# Initialization-dependent BFGS trial rates for tilted absolute values

arXiv: https://arxiv.org/abs/2609.09872 · HTML: https://mlsanity.com/paper/2609.09872/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 Sep 2026.

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

- [Euler: Towards Human-Centered Multi-Agent Mathematical Discovery](https://mlsanity.com/paper/2610.19864/) (2026)

## Abstract (arXiv)

Lewis and Overton conjectured in 2008 that, for BFGS applied to the one-dimensional tilted absolute value with their Armijo--Wolfe line search, every nonterminating execution converges at a trial-normalized rate determined only by the tilt parameter and independent of the initial data. We prove that this initialization-independent rate assertion is false. In fact, for every tilt parameter in an explicit open interval, we construct two nonterminating executions with different sharp trial-normalized convergence rates. The proof reduces the line search to a scale-free state consisting of the iterate sign and the zero-crossing stepsize, and verifies two periodic state cycles by elementary rational inequalities. This yields an explicit family of counterexamples and shows that the asymptotic trial-normalized behavior of nonsmooth BFGS can depend essentially on the initialization even for this canonical one-dimensional model.