Non-asymptotic global convergence rates of BFGS with exact line search

• 成果类型: Article; Early Access
• 署名作者: Jin, Qiujiang; Jiang, Ruichen; Mokhtari, Aryan
• 署名单位: University of Texas System; University of Texas Austin
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-025-02256-7
• 发表日期: 2025
• 关键词: quasi-newton methods; superlinear convergence; extragradient; algorithm

摘要

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.