Mathematical exploration and discovery at scale

- Context: Terence Tao is one of the best mathematician alive.

- Context: AlphaEvolve is an optimization tool from Google. It differs from traditional tools because the search is guided by an LLM, whose job is to mutate a program written in a normal programming language (they used Python). Hallucinations are not a problem because the LLM is only a part of the optimization loop. If the LLM fucks up, that branch is cut.

- They tested this over a set of 67 problems, including both solved and unsolved ones.

- They find that in many cases AlphaEvolve achieves similar results to what an expert human could do with a traditional optimization software package.

- The main advantages they find are: ability to work at scale, "robustness", i.e. no need to tune the algorithm to work on different problems, better interpretability of results.

- Unsurprisingly, well-known problems likely to be in the training set quickly converged to the best known solution.

- Similarly unsurprisingly, the system was good at "exploiting bugs" in the problem specification. Imagine an underspecified unit test that the system would maliciously comply to. They note that it takes significant human effort to construct an objective function that can't be exploited in this way.

- They find the system doesn't perform as well on some areas of mathematics like analytic number theory. They conjecture that this is because those problems are less amenable to an evolutionary approach.

- In one case they could use the tool to very slightly beat an existing bound.

- In another case they took inspiration from an inferior solution produced by the tool to construct a better (entirely human-generated) one.

It's not doing the job of a mathematician by any stretch of the imagination, but to my (amateur) eye it's very impressive. Google is cooking.

Can you explain more on this? How on earth are we supposed to know LLM is hallucinating?

AI will be better than almost all mathematicians in a few years.