High-Dimensional $L_2$Boosting: Rate of Convergence

TL;DR

This paper analyzes the convergence rates of $L_2$Boosting in high-dimensional regression, introduces post-Boosting and orthogonal boosting variants, and compares their performance to LASSO through theory and simulations.

Contribution

It provides new theoretical convergence results for $L_2$Boosting, introduces practical variants like post-Boosting and orthogonal boosting, and compares them to LASSO in high-dimensional settings.

Findings

Post-$L_2$Boosting outperforms LASSO in simulations.

Both post-$L_2$Boosting and orthogonal boosting achieve LASSO's convergence rate.

New approximation results for greedy algorithms are derived.

Abstract

Boosting is one of the most significant developments in machine learning. This paper studies the rate of convergence of $L_2$Boosting, which is tailored for regression, in a high-dimensional setting. Moreover, we introduce so-called \textquotedblleft post-Boosting\textquotedblright. This is a post-selection estimator which applies ordinary least squares to the variables selected in the first stage by $L_2$Boosting. Another variant is \textquotedblleft Orthogonal Boosting\textquotedblright\ where after each step an orthogonal projection is conducted. We show that both post-$L_2$Boosting and the orthogonal boosting achieve the same rate of convergence as LASSO in a sparse, high-dimensional setting. We show that the rate of convergence of the classical $L_2$Boosting depends on the design matrix described by a sparse eigenvalue constant. To show the latter results, we derive new approximation results for the pure greedy algorithm, based on analyzing the revisiting behavior of $L_2$Boosting. We also introduce feasible rules for early stopping, which can be easily implemented and used in applied work. Our results also allow a direct comparison between LASSO and boosting which has been missing from the literature. Finally, we present simulation studies and applications to illustrate the relevance of our theoretical results and to provide insights into the practical aspects of boosting. In these simulation studies, post-$L_2$Boosting clearly outperforms LASSO.