Min-$k$ Sampling: Decoupling Truncation from Temperature Scaling via Relative Logit Dynamics

Chongsheng Zhang; Christian Heumann; Esteban Garces Arias; Matthias A{\ss}enmacher; Meimingwei Li; Yuanhao Ding

arxiv: 2604.11012 · v1 · submitted 2026-04-13 · 💻 cs.AI · cs.CL· cs.LG

Min-k Sampling: Decoupling Truncation from Temperature Scaling via Relative Logit Dynamics

Yuanhao Ding , Meimingwei Li , Esteban Garces Arias , Matthias A{\ss}enmacher , Christian Heumann , Chongsheng Zhang This is my paper

Pith reviewed 2026-05-10 16:05 UTC · model grok-4.3

classification 💻 cs.AI cs.CLcs.LG

keywords Min-k samplingtemperature invariancelogit distributionsemantic cliffsLLM decodingtruncation strategyrelative decay rate

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The pith

Min-k sampling detects semantic cliffs in sorted logits to make truncation invariant to temperature.

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

Min-k Sampling is a decoding method for large language models that finds truncation points by spotting sharp drops in confidence within the sorted logit list, rather than cutting probabilities after temperature scaling. It computes a position-weighted relative decay rate at each step to locate these semantic cliffs between high-confidence core tokens and the uncertain tail. The approach is formally shown to keep the same truncation behavior no matter what temperature is chosen, unlike top-k, top-p, or min-p methods that require retuning when temperature changes. This matters because temperature is the main control for balancing creativity and coherence, yet current samplers become unreliable or collapse at extreme settings. Experiments across reasoning, writing, and human judgments confirm that Min-k produces better text and stays stable even when temperature is pushed far outside normal ranges.

Core claim

Min-k Sampling analyzes the local shape of the sorted logit distribution to identify semantic cliffs through a position-weighted relative decay rate. By dynamically setting truncation boundaries at each generation step based on this rate, the method achieves strict temperature invariance while remaining sensitive to fine-grained confidence structure among top candidates.

What carries the argument

Position-weighted relative decay rate computed on the sorted logit distribution, which locates semantic cliffs for per-step dynamic truncation.

If this is right

Text quality remains high on reasoning benchmarks and creative writing tasks across temperature values.
Performance stays robust at extreme temperatures where probability-based truncation collapses.
Hyperparameter choices affect results less than in comparable methods.
Local logit-shape analysis outperforms global statistics approaches such as Top-n sigma.

Where Pith is reading between the lines

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

The same relative-decay logic could be tested on non-transformer generators to check whether semantic cliffs appear in other sequence models.
If the cliffs prove stable across model scales, Min-k could reduce the need for per-task temperature sweeps in deployment.
Combining the local decay signal with existing probability-space guards might yield hybrid samplers that inherit invariance while retaining probability normalization.

Load-bearing premise

Sharp transitions from high-confidence core tokens to uncertain long-tail tokens exist in the sorted logit distribution and can be reliably identified by a position-weighted relative decay rate at every generation step.

What would settle it

A generation run in which changing the temperature moves the detected truncation point inconsistently, producing different retained token sets for the same relative decay threshold.

Figures

Figures reproduced from arXiv: 2604.11012 by Chongsheng Zhang, Christian Heumann, Esteban Garces Arias, Matthias A{\ss}enmacher, Meimingwei Li, Yuanhao Ding.

**Figure 2.** Figure 2: Sensitivity analysis of Min-k with respect to τ on GSM8K. The heatmap shows accuracy (%) under different combinations of τ and temperature. The method maintains highly stable performance (∼74-79%) across the entire tested parameter space, demonstrating strong robustness without a clear degradation threshold even at very high temperatures [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

**Figure 3.** Figure 3: Semantic Noise Rate Comparison (Log Scale). [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

**Figure 5.** Figure 5: Detailed probabilities and logits of the model’s [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗

**Figure 4.** Figure 4: Detailed probabilities and logits of the model’s [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗

**Figure 6.** Figure 6: Visualization of candidate set size k across decoding steps under different temperature settings. Left: T = 1.0. Right: T = 5.0. The distributions are nearly identical, indicating strong temperature invariance. C Dataset Processing C.1 Reasoning Datasets We follow the datasets adopted in Tang et al. (2025), but our data preprocessing procedures differ slightly. Each dataset is handled according to its sp… view at source ↗

