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While classic work in convex-concave min-max optimization relies onaverage-iterate convergence results, the emergence of nonconvex applicationssuch as training Generative Adversarial Networks has led to renewed interest inlast-iterate convergence guarantees. Proving last-iterate convergence ischallenging because many natural algorithms, such as Simultaneous GradientDescent/Ascent, provably diverge or cycle even in simple convex-concave min-maxsettings, and previous work on global last-iterate convergence rates has beenlimited to the bilinear and convex-strongly concave settings. In this work, weshow that the Hamiltonian Gradient Descent (HGD) algorithm achieves linearconvergence in a variety of more general settings, including convex-concaveproblems that satisfy a "sufficiently bilinear" condition. We also provesimilar convergence rates for the Consensus Optimization (CO) algorithm of[MNG17] for some parameter settings of CO.

Last-iterate convergence rates for min-max optimization