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Infinite-Dimensional Operator/Block Kaczmarz Algorithms: Regret Bounds and λ-Effectiveness
Journal article   Peer reviewed

Infinite-Dimensional Operator/Block Kaczmarz Algorithms: Regret Bounds and λ-Effectiveness

Halyun Jeong, Palle E.T. Jorgensen, Hyun Kyoung Kwon and Myung Sin Song
Analysis and applications
02/06/2026
DOI: 10.1142/S021953052650034X

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Abstract

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.
 Regret bounds Kaczmarz algorithms projections bounded linear operators Hardy space inner functions machine learning signal processing

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