Lower Bounds for Stochastic First-Order Algorithms with Variance Reduction in Nonconvex--Concave Minimax Optimization

2026-10-01

AI Summary

This study addresses the computational complexity lower bounds for nonconvex-concave and strongly-concave minimax optimization when quantization permits variance reduction. By employing a stochastic first-order oracle model and a zero-respecting algorithm framework, the authors conduct theoretical derivations under mean-squared smoothness and unbiased gradient assumptions, thereby overcoming existing limitations to establish the universality of specific lower bounds. The primary contribution lies in deriving complexity lower bounds that explicitly capture dependencies on accuracy, dual domain radius, and noise—for instance, Ω(L²D_YΔε⁻³+...). These results rigorously characterize convergence rate barriers and fundamental theoretical performance limits across various settings, effectively filling a critical gap in the existing literature on minimax optimization theory.

Abstract

We establish complexity lower bounds for stochastic first-order algorithms in nonconvex--concave minimax optimization, allowing algorithms to use variance reduction. Our main contribution is a lower bound for a zero-respecting algorithm class that permits variance reduction, extending beyond the algorithmic restrictions imposed by some existing lower bounds. We consider objectives with an $L$-Lipschitz continuous joint gradient, a compact convex dual domain of Euclidean radius at most $D_Y$, and a primal value function, defined by maximizing the objective over the dual variable, with initial suboptimality at most $Δ$. The target accuracy $\varepsilon$ is measured by the gradient norm of the Moreau envelope of the constrained primal value function with parameter $1/(2L)$. Under an unbiased stochastic first-order oracle with variance at most $σ^2$ and mean-square smoothness, we prove the lower bound $Ω\!\left(L^2D_YΔ\varepsilon^{-3}+L^3D_Y^2Δσ^2\varepsilon^{-6}\right)$. This result quantifies the dependence on accuracy, dual-domain radius, and oracle noise even when variance reduction is allowed. We also establish complementary lower bounds for nonconvex--strongly-concave minimax optimization. With dual strong-concavity parameter $μ>0$ and condition number $κ:=L/μ$, we obtain $Ω\!\left(LΔ\sqrtκ\,\varepsilon^{-2}+LΔκσ^2\varepsilon^{-4}\right)$ under the bounded-variance oracle model. Under the additional mean-square smoothness condition with constant $\bar L$, we obtain $Ω\!\left(LΔ\sqrtκ\,\varepsilon^{-2}+Δ\bar Lσκ^{3/2}\varepsilon^{-3}\right)$. Together, these results identify complexity barriers across the concave and strongly concave regimes, with the main nonconvex--concave bound remaining valid for algorithms that use variance reduction.

Problem

• minimax optimization
• complexity lower bounds
• stochastic first-order algorithms
• variance reduction
• nonconvex-concave

Innovation

• minimax optimization
• variance reduction
• complexity lower bounds
• stochastic first-order algorithms
• nonconvex-concave