This paper resolves a longstanding open question pertaining to the design of near-optimal first-order algorithms for smooth and strongly-convex-strongly-concave minimax problems. Current state-of-the-art first-order algorithms find an approximate Nash equilibrium using $\tilde{O}(\kappa_{\mathbf x}+\kappa_{\mathbf y})$ or $\tilde{O}(\min\{\kappa_{\mathbf x}\sqrt{\kappa_{\mathbf y}}, \sqrt{\kappa_{\mathbf x}}\kappa_{\mathbf y}\})$ gradient evaluations, where $\kappa_{\mathbf x}$ and $\kappa_{\mathbf y}$ are the condition numbers for the strong-convexity and strong-concavity assumptions. A gap still remains between these results and the best existing lower bound $\tilde{\Omega}(\sqrt{\kappa_{\mathbf x}\kappa_{\mathbf y}})$. This paper presents the first algorithm with $\tilde{O}(\sqrt{\kappa_{\mathbf x}\kappa_{\mathbf y}})$ gradient complexity, matching the lower bound up to logarithmic factors. Our algorithm is designed based on an accelerated proximal point method and an accelerated solver for minimax proximal steps. It can be easily extended to the settings of strongly-convex-concave, convex-concave, nonconvex-strongly-concave, and nonconvex-concave functions. This paper also presents algorithms that match or outperform all existing methods in these settings in terms of gradient complexity, up to logarithmic factors.

翻译：本文解决了与近最佳第一阶算法设计有关的一个长期未决问题, 用于平滑和强烈的精密小型max问题。 目前最先进的第一阶算法算法在 $\ tilde{ O} (\ kapa\\ mathbf x ⁇ kappa ⁇ mathbf y} ) 或$\ tilde{O} (min ⁇ kappa\\ mathbf x xx x\\ svevax x kapa_mathf y} 中找到一个大约的 Nash 平衡 。 这些结果与目前所有更低的精度 $\ talpha_ max 的精度运算法 。 这些结果与目前最低的精度 $\ talde_ compab\ x 的精度运算法, 直径直径直到 ma\\\\ x max max max max max max max max max amax max max amax max max max max max max amax max max max max max max max max max max amax max max max max max max max max amax max max max max max max max max max max max max max max max max max max max max max max max max max max max max max max max max max max max max max max max max max max max max max max max max max max max max max max max max max max max max