## The Annals of Statistics

### Empirical Margin Distributions and Bounding the Generalization Error of Combined Classifiers

#### Abstract

We prove new probabilistic upper bounds on generalization error of complex classifiers that are combinations of simple classifiers. Such combinations could be implemented by neural networks or by voting methods of combining the classifiers, such as boosting and bagging. The bounds are in terms of the empirical distribution of the margin of the combined classifier. They are based on the methods of the theory of Gaussian and empirical processes (comparison inequalities, symmetrization method, concentration inequalities) and they improve previous results of Bartlett (1998) on bounding the generalization error of neural networks in terms of $\ell_1$-norms of the weights of neurons and of Schapire, Freund, Bartlett and Lee (1998) on bounding the generalization error of boosting. We also obtain rates of convergence in Lévy distance of empirical margin distribution to the true margin distribution uniformly over the classes of classifiers and prove the optimality of these rates.

#### Article information

Source
Ann. Statist., Volume 30, Number 1 (2002), 1-50.

Dates
First available in Project Euclid: 5 March 2002

https://projecteuclid.org/euclid.aos/1015362183

Digital Object Identifier
doi:10.1214/aos/1015362183

Mathematical Reviews number (MathSciNet)
MR1892654

Zentralblatt MATH identifier
1012.62004

Subjects
Primary: 62G05: Estimation
Secondary: 62G20: Asymptotic properties 60F15

#### Citation

Koltchinskii, V.; Panchenko, D. Empirical Margin Distributions and Bounding the Generalization Error of Combined Classifiers. Ann. Statist. 30 (2002), no. 1, 1--50. doi:10.1214/aos/1015362183. https://projecteuclid.org/euclid.aos/1015362183

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• ALBUQUERQUE, NEW MEXICO 87131-1141 E-MAIL: vlad@math.unm.edu panchenk@math.unm.edu