Finite Model Approximations for Partially Observed Markov Decision Processes with Discounted Cost

TL;DR

This paper develops finite model approximations for partially observed Markov decision processes (POMDPs) with discounted costs, demonstrating near-optimal policies through quantization of the belief space and establishing convergence under mild conditions.

Contribution

It introduces a method for approximating POMDPs via quantization of belief space, providing explicit procedures and convergence guarantees for near-optimal policies.

Findings

Finite models yield nearly optimal policies for POMDPs.

Explicit quantization procedures for belief space are provided.

Convergence of policies is established under mild assumptions.

Abstract

We consider finite model approximations of discrete-time partially observed Markov decision processes (POMDPs) under the discounted cost criterion. After converting the original partially observed stochastic control problem to a fully observed one on the belief space, the finite models are obtained through the uniform quantization of the state and action spaces of the belief space Markov decision process (MDP). Under mild assumptions on the components of the original model, it is established that the policies obtained from these finite models are nearly optimal for the belief space MDP, and so, for the original partially observed problem. The assumptions essentially require that the belief space MDP satisfies a mild weak continuity condition. We provide examples and introduce explicit approximation procedures for the quantization of the set of probability measures on the state space of POMDP (i.e., belief space).