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We formally study how ensemble of deep learning models can improve testaccuracy, and how the superior performance of ensemble can be distilled into asingle model using knowledge distillation. We consider the challenging casewhere the ensemble is simply an average of the outputs of a few independentlytrained neural networks with the SAME architecture, trained using the SAMEalgorithm on the SAME data set, and they only differ by the random seeds usedin the initialization.  We show that ensemble/knowledge distillation in Deep Learning works verydifferently from traditional learning theory (such as boosting or NTKs, neuraltangent kernels). To properly understand them, we develop a theory showing thatwhen data has a structure we refer to as ``multi-view'', then ensemble ofindependently trained neural networks can provably improve test accuracy, andsuch superior test accuracy can also be provably distilled into a single modelby training a single model to match the output of the ensemble instead of thetrue label. Our result sheds light on how ensemble works in deep learning in away that is completely different from traditional theorems, and how the ``darkknowledge'' is hidden in the outputs of the ensemble and can be used indistillation. In the end, we prove that self-distillation can also be viewed asimplicitly combining ensemble and knowledge distillation to improve testaccuracy.

Towards Understanding Ensemble, Knowledge Distillation and Self-Distillation in Deep Learning