This is a self review. It's been about 600 days since this was posted and I'm still happy and proud about this post. In terms of what I view as the important message to the readership, the main thing is introducing a framework and way of thinking that connects what is a pretty fuzzy notion of "world model" to the concrete internal structure of neural networks. It does this in a way that is both theoretically clear and amenable to experiments. It provides a way to think about representations in transformers in a general sense, that is quite different than the normal tactic taken in interpretability. One interesting thing given how interp has progressed since then, is that it provides a way to think about geometric structures in the activations of these networks, which is something the field seems to be moving towards (albeit not guided by theory the way this work is).
I spent a lot of time/effort on making the presentation both easy to follow while actually containing the important lessons. I feel this post does a great job at that. (and in fact, others around me were telling me I was taking too long and that I just needed to hit submit - I'm glad I didn't follow that advice :)
This work has been expanded on in a number of directions (a recent post summarizes three of those direction). We now have Simplex, an entire org that would likely not exist in the way that it does without this post (we got seed funding right before this post, but the ability to fundraise and attract talent were highly dependent on this post)! We are ~10 people and expanding. Starting this org, watching it grow, being part of the building up of a new way to think about/do interpretability has been the greatest intellectual achievement of my life, and again, I do not think this would have occured (at least not the way it did), without this post.
Some of the things we are working on is extending the framework to more complicated (factorizable) generative structures, which gives way to both a very nice story explaining how it is neural nets can cram so much into their activations, and the corresponding geometric relationship to computational structure. We also have a team dedicated to red-teaming the theory and application to experiments in a number of ways, one of which is more precisely testing the distinction between representing next-token vs. far future representaitons. We are also looking for the types of geometric structures our theory predicts in LLMs, and building tools to do unsuprvised finding of these geometric structures in LLMs.
In retrospect there are a few things I would have emphasized or done differently. The most concrete one would have been to include experiments on RRXOR in this writeup, which more directly test/show representations that have information beyond the next token, and that are predicted by the theory.
Another regret is that I haven't done a good job of keeping up with the public facing side of our work since this post. We made one post about our work since then, and it really does hide a lot of the insights we've made internally. This is not great, since I do believe that we have refined our way of thinking about representations and computations in these systems, in a way that the interp community needs.
More personally (and perhaps this isn't relevant for a review since it really is just about me), this post came after switching fields from neuroscience, where I was for a decade, to interpretability. There was a lot of uncertainty and stress about that move, and quite a bit of wondering if I could actually contribute. So it was a big deal for me to see the community react positively to this work. Additionally, I had been interested in computational mechanics for a decade from within neuroscience, since it felt like it could say something about computation in biological neural nets, but I could never figure out how to get it to do anything useful. So it's also a privilege to see the dream play out and be realized in neural nets, and to start a research org dedicated to that vision!
The start of the research program presented here is in a real sense a bet, in that it could fail to be useful at scale, for a few different reasons. But the fact that we are building up with both theory and experiment in a way where each informs the other, and that we were able to show the community a small part of what this looks like, is a lesson I hope the readership takes, even beyond the specific content of this post. (hot take) I think a lot of great work happens within this community, but too often the more theoretical and the more empirical approaches don't talk to eachother as much as they should.
Pre-registering a prediction for experiment results testing why and when and how attention heads share information. We (simplex) will train transformers on data generated from the cartesian product of sequences generated from 2 independent Mess3 processes. If the degenerate eigenvalue of each is the same and positive (e.g. lambda=0.7 for both) then a transformer with a single attention head will learn it, and the attention patterns will be 0.7^(d-s) where d-s is the number of context positions between the source and desitination. If instead one of the processes has lambda = 0.7 and another lambda = 0.3, then this will seperate info. between two heads, where one has attention pattersn of 0.7^(d-s) and the other 0.3^(d-s). If instead we have one process with lambda=0.7 and the other with lambda = -0.3, then this will require 3 heads, with .7^(d-s) and the 0.3^(d-s) seperating between two heads to account for the fact that attention has to be positive. BUT if we have one process with lambda = 0.7 and the other with lambda = -0.7, then this will require 2 heads! With the 0.7^(d-s) being able to be shared on one head for the lambda=0.7 process, and for the -0.7 process when d-s is even, and then the odd d-s cases will be on the other head.
This looks great! Thanks for making the videos public.
Any chance the homework assignments/experiments can be made public?
A fun side note, that probably isn't useful - I think if you shuffle the data across neurons (which effectively gets rid of the covariance amongst neurons), and then do linear regression, you will get theta_t.
This is a somewhat common technique in neuroscience analysis when studying correlation structure in neural reps and separability.
Somewhat related, in our recent preprint we showed how Bayesian updates work over quantum generators of stochastic processes. It's a different setup than the one you show here, but does give a generalization of Bayes to the quantum and even post-quantum setting. We also show that the quantum (and post-quantum) Bayesian belief states are what transformers and other neural nets learn to represent during pre-training. This happens because to predict the future of a sequence optimaly given some history of that sequence, the best you can do is to perform Bayes on the hidden latent states of the (sometimes quantum) generator of the data.
2Beautiful post, thank you!
In some sense I agree but I do think it's more nuanced than that in practice. Once you add in cross-entropy loss on next token prediction, alongside causal masking, you really do get a strong sense of "operating on sequences". This is because next token prediction is fundamentally sequential in nature in that the entire task is to make use of the correlational structure of sequences of data in order to predict future sequences.
There's actually a market that's even more directly relevant!
This all sounds very reasonable to me! Thanks for the response. I agree that we are likely quite aligned about a lot of these issues.
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Why do you think finding the true features should make the network look sparse and modular?