Uniform Convergence and Generalization for Nonconvex Stochastic Minimax Problems

This paper studies the uniform convergence and generalization bounds for nonconvex-(strongly)-concave (NC-SC/NC-C) stochastic minimax optimization. We first establish the uniform convergence between the empirical minimax problem and the population minimax problem and show the $\tilde{\mathcal{O}}(d\kappa^2\epsilon^{-2})$ and $\tilde{\mathcal{O}}(d\epsilon^{-4})$ sample complexities respectively for the NC-SC and NC-C settings, where $d$ is the dimension number and $\kappa$ is the condition number. To the best of our knowledge, this is the first uniform convergence measured by the first-order stationarity in stochastic minimax optimization. Based on the uniform convergence, we shed light on the sample and gradient complexities required for finding an approximate stationary point for stochastic minimax optimization in the NC-SC and NC-C settings.