Last-iterate convergence rates for min-max optimization [article]

Jacob Abernethy, Kevin A. Lai, Andre Wibisono
2019 arXiv   pre-print
While classic work in convex-concave min-max optimization relies on average-iterate convergence results, the emergence of nonconvex applications such as training Generative Adversarial Networks has led to renewed interest in last-iterate convergence guarantees. Proving last-iterate convergence is challenging because many natural algorithms, such as Simultaneous Gradient Descent/Ascent, provably diverge or cycle even in simple convex-concave min-max settings, and previous work on global
more » ... ate convergence rates has been limited to the bilinear and convex-strongly concave settings. In this work, we show that the Hamiltonian Gradient Descent (HGD) algorithm achieves linear convergence in a variety of more general settings, including convex-concave problems that satisfy a "sufficiently bilinear" condition. We also prove similar convergence rates for the Consensus Optimization (CO) algorithm of [MNG17] for some parameter settings of CO.
arXiv:1906.02027v3 fatcat:sfbwjzoutzdcbgum5obh6iuj2m