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This research advances the theoretical foundations of adversarial training by leveraging optimal transport geometry to enforce cyclical monotonicity in distributionally robust optimization. By reformulating the adversary's problem over transport maps and introducing input-convex neural network parameterizations, the authors demonstrate significant improvements in robustness and generalization across regression, classification, and control tasks. These findings offer a principled approach to mitigating distribution shift without relying on overly restrictive assumptions about the adversary.
Brenier Meets Adversarial Training: Optimal Transport Geometry for Robust Learning
arXiv:2609.31363v1 Announce Type: new Abstract: 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.
Abstract: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.
Machine Learning (cs.LG); Optimization and Control (math.OC); Machine Learning (stat.ML)
arXiv:2609.31363 [cs.LG]
(or arXiv:2609.31363v1 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2609.31363 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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