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Bounds of Chain-of-Thought Robustness: Reasoning Steps, Embed Norms, and Beyond

ICLR 2026 · Large language models › Reasoning and chain of thought · approach: Chain-of-thought training and analysis · 0 citations (OpenAlex)

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Theoretically bounds how input perturbations propagate through multi-step CoT reasoning and Linear Self-Attention, showing more steps shrink but never eliminate the tolerable perturbation, and validates on MATH, MMLU-Pro, GPQA with four LLMs.

Problem

Why and how input perturbations affect the output fluctuation of chain-of-thought reasoning in LLMs lacks theoretical explanation, limiting understanding and prompt optimization design.

Approach

Models CoT as a multistep iterative map under Lipschitz continuity, derives an upper bound on output fluctuation and a certified tolerable input-perturbation radius that depends on step count K, and specializes the constants to a linear self-attention model, linking the radius to embedding and hidden-state norms.

Key results

• Output fluctuation rises with input perturbation magnitude (Pearson correlation 0.619), supporting the theoretical bound.
• Qwen3-8b reaches the highest EM (77.2 on MATH, 46.9 on MMLU-Pro, 37.3 on GPQA) and lowest OF (0.097, 0.162, 0.214).
• Selecting prompts by the derived bound gives EM gains over TextGrad/OPRO/CFPO, e.g., Llama3.1-8b GPQA 32.3 vs 27.6 for CFPO and Qwen3-8b MMLU-Pro 49.2 vs 45.8.

Surprising findings

• Larger models can show larger absolute EM fluctuation while having lower output fluctuation, so EM and OF can disagree.
• Fitted gamma is below 1 in most settings but for Llama3.1-8b and Qwen3-8b on GPQA the fit is near 0.01-0.02.

Limitations

• Lipschitz constants C and gamma are bounded for LSA only, and the non-linear Transformer effects are only discussed intuitively without proof.
• Gamma and the data covariance Gamma/residual eta are not empirically verified.
• The prompt selection method is evaluated against few baselines and lacks efficiency analysis beyond an O(M*T) argument.

Experimental setup

Datasets: MATH MMLU-Pro GPQA Amazon Review FinQA ToolE

Benchmarks: MATH MMLU-Pro GPQA Amazon Review FinQA ToolE

Metrics: exact match output fluctuation (normalized entropy) Pearson correlation

Baselines: TextGrad OPRO CFPO base prompt

Builds on: Linear Self-Attention in-context learning analysis TextGrad OPRO CFPO

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