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# Sharper Rates for Separable Minimax and Finite Sum Optimization via Primal-Dual Extragradient Methods

*Proceedings of Thirty Fifth Conference on Learning Theory*, PMLR 178:4362-4415, 2022.

#### Abstract

We design accelerated algorithms with improved rates for several fundamental classes of optimization problems. Our algorithms all build upon techniques related to the analysis of primal-dual extragradient methods via relative Lipschitzness proposed recently by Cohen, Sidford, and Tian ’21. (1) We study separable minimax optimization problems of the form $\min_x \max_y f(x) - g(y) + h(x, y)$, where $f$ and $g$ have smoothness and strong convexity parameters $(L^x, \mu^x)$, $(L^y, \mu^y)$, and h is convex-concave with a $(\Lambda^{xx}, \Lambda^{xy}, \Lambda^{yy})$-blockwise operator norm bounded Hessian. We provide an algorithm using $\tilde{O}(\sqrt{\frac{L^x}{\mu^x}} + \sqrt{\frac{L^y}{\mu^y}} + \frac{\Lambda^{xx}}{\mu^x} + \frac{\Lambda^{xy}}{\sqrt{\mu^x\mu^y}} + \frac{\Lambda^{yy}}{\mu^y})$ gradient queries. Notably, for convex-concave minimax problems with bilinear coupling (e.g. quadratics), where $\Lambda^{xx} = \Lambda^{yy} = 0$, our rate matches a lower bound of Zhang, Hong, and Zhang ’19. (2) We study finite sum optimization problems of the form $\min_x \frac 1 n \sum_{i \in [n]} f_i(x)$, where each $f_i$ is $L_i$-smooth and the overall problem is $\mu$-strongly convex. We provide an algorithm using $\tilde{O}(n + \sum_{i \in [n]} \sqrt{\frac{L_i}{n\mu}} )$ gradient queries. Notably, when the smoothness bounds $\{L_i\}_{i\in[n]}$ are non-uniform, our rate improves upon accelerated SVRG (Lin et al., Frostig et al. ’15) and Katyusha (Allen-Zhu ’17) by up to a $\sqrt{n}$ factor. (3) We generalize our algorithms for minimax and finite sum optimization to solve a natural family of minimax finite sum optimization problems at an accelerated rate, encapsulating both above results up to a logarithmic factor.