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Donate arxiv logo > cs > arXiv:2509.21519 [ ] Help | Advanced Search [All fields ] Search arXiv logo Cornell University Logo [ ] GO quick links * Login * Help Pages * About Computer Science > Machine Learning arXiv:2509.21519 (cs) [Submitted on 25 Sep 2025 (v1), last revised 30 Sep 2025 (this version, v3)] Title:Provable Scaling Laws of Feature Emergence from Learning Dynamics of Grokking Authors:Yuandong Tian View a PDF of the paper titled Provable Scaling Laws of Feature Emergence from Learning Dynamics of Grokking, by Yuandong Tian View PDF HTML (experimental) Abstract:While the phenomenon of grokking, i.e., delayed generalization, has been studied extensively, it remains an open problem whether there is a mathematical framework that characterizes what kind of features will emerge, how and in which conditions it happens, and is closely related to the gradient dynamics of the training, for complex structured inputs. We propose a novel framework, named $\mathbf{Li_2}$, that captures three key stages for the grokking behavior of 2-layer nonlinear networks: (I) \underline{\textbf{L}}azy learning, (II) \underline {\textbf{i}}ndependent feature learning and (III) \underline{\ textbf{i}}nteractive feature learning. At the lazy learning stage, top layer overfits to random hidden representation and the model appears to memorize. Thanks to lazy learning and weight decay, the \emph{backpropagated gradient} $G_F$ from the top layer now carries information about the target label, with a specific structure that enables each hidden node to learn their representation \emph{independently}. Interestingly, the independent dynamics follows exactly the \emph{gradient ascent} of an energy function $E$, and its local maxima are precisely the emerging features. We study whether these local-optima induced features are generalizable, their representation power, and how they change on sample size, in group arithmetic tasks. When hidden nodes start to interact in the later stage of learning, we provably show how $G_F$ changes to focus on missing features that need to be learned. Our study sheds lights on roles played by key hyperparameters such as weight decay, learning rate and sample sizes in grokking, leads to provable scaling laws of feature emergence, memorization and generalization, and reveals the underlying cause why recent optimizers such as Muon can be effective, from the first principles of gradient dynamics. Our analysis can be extended to multi-layer architectures. Subjects: Machine Learning (cs.LG); Artificial Intelligence (cs.AI) Cite as: arXiv:2509.21519 [cs.LG] (or arXiv:2509.21519v3 [cs.LG] for this version) https://doi.org/10.48550/arXiv.2509.21519 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Yuandong Tian [view email] [v1] Thu, 25 Sep 2025 20:08:09 UTC (349 KB) [v2] Mon, 29 Sep 2025 17:29:44 UTC (404 KB) [v3] Tue, 30 Sep 2025 17:43:09 UTC (405 KB) Full-text links: Access Paper: View a PDF of the paper titled Provable Scaling Laws of Feature Emergence from Learning Dynamics of Grokking, by Yuandong Tian * View PDF * HTML (experimental) * TeX Source license icon view license Current browse context: cs.LG < prev | next > new | recent | 2025-09 Change to browse by: cs cs.AI References & Citations * NASA ADS * Google Scholar * Semantic Scholar export BibTeX citation Loading... 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