Elastic-Net Regularization in Learning Theory

TL;DR

This paper provides a theoretical analysis of elastic-net regularization in statistical learning, demonstrating its consistency for prediction and feature selection in high-dimensional, sparse settings, and introduces an iterative algorithm for its computation.

Contribution

It establishes the statistical properties and consistency of elastic-net regularization in a general, infinite-dimensional setting, and proposes a new iterative thresholding algorithm for its implementation.

Findings

Elastic-net estimator is consistent for prediction and feature selection as data increases.

Finite-sample bounds are derived for the elastic-net regularization.

An iterative thresholding algorithm for elastic-net computation is proposed.

Abstract

Within the framework of statistical learning theory we analyze in detail the so-called elastic-net regularization scheme proposed by Zou and Hastie for the selection of groups of correlated variables. To investigate on the statistical properties of this scheme and in particular on 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 combination of elements ({\em 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 ``{\em 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 by Zou and Hastie