Repeated examples help learn arithmetic

We study small transformers trained on two problems of arithmetic: the greatest common divisor (GCD) and modular multiplication, and show that models trained on a limited set of repeated examples achieve better performance than models trained from unlimited data. In fact, modular multiplication is only learned on small training sets. We also demonstrate that two-set training - repeated use of a small random subset of examples, along normal sampling on the rest of the training set - provides for faster learning and better performance.These experiments highlight that the benefits of repetition can outweigh those of data diversity; and shed light on the still poorly understood interplay between generalization and memorization in deep learning.