  Ensembles of machine learning models yield improved system performance as
well as robust and interpretable uncertainty estimates; however, their
inference costs may often be prohibitively high. \emph{Ensemble Distribution
Distillation} is an approach that allows a single model to efficiently capture
both the predictive performance and uncertainty estimates of an ensemble. For
classification, this is achieved by training a Dirichlet distribution over the
ensemble members' output distributions via the maximum likelihood criterion.
Although theoretically principled, this criterion exhibits poor convergence
when applied to large-scale tasks where the number of classes is very high. In
our work, we analyze this effect and show that the Dirichlet log-likelihood
criterion classes with low probability induce larger gradients than
high-probability classes. This forces the model to focus on the distribution of
the ensemble tail-class probabilities. We propose a new training objective that
minimizes the reverse KL-divergence to a \emph{Proxy-Dirichlet} target derived
from the ensemble. This loss resolves the gradient issues of Ensemble
Distribution Distillation, as we demonstrate both theoretically and empirically
on the ImageNet and WMT17 En-De datasets containing 1000 and 40,000 classes,
respectively.
