Elastic-net regularization in learning theory

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. © 2009 Elsevier Inc. All rights reserved. ∗ Corresponding author at: Center for Biological and Computational Learning, Massachusetts Institute of Technology, 43 Vassar Street, Cambridge, MA 02139, United States. E-mail addresses: [email protected] (C. De Mol), [email protected] (E. De Vito), [email protected] (L. Rosasco). 0885-064X/$ – see front matter© 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jco.2009.01.002 202 C. De Mol et al. / Journal of Complexity 25 (2009) 201–230