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33Kaarel
9Alexander Gietelink Oldenziel
2Alexander Gietelink Oldenziel
10Kaarel
12Jacob Pfau
3Kaarel
7Kaarel
5Kaarel
3Mateusz Bagiński
4Kaarel
2Kaarel
4Kaarel
3Kaarel
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1Kaarel
a few thoughts on hyperparams for a better learning theory (for understanding what happens when a neural net is trained with gradient descent)
Having found myself repeating the same points/claims in various conversations about what NN learning is like (especially around singular learning theory), I figured it's worth writing some of them down. My typical confidence in a claim below is like 95%[1]. I'm not claiming anything here is significantly novel. The claims/points:
- local learning (eg gradient descent) strongly does not find global optima. insofar as running a local learning process from many seeds produces outputs with 'similar' (train or test) losses, that's a law of large numbers phenomenon[2], not a consequence of always finding the optimal neural net weights.[3][4]
- if your method can't produce better weights: were you trying to produce better weights by running gradient descent from a bunch of different starting points? getting similar losses this way is a LLN phenomenon
- maybe this is a crisp way to see a counterexample instead: train, then identify a 'lottery ticket' subnetwork after training like done in that literature. now get rid of all other edges in the network, and retrain that subnetwork either from the previous initialization or from a new initialization — i think this literature says that you get a much worse loss in the latter case. so training from a random initialization here gives a much worse loss than possible
- dynamics (kinetics) matter(s). the probability of getting to a particular training endpoint is highly dependent not just on stuff that is evident from the neighborhood of that point, but on there being a way to make those structures incrementally, ie by a sequence of local moves each of which is individually useful.[5][6][7] i think that this is not an academic correction, but a major one — the structures found in practice are very massively those with sensible paths into them and not other (naively) similarly complex structures. some stuff to consider:
- the human eye evolving via a bunch of individually sensible steps, https://en.wikipedia.org/wiki/Evolution_of_the_eye
- (given a toy setup and in a certain limit,) the hardness of learning a boolean function being characterized by its leap complexity, ie the size of the 'largest step' between its fourier terms, https://arxiv.org/pdf/2302.11055
- imagine a loss function on a plane which has a crater somewhere and another crater with a valley descending into it somewhere else. the local neighborhoods of the deepest points of the two craters can look the same, but the crater with a valley descending into it will have a massively larger drainage basin. to say more: the crater with a valley is a case where it is first loss-decreasing to build one simple thing, (ie in this case to fix the value of one parameter), and once you've done that loss-decreasing to build another simple thing (ie in this case to fix the value of another parameter); getting to the isolated crater is more like having to build two things at once. i think that with a reasonable way to make things precise, the drainage basin of a 'k-parameter structure' with no valley descending into it will be exponentially smaller than that of eg a 'k-parameter structure' with 'a k/2-parameter valley' descending into it, which will be exponentially smaller still than a 'k-parameter structure' with a sequence of valleys of slowly increasing dimension descending into it
- it seems plausible to me that the right way to think about stuff will end up revealing that in practice there are basically only systems of steps where a single [very small thing]/parameter gets developed/fixed at a time
- i'm further guessing that most structures basically have 'one way' to descend into them (tho if you consider sufficiently different structures to be the same, then this can be false, like in examples of convergent evolution) and that it's nice to think of the probability of finding the structure as the product over steps of the probability of making the right choice on that step (of falling in the right part of a partition determining which next thing gets built)
- one correction/addition to the above is that it's probably good to see things in terms of there being many 'independent' structures/circuits being formed in parallel, creating some kind of ecology of different structures/circuits. maybe it makes sense to track the 'effective loss' created for a structure/circuit by the global loss (typically including weight norm) together with the other structures present at a time? (or can other structures do sufficiently orthogonal things that it's fine to ignore this correction in some cases?) maybe it's possible to have structures which were initially independent be combined into larger structures?[8]
- everything is a loss phenomenon. if something is ever a something-else phenomenon, that's logically downstream of a relation between that other thing and loss (but this isn't to say you shouldn't be trying to find these other nice things related to loss)
- grokking happens basically only in the presence of weight regularization, and it has to do with there being slower structures to form which are eventually more efficient at making logits high (ie more logit bang for weight norm buck)
- in the usual case that generalization starts to happen immediately, this has to do with generalizing structures being stronger attractors even at initialization. one consideration at play here is that
- nothing interesting ever happens during a random walk on a loss min surface
- it's not clear that i'm conceiving of structures/circuits correctly/well in the above. i think it would help a library of like >10 well-understood toy models (as opposed to like the maybe 1.3 we have now), and to be very closely guided by them when developing an understanding of neural net learning
some related (more meta) thoughts
- to do interesting/useful work in learning theory (as of 2024), imo it matters a lot that you think hard about phenomena of interest and try to build theory which lets you make sense of them, as opposed to holding fast to an existing formalism and trying to develop it further / articulate it better / see phenomena in terms of it
- this is somewhat downstream of current formalisms imo being bad, it imo being appropriate to think of them more as capturing preliminary toy cases, not as revealing profound things about the phenomena of interest, and imo it being feasible to do better
- but what makes sense to do can depend on the person, and it's also fine to just want to do math lol
- and it's certainly very helpful to know a bunch of math, because that gives you a library in terms of which to build an understanding of phenomena
