  Many real-world recognition problems are characterized by long-tailed label
distributions. These distributions make representation learning highly
challenging due to limited generalization over the tail classes. If the test
distribution differs from the training distribution, e.g. uniform versus
long-tailed, the problem of the distribution shift needs to be addressed. A
recent line of work proposes learning multiple diverse experts to tackle this
issue. Ensemble diversity is encouraged by various techniques, e.g. by
specializing different experts in the head and the tail classes. In this work,
we take an analytical approach and extend the notion of logit adjustment to
ensembles to form a Balanced Product of Experts (BalPoE). BalPoE combines a
family of experts with different test-time target distributions, generalizing
several previous approaches. We show how to properly define these distributions
and combine the experts in order to achieve unbiased predictions, by proving
that the ensemble is Fisher-consistent for minimizing the balanced error. Our
theoretical analysis shows that our balanced ensemble requires calibrated
experts, which we achieve in practice using mixup. We conduct extensive
experiments and our method obtains new state-of-the-art results on three
long-tailed datasets: CIFAR-100-LT, ImageNet-LT, and iNaturalist-2018. Our code
is available at https://github.com/emasa/BalPoE-CalibratedLT.
