Revisiting Inexact Fixed-Point Iterations for Min-Max Problems: Stochasticity and Structured Nonconvexity
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...
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Published in | arXiv.org |
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Main Authors | , , |
Format | Paper |
Language | English |
Published |
Ithaca
Cornell University Library, arXiv.org
12.08.2024
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Subjects | |
Online Access | Get full text |
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Summary: | 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. |
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ISSN: | 2331-8422 |