- it's imo especially great if you're picking phenomena to be interested in with the future going well around ai in mind
(* but it looks to me like learning theory is unfortunately hard to make relevant to ai alignment[9])
acknowledgments
these thoughts are sorta joint with Jake Mendel and Dmitry Vaintrob (though i'm making no claim about whether they'd endorse the claims). also thank u for discussions: Sam Eisenstat, Clem von Stengel, Lucius Bushnaq, Zach Furman, Alexander Gietelink Oldenziel, Kirke Joamets
with the important caveat that, especially for claims involving 'circuits'/'structures', I think it's plausible they are made in a frame which will soon be superseded or at least significantly improved/clarified/better-articulated, so it's a 95% given a frame which is probably silly ↩︎
train loss in very overparametrized cases is an exception. in this case it might be interesting to note that optima will also be off at infinity if you're using cross-entropy loss, https://arxiv.org/pdf/2006.06657 ↩︎
also, gradient descent is very far from doing optimal learning in some solomonoff sense — though it can be fruitful to try to draw analogies between the two — and it is also very far from being the best possible practical learning algorithm ↩︎
by it being a law of large numbers phenomenon, i mean sth like: there are a bunch of structures/circuits/pattern-completers that could be learned, and each one gets learned with a certain probability (or maybe a roughly given total number of these structures gets learned), and loss is roughly some aggregation of indicators for whether each structure gets learned — an aggregation to which the law of large numbers applies ↩︎
to say more: any concept/thinking-structure in general has to be invented somehow — there in some sense has to be a 'sensible path' to that concept — but any local learning process is much more limited than that still — now we're forced to have a path in some (naively seen) space of possible concepts/thinking-structures, which is a major restriction. eg you might find the right definition in mathematics by looking for a thing satisfying certain constraints (eg you might want the definition to fit into theorems characterizing something you want to characterize), and many such definitions will not be findable by doing sth like gradient descent on definitions ↩︎
ok, (given an architecture and a loss,) technically each point in the loss landscape will in fact have a different local neighborhood, so in some sense we know that the probability of getting to a point is a function of its neighborhood alone, but what i'm claiming is that it is not nicely/usefully a function of its neighborhood alone. to the extent that stuff about this probability can be nicely deduced from some aspect of the neighborhood, that's probably 'logically downstream' of that aspect of the neighborhood implying something about nice paths to the point. ↩︎
also note that the points one ends up at in LLM training are not local minima — LLMs aren't trained to convergence ↩︎
i think identifying and very clearly understanding any toy example where this shows up would plausibly be better than anything else published in interp this year. the leap complexity paper does something a bit like this but doesn't really do this ↩︎
i feel like i should clarify here though that i think basically all existing alignment research fails to relate much to ai alignment. but then i feel like i should further clarify that i think each particular thing sucks at relating to alignment after having thought about how that particular thing could help, not (directly) from some general vague sense of pessimism. i should also say that if i didn't think interp sucked at relating to alignment, i'd think learning theory sucks less at relating to alignment (ie, not less than interp but less than i currently think it does). but then i feel like i should further say that fortunately you can just think about whether learning theory relates to alignment directly yourself :) ↩︎
Simon-Pepin Lehalleur weighs in on the DevInterp Discord:
I think his overall position requires taking degeneracies seriously: he seems to be claiming that there is a lot of path dependency in weight space, but very little in function space 😄
In general his position seems broadly compatible with DevInterp:
- models learn circuits/algorithmic structure incrementally
- the development of structures is controlled by loss landscape geometry
- and also possibly in more complicated cases by the landscapes of "effective losses" corresponding to subcircuits...
This perspective certainly is incompatible with a naive SGD = Bayes = Watanabe's global SLT learning process, but I don't think anyone has (ever? for a long time?) made that claim for non toy models.
It seems that the difference with DevInterp is that
- we are more optimistic that it is possible to understand which geometric observables of the landscape control the incremental development of circuits
- we expect, based on local SLT considerations, that those observables have to do with the singularity theory of the loss and also of sub/effective losses, with the LLC being the most important but not the only one
- we dream that it is possible to bootstrap this to a full fledged S4 correspondence, or at least to get as close as we can.
Ok, no pb. You can also add the following :
I am sympathetic but also unsatisfied with a strong empiricist position about deep learning. It seems to me that it is based on a slightly misapplied physical, and specifically thermodynamical intuition. Namely that we can just observe a neural network and see/easily guess what the relevant "thermodynamic variables" of the system.
For ordinary 3d physical systems, we tend to know or easily discover those thermodynamic variables through simple interactions/observations. But a neural network is an extremely high-dimensional system which we can only "observe" through mathematical tools. The loss is clearly one such thermodynamic variable, but if we expect NN to be in some sense stat mech systems it can't be the only one (otherwise the learning process would be much more chaotic and unpredictable). One view of DevInterp is that we are "just" looking for those missing variables...
I'd be curious about hearing your intuition re " i'm further guessing that most structures basically have 'one way' to descend into them"
I designed a pro-human(ity)/anti-(non-human-)AI flag:
- The red-black circle is HAL's eye; it represents the non-human in-all-ways-super-human AI(s) that the world's various AI capability developers are trying to create, that will imo by default render all remotely human beings completely insignificant and cause humanity to completely lose control over what happens :(.
- The white star covering HAL's eye has rays at the angles of the limbs of Leonardo's Vitruvian Man; it represents humans/humanity remaining more capable than non-human AI (by banning AGI development and by carefully self-improving).
- The blue background represents our potential self-made ever-better future, involving global governance/cooperation/unity in the face of AI.
