  Deep learning, in the form of artificial neural networks, has achieved
remarkable practical success in recent years, for a variety of difficult
machine learning applications. However, a theoretical explanation for this
remains a major open problem, since training neural networks involves
optimizing a highly non-convex objective function, and is known to be
computationally hard in the worst case. In this work, we study the
\emph{geometric} structure of the associated non-convex objective function, in
the context of ReLU networks and starting from a random initialization of the
network parameters. We identify some conditions under which it becomes more
favorable to optimization, in the sense of (i) High probability of initializing
at a point from which there is a monotonically decreasing path to a global
minimum; and (ii) High probability of initializing at a basin (suitably
defined) with a small minimal objective value. A common theme in our results is
that such properties are more likely to hold for larger ("overspecified")
networks, which accords with some recent empirical and theoretical
observations.
