A Fisher-Rao gradient flow for entropy-regularised Markov decision processes in Polish spaces

TL;DR

This paper analyzes a Fisher-Rao policy gradient flow for entropy-regularised Markov decision processes in Polish spaces, establishing its convergence and stability, and providing a theoretical basis for policy gradient algorithms.

Contribution

It introduces a continuous-time Fisher-Rao gradient flow for entropy-regularised MDPs, proving its global convergence and stability, and links it to discrete policy gradient methods.

Findings

Proves exponential convergence of the flow to the optimal policy

Demonstrates stability of the flow with respect to gradient evaluation

Provides a theoretical foundation for policy gradient algorithms

Abstract

We study the global convergence of a Fisher-Rao policy gradient flow for infinite-horizon entropy-regularised Markov decision processes with Polish state and action space. The flow is a continuous-time analogue of a policy mirror descent method. We establish the global well-posedness of the gradient flow and demonstrate its exponential convergence to the optimal policy. Moreover, we prove the flow is stable with respect to gradient evaluation, offering insights into the performance of a natural policy gradient flow with log-linear policy parameterisation. To overcome challenges stemming from the lack of the convexity of the objective function and the discontinuity arising from the entropy regulariser, we leverage the performance difference lemma and the duality relationship between the gradient and mirror descent flows. Our analysis provides a theoretical foundation for developing various discrete policy gradient algorithms.