Singular learning theory, or something like it, is probably a necessary foundational tool here. It doesn’t directly answer the core question about how environment structure gets represented in the net, but it does give us the right mental picture for thinking about things being “learned by the net” at all. (Though if you just want to understand the mental picture, this video is probably more helpful than reading a bunch of SLT.)

I think this is probably wrong. Vanilla SLT describes a toy case of how Bayesian learning on neural networks works. I think there is a big difference between Bayesian learning, which requires visiting every single point in the loss landscape and trying them all out on every data point, and local learning algorithms, such as evolution, stochastic gradient descent, AdamW, etc., which try to find a good solution using information from just a small number of local neighbourhoods in the loss landscape. Those local learning algorithms are the ones I'd expect to be used by real minds, because they're much more compute efficient.

I think this locality property matters a lot. It introduces additional, important constraints on what nets can feasibly learn. It's where path dependence in learning comes from. I think vanilla SLT was probably a good tutorial for us before delving into the more realistic and complicated local learning case, but there's still work to do to get us to an actually roughly accurate model of how nets learn things.

If a solution consists of $1000$ internal pieces of machinery that need to be arranged exactly right to do anything useful at all, a local algorithm will need something like $O\left(e^{1000c}\right)$ update steps to learn it.^[1] In other words, it won't do better than a random walk that aimlessly wanders around the loss landscape until it runs into a point with low loss by sheer chance. But if a solution with $1000$ internal pieces of machinery can instead be learned in small chunks that each individually decrease the loss a little bit, the leading term in the number of update steps required to find that solution scales exponentially with the size of the single biggest solution chunk, rather than with the size of the whole solution. So, if the biggest chunk had size $50$, the total learning time will be around $O\left(e^{50c}\right)$.^[2] 

For an example where the solution cannot be learned in chunks like this, see the subset parity learning problem, where SGD really does need a number of update steps exponential in the effective parameter count of the whole solution to learn. Which for most practical purposes means it cannot learn the solution at all.

For a net to learn a big and complicated solution with high Local Learning Coefficient (LLC), it needs a learning story to find the solution's basin in the loss landscape in a feasible timeframe. It can't just rely on random walking, that takes too long. The expected total time it takes the net to get to a basin is, I think, determined mostly by the dimensionality of the mode connections from that basin to the rest of the landscape. Not just by the dimensionality of the basin itself, as would be the case for the sort of global, Bayesian learning modelled by vanilla SLT. The geometry of those connections is the core mathematical object that reflects the structure of the learning process and determines the learnability of a solution.^[3] Learning a big solution chunk that increases the total LLC by a lot in one go means needing to find a very low-dimensional mode connection to traverse. This takes a long time, because the connection interface is very small compared to the size of the search space. To learn a smaller chunk that increases the total LLC by less, the net only needs to reach a higher-dimensional mode connection, which will have an exponentially larger interface that is thus exponentially quicker to find.^[4] 

I agree that vanilla SLT seems like a useful tool for developing the right mental picture of how nets learn things, but it is not itself that picture. The simplified Bayesian learning case is instructive for illuminating the connection between learning and loss landscape geometry in the most basic setting, but taken on its own it's still failing to capture a lot of the structure of learning in real minds.

1. ^{^}

   Where $c$ is some constant which probably depends on the details of the update algorithm.

2. ^{^}

   I'm not going to add "I think" and "I suspect" to every sentence in this comment, but you should imagine them being there. I haven't actually worked this out in math properly or tested it.

3. ^{^}

   At least for a specific dataset and architecture. Modelling changes in the geometry of the loss landscape if we allow dataset and architecture to vary based on the mind's own decisions as it learns might be yet another complication we'll need to deal with in the future, once we start thinking about theories of learning for RL agents with enough freedom and intelligence to pick their learning curricula themselves.

