Convergence of Finite Memory Q Learning for POMDPs and Near Optimality of Learned Policies Under Filter Stability

• 成果类型: Article
• 署名作者: Kara, Ali Devran; Yuksel, Serdar
• 署名单位: University of Michigan System; University of Michigan; Queens University - Canada
• 刊物名称: MATHEMATICS OF OPERATIONS RESEARCH
• ISSN/ISSBN: 0364-765X
• DOI: 10.1287/moor.2022.1331
• 发表日期: 2023
• 页码: 2066-2093
• 关键词: markov decision-processes; Asymptotic Optimality; value-iteration; approximations; state; INFORMATION; reduction; MODEL

摘要

In this paper, for partially observed Markov decision problems (POMDPs), we provide the convergence of a Q learning algorithm for control policies using a finite history of past observations and control actions, and consequentially, we establish near optimality of such limit Q functions under explicit filter stability conditions. We present explicit error bounds relating the approximation error to the length of the finite history window. We establish the convergence of such Q learning iterations under mild ergodicity assumptions on the state process during the exploration phase. We further show that the limit fixed point equation gives an optimal solution for an approximate belief Markov decision problem (MDP). We then provide bounds on the performance of the policy obtained using the limit Q values compared with the performance of the optimal policy for the POMDP, in which we also present explicit conditions using recent results on filter stability in controlled POMDPs. Whereas there exist many experimental results, (i) the rigorous asymptotic convergence (to an approximate MDP value function) for such finite memory Q learning algorithms and (ii) the near optimality with an explicit rate of convergence (in the memory size) under filter stability are results that are new to the literature to our knowledge.