The First Optimal Algorithm for Smooth and Strongly-Convex-Strongly-Concave Minimax Optimization

In this paper, we revisit the smooth and strongly-convex-strongly-concave minimax optimization problem. Zhang et al. (2021) and Ibrahim et al. (2020) established the lower bound Omegaleft(√kappaxkappay log frac1εright) on the number of gradient evaluations required to find an ε-accurate solution, where kappax and kappay are condition numbers for the strong convexity and strong concavity assumptions. However, the existing state-of-the-art methods do not match this lower bound: algorithms of Lin et al. (2020) and Wang and Li (2020) have gradient evaluation complexity Oleft( √kappaxkappaylog3frac1εright) and Oleft( √kappaxkappaylog3 (kappaxkappay)logfrac1εright), respectively. We fix this fundamental issue by providing the first algorithm with Oleft(√kappaxkappaylogfrac1εright) gradient evaluation complexity. We design our algorithm in three steps: (i) we reformulate the original problem as a minimization problem via the pointwise conjugate function; (ii) we apply a specific variant of the proximal point algorithm to the reformulated problem; (iii) we compute the proximal operator inexactly using the optimal algorithm for operator norm reduction in monotone inclusions.