542, + 384, + 552 + ], + "score": 0.59, + "content": "\\mathrm { P W I + B A }", + "type": "inline_equation" + }, + { + "bbox": [ + 384, + 541, + 506, + 555 + ], + "score": 1.0, + "content": "accumulates marginally lower", + "type": "text" + } + ], + "index": 37, + "is_list_start_line": true + }, + { + "bbox": [ + 85, + 551, + 505, + 566 + ], + "spans": [ + { + "bbox": [ + 85, + 554, + 100, + 564 + ], + "score": 1.0, + "content": "313", + "type": "text" + }, + { + "bbox": [ + 104, + 551, + 159, + 566 + ], + "score": 1.0, + "content": "reward while", + "type": "text" + }, + { + "bbox": [ + 160, + 552, + 200, + 563 + ], + "score": 0.63, + "content": "\\mathrm { C W I + G A }", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 551, + 505, + 566 + ], + "score": 1.0, + "content": "performs poorly in terms of fairness, hence asserting the importance of using", + "type": "text" + } + ], + "index": 38, + "is_list_start_line": true + }, + { + "bbox": [ + 86, + 563, + 264, + 576 + ], + "spans": [ + { + "bbox": [ + 86, + 566, + 100, + 575 + ], + "score": 1.0, + "content": "314", + "type": "text" + }, + { + "bbox": [ + 106, + 564, + 146, + 574 + ], + "score": 0.72, + "content": "\\mathrm { C W I + B A }", + "type": "inline_equation" + }, + { + "bbox": [ + 147, + 563, + 264, + 576 + ], + "score": 1.0, + "content": "for the MWRAMB problem.", + "type": "text" + } + ], + "index": 39, + "is_list_start_line": true + } + ], + "index": 33, + "bbox_fs": [ + 85, + 432, + 506, + 576 + ] + }, + { + "type": "text", + "bbox": [ + 102, + 579, + 505, + 646 + ], + "lines": [ + { + "bbox": [ + 107, + 579, + 506, + 593 + ], + "spans": [ + { + "bbox": [ + 107, + 579, + 173, + 593 + ], + "score": 1.0, + "content": "Fig 3 shows that", + "type": "text" + }, + { + "bbox": [ + 173, + 579, + 214, + 590 + ], + "score": 0.78, + "content": "\\mathbf { C W I + B A }", + "type": "inline_equation" + }, + { + "bbox": [ + 215, + 579, + 506, + 593 + ], + "score": 1.0, + "content": "is significantly faster than OPT-fair (the optimal MWRMAB solution),", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 590, + 506, + 604 + ], + "spans": [ + { + "bbox": [ + 105, + 590, + 273, + 604 + ], + "score": 1.0, + "content": "with an execution time improvement of", + "type": "text" + }, + { + "bbox": [ + 273, + 591, + 293, + 601 + ], + "score": 0.87, + "content": "3 3 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 590, + 297, + 604 + ], + "score": 1.0, + "content": ",", + "type": "text" + }, + { + "bbox": [ + 297, + 591, + 317, + 602 + ], + "score": 0.86, + "content": "7 8 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 318, + 590, + 336, + 604 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 337, + 591, + 357, + 601 + ], + "score": 0.88, + "content": "8 3 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 357, + 590, + 506, + 604 + ], + "score": 1.0, + "content": "for the three domains, respectively,", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 600, + 506, + 615 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 129, + 615 + ], + "score": 1.0, + "content": "when", + "type": "text" + }, + { + "bbox": [ + 130, + 602, + 149, + 612 + ], + "score": 0.7, + "content": "{ \\Nu } { = } 5", + "type": "inline_equation" + }, + { + "bbox": [ + 149, + 600, + 269, + 615 + ], + "score": 1.0, + "content": ". Moreover, for instances with", + "type": "text" + }, + { + "bbox": [ + 269, + 602, + 294, + 612 + ], + "score": 0.84, + "content": "{ \\Nu } { = } 1 0", + "type": "inline_equation" + }, + { + "bbox": [ + 294, + 600, + 506, + 615 + ], + "score": 1.0, + "content": "onwards, both OPT and OPT-fair ran out of memory", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 612, + 506, + 626 + ], + "spans": [ + { + "bbox": [ + 105, + 612, + 506, + 626 + ], + "score": 1.0, + "content": "because the execution of the optimal algorithms required exponentially larger memory. 