Empirical Comparison of Boosting

Methods for voting classiication algorithms, such as Bagging and AdaBoost, have been shown to be very successful in improving the accuracy of certain classiiers for artiicial and real-world datasets. We review these algorithms and describe a large empirical study comparing several variants in conjunction with a decision tree inducer (three variants) and a Naive-Bayes inducer. The purpose of the study is to improve our understanding of why and when these algorithms, which use perturbation, reweighting, and combination techniques, aaect classiication error. We provide a bias and variance decomposition of the error to show how diierent methods and variants innuence these two terms. This allowed us to determine that Bagging reduced variance of unstable methods, while boosting methods (AdaBoost and Arc-x4) reduced both the bias and variance of unstable methods but increased the variance for Naive-Bayes, which was very stable. We observed that Arc-x4 behaves diierently than AdaBoost if reweighting is used instead of resampling, indicating a fundamental diierence. Voting variants, some of which are introduced in this paper, include: pruning versus no pruning, use of probabilistic estimates, weight perturbations (Wagging), and backktting of data. We found that Bagging improves when probabilistic estimates in conjunction with no-pruning are used, as well as when the data was backkt. We measure tree sizes and show an interesting positive correlation between the increase in the average tree size in AdaBoost trials and its success in reducing the error. We compare the mean-squared error of voting methods to non-voting methods and show that the voting methods lead to large and signiicant reductions in the mean-squared errors. Practical problems that arise in implementing boosting algorithms are explored, including numerical instabilities and underrows. We use scatterplots that graphically show how AdaBoost reweights instances, emphasizing not only \hard" areas but also outliers and noise.