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

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 DY, and a primal value function, defined by maximizing the objective over the dual variable, with initial suboptimality at most Delta. 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 σ2 and mean-square smoothness, we prove the lower bound Omega!left(L2DYDeltavarepsilon-3+L3DY2Deltaσ2varepsilon-6right). 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 kappa:=L/μ, we obtain Omega!left(LDelta√kappa,varepsilon-2+LDeltakappaσ2varepsilon-4right) under the bounded-variance oracle model. Under the additional mean-square smoothness condition with constant bar L, we obtain Omega!left(LDelta√kappa,varepsilon-2+Deltabar Lσkappa3/2varepsilon-3right). 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.