  Knowledge distillation is classically a procedure where a neural network is
trained on the output of another network along with the original targets in
order to transfer knowledge between the architectures. The special case of
self-distillation, where the network architectures are identical, has been
observed to improve generalization accuracy. In this paper, we consider an
iterative variant of self-distillation in a kernel regression setting, in which
successive steps incorporate both model outputs and the ground-truth targets.
This allows us to provide the first theoretical results on the importance of
using the weighted ground-truth targets in self-distillation. Our focus is on
fitting nonlinear functions to training data with a weighted mean square error
objective function suitable for distillation, subject to $\ell_2$
regularization of the model parameters. We show that any such function obtained
with self-distillation can be calculated directly as a function of the initial
fit, and that infinite distillation steps yields the same optimization problem
as the original with amplified regularization. Furthermore, we provide a closed
form solution for the optimal choice of weighting parameter at each step, and
show how to efficiently estimate this weighting parameter for deep learning and
significantly reduce the computational requirements compared to a grid search.
