Thank you for your comment!

What you're saying seems more galaxy-brained than what I was saying in my notes, and I'm probably not understanding it well. Maybe I'll try to just briefly (re)state some of my claims that seem most relevant to what you're saying here (with not much justification for my claims provided in my present comment, but there's some in the post), and then if it looks to you like I'm missing your point, feel very free to tell me that and I can then put some additional effort into understanding you.

• So, first, math is this richly infinite thing that will never be mostly done.
• If one is a certain kind of guy doing alignment, one might hope that one could understand how e.g. mathematical thinking works (or could work), and then make like an explicit math AI one can understand (one would probably really want this for science or for doing stuff in general^[1], but a fortiori one would need to be able to do this for math).^[2]
• But oops, this is very cursed, because thinking is an infinitely rich thing, like math!
• I think a core idea here is that thinking is a technological thing. Like, one aim of notes 1–6 (and especially 3 and 4) is to "reprogram" the reader into thinking this way about thinking. That is, the point is to reprogram the reader away from sth like "Oh, how does thinking, the definite thing, work? Yea, this is an interesting puzzle that we haven't quite cracked yet. You probably have to, like, combine logical deduction with some probability stuff or something, and then like also the right decision theory (which still requires some work but we're getting there), and then maybe a few other components that we're missing, but bro we will totally get there with a few ideas about how to add search heuristics, or once we've figured out a few more details about how abstraction works, or something."
• Like, a core intuition is to think of thinking like one would think of, like, the totality of humanity's activities, or about human technology. There's a great deal going on! It's a developing sort of thing! It's the sort of thing where you need/want to have genuinely new inventions! There is a rich variety of useful thinking-structures, just like there is a rich variety of useful technological devices/components, just like there is a rich variety of mathematical things!
• Given this, thinking starts to look a lot like math — in particular, the endeavor to understand thinking will probably always be mostly unfinished. It's the sort of thing that calls for an infinite library of textbooks to be written.
• In alignment, we're faced with an infinitely rich domain — of ways to think, or technologies/components/ideas for thinking, or something. This infinitely rich domain again calls for textbooks to keep being written as one proceeds.
• Also, the thing/thinker/thought writing these textbooks will itself need to be rich and developing as well, just like the math AI will need to be rich and developing.
• Generally, you can go meta more times, but on each step, you'll just be asking "how do I think about this infinitely rich domain?", answering which will again be an infinite endeavor.
• You could also try to make sense of climbing to higher infinite ordinal levels, I guess?

(* Also, there's something further to be said also about how [[doing math] and [thinking about how one should do math]] are not that separate.)

I'm at like inside-view p=0.93 that the above presents the right vibe to have about thinking (like, maybe genuinely about its potential development forever, but if it's like technically only the right vibe wrt the next ${10}^{12}$ years of thinking (at a 2024 rate) or something, then I'm still going to count that as thinking having this infinitary vibe for our purposes).^[3]

However, the question about whether one can in principle make a math AI that is in some sense explicit/understandable anyway (that in fact proves impressive theorems with a non-galactic amount of compute) is less clear. Making progress on this question might require us to clarify what we want to mean by "explicit/understandable". We could get criteria on this notion from thinking through what we want from it in the context of making an explicit/understandable AI that makes mind uploads (and "does nothing else"). I say some more stuff about this question in 4.4.

not really an answer but i wanted to communicate that the vibe of this question feels off to me because: surely one's criteria on what to be up to are/[should be] rich and developing. that is, i think things are more like: currently i have some projects i'm working on and other things i'm up to, and then later i'd maybe decide to work on some new projects and be up to some new things, and i'd expect to encounter many choices on the way (in particular, having to do with whom to become) that i'd want to think about in part as they come up. should i study A or B? should i start job X? should i 2x my neuron count using such and such a future method? these questions call for a bunch of thought (of the kind given to them in usual circumstances, say), and i would usually not want to be making these decisions according to any criterion i could articulate ahead of time (though it could be helpful to tentatively state some general principles like "i should be learning" and "i shouldn't do psychedelics", but these obviously aren't supposed to add up to some ultimate self-contained criterion on a good life)

• make humans (who are) better at thinking (imo maybe like continuing this way $\approx$forever, not until humans can "solve AI alignment")
• think well. do math, philosophy, etc.. learn stuff. become better at thinking
• live a good life

