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.

通过证明，使用具有分段线性单元、单输出和二次损失的一层隐藏层的 MNN，在标准正常输入和更现实的数量的隐藏单元情况下，可以消失指数数量的不同可微区域的的局部最小值，以及通过数值演示达到的结果，发现在 CIFAR 上只有 16 个隐藏神经元时可以达到 0％的二进制分类训练误差。