On the doubt about margin explanation of boosting

Margin theory provides one of the most popular explanations to the success of AdaBoost, where the central point lies in the recognition that margin is the key for characterizing the performance of AdaBoost. This theory has been very influential, e.g., it has been used to argue that AdaBoost usually does not overfit since it tends to enlarge the margin even after the training error reaches zero. Previously the minimum margin bound was established for AdaBoost, however, Breiman [10] pointed out that maximizing the minimum margin does not necessarily lead to a better generalization. Later, Reyzin and Schapire [34] emphasized that the margin distribution rather than minimum margin is crucial to the performance of AdaBoost. In this paper, we show that previous margin bounds are special cases of the kth margin bound, and none of them is really based on the whole margin distribution. Then, we improve the empirical Bernstein bound given by Maurer and Pontil [28]. Based on this result, we defend the margin-based explanation against Breiman’s doubt by proving a new generalization error bound that considers exactly the same factors as Schapire et al. [35] but is uniformly tighter than Breiman [10]’s bound. We also provide a lower bound for generalization error of voting classifiers, and by incorporating factors such as average margin and variance, we present a generalization error bound that is heavily related to the whole margin distribution. Finally, we provide empirical evidence to verify our theory.