Tight Stochastic Condition-Number Dependence in Nonconvex-Strongly-Concave Minimax Optimization

Abstract

We study whether the linear condition-number dependence in the stochastic complexity of SAPD+ is necessary for nonconvex-strongly-concave minimax optimization. For jointly $L$-smooth objectives with dual strong-concavity parameter $\mu$, we prove a lower bound that matches the SAPD+ upper bound under the same Moreau-envelope stationarity criterion and the same primal-dual initialization gap. Specifically, when $\sigma\ge\varepsilon$, the worst-case complexity of zero-respecting algorithms is $\Theta(\kappa LG\sigma^2\varepsilon^{-4})$ in the stated accuracy regime, where $\kappa=L/\mu$, $G$ bounds the initial primal-dual gap, and $\sigma^2$ bounds the variance of a general unbiased first-order oracle. The lower bound is realized on a smooth problem class with a bounded dual box. Our construction routes each link of a nonconvex zero-chain through a dual gradient of magnitude proportional to $\varepsilon/\sqrt{\kappa}$, while an undiscovered primal coordinate prevents stationarity. It also yields the primal-gradient lower bound $\Omega(L\Delta(\sqrt{\kappa}\varepsilon^{-2}+\kappa\sigma^2\varepsilon^{-4}))$ after combination with the known deterministic bound, where $\Delta$ bounds the initial primal function gap.