We study three models of the problem of adversarial training in multiclass classification designed to construct robust classifiers against adversarial perturbations of data in the agnostic-classifier setting. We prove the existence of Borel measurable robust classifiers in each model and provide a unified perspective of the adversarial training problem, expanding the connections with optimal transport initiated by the authors in previous work and developing new connections between adversarial training in the multiclass setting and total variation regularization. As a corollary of our results, we prove the existence of Borel measurable solutions to the agnostic adversarial training problem in the binary classification setting, a result that improves results in the literature of adversarial training, where robust classifiers were only known to exist within the enlarged universal $\sigma$-algebra of the feature space.

本文研究了三种拟建立鲁棒分类器以对抗对数据的恶意扰动的多分类分类问题模型，证明了每个模型中存在Borel可测的鲁棒分类器，并提供了对抗训练问题的统一视角，扩展了与作者在先前工作中启动的最优传输方面的关联，并发展了在多类设置中进行对抗训练和总变差规则化之间的新关联。

多类分类对抗训练解的存在性