  Depth separation results propose a possible theoretical explanation for the
benefits of deep neural networks over shallower architectures, establishing
that the former possess superior approximation capabilities. However, there are
no known results in which the deeper architecture leverages this advantage into
a provable optimization guarantee. We prove that when the data are generated by
a distribution with radial symmetry which satisfies some mild assumptions,
gradient descent can efficiently learn ball indicator functions using a depth 2
neural network with two layers of sigmoidal activations, and where the hidden
layer is held fixed throughout training. By building on and refining existing
techniques for approximation lower bounds of neural networks with a single
layer of non-linearities, we show that there are $d$-dimensional radial
distributions on the data such that ball indicators cannot be learned
efficiently by any algorithm to accuracy better than $\Omega(d^{-4})$, nor by a
standard gradient descent implementation to accuracy better than a constant.
These results establish what is to the best of our knowledge, the first
optimization-based separations where the approximation benefits of the stronger
architecture provably manifest in practice. Our proof technique introduces new
tools and ideas that may be of independent interest in the theoretical study of
both the approximation and optimization of neural networks.
