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

author: Alex Gittens, Department of Computing and Mathematical Sciences, California Institute of Technology (Caltech)
published: Sept. 27, 2015,   recorded: July 2015,   views: 1368

See Also:

Download slides icon Download slides: icml2015_gittens_ordinary_least_squares_01.pdf (200.6┬áKB)

Help icon Streaming Video Help

Related Open Educational Resources

Related content

Report a problem or upload files

If you have found a problem with this lecture or would like to send us extra material, articles, exercises, etc., please use our ticket system to describe your request and upload the data.
Enter your e-mail into the 'Cc' field, and we will keep you updated with your request's status.
Lecture popularity: You need to login to cast your vote.


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.

Link this page

Would you like to put a link to this lecture on your homepage?
Go ahead! Copy the HTML snippet !

Write your own review or comment:

make sure you have javascript enabled or clear this field: