Inside OpenAI’s Breakthroughs in Mathematical Reasoning

0:00 often as a practicing mathematician you have an idea and then you kind of think it might work then you try for a few hours a few weeks and at some point you give up >> whereas for GPT like okay a human told me to do this let's let's just do this and so that's why we're sort of in this renaissance of like reachable results >> this is the best part about this problem which is really nobody has any idea >> is the model just guessing in some insane way >> it doesn't seem like there's a limit so far but it doesn't have that context yet >> it'd be nice for the world if applied mathematics went a lot faster The ceiling for difficulty of a math problem is pretty high. Even if AI continues getting like exponentially better at math, plausible will never solve something like P versus >> what's the ideal way that this is being taken up by the math community? Probably most at this point are like, okay, AI is obviously doing some non-trivial stuff.

0:46 >> so >> well, thank you guys for coming. This is really exciting because I think math has been moving so fast with AI. I'd just love to, you know, get to both practicing mathematicians and who work at OpenAI to chat on some of these results. so, you know, we have with us Mark Selki and Matab Swani. We're connected actually because EU was actually your adviser. And so both of you guys have worked much more deeply in math since I've like quit many many like over a decade ago. and so this is very exciting to kind of hear a download of your thoughts on how OpenAI has been sort of approaching this and also just like where you think math is going with the incredibly rapid advance of of how AI has been helping. so yeah I don't know maybe we can start off with some very basic questions of like you know you guys what what do you do to the extent that you can of course share and and how did you come from you know being a practicing mathematician to working at open AAI.

1:57 >> Yeah I mean I guess we we both you know broadly got excited last year when the model started to really take off in math. >> Yeah. so I I joined a little bit before Matab. I saw the IMO gold medal last summer basically and I I thought, you know, this is this is amazing. You know, I I want to see what the heck they did. Let me let me go see. and then yeah, I guess in the fall we Mark gave me a G Mark gave me a GP5 account and then I started playing with models and very quickly became convinced that yeah, it was extremely exciting to play with them. Yeah, >> you two were collaborating before this.

2:37 >> Yeah, you've known each other for a while. >> Yeah, >> we had like one paper we actually wrote jointly. >> Yeah. >> Yeah. >> So, GPG5 was your conversion. >> What was the magic that sort of I don't know as you're doing like what question do you throw at it? >> What process? >> Yeah. So, I think actually Yeah. So, I think how this started was u at least for me the starting moment was something like there's a collection of problems called so Paul Erdish is a very famous mathematician. posed a bunch of problems and so they've now all been collected on this site and I so I specifically worked in combinotaurics and a lot of these questions are among the most important. So it's always fun to flick through the site but one thing that often happened to me that was extremely frustrating was I would look at a question see that it's marked as open and then not actually know if it's correct not actually know if it had was still unsolved because the literature is often quite hard to search.

3:30 >> yeah. >> And I one instance I just plugged it into GPD5 and like five minutes later it found a reference and and this was aa this was a case where a few few of my friends had actually started thinking about the problem on the site. I was talking with them and I mean we had spent a few hours didn't seem wasn't clear if the problem was within reach and it was just very nice okay to be told yes this is in reach here's how you do it and yeah GPD5 told me this and then I told Mark about this and this yeah this is sort of yeah this for me was quite a surprising moment. Yeah. And then we we looked more into it and we found like 10 more cases sort of like this >> at the time. I feel like you know being better at maybe making connections between wide or even just like as you're saying like the the search for whether there's been a result or a related thing earlier is just like kind of humanely hard but maybe better for machine. But I imagine as the progress has happened in the last year what has been impressive has kind of reached beyond that and maybe you know through talking about it more abstractly or if it's more natural to talk about it through like one of the problems that has been recently announced through you know Astra you can kind of enlighten me as to like how how the recent progress has been a lot more than just like you know searching through more areas making these connections between the field and perhaps just like actually deeper more mathematical reasoning that's similar to a working mathematician.

4:57 >> Yeah. I mean I I I think this like search point of being you know familiar with everything is still definitely like a relative strength that yeah >> maybe informs like the types of problems that AI is solving. Now >> I think there's there there are some other relative strengths and weaknesses. another relative strength that's pretty noticeable is just like it's very good at executing on some like idea once it once it has it like you know when whenever you have an idea there's like there's there's usually some amount of you know getting everything lined up like you know is epsilon like smaller than delta this this kind of of thing you have to get everything correct and like for a human you know you it's easy to get lost in these kinds of details and the AIS just kind of always nail these kinds of arguments I find.

5:54 >> Yeah. Is it usually just I mean I feel like you guys will know more detail on this but like for the unit distance problem it was just like the approach there was definitely contributions from you know the open AI but like the approach perhaps was suggested even you know originally by Erdos and then it's just that the actual reasoning was a very very like momentous like feat and so for a human you're like well I only have a limit amount of time and if after so many you know steps it is still not clear I mean maybe you're like Andrew Wilds and you actually spend 10 years alone and do something but like it's not clear then it's just it it's it doesn't become like the riskreward is not good enough whereas for GPT like okay I'll like a human told me to do this let's just do this and so that's why we're sort of in this renaissance of like reachable results does that track and did you feel like with the Astro results is that sort of like where the strengths have been primarily or there's an extra ingredient or magic here.

7:04 >> I I feel I think the unit distance examples about it's quite telling in the sense of maybe the exact construction you can kind of you can make it look very similar to what people had tried before. But I think what what you can often I mean often as a practicing mathematician you you have an idea and then you kind of think it might work then you try for a few hours a few days a few weeks and at some point you give up and then a not so uncommon experience is that you find out a year or two later that somebody else got the idea to work that you thought that didn't work. So somehow getting an idea to work is it it can even be a large portion of the battle. And I think what the especially in the case of the unit distance conjecture there's just a lot of extraordinarily finicky details and very often when you're doing mathematics it's it's you're kind of gambling against the problem. You're like maybe I should try this approach but it seems really unlikely and just not worth my time. And the model I think in several of these cases both by combining what it knew and sort of having good taste kind of made the correct bet. And you can kind of see this in the the summarized chain of thought we released. can sort of look at it and it's reasoning like a mathematician and because it knows a few very correct bits, it makes the right decisions and is eventually able to prune the search tree. It's it's not really trying everything. It tries a lot of different things. It's extremely dogged, but it kind of >> I mean it can't try every idea. It has to try a limited set of ideas and it's able to kind of use its knowledge plus good good mathematical judgment and find the right path to go along. So I mean that for me was like because this was a problem which a lot of people had thought about and yeah >> I mean the idea the fact that the idea is not so foreign probably indicates that a lot of people had tried it or at least a few very serious mathematicians have tried and I think that's what made it really interesting to see. I I think something else that's that like I feel when I see these proofs is like like if I have an idea and I'm trying to execute like it might be that I have some like wrong plan for like how to get things to work. And like as a human if you have some like wrong path you go down for a while it can be hard to like rewire your brain to like start over and like try a different path. like you're you're kind of the initial idea is kind of linked in your brain with these other things that ended up not working. You know, it's sort of like your context window is like like a little polluted and like you know you can't just make another clone of yourself from like last week and say you know don't do this try something else build your intuition another direction >> but you know it's very easy to do this with an AI. So, I think this is another reason that it's like >> it like like getting the details right once you have some good general direction is like much less of a barrier all of a sudden.

9:51 >> And when you say it's much easier to do with AI, it's like it's not actually being directed with human interference too. It's like as you were saying in the reasoning traces, it's like making these choices. Maybe it backtracks, but then it's able to not be distracted by maybe like the context in which it's thinking about the problem like via these machinery and it's like go go back to like do you see it kind of go back as well or is it just like making good choices like is it a lucky sample or is it like actually reasoning like a mathematician where okay it doesn't it you know doesn't do well in this path but it goes back but then it doesn't let that pollute.

10:22 >> No, I mean it definitely makes mistakes and then it goes back and thinks about it. I think it's somehow very calculating very correct. I mean, yeah, >> as human mathematicians, you're not always perfect at making these decisions and like >> like the first time something doesn't work, you you automatically kind of downgrade how likely this approach is to work and you keep doing this a few times. The model somehow is much better able to like it seems for several of the solutions we've seen somehow seem it seems much better able to update the sol like how likely the path is to work like yeah versus rejecting a path versus a human doing it.

10:56 >> but but I think even if it weren't the fact that you could just start another model session over means that like you know it's always going to be the case that >> got it here. So in some sense it is like still leveraging the fact that you could like run kind of parallel you know agents on the problem. but if it were kind of backtracking then does make it seem much more like a human you know mathematician and and perhaps it is kind of doing some of that stuff too because like obviously like we have to like make mistakes in in order to like even gain intuition for like why that solution space is like not you know not in the in the set of paths that it could be in.

11:30 >> I mean, I think this kind of thing happens with humans, too, where like if you get stuck on some approach, you might tell another human your kind of general idea and then they'll come back and like figure out how to get it to work and you know, it's just >> yeah, >> it takes more time to do this like with humans. I wonder I mean you know maybe this gets to extent that you can actually talk about sort of like obviously don't talk about the training recipes or whatever but like it's it's interesting that if you're just studying for instance for math papers it's like a very poor training set like a priority for math because I mean maybe math textbooks are even a pure example of this. It's like really bad at actually reconstructing the motivation for why things were you know it's like don't I mean maybe some people like it but don't learn real analysis from Ruden. it just like it's very clean already and crisp.

12:16 and I think that that's bad because it doesn't show the struggle that made us formulate definitions in a certain way. Like why do we even need to have real numbers be defined in this like super abstract way etc. And so you know I think papers also I mean unless you're most people don't write papers with the the context of I need to educate somebody to to be a mathematician and so like the actual maybe curriculum of like learning math is not inherent in like a lot of our artifacts as mathematicians. So maybe another way to ask this question is if the reasoning tracers are actually producing things that it's like okay this is actually more close to mathematical thought like how does that arise?

12:59 I mean I yeah I guess OpenAI has been like the pioneer of reasoning models and yeah you know teaching AI to reason in this way. so you know we're we're doing a lot of work at kind of kind of all possible directions on on you know teaching models to reason better and for longer and and you know in all kinds of different domains. yeah, I mean I think that I I think we're training general purpose reasoning models and if yeah kind of one a lot of these behaviors that we're describing mathematically like backtracking or kind of starting again.

13:35 I mean these are these are not really specific to mathemat I mean we're seeing them specifically in mathematics in these examples but kind of they're general purpose tools for reasoning and I think if you work hard at reasoning you should see these patterns eventually. Mhm. >> So it's this emergent because it's I mean I do think that's why the open AI approach was so I mean it's like it doesn't rely on you you know doing auto formalization in order to like guide the reasoning and I think that's like obviously more like us but it's just like so not obvious that if you're just like training on say a corpus of like math proofs maybe autoformalized and lean that you get get the sort of like projection of like how to think well like put another way like with maybe maybe if we think about it with code like code is such a good corpus to train on because it's one of the few data sets that has such large context you just like I mean maybe you see this kind of with books but they're less structurally interconnected there's just like less structure there I think it's safe to say like on average in a in a book compared to like a piece of code and so like with math papers I feel like But maybe what what we're still bad at with coding models is stuff that that data set doesn't contain which is like kind of the semantics like the syntax is there but there's a little bit of like the higher level semantics of what produced like why do I have to write it this way is not I'm kind of getting to too much in the philosophical but it is just like really interesting how it's still emergent that it's doing good mathematics and we'll probably get into this in more detail if you guys you know wanted to talk in more detail about some of the problems which is just like it's it's not just doing like the expected like we'll push the brute force thing like you clearly are impressed with some of the reasoning traces and it's just not obvious that's gleaned from you know what we would imagine would be the easy training set here.

15:28 >> Yeah, absolutely. I mean, I think this kind of thing is one reason we decided it was important to release like these summarized chains of thought for these kinds of results because >> if you if you've never seen these and you just see all these proofs coming out, you're kind of >> you're not sure what it means like like is the model just guessing in some insane way like is it thinking in some totally foreign like what what's going on? But but actually it it's it's reasoning kind of shockingly like an expert human would.

