Gradient Flows for Regularized Stochastic Control Problems

TL;DR

This paper develops a gradient flow framework for measure-valued stochastic control problems with entropy regularization, providing convergence guarantees and a Bayesian interpretation, thus advancing theoretical understanding and algorithmic approaches in reinforcement learning.

Contribution

It introduces a gradient flow approach on measure spaces for entropy-regularized stochastic control, establishing convergence and linking to Bayesian methods.

Findings

Gradient flow decreases the cost functional over time.

Invariant measures satisfy Pontryagin's optimality principle.

Exponential convergence occurs under convexity.

Abstract

This paper studies stochastic control problems with the action space taken to be probability measures, with the objective penalised by the relative entropy. We identify suitable metric space on which we construct a gradient flow for the measure-valued control process, in the set of admissible controls, along which the cost functional is guaranteed to decrease. It is shown that any invariant measure of this gradient flow satisfies the Pontryagin optimality principle. If the problem we work with is sufficiently convex, the gradient flow converges exponentially fast. Furthermore, the optimal measure-valued control process admits a Bayesian interpretation which means that one can incorporate prior knowledge when solving such stochastic control problems. This work is motivated by a desire to extend the theoretical underpinning for the convergence of stochastic gradient type algorithms widely employed in the reinforcement learning community to solve control problems.