Complexity Lower Bounds for Nonconvex-Strongly-Concave Min-Max Optimization

We provide a first-order oracle complexity lower bound for finding stationary points of min-max optimization problems where the objective function is smooth, nonconvex in the minimization variable, and strongly concave in the maximization variable. We establish a lower bound of $Omegaleft(sqrt{kappa}epsilon^{-2}right)$ for deterministic oracles, where $epsilon$ defines the level of approximate stationarity and $kappa$ is the condition number. Our analysis shows that the upper bound achieved in (Lin et al., 2020b) is optimal in the $epsilon$ and $kappa$ dependence up to logarithmic factors. For stochastic oracles, we provide a lower bound of $Omegaleft(sqrt{kappa}epsilon^{-2} + kappa^{1/3}epsilon^{-4}right)$. It suggests that there is a significant gap between the upper bound $mathcal{O}(kappa^3 epsilon^{-4})$ in (Lin et al., 2020a) and our lower bound in the condition number dependence.