  Ensemble algorithms offer state of the art performance in many machine
learning applications. A common explanation for their excellent performance is
due to the bias-variance decomposition of the mean squared error which shows
that the algorithm's error can be decomposed into its bias and variance. Both
quantities are often opposed to each other and ensembles offer an effective way
to manage them as they reduce the variance through a diverse set of base
learners while keeping the bias low at the same time. Even though there have
been numerous works on decomposing other loss functions, the exact mathematical
connection is rarely exploited explicitly for ensembling, but merely used as a
guiding principle. In this paper, we formulate a generalized bias-variance
decomposition for arbitrary twice differentiable loss functions and study it in
the context of Deep Learning. We use this decomposition to derive a Generalized
Negative Correlation Learning (GNCL) algorithm which offers explicit control
over the ensemble's diversity and smoothly interpolates between the two
extremes of independent training and the joint training of the ensemble. We
show how GNCL encapsulates many previous works and discuss under which
circumstances training of an ensemble of Neural Networks might fail and what
ensembling method should be favored depending on the choice of the individual
networks. We make our code publicly available under
https://github.com/sbuschjaeger/gncl
