Malgorzata Bogdan
Professor
False discoveries occur early on the lasso path
Författare
Summary, in English
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.
Publiceringsår
2017-10
Språk
Engelska
Sidor
2133-2150
Publikation/Tidskrift/Serie
Annals of Statistics
Volym
45
Issue
5
Dokumenttyp
Artikel i tidskrift
Förlag
Institute of Mathematical Statistics
Ämne
- Probability Theory and Statistics
Nyckelord
- Adaptive selection of parameters
- Approximate message passing (AMP)
- False discovery rate
- False negative rate
- Lasso
- Lasso path
- Power
Status
Published
ISBN/ISSN/Övrigt
- ISSN: 0090-5364