Statistical Guarantees for Distributionally Robust Optimization with Optimal Transport and OT-Regularized Divergences

Distributionally robust optimization (DRO) provides a principled framework for learning under distribution shift, but its practical use is hindered by the difficulty of evaluating worst-case risks for nonconvex loss functions. We study a penalized DRO formulation in which the adversary may choose any distribution but incurs a Wasserstein penalty for deviating from the empirical distribution. We show that the adversary's problem can be reformulated as an optimization problem over transport maps that push empirical samples to adversarial ones, and we prove that optimal maps are cyclically monotone. We also show that standard adversarial training---based on per-sample local optimization---violates cyclical monotonicity and wastes transport costs unless the adversary is severely restricted. We propose two remedies. First, we introduce multi-start particle ascent, which alternates parallel gradient ascent with reassignment to enforce cyclical monotonicity across samples. Second, we parameterize adversarial maps as gradients of input-convex neural networks, which guarantees cyclical monotonicity by construction. Experiments on robust regression, image classification, and robust control show that our methods consistently outperform standard adversarial training and state-of-the-art baselines, achieving improved robustness and better generalization under distribution shift.

Distributionally robust optimization (DRO) studies parameter estimation under uncertainty in the underlying probability distribution and has emerged as a principled framework for analyzing robustness and generalization. In particular, Wasserstein DRO, with distributional uncertainty induced by the Wasserstein distance, generalizes several popular regularizers. This paper studies Wasserstein DRO linear regression, unifying square-root Lasso and adversarial linear regression as important special cases. We prove that many properties of these two special cases carry over to this general method. In particular, we show (i) deterministic and non-asymptotic in-sample error bounds $O(n^{-1/2})$ in general and $O(n^{-1})$ under design matrix and sparsity conditions; (ii) insensitivity to the noise level, also known as the pivotal property; and (iii) solution equivalences for small and large ambiguity sets. The key proof step is to recast the method into a quadratic form, mimicking adversarial linear regression. We also show that the method can be solved efficiently, and we validate our findings through numerical simulations.

We study distributionally robust optimization (DRO) for robust inference when the worst-case distribution is continuous, leading to significant computational challenges due to the infinite-dimensional nature of the optimization problem. Unlike traditional discrete DRO approaches, which often suffer from scalability issues, limited generalization, and costly worst-case inference, our framework exploits Brenier's theorem to characterize the least favorable distribution as the pushforward of a transport map from a continuous reference measure. This characterization motivates our study of the minimax problem in Wasserstein space. We propose an iterative algorithmic framework with multiple variants and establish global convergence guarantees under mild assumptions, deriving complexity bounds in terms of subgradient evaluations and inexact Jordan-Kinderlehrer-Otto updates. Numerical results with neural network-based transport maps demonstrate that the proposed method enables both stable training of robust classifiers and effective worst-case inference for classification tasks.