AI maths whiz creates tough new problems for humans to solve | Hacker News Reader

AI maths whiz creates tough new problems for humans to solve

nature.com

145 pointstzury79 comments

The praise from popular press and the promotion by the authors should be put into context of what mathematicians think of it.

Two blog posts by a professor at U. Chicago, qualifying it of intellectual fraud:

https://www.galoisrepresentations.com/2019/07/17/the-ramanuj...

https://www.galoisrepresentations.com/2019/07/07/en-passant-...

This is shocking stuff. I encourage everyone to read these two links.

Man, this is an example of how difficult it is to know what is BS or not if you're not an expert on the subject. On one hand, this article was published in Nature, which I thought was trustworthy. On the other, there's this comment on a social media platform that links to a blog that also seems legit. No wonder misinformation spreads so fast. Even after reading both, I don't know what to make of it. The reaction and comments here just confuse me more.

This phenomenon has a name: epistemic learned helplessness.

https://slatestarcodex.com/2019/06/03/repost-epistemic-learn...

Nature in particular seems vulnerable to the academic equivalent of click-bait articles. I think the top journals within a specific field are more reliable.

Nature has published some very questionable papers in AI/ML that are filled with malpractices. Another bogus paper that comes to mind was predicting earthquakes with a deep(read huge) neural network that appears to have information leakage and was fuelled with the hype of DL when a simple logistic regression (i.e. single neuron) could perform just as well [1,2,3].

[1] https://www.reddit.com/r/MachineLearning/comments/c4ylga/d_m...

[2] https://www.reddit.com/r/MachineLearning/comments/c8zf14/d_w...

[3] https://www.nature.com/articles/s41586-019-1582-8 / https://arxiv.org/pdf/1904.01983.pdf

This is a frighteningly common practice in DL research. Baselines are rarely taken with resect to alternate techniques, largely due to publication bias.

On one hand papers about DL applications are of interest to the DL community, and useful to see if there is promise in the technique. On the other hand, they may not be particularly useful to industry, or to forwarding broader research goals.

A good rule of thumb is to be slightly more suspicious of "DL for X" unless X was part of the AI/ML umbrella in the 2000s. If no one was publishing about X in AAAI/NIPS/ICML before 2013 or so then there's a pretty good chance that "DL for X" is ignoring 30+ years of work on X. This is becoming less true if one of the paper's senior author comes from the field where "X" is traditionally studied.

Another good rule of thumb is that physicists writing DL papers about "DL for X" where X is not physics are especially terrible about arrogantly ignoring 30+ years of deeply related research. I don't quite understand why, but there's an epidemic of physicists dabbling in CS/AI and hyping it way the hell up.

Anecdotally, having come from a physics background myself - DL is more similar to the math that physicists are used to than traditional ML techniques or even standard comp-sci approaches are. In combination with the universal approximation proofs of DL, it's easy to get carried away and think that DL should be the supervised ML technique.

Curiously, having also spent heavy time on traditional data-structures and algorithms gave me an appreciation for how stupendously inefficient a neural net is and part of me cringes whenever I see a one-hot encoding starting point...

Re: similar to the math they know, this makes sense.

I don't understand why over-hyping and over-selling is so common with AI/ML/DL work (to be fair, over-hyping is more related to AI than physicists in particular. But people from non-CS fields get themselves into extra trouble perhaps because they don't realize there are old-ish subfields dedicated to very similar problems to the ones they're working on.)

I think the answer is related to the startup culture, and that is to gather funding.

That's the confusing thing. Appealing to rich twitter users/journalists/the public doesn't strike me as a particularly good strategy for raising research funds!

Random rich people rarely fund individual researchers. More common for them to fund an institute (perhaps even by starting a new one). The institute then awards grants based on recommendations from a panel of experts. This was true before Epstein scandals, and now I cannot imagine a decent university signing off on random one-off funding streams from rich people.

All gov funding goes through panels of experts.

Listening to random rich people or journalists or the public just isn't how those panels of experts work. Over-hyping work by eg tweeting at rich/famous people or getting a bunch of news articles published is in fact a good way to turn off exactly the people who decide which scientists get money.

Maybe a particularly clueless/hapless PR person at the relevant university (or at Nature) is creating a mess for the authors?

While I agree with you, the nature paper that I linked above was published by the folks at google of all places. I think a valid hypothesis is that work done during internships (or even residencies) may not be on-par with what NeurIPS/ICLR/etc require but they give publicity and thus the PR teams push for that kind of papers.

However, it still does note explain why this kind of sloppy work done and published by publicly funded research labs, except perhaps as a form of advertisement.

