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

The Krasnosel'skiĭ--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 $CAT(κ)$ 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 twofold: first, we extend to complete $CAT(0)$ spaces proof techniques originally developed in the linear setting of Banach and Hilbert spaces, thereby recovering the same asymptotic regularity bounds; second, we introduce a Halpern--type optimizer for hyperbolic optimization as a nonlinear counterpart of the Euclidean HalpernSGD scheme.