A few quick observations (each with like $90\%$ confidence; I won't provide detailed arguments atm, but feel free to LW-msg me for more details):

• Any finite number of iterates just gives you the solomonoff distribution up to at most a const multiplicative difference (with the const depending on how many iterates you do). My other points will be about the limit as we iterate many times.
• The quines will have mass at least their prior, upweighted by some const because of programs which do not produce an infinite output string. They will generally have more mass than that, and some will gain mass by a larger multiplicative factor than others, but idk how to say something nice about this further.
• Yes, you can have quine-cycles. Relevant tho not exactly this: https://github.com/mame/quine-relay
• As you do more and more iterates, there's not convergence to a stationary distribution, at least in total variation distance. One reason is that you can write a quine which adds a string to itself (and then adds the same string again next time, and so on)^[1], creating "a way for a finite chunk of probability to escape to infinity". So yes, some mass diverges.
• Quine-cycles imply (or at least very strongly suggest) probabilities also do not converge pointwise.
• What about pointwise convergence when we also average over the number of iterates? It seems plausible you get convergence then, but not sure (and not sure if this would be an interesting claim). It would be true if we could somehow think of the problem as living on a directed graph with countably many vertices, but idk how to do that atm.
• There are many different stationary distributions — e.g. you could choose any distribution on the quines.

I think AlphaProof is pretty far from being just RL from scratch:

• they use a pretrained language model; I think the model is trained on human math in particular ( https://archive.is/Cwngq#selection-1257.0-1272.0:~:text=Dr. Hubert’s team,frequency was reduced. )
• do we have good reason to think they didn't specifically train it on human lean proofs? it seems plausible to me that they did but idk
• the curriculum of human problems teaches it human tricks
• lean sorta "knows" a bunch of human tricks

We could argue about whether AlphaProof "is mostly human imitation or mostly RL", but I feel like it's pretty clear that it's more analogous to AlphaGo than to AlphaZero.

(a relevant thread: https://www.lesswrong.com/posts/sTDfraZab47KiRMmT/views-on-when-agi-comes-and-on-strategy-to-reduce?commentId=ZKuABGnKf7v35F5gp )

I didn't express this clearly, but yea I meant no pretraining on human text at all, and also nothing computer-generated which "uses human mathematical ideas" (beyond what is in base ZFC), but I'd probably allow something like the synthetic data generation used for AlphaGeometry (Fig. 3) except in base ZFC and giving away very little human math inside the deduction engine. I agree this would be very crazy to see. The version with pretraining on non-mathy text is also interesting and would still be totally crazy to see. I agree it would probably imply your "come up with interesting math concepts". But I wouldn't be surprised if like $>20\%$ of the people on LW who think A[G/S]I happens in like $2-3$ years thought that my thing could totally happen in 2025 if the labs were aiming for it (though they might not expect the labs to aim for it), with your things plausibly happening later. E.g. maybe such a person would think "AlphaProof is already mostly RL/search and one could replicate its performance soon without human data, and anyway, AlphaGeometry already pretty much did this for geometry (and AlphaZero did it for chess)" and "some RL+search+self-play thing could get to solving major open problems in math in 2 years, and plausibly at that point human data isn't so critical, and IMO problems are easier than major open problems, so plausibly some such thing gets to IMO problems in 1 year". But also idk maybe this doesn't hang together enough for such people to exist. I wonder if one can use this kind of idea to get a different operationalization with parties interested in taking each side though. Like, maybe whether such a system would prove Cantor's theorem (stated in base ZFC) (imo this would still be pretty crazy to see)? Or whether such a system would get to IMO combos relying moderately less on human data?

¿ thoughts on the following:

• solving >95% of IMO problems while never seeing any human proofs, problems, or math libraries (before being given IMO problems in base ZFC at test time). like alphaproof except not starting from a pretrained language model and without having a curriculum of human problems and in base ZFC with no given libraries (instead of being in lean), and getting to IMO combos

some afaik-open problems relating to bridging parametrized bayes with sth like solomonoff induction

I think that for each NN architecture+prior+task/loss, conditioning the initialization prior on train data (or doing some other bayesian thing) is typically basically a completely different learning algorithm than (S)GD-learning, because local learning is a very different thing, which is one reason I doubt the story in the slides as an explanation of generalization in deep learning^[1].^[2] But setting this aside (though I will touch on it again briefly in the last point I make below), I agree it would be cool to have a story connecting the parametrized bayesian thing to something like Solomonoff induction. Here's an outline of an attempt to give a more precise story extending the one in Lucius's slides, with a few afaik-open problems:

