New preprint/tweeprint! 🧵👇
— Dan Goodman (@neuralreckoning) June 16, 2021
Modularity can be structural (what connects to what) or functional (specialised groups of neurons). Are these related? Yes, but more weakly than you might guess.
Work by PhD student Gabriel Béna - feedback appreciated!https://t.co/h70TXa7jFT
TLDR: enforcing structural modularity in the architecture of a NN trained on a task naturally composed of subtasks leads to module specialisation on subtasks, but only at extreme levels. Even quite high levels of structural modularity lead to no functional specialisation.
— Dan Goodman (@neuralreckoning) June 16, 2021
We looked at the simplest possible case of two modules, each densely connected with sparse connections between them. This lets us precisely control structural modularity from maximum (single interconnect) to no modularity (fully interconnected).
— Dan Goodman (@neuralreckoning) June 16, 2021
Each module gets a separate input MNIST digit. The whole network has to return one or the other of the digits depending on whether two digit parities same or different. This forces modules to share some info, but allows for specialisation on subtask of classifying each digit.
— Dan Goodman (@neuralreckoning) June 16, 2021
Now the tricky part: how do we measure whether or not the modules have specialised? This is a surprisingly complicated question - see the replies in this thread: https://t.co/vS89qj08xy
— Dan Goodman (@neuralreckoning) June 16, 2021
Thanks to everyone who replied in that thread - your ideas really helped!
We couldn't find a single satisfactory answer, so we tried three separate measures of functional specialisation (based on informational bottleneck, weight masks, correlation). Fortunately, they all qualitatively agreed in this case, so it seems they are measuring something real.
— Dan Goodman (@neuralreckoning) June 16, 2021
Results: functional specialisation increases monotonically with structural modularity Q, but only becomes substantial when Q is close to max (Q=0.5). If there are more than a handful of connections between modules, they become functionally entangled. pic.twitter.com/MFAKV3WAoN
— Dan Goodman (@neuralreckoning) June 16, 2021
So what's the take home message here? Firstly, it's great that these three very different measures broadly agree. Secondly, if you're looking to use structural modularity as a proxy for functional modularity, beware! You only get that at extreme levels.
— Dan Goodman (@neuralreckoning) June 16, 2021
For reference, a network with a structural modularity Q=0.35 is usually described as "highly modular", but in all of our measures doesn't give rise to much functional specialisation at all. Does this pose problems for the connectomics project?
— Dan Goodman (@neuralreckoning) June 16, 2021
One thing we'd like to look into in future is how this changes when the number of bits of information that the modules need to share changes. In our task, they only need to share one bit.
— Dan Goodman (@neuralreckoning) June 16, 2021
Any other ideas or comments/questions?
Thanks for reading!
Also, I'm using this as an opportunity to share the "notpaper" version of this preprint. It's an experimental new way of reading papers I'm working on as a side project. Would be interested in feedback on this too - do you find it helpful?https://t.co/zx5ifDT3do
— Dan Goodman (@neuralreckoning) June 16, 2021
Extreme sparsity gives rise to functional specialization
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