Efficient Algorithms for Smooth Minimax Optimization

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 direct optimal algorithm combining Mirror-Prox and Nesterov's AGD, and show that it can find global optimum in $\widetilde{O}\left(1/k^2 \right)$ 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 $\widetilde{O}\left(1/k^{1/3} \right)$ 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 $O(m^{1/3}\sqrt{\log m}/k^{1/3})$.

Authors

• Kiran Thekumparampil
• Prateek Jain
• Praneeth Netrapalli
• Sewoong Oh

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