15:58 >> Yeah. Yeah. Yeah, it's it's very much like reading a colleagueu's like notes. I mean, it's a little more disorganized in some way. They kind of like especially you work close enough with the collaborators, sometimes you'll just see them like spill out their thoughts in an email to you and it kind of it feels like reading a lot of those chained together. So, it's yeah, it's it's very it's quite surprising the first few times. were you two sort of very involved in choosing the problems to to release in this like last 10 problem set that Astros applied to?

16:30 >> >> which was your favorite? >> Yeah, we were definitely involved. do you want to start on? >> Yeah, I mean packing maybe. >> Yeah, I guess. Yeah. So, I guess my personal favorite among these problems is the following. It's it's extremely simple question which is just like it's just about how efficiently can you put a bunch my circles are not very good and they're not all the same size but >> but we're assuming they are.

16:53 >> Yeah. So the question is just like how dense can you place a bunch of so you have a bunch of spheres you have a bunch of spheres of radius one in dimensions. So the question is how densely can they pack and so yeah so so in two dimensions it's kind of like the so so d equals one this is not an interesting question kind of it's just the real line and yeah you can cut it up and a sphere in dimension one is just a unit segment so okay you can cover everything so in d equals 2 it's kind of the picture that you know that everybody loves it's just it's just a bunch of sphere which sort of form like a hexagonal lattice.

17:38 >> Hopefully I've drawn it well enough that I can draw the hexagon. >> Kind of betraying my naite on this problem. Is that like obvious? Is like a very elegant proof that it's lattice. >> Yeah, it's not so obvious that this should work. It was only proven in the 60s, I think. there's a short argument, but it's not it's not so easy. >> Yeah. >> where's the intuition? Like what is kind of like the machinery of the argument?

18:01 >> I mean it kind of like >> I mean it definitely looks like it should work. That's why we're >> Yeah. So I think >> but so did >> Yeah. I think this is the best part about this problem which is really nobody has any idea this problem. So yeah so I mean honestly the best intuition I have for this is that like bees do this and if there was a more efficient way then probably bees would pack honeycomb some other way.

18:21 >> Evolution is efficient. >> Yeah. >> I think beyond that like I don't have a great argument. I mean, and I think how little we know is demonstrated by the fact so, okay, D equals three. The the answer is just like it's what it's how you pack like oranges in a grocery store. >> And this was only this was proved by Hails sometime in the 2000s and like and we don't have a short proof of this like I think the shortest proof is like a few hundred pages.

18:48 >> and what area does it like draw from in order to >> so it's a lot of linear programming arguments and it's very delicate like geometry. It's It's quite ugly actually. >> It's like >> this is like a famously ugly argument. >> Oh no. >> And then the two most famous results are d= 8 and 24 >> 8 and 24. There must be some like weird subspace. >> Yeah. Yeah. Exactly. So this was done in >> Yeah. So this was done in 2017.

19:17 >> Sounds slightly prettier though. I hope. So it so the reason it works out in these two very special dimensions is that so this is called a lattice packing. So it's like kind of very regular and it turns out in these two dimensions there are two very special lattises. They're called the E8 and leech lattice and they're very nice and they're like unusually dense like kind of >> they're just very very pretty structures coming from other areas of math and it turns out that they're the optimal structures >> but they're still like regular. Yeah, they're very regular. But I mean beyond this, so we don't know any more exact dimensions. We know these five dimensions and we kind of don't know anything else. and I mean like to give an indication of how little we know. So there are two very surprising things about this. So you can define like delta d to be like the densest sphere packed in dimensions. So there's kind of an easy lower bound of like 2 to the minusd.

20:15 basic. Yeah, this is >> this is not so hard to show. Basically, any packing where you can't put in another sphere has to have this density. So, okay, it's not ridiculously small. And we know that it has to decay exponentially. So, it has to decay like it grows like 1 minus C for some at least for some constant. So, in large dimensions, you can only cover cover like a vanishingly small portion. but we know like basically nothing else. and so forth.

20:44 >> That's just because like the high dimensional sphere like thing where it occupies it just like Yeah. Yeah. The volume behavior is weird. >> Yeah. So basically I mean basically they don't want to touch next to each other. I I don't know if I don't think there's a particularly short way to see that that it's like exponentially small but it's known to be exponentially small. And for a long long time the best bound was something like this funny number like 2us.599 d. This was proved by two mathematicians in the 70s capski.

21:18 >> Okay, that's a weird number. Where's that spit out of? >> It's not it's like cominatorial or >> it's a it's the answer to some extremely ugly optimization problem. >> There's there's like a nice underlying strategy. >> Okay. >> Yeah. Yeah, I'll say one last thing about this. Yeah. So, yeah, I mean these were these two Russian mathematicians in the 70s. Yeah, it's actually very hard to find their paper. It's like one page. It's like two pages long. Yeah, they don't write very many details because paper was flared. But yeah, >> two negative D is just like the square lattice like the dumb one or is that >> so? Yeah, it's actually not so easy to So the argument for this is as follows.

21:58 Basically imagine that you have a set of spheres and I tell you can construct a set of spheres so that like you can't put down another sphere because >> if you could put down an extra sphere, you just keep putting it down. So you have a set of spheres so that there's no other sphere which you can put down. >> >> that's your it's like almost like a >> just Yeah. Just take any such packing. >> Mhm. Okay. Yeah. Yeah.

22:19 >> And I claim that this has to cover at least 2 to the minus d fraction. The reason is that like if you blew up each of these spheres by a factor of two then like they have to cover every point in space. >> And the reason is otherwise you could put down if there was any empty space you could put down a sphere there. >> Yeah. So I guess if you take the usual lettuce, there are actually like more places you can put things kind of diagonally. Yeah.

22:42 >> Okay. Yeah. Yeah. So that's actually not it's like a worse bound. >> Yeah. You can just keep plopping things in. >> Yeah. Yeah. This is >> So this is like more Okay. Okay. >> Yeah. This is related to this like really funny fact where you like put a sphere on every point on. If you take a cube in high dimensions, you put a sphere on every point. >> It's like vanishingly small. >> It's vanishingly small. It's so small you can put another sphere in the middle >> and it fits.

