@inproceedings{
zhou2024pretrained,
title={Pre-trained Large Language Models Use Fourier Features to Compute Addition},
author={Tianyi Zhou and Deqing Fu and Vatsal Sharan and Robin Jia},
booktitle={The Thirty-eighth Annual Conference on Neural Information Processing Systems},
year={2024},
url={https://openreview.net/forum?id=i4MutM2TZb}
}
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.
Pre-trained Large Language Models Use Fourier Features to Compute Addition
Tianyi Zhou Deqing Fu Vatsal Sharan Robin Jia