# Grokking as the Transition from Lazy to Rich Training Dynamics

arXiv: https://arxiv.org/abs/2310.06110 · HTML: https://mlsanity.com/paper/2310.06110/

## Key facts

- Cited by 108 papers in the map (data through 14 Oct 2026).
- Builds on 38 of its 45 references that are in the map.
- First posted on arXiv 9 Oct 2023.

## Builds on (most cited first, 30 of 38)

- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [Similarity of Neural Network Representations Revisited](https://mlsanity.com/paper/1905.00414/) (2019)
- [Progress measures for grokking via mechanistic interpretability](https://mlsanity.com/paper/2301.05217/) (2023)
- [Wide Neural Networks of Any Depth Evolve as Linear Models Under Gradient Descent](https://mlsanity.com/paper/1902.06720/) (2019)
- [On Lazy Training in Differentiable Programming](https://mlsanity.com/paper/1812.07956/) (2018)
- [Grokking: Generalization Beyond Overfitting on Small Algorithmic Datasets](https://mlsanity.com/paper/2201.02177/) (2022)
- [Fine-Grained Analysis of Optimization and Generalization for Overparameterized Two-Layer Neural Networks](https://mlsanity.com/paper/1901.08584/) (2019)
- [Are Emergent Abilities of Large Language Models a Mirage?](https://mlsanity.com/paper/2304.15004/) (2023)
- [A Mean Field View of the Landscape of Two-Layers Neural Networks](https://mlsanity.com/paper/1804.06561/) (2018)
- [Gradient Descent on Neural Networks Typically Occurs at the Edge of Stability](https://mlsanity.com/paper/2103.00065/) (2021)
- [Towards Understanding Grokking: An Effective Theory of Representation Learning](https://mlsanity.com/paper/2205.10343/) (2022)
- [Spectrum Dependent Learning Curves in Kernel Regression and Wide Neural Networks](https://mlsanity.com/paper/2002.02561/) (2020)
- [Hidden Progress in Deep Learning: SGD Learns Parities Near the Computational Limit](https://mlsanity.com/paper/2207.08799/) (2022)
- [Spectral Bias and Task-Model Alignment Explain Generalization in Kernel Regression and Infinitely Wide Neural Networks](https://mlsanity.com/paper/2006.13198/) (2020)
- [Linearized two-layers neural networks in high dimension](https://mlsanity.com/paper/1904.12191/) (2019)
- [High-dimensional Asymptotics of Feature Learning: How One Gradient Step Improves the Representation](https://mlsanity.com/paper/2205.01445/) (2022)
- [Omnigrok: Grokking Beyond Algorithmic Data](https://mlsanity.com/paper/2210.01117/) (2022)
- [Online stochastic gradient descent on non-convex losses from high-dimensional inference](https://mlsanity.com/paper/2003.10409/) (2020)
- [Explaining grokking through circuit efficiency](https://mlsanity.com/paper/2309.02390/) (2023)
- [The Slingshot Mechanism: An Empirical Study of Adaptive Optimizers and the Grokking Phenomenon](https://mlsanity.com/paper/2206.04817/) (2022)
- [Learning Single-Index Models with Shallow Neural Networks](https://mlsanity.com/paper/2210.15651/) (2022)
- [Limitations of Lazy Training of Two-layers Neural Networks](https://mlsanity.com/paper/1906.08899/) (2019)
- [Grokking modular arithmetic](https://mlsanity.com/paper/2301.02679/) (2023)
- [Classifying high-dimensional Gaussian mixtures: Where kernel methods fail and neural networks succeed](https://mlsanity.com/paper/2102.11742/) (2021)
- [Smoothing the Landscape Boosts the Signal for SGD: Optimal Sample Complexity for Learning Single Index Models](https://mlsanity.com/paper/2305.10633/) (2023)
- [Dynamical mean-field theory for stochastic gradient descent in Gaussian mixture classification](https://mlsanity.com/paper/2006.06098/) (2020)
- [Learning time-scales in two-layers neural networks](https://mlsanity.com/paper/2303.00055/) (2023)
- [Optimization and Generalization of Shallow Neural Networks with Quadratic Activation Functions](https://mlsanity.com/paper/2006.15459/) (2020)
- [Unifying Grokking and Double Descent](https://mlsanity.com/paper/2303.06173/) (2023)
- [Spectrum of inner-product kernel matrices in the polynomial regime and multiple descent phenomenon in kernel ridge regression](https://mlsanity.com/paper/2204.10425/) (2022)

