theoretical ml

Dimitris Papailiop... @DimitrisP... · 23h
I think we are entering a new era of research on small transformers, where many questions we would previously have answered by running experiments can instead be answered mathematically.

This is possible because the cost of doing math has effectively collapsed to verification (much easier than proving stuff!).

Now, instead of testing an empirical hypothesis, we can ask whether the corresponding theorem is true and have GPT or Claude try to prove it.

Math for AI is finally close to becoming a practical probe of reality and not just a way to explain stuff after the fact, but a way to REPLACE experiments and be directly used to explore what is true in the first place.

Kind of incredible!

[quoted tweet]
Dimitris Papailiop... @DimitrisP... · 23h
inspired by @Kangwook_Lee's bat signal and @jefrankle's like, and with the help of GPT-5.6 Sol you can actually prove it :)

In fact it is true that any function f(a,b) -> C ca...

[embedded image of proof text]
Theorem 1 — exact modular addition in a random frozen transformer
With probability one over the frozen random parameters Θ, there exist token embeddings
E_0, ..., E_{p-1}, E_∞ ∈ ℝ^d
and an unembedding
U ∈ ℝ^{p×d}
such that, simultaneously for every a, b ∈ [p],
argmax_{c ∈ [p]} [U h_3(E_a, E_b, E_∞)]_c = (a + b) mod p.
Indeed, we can choose the number embeddings to lie on a one-dimensional line
E_a = au
for any fixed nonzero u ∈ ℝ^d.
Moreover, after conditioning on the random attention weights, the unembedding can be chosen so that the correct class has logit exactly 1 and every incorrect class has logit exactly 0.
Thus the classification margin is exactly 1.

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