Homogeneous second-order descent framework: a fast alternative to Newton-type methods

• 成果类型: Article; Early Access
• 署名作者: He, Chang; Jiang, Yuntian; Zhang, Chuwen; Ge, Dongdong; Jiang, Bo; Ye, Yinyu
• 署名单位: Shanghai University of Finance & Economics; Shanghai Jiao Tong University; Stanford University
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-025-02230-3
• 发表日期: 2025
• 关键词: cubic regularization; EIGENVALUE; complexity; algorithm

摘要

This paper proposes a homogeneous second-order descent framework (HSODF) for nonconvex and convex optimization based on the generalized homogeneous model (GHM). In comparison to the Newton steps, the GHM can be solved by extremal symmetric eigenvalue procedures and thus grant an advantage in ill-conditioned problems. Moreover, GHM extends the ordinary homogeneous model (Zhang et al. A homogenous second-order descent method for nonconvex optimization, 2022. arXiv:2211.08212 [math]) to allow adaptiveness in the construction of the aggregated matrix. Consequently, HSODF is able to recover some well-known second-order methods, such as trust-region methods and gradient regularized methods, while maintaining comparable iteration complexity bounds. We also study two specific realizations of HSODF. One is adaptive HSODM, which has a parameter-free O(& varepsilon;-3/2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(\epsilon <^>{-3/2})$$\end{document} global complexity bound for nonconvex second-order Lipschitz continuous objective functions. The other is homotopy HSODM, which is proven to have a global linear rate of convergence without strong convexity. The efficiency of our approach to high-dimensional and ill-conditioned problems is justified by some preliminary numerical results.