TLDR; In this sequence I distill Sumio Watanabe's Singular 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 t...

22

DSLT 1. The RLCT Measures the Effective Dimension of Neural Networks

22