  In this paper we study the problem of learning a shallow artificial neural
network that best fits a training data set. We study this problem in the
over-parameterized regime where the number of observations are fewer than the
number of parameters in the model. We show that with quadratic activations the
optimization landscape of training such shallow neural networks has certain
favorable characteristics that allow globally optimal models to be found
efficiently using a variety of local search heuristics. This result holds for
an arbitrary training data of input/output pairs. For differentiable activation
functions we also show that gradient descent, when suitably initialized,
converges at a linear rate to a globally optimal model. This result focuses on
a realizable model where the inputs are chosen i.i.d. from a Gaussian
distribution and the labels are generated according to planted weight
coefficients.
