by S. Zhang, Y. Hu, L. Zhang, N. He

Abstract:

This paper studies the generalization performance 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 of the primal variable 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 stability and generalization. As a result, we establish algorithm-dependent generalization bounds for stochastic gradient descent ascent (SGDA) and the more general sampling-determined algorithms (SDA).

Reference:

Generalization Bounds of Nonconvex-(Strongly)-Concave Stochastic Minimax Optimization S. Zhang, Y. Hu, L. Zhang, N. HeIn Proc. International Conference on Artificial Intelligence and Statistics (AISTATS), PMLR, volume 238, 2024

Bibtex Entry:

@inproceedings{zhang2024generalization}(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 of the primal variable 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 stability and generalization. As a result, we establish algorithm-dependent generalization bounds for stochastic gradient descent ascent (SGDA) and the more general sampling-determined algorithms (SDA). },
	author = {Zhang, Siqi and Hu, Yifan and Zhang, Liang and He, Niao},
	booktitle = {Proc. International Conference on Artificial Intelligence and Statistics (AISTATS)},
	month = {02--04 May},
	pages = {694--702},
	pdf = {https://proceedings.mlr.press/v238/zhang24c/zhang24c.pdf},
	publisher = {PMLR},
	series = {Proceedings of Machine Learning Research},
	title = {Generalization Bounds of Nonconvex-(Strongly)-Concave Stochastic Minimax Optimization},
	volume = {238},
	year = {2024}}