Clustering and Alignment: Understanding the Training Dynamics in Modular Addition

Recent studies have revealed that neural networks learn interpretable algorithms for many simple problems. However, little is known about how these algorithms emerge during training. In this article, we study the training dynamics of a simplified transformer with 2-dimensional embeddings on the problem of modular addition. We observe that embedding vectors tend to organize into two types of structures: grids and circles. We study these structures and explain their emergence as a result of two simple tendencies exhibited by pairs of embeddings: clustering and alignment. We propose explicit formulae for these tendencies as interaction forces between different pairs of embeddings. To show that our formulae can fully account for the emergence of these structures, we construct an equivalent particle simulation where we find that identical structures emerge. We use our insights to discuss the role of weight decay and reveal a new mechanism that links regularization and training dynamics. We also release an interactive demo to support our findings: https://modular-addition.vercel.app/.