• Let's focus on boolean functions (because that's easy to think about — but feel free to make a different choice). Let's take a learner to be shown certain input-output pairs (that's "training it"), and having to predict outputs on new inputs (that's "test time"). Let's say we're interested in understanding something about which learning setups "generalize well" to these new inputs.
• What should we mean by "generalizing well" in this context? This isn't so clear to me — we could e.g. ask that it does well on problems "like this" which come up in practice, but to solve such problems, one would want to look at what situation gave us the problem and so on, which doesn't seem like the kind of data we want to include in the problem setup here; we could imagine simply removing such data and asking for something that would work well in practice, but this still doesn't seem like such a clean criterion.
• But anyway, the following seems like a reasonable Solomonoff-like thing:
  • There's some complexity (i.e., size/[description length], probably) prior on boolean circuits. There can be multiple reasonable choices of [types of circuits admitted] and/or [description language] giving probably genuinely different priors here, but make some choice (it seems fine to make whatever reasonable choice which will fit best with the later parts of the story we're attempting to build).
  • Think of all the outputs (i.e. train and test) as being generated by taking a circuit from this prior and running the inputs through it.
  • To predict outputs on new inputs, just do the bayesian thing (ie condition the induced prior on functions on all the outputs you've seen).
• My suggestion is that to explain why another learning setup (for boolean functions) has good generalization properties, we could be sort of happy with building a bridge between it and the above simplicity-prior-circuit-solomonoff thing. (This could let us bypass having to further specify what it is to generalize well.)^[3]
• One key step in the present attempt at building a bridge from NN-bayes to simplicity-prior-circuit-solomonoff is to get from simplicity-prior-circuit-solomonoff to a setup with a uniform prior over circuits — the story would like to say that instead of picking circuits from a simplicity prior, you can pick circuits uniformly at random from among all circuits of up to a certain size. The first main afaik-open problem I want to suggest is to actually work out this step: to provide a precise setup where the uniform prior on boolean circuits up to a certain size is like the simplicity prior on boolean circuits (and to work out the correspondence). (It could also be interesting and [sufficient for building a bridge] to argue that the uniform prior on boolean circuits has good generalization properties in some other way.) I haven't thought about this that much, but my initial sense is that this could totally be false unless one is careful about getting the right setup (for example: given inputs-outputs from a particular boolean function with a small circuit, maybe it would work up to a certain upper bound on the size of the circuits on which we have a uniform prior, and then stop working; and/or maybe it depends more precisely on our [types of circuits admitted] and/or [description language]). (I know there is this story with programs, but idk how to get such a correspondence for circuits from that, and the correspondence for circuits seems like what we actually need/want.)
• The second afaik-open problem I'm suggesting is to figure out in much more detail how to get from e.g. the MLP with a certain prior to boolean circuits with a uniform prior.
• One reason I'm stressing these afaik-open problems (particularly the second one) is that I'm pretty sure many parametrized bayesian setups do not in fact give good generalization behavior — one probably needs some further things (about the architecture+prior, given the task) to go right to get good generalization (in fact, I'd guess that it's "rare" to get good generalization without these further unclear hyperparams taking on the right values), and one's attempt at building a bridge should probably make contact with these further things (so as to not be "explaining" a falsehood).
  • One interesting example is given by MLPs in the NN gaussian process limit (i.e. a certain kind of initialization + taking the width to infinity) learning boolean functions (edit: I've realized I should clarify that I'm (somewhat roughly speaking) assuming the $\{-1,1{\}}^n\to \{-1,1\}$ convention, not the $\{0,1\}$ convention), which I think ends up being equivalent to kernel ridge regression with the fourier basis on boolean functions as the kernel features (with certain $L^2$ weights depending on the size of the XOR), which I think doesn't have great generalization properties — in particular, it's quite unlike simplicity-prior-circuit-solomonoff, and it's probably fair to think of it as doing sth more like a polyfit in some sense. I think this also happens for the NTK, btw. (But I should say I'm going off some only loosely figured out calculations (joint with Dmitry Vaintrob and o1-preview) here, so there's a real chance I'm wrong about this example and you shouldn't completely trust me on it currently.) But I'd guess that deep learning can do somewhat better than this. (speculation: Maybe a major role in getting bad generalization here is played by the NNGP and NTK not "learning intermediate variables", preventing any analogy with boolean circuits with some depth going through, whereas deep learning can learn intermediate variables to some extent.) So if we want to have a correct solomonoff story which explains better generalization behavior than that of this probably fairly stupid kernel thing, then we would probably want the story to make some distinction which prevents it from also applying in this NNGP limit. (Anyway, even if I'm wrong about the NNGP case, I'd guess that most setups provide examples of fairly poor generalization, so one probably really needn't appeal to NNGP calculations to make this point.)

