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arXiv · 2608.15966

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

Ege C. Kaya·Arda Fazla·M. Berk Sahin·Abolfazl Hashemi
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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.

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Ege C. Kaya, Arda Fazla, M. Berk Sahin, Abolfazl Hashemi. 2026-08-16. A Banach-Space Theory of Markovian Halpern Iteration for Non-Expansive Maps. https://arxiv.org/abs/2608.15966

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