MULTI-ARMED BANDITS UNDER GENERAL DEPRECIATION AND COMMITMENT
Generally, the multi-armed has been studied under the setting that at each time step over an infinite horizon a controller chooses to activate a single process or bandit out of a finite collection of independent processes (statistical experiments, populations, etc.) for a single period, receiving a...
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| Published in | Probability in the engineering and informational sciences Vol. 29; no. 1; pp. 51 - 76 |
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| Main Authors | , |
| Format | Journal Article |
| Language | English |
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New York, USA
Cambridge University Press
01.01.2015
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| Abstract | Generally, the multi-armed has been studied under the setting that at each time step over an infinite horizon a controller chooses to activate a single process or bandit out of a finite collection of independent processes (statistical experiments, populations, etc.) for a single period, receiving a reward that is a function of the activated process, and in doing so advancing the chosen process. Classically, rewards are discounted by a constant factor β∈(0, 1) per round. In this paper, we present a solution to the problem, with potentially non-Markovian, uncountable state space reward processes, under a framework in which, first, the discount factors may be non-uniform and vary over time, and second, the periods of activation of each bandit may be not be fixed or uniform, subject instead to a possibly stochastic duration of activation before a change to a different bandit is allowed. The solution is based on generalized restart-in-state indices, and it utilizes a view of the problem not as “decisions over state space” but rather “decisions over time”. |
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| AbstractList | Generally, the multi-armed has been studied under the setting that at each time step over an infinite horizon a controller chooses to activate a single process or bandit out of a finite collection of independent processes (statistical experiments, populations, etc.) for a single period, receiving a reward that is a function of the activated process, and in doing so advancing the chosen process. Classically, rewards are discounted by a constant factor [beta]∈(0, 1) per round. In this paper, we present a solution to the problem, with potentially non-Markovian, uncountable state space reward processes, under a framework in which, first, the discount factors may be non-uniform and vary over time, and second, the periods of activation of each bandit may be not be fixed or uniform, subject instead to a possibly stochastic duration of activation before a change to a different bandit is allowed. The solution is based on generalized restart-in-state indices, and it utilizes a view of the problem not as "decisions over state space" but rather "decisions over time". Generally, the multi-armed has been studied under the setting that at each time step over an infinite horizon a controller chooses to activate a single process or bandit out of a finite collection of independent processes (statistical experiments, populations, etc.) for a single period, receiving a reward that is a function of the activated process, and in doing so advancing the chosen process. Classically, rewards are discounted by a constant factor β∈(0, 1) per round. In this paper, we present a solution to the problem, with potentially non-Markovian, uncountable state space reward processes, under a framework in which, first, the discount factors may be non-uniform and vary over time, and second, the periods of activation of each bandit may be not be fixed or uniform, subject instead to a possibly stochastic duration of activation before a change to a different bandit is allowed. The solution is based on generalized restart-in-state indices, and it utilizes a view of the problem not as “decisions over state space” but rather “decisions over time”. Generally, the multi-armed has been studied under the setting that at each time step over an infinite horizon a controller chooses to activate a single process or bandit out of a finite collection of independent processes (statistical experiments, populations, etc.) for a single period, receiving a reward that is a function of the activated process, and in doing so advancing the chosen process. Classically, rewards are discounted by a constant factor beta (0, 1) per round. In this paper, we present a solution to the problem, with potentially non-Markovian, uncountable state space reward processes, under a framework in which, first, the discount factors may be non-uniform and vary over time, and second, the periods of activation of each bandit may be not be fixed or uniform, subject instead to a possibly stochastic duration of activation before a change to a different bandit is allowed. The solution is based on generalized restart-in-state indices, and it utilizes a view of the problem not as "decisions over state space" but rather "decisions over time". Generally, the multi-armed has been studied under the setting that at each time step over an infinite horizon a controller chooses to activate a single process or bandit out of a finite collection of independent processes (statistical experiments, populations, etc.) for a single period, receiving a reward that is a function of the activated process, and in doing so advancing the chosen process. Classically, rewards are discounted by a constant factor β∈(0, 1) per round. In this paper, we present a solution to the problem, with potentially non-Markovian, uncountable state space reward processes, under a framework in which, first, the discount factors may be non-uniform and vary over time, and second, the periods of activation of each bandit may be not be fixed or uniform, subject instead to a possibly stochastic duration of activation before a change to a different bandit is allowed. The solution is based on generalized restart-in-state indices, and it utilizes a view of the problem not as “decisions over state space” but rather “decisions over time”. |
