# Margin-based Ranking and an Equivalence between AdaBoost and RankBoost

Cynthia Rudin, Robert E. Schapire.

Year: 2009, Volume: 10, Issue: 77, Pages: 2193−2232

#### Abstract

We study boosting algorithms for learning to rank. We give a general margin-based bound for ranking based on covering numbers for the hypothesis space. Our bound suggests that algorithms that maximize the ranking margin will generalize well. We then describe a new algorithm, smooth margin ranking, that precisely converges to a maximum ranking-margin solution. The algorithm is a modification of RankBoost, analogous to "approximate coordinate ascent boosting." Finally, we prove that AdaBoost and RankBoost are equally good for the problems of bipartite ranking and classification in terms of their asymptotic behavior on the training set. Under natural conditions, AdaBoost achieves an area under the ROC curve that is equally as good as RankBoost's; furthermore, RankBoost, when given a specific intercept, achieves a misclassification error that is as good as AdaBoost's. This may help to explain the empirical observations made by Cortes and Mohri, and Caruana and Niculescu-Mizil, about the excellent performance of AdaBoost as a bipartite ranking algorithm, as measured by the area under the ROC curve.