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 ρ-cohypomonotonicity or admitting a solution to the ρ-weakly Minty Variational Inequality (MVI), where larger values of the parameter ρ>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 ρ no larger than frac1L, but existing results in the literature have stagnated at the tighter requirement ρ < frac12L. With a simple argument, we obtain optimal or best-known complexity guarantees with cohypomonotonicity or weak MVI conditions for ρ < frac1L. First main insight for the improvements in the convergence analyses is to harness the recently proposed textitconic 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'skiui-Mann iteration with a multilevel Monte Carlo estimator when the assumptions only hold with respect to a solution.