Robust and Sparse Regression via $\gamma$-divergence

TL;DR

This paper introduces a robust sparse regression method based on the $oldsymbol{\gamma}$-divergence, demonstrating strong robustness against outliers and heterogeneity in high-dimensional data, with an efficient algorithm and superior empirical performance.

Contribution

It extends the $oldsymbol{\gamma}$-divergence to regression, proposing a new robust sparse estimation method with an efficient update algorithm and theoretical robustness guarantees.

Findings

Outperforms existing robust sparse methods in experiments

Maintains robustness under heavy and heterogeneous contamination

Provides an efficient algorithm with monotone loss decrease

Abstract

In high-dimensional data, many sparse regression methods have been proposed. However, they may not be robust against outliers. Recently, the use of density power weight has been studied for robust parameter estimation and the corresponding divergences have been discussed. One of such divergences is the $\gamma$-divergence and the robust estimator using the $\gamma$-divergence is known for having a strong robustness. In this paper, we consider the robust and sparse regression based on $\gamma$-divergence. We extend the $\gamma$-divergence to the regression problem and show that it has a strong robustness under heavy contamination even when outliers are heterogeneous. The loss function is constructed by an empirical estimate of the $\gamma$-divergence with sparse regularization and the parameter estimate is defined as the minimizer of the loss function. To obtain the robust and sparse estimate, we propose an efficient update algorithm which has a monotone decreasing property of the loss function. Particularly, we discuss a linear regression problem with $L_1$ regularization in detail. In numerical experiments and real data analyses, we see that the proposed method outperforms past robust and sparse methods.