Revisiting Inexact Fixed-Point Iterations for Min-Max Problems: Stochasticity and Structured Nonconvexity

• Published in: arXiv.org
• Main Authors: Alacaoglu, Ahmet, Kim, Donghwan, Wright, Stephen J
• Format: Paper
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 12.08.2024
• Subjects: Algorithms; Complexity; First order algorithms
• ISSN: 2331-8422

We focus on constrained, \(L\)-smooth, potentially stochastic and nonconvex-nonconcave min-max problems either satisfying \(\rho\)-cohypomonotonicity or admitting a solution to the \(\rho\)-weakly Minty Variational Inequality (MVI), where larger values of the parameter \(\rho>0\) correspond to a greater degree of nonconvexity. These problem classes include examples in two player reinforcement learning, interaction dominant min-max problems, and certain synthetic test problems on which classical min-max algorithms fail. It has been conjectured that first-order methods can tolerate a value of \(\rho\) no larger than \(\frac{1}{L}\), but existing results in the literature have stagnated at the tighter requirement \(\rho < \frac{1}{2L}\). With a simple argument, we obtain optimal or best-known complexity guarantees with cohypomonotonicity or weak MVI conditions for \(\rho < \frac{1}{L}\). First main insight for the improvements in the convergence analyses is to harness the recently proposed \(\textit{conic nonexpansiveness}\) property of operators. Second, we provide a refined analysis for inexact Halpern iteration that relaxes the required inexactness level to improve some state-of-the-art complexity results even for constrained stochastic convex-concave min-max problems. Third, we analyze a stochastic inexact Krasnosel'ski\uı-Mann iteration with a multilevel Monte Carlo estimator when the assumptions only hold with respect to a solution.