Empirical Bayes Estimation for Lasso-Type Regularizers: Analysis of Automatic Relevance Determination

TL;DR

This paper analyzes the properties of empirical Bayes estimators in sparse linear regression models with regularizers like lasso and group lasso, revealing conditions for automatic relevance determination (ARD) to occur.

Contribution

It derives empirical Bayes estimators for group lasso models, explains when they diverge to induce ARD, and generalizes these findings to other regularizers.

Findings

Empirical Bayes estimators diverge under specific conditions, leading to ARD.

The paper clarifies when regularizers like ridge, lasso, and group lasso induce ARD.

Conditions for ARD occurrence are explicitly characterized.

Abstract

This paper focuses on linear regression models with non-conjugate sparsity-inducing regularizers such as lasso and group lasso. Although the empirical Bayes approach enables us to estimate the regularization parameter, little is known on the properties of the estimators. In particular, many aspects regarding the specific conditions under which the mechanism of automatic relevance determination (ARD) occurs remain unexplained. In this paper, we derive the empirical Bayes estimators for the group lasso regularized linear regression models with limited parameters. It is shown that the estimators diverge under a specific condition, giving rise to the ARD mechanism. We also prove that empirical Bayes methods can produce the ARD mechanism in general regularized linear regression models and clarify the conditions under which models such as ridge, lasso, and group lasso can do so.