[yorwba]

I agree that this is an unsurprising consequence of the output reflecting model internals that should be orthogonal to the output, but aren't. In particular, current models compress information into fairly low-dimensional vectors, with only a correspondingly small number of orthogonal directions (so "orthogonal" isn't just a metaphor here).

Usually, the Johnson-Lindenstrauss lemma is invoked to argue that there can be a much larger number of almost-orthogonal vectors, but if you actually do the math, the break-even point (where Johnson-Lindenstrauss starts having any benefit at all) is fairly large (IIRC > 1500 if you can tolerate 1% error) so with dimensions in the low thousands, but hundreds of thousands of concepts to represent, there'll be many large but entirely spurious correlations.

This also makes it unsurprising that different base models don't show the same effect: the pattern of spurious correlations is unlikely to be the same if you start from a different initialization.

[Vetch]

That math is for random projections? Note that JL lemma is a worst case guarantee and in practice, there's a lot more distortion tolerance than the given bounds would suggest. Concepts tend to live in a space of much lower intrinsic dimensionality than the data's and we often care more about neighbor and rank information than precise pair-wise distances.

Also, JL is only a part of the story for the transformers.

[yorwba]

Johnson-Lindenstrauss is an example of a probabilistic existence argument: the probability of a random projection having low error is nonzero, therefore a low-error projection must exist. That doesn't mean any given random projection can be expected to have low error, although if you keep rerolling often enough, you'll eventually find one.

The existence argument does only provide a lower bound on the number of dimensions that can be represented with low error, but there's not necessarily much room for improvement left.

[jsrozner]

Interesting. I have been thinking that with these high dimensional representations that we have nearly infinite nearly orthogonal dimensions.

One thing that's interesting to me is where / how the model stores the info about a preference for a particular animal, and that this (presumably small) weights change leads to a difference in random numbers that then leaks into a student model.

The fact that this does not happen on models that are separately initialized/ trained could be seen to provide counter evidence to the recently published Platonic hypothesis paper.