The First Optimal Algorithm for Smooth and Strongly-Convex-Strongly-Concave Minimax Optimization

• URL: http://arxiv.org/abs/2205.05653v1
• Date: Wed, 11 May 2022 17:33:07 GMT
• Title: The First Optimal Algorithm for Smooth and Strongly-Convex-Strongly-Concave Minimax Optimization
• Authors: Dmitry Kovalev, Alexander Gasnikov
• Abstract summary: In this paper, we revisit the smooth and strongly-strongly-concave minimax optimization problem. Existing state-of-the-art methods do not match lower bound $Omegaleft(sqrtkappa_xkappa_ylog3 (kappa_xkappa_y)logfrac1epsilonright)$. We fix this fundamental issue by providing the first algorithm with $mathcalOleft( sqrtkappa_xkappa_ylog
• Score: 88.91190483500932
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: In this paper, we revisit the smooth and strongly-convex-strongly-concave minimax optimization problem. Zhang et al. (2021) and Ibrahim et al. (2020) established the lower bound $\Omega\left(\sqrt{\kappa_x\kappa_y} \log \frac{1}{\epsilon}\right)$ on the number of gradient evaluations required to find an $\epsilon$-accurate solution, where $\kappa_x$ and $\kappa_y$ are condition numbers for the strong convexity and strong concavity assumptions. However, the existing state-of-the-art methods do not match this lower bound: algorithms of Lin et al. (2020) and Wang and Li (2020) have gradient evaluation complexity $\mathcal{O}\left( \sqrt{\kappa_x\kappa_y}\log^3\frac{1}{\epsilon}\right)$ and $\mathcal{O}\left( \sqrt{\kappa_x\kappa_y}\log^3 (\kappa_x\kappa_y)\log\frac{1}{\epsilon}\right)$, respectively. We fix this fundamental issue by providing the first algorithm with $\mathcal{O}\left(\sqrt{\kappa_x\kappa_y}\log\frac{1}{\epsilon}\right)$ gradient evaluation complexity. We design our algorithm in three steps: (i) we reformulate the original problem as a minimization problem via the pointwise conjugate function; (ii) we apply a specific variant of the proximal point algorithm to the reformulated problem; (iii) we compute the proximal operator inexactly using the optimal algorithm for operator norm reduction in monotone inclusions.

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