Constraints and Conditions: the Lasso Oracle-inequalities

TL;DR

This paper reviews and simplifies the derivation of oracle inequalities for the Lasso estimator in high-dimensional sparse regression, focusing on constraints on coefficients and design matrices to improve understanding of prediction and variable selection accuracy.

Contribution

It provides a clearer, more detailed derivation of oracle inequalities for Lasso under various constraints, along with illustrative examples, enhancing theoretical understanding.

Findings

Simplified derivations of oracle inequalities for Lasso

Conditions on coefficients and design matrices for improved bounds

Illustrated examples demonstrating theoretical results

Abstract

We study various constraints and conditions on the true coefficient vector and on the design matrix to establish non-asymptotic oracle inequalities for the prediction error, estimation accuracy and variable selection for the Lasso estimator in high dimensional sparse regression models. We review results from the literature and we provide simpler and detailed derivation for several boundedness theorems. In addition, we complement the theory with illustrated examples.