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Group maintenance: A restless bandits approach. INFORMS", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 111, + 100, + 302, + 116 + ], + "spans": [ + { + "bbox": [ + 111, + 100, + 302, + 116 + ], + "score": 1.0, + "content": "Journal on Computing, 31(4):719–731, 2019.", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 88, + 121, + 505, + 135 + ], + "spans": [ + { + "bbox": [ + 88, + 121, + 505, + 135 + ], + "score": 1.0, + "content": "341 Daniel Adelman and Adam J. Mersereau. Relaxations of weakly coupled stochastic dynamic", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 114, + 132, + 338, + 145 + ], + "spans": [ + { + "bbox": [ + 114, + 132, + 338, + 145 + ], + "score": 1.0, + "content": "programs. Operations Research, 56(3):712–727, 2008.", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 88, + 152, + 506, + 166 + ], + "spans": [ + { + "bbox": [ + 88, + 152, + 506, + 166 + ], + "score": 1.0, + "content": "343 N. Akbarzadeh and A. Mahajan. Restless bandits with controlled restarts: Indexability and computa-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 115, + 164, + 463, + 176 + ], + "spans": [ + { + "bbox": [ + 115, + 164, + 463, + 176 + ], + "score": 1.0, + "content": "tion of whittle index. In 2019 IEEE Conference on Decision and Control. IEEE, 2019.", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 88, + 181, + 508, + 198 + ], + "spans": [ + { + "bbox": [ + 88, + 181, + 508, + 198 + ], + "score": 1.0, + "content": "345 Arpita Biswas and Siddharth Barman. Fair division under cardinality constraints. In Proceedings of", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 114, + 194, + 460, + 208 + ], + "spans": [ + { + "bbox": [ + 114, + 194, + 460, + 208 + ], + "score": 1.0, + "content": "the 27th International Joint Conference on Artificial Intelligence, pages 91–97, 2018.", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 87, + 212, + 507, + 228 + ], + "spans": [ + { + "bbox": [ + 87, + 212, + 507, + 228 + ], + "score": 1.0, + "content": "347 Felix Brandt, Vincent Conitzer, Ulle Endriss, Jérôme Lang, and Ariel D Procaccia. Handbook of", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 115, + 226, + 426, + 237 + ], + "spans": [ + { + "bbox": [ + 115, + 226, + 426, + 237 + ], + "score": 1.0, + "content": "computational social choice, Chapter 12. Cambridge University Press, 2016.", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 245, + 505, + 257 + ], + "spans": [ + { + "bbox": [ + 105, + 245, + 505, + 257 + ], + "score": 1.0, + "content": "Eric Budish. The combinatorial assignment problem: Approximate competitive equilibrium from", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 114, + 255, + 406, + 269 + ], + "spans": [ + { + "bbox": [ + 114, + 255, + 406, + 269 + ], + "score": 1.0, + "content": "equal incomes. Journal of Political Economy, 119(6):1061–1103, 2011.", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 273, + 507, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 273, + 507, + 288 + ], + "score": 1.0, + "content": "Yifang Chen, Alex Cuellar, Haipeng Luo, Jignesh Modi, Heramb Nemlekar, and Stefanos Nikolaidis.", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 114, + 285, + 506, + 299 + ], + "spans": [ + { + "bbox": [ + 114, + 285, + 506, + 299 + ], + "score": 1.0, + "content": "Fair contextual multi-armed bandits: Theory and experiments. In Conference on Uncertainty in", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 113, + 296, + 329, + 309 + ], + "spans": [ + { + "bbox": [ + 113, + 296, + 329, + 309 + ], + "score": 1.0, + "content": "Artificial Intelligence, pages 181–190. PMLR, 2020.", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 103, + 316, + 506, + 330 + ], + "spans": [ + { + "bbox": [ + 103, + 316, + 506, + 330 + ], + "score": 1.0, + "content": "Kevin D. Glazebrook, David J. Hodge, and Christopher Kirkbride. General notions of indexability for", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 114, + 326, + 507, + 342 + ], + "spans": [ + { + "bbox": [ + 114, + 326, + 507, + 342 + ], + "score": 1.0, + "content": "queueing control and asset management. The Annals of Applied Probability, 21(3):876–907, 2011.", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 89, + 348, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 89, + 348, + 505, + 360 + ], + "score": 1.0, + "content": "Yasin Gocgun and Archis Ghate. Lagrangian relaxation and constraint generation for allocation and", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 115, + 359, + 453, + 371 + ], + "spans": [ + { + "bbox": [ + 115, + 359, + 453, + 371 + ], + "score": 1.0, + "content": "advanced scheduling. Computers & Operations Research, 39(10):2323–2336, 2012.", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 88, + 378, + 505, + 391 + ], + "spans": [ + { + "bbox": [ + 88, + 378, + 505, + 391 + ], + "score": 1.0, + "content": "358 Jeffrey Thomas Hawkins. A Langrangian decomposition approach to weakly coupled dynamic", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 114, + 388, + 507, + 405 + ], + "spans": [ + { + "bbox": [ + 114, + 388, + 507, + 405 + ], + "score": 1.0, + "content": "optimization problems and its applications. PhD thesis, Massachusetts Institute of Technology,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 114, + 399, + 144, + 414 + ], + "spans": [ + { + "bbox": [ + 114, + 399, + 144, + 414 + ], + "score": 1.0, + "content": "2003.", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 88, + 419, + 507, + 433 + ], + "spans": [ + { + "bbox": [ + 88, + 419, + 507, + 433 + ], + "score": 1.0, + "content": "361 Christine Herlihy, Aviva Prins, Aravind Srinivasan, and John Dickerson. Planning to fairly allocate:", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 114, + 430, + 482, + 444 + ], + "spans": [ + { + "bbox": [ + 114, + 430, + 482, + 444 + ], + "score": 1.0, + "content": "Probabilistic fairness in the restless bandit setting. arXiv preprint arXiv:2106.07677, 2021.", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 90, + 449, + 506, + 464 + ], + "spans": [ + { + "bbox": [ + 90, + 449, + 506, + 464 + ], + "score": 1.0, + "content": "63 Matthew Joseph, Michael Kearns, Jamie H Morgenstern, and Aaron Roth. Fairness in learning:", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 114, + 460, + 507, + 475 + ], + "spans": [ + { + "bbox": [ + 114, + 460, + 507, + 475 + ], + "score": 1.0, + "content": "Classic and contextual bandits. Advances in Neural Information Processing Systems, 29:325��333,", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 114, + 470, + 144, + 487 + ], + "spans": [ + { + "bbox": [ + 114, + 470, + 144, + 487 + ], + "score": 1.0, + "content": "2016.", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 103, + 491, + 505, + 506 + ], + "spans": [ + { + "bbox": [ + 103, + 491, + 505, + 506 + ], + "score": 1.0, + "content": "Aditya Mate, Jackson A Killian, Haifeng Xu, Andrew Perrault, and Milind Tambe. Collapsing", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 114, + 502, + 505, + 516 + ], + "spans": [ + { + "bbox": [ + 114, + 502, + 505, + 516 + ], + "score": 1.0, + "content": "bandits and their application to public health interventions. In Advances in Neural Information", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 114, + 514, + 225, + 528 + ], + "spans": [ + { + "bbox": [ + 114, + 514, + 225, + 528 + ], + "score": 1.0, + "content": "Processing Systems, 2020.", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 88, + 533, + 506, + 547 + ], + "spans": [ + { + "bbox": [ + 88, + 533, + 506, + 547 + ], + "score": 1.0, + "content": "369 Rahul Meshram and Kesav Kaza. Simulation based algorithms for markov decision processes and", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 115, + 545, + 396, + 557 + ], + "spans": [ + { + "bbox": [ + 115, + 545, + 396, + 557 + ], + "score": 1.0, + "content": "multi-action restless bandits. arXiv preprint arXiv:2007.12933, 2020.", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 103, + 564, + 506, + 578 + ], + "spans": [ + { + "bbox": [ + 103, + 564, + 506, + 578 + ], + "score": 1.0, + "content": "Rahul Meshram, D Manjunath, and Aditya Gopalan. A restless bandit with no observable states for", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 116, + 576, + 505, + 588 + ], + "spans": [ + { + "bbox": [ + 