Lower Bounds for Nonconvex-PŁ Minimax Optimization

Abstract

We study the deterministic first-order oracle complexity of finding stationary points of the value function in smooth nonconvex-Polyak-Łojasiewicz (NC-PŁ) minimax optimization. We assume that the objective is jointly $\ell$-smooth and satisfies the $μ$-PŁ condition in the dual variable, and that its value function $Φ(x):=\max_y f(x;y)$ satisfies $Φ(0)-\inf_xΦ(x)\leqΔ$. When $κ:=\ell/μ\gtrsim 1$ and $0<ε^2\lesssim\ellΔ$, we prove that every deterministic first-order method requires $Ω(\ellΔκ/ε^2)$ oracle queries in the worst case to find $x$ satisfying $\|\nablaΦ(x)\|\leqε$. This rate matches the known upper bound in its dependence on $(\ell,Δ,κ,ε)$ [Yang et al., 2022] and shows that the linear dependence on $κ$ is unavoidable for deterministic first-order methods.