April 2024 Early stopping for L2-boosting in high-dimensional linear models
Bernhard Stankewitz
Author Affiliations +
Ann. Statist. 52(2): 491-518 (April 2024). DOI: 10.1214/24-AOS2356


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 L2-boosting via orthogonal matching pursuit in a high-dimensional linear model and analyze a data-driven early stopping time τ of the algorithm, which is sequential in the sense that its computation is based on the first τ iterations only. This approach is much less costly than established model selection criteria, that require the computation of the full boosting path, which may even be computationally infeasible in truly high-dimensional applications. 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 a significantly reduced computational cost, the performance of early stopping 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.

Funding Statement

The research of the author has been partially funded by the Deutsche Forschungsgemeinschaft (DFG)—Project-ID 318763901-SFB1294. Cofunded by the European Union (ERC, BigBayesUQ, project number: 101041064). Views and opinions expressed are, however, those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council. Neither the European Union nor the granting authority can be held responsible for them.


The author is very grateful for the discussions with Markus Reiß and Martin Wahl that were indispensable during the preparation of this paper. Further, the author would like to thank Botond Szabo, Martin Spindler, Richard Nickl, the Associate Editor and two anonymous referees for very valuable feedback during the revision process.


Download Citation

Bernhard Stankewitz. "Early stopping for L2-boosting in high-dimensional linear models." Ann. Statist. 52 (2) 491 - 518, April 2024. https://doi.org/10.1214/24-AOS2356


Received: 1 October 2022; Revised: 1 October 2023; Published: April 2024
First available in Project Euclid: 9 May 2024

Digital Object Identifier: 10.1214/24-AOS2356

Primary: 62G05 , 62J07
Secondary: 62F35

Keywords: adaptive estimation , discrepancy principle , early stopping , L2-boosting , Oracle inequalities , orthogonal matching pursuit

Rights: Copyright © 2024 Institute of Mathematical Statistics


This article is only available to subscribers.
It is not available for individual sale.

Vol.52 • No. 2 • April 2024
Back to Top