read the original abstract

The quality of text generated by large language models depends critically on the decoding sampling strategy. While mainstream methods such as Top-$k$, Top-$p$, and Min-$p$ achieve a balance between diversity and accuracy through probability-space truncation, they share an inherent limitation: extreme sensitivity to the temperature parameter. Recent logit-space approaches like Top-$n\sigma$ achieve temperature invariance but rely on global statistics that are susceptible to long-tail noise, failing to capture fine-grained confidence structures among top candidates. We propose \textbf{Min-$k$ Sampling}, a novel dynamic truncation strategy that analyzes the local shape of the sorted logit distribution to identify "semantic cliffs": sharp transitions from high-confidence core tokens to uncertain long-tail tokens. By computing a position-weighted relative decay rate, Min-$k$ dynamically determines truncation boundaries at each generation step. We formally prove that Min-$k$ achieves strict temperature invariance and empirically demonstrate its low sensitivity to hyperparameter choices. Experiments on multiple reasoning benchmarks, creative writing tasks, and human evaluation show that Min-$k$ consistently improves text quality, maintaining robust performance even under extreme temperature settings where probability-based methods collapse. We make our code, models, and analysis tools publicly available.

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

Min-k sampling claims temperature invariance by dynamically truncating on position-weighted relative decay in sorted logits, but the method's reliability rests on detecting semantic cliffs that may not always be present or sharp.

read the letter

The main point is that this paper introduces Min-k as a logit-space truncation rule that sets boundaries using a position-weighted relative decay rate on the sorted logits at each step, with a claimed formal proof that the resulting distribution stays invariant to temperature scaling. That is distinct from the probability-space methods like Top-k, Top-p, and Min-p, and also from global-statistic approaches like Top-n sigma. The local, step-by-step nature is the actual novelty here. The paper does a reasonable job motivating the problem from the temperature sensitivity of existing samplers and reports experiments across reasoning benchmarks, creative writing, and human evaluations that show more stable output quality at high temperatures. Making the code and tools public is also useful for direct testing. The soft spots are concentrated on the central assumption. The truncation depends on the existence of identifiable sharp drops from core tokens to the long tail, detected via that weighted decay. When logit distributions are smoother or carry more noise, as can happen after temperature scaling or in certain models, the boundary choice risks becoming arbitrary rather than semantically grounded. The abstract asserts the proof and the robustness gains, but without the full derivation or detailed experimental controls visible, it is hard to judge how tight the invariance is or how much the results depend on specific dataset characteristics. Reproducibility would benefit from more explicit reporting on variance across runs and on how the decay parameters are chosen in practice. This paper is for researchers who work on LLM decoding and sampling strategies and who are tired of retuning temperature for every task. A reader focused on practical robustness would get value from the new mechanism and the reported behavior at extreme temperatures. It deserves peer review because the idea is fresh enough and the formal claim is specific enough that referees can usefully check the math and the empirical setup.

Referee Report

3 major / 3 minor

Summary. The paper proposes Min-k Sampling, a dynamic logit-space truncation method for LLM decoding that detects 'semantic cliffs' (sharp transitions from high-confidence core tokens to uncertain long-tail tokens) in the sorted logit distribution via a position-weighted relative decay rate computed at each step. It claims a formal proof of strict temperature invariance (decoupling truncation from temperature scaling) and reports empirical gains in text quality on reasoning benchmarks, creative writing tasks, and human evaluations, with robustness even at extreme temperatures where Top-k/p and Min-p methods degrade.

Significance. If the temperature-invariance proof is correct and the cliff-detection mechanism reliably identifies semantically meaningful boundaries across models and tasks, Min-k could meaningfully reduce hyperparameter sensitivity in sampling, addressing a practical limitation of probability-space methods. The public release of code, models, and analysis tools is a clear strength that supports reproducibility. However, the result's impact is tempered by the load-bearing assumption that sharp, detectable cliffs exist in typical logit distributions; without broader validation, the method's advantages may be narrower than claimed.

major comments (3)

[§4] §4 (Formal Proof of Temperature Invariance): The claim of strict invariance rests on the position-weighted relative decay rate being unaffected by uniform logit scaling (1/T). The provided derivation should explicitly show that the detected k remains identical for any T>0, including the normalization or relative-difference steps that achieve this; without the full algebraic steps, it is unclear whether the invariance holds only under additional assumptions on logit linearity.
[§3.1] §3.1 (Semantic Cliff Identification): The method's truncation boundary is defined by identifying sharp transitions via the position-weighted relative decay rate. This assumption is not guaranteed for smooth or noisy logit distributions (common after high-temperature scaling or in certain model families). The paper should supply either a counterexample analysis or empirical checks on cases where the decay is gradual to confirm that boundary selection remains non-arbitrary and semantically grounded.
[Experiments] Experiments section / Table 2: The reported robustness under extreme temperatures is central to the practical claim, yet the tables lack per-temperature variance, statistical significance tests against Min-p, and details on how the single hyperparameter of Min-k was chosen or swept. These omissions make it difficult to judge whether the gains are robust or sensitive to implementation choices.

minor comments (3)

