How Numerical Precision Affects Arithmetical Reasoning Capabilities of LLMs

TL;DR

This paper investigates how the numerical precision used in large language models affects their ability to perform arithmetic tasks, combining theoretical analysis with empirical experiments to identify key factors influencing mathematical reasoning.

Contribution

It provides a theoretical framework linking numerical precision to arithmetic performance in LLMs and demonstrates the importance of standard precision for efficient reasoning.

Findings

Low numerical precision impairs arithmetic capabilities unless models are super-polynomially large.

Standard numerical precision enables smaller models to perform arithmetic tasks effectively.

Empirical results confirm the theoretical impact of numerical precision on LLMs' mathematical reasoning.

Abstract

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.