Well, yeah, corporate research is what it is. A lot of the value add is marketing.

>Random rich people rarely fund individual researchers.

Yes and no. There are private foundations that, if someone donates a reasonably large amount, say at least the amount of their typical grant, they will match the donor with a particular researcher, and the researcher will have lunch, give a tour, and send them a letter later about about the conclusions (more research is needed).

That doesn't mean the donor gets input into which proposals are accepted; that is indeed done by a panel of experts as far as I know. It's more of a thing to keep them engaged and relating to where the money goes when there are emotional reasons for supporting e.g. medical research.

Even less than that now. The ability of specific donors to direct funds to specific academic groups for specific research is WAY more constrained now than it was pre-Epstein. Institutions want that that extra layer of indirection.

I agree with the sibling comment that whenever ML has not been used before on in a field and DL and especially DRL (deep reinforcement learning) are used, it is likely that the authors are ignoring decades of good research in ML.

After a very theoretical grad course in ML, I have come to appreciate other tools that come with many theoretical guarantees and even built-in regularization that are less Grad Student Descent and more understanding the field.

I think that the hype that was used to gather funding in DL is getting projected onto other fields, if only to gather more funding.

The articles don't contradict each other when it comes to cited facts - you can believe both!

I suppose its all in the implications though, which are contradicting as the nature article implies it is a big deal. The nature article doesn't give any examples of interesting conjectures, or examples of interesting consequences if any of the conjectures should be true. They talk a lot about alternate formulae to calculate things we already know how to calculate. Why would we care? Do they have a smaller big-oh? Nature references the theory of links between other areas of math, if true that's great, but if its true surely they would have mentioned an example of such a link? Anyways I lean towards this not being that interesting, even if you base that just on what the nature article said.

Re why would we care: this is a search algorithm for numerical coincidences. Most numerical coincidences are trivial, for example can be derived from hypergeometric function relation which was known to Gauss. In fact it would be interesting to automatically filter formulae which can be derived from hypergeometric function relation... On the other hand, numerical coincidences can lead to deep theory, monstrous moonshine is a prime example. Hope is that by searching for numerical coincidences, we can discover one leading to deep theory without already knowing that deep theory. This seems reasonable.

That's a very good point - and it really motivates these kinds of computer searches.

The Nature paper has quite a lot of detail in its supplementary

https://static-content.springer.com/esm/art%3A10.1038%2Fs415...

Table 3 inside also shows new conjectures for constants such as Catalan's and zeta(3). These results do not seem to trivially arise from known knowledge.

Sometimes popular science is itself the misinformation. The authors stretch the findings to land in prestigious journals. The news stretches the findings further to sell clicks (c.f. Gell-Mann Amnesia effect). The people on the internet selectively quote articles and selectively ignore others. The algorithm tries to only show you content that you like.

The truth doesn't have a chance.

Science/Nature are prestigious, but the quality of their articles are often questionable. Part of the problem is the short format, which makes it difficult to include a lot of context and sanity-checking. Another issue is that they prioritize the “sexiness” of the research over pretty much everything else.

I'll never understand why Science/Nature carry any currency in CS and Math. TBH I consider them negative signals in these fields, and I encourage others to do the same when hiring -- the same way that a prestigious newspaper or magazine would treat someone with a bunch of (non-news) Buzzfeed bylines.

There are some exceptions. E.g., a Science/Nature paper summarizing several years worth of papers published in "real" venues. Truly novel work that's reported on for the first time in Nature/Science is almost universally garbage. At least in CS/Math.

In my experience, at least in pure math, publishing in Nature/Science doesn’t carry any weight. The most prestigious journals for a given subfield usually specialize in that subfield (with a name like Journal of Geometric Topology), with a few exceptions like Annals and JAMS. Even those are still focused heavily on pure math; I can’t think of any which are cross-disciplinary outside of math.

This is why I have empathy for conspiracy believers. From their perspective, their understanding of the world is accurate.

This is also why I see the inevitable failure of social media platforms in regulating truth-vs-non-truth.

FWIW that blog is written by one of the top leading number theorists in the US today. Of course, his opinions are his and you’re free to form your own, but just wanted to clarify that the blog is very much legit.

It seems like Calegari is chasing after PR and may be angry at computer scientists getting into his field.

His criticism was discussed and found incorrect by the peer review process:

https://static-content.springer.com/esm/art%3A10.1038%2Fs415...

The authors tweeted at Elon Musk and Yuri Milner, so it's obvious who is chasing PR (and "dumb money")

Meanwhile, the blog author congratulated Mathematica for being for being good at solving continued fractions

I'd ask you where the criticism was "found to be incorrect", but I know that's absurd (aka, not even wrong), as peer review comments are not in the business of "finding criticism to be incorrect".

