Pruned Neural Networks are Surprisingly Modular

TL;DR

This paper introduces a measure of modularity in neural networks, revealing that training and pruning lead to more modular structures, especially with dropout, which may enhance interpretability.

Contribution

It proposes a new modularity measure for MLPs, demonstrating that training and pruning increase modularity, and explores the relationship between modules and network performance.

Findings

Pruned and trained MLPs are more modular than random or untrained networks.

Dropout during training increases network modularity.

Modules have varying importance and interdependence affecting performance.

Abstract

The learned weights of a neural network are often considered devoid of scrutable internal structure. To discern structure in these weights, we introduce a measurable notion of modularity for multi-layer perceptrons (MLPs), and investigate the modular structure of MLPs trained on datasets of small images. Our notion of modularity comes from the graph clustering literature: a "module" is a set of neurons with strong internal connectivity but weak external connectivity. We find that training and weight pruning produces MLPs that are more modular than randomly initialized ones, and often significantly more modular than random MLPs with the same (sparse) distribution of weights. Interestingly, they are much more modular when trained with dropout. We also present exploratory analyses of the importance of different modules for performance and how modules depend on each other. Understanding the modular structure of neural networks, when such structure exists, will hopefully render their inner workings more interpretable to engineers. Note that this paper has been superceded by "Clusterability in Neural Networks", arxiv:2103.03386 and "Quantifying Local Specialization in Deep Neural Networks", arxiv:2110.08058!