Sparse-Input Neural Networks for High-dimensional Nonparametric Regression and Classification

TL;DR

This paper introduces a sparse-input neural network approach with regularization that effectively handles high-dimensional nonparametric regression and classification by selecting relevant features and achieving favorable convergence properties.

Contribution

The authors propose a novel neural network method with a sparse group lasso penalty on input weights, enabling feature selection and improved performance in high-dimensional settings.

Findings

Sparse-input neural networks outperform existing methods in complex high-dimensional data.

The method's excess risk grows only logarithmically with the number of features.

Irrelevant feature weights converge to zero, ensuring effective feature selection.

Abstract

Neural networks are usually not the tool of choice for nonparametric high-dimensional problems where the number of input features is much larger than the number of observations. Though neural networks can approximate complex multivariate functions, they generally require a large number of training observations to obtain reasonable fits, unless one can learn the appropriate network structure. In this manuscript, we show that neural networks can be applied successfully to high-dimensional settings if the true function falls in a low dimensional subspace, and proper regularization is used. We propose fitting a neural network with a sparse group lasso penalty on the first-layer input weights. This results in a neural net that only uses a small subset of the original features. In addition, we characterize the statistical convergence of the penalized empirical risk minimizer to the optimal neural network: we show that the excess risk of this penalized estimator only grows with the logarithm of the number of input features; and we show that the weights of irrelevant features converge to zero. Via simulation studies and data analyses, we show that these sparse-input neural networks outperform existing nonparametric high-dimensional estimation methods when the data has complex higher-order interactions.