# Explaining grokking through circuit efficiency

arXiv: https://arxiv.org/abs/2309.02390 · HTML: https://mlsanity.com/paper/2309.02390/

## Key facts

- Cited by 111 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 45 references that are in the map.
- First posted on arXiv 5 Sep 2023.

## Builds on (most cited first, 24 of 24)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Scaling Laws for Neural Language Models](https://mlsanity.com/paper/2001.08361/) (2020)
- [Transformer Feed-Forward Layers Are Key-Value Memories](https://mlsanity.com/paper/2012.14913/) (2020)
- [Interpretability in the Wild: a Circuit for Indirect Object Identification in GPT-2 small](https://mlsanity.com/paper/2211.00593/) (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)
- [The Implicit Bias of Gradient Descent on Separable Data](https://mlsanity.com/paper/1710.10345/) (2017)
- [Grokking: Generalization Beyond Overfitting on Small Algorithmic Datasets](https://mlsanity.com/paper/2201.02177/) (2022)
- [Deep Learning Scaling is Predictable, Empirically](https://mlsanity.com/paper/1712.00409/) (2017)
- [Emergent World Representations: Exploring a Sequence Model Trained on a Synthetic Task](https://mlsanity.com/paper/2210.13382/) (2022)
- [Gradient Descent Maximizes the Margin of Homogeneous Neural Networks](https://mlsanity.com/paper/1906.05890/) (2019)
- [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)
- [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 Quantization Model of Neural Scaling](https://mlsanity.com/paper/2303.13506/) (2023)
- [A Bayesian Perspective on Generalization and Stochastic Gradient Descent](https://mlsanity.com/paper/1710.06451/) (2017)
- [Three Mechanisms of Weight Decay Regularization](https://mlsanity.com/paper/1810.12281/) (2018)
- [The Slingshot Mechanism: An Empirical Study of Adaptive Optimizers and the Grokking Phenomenon](https://mlsanity.com/paper/2206.04817/) (2022)
- [A Tale of Two Circuits: Grokking as Competition of Sparse and Dense Subnetworks](https://mlsanity.com/paper/2303.11873/) (2023)
- [Unifying Grokking and Double Descent](https://mlsanity.com/paper/2303.06173/) (2023)
- [The Implicit Bias for Adaptive Optimization Algorithms on Homogeneous Neural Networks](https://mlsanity.com/paper/2012.06244/) (2020)
- [Predicting Grokking Long Before it Happens: A look into the loss landscape of models which grok](https://mlsanity.com/paper/2306.13253/) (2023)

## Built on by (most cited first, 30 of 111)

- [Mechanistic Interpretability for AI Safety -- A Review](https://mlsanity.com/paper/2404.14082/) (2024)
- [Towards Best Practices of Activation Patching in Language Models: Metrics and Methods](https://mlsanity.com/paper/2309.16042/) (2023)
- [Emergent Misalignment: Narrow finetuning can produce broadly misaligned LLMs](https://mlsanity.com/paper/2502.17424/) (2025)
- [A Practical Review of Mechanistic Interpretability for Transformer-Based Language Models](https://mlsanity.com/paper/2407.02646/) (2024)
- [Grokking as the Transition from Lazy to Rich Training Dynamics](https://mlsanity.com/paper/2310.06110/) (2023)
- [Arithmetic Without Algorithms: Language Models Solve Math With a Bag of Heuristics](https://mlsanity.com/paper/2410.21272/) (2024)
- [A Primer on the Inner Workings of Transformer-based Language Models](https://mlsanity.com/paper/2405.00208/) (2024)
- [Dichotomy of Early and Late Phase Implicit Biases Can Provably Induce Grokking](https://mlsanity.com/paper/2311.18817/) (2023)
- [Grokked Transformers are Implicit Reasoners: A Mechanistic Journey to the Edge of Generalization](https://mlsanity.com/paper/2405.15071/) (2024)
- [All Roads Lead to Likelihood: The Value of Reinforcement Learning in Fine-Tuning](https://mlsanity.com/paper/2503.01067/) (2025)
- [Deep Networks Always Grok and Here is Why](https://mlsanity.com/paper/2402.15555/) (2024)
- [Grokking as a First Order Phase Transition in Two Layer Networks](https://mlsanity.com/paper/2310.03789/) (2023)
- [Grokking at the Edge of Numerical Stability](https://mlsanity.com/paper/2501.04697/) (2025)
- [Why Do You Grok? A Theoretical Analysis of Grokking Modular Addition](https://mlsanity.com/paper/2407.12332/) (2024)
- [Benign Overfitting and Grokking in ReLU Networks for XOR Cluster Data](https://mlsanity.com/paper/2310.02541/) (2023)
- [Predicting Emergent Abilities with Infinite Resolution Evaluation](https://mlsanity.com/paper/2310.03262/) (2023)
- [Emergence in non-neural models: grokking modular arithmetic via average gradient outer product](https://mlsanity.com/paper/2407.20199/) (2024)
- [The Same But Different: Structural Similarities and Differences in Multilingual Language Modeling](https://mlsanity.com/paper/2410.09223/) (2024)
- [Grokking Group Multiplication with Cosets](https://mlsanity.com/paper/2312.06581/) (2023)
- [Critical Data Size of Language Models from a Grokking Perspective](https://mlsanity.com/paper/2401.10463/) (2024)
- [Learning Syntax Without Planting Trees: Understanding Hierarchical Generalization in Transformers](https://mlsanity.com/paper/2404.16367/) (2024)
- [Unified View of Grokking, Double Descent and Emergent Abilities: A Perspective from Circuits Competition](https://mlsanity.com/paper/2402.15175/) (2024)
- [Emergent Misalignment is Easy, Narrow Misalignment is Hard](https://mlsanity.com/paper/2602.07852/) (2026)
- [Questioning Representational Optimism in Deep Learning: The Fractured Entangled Representation Hypothesis](https://mlsanity.com/paper/2505.11581/) (2025)
- [σ-GPTs: A New Approach to Autoregressive Models](https://mlsanity.com/paper/2404.09562/) (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)
- [Grokking as Compression: A Nonlinear Complexity Perspective](https://mlsanity.com/paper/2310.05918/) (2023)
- [Towards Empirical Interpretation of Internal Circuits and Properties in Grokked Transformers on Modular Polynomials](https://mlsanity.com/paper/2402.16726/) (2024)
- [Let Me Grok for You: Accelerating Grokking via Embedding Transfer from a Weaker Model](https://mlsanity.com/paper/2504.13292/) (2025)

## Abstract (arXiv)

One of the most surprising puzzles in neural network generalisation is grokking: a network with perfect training accuracy but poor generalisation will, upon further training, transition to perfect generalisation. We propose that grokking occurs when the task admits a generalising solution and a memorising solution, where the generalising solution is slower to learn but more efficient, producing larger logits with the same parameter norm. We hypothesise that memorising circuits become more inefficient with larger training datasets while generalising circuits do not, suggesting there is a critical dataset size at which memorisation and generalisation are equally efficient. We make and confirm four novel predictions about grokking, providing significant evidence in favour of our explanation. Most strikingly, we demonstrate two novel and surprising behaviours: ungrokking, in which a network regresses from perfect to low test accuracy, and semi-grokking, in which a network shows delayed generalisation to partial rather than perfect test accuracy.