A Theoretical Analysis of Generalization Dynamics in Neural Networks under Gradient Descent with Weight Decay

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.

Delayed generalization, or grokking, remains poorly understood despite extensive empirical study. We identify an exactly solvable late-time relaxation mechanism for grokking in linear models trained with full-batch heavy-ball optimization and weight decay, together with a locally quadratic extension to nonlinear neural networks. Our analysis reveals a distinguished population-active component of the empirical null space, which we call the grokking subspace. Along this subspace, the training predictions remain unchanged, leaving weight decay as the sole restoring force and giving rise to a slow dissipative relaxation governed by an exact discrete-time and continuous-time law. We show that only this subspace contributes to the slow asymptotic decay of the population risk and derive explicit iteration-scale predictions for the grokking time, recovering the familiar $(1-β)/(ηλ)$ scaling in the weak-regularization regime. The theory further predicts distinct effects of optimizer choice, distinguishing coupled $L_2$ regularization from decoupled weight decay, and yields causal predictions for interventions that modify the grokking component. We verify all theoretical identities without fitted parameters in a synthetic model where every subspace and relaxation rate is computable in closed form. We further observe genuine delayed generalization in modular addition, where the measured delay follows the predicted scaling and the late-time relaxation agrees closely with the theoretical clock.

In grokking an early fit to the training data separates from a much later improvement in generalization. During this delay, training can move from a fixed neural tangent kernel (NTK) regime to one in which task-relevant kernel eigendirections continue to evolve. We provide a quantitative theory for how this transition from lazy to rich learning can produce delayed generalization. For homogeneous networks trained with squared loss and $L_2$ weight decay, we show that a finite residual remains after memorization, with larger residual fractions in target components associated with smaller NTK eigenvalues. These residuals feed back into the dynamics of the NTK itself, and projecting the resulting dynamics onto task-relevant spectral directions yields a reduced system in which residual-driven kernel growth competes with weight decay. This system predicts that the grokking timescale is controlled by the product of learning rate and weight decay, that feature learning slows logarithmically near a critical decay above which task-aligned NTK structure can no longer support generalization, and that stronger decay can prevent fitting altogether. We test these predictions in modular addition. In a homogeneous MLP, task-aligned Fourier structure continues to emerge in the NTK after training accuracy has saturated, and an 84$\times$90-grid of trained networks across varying learning rate and weight decay recovers the predicted phase geometry and inverse-product scaling of the generalization time with learning rate and weight decay. A one-block Transformer shows similar macroscopic phase structure in a 42$\times$45-grid, as well as the same transition-time scaling despite violating exact homogeneity. Together, these results provide a mechanistic derivation connecting post-fit feature learning to both the onset of generalization and its phase structure in the learning rate and weight decay plane.