An Iterative, Sketching-based Framework for Ridge Regression
Agniva Chowdhury, Jiasen Yang, Petros Drineas
Abstract
Ridge regression is a variant of regularized least squares regression that is particularly suitable in settings where the number of predictor variables greatly exceeds the number of observations. We present a simple, iterative, sketching-based algorithm for ridge regression that guarantees high-quality approximations to the optimal solution vector. Our analysis builds upon two simple structural results that boil down to randomized matrix multiplication, a fundamental and well-understood primitive of randomized linear algebra. An important contribution of our work is the analysis of the behavior of subsampled ridge regression problems when the ridge leverage scores are used: we prove that accurate approximations can be achieved by a sample whose size depends on the degrees of freedom of the ridge-regression problem rather than the dimensions of the design matrix. Our experimental evaluations verify our theoretical results on both real and synthetic data.
BibTeX
@InProceedings{pmlr-v80-chowdhury18a,
title = {An Iterative, Sketching-based Framework for Ridge Regression},
author = {Chowdhury, Agniva and Yang, Jiasen and Drineas, Petros},
booktitle = {Proceedings of the 35th International Conference on Machine Learning},
pages = {989--998},
year = {2018},
editor = {Dy, Jennifer and Krause, Andreas},
volume = {80},
series = {Proceedings of Machine Learning Research},
month = {10--15 Jul},
publisher = {PMLR},
pdf = {http://proceedings.mlr.press/v80/chowdhury18a/chowdhury18a.pdf},
url = {https://proceedings.mlr.press/v80/chowdhury18a.html},
abstract = {Ridge regression is a variant of regularized least squares regression that is particularly suitable in settings where the number of predictor variables greatly exceeds the number of observations. We present a simple, iterative, sketching-based algorithm for ridge regression that guarantees high-quality approximations to the optimal solution vector. Our analysis builds upon two simple structural results that boil down to randomized matrix multiplication, a fundamental and well-understood primitive of randomized linear algebra. An important contribution of our work is the analysis of the behavior of subsampled ridge regression problems when the ridge leverage scores are used: we prove that accurate approximations can be achieved by a sample whose size depends on the degrees of freedom of the ridge-regression problem rather than the dimensions of the design matrix. Our experimental evaluations verify our theoretical results on both real and synthetic data.}
}