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

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 $σ<sup>2$ and mean-square smoothness, we prove the lower bound $$Ω!\left(L<sup>2D_YΔ\varepsilon<sup>{-3}+L<sup>3D_Y<sup>2Δσ<sup>2\varepsilon<sup>{-6}\right)$$. 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 $μ&gt;0$ and condition number $κ:=L/μ$, we obtain $$Ω!\left(LΔ\sqrtκ\,\varepsilon<sup>{-2}+LΔκσ<sup>2\varepsilon<sup>{-4}\right)$$ under the bounded-variance oracle model. Under the additional mean-square smoothness condition with constant $\bar L$, we obtain $$Ω!\left(LΔ\sqrtκ\,\varepsilon^{-2}+Δ\bar Lσκ<sup>{3/2}\varepsilon<sup>{-3}\right)$$. 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.