# New Results on Superlinear Convergence of Classical Quasi-Newton Methods

arXiv: https://arxiv.org/abs/2004.14866 · HTML: https://mlsanity.com/paper/2004.14866/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Apr 2020.

## Built on by (most cited first, 15 of 15)

- [Second-order optimization with lazy Hessians](https://mlsanity.com/paper/2212.00781/) (2022)
- [Hessian Averaging in Stochastic Newton Methods Achieves Superlinear Convergence](https://mlsanity.com/paper/2204.09266/) (2022)
- [Online Learning Guided Curvature Approximation: A Quasi-Newton Method with Global Non-Asymptotic Superlinear Convergence](https://mlsanity.com/paper/2302.08580/) (2023)
- [Quasi-Newton Methods for Saddle Point Problems and Beyond](https://mlsanity.com/paper/2111.02708/) (2021)
- [Optimization for Classical Machine Learning Problems on the GPU](https://mlsanity.com/paper/2203.16340/) (2022)
- [Accelerated Quasi-Newton Proximal Extragradient: Faster Rate for Smooth Convex Optimization](https://mlsanity.com/paper/2306.02212/) (2023)
- [Approximate Newton policy gradient algorithms](https://mlsanity.com/paper/2110.02398/) (2021)
- [Multiple Greedy Quasi-Newton Methods for Saddle Point Problems](https://mlsanity.com/paper/2408.00241/) (2024)
- [Gradient Methods with Online Scaling Part I. Theoretical Foundations](https://mlsanity.com/paper/2505.23081/) (2025)
- [Gradient-Normalized Smoothness for Optimization with Approximate Hessians](https://mlsanity.com/paper/2506.13710/) (2025)
- [Exploiting Local Convergence of Quasi-Newton Methods Globally: Adaptive Sample Size Approach](https://mlsanity.com/paper/2106.05445/) (2021)
- [qNBO: quasi-Newton Meets Bilevel Optimization](https://mlsanity.com/paper/2502.01076/) (2025)
- [Sharpened Lazy Incremental Quasi-Newton Method](https://mlsanity.com/paper/2305.17283/) (2023)
- [Simple Stepsize for Quasi-Newton Methods with Global Convergence Guarantees](https://mlsanity.com/paper/2508.19712/) (2025)
- [Primal Acceleration of Newton's Method](https://mlsanity.com/paper/2608.21359/) (2026)

## Abstract (arXiv)

We present a new theoretical analysis of local superlinear convergence of classical quasi-Newton methods from the convex Broyden class. As a result, we obtain a significant improvement in the currently known estimates of the convergence rates for these methods. In particular, we show that the corresponding rate of the Broyden-Fletcher-Goldfarb-Shanno method depends only on the product of the dimensionality of the problem and the logarithm of its condition number.