Wasserstein Formulation of Reinforcement Learning. An Optimal Transport Perspective on Policy Optimization

TL;DR

This paper introduces a geometric Wasserstein space framework for reinforcement learning, utilizing optimal transport theory to analyze and optimize policies with a focus on policy gradients and second-order analysis.

Contribution

It develops a Riemannian geometric structure for policies in Wasserstein space, formalizes gradient flows using Otto's calculus, and applies the framework to neural network policies.

Findings

Established a Riemannian structure on policy space

Derived gradient and Hessian expressions for policy optimization

Demonstrated the approach with numerical examples for low and high-dimensional problems

Abstract

We present a geometric framework for Reinforcement Learning (RL) that views policies as maps into the Wasserstein space of action probabilities. First, we define a Riemannian structure induced by stationary distributions, proving its existence in a general context. We then define the tangent space of policies and characterize the geodesics, specifically addressing the measurability of vector fields mapped from the state space to the tangent space of probability measures over the action space. Next, we formulate a general RL optimization problem and construct a gradient flow using Otto's calculus. We compute the gradient and the Hessian of the energy, providing a formal second-order analysis. Finally, we illustrate the method with numerical examples for low-dimensional problems, computing the gradient directly from our theoretical formalism. For high-dimensional problems, we parameterize the policy using a neural network and optimize it based on an ergodic approximation of the cost.