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

Delayed generalization (\ie~grokking) refers to the phenomenon in which a neural network fits its training data early in training but only begins to generalize after a prolonged delay, often through an abrupt transition. Despite extensive empirical study, its underlying mechanism remains poorly understood. In this work, we first theoretically characterize a shell--core topological configuration of the reachable solution space induced by Adam's optimization dynamics with weight-shrinkage regularization, supported by empirical evidence. This optimization-induced topological configuration gives rise to grokking. In model's parameter space, random initialization solutions concentrate on a thin outer spherical shell, enclosing another spherical shell of memorization solutions, which in turn contains a core corresponding to the generalization solutions. Leveraging stopping-time theory, we then analyze the geometry of this topological configuration and the solution transition time at which optimization trajectories escape the memorization manifold and first reach the boundary of the generalization manifold. Our theoretical analysis derives grokking scaling laws for the learning rate, batch size, and $\ell_2$ regularization coefficient, which are further validated through experiments and shown to recover results from prior literature.

Existing accounts of grokking explain the phenomena in terms of mechanistic frameworks such as circuit efficiency or lazy-to-rich transitions. However, despite a known dependence between grokking and model size, how model capacity shapes grokking remains an open question. We give an information-theoretic account of this relationship on the task of modular arithmetic, showing that grokking does not immediately occur when a model becomes large enough to memorise the training set, but rather emerges as the outcome of a competition between two measurable timescales: a memorisation speed $T_{\text{mem}}(P)$ and a generalisation speed $T_{\text{gen}}(P)$, both of which are functions of model parameter count $P$. Adapting the information capacity framework of Morris et al. (2025), we estimate $T_{\text{mem}}(P)$ on random-label data of equivalent complexity and $T_{\text{gen}}(P)$ on the modular task itself, and show that grokking emerges close to the parameter scale where these timescales intersect. The framework also suggests an empirical model for predicting memorisation speed given model capacity and dataset complexity, recovering the previously reported empirical observation that larger models memorise faster. Overall, we motivate the formalisation of different learning timescales as important abstractions to study when explaining how model capacity shapes grokking on algorithmic tasks.

Deep neural networks (DNNs) exhibit first order phase transitions under variations of the L2 regularization strength, with each transition marking the onset of a new learnable feature. Below a critical regularization strength, all features are in principle learnable, but coexisting metastable states, separated by energy barriers, can trap the network and impede convergence. A strength of DNNs is their ability to generalize. But many open questions remain, among them the origin of so called grokking: the abrupt, delayed onset of generalization after prolonged apparent overfitting. We show for linear DNNs that grokking is consistent with hysteresis in first-order L2 phase transitions: using L2 regularization to engineer deliberate trapping, we demonstrate that a model in a low-accuracy metastable state escapes only when SGD noise drives it across an energy barrier, with escape times following Arrhenius scaling. We reproduce grokking-like delayed convergence across two orders of magnitude in escape time by deliberately trapping models in metastable phases. Using sparse sub-sampling we also reproduce the canonical grokking curve where test error eventually approaches the final training error. Our work suggests that the number of metastable states equals the number of learnable features -- one per singular value of the data covariance -- the potential for hysteresis grows naturally with task complexity. We provide evidence that the same mechanism likely operates in general nonlinear DNNs. Our results provide routes toward more efficient learning schemes.