Non-asymptotic Global Convergence Rates of BFGS with Exact Line Search

TL;DR

This paper establishes non-asymptotic convergence rates for BFGS with exact line search in strongly convex optimization, revealing detailed phase behavior and trade-offs influenced by initial Hessian choices.

Contribution

It provides the first detailed non-asymptotic analysis of BFGS's convergence, including phase-specific rates and effects of initial Hessian modifications.

Findings

Three-phase convergence process identified

Linear and superlinear rates depend on iteration stage

Initial Hessian choice affects convergence trade-offs

Abstract

In this paper, we explore the non-asymptotic global convergence rates of the Broyden-Fletcher-Goldfarb-Shanno (BFGS) method implemented with exact line search. Notably, due to Dixon's equivalence result, our findings are also applicable to other quasi-Newton methods in the convex Broyden class employing exact line search, such as the Davidon-Fletcher-Powell (DFP) method. Specifically, we focus on problems where the objective function is strongly convex with Lipschitz continuous gradient and Hessian. Our results hold for any initial point and any symmetric positive definite initial Hessian approximation matrix. The analysis unveils a detailed three-phase convergence process, characterized by distinct linear and superlinear rates, contingent on the iteration progress. Additionally, our theoretical findings demonstrate the trade-offs between linear and superlinear convergence rates for BFGS when we modify the initial Hessian approximation matrix, a phenomenon further corroborated by our numerical experiments.