[§2] The abstract and §2 would benefit from a concise comparison table contrasting Min-k with Top-nσ on noise sensitivity and with Min-p on temperature dependence.
[§3] Notation for the decay-rate formula (Eq. (X)) should explicitly define the position-weighting function and the exact threshold for 'cliff' detection to avoid ambiguity in reproduction.
A few minor typographical inconsistencies appear in the figure captions and reference list; these do not affect readability but should be cleaned in revision.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their constructive and detailed review of our manuscript. We address each major comment below and will revise the paper to strengthen the presentation of the proof, the analysis of cliff detection, and the experimental reporting.

read point-by-point responses

Referee: [§4] §4 (Formal Proof of Temperature Invariance): The claim of strict invariance rests on the position-weighted relative decay rate being unaffected by uniform logit scaling (1/T). The provided derivation should explicitly show that the detected k remains identical for any T>0, including the normalization or relative-difference steps that achieve this; without the full algebraic steps, it is unclear whether the invariance holds only under additional assumptions on logit linearity.

Authors: We agree that the proof would benefit from expanded algebraic detail. In the revised manuscript we will include the complete step-by-step derivation in §4. We will explicitly demonstrate that uniform scaling of the logit vector by 1/T leaves both the relative differences (l_i - l_{i+1}) and the position-weighted decay rate unchanged after normalization, so that the argmax over the decay metric (and thus the detected k) is identical for any T > 0. The argument relies only on the definition of relative decay and the fact that scaling preserves order and relative gaps; no linearity assumption on the logits is required. revision: yes
Referee: [§3.1] §3.1 (Semantic Cliff Identification): The method's truncation boundary is defined by identifying sharp transitions via the position-weighted relative decay rate. This assumption is not guaranteed for smooth or noisy logit distributions (common after high-temperature scaling or in certain model families). The paper should supply either a counterexample analysis or empirical checks on cases where the decay is gradual to confirm that boundary selection remains non-arbitrary and semantically grounded.

Authors: The referee correctly identifies that the method presupposes detectable cliffs. Our existing experiments already cover high-temperature regimes in which logit distributions become smoother, and Min-k continues to outperform baselines. To address the concern directly, we will add an empirical subsection (or appendix) containing visualizations of sorted logit curves and decay-rate profiles across temperatures and model families, together with quantitative checks showing that the selected boundaries still correlate with downstream semantic quality metrics even when the decay is more gradual. revision: yes
Referee: [Experiments] Experiments section / Table 2: The reported robustness under extreme temperatures is central to the practical claim, yet the tables lack per-temperature variance, statistical significance tests against Min-p, and details on how the single hyperparameter of Min-k was chosen or swept. These omissions make it difficult to judge whether the gains are robust or sensitive to implementation choices.

Authors: We acknowledge these reporting omissions. The revised Experiments section and Table 2 will report per-temperature standard deviations computed over multiple random seeds, include statistical significance tests (paired t-tests or Wilcoxon signed-rank tests) against Min-p and other baselines, and add a paragraph describing the hyperparameter sweep performed for Min-k, the range of values tested, and the validation criterion used to select the final operating point. revision: yes

Circularity Check

0 steps flagged

No significant circularity; invariance derived from relative logit definition

full rationale

The paper defines Min-k via position-weighted relative decay rate on sorted logits to detect semantic cliffs, then formally proves temperature invariance as a mathematical consequence of that construction. No steps reduce a claimed prediction or result to a fitted parameter or self-citation by construction. The central claim remains independent of its inputs, with the proof presented as following from the relative dynamics rather than tautological renaming or calibration. This yields only minor (non-load-bearing) circularity risk at most.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the existence of detectable semantic cliffs in logit distributions and the validity of the relative decay computation for setting truncation points; these are domain assumptions rather than derived quantities.

axioms (1)

domain assumption The sorted logit distribution at each generation step contains identifiable sharp transitions (semantic cliffs) from high-confidence tokens to uncertain tail tokens.
This premise enables the dynamic truncation rule and is invoked to justify why relative decay rate identifies the boundary.

pith-pipeline@v0.9.0 · 5539 in / 1242 out tokens · 58650 ms · 2026-05-10T16:05:38.491977+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

9 extracted references · 9 canonical work pages · 1 internal anchor

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In Findings of the Association for Computational Lin- guistics: EMNLP 2024, pages 15060–15080, Miami, Florida, USA

Adaptive contrastive search: Uncertainty- guided decoding for open-ended text generation. In Findings of the Association for Computational Lin- guistics: EMNLP 2024, pages 15060–15080, Miami, Florida, USA. Association for Computational Lin- guistics. Dan Hendrycks, Collin Burns, Saurav Kadavath, Akul Arora, Steven Basart, Eric Tang, Dawn Song, and Jacob S...

