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

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 μ, 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 σ epsilon, the worst-case complexity of zero-respecting algorithms is Θ(κLGσ 2 epsilon-4) in the stated accuracy regime, where κ=L/μ, G bounds the initial primal-dual gap, and σ squared 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 epsilon/sqrtκ, while an undiscovered primal coordinate prevents stationarity. It also yields the primal-gradient lower bound Ω(LΔ(sqrtκ epsilon-2+κσ 2 epsilon-4)) after combination with the known deterministic bound, where Δ bounds the initial primal function gap.