Crossposted from the AI Alignment Forum. May contain more technical jargon than usual.

DSLT 0. Distilling Singular Learning Theory

4Ege Erdil

2Liam Carroll

2Nathaniel Monson

11Liam Carroll

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I'm kind of puzzled by the amount of machinery that seems to be going into these arguments, because it seems to me that there is a discrete analog of the same arguments which is probably both more realistic (as neural networks are not actually continuous, especially with people constantly decreasing the precision of the floating point numbers used in implementation) and simpler to understand.

Suppose you represent a neural network architecture as a map where and is the set of all possible computable functions from the input and output space you're considering. In thermodynamic terms, we could identify elements of as "microstates" and the corresponding functions that the NN architecture maps them to as "macrostates".

Furthermore, suppose that comes together with a loss function evaluating how good or bad a particular function is. Assume you optimize using something like stochastic gradient descent on the function with a particular learning rate.

Then, in general, we have the following results:

- SGD defines a Markov chain structure on the space whose stationary distribution is proportional to on parameters for some positive constant . This is just a basic fact about the Langevin dynamics that SGD would induce in such a system.
- In general is not injective, and we can define the "-complexity" of any function as . Then, the probability that we arrive at the macrostate is going to be proportional to .
- When is some kind of negative log likelihood, this approximates Solomonoff induction in a tempered Bayes paradigm insofar as the -complexity is a good approximation for the Kolmogorov complexity of the function , which will happen if the function approximator defined by is sufficiently well-behaved.

Is there some additional content of singular value theory that goes beyond the above insights?

**Edit:** I've converted this comment to a post, which you can find here.

**Edit**: Originally the sequence was going to contain a post about SLT for Alignment, but this can now be found here instead, where a new research agenda, Developmental Interpretability, is introduced. I have also now included references to the lectures from the recent SLT for Alignment Workshop in June 2023.

I'm looking forward to both this series, and the workshop!

I think I (and probably many other people) would find it helpful if there was an entry in this sequence which was purely the classical story told in a way/with language which makes its deficiencies clear and the contrasts with the Watanabe version very easy to point out. (Maybe a -1 entry, since 0 is already used?)

The way I've structured the sequence means these points are interspersed throughout the broader narrative, but its a great question so I'll provide a brief summary here, and as they are released I will link to the relevant sections in this comment.

- In regular model classes, the set of true parameters that minimise the loss is a single point. In singular model classes, it can be
*significantly*more than a single point. Generally, it is a higher-dimensional structure. See here in DSLT1. - In regular model classes, every point on can be approximated by a quadratic form. In singular model classes, this is not true. Thus, asymptotic normality of regular models (i.e. every regular posterior is just a Gaussian as ) breaks down. See here in DSLT1, or here in DSLT2.
- Relatedly, the Bayesian Information Criterion (BIC) doesn't hold in singular models, because of a similar problem of non-quadraticness. Watanabe generalises the BIC to singular models with the WBIC, which shows complexity is measured by the RLCT , not the dimension of parameter space . The RLCT satisfies in general, and when its regular. See here in DSLT2.
- In regular model classes, every parameter has the same complexity . In singular model classes, different parameters have
*different*complexities according to their RLCT. See here in DSLT2. - If you have a fixed model class, the BIC can only be minimised by optimising the accuracy (see here). But in a singular model class, the WBIC can be minimised according to an accuracy-complexity tradeoff. So "simpler" models exist on singular loss landscapes, but every model is equally complex in regular models. See here in DSLT2.
- With this latter point in mind, phase transitions are anticipated in singular models because the free energy is comprised of accuracy and complexity, which is different across parameter space. In regular models, since complexity is fixed, phase transitions are far less natural or interesting, in general.

TLDR; In this sequence I distill Sumio Watanabe'sSingular Learning Theory (SLT)by explaining the essence of its main theorem - Watanabe's Free Energy Formula for Singular Models - and illustrating its implications with intuition-building examples. I then show why neural networks are singular models, and demonstrate how SLT provides a framework for understanding phases and phase transitions in neural networks.Epistemic status:The core theorems of Singular Learning Theory have been rigorously proven and published by Sumio Watanabe across 20 years of research. Precisely what it says about modern deep learning, and its potential application to alignment, is still speculative.Acknowledgements:This sequence has been produced with the support of a grant from the Long Term Future Fund. I'd like to thank all of the people that have given me feedback on each post: Ben Gerraty, @Jesse Hoogland , @mfar, @LThorburn , Rumi Salazar, Guillaume Corlouer, and in particular my supervisor and editor-in-chief Daniel Murfet.Theory vs Examples:The sequence is a mixture of synthesising the main theoretical results of SLT, and providing simple examples and animations that illustrate its key points. As such, some theory-based sections are slightly more technical. Some readers may wish to skip ahead to the intuitive examples and animations before diving into the theory - these are clearly marked in the table of contents of each post.Prerequisites:Anybody with a basic grasp of Bayesian statistics and multivariable calculus should have no problems understanding the key points. Importantly, despite SLT pointing out the relationship between algebraic geometry and statistical learning, no prior knowledge of algebraic geometry is required to understand this sequence - I will merely gesture at this relationship. Jesse Hoogland wrote an excellent introduction to SLT which serves as a high level overview of the ideas that I will discuss here, and is thus recommended pre-reading to this sequence.SLT for Alignment Workshop:This sequence was prepared in anticipation of the SLT for Alignment Workshop 2023 and serves as a useful companion piece to the material covered in the Primer Lectures.Thesis:The sequence is derived from my recent masters thesis which you can read about at my website.Developmental Interpretability:Originally the sequence was going to contain a short outline of a new research agenda, but this can now be found here instead.## Introduction

