Revisiting Inexact Fixed-Point Iterations for Min-Max Problems: Stochasticity and Structured Nonconvexity
<|reference_start|>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 $\rho$-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.<|reference_end|>
@article{alacaoglu2024revisiting,
    title={Revisiting Inexact Fixed-Point Iterations for Min-Max Problems:
  Stochasticity and Structured Nonconvexity},
    author={Ahmet Alacaoglu, Donghwan Kim, Stephen J. Wright},
    journal={Proceedings of the International Conference on Machine Learning
  (ICML) 2024},
    year={2024},
    archivePrefix={arXiv},
    eprint={2402.05071},
    primaryClass={math.OC cs.LG stat.ML}
}