## The Annals of Statistics

### False discoveries occur early on the Lasso path

#### 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.

#### Article information

Source
Ann. Statist., Volume 45, Number 5 (2017), 2133-2150.

Dates
Revised: September 2016
First available in Project Euclid: 31 October 2017

https://projecteuclid.org/euclid.aos/1509436830

Digital Object Identifier
doi:10.1214/16-AOS1521

Mathematical Reviews number (MathSciNet)
MR3718164

Zentralblatt MATH identifier
06821121

Subjects
Primary: 62F03: Hypothesis testing
Secondary: 62J07: Ridge regression; shrinkage estimators 62J05: Linear regression

#### Citation

Su, Weijie; Bogdan, Małgorzata; Candès, Emmanuel. False discoveries occur early on the Lasso path. Ann. Statist. 45 (2017), no. 5, 2133--2150. doi:10.1214/16-AOS1521. https://projecteuclid.org/euclid.aos/1509436830

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#### Supplemental materials

• Supplement to “False discoveries occur early on the Lasso path”. The supplementary materials contain proofs of some technical results in this paper.