# Explicit Superlinear Convergence Rates of The SR1 Algorithm

arXiv: https://arxiv.org/abs/2105.07162 · HTML: https://mlsanity.com/paper/2105.07162/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 15 May 2021.

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

- [Quasi-Newton Methods for Saddle Point Problems and Beyond](https://mlsanity.com/paper/2111.02708/) (2021)
- [Multiple Greedy Quasi-Newton Methods for Saddle Point Problems](https://mlsanity.com/paper/2408.00241/) (2024)
- [Incremental Quasi-Newton Methods with Faster Superlinear Convergence Rates](https://mlsanity.com/paper/2402.02359/) (2024)

## Abstract (arXiv)

We study the convergence rate of the famous Symmetric Rank-1 (SR1) algorithm which has wide applications in different scenarios. Although it has been extensively investigated, SR1 still lacks a non-asymptotic superlinear rate compared with other quasi-Newton methods such as DFP and BFGS. In this paper we address this problem. Inspired by the recent work on explicit convergence analysis of quasi-Newton methods, we obtain the first explicit non-asymptotic rates of superlinear convergence for the vanilla SR1 methods with correction strategy to achieve the numerical stability. Specifically, the vanilla SR1 with the correction strategy achieves the rates of the form left(frac4nln(ekappa) kright)k/2 for general smooth strongly-convex functions where k is the iteration counter, kappa is the condition number of the objective function and n is the dimension of the problem. For the quadratic function, the vanilla SR1 algorithm can find the optima of the objective function at most n steps.