  Wasserstein distributionally robust estimators have emerged as powerful
models for prediction and decision-making under uncertainty. These estimators
provide attractive generalization guarantees: the robust objective obtained
from the training distribution is an exact upper bound on the true risk with
high probability. However, existing guarantees either suffer from the curse of
dimensionality, are restricted to specific settings, or lead to spurious error
terms. In this paper, we show that these generalization guarantees actually
hold on general classes of models, do not suffer from the curse of
dimensionality, and can even cover distribution shifts at testing. We also
prove that these results carry over to the newly-introduced regularized
versions of Wasserstein distributionally robust problems.
