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

arXiv:2609.30877v1 Announce Type: cross 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 sigmagearepsilon, the worst-case complexity of zero-respecting algorithms is Theta(kappa LGsigma^2arepsilon^{-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 arepsilon/sqrt{kappa}, while an undiscovered primal coordinate prevents stationarity. It also yields the primal-gradient lower bound Omega(LDelta(sqrt{kappa}arepsilon^{-2}+kappasigma^2arepsilon^{-4})) after combination with the known deterministic bound, where Delta bounds the initial primal function gap.

Related

• The Condition-Number Barrier in Sparse Least Squares
• A Tight Lower Bound for Smooth Nonconvex Stochastic Optimization with Bounded Gradient Noise
• Beyond Bounded Variance: Variance-Reduced Normalized Methods for Nonconvex Optimization under Blum-Gladyshev Noise

Source: arXiv cs.LG | 2026-09-28