  This paper studies the complexity for finding approximate stationary points
of nonconvex-strongly-concave (NC-SC) smooth minimax problems, in both general
and averaged smooth finite-sum settings. We establish nontrivial lower
complexity bounds of $\Omega(\sqrt{\kappa}\Delta L\epsilon^{-2})$ and
$\Omega(n+\sqrt{n\kappa}\Delta L\epsilon^{-2})$ for the two settings,
respectively, where $\kappa$ is the condition number, $L$ is the smoothness
constant, and $\Delta$ is the initial gap. Our result reveals substantial gaps
between these limits and best-known upper bounds in the literature. To close
these gaps, we introduce a generic acceleration scheme that deploys existing
gradient-based methods to solve a sequence of crafted
strongly-convex-strongly-concave subproblems. In the general setting, the
complexity of our proposed algorithm nearly matches the lower bound; in
particular, it removes an additional poly-logarithmic dependence on accuracy
present in previous works. In the averaged smooth finite-sum setting, our
proposed algorithm improves over previous algorithms by providing a
nearly-tight dependence on the condition number.