23:02 >> high dimensional sphere behavior is weird. >> Yeah. very weird. and so okay so the great part is that so the model shows the following. So I'll write two things. So this is Astra I guess probably the right way to refer to this. >> so okay I'm going to write something called the LP bound. I'm not I'll explain this in a second. And it shows that it's smaller than this very nice number. do you want to say equals?

23:33 yeah, it's equal essentially e to the 2 pi plus little01 to the d. And if you you can work out what this number is, it's like >> it's like roughly something like 2 to the minus.6 >> close. >> Yeah, it's surprising. >> But this one, you know, e to you're like, oh, maybe there's some like nicer kind of like structure there that that that fell out. It's at most This is like roughly something like 2 to the minus.61 dot dot dot D.

24:07 >> That's the numeric. >> I thought it was 604. >> Great. >> yeah, this shows mine. Yeah. so okay, so there are a couple of things. So first, what is this LP of D? So VZOs's work actually builds on some earlier work. it turns out that there's a way to attack sphere packing via what's called a linear programming ground. So LP just stands for linear programming. So Conan Elkis gave an approach for super packing based on linear programming. so it's like a it's like a linear optimization problem over a convex set but it's all kind of infinite dimensional here.

24:46 >> And basically what this reduces down to is you try to understand the follow. So what you try to show is you basically you construct a function f. So this is in dimensions and it's mapping to r and it has the following properties. So first f ofx so this is a function in dimensions. So it's always less than zero if like the size of x is bigger than one and you second have that the fouryear transform of x this is always non- negative. So this is just this is a linear program because the for transform is a linear operator and night race.

25:27 >> So you're taking just some arbitrary f that satisfies this property. >> Yeah. So you can take any f that satisfies these properties. And what they prove is that delta d is bounded by the ratio of the foyer transform at zero to its to the the the for the for transform at zero of f of 0 to the over the for transform of zero times the volume of the ball of radius 1/2 in data matches. and so okay this this proof is not so short for experience mathematician it's like it took half a paragraph to prove it but it's a little bit tricky.

26:04 >> and the point is so it turns out so this is a relaxation problem. There's no guarantee that taking the optimal f will give you a good bound on delta d. but so what Viazosa did and this was sort of the key I mean a large part of the reason she won a fuels medal in in 2020 was that or in 2022 was that she constructed a function in 8 and 24 dimensions such that this upper bound matches exactly these two very special lises >> and these are kind of miracles of nature that both you can construct this function and that it gives you the optimal bound. but you can just this is a very very natural problem. I mean it's a function with a very two very simple properties and you just want to understand how this behaves in for large dimensions d and that was a big mystery.

26:55 we there was a numeric paper by con and several others which conjectured that just based on doing numeric that this was the answer but they had no idea why this would be the answer. M >> and what the model shows is that actually the linear programming bound in large dimensions has this extremely nice asintoic behavior and the proof kind of explains where this is coming from and because you understand this LP bound perfectly this actually just gives a better bound on delta D. It turns out that this old bound can be kind of reinterpreted in this framework and what the model does it shows you the best possible bound you can get by this framework.

27:34 >> So the model sort of made the connection and Yes. What is the sort of like what do yeah what I think so the model gives a function f which so first it constructs a function f which gives you this bound and then it shows that there's no function f which does any better. So it's an equality which is quite strong. So in like sort of we now understand this this problem in high dimensions very well >> and that's pretty remarkable. and the model was just kind of told like analyze this linear program in high dimensions, you know, go have fun.

28:08 >> Got it. >> And and to give an indication of like how little I think this conjecture was based basically only by doing numeric extremely clever numeric but numeric and so >> okay >> yeah you have to kind of figure out why this is the right thing to aim for and it does and that was pretty remarkable. Yeah, I mean I I had actually thought about this problem for about six months at some point when I was a graduate student and yeah just I remember making like absolutely zero progress on it. So it was very nice to be like explained why it was yeah why it was true. so this was that was a pleasant experience.

28:41 >> I think also in general it was it was one of these solutions which I I knew several people had tried the problem. It's pretty remarkable because like the the model solution especially for this being like that the LP can't do better than this was like quite short. It's it's a few pages of like complex analysis, but it's kind of exactly the right approach. Like once you see it, it's kind of it's like unbelievable like why why hadn't somebody done this before? It was it was like there are many types of good mathematics, but I think one of them is just like you see it and you're like, oh man, why didn't I think of this? And it was really fun and it I mean I sort of knew why I didn't think of it, but it was quite nice to see it and it was fun to see. That's why I like this problem a lot.

29:19 >> Yeah. So so this is the first of the 10 problems that Astra saw. Mhm. >> but the second is actually closely related. >> so this was sphere packing. The second one is spherical and binary codes. >> Mhm. >> so what's you should draw a code. You drew a packing. It's it's going to be the same picture. Okay. Sure. Sure. >> Otherwise, we're going to have his picture packing in our mind. >> Yeah. A spherical code is literally just a sphere packing but on another sphere.

29:57 >> yeah. So, I mean, a spherical code is basically just a sphere packing on the surface of another sphere. So, yeah. >> Yeah, it looks like a sphere. >> Yeah. Okay. So, same same picture as before, except you're kind of on a curved surface. >> Okay. Okay. >> Okay. So, why is it called a code? Well, you can I I guess the the reason is because of binary codes, which is again the same sort of thing, but now it's on a cube.

30:30 Okay. Yeah, fine. Let me let me draw a picture of a a cube and some like simplest possible code on it. So like when you're when you're like sending so so this is really like about error correcting codes. So, so what are error correcting codes? So, you know, it's like I send you some string of bits, right? like and maybe I'm I'm like worried that some of the bits I send you get corrupted, right? So, so maybe like just because of some errors in my system like this one gets gets changed and we want some communication protocol so that like you can decode this like small amount of error and like recover what I was trying to tell you. and you know like normal English language kind of has this sort of property, right? If I make a few typos, you know, you're going to be able to understand what I'm saying. But if I if we have some like really brittle communication scheme, it's not going to work. So, codes are kind of the the way you you solve this. And mathematically, it just means like, you know, what's a binary string like this of a fixed length? It's like a point on some hyper cube.

