The Kaczmarz method and its variants, which are types of stochastic gradient descent (SGD) methods, have been extensively studied for their simplicity and efficiency in solving linear systems. Random reshuffling (RR), also known as SGD without replacement, is typically faster in practice than traditional SGD method. Although some convergence analysis results for RR apply to the reshuffling Kaczmarz method, they do not comprehensively characterize its convergence. In this paper, we present a new convergence analysis of the reshuffling Kaczmarz method and demonstrate that it can converge linearly to the unique least-norm solution of the linear system. Furthermore, the convergence upper bound is tight and does not depend on the dimension of the coefficient matrix.

翻译：Kaczmarz方法及其变体作为随机梯度下降（SGD）方法的一类，因其求解线性系统的简洁性与高效性而被广泛研究。随机重排（RR），亦称无放回随机梯度下降，在实践中通常比传统SGD方法收敛更快。尽管已有一些针对RR的收敛性分析结果适用于重排Kaczmarz方法，但这些分析未能全面刻画其收敛特性。本文提出了一种针对重排Kaczmarz方法的新收敛性分析，证明该方法能以线性收敛速度逼近线性系统的唯一最小范数解。此外，所得收敛上界是紧致的，且不依赖于系数矩阵的维度。