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

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 $O(\log N\mathrm{/}N)$, but accrues a substantive complexity of $\tilde{O}({\epsilon }^{-5})$ 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 $O({\epsilon }^{-3})$ 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 $\tilde{O}({\epsilon }^{-3})$ 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.