Exponentially vanishing sub-optimal local minima in multilayer neural networks

Abstract

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.