Pre-trained Language Models Learn Remarkably Accurate Representations of Numbers (2506.08966v1)

Abstract: Pretrained LLMs (LMs) are prone to arithmetic errors. Existing work showed limited success in probing numeric values from models' representations, indicating that these errors can be attributed to the inherent unreliability of distributionally learned embeddings in representing exact quantities. However, we observe that previous probing methods are inadequate for the emergent structure of learned number embeddings with sinusoidal patterns. In response, we propose a novel probing technique that decodes numeric values from input embeddings with near-perfect accuracy across a range of open-source LMs. This proves that after the sole pre-training, LMs represent numbers with remarkable precision. Finally, we find that the embeddings' preciseness judged by our probe's accuracy explains a large portion of LM's errors in elementary arithmetic, and show that aligning the embeddings with the pattern discovered by our probe can mitigate these errors.

An Investigation into Numeric Representations within Pre-trained LLMs

The paper "Pre-trained LLMs Learn Remarkably Accurate Representations of Numbers" by Marek Kadlčík et al. explores the numerical embedding capabilities of pre-trained LLMs (LMs), focusing on their ability to accurately encode and retrieve numeric values. This paper provides significant insights into the underlying structure of number embeddings in open-source LLMs, challenging previous assumptions about their inherent imprecision.

Background and Motivation

Historically, pre-trained LLMs have exhibited limitations in handling arithmetic tasks, often attributed to the imprecise representations of numeric values derived from distributional embeddings. Conventional probing techniques have only achieved limited success in interpreting these numeric embeddings from LMs, thus suggesting a need for more refined methodologies.

Methodology and Approach

The authors propose a novel probing technique designed to decode numeric embeddings with near-perfect accuracy, capitalizing on the sinusoidal patterns inherent in the learned embeddings. This approach is applied across various models, including Llama 3, Phi 4, and OLMo 2, with sizes ranging from 1 billion to 72 billion parameters.

To demonstrate the efficacy of their method, four different probe architectures were utilized, including linear, logarithmic-linear, sinusoidal, and binary encoding schemes. Each probe's accuracy was assessed through a cross-validation setup, ensuring robustness and generalizability of the findings.

Key Findings

1. Sinusoidal Basis in Numeric Representations: The models demonstrated a strong sinusoidal pattern in their number embeddings, particularly when observed through PCA reductions. The sinusoidal probe consistently outperformed other architectures, achieving nearly perfect retrieval accuracy across tested models, thus challenging prior beliefs regarding linear encodings in LMs.
2. Impact on Arithmetic Tasks: The research showed that errors in arithmetic tasks, such as addition and subtraction, could often be traced back to issues in numeric representation. The probe's accuracy effectively explained a considerable portion of these errors, suggesting that aligning number embeddings with the discovered sinusoidal pattern could enhance arithmetic reasoning and reduce errors.
3. Model-Specific Variations: Interestingly, the paper highlighted OLMo 2 32B as an anomaly, where embeddings deviated from the sinusoidal pattern despite high arithmetic task success rates. This discovery calls for further investigation into model-specific embedding strategies and their impact on computational tasks.

Implications and Future Directions

The paper's findings have both theoretical and practical implications. Theoretically, they offer a refined understanding of how pre-trained LMs encode numeric information, revealing a hidden, more precise structure than previously assumed. This knowledge can guide the development of more accurate and efficient probing techniques for understanding neural model internals.

Practically, aligning number embeddings with recognized sinusoidal patterns can enhance LMs' arithmetic capabilities, offering a promising direction for improving numeric reasoning in AI applications. Future research could expand on these findings by exploring more complex numerical tasks and extending this approach to different model architectures and training regimes.

Conclusion

The research conducted by Kadlčík et al. significantly enhances our understanding of numeric representations in LLMs, demonstrating their remarkable precision when accurately probed. By leveraging sinusoidal-based probing strategies, the paper provides a robust framework for decoding and improving numeric embeddings, potentially transforming arithmetic reasoning capabilities in LLMs moving forward.

This exploration into numeric representation emphasizes the importance of accurately identifying and utilizing embedded structures within LMs, setting a benchmark for future interpretability studies in the domain of AI and numerical reasoning.