# Properties of the geometry of solutions and capacity of multi-layer neural networks with Rectified Linear Units activations

## TL;DR

This paper analytically investigates how ReLU activations influence the capacity and geometry of solution spaces in two-layer neural networks, revealing finite capacity and unique solution clustering properties.

## Contribution

It provides the first analytical insights into ReLU effects on network capacity and solution landscape, contrasting with threshold units and highlighting robustness features.

## Key findings

- Network capacity remains finite as hidden layer size increases.
- Existence of dense, robust solution regions in the solution space.
- Solutions are mostly isolated but some form large, stable clusters.

## Abstract

Rectified Linear Units (ReLU) have become the main model for the neural units in current deep learning systems. This choice has been originally suggested as a way to compensate for the so called vanishing gradient problem which can undercut stochastic gradient descent (SGD) learning in networks composed of multiple layers. Here we provide analytical results on the effects of ReLUs on the capacity and on the geometrical landscape of the solution space in two-layer neural networks with either binary or real-valued weights. We study the problem of storing an extensive number of random patterns and find that, quite unexpectedly, the capacity of the network remains finite as the number of neurons in the hidden layer increases, at odds with the case of threshold units in which the capacity diverges. Possibly more important, a large deviation approach allows us to find that the geometrical landscape of the solution space has a peculiar structure: while the majority of solutions are close in distance but still isolated, there exist rare regions of solutions which are much more dense than the similar ones in the case of threshold units. These solutions are robust to perturbations of the weights and can tolerate large perturbations of the inputs. The analytical results are corroborated by numerical findings.