Robust Reinforcement Learning aims to find the optimal policy with some extent of robustness to environmental dynamics. Existing learning algorithms usually enable the robustness through disturbing the current state or simulating environmental parameters in a heuristic way, which lack quantified robustness to the system dynamics (i.e. transition probability). To overcome this issue, we leverage Wasserstein distance to measure the disturbance to the reference transition kernel. With Wasserstein distance, we are able to connect transition kernel disturbance to the state disturbance, i.e. reduce an infinite-dimensional optimization problem to a finite-dimensional risk-aware problem. Through the derived risk-aware optimal Bellman equation, we show the existence of optimal robust policies, provide a sensitivity analysis for the perturbations, and then design a novel robust learning algorithm--Wasserstein Robust Advantage Actor-Critic algorithm (WRAAC). The effectiveness of the proposed algorithm is verified in the Cart-Pole environment.
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