31:47 >> Mhm. >> And we want a dictionary of allowable code words that are like separated from each other. So, like in this case, if I don't want any two to be adjacent, I would kind of take these four vertices, kind of the like even ones if you sum up the the digits, right? and and like okay I I guess okay in in this case I guess if I if I have an error you can't tell which one it's from but at least you can tell it's like not at least you can tell there was an error.

32:16 >> Oh I see. I see. Yeah. Yeah. Because it's like kind of sparse in the Yeah. It's like it's not two adjacent. So that like is this is like when the hamming distance is not >> Yeah. Yeah. Right. Right. You want Yeah. So you want like a large Hamming distance between any distinct in your in your dictionary. >> And Yeah. I guess if you take two opposite corners then if I have like a single bit error I can always like recover which point it was coming from >> the one that it's definitely closest to.

32:41 >> Yeah. So there's there's kind of the you know same question in both of these cases like in a very highdimensional setting what kind of rate can you get? >> And like for binary codes it's really like you know an extremely practical question. It's it's sort of like if I send you like an n bit string and there's like you know 1% error rate like how much longer does my message have to become to tolerate that amount of errors >> right this like some fundamental fundamental information theoretic limit of like like you know send like communication and you know but but you can see like like certainly this this spherical case is like it looks very much like sphere packing for example if you like If you make all these little spheres really small, then like the curvature of the big sphere is kind of not going to matter so much and it it looks like just packing spheres in full space.

33:33 >> and in fact, yeah, like these these problems turned out to be very related. so so for for these problems, there was like there were similar bounds coming from these KL authors and and like there's like something for the sphere and something for the cube, but it's it's all kind of the same stuff. Mhm. >> And our models found better bounds for for these cases as well. And like it you you I mean the the techniques look pretty different actually if you if you like write them out. So so this this full space analysis of this linear programming was using like just complex analysis.

34:14 >> but if you like the the method for for these cases we're using representation theory. Oh, >> like like both the sphere and the cube have a lot of symmetry >> and basically the the idea of the proof was to really leverage this symmetry. >> like there's some amount of this in the the previous like existing method and and really the improvement is to like lean into the representation theory like really hard and and kind of make the the algebraic symmetry like enter in a more sophisticated way. and then it like turns out that from the representation theory formulas if you kind of take this like like small sphere limit in the spherical code case you you recover like >> part of this result and you recover this this value. so like this result isn't a special case. You kind of only went one direction of the bound from from looking at it from the code's point of view. But like there's like a very close connection.

35:10 >> Okay. >> Yeah. >> Yeah. Yeah. So you you guys let this like run in parallel. So it's like kind of discovering or because you're not sort of like feeding it. >> So so actually this was the one case where there was some interactivity involved. Interesting. So for for all of so except for this pair it was just you know we had some problems we we fed them in and we you know the model the model came back with some solutions.

35:32 >> what happened here was actually pretty interesting. So so we first asked it to improve the bound for the codes. >> Mhm. And it came back with an improvement that like used some amount of representation theory. >> And then we kind of asked it, hey, can you like push this further like you know what happens? And then it came back with some like much more sophisticated representation theory and like it turned out that you got this conjectured value for full space sphere packing like out of that method by pushing it as far as it can go. So then we we kind of ask to directly analyze the sky and try to complete the picture.

36:13 >> Okay. >> Yeah. >> So so the relationship like isn't a coincidence. >> Yeah. Yeah. >> It's like interesting when you're saying the first prompt which is you know maybe so basic which is like can you push this further? It does require some judgment from mathematicians and but like eventually you would imagine by scaling the models you don't need to do that or there's another view that the harness actually does matter and this is kind of part of the harness apparatus. Do you guys have any views on that with your working with Astra, especially generations of models and how how much you have to kind of input or how much the harness matters versus not.

36:48 >> I mean, yeah, I I guess there have been some like funny quirks like this that just come from like exactly what you asked the model to do basically like like in in this case, what the model was asked to do originally for codes was to improve the bounds >> by like some exponential factor. So it it really like shows up in this like leading constant up here. >> Mhm. >> and you know it it improved the bounds and it it didn't try to push things too much further. Like sometimes you see it do but sometimes it just doesn't bother but yeah you know you just ask it again and it it goes further. So it it wasn't like a capabilities issue. It just >> kind of didn't feel like it at the time.

37:24 >> Do you call that judgment or like what is the because like there there is a Yeah. What do you call that? >> I think models tend to be pretty task oriented. If you if you tell it to do a task and it accomplishes the task, >> it's pretty happy. >> So yeah, like the task orientedness it's like but do we expect that level to kind of ascend up to it's not that they will be less good at being task oriented. It's like they'll ascend to the level of like okay no let's let's go in this direction. You'll have the judgment because you guys had the judgment. You're like okay this is pretty promising. Looks like you're using a lot of representation theory. It doesn't seem like there's a limit so far. but it doesn't have that context yet. But like I guess what I'm trying to say is like this one it's hard to maybe extra harder to extrapolate.

