# A family of spectral gradient methods for optimization

arXiv: https://arxiv.org/abs/1812.02974 · HTML: https://mlsanity.com/paper/1812.02974/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Dec 2018.

## Built on by (most cited first, 3 of 3)

- [On a Faster R-Linear Convergence Rate of the Barzilai-Borwein Method](https://mlsanity.com/paper/2101.00205/) (2021)
- [First-ish Order Methods: Hessian-aware Scalings of Gradient Descent](https://mlsanity.com/paper/2502.03701/) (2025)
- [Barzilai-Borwein Fails Superlinear Convergence on an Open Set of Quadratics for Every Dimension n≥ 4](https://mlsanity.com/paper/2607.21579/) (2026)

## Abstract (arXiv)

We propose a family of spectral gradient methods, whose stepsize is determined by a convex combination of the long Barzilai-Borwein (BB) stepsize and the short BB stepsize. Each member of the family is shown to share certain quasi-Newton property in the sense of least squares. The family also includes some other gradient methods as its special cases. We prove that the family of methods is R-superlinearly convergent for two-dimensional strictly convex quadratics. Moreover, the family is R-linearly convergent in the any-dimensional case. Numerical results of the family with different settings are presented, which demonstrate that the proposed family is promising.