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

What the math gives to ML

The paper's transferable asset is a stochastic fixed-point solver for non-expansive maps when samples arrive along a correlated Markov trajectory rather than as independent minibatches. Halpern's anchored averaging avoids the lack of contraction drift, while PAGE-style refresh/difference estimators reduce the number of expensive Markovian oracle samples; the Poisson-equation analysis is specifically designed to control temporal correlation. The most credible neural-network use is a deep-equilibrium or recurrent module whose state update is constrained to be non-expansive, replacing ordinary noisy fixed-point iteration with an anchored, variance-reduced solver.