Generalization Bounds of Nonconvex-(Strongly)-Concave Stochastic Minimax Optimization

TL;DR

This paper systematically investigates the generalization bounds of algorithms for nonconvex-(strongly)-concave stochastic minimax optimization, providing both algorithm-agnostic and algorithm-dependent bounds with novel stability concepts.

Contribution

It introduces a comprehensive analysis of generalization bounds for nonconvex stochastic minimax problems, including new stability notions and bounds for SGDA and related algorithms.

Findings

Sample complexity for NC-SC is (d\u00b7\u03ba^2\u00b7\u03b5^{-2})

Sample complexity for NC-C is (d\u00b7\u03b5^{-4})

Established stability-based generalization bounds for SGDA

Abstract

This paper takes an initial step to systematically investigate the generalization bounds of algorithms for solving nonconvex-(strongly)-concave (NC-SC/NC-C) stochastic minimax optimization measured by the stationarity of primal functions. We first establish algorithm-agnostic generalization bounds via uniform convergence between the empirical minimax problem and the population minimax problem. The sample complexities for achieving $\epsilon$-generalization are $\tilde{\mathcal{O}}(d\kappa^2\epsilon^{-2})$ and $\tilde{\mathcal{O}}(d\epsilon^{-4})$ for NC-SC and NC-C settings, respectively, where $d$ is the dimension and $\kappa$ is the condition number. We further study the algorithm-dependent generalization bounds via stability arguments of algorithms. In particular, we introduce a novel stability notion for minimax problems and build a connection between generalization bounds and the stability notion. As a result, we establish algorithm-dependent generalization bounds for stochastic gradient descent ascent (SGDA) algorithm and the more general sampling-determined algorithms.