  The goal of this work is to shed light on the remarkable phenomenon of
transition to linearity of certain neural networks as their width approaches
infinity. We show that the transition to linearity of the model and,
equivalently, constancy of the (neural) tangent kernel (NTK) result from the
scaling properties of the norm of the Hessian matrix of the network as a
function of the network width. We present a general framework for understanding
the constancy of the tangent kernel via Hessian scaling applicable to the
standard classes of neural networks. Our analysis provides a new perspective on
the phenomenon of constant tangent kernel, which is different from the widely
accepted "lazy training". Furthermore, we show that the transition to linearity
is not a general property of wide neural networks and does not hold when the
last layer of the network is non-linear. It is also not necessary for
successful optimization by gradient descent.
