  The growing size of available data has attracted increasing interest in
solving minimax problems in a decentralized manner for various machine learning
tasks. Previous theoretical research has primarily focused on the convergence
rate and communication complexity of decentralized minimax algorithms, with
little attention given to their generalization. In this paper, we investigate
the primal-dual generalization bound of the decentralized stochastic gradient
descent ascent (D-SGDA) algorithm using the approach of algorithmic stability
under both convex-concave and nonconvex-nonconcave settings. Our theory refines
the algorithmic stability in a decentralized manner and demonstrates that the
decentralized structure does not destroy the stability and generalization of
D-SGDA, implying that it can generalize as well as the vanilla SGDA in certain
situations. Our results analyze the impact of different topologies on the
generalization bound of the D-SGDA algorithm beyond trivial factors such as
sample sizes, learning rates, and iterations. We also evaluate the optimization
error and balance it with the generalization gap to obtain the optimal
population risk of D-SGDA in the convex-concave setting. Additionally, we
perform several numerical experiments which validate our theoretical findings.