Assuming that it's the PI/senior author doing all of this shameless promotional work, I feel really REALLY bad for the grad student(s) on this paper... what a way to start your academic career :(

The paper is actually really nice work, but holy jesus someone on that author list is making a complete ass out of themselves.

Academia isn't startup world. The community is small, people have long memories, and I've rarely seen the strategy being deployed here work out. It does work sometimes, but more often it backfires. Especially for folks who aren't yet on a tenure track.

This is an incredibly strange slate of reviewers. Only the first seems to really understand the mathematical context. It's odd to appeal to the peer review process when the "peers" are not suited to complete the review.

I assure you that Calegari knows more about number theory than any of those referees, and the reasons why the paper is bad are well-explained on his blog (cf. the two links above) and by referee #1. Speaking of "peer review," look at how all the excellent mathematicians commenting on that blog agree with him!

Calegari gets to cherry-pick comments he approves or rejects on his blog, so calling it "peer review" is taking the concept out of context =).

True! I tried several times to comment on his blog - but Calegari didn't confirm my comments

It's hypocritical to criticize but to avoid criticism ...

I agree. Without being an expert in the field, the 2nd and 3rd reviews "smell funny"; they clearly lack depth, are very enthusiastic, and don't seem to make a good and comprehensive point as to _why_ this paper is supposed to be as great as they claim it is. In a strong community, any editor should consider these reviews disappointing, and any author should at least have mixed feelings about them.

As a thorough non-expert, I don't take headlines in the style of The Register seriously, even if the article is in Nature.

Although, if it was really from The Register it probably would have said "boffins" rather than "humans".

> Well … OK I guess? But, pretty much exactly as pointed out last time, not only is the proof only one line, but the nature of the proof makes clear exactly how unoriginal this is to mathematics

This is what I was wondering about while reading the article. If the AI only generates formula for which proofs involve only a few trivial steps back to something that is known, then it doesn’t feel useful. But I feel like the question “what makes a good conjecture?” in its own right makes for a very interesting discussion.

Wouldn't a good conjecture be anything that's interesting if true. General bonus points for if intuitively it seems like it should be obviously true (or false) but yet is hard to prove or if proving it is true would allow you to prove lots of other interesting statements.

Sure. Define "interesting".

What's interesting to me is probably in a standard textbook already.

If it is in a standard textbook, than its almost certainly interesting (although probably not a conjecture unless its a pretty advanced textbook)

Mathematical physicist Robbert Dijkgraaf has got you covered:

https://www.quantamagazine.org/the-subtle-art-of-the-mathema...

I agree with the article you linked. Mathematical knowledge is about compression. Most if not all of these formulae are just specializations of known formulae. So the value of this approach is questionable. Generating these forms can possibly be done in a much simpler way.

who else holds the compression view ?

I understand his frustration. But calling it a fraud is a little bit too much.

Ah yes, the good old "SV culture disrupts X! Revolution at 8 o'clock!"

There's an arms race:

* People are evolving memetic resistance to the incessant BS, ads and bombastic headlines.

* The SV/startup culture is evolving to inject authenticity to overcome people's BS defenses, convince them they need a change.

Honestly, do you still get excited when you read "AI solves X!"?

Probably another huckster peddling empty air, cutting corners, externalizing costs. The whole game is tired, and people are taking note. Not everything that exists requires a radical change.

I think one interesting lesson from this nice qualification here is that at the moment ML methods to learn mathematics may look trivial from a professional mathematician (ie the results are unoriginal or trivial) but perhaps the target audience of this method may be for non professional mathematicians or students training to be mathematicians. I could still see this ML tool as a way to automate the work of some more “trivial” (from the POV of an expert) mathematics, although not the work of professional mathematicians.

The knowledge gap in mathematics between professional mathematicians and non professionals is vast, and this tool could narrow the gap.

I would bet the majority of readers of nature would not be able to point out that the outputs of the ML tool were trivial. So there is need to narrow this gap.

Simple results in almost any specialist field would stump most readers of Nature. That's not a reason to publish in Nature.

Look at the comments:

> The paper is amazingly bad. None of the authors are mathematicians as far as I can see. I think the word “new” appears 50+ times in the paper. Looks like they updated the paper to include your observation from last time about the Gauss continued fraction without mentioning the source (the authors admit here they read your blog: http://www.ramanujanmachine.com/idea/some-well-known-results...). Classy!

Just some light plagiarism/academic misconduct!

