Random deep neural networks are biased towards simple functions

TL;DR

This paper proves that random wide deep neural networks with ReLU activation tend to produce simple functions, which helps explain their strong generalization properties and provides insights into their prior distribution in PAC-Bayesian bounds.

Contribution

It establishes that random deep neural networks are biased towards simple functions, confirming a conjecture and advancing understanding of their generalization behavior.

Findings

Average Hamming distance to different classification is at least sqrt(n / (2π log n))

Number of flips to change classification grows linearly with n

Results confirmed by numerical experiments on networks with two hidden layers

Abstract

We prove that the binary classifiers of bit strings generated by random wide deep neural networks with ReLU activation function are biased towards simple functions. The simplicity is captured by the following two properties. For any given input bit string, the average Hamming distance of the closest input bit string with a different classification is at least sqrt(n / (2{\pi} log n)), where n is the length of the string. Moreover, if the bits of the initial string are flipped randomly, the average number of flips required to change the classification grows linearly with n. These results are confirmed by numerical experiments on deep neural networks with two hidden layers, and settle the conjecture stating that random deep neural networks are biased towards simple functions. This conjecture was proposed and numerically explored in [Valle P\'erez et al., ICLR 2019] to explain the unreasonably good generalization properties of deep learning algorithms. The probability distribution of the functions generated by random deep neural networks is a good choice for the prior probability distribution in the PAC-Bayesian generalization bounds. Our results constitute a fundamental step forward in the characterization of this distribution, therefore contributing to the understanding of the generalization properties of deep learning algorithms.