Near Optimal Approximations and Finite Memory Policies for POMPDs with Continuous Spaces

TL;DR

This paper introduces a rigorous approximation method for continuous-space POMDPs by discretizing observations and constructing finite MDPs, enabling near-optimal policies and a convergent Q-learning algorithm with finite memory.

Contribution

It generalizes recent work to provide a practical discretization approach for POMDPs with continuous spaces, ensuring near-optimality and convergence of learning algorithms.

Findings

Discretizing observations yields near-optimal policies under regularity assumptions.

The proposed Q-learning algorithm converges to the optimality equation using finite memory.

Refined filter stability conditions improve approximation accuracy.

Abstract

We study an approximation method for partially observed Markov decision processes (POMDPs) with continuous spaces. Belief MDP reduction, which has been the standard approach to study POMDPs requires rigorous approximation methods for practical applications, due to the state space being lifted to the space of probability measures. Generalizing recent work, in this paper we present rigorous approximation methods via discretizing the observation space and constructing a fully observed finite MDP model using a finite length history of the discrete observations and control actions. We show that the resulting policy is near-optimal under some regularity assumptions on the channel, and under certain controlled filter stability requirements for the hidden state process. Furthermore, by quantizing the measurements, we are able to utilize refined filter stability conditions. We also provide a Q learning algorithm that uses a finite memory of discretized information variables, and prove its convergence to the optimality equation of the finite fully observed MDP constructed using the approximation method.