  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.
