  In this paper, we consider nonconvex minimax optimization, which is gaining
prominence in many modern machine learning applications such as GANs.
Large-scale edge-based collection of training data in these applications calls
for communication-efficient distributed optimization algorithms, such as those
used in federated learning, to process the data. In this paper, we analyze
Local stochastic gradient descent ascent (SGDA), the local-update version of
the SGDA algorithm. SGDA is the core algorithm used in minimax optimization,
but it is not well-understood in a distributed setting. We prove that Local
SGDA has \textit{order-optimal} sample complexity for several classes of
nonconvex-concave and nonconvex-nonconcave minimax problems, and also enjoys
\textit{linear speedup} with respect to the number of clients. We provide a
novel and tighter analysis, which improves the convergence and communication
guarantees in the existing literature. For nonconvex-PL and
nonconvex-one-point-concave functions, we improve the existing complexity
results for centralized minimax problems. Furthermore, we propose a
momentum-based local-update algorithm, which has the same convergence
guarantees, but outperforms Local SGDA as demonstrated in our experiments.
