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To Grok Grokking: Provable Grokking in Ridge Regression
[Submitted on 27 Jan 2026 (v1), last revised 16 Jul 2026 (this v · 2026-01-28 · via cs.LG updates on arXiv.org

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Abstract: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 decay. Specifically, we prove that the following stages occur: (i) the model overfits the training data early during training; (ii) poor generalization persists long after overfitting has manifested; and (iii) the generalization error eventually becomes arbitrarily small. Moreover, we show, both theoretically and empirically, that grokking can be amplified or eliminated in a principled manner through proper hyperparameter tuning. To the best of our knowledge, these are the first rigorous quantitative bounds on the generalization delay (which we refer to as the "grokking time") in terms of training hyperparameters. Lastly, going beyond the linear setting, we empirically demonstrate that our quantitative bounds also capture the behavior of grokking on non-linear neural networks. Our results suggest that grokking is not an inherent failure mode of deep learning, but rather a consequence of specific training conditions, and thus does not require fundamental changes to the model architecture or learning algorithm to avoid.

Submission history

From: Mingyue Xu [view email]
[v1] Tue, 27 Jan 2026 16:52:04 UTC (1,520 KB)
[v2] Thu, 5 Feb 2026 19:04:09 UTC (1,521 KB)
[v3] Thu, 28 May 2026 21:17:02 UTC (2,128 KB)
[v4] Thu, 16 Jul 2026 15:51:18 UTC (2,128 KB)

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