  Wasserstein distributionally robust optimization (DRO) aims to find robust
and generalizable solutions by hedging against data perturbations in
Wasserstein distance. Despite its recent empirical success in operations
research and machine learning, existing performance guarantees for generic loss
functions are either overly conservative due to the curse of dimensionality, or
plausible only in large sample asymptotics. In this paper, we develop a
non-asymptotic framework for analyzing the out-of-sample performance for
Wasserstein robust learning and the generalization bound for its related
Lipschitz and gradient regularization problems. To the best of our knowledge,
this gives the first finite-sample guarantee for generic Wasserstein DRO
problems without suffering from the curse of dimensionality. Our results
highlight that Wasserstein DRO, with a properly chosen radius, balances between
the empirical mean of the loss and the variation of the loss, measured by the
Lipschitz norm or the gradient norm of the loss. Our analysis is based on two
novel methodological developments that are of independent interest: 1) a new
concentration inequality controlling the decay rate of large deviation
probabilities by the variation of the loss and, 2) a localized Rademacher
complexity theory based on the variation of the loss.
