kaarelh AT gmail DOT com
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 :)
I'd be very interested in a concrete construction of a (mathematical) universe in which, in some reasonable sense that remains to be made precise, two 'orthogonal pattern-universes' (preferably each containing 'agents' or 'sophisticated computational systems') live on 'the same fundamental substrate'. One of the many reasons I'm struggling to make this precise is that I want there to be some condition which meaningfully rules out trivial constructions in which the low-level specification of such a universe can be decomposed into a pair such that and are 'independent', everything in the first pattern-universe is a function only of , and everything in the second pattern-universe is a function only of . (Of course, I'd also be happy with an explanation why this is a bad question :).)
I find [the use of square brackets to show the merge structure of [a linguistic entity that might otherwise be confusing to parse]] delightful :)
I'd be quite interested in elaboration on getting faster alignment researchers not being alignment-hard — it currently seems likely to me that a research community of unupgraded alignment researchers with a hundred years is capable of solving alignment (conditional on alignment being solvable). (And having faster general researchers, a goal that seems roughly equivalent, is surely alignment-hard (again, conditional on alignment being solvable), because we can then get the researchers to quickly do whatever it is that we could do — e.g., upgrading?)
I was just claiming that your description of pivotal acts / of people that support pivotal acts was incorrect in a way that people that think pivotal acts are worth considering would consider very significant and in a way that significantly reduces the power of your argument as applying to what people mean by pivotal acts — I don't see anything in your comment as a response to that claim. I would like it to be a separate discussion whether pivotal acts are a good idea with this in mind.
Now, in this separate discussion: I agree that executing a pivotal act with just a narrow, safe, superintelligence is a difficult problem. That said, all paths to a state of safety from AGI that I can think of seem to contain difficult steps, so I think a more fine-grained analysis of the difficulty of various steps would be needed. I broadly agree with your description of the political character of pivotal acts, but I disagree with what you claim about associated race dynamics — it seems plausible to me that if pivotal acts became the main paradigm, then we'd have a world in which a majority of relevant people are willing to cooperate / do not want to race that much against others in the majority, and it'd mostly be a race between this group and e/acc types. I would also add, though, that the kinds of governance solutions/mechanisms I can think of that are sufficient to (for instance) make it impossible to perform distributed training runs on consumer devices also seem quite authoritarian.
In this comment, I will be assuming that you intended to talk of "pivotal acts" in the standard (distribution of) sense(s) people use the term — if your comment is better described as using a different definition of "pivotal act", including when "pivotal act" is used by the people in the dialogue you present, then my present comment applies less.
I think that this is a significant mischaracterization of what most (? or definitely at least a substantial fraction of) pivotal activists mean by "pivotal act" (in particular, I think this is a significant mischaracterization of what Yudkowsky has in mind). (I think the original post also uses the term "pivotal act" in a somewhat non-standard way in a similar direction, but to a much lesser degree.) Specifically, I think it is false that the primary kinds of plans this fraction of people have in mind when talking about pivotal acts involve creating a superintelligent nigh-omnipotent infallible FOOMed properly aligned ASI. Instead, the kind of person I have in mind is very interested in coming up with pivotal acts that do not use a general superintelligence, often looking for pivotal acts that use a narrow superintelligence (for instance, a narrow nanoengineer) (though this is also often considered very difficult by such people (which is one of the reasons they're often so doomy)). See, for instance, the discussion of pivotal acts in https://www.lesswrong.com/posts/7im8at9PmhbT4JHsW/ngo-and-yudkowsky-on-alignment-difficulty.
A few notes/questions about things that seem like errors in the paper (or maybe I'm confused — anyway, none of this invalidates any conclusions of the paper, but if I'm right or at least justifiably confused, then these do probably significantly hinder reading the paper; I'm partly posting this comment to possibly prevent some readers in the future from wasting a lot of time on the same issues):
1) The formula for here seems incorrect:
This is because W_i is a feature corresponding to the i'th coordinate of x (this is not evident from the screenshot, but it is evident from the rest of the paper), so surely what shows up in this formula should not be W_i, but instead the i'th row of the matrix which has columns W_i (this matrix is called W later). (If one believes that W_i is a feature, then one can see this is wrong already from the dimensions in the dot product not matching.)
