Exponentially vanishing sub-optimal local minima in multilayer neural networks

We examine a multilayer neural network with piecewise linear units, input of dimension $d_{0}$, one hidden layer of width $d_{1}$, a single output, and a quadratic loss, trained on $N$ datapoints. We prove that in the limit that $N\rightarrow\infty$, the volume of differentiable regions of the loss containing sub-optimal differentiable local minima is exponentially vanishing in comparison with the same volume of global minima, given standard normal input, $d_{0}\left(N\right)=\tilde{\Omega}\left(\sqrt{N}\right)$ and an asymptotically "mild" over-parameterization: $\#\mathrm{parameters=\,}\tilde{\Omega}\left(N\right)$. Previous results on vanishing local minima so far required many more parameters: $\#\mathrm{parameters=\,}\Omega\left(Nd_{0}\left(N\right)\right)$, which is typically worse.