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.

翻译：我们关注受约束的、$L$-光滑的非凸-非凹极小-极大问题，这类问题或满足$\rho$-协单调性，或允许$\rho$-弱Minty变分不等式（MVI）存在解，其中参数$\rho>0$越大对应非凸程度越高。此类问题包括双人强化学习中的例子、交互主导的极小-极大问题，以及经典极小-极大算法失效的特定合成测试问题。已有猜想认为一阶方法能容忍的$\rho$值不超过$\frac{1}{L}$，但现有文献结果停滞于更严格的条件$\rho < \frac{1}{2L}$。通过一个简单论证，我们在$\rho < \frac{1}{L}$条件下，获得了协单调性或弱MVI假设下的最优或已知最佳复杂度保证。所分析的算法是Halpern迭代和Krasnosel'ski\u{\i}-Mann（KM）迭代的不精确变体。我们还在随机情形的相同$\rho$范围内提供了算法与复杂度保证。收敛性分析改进的核心见解是采用算子近期提出的"锥非扩张性"性质。作为副产品，我们给出了不精确Halpern迭代的精细化分析，并提出了一种结合多层蒙特卡洛估计器的随机KM迭代。