金融杠杆的统计透视
A Statistical Perspective on Algorithmic Leveraging
摘要：
处理大规模数据集的一种流行方法是抽样。例如，通过使用经验统计杠杆分数作为重要性抽样分布，算法杠杆的方法对数据矩阵的行/列进行抽样和重新缩放，以在对子问题执行计算之前减小数据大小。这种方法已成功地提高了算法的计算效率，如最小二乘近似，最小绝对偏差近似，低秩矩阵近似矩阵问题。现有的工作主要集中在算法问题上，如最坏情况下的运行时间和与提供高质量实现相关的数值问题，但没有一个涉及这种方法的统计方面。在本文中，我们提供了一个简单而有效的框架来评估算法杠杆的统计特性的上下文中估计参数的线性回归模型与固定数量的预测。我们表明，从统计的角度来看，偏差和方差，无论是基于抽样的均匀抽样或占主导地位的其他。这个结果是特别惊人的，鉴于众所周知的结果，从算法的角度来看，最坏情况下的分析，基于采样的采样提供了一致优于最坏情况下的算法结果相比，均匀采样。基于这些理论结果，我们提出并分析了两个新的杠杆算法。对现有的基于存储的方法以及这两种新方法进行了详细的实证评估，在合成和真实数据集上。实证结果表明，我们的理论是一个很好的预测现有的和新的基于数据的算法的实际性能，新的算法实现了改进的性能。
英文摘要：
One popular method for dealing with large-scale data sets is sampling. For example, by using the empirical statistical leverage scores as an importance sampling distribution, the method of algorithmic leveraging samples and rescales rows/columns of data matrices to reduce the data size before performing computations on the subproblem. This method has been successful in improving computational efficiency of algorithms for matrix problems such as least-squares approximation, least absolute deviations approximation, and low-rank matrix approximation. Existing work has focused on algorithmic issues such as worst-case running times and numerical issues associated with providing high-quality implementations, but none of it addresses statistical aspects of this method. In this paper, we provide a simple yet effective framework to evaluate the statistical properties of algorithmic leveraging in the context of estimating parameters in a linear regression model with a fixed number of predictors. We show that from the statistical perspective of bias and variance, neither leverage-based sampling nor uniform sampling dominates the other. This result is particularly striking, given the well-known result that, from the algorithmic perspective of worst-case analysis, leverage-based sampling provides uniformly superior worst-case algorithmic results, when compared with uniform sampling. Based on these theoretical results, we propose and analyze two new leveraging algorithms. A detailed empirical evaluation of existing leverage-based methods as well as these two new methods is carried out on both synthetic and real data sets. The empirical results indicate that our theory is a good predictor of practical performance of existing and new leverage-based algorithms and that the new algorithms achieve improved performance.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
