Sharpened Quasi-Newton Methods: Faster Superlinear Rate and Larger Local Convergence Neighborhood

TL;DR

This paper introduces a new BFGS variant that combines the advantages of BFGS and Greedy-BFGS, achieving faster superlinear convergence and larger local convergence neighborhoods, supported by theoretical analysis and numerical experiments.

Contribution

A novel BFGS method that simultaneously approximates the Newton direction and Hessian, outperforming existing methods in convergence speed and neighborhood size.

Findings

Outperforms BFGS and Greedy-BFGS in convergence rate.

Achieves quadratic convergence with fewer steps than Greedy-BFGS.

Numerical experiments confirm theoretical advantages.

Abstract

Non-asymptotic analysis of quasi-Newton methods have gained traction recently. In particular, several works have established a non-asymptotic superlinear rate of $\mathcal{O}((1/\sqrt{t})^t)$ for the (classic) BFGS method by exploiting the fact that its error of Newton direction approximation approaches zero. Moreover, a greedy variant of BFGS was recently proposed which accelerates its convergence by directly approximating the Hessian, instead of the Newton direction, and achieves a fast local quadratic convergence rate. Alas, the local quadratic convergence of Greedy-BFGS requires way more updates compared to the number of iterations that BFGS requires for a local superlinear rate. This is due to the fact that in Greedy-BFGS the Hessian is directly approximated and the Newton direction approximation may not be as accurate as the one for BFGS. In this paper, we close this gap and present a novel BFGS method that has the best of both worlds in that it leverages the approximation ideas of both BFGS and Greedy-BFGS to properly approximate the Newton direction and the Hessian matrix simultaneously. Our theoretical results show that our method out-performs both BFGS and Greedy-BFGS in terms of convergence rate, while it reaches its quadratic convergence rate with fewer steps compared to Greedy-BFGS. Numerical experiments on various datasets also confirm our theoretical findings.