Gradient Flows for Regularized Stochastic Control Problems

This paper studies stochastic control problems with the action space taken to be the space of measures, regularized by the relative entropy. We identify suitable metric space on which we construct a gradient flow for the measure-valued control process 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 admits Bayesian interpretation which means that one can incorporate prior knowledge when solving stochastic control problem. This work is motivated by a desire to extend the theoretical underpinning for the convergence of stochastic gradient type algorithms widely used in the reinforcement learning community to solve control problems.