Near Optimality of Finite Memory Feedback Policies in Partially Observed Markov Decision Processes

TL;DR

This paper demonstrates that finite memory feedback policies can be nearly optimal in POMDPs with finite actions and measurements, providing explicit convergence rates under mild stability conditions, thus advancing theoretical understanding and practical approximation methods.

Contribution

The paper introduces a rigorous analysis of finite window policies in POMDPs, establishing near optimality and explicit convergence rates under mild conditions, which was previously lacking in the literature.

Findings

Finite window policies are nearly optimal in POMDPs.

Explicit exponential convergence rates are established.

The results apply under mild filter stability conditions.

Abstract

In the theory of Partially Observed Markov Decision Processes (POMDPs), existence of optimal policies have in general been established via converting the original partially observed stochastic control problem to a fully observed one on the belief space, leading to a belief-MDP. However, computing an optimal policy for this fully observed model, and so for the original POMDP, using classical dynamic or linear programming methods is challenging even if the original system has finite state and action spaces, since the state space of the fully observed belief-MDP model is always uncountable. Furthermore, there exist very few rigorous value function approximation and optimal policy approximation results, as regularity conditions needed often require a tedious study involving the spaces of probability measures leading to properties such as Feller continuity. In this paper, we study a planning problem for POMDPs where the system dynamics and measurement channel model are assumed to be known. We construct an approximate belief model by discretizing the belief space using only finite window information variables. We then find optimal policies for the approximate model and we rigorously establish near optimality of the constructed finite window control policies in POMDPs under mild non-linear filter stability conditions and the assumption that the measurement and action sets are finite (and the state space is real vector valued). We also establish a rate of convergence result which relates the finite window memory size and the approximation error bound, where the rate of convergence is exponential under explicit and testable exponential filter stability conditions. While there exist many experimental results and few rigorous asymptotic convergence results, an explicit rate of convergence result is new in the literature, to our knowledge.