116, + 576, + 505, + 588 + ], + "score": 1.0, + "content": "recommendation systems and communication link scheduling. In 2015 54th IEEE Conference on", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 114, + 586, + 365, + 600 + ], + "spans": [ + { + "bbox": [ + 114, + 586, + 365, + 600 + ], + "score": 1.0, + "content": "Decision and Control (CDC), pages 7820–7825. IEEE, 2015.", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 606, + 506, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 506, + 618 + ], + "score": 1.0, + "content": "Christos H Papadimitriou and John N Tsitsiklis. The complexity of optimal queueing network control.", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 114, + 617, + 506, + 631 + ], + "spans": [ + { + "bbox": [ + 114, + 617, + 506, + 631 + ], + "score": 1.0, + "content": "In Proceedings of IEEE 9th Annual Conference on Structure in Complexity Theory, pages 318–322.", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 114, + 627, + 169, + 641 + ], + "spans": [ + { + "bbox": [ + 114, + 627, + 169, + 641 + ], + "score": 1.0, + "content": "IEEE, 1994.", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 104, + 647, + 505, + 661 + ], + "spans": [ + { + "bbox": [ + 104, + 647, + 505, + 661 + ], + "score": 1.0, + "content": "Vishakha Patil, Ganesh Ghalme, Vineet Nair, and Y Narahari. Achieving fairness in the stochastic", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 115, + 658, + 506, + 672 + ], + "spans": [ + { + "bbox": [ + 115, + 658, + 506, + 672 + ], + "score": 1.0, + "content": "multi-armed bandit problem. In Proceedings of the AAAI Conference on Artificial Intelligence,", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 113, + 668, + 265, + 682 + ], + "spans": [ + { + "bbox": [ + 113, + 668, + 265, + 682 + ], + "score": 1.0, + "content": "volume 34, pages 5379–5386, 2020.", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 89, + 688, + 506, + 702 + ], + "spans": [ + { + "bbox": [ + 89, + 688, + 506, + 702 + ], + "score": 1.0, + "content": "380 Y. Qian, C. Zhang, B. Krishnamachari, and B. Tambe. Restless poachers: Handling exploration-", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 114, + 699, + 506, + 714 + ], + "spans": [ + { + "bbox": [ + 114, + 699, + 506, + 714 + ], + "score": 1.0, + "content": "exploitation tradeoffs in security domains. In International Joint Conference on Autonomous", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 113, + 711, + 361, + 724 + ], + "spans": [ + { + "bbox": [ + 113, + 711, + 361, + 724 + ], + "score": 1.0, + "content": "Agents and Multi-Agent Systems, AAMAS. IFAAMAS, 2016.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 22 + } + ], + "page_idx": 9, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 300, + 741, + 311, + 750 + ], + "lines": [ + { + "bbox": [ + 299, + 740, + 313, + 754 + ], + "spans": [ + { + "bbox": [ + 299, + 740, + 313, + 754 + ], + "score": 1.0, + "content": "10", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 85, + 76, + 102, + 734 + ], + "lines": [ + { + "bbox": [ + 83, + 102, + 102, + 117 + ], + "spans": [ + { + "bbox": [ + 83, + 102, + 102, + 117 + ], + "score": 1.0, + "content": "340", + "type": "text" + } + ] + }, + { + "bbox": [ + 83, + 133, + 101, + 148 + ], + "spans": [ + { + "bbox": [ + 83, + 133, + 101, + 148 + ], + "score": 1.0, + "content": "342", + "type": "text" + } + ] + }, + { + "bbox": [ + 83, + 164, + 101, + 180 + ], + "spans": [ + { + "bbox": [ + 83, + 164, + 101, + 180 + ], + "score": 1.0, + "content": "344", + "type": "text" + } + ] + }, + { + "bbox": [ + 83, + 194, + 101, + 209 + ], + "spans": [ + { + "bbox": [ + 83, + 194, + 101, + 209 + ], + "score": 1.0, + "content": "346", + "type": "text" + } + ] + }, + { + "bbox": [ + 83, + 226, + 101, + 241 + ], + "spans": [ + { + "bbox": [ + 83, + 226, + 101, + 241 + ], + "score": 1.0, + "content": "348", + "type": "text" + } + ] + }, + { + "bbox": [ + 84, + 246, + 100, + 257 + ], + "spans": [ + { + "bbox": [ + 84, + 246, + 100, + 257 + ], + "score": 1.0, + "content": "349", + "type": "text" + } + ] + }, + { + "bbox": [ + 