Sharp Fresh-Gradient Complexity of Nonconvex-Strongly-Concave Minimax Optimization

Abstract

We characterize the fresh-gradient oracle complexity of smooth nonconvex-strongly-concave minimax optimization, with matching upper and lower bounds up to logarithmic factors. Let $\Phi(x)=\max_y f(x,y)$, where $f$ is jointly $L$-smooth and $\mu$-strongly concave in $y$ on unconstrained Euclidean domains, and set $\kappa=L/\mu$. Each query returns a fresh unbiased joint gradient with conditional variance at most $\sigma^2$. Given initial primal gap at most $\Delta$ and dual residual $\|\nabla_y f(x_0,y_0)\|\le G$, the fixed-budget complexity of finding $\|\nabla\Phi(\widehat x)\|\le\epsilon$ with probability at least $2/3$ is $\widetilde{\Theta}(\sqrt{\kappa}L\Delta/\epsilon^2+\kappa L\Delta\sigma^2/\epsilon^4+\kappa^2\sigma^2/\epsilon^2+\sqrt{\kappa}\log_+(G/(\epsilon\sqrt{\kappa})))$, where $\log_+u=\log\max\{1,u\}$. This characterization holds when $\kappa$ and $L\Delta/\epsilon^2$ exceed universal constants, uniformly over finite dimensions, and the suppressed logarithms are independent of $G$. It establishes the necessity of linear condition-number dependence in the global stochastic cost and identifies a separate statistical refinement cost. A proximal method separates coarse primal progress from one final refinement, while the lower bounds apply to arbitrary adaptive randomized algorithms. Dual initialization contributes only an additive logarithmic cost, yet removing its control eliminates every finite dimension-free complexity bound, even with exact gradients.