  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.
