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

TL;DR

This paper proves the convergence of finite-memory Q-learning algorithms for POMDPs and demonstrates that learned policies are near optimal under certain filter stability conditions, providing explicit error bounds and convergence rates.

Contribution

It introduces the first rigorous proof of asymptotic convergence and near optimality of finite-memory Q-learning in POMDPs with explicit error bounds and convergence rates.

Findings

Convergence of finite-memory Q-learning under mild ergodicity assumptions.

Explicit error bounds relating history length to approximation accuracy.

Near optimality of learned policies with convergence rates in memory size.

Abstract

In this paper, for 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-MDP. We then provide bounds on the performance of the policy obtained using the limit Q values compared to the performance of the optimal policy for the POMDP, where we also present explicit conditions using recent results on filter stability in controlled POMDPs. While 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) are results that are new to the literature, to our knowledge.