Extending the Reach of First-Order Algorithms for Nonconvex Min-Max Problems with Cohypomonotonicity

We focus on constrained, $L$-smooth, 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 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 $\rho<\frac{1}{L}$. The algorithms we analyze are inexact variants of Halpern and Krasnosel'ski\u{\i}-Mann (KM) iterations. We also provide algorithms and complexity guarantees in the stochastic case with the same range on $\rho$. Our main insight for the improvements in the convergence analyses is to harness the recently proposed"conic nonexpansiveness"property of operators. As byproducts, we provide a refined analysis for inexact Halpern iteration and propose a stochastic KM iteration with a multilevel Monte Carlo estimator.