We study a family of adversarial multiclass classification problems and provide equivalent reformulations in terms of: 1) a family of generalized barycenter problems introduced in the paper and 2) a family of multimarginal optimal transport problems where the number of marginals is equal to the number of classes in the original classification problem. These new theoretical results reveal a rich geometric structure of adversarial learning problems in multiclass classification and extend recent results restricted to the binary classification setting. A direct computational implication of our results is that by solving either the barycenter problem and its dual, or the MOT problem and its dual, we can recover the optimal robust classification rule and the optimal adversarial strategy for the original adversarial problem. Examples with synthetic and real data illustrate our results.

翻译：我们研究了一类对抗性多类分类问题，并提供了等价重述形式，具体包括：1）本文提出的一类广义重心问题，以及2）一类边际数量等于原始分类问题中类别数量的多边际最优输运问题。这些新的理论结果揭示了多类分类中对抗学习问题丰富的几何结构，并推广了近期仅限于二分类设置的研究成果。我们结果的直接计算意义在于，通过求解重心问题及其对偶问题，或求解多边际最优输运问题及其对偶问题，可以恢复原始对抗问题的鲁棒最优分类规则和最优对抗策略。通过合成数据与真实数据的实例验证了我们的结果。