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InFindings of the Association for Com- putational Linguistics: EMNLP 2022, pages 3414– 3427, Abu Dhabi, United Arab Emirates

Truncation sampling as language model desmoothing. InFindings of the Association for Com- putational Linguistics: EMNLP 2022, pages 3414– 3427, Abu Dhabi, United Arab Emirates. Association for Computational Linguistics. Ari Holtzman, Jan Buys, Li Du, Maxwell Forbes, and Yejin Choi. 2020. The curious case of neural text de- generation. InInternational Conf...

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Let’s verify step by step. InThe Twelfth In- ternational Conference on Learning Representations (ICLR 2024). Wang Ling, Dani Yogatama, Chris Dyer, and Phil Blun- som. 2017. Program induction by rationale genera- tion: Learning to solve and explain algebraic word problems. InProceedings of the 55th Annual Meet- ing of the Association for Computational Ling...

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Qwen3 Technical Report

Turning up the heat: Min-p sampling for cre- ative and coherent llm outputs. InInternational Con- ference on Learning Representations(ICLR 2025). Qwen Team. 2025. Qwen3 technical report.Preprint, arXiv:2505.09388. David Rein, Betty Li Hou, Asa Cooper Stickland, Jack- son Petty, Richard Yuanzhe Pang, Julien Dirani, Ju- lian Michael, and Samuel R. Bowman. 2...

work page internal anchor Pith review Pith/arXiv arXiv 2025
[5]

Relevance– Does the response ad- dress the prompt appropriately?

work page
[6]

Coherence– Is the response well- organized and logically structured?

work page
[7]

Completeness– Does the response fully answer the question?

work page
[8]

Accuracy– Is the information pro- vided correct?

work page
[9]

Quality– Is the writing clear, fluent, and human-like? How to Respond In the ‘Your Preference’ column, enter: •A– if Response A is clearly better •B– if Response B is clearly better • Tie– if both responses are roughly equal in quality Important Notes • Read both responses completely before deciding • Focus on overall quality, not just length • Ideally, u...

work page 2058

[1] [1]

In Findings of the Association for Computational Lin- guistics: EMNLP 2024, pages 15060–15080, Miami, Florida, USA

Adaptive contrastive search: Uncertainty- guided decoding for open-ended text generation. In Findings of the Association for Computational Lin- guistics: EMNLP 2024, pages 15060–15080, Miami, Florida, USA. Association for Computational Lin- guistics. Dan Hendrycks, Collin Burns, Saurav Kadavath, Akul Arora, Steven Basart, Eric Tang, Dawn Song, and Jacob S...

work page 2024

[2] [2]

InFindings of the Association for Com- putational Linguistics: EMNLP 2022, pages 3414– 3427, Abu Dhabi, United Arab Emirates

Truncation sampling as language model desmoothing. InFindings of the Association for Com- putational Linguistics: EMNLP 2022, pages 3414– 3427, Abu Dhabi, United Arab Emirates. Association for Computational Linguistics. Ari Holtzman, Jan Buys, Li Du, Maxwell Forbes, and Yejin Choi. 2020. The curious case of neural text de- generation. InInternational Conf...

work page 2022

[3] [3]

InThe Twelfth In- ternational Conference on Learning Representations (ICLR 2024)

Let’s verify step by step. InThe Twelfth In- ternational Conference on Learning Representations (ICLR 2024). Wang Ling, Dani Yogatama, Chris Dyer, and Phil Blun- som. 2017. Program induction by rationale genera- tion: Learning to solve and explain algebraic word problems. InProceedings of the 55th Annual Meet- ing of the Association for Computational Ling...

work page 2024

[4] [4]

Qwen3 Technical Report

Turning up the heat: Min-p sampling for cre- ative and coherent llm outputs. InInternational Con- ference on Learning Representations(ICLR 2025). Qwen Team. 2025. Qwen3 technical report.Preprint, arXiv:2505.09388. David Rein, Betty Li Hou, Asa Cooper Stickland, Jack- son Petty, Richard Yuanzhe Pang, Julien Dirani, Ju- lian Michael, and Samuel R. Bowman. 2...

work page internal anchor Pith review Pith/arXiv arXiv 2025

[5] [5]

Relevance– Does the response ad- dress the prompt appropriately?

work page

[6] [6]

Coherence– Is the response well- organized and logically structured?

work page

[7] [7]

Completeness– Does the response fully answer the question?

work page

[8] [8]

Accuracy– Is the information pro- vided correct?

work page

[9] [9]

Quality– Is the writing clear, fluent, and human-like? How to Respond In the ‘Your Preference’ column, enter: •A– if Response A is clearly better •B– if Response B is clearly better • Tie– if both responses are roughly equal in quality Important Notes • Read both responses completely before deciding • Focus on overall quality, not just length • Ideally, u...

work page 2058