Pre-trained Large Language Models Use Fourier Features to Compute Addition
<|reference_start|>Pre-trained Large Language Models Use Fourier Features to Compute Addition: 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.<|reference_end|>
@article{zhou2024pre-trained,
    title={Pre-trained Large Language Models Use Fourier Features to Compute
  Addition},
    author={Tianyi Zhou, Deqing Fu, Vatsal Sharan, Robin Jia},
    journal={arXiv preprint arXiv:2406.03445},
    year={2024},
    archivePrefix={arXiv},
    eprint={2406.03445},
    primaryClass={cs.LG cs.CL}
}