重新审视极小极大问题的不精确不动点迭代：随机性与结构化非凸性
Revisiting Inexact Fixed-Point Iterations for Min-Max Problems: Stochasticity and Structured Nonconvexity
AI总结：
本文通过锥非扩张性与精细化分析，将一阶方法在非凸-非凹极小极大问题中可容忍的$\ ho$值从$\ rac{1}{2L}$提升至$\ rac{1}{L}$，并改进随机凸-凹情形下的复杂度结果。
摘要：
我们关注受约束的、$L$-光滑的、可能具有随机性且非凸-非凹的极小极大问题，这些问题要么满足$\ ho$-余次单调性，要么承认$\ ho$-弱Minty变分不等式（MVI）的解，其中参数$\ ho>0$的值越大，对应非凸程度越高。这些问题类别包括双人强化学习、交互主导的极小极大问题以及某些经典极小极大算法失效的合成测试问题。已有猜想认为一阶方法能够容忍的$\ ho$值不超过$\ rac{1}{L}$，但文献中的现有结果停滞在更严格的要求$\ ho < \ rac{1}{2L}$。通过一个简单的论证，我们在$\ ho < \ rac{1}{L}$的条件下，利用余次单调性或弱MVI条件获得了最优或已知最佳的复杂度保证。改进收敛性分析的第一个主要见解是利用最近提出的算子$\ extit{锥非扩张性}$性质。其次，我们提供了对不精确Halpern迭代的精细化分析，放宽了所需的不精确度水平，从而改进了即使对于受约束的随机凸-凹极小极大问题的一些最先进的复杂度结果。第三，当假设仅相对于某个解成立时，我们分析了带有多层蒙特卡洛估计器的随机不精确Krasnosel'ski\u1d4D-Mann迭代。
英文摘要：
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 $\frac{1}{L}$, but existing results in the literature have stagnated at the tighter requirement $ρ< \frac{1}{2L}$. With a simple argument, we obtain optimal or best-known complexity guarantees with cohypomonotonicity or weak MVI conditions for $ρ< \frac{1}{L}$. 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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
