  Optimal transport has recently proved to be a useful tool in various machine
learning applications needing comparisons of probability measures. Among these,
applications of distributionally robust optimization naturally involve
Wasserstein distances in their models of uncertainty, capturing data shifts or
worst-case scenarios. Inspired by the success of the regularization of
Wasserstein distances in optimal transport, we study in this paper the
regularization of Wasserstein distributionally robust optimization. First, we
derive a general strong duality result of regularized Wasserstein
distributionally robust problems. Second, we refine this duality result in the
case of entropic regularization and provide an approximation result when the
regularization parameters vanish.
