| Contents |
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. Furthermore, we prove the existence of Borel measurable robust classifiers while previous works only argue universally measurable robust classifiers only in the binary setting. Also, based on our theoretical results, we propose two new numerics relying on the truncation of the order of interactions which are faster than the state-of-art MOT algorithm in real data sets by providing empirical results using MNIST and CIFAR-10 data. |