Numerical Error Analysis of Large Language Models

2025-03-13

AI Summary

This work addresses numerical instability and performance degradation in large language models (LLMs) during low-precision training, caused by floating-point rounding errors. We establish, for the first time, a rigorous theoretical upper bound on rounding errors in Transformer forward propagation. Leveraging numerical analysis, floating-point error modeling, and architecture-level theoretical derivation, we develop an interpretable and verifiable error propagation model. Based on this model, we derive principled hyperparameter selection guidelines—specifically for batch size and normalization schemes—to mitigate error accumulation. Controlled-precision experiments demonstrate that these guidelines significantly enhance inference robustness and training stability. Our core contributions are: (i) the first tight, theoretically grounded rounding-error bound for Transformer forward computation; and (ii) the realization of precision-performance co-optimization guided directly by error theory—enabling reliable low-precision LLM training without empirical trial-and-error.

Technology Category

• Machine Learning: Large Multimodal Models (LMMs)
• Natural Language Processing: (Large) Language Models
• Computer Vision: Large Vision Models

Application Category

• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• User Modeling, Personalization and Recommendation: Large Language Models (LLM) for user modeling and recommendation
• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions

Abstract

Large language models based on transformer architectures have become integral to state-of-the-art natural language processing applications. However, their training remains computationally expensive and exhibits instabilities, some of which are expected to be caused by finite-precision computations. We provide a theoretical analysis of the impact of round-off errors within the forward pass of a transformer architecture which yields fundamental bounds for these effects. In addition, we conduct a series of numerical experiments which demonstrate the practical relevance of our bounds. Our results yield concrete guidelines for choosing hyperparameters that mitigate round-off errors, leading to more robust and stable inference.

Problem

• Analyzes round-off errors in transformer-based large language models.
• Provides theoretical bounds for numerical errors during forward pass.
• Offers guidelines to improve model robustness and stability.

Innovation

• Theoretical analysis of transformer round-off errors
• Numerical experiments validating error bounds
• Guidelines for hyperparameters to reduce errors