# Stabilized Barzilai-Borwein method

arXiv: https://arxiv.org/abs/1907.06409 · HTML: https://mlsanity.com/paper/1907.06409/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 15 Jul 2019.

## Built on by (most cited first, 10 of 10)

- [FedExP: Speeding Up Federated Averaging via Extrapolation](https://mlsanity.com/paper/2301.09604/) (2023)
- [Adaptive proximal algorithms for convex optimization under local Lipschitz continuity of the gradient](https://mlsanity.com/paper/2301.04431/) (2023)
- [Non-Uniform Smoothness for Gradient Descent](https://mlsanity.com/paper/2311.08615/) (2023)
- [Nesterov Finds GRAAL: Optimal and Adaptive Gradient Method for Convex Optimization](https://mlsanity.com/paper/2507.09823/) (2025)
- [First-ish Order Methods: Hessian-aware Scalings of Gradient Descent](https://mlsanity.com/paper/2502.03701/) (2025)
- [An Adaptive Incremental Gradient Method With Support for Non-Euclidean Norms](https://mlsanity.com/paper/2205.02273/) (2022)
- [Safeguarding adaptive methods: global convergence of Barzilai-Borwein and other stepsize choices](https://mlsanity.com/paper/2404.09617/) (2024)
- [Recent Advances in Non-convex Smoothness Conditions and Applicability to Deep Linear Neural Networks](https://mlsanity.com/paper/2409.13672/) (2024)
- [Barzilai-Borwein Fails Superlinear Convergence on an Open Set of Quadratics for Every Dimension n≥ 4](https://mlsanity.com/paper/2607.21579/) (2026)
- [Adaptive Bregman Proximal Stochastic Gradient with a Stabilized Barzilai--Borwein Step Size](https://mlsanity.com/paper/2608.12009/) (2026)

## Abstract (arXiv)

The Barzilai-Borwein (BB) method is a popular and efficient tool for solving large-scale unconstrained optimization problems. Its search direction is the same as for the steepest descent (Cauchy) method, but its stepsize rule is different. Owing to this, it converges much faster than the Cauchy method. A feature of the BB method is that it may generate too long steps, which throw the iterates too far away from the solution. Moreover, it may not converge, even when the objective function is strongly convex. In this paper, a stabilization technique is introduced. It consists in bounding the distance between each pair of successive iterates, which often allows for decreasing the number of BB iterations. When the BB method does not converge, our simple modification of this method makes it convergent. Under suitable assumptions, we prove its global convergence, despite the fact that no line search is involved, and only gradient values are used. Since the number of stabilization steps is proved to be finite, the stabilized version inherits the fast local convergence of the BB method. The presented results of extensive numerical experiments show that our stabilization technique often allows the BB method to solve problems in a fewer iterations, or even to solve problems where the latter fails.