The Weight Norm Sets the Grokking Timescale: A Causal Delay Law
A new paper resolves the debate over whether weight norm causes grokking in neural networks by directly intervening on the norm during training. Under free training, grokking occurs when the norm reaches a critical value Wc that is highly consistent across seeds and learning rates. When the norm is clamped, delay follows an exponential law T_grok ∝ exp(αρ) with α≈7.5. LayerNorm eliminates this dependence.
[2606.13753] The Weight Norm Sets the Grokking Timescale: A Causal Delay Law
[Submitted on 11 Jun 2026]
Title:The Weight Norm Sets the Grokking Timescale: A Causal Delay Law
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Abstract:Grokking is the delayed onset of generalization in neural networks, arising long after they fit the training data. Whether the weight norm causes this delay is disputed: some studies report a critical norm at the transition, others observe grokking with no fixed norm at all. We settle this by intervening on the norm during training rather than only observing it. Under free training with weight decay, networks grok when the weight norm reaches a value Wc that varies little across seeds and learning rates (CV 1 to 2 percent) and grows with the modular base as a power law. When we instead clamp the norm to a fixed multiple rho of Wc and hold it there, the network still groks, but the delay follows T_grok proportional to exp(alpha rho). One exponent, alpha near 7.5, fits this delay across four moduli (R^2 = 0.996). Over the swept ranges the held norm moves the delay by about 19x and the learning rate by only about 2x, and holding the norm above Wc slows grokking rather than preventing it. A final LayerNorm removes the dependence by decoupling weight scale from the network function; without it the exponential law returns. This pinned-norm delay is the exponential counterpart to the logarithmic delay predicted for a freely contracting norm.
Comments: 14 papges, 9 figs and 3 tables
Subjects:
Machine Learning (cs.LG); Artificial Intelligence (cs.AI)
Cite as: arXiv:2606.13753 [cs.LG]
(or arXiv:2606.13753v1 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2606.13753
arXiv-issued DOI via DataCite
Submission history
From: Xuan Khanh Truong [view email] [v1] Thu, 11 Jun 2026 15:36:10 UTC (582 KB)
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