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We consider differentiable games where the goal is to find a Nashequilibrium. The machine learning community has recently started using variantsof the gradient method (GD). Prime examples are extragradient (EG), theoptimistic gradient method (OG) and consensus optimization (CO), which enjoylinear convergence in cases like bilinear games, where the standard GD fails.The full benefits of theses relatively new methods are not known as there is nounified analysis for both strongly monotone and bilinear games. We provide newanalyses of the EG's local and global convergence properties and use is to geta tighter global convergence rate for OG and CO. Our analysis covers the wholerange of settings between bilinear and strongly monotone games. It reveals thatthese methods converge via different mechanisms at these extremes; in between,it exploits the most favorable mechanism for the given problem. We then provethat EG achieves the optimal rate for a wide class of algorithms with anynumber of extrapolations. Our tight analysis of EG's convergence rate in gamesshows that, unlike in convex minimization, EG may be much faster than GD.

A Tight and Unified Analysis of Gradient-Based Methods for a Whole Spectrum of Games