# Noise-Driven Escape from Metastable Phases explains Grokking in Deep Neural Networks

arXiv: https://arxiv.org/abs/2606.17120 · HTML: https://mlsanity.com/paper/2606.17120/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 24 references that are in the map.
- First posted on arXiv 15 Jun 2026.

## Builds on (most cited first, 15 of 15)

- [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)
- [Grokking: Generalization Beyond Overfitting on Small Algorithmic Datasets](https://mlsanity.com/paper/2201.02177/) (2022)
- [Essentially No Barriers in Neural Network Energy Landscape](https://mlsanity.com/paper/1803.00885/) (2018)
- [Don't Decay the Learning Rate, Increase the Batch Size](https://mlsanity.com/paper/1711.00489/) (2017)
- [Stochastic Gradient Descent as Approximate Bayesian Inference](https://mlsanity.com/paper/1704.04289/) (2017)
- [Omnigrok: Grokking Beyond Algorithmic Data](https://mlsanity.com/paper/2210.01117/) (2022)
- [Dichotomy of Early and Late Phase Implicit Biases Can Provably Induce Grokking](https://mlsanity.com/paper/2311.18817/) (2023)
- [Grokking as a First Order Phase Transition in Two Layer Networks](https://mlsanity.com/paper/2310.03789/) (2023)
- [Power-law escape rate of SGD](https://mlsanity.com/paper/2105.09557/) (2021)
- [Implicit bias of SGD in L2-regularized linear DNNs: One-way jumps from high to low rank](https://mlsanity.com/paper/2305.16038/) (2023)
- [Grokking in Linear Estimators -- A Solvable Model that Groks without Understanding](https://mlsanity.com/paper/2310.16441/) (2023)
- [Provable Scaling Laws of Feature Emergence from Learning Dynamics of Grokking](https://mlsanity.com/paper/2509.21519/) (2025)
- [Is Grokking a Computational Glass Relaxation?](https://mlsanity.com/paper/2505.11411/) (2025)
- [Phase transitions reveal hierarchical structure in deep neural networks](https://mlsanity.com/paper/2512.11866/) (2025)

## Built on by (most cited first, 5 of 5)

- [At-Grok Is Not Converged:A Measurement-Validity Audit for Grokking Representation Metrics](https://mlsanity.com/paper/2607.06639/) (2026)
- [The Orthogonalized Read Is a Removable Training Scaffold for Recurrent Memory](https://mlsanity.com/paper/2607.19390/) (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)
- [Cascading Through the Hierarchy: Regularizer-Induced Feature Detection as Phase Transitions in Deep Linear Neural Networks](https://mlsanity.com/paper/2608.06597/) (2026)
- [Force without transmission: a depth-induced rank collapse that no loss on the representation reopens](https://mlsanity.com/paper/2610.09958/) (2026)

## Abstract (arXiv)

Deep neural networks (DNNs) exhibit first order phase transitions under variations of the L2 regularization strength, with each transition marking the onset of a new learnable feature. Below a critical regularization strength, all features are in principle learnable, but coexisting metastable states, separated by energy barriers, can trap the network and impede convergence. A strength of DNNs is their ability to generalize. But many open questions remain, among them the origin of so called grokking: the abrupt, delayed onset of generalization after prolonged apparent overfitting. We show for linear DNNs that grokking is consistent with hysteresis in first-order L2 phase transitions: using L2 regularization to engineer deliberate trapping, we demonstrate that a model in a low-accuracy metastable state escapes only when SGD noise drives it across an energy barrier, with escape times following Arrhenius scaling. We reproduce grokking-like delayed convergence across two orders of magnitude in escape time by deliberately trapping models in metastable phases. Using sparse sub-sampling we also reproduce the canonical grokking curve where test error eventually approaches the final training error. Our work suggests that the number of metastable states equals the number of learnable features -- one per singular value of the data covariance -- the potential for hysteresis grows naturally with task complexity. We provide evidence that the same mechanism likely operates in general nonlinear DNNs. Our results provide routes toward more efficient learning schemes.