Quantifying the Memorization-to-Generalization Transition: Scaling Laws and Phase Structure in Grokking

论文概要

研究领域: ML 作者: Anish Kataria 发布时间: 2026-09-11 arXiv: 2509.05824

中文摘要

过拟合记忆后训练的神经网络常经历向泛化的延迟过渡，这一现象称为grokking。尽管对其发生原因已有理论进展，但该过渡在超参数空间中何时发生的定量结构仍未被刻画。我们在模运算上针对384种双层MLP配置绘制记忆-泛化边界，拟合泛化 onset 时间的幂律缩放关系：\(T_{\mathrm{grok}} \propto H^{-0.27} D^{-2.04} \eta^{-0.50} \lambda^{-0.64}\) (\(R^2=0.732\)；含交互项$0.821$)。指数层级揭示数据复杂度(\(D^{-2.04}\))是状态转换的主导驱动，而非模型容量(\(H^{-0.27}\))：数据翻倍加速泛化约4倍，而宽度翻倍仅约1.2倍。权重衰减 \(\lambda \gtrsim 1.0\) 处存在尖锐相边界，分隔grokking与非grokking配置，权重范数轨迹显示过渡期间单调压缩，与隐式正则化选择低复杂度解一致。这些结果为预测和控制过参数化网络中的状态转换提供定量基础。

原文摘要

Neural networks trained past memorization frequently undergo a delayed transition to generalization, a phenomenon known as grokking. Despite theoretical progress on \emph{why} this transition occurs, the quantitative structure of \emph{when} it occurs in hyperparameter space remains uncharacterized. We map the memorization-to-generalization boundary across 384 configurations of two-hidden-layer MLPs on modular arithmetic, fitting a power-law scaling relation for generalization onset time: \(T_{\mathrm{grok}} \propto H^{-0.27}\, D^{-2.04}\, \eta^{-0.50}\, \lambda^{-0.64}\) (\(R^2 = 0.732\); $0.821$ with interactions). The exponent hierarchy reveals that data复杂度 (\(D^{-2.04}\)) is the dominant driver of regime transition, not model capacity (\(H^{-0.27}\)): doubling data accelerates generalization b...

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