Analysis of boosting algorithms using the smooth margin function

Details Analysis of boosting algorithms using the smooth margin function Analysis of boosting algorithms using the smooth margin function

2008-03-28

arxiv.org/abs/0803.4092v1

Annals of Statistics 2007, Vol. 35, No. 6, 2723-2768

Statistics

Machine Learning

Published in at http://dx.doi.org/10.1214/009053607000000785 the Annals of Statistics (http://www.imstat.org/aos/) by the Inst

Scientific paper

10.1214/009053607000000785

We introduce a useful tool for analyzing boosting algorithms called the ``smooth margin function,'' a differentiable approximation of the usual margin for boosting algorithms. We present two boosting algorithms based on this smooth margin, ``coordinate ascent boosting'' and ``approximate coordinate ascent boosting,'' which are similar to Freund and Schapire's AdaBoost algorithm and Breiman's arc-gv algorithm. We give convergence rates to the maximum margin solution for both of our algorithms and for arc-gv. We then study AdaBoost's convergence properties using the smooth margin function. We precisely bound the margin attained by AdaBoost when the edges of the weak classifiers fall within a specified range. This shows that a previous bound proved by R\"{a}tsch and Warmuth is exactly tight. Furthermore, we use the smooth margin to capture explicit properties of AdaBoost in cases where cyclic behavior occurs.

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