A Training-Time Diagnostic for Generalization via the Log-Alignment Ratio

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: $T_{\mathrm{grok}} \propto H^{-0.27}\, D^{-2.04}\, \eta^{-0.50}\, \lambda^{-0.64}$ ($R^2 = 0.732$; $0.821$ 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 $\lambda \gtrsim 1.0$ 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.

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.

We derive a differential equation that governs the evolution of the generalization gap when a model is trained by gradient descent-based methods. This differential equation is driven by two key quantities, a contraction factor that brings together trajectories corresponding to slightly different datasets, and a perturbation factor that accounts for them training on different datasets. The coupled decay of contraction and perturbation guarantees a controlled accumulation of generalization gap during training. We analyze this differential equation to show that the generalization gap is given by a quadratic form that consists of an ``effective Gram matrix'' that depends upon the training trajectory and a certain residual of the predictor at initialization. Our framework is applicable to general deep networks and smooth loss functions. In numerical experiments on different neural network architectures, datasets and sample sizes, we show that this quadratic form accurately captures the actual generalization gap. We also show how to instantiate our framework in a number of examples via analytical calculations. For example, for high-dimensional linear regression, our framework matches existing calculations of generalization gap in the literature exactly in under-parameterized, over-parameterized and critical regimes.