  The stunning empirical successes of neural networks currently lack rigorous
theoretical explanation. What form would such an explanation take, in the face
of existing complexity-theoretic lower bounds? A first step might be to show
that data generated by neural networks with a single hidden layer, smooth
activation functions and benign input distributions can be learned efficiently.
We demonstrate here a comprehensive lower bound ruling out this possibility:
for a wide class of activation functions (including all currently used), and
inputs drawn from any logconcave distribution, there is a family of
one-hidden-layer functions whose output is a sum gate, that are hard to learn
in a precise sense: any statistical query algorithm (which includes all known
variants of stochastic gradient descent with any loss function) needs an
exponential number of queries even using tolerance inversely proportional to
the input dimensionality. Moreover, this hard family of functions is realizable
with a small (sublinear in dimension) number of activation units in the single
hidden layer. The lower bound is also robust to small perturbations of the true
weights. Systematic experiments illustrate a phase transition in the training
error as predicted by the analysis.
