We study the problem of finding min-max solutions for smooth two-input objective functions. While classic results show average-iterate convergence rates for various algorithms, nonconvex applications such as training Generative Adversarial Networks require last-iterate convergence guarantees, which are more difficult to prove. It has been an open problem as to whether any algorithm achieves non-asymptotic last-iterate convergence in settings beyond the bilinear and convex-strongly concave settings. In this paper, we study the HAMILTONIAN GRADIENT DESCENT (HGD) algorithm, and we show that HGD exhibits a linear convergence rate in a variety of more general settings, including convex-concave settings that are "sufficiently bilinear." We also prove similar convergence rates for the Consensus Optimization (CO) algorithm of [MNG17] for some parameter settings of CO.
REF recently showed the linear convergence of a second-order iterative algorithm, called Hamiltonian gradient descent (HGD), for a subclass of "sufficiently bilinear" functions.
Last-iterate convergence rates for min-max optimization
ArXiv
ArXiv
Mathematics Computer Science