  Background: Statistical mechanics results (Dauphin et al. (2014); Choromanska
et al. (2015)) suggest that local minima with high error are exponentially rare
in high dimensions. However, to prove low error guarantees for Multilayer
Neural Networks (MNNs), previous works so far required either a heavily
modified MNN model or training method, strong assumptions on the labels (e.g.,
"near" linear separability), or an unrealistic hidden layer with
$\Omega\left(N\right)$ units.
  Results: We examine a MNN with one hidden layer of piecewise linear units, a
single output, and a quadratic loss. We prove that, with high probability in
the limit of $N\rightarrow\infty$ datapoints, the volume of differentiable
regions of the empiric loss containing sub-optimal differentiable local minima
is exponentially vanishing in comparison with the same volume of global minima,
given standard normal input of dimension
$d_{0}=\tilde{\Omega}\left(\sqrt{N}\right)$, and a more realistic number of
$d_{1}=\tilde{\Omega}\left(N/d_{0}\right)$ hidden units. We demonstrate our
results numerically: for example, $0\%$ binary classification training error on
CIFAR with only $N/d_{0}\approx 16$ hidden neurons.
