Revisiting Inexact Fixed-Point Iterations for Min-Max Problems: Stochasticity and Structured Nonconvexity

TL;DR

This paper extends first-order methods for nonconvex-nonconcave min-max problems by leveraging conic nonexpansiveness to enlarge the admissible nonconvexity range to $\rho<1/L$ under $\rho$-cohypomonotonicity or $\rho$-weak MVI. It develops a double-loop framework combining Halpern (or KM) iterations with inexact resolvent computations, where the resolvent of $\eta(F+G)$ is approximated via inner FBF steps and, in stochastic settings, via unbiased or MLMC estimators. The deterministic results achieve near-optimal $\tilde{O}(\varepsilon^{-1})$ (cohypomonotone) or $\tilde{O}(\varepsilon^{-2})$ (weak MVI) rates in suitable measures, while the stochastic extensions yield $\tilde{O}(\varepsilon^{-4})$ complexities, aided by MLMC in the weak MVI case. The modular approach accommodates stochastic access to $F$ and provides a pathway to practical algorithms for structured nonconvex-nonconcave min-max problems, including RL and interaction-dominant scenarios, albeit with log overhead from the double-loop design. Open questions include single-loop alternatives to remove logs and broader $\rho$-ranges beyond the current structured settings.

Abstract

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'skiĭ-Mann iteration with a multilevel Monte Carlo estimator when the assumptions only hold with respect to a solution.

Theorems & Definitions (65)

• Example 1.1
• Example 1.2
• Theorem 2.1
• Corollary 2.2
• Remark 2.3
• Remark 2.4
• Lemma 2.5
• Remark 2.6
• Theorem 2.7
• proof : Proof sketch of \ref{['th: cohypo_det']}