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TokUR: Token-Level Uncertainty Estimation for Large Language Model Reasoning

ICLR 2026 · Large language models › Reasoning and chain of thought · approach: Test-time compute scaling

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Proposes TokUR, a training-free token-level uncertainty estimator that perturbs low-rank attention weights at decoding to decompose total uncertainty into aleatoric and epistemic parts, and uses these for math-reasoning error detection and test-time selection.

Problem

Sequence-level uncertainty estimation for long-form LLM reasoning is hard: query-level Bayesian methods require intractable marginalization over outputs, while log-probability response-level heuristics lack theoretical grounding and fail to capture response-specific uncertainty.

Approach

Apply random low-rank Gaussian perturbations to the query and key projection matrices during decoding to obtain stepwise predictive distributions, then compute token-level TU/AU/EU and sum them over the response as a principled response-level uncertainty (shown to be an unbiased estimator of query-level uncertainty).

Key results

• On MATH500 with Llama-3.2-1B-Instruct, TokUR (TU) reaches 80.64% AUROC vs 71.77% for DeepConf (best baseline).
• On MATH500 with Llama-3.1-8B-Instruct, TokUR (EU) reaches 82.86% AUROC and 81.35% AUPRC, versus 76.41% AUROC for Self-Certainty.
• In test-time selection with N=16 on MATH500 (Llama-3.2-1B), TokUR (EU) with Maj@N achieves 28.28% vs 25.68% for DeepConf and 26.42% for LL.

Surprising findings

• Response-level (sequence-summed) EU/TU gives strong hallucination detection without any external NLI model, outperforming semantic-entropy-style baselines that use one.
• AUROC rises again at the highest question difficulty level (10), which the authors attribute to class imbalance rather than genuine signal.

Limitations

• Repeated weight-perturbation sampling at inference adds cost, limiting real-time use.
• Token-level aggregation may miss higher-level semantic or logical inconsistencies across reasoning steps.
• High variance of the uncertainty estimates remains unresolved.

Experimental setup

Datasets: GSM8K MATH500 DeepScaleR math-orz Reasoning Gym HumanEval FactScore

Benchmarks: MATH500 GSM8K DeepScaleR Reasoning Gym (Zebra Puzzles, Leg Counting, Color Cube) HumanEval FactScore

Metrics: AUROC AUPRC Top-50% ACC Precision* Maj@N accuracy WBoN accuracy

Baselines: Self-Certainty DeepConf LLM-Check P(True) Predictive Entropy Log-Likelihood Semantic Entropy SAR INSIDE Degree Matrix

Builds on: LoRA BLoB TFB Semantic Entropy DeepConf Self-Certainty

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