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

2026-04-16 · Source: cs.AI updates on arXiv.org · Field: Technology & Digital — Artificial Intelligence & Machine Learning, Robotics & Autonomous Systems · Depth: Expert, extended

Summary

Large Language Models (LLMs) exhibit significant unpredictability due to numerical instability, particularly when integrated into multi-agent workflows. This instability stems from finite numerical precision in floating-point representations, causing rounding errors to propagate and amplify through Transformer layers. The research identifies a chaotic "avalanche effect" in early layers, where minor perturbations lead to rapid amplification or complete attenuation. LLMs demonstrate universal, scale-dependent chaotic behaviors characterized by three regimes: a stable regime where perturbations vanish, a chaotic regime where rounding errors dominate output divergence, and a signal-dominated regime where true input variations override numerical noise. These findings were validated across Llama-3.1-8B and GPT-OSS-20B architectures using TruthfulQA and AdvBench datasets, and across BFloat16, FP32, and FP64 precisions.

Key takeaway

For NLP engineers and research scientists building or deploying multi-agent LLM systems, understanding numerical instability is crucial for ensuring reproducibility and reliability. Your systems may exhibit unpredictable failures (23-31% reported) even with fixed random seeds due to floating-point errors. Consider implementing noise averaging during inference to stabilize outputs and accurately measure true model sensitivity, especially in safety-critical applications where chaotic decision boundaries can lead to erratic behavior.

Key insights

LLMs exhibit universal, scale-dependent chaotic behaviors driven by floating-point precision and error propagation.

Principles

• Directional sensitivity is scale-driven, not Jacobian spectrum-driven.
• Early-layer "avalanche effects" cause microscopic errors to cascade.
• Increasing precision shifts, but does not eliminate, chaotic regime boundaries.

Method

The study quantifies LLM stability using the absolute directional condition number, $|J(x)v|_{2}$, and analyzes layer-wise error propagation and decision boundary fragmentation across various perturbation scales and singular vector planes.

In practice

• Averaging multiple forward passes with injected noise mitigates instability.
• GPT-OSS-20B shows larger Constant Regions than Llama-3.1-8B.
• Microscopic perturbations can cause erratic decision flips near tie points.

Topics

• Numerical Instability
• Large Language Models
• Floating-Point Precision
• Chaotic Dynamics
• Transformer Architectures

Best for: NLP Engineer, Research Scientist, AI Scientist, Machine Learning Engineer, AI Engineer

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Editorial summary, takeaway, and curation by AIssential. Original article published by cs.AI updates on arXiv.org.