A Unified Analysis of Extra-gradient and Optimistic Gradient Methods for Saddle Point Problems: Proximal Point Approach

We consider solving convex-concave saddle point problems. We focus on two variants of gradient decent-ascent algorithms, Extra-gradient (EG) and Optimistic Gradient (OGDA) methods, and show that they admit a unified analysis as approximations of the classical proximal point method for solving saddle-point problems. This viewpoint enables us to generalize EG (in terms of extrapolation steps) and OGDA (in terms of parameters) and obtain new convergence rate results for these algorithms for the bilinear case as well as the strongly convex-concave case.