A Sharper Computational Tool for $\text{L}_2\text{E}$ Regression

TL;DR

This paper introduces a faster, more efficient algorithm for robust structured regression using the $ ext{L}_2 ext{E}$ criterion, incorporating majorization-minimization, reparameterization, and distance-to-set penalties.

Contribution

It presents a novel, sharper majorization technique and a reparameterized estimation approach that enhance convergence speed and estimation accuracy in $ ext{L}_2 ext{E}$ regression.

Findings

Faster convergence compared to previous algorithms

Improved coefficient estimation and structure recovery

Effective constrained estimation with nonconvex sets

Abstract

Building on previous research of Chi and Chi (2022), the current paper revisits estimation in robust structured regression under the $\text{L}_2\text{E}$ criterion. We adopt the majorization-minimization (MM) principle to design a new algorithm for updating the vector of regression coefficients. Our sharp majorization achieves faster convergence than the previous alternating proximal gradient descent algorithm (Chi and Chi, 2022). In addition, we reparameterize the model by substituting precision for scale and estimate precision via a modified Newton's method. This simplifies and accelerates overall estimation. We also introduce distance-to-set penalties to allow constrained estimation under nonconvex constraint sets. This tactic also improves performance in coefficient estimation and structure recovery. Finally, we demonstrate the merits of our improved tactics through a rich set of simulation examples and a real data application.