Separately from the above bridge attempt, it is not at all obvious to me that parametrized bayes in fact has such good generalization behavior at all (i.e., "at least as good as deep learning", whatever that means, let's say)^[4]; here's some messages on this topic I sent to [the group chat in which the posted discussion happened] later:

"i'd be interested in hearing your reasons to think that NN-parametrized bayesian inference with a prior given by canonical initialization randomization (or some other reasonable prior) generalizes well (for eg canonical ML tasks or boolean functions), if you think it does — this isn't so clear to me at all

some practical SGD-NNs generalize decently, but that's imo a sufficiently different learning process to give little evidence about the bayesian case (but i'm open to further discussion of this). i have some vague sense that the bayesian thing should be better than SGD, but idk if i actually have good reason to believe this?

i assume that there are some other practical ML things inspired by bayes which generalize decently but it seems plausible that those are still pretty local so pretty far from actual bayes and maybe even closer to SGD than to bayes, tho idk what i should precisely mean by that. but eg it seems plausible from 3 min of thinking that some MCMC (eg SGLD) setup with a non-galactic amount of time on a NN of practical size would basically walk from init to a local likelihood max and not escape it in time, which sounds a lot more like SGD than like bayes (but idk maybe some step size scheduling makes the mixing time non-galactic in some interesting case somehow, or if it doesn't actually do that maybe it can give a fine approximation of the posterior in some other practical sense anyway? seems tough). i haven't thought about variational inference much tho — maybe there's something practical which is more like bayes here and we could get some relevant evidence from that

maybe there's some obvious answer and i'm being stupid here, idk :)

one could also directly appeal to the uniformly random program analogy but the current version of that imo doesn't remotely constitute sufficiently good reason to think that bayesian NNs generalize well on its own"

(edit: this comment suggests https://arxiv.org/pdf/2002.02405 as evidence that bayes-NNs generalize worse than SGD-NNs. but idk — I haven't looked at the paper yet — ie no endorsement of it one way or the other from me atm)

you say "Human ingenuity is irrelevant. Lots of people believe they know the one last piece of the puzzle to get AGI, but I increasingly expect the missing pieces to be too alien for most researchers to stumble upon just by thinking about things without doing compute-intensive experiments." and you link https://tsvibt.blogspot.com/2024/04/koan-divining-alien-datastructures-from.html for "too alien for most researchers to stumble upon just by thinking about things without doing compute-intensive experiments"

i feel like that post and that statement are in contradiction/tension or at best orthogonal

there's imo probably not any (even-nearly-implementable) ceiling for basically any rich (thinking-)skill at all^[1] — no cognitive system will ever be well-thought-of as getting close to a ceiling at such a skill — it's always possible to do any rich skill very much better (I mean these things for finite minds in general, but also when restricting the scope to current humans)

(that said, (1) of course, it is common for people to become better at particular skills up to some time and to become worse later, but i think this has nothing to do with having reached some principled ceiling; (2) also, we could perhaps eg try to talk about 'the artifact that takes at most $n$ bits to specify (in some specification-language) which figures out $x$ units of math the quickest (for some $x$ sufficiently large compared to $n$)', but even if we could make sense of that, it wouldn't be right to think of it as being at some math skill ceiling to begin with, because it will probably very quickly change very much about its thinking (i.e. reprogram itself, imo plausibly indefinitely many times, including indefinitely many times in important ways, until the heat death of the universe or whatever); (3) i admit that there can be some purposes for which there is an appropriate way to measure goodness at some rich skill with a score in $[0,1]$, and for such a purpose potential goodness at even a rich skill is of course appropriate to consider bounded and optimal performance might be rightly said to be approachable, but this somehow feels not-that-relevant in the present context)