# Dynamical versus Bayesian Phase Transitions in a Toy Model of Superposition

arXiv: https://arxiv.org/abs/2310.06301 · HTML: https://mlsanity.com/paper/2310.06301/

## Key facts

- Cited by 29 papers in the map (data through 14 Oct 2026).
- Placed in Sparse autoencoder training and evaluation, one of 13 directions in Mechanistic interpretability.
- Builds on 22 of its 38 references that are in the map.
- First posted on arXiv 10 Oct 2023.

## Placed in

- Mechanistic interpretability / [Sparse autoencoder training and evaluation](https://mlsanity.com/mechinterp/sparse-autoencoder-training-and-evaluation/)

## Builds on (most cited first, 22 of 22)

- [Emergent Abilities of Large Language Models](https://mlsanity.com/paper/2206.07682/) (2022)
- [Toy Models of Superposition](https://mlsanity.com/paper/2209.10652/) (2022)
- [In-context Learning and Induction Heads](https://mlsanity.com/paper/2209.11895/) (2022)
- [Progress measures for grokking via mechanistic interpretability](https://mlsanity.com/paper/2301.05217/) (2023)
- [Exact solutions to the nonlinear dynamics of learning in deep linear neural networks](https://mlsanity.com/paper/1312.6120/) (2013)
- [Implicit Regularization in Deep Matrix Factorization](https://mlsanity.com/paper/1905.13655/) (2019)
- [High-dimensional dynamics of generalization error in neural networks](https://mlsanity.com/paper/1710.03667/) (2017)
- [Deep Learning without Poor Local Minima](https://mlsanity.com/paper/1605.07110/) (2016)
- [Towards Understanding Grokking: An Effective Theory of Representation Learning](https://mlsanity.com/paper/2205.10343/) (2022)
- [Hidden Progress in Deep Learning: SGD Learns Parities Near the Computational Limit](https://mlsanity.com/paper/2207.08799/) (2022)
- [A mathematical theory of semantic development in deep neural networks](https://mlsanity.com/paper/1810.10531/) (2018)
- [A Convergence Analysis of Gradient Descent for Deep Linear Neural Networks](https://mlsanity.com/paper/1810.02281/) (2018)
- [Towards Resolving the Implicit Bias of Gradient Descent for Matrix Factorization: Greedy Low-Rank Learning](https://mlsanity.com/paper/2012.09839/) (2020)
- [SGD learning on neural networks: leap complexity and saddle-to-saddle dynamics](https://mlsanity.com/paper/2302.11055/) (2023)
- [Implicit Regularization of Discrete Gradient Dynamics in Linear Neural Networks](https://mlsanity.com/paper/1904.13262/) (2019)
- [Acquisition of Chess Knowledge in AlphaZero](https://mlsanity.com/paper/2111.09259/) (2021)
- [Saddle-to-Saddle Dynamics in Deep Linear Networks: Small Initialization Training, Symmetry, and Sparsity](https://mlsanity.com/paper/2106.15933/) (2021)
- [The Implicit Bias of Depth: How Incremental Learning Drives Generalization](https://mlsanity.com/paper/1909.12051/) (2019)
- [The Local Learning Coefficient: A Singularity-Aware Complexity Measure](https://mlsanity.com/paper/2308.12108/) (2023)
- [Deep Learning is Singular, and That's Good](https://mlsanity.com/paper/2010.11560/) (2020)
- [Energy-entropy competition and the effectiveness of stochastic gradient descent in machine learning](https://mlsanity.com/paper/1803.01927/) (2018)
- [Training Linear Neural Networks: Non-Local Convergence and Complexity Results](https://mlsanity.com/paper/2002.09852/) (2020)

