# Beyond Progress Measures: Theoretical Insights into the Mechanism of Grokking

arXiv: https://arxiv.org/abs/2504.03162 · HTML: https://mlsanity.com/paper/2504.03162/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 26 of its 30 references that are in the map.
- First posted on arXiv 4 Apr 2025.

## Builds on (most cited first, 26 of 26)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Progress measures for grokking via mechanistic interpretability](https://mlsanity.com/paper/2301.05217/) (2023)
- [Grokking: Generalization Beyond Overfitting on Small Algorithmic Datasets](https://mlsanity.com/paper/2201.02177/) (2022)
- [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)
- [Complexity of Linear Regions in Deep Networks](https://mlsanity.com/paper/1901.09021/) (2019)
- [The Slingshot Mechanism: An Empirical Study of Adaptive Optimizers and the Grokking Phenomenon](https://mlsanity.com/paper/2206.04817/) (2022)
- [Grokking modular arithmetic](https://mlsanity.com/paper/2301.02679/) (2023)
- [Grokked Transformers are Implicit Reasoners: A Mechanistic Journey to the Edge of Generalization](https://mlsanity.com/paper/2405.15071/) (2024)
- [Dichotomy of Early and Late Phase Implicit Biases Can Provably Induce Grokking](https://mlsanity.com/paper/2311.18817/) (2023)
- [Deep Networks Always Grok and Here is Why](https://mlsanity.com/paper/2402.15555/) (2024)
- [Get rich quick: exact solutions reveal how unbalanced initializations promote rapid feature learning](https://mlsanity.com/paper/2406.06158/) (2024)
- [Grokfast: Accelerated Grokking by Amplifying Slow Gradients](https://mlsanity.com/paper/2405.20233/) (2024)
- [Learning to grok: Emergence of in-context learning and skill composition in modular arithmetic tasks](https://mlsanity.com/paper/2406.02550/) (2024)
- [Why Do You Grok? A Theoretical Analysis of Grokking Modular Addition](https://mlsanity.com/paper/2407.12332/) (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)
- [Critical Data Size of Language Models from a Grokking Perspective](https://mlsanity.com/paper/2401.10463/) (2024)
- [Towards Uncovering How Large Language Model Works: An Explainability Perspective](https://mlsanity.com/paper/2402.10688/) (2024)
- [Unified View of Grokking, Double Descent and Emergent Abilities: A Perspective from Circuits Competition](https://mlsanity.com/paper/2402.15175/) (2024)
- [Information-Theoretic Progress Measures reveal Grokking is an Emergent Phase Transition](https://mlsanity.com/paper/2408.08944/) (2024)
- [Acceleration of Grokking in Learning Arithmetic Operations via Kolmogorov-Arnold Representation](https://mlsanity.com/paper/2405.16658/) (2024)
- [A rationale from frequency perspective for grokking in training neural network](https://mlsanity.com/paper/2405.17479/) (2024)
- [Unveiling the Dynamics of Information Interplay in Supervised Learning](https://mlsanity.com/paper/2406.03999/) (2024)
- [Measuring Sharpness in Grokking](https://mlsanity.com/paper/2402.08946/) (2024)

## Built on by (most cited first, 2 of 2)

- [Intrinsic Task Symmetry Drives Generalization in Algorithmic Tasks](https://mlsanity.com/paper/2603.01968/) (2026)
- [Is Grokking Worthwhile? Functional Analysis and Transferability of Generalization Circuits in Transformers](https://mlsanity.com/paper/2601.09049/) (2026)

## Abstract (arXiv)

Grokking, referring to the abrupt improvement in test accuracy after extended overfitting, offers valuable insights into the mechanisms of model generalization. Existing researches based on progress measures imply that grokking relies on understanding the optimization dynamics when the loss function is dominated solely by the weight decay term. However, we find that this optimization merely leads to token uniformity, which is not a sufficient condition for grokking. In this work, we investigate the grokking mechanism underlying the Transformer in the task of prime number operations. Based on theoretical analysis and experimental validation, we present the following insights: (i) The weight decay term encourages uniformity across all tokens in the embedding space when it is minimized. (ii) The occurrence of grokking is jointly determined by the uniformity of the embedding space and the distribution of the training dataset. Building on these insights, we provide a unified perspective for understanding various previously proposed progress measures and introduce a novel, concise, and effective progress measure that could trace the changes in test loss more accurately. Finally, to demonstrate the versatility of our theoretical framework, we design a dedicated dataset to validate our theory on ResNet-18, successfully showcasing the occurrence of grokking. The code is released at https://github.com/Qihuai27/Grokking-Insight.