# Limited-Memory BFGS with Displacement Aggregation

arXiv: https://arxiv.org/abs/1903.03471 · HTML: https://mlsanity.com/paper/1903.03471/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 8 Mar 2019.

## Built on by (most cited first, 4 of 4)

- [Federated Learning Aggregation: New Robust Algorithms with Guarantees](https://mlsanity.com/paper/2205.10864/) (2022)
- [Limited-Memory Greedy Quasi-Newton Method with Non-asymptotic Superlinear Convergence Rate](https://mlsanity.com/paper/2306.15444/) (2023)
- [Tackling Computational Heterogeneity in FL: A Few Theoretical Insights](https://mlsanity.com/paper/2307.06283/) (2023)
- [Closing the Curvature Gap: Full Transformer Hessians](https://mlsanity.com/paper/2510.16927/) (2025)

## Abstract (arXiv)

A displacement aggregation strategy is proposed for the curvature pairs stored in a limited-memory BFGS (a.k.a. L-BFGS) method such that the resulting (inverse) Hessian approximations are equal to those that would be derived from a full-memory BFGS method. This means that, if a sufficiently large number of pairs are stored, then an optimization algorithm employing the limited-memory method can achieve the same theoretical convergence properties as when full-memory (inverse) Hessian approximations are stored and employed, such as a local superlinear rate of convergence under assumptions that are common for attaining such guarantees. To the best of our knowledge, this is the first work in which a local superlinear convergence rate guarantee is offered by a quasi-Newton scheme that does not either store all curvature pairs throughout the entire run of the optimization algorithm or store an explicit (inverse) Hessian approximation. Numerical results are presented to show that displacement aggregation within an adaptive L-BFGS scheme can lead to better performance than standard L-BFGS.