Grokking, or delayed generalization, is an intriguing learning phenomenon where a model's test set performance improves sharply but only long after its training performance has converged. This challenges conventional understanding of training dynamics in deep learning. In this paper, we formalize and investigate grokking, highlighting that a distribution shift between training and test data is a key factor in its emergence. We introduce two novel synthetic datasets specifically designed to systematically induce and analyze the phenomenon. By controlling the imbalance in the sampling of data sub-categories, we can reliably reproduce grokking, demonstrating that while small sample sizes are associated with grokking, they are primarily only a mechanism for achieving the necessary distribution shift. We show that when classes possess a relational structure, such as in an equivariant mapping, the model can leverage this information to generalize to entirely unseen subclasses. To explore the limits of this phenomenon, we extend our analysis to the MNIST dataset. Our findings indicate that while distribution shifts can systematically cause grokking in structured synthetic settings, grokking may not emerge as readily in more complex real-world scenarios, suggesting that grokking may depend on the interplay between data distribution and the dataset's intrinsic structure.

Recent work on neuro-symbolic inductive logic programming has led to promising approaches that can learn explanatory rules from noisy, real-world data. While some proposals approximate logical operators with differentiable operators from fuzzy or real-valued logic that are parameter-free thus diminishing their capacity to fit the data, other approaches are only loosely based on logic making it difficult to interpret the learned ``rules". In this paper, we propose learning rules with the recently proposed logical neural networks (LNN). Compared to others, LNNs offer a strong connection to classical Boolean logic thus allowing for precise interpretation of learned rules while harboring parameters that can be trained with gradient-based optimization to effectively fit the data. We extend LNNs to induce rules in first-order logic. Our experiments on standard benchmarking tasks confirm that LNN rules are highly interpretable and can achieve comparable or higher accuracy due to their flexible parameterization.

Prithviraj Sen, Breno William Carvalho, Ibrahim Abdelaziz, Pavan Kapanipathi, Salim Roukos, Alexander Gray. Proceedings of the 2022 Conference on Empirical Methods in Natural Language Processing. 2022.