  In this work we consider a model problem of deep neural learning, namely the
learning of a given function when it is assumed that we have access to its
point values on a finite set of points. The deep neural network interpolant is
the the resulting approximation of f, which is obtained by a typical machine
learning algorithm involving a given DNN architecture and an optimisation step,
which is assumed to be solved exactly. These are among the simplest regression
algorithms based on neural networks. In this work we introduce a new approach
to the estimation of the (generalisation) error and to convergence. Our results
include (i) estimates of the error without any structural assumption on the
neural networks and under mild regularity assumptions on the learning function
f (ii) convergence of the approximations to the target function f by only
requiring that the neural network spaces have appropriate approximation
capability.
