Wasserstein Formulation of Reinforcement Learning. An Optimal Transport Perspective on Policy Optimization
For RL researchers, this provides a novel geometric perspective on policy optimization, but the numerical results are limited to low-dimensional examples and the high-dimensional method is a standard neural network approach, making the contribution incremental.
This paper introduces a geometric framework for reinforcement learning by viewing policies as maps into the Wasserstein space of action probabilities, deriving a Riemannian structure and gradient flow for policy optimization. Numerical examples on low-dimensional problems and a neural network parameterization for high-dimensional problems demonstrate the approach.
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.