  Boosting and other ensemble methods combine a large number of weak
classifiers through weighted voting to produce stronger predictive models. To
explain the successful performance of boosting algorithms, Schapire et al.
(1998) showed that AdaBoost is especially effective at increasing the margins
of the training data. Schapire et al. (1998) also developed an upper bound on
the generalization error of any ensemble based on the margins of the training
data, from which it was concluded that larger margins should lead to lower
generalization error, everything else being equal (sometimes referred to as the
``large margins theory''). Tighter bounds have been derived and have reinforced
the large margins theory hypothesis. For instance, Wang et al. (2011) suggest
that specific margin instances, such as the equilibrium margin, can better
summarize the margins distribution. These results have led many researchers to
consider direct optimization of the margins to improve ensemble generalization
error with mixed results. We show that the large margins theory is not
sufficient for explaining the performance of voting classifiers. We do this by
illustrating how it is possible to improve upon the margin distribution of an
ensemble solution, while keeping the complexity fixed, yet not improve the test
set performance.
