# More Recent Progress in the Theory of Neural Networks

*Thanks to Dan Roberts and Sho Yaida for comments on a draft of this post.*

In this post, I would like to draw attention to the book [Principles of Deep Learning Theory](https://deeplearningtheory.com/) (PDLT), which I think represents a significant advance in our understanding of how neural networks work [^p36imn6ssl]. Among other things, this book explains how to write a closed-form formula for the function learned by a *realistic, finite-width* neural network at the end of training [^4cmkemwtxpf] to an order of approximation that suffices to describe representation learning, and how that formula can be interpreted as the solution to a regression model. This makes manifest the intuition that NNs are doing something like regression, but where they learn the features appropriate for a given dataset rather than having them be hand-engineered from the start.

I've condensed some main points from the 400-page book into an 8-page summary here:

[Review of select results from PDLT](https://www.dropbox.com/s/0fq3u4wjhkeyb48/Review%20of%20PDLT%20v2.pdf?dl=0)

(Other good places to learn about the book, though perhaps with less of a focus on AI-safety-relevant parts, include this [series of five lectures](https://deeplearningtheory.com/lectures/) given by the authors at a deep learning summer school or this [one-hour lecture](https://www.youtube.com/watch?v=YzR2gZrsdJc) for a non-expert audience.)

For those who have been following the discussions of ML theory on this forum, the method used in the book is to go to the next-to-leading order in a 1/width expansion. It thus builds on recent studies of infinitely wide NNs that were reviewed in the AF post [Recent Progress in the Theory of Neural Networks](https://www.alignmentforum.org/posts/KrQvZM8uFjSTJ7hq3/recent-progress-in-the-theory-of-neural-networks-1) [^153qvapio6r].  However, by going beyond the leading order, the authors of PDLT are able to get around a key qualitative shortcoming of the earlier work in that [infinitely wide NNs can't learn features](https://www.alignmentforum.org/posts/76cReK4Mix3zKCWNT/ntk-gp-models-of-neural-nets-can-t-learn-features).  The next-to-leading order formula also introduces a sum over many steps of gradient descent, getting around an objection [^lhd6zqd4ws] that the NTK/infinite width limit may not be applicable to realistic models since in that limit, we can land on the fully trained model after just one fine-tuned training step. 

I think that this work could have significant implications for AGI forecasting and safety (via [interpretability](https://www.alignmentforum.org/posts/QirLfXhDPYWCP8PK5/transparency-and-agi-safety)), and deserves to be better appreciated in this community. For example, 

*   In AGI forecasting, an important open question is whether the [strong scaling hypothesis](https://www.lesswrong.com/posts/kpK6854ArgwySuv7D/probability-that-other-architectures-will-scale-as-well-as#PEqsLsDswcRNKRhje) holds for any modern architectures. (For example, the forecasts in Ajeya Cotra's [Bio-Anchors report](https://www.lesswrong.com/posts/KrJfoZzpSDpnrv9va/draft-report-on-ai-timelines) are conditioned on assuming that 2020 algorithms can scale to TAI.) A longstanding challenge for this field is that as long as we treat neural networks as black boxes or random program search, it's hard to reason about this question in a principled way. But I think that by identifying a space of functions that realistic NNs end up learning in practice (<< the space of all neural networks with finely-tuned weights!), the approach of PDLT gives us a way to start to reason about it. For example, despite the existence of the universal approximation theorem, I think the results of PDLT can be used to rule out the (strawmannish) hypothesis that feedforward MLPs can scale to AGI (see my [review of the Bio-Anchors report](https://www.lesswrong.com/posts/TMHWfRE7zZkzgFDSo/a-review-of-the-bio-anchors-report) for more on this point). As such, it could be really interesting to generalize PDLT to other architectures. 
*   In mechanistic interpretability, a basic open question is what the fundamental degrees of freedom are that we should be trying to interpret. A lot of work has been done under the assumption that we should look at the activations of individual neurons, but there's naively no reason that semantically meaningful properties of a dataset must align with individual neurons after training, and even some interesting counterexamples [^16390ni4s02]. By finding a dual description of a trained NN as a trained regression model, PDLT seems to hint that a (related, but) different set of degrees of freedom -- the effective features (32) in the above-linked note -- *may* be more natural objects to look at. It would be really interesting to see if this turns out to be the case [^v21dr8zvwsr].
*    More generally, a dream for interpretability research would be if we could [reverse-engineer our future AI systems into human-understandable code](https://www.lesswrong.com/posts/X2i9dQQK3gETCyqh2/chris-olah-s-views-on-agi-safety). If we take this dream seriously, it may be helpful to split it into two parts: first understanding what "programming language" an architecture + learning algorithm will end up using at the end of training, and then what "program" a particular training regimen will lead to in that language  [^3n7jk2sijta]. It seems to me that by focusing on specific trained models, most interpretability research discussed here is of the second type. But by constructing an effective theory for an entire class of architecture that's agnostic to the choice of dataset, PDLT is a rare example of the first type. So it could be not only useful but also totally complementary to other agendas to try to develop it further and/or generalize it to new architectures as they come along.

[^p36imn6ssl]: See here for an earlier, though shorter discussion of this book on LW. 

[^4cmkemwtxpf]: As a function of its architecture + weight initialization + training set + learning algorithm. The formula is (\(\infty\).154) on page 1 of the note that I link to below. 

[^153qvapio6r]: See also yesterday's post Neural Tangent Kernel Distillation. 

[^lhd6zqd4ws]: As raised e.g. by Paul Christiano here. 

[^16390ni4s02]: See the recent Anthropic paper Toy Models of Superposition for a discussion of this and related issues. 

[^v21dr8zvwsr]: For example, one could try to see which dataset examples or synthetic data would maximally activate the effective features, generalizing the experiments done on neurons in the Circuits thread. (However, a caveat for this project idea is that there are a huge number of effective features in PDLT, scaling as the number of weights instead of the number of neurons! So one might have to start with a really small toy model, or be clever about picking a subset / combination of features to visualize.) It could also be interesting to understand how the PDLT effective theory fits conceptually with other ideas from the interpretability and broader "science of ML" literature. For example, how should we think of compositional circuits in the dual frame, which seems to put all effective features on the same footing instead of having some be built from others? (Perhaps this isn't a meaningful question to ask since the compositional circuits in vision models are themselves a fuzzy emergent description. But in that case, can we generalize PDLT to transformers, and then understand crisp emergent circuits like modular arithmetic circuits in the dual frame?) Or since the dual model has a huge number of effective features, might there be a lottery ticket hypothesis at the level of the features? 

[^3n7jk2sijta]: Or to put it another way, if we want to understand what cognition a trained AI system is performing at inference time, there's both a kinematic aspect of claiming that some degrees of freedom in the system (or the way that they activate or something) approximately encode some human-understandable concepts in a Platonic latent space (e.g. paperclips), and a dynamical aspect of how those concepts get put together (e.g. into a plan to turn us into paperclips). The space of allowed dynamics is what I'm calling the "programming language" per Chris's metaphor.