Early stopping for $ L^2 $-boosting in high-dimensional linear models

TL;DR

This paper proposes a data-driven early stopping method for $L^2$-boosting in high-dimensional linear models, achieving statistical optimality with significantly reduced computational cost compared to full path methods.

Contribution

It introduces a sequential early stopping rule for $L^2$-boosting that maintains statistical guarantees while being computationally more efficient than existing full-path criteria.

Findings

Early stopping preserves statistical optimality.

Method performs comparably to state-of-the-art algorithms.

Significant reduction in computational cost.

Abstract

Increasingly high-dimensional data sets require that estimation methods do not only satisfy statistical guarantees but also remain computationally feasible. In this context, we consider $ L^{2} $-boosting via orthogonal matching pursuit in a high-dimensional linear model and analyze a data-driven early stopping time $ \tau $ of the algorithm, which is sequential in the sense that its computation is based on the first $ \tau $ iterations only. This approach is much less costly than established model selection criteria, that require the computation of the full boosting path. We prove that sequential early stopping preserves statistical optimality in this setting in terms of a fully general oracle inequality for the empirical risk and recently established optimal convergence rates for the population risk. Finally, an extensive simulation study shows that at an immensely reduced computational cost, the performance of these type of methods is on par with other state of the art algorithms such as the cross-validated Lasso or model selection via a high dimensional Akaike criterion based on the full boosting path.