Grokking refers to a delayed generalization following overfitting when optimizing artificial neural networks with gradient-based methods. In this work, we demonstrate that grokking can be induced by regularization, either explicit or implicit. More precisely, we show that when there exists a model with a property $P$ (e.g., sparse or low-rank weights) that generalizes on the problem of interest, gradient descent with a small but non-zero regularization of $P$ (e.g., $\ell_1$ or nuclear norm regularization) results in grokking. This extends previous work showing that small non-zero weight decay induces grokking. Moreover, our analysis shows that over-parameterization by adding depth makes it possible to grok or ungrok without explicitly using regularization, which is impossible in shallow cases. We further show that the $\ell_2$ norm is not a reliable proxy for generalization when the model is regularized toward a different property $P$, as the $\ell_2$ norm grows in many cases where no weight decay is used, but the model generalizes anyway. We also show that grokking can be amplified solely through data selection, with any other hyperparameter fixed.

Vector Quantization (VQ) is a method for discretizing latent representations and has become a major part of the deep learning toolkit. It has been theoretically and empirically shown that discretization of representations leads to improved generalization, including in reinforcement learning where discretization can be used to bottleneck multi-agent communication to promote agent specialization and robustness. The discretization tightness of most VQ-based methods is defined by the number of discrete codes in the representation vector and the codebook size, which are fixed as hyperparameters. In this work, we propose learning to dynamically select discretization tightness conditioned on inputs, based on the hypothesis that data naturally contains variations in complexity that call for different levels of representational coarseness which is observed in many heterogeneous data sets. We show that dynamically varying tightness in communication bottlenecks can improve model performance on visual reasoning and reinforcement learning tasks with heterogeneity in representations.