In 2009, Sumio Watanabe wrote these two profound statements in his groundbreaking book Algebraic Geometry and Statistical Learning where he proved the first main results of Singular Learning Theory (SLT). Up to this point, this work has gone largely under-appreciated by the AI community, probably because it is rooted in highly technical algebraic geometry and distribution theory. On top of this, the theory is framed in the Bayesian setting, which contrasts the SGD-based setting of modern deep learning.

But this is a crying shame, because SLT has a

lotto say about why neural networks, which are singular models, are able to generalise well in the Bayesian setting, and it is very possible that these insights carry over to modern deep learning.At its core, SLT shows that the loss landscape of singular models, the KL divergence K(w), is fundamentally different to that of regular models like linear regression, consisting of flat valleys instead of broad parabolic basins. Correspondingly, the measure of effective dimension (complexity) in singular models is a rational quantity called the RLCT

^{[1]}, which can be less than half the total number of parameters. This fact means that classical results of Bayesian statistics like asymptotic normality break down, but what Watanabe shows is that this is actually a feature and not a bug: different regions of the loss landscape have different tradeoffs between accuracy and complexity because of their differing information geometry. This is the content of Watanabe's Free Energy Formula, from which the Widely Applicable Bayesian Information Criterion (WBIC) is derived, a generalisation of the standard Bayesian Information Criterion (BIC) for singular models.With this in mind, SLT provides a framework for understanding

phasesandphase transitionsin neural networks. It has been mooted that understanding phase transitions in deep learning may be a key part of mechanistic interpretability, for example in Induction Heads, Toy Models of Superposition, and Progress Measures for Grokking via Mechanistic Interpretability, which relate phase transitions to the formation of circuits. Furthermore, the existence of scaling laws and other critical phenomena in neural networks suggests that there is a natural thermodynamic perspective on deep learning. As it stands there is no agreed-upon theory that connects all of this, but in this sequence we will introduce SLT as a bedrock for a theory that can tie these concepts together.In particular, I will demonstrate the existence of first and second order phase transitions in simple two layer feedforward ReLU neural networks which we can understand precisely through the lens of SLT. By the end of this sequence, the reader will understand why the following phase transition in the Bayesian posterior corresponds to a changing accuracy-complexity tradeoff of the different phases in the loss landscape:

## Key Points of the Sequence

To understand phase transitions in neural networks from the point of view of SLT, we need to understand how different regions of parameter space can have different accuracy-complexity tradeoffs, a feature of singular models that is not present in regular models. Here is the outline of how these posts get us there:

DSLT 1. The RLCT Measures the Effective Dimension of Neural NetworksDSLT 2. Why Neural Networks obey Occam's RazorDSLT 3. Neural Networks are SingularDSLT 4. Phase Transitions in Neural Networks(

Edit: Originally the sequence was going to contain a post about SLT for Alignment, but this can now be found here instead, where a new research agenda, Developmental Interpretability, is introduced).## Resources

Though these resources are relatively sparse for now, expanding the reach of SLT and encouraging new research is the primary longterm goal of this sequence.

## SLT Workshop for Alignment Primer

In June 2023, a summit, "SLT for Alignment", was held, which produced over 20hrs of lectures. The details of these talks can be found here, with recordings found here.

## Research groups

Research groups I know of working on SLT:

## Literature

The two canonical textbooks due to Watanabe are:

The grey book:S. Watanabe Algebraic Geometry and Statistical Learning Theory 2009The green book:S. Watanabe Mathematical Theory of Bayesian Statistics 2018The two main papers that were precursors to these books:

This sequence is based on my recent thesis:

MDLG recently wrote an introduction to SLT:

Other theses studying SLT:

Other introductory blogs:

^{^}Short for the algebro-geometric Real Log Canonical Threshold, which I define in DSLT1.