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There aren’t enough simulators above us that care enough about us-in-particular to pay in paperclips. There are so many things to care about! Why us, rather than giant gold obelisks?

What about neighboring Everett branches where humanity succeeds at alignment? If you think alignment isn't completely impossible, it seems such branches should have at least roughly comparable weight to branches where we fail, so trade could be possible.

Yeah. I think this sort of thing is why Eliezer thinks we're doomed

Hmm, interesting...but wasn't he more optimistic a few years ago, when his plan was still "pull off a pivotal act with a limited AI"? I thought the thing that made him update towards doom was the apparent difficulty of safely making even a limited AI, plus shorter timelines.

other gestured-example I've heard is "upload aligned people who think hard for 1000 subjective years and hopefully figure something out."

Ah, that actually seems like it might work. I guess the problem is that an AI that can competently do neuroscience well enough to do this would have to be pretty general. Maybe a more realistic plan along the same lines might be to try using ML to replicate the functional activity of various parts of the human brain and create 'pseudo-uploads'. Or just try to create an AI with similar architecture and roughly-similar reward function to us, hoping that human values are more generic than they might appear.

Oh, melting the GPUs would not actually be a pivotal act

Well yeah, that's my point. It seems to me that any pivotal act worthy of the name would essentially require the AI team to become an AGI-powered world government, which seems pretty darn difficult to pull off safely. The superpowered-AI-propaganda plan falls under this category. The long-lasting nanomachines idea is cute, but I bet people would just figure out ways to evade the nanomachines' definition of 'GPU'.

Note that these aren't intended to be very good/realistic suggestions, they're just meant to point to different dimensions of the possibility space

Fair enough...but if the pivotal act plan is workable, there should be some member of that space which actually is good/seems like it has a shot of working out in reality(and which wouldn't require a full FAI). I've never heard any and am having a hard time thinking of one. Now it could be that MIRI or others think they have a workable plan which they don't want to share the details of due to infohazard concerns. But as an outside observer, I have to assign a certain amount of probability to that being self-delusion.

it's much more likely that someone could actually perform a unilateral pivotal act; it is a far easier problem, even after accounting for the problems the OP mentions in Part 1.

What I've never understood about the pivotal act plan is exactly what the successful AGI team is supposed to do after melting the GPUs or whatever. Every government on Earth will now consider them their enemy; they will immediately be destroyed unless they can defend themselves militarily, then countries will simply rebuild the GPU factories and continue on as before(except now in a more combative, disrupted, AI-race-encouraging geopolitical situation). So any pivotal act seems to require, at a minimum, an AI capable of militarily defeating all countries' militaries. Then in order to not have society collapse, you probably need to become the government yourself, or take over or persuade existing governments to go along with your agenda. But an AGI that would be capable of doing all this safely seems...not much easier to create than a full-on FAI? It's not like you could get by with an AI that was freakishly skilled at designing nanomachines but nothing else, you'd need something much more general. But isn't the whole idea of the pivotal act plan that you don't need to solve alignment in full generality to execute a pivotal act? For these reasons, executing a unilateral pivotal act(that actually results in an x-risk reduction) does not seem obviously easier than convincing governments to me.

Regarding your first point, I think when people say that language models "don't bring us closer to full code automation" they mean there's no way of improving/upgrading language models such that they implement full code automation. I think it would be better to argue against that claim directly instead of bringing up language model's productivity-boosting effects. There are many things that could potentially boost programmers' productivity -- better nootropics, say -- but it seems overly broad to say that they all "bring us closer to full code automation", even if it might be causally true that they reduce the time to automation in expectation.

For reasons elaborated upon in this post and its comments, I'm kinda skeptical of these results. Basically the claims made are

(A) That the parameter->function map is "biased towards simple functions". It's important to distinguish simple --> large volume and large volume --> simple. Simple --> large volume is the property that Solomonoff induction has and what makes it universal, but large volume-->simple is what is proven in these papers(plus some empirical evidence of unclear import)

(B) SGD being equivalent to random selection. The evidence is empirical performance of Gaussian processes being similar to neural nets on simple tasks. But this may break down on more difficult problems(link is about the NTK, not GP, but they tend to perform similarly, indeed NTK usually performs better than GP)

Overall I think it's likely we'll need to actually analyze SGD in a non-kernel limit to get a satisfactory understanding of "what's really going on" with neural nets.

There's an important distinction[1] to be made between these two claims:

A) Every function with large volume in parameter-space is simple

B) Every simple function has a large volume in parameter space

For a method of inference to qualify as a 'simplicity prior', you want both claims to hold. This is what lets us derive bounds like 'Solomonoff induction matches the performance of any computable predictor', since all of the simple, computable predictors have relatively large volume in the Solomonoff measure, so they'll be picked out after boundedly many mistakes. In particular, you want there to be an implication like, if a function has complexity , it will have parameter-volume at least .

Now, the Mingard results, at least the ones that have mathematical proof, rely on the Levin bound. This only shows (A), which is the direction that is much easier to prove -- it automatically holds for any mapping from parameter-space to functions with bounded complexity. They also have some empirical results that show there is substantial 'clustering', that is, there are some simple functions that have large volumes. But this still doesn't show that all of them do, and indeed is compatible with the learnable function class being extremely limited. For instance, this could easily be the case even if NTK/GP was only able to learn linear functions. In reality the NTK/GP is capable of approximating arbitrary functions on finite-dimensional inputs but, as I argued in another comment, this is not the right notion of 'universality' for classification problems. I strongly suspect[2] that the NTK/GP can be shown to not be 'universally data-efficient' as I outlined there, but as far as I'm aware no one's looked into the issue formally yet. Empirically, I think the results we have so far suggest that the NTK/GP is a decent first-order approximation for simple tasks that tends to perform worse on the more difficult problems that require non-trivial feature learning/efficiency.

