Grokking From Abstraction to Intelligence

Grokking refers to the delayed emergence of validation-set generalization after a model has already overfit the training set. Although first observed in small algorithmic tasks trained with transformers, its underlying mechanism remains unsettled. In this work, we develop an analysis framework based on mode connectivity and the geometry of low-loss regions. The framework predicts that the standard modular-arithmetic setting does not always produce grokking: under a symmetry-preserving train/validation split, we observe a stable anti-grokking case in which validation performance does not recover. This counterexample challenges several existing correlational explanations of grokking. More broadly, our analysis framework and results further suggest that grokking arises when the low-loss regions induced by the training and validation partitions are misaligned. Once these regions become well aligned, training hyperparameters alone cannot produce grokking and the observed dynamics collapse to either trainable or non-trainable behavior.

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.

Neural networks trained past memorization frequently undergo a delayed transition to generalization, a phenomenon known as grokking. Despite theoretical progress on why this transition occurs, the quantitative structure of 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.