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