  As opposed to standard empirical risk minimization (ERM), distributionally
robust optimization aims to minimize the worst-case risk over a larger
ambiguity set containing the original empirical distribution of the training
data. In this work, we describe a minimax framework for statistical learning
with ambiguity sets given by balls in Wasserstein space. In particular, we
prove generalization bounds that involve the covering number properties of the
original ERM problem. As an illustrative example, we provide generalization
guarantees for transport-based domain adaptation problems where the Wasserstein
distance between the source and target domain distributions can be reliably
estimated from unlabeled samples.
