Spurious Local Minima are Common in Two-Layer ReLU Neural Networks

TL;DR

This paper demonstrates that spurious local minima are prevalent in training simple two-layer ReLU neural networks, especially in high dimensions, and that over-parameterization can significantly mitigate this issue.

Contribution

It provides a computer-assisted proof of the ubiquity of spurious local minima in two-layer ReLU networks and highlights the role of over-parameterization in avoiding them.

Findings

Spurious local minima exist for networks with 6 to 20 hidden units.

High-dimensional input data makes spurious minima nearly unavoidable.

Over-parameterization reduces the likelihood of encountering spurious minima.

Abstract

We consider the optimization problem associated with training simple ReLU neural networks of the form $\mathbf{x}\mapsto \sum_{i=1}^{k}\max\{0,\mathbf{w}_i^\top \mathbf{x}\}$ with respect to the squared loss. We provide a computer-assisted proof that even if the input distribution is standard Gaussian, even if the dimension is arbitrarily large, and even if the target values are generated by such a network, with orthonormal parameter vectors, the problem can still have spurious local minima once $6\le k\le 20$. By a concentration of measure argument, this implies that in high input dimensions, \emph{nearly all} target networks of the relevant sizes lead to spurious local minima. Moreover, we conduct experiments which show that the probability of hitting such local minima is quite high, and increasing with the network size. On the positive side, mild over-parameterization appears to drastically reduce such local minima, indicating that an over-parameterization assumption is necessary to get a positive result in this setting.