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

• URL: http://arxiv.org/abs/2402.05071v2
• Date: Mon, 12 Aug 2024 23:43:59 GMT
• Title: Revisiting Inexact Fixed-Point Iterations for Min-Max Problems: Stochasticity and Structured Nonconvexity
• Authors: Ahmet Alacaoglu, Donghwan Kim, Stephen J. Wright,
• Abstract summary: We show that $1L 1$ can be used to improve some state-of-the-art problems even for a multilevel Carlo iteration. We provide an analysis for inexact Halperness estimators for $1L 1$ when the only hold with respect to a solution is a new $1L 1$ theory.
• Score: 18.427215139020632
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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\u{\i}-Mann iteration with a multilevel Monte Carlo estimator when the assumptions only hold with respect to a solution.

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