Brenier Meets Adversarial Training: Optimal Transport Geometry for Robust Learning
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.
Related
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BibTeX
@misc{abdollahpoorrostam2026brenier,
title = {{Brenier Meets Adversarial Training: Optimal Transport Geometry for Robust Learning}},
author = {Alireza Abdollahpoorrostam and Ehsan Sharifian and Buse Şen and Marco Cuturi and Daniel Kuhn},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.31363},
eprint = {2609.31363},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.31363},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Brenier Meets Adversarial Training: Optimal Transport Geometry for Robust Learning
AU - Alireza Abdollahpoorrostam
AU - Ehsan Sharifian
AU - Buse Şen
AU - Marco Cuturi
AU - Daniel Kuhn
PY - 2026
DA - 2026/09/25
PB - arXiv
DO - 10.48550/arxiv.2609.31363
UR - https://arxiv.org/abs/2609.31363
AB - 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.
ER -