Feel free to suggest improvements to the flag. Here's latex to generate it:
% written mostly by o3 and o4-mini-high, given k's prompting
% an anti-AI flag. a HAL "eye" (?) is covered by a vitruvian man star
\documentclass[tikz]{standalone}
\usetikzlibrary{calc}
\usepackage{xcolor} % for \definecolor
\definecolor{UNBlue}{HTML}{5B92E5}
\begin{document}
\begin{tikzpicture}
%--------------------------------------------------------
% flag geometry
%--------------------------------------------------------
\def\flagW{6cm} % width -> 2 : 3 aspect
\def\flagH{4cm} % height
\def\eyeR {1.3cm} % HAL-eye radius
% light-blue background
\fill[UNBlue] (0,0) rectangle (\flagW,\flagH);
%--------------------------------------------------------
% concentric “HAL eye” (outer-most ring first)
%--------------------------------------------------------
\begin{scope}[shift={(\flagW/2,\flagH/2)}] % centre of the flag
\foreach \f/\c in {%
1.00/black,
.68/{red!50!black},
.43/{red!80!orange},
.1/orange,
.05/yellow}%
{%
\fill[fill=\c,draw=none] circle ({\f*\eyeR});
}
%── parameters ───────────────────────────────────────
\def\R{\eyeR} % distance from centre to triangle’s tip
\def\Alpha{10} % full apex angle (°)
%── compute half-angle & half-base once ─────────────
\pgfmathsetmacro\halfA{\Alpha/2}
\pgfmathsetlengthmacro\halfside{\R*tan(\halfA)}
%── loop over Vitruvian‐man angles ───────────────────
\foreach \Beta in {0,30,90,150,180,240,265,275,300} {%
% apex on the eye‐rim
\coordinate (A) at (\Beta:\R);
% base corners offset ±90°
\coordinate (B) at (\Beta+90:\halfside);
\coordinate (C) at (\Beta-90:\halfside);
% fill the spike
\path[fill=white,draw=none] (A) -- (B) -- (C) -- cycle;
}
\end{scope}
\end{tikzpicture}
\end{document}
I like the concept; on the other hand the flag feels strongly fascist to me.
Ran it by the AIs and 2 out of 3 had "authoritarian" as their first descriptor responding to "What political alignment does the aesthetic of this flag evoke?" FWIW.
Hmm, thanks for telling me, I hadn't considered that. I think I didn't notice this in part because I've been thinking of the red-black circle as being "canceled out"/"negated" on the flag, as opposed to being "asserted". But this certainly wouldn't be obvious to someone just seeing the flag.
on seeing the difference between profound and meaningless radically alien futures
Here's a question that came up in a discussion about what kind of future we should steer toward:
- Okay, a future in which all remotely human entities promptly get replaced by alien AIs would soon look radically incomprehensible and void to us — like, imagine our current selves seeing videos from this future world, and the world in these videos mostly not making sense to them, and to an even greater extent not seeming very meaningful in the ethical sense. But a future in which [each human]/humanity has spent a million years growing into a galaxy-being would also look radically incomprehensible/weird/meaningless to us.[1] So, if we were to ignore near-term stuff, would we really still have reason to strive for the latter future over the former?
a couple points in response:
- The world in which we are galaxy-beings will in fact probably seem more ethically meaningful to us in many fairly immediate ways. Related: (for each past time ) a modern species typically still shares meaningfully more with its ancestors from time than it does with other species that were around at time (that diverged from the ancestral line of the species way before ).
- A specific case: we currently already have many projects we care about — understanding things, furthering research programs, creating technologies, fashioning families and friendships, teaching, etc. — some of which are fairly short-term, but others of which could meaningfully extend into the very far future. Some of these will be meaningfully continuing in the world in which we are galaxy-beings, in a way that is not too hard to notice. That said, they will have grown into crazy things, yes, with many aspects that one isn't going to immediately consider cool; I think there is in fact a lot that's valuable here as well; I'll argue for this in item 3.
- The world in which we have become galaxy-beings had our own (developing) sense/culture/systems/laws guide decision-making and development (and their own development in particular), and we to some extent just care intrinsically/terminally about this kinda meta thing in various ways.
- However, more importantly: I think we mostly care about [decisions being made and development happening] according to our own sense/culture/systems/laws not intrinsically/terminally, but because our own sense/culture/systems/laws is going to get things right (or well, more right than alternatives) — for instance, it is going to lead us more to working on projects that really are profound. However, that things are going well is not immediately obvious from looking at videos of a world — as time goes on, it takes increasingly more thought/development to see that things are going well.
- I think one is making a mistake when looking at videos from the future and quickly being like "what meaningless nonsense!". One needs to spend time making sense of the stuff that's going on there to properly evaluate it — one doesn't have immediate access to one's true preferences here. If development has been thoughtful in this world, very many complicated decisions have been made to get to what you're now seeing in these videos. When evaluating this future, you might want to (for instance) think through these decisions for yourself in the order in which they were made, understanding the context in which each decision was made, hearing the arguments that were made, becoming smart enough to understand them, maybe trying out some relevant experiences, etc.. Or you might do other kinds of thinking that gets you into a position from which you can properly understand the world and judge it. After a million years[2] of this, you might see much more value in this human(-induced) world than before.
- But maybe you'll still find that world quite nonsensical? If you went about your thinking and learning in a great deal of isolation, without much attempting to do something together with the beings in that world, then imo you probably will indeed find that world quite bad/empty compared to what it could have been[3] [4] (though I'd guess that you would similarly also find other isolated rollouts of your own reflection quite silly[5], and that different independent sufficiently long further rollouts from your current position would again find each other silly, and so on). However, note that like the galaxy-you that came out of this reflection, this world you're examining has also gone through an [on most steps ex ante fairly legitimate] process of thoughtful development (by assumption, I guess), and the being(s) in that world now presumably think there's a lot of extremely cool stuff happening in it. In fact, we could suppose that a galaxy-you is living in that world, and that they contributed to its development throughout its history, and that they now think that their world (or their corner of their world) is extremely based.[6]
- Am I saying that the galaxy-you looking at this world from the outside is actually supposed to think it's really cool, because it's supposed to defer to the beings in that world, or because it's supposed to think any development path consisting of ex ante reasonable-seeming steps is fine, or because some sort of relativism is right, or something? I think this isn't right, and so I don't want to say that — I think it's probably fine for the galaxy-you to think stuff has really gone off the rails in that world. But I do want to say that when we ourselves are making this decision of which kind of future to have from our own embedded point of view, we should expect there to be a great deal of incomprehensible coolness in a human future (if things go right) — for instance, projects whose worth we wouldn't see yet, but which we would come to correctly consider really profound in such a future (indeed, we would be tracking what's worthwhile and coming up with new worthwhile things and doing those) — whereas we should expect there to instead be a great deal of incomprehensible valueless nonsense in an alien future.
