How to Tame Grokking: Representation Geometry as a Control Signal

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.

Grokking describes a delayed generalization phenomenon in which a neural network achieves perfect training accuracy long before validation accuracy improves, followed by an abrupt transition to strong generalization. Existing detection signals are indirect: weight norm reflects parameter-space regularization and consistently lags the transition, while GrokFast's slow gradient EMA, used without gradient amplification, is unstable across seeds with standard deviation exceeding mean lead time. We propose the Inter/Intra-class Distance Ratio (ILDR), a geometric metric computed on second-to-last layer representations as the ratio of inter-class centroid separation to intra-class scatter. ILDR provides an early detection signal: it rises and crosses a threshold at 2.5 times its baseline before the grokking transition appears in validation accuracy, indicating early geometric reorganization in representation space. Grounded in Fisher's linear discriminant criterion, ILDR requires no eigendecomposition and runs in O(|C|^2 + N). It is evaluated exclusively on held-out data, making it robust to memorization effects. Across modular arithmetic and permutation group composition (S5), ILDR leads the grokking transition by 9 to 73 percent of the training budget, with lead time increasing with task algebraic complexity. Over eight random seeds, ILDR leads by 950 +/- 250 steps with a coefficient of variation of 26 percent, and post-grokking variance drops by 1696 times, consistent with a sharp phase transition in representation space. Using ILDR as an early stopping trigger reduces training by 18.6 percent on average. Optimizer interventions triggered at the ILDR threshold demonstrate bidirectional control over the transition, suggesting ILDR tracks representational conditions underlying generalization rather than a downstream correlate.

Why do neural networks memorize algorithmic training data long before they generalize? We present a geometric case study demonstrating that, on tasks where generalization requires discovering structured low-dimensional circuits, the memorization-generalization delay is driven by radial inflation of hidden representations under cross-entropy optimization. We formalize a radial-angular decomposition of activation-space dynamics and derive three testable propositions: (i) that penalizing radial inflation induces anisotropic, data-dependent weight regularization; (ii) that it suppresses radial gradient energy below the isotropic random baseline, forcing predominantly angular updates; and (iii) that it biases convergence toward flatter minima. To empirically validate these propositions, we study a single-hyperparameter norm penalty that softly constrains activations to a sqrt(d)-radius hypersphere. On modular arithmetic, this penalty accelerates grokking up to 6x across MLPs and Transformers, and halves training steps for a 10M-parameter nanoGPT on 3-digit addition.