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Check out our latest preprint (with @maithra_raghu and Samy Bengio) on the relationship between representational similarity and generalization and optimization in CNNs and training and sequence dynamics in RNNs! 1/N
Blog: https://ai.googleblog.com/2018/06/how-can-neural-network-similarity-help.html …
Paper: http://arxiv.org/abs/1806.05759
Building off @maithra_raghu's awesome previous work on SVCCA (https://arxiv.org/abs/1706.05806 ) we first develop a new weighted version of CCA distance which better distinguishes between signal and noise in representational similarity.pic.twitter.com/tz20LpOGMx
Applying CCA to generalizing and memorizing networks, we found that generalizing networks converged to more similar solutions than memorizing networks, especially in later layers, suggesting that while there are lots of ways to memorize, there are only a few ways to generalize.pic.twitter.com/IRnAxumSMJ
Inspired by the lottery ticket hypothesis (https://arxiv.org/abs/1803.03635 from @jefrankle), we reasoned that wider networks should converge to more similar solutions than narrow networks since larger networks are more likely to "win the initialization lottery," and indeed, they do!pic.twitter.com/1vrCZ0Kr4e
CCA also reveals clusters of solutions across different initializations and learning rates, and that these nearly perfectly match the clusters found in my previous work (https://arxiv.org/abs/1803.06959 ), suggesting that nets with similar performance often learn highly diverse solutions.pic.twitter.com/I51KXIwbOm
Moving to RNNs, we found that like CNNs, RNNs exhibit bottom-up convergence dynamics, in which layers close to the input converge to their final representations earlier in training than layer close to the output.pic.twitter.com/JBFbka83zl
Because representations in RNNs change over a sequence, CCA is an ideal tool to study the extent to which they vary non-linearly. Interestingly, we found that the non-linearity of recurrence depends on the number of past inputs to an RNN. Lots more to do in this direction though!pic.twitter.com/E0UOHZrtVn
Check out the paper for more detail and controls with other distance metrics, etc. The code for CCA is also open-source (https://github.com/google/svcca/ ), so apply CCA to your networks and see what you find!
Looks really interesting Ari! In the RNN case, are you suggesting that these task-optimized networks are not necessarily falling into “attractor states” over the course of the input sequence?
Thanks, Dan! The RNN over the course of a sequence work is definitely the least fleshed out, so I'm hesitant to draw any strong conclusions, but it does seem like the hidden states over the course of a sequence aren't that similar subject to a linear transform.
In fact, the similarity between CCA, cosine and Euclidean distance suggests that there isn't necessarily even a strong rotation/scale component at all.
To try to work out if this was due to the input or recurrent weights, we tried repeating the same input and measuring distance between states. CCA then shows the states as similar, suggesting that the recurrent dynamics are not inherently non-linear.
But, oddly, this result changes depending on how many previous inputs the RNN has seen over inference, so if you've shown the RNN 30,000 previous tokens, this effect decreases substantially. This seems to suggest that the non-linearity of the dynamics depends on network past.
Not really sure what to make of that result, but definitely an interesting question for future work!
Looking forward to close reading!
Thanks for the great work! Have you tried inverting the currently common design scheme of increasing layer width with depth , i.e. larger early widths with monotical decrease as is common in MLPs. I've found the correlation to flip & performance to be unaffected in these cases.
No, we haven't tried that, but it would be interesting to see what happens in that case!
In your larger width networks, how much of the increasing similarity do you think is due to a greater subset not deviating much from the initial state & thus a question of how similar sampling of initialization scheme is? Or is this what you mean with "winning the lottery"?
In the Appendix, we have a random weights control which shows that the increase in similarity is not merely due to increased layer size. The lottery ticket hypothesis frames it as a larger networks having a higher chance of a lucky init in a subnetwork.
Ah thanks! I should have taken a look at the appendix :)
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