Faster Stochastic Algorithms for Minimax Optimization under Polyak--{\L}ojasiewicz Conditions

• URL: http://arxiv.org/abs/2307.15868v1
• Date: Sat, 29 Jul 2023 02:26:31 GMT
• Title: Faster Stochastic Algorithms for Minimax Optimization under Polyak--{\L}ojasiewicz Conditions
• Authors: Lesi Chen, Boyuan Yao, Luo Luo
• Abstract summary: We propose SPIDER-GDA for solving the finite-sum problem of the form $min_x max_y f(x,y)triqangle frac1n sum_i=1n f_i(x,y)$. We prove SPIDER-GDA could find an $epsilon$-optimal solution within $mathcal Oleft((n + sqrtn,kappa_xkappa_y2)log (1/epsilon)
• Score: 12.459354707528819
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: This paper considers stochastic first-order algorithms for minimax optimization under Polyak--{\L}ojasiewicz (PL) conditions. We propose SPIDER-GDA for solving the finite-sum problem of the form $\min_x \max_y f(x,y)\triangleq \frac{1}{n} \sum_{i=1}^n f_i(x,y)$, where the objective function $f(x,y)$ is $\mu_x$-PL in $x$ and $\mu_y$-PL in $y$; and each $f_i(x,y)$ is $L$-smooth. We prove SPIDER-GDA could find an $\epsilon$-optimal solution within ${\mathcal O}\left((n + \sqrt{n}\,\kappa_x\kappa_y^2)\log (1/\epsilon)\right)$ stochastic first-order oracle (SFO) complexity, which is better than the state-of-the-art method whose SFO upper bound is ${\mathcal O}\big((n + n^{2/3}\kappa_x\kappa_y^2)\log (1/\epsilon)\big)$, where $\kappa_x\triangleq L/\mu_x$ and $\kappa_y\triangleq L/\mu_y$. For the ill-conditioned case, we provide an accelerated algorithm to reduce the computational cost further. It achieves $\tilde{{\mathcal O}}\big((n+\sqrt{n}\,\kappa_x\kappa_y)\log^2 (1/\epsilon)\big)$ SFO upper bound when $\kappa_y \gtrsim \sqrt{n}$. Our ideas also can be applied to the more general setting that the objective function only satisfies PL condition for one variable. Numerical experiments validate the superiority of proposed methods.

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