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

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, providing rigorous guarantees for their existence in a general context. We then define the tangent space of policies and characterize the geodesics, specifically addressing the measurability of vector fields mapping 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 (i.e., the expected cumulative cost), providing a formal second-order analysis. Finally, we illustrate the method with numerical examples: we compute the gradient directly from our theoretical formalism for low-dimensional problems, and we illustrate the numerical applicability of our formal framework to high-dimensional continuous control by parameterizing the policy with a neural network optimized via an ergodic approximation of the cost.