A Banach-Space Theory of Markovian Halpern Iteration for Non-Expansive Maps

2026-08-16

AI Summary

This study addresses the excessive sample complexity in stochastic approximation of fixed points for non-expansive operators under Markovian trajectories by proposing a variance-reduced PAGE-Halpern method. Leveraging Poisson equation analysis, block differences, norm smoothing techniques, and shift-level Halpern bounds, this work extends existing Hilbert space results to general finite-dimensional Banach spaces. The proposed approach achieves an optimal sample complexity of Õ(ε⁻³) with respect to the non-expansive norm and establishes high-probability convergence guarantees with matching precision dependence. These contributions significantly enhance both the theoretical efficiency and reliability of fixed-point iterations in Markovian noise environments, providing a rigorous foundation for stochastic approximation under dependent data structures.

Abstract

We study stochastic approximation of fixed points of a non-expansive operator when the oracle samples originate from a continuing Markovian trajectory. A direct block-minibatch implementation of Halpern iteration attains an expected last-iterate residual of order $O(\log N/N)$, but accrues a substantive complexity of $\tilde O(ε^{-5})$ Markovian samples. We therefore introduce a variance-reduced Markovian PAGE-Halpern method whose refresh and same-state difference blocks are analyzed through the Poisson equation. In Hilbert spaces, the cocoercivity of $I-T$ results in an $O(ε^{-3})$ sample complexity. Our main result extends this construction to a general finite-dimensional Banach space. A displacement-level Halpern bound replaces the Hilbert-space potential and yields $\tilde O(ε^{-3})$ sample complexity in the original non-expansiveness norm. We also establish a high-probability guarantee with the same leading accuracy dependence by measuring the estimator in an auxiliary smooth norm. Non-smooth sup and block-sup geometries are covered through norm smoothing.

Problem

• Non-expansive maps
• Markovian Halpern iteration
• Fixed point approximation
• Sample complexity
• Banach space

Innovation

• Variance-Reduced PAGE-Halpern
• Banach Space Theory
• Markovian Sampling
• Poisson Equation Analysis
• Norm Smoothing