To conduct a more in-depth investigation of randomized solvers for solving linear systems, we adopt a unified randomized batch-sampling Kaczmarz framework with per-iteration costs as low as cyclic block methods, and develop a general analysis technique to establish its convergence guarantee. With concentration inequalities, we derive new expected linear convergence rate bounds. The analysis applies to any randomized non-extended block Kaczmarz methods with arbitrary static stochastic samplings. In addition, the new rate bounds are scale-invariant, which eliminate the dependence on the magnitude of the data matrix. In most experiments, the new bounds are significantly tighter than existing ones and better reflect the empirical convergence behavior of block methods. Within this new framework, the batch-sampling distribution, as a learnable parameter, provides the possibility for block methods to achieve efficient performance in specific application scenarios, which deserves further investigation.

翻译：为了更深入地研究求解线性方程组的随机求解器，我们采用统一化的随机批量采样Kaczmarz框架，其单次迭代成本与循环分块方法相当，并开发出一种通用分析技术以建立其收敛性保证。利用浓度不等式，我们推导出新的期望线性收敛速率界。该分析适用于任意具有任意静态随机采样的随机非扩展分块Kaczmarz方法。此外，新的速率界具有尺度不变性，消除了对数据矩阵幅度的依赖性。在大多数实验中，新速率界显著优于现有结果，并能更好地反映分块方法的实际收敛行为。在此新框架下，批量采样分布作为可学习参数，为分块方法在特定应用场景中实现高效性能提供了可能性，值得进一步研究。