The paper “Wasserstein distributionally robust stochastic control: A data-driven approach,” authored by Insoon Yang, has been accepted for publication in the IEEE Transactions on Automatic Control. This paper proposes the theory, algorithms, and applications of distributionally robust stochastic control with Wasserstein distance. In particular, we show that the contraction property of associated Bellman operators extends a single-stage out-of-sample performance guarantee, obtained using a measure concentration inequality, to the corresponding multi-stage guarantee without any degradation in the confidence level.

Abstract: Standard stochastic control methods assume that the probability distribution of uncertain variables is available. Unfortunately, in practice, obtaining accurate distribution information is a challenging task. To resolve this issue, we investigate the problem of designing a control policy that is robust against errors in the empirical distribution obtained from data. This problem can be formulated as a two-player zero-sum dynamic game problem, where the action space of the adversarial player is a Wasserstein ball centered at the empirical distribution. A dynamic programming solution is provided exploiting the reformulation techniques for Wasserstein distributionally robust optimization. We show that the contraction property of associated Bellman operators extends a single-stage out-of-sample performance guarantee, obtained using a measure concentration inequality, to the corresponding multi-stage guarantee without any degradation in the confidence level. Furthermore, we characterize an explicit form of the optimal control policy and the worst-case distribution policy for linear-quadratic problems with Wasserstein penalty.