DRSOM: A Dimension Reduced Second-Order Method

• URL: http://arxiv.org/abs/2208.00208v3
• Date: Sun, 2 Jul 2023 16:11:27 GMT
• Title: DRSOM: A Dimension Reduced Second-Order Method
• Authors: Chuwen Zhang, Dongdong Ge, Chang He, Bo Jiang, Yuntian Jiang, Yinyu Ye
• Abstract summary: Under a trust-like framework, our method preserves the convergence of the second-order method while using only information in a few directions. Theoretically, we show that the method has a local convergence and a global convergence rate of $O(epsilon-3/2)$ to satisfy the first-order and second-order conditions.
• Score: 13.778619250890406
• License: http://creativecommons.org/licenses/by-nc-nd/4.0/
• Abstract: In this paper, we propose a Dimension-Reduced Second-Order Method (DRSOM) for convex and nonconvex (unconstrained) optimization. Under a trust-region-like framework, our method preserves the convergence of the second-order method while using only curvature information in a few directions. Consequently, the computational overhead of our method remains comparable to the first-order such as the gradient descent method. Theoretically, we show that the method has a local quadratic convergence and a global convergence rate of $O(\epsilon^{-3/2})$ to satisfy the first-order and second-order conditions if the subspace satisfies a commonly adopted approximated Hessian assumption. We further show that this assumption can be removed if we perform a corrector step using a Krylov-like method periodically at the end stage of the algorithm. The applicability and performance of DRSOM are exhibited by various computational experiments, including $L_2 - L_p$ minimization, CUTEst problems, and sensor network localization.

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