  Ensembling has a long history in statistical data analysis, with many
impactful applications. However, in many modern machine learning settings, the
benefits of ensembling are less ubiquitous and less obvious. We study, both
theoretically and empirically, the fundamental question of when ensembling
yields significant performance improvements in classification tasks.
Theoretically, we prove new results relating the \emph{ensemble improvement
rate} (a measure of how much ensembling decreases the error rate versus a
single model, on a relative scale) to the \emph{disagreement-error ratio}. We
show that ensembling improves performance significantly whenever the
disagreement rate is large relative to the average error rate; and that,
conversely, one classifier is often enough whenever the disagreement rate is
low relative to the average error rate. On the way to proving these results, we
derive, under a mild condition called \emph{competence}, improved upper and
lower bounds on the average test error rate of the majority vote classifier. To
complement this theory, we study ensembling empirically in a variety of
settings, verifying the predictions made by our theory, and identifying
practical scenarios where ensembling does and does not result in large
performance improvements. Perhaps most notably, we demonstrate a distinct
difference in behavior between interpolating models (popular in current
practice) and non-interpolating models (such as tree-based methods, where
ensembling is popular), demonstrating that ensembling helps considerably more
in the latter case than in the former.
