A Statistical Perspective on Algorithmic Leveraging

TL;DR

This paper evaluates the statistical properties of algorithmic leveraging in linear regression, revealing that leverage-based and uniform sampling have comparable bias and variance, and introduces two improved algorithms with empirical validation.

Contribution

It provides a statistical framework for analyzing leverage-based sampling, showing its bias and variance are comparable to uniform sampling, and proposes two new leveraging algorithms with empirical performance improvements.

Findings

Leverage-based and uniform sampling have similar bias and variance in linear regression.

Theoretical results predict practical performance of leverage-based algorithms.

New algorithms outperform existing methods in empirical tests.

Abstract

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.