{"title":"Primal-Dual Regression Approach for Markov Decision Processes with General State and Action Spaces","authors":"Denis Belomestny, John Schoenmakers","doi":"10.1137/22m1526010","DOIUrl":null,"url":null,"abstract":"SIAM Journal on Control and Optimization, Volume 62, Issue 1, Page 650-679, February 2024. <br/> Abstract. We develop a regression-based primal-dual martingale approach for solving discrete time, finite-horizon MDPs. The state and action spaces may be finite or infinite (but regular enough) subsets of Euclidean space. Consequently, our method allows for the construction of tight upper and lower-biased approximations of the value functions, providing precise estimates of the optimal policy. Importantly, we prove error bounds for the estimated duality gap featuring polynomial dependence on the time horizon. Additionally, we observe sublinear dependence of the stochastic part of the error on the cardinality/dimension of the state and action spaces. From a computational perspective, our proposed method is efficient. Unlike typical duality-based methods for optimal control problems in the literature, the Monte Carlo procedures involved here do not require nested simulations.","PeriodicalId":49531,"journal":{"name":"SIAM Journal on Control and Optimization","volume":null,"pages":null},"PeriodicalIF":2.2000,"publicationDate":"2024-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"SIAM Journal on Control and Optimization","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1137/22m1526010","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"AUTOMATION & CONTROL SYSTEMS","Score":null,"Total":0}
引用次数: 0
Abstract
SIAM Journal on Control and Optimization, Volume 62, Issue 1, Page 650-679, February 2024. Abstract. We develop a regression-based primal-dual martingale approach for solving discrete time, finite-horizon MDPs. The state and action spaces may be finite or infinite (but regular enough) subsets of Euclidean space. Consequently, our method allows for the construction of tight upper and lower-biased approximations of the value functions, providing precise estimates of the optimal policy. Importantly, we prove error bounds for the estimated duality gap featuring polynomial dependence on the time horizon. Additionally, we observe sublinear dependence of the stochastic part of the error on the cardinality/dimension of the state and action spaces. From a computational perspective, our proposed method is efficient. Unlike typical duality-based methods for optimal control problems in the literature, the Monte Carlo procedures involved here do not require nested simulations.
期刊介绍：
SIAM Journal on Control and Optimization (SICON) publishes original research articles on the mathematics and applications of control theory and certain parts of optimization theory. Papers considered for publication must be significant at both the mathematical level and the level of applications or potential applications. Papers containing mostly routine mathematics or those with no discernible connection to control and systems theory or optimization will not be considered for publication. From time to time, the journal will also publish authoritative surveys of important subject areas in control theory and optimization whose level of maturity permits a clear and unified exposition.
The broad areas mentioned above are intended to encompass a wide range of mathematical techniques and scientific, engineering, economic, and industrial applications. These include stochastic and deterministic methods in control, estimation, and identification of systems; modeling and realization of complex control systems; the numerical analysis and related computational methodology of control processes and allied issues; and the development of mathematical theories and techniques that give new insights into old problems or provide the basis for further progress in control theory and optimization. Within the field of optimization, the journal focuses on the parts that are relevant to dynamic and control systems. Contributions to numerical methodology are also welcome in accordance with these aims, especially as related to large-scale problems and decomposition as well as to fundamental questions of convergence and approximation.
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