# Limited-Memory BFGS with Displacement Aggregation

## TL;DR

This paper introduces a displacement aggregation strategy for limited-memory BFGS that achieves full-memory convergence properties with fewer stored pairs, improving optimization efficiency and performance.

## Contribution

It proposes a novel displacement aggregation method that enables L-BFGS to attain superlinear convergence without storing all curvature pairs.

## Key findings

- Displacement aggregation achieves full-memory BFGS convergence properties.
- Numerical results show improved performance over standard L-BFGS.
- The method guarantees local superlinear convergence under common assumptions.

## Abstract

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.