## The Annals of Statistics

### Boosting with early stopping: Convergence and consistency

#### Abstract

Boosting is one of the most significant advances in machine learning for classification and regression. In its original and computationally flexible version, boosting seeks to minimize empirically a loss function in a greedy fashion. The resulting estimator takes an additive function form and is built iteratively by applying a base estimator (or learner) to updated samples depending on the previous iterations. An unusual regularization technique, early stopping, is employed based on CV or a test set.

This paper studies numerical convergence, consistency and statistical rates of convergence of boosting with early stopping, when it is carried out over the linear span of a family of basis functions. For general loss functions, we prove the convergence of boosting’s greedy optimization to the infinimum of the loss function over the linear span. Using the numerical convergence result, we find early-stopping strategies under which boosting is shown to be consistent based on i.i.d. samples, and we obtain bounds on the rates of convergence for boosting estimators. Simulation studies are also presented to illustrate the relevance of our theoretical results for providing insights to practical aspects of boosting.

As a side product, these results also reveal the importance of restricting the greedy search step-sizes, as known in practice through the work of Friedman and others. Moreover, our results lead to a rigorous proof that for a linearly separable problem, AdaBoost with ɛ→0 step-size becomes an L1-margin maximizer when left to run to convergence.

#### Article information

Source
Ann. Statist., Volume 33, Number 4 (2005), 1538-1579.

Dates
First available in Project Euclid: 5 August 2005

Permanent link to this document
https://projecteuclid.org/euclid.aos/1123250222

Digital Object Identifier
doi:10.1214/009053605000000255

Mathematical Reviews number (MathSciNet)
MR2166555

Zentralblatt MATH identifier
1078.62038

Subjects
Primary: 62G05: Estimation 62G08: Nonparametric regression

#### Citation

Zhang, Tong; Yu, Bin. Boosting with early stopping: Convergence and consistency. Ann. Statist. 33 (2005), no. 4, 1538--1579. doi:10.1214/009053605000000255. https://projecteuclid.org/euclid.aos/1123250222

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