  Ensemble methods are a cornerstone of modern machine learning. The
performance of an ensemble depends crucially upon the level of diversity
between its constituent learners. This paper establishes a connection between
diversity and degrees of freedom (i.e. the capacity of the model), showing that
diversity may be viewed as a form of inverse regularisation. This is achieved
by focusing on a previously published algorithm Negative Correlation Learning
(NCL), in which model diversity is explicitly encouraged through a diversity
penalty term in the loss function. We provide an exact formula for the
effective degrees of freedom in an NCL ensemble with fixed basis functions,
showing that it is a continuous, convex and monotonically increasing function
of the diversity parameter. We demonstrate a connection to Tikhonov
regularisation and show that, with an appropriately chosen diversity parameter,
an NCL ensemble can always outperform the unregularised ensemble in the
presence of noise. We demonstrate the practical utility of our approach by
deriving a method to efficiently tune the diversity parameter. Finally, we use
a Monte-Carlo estimator to extend the connection between diversity and degrees
of freedom to ensembles of deep neural networks.
