# Analysis of the BFGS Method with Errors

arXiv: https://arxiv.org/abs/1901.09063 · HTML: https://mlsanity.com/paper/1901.09063/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 25 Jan 2019.

## Built on by (most cited first, 5 of 5)

- [Adaptive Sampling Quasi-Newton Methods for Zeroth-Order Stochastic Optimization](https://mlsanity.com/paper/2109.12213/) (2021)
- [Quasi-Newton Quasi-Monte Carlo for variational Bayes](https://mlsanity.com/paper/2104.02865/) (2021)
- [Diffusion differentiable resampling](https://mlsanity.com/paper/2512.10401/) (2025)
- [An Efficient Nonlinear Acceleration method that Exploits Symmetry of the Hessian](https://mlsanity.com/paper/2210.12573/) (2022)
- [Curvature-aware dynamic precision approach for physics-informed neural networks](https://mlsanity.com/paper/2606.04736/) (2026)

## Abstract (arXiv)

The classical convergence analysis of quasi-Newton methods assumes that the function and gradients employed at each iteration are exact. In this paper, we consider the case when there are (bounded) errors in both computations and establish conditions under which a slight modification of the BFGS algorithm with an Armijo-Wolfe line search converges to a neighborhood of the solution that is determined by the size of the errors. One of our results is an extension of the analysis presented in Byrd, R. H., & Nocedal, J. (1989), which establishes that, for strongly convex functions, a fraction of the BFGS iterates are good iterates. We present numerical results illustrating the performance of the new BFGS method in the presence of noise.