On the Rate of Convergence of Iterative Methods for Nonexpansive Mappings in CAT(0) Spaces and Hyperbolic Optimization

Abstract: The Krasnosel'ski\u{\i} Mann and Halpern iterations are classical schemes for approximating fixed points of nonexpansive mappings in Banach spaces, and have been widely studied in more general frameworks such as $CAT(\kappa)$ and, more generally, geodesic spaces. Convergence results and convergence rate estimates in these nonlinear settings are already well established. The contribution of this paper is to extend to complete $CAT(0)$ spaces the proof techniques originally developed in the linear setting of Banach and Hilbert spaces, thereby recovering the same asymptotic regularity bounds and to introduce a novel optimizer for Hyperbolic Deep learning based on Halpern Iteration similarly to HalpernSGD \cite{foglia2024halpernsgd,colao2025optimizer} in Euclidean setting.