- If you've read the above and still think a galaxy-human future wouldn't be based, let me try one more story on you. I think this looking-at-videos-of-a-distant-world framing of the question makes one think in terms of sth like assigning value to spacetime blocks "from the outside", and this is a framing of ethical decisions which is imo tricky to handle well, and in particular can make one forget how much one cares about stuff. Like, I think it's common to feel like your projects matter a lot while simultaneously feeling that [there being a universe in which there is a you-guy that is working on those projects] isn't so profound; maybe you really want to have a family, but you're confused about how much you want to make there be a spacetime block in which there is a such-and-such being with a family. This could even turn an ordinary ethical decision that you can handle just fine into something you're struggling to make sense of — like, wait, what kind of guy needs to live in this spacetime block (and what relation do they need to have to me-now-answering-this-question); also, what does it even mean for a spacetime block to exist (what if we should say that all possible spacetime blocks exist?)? One could adopt the point of view that the spacetime block question is supposed to just be a rephrasing of the ordinary ethical question, and so one should have the same answer for it, and feel no more confused about what it means. One could probably spend some time thinking of one's ordinary ethical decisions in terms of spacetime-block-making and perhaps then come to have one's answers be reasonably coherent under having (arguably) the same decision problem presented in the ordinary way vs in some spacetime block way.[7] But I think this sort of thing is very far from being set up in almost any current human. So: you might feel like saying "whatever" way too much when ethical questions are framed in terms of spacetime-block-making, and the situation we're considering could push one toward that frame; I want to alert you that maybe this is happening, maybe you really care more than it seems in that frame, and that maybe you should somehow imagine yourself being more embedded in this world when evaluating it.
I guess one could imagine a future in which someone tiles the world with happy humans of the current year variety or something, but imo this is highly unlikely even conditional on the future being human-shaped, and also much worse than futures in which a wild variety of galaxy-human stuff is going on. Background context: imo we should probably be continuously growing more capable/intelligent ourselves for a very long time (and maybe forever), with the future being determined by us "from inside human life", as opposed to ever making an artificial system that is more capable than humanity and fairly separate/distinct from humanity that would "design human affairs from the outside" (really, I think we shouldn't be making [AIs more generally capable than individual humans] of any kind, except for ones that just are smarter versions of individual humans, for a long time (and maybe forever); see this for some of my thoughts on these topics). ↩︎
maybe we should pick a longer time here, to be comparing things which are more alike? ↩︎
I think this is probably true even if we condition the rollout on you coming to understand the world in the videos quite well. ↩︎
But if you disagree here, then I think I've already finished [the argument that the human far future is profoundly better] which I want to give to you, so you could stop reading here — the rest of this note just addresses a supposed complication you don't believe exists. ↩︎
much like you could grow up from a kid into a mathematician or a philosopher or an engineer or a composer, thinking in each case that the other paths would have been much worse ↩︎
Unlike you growing up in isolation, that galaxy-you's activities and judgment and growth path will be influenced by others; maybe it has even merged with others quite fully. But that's probably how things should be, anyway — we probably should grow up together; our ordinary valuing is already done together to a significant extent (like, for almost all individuals, the process determining (say) the actions of that individual already importantly involves various other individuals, and not just in a way that can easily be seen as non-ethical). ↩︎
There might be some stuff that's really difficult to make sense of here — it is imo plausible that the ethical cognition that a certain kind of all-seeing spacetime-block-chooser would need to have to make good choices is quite unlike any ethical cognition that exists (or maybe even could exist) in our universe. That said, we can imagine a more mundane spacetime-block-chooser, like a clone of you that gets to make a single life choice for you given ordinary information about the decision and that gets deleted after that; it is easier to imagine this clone having ethical cognition that leads to it making reasonably good decisions. ↩︎
a chat with Towards_Keeperhood on what it takes for sentences/phrases/words to be meaningful
- you could define "mother(x,y)" as "x gave birth to y", and then "gave birth" as some more precise cluster of observations, which eventually need to be able to be identified from visual inputs
Kaarel:
- if i should read this as talking about a translation of "x is the mother of y", then imo this is a bad idea.
- in particular, i think there is the following issue with this: saying which observations "x gave birth to y" corresponds to intuitively itself requires appealing to a bunch of other understanding. it's like: sure, your understanding can be used to create visual anticipations, but it's not true that any single sentence alone could be translated into visual anticipations — to get a typical visual anticipation, you need to rely on some larger segment of your understanding. a standard example here is "the speed of light in vacuum is 3*10^8 m/s" creating visual anticipations in some experimental setups, but being able to derive those visual anticipations depends on a lot of further facts about how to create a vacuum and properties of mirrors and interferometers and so on (and this is just for one particular setup — if we really mean to make a universally quantified statement, then getting the observation sentences can easily end up requiring basically all of our understanding). and it seems silly to think that all this crazy stuff was already there in what you meant when you said "the speed of light in vacuum is m/s". one concrete reason why you don't want this sentence to just mean some crazy AND over observation sentences or whatever is because you could be wrong about how some interferometer works and then you'd want it to correspond to different observation sentences
- this is roughly https://en.wikipedia.org/wiki/Confirmation_holism as a counter to https://en.wikipedia.org/wiki/Verificationism
- that said, i think there is also something wrong with some very strong version of holism: it's not really like our understanding is this unitary thing that only outputs visual anticipations using all the parts together, either — the real correspondence is somewhat more granular than that
TK:
- On reflection, I think my "mother" example was pretty sloppy and perhaps confusing. I agree that often quite a lot of our knowledge is needed to ground a statement in anticipations. And yeah actually it doesn't always ground out in that, e.g. for parsing the meaning of counterfactuals. (See "Mixed Reference: The great reductionist project".)
