Metrics for Markov Decision Processes with Infinite State Spaces

TL;DR

This paper introduces metrics for quantifying state similarity in infinite-state Markov decision processes, enabling stable approximations and demonstrating the continuity of the value function with respect to these metrics.

Contribution

It proposes a new class of metrics for infinite-state MDPs that generalize bisimulation and support approximation methods.

Findings

Value functions are continuous under the proposed metrics.

Metrics extend bisimulation to infinite and continuous state spaces.

Facilitates stable MDP approximations.

Abstract

We present metrics for measuring state similarity in Markov decision processes (MDPs) with infinitely many states, including MDPs with continuous state spaces. Such metrics provide a stable quantitative analogue of the notion of bisimulation for MDPs, and are suitable for use in MDP approximation. We show that the optimal value function associated with a discounted infinite horizon planning task varies continuously with respect to our metric distances.