| Author | Katehakis, Michael N. Cowan, Wesley |
| Author_xml | – sequence: 1 givenname: Wesley surname: Cowan fullname: Cowan, Wesley email: cwcowan@mah.rutgers.edu organization: Department of Mathematics, Rutgers University, 110 Frelinghuysen Road, Piscataway, NJ 08854, USA E-mail: cwcowan@mah.rutgers.edu – sequence: 2 givenname: Michael N. surname: Katehakis fullname: Katehakis, Michael N. email: mnk@rutgers.edu organization: Department of Management Science and Information Systems, Rutgers Business School, Newark and New Brunswick, 100 Rockafeller Road, Piscataway, NJ 08854, USA E-mail: mnk@rutgers.edu |
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| References_xml | – ident: S0269964814000217_ref11 doi: 10.1007/978-1-4757-1776-1 – ident: S0269964814000217_ref2 doi: 10.1109/ROBOT.2008.4543563 – volume: II volume-title: Dynamic programming and optimal control year: 2011 ident: S0269964814000217_ref4 – ident: S0269964814000217_ref50 doi: 10.1109/TAC.1985.1103989 – ident: S0269964814000217_ref38 doi: 10.1287/moor.1050.0165 – volume-title: Cyrus Derman Memorial Volume I: Optimization under Uncertainty: Costs, Risks and Revenues year: 2013 ident: S0269964814000217_ref12 – ident: S0269964814000217_ref10 doi: 10.1017/S0001867800017456 – ident: S0269964814000217_ref24 doi: 10.1287/opre.1070.0444 – ident: S0269964814000217_ref9 doi: 10.1017/S0269964810000021 – start-page: 67 volume-title: COLT year: 2010 ident: S0269964814000217_ref26 – ident: S0269964814000217_ref34 – ident: S0269964814000217_ref46 doi: 10.1109/MCOM.2012.6257528 – volume: 41 start-page: 335 year: 1979 ident: S0269964814000217_ref21 article-title: Bandit processes and dynamic allocation indices (with discussion) publication-title: Journal of Royal Statistics Society, Series B doi: 10.1111/j.2517-6161.1979.tb01068.x – ident: S0269964814000217_ref5 doi: 10.1561/9781601986276 – volume: 17 start-page: 157 year: 2002 ident: S0269964814000217_ref7 article-title: Asymptotic Bayes analysis for the finite horizon one armed bandit problem publication-title: Probability in the Engineering and Informational Science – ident: S0269964814000217_ref33 doi: 10.1287/moor.12.2.262 – ident: S0269964814000217_ref13 doi: 10.1002/nav.3800070429 – ident: S0269964814000217_ref32 doi: 10.1073/pnas.92.19.8584 – volume-title: Cyrus Derman Memorial Volume II: Optimization under Uncertainty: Costs, Risks and Revenues year: 2014 ident: S0269964814000217_ref45 – ident: S0269964814000217_ref37 doi: 10.1007/978-0-387-49819-5_6 – start-page: 1505 volume-title: Advances in Neural Information Processing Systems year: 2007 ident: S0269964814000217_ref48 – ident: S0269964814000217_ref42 doi: 10.1090/S0002-9947-1952-0050209-9 – ident: S0269964814000217_ref52 doi: 10.1017/S0021900200039176 – ident: S0269964814000217_ref1 doi: 10.1017/S0269964811000015 – ident: S0269964814000217_ref6 doi: 10.1287/moor.22.1.222 – volume: 8 start-page: 1270 year: 1998 ident: S0269964814000217_ref28 article-title: Multi-armed bandits in discrete and continuous time publication-title: The Annals of Applied Probability doi: 10.1214/aoap/1028903380 – ident: S0269964814000217_ref43 doi: 10.1016/j.spl.2008.01.049 – volume-title: Multi-armed bandit allocation indices year: 1989 ident: S0269964814000217_ref22 – ident: S0269964814000217_ref41 – ident: S0269964814000217_ref8 doi: 10.1006/aama.1996.0007 – ident: S0269964814000217_ref27 doi: 10.1007/BF02191765 – ident: S0269964814000217_ref14 doi: 10.1073/pnas.90.4.1232 – ident: S0269964814000217_ref35 doi: 10.1016/0196-8858(85)90002-8 – ident: S0269964814000217_ref16 doi: 10.1109/ALLERTON.2010.5706896 – start-page: 241 volume-title: Progress in statistics year: 1974 ident: S0269964814000217_ref20 – volume-title: Cyrus Derman Memorial Volume II: Optimization under Uncertainty: Costs, Risks and Revenues year: 2014 ident: S0269964814000217_ref18 – ident: S0269964814000217_ref25 doi: 10.1080/01966324.1991.10737307 – ident: S0269964814000217_ref40 – ident: S0269964814000217_ref17 doi: 10.5711/morj.14.2.41 – ident: S0269964814000217_ref29 doi: 10.1214/lnms/1215540286 – ident: S0269964814000217_ref23 doi: 10.1214/10-AAP705 – ident: S0269964814000217_ref19 doi: 10.1002/9780470980033 – volume: 6 start-page: 1024 year: 1996 ident: S0269964814000217_ref30 article-title: Finite state multi-armed bandit problems: sensitive-discount, average-reward and average-overtaking optimality publication-title: The Annals of Applied Probability doi: 10.1214/aoap/1034968239 – ident: S0269964814000217_ref36 doi: 10.1109/TSP.2010.2041600 – volume: 83 start-page: 405 year: 2011 ident: S0269964814000217_ref44 article-title: Optimal stopping of Markov chains and three abstract optimization problems publication-title: Stochastics doi: 10.1080/17442508.2010.514051 – ident: S0269964814000217_ref47 – ident: S0269964814000217_ref3 doi: 10.1080/17442509008833627 – ident: S0269964814000217_ref39 doi: 10.1109/ACSSC.2012.6489015 – ident: S0269964814000217_ref31 doi: 10.1007/s10479-013-1430-4 – ident: S0269964814000217_ref51 doi: 10.1214/aoap/1177005588 – ident: S0269964814000217_ref49 doi: 10.1214/aoap/1177005207 – ident: S0269964814000217_ref15 doi: 10.1109/9.222316 |
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| SubjectTerms | Activation Collection Constants Decisions Depreciation Discounts Mathematical analysis Mathematical models Probability Receiving Stochasticity |
| Title | MULTI-ARMED BANDITS UNDER GENERAL DEPRECIATION AND COMMITMENT |
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