K:
- i wouldn't say a sentence is grounded in anticipations with a lot of our knowledge, because that makes it sound like in the above example, "the speed of light is m/s" is somehow privileged compared to our understanding of mirrors and interferometers even though it's just all used together to create anticipations; i'd instead maybe just say that a bunch of our knowledge together can create a visual anticipation
TK:
- thx. i wanted to reply sth like "a true statement can either be tautological (e.g. math theorems) or empirical, and for it to be an empirical truth there needs to be some entanglement between your belief and reality, and entanglement happens through sensory anticipations. so i feel fine with saying that the sentence 'the speed of light is m/s' still needs to be grounded in sensory anticipations". but i notice that the way i would use "grounded" here is different from the way I did in my previous comment, so perhaps there are two different concepts that need to be disentangled.
K:
- here's one thing in this vicinity that i'm sympathetic to: we should have as a criterion on our words, concepts, sentences, thoughts, etc. that they play some role in determining our actions; if some mental element is somehow completely disconnected from our lives, then i'd be suspicious of it. (and things can be connected to action via creating visual anticipations, but also without doing that.)
- that said, i think it can totally be good to be doing some thinking with no clear prior sense about how it could be connected to action (or prediction) — eg doing some crazy higher math can be good, imagining some crazy fictional worlds can be good, games various crazy artists and artistic communities are playing can be good, even crazy stuff religious groups are up to can be good. also, i think (thought-)actions in these crazy domains can themselves be actions one can reasonably be interested in supporting/determining, so this version of entanglement with action is really a very weak criterion
- generally it is useful to be able to "run various crazy programs", but given this, it seems obvious that not all variables in all useful programs are going to satisfy any such criterion of meaningfulness? like, they can in general just be some arbitrary crazy things (like, imagine some memory bit in my laptop or whatever) playing some arbitrary crazy role in some context, and this is fine
- and similarly for language: we can have some words or sentences playing some useful role without satisfying any strict meaningfulness criterion (beyond maybe just having some relation to actions or anticipations which can be of basically arbitrary form)
- a different point: in human thinking, the way "2+2=4" is related to visual anticipations is very similar to the way "the speed of light is m/s" is related to visual anticipations
TK:
- Thanks!
- I agree that e.g. imagining fictional worlds like HPMoR can be useful.
- I think I want to expand my notion of "tautological statements" to include statements like "In the HPMoR universe, X happens". You can also pick any empirical truth "X" and turn it into a tautological one by saying "In our universe, X". Though I agree it seems a bit weird.
- Basically, mathematics tells you what's true in all possible worlds, so from mathematics alone you never know in which world you may be in. So if you want to say something that's true about your world specifically (but not across all possible worlds), you need some observations to pin down what world you're in.
- I think this distinction is what Eliezer means in his highly advanced epistemology sequence when he uses "logical pinpointing" and "physical pinpointing".
- You can also have a combination of the two. (I'd say as soon as some physical pinpointing is involved I'd call it an empirical fact.)
- Commented about that. (I actually changed my model slightly): https://www.lesswrong.com/posts/bTsiPnFndZeqTnWpu/mixed-reference-the-great-reductionist-project?commentId=HuE78qSkZJ9MxBC8p
K:
- the imo most important thing in my messages above is the argument against [any criterion of meaningfulness which is like what you’re trying to state] being reasonable
- in brief, because it’s just useful to be allowed to have arbitrary “variables" in "one’s mental circuits”
- just like there’s no such meaningfulness criterion on a bit in your laptop’s memory
- if you want to see from the outside the way the bit is “connected to the world”, one thing you could do is to say that the bit is 0 in worlds which are such-and-such and 1 in worlds which are such-and-such, or, if you have a sense of what the laptop is supposed to be doing, you could say in which worlds the bit "should be 0" and in which worlds the bit "should be 1", but it’s not like anything like this crazy god’s eye view picture is (or even could explicitly be) present inside the laptop
- our sentences and terms don’t have to have meanings “grounded in visual anticipations”, just like the bit in the laptop doesn’t
- except perhaps in the very weak sense that it should be possible for a sentence to be involved in determining actions (or anticipations) in some potentially arbitrarily remote way
- the following is mostly a side point: one problem with seeing from the inside what your bits (words, sentences) are doing (especially in the context of pushing the frontier of science, math, philosophy, tech, or generally doing anything you don’t know how to do yet, but actually also just basically all the time) is that you need to be open to using your bits in new ways; the context in which you are using your bits usually isn’t clear to you
- btw, this is a sort of minor point but i'm stating it because i'm hoping it might contribute to pushing you out of a broader imo incorrect view: even when one is stating formal mathematical statements, one should be allowed to state sentences with no regard for whether they are tautologies/contradictions (that is, provable/disprovable) or not — ie, one should be allowed to state undecidable sentences, right? eg you should be allowed to state a proof that has the structure "if P, then blabla, so Q; but if not-P, then other-blabla, but then also Q; therefore, Q", without having to pay any attention to whether P itself is tautological/contradictory or undecidable
- so, if what you want to do with your criterion of meaningfulness involves banning saying sentences which are not "meaningful", then even in formal math, you should consider non-tautological/contradictory sentences meaningful. (if you don't want to ban the "meaningless" sentences, then idk what we're even supposed to be doing with this notion of meaningfulness.)
TK:
- Thx. I definitely agree one should be able to state all mathematical statements (including undecidable ones), and that for proofs you shouldn't need to pay attention to whether a statement is undecidable or not. (I'm having sorta constructivist tendencies though, where "if P, then blabla, so Q; but if not-P, then other-blabla, but then also Q; therefore, Q" wouldn't be a valid proof because we don't assume the law of excluded middle.)
