Statistical Inference for Adversarial Training: Central Limit Theorems via Optimal Transport

Kim Jakwang; Kwon Dohyun

cs.LGSep 4, 2026

Statistical Inference for Adversarial Training: Central Limit Theorems via Optimal Transport

Authors: Kim Jakwang, Kwon Dohyun

Abstract

The purpose of this paper is to rigorously quantify the statistical and learning-theoretic properties of adversarial training models for classification. Equivalently, we establish the statistical properties of empirical optimal partial transport. Precisely, first we provide two types of central limit theorems (CLT): CLT centered at the expected empirical value, and CLT centered at the population one with smoothing. These results are based on the uniqueness of optimal potential for various equivalent optimal transport formulations, and the empirical process theory argument. For the binary setting, we indeed prove the uniqueness of optimal potential by leveraging the connection between optimal partial transport and the derived multi-marginal optimal transport formula. As byproducts, we also obtain the stability of a saddle point of the adversarial training model, and the sample complexity and concentration probability of the generalization error.

Figures & tables

Panel of Figure 1 — Figure 1. Geometry of the closed $\ell^{1}$ -balls of radius $\varepsilon=1$ . In the limiting configuration, all three classes can be transported to $(0,0)$ . After replacing $a_{3}$ by $\tilde{a}^{n}_{3}=(-1-\frac{1}{n},0)$ , the third ball is separated from the other two, while classes $1$ and $2$ still interact along $B_{12}$ .

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

Jeremiah Birrell, Xiaoxi Shen

Department of Mathematics Texas State University San Marcos, TX, USA

Distributionally Robust Optimization Wasserstein Distributionally Robust Optimization Statistical Learning Theory Adversarial Training Optimal Transport

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We study finite-sample statistical performance guarantees for distributionally robust optimization (DRO) with optimal transport (OT) and OT-regularized divergence model neighborhoods. Specifically, we derive concentration inequalities for supervised learning via DRO-based adversarial training, as commonly employed to enhance the adversarial robustness of machine learning models. Our results apply to a wide range of OT cost functions, beyond the

p

-Wasserstein case studied by previous authors. In particular, our results are the first to: 1) cover soft-constraint norm-ball OT cost functions; soft-constraint costs have been shown empirically to enhance robustness when used in adversarial training, 2) apply to the combination of adversarial sample generation and adversarial reweighting that is induced by using OT-regularized

f

-divergence model neighborhoods; the added reweighting mechanism has also been shown empirically to further improve performance. In addition, even in the

p

-Wasserstein case, our bounds exhibit better behavior as a function of the DRO neighborhood size than previous results when applied to the adversarial setting.

Abstract

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