[nybsjytm]

> Many AI researchers are mathematicians. Any theoretical AI research paper will typically be filled with eye-wateringly dense math. AI dissolves into math the closer you inspect it. It's math all the way down.

There is a major caveat here. Most 'serious math' in AI papers is wrong and/or irrelevant!

It's even the case for famous papers. Each lemma in Kingma and Ba's ADAM optimization paper is wrong, the geometry in McInnes and Healy's UMAP paper is mostly gibberish, etc...

I think it's pretty clear that AI researchers (albeit surely with some exceptions) just don't know how to construct or evaluate a mathematical argument. Moreover the AI community (at large, again surely with individual exceptions) seems to just have pretty much no interest in promoting high intellectual standards.

[zipy124]

I'd be interested to read about the gibberish in UMAP, I know the paper "An improvement of the convergence proof of the ADAM-Optimizer" for the lemma problem in the original ADAM but hadn't heard of the second one. Do you have any further info on it?

[Xcelerate]

> Each lemma in Kingma and Ba's ADAM optimization paper is wrong

Wrong in the strict formal sense or do you mean even wrong in “spirit”?

Physicists are well-known for using “physicist math” that isn’t formally correct but can easily be made as such in a rigorous sense with the help of a mathematician. Are you saying the papers of the AI community aren’t even correct “in spirit”?

[nybsjytm]

Much physicist math can't be made rigorous so easily! Which isn't to say that much of it doesn't still have great value.

However the math in AI papers is indeed different. For example, Kingma and Ba's paper self-presents as having a theorem with a rigorous proof via a couple of lemmas proved by a chain of inequalities. The key thing is that the mathematical details are purportedly all present, but are just wrong.

This isn't at all like what you see in physics papers, which might just openly lack detail, or might use mathematical objects whose existence or definition remain conjectural. There can be some legitimate problems with that, but at least in the best cases it can be very visionary. (Mirror symmetry is a standard example.) By contrast I'm not sure what 'spirit' is even possible in a detailed couple-page 'proof' that its authors probably don't even fully understand. In most cases, the 'theorem' isn't remotely interesting enough as pure mathematics and is also not of any serious relevance to the empirical problem at hand. It just adds an impressive-looking section to the paper.

I do think it's possible that in the future there will be very interesting pure mathematics inspired by AI. But it hasn't been found yet, and I'm very certain it won't come from reconsidering these kinds of badly-written theorems and proofs.

[skinner_]

Amazing! I looked into your ADAM claim, and it checks out. Thanks! Now I'm curious. I you have the time, could you please follow up with the 'etc...'?

[nybsjytm]

There's a related section about 'mathiness' in section 3.3 of the article "Troubling Trends in Machine Learning Scholarship" https://arxiv.org/abs/1807.03341. I would say the situation has only gotten worse since that paper was written (2018).

However the discussion there is more about math which is unnecessary to a paper, not so much about the problem of math which is unintelligible or, if intelligible, then incorrect. I don't have other papers off the top of my head, although by now it's my default expectation when I see a math-centric AI paper. If you have any such papers in mind, I could tell you my thoughts on it.