How Numerical Precision Affects Arithmetical Reasoning Capabilities of LLMs
This work addresses the challenge of enhancing mathematical capabilities in LLMs, which is crucial for applications requiring reliable numerical reasoning, though it is incremental as it builds on existing theoretical analyses of Transformers.
The paper tackles the problem of limited mathematical reasoning in large language models by identifying numerical precision as a key factor affecting their performance on arithmetic tasks like iterated addition and integer multiplication. The results show that low-precision Transformers require super-polynomial model size growth to handle these tasks, while standard-precision ones do so efficiently with smaller sizes.
Despite the remarkable success of Transformer-based large language models (LLMs) across various domains, understanding and enhancing their mathematical capabilities remains a significant challenge. In this paper, we conduct a rigorous theoretical analysis of LLMs' mathematical abilities, with a specific focus on their arithmetic performances. We identify numerical precision as a key factor that influences their effectiveness in arithmetical tasks. Our results show that Transformers operating with low numerical precision fail to address arithmetic tasks, such as iterated addition and integer multiplication, unless the model size grows super-polynomially with respect to the input length. In contrast, Transformers with standard numerical precision can efficiently handle these tasks with significantly smaller model sizes. We further support our theoretical findings through empirical experiments that explore the impact of varying numerical precision on arithmetic tasks, providing valuable insights for improving the mathematical reasoning capabilities of LLMs.