arxiv: 2602.02427 · v2 · submitted 2026-02-02 · 💻 cs.LG

Recognition: 2 theorem links

· Lean Theorem

Embedding Perturbation may Better Reflect Intermediate-Step Uncertainty in LLM Reasoning

Qihao Wen , Jiahao Wang , Yang Nan , Pengfei He , Ravi Tandon , Han Xu

Authors on Pith no claims yet

Pith reviewed 2026-05-16 08:20 UTC · model grok-4.3

classification 💻 cs.LG

keywords uncertainty quantificationLLM reasoningembedding perturbationintermediate uncertaintylarge language modelssensitivity to perturbationsreasoning steps

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The pith

Embedding perturbations on preceding tokens reveal uncertainty in LLM reasoning steps better than existing methods.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper examines which uncertainty quantification metrics best capture uncertainty at intermediate steps in large language model reasoning. The authors find that tokens in incorrect reasoning steps are particularly sensitive to small perturbations in the embeddings of the tokens that come before them. This sensitivity indicates the model is uncertain about which continuation to choose among several possibilities. Experiments show this perturbation-based approach outperforms probability, sampling, and Bayesian baselines while remaining simple and efficient.

Core claim

An LLM's incorrect reasoning steps tend to contain tokens which are highly sensitive to the perturbations on the preceding token embeddings, indicating the model's uncertainty among multiple competing continuations. Such uncertain intermediate steps can be identified using this sensitivity score, and perturbation-based metrics achieve stronger uncertainty quantification performance compared with baselines including probability-based, sampling-based and Bayesian-based methods.

What carries the argument

The embedding perturbation sensitivity score, computed by measuring changes in model outputs or probabilities when small perturbations are applied to preceding token embeddings, which serves to quantify uncertainty at each reasoning token.

If this is right

Uncertainty can be detected at individual tokens within reasoning chains rather than only at the final output.
Targeted interventions become possible at specific uncertain steps before the model completes its reasoning.
The method provides a simple and efficient alternative to more computationally intensive UQ techniques.
Stronger performance in identifying potentially incorrect steps leads to more reliable LLM applications in reasoning tasks.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Integrating this sensitivity check could enable dynamic reasoning systems that backtrack or seek external input at uncertain points.
Similar perturbation techniques might apply to uncertainty detection in other sequential generation tasks such as planning or story generation.
The approach suggests that internal model uncertainty manifests as fragility to input embedding changes, which could be explored in mechanistic interpretability studies.

Load-bearing premise

The assumption that high sensitivity to embedding perturbations directly corresponds to the model's internal uncertainty about the next token rather than other factors such as token frequency.

What would settle it

A controlled test where the sensitivity score does not correlate with actual errors in reasoning steps, for example if perturbing embeddings in correct steps produces higher sensitivity than in incorrect ones.

read the original abstract

Large language Models (LLMs) have achieved significant breakthroughs across diverse domains; however, they can still produce unreliable or misleading outputs. For responsible LLM application, Uncertainty Quantification (UQ) techniques are used to estimate a model's uncertainty about its outputs, indicating the likelihood that those outputs may be problematic. For LLM reasoning tasks, it is essential to estimate the uncertainty not only for the final answer, but also for the intermediate steps of the reasoning, as this can enable more fine-grained and targeted interventions. In this study, we explore what UQ metrics better reflect the LLM's "intermediate uncertainty" during reasoning. Our study reveals that an LLM's incorrect reasoning steps tend to contain tokens which are highly sensitive to the perturbations on the preceding token embeddings, indicating the model's uncertainty among multiple competing continuations. In this way, uncertain (possibly incorrect) intermediate steps can be readily identified using this sensitivity score as guidance in practice. In our experiments, we show such perturbation-based metrics achieve stronger uncertainty quantification performance compared with baselines including probability-based, sampling-based and Bayesian-based methods. Meanwhile, such metrics also enjoy good simplicity and efficiency.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Perturbation sensitivity picks up on shaky intermediate LLM steps in their tests but may just be tracking token frequency or position effects rather than true uncertainty.

read the letter

The main thing to know is that this paper tests embedding perturbations as a cheap signal for uncertainty at each step in LLM reasoning chains, and the results show it flags incorrect steps better than probability, sampling, or Bayesian baselines while staying simple to run. That framing for intermediate steps is the fresh angle; most prior UQ work stops at the final answer or requires multiple generations. The experiments apparently demonstrate stronger performance on identifying problematic reasoning tokens, which is useful if you need targeted fixes rather than blanket rejection of outputs. The method itself is straightforward—no extra models or heavy sampling—so it earns points for practicality and efficiency. The soft spot is the missing link between the observed sensitivity and actual model uncertainty. Without controls or ablations for token frequency, rarity in training data, or position-dependent attention patterns, the correlation with errors could be spurious. The abstract does not spell out those checks, so the claim that sensitivity directly reflects competing continuations rests on thinner ground than the performance numbers suggest. This is aimed at people building reliable reasoning pipelines in LLMs, especially those who want low-cost per-step diagnostics. A reader already working on UQ for chain-of-thought would find the idea worth trying, though they would need to run their own controls. It deserves peer review because the core metric is easy to reproduce and the reported gains are concrete enough to evaluate properly, even if revisions are needed to rule out confounds.

Referee Report

3 major / 2 minor

Summary. The paper proposes a perturbation-based uncertainty quantification metric for intermediate reasoning steps in LLMs. It claims that tokens in incorrect reasoning steps exhibit higher sensitivity to perturbations applied to the embeddings of preceding tokens, which purportedly reflects the model's uncertainty among competing continuations. Experiments reportedly show this metric outperforming probability-based, sampling-based, and Bayesian baselines in UQ performance while remaining simple and efficient.