Looking at his post, the main criticism is "that the program has not yet generated anything new", but the post does not refer to the actual results (like formulas for Catalan's and Apery's constants).

Full disclosure- I am one of the authors of the paper.

Note that the blog you're citing was written a year and a half ago. It refers to a select few conjectures, and naturally has no references to the developments in the past year and half (which were the main reason the paper got published).

Furthermore, the author of the blog didn't respond to multiple emails we sent him, attempting to discuss the actual mathematics.

So basically the vast majority of the criticism here, is based on a single, outdated blog, by a professor (respected as he may be) who has not revisited the issues and new results since first posting the blog, and has not given any mathematical argument as to why the results shown in the paper (the actual updated paper that was published) are supposably unimportant.

Would appreciate your opinions on the matter.

I think the vast majority of criticism here does not target the research per se, but rather the way the results are "hyped" and presented as a massive break-through. I agree with this criticism, and also think that the two positive Nature reviews seem rather shallow, at least from a non-expert's perspective (this is not your fault, of course). When it comes to long term impact, I'd find it interesting to discuss how your work can (ideally) interact with proof assistants like Lean. Also, the work around Lean is a good example of a "hyped" topic that is presented by its contributors with caution and modesty.

I don't really see much value in debating the procedural aspects (Nature review process etc). I see a lot of value discussing the research and its content. We think the results shown in the paper are significant and of some importance, and so do others who reviewed our work. This is where I think the focus should be.

Please read our paper and not only the blogs criticizing it:) There is a link to access it here: https://rdcu.be/ceH4i

Then don't take offense by the discussion here, because it's mostly on some "meta" aspects of science communication, and you are probably not responsible for any of the aspects that have been critizised.

Regarding the research itself, I am not an expert, but I am curious to learn how this line of research (automated conjecture generation) intersects with proof automation/proof assistants, and in particular with the work that the Lean community is doing (creating an "executable" collection of mathematical knowledge). Perhaps there are some works you can point to.

Not the person you're replying to, but I admit to characterizing your paper as "garbage" in another comment thread. Since you're inviting discourse, which I greatly appreciate, I'm compelled to reply.

1) To anyone who's studied algebra, it is clear that identities of the form LHS = RHS can be obtained by a nested application of transformations and substitutions in a consistent manner.

2) Of course, arriving at a new, insightful result often involves taking mundane steps. However, in this case, the new mathematical discoveries based on the output tableaus of your algorithm are hypothetical. Whereas the manuscript (and the authors) have already pocketed one of the premium accolades in sciences in the form of a Nature publication.

3) To drive the point above home, do you think the resulting mathematical insights themselves, without riding on the "AI" novelty aspect, would clear the bar for a Nature (or similar high-impact) publication? To be clear, I'm not a mathematican, but I believe the answer would be no. Contrast this with another AI/ML advance published in Nature quite recently: AlphaGo. Note how the gist of their paper, superhuman performance in Go, is a self-standing achievement that merely makes use of machine learning techniques.

"garbage" and "fraud" are really strong words.

I would give the actual work behind this paper a "strong accept" if the claims were properly scoped, perhaps with a weak/borderline score on "significance/impact" since I'm not really sure why anyone cares about discovering discovering these sorts of identities. Probably a Conditional Accept in its current form because of the mismatch between actual results + reasonable expectation of potential vs. what's claimed.

So, "over-hyped" and "claims wildly out of line with actual results" are definitely more than fair statements. "Fraud" or "garbage" are way too strong.

Re: Nature, I don't really understand it or care. I can say that in my own input to hiring committees I tend to treat Nature papers in CS/Math as red flags unless they're consolidations of a bunch of other work published in top sub-field journals/conferences.

For some reason Nature really loves these "automated discovery of random mathematical facts" type of papers. I don't understand it. I tend to assume it's click-through-rate-driven editorial decision making.

I appreciate your views otherwise, but I did not use or imply "fraud" in my comment(s).

No, I know. I was referring to other comments on this story. I think garbage is also strong.

Irrelevant.

The Nature paper presents several new conjectures related to the Catalan constant, pi^2, and zeta(3) (Apéry's constant):

http://www.ramanujanmachine.com/wp-content/uploads/2020/06/c... http://www.ramanujanmachine.com/wp-content/uploads/2020/06/p... http://www.ramanujanmachine.com/wp-content/uploads/2020/06/z...

The main criticism of the blog is "that the program has not yet generated anything new", but the post does not refer to these results. So it seems that this blog post is currently irrelevant and out-dated compared to the Nature publication.

