Sketch-and-solve (SAS) is a very successful method to efficiently estimate the solution of heavily overdetermined large linear least squares problems. It uses random sketching to reduce the size of the problem, hence reducing the computational cost. Several authors have shown that averaging several solutions from SAS further improves the accuracy, which is measured by the residual associated to the approximate solution. Going further, we combine solutions from sketch-and-solve in a multilevel manner, such that the approximate solution is a combination of SAS samples obtained from small sketches and more accurate correction terms obtained from larger sketches. We first consider the variance of the estimator, which depends on the variance of the coarse samples and the correction terms. We show that the variance of the correction terms on each level follows a trend and decreases faster than the variance of the simple SAS estimator. However, we then show that the overall computational cost of our multilevel framework is slightly higher than that of the simple average estimator, so a naive application of multilevel methods appears unattractive for least squares problems.

翻译：草图求解（SAS）是一种高效求解大规模超定线性最小二乘问题的有效方法。该方法通过随机草图技术降低问题规模，从而减少计算成本。多位学者已证明，对多次SAS求解结果进行平均可进一步提升精度（通过近似解对应的残差衡量）。在此基础上，我们提出以多层方式组合SAS求解结果：近似解由从小规模草图获得的SAS样本与更精确的校正项（来自大规模草图）共同构成。首先分析估计量的方差，该方差受粗尺度样本方差与校正项方差共同影响。研究表明，各层校正项的方差具有递减趋势，其衰减速度快于简单SAS估计量的方差。然而，多层框架的总计算成本略高于简单平均估计量，因此对最小二乘问题直接应用多层方法似乎不具备显著优势。