Towards Empirical Interpretation of Internal Circuits and Properties in Grokked Transformers on Modular Polynomials

Grokking has been actively explored to reveal the mystery of delayed generalization and identifying interpretable representations and algorithms inside the grokked models is a suggestive hint to understanding its mechanism. Grokking on modular addition has been known to implement Fourier representation and its calculation circuits with trigonometric identities in Transformers. Considering the periodicity in modular arithmetic, the natural question is to what extent these explanations and interpretations hold for the grokking on other modular operations beyond addition. For a closer look, we first hypothesize that any modular operations can be characterized with distinctive Fourier representation or internal circuits, grokked models obtain common features transferable among similar operations, and mixing datasets with similar operations promotes grokking. Then, we extensively examine them by learning Transformers on complex modular arithmetic tasks, including polynomials. Our Fourier analysis and novel progress measure for modular arithmetic, Fourier Frequency Density and Fourier Coefficient Ratio, characterize distinctive internal representations of grokked models per modular operation; for instance, polynomials often result in the superposition of the Fourier components seen in elementary arithmetic, but clear patterns do not emerge in challenging non-factorizable polynomials. In contrast, our ablation study on the pre-grokked models reveals that the transferability among the models grokked with each operation can be only limited to specific combinations, such as from elementary arithmetic to linear expressions. Moreover, some multi-task mixtures may lead to co-grokking -- where grokking simultaneously happens for all the tasks -- and accelerate generalization, while others may not find optimal solutions. We provide empirical steps towards the interpretability of internal circuits.