# Acceleration of Grokking in Learning Arithmetic Operations via Kolmogorov-Arnold Representation

arXiv: https://arxiv.org/abs/2405.16658 · HTML: https://mlsanity.com/paper/2405.16658/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 37 references that are in the map.
- First posted on arXiv 26 May 2024.

## Builds on (most cited first, 20 of 20)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Deep Sets](https://mlsanity.com/paper/1703.06114/) (2017)
- [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)
- [The Clock and the Pizza: Two Stories in Mechanistic Explanation of Neural Networks](https://mlsanity.com/paper/2306.17844/) (2023)
- [Omnigrok: Grokking Beyond Algorithmic Data](https://mlsanity.com/paper/2210.01117/) (2022)
- [A Toy Model of Universality: Reverse Engineering How Networks Learn Group Operations](https://mlsanity.com/paper/2302.03025/) (2023)
- [The Slingshot Mechanism: An Empirical Study of Adaptive Optimizers and the Grokking Phenomenon](https://mlsanity.com/paper/2206.04817/) (2022)
- [The Kolmogorov-Arnold representation theorem revisited](https://mlsanity.com/paper/2007.15884/) (2020)
- [Dichotomy of Early and Late Phase Implicit Biases Can Provably Induce Grokking](https://mlsanity.com/paper/2311.18817/) (2023)
- [Neural Arithmetic Logic Units](https://mlsanity.com/paper/1808.00508/) (2018)
- [Deep Networks Always Grok and Here is Why](https://mlsanity.com/paper/2402.15555/) (2024)
- [Learning the greatest common divisor: explaining transformer predictions](https://mlsanity.com/paper/2308.15594/) (2023)
- [The Goldilocks zone: Towards better understanding of neural network loss landscapes](https://mlsanity.com/paper/1807.02581/) (2018)
- [On the Kolmogorov neural networks](https://mlsanity.com/paper/2311.00049/) (2023)
- [Towards Empirical Interpretation of Internal Circuits and Properties in Grokked Transformers on Modular Polynomials](https://mlsanity.com/paper/2402.16726/) (2024)
- [Understanding Grokking Through A Robustness Viewpoint](https://mlsanity.com/paper/2311.06597/) (2023)
- [Visual Learning of Arithmetic Operations](https://mlsanity.com/paper/1506.02264/) (2015)

## Built on by (most cited first, 3 of 3)

- [Beyond Progress Measures: Theoretical Insights into the Mechanism of Grokking](https://mlsanity.com/paper/2504.03162/) (2025)
- [Is Grokking Worthwhile? Functional Analysis and Transferability of Generalization Circuits in Transformers](https://mlsanity.com/paper/2601.09049/) (2026)
- [Fast Generalization after Interpolation via Critically Damped Momentum Optimization](https://mlsanity.com/paper/2606.01521/) (2026)

## Abstract (arXiv)

We propose novel methodologies aimed at accelerating the grokking phenomenon, which refers to the rapid increment of test accuracy after a long period of overfitting as reported in~citepower2022grokking. Focusing on the grokking phenomenon that arises in learning arithmetic binary operations via the transformer model, we begin with a discussion on data augmentation in the case of commutative binary operations. To further accelerate, we elucidate arithmetic operations through the lens of the Kolmogorov-Arnold (KA) representation theorem, revealing its correspondence to the transformer architecture: embedding, decoder block, and classifier. Observing the shared structure between KA representations associated with binary operations, we suggest various transfer learning mechanisms that expedite grokking. This interpretation is substantiated through a series of rigorous experiments. In addition, our approach is successful in learning two nonstandard arithmetic tasks: composition of operations and a system of equations. Furthermore, we reveal that the model is capable of learning arithmetic operations using a limited number of tokens under embedding transfer, which is supported by a set of experiments as well.