## Built on by (most cited first, 30 of 108)

- [The Evolution of Statistical Induction Heads: In-Context Learning Markov Chains](https://mlsanity.com/paper/2402.11004/) (2024)
- [Dichotomy of Early and Late Phase Implicit Biases Can Provably Induce Grokking](https://mlsanity.com/paper/2311.18817/) (2023)
- [Grokking at the Edge of Numerical Stability](https://mlsanity.com/paper/2501.04697/) (2025)
- [Get rich quick: exact solutions reveal how unbalanced initializations promote rapid feature learning](https://mlsanity.com/paper/2406.06158/) (2024)
- [Why Do You Grok? A Theoretical Analysis of Grokking Modular Addition](https://mlsanity.com/paper/2407.12332/) (2024)
- [Emergence of Hidden Capabilities: Exploring Learning Dynamics in Concept Space](https://mlsanity.com/paper/2406.19370/) (2024)
- [Approaching Deep Learning through the Spectral Dynamics of Weights](https://mlsanity.com/paper/2408.11804/) (2024)
- [Emergence in non-neural models: grokking modular arithmetic via average gradient outer product](https://mlsanity.com/paper/2407.20199/) (2024)
- [Grokking Group Multiplication with Cosets](https://mlsanity.com/paper/2312.06581/) (2023)
- [Towards Uncovering How Large Language Model Works: An Explainability Perspective](https://mlsanity.com/paper/2402.10688/) (2024)
- [Information-Theoretic Progress Measures reveal Grokking is an Emergent Phase Transition](https://mlsanity.com/paper/2408.08944/) (2024)
- [On Catastrophic Inheritance of Large Foundation Models](https://mlsanity.com/paper/2402.01909/) (2024)
- [Provable Scaling Laws of Feature Emergence from Learning Dynamics of Grokking](https://mlsanity.com/paper/2509.21519/) (2025)
- [Towards Empirical Interpretation of Internal Circuits and Properties in Grokked Transformers on Modular Polynomials](https://mlsanity.com/paper/2402.16726/) (2024)
- [Sanity Checks for Sparse Autoencoders: Do SAEs Beat Random Baselines?](https://mlsanity.com/paper/2602.14111/) (2026)
- [The Optimization Landscape of SGD Across the Feature Learning Strength](https://mlsanity.com/paper/2410.04642/) (2024)
- [Do We Always Need the Simplicity Bias? Looking for Optimal Inductive Biases in the Wild](https://mlsanity.com/paper/2503.10065/) (2025)
- [Let Me Grok for You: Accelerating Grokking via Embedding Transfer from a Weaker Model](https://mlsanity.com/paper/2504.13292/) (2025)
- [A Theoretical Framework for Grokking: Interpolation followed by Riemannian Norm Minimisation](https://mlsanity.com/paper/2505.20172/) (2025)
- [On Regularization via Early Stopping for Least Squares Regression](https://mlsanity.com/paper/2406.04425/) (2024)
- [Low-Dimensional and Transversely Curved Optimization Dynamics in Grokking](https://mlsanity.com/paper/2602.16746/) (2026)
- [Why Larger Models Learn More: Effects of Capacity, Interference, and Rare-Task Retention](https://mlsanity.com/paper/2605.29548/) (2026)
- [Deep Learning Through A Telescoping Lens: A Simple Model Provides Empirical Insights On Grokking, Gradient Boosting & Beyond](https://mlsanity.com/paper/2411.00247/) (2024)
- [Position: Solve Layerwise Linear Models First to Understand Neural Dynamical Phenomena (Neural Collapse, Emergence, Lazy/Rich Regime, and Grokking)](https://mlsanity.com/paper/2502.21009/) (2025)
- [Feature Learning beyond the Lazy-Rich Dichotomy: Insights from Representational Geometry](https://mlsanity.com/paper/2503.18114/) (2025)
- [The Geometry of Multi-Task Grokking: Transverse Instability, Superposition, and Weight Decay Phase Structure](https://mlsanity.com/paper/2602.18523/) (2026)
- [Is Grokking a Computational Glass Relaxation?](https://mlsanity.com/paper/2505.11411/) (2025)
- [What Can Grokking Teach Us About Learning Under Nonstationarity?](https://mlsanity.com/paper/2507.20057/) (2025)
- [Evolution of Concepts in Language Model Pre-Training](https://mlsanity.com/paper/2509.17196/) (2025)
- [Flatness is Necessary, Neural Collapse is Not: Rethinking Generalization via Grokking](https://mlsanity.com/paper/2509.17738/) (2025)

## Abstract (arXiv)

We propose that the grokking phenomenon, where the train loss of a neural network decreases much earlier than its test loss, can arise due to a neural network transitioning from lazy training dynamics to a rich, feature learning regime. To illustrate this mechanism, we study the simple setting of vanilla gradient descent on a polynomial regression problem with a two layer neural network which exhibits grokking without regularization in a way that cannot be explained by existing theories. We identify sufficient statistics for the test loss of such a network, and tracking these over training reveals that grokking arises in this setting when the network first attempts to fit a kernel regression solution with its initial features, followed by late-time feature learning where a generalizing solution is identified after train loss is already low. We find that the key determinants of grokking are the rate of feature learning -- which can be controlled precisely by parameters that scale the network output -- and the alignment of the initial features with the target function y(x). We argue this delayed generalization arises when (1) the top eigenvectors of the initial neural tangent kernel and the task labels y(x) are misaligned, but (2) the dataset size is large enough so that it is possible for the network to generalize eventually, but not so large that train loss perfectly tracks test loss at all epochs, and (3) the network begins training in the lazy regime so does not learn features immediately. We conclude with evidence that this transition from lazy (linear model) to rich training (feature learning) can control grokking in more general settings, like on MNIST, one-layer Transformers, and student-teacher networks.