On the Linear Convergence of Extra-Gradient Methods for Nonconvex-Nonconcave Minimax Problems

Abstract: Recently, minimax optimization received renewed focus due to modern applications in machine learning, robust optimization, and reinforcement learning. The scale of these applications naturally leads to the use of first-order methods. However, the nonconvexities and nonconcavities present in these problems, prevents the application of typical Gradient Descent-Ascent, which is known to diverge even in bilinear problems. Recently, it was shown that the Proximal Point Method (PPM) converges linearly for a family of nonconvex-nonconcave problems. In this paper, we study the convergence of a damped version of Extra-Gradient Method (EGM) which avoids potentially costly proximal computations, only relying on gradient evaluation. We show that EGM converges linearly for smooth minimax optimization problem satisfying the same nonconvex-nonconcave condition needed by PPM.