83, + 256, + 101, + 272 + ], + "spans": [ + { + "bbox": [ + 83, + 256, + 101, + 272 + ], + "score": 1.0, + "content": "350", + "type": "text" + } + ] + }, + { + "bbox": [ + 84, + 276, + 100, + 288 + ], + "spans": [ + { + "bbox": [ + 84, + 276, + 100, + 288 + ], + "score": 1.0, + "content": "351", + "type": "text" + } + ] + }, + { + "bbox": [ + 84, + 287, + 100, + 299 + ], + "spans": [ + { + "bbox": [ + 84, + 287, + 100, + 299 + ], + "score": 1.0, + "content": "352", + "type": "text" + } + ] + }, + { + "bbox": [ + 83, + 298, + 101, + 313 + ], + "spans": [ + { + "bbox": [ + 83, + 298, + 101, + 313 + ], + "score": 1.0, + "content": "353", + "type": "text" + } + ] + }, + { + "bbox": [ + 83, + 317, + 102, + 331 + ], + "spans": [ + { + "bbox": [ + 83, + 317, + 102, + 331 + ], + "score": 1.0, + "content": "354", + "type": "text" + } + ] + }, + { + "bbox": [ + 83, + 328, + 101, + 345 + ], + "spans": [ + { + "bbox": [ + 83, + 328, + 101, + 345 + ], + "score": 1.0, + "content": "355", + "type": "text" + } + ] + }, + { + "bbox": [ + 83, + 348, + 102, + 361 + ], + "spans": [ + { + "bbox": [ + 83, + 348, + 102, + 361 + ], + "score": 1.0, + "content": "356", + "type": "text" + } + ] + }, + { + "bbox": [ + 85, + 360, + 101, + 374 + ], + "spans": [ + { + "bbox": [ + 85, + 360, + 101, + 374 + ], + "score": 1.0, + "content": "357", + "type": "text" + } + ] + }, + { + "bbox": [ + 84, + 390, + 100, + 402 + ], + "spans": [ + { + "bbox": [ + 84, + 390, + 100, + 402 + ], + "score": 1.0, + "content": "359", + "type": "text" + } + ] + }, + { + "bbox": [ + 83, + 401, + 101, + 416 + ], + "spans": [ + { + "bbox": [ + 83, + 401, + 101, + 416 + ], + "score": 1.0, + "content": "360", + "type": "text" + } + ] + }, + { + "bbox": [ + 83, + 432, + 101, + 448 + ], + "spans": [ + { + "bbox": [ + 83, + 432, + 101, + 448 + ], + "score": 1.0, + "content": "362", + "type": "text" + } + ] + }, + { + "bbox": [ + 83, + 450, + 102, + 465 + ], + "spans": [ + { + "bbox": [ + 83, + 450, + 102, + 465 + ], + "score": 1.0, + "content": "3", + "type": "text" + } + ] + }, + { + "bbox": [ + 84, + 462, + 100, + 475 + ], + "spans": [ + { + "bbox": [ + 84, + 462, + 100, + 475 + ], + "score": 1.0, + "content": "364", + "type": "text" + } + ] + }, + { + "bbox": [ + 83, + 473, + 101, + 487 + ], + "spans": [ + { + "bbox": [ + 83, + 473, + 101, + 487 + ], + "score": 1.0, + "content": "365", + "type": "text" + } + ] + }, + { + "bbox": [ + 83, + 492, + 102, + 507 + ], + "spans": [ + { + "bbox": [ + 83, + 492, + 102, + 507 + ], + "score": 1.0, + "content": "366", + "type": "text" + } + ] + }, + { + "bbox": [ + 84, + 505, + 100, + 516 + ], + "spans": [ + { + "bbox": [ + 84, + 505, + 100, + 516 + ], + "score": 1.0, + "content": "367", + "type": "text" + } + ] + }, + { + "bbox": [ + 83, + 515, + 101, + 529 + ], + "spans": [ + { + "bbox": [ + 83, + 515, + 101, + 529 + ], + "score": 1.0, + "content": "368", + "type": "text" + } + ] + }, + { + "bbox": [ + 83, + 546, + 101, + 561 + ], + "spans": [ + { + "bbox": [ + 83, + 546, + 101, + 561 + ], + "score": 1.0, + "content": "370", + "type": "text" + } + ] + }, + { + "bbox": [ + 84, + 565, + 100, + 577 + ], + "spans": [ + { + "bbox": [ + 84, + 565, + 100, + 577 + ], + "score": 1.0, + "content": "371", + "type": "text" + } + ] + }, + { + "bbox": [ + 84, + 577, + 101, + 588 + ], + "spans": [ + { + "bbox": [ + 84, + 577, + 101, + 588 + ], + "score": 1.0, + "content": "372", + "type": "text" + } + ] + }, + { + "bbox": [ + 83, + 587, + 102, + 601 + ], + "spans": [ + { + "bbox": [ + 83, + 587, + 102, + 601 + ], + "score": 1.0, + "content": "373", + "type": "text" + } + ] + }, + { + "bbox": [ + 83, + 606, + 102, + 620 + ], + "spans": [ + { + "bbox": [ + 83, + 606, + 102, + 620 + ], + "score": 1.0, + "content": "374", + "type": "text" + } + ] + }, + { + "bbox": [ + 84, + 619, + 101, + 630 + ], + "spans": [ + { + "bbox": [ + 84, + 619, + 101, + 630 + ], + "score": 1.0, + "content": "375", + "type": "text" + } + ] + }, + { + "bbox": [ + 83, + 629, + 102, + 642 + ], + "spans": [ + { + "bbox": [ + 83, + 629, + 102, + 642 + ], + "score": 