  Wasserstein distributionally robust optimization (DRO) has found success in
operations research and machine learning applications as a powerful means to
obtain solutions with favourable out-of-sample performances. Two compelling
explanations for the success are the generalization bounds derived from
Wasserstein DRO and the equivalency between Wasserstein DRO and the
regularization scheme commonly applied in machine learning. Existing results on
generalization bounds and the equivalency to regularization are largely limited
to the setting where the Wasserstein ball is of a certain type and the decision
criterion takes certain forms of an expected function. In this paper, we show
that by focusing on Wasserstein DRO problems with affine decision rules, it is
possible to obtain generalization bounds and the equivalency to regularization
in a significantly broader setting where the Wasserstein ball can be of a
general type and the decision criterion can be a general measure of risk, i.e.,
nonlinear in distributions. This allows for accommodating many important
classification, regression, and risk minimization applications that have not
been addressed to date using Wasserstein DRO. Our results are strong in that
the generalization bounds do not suffer from the curse of dimensionality and
the equivalency to regularization is exact. As a byproduct, our regularization
results broaden considerably the class of Wasserstein DRO models that can be
solved efficiently via regularization formulations.
