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

量化从记忆到泛化的转变：Grokking 中的缩放定律与相结构

Abstract: 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.

摘要： 神经网络在经过记忆阶段后的训练中，经常会经历一个延迟的泛化转变，这一现象被称为“Grokking”（顿悟）。尽管在理论上已经解释了这一转变“为何”发生，但关于它在超参数空间中“何时”发生的定量结构仍未得到充分表征。

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).

我们通过模运算任务，对 384 种双隐藏层多层感知机（MLP）配置进行了映射，确定了从记忆到泛化的边界，并拟合出了泛化起始时间的幂律缩放关系：$T_{\mathrm{grok}} \propto H^{-0.27}, D^{-2.04}, \eta^{-0.50}, \lambda^{-0.64}$（$R^2 = 0.732$；考虑交互作用后为 $0.821$）。

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 ${\sim}1.2\times$.

指数层级显示，数据复杂度（$D^{-2.04}$）是驱动机制转变的主导因素，而非模型容量（$H^{-0.27}$）：数据量翻倍可使泛化速度加快约 4 倍，而网络宽度翻倍仅能带来约 1.2 倍的提升。

A sharp phase boundary at weight decay $\lambda \gtrsim 1.0$ 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.

在权重衰减 $\lambda \gtrsim 1.0$ 处存在一个明显的相边界，将 Grokking 配置与非 Grokking 配置区分开来。权重范数的轨迹在转变过程中表现出单调压缩，这与隐式正则化倾向于选择低复杂度解的理论相一致。这些结果为预测和控制过参数化网络中的机制转变提供了定量基础。