  1. I actually posted basically the same thing underneath another one of your comments a few weeks ago, but maybe you didn't see it because it was only posted on LW, not the alignment forum ↩︎

  2. Basically, because in the NTK/GP limit the functions for all the neurons in a given layer are sampled from a single computable distribution, so I think you can show that the embedding is 'effectively finite' in some sense(although note it is a universal approximator for fixed input dimension) ↩︎

Hmm, so regarding the linear combinations, it's true that there are some linear combinations that will change by in the large-width limit -- just use the vector of partial derivatives of the output at some particular input, this sum will change by the amount that the output function moves during the regression. Indeed, I suspect(but don't have a proof) that these particular combinations will span the space of linear combinations that change non-trivially during training. I would dispute "we expect most linear combinations to change" though -- the CLT argument implies that we should expect almost all combinations to not appreciably change. Not sure what effect this would have on the PCA and still think it's plausible that it doesn't change at all(actually, I think Greg Yang states that it doesn't change in section 9 of his paper, haven't read that part super carefully though)

And the tangent kernel not changing does not imply that transfer learning won’t work

So I think I was a bit careless in saying that the NTK can't do transfer learning at all -- a more exact statement might be "the NTK does the minimal amount of transfer learning possible". What I mean by this is, any learning algorithm can do transfer learning if the task we are 'transferring' to is sufficiently similar to the original task -- for instance, if it's just the exact same task but with a different data sample. I claim that the 'transfer learning' the NTK does is of this sort. As you say, since the tangent kernel doesn't change at all, the net effect is to move where the network starts in the tangent space. Disregarding convergence speed, the impact this has on generalization is determined by the values set by the old function on axes of the NTK outside of the span of the partial derivatives at the new function's data points. This means that, for the NTK to transfer anything from one task to another, it's not enough for the tasks to both feature, for instance, eyes. It's that the eyes have to correlate with the output in the exact same way in both tasks. Indeed, the transfer learning could actually hurt the generalization. Nor is its effect invariant under simple transformations like flipping the sign of the target function(this would change beneficial transfer to harmful). By default, for functions that aren't simple multiples, I expect the linear correlation between values on different axes to be about 0, even if the functions share many meaningful features. So while the NTK can do 'transfer learning' in a sense, it's about as weak as possible, and I strongly doubt that this sort of transfer is sufficient to explain transfer learning's successes in practice(but don't have empirical proof).

I do think the empirical results pretty strongly suggest that the NTK/GP model captures everything important about neural nets, at least in terms of their performance on the original task.

It's true that NTK/GP perform pretty closely to finite nets on the tasks we've tried them on so far, but those tasks are pretty simple and we already had decent non-NN solutions. Generally the pattern is '"GP matches NNs on really simple tasks, NTK on somewhat harder ones". I think the data we have is consistent with this breaking down as we move to the harder problems that have no good non-NN solutions. I would be very interested in seeing an experiment with NTK on, say, ImageNet for this reason, but as far as I know no one's done so because of the prohibitive computational cost.

I only found one directly-relevant study, which is on way too small and simple a system for me to draw much of a conclusion from it, but it does seem to have worked.

Thanks for the link -- will read this tomorrow.

BTW, thanks for humoring me throughout this thread. This is really useful, and my understanding is updating considerably.

And thank you for engaging in detail -- I have also found this very helpful in forcing me to clarify(partially to myself) what my actual beliefs are.

I don't think taking linear combinations will help, because adding terms to the linear combination will also increase the magnitude of the original activation vector -- e.g. if you add together units, the magnitude of the sum of their original activations will with high probability be , dwarfing the O(1) change due to change in the activations. But regardless, it can't help with transfer learning at all, since the tangent kernel(which determines learning in this regime) doesn't change by definition.

What empirical results do you think are being contradicted? As far as I can tell, the empirical results we have are 'NTK/GP have similar performance to neural nets on some, but not all, tasks'. I don't think transfer/feature learning is addressed at all. You might say these results are suggestive evidence that NTK/GP captures everything important about neural nets, but this is precisely what is being disputed with the transfer learning arguments.

I can imagine doing an experiment where we find the 'empirical tangent kernel' of some finite neural net at initialization, solve the linear system, and then analyze the activations of the resulting network. But it's worth noting that this is not what is usually meant by 'NTK' -- that usually includes taking the infinite-width limit at the same time. And to the extent that we expect the activations to change at all, we no longer have reason to think that this linear system is a good approximation of SGD. That's what the above mathematical results mean -- the same mathematical analysis that implies that network training is like solving a linear system, also implies that the activations don't change at all.

The result that NTK does not learn features in the large N limit is not in dispute at all -- it's right there on page 15 of the original NTK paper, and indeed holds after arbitrarily many steps of backprop. I don't think that there's really much room for loopholes in the math here. See Greg Yang's paper for a lengthy proof that this holds for all architectures. Also worth noting that when people 'take the NTK limit' they often don't initialize an actual net at all, they instead use analytical expressions for what the inner product of the gradients would be at N=infinity to compute the kernel directly.

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