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 $\Omega\left(\sqrt{\kappax\kappay} \log \frac{1}{\epsilon}\right)$ on the number of gradient evaluations required to find an $\epsilon$-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 $\mathcal{O}\left( \sqrt{\kappax\kappay}\log^{3\frac{1}{\epsilon}\right)$} and $\mathcal{O}\left( \sqrt{\kappax\kappay}\log³ (\kappax\kappay)\log\frac{1}{\epsilon}\right)$, respectively. We fix this fundamental issue by providing the first algorithm with $\mathcal{O}\left(\sqrt{\kappax\kappay}\log\frac{1}{\epsilon}\right)$ 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.