Minimax-optimal Halpern iterations for Lipschitz maps

TL;DR

This paper derives tight, minimax-optimal bounds for Halpern fixed-point iterations applied to Lipschitz maps, revealing optimal schemes and adaptive methods that outperform existing approaches in various settings.

Contribution

It introduces the first minimax-optimal Halpern iteration scheme for Lipschitz maps, including adaptive variants and bounds for unbounded domains, advancing fixed-point iteration theory.

Findings

Derived tight, non-asymptotic residual bounds for Lipschitz maps.

Identified minimax-optimal Halpern iteration schemes for contractions and expansive maps.

Designed adaptive algorithms with residuals smaller than the minimax bounds.

Abstract

This paper investigates the minimax-optimality of Halpern fixed-point iterations for Lipschitz maps in general normed spaces. Starting from an a priori bound on the orbit of iterates, we derive non-asymptotic estimates for the fixed-point residuals. These bounds are tight, meaning that they are attained by a suitable Lipschitz map and an associated Halpern sequence. By minimizing these tight bounds we identify the minimax-optimal Halpern scheme. For contractions, the optimal iteration exhibits a transition from an initial Halpern phase to the classical Banach-Picard iteration and, as the Lipschitz constant approaches one, we recover the known convergence rate for nonexpansive maps. For expansive maps, the algorithm is purely Halpern with no Banach-Picard phase; moreover, on bounded domains, the residual estimates converge to the minimal displacement bound. Inspired by the minimax-optimal iteration, we design an adaptive scheme whose residuals are uniformly smaller than the minimax-optimal bounds, and can be significantly sharper in practice. Finally, we extend the analysis by introducing alternative bounds based on the distance to a fixed point, which allow us to handle mappings on unbounded domains; including the case of affine maps for which we also identify the minimax-optimal iteration.