High-Dimensional Generalized Linear Models and the Lasso

We consider high-dimensional generalized linear models with Lipschitz loss functions, and prove a nonasymptotic oracle inequality for the empirical risk minimizer with Lasso penalty. The penalty is based on the coefficients in the linear predictor, after normalization with the empirical norm. The ex...

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Bibliographic Details
Published in:The Annals of statistics 2008-04, Vol.36 (2), p.614-645
Main Author: van de Geer, Sara A.
Format: Article
Language:English
Subjects:
62G08
Asymptotic methods
Density estimation
Estimators
Exact sciences and technology
General topics
Generalized linear models
Lasso
Linear inference, regression
Logical givens
Logistic regression
Mathematical constants
Mathematical functions
Mathematical inequalities
Mathematics
Multivariate analysis
Nonparametric inference
oracle inequality
Oracles
Probability and statistics
Random variables
Sciences and techniques of general use
sparsity
Statistical analysis
Statistics
Studies
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Description
Summary:We consider high-dimensional generalized linear models with Lipschitz loss functions, and prove a nonasymptotic oracle inequality for the empirical risk minimizer with Lasso penalty. The penalty is based on the coefficients in the linear predictor, after normalization with the empirical norm. The examples include logistic regression, density estimation and classification with hinge loss. Least squares regression is also discussed.
ISSN:0090-5364
2168-8966
DOI:10.1214/009053607000000929