The minimax problems arise throughout machine learning applications, ranging from adversarial training and policy evaluation in reinforcement learning to AUROC maximization. To address the large-scale data challenges across multiple clients with communication-efficient distributed training, federated learning (FL) is gaining popularity. Many optimization algorithms for minimax problems have been developed in the centralized setting (\emph{i.e.} single-machine). Nonetheless, the algorithm for minimax problems under FL is still underexplored. In this paper, we study a class of federated nonconvex minimax optimization problems. We propose FL algorithms (FedSGDA+ and FedSGDA-M) and reduce existing complexity results for the most common minimax problems. For nonconvex-concave problems, we propose FedSGDA+ and reduce the communication complexity to $O(\varepsilon^{-6})$. Under nonconvex-strongly-concave and nonconvex-PL minimax settings, we prove that FedSGDA-M has the best-known sample complexity of $O(\kappa^{3} N^{-1}\varepsilon^{-3})$ and the best-known communication complexity of $O(\kappa^{2}\varepsilon^{-2})$. FedSGDA-M is the first algorithm to match the best sample complexity $O(\varepsilon^{-3})$ achieved by the single-machine method under the nonconvex-strongly-concave setting. Extensive experimental results on fair classification and AUROC maximization show the efficiency of our algorithms.

翻译：极小极大问题广泛出现在机器学习应用中，涵盖对抗训练、强化学习中的策略评估以及AUROC最大化等场景。为应对跨多个客户端的大规模数据挑战并实现通信高效分布式训练，联邦学习正日益受到关注。目前已有许多针对极小极大问题的优化算法在集中式设置（即单机环境）中被开发出来，然而联邦学习下的极小极大问题算法仍探索不足。本文研究了一类联邦非凸极小极大优化问题，提出联邦学习算法（FedSGDA+和FedSGDA-M），并降低了现有常见极小极大问题的复杂度结果。对于非凸-凹问题，我们提出FedSGDA+，将通信复杂度降低至$O(\varepsilon^{-6})$。在非凸-强凹和非凸-PL极小极大设置下，我们证明FedSGDA-M具有$O(\kappa^{3} N^{-1}\varepsilon^{-3})$的最优样本复杂度和$O(\kappa^{2}\varepsilon^{-2})$的最优通信复杂度。FedSGDA-M是首个在非凸-强凹设置下匹配单机方法最优样本复杂度$O(\varepsilon^{-3})$的算法。在公平分类和AUROC最大化上的大量实验结果展示了我们算法的高效性。