The Complexity of Nonconvex-Strongly-Concave Minimax Optimization

• URL: http://arxiv.org/abs/2103.15888v1
• Date: Mon, 29 Mar 2021 18:53:57 GMT
• Title: The Complexity of Nonconvex-Strongly-Concave Minimax Optimization
• Authors: Siqi Zhang, Junchi Yang, Crist\'obal Guzm\'an, Negar Kiyavash, Niao He
• Abstract summary: This paper establishes the complexity for finding approximate stationary points of non-strongly-concave (NC-SC) smooth minimax problems. We deploy a proposed sequence of $Omega-strong$lyconcave sub-2 problems in both general complexity and averaged complexity. In our proposed finite-sum setting, our proposed algorithm provides a nearly-tight dependence on the condition number.
• Score: 43.07732143522183
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: This paper studies the complexity for finding approximate stationary points of nonconvex-strongly-concave (NC-SC) smooth minimax problems, in both general and averaged smooth finite-sum settings. We establish nontrivial lower complexity bounds of $\Omega(\sqrt{\kappa}\Delta L\epsilon^{-2})$ and $\Omega(n+\sqrt{n\kappa}\Delta L\epsilon^{-2})$ for the two settings, respectively, where $\kappa$ is the condition number, $L$ is the smoothness constant, and $\Delta$ is the initial gap. Our result reveals substantial gaps between these limits and best-known upper bounds in the literature. To close these gaps, we introduce a generic acceleration scheme that deploys existing gradient-based methods to solve a sequence of crafted strongly-convex-strongly-concave subproblems. In the general setting, the complexity of our proposed algorithm nearly matches the lower bound; in particular, it removes an additional poly-logarithmic dependence on accuracy present in previous works. In the averaged smooth finite-sum setting, our proposed algorithm improves over previous algorithms by providing a nearly-tight dependence on the condition number.

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