To improve the convergence property of the randomized Kaczmarz (RK) method for solving linear systems, Bai and Wu (SIAM J. Sci. Comput., 40(1):A592--A606, 2018) originally introduced a greedy probability criterion for effectively selecting the working row from the coefficient matrix and constructed the greedy randomized Kaczmarz (GRK) method. Due to its simplicity and efficiency, this approach has inspired numerous subsequent works in recent years, such as the capped adaptive sampling rule, the greedy augmented randomized Kaczmarz method, and the greedy randomized coordinate descent method. Since the iterates of the GRK method are actually random variables, existing convergence analyses are all related to the expectation of the error. In this note, we prove that the linear convergence rate of the GRK method is deterministic, i.e. not in the sense of expectation. Moreover, the Polyak's heavy ball momentum technique is incorporated to improve the performance of the GRK method. We propose a refined convergence analysis, compared with the technique used in Loizou and Richt\'{a}rik (Comput. Optim. Appl., 77(3):653--710, 2020), of momentum variants of randomized iterative methods, which shows that the proposed GRK method with momentum (mGRK) also enjoys a deterministic linear convergence. Numerical experiments show that the mGRK method is more efficient than the GRK method.

翻译：为改善求解线性系统的随机Kaczmarz（RK）方法的收敛性，Bai与Wu（SIAM J. Sci. Comput., 40(1):A592--A606, 2018）首次引入了一种贪婪概率准则，用于有效选取系数矩阵中的工作行，进而构建了贪婪随机Kaczmarz（GRK）方法。由于该方法简便高效，近年来启发了一系列后续工作，例如截断自适应采样准则、贪婪增广随机Kaczmarz方法及贪婪随机坐标下降法。由于GRK方法的迭代结果本质上是随机变量，现有收敛性分析均与误差的期望相关。本文证明了GRK方法的线性收敛速率具有确定性，即并非在期望意义下成立。此外，我们引入了Polyak重球动量技术以提升GRK方法的性能。相较于Loizou与Richtárik（Comput. Optim. Appl., 77(3):653--710, 2020）对动量变体随机迭代方法所采用的分析技术，我们提出了更精细的收敛性分析，表明所提出的含动量GRK（mGRK）方法同样具有确定性线性收敛性。数值实验表明，mGRK方法较GRK方法具有更高效率。