The First Optimal Algorithm for Smooth and Strongly-Convex-Strongly-Concave Minimax Optimization
This solves a fundamental issue in optimization theory by closing the gap between existing methods and the theoretical lower bound for this class of problems.
The paper tackles the problem of smooth and strongly-convex-strongly-concave minimax optimization by providing the first algorithm that matches the known lower bound, achieving an optimal gradient evaluation complexity of O(√(κ_xκ_y) log(1/ε)).
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 $Ω\left(\sqrt{κ_xκ_y} \log \frac{1}ε\right)$ on the number of gradient evaluations required to find an $ε$-accurate solution, where $κ_x$ and $κ_y$ 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 $\mathcal{O}\left( \sqrt{κ_xκ_y}\log^3\frac{1}ε\right)$ and $\mathcal{O}\left( \sqrt{κ_xκ_y}\log^3 (κ_xκ_y)\log\frac{1}ε\right)$, respectively. We fix this fundamental issue by providing the first algorithm with $\mathcal{O}\left(\sqrt{κ_xκ_y}\log\frac{1}ε\right)$ 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.