Distributional Robustness and Regularization in Reinforcement Learning

Distributionally Robust Optimization (DRO) has enabled to prove the equivalence between robustness and regularization in classification and regression, thus providing an analytical reason why regularization generalizes well in statistical learning. Although DRO's extension to sequential decision-making overcomes $\textit{external uncertainty}$ through the robust Markov Decision Process (MDP) setting, the resulting formulation is hard to solve, especially on large domains. On the other hand, existing regularization methods in reinforcement learning only address $\textit{internal uncertainty}$ due to stochasticity. Our study aims to facilitate robust reinforcement learning by establishing a dual relation between robust MDPs and regularization. We introduce Wasserstein distributionally robust MDPs and prove that they hold out-of-sample performance guarantees. Then, we introduce a new regularizer for empirical value functions and show that it lower bounds the Wasserstein distributionally robust value function. We extend the result to linear value function approximation for large state spaces. Our approach provides an alternative formulation of robustness with guaranteed finite-sample performance. Moreover, it suggests using regularization as a practical tool for dealing with $\textit{external uncertainty}$ in reinforcement learning methods.