38:08 But from like previous generations when you had to give it more maybe prompting more of that harness work but eventually you probably had to give it less. So it probably gives you some confidence that there's this like really fast ascension and and do you see yeah like someow somehow solving a harder math problem is like you have to solve many smaller like somewhat less hard math problems and the fact that the math problems are getting harder is kind of an indication that you're the model is able to take on more and more work in like a single continuous unit. and I think that's that's the thing that looks very promising somehow like any of these solutions it's not like one idea and then you're kind of home free. You need several you need several pieces to kind of interact and talk to each other. And >> the models I the model doesn't come up with all the ideas at once, right? It doesn't pull everything out of in in an instance of kind of the fact that it needs to sort of see how this piece interacts with another piece. That's kind of like solving a problem in itself or piecing together many problems in itself. It could just be that okay when you're telling it okay push this even further that was of the same order of like magnitude as like all the smaller things it's solving as well in in between and so you don't think that's kind of like a privilege direction it's just sort of like hey let's give it like one one more help or you actually think that there's I guess what I'm trying to get at a bigger question is like is there a good sense of like you know taste because like when people talk about for instance how well the models are getting at like doing research church for instance that's what we you know we want a little bit of RSI and and sort of like there's surprising things about how that improves and then there's like the oh you know maybe right now it's at a level of still like a junior researcher it's like not really asking like the right problems and so I'm just trying to get like a maybe a sense of like where you're seeing that progress through the model advancements each generation I mean what is taste even >> yeah I I think I tend to be pretty utilitarian in my view of taste and like if you're able to solve problems faster by making better judgments like I think that's like the best like general proxy I have for taste and somehow the fact that solving harder problems means it has kind of by definition means it has better taste. I think there there are these no yeah I think occasion occasionally because they are task oriented you do occasionally get these these symptoms of like oh it clearly has made a breakthrough it kind of understands it's made a breakthrough and then it doesn't kind of push all the way to the limit because that's not what you asked but that seems yeah that seems seems rather minor compared to like the set the the state of progress we've seen so far >> okay yeah I think it's a pretty clear >> like I think it's it's like maybe you're liable to get confused if you're trying to like do a concrete long horizon task and show taste kind of at the same time.

40:59 But like you know if if you if you have like one model that's responsible for taste and one model that's responsible for going out and like you know working for a long time at solving a hard problem kind of as the as the like you know underling of the of the the supervising AI. I I feel like that kind of going to be fine currently. >> Oh, interesting. Because that is like saying that they these two things are somewhat sep if not separate at least they shouldn't kind of pollute each other's contexts which is a little bit I mean it could be potentially like a strong stronger statement than I guess. Yeah. Know it's just kind of interesting because it's it it it might just be like to your point it's you know let's take the utilitarian answer is solving harder and harder problems. is it's doing a lot more than just like you know brute forcing something. It's making choices.

41:52 It's like pruning you know a vastly large space of possible paths into something that's like really you know it's both tractable but then ends up being like it's a diminishingly small path within that space. but like having like why would would be like a separate model a se separate generation or something that's a different version of the model that would contribute to taste or maybe that's totally like it's too abstract doesn't make any sense you know we should just let the actual this might like the related question be like you know what is what is the the thing that gets us to a better version of intelligence the harness and the model is it just the model and it's like we we see this in you know at least in applied AI or you know startups where it's like it's a continual battle of like you need the harness but then the harness adapts very poorly to a new model because sometimes like a very very minimal harness is still the best way to expose to the raw power of the model but then now we also have these like training regimes where we require the harness to be you know trained with I mean maybe part part of this is to keep things more proprietary and harder to harder for other people to use it but I think partially it's it's maybe actually that it helps have more control on like the reasoning traces you care about. It's a long rambling way of saying it's like, yeah, I I don't actually like this is so interesting to see how the models have gotten better at math and maybe something that's like very abstract and hard to describe like taste is a way to tease out like what is actually necessary here.

43:22 I think my only like non-trivial thought here is that like when you're working I mean just when you're doing any task occasionally you get pigeon holed and you like work really hard and just having a friend look over your shoulder and be like >> what are you doing and then just like just having that one bit of like step back for 10 seconds like this is often very useful. >> Yeah. Yeah. I mean, I see no reason why humans would be so different than models somehow.

43:45 >> Having or models would be so different than humans. Having >> having a few humans working together is often more powerful than just having one. >> Yeah. It's like in this kind of collaborative thing, you actually you kind of Yeah. artificially created it, but it's very similar and dynamic. >> But I think a lot of taste is also like having a sense of what problems you or like some method you have in mind are going to be good at solving. Mhm.

44:08 >> Like it's I mean certainly there's some amount of like absolute aesthetic point, right? But there there's also just like you know having a nose for what what you might want to pursue because you'll be able to make progress. and you know I think for that like there's you know you would expect that as a side product of being good at completing tasks you would you would get there sort of right. Let me know if we still want to do like a section on sophetic groups because I think you know up to you guys it's definitely super interesting.

44:40 >> So maybe the the first question is what is a group? Let's remind ourselves. So a group is a set of elements with some multiplication operation. And basically this is how mathematicians think about symmetry.

45:11 So, so you're like basically like if G and H are elements of your group, then GH has some is some other well- definfined element of your group and you have associativity and you have an inverse So for every G there's some inverse and there's some like specific element in the group that is is kind of the identity. okay so you know it's it's some like abstraction of like composing operations. So these could be like numbers they could be like multiplying matrices they could be like like rotating something which is you know a special case of multiplying matrices. and a group is sofic.

46:12 What if well there's some you know precise definition but you know roughly it means it so so I should say like groups that can be finite or infinite. So like you know if if you have like a square like all the rotations of it form a group with like four elements. >> If you have like a circle then the rotations form a group with like uncountably any many elements.

46:42 >> and so sophet groups are either finite or countable. You should think of them as being countably infinite. So there's like the same number of elements as like the integers. and if it's sofic if in some sense it can be approximated by finite groups. So we didn't know if there was a non-sulfic group. So so the the the result that Astra proved >> is simply that there exists a non-sic group.

47:14 >> Yeah. And without like I mean we can you know before going to to that proof it is like you know it's like I feel like a lot of the programs of math is like okay we are such finite creatures let's see how well our finite approximations are you know do and in this case especially for the countable case it's like maybe you'll be relating it to like the eldest Leon's thing. It's just like it helps kind of anchor the picture of like it seems like such a I mean it's a nice result if it were true, but it it's not and it seems almost like reasonable.

47:45 And so yeah, I I actually didn't didn't go I would love to hear the explanation of like how it it found a counter example. Yeah, I mean I I would say that like you know the the hope that there was no non-s so fit group so every group has this kind of approximation like maybe this is sort of like people hoping that there's a miracle >> because it turns out that groups like this have a lot of nice properties because you can run certain proofs for finite groups and then you know kind of approximate them in whatever way the definition of being sofic lets you approximate them and get the result. So, so like there's this notion of being a subjunctive group. So there's there's some fact that any group which is sofic is also surjunctive.

