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This paper studies emergent phenomena in neural networks by focusing on grokking where models suddenly generalize after delayed memorization. To understand this phase transition, we utilize higher-order mutual information to analyze the…

Machine Learning · Computer Science 2024-08-20 Kenzo Clauw , Sebastiano Stramaglia , Daniele Marinazzo

In the realm of fractal geometry, intricate structures emerge from simple iterative processes that partition parameter spaces into regions of stability and instability. Likewise, training large language models involves iteratively applying…

Machine Learning · Computer Science 2025-02-18 Bahman Torkamandi

Understanding how biological constraints shape neural computation is a central goal of computational neuroscience. Spatially embedded recurrent neural networks provide a promising avenue to study how modelled constraints shape the combined…

Neural and Evolutionary Computing · Computer Science 2024-09-27 Cornelia Sheeran , Andrew S. Ham , Duncan E. Astle , Jascha Achterberg , Danyal Akarca

Training modern neural networks often relies on large learning rates, operating at the edge of stability, where the optimization dynamics exhibit oscillatory and chaotic behavior. Empirically, this regime often yields improved…

Machine Learning · Computer Science 2026-04-22 Mario Tuci , Caner Korkmaz , Umut Şimşekli , Tolga Birdal

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

Grokking -- the sudden generalisation that appears long after a model has perfectly memorised its training data -- has been widely observed but lacks a quantitative theory explaining the length of the delay. We show that grokking is a…

Artificial Intelligence · Computer Science 2026-05-05 Truong Xuan Khanh , Truong Quynh Hoa , Luu Duc Trung , Phan Thanh Duc

Grokking, the sudden generalization that occurs after prolonged overfitting, is a surprising phenomenon challenging our understanding of deep learning. Although significant progress has been made in understanding grokking, the reasons…

Machine Learning · Computer Science 2025-05-20 Lucas Prieto , Melih Barsbey , Pedro A. M. Mediano , Tolga Birdal

Understanding the dynamics of neural networks in different width regimes is crucial for improving their training and performance. We present an exact solution for the learning dynamics of a one-hidden-layer linear network, with…

Machine Learning · Computer Science 2025-02-24 Yizhou Xu , Liu Ziyin

Neural networks sometimes exhibit grokking, a phenomenon where perfect or near-perfect performance is achieved on a validation set well after the same performance has been obtained on the corresponding training set. In this workshop paper,…

Machine Learning · Computer Science 2024-02-15 Jack Miller , Patrick Gleeson , Charles O'Neill , Thang Bui , Noam Levi

Recent works have demonstrated that the sample complexity of gradient-based learning of single index models, i.e. functions that depend on a 1-dimensional projection of the input data, is governed by their information exponent. However,…

Machine Learning · Statistics 2023-09-08 Alireza Mousavi-Hosseini , Denny Wu , Taiji Suzuki , Murat A. Erdogdu

Grokking is a phenomenon where a model trained on an algorithmic task first overfits but, then, after a large amount of additional training, undergoes a phase transition to generalize perfectly. We empirically study the internal structure…

Machine Learning · Computer Science 2023-03-22 William Merrill , Nikolaos Tsilivis , Aman Shukla

Recent work by Power et al. (2022) highlighted a surprising "grokking" phenomenon in learning arithmetic tasks: a neural net first "memorizes" the training set, resulting in perfect training accuracy but near-random test accuracy, and after…

Machine Learning · Computer Science 2024-04-03 Kaifeng Lyu , Jikai Jin , Zhiyuan Li , Simon S. Du , Jason D. Lee , Wei Hu

We propose that learning in deep neural networks proceeds in two phases: a rapid curve fitting phase followed by a slower compression or coarse graining phase. This view is supported by the shared temporal structure of three phenomena:…

High Energy Physics - Theory · Physics 2025-07-28 Robert de Mello Koch , Animik Ghosh

A principled understanding of generalization in deep learning may require unifying disparate observations under a single conceptual framework. Previous work has studied \emph{grokking}, a training dynamic in which a sustained period of…

Machine Learning · Computer Science 2023-03-14 Xander Davies , Lauro Langosco , David Krueger

Despite their ability to represent highly expressive functions, deep learning models seem to find simple solutions that generalize surprisingly well. Spectral bias -- the tendency of neural networks to prioritize learning low frequency…

Machine Learning · Computer Science 2022-09-30 Sara Fridovich-Keil , Raphael Gontijo-Lopes , Rebecca Roelofs

Speculative decoding accelerates LLM inference by drafting a tree of candidate continuations and verifying it in one target forward. Existing drafters fall into two camps with opposite weaknesses. Autoregressive drafters such as EAGLE-3…

Computation and Language · Computer Science 2026-05-22 Weijie Shi , Qiang Xu , Fan Deng , Yaguang Wu , Jiarun Liu , Yehong Xu , Hao Chen , Jia Zhu , Jiajie Xu , Xiangjun Huang , Jian Yang , Xiaofang Zhou

''Grokking'' is a phenomenon where a neural network first memorizes training data and generalizes poorly, but then suddenly transitions to near-perfect generalization after prolonged training. While intriguing, this delayed generalization…

Machine Learning · Computer Science 2025-04-21 Zhiwei Xu , Zhiyu Ni , Yixin Wang , Wei Hu

One puzzling artifact in machine learning dubbed grokking is where delayed generalization is achieved tenfolds of iterations after near perfect overfitting to the training data. Focusing on the long delay itself on behalf of machine…

Machine Learning · Computer Science 2024-06-06 Jaerin Lee , Bong Gyun Kang , Kihoon Kim , Kyoung Mu Lee

Learning-based edge detection has hereunto been strongly supervised with pixel-wise annotations which are tedious to obtain manually. We study the problem of self-training edge detection, leveraging the untapped wealth of large-scale…

Computer Vision and Pattern Recognition · Computer Science 2023-07-06 Yunfan Ye , Renjiao Yi , Zhiping Cai , Kai Xu

Grokking-the phenomenon where validation accuracy of neural networks on modular addition of two integers rises long after training data has been memorized-has been characterized in previous works as producing sinusoidal input weight…

Machine Learning · Computer Science 2026-03-26 Anand Swaroop