2) Even though you say in the text at the beginning of Section 3 that the input features are independent, the first sentence below made me make a pragmatic inference that you are not assuming that the coordinates are independent for this particular claim about how the loss simplifies (in part because if you were assuming independence, you could replace the covariance claim with a weaker variance claim, since the 0 covariance part is implied by independence):
However, I think you do use the fact that the input features are independent in the proof of the claim (at least you say "because the x's are independent"):
Additionally, if you are in fact just using independence in the argument here and I'm not missing something, then I think that instead of saying you are using the moment-cumulants formula here, it would be much much better to say that independence implies that any term with an unmatched index is . If you mean the moment-cumulants formula here https://en.wikipedia.org/wiki/Cumulant#Joint_cumulants , then (while I understand how to derive every equation of your argument in case the inputs are independent), I'm currently confused about how that's helpful at all, because one then still needs to analyze which terms of each cumulant are 0 (and how the various terms cancel for various choices of the matching pattern of indices), and this seems strictly more complicated than problem before translating to cumulants, unless I'm missing something obvious.
3) I'm pretty sure this should say x_i^2 instead of x_i x_j, and as far as I can tell the LHS has nothing to do with the RHS:
(I think it should instead say sth like that the loss term is proportional to the squared difference between the true and predictor covariance.)
At least ignoring legislation, an exchange could offer a contract with the same return as S&P 500 (for the aggregate of a pair of traders entering a Kalshi-style event contract); mechanistically, this index-tracking could be supported by just using the money put into a prediction market to buy VOO and selling when the market settles. (I think.)
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:
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 hi(x)∈Rk denote the activation vector in layer i on input x∈Rd, with the input layer being at index i=1, so h1(x)=x. Let Wi be the weight matrix after activation layer i. Let fi be the function that maps from the ith activation layer to the output. Then their Deep Neural Feature Ansatz says that WTiWi∝∼1|D|∑x∈D∇fi(hi(x))∇fi(hi(x))T (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 Wi=UΣVT be a SVD of Wi, we note that this is equivalent to VΣ2VT∝∼1|D|∑x∈D∇fi(hi(x))∇fi(hi(x))T, i.e., that the eigenvectors of the matrix M 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 Wi are the highest orthonormal vTMv directions for the matrix M on the RHS. Plugging in the definition of M, this is equivalent to saying that the right singular vectors are the sequence of highest-variance directions of the data set of gradients ∇fi(hi(x)).
(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 Wi dependence on the RHS?
Assuming there isn't an off-by-one error in the paper, we can pull some Wi term out of the RHS maybe? This is because applying the chain rule to the Jacobians of the transitions i→i+1→end gives ∇fi(hi(x))T=∇fi+1(hi+1(x))TWi, so 1|D|∑x∈D∇fi(hi(x))∇fi(hi(x))T=1|D|∑x∈DWTi∇fi+1(hi+1(x))∇fi+1(hi+1(x))TWi.
Wait, so the claim is just WTiWi∝∼WTi(∑x∈D∇fi+1(hi+1(x))∇fi+1(hi+1(x))T)Wi which, assuming Wi is invertible, should be the same as ∑x∈D∇fi+1(hi+1(x))∇fi+1(hi+1(x))T∝∼I. But also, they claim that it is WTi+1Wi+1? 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 ∇fi(hi(x))T=∇fi+1(hi+1(x))TWi, i.e., it should really say ∇fi(hi(x))T=∇fi+1(hi+1(x))TDi+1(x)Wi. We now instead get WTiWi∝∼WTi(∑x∈DDi+1(x)∇fi+1(hi+1(x))∇fi+1(hi+1(x))TDi+1(x))Wi. This should be the same as ∑x∈DDi+1(x)∇fi+1(hi+1(x))∇fi+1(hi+1(x))TDi+1(x)∝∼I which, with pi denoting preactivations in layer i and fp,i denoting the function from these preactivations to the output, is the same as ∑x∈D∇fp,i+1(pi+1(x))∇fp,i+1(pi+1(x))T∝∼I. 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 L2 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 L2 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 ∑x∈D∇fp,i+1(pi+1(x))∇fp,i+1(pi+1(x))T∝∼Iclearly implies that the L2 mass in any direction is the same, but also the L2 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 U′Σ′V′T, the above is U′Σ′Σ′TU′T=σ2I, where we used the fact that Σ=σI).
Some questions
typos in the paper
indexing error in the first displaymath in Sec 2: it probably should say 'WL', not 'WL+1'