  Empirically it has been observed that the performance of deep neural networks
steadily improves as we increase model size, contradicting the classical view
on overfitting and generalization. Recently, the double descent phenomena has
been proposed to reconcile this observation with theory, suggesting that the
test error has a second descent when the model becomes sufficiently
overparameterized, as the model size itself acts as an implicit regularizer. In
this paper we add to the growing body of work in this space, providing a
careful study of learning dynamics as a function of model size for the least
squares scenario. We show an excess risk bound for the gradient descent
solution of the least squares objective. The bound depends on the smallest
non-zero eigenvalue of the covariance matrix of the input features, via a
functional form that has the double descent behavior. This gives a new
perspective on the double descent curves reported in the literature. Our
analysis of the excess risk allows to decouple the effect of optimization and
generalization error. In particular, we find that in case of noiseless
regression, double descent is explained solely by optimization-related
quantities, which was missed in studies focusing on the Moore-Penrose
pseudoinverse solution. We believe that our derivation provides an alternative
view compared to existing work, shedding some light on a possible cause of this
phenomena, at least in the considered least squares setting. We empirically
explore if our predictions hold for neural networks, in particular whether the
covariance of intermediary hidden activations has a similar behavior as the one
predicted by our derivations.
