Nonparametric sparsity and regularization

TL;DR

This paper introduces a nonparametric sparsity measure and regularization method for supervised learning that identifies relevant variables using partial derivatives, leveraging kernel methods and proximal algorithms.

Contribution

It proposes a novel nonparametric sparsity concept and a regularization scheme that effectively performs variable selection without assuming linearity or additivity.

Findings

Method performs favorably compared to state-of-the-art techniques.

Algorithm is provably convergent and consistent for prediction and variable selection.

Extensive empirical analysis validates the approach.

Abstract

In this work we are interested in the problems of supervised learning and variable selection when the input-output dependence is described by a nonlinear function depending on a few variables. Our goal is to consider a sparse nonparametric model, hence avoiding linear or additive models. The key idea is to measure the importance of each variable in the model by making use of partial derivatives. Based on this intuition we propose a new notion of nonparametric sparsity and a corresponding least squares regularization scheme. Using concepts and results from the theory of reproducing kernel Hilbert spaces and proximal methods, we show that the proposed learning algorithm corresponds to a minimization problem which can be provably solved by an iterative procedure. The consistency properties of the obtained estimator are studied both in terms of prediction and selection performance. An extensive empirical analysis shows that the proposed method performs favorably with respect to the state-of-the-art methods.