  Boosting is one of the most successful ideas in machine learning. The most
well-accepted explanations for the low generalization error of boosting
algorithms such as AdaBoost stem from margin theory. The study of margins in
the context of boosting algorithms was initiated by Schapire, Freund, Bartlett
and Lee (1998) and has inspired numerous boosting algorithms and generalization
bounds. To date, the strongest known generalization (upper bound) is the $k$th
margin bound of Gao and Zhou (2013). Despite the numerous generalization upper
bounds that have been proved over the last two decades, nothing is known about
the tightness of these bounds. In this paper, we give the first margin-based
lower bounds on the generalization error of boosted classifiers. Our lower
bounds nearly match the $k$th margin bound and thus almost settle the
generalization performance of boosted classifiers in terms of margins.
