论文概要
研究领域: ML 作者: Nikita Doikov 发布时间: 2026-08-21 arXiv: 2608.21359
中文摘要
我们开发了一种新的直接加速牛顿法，用于最小化具有Lipschitz连续Hessian的凸函数。该算法仅使用原始变量，每次迭代仅执行一次线性求解。通过简单的预定参数选择，它在函数残差方面实现了O(1/k³)的全局收敛速率。据我们所知，这是首个达到该速率的二阶方法，且每次迭代仅依赖一次线性系统求解（无需求解辅助非线性正则化子问题，如三次正则化，无需执行非线性参数搜索，也无需使用对偶外梯度修正）。我们的方法可以以无Hessian方式实现，使用不精确的线性系统求解器，同时保持快速的全局速率。我们进一步将构造推广到任意几何情形（通过Bregman散度）和复合优化问题。
原文摘要
We develop a new direct accelerated Newton method for minimizing convex functions with Lipschitz continuous Hessian. The algorithm uses only primal variables and performs just one linear solve per iteration. With a simple predetermined choice of parameters, it achieves the global convergence rate of 
in terms of the functional residual. To the best of our knowledge, this is the first second-order method for this problem class attaining this rate while relying solely on one linear system solve per iteration (without solving auxiliary nonlinear regularized subproblems, such as cubic regularization, performing nonlinear parameter searches, or using dual extragradient corrections). Our method can be implemented in a Hessian-free way, using an inexact linear system solver, while prese…
— 自动采集于 2026-08-25
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