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Nesterov Meets Optimism: Rate-Optimal Separable Minimax Optimization. (arXiv:2210.17550v2 [math.OC] UPDATED)

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We propose a new first-order optimization algorithm --
AcceleratedGradient-OptimisticGradient (AG-OG) Descent Ascent -- for separable
convex-concave minimax optimization. The main idea of our algorithm is to
carefully leverage the structure of the minimax problem, performing Nesterov
acceleration on the individual component and optimistic gradient on the
coupling component. Equipped with proper restarting, we show that AG-OG
achieves the optimal convergence rate (up to a constant) for a variety of
settings, including bilinearly coupled strongly convex-strongly concave minimax
optimization (bi-SC-SC), bilinearly coupled convex-strongly concave minimax
optimization (bi-C-SC), and bilinear games. We also extend our algorithm to the
stochastic setting and achieve the optimal convergence rate in both bi-SC-SC
and bi-C-SC settings. AG-OG is the first single-call algorithm with optimal
convergence rates in both deterministic and stochastic settings for bilinearly
coupled minimax optimization problems.

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