How Correlations Influence Lasso Prediction

TL;DR

This paper investigates how correlations in the design matrix affect Lasso prediction, emphasizing the importance of tuning parameter selection and demonstrating that proper tuning can mitigate correlation effects.

Contribution

It provides a theoretical and empirical analysis showing the impact of correlations on Lasso tuning and prediction accuracy, highlighting the need for data-dependent tuning parameters.

Findings

Higher correlations require smaller optimal tuning parameters

Standard tuning parameters are often suboptimal for correlated data

Proper tuning enables Lasso to perform well regardless of correlation levels

Abstract

We study how correlations in the design matrix influence Lasso prediction. First, we argue that the higher the correlations are, the smaller the optimal tuning parameter is. This implies in particular that the standard tuning parameters, that do not depend on the design matrix, are not favorable. Furthermore, we argue that Lasso prediction works well for any degree of correlations if suitable tuning parameters are chosen. We study these two subjects theoretically as well as with simulations.