## Built on by (most cited first, 29 of 29)

- [Mechanistic Interpretability for AI Safety -- A Review](https://mlsanity.com/paper/2404.14082/) (2024)
- [Loss Landscape Degeneracy and Stagewise Development in Transformers](https://mlsanity.com/paper/2402.02364/) (2024)
- [Differentiation and Specialization of Attention Heads via the Refined Local Learning Coefficient](https://mlsanity.com/paper/2410.02984/) (2024)
- [Dynamics of Transient Structure in In-Context Linear Regression Transformers](https://mlsanity.com/paper/2501.17745/) (2025)
- [Using Degeneracy in the Loss Landscape for Mechanistic Interpretability](https://mlsanity.com/paper/2405.10927/) (2024)
- [Beyond Interpretability: The Gains of Feature Monosemanticity on Model Robustness](https://mlsanity.com/paper/2410.21331/) (2024)
- [The Loss Kernel: A Geometric Probe for Deep Learning Interpretability](https://mlsanity.com/paper/2509.26537/) (2025)
- [Influence Dynamics and Stagewise Data Attribution](https://mlsanity.com/paper/2510.12071/) (2025)
- [Estimating the Local Learning Coefficient at Scale](https://mlsanity.com/paper/2402.03698/) (2024)
- [From Global to Local: A Scalable Benchmark for Local Posterior Sampling](https://mlsanity.com/paper/2507.21449/) (2025)
- [Compressibility Measures Complexity: Minimum Description Length Meets Singular Learning Theory](https://mlsanity.com/paper/2510.12077/) (2025)
- [Stagewise Reinforcement Learning and the Geometry of the Regret Landscape](https://mlsanity.com/paper/2601.07524/) (2026)
- [A Basin-Selection Perspective on Grokking via Singular Learning Theory](https://mlsanity.com/paper/2603.01192/) (2026)
- [The Persian Rug: solving toy models of superposition using large-scale symmetries](https://mlsanity.com/paper/2410.12101/) (2024)
- [Modes of Sequence Models and Learning Coefficients](https://mlsanity.com/paper/2504.18048/) (2025)
- [SAFR: Neuron Redistribution for Interpretability](https://mlsanity.com/paper/2501.16374/) (2025)
- [Grokking vs. Learning: Same Features, Different Encodings](https://mlsanity.com/paper/2502.01739/) (2025)
- [Almost Bayesian: The Fractal Dynamics of Stochastic Gradient Descent](https://mlsanity.com/paper/2503.22478/) (2025)
- [Emergence of Computational Structure in a Neural Network Physics Simulator](https://mlsanity.com/paper/2504.11830/) (2025)
- [Dead Directions: Geometric Singular Learning](https://mlsanity.com/paper/2606.05957/) (2026)
- [Decomposing Prediction Mechanisms for In-Context Recall](https://mlsanity.com/paper/2507.01414/) (2025)
- [Using physics-inspired Singular Learning Theory to understand grokking & other phase transitions in modern neural networks](https://mlsanity.com/paper/2512.00686/) (2025)
- [Structure and Scale in Simplicial Sequence Modelling](https://mlsanity.com/paper/2606.01302/) (2026)
- [Discovering Millions of Interpretable Features with Sparse Autoencoders](https://mlsanity.com/paper/2606.26620/) (2026)
- [Local Redundancy: An Information-Theoretic Measure of Plasticity from Synthetic Memorization](https://mlsanity.com/paper/2607.13432/) (2026)
- [The Kinetics of Training: A Driven-Nucleation Rate Law for Emergence, Plasticity Loss, and Circuit Control in Language Models](https://mlsanity.com/paper/2607.27281/) (2026)
- [Langevin dynamics along the zero set of real-analytic potentials](https://mlsanity.com/paper/2608.09840/) (2026)
- [Exact Algebraic Computation of Learning Coefficients for Two-Dimensional Singular Models](https://mlsanity.com/paper/2608.20183/) (2026)
- [Emergent phases of superposition: from partial to full representation](https://mlsanity.com/paper/2609.36455/) (2026)

## Abstract (arXiv)

We investigate phase transitions in a Toy Model of Superposition (TMS) using Singular Learning Theory (SLT). We derive a closed formula for the theoretical loss and, in the case of two hidden dimensions, discover that regular k-gons are critical points. We present supporting theory indicating that the local learning coefficient (a geometric invariant) of these k-gons determines phase transitions in the Bayesian posterior as a function of training sample size. We then show empirically that the same k-gon critical points also determine the behavior of SGD training. The picture that emerges adds evidence to the conjecture that the SGD learning trajectory is subject to a sequential learning mechanism. Specifically, we find that the learning process in TMS, be it through SGD or Bayesian learning, can be characterized by a journey through parameter space from regions of high loss and low complexity to regions of low loss and high complexity.