The Annals of Statistics

Statistical behavior and consistency of classification methods based on convex risk minimization

Tong Zhang

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Abstract

We study how closely the optimal Bayes error rate can be approximately reached using a classification algorithm that computes a classifier by minimizing a convex upper bound of the classification error function. The measurement of closeness is characterized by the loss function used in the estimation. We show that such a classification scheme can be generally regarded as a (nonmaximum-likelihood) conditional in-class probability estimate, and we use this analysis to compare various convex loss functions that have appeared in the literature. Furthermore, the theoretical insight allows us to design good loss functions with desirable properties. Another aspect of our analysis is to demonstrate the consistency of certain classification methods using convex risk minimization. This study sheds light on the good performance of some recently proposed linear classification methods including boosting and support vector machines. It also shows their limitations and suggests possible improvements.

Article information

Source
Ann. Statist. Volume 32, Number 1 (2004), 56-85.

Dates
First available in Project Euclid: 12 March 2004

Permanent link to this document
http://projecteuclid.org/euclid.aos/1079120130

Digital Object Identifier
doi:10.1214/aos/1079120130

Mathematical Reviews number (MathSciNet)
MR2051001

Zentralblatt MATH identifier
02113743

Subjects
Primary: 62G05: Estimation G2H30 68T05: Learning and adaptive systems [See also 68Q32, 91E40]

Keywords
Classification consistency boosting large margin methods kernel methods

Citation

Zhang, Tong. Statistical behavior and consistency of classification methods based on convex risk minimization. Ann. Statist. 32 (2004), no. 1, 56--85. doi:10.1214/aos/1079120130. http://projecteuclid.org/euclid.aos/1079120130.


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