1.0, + "content": "376", + "type": "text" + } + ] + }, + { + "bbox": [ + 83, + 648, + 101, + 662 + ], + "spans": [ + { + "bbox": [ + 83, + 648, + 101, + 662 + ], + "score": 1.0, + "content": "377", + "type": "text" + } + ] + }, + { + "bbox": [ + 84, + 660, + 101, + 671 + ], + "spans": [ + { + "bbox": [ + 84, + 660, + 101, + 671 + ], + "score": 1.0, + "content": "378", + "type": "text" + } + ] + }, + { + "bbox": [ + 83, + 670, + 101, + 684 + ], + "spans": [ + { + "bbox": [ + 83, + 670, + 101, + 684 + ], + "score": 1.0, + "content": "379", + "type": "text" + } + ] + }, + { + "bbox": [ + 83, + 689, + 102, + 703 + ], + "spans": [ + { + "bbox": [ + 83, + 689, + 102, + 703 + ], + "score": 1.0, + "content": "380", + "type": "text" + } + ] + }, + { + "bbox": [ + 83, + 700, + 101, + 715 + ], + "spans": [ + { + "bbox": [ + 83, + 700, + 101, + 715 + ], + "score": 1.0, + "content": "381", + "type": "text" + } + ] + }, + { + "bbox": [ + 83, + 712, + 102, + 726 + ], + "spans": [ + { + "bbox": [ + 83, + 712, + 102, + 726 + ], + "score": 1.0, + "content": "382", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "list", + "bbox": [ + 92, + 57, + 507, + 729 + ], + "lines": [ + { + "bbox": [ + 88, + 70, + 166, + 86 + ], + "spans": [ + { + "bbox": [ + 88, + 70, + 166, + 86 + ], + "score": 1.0, + "content": "338 References", + "type": "text" + } + ], + "index": 0, + "is_list_start_line": true, + "is_list_end_line": true + }, + { + "bbox": [ + 86, + 89, + 507, + 105 + ], + "spans": [ + { + "bbox": [ + 86, + 89, + 507, + 105 + ], + "score": 1.0, + "content": "339 Abderrahmane Abbou and Viliam Makis. Group maintenance: A restless bandits approach. INFORMS", + "type": "text" + } + ], + "index": 1, + "is_list_start_line": true + }, + { + "bbox": [ + 111, + 100, + 302, + 116 + ], + "spans": [ + { + "bbox": [ + 111, + 100, + 302, + 116 + ], + "score": 1.0, + "content": "Journal on Computing, 31(4):719–731, 2019.", + "type": "text" + } + ], + "index": 2, + "is_list_end_line": true + }, + { + "bbox": [ + 88, + 121, + 505, + 135 + ], + "spans": [ + { + "bbox": [ + 88, + 121, + 505, + 135 + ], + "score": 1.0, + "content": "341 Daniel Adelman and Adam J. Mersereau. Relaxations of weakly coupled stochastic dynamic", + "type": "text" + } + ], + "index": 3, + "is_list_start_line": true + }, + { + "bbox": [ + 114, + 132, + 338, + 145 + ], + "spans": [ + { + "bbox": [ + 114, + 132, + 338, + 145 + ], + "score": 1.0, + "content": "programs. Operations Research, 56(3):712–727, 2008.", + "type": "text" + } + ], + "index": 4, + "is_list_end_line": true + }, + { + "bbox": [ + 88, + 152, + 506, + 166 + ], + "spans": [ + { + "bbox": [ + 88, + 152, + 506, + 166 + ], + "score": 1.0, + "content": "343 N. Akbarzadeh and A. Mahajan. Restless bandits with controlled restarts: Indexability and computa-", + "type": "text" + } + ], + "index": 5, + "is_list_start_line": true + }, + { + "bbox": [ + 115, + 164, + 463, + 176 + ], + "spans": [ + { + "bbox": [ + 115, + 164, + 463, + 176 + ], + "score": 1.0, + "content": "tion of whittle index. In 2019 IEEE Conference on Decision and Control. IEEE, 2019.", + "type": "text" + } + ], + "index": 6, + "is_list_end_line": true + }, + { + "bbox": [ + 88, + 181, + 508, + 198 + ], + "spans": [ + { + "bbox": [ + 88, + 181, + 508, + 198 + ], + "score": 1.0, + "content": "345 Arpita Biswas and Siddharth Barman. Fair division under cardinality constraints. In Proceedings of", + "type": "text" + } + ], + "index": 7, + "is_list_start_line": true + }, + { + "bbox": [ + 114, + 194, + 460, + 208 + ], + "spans": [ + { + "bbox": [ + 114, + 194, + 460, + 208 + ], + "score": 1.0, + "content": "the 27th International Joint Conference on Artificial Intelligence, pages 91–97, 2018.", + "type": "text" + } + ], + "index": 8, + "is_list_end_line": true + }, + { + "bbox": [ + 87, + 212, + 507, + 228 + ], + "spans": [ + { + "bbox": [ + 87, + 212, + 507, + 228 + ], + "score": 1.0, + "content": "347 Felix Brandt, Vincent Conitzer, Ulle Endriss, Jérôme Lang, and Ariel D Procaccia. Handbook of", + "type": "text" + } + ], + "index": 9, + "is_list_start_line": true + }, + { + "bbox": [ + 115, + 