提出无限维空间的投影算法，量化其性能与最优解差距。
Infinite-Dimensional Operator/Block Kaczmarz Algorithms: Regret Bounds and $λ$-Effectiveness
- 基于广义Kaczmarz算法，引入松弛参数分析性能
- 给出显式依赖λ的后悔边界，适用于噪声数据
- 适合研究机器学习算法性能与理论分析者
我们提出一系列基于投影的线性回归算法，聚焦现代机器学习模型及其算法性能。研究了广义Kaczmarz算法中松弛参数的作用，建立了具有明确λ-依赖性的先验后悔边界，用于量化算法性能偏离最优性能的程度。提供了对松弛参数的详细分析。应用包括：Kaczmarz算法框架的显式后悔边界、非正交傅里叶展开，以及在现代机器学习模型（如含噪声数据）中的后悔估计，即噪声Kaczmarz算法的后悔边界。受机器学习实践启发，我们的框架扩展至无限维希尔伯特空间上的有界算子，通过（块）Kaczmarz更新实现，带来新且通用的结果。
原文摘要 · Abstract (English)
We present a variety of projection-based linear regression algorithms with a focus on modern machine-learning models and their algorithmic performance. We study the role of the relaxation parameter in generalized Kaczmarz algorithms and establish a priori regret bounds with explicit $λ$-dependence to quantify how much an algorithm's performance deviates from its optimal performance. A detailed analysis of relaxation parameter is also provided. Applications include: explicit regret bounds for the framework of Kaczmarz algorithm models, non-orthogonal Fourier expansions, and the use of regret estimates in modern machine learning models, including for noisy data, i.e., regret bounds for the noisy Kaczmarz algorithms. Motivated by machine-learning practice, our wider framework treats bounded operators (on infinite-dimensional Hilbert spaces), with updates realized as (block) Kaczmarz algorithms, leading to new and versatile results.
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