Near Optimal Stochastic Algorithms for Finite-Sum Unbalanced Convex-Concave Minimax Optimization

• URL: http://arxiv.org/abs/2106.01761v1
• Date: Thu, 3 Jun 2021 11:30:32 GMT
• Title: Near Optimal Stochastic Algorithms for Finite-Sum Unbalanced Convex-Concave Minimax Optimization
• Authors: Luo Luo, Guangzeng Xie, Tong Zhang, Zhihua Zhang
• Abstract summary: This paper considers first-order algorithms for convex-con minimax problems of the form $min_bf xmax_yf(bfbf y) simultaneously. Our methods can be used to solve more general unbalanced minimax problems and are also near optimal.
• Score: 41.432757205864796
• License: http://creativecommons.org/publicdomain/zero/1.0/
• Abstract: This paper considers stochastic first-order algorithms for convex-concave minimax problems of the form $\min_{\bf x}\max_{\bf y}f(\bf x, \bf y)$, where $f$ can be presented by the average of $n$ individual components which are $L$-average smooth. For $\mu_x$-strongly-convex-$\mu_y$-strongly-concave setting, we propose a new method which could find a $\varepsilon$-saddle point of the problem in $\tilde{\mathcal O} \big(\sqrt{n(\sqrt{n}+\kappa_x)(\sqrt{n}+\kappa_y)}\log(1/\varepsilon)\big)$ stochastic first-order complexity, where $\kappa_x\triangleq L/\mu_x$ and $\kappa_y\triangleq L/\mu_y$. This upper bound is near optimal with respect to $\varepsilon$, $n$, $\kappa_x$ and $\kappa_y$ simultaneously. In addition, the algorithm is easily implemented and works well in practical. Our methods can be extended to solve more general unbalanced convex-concave minimax problems and the corresponding upper complexity bounds are also near optimal.

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