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 Φ(x)=maxy f(x,y), where f is jointly L-smooth and μ-strongly concave in y on unconstrained Euclidean domains, and set κ=L/μ. Each query returns a fresh unbiased joint gradient with conditional variance at most σ2. Given initial primal gap at most Δ and dual residual |∇y f(x0,y0)|≤ G, the fixed-budget complexity of finding |∇Φ(x̂)|≤ε with probability at least 2/3 is Θ̃(√κLΔ/ε2+κLΔσ2/ε4+κ2σ2/ε2+√κlog+(G/(ε√κ))), where log+u=logmax{1,u}. This characterization holds when κ and LΔ/ε2 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.
Related
- Near-Optimal Complexity of Finite-Sum Nonconvex-Strongly-Concave Minimax Optimization
- Matching Full-Hessian Oracle Complexity for Nonconvex--Strongly-Concave Minimax Optimization
- Tight Stochastic Condition-Number Dependence in Nonconvex-Strongly-Concave Minimax Optimization
- Tight Lower Bounds for Stochastic Nonconvex-Strongly-Concave Minimax Optimization
- Envelope Cubic Regularized Newton Method for Nonconvex-Strongly-Concave Minimax Optimization
BibTeX
@misc{wu2026sharp,
title = {{Sharp Fresh-Gradient Complexity of Nonconvex-Strongly-Concave Minimax Optimization}},
author = {Wendao Wu and Haihan Zhang and Chenheng Zhang and Yanyi Li and Chunyuan Zheng and Cong Fang and Haoxuan Li and Zhouchen Lin},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.00229},
eprint = {2610.00229},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.00229},
}
RIS
TY - UNPB
TI - Sharp Fresh-Gradient Complexity of Nonconvex-Strongly-Concave Minimax Optimization
AU - Wendao Wu
AU - Haihan Zhang
AU - Chenheng Zhang
AU - Yanyi Li
AU - Chunyuan Zheng
AU - Cong Fang
AU - Haoxuan Li
AU - Zhouchen Lin
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.00229
UR - https://arxiv.org/abs/2610.00229
AB - 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 Φ(x)=max_y f(x,y), where f is jointly L-smooth and μ-strongly concave in y on unconstrained Euclidean domains, and set κ=L/μ. Each query returns a fresh unbiased joint gradient with conditional variance at most σ^2. Given initial primal gap at most Δ and dual residual |∇_y f(x_0,y_0)|≤ G, the fixed-budget complexity of finding |∇Φ(x̂)|≤ε with probability at least 2/3 is Θ̃(√κLΔ/ε^2+κLΔσ^2/ε^4+κ^2σ^2/ε^2+√κlog_+(G/(ε√κ))), where log_+u=logmax{1,u}. This characterization holds when κ and LΔ/ε^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.
ER -