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Neural networks trained to solve modular arithmetic tasks exhibit grokking, a phenomenon where the test accuracy starts improving long after the model achieves 100% training accuracy in the training process. It is often taken as an example…

We present a theoretical explanation of the ``grokking'' phenomenon, where a model generalizes long after overfitting,for the originally-studied problem of modular addition. First, we show that early in gradient descent, when the ``kernel…

Machine Learning · Computer Science 2024-07-18 Mohamad Amin Mohamadi , Zhiyuan Li , Lei Wu , Danica J. Sutherland

Training Neural Networks (NNs) without overfitting is difficult; detecting that overfitting is difficult as well. We present a novel Random Matrix Theory method that detects the onset of overfitting in deep learning models without access to…

Machine Learning · Computer Science 2026-05-15 Hari K. Prakash , Charles H Martin

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…

Machine Learning · Statistics 2024-04-12 Tanishq Kumar , Blake Bordelon , Samuel J. Gershman , Cengiz Pehlevan

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…

Machine Learning · Computer Science 2025-04-15 Zihan Gu , Ruoyu Chen , Hua Zhang , Yue Hu , Xiaochun Cao

We attribute grokking, the phenomenon where generalization is much delayed after memorization, to compression. To do so, we define linear mapping number (LMN) to measure network complexity, which is a generalized version of linear region…

Machine Learning · Computer Science 2023-10-10 Ziming Liu , Ziqian Zhong , Max Tegmark

Predictive Coding Networks (PCNs) aim to learn a generative model of the world. Given observations, this generative model can then be inverted to infer the causes of those observations. However, when training PCNs, a noticeable pathology is…

Artificial Intelligence · Computer Science 2022-09-02 Paul F Kinghorn , Beren Millidge , Christopher L Buckley

We study the well-known grokking phenomena in neural networks (NNs) using a 3-layer MLP trained on 1 k-sample subset of MNIST, with and without weight decay, and discover a novel third phase -- \emph{anti-grokking} -- that occurs very late…

Machine Learning · Computer Science 2025-06-06 Hari K. Prakash , Charles H. Martin

Grokking, the unusual phenomenon for algorithmic datasets where generalization happens long after overfitting the training data, has remained elusive. We aim to understand grokking by analyzing the loss landscapes of neural networks,…

Machine Learning · Computer Science 2023-03-24 Ziming Liu , Eric J. Michaud , Max Tegmark

Understanding neural network's (NN) generalizability remains a central question in deep learning research. The special phenomenon of grokking, where NNs abruptly generalize long after the training performance reaches a near-perfect level,…

Machine Learning · Computer Science 2026-01-06 Xiaotian Zhang , Yue Shang , Entao Yang , Ge Zhang

Neural collapse, i.e., the emergence of highly symmetric, class-wise clustered representations, is frequently observed in deep networks and is often assumed to reflect or enable generalization. In parallel, flatness of the loss landscape…

Machine Learning · Computer Science 2026-02-05 Ting Han , Linara Adilova , Henning Petzka , Jens Kleesiek , Michael Kamp

Grokking is a puzzling phenomenon in neural networks where full generalization occurs only after a substantial delay following the complete memorization of the training data. Previous research has linked this delayed generalization to…

Machine Learning · Computer Science 2026-01-12 Tiberiu Musat

We investigate the phenomenon of grokking -- delayed generalization accompanied by non-monotonic test loss behavior -- in a simple binary logistic classification task, for which "memorizing" and "generalizing" solutions can be strictly…

Machine Learning · Statistics 2025-07-22 Alon Beck , Noam Levi , Yohai Bar-Sinai

Grokking in modular arithmetic has established itself as the quintessential fruit fly experiment, serving as a critical domain for investigating the mechanistic origins of model generalization. Despite its significance, existing research…

Artificial Intelligence · Computer Science 2026-04-01 Junjie Zhang , Zhen Shen , Gang Xiong , Xisong Dong

Grokking -- the abrupt transition from memorization to generalization after prolonged training -- has been linked to confinement on low-dimensional execution manifolds in modular arithmetic. Whether this mechanism extends beyond arithmetic…

Machine Learning · Computer Science 2026-04-06 Yongzhong Xu

This paper proposes to study neural networks through neuronal correlation, a statistical measure of correlated neuronal activity on the penultimate layer. We show that neuronal correlation can be efficiently estimated via weight matrix, can…

Machine Learning · Computer Science 2022-01-25 Gaojie Jin , Xinping Yi , Xiaowei Huang

Grokking is the phenomenon where neural networks NNs initially fit the training data and later generalize to the test data during training. In this paper, we empirically provide a frequency perspective to explain the emergence of this…

Machine Learning · Computer Science 2024-05-29 Zhangchen Zhou , Yaoyu Zhang , Zhi-Qin John Xu

This paper investigates the grokking phenomenon, which refers to the sudden transition from a long memorization to generalization observed during neural networks training, in the context of learning multiplication in finite-dimensional…

Machine Learning · Computer Science 2026-05-15 Pascal Jr Tikeng Notsawo , Guillaume Dumas , Guillaume Rabusseau

We study grokking, the onset of generalization long after overfitting, in a classical ridge regression setting. We prove end-to-end grokking results for learning over-parameterized linear regression models using gradient descent with weight…

Machine Learning · Computer Science 2026-02-09 Mingyue Xu , Gal Vardi , Itay Safran

Maintaining the ability to fire sparsely is crucial for information encoding in neural networks. Additionally, spiking homeostasis is vital for spiking neural networks with changing numbers of weights and neurons. We discuss a range of…

Neural and Evolutionary Computing · Computer Science 2019-10-02 Katarzyna Kozdon , Peter Bentley