This paper investigates how pre-trained large language models (LLMs) perform integer addition. It finds that LLMs, despite lacking explicit training on arithmetic, learn to leverage positional encoding based on Fourier features to represent numbers internally. This allows them to achieve surprisingly good accuracy on addition tasks, particularly within the range of numbers present in their training data. The authors demonstrate this by analyzing attention patterns and comparing LLM performance with models using alternative positional encodings. They also show how manipulating or ablating these Fourier features directly impacts the models' ability to add, strongly suggesting that LLMs have implicitly learned a form of Fourier-based arithmetic.
The preprint "Pre-Trained Large Language Models Use Fourier Features for Addition (2024)" by Michael Petrov, Hritik Bansal, and Micah Goldblum delves into the inner workings of pre-trained large language models (LLMs) and how they perform arithmetic operations, specifically focusing on addition. The authors hypothesize that LLMs leverage a mechanism similar to Fourier features, commonly used in signal processing and computer graphics, to represent and manipulate numerical information. This hypothesis stems from the observation that LLMs exhibit wave-like oscillatory behavior in their activation patterns when processing numbers.
The research centers around analyzing the activations within LLMs, which are the internal representations of information as the model processes data. By probing these activations, the authors attempt to decode the internal mechanisms the model employs. They introduce a novel probing method specifically designed to detect the presence of Fourier features within the activations. This method involves fitting linear models to the activations and examining the frequency components present in these linear models. The presence of specific, predictable frequencies would suggest the utilization of a Fourier-like mechanism.
Their experimental results across several popular LLMs, including Llama-2, GPT-NeoX, and Pythia, provide compelling evidence supporting their hypothesis. They demonstrate that the activations within these models, particularly in layers associated with numerical processing, indeed exhibit patterns consistent with the use of Fourier features. Furthermore, the observed frequencies within these activations correlate with the numerical values being processed, indicating a direct link between the Fourier-like representation and the actual arithmetic operations.
The paper also explores the potential implications of these findings. The authors suggest that this Fourier-based representation might explain certain limitations observed in LLMs when dealing with large numbers or complex arithmetic tasks. The inherent periodicity of Fourier features might introduce ambiguities or inaccuracies when representing numbers outside a certain range or performing operations that require high precision. Understanding these limitations could pave the way for developing more robust and accurate LLMs for numerical reasoning.
Finally, the study touches upon the broader significance of these discoveries within the context of understanding how LLMs represent and process information. The emergence of Fourier-like features, a concept borrowed from signal processing, suggests that LLMs might be developing internal representations that are surprisingly analogous to methods used in other fields. This unexpected connection could provide valuable insights into the underlying principles governing the learning and representation capabilities of these powerful models. The findings contribute to the ongoing effort to unravel the “black box” nature of LLMs and move towards a deeper understanding of their internal workings.
Summary of Comments ( 15 )
https://news.ycombinator.com/item?id=42960989
Hacker News users discussed the surprising finding that LLMs appear to use Fourier features internally to perform addition, as indicated by the linked paper. Several commenters expressed fascination with this emergent behavior, highlighting how LLMs discover and utilize mathematical concepts without explicit instruction. Some questioned the paper's methodology and the strength of its conclusions, suggesting alternative explanations or calling for further research to solidify the claims. A few users also discussed the broader implications of this discovery for understanding how LLMs function and how they might be improved. The potential link to the Fourier-based positional encoding used in Transformer models was also noted as a possible contributing factor.
The Hacker News post titled "Pre-Trained Large Language Models Use Fourier Features for Addition (2024)" linking to the arXiv paper has generated a moderate amount of discussion with a few interesting threads.
Several commenters focus on the implications of LLMs appearing to use Fourier transforms for addition. One commenter expresses surprise, stating they wouldn't have guessed this mechanism and questioning if it's a learned behavior or an emergent property of the architecture. This sparks further discussion about whether this behavior is specifically trained or a consequence of the training data's statistical properties. Some suggest it could be related to the positional encoding mechanisms already employed in transformer models, which use sinusoidal functions. Another commenter wonders if this Fourier-based approach to addition might offer advantages in terms of computational efficiency or generalization.
Another thread delves into the limitations of the research. One commenter points out that the paper focuses specifically on addition and questions whether similar mechanisms are used for other arithmetic operations. They suggest investigating multiplication next. Another commenter questions the significance of the findings, arguing that demonstrating LLMs use Fourier transforms for addition doesn't necessarily reveal anything profound about their understanding of arithmetic. They argue it could simply be a pattern-matching technique that happens to be effective for addition.
There's also a discussion about the interpretability of LLMs. One commenter expresses hope that research like this will eventually lead to a better understanding of how LLMs function internally. Another, however, is more skeptical, suggesting that even if we can identify specific mechanisms like the use of Fourier transforms, it might not provide a satisfying explanation of the overall emergent behavior of these complex models.
Finally, a few comments offer tangential observations. One commenter notes the increasing prevalence of papers analyzing the internal workings of LLMs, highlighting the growing interest in this area of research. Another points out the connection to older research on neural networks and their ability to approximate functions, suggesting this work builds upon those foundations.