On the Complexity of Finite-Sum Nonconvex-Strongly-Concave Minimax Optimization with Exact Proximal Oracles

Finite-sum nonconvex-strongly-concave minimax problems arise in adversarial training and distributionally robust learning. Their known lower bounds cover only linear-span algorithms, zero-respecting algorithms, or a restricted proximal protocol. They leave open the power of exact component proximal steps, whose responses are global saddle points of regularized component problems and can expose directions that gradients do not reveal. We show that neither unrestricted adaptivity nor exact proximal access improves the polynomial complexity. We consider n components with joint mean-squared smoothness L and condition number $κ$, started from a point with primal gap at most $Δ$ and joint gradient norm at most √2LΔ. For ε²/(LΔ) below a universal constant, every randomized adaptive algorithm with exact component values, gradients, and proximal responses needs Ω(n+min\√ n\,κ,n3/4√κΔ/ε²) component queries in the worst case to find an ε-stationary point of the value function. A method using only component gradients attains this rate up to a factor log²(2+κ), including initialization. Zhou (arXiv:2610.04944) concurrently proves the same rate for zero-respecting first-order methods. In our hard instances, every component admits proximal steps with the same step sizes, and a response for a component reveals hidden directions only up to the first one assigned to another component.