Uniform Localized Convergence and Sharper Generalization Bounds for Minimax Problems

Bowei Zhu, Shaojie Li, Yong Liu

Primary Area: learning theory

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Keywords: generalization analysis; minimax problems

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Abstract: Minimax problems have achieved widely success in machine learning such as adversarial training, robust optimization, reinforcement learning. Existing studies focus on minimax problems with specific \text{algorithms} in stochastic optimization, with only a few work on generalization performance. Current generalization bounds almost all depend on stability, which need case-by-case analyses for specific \text{algorithms}. Additionally, recent work provides the $O(\sqrt{d/n})$ generalization bound in expectation based on uniform convergence. In this paper, we study the generalization bounds measured by the gradients of primal functions using the uniform localized convergence. We relax the Lipschitz continuity assumption and give a sharper high probability generalization bound for nonconvex-strongly-concave (NC-SC) stochastic minimax problems considering the localized information. Furthermore, we provide dimension-independent results under Polyak-Lojasiewicz condition for the outer layer. Based on the uniform localized convergence, we analyze some popular \text{algorithms} such as the empirical saddle point (ESP), gradient descent ascent (GDA) and stochastic gradient descent ascent (SGDA) and improve the generalization bounds for primal functions. We can even gain approximate $O(1/n^2)$ excess primal risk bounds with further assumptions that the optimal population risks are small, which, to the best of our knowledge, are the sharpest results in minimax problems.

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Submission Number: 5348