Distributionally and Adversarially Robust Logistic Regression via Intersecting Wasserstein Balls

Adversarially robust optimization (ARO) has become the de facto standard for training models to defend against adversarial attacks during testing. However, despite their robustness, these models often suffer from severe overfitting. To mitigate this issue, several successful approaches have been proposed, including replacing the empirical distribution in training with: (i) a worst-case distribution within an ambiguity set, leading to a distributionally robust (DR) counterpart of ARO; or (ii) a mixture of the empirical distribution with one derived from an auxiliary dataset (e.g., synthetic, external, or out-of-domain). Building on the first approach, we explore the Wasserstein DR counterpart of ARO for logistic regression and show it admits a tractable convex optimization reformulation. Adopting the second approach, we enhance the DR framework by intersecting its ambiguity set with one constructed from an auxiliary dataset, which yields significant improvements when the Wasserstein distance between the data-generating and auxiliary distributions can be estimated. We analyze the resulting optimization problem, develop efficient solutions, and show that our method outperforms benchmark approaches on standard datasets.