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.

我们研究了分布式随机梯度上升下降（D-SGDA）算法的原始 - 对偶广义界限，通过算法稳定性方法，在凸凹和非凸非凹环境下对分布式最小最大算法的广义界限进行了改进。我们的理论研究表明，分布式结构不会破坏 D-SGDA 的稳定性和广义化能力，在某些情况下可以实现和普通 SGDA 相同的广义化能力。此外，我们还评估了凸凹设定下 D-SGDA 算法的优化误差，并将其与广义间隙相平衡，以获得最佳的总体风险。最后，我们进行了多项数值实验来验证我们的理论发现。