The GitHub repo contains more details on how the algorithm works: https://github.com/AnonGit90210/RamanujanMachine

> We have continued fractions which are generated by polynomials. What does that mean? object of the type: a(0)+b(1)/(a(1)+b(2)/...)) where a,b are polynomials with integer coefficients. This is the RHS. On the LHS we have a function of some constant, either rational or ULCD. A ULCD function is of the type f(const)=(u/const + const/l + c)/d while a rational function is the ratio of two polynomials. We seek for a match between some LHS function on a constant to a continued fraction. To do this, we enumerate over the coefficients of a,b and those of the rational func on the LHS/the ulcd parameters. They're all integers. We actually don't compare between the LHS to a continued fraction, but to a continued fraction after we applied some function to it, e.g. contfrac^2, sqrt(contfrac), 1/contfrac etc. In order to enhance the algorithm's complexity, we first enumerate over the parameters of the RHS, saving the results to a hashtable, then enumerate over the LHS and look for a match in the table. This is a TMTO (time-memory tradeoff), it's called MITM (meet in the middle), you can look it up if it's not clear. We first find matches, then filter redundant results (discussed later on), filter only continued fractions that converge fastly, find how fast they are converging, and how many iterations we should calculate to get the desired precision. We calculate them to that precision, and filter again.

This doesn't seems that complicated, but at the same time probably gives no mathematical insights as to how to derive those formulaes. I can see how this would look pointless to some people.

The Nature article is about [1]:

> The Ramanujan Machine is a computer algorithm, named after the Indian mathematician Srinivas Ramanujan, which generates conjectures similar to what proposed by Ramanujan.

The #Criticism section states:

> Prof. Frank Calegari criticized the Ramanujan Machine claiming that the program has not yet generated anything new and dubbed it as "absurd" and an "intellectual fraud".

[1] https://en.wikipedia.org/wiki/Ramanujan_Machine

Funny that no other name other than Frank Calegari's appear in the Wikipedia page

This page appears as if it was written by the authors or the PR department of the university...

no examples of formula's generated by AI and if they are correct or proven incorrect? Without examples this seems an advertisement to non-existing product. Anyone can create AI then with simple function pick_N_random(=, +, -, *, /, pi, e, alpha, tetta, 1,2,3,4,5,6,...)

AI? Much overused buzzword these days. Don't we have metaheuristics (genetic algorithms etc.) which already do exactly what the article states as novel?

Meta-heuristics aren’t the current buzzword, likewise evolutionary algorithms won’t be front paging anytime soon... unless it’s neuroevolution disguised as a novel contribution in 2021. Sprinkle on some blockchain and you’re onto a winner!

the link to the project is in the article

> “Until I can detect a well-developed ‘sense of mathematical taste’ in AI, I expect its role to be that of an important auxiliary tool, not that of independent discoverer.”

AI, no, but ML, yes. ML can only make estimates, e.g. compute functions, it doesn’t have judgement. That requires a human decision maker. It goes back to what I learned in primary school about computers: “a computer can only do what you tell it to do.”

To be fair creating mathematical problems that are hard or impossible to solve is not really very hard.

So this is essentially AI-guided fuzzing for math equations?

THIS is mind-bendingly awesome. What are the implications of this for technological advancement? Will we be getting to a point where machines are inventing new things that revolutionize life as we know it? I'm thinking about the difference in technology from now vs. 1920s. Could we be moving toward another advancement leap of such a proportion?

Just consider for a moment how far we've come. In the past, all our machines were purely mechanical---steam engines, water cranks, etc. Now we're creating machines that discover math. Amazing!

It does seem like an amazing project, but I'm struggling to see how this has any significant implications for technological advancement.

Its just a continued fraction generator that looks for fractions equal to constants. Not an AI, more like search function with optimization.

The singularity is not yet upon us, and we have had machines generating math since the 70s with the proof of the 4 colour theorem.

Yaay! I fucking love science™ too!

"Don't be snarky."

https://news.ycombinator.com/newsguidelines.html

TLDR;

AI machine called: Ramanujan Machine.

Attempts at finding formula. Tries to determine whether the following constants are "irrational" or "rational" number.

Constants mentioned: Catalan’s constant, Apery’s constant

Unrelated: I never heard of the above 2 constants before. Although, I have met someone who speaks Catalan and from Catalonia.

That names comes from the Belgian mathematician Eugène Charles Catalan.

The two blog posts by Frank Calegari are from July 2019 and irrelevant to the paper in Nature (from February 2021). The paper went through peer review by mathematicians and experts in the field.

What was changed in the new paper? Hard to say "irrelevant" when the blog post directly influenced a rewrite of the paper.

There are several new conjectures related to the Catalan constant, pi^2, and zeta(3) (Apéry's constant):