226, + 426, + 237 + ], + "spans": [ + { + "bbox": [ + 115, + 226, + 426, + 237 + ], + "score": 1.0, + "content": "computational social choice, Chapter 12. Cambridge University Press, 2016.", + "type": "text" + } + ], + "index": 10, + "is_list_end_line": true + }, + { + "bbox": [ + 105, + 245, + 505, + 257 + ], + "spans": [ + { + "bbox": [ + 105, + 245, + 505, + 257 + ], + "score": 1.0, + "content": "Eric Budish. The combinatorial assignment problem: Approximate competitive equilibrium from", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 114, + 255, + 406, + 269 + ], + "spans": [ + { + "bbox": [ + 114, + 255, + 406, + 269 + ], + "score": 1.0, + "content": "equal incomes. Journal of Political Economy, 119(6):1061–1103, 2011.", + "type": "text" + } + ], + "index": 12, + "is_list_end_line": true + }, + { + "bbox": [ + 105, + 273, + 507, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 273, + 507, + 288 + ], + "score": 1.0, + "content": "Yifang Chen, Alex Cuellar, Haipeng Luo, Jignesh Modi, Heramb Nemlekar, and Stefanos Nikolaidis.", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 114, + 285, + 506, + 299 + ], + "spans": [ + { + "bbox": [ + 114, + 285, + 506, + 299 + ], + "score": 1.0, + "content": "Fair contextual multi-armed bandits: Theory and experiments. In Conference on Uncertainty in", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 113, + 296, + 329, + 309 + ], + "spans": [ + { + "bbox": [ + 113, + 296, + 329, + 309 + ], + "score": 1.0, + "content": "Artificial Intelligence, pages 181–190. PMLR, 2020.", + "type": "text" + } + ], + "index": 15, + "is_list_end_line": true + }, + { + "bbox": [ + 103, + 316, + 506, + 330 + ], + "spans": [ + { + "bbox": [ + 103, + 316, + 506, + 330 + ], + "score": 1.0, + "content": "Kevin D. Glazebrook, David J. Hodge, and Christopher Kirkbride. General notions of indexability for", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 114, + 326, + 507, + 342 + ], + "spans": [ + { + "bbox": [ + 114, + 326, + 507, + 342 + ], + "score": 1.0, + "content": "queueing control and asset management. The Annals of Applied Probability, 21(3):876–907, 2011.", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 89, + 348, + 505, + 360 + ], + "spans": [ + { + "bbox": [ + 89, + 348, + 505, + 360 + ], + "score": 1.0, + "content": "Yasin Gocgun and Archis Ghate. Lagrangian relaxation and constraint generation for allocation and", + "type": "text" + } + ], + "index": 18, + "is_list_start_line": true + }, + { + "bbox": [ + 115, + 359, + 453, + 371 + ], + "spans": [ + { + "bbox": [ + 115, + 359, + 453, + 371 + ], + "score": 1.0, + "content": "advanced scheduling. Computers & Operations Research, 39(10):2323–2336, 2012.", + "type": "text" + } + ], + "index": 19, + "is_list_end_line": true + }, + { + "bbox": [ + 88, + 378, + 505, + 391 + ], + "spans": [ + { + "bbox": [ + 88, + 378, + 505, + 391 + ], + "score": 1.0, + "content": "358 Jeffrey Thomas Hawkins. A Langrangian decomposition approach to weakly coupled dynamic", + "type": "text" + } + ], + "index": 20, + "is_list_start_line": true + }, + { + "bbox": [ + 114, + 388, + 507, + 405 + ], + "spans": [ + { + "bbox": [ + 114, + 388, + 507, + 405 + ], + "score": 1.0, + "content": "optimization problems and its applications. PhD thesis, Massachusetts Institute of Technology,", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 114, + 399, + 144, + 414 + ], + "spans": [ + { + "bbox": [ + 114, + 399, + 144, + 414 + ], + "score": 1.0, + "content": "2003.", + "type": "text" + } + ], + "index": 22, + "is_list_end_line": true + }, + { + "bbox": [ + 88, + 419, + 507, + 433 + ], + "spans": [ + { + "bbox": [ + 88, + 419, + 507, + 433 + ], + "score": 1.0, + "content": "361 Christine Herlihy, Aviva Prins, Aravind Srinivasan, and John Dickerson. Planning to fairly allocate:", + "type": "text" + } + ], + "index": 23, + "is_list_start_line": true + }, + { + "bbox": [ + 114, + 430, + 482, + 444 + ], + "spans": [ + { + "bbox": [ + 114, + 430, + 482, + 444 + ], + "score": 1.0, + "content": "Probabilistic fairness in the restless bandit setting. arXiv preprint