  In the context of neural network models, overparametrization refers to the
phenomena whereby these models appear to generalize well on the unseen data,
even though the number of parameters significantly exceeds the sample sizes,
and the model perfectly fits the in-training data. A conventional explanation
of this phenomena is based on self-regularization properties of algorithms used
to train the data. In this paper we prove a series of results which provide a
somewhat diverging explanation. Adopting a teacher/student model where the
teacher network is used to generate the predictions and student network is
trained on the observed labeled data, and then tested on out-of-sample data, we
show that any student network interpolating the data generated by a teacher
network generalizes well, provided that the sample size is at least an explicit
quantity controlled by data dimension and approximation guarantee alone,
regardless of the number of internal nodes of either teacher or student
network.
  Our claim is based on approximating both teacher and student networks by
polynomial (tensor) regression models with degree depending on the desired
accuracy and network depth only. Such a parametrization notably does not depend
on the number of internal nodes. Thus a message implied by our results is that
parametrizing wide neural networks by the number of hidden nodes is misleading,
and a more fitting measure of parametrization complexity is the number of
regression coefficients associated with tensorized data. In particular, this
somewhat reconciles the generalization ability of neural networks with more
classical statistical notions of data complexity and generalization bounds. Our
empirical results on MNIST and Fashion-MNIST datasets indeed confirm that
tensorized regression achieves a good out-of-sample performance, even when the
degree of the tensor is at most two.
