Matching Multi-Loop Complexities with a Single Loop: Optimal Optimization Stationarity and Best-Known Game Stationarity in Nonconvex--Concave Minimax Optimization

arXiv:2609.17973v1 Announce Type: cross Abstract: We introduce a new single-loop algorithmic framework for smooth nonconvex--concave minimax optimization. The resulting projected damped extragradient method combines projected extragradient updates, dual momentum, and a moving proximal center. Under both the optimization-stationarity and game-stationarity criteria, our method achieves the best-known complexity among single-loop first-order methods. For optimization stationarity, our method achieves a gradient complexity of O(L^2D_YarDelta_0arepsilon^{-3}), where L is the gradient Lipschitz constant, D_Y bounds the diameter of the dual feasible set, and arDelta_0 is an initialization quantity involving the value-function gap and the initial gradients. Moreover, by incorporating a fixed-center warm-up phase, the complexity can be improved to O(L^2D_YDelta_phiarepsilon^{-3}), up to an additive lower-order cost, where Delta_phi:=phi(x_0)-inf_xphi(x). We further establish a lower bound of Omega(L^2D_YDelta_phiarepsilon^{-3}) for optimization stationarity over projected zero-respecting first-order methods. This lower bound proves that the warm-started version of our algorithm is optimal up to a constant factor for optimization stationarity within this oracle class. For game stationarity, our method achieves O!(L^{3/2}D_Y^{1/2}Delta_phiarepsilon^{-5/2}) gradient complexity. This matches the best-known complexity of multi-loop first-order methods, thereby establishing the same complexity with a single-loop algorithmic structure. Under dual strong concavity, the proposed framework achieves O!(sqrt{kappa},LDelta_phiarepsilon^{-2}) leading complexity for both stationarity criteria, where kappa=L/mu is the dual condition number, up to an additive initialization cost. The arepsilon^{-2} accuracy dependence is optimal under fixed regularity and initialization bounds.

Source: arXiv cs.LG | 2026-09-17