Prediction Error Bounds for Linear Regression With the TREX

TL;DR

This paper establishes the first prediction error bounds for the TREX, a tuning-parameter-free sparse linear regression method, and extends its theoretical understanding to more general penalty functions.

Contribution

It provides the first theoretical prediction error bounds for TREX and extends the method to a broader class of penalties, enhancing understanding of penalized regression.

Findings

First prediction error bounds for TREX established.

Extensions of TREX to general penalty functions introduced.

Theoretical insights deepen understanding of sparse linear regression.

Abstract

The TREX is a recently introduced approach to sparse linear regression. In contrast to most well-known approaches to penalized regression, the TREX can be formulated without the use of tuning parameters. In this paper, we establish the first known prediction error bounds for the TREX. Additionally, we introduce extensions of the TREX to a more general class of penalties, and we provide a bound on the prediction error in this generalized setting. These results deepen the understanding of TREX from a theoretical perspective and provide new insights into penalized regression in general.