We propose a novel randomized framework for the estimation problem of large-scale linear statistical models, namely Sequential Least-Squares Estimators with Fast Randomized Sketching (SLSE-FRS), which integrates Sketch-and-Solve and Iterative-Sketching methods for the first time. By iteratively constructing and solving sketched least-squares (LS) subproblems with increasing sketch sizes to achieve better precisions, SLSE-FRS gradually refines the estimators of the true parameter vector, ultimately producing high-precision estimators. We analyze the convergence properties of SLSE-FRS, and provide its efficient implementation. Numerical experiments show that SLSE-FRS outperforms the state-of-the-art methods, namely the Preconditioned Conjugate Gradient (PCG) method, and the Iterative Double Sketching (IDS) method.

翻译：我们提出了一种新颖的随机化框架用于大规模线性统计模型的估计问题，即快速随机草图化序贯最小二乘估计器（SLSE-FRS），该框架首次整合了“草图-求解”与“迭代草图”方法。通过迭代构建并求解草图尺寸逐渐增大的草图化最小二乘子问题以提升精度，SLSE-FRS逐步优化真实参数向量的估计值，最终生成高精度估计器。我们分析了SLSE-FRS的收敛性质，并给出了其高效实现方案。数值实验表明，SLSE-FRS在性能上优于当前最优方法，即预处理共轭梯度法（PCG）与迭代双重草图法（IDS）。