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2018 On the prediction loss of the lasso in the partially labeled setting
Pierre C. Bellec, Arnak S. Dalalyan, Edwin Grappin, Quentin Paris
Electron. J. Statist. 12(2): 3443-3472 (2018). DOI: 10.1214/18-EJS1457

Abstract

In this paper we revisit the risk bounds of the lasso estimator in the context of transductive and semi-supervised learning. In other terms, the setting under consideration is that of regression with random design under partial labeling. The main goal is to obtain user-friendly bounds on the off-sample prediction risk. To this end, the simple setting of bounded response variable and bounded (high-dimensional) covariates is considered. We propose some new adaptations of the lasso to these settings and establish oracle inequalities both in expectation and in deviation. These results provide non-asymptotic upper bounds on the risk that highlight the interplay between the bias due to the mis-specification of the linear model, the bias due to the approximate sparsity and the variance. They also demonstrate that the presence of a large number of unlabeled features may have significant positive impact in the situations where the restricted eigenvalue of the design matrix vanishes or is very small.

Citation

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Pierre C. Bellec. Arnak S. Dalalyan. Edwin Grappin. Quentin Paris. "On the prediction loss of the lasso in the partially labeled setting." Electron. J. Statist. 12 (2) 3443 - 3472, 2018. https://doi.org/10.1214/18-EJS1457

Information

Received: 1 January 2018; Published: 2018
First available in Project Euclid: 16 October 2018

zbMATH: 06970009
MathSciNet: MR3864589
Digital Object Identifier: 10.1214/18-EJS1457

Subjects:
Primary: 62H30
Secondary: 62G08

JOURNAL ARTICLE
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Vol.12 • No. 2 • 2018
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