4. ^{^}

   To get the rough idea across I'm focusing here on the very basic case where the "chunks" are literal pieces of the final solution and each of them lowers the loss a little and increases the total LLC a little. In general, this doesn't have to be true though. For example, a solution D with effective parameter count 120 might be learned by first learning independent chunks A and B, each with effective parameter count 50, then learning a chunk C with effective parameter count 30 which connects the formerly independent A and B together into a single mechanistic whole to form solution D. The expected number of update steps in this learning story would be $\approx e^{50c}+e^{50c}+e^{30c}=O\left(e^{50c}\right)$.

This was my favourite solstice to date. Thank you.

I just meant that if an oracle told me ASI was coming in two years, I probably couldn't spend down energy reserves to get more done within that timeframe compared to being told it'll take ten years. I might feel a greater sense of urgency than I already am and perhaps end up working longer hours as a result of that, but if so that'd probably be an unendorsed emotional response I couldn't help more than a considered plan. I kind of doubt I'd actually get more done that way. Some slack for curiosity and play is required for me to do my job well. 

The stakes are already so high and time so short that varying either within an order of magnitude up or down really doesn't change things all that much. 

I guess figuring out whether we’re “in a bubble” just hasn’t seemed very important to me, relative to how hard it seems to determine? What effects on the strategic calculus do you think it has?

E.g. my current best guess is that I personally should just do what I can to help build the science of interpretability and learning as fast as possible, so we can get to a point where we can start doing proper alignment research and reason more legibly about why alignment might be very hard and what could go wrong. Whether we’re in a bubble or not mostly matters for that only insofar as it’s one factor influencing how much time we have left to do that research. 

But I’m already going about as fast as I can anyway, so having a better estimate of timelines isn’t very action-relevant for me. And “bubble vs. no bubble” doesn’t even seem like a leading-order term in timeline uncertainty anyway. 

Yeah, the observation that the universe seems maybe well-predicted by a program running on some UTM is a subset of the observation that the universe seems amendable to mathematical description and compression. So the former observation isn't really an explanation for the latter, just a kind of restatement. We'd need an argument for why a prior over random programs running on an UTM should be preferred over a prior over random strings. Why does the universe have structure? The Universal Prior isn't an answer to that question. It's just an attempt to write down a sensible prior that takes the observation that the universe is structured and apparently predictable into account.

See footnote. Since this permutation freedom always exists no matter what the learned algorithm is, it can't tell us anything about the learned algorithm.

... Wait, are you saying we're not propagating updates into $\nu$ to change the mass it puts on inputs $0$ vs. $1$?

My viewpoint is that the prior distributions giving weight $1/3$ to each of the three hypotheses is different from the one giving weight $1/2$ to each of ${\nu }_0$ and ${\nu }_1$, even if their mixture distributions are exactly the same.

That's pretty unintuitive to me. What does it matter whether we happen to write out our belief state one way or the other? So long as the predictions come out the same, what we do and don't choose to call our 'hypotheses' doesn't seem particularly relevant for anything? 

We made our choice when we settled on $M$ as the prior. Everything past that point just seems like different choices of notation to me? If our induction procedure turned out to be wrong or suboptimal, it'd be because $M$ was a bad prior to pick, not because we happened to write $M$ down in a weird way, right?

If they have the same prior on sequences/histories, then in what relevant sense are they not the same prior on hypotheses? If they both sum to $M\left(x\right)$, how can their predictions come to differ?

I'm confused. Isn't one of the standard justifications for the Solomonoff prior that you can get it without talking about K-complexity, just by assuming a uniform prior over programs of length $l$ on a universal monotone Turing machine and letting $l$ tend to infinity?^[1] How is that different from your $P_{ap}\left(\nu \right)$? It's got to be different right, since you say that $P_{ap}\left(\nu \right)$ is not equivalent to the Solomonoff prior.

1. ^{^}

   See e.g. An Introduction to Universal Artifical Intelligence, pages 145 and 146.