Elastic-net regularization in learning theory
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Mol-2009-Elastic-net regulari.pdf
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Author(s)
De Mol, Christine
De Vito, Ernesto
Rosasco, Lorenzo Andrea
Date Issued
January 2009
Journal
Journal of Complexity
Publisher
Elsevier
Citation
De Mol, Christine, Ernesto De Vito, and Lorenzo Rosasco. “Elastic-Net Regularization in Learning Theory.” Journal of Complexity 25, no. 2 (April 2009): 201–230. © 2009 Elsevier Inc.
Version
Final published version
Abstract
Within the framework of statistical learning theory we analyze in detail the so-called elastic-net regularization scheme proposed by Zou and Hastie [H. Zou, T. Hastie, Regularization and variable selection via the elastic net, J. R. Stat. Soc. Ser. B, 67(2) (2005) 301–320] for the selection of groups of correlated variables. To investigate the statistical properties of this scheme and in particular its consistency properties, we set up a suitable mathematical framework. Our setting is random-design regression where we allow the response variable to be vector-valued and we consider prediction functions which are linear combinations of elements (features) in an infinite-dimensional dictionary. Under the assumption that the regression function admits a sparse representation on the dictionary, we prove that there exists a particular “elastic-net representation” of the regression function such that, if the number of data increases, the elastic-net estimator is consistent not only for prediction but also for variable/feature selection. Our results include finite-sample bounds and an adaptive scheme to select the regularization parameter. Moreover, using convex analysis tools, we derive an iterative thresholding algorithm for computing the elastic-net solution which is different from the optimization procedure originally proposed in the above-cited work.
MIT Department
Massachusetts Institute of Technology. Center for Biological & Computational Learning
McGovern Institute for Brain Research at MIT
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Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use.
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DOI of Published Version
https://doi.org/10.1016/j.jco.2009.01.002