- Ok yeah thx I think the way I previously used "meaningfully" was pretty confused. I guess I don't really want to rule out any sentences people use.
- I think sth is not meaningful if there's no connection between a belief to your main belief pool. So "a puffy is a flippo" is perhaps not meaningful to you because those concepts don't relate to anything else you know? (But that's a different kind of meaningful from what errors people mostly make.)
K:
- yea. tho then we could involve more sentences about puffies and flippos and start playing some game involving saying/thinking those sentences and then that could be fun/useful/whatever
TK:
- maybe. idk.
so this version of entanglement with action is really a very weak criterion
Yeah, exactly, and hence the question: what are some counterexamples, ~concepts that clearly are not tied to action in any way? E.g., I could imagine metaphysical philosophizing to connect to action via contributing to a line of thinking that eventually produces a useful insight on how to do science or something. Is it about "being/remaining open to using it in new ways"?
I think I want to expand my notion of "tautological statements" to include statements like "In the HPMoR universe, X happens". You can also pick any empirical truth "X" and turn it into a tautological one by saying "In our universe, X". Though I agree it seems a bit weird.
I'm inclined to think that your generalized tautological statements are about something like "playing games according to ~rules in (~confined to) some mind-like system". This is in contrast to (canonically) empirical statements that involve throwing a referential bridge across the boundary of the system.
- I think sth is not meaningful if there's no connection between a belief to your main belief pool. So "a puffy is a flippo" is perhaps not meaningful to you because those concepts don't relate to anything else you know? (But that's a different kind of meaningful from what errors people mostly make.)
K:
- yea. tho then we could involve more sentences about puffies and flippos and start playing some game involving saying/thinking those sentences and then that could be fun/useful/whatever
[Thinking out loud.]
Intuitively, it does seem to me that if you start with a small set of elements isolated from the rest of your understanding, then they are meaningless, but then, as you grow this set of elements and add more relations/functions/rules/propositions with high implicative potential, this network becomes increasingly meaningful, even though it's completely disconnected from the rest of understanding and our lives except for playing this domain/subnetwork-specific game.
Is it (/does it seem) meaningful just because I could throw a bridge between it and the rest of my understanding? Well, one could build a computer with this game installed only (+ ofc bare minimum to make it work: OS and stuff) and I would still be inclined to think it meaningful, although perhaps I would be imposing, and the meaningfulness would be co-created by the eye/mind of the beholder.
This leads to the question: What criteria do we want our (explicated) notion of meaningfulness to satisfy?
[For completeness, the concept of meaningfulness may need to be splintered or even eliminated (/factored out in a way that doesn't leave anything clearly serving its role), though I think the latter rather unlikely.]
A thread into which I'll occasionally post notes on some ML(?) papers I'm reading
I think the world would probably be much better if everyone made a bunch more of their notes public. I intend to occasionally copy some personal notes on ML(?) papers into this thread. While I hope that the notes which I'll end up selecting for being posted here will be of interest to some people, and that people will sometimes comment with their thoughts on the same paper and on my thoughts (please do tell me how I'm wrong, etc.), I expect that the notes here will not be significantly more polished than typical notes I write for myself and my reasoning will be suboptimal; also, I expect most of these notes won't really make sense unless you're also familiar with the paper — the notes will typically be companions to the paper, not substitutes.
I expect I'll sometimes be meaner than some norm somewhere in these notes (in fact, I expect I'll sometimes be simultaneously mean and wrong/confused — exciting!), but I should just say to clarify that I think almost all ML papers/posts/notes are trash, so me being mean to a particular paper might not be evidence that I think it's worse than some average. If anything, the papers I post notes about had something worth thinking/writing about at all, which seems like a good thing! In particular, they probably contained at least one interesting idea!
So, anyway: I'm warning you that the notes in this thread will be messy and not self-contained, and telling you that reading them might not be a good use of your time :)
The Deep Neural Feature Ansatz
@misc{radhakrishnan2023mechanism, title={Mechanism of feature learning in deep fully connected networks and kernel machines that recursively learn features}, author={Adityanarayanan Radhakrishnan and Daniel Beaglehole and Parthe Pandit and Mikhail Belkin}, year={2023}, url = { https://arxiv.org/pdf/2212.13881.pdf } }
The ansatz from the paper
Let denote the activation vector in layer on input , with the input layer being at index , so . Let be the weight matrix after activation layer . Let be the function that maps from the th activation layer to the output. Then their Deep Neural Feature Ansatz says that (I'm somewhat confused here about them not mentioning the loss function at all — are they claiming this is reasonable for any reasonable loss function? Maybe just MSE? MSE seems to be the only loss function mentioned in the paper; I think they leave the loss unspecified in a bunch of places though.)
A singular vector version of the ansatz
Letting be a SVD of , we note that this is equivalent to i.e., that the eigenvectors of the matrix on the RHS are the right singular vectors. By the variational characterization of eigenvectors and eigenvalues (Courant-Fischer or whatever), this is the same as saying that right singular vectors of are the highest orthonormal directions for the matrix on the RHS. Plugging in the definition of , this is equivalent to saying that the right singular vectors are the sequence of highest-variance directions of the data set of gradients .
(I have assumed here that the linearity is precise, whereas really it is approximate. It's probably true though that with some assumptions, the approximate initial statement implies an approximate conclusion too? Getting approx the same vecs out probably requires some assumption about gaps in singular values being big enough, because the vecs are unstable around equality. But if we're happy getting a sequence of orthogonal vectors that gets variances which are nearly optimal, we should also be fine without this kind of assumption. (This is guessing atm.))
Getting rid of the dependence on the RHS?
