Numerical Instability and Chaos: Quantifying the Unpredictability of Large Language Models
Chashi Mahiul Islam, Alan Villarreal, Mao Nishino, Shaeke Salman, Xiuwen Liu
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.
AI Impact Assessments
(3 models)Scientific Impact Assessment
Core Contribution
This paper investigates the numerical instability of Large Language Models (LLMs) arising from floating-point arithmetic, framing it through the lens of chaotic dynamics. The central claim is that LLMs exhibit three distinct operating regimes with respect to perturbation magnitude: (1) a stable/constant regime where perturbations below a threshold produce bitwise-identical outputs, (2) a chaotic regime where floating-point rounding errors dominate and cause erratic output changes, and (3) a signal-dominated regime where genuine input variations override numerical noise. The paper introduces the concept of "directional absolute condition numbers" to quantify sensitivity, identifies an "avalanche effect" in early transformer layers, and proposes noise averaging as a mitigation strategy.
Methodological Rigor
The methodology has both strengths and notable concerns:
Strengths: The use of directional condition numbers rather than worst-case spectral norms is well-motivated for high-dimensional systems. The layer-wise propagation analysis (Figures 2-3) effectively illustrates how perturbation behavior differs across scales. The decision boundary visualization (Figure 5) is compelling, showing salt-and-pepper fragmentation near logit ties. The ULP-precise binary search for stability boundaries (Section IV-D) demonstrates careful numerical methodology.
Concerns: The experimental scope is limited — only two models (Llama-3.1-8B and GPT-OSS-20B) with 100 and 10 prompts respectively. The GPT-OSS-20B experiments run on CPU rather than GPU, introducing a confound since CPU and GPU floating-point behavior differ substantially — the very point the paper makes about hardware heterogeneity. The paper claims "universal" phenomena but validates on a narrow set of configurations.
The construction of "near-tie scenarios" (Section IV-D) is somewhat artificial. By deliberately engineering inputs where the top two logits are nearly equal, the authors create maximally unstable conditions. While this demonstrates existence of chaotic boundaries, it doesn't establish how frequently natural inputs encounter such regions. The logit margins reported in Table I (0.5-0.98) suggest that typical inputs have comfortable margins, which would place them in the stable regime most of the time.
The mitigation via noise averaging (Section IV-F) is presented as novel but is essentially Monte Carlo smoothing — a well-established technique. The convergence to the theoretical singular value with n=100 samples is interesting but the practical implications are unclear, since averaging forward passes multiplies inference cost.
Potential Impact
The paper addresses a genuinely important problem. As LLMs are deployed in multi-agent systems and safety-critical applications, understanding the boundaries of reproducibility is essential. The three-regime characterization provides a useful conceptual framework.
However, the practical impact may be limited by several factors:
Timeliness & Relevance
The timing is appropriate. Multi-agent LLM systems are proliferating, and reproducibility concerns are increasingly recognized. The concurrent work by Yuan et al. (2025) at NeurIPS addresses related issues from a more engineering-focused perspective, while this paper attempts a more theoretical characterization. The framing through chaos theory and condition numbers is timely given growing interest in LLM reliability.
Strengths
1. Novel analytical framework: Applying directional condition number analysis to LLM inference at the floating-point level is original and provides genuine insight into the structure of instability.
2. Compelling visualizations: The decision boundary maps (Figure 5), angular stability profiles (Figure 6), and layer-wise propagation plots effectively communicate the phenomena.
3. Cross-precision validation: Demonstrating that regime structure persists across BFloat16, Float32, and Float64 strengthens the universality claim.
4. The "spectrum collapse" finding: The observation that at small ε, directional sensitivity becomes independent of singular value ranking is a genuinely interesting result that contradicts classical conditioning intuitions.
Limitations
1. Limited experimental scale: Two models, two datasets, 10-100 prompts is insufficient for claims of "universality." No experiments on truly large-scale models (70B+) or diverse architectures (mixture-of-experts, state-space models).
2. Gap between analysis and real-world impact: The paper doesn't bridge the gap between demonstrating that chaos exists at machine-epsilon scales and showing this matters for practical deployment. Token-level accuracy changes are not measured.
3. Missing baselines: No comparison with Yuan et al.'s mitigation strategies or standard deterministic execution modes. The paper doesn't benchmark against existing approaches to numerical reproducibility.
4. Theoretical depth: Despite invoking chaos theory terminology ("Lyapunov-like" behavior, "avalanche effects"), the paper doesn't compute actual Lyapunov exponents or provide formal proofs. The analysis remains largely empirical.
5. Causal claims: The paper implies numerical instability causes multi-agent failures but provides only circumstantial evidence. The cited failure rates could stem from prompt sensitivity, sampling stochasticity, or other factors.
6. Writing precision: Some claims are overclaimed relative to evidence — "universal phenomena" from two models is a stretch. The term "chaotic" is used loosely without rigorous mathematical definition in this context.
