Neuroscientist turned Interpretability Researcher. Starting Simplex, an AI Safety Research Org.

# Adam Shai

# Computational Mechanics Hackathon (June 1 & 2)

This is a great question, and one of the things I’m most excited about using this framework to study in the future! I have a few ideas but nothing to report yet.

But I will say that I think we should be able to formalize exactly what it would mean for a transformer to create/discover new knowledge, and also to apply the structure from one dataset and apply it to another, or to mix two abstract structures together, etc. I want to have an entire theory of cognitive abilities and the geometric internal structures that support them.

If I’m understanding your question correctly, then the answer is yes, though in practice it might be difficult (I’m actually unsure how computationally intensive it would be, haven’t tried anything along these lines yet). This is definitely something to look into in the future!

It’s surprising for a few reasons:

The structure of the points in the simplex is NOT

The next token prediction probabilities (ie. the thing we explicitly train the transformer to do)

The structure of the data generating model (ie. the thing the good regulator theorem talks about, if I understand the good regulator theorem, which I might not)

The first would be not surprising because it’s literally what our loss function asks for, and the second might not be that surprising since this is the intuitive thing people often think about when we say “model of the world.” But the MSP structure is neither of those things. It’s the structure of inference over the model of the world, which is quite a different beast than the model of the world.

Others might not find it as surprising as I did—everyone is working off their own intuitions.

edit: also I agree with what Kave said about the linear representation.

A neglected problem in AI safety technical research is teasing apart the mechanisms of dangerous capabilities exhibited by current LLMs. In particular, I am thinking that for any

*model organism*( see Model Organisms of Misalignment: The Case for a New Pillar of Alignment Research) of dangerous capabilities (e.g. sleeper agents paper), we don’t know how much of the phenomenon depends on the particular semantics of terms like “goal” and “deception” and “lie” (insofar as they are used in the scratchpad or in prompts or in finetuning data) or if the same phenomenon could be had by subbing in more or less any word. One approach to this is to make small toy models of these type of phenomenon where we can more easily control data distributions and yet still get analogous behavior. In this way we can really control for any particular aspect of the data and figure out, scientifically, the nature of these dangers. By small toy model I’m thinking of highly artificial datasets (perhaps made of binary digits with specific correlation structure, or whatever the minimum needed to get the phenomenon at hand).

# Adam Shai’s Shortform

This all looks correct to me! Thanks for this.

Thanks John and David for this post! This post has really helped people to understand the full story. I’m especially interested in thinking more about plans for how this type of work can be helpful for AI safety. I do think the one you presented here is a great one, but I hope there are other potential pathways. I have some ideas, which I’ll present in a post soon, but my views on this are still evolving.

Thanks! I’ll have more thorough results to share about layer-wise reprsentations of the MSP soon. I’ve already run some of the analysis concatenating over all layers residual streams with RRXOR process and it is quite interesting. It seems there’s a lot more to explore with the relationship between number of states in the generative model, number of layers in the transformer, residual stream dimension, and token vocab size. All of these (I think) play some role in how the MSP is represented in the transformer. For RRXOR it is the case that things look crisper when concatenating.

Even for cases where redundant info is discarded, we

*should*be able to see the distinctions*somewhere*in the transformer. One thing I’m keen on really exploring is such a case, where we can very concretely follow the path/circuit through which redundant info is first distinguished and then is collapsed.

That is a fair summary.

Thanks!

one way to construct an HMM is by finding all past histories of tokens that condition the future tokens with the same probablity distribution, and make that equivalence class a hidden state in your HMM. Then the conditional distributions determine the arrows coming out of your state and which state you go to next. This is called the “epsilon machine” in Comp Mech, and it is unique. It is one

*presentation*of the data generating process, but in general there are an infinite number of HMM presntations that would generate the same data. The epsilon machine is a particular type of HMM presentation—it is the smallest one where the hidden states are the minimal sufficient statistics for predicting the future based on the past. The epsilon machine is one of the most fundamental things in Comp Mech but I didn’t talk about it in this post. In the future we plan to make a more generic Comp Mech primer that will go through these and other concepts.The interpretability of these simplexes is an issue that’s in my mind a lot these days. The short answer is I’m still wrestling with it. We have a rough experimental plan to go about studying this issue but for now, here are some related questions I have in my mind:

What is the relationship between the belief states in the simplex and what mech interp people call “features”?

What are the information theoretic aspects of natural language (or coding databases or some other interesting training data) that we can instantiate in toy models and then use our understanding of these toy systems to test if similar findings apply to real systems.

For something like situational awareness, I have the beginnings of a story in my head but it’s too handwavy to share right now. For something slightly more mundane like out-of-distribution generaliztion or transfer learning or abstraction, the idea would be to use our ability to formalize data-generating structure as HMMs, and then do theory and experiments on what it would mean for a transformer to

*understand*that e.g. two HMMs have similar hidden/abstract structure but different vocabs.Hopefully we’ll have a lot more to say about this kind of thing soon!

Oh wait one thing that looks not quite right is the initial distribution. Instead of starting randomly we begin with the optimal initial distribution, which is the steady-state distribution. Can be computed by finding the eigenvector of the transition matrix that has an eigenvalue of 1. Maybe in practice that doesn’t matter that much for mess3, but in general it could.

I should have explained this better in my post.

For every input into the transformer (of every length up to the context window length), we know the ground truth belief state that comp mech says an observer should have over the HMM states. In this case, this is 3 numbers. So for each input we have a 3d ground truth vector. Also, for each input we have the residual stream activation (in this case a 64D vector). To find the projection we just use standard Linear Regression (as implemented in sklearn) between the 64D residual stream vectors and the 3D (really 2D) ground truth vectors. Does that make sense?

Everything looks right to me! This is the annoying problem that people forget to write the actual parameters they used in their work (sorry).

Try x=0.05, alpha=0.85. I’ve edited the footnote with this info as well.

That sounds interesting. Do you have a link to the apperception paper?

That’s an interesting framing. From my perspective that is

*still*just local next-token accuracy (cross-entropy more precisely), but averaged over all subsets of the data up to the context length. That is distinct from e.g. an objective function that explicitly mentioned not just next-token prediction, but multiple future tokens in what was needed to minimize loss. Does that distinction make sense?One conceptual point I’d like to get across is that even though the equation for the predictive cross-entropy loss

*only*has the next token at a given context window position in it, the states internal to the transformer have the information for predictions into the infinite future.This is a slightly different issue than how one averages over training data, I think.

Thanks! I appreciate the critique. From this comment and from John’s it seems correct and I’ll keep it in mind for the future.

On the question, by optimize the representation do you mean causally intervene on the residual stream during inference (e.g. a patching experiment)? Or do you mean something else that involves backprop? If the first, then we haven’t tried, but definitely want to! It could be something someone does at the Hackathon, if interested ;)

Cool question. This is one of the things we’d like to explore more going forward. We are pretty sure this is pretty nuanced and has to do with the relationship between the (minimal) state of the generative model, the token vocab size, and the residual stream dimensionality.

One your last question, I believe so but one would have to do the experiment! It totally should be done. check out the Hackathon if you are interested ;)

this looks highly relevant! thanks!

Lengthening from what to what?