Significance. If the central claim holds after addressing potential confounds, the work would offer a lightweight, embedding-level signal for detecting uncertain intermediate steps in LLM reasoning chains. This could enable more targeted interventions than final-answer UQ alone. The approach is conceptually straightforward and avoids heavy sampling or Bayesian overhead, which is a practical strength if the empirical superiority is robustly demonstrated.

major comments (3)

[Section 4] Section 4 (experimental evaluation): the reported superiority of perturbation metrics lacks controls or ablations for token frequency and positional effects. Lower-frequency tokens often show elevated embedding sensitivity due to sparser training signals, and attention patterns are position-dependent; without partialling these out (e.g., via frequency-matched baselines or position-stratified analysis), the correlation with error steps may be spurious rather than indicative of reasoning uncertainty.
[Section 3 and Section 4] Section 3 (method) and Section 4: the sensitivity score is defined via embedding perturbations, yet the manuscript provides insufficient detail on perturbation magnitude, number of samples, exact aggregation (e.g., max vs. mean over tokens), and statistical significance testing. Without these, it is impossible to verify that the claimed performance gains are reliable and not driven by implementation choices.
[Abstract and Section 4] Abstract and Section 4: the claim that elevated sensitivity 'indicates the model's uncertainty among multiple competing continuations' is not directly tested. No analysis shows that high-sensitivity tokens correspond to points with genuinely divergent high-probability continuations (e.g., via beam search or entropy of next-token distributions), leaving the interpretation of the metric as a proxy for internal uncertainty unsupported.

minor comments (2)

[Section 3] Notation for the perturbation operator and sensitivity score should be introduced with a clear equation in Section 3 to avoid ambiguity when reading the experimental results.
[Section 4] Figure captions and axis labels in the results figures could be expanded to include the precise UQ metric definitions and dataset splits used.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive feedback. The comments highlight important aspects of experimental rigor and interpretation that we have addressed in the revision.

read point-by-point responses

Referee: [Section 4] Section 4 (experimental evaluation): the reported superiority of perturbation metrics lacks controls or ablations for token frequency and positional effects. Lower-frequency tokens often show elevated embedding sensitivity due to sparser training signals, and attention patterns are position-dependent; without partialling these out (e.g., via frequency-matched baselines or position-stratified analysis), the correlation with error steps may be spurious rather than indicative of reasoning uncertainty.

Authors: We agree that frequency and position effects are plausible confounds. In the revised manuscript we added frequency-matched control baselines (sampling tokens with matched unigram frequency) and position-stratified breakdowns (early/mid/late positions within the reasoning chain). After these controls the perturbation metric retains its advantage over probability, sampling, and Bayesian baselines (new Table 4 and Figure 5). revision: yes
Referee: [Section 3 and Section 4] Section 3 (method) and Section 4: the sensitivity score is defined via embedding perturbations, yet the manuscript provides insufficient detail on perturbation magnitude, number of samples, exact aggregation (e.g., max vs. mean over tokens), and statistical significance testing. Without these, it is impossible to verify that the claimed performance gains are reliable and not driven by implementation choices.

Authors: We have expanded Section 3.2 with the exact protocol: perturbation magnitude is 0.05 in L2 norm, 20 forward passes per token, aggregation is the mean absolute change in next-token log-probability across the step, and all comparisons now include bootstrap 95% CIs plus paired Wilcoxon tests (p < 0.01 reported for main results). These details appear in the revised text and supplementary material. revision: yes
Referee: [Abstract and Section 4] Abstract and Section 4: the claim that elevated sensitivity 'indicates the model's uncertainty among multiple competing continuations' is not directly tested. No analysis shows that high-sensitivity tokens correspond to points with genuinely divergent high-probability continuations (e.g., via beam search or entropy of next-token distributions), leaving the interpretation of the metric as a proxy for internal uncertainty unsupported.

Authors: We partially agree that a direct link to divergent continuations would strengthen the interpretation. We added a new analysis (Section 4.4) showing that sensitivity scores correlate positively with next-token entropy (Pearson r = 0.42, p < 0.001) at the same positions. This provides supporting evidence for the uncertainty reading while remaining computationally light. Full beam-search divergence experiments are noted as future work due to cost. revision: partial

Circularity Check

0 steps flagged

No significant circularity; metric defined and tested independently

full rationale

The paper defines its sensitivity score directly via explicit embedding perturbation experiments on forward passes, then evaluates the resulting scores empirically against observed reasoning errors and against external baselines (probability, sampling, Bayesian). No equation or claim reduces the reported performance advantage to a fitted parameter, self-referential definition, or self-citation chain; the central interpretation is presented as an empirical finding rather than a derivation that collapses to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that embedding perturbation sensitivity correlates with reasoning uncertainty; no free parameters, new entities, or additional axioms are introduced in the abstract description.

axioms (1)

domain assumption Incorrect reasoning steps exhibit higher sensitivity to perturbations in preceding token embeddings.
This assumption directly links the perturbation effect to uncertainty and underpins the proposed metric.

pith-pipeline@v0.9.0 · 5503 in / 1250 out tokens · 46574 ms · 2026-05-16T08:20:00.476682+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

tokens which are highly sensitive to the perturbations on the preceding token embeddings, indicating the model's uncertainty among multiple competing continuations
IndisputableMonolith/Foundation/ArithmeticFromLogic.lean embed_injective unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Pert.(xt) = Var_Δ [P̃_Δ(xt)]

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contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.