# Products of Many Large Random Matrices and Gradients in Deep Neural Networks

## TL;DR

This paper analyzes the behavior of products of large random matrices and their impact on gradients in deep neural networks, revealing Gaussian fluctuations and providing insights into gradient stability issues.

## Contribution

It introduces a new asymptotic Gaussian limit for the log-norm of matrix products and applies this to quantify gradient stability in deep neural networks.

## Key findings

- Logarithm of matrix product norms is asymptotically Gaussian.
- Explicit error bounds for moments and Gaussian approximation.
- Quantitative assessment of gradient explosion and vanishing in neural networks.

## Abstract

We study products of random matrices in the regime where the number of terms and the size of the matrices simultaneously tend to infinity. Our main theorem is that the logarithm of the $\ell_2$ norm of such a product applied to any fixed vector is asymptotically Gaussian. The fluctuations we find can be thought of as a finite temperature correction to the limit in which first the size and then the number of matrices tend to infinity. Depending on the scaling limit considered, the mean and variance of the limiting Gaussian depend only on either the first two or the first four moments of the measure from which matrix entries are drawn. We also obtain explicit error bounds on the moments of the norm and the Kolmogorov-Smirnov distance to a Gaussian. Finally, we apply our result to obtain precise information about the stability of gradients in randomly initialized deep neural networks with ReLU activations. This provides a quantitative measure of the extent to which the exploding and vanishing gradient problem occurs in a fully connected neural network with ReLU activations and a given architecture.