Pre-trained Large Language Models Use Fourier Features to Compute Addition
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866914825490137088 |
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| author | Zhou, Tianyi Fu, Deqing Sharan, Vatsal Jia, Robin |
| author_facet | Zhou, Tianyi Fu, Deqing Sharan, Vatsal Jia, Robin |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_03445 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Pre-trained Large Language Models Use Fourier Features to Compute Addition Zhou, Tianyi Fu, Deqing Sharan, Vatsal Jia, Robin Machine Learning Computation and Language 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. |
| title | Pre-trained Large Language Models Use Fourier Features to Compute Addition |
| topic | Machine Learning Computation and Language |
| url | https://arxiv.org/abs/2406.03445 |