A Kernel Perspective on Behavioural Metrics for Markov Decision Processes

TL;DR

This paper introduces a new kernel-based approach to behavioural metrics in Markov decision processes, providing theoretical guarantees and empirical evidence for improved reinforcement learning representations.

Contribution

It presents a novel kernel perspective on behavioural metrics, establishing equivalence with MICo distance and deriving new theoretical bounds and embedding properties.

Findings

Bounded value function differences using the new metric

Proved low-distortion Euclidean embedding of the metric

Empirical results show improved reinforcement learning performance

Abstract

Behavioural metrics have been shown to be an effective mechanism for constructing representations in reinforcement learning. We present a novel perspective on behavioural metrics for Markov decision processes via the use of positive definite kernels. We leverage this new perspective to define a new metric that is provably equivalent to the recently introduced MICo distance (Castro et al., 2021). The kernel perspective further enables us to provide new theoretical results, which has so far eluded prior work. These include bounding value function differences by means of our metric, and the demonstration that our metric can be provably embedded into a finite-dimensional Euclidean space with low distortion error. These are two crucial properties when using behavioural metrics for reinforcement learning representations. We complement our theory with strong empirical results that demonstrate the effectiveness of these methods in practice.