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This paper introduces a novel geometric framework for RL by viewing policies as mappings into the Wasserstein space of action probabilities and defining a Riemannian structure induced by stationary distributions. It formulates a general RL optimization problem and constructs a gradient flow using Otto's calculus, providing a formal second-order analysis by computing the gradient and Hessian of the energy. The approach is validated through numerical examples, demonstrating the computation of gradients from the theoretical formalism and neural network-based policy optimization for high-dimensional problems.
Reinforcement learning gets a Riemannian makeover, offering a new geometric perspective on policy optimization via Wasserstein space and Otto's calculus.
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.