Pre-trained Large Language Models Use Fourier Features to Compute Addition

Tianyi Zhou, Deqing Fu, Vatsal Sharan, Robin Jia

Keywords: large language models, arithmetic task, Fourier analysis

Abstract:

Pre-trained large language models (LLMs) exhibit impressive mathematical reasoning capabilities, yet how they compute basic arithmetic, such as addition, remains unclear.
This paper shows that pre-trained LLMs add numbers using Fourier features---dimensions in the hidden state that represent numbers via a set of features sparse in the frequency domain.
Within the model, MLP and attention layers use Fourier features in complementary ways: MLP layers primarily approximate the magnitude of the answer using low-frequency features, while attention layers primarily perform modular addition (e.g., computing whether the answer is even or odd) using high-frequency features.
Pre-training is crucial for this mechanism: models trained from scratch to add numbers only exploit low-frequency features, leading to lower accuracy.
Introducing pre-trained token embeddings to a randomly initialized model rescues its performance.
Overall, our analysis demonstrates that appropriate pre-trained representations (e.g., Fourier features) can unlock the ability of Transformers to learn precise mechanisms for algorithmic tasks.

Primary Area: Interpretability and explainability

Submission Number: 1350