# Higher-Order Derivatives Do Not Accelerate the Computation of Fixed Points

arXiv: https://arxiv.org/abs/2607.05947 · HTML: https://mlsanity.com/paper/2607.05947/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Jul 2026.

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

- [Anchored Extra-Proximal Methods: Optimal Higher-Order Methods for Monotone Inclusion Problems](https://mlsanity.com/paper/2609.30212/) (2026)

## Abstract (arXiv)

The Picard iteration converges to the unique fixed point of a q-contractive operator at a linear rate qN, and a lower bound with an affine construction shows that no deterministic method querying only operator values can do better. But what about higher-order methods that query derivatives? A single Jacobian evaluation reveals an affine map entirely, so the affine construction says nothing about higher-order methods. In this work, we show that finite-order derivative information still does not accelerate the worst-case complexity for smooth contractive fixed-point computation. This contrasts with higher-order smooth minimization, where higher-order derivatives do improve worst-case rates for convex and non-convex minimization.