48:40 surjunctive is some property of like dynamical systems on the group >> and I guess the original question was whether every group is surjunctive. This is some question of gotch from the 70s. and this this fact that follows this like pattern of prove it for finite groups and then do this approximation is is what motivated the question about if there's a non-sulfic group. >> Mhm. yeah may maybe I'll say a little bit about this I'll just lions conjecture and >> yeah sure yeah yeah >> so I I guess I had like heard of this a little bit beforehand because there's a related stronger conjecture in probability that was made popular by Aldos and Lions this this conjecture roughly what it says is like any like infinite graph with some nice property called unodularity.

49:47 a random in modular random graph can be approximated by large finite graphs. So maybe the the way to like explain what these kinds of things are trying to say without without getting into technical weeds is to say what they mean about the integers. >> So so like how would I draw the integers as a graph so this is called like hilly graph you're just going to connect nearest neighbors. Okay so there's some kind of canonical way in which this is like the graph that represents the integers.

50:31 Okay. And there's some sense in which you can approximate this by finite graphs. why? Well, if you look at integers mod n then you kind of get the same picture but like you have like a big circle instead of an infinite line. and the the point is if you like look at any point here and any point here like in nearby things look the same. You have to go like very far away to kind of see this global geometric structure that you have a circle and not a line.

51:09 >> Mhm. >> and in fact the integers and integers mod n are both groups just by like adding numbers or adding numbers mod n. >> So these integers mod n are like sofic approximations for the it full integers. So like this approximation is kind of why the integers are a sofic group. >> So so the the sufficity the the statement that every group is sofic is sort of a generalization of the fact that you can do this approximation with groups and this also lions conjecture is kind of a broader conjecture that like any network you can do this and it like you don't require as much algebraic structure roughly. So it's it's kind of a broader conjecture.

51:55 >> So so this conjecture was disproved earlier like two years ago and it was kind of a really to force work like it was like 250 pages building on another 200 pages. It uses like like quantum complexity theory. So it really builds this like >> you know very complicated bridge and like you know I I think not very many people could understand this. right. So since this is a stronger conjecture, the disproof like is weaker than disproving this statement that all groups are sofic.

52:28 >> but it turns out that the direct proof that all that there's a non-sopic group was like much shorter and and easier than this than this really amazing disproof of the all this lions conjecture. It's like like 15 pages maybe and it doesn't have any of this like very complicated connection with quantum complexity. just kind of stays in group theory land. I mean it uses some important like you know existing results by other mathematicians like Kun and and Tom but it's like it's it's like a it's like a very reasonable normal kind of proof.

53:01 >> Yeah. And and kind of spelling maybe out the obvious but like the the connection between the sof group statement is just you take the kay graph and that's the one that is like what they use or for eldest Leon. >> Yeah. >> Yeah. And so that's why it's like a you know a subset of >> right so yeah basically what happens is yeah so for a group you can take exactly a kaly graph so you take some like elements that like generate the group and you kind of connect elements that are that are >> adjacent so so in this case like this is a kaly graph of the integers >> so so right so when you do that from a group you get like a deterministic graph >> right you just get like a single graph the graph you fix some set of generators so this conjecture is stronger basically because it allows a broader set of graphs that aren't deterministic. It allows them to be random but have some extra you know you know modularity property that constrains exactly how it can be random.

53:58 >> But yeah, basically that's that's the difference. like here you kind of have to give a deterministic network instead of a random one. >> Yeah. Anything kind of interesting surprising about the result? I mean you mentioned some things which is like it's it's stayed within group theory the techniques. >> I mean I I I think maybe it's like a a nice example of this general pattern that theorems produced by AI have generally been like like the proofs are pretty short generally. They're like >> >> like with the counter examples so far.

54:36 >> Yeah. Yeah, but I mean this one it's like okay it's sort of a counter example but like there's some you know there's some like >> stuff you have to to do to to analyze things. The difficult part here is that like >> the property of being a Sophie group is not so easy to get your hands on. So you have to find like a concrete way of saying like like producing a producing a way of saying this group cannot be so and like yeah >> and the proof is actually it's very short. It's like it's almost it's a combinatorics argument, but it's like a very very delicate combinatorics argument and the model like somehow you need to both have the right statement and know what pieces in the literature and then execute it correctly and that's very nice.

55:13 >> like the difficulty of this problem is that like it's just a really really hard it's like very hard to get your hands on like being approximated by any possible finite. >> I was going to say it's like what is happening at that countable infinity that's like resisting this approximation? Like do you guys kind of did it give a sense of like when do you do postmortem when you're like okay Astra explain to me like what is the >> oh yeah we did that all right what was a good explanation you got out of it >> I think there's like some concrete like cominatorial obstruction basically it's like >> it's hard to explain but there's like some concrete cominatorics obstruction which if you read the previous papers you realize that that's what they couldn't rule out and Astra found a way to kind of >> say okay no no if you add this one extra algebraic fact this this like weird conspiracy can't happen. it's like very clearly trying to rule out conspiracy in the sort of previous authors had implicitly written about.

56:05 >> Yeah. >> And those were the actual suspects it turned out. So they were sort of on the right track and then this did the last mile of well whatever however you quantify that. But but I I think it's like like a year ago I I would have been very surprised to learn that like all of these AI proofs are like very short and elegant. >> Yeah. >> Like they're you know you you're kind of like afraid that they're going to like generate all these thousand page things and you're just like never going to be able to understand it. But it's been kind of the opposite. Yeah. Like only humans can generate like >> 200page proofs right now.

56:41 >> Yeah. Well, and also I was like asking I'm like if you do that postmortem it ends up usually engendering more mathematics because when you do that with humans like that's what you know breeds new mathematics. So maybe maybe if you kind of alter the prompt a little bit and be like how would you you know generalize this or something like yeah I don't know if that's been a technique for you guys to like have it explore and exploit what it has already developed.