Assuming there isn't an off-by-one error in the paper, we can pull some term out of the RHS maybe? This is because applying the chain rule to the Jacobians of the transitions gives , so
Wait, so the claim is just which, assuming is invertible, should be the same as . But also, they claim that it is ? Are they secretly approximating everything with identity matrices?? This doesn't seem to be the case from their Figure 2 though.
Oh oops I guess I forgot about activation functions here! There should be extra diagonal terms for jacobians of preactivations->activations in , i.e., it should really say We now instead get This should be the same as which, with denoting preactivations in layer and denoting the function from these preactivations to the output, is the same as This last thing also totally works with activation functions other than ReLU — one can get this directly from the Jacobian calculation. I made the ReLU assumption earlier because I thought for a bit that one can get something further in that case; I no longer think this, but I won't go back and clean up the presentation atm.
Anyway, a takeaway is that the Deep Neural Feature Ansatz is equivalent to the (imo cleaner) ansatz that the set of gradients of the output wrt the pre-activations of any layer is close to being a tight frame (in other words, the gradients are in isotropic position; in other words still, the data matrix of the gradients is a constant times a semi-orthogonal matrix). (Note that the closeness one immediately gets isn't in to a tight frame, it's just in the quantity defining the tightness of a frame, but I'd guess that if it matters, one can also conclude some kind of closeness in from this (related).) This seems like a nicer fundamental condition because (1) we've intuitively canceled terms and (2) it now looks like a generic-ish condition, looks less mysterious, though idk how to argue for this beyond some handwaving about genericness, about other stuff being independent, sth like that.
proof of the tight frame claim from the previous condition: Note that clearly implies that the mass in any direction is the same, but also the mass being the same in any direction implies the above (because then, letting the SVD of the matrix with these gradients in its columns be , the above is , where we used the fact that ).
Some questions
- Can one come up with some similar ansatz identity for the left singular vectors of ? One point of tension/interest here is that an ansatz identity for would constrain the left singular vectors of together with its singular values, but the singular values are constrained already by the deep neural feature ansatz. So if there were another identity for in terms of some gradients, we'd get a derived identity from equality between the singular values defined in terms of those gradients and the singular values defined in terms of the Deep Neural Feature Ansatz. Or actually, there probably won't be an interesting identity here since given the cancellation above, it now feels like nothing about is really pinned down by 'gradients independent of ' by the DNFA? Of course, some -dependence remains even in the gradients because the preactivations at which further gradients get evaluated are somewhat -dependent, so I guess it's not ruled out that the DNFA constrains something interesting about ? But anyway, all this seems to undermine the interestingness of the DNFA, as well as the chance of there being an interesting similar ansatz for the left singular vectors of .
- Can one heuristically motivate that the preactivation gradients above should indeed be close to being in isotropic position? Can one use this reduction to provide simpler proofs of some of the propositions in the paper which say that the DNFA is exactly true in certain very toy cases?
- The authors claim that the DNFA is supposed to somehow elucidate feature learning (indeed, they claim it is a mechanism of feature learning?). I take 'feature learning' to mean something like which neuronal functions (from the input) are created or which functions are computed in a layer in some broader sense (maybe which things are made linearly readable?) or which directions in an activation space to amplify or maybe less precisely just the process of some internal functions (from the input to internal activations) being learned of something like that, which happens in finite networks apparently in contrast to infinitely wide networks or NTK models or something like that which I haven't yet understood? I understand that their heuristic identity on the surface connects something about a weight matrix to something about gradients, but assuming I've not made some index-off-by-one error or something, it seems to probably not really be about that at all, since the weight matrix sorta cancels out — if it's true for one , it would maybe also be true with any other replacing it, so it doesn't really pin down ? (This might turn out to be false if the isotropy of preactivation gradients is only true for a very particular choice of .) But like, ignoring that counter, I guess their point is that the directions which get stretched most by the weight matrix in a layer are the directions along which it would be the best to move locally in that activation space to affect the output? (They don't explain it this way though — maybe I'm ignorant of some other meaning having been attributed to in previous literature or something.) But they say "Informally, this mechanism corresponds to the approach of progressively re-weighting features in proportion to the influence they have on the predictions.". I guess maybe this is an appropriate description of the math if they are talking about reweighting in the purely linear sense, and they take features in the input layer to be scaleless objects or something? (Like, if we take features in the input activation space to each have some associated scale, then the right singular vector identity no longer says that most influential features get stretched the most.) I wish they were much more precise here, or if there isn't a precise interesting philosophical thing to be deduced from their math, much more honest about that, much less PR-y.
- So, in brief, instead of "informally, this mechanism corresponds to the approach of progressively re-weighting features in proportion to the influence they have on the predictions," it seems to me that what the math warrants would be sth more like "The weight matrix reweights stuff; after reweighting, the activation space is roughly isotropic wrt affecting the prediction (ansatz); so, the stuff that got the highest weight has most effect on the prediction now." I'm not that happy with this last statement either, but atm it seems much more appropriate than their claim.
- I guess if I'm not confused about something major here (plausibly I am), one could probably add 1000 experiments (e.g. checking that the isotropic version of the ansatz indeed equally holds in a bunch of models) and write a paper responding to them. If you're reading this and this seems interesting to you, feel free to do that — I'm also probably happy to talk to you about the paper.
typos in the paper
indexing error in the first displaymath in Sec 2: it probably should say '', not ''
An attempt at a specification of virtue ethics
I will be appropriating terminology from the Waluigi post. I hereby put forward the hypothesis that virtue ethics endorses an action iff it is what the better one of Luigi and Waluigi would do, where Luigi and Waluigi are the ones given by the posterior semiotic measure in the given situation, and "better" is defined according to what some [possibly vaguely specified] consequentialist theory thinks about the long-term expected effects of this particular Luigi vs the long-term effects of this particular Waluigi. One intuition here is that a vague specification could be more fine if we are not optimizing for it very hard, instead just obtaining a small amount of information from it per decision.
