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October 2017 False discoveries occur early on the Lasso path
Weijie Su, Małgorzata Bogdan, Emmanuel Candès
Ann. Statist. 45(5): 2133-2150 (October 2017). DOI: 10.1214/16-AOS1521

Abstract

In regression settings where explanatory variables have very low correlations and there are relatively few effects, each of large magnitude, we expect the Lasso to find the important variables with few errors, if any. This paper shows that in a regime of linear sparsity—meaning that the fraction of variables with a nonvanishing effect tends to a constant, however small—this cannot really be the case, even when the design variables are stochastically independent. We demonstrate that true features and null features are always interspersed on the Lasso path, and that this phenomenon occurs no matter how strong the effect sizes are. We derive a sharp asymptotic trade-off between false and true positive rates or, equivalently, between measures of type I and type II errors along the Lasso path. This trade-off states that if we ever want to achieve a type II error (false negative rate) under a critical value, then anywhere on the Lasso path the type I error (false positive rate) will need to exceed a given threshold so that we can never have both errors at a low level at the same time. Our analysis uses tools from approximate message passing (AMP) theory as well as novel elements to deal with a possibly adaptive selection of the Lasso regularizing parameter.

Citation

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Weijie Su. Małgorzata Bogdan. Emmanuel Candès. "False discoveries occur early on the Lasso path." Ann. Statist. 45 (5) 2133 - 2150, October 2017. https://doi.org/10.1214/16-AOS1521

Information

Received: 1 June 2016; Revised: 1 September 2016; Published: October 2017
First available in Project Euclid: 31 October 2017

zbMATH: 06821121
MathSciNet: MR3718164
Digital Object Identifier: 10.1214/16-AOS1521

Subjects:
Primary: 62F03
Secondary: 62J05, 62J07

Rights: Copyright © 2017 Institute of Mathematical Statistics

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Vol.45 • No. 5 • October 2017
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