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Universal One-third Time Scaling in Learning Peaked Distributions

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arxiv 2602.03685 v2 pith:NPBXIYI3 submitted 2026-02-03 cs.LG cs.AIstat.ML

Universal One-third Time Scaling in Learning Peaked Distributions

classification cs.LG cs.AIstat.ML
keywords distributionspower-lawscalinglearningllmslossmodelspeaked
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved
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Training large language models (LLMs) is computationally expensive, partly because the loss exhibits slow power-law convergence whose origin remains debatable. Through systematic analysis of toy models and empirical evaluation of LLMs, we show that this behavior can arise intrinsically from the use of softmax and cross-entropy. When learning peaked probability distributions, e.g., next-token distributions, these components generically yield power-law vanishing losses and gradients, regardless of many microscopic details, creating a fundamental optimization bottleneck. This ultimately leads to power-law time scaling of the loss with a universal exponent of $1/3$. Our results provide a mechanistic explanation for observed neural scaling and suggest new directions for improving LLM training efficiency.

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Cited by 4 Pith papers

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