| Issue |
ESAIM: PS
Volume 28, 2024
|
|
|---|---|---|
| Page(s) | 227 - 257 | |
| DOI | https://doi.org/10.1051/ps/2024006 | |
| Published online | 24 May 2024 | |
The vanishing learning rate asymptotic for linear L2-boosting
1
Université de Franche-Comté, CNRS, LmB, F-25000 Besançon, France
2
Le Mans Université, Laboratoire Manceau de Mathématiques, Avenue Olivier Messiaen, 72085 Le Mans Cedex 09, France
* Corresponding author: Youssef.Esstafa@univ-lemans.fr
Received:
27
February
2023
Accepted:
25
March
2024
We investigate the asymptotic behaviour of gradient boosting algorithms when the learning rate converges to zero and the number of iterations is rescaled accordingly. We mostly consider L2-boosting for regression with linear base learner as studied in P. Bühlmann and B. Yu, J. Am. Statist. Assoc. 98 (2003) 324–339 and analyze also a stochastic version of the model where subsampling is used at each step (J.H. Friedman, Computat. Statist. Data Anal. 38 (2002) 367–378). We prove a deterministic limit in the vanishing learning rate asymptotic and characterize the limit as the unique solution of a linear differential equation in an infinite dimensional function space. Besides, the training and test error of the limiting procedure are thoroughly analyzed. We finally illustrate and discuss our result on a simple numerical experiment where the linear L2-boosting operator is interpreted as a smoothed projection and time is related to its number of degrees of freedom.
Mathematics Subject Classification: 62G08 / 60J20
Key words: Boosting / non parametric regression / statistical learning / stochastic algorithm / Markov chain / convergence of stochastic process
© The authors. Published by EDP Sciences, SMAI 2024
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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