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 $\Omega\left(\sqrt{\kappa}\epsilon^{-2}\right)$ 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 $\Omega\left(\sqrt{\kappa}\epsilon^{-2} + \kappa^{1/3}\epsilon^{-4}\right)$. 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.

我们通过使用第一阶oracle及条件数,提供了寻找min-max优化问题中目标函数在最小化变量上是非凸及在最大化变量上是强凸时的稳定点的复杂度的下界，这既适用于确定性oracle也适用于随机oracle，并提供了各自的下界，并与其他文献的上界进行了比较。

非凸强凹Min-Max优化的复杂度下界