  Distillation is a method to transfer knowledge from one model to another and
often achieves higher accuracy with the same capacity. In this paper, we aim to
provide a theoretical understanding on what mainly helps with the distillation.
Our answer is "early stopping". Assuming that the teacher network is
overparameterized, we argue that the teacher network is essentially harvesting
dark knowledge from the data via early stopping. This can be justified by a new
concept, {Anisotropic Information Retrieval (AIR)}, which means that the neural
network tends to fit the informative information first and the non-informative
information (including noise) later. Motivated by the recent development on
theoretically analyzing overparameterized neural networks, we can characterize
AIR by the eigenspace of the Neural Tangent Kernel(NTK). AIR facilities a new
understanding of distillation. With that, we further utilize distillation to
refine noisy labels. We propose a self-distillation algorithm to sequentially
distill knowledge from the network in the previous training epoch to avoid
memorizing the wrong labels. We also demonstrate, both theoretically and
empirically, that self-distillation can benefit from more than just early
stopping. Theoretically, we prove convergence of the proposed algorithm to the
ground truth labels for randomly initialized overparameterized neural networks
in terms of $\ell_2$ distance, while the previous result was on convergence in
$0$-$1$ loss. The theoretical result ensures the learned neural network enjoy a
margin on the training data which leads to better generalization. Empirically,
we achieve better testing accuracy and entirely avoid early stopping which
makes the algorithm more user-friendly.
