Distributionally robust optimization (DRO) is an effective approach for
data-driven decision-making in the presence of uncertainty. Geometric
uncertainty due to sampling or localized perturbations of data points is
captured by Wasserstein DRO (WDRO), which seeks to learn a model that performs
uniformly well over a Wasserstein ball centered around the observed data
distribution. However, WDRO fails to account for non-geometric perturbations
such as adversarial outliers, which can greatly distort the Wasserstein
distance measurement and impede the learned model. We address this gap by
proposing a novel outlier-robust WDRO framework for decision-making under both
geometric (Wasserstein) perturbations and non-geometric (total variation (TV))
contamination that allows an $\varepsilon$-fraction of data to be arbitrarily
corrupted. We design an uncertainty set using a certain robust Wasserstein ball
that accounts for both perturbation types and derive minimax optimal excess
risk bounds for this procedure that explicitly capture the Wasserstein and TV
risks. We prove a strong duality result that enables tractable convex
reformulations and efficient computation of our outlier-robust WDRO problem.
When the loss function depends only on low-dimensional features of the data, we
eliminate certain dimension dependencies from the risk bounds that are
unavoidable in the general setting. Finally, we present experiments validating
our theory on standard regression and classification tasks.

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