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### Arcing classifier (with discussion and a rejoinder by the author)

Leo Breiman
Source: Ann. Statist. Volume 26, Number 3 (1998), 801-849.

#### Abstract

Recent work has shown that combining multiple versions of unstable classifiers such as trees or neural nets results in reduced test set error. One of the more effective is bagging. Here, modified training sets are formed by resampling from the original training set, classifiers constructed using these training sets and then combined by voting. Freund and Schapire propose an algorithm the basis of which is to adaptively resample and combine (hence the acronym “arcing”) so that the weights in the resampling are increased for those cases most often misclassified and the combining is done by weighted voting. Arcing is more successful than bagging in test set error reduction. We explore two arcing algorithms, compare them to each other and to bagging, and try to understand how arcing works. We introduce the definitions of bias and variance for a classifier as components of the test set error. Unstable classifiers can have low bias on a large range of data sets. Their problem is high variance. Combining multiple versions either through bagging or arcing reduces variance significantly.

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Primary Subjects: 62H30
Full-text: Open access

Permanent link to this document: http://projecteuclid.org/euclid.aos/1024691079
Mathematical Reviews number (MathSciNet): MR1635406
Digital Object Identifier: doi:10.1214/aos/1024691079

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FLORHAM PARK, NEW JERSEY 07932-0971 E-MAIL: yoav@research.att.com schapire@research.att.com
Q I, where X is one component of a fixed-length feature vector and m Ä X c4 i i c is a constant. One could of course use multivariate functions or trans-gen erated features'' 7. Suppose we examine some of the queries by constructing a single binary tree TT by the usual data-driven induction method: stepwise entropy reduction estimated from a training set LL. Since we cannot entertain all possible splits at each node, we exploit a natural partial ordering on the set Q and examine only a tiny fraction of them. Basically we incrementally grow the geometric arrangements as we proceed down the tree. The classifier based on Z. Z. TT is then C Q, LL arg max P Y j TT. If the depths of the leaves of TT j Z. are far smaller than M, then evidently C Q, LL is not the Bay es classifier. However, for depths on the order of hundreds or thousands we could expect that P Y j TT P Y j Q, Z. Z. Z. the difference in some appropriate norm being a kind of approximation error.'' Of course, we cannot actually create or store a tree of such depth, and Z the best classification rate we obtained with a single tree of average depth. around ten was about 90% on test sets similar to the one discussed by Leo Breiman.
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TT, P Y k TT Y j, 1 n, m N. Naturally, small covariances lead to n m small errors. More generally, if the trees are produced with some sampling mechanism from the population of trees, involving either resampling from LL or random restrictions on the queries, then the quantities above can be analyzed by taking expectations relative to the space of trees. In regard to arcing, probably the deterministic reweighting on misclassified examples produces new trees which are quite different from the previous ones. Moreover, the errors induced on data points which were correctly classified by the existing trees are sufficiently randomized to avoid any sy stematic deterioration.
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Zentralblatt MATH: 0929.62069
Digital Object Identifier: doi:10.1214/aos/1024691352
Project Euclid: euclid.aos/1024691352
VAPNIK, V. N. 1995. The Nature of Statistical Learning Theory. Springer, New York.
Mathematical Reviews (MathSciNet): MR98a:68159
BERKELEY, CALIFORNIA 94720-3860 E-MAIL: leo@stat.berkeley.edu
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