A Unified Theory of Diversity in Ensemble Learning

We present a theory of ensemble diversity, explaining the nature and effect of diversity for a wide range of supervised learning scenarios. This challenge, of understanding ensemble diversity, has been referred to as the holy grail of ensemble learning, an open question for over 30 years. Our framework reveals that diversity is in fact a hidden dimension in the bias-variance decomposition of an ensemble. In particular, we prove a family of exact bias-variance-diversity decompositions, for both classification and regression losses, e.g., squared, and cross-entropy. The framework provides a methodology to automatically identify the combiner rule enabling such a decomposition, specific to the loss. The formulation of diversity is therefore dependent on just two design choices: the loss, and the combiner. For certain choices (e.g., 0-1 loss with majority voting) the effect of diversity is necessarily dependent on the target label. Experiments illustrate how we can use our framework to understand the diversity-encouraging mechanisms of popular ensemble methods: Bagging, Boosting, and Random Forests.