Statistical and Algorithmic Perspectives on Randomized Sketching for Ordinary Least-Squares -- ICML

TL;DR

This paper compares the statistical and algorithmic perspectives on randomized sketching algorithms for large-scale least-squares problems, providing a unified framework and bounds on their efficiency and error.

Contribution

It introduces a unified framework for analyzing both perspectives and derives bounds for statistical prediction and residual efficiencies of sketched least-squares estimators.

Findings

Residual efficiency can be bounded with small sample sizes.

Prediction efficiency generally requires larger sample sizes.

Upper bounds on prediction efficiency are shown to be tight.

Abstract

We consider statistical and algorithmic aspects of solving large-scale least-squares (LS) problems using randomized sketching algorithms. Prior results show that, from an \emph{algorithmic perspective}, when using sketching matrices constructed from random projections and leverage-score sampling, if the number of samples $r$ much smaller than the original sample size $n$, then the worst-case (WC) error is the same as solving the original problem, up to a very small relative error. From a \emph{statistical perspective}, one typically considers the mean-squared error performance of randomized sketching algorithms, when data are generated according to a statistical linear model. In this paper, we provide a rigorous comparison of both perspectives leading to insights on how they differ. To do this, we first develop a framework for assessing, in a unified manner, algorithmic and statistical aspects of randomized sketching methods. We then consider the statistical prediction efficiency (PE) and the statistical residual efficiency (RE) of the sketched LS estimator; and we use our framework to provide upper bounds for several types of random projection and random sampling algorithms. Among other results, we show that the RE can be upper bounded when $r$ is much smaller than $n$, while the PE typically requires the number of samples $r$ to be substantially larger. Lower bounds developed in subsequent work show that our upper bounds on PE can not be improved.