张量方法用于找到凸函数的近似静止点
G N Grapiglia1, Yurii Nesterov2
1Departamento de Matemática, Universidade Federal do Paraná, Curitiba, Brazil.
Optimization methods & software
|February 27, 2025
概括
本研究介绍了探子方法,用于在凸函数中找到近似的静止点. 它为加速和非加速方案建立了新的代复杂度界限,提高了优化效率.
科学领域:
- 优化理论 优化理论
- 数字分析 数字分析
- 凸的分析 凸的分析
背景情况:
- 找到近似的静止点对于解决优化问题至关重要.
- 有 p-倍微分和 nu-Hölder 连续 pth 导数的凸函数在机器学习和应用数学中很常见.
- 现有的方法可能缺乏高阶衍生品的效率.
研究的目的:
- 开发和分析张量方法,以寻找凸函数的epsilon-近似静止点.
- 为这些方法建立改进的代复杂度极限,考虑加速和非加速方案.
- 调查知道或不知道霍尔德参数数的影响.
主要方法:
- 开发非加速张量方法.
- 发展加速张量方案. 加速张量方案.
- 基于函数属性 (p-倍可微分性,nu-Hölder连续性) 的代复杂度边界的分析.
主要成果:
- 非加速方案在最多的O(1/epsilon^(2/p)) 代中实现了低于epsilon的梯度规范减小.
- 加速张量方案在已知nu时,可以达到O(1/epsilon^(1/p)) 的更好的复杂度极限.
- 一个通用加速方案在nu未知时达到O(1/epsilon^(2/(2p-1))) 复杂度,并且确定了O(1/epsilon^(1/p)) 的下界.
结论:
- 拟议的张量法为找到凸函数的近似静止点提供了有效的解决方案.
- 建立的复杂度极限显示出显著的改进,特别是在加速方案中.
- 这项研究为这些优化算法的性能提供了理论上的保证.
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