Near-Optimal Algorithms for Minimax Optimization

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 Õ(κ_x+κ_y) (Tseng, 1995) or Õ(min{κ_x √κ_y, √κ_xκ_y}) (Alkousa et al., 2019) gradient evaluations, where κ_x and κ_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 Ω̃(√κ_xκ_y) (Ibrahim et al., 2019; Zhang et al., 2019). This paper presents the first algorithm with Õ(√κ_xκ_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.