In this sense, virtue ethics literally equals continuously choosing actions as if coming from a good character. Furthermore, considering the new posterior semiotic measure after a decision, in this sense, virtue ethics is about cultivating a virtuous character in oneself. Virtue ethics is about rising to the occasion (i.e. the situation, the context). It's about constantly choosing the Luigi in oneself over the Waluigi in oneself (or maybe the Waluigi over the Luigi if we define "Luigi" as the more likely of the two and one has previously acted badly in similar cases or if the posterior semiotic measure is otherwise malign). I currently find this very funny, and, if even approximately correct, also quite cool.
Here are some issues/considerations/questions that I intend to think more about:
- What's a situation? For instance, does it encompass the agent's entire life history, or are we to make it more local?
- Are we to use the agent's own semiotic measure, or some objective semiotic measure?
- This grounds virtue ethics in consequentialism. Can we get rid of that? Even if not, I think this might be useful for designing safe agents though.
- Does this collapse into cultivating a vanilla consequentialist over many choices? Can we think of examples of prompting regimes such that collapse does not occur? The vague motivating hope I have here is that in the trolley problem case with the massive man, the Waluigi pushing the man is a corrupt psycho, and not a conflicted utilitarian.
- Even if this doesn't collapse into consequentialism from these kinds of decisions, I'm worried about it being stable under reflection, I guess because I'm worried about the likelihood of virtue ethics being part of an agent in reflective equilibrium. It would be sad if the only way to make this work would be to only ever give high semiotic measure to agents that don't reflect much on values.
- Wait, how exactly do we get Luigi and Waluigi from the posterior semiotic measure? Can we just replace this with picking the best character from the most probable few options according to the semiotic measure? Wait, is this just quantilization but funnier? I think there might be some crucial differences. And regardless, it's interesting if virtue ethics turns out to be quantilization-but-funnier.
- More generally, has all this been said already?
- Is there a nice restatement of this in shard theory language?
A small observation about the AI arms race in conditions of good infosec and collaboration
Suppose we are in a world where most top AI capabilities organizations are refraining from publishing their work (this could be the case because of safety concerns, or because of profit motives) + have strong infosec which prevents them from leaking insights about capabilities in other ways. In this world, it seems sort of plausible that the union of the capabilities insights of people at top labs would allow one to train significantly more capable models than the insights possessed by any single lab alone would allow one to train. In such a world, if the labs decide to cooperate once AGI is nigh, this could lead to a significantly faster increase in capabilities than one might have expected otherwise.
(I doubt this is a novel thought. I did not perform an extensive search of the AI strategy/governance literature before writing this.)
I'm updating my estimate of the return on investment into culture wars from being an epsilon fraction compared to canonical EA cause areas to epsilon+delta. This has to do with cases where AI locks in current values extrapolated "correctly" except with too much weight put on the practical (as opposed to the abstract) layer of current preferences. What follows is a somewhat more detailed status report on this change.
For me (and I'd guess for a large fraction of autistic altruistics multipliers), the general feels regarding [being a culture war combatant in one's professional capacity] seem to be that while the questions fought over have some importance, the welfare-produced-per-hour-worked from doing direct work is at least an order of magnitude smaller than the same quantities for any canonical cause area (also true for welfare/USD). I'm fairly certain one can reach this conclusion from direct object-level estimates, as I imagine e.g. OpenPhil has done, although I admit I haven't carried out such calculations with much care myself. Considering the incentives of various people involved should also support this being a lower welfare-per-hour-worked cause area (whether an argument along these lines gives substantive support to the conclusion that there is an order-of-magnitude difference appears less clear).
So anyway, until today part of my vague cloud of justification for these feels was that "and anyway, it's fine if this culture war stuff is fixed in 30 years, after we have dealt with surviving AGI". The small realization I had today was that maybe a significant fraction of the surviving worlds are those where something like corrigibility wasn't attainable but AI value extrapolation sort of worked out fine, i.e. with the values that got locked in being sort of fine, but the relative weights of object-level intuitions/preferences was kinda high compared to the weight on simplicity/[meta-level intuitions], like in particular maybe the AI training did some Bayesian-ethics-evidential-double-counting of object-level intuitions about 10^10 similar cases (I realize it's quite possible that this last clause won't make sense to many readers, but unfortunately I won't provide an explanation here; I intend to write about a few ideas on this picture of Bayesian ethics at some later time, but I want to read Beckstead's thesis first, which I haven't done yet; anyway the best I can offer is that I estimate a 75% of you understanding the rough idea I have in mind (which does not necessarily imply that the idea can actually be unfolded into a detailed picture that makes sense), conditional on understanding my writing in general and conditional on not having understood this clause yet, after reading Beckstead's thesis; also: woke: Bayesian ethics, bespoke: INFRABAYESIAN ETHICS, am I right folks).
So anyway, finally getting to the point of all this at the end of the tunnel, in such worlds we actually can't fix this stuff later on, because all the current opinions on culture war issues got locked in.
(One could argue that we can anyway be quite sure that this consideration matters little, because most expected value is not in such kinda-okay worlds, because even if these were 99% percent of the surviving worlds, assuming fun theory makes sense or simulated value-bearing minds are possible, there will be amazingly more value in each world where AGI worked out really well, as compared to a world tiled with Earth society 2030. But then again, this counterargument could be iffy to some, in sort of the same way in which fanaticism (in Bostrom's sense) or the St. Petersburg paradox feel iffy to some, or perhaps in another way. I won't be taking a further position on this at the moment.)
I proposed a method for detecting cheating in chess; cross-posting it here in the hopes of maybe getting better feedback than on reddit: https://www.reddit.com/r/chess/comments/xrs31z/a_proposal_for_an_experiment_well_data_analysis/