Constrained Belief Updates and Geometric Structures in Transformer Representations for the RRXOR Process

Very nice work! With more time, it would be interesting to explore the attention pattern if the architecture is constrained to have fewer heads, and indeed to see how many heads are necessary (I assume at least two because of non-positive eigenvalues). Fewer heads may also reveal more interpretable OV vectors. Relaxing to more heads may then allow for a larger degenerate space of solutions.

Fig. 3 caption suggests that the rate of exponential decay is determined by the real part of the complex eigenvalues of the transition matrix of the HMM. Rather, the magnitude of this eigenvalue should determine the decay rate of an envelope, while there will be a decaying oscillation within this envelope as determined by the angle of this eigenvalue in the complex plane.

In follow-up errata, the author mentions that the transition matrix is diagonalizable but singular. This seems fine though, since this does not preclude a spectral analysis.

This is a well-scoped extension of Piotrowski et al's belief state geometry work to the RRXOR process. The authors got a lot up and running over the course of the hackathon.

Interestingly, they report that in order to extract RRXOR belief state geometry via PCA, they had to concatenate across layers. If the reasons behind this were better understood, perhaps RRXOR could serve as a tractable, minimal example of "depth separation" in transformers. This is exactly the sort of result one should aim for in a hackathon.

I like this project not because it made tremendous gains, but because it set a clearly scoped and well-contextualized project that i) has a physics connection and AI safety implication, ii) could be more-or-less completed within the hackathon period, and iii) demonstrates an understanding of limitations and how these could be addressed. This project aims to reproduce Simplex’s work in a slightly more complex setting, namely the tension between the parallel distribution of transformer attention and belief states computed by optimal Bayesian updating. The plot produced by PCA does seem to have some structure, but this could have been explained a bit better. Overall, the author acknowledged gaps in both the set up (e.g., differences between RRXOR and natural language data, the possibility of non-linear belief states, lack of layernorm), analysis (e.g., ‘exponential’ decay of the attention weight with distance was approximate, and results (e.g., potential lack of structure in OV circuit, in contrast with original paper). All of this lends credibility to the author being able to carry out the future work as laid out.

This project extends recent results in computational mechanics to find preliminary evidence that transformers trained on the RRXOR process perform constrained belief state updating. The empirical and theoretical results are novel and interesting. It would be great to see this project extended in future work.