Distributional Reinforcement Learning by Sinkhorn Divergence

The empirical success of distributional reinforcement learning~(RL) highly depends on the distribution representation and the choice of distribution divergence. In this paper, we propose \textit{Sinkhorn distributional RL~(SinkhornDRL)} that learns unrestricted statistics from return distributions and leverages Sinkhorn divergence to minimize the difference between current and target Bellman return distributions. Theoretically, we prove the contraction properties of SinkhornDRL, consistent with the interpolation nature of Sinkhorn divergence between Wasserstein distance and Maximum Mean Discrepancy~(MMD). We also establish the equivalence between Sinkhorn divergence and a regularized MMD with a regularized Moment Matching behavior, contributing to explaining the superiority of SinkhornDRL. Empirically, we show that SinkhornDRL is consistently better or comparable to existing algorithms on the Atari games suite.