Exponentially vanishing sub-optimal local minima in multilayer neural networks

TL;DR

This paper proves that in high-dimensional multilayer neural networks, sub-optimal local minima are exponentially rare compared to global minima, especially as the number of data points grows, with supporting numerical experiments.

Contribution

It provides a rigorous proof that sub-optimal local minima are exponentially unlikely in large neural networks under realistic conditions, improving understanding of loss landscape geometry.

Findings

Sub-optimal local minima volume vanishes exponentially with data size.

High probability of global minima presence in large networks.

Numerical validation on CIFAR dataset with few hidden neurons.

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.