Acceleration of grokking in learning arithmetic operations via Kolmogorov–Arnold representation

We propose novel methodologies aimed at accelerating the grokking phenomenon, which refers to the rapid increment of test accuracy after a long period of overfitting as reported by Power et al. (2022). Focusing on the grokking phenomenon that arises in learning arithmetic operations via the transformer model, we begin with a discussion on data augmentation in the case of commutative arithmetic operations. To further accelerate, we elucidate arithmetic operations through the lens of the Kolmogorov–Arnold (KA) representation theorem, revealing its correspondence to the transformer architecture: embedding, decoder block, and classifier. Observing the shared structure between KA representations associated with arithmetic operations, we suggest various transfer learning mechanisms that expedite grokking. This interpretation is substantiated through a series of rigorous experiments. Our study demonstrates that a simple data augmentation makes the grokking phenomenon nearly two times faster, while the transfer approach speeds it up by almost five times. In addition, our approach is successful in learning two nonstandard arithmetic tasks: composition of operations and solving a system of equations. Furthermore, we reveal that the model is capable of learning arithmetic operations using a limited number of tokens under embedding transfer, which is supported by a set of experiments as well.