Overall Assessment
This paper identifies a real and understudied phenomenon — the interaction between floating-point arithmetic and LLM inference — and provides a useful conceptual framework (three regimes) for reasoning about it. The directional condition number analysis and spectrum collapse findings are genuinely interesting. However, the experimental validation is too narrow for the universal claims made, the practical significance remains unclear, and the theoretical depth is insufficient for the chaos-theoretic framing employed. The paper opens an interesting research direction but requires substantially more evidence to establish the claimed impact.
Generated Apr 16, 2026
Comparison History (83)
Paper 2 introduces a novel, practical framework (GSS) that unifies generative models with physics-based structure search for molecular and materials discovery—a field with enormous real-world impact in drug design and materials engineering. It demonstrates concrete efficiency gains (>10x cost reduction) and generalization beyond training data. Paper 1 provides valuable theoretical analysis of LLM numerical instability, but is more diagnostic than constructive. Paper 2's methodological innovation, broad applicability across chemistry and materials science, and immediate practical utility give it higher potential scientific impact.
Paper 2 has higher impact potential due to its strong timeliness for deployed agentic systems, clear real-world utility (unsupervised stage-level diagnosis of silent memory failures), and methodological innovation in circuit tracing across scales and frameworks with causal/steerability distinctions. Its findings generalize across model sizes and memory implementations and connect interpretability to reliability engineering, widening cross-field relevance (mechanistic interpretability, alignment, systems, HCI). Paper 1 is novel and important for numerical reliability, but may be narrower in immediate applicability and actionable diagnostics compared to Paper 2’s deployable insights.
Paper 1 addresses a fundamental and poorly understood problem—numerical instability and chaos in LLMs—providing rigorous theoretical analysis of how floating-point rounding errors propagate through Transformer layers. Its identification of universal, scale-dependent chaotic regimes is highly novel and has broad implications for LLM reliability, reproducibility, and deployment across all applications. Paper 2 presents a useful but more incremental contribution to RL credit assignment for LLM agents. While practically valuable, Paper 1's foundational insights into the inherent unpredictability of LLMs have wider cross-disciplinary impact and deeper theoretical significance.
Paper 2 addresses a fundamental and universal issue in modern AI (numerical instability and unpredictability in LLMs), offering deep theoretical insights that will likely impact model reliability, hardware design, and quantization across the field. In contrast, Paper 1 offers a valuable but more specialized algorithmic improvement for RL training in LLM agents.
Paper 2 addresses a fundamental and broadly applicable problem—understanding the numerical instability and chaotic behavior inherent in LLM architectures. Its rigorous theoretical analysis of how floating-point rounding errors propagate through Transformer layers reveals universal, scale-dependent behaviors relevant to all LLM applications, not just one domain. This foundational understanding of LLM reliability has implications across every field using LLMs, from agentic workflows to scientific computing. Paper 1, while valuable, is a domain-specific application framework for climate science with incremental engineering contributions rather than fundamental scientific insights.
Paper 2 investigates a fundamental, structural issue within LLMs—numerical instability and chaotic error propagation. While Paper 1 provides a practical training methodology for reasoning, Paper 2's theoretical insights into the deterministic chaos of Transformers bridge numerical analysis and deep learning. This foundational understanding could broadly influence future model architectures, quantization strategies, hardware design, and reliability protocols for agentic workflows, leading to a deeper and more lasting scientific impact.
Paper 1 has higher potential scientific impact due to deeper novelty and breadth: it targets fundamental numerical/chaotic mechanisms behind LLM unpredictability, likely affecting reliability, evaluation, and deployment across many models and domains. The proposed universal regimes and layerwise error-propagation analysis could reshape understanding of determinism, reproducibility, and safety in large-scale ML systems. Paper 2 is timely and practically valuable for cost-saving in production via surrogate routing, but the core idea (distillation/cascades with abstention/thresholding) is more incremental and its impact is narrower to classification serving workflows.
Paper 1 addresses a fundamental, theoretical issue affecting all Large Language Models—numerical instability and error propagation—providing insights into their fundamental limits and behaviors. This has broad implications across the entire field of AI. In contrast, Paper 2 focuses on a narrow, domain-specific application (GDPR formalization) using existing LLM capabilities, which offers practical value but significantly less foundational scientific novelty and narrower cross-disciplinary impact.
Paper 1 likely has higher scientific impact due to stronger novelty and cross-field breadth: it proposes a general, mechanism-level theory of LLM unpredictability rooted in floating-point chaos with universal regimes, relevant to reliability, safety, evaluation, and hardware/software co-design across essentially all Transformer deployments. If rigorously validated, it could influence core ML practice and standards. Paper 2 is timely and application-rich for digital health, but impact may be narrower and incremental (modest ΔR^2, cohort-specific biomarkers) and more dependent on clinical translation and external validation.
Paper 1 introduces a novel and practically useful method (Introspection Adapters) for auditing fine-tuned LLMs by enabling self-reporting of learned behaviors. This addresses a critical AI safety challenge with immediate real-world applications in model governance and security. The approach is scalable, generalizable, and achieves state-of-the-art results on relevant benchmarks. Paper 2 provides valuable theoretical analysis of numerical instability in LLMs, but its findings are more descriptive/analytical in nature, characterizing known reliability issues rather than offering actionable solutions. Paper 1's direct applicability to AI safety auditing gives it broader and more immediate impact.