arXiv:2106.07677, 2021.", + "type": "text" + } + ], + "index": 24, + "is_list_end_line": true + }, + { + "bbox": [ + 90, + 449, + 506, + 464 + ], + "spans": [ + { + "bbox": [ + 90, + 449, + 506, + 464 + ], + "score": 1.0, + "content": "63 Matthew Joseph, Michael Kearns, Jamie H Morgenstern, and Aaron Roth. Fairness in learning:", + "type": "text" + } + ], + "index": 25, + "is_list_start_line": true + }, + { + "bbox": [ + 114, + 460, + 507, + 475 + ], + "spans": [ + { + "bbox": [ + 114, + 460, + 507, + 475 + ], + "score": 1.0, + "content": "Classic and contextual bandits. Advances in Neural Information Processing Systems, 29:325–333,", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 114, + 470, + 144, + 487 + ], + "spans": [ + { + "bbox": [ + 114, + 470, + 144, + 487 + ], + "score": 1.0, + "content": "2016.", + "type": "text" + } + ], + "index": 27, + "is_list_end_line": true + }, + { + "bbox": [ + 103, + 491, + 505, + 506 + ], + "spans": [ + { + "bbox": [ + 103, + 491, + 505, + 506 + ], + "score": 1.0, + "content": "Aditya Mate, Jackson A Killian, Haifeng Xu, Andrew Perrault, and Milind Tambe. Collapsing", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 114, + 502, + 505, + 516 + ], + "spans": [ + { + "bbox": [ + 114, + 502, + 505, + 516 + ], + "score": 1.0, + "content": "bandits and their application to public health interventions. In Advances in Neural Information", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 114, + 514, + 225, + 528 + ], + "spans": [ + { + "bbox": [ + 114, + 514, + 225, + 528 + ], + "score": 1.0, + "content": "Processing Systems, 2020.", + "type": "text" + } + ], + "index": 30, + "is_list_end_line": true + }, + { + "bbox": [ + 88, + 533, + 506, + 547 + ], + "spans": [ + { + "bbox": [ + 88, + 533, + 506, + 547 + ], + "score": 1.0, + "content": "369 Rahul Meshram and Kesav Kaza. Simulation based algorithms for markov decision processes and", + "type": "text" + } + ], + "index": 31, + "is_list_start_line": true + }, + { + "bbox": [ + 115, + 545, + 396, + 557 + ], + "spans": [ + { + "bbox": [ + 115, + 545, + 396, + 557 + ], + "score": 1.0, + "content": "multi-action restless bandits. arXiv preprint arXiv:2007.12933, 2020.", + "type": "text" + } + ], + "index": 32, + "is_list_end_line": true + }, + { + "bbox": [ + 103, + 564, + 506, + 578 + ], + "spans": [ + { + "bbox": [ + 103, + 564, + 506, + 578 + ], + "score": 1.0, + "content": "Rahul Meshram, D Manjunath, and Aditya Gopalan. A restless bandit with no observable states for", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 116, + 576, + 505, + 588 + ], + "spans": [ + { + "bbox": [ + 116, + 576, + 505, + 588 + ], + "score": 1.0, + "content": "recommendation systems and communication link scheduling. In 2015 54th IEEE Conference on", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 114, + 586, + 365, + 600 + ], + "spans": [ + { + "bbox": [ + 114, + 586, + 365, + 600 + ], + "score": 1.0, + "content": "Decision and Control (CDC), pages 7820–7825. IEEE, 2015.", + "type": "text" + } + ], + "index": 35, + "is_list_end_line": true + }, + { + "bbox": [ + 105, + 606, + 506, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 506, + 618 + ], + "score": 1.0, + "content": "Christos H Papadimitriou and John N Tsitsiklis. The complexity of optimal queueing network control.", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 114, + 617, + 506, + 631 + ], + "spans": [ + { + "bbox": [ + 114, + 617, + 506, + 631 + ], + "score": 1.0, + "content": "In Proceedings of IEEE 9th Annual Conference on Structure in Complexity Theory, pages 318–322.", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 114, + 627, + 169, + 641 + ], + "spans": [ + { + "bbox": [ + 114, + 627, + 169, + 641 + ], + "score": 1.0, + "content": "IEEE, 1994.", + "type": "text" + } + ], + "index": 38, + "is_list_end_line": true + }, + { + "bbox": [ + 104, + 647, + 505, + 661 + ], + "spans": [ + { + "bbox": [ + 104, + 647, + 505, + 661 + ], + "score": 1.0, + "content": "Vishakha Patil, Ganesh Ghalme, Vineet Nair, and Y Narahari. 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