Numerical Instability and Chaos: Quantifying the Unpredictability of Large Language Models

2026-04-14

AI Summary

This study addresses the unpredictable outputs of large language models in agent workflows, which often stem from numerical instabilities whose underlying mechanisms remain poorly understood. By tracing the propagation and amplification of floating-point rounding errors across Transformer layers, this work reveals for the first time a chaotic “avalanche effect” originating in early layers. The authors identify three universal dynamical regimes—stable, chaotic, and signal-dominated—that commonly govern large models’ behavior. Through an integrated approach combining numerical error tracking, chaos dynamics analysis, and cross-architecture experiments, they validate this classification across diverse models and datasets. The findings clarify the relative contributions of numerical noise and input signals to output divergence, offering a novel perspective for understanding and controlling the behavior of large language models.

Technology Category

• Natural Language Processing: (Large) Language Models
• Machine Learning: Large Multimodal Models (LMMs)
• Planning, Routing, and Scheduling: Planning with Language Models

Application Category

• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• Social Networks and Social Media: Generative AI / large language models and their impact on social systems
• User Modeling, Personalization and Recommendation: Large Language Models (LLM) for user modeling and recommendation

Abstract

As Large Language Models (LLMs) are increasingly integrated into agentic workflows, their unpredictability stemming from numerical instability has emerged as a critical reliability issue. While recent studies have demonstrated the significant downstream effects of these instabilities, the root causes and underlying mechanisms remain poorly understood. In this paper, we present a rigorous analysis of how unpredictability is rooted in the finite numerical precision of floating-point representations, tracking how rounding errors propagate, amplify, or dissipate through Transformer computation layers. Specifically, we identify a chaotic "avalanche effect" in the early layers, where minor perturbations trigger binary outcomes: either rapid amplification or complete attenuation. Beyond specific error instances, we demonstrate that LLMs exhibit universal, scale-dependent chaotic behaviors characterized by three distinct regimes: 1) a stable regime, where perturbations fall below an input-dependent threshold and vanish, resulting in constant outputs; 2) a chaotic regime, where rounding errors dominate and drive output divergence; and 3) a signal-dominated regime, where true input variations override numerical noise. We validate these findings extensively across multiple datasets and model architectures.

Problem

• numerical instability
• large language models
• chaos
• unpredictability
• floating-point precision

Innovation

• numerical instability
• chaos theory
• floating-point precision
• Transformer dynamics
• LLM unpredictability