Analysis of boosting algorithms using the smooth margin function

TL;DR

This paper introduces the smooth margin function as a differentiable tool for analyzing boosting algorithms, proposes new boosting methods based on it, and provides convergence analysis and bounds for AdaBoost.

Contribution

It presents the smooth margin function, develops two new boosting algorithms, and offers convergence rates and margin bounds for AdaBoost and related algorithms.

Findings

Convergence rates to maximum margin for proposed algorithms and arc-gv.

Exact tightness of previous margin bounds for AdaBoost.

Explicit properties of AdaBoost in cyclic behavior cases.

Abstract

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.