Deep-layered machines have a built-in Occam's razor

TL;DR

This paper provides an exact theoretical analysis showing that deep-layered Boolean networks inherently bias their outputs towards simplicity as depth increases, explaining the prevalence of simple solutions in complex systems.

Contribution

It introduces an exact theory for output distributions in deep Boolean networks and confirms the bias towards simplicity through extensive experiments.

Findings

Distribution becomes exponentially biased towards simple outputs with increased depth

Deep-layered machines inherently favor simplicity in generated models

Theoretical predictions are validated by computer experiments

Abstract

Input-output maps are prevalent throughout science and technology. They are empirically observed to be biased towards simple outputs, but we don't understand why. To address this puzzle, we study the archetypal input-output map: a deep-layered machine in which every node is a Boolean function of all the nodes below it. We give an exact theory for the distribution of outputs, and we confirm our predictions through extensive computer experiments. As the network depth increases, the distribution becomes exponentially biased towards simple outputs. This suggests that deep-layered machines and other learning methodologies may be inherently biased towards simplicity in the models that they generate.