Acceleration of Grokking in Learning Arithmetic Operations via Kolmogorov-Arnold Representation

TL;DR

This paper introduces methods to accelerate the grokking phenomenon in learning arithmetic with transformers by leveraging Kolmogorov-Arnold representations and transfer learning, achieving faster convergence and success on complex tasks.

Contribution

It presents a novel interpretation of arithmetic learning via Kolmogorov-Arnold representations and proposes transfer learning techniques to speed up grokking in transformer models.

Findings

Accelerated grokking using Kolmogorov-Arnold representations.

Successful learning of nonstandard arithmetic tasks.

Effective transfer learning with limited tokens.

Abstract

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 in~\cite{power2022grokking}. Focusing on the grokking phenomenon that arises in learning arithmetic binary operations via the transformer model, we begin with a discussion on data augmentation in the case of commutative binary 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 binary operations, we suggest various transfer learning mechanisms that expedite grokking. This interpretation is substantiated through a series of rigorous experiments. In addition, our approach is successful in learning two nonstandard arithmetic tasks: composition of operations and 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.