A note on convergence of Wasserstein policy optimization

TL;DR

This paper provides a theoretical analysis demonstrating that Wasserstein Policy Optimization (WPO) converges linearly in entropy-regularized Markov Decision Processes, supported by mean-field analysis and log-Sobolev inequalities.

Contribution

It establishes the first theoretical convergence guarantee for WPO in continuous spaces using mean-field and log-Sobolev techniques.

Findings

WPO converges linearly in entropy-regularized MDPs.

Monotonic energy dissipation along the gradient flow.

Existence of a local log-Sobolev inequality supports convergence.

Abstract

Wasserstein Policy Optimization (WPO) is a recently proposed reinforcement learning algorithm that leverages Wasserstein gradient flows to optimize stochastic policies in continuous action spaces. Despite its empirical success, the theoretical convergence properties of WPO in environments with continuous state and action spaces have yet to be fully established. In this note, we argue that WPO within the framework of entropy-regularised Markov Decision Processes converges linearly. This is done by leveraging recent advances in mean-field analysis for convergence of gradient flows using log-Sobole inequalities. Assuming existence of sufficiently regular solution to the gradient flow equation we demonstrate monotonic energy dissipation along the flow and establish a local log-Sobolev inequality. Ultimately, these properties allow us to argue that the value function should converge linearly to the global optimum.