发表机构
Princeton University（普林斯顿大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究通过384种配置的模算术MLP实验，量化了grokking中记忆化到泛化转变的标度律，发现数据复杂度是主导因素，并识别出权重衰减的尖锐相边界，为预测和调控过参数化网络的状态转变提供了定量基础。
AI 中文摘要
经过记忆化训练的神经网络经常经历一个延迟的泛化转变，这一现象被称为grokking。尽管关于这一转变为何发生的理论已有进展，但其在超参数空间中何时发生的定量结构仍未得到表征。我们绘制了在模算术上384种两层隐藏层MLP配置下的记忆化到泛化边界，拟合了泛化起始时间的幂律标度关系：$T_{\mathrm{grok}} \propto H^{-0.27}\\, D^{-2.04}\\, \eta^{-0.50}\\, \lambda^{-0.64}$（$R^2 = 0.732$；含交互项时为$0.821$）。指数层级揭示数据复杂度（$D^{-2.04}$）是状态转变的主要驱动因素，而非模型容量（$H^{-0.27}$）：数据量加倍使泛化加速约$4\times$，而宽度加倍仅带来约$1.2\times$的加速。在权重衰减$\lambda \gtrsim 1.0$处存在一个尖锐的相边界，将grokking配置与非grokking配置分开，且权重范数轨迹在转变期间表现出单调压缩，这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变提供了定量基础。
英文摘要
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: $T_{\mathrm{grok}} \propto H^{-0.27}\, D^{-2.04}\, η^{-0.50}\, λ^{-0.64}$ ($R^2 = 0.732$; $0.821$ with interactions). The exponent hierarchy reveals that data complexity ($D^{-2.04}$) is the dominant driver of regime transition, not model capacity ($H^{-0.27}$): doubling data accelerates generalization by ${\sim}4\times$, while doubling width yields only ${\sim}1.2\times$. A sharp phase boundary at weight decay $λ\gtrsim 1.0$ separates grokking from non-grokking configurations, and weight norm trajectories show monotonic compression during the transition, consistent with implicit regularization selecting low-complexity solutions. These results provide a quantitative foundation for predicting and controlling regime transitions in overparameterized networks.
