# Minimax-optimal Halpern iterations for Lipschitz maps

arXiv: https://arxiv.org/abs/2601.15996 · HTML: https://mlsanity.com/paper/2601.15996/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 22 Jan 2026.

## Built on by (most cited first, 1 of 1)

- [Learning Policy from a Single Trajectory in Average-Reward Markov Decision Process](https://mlsanity.com/paper/2606.16729/) (2026)

## Abstract (arXiv)

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.