A novel nonconvex, smooth-at-origin penalty for statistical learning

TL;DR

This paper introduces a new nonconvex, smooth-at-origin penalty for statistical learning, demonstrating its theoretical advantages and empirical performance improvements in deep learning models across multiple datasets.

Contribution

The paper proposes a novel nonconvex penalty that is smooth at the origin, with asymptotic properties and empirical benefits in deep neural network regularization.

Findings

Asymptotic bias vanishes exponentially fast with the new penalty.

Better performance in 5 out of 7 datasets using deep neural networks.

The new penalty outperforms existing methods in certain high-dimensional settings.

Abstract

Nonconvex penalties are utilized for regularization in high-dimensional statistical learning algorithms primarily because they yield unbiased or nearly unbiased estimators for the parameters in the model. Nonconvex penalties existing in the literature such as SCAD, MCP, Laplace and arctan have a singularity at origin which makes them useful also for variable selection. However, in several high-dimensional frameworks such as deep learning, variable selection is less of a concern. In this paper, we present a nonconvex penalty which is smooth at origin. The paper includes asymptotic results for ordinary least squares estimators regularized with the new penalty function, showing asymptotic bias that vanishes exponentially fast. We also conducted an empirical study employing deep neural network architecture on three datasets and convolutional neural network on four datasets. The empirical study showed better performance for the new regularization approach in five out of the seven datasets.