  It has been hypothesized that some form of "modular" structure in artificial
neural networks should be useful for learning, compositionality, and
generalization. However, defining and quantifying modularity remains an open
problem. We cast the problem of detecting functional modules into the problem
of detecting clusters of similar-functioning units. This begs the question of
what makes two units functionally similar. For this, we consider two broad
families of methods: those that define similarity based on how units respond to
structured variations in inputs ("upstream"), and those based on how variations
in hidden unit activations affect outputs ("downstream"). We conduct an
empirical study quantifying modularity of hidden layer representations of
simple feedforward, fully connected networks, across a range of
hyperparameters. For each model, we quantify pairwise associations between
hidden units in each layer using a variety of both upstream and downstream
measures, then cluster them by maximizing their "modularity score" using
established tools from network science. We find two surprising results: first,
dropout dramatically increased modularity, while other forms of weight
regularization had more modest effects. Second, although we observe that there
is usually good agreement about clusters within both upstream methods and
downstream methods, there is little agreement about the cluster assignments
across these two families of methods. This has important implications for
representation-learning, as it suggests that finding modular representations
that reflect structure in inputs (e.g. disentanglement) may be a distinct goal
from learning modular representations that reflect structure in outputs (e.g.
compositionality).
