The sparsity and bias of the Lasso selection in high-dimensional linear regression

TL;DR

This paper analyzes the sparsity and bias of the Lasso in high-dimensional linear regression, showing it can select models of correct size and control bias under certain conditions, even when the number of variables exceeds the sample size.

Contribution

It extends existing results by proving rate consistency of the Lasso in model selection with small nonzero coefficients and high-dimensional data under a sparse Riesz condition.

Findings

Lasso selects models of correct order of dimensionality.

Bias of the selected model is controlled by small coefficients and threshold bias.

Error measures converge at optimal rates under specified conditions.

Abstract

Meinshausen and Buhlmann [Ann. Statist. 34 (2006) 1436--1462] showed that, for neighborhood selection in Gaussian graphical models, under a neighborhood stability condition, the LASSO is consistent, even when the number of variables is of greater order than the sample size. Zhao and Yu [(2006) J. Machine Learning Research 7 2541--2567] formalized the neighborhood stability condition in the context of linear regression as a strong irrepresentable condition. That paper showed that under this condition, the LASSO selects exactly the set of nonzero regression coefficients, provided that these coefficients are bounded away from zero at a certain rate. In this paper, the regression coefficients outside an ideal model are assumed to be small, but not necessarily zero. Under a sparse Riesz condition on the correlation of design variables, we prove that the LASSO selects a model of the correct order of dimensionality, controls the bias of the selected model at a level determined by the contributions of small regression coefficients and threshold bias, and selects all coefficients of greater order than the bias of the selected model. Moreover, as a consequence of this rate consistency of the LASSO in model selection, it is proved that the sum of error squares for the mean response and the $\ell_{\alpha}$-loss for the regression coefficients converge at the best possible rates under the given conditions. An interesting aspect of our results is that the logarithm of the number of variables can be of the same order as the sample size for certain random dependent designs.