My name is pronounced "YOO-ar SKULL-se".
I'm a DPhil Scholar at the Future of Humanity Institute in Oxford.
If a universality statement like the above holds for neural networks, it would tell us that most of the details of the parameter-function map are irrelevant.
I suppose this depends on what you mean by "most". DNNs and CNNs have noticeable and meaningful differences in their (macroscopic) generalisation behaviour, and these differences are due to their parameter-function map. This is also true of LSTMs vs transformers, and so on. I think it's fairly likely that these kinds of differences could have a large impact on the probability that a given type model will learn to exhibit goal-directed behaviour in a given training setup, for example.
The ambitious statement here might be that all the relevant information you might care about (in terms of understanding universality) are already contained in the loss landscape.
Do you mean the loss landscape in the limit of infinite data, or the loss landscape for a "small" amount of data? In the former case, the loss landscape determines the parameter-function map over the data distribution. In the latter case, my guess would be that the statement probably is false (though I'm not sure).
You're right, I put the parameters the wrong way around. I have fixed it now, thanks!
I could have changed it to Why Neural Networks can obey Occam's Razor, but I think this obscures the main point.
I think even this would be somewhat inaccurate (in my opinion). If a given parametric Bayesian learning machine does obey (some version of) Occam's razor, then this must be because of some facts related to its prior, and because of some facts related to its parameter-function map. SLT does not say very much about either of these two things. What the post is about is primarily the relationship between the RLCT and posterior probability, and how this relationship can be used to reason about training dynamics. To connect this to Occam's razor (or inductive bias more broadly), further assumptions and claims would be required.
At the time of writing, basically nobody knew anything about SLT
Yes, thank you so much for taking the time to write those posts! They were very helpful for me to learn the basics of SLT.
As we discussed at Berkeley, I do like the polynomial example you give and this whole discussion has made me think more carefully about various aspects of the story, so thanks for that.
I'm very glad to hear that! :)
My inclination is that the polynomial example is actually quite pathological and that there is a reasonable correlation between the RLCT and Kolmogorov complexity in practice
Yes, I also believe that! The polynomial example is definitely pathological, and I do think that low λ almost certainly is correlated with simplicity in the case of neural networks. My point is more that the mathematics of SLT does not explain generalisation, and that additional assumptions definitely will be needed to derive specific claims about the inductive bias of neural networks.
Well neural networks do obey Occam's razor, at least according to the formalisation of that statement that is contained in the post (namely, neural networks when formulated in the context of Bayesian learning obey the free energy formula, a generalisation of the BIC which is often thought of as a formalisation of Occam's razor).
Would that not imply that my polynomial example also obeys Occam's razor?
However, I accept your broader point, which I take to be: readers of these posts may naturally draw the conclusion that SLT currently says something profound about (ii) from my other post, and the use of terms like "generalisation" in broad terms in the more expository parts (as opposed to the technical parts) arguably doesn't make enough effort to prevent them from drawing these inferences.
Yes, I think this probably is the case. I also think the vast majority of readers won't go deep enough into the mathematical details to get a fine-grained understanding of what the maths is actually saying.
I'm often critical of the folklore-driven nature of the ML literature and what I view as its low scientific standards, and especially in the context of technical AI safety I think we need to aim higher, in both our technical and more public-facing work.
Yes, I very much agree with this too.
Does that sound reasonable?
At least right now, the value proposition I see of SLT lies not in explaining the "generalisation puzzle" but in understanding phase transitions and emergent structure; that might end up circling back to say something about generalisation, eventually.
I also think that SLT probably will be useful for understanding phase shifts and training dynamics (as I also noted in my post above), so we have no disagreements there either.
I think I recall reading that, but I'm not completely sure.Note that the activation function affects the parameter-function map, and so the influence of the activation function is subsumed by the general question of what the parameter-function map looks like.
I'm not sure, but I think this example is pathological.
Yes, it's artificial and cherry-picked to make a certain rhetorical point as simply as possible.
This is the more relevant and interesting kind of symmetry, and it's easier to see what this kind of symmetry has to do with functional simplicity: simpler functions have more local degeneracies.¨
This is probably true for neural networks in particular, but mathematically speaking, it completely depends on how you parameterise the functions. You can create a parameterisation in which this is not true.
You can make the same critique of Kolmogorov complexity.
Yes, I have been using "Kolmogorov complexity" in a somewhat loose way here.
Wild conjecture: [...]
Is this not satisfied trivially due to the fact that the RLCT has a certain maximum and minimum value within each model class? (If we stick to the assumption that Θ is compact, etc.)
Will do, thank you for the reference!
Yes, I completely agree. The theorems that have been proven by Watanabe are of course true and non-trivial facts of mathematics; I do not mean to dispute this. What I do criticise is the magnitude of the significance of these results for the problem of understanding the behaviour of deep learning systems.
Thank you for this -- I agree with what you are saying here. In the post, I went with a somewhat loose equivocation between "good priors" and "a prior towards low Kolmogorov complexity", but this does skim past a lot of nuance. I do also very much not want to say that the DNN prior is exactly towards low Kolmogorov complexity (this would be uncomputable), but only that it is mostly correlated with Kolmogorov complexity for typical problems.
Yes, I mostly just mean "low test error". I'm assuming that real-world problems follow a distribution that is similar to the Solomonoff prior (i.e., that data generating functions are more likely to have low Kolmogorov complexity than high Kolmogorov complexity) -- this is where the link is coming from. This is an assumption about the real world, and not something that can be established mathematically.