Near-Optimal Complexity of Finite-Sum Nonconvex-Strongly-Concave Minimax Optimization
Qihao Zhou
Abstract
We characterize, up to logarithmic factors, the minimax expected query complexity of finite-sum nonconvex-strongly-concave optimization under mean-squared averaged smoothness. For randomized zero-respecting incremental first-order algorithms, the complexity is $\tildeΘ(n+\min\{\sqrt{n}\,κ,n^{3/4}\sqrtκ\}\,LΔ\varepsilon^{-2})$ throughout $κ=L/μ\ge1$, under exact dual initialization and for $0<\varepsilon^2\le c_0LΔ$, where $c_0>0$ is universal. This characterization is our main result. A dual-chain construction provides the lower bound and matches the existing Catalyst upper bound for $κ\ge\sqrt{n}$. For $1\leκ\le\sqrt{n}$, we introduce recursive proximal descent-ascent (RPDA), which retains a recursive gradient estimator across proximal subproblems and attains $\tilde{O}(n+\sqrt{n}\,κLΔ\varepsilon^{-2})$ with a fixed query budget. Both upper bounds guarantee an expected squared primal gradient at most $\varepsilon^2$. Under individual smoothness, we also prove $Ω(n+\min\{κ,\sqrt{nκ}\}\,LΔ\varepsilon^{-2})$ for every $κ\ge1$. It matches the polynomial upper rate for a bilinear subclass when $κ\ge n$; the intermediate-condition-number gap remains open.