Accelerated Primal-Dual Gradient Method for Smooth and Convex-Concave Saddle-Point Problems with Bilinear Coupling

• URL: http://arxiv.org/abs/2112.15199v1
• Date: Thu, 30 Dec 2021 20:31:46 GMT
• Title: Accelerated Primal-Dual Gradient Method for Smooth and Convex-Concave Saddle-Point Problems with Bilinear Coupling
• Authors: Dmitry Kovalev, Alexander Gasnikov, Peter Richt\'arik
• Abstract summary: We study a linear convex-concave saddle-point problem $min_xmax_y f(x) ytopmathbfA x - g(y)
• Score: 84.47780064014262
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: In this paper we study a convex-concave saddle-point problem $\min_x\max_y f(x) + y^\top\mathbf{A} x - g(y)$, where $f(x)$ and $g(y)$ are smooth and convex functions. We propose an Accelerated Primal-Dual Gradient Method for solving this problem which (i) achieves an optimal linear convergence rate in the strongly-convex-strongly-concave regime matching the lower complexity bound (Zhang et al., 2021) and (ii) achieves an accelerated linear convergence rate in the case when only one of the functions $f(x)$ and $g(y)$ is strongly convex or even none of them are. Finally, we obtain a linearly-convergent algorithm for the general smooth and convex-concave saddle point problem $\min_x\max_y F(x,y)$ without requirement of strong convexity or strong concavity.

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