  This paper studies first order methods for solving smooth minimax
optimization problems $\min_x \max_y g(x,y)$ where $g(\cdot,\cdot)$ is smooth
and $g(x,\cdot)$ is concave for each $x$. In terms of $g(\cdot,y)$, we consider
two settings -- strongly convex and nonconvex -- and improve upon the best
known rates in both. For strongly-convex $g(\cdot, y),\ \forall y$, we propose
a new algorithm combining Mirror-Prox and Nesterov's AGD, and show that it can
find global optimum in $\tilde{O}(1/k^2)$ iterations, improving over current
state-of-the-art rate of $O(1/k)$. We use this result along with an inexact
proximal point method to provide $\tilde{O}(1/k^{1/3})$ rate for finding
stationary points in the nonconvex setting where $g(\cdot, y)$ can be
nonconvex. This improves over current best-known rate of $O(1/k^{1/5})$.
Finally, we instantiate our result for finite nonconvex minimax problems, i.e.,
$\min_x \max_{1\leq i\leq m} f_i(x)$, with nonconvex $f_i(\cdot)$, to obtain
convergence rate of $O(m(\log m)^{3/2}/k^{1/3})$ total gradient evaluations for
finding a stationary point.
