通过固定点方程进行里曼尼暗示差异化,以实现里曼尼双级优化
IEEE transactions on neural networks and learning systems
|November 14, 2025
概括
本研究介绍了一种统一的里曼隐含差异化方法,用于两级优化问题. 它简化了复杂的导数,使得在里曼的元优化和元学习任务中能够得到更广泛的应用.
科学领域:
- 优化理论 优化理论
- 机器学习 机器学习
- 不同几何学微分几何学
背景情况:
- 里曼的元优化 (RMO) 和元学习通常是双层优化问题.
- 隐式分化通过脱外部梯度来有效地解决RMO,但需要专家推导新任务.
- 将隐式差异化扩展到多种不同的里曼两级优化任务是具有挑战性的,因为每个案例都需要推导.
研究的目的:
- 提出一个统一的里曼尼隐性差异化方法,以便在各种里曼尼双级优化任务中灵活应用.
- 为了减少通常在新的优化场景中导出梯度所需的专家参与.
- 为解决里曼两级优化问题提供一个一般的框架.
主要方法:
- 制定内部级优化作为固定点方程的根查找过程,以实现统一的任务表示.
- 通过微分固定点方程来导出外部梯度的统一表达式,从而避免特定任务的导出.
- 进行趋同和近似误差分析,以验证该方法的理论保证.
主要成果:
- 介绍了一种新的里曼隐式微分法,用于外部梯度的统一表达式.
- 该方法在减少专家参与的情况下,可以灵活地应用于各种里曼优化任务.
- 收和近似误差分析证实了该方法的有效性.
结论:
- 提议的里曼隐式微分法提供了一种统一而灵活的方法来解决在里曼的多元体上的双层优化问题.
- 这种方法显著降低了将隐式分化应用于新的和多样化的里曼优化任务的障碍.
- 实验验证证证实了该方法在现场的有效性和广泛适用性.
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