When Does the Best Sampling Temperature Rise with the Budget? Sufficient Conditions for Pass@k

[2608.14665] When Does the Best