Paper 1 addresses a fundamental and pervasive issue (numerical instability and unpredictability) in Large Language Models, which are currently deployed across virtually all domains. Its insights into the chaotic behaviors of Transformer layers have broader implications for model reliability, reproducibility, and safety compared to Paper 2, which focuses on the more specialized subfield of Neuro-Symbolic Reinforcement Learning.
Paper 1 likely has higher impact due to its foundational, broadly applicable analysis of numerical precision–induced chaos in Transformer computations, a concern spanning essentially all LLM deployments and influencing reliability, reproducibility, and safety. Its identification of universal regimes and mechanistic propagation across layers suggests strong novelty and potential to inform hardware, inference, training, and evaluation practices across models and tasks. Paper 2 is timely and useful for omni-modal systems with an actionable benchmark/tool, but its scope is narrower (modality preference in OLLMs) and more contingent on rapidly changing model families.
Paper 1 addresses a fundamental, universal issue in LLM computation: numerical instability and chaotic behavior. By bridging numerical analysis and deep learning, it offers profound insights into LLM reliability and reproducibility, impacting almost all applications. While Paper 2 provides valuable insights for multi-modal systems, Paper 1's focus on the underlying computational mechanics of Transformers offers deeper theoretical novelty, broader implications across all text-based and agentic workflows, and a more rigorous methodological approach to understanding model unpredictability.
Paper 1 addresses a fundamental and broadly impactful problem—understanding the numerical instability and chaotic behavior inherent in LLM computations. By rigorously characterizing three distinct regimes of chaotic behavior rooted in floating-point precision, it provides foundational insights relevant to all LLM deployments, especially as agentic workflows scale. Its breadth of impact spans reliability engineering, numerical analysis, and AI safety. Paper 2, while novel in automating concept grounding for neuro-symbolic RL, addresses a narrower problem with demonstrations limited to specific Atari games, limiting its broader scientific impact.
Paper 2 likely has higher impact: it introduces a practically deployable, scalable method (introspection adapters) with clear real-world applications in auditing, safety, and security of fine-tuned LLMs, and reports strong generalization plus SOTA results on a benchmark and attack detection. Its breadth spans alignment, ML security, and governance, and it is timely given widespread fine-tuning and API risks. Paper 1 is novel and rigorous, but its primary contributions are diagnostic/theoretical; translating chaos/precision analyses into broadly adopted mitigation practices may be slower and narrower in immediate cross-field uptake.
Paper 1 addresses a fundamental, low-level mathematical issue (numerical instability and chaos) that affects the reliability and reproducibility of all Transformer-based models. Its rigorous quantification of error propagation and discovery of universal chaotic regimes provide deep insights into foundation model mechanics. While Paper 2 offers a highly creative and interdisciplinary approach to multi-agent orchestration, Paper 1's findings have broader, more fundamental implications for AI safety, hardware design, and model architecture, giving it a higher potential for widespread scientific impact.
Paper 1 investigates a fundamental, universal property of LLMs—numerical instability and chaotic behavior. Its foundational nature bridges theoretical ML, hardware precision, and reliability, offering profound implications for how all transformer models are executed and evaluated. While Paper 2 presents a strong practical innovation for generation and alignment, Paper 1's theoretical insights into LLM predictability address critical reliability bottlenecks, promising broader cross-disciplinary scientific impact.
Paper 1 (OLLM) introduces a novel architectural paradigm that replaces single next-token prediction with learned options indexed by discrete latent variables, offering a practical plug-in for pretrained LLMs with significant performance gains (51%→70% on math reasoning). This has broad applicability to LLM alignment, controllability, and RL-based optimization. Paper 2 provides valuable theoretical analysis of numerical instability in LLMs, but is primarily diagnostic rather than prescriptive. OLLM's combination of architectural innovation, practical applicability, sample-efficient RL, and structural alignment gives it higher transformative potential across multiple LLM research directions.
Paper 2 addresses the fundamental question of whether AI agents can truly perform scientific reasoning, finding systematic epistemic failures across 25,000+ runs and 8 domains. Its implications are broader—affecting the entire field of AI-driven science—and more immediately actionable, arguing that scaffold engineering is insufficient and reasoning must become a training target. While Paper 1 provides rigorous technical analysis of numerical instability in LLMs (important for reliability), Paper 2's findings challenge the foundational assumptions of autonomous AI science, a rapidly growing and high-stakes field, giving it wider cross-disciplinary impact and policy relevance.
Paper 2 has higher likely impact due to its broad, timely implications for autonomous AI research across many scientific domains, backed by a large-scale empirical evaluation (25,000+ runs) and clear quantitative findings about epistemic failure modes that affect reliability and governance. Its conclusions directly inform deployment, evaluation methodology, and training objectives, influencing ML, HCI, meta-science, and research policy. Paper 1 is novel and useful for LLM reliability/precision analysis, but its impact is narrower (numerical stability/chaos) and more engineering-focused, with less immediate cross-field reach.