57:06 Well, well, there has been some there has been followup on this already actually by by and Tom who this was built on. So they they like the >> math community is coming on. >> Yeah. Yeah. Yeah. Which is kind of what what we're hoping, right? You know, we don't want to be you know writing lots of followup papers ourselves, but if if there's some interesting followup that you know it's like we're very very happy that that there's some there's some followup building out these ideas more and giving like more examples of non-sitic groups in this case.

57:32 >> Yeah. Well, actually maybe that's a great segue into like how you know what's the ideal way that this is being taken up by the math community because I feel like there's a spectrum of answers from working mathematicians sense of like some you know probably most at this point are like okay AI is obviously doing some non-trivial stuff. it would be a disadvantage not to admit that in my workflow. I've definitely heard some stories where people of, you know, would find it hard to either take AI as a co-author or like how do you even do kind of attribution this way? But I don't know like what maybe to paint the the more optimistic picture. So you're saying you want the mathematicians to be building on this these results. It definitely generates a lot more results to be verified. So it you know it puts pressure in on the community and the profession like how do you how do you kind of expect the evolution of kind of uptake and and collaboration with mathematicians.

58:28 >> I mean given that the fact that the models can produce sophisticated mathematics means that they can help you understand like sophisticated mathematics. I mean like I don't know occasionally I like I enjoy looking at the archive and I want to understand some proof and like I could read the introduction but in practice it's just much faster take the PDF put it into put into my favorite model and then like get a get an output of like what is the rough proof strategy and somehow this like >> along I mean of course models are going to help us produce exponentially more mathematics but they also make it much easier to absorb it. yeah. And right now, okay, it's still a bit of a challenge back and forth, but I think it's for me at least a much much faster at understanding it's much much faster to understand a piece of mathematics with a model than without it. so it's helping solve the problem it creates anyways.

59:18 >> Yeah, I feel like that at least and it's, you know, I I don't I don't view it as like creating much more problem, but again, like I don't have such you know, high stakes in like, okay, I'm I'm going to get I'm not going to get tenure, etc. So like I I I agree like making it more accessible. Like if I'm not spending so much time absorbing an area, I can like put it into chat GPT and then expect to I mean you guys have an even more powerful model hopefully releasing for other people to enjoy as well. But like it's I think like the the positive version of that is actually more people can participate mathematics. It's like people might be coming with other intuitions and they could actually maybe generate good mathematics. Is that sort of like closer to the vision of what you're hoping this is, you know, pushing towards? or like what what things do you think to be wary of to to kind of adapt fast enough to take advantage of AI?

60:09 >> Yeah, I mean I I think certainly there will be a lot of changes, right? Like I I guess in math like there are there are a lot of things that are kind of important for for like a given result, right? You need someone to come up with it, but you also need people to understand and absorb it and like, you know, internalize it enough to to do more with it and and like figure out where it fits into like humanity's understanding, right? And like a a couple of years ago like the proving the result was like so hard that kind of the the other stuff was just kind of coming along for the ride, right? you know, like if you if you manage to like prove this thing yourself, you're automatically going to understand it quite well. You're kind of responsible for like maintaining it and in some sense and like you know, explaining it to other people. and yeah, now this kind of what was the main bottleneck before is kind of much less of a bottleneck and you know, these other kind of constraints come into play.

61:15 so it's yeah the the like optimal structuring for you know organizing the knowledge could could look rather different. >> Yeah. How does that look? I mean does this make the field a lot more kind of empirical? Will people do sort of the hard like the first thing that was scarce which is like all the reasoning and then more I mean not that it's like a bad thing to make it empirical but it's almost like it functions as a very different discipline. like a lot of the fun stuff is understanding you know and so understanding communicating maybe assembling having still the human taste does that sort of remain rarified and and that's how you know current mathematicians need to adapt and and reward you know contributions or or is this too much of a caricature it's like something else >> I think certainly understanding how to put as we get more and more mathematics put it in like a proper framework and sort of how sort of like being able to explain it to other humans so that they can also appreciate it. I mean somehow implicitly we valued this but it was usually because you were the person proving the result that gave everybody else the understanding but I think increasingly it would be a function of like you're sort of helping you're the human who can sort of give this understanding to other people and sort of help them with it. I think that more of that communal understanding will I think become it was much more implicit in how we view math in general but I think it will be an increasingly more explicit and valuable part of the subject.

62:43 >> I mean a nice thing about math is that the the ceiling for difficulty of a math problem is pretty high. So even if you know kind of even if AI get you know continues getting like exponentially better at math like it might you know it's plausible we'll never solve something like P versus NP >> and it could be that like the the field kind of becomes more >> you know attached to like like these big mysteries and and less to like smaller mysteries that are more like routine. I mean know.

63:21 >> Yeah. Yeah. I think that's a positive vision of the >> I mean also like I don't know there are things I spent like months or years of my life wondering about not getting to know and hope >> we get what a joy. >> Yeah. Some portion of them I'll get to know the answer to. I'm pretty happy about that. >> No, exactly. No, I'm I'm excited about this like renaissance of results and understanding and I feel like I mean this is such an an infinite, you know, field like no pun intended, but like it's just like it's it's it's just there's so much that you can actually create here. so I mean especially for somebody like me who's not going to have the time to actually like practice mathematics now there's like a lot more that you can actually do in the activity of math. So yeah.

64:02 >> Yeah. I think the the like the ability of someone who's not working on math is like their literal job all the time to like understand what's going on and like you know learn about some of the mysteries they might have wondered about will will go up quite a lot. also you know if you're if you're like if you're working on something that requires some math >> you know suddenly you you don't need to like find a world expert on this topic to to be able to you know use it in your own work. you could >> sorry mathematicians.

64:31 Yeah. Yeah. No, it's true. I mean, I think there was just like a der of actual like people who could could do that and so I think this is helpful. Maybe it's helpful for the theoretical physics. Like we'll see. but a lot of other applied areas as well. >> It'd be nice for the world if applied mathematics went a lot faster. >> Yes. I mean I'm of that opinion. Well, thank you guys for joining. This is a lot of fun and I'm, you know, just so excited for how much the models are advancing. So maybe we'll have you guys back soon. Yeah, thanks so much for having us.

65:00 >> Yeah, thanks for having us.