Dynamic Programming for POMDP with Jointly Discrete and Continuous State-Spaces

TL;DR

This paper develops dynamic programming algorithms for POMDPs with mixed discrete and continuous states, proving convergence to near-optimal solutions and laying groundwork for future reinforcement learning methods.

Contribution

It introduces a DP framework for systems with coupled discrete and continuous states, with convergence proofs and potential for reinforcement learning applications.

Findings

DP algorithms converge to a bounded set near the optimal solution

Applicable to systems like Markovian jump linear systems and human-interacting physical systems

Provides theoretical foundation for future RL algorithms in mixed state spaces

Abstract

In this work, we study dynamic programming (DP) algorithms for partially observable Markov decision processes with jointly continuous and discrete state-spaces. We consider a class of stochastic systems which have coupled discrete and continuous systems, where only the continuous state is observable. Such a family of systems includes many real world systems, for example, Markovian jump linear systems and physical systems interacting with humans. A finite history of observations is used as a new information state, and the convergence of the corresponding DP algorithms is proved. In particular, we prove that the DP iterations converge to a certain bounded set around an optimal solution. Although deterministic DP algorithms are studied in this paper, it is expected that this fundamental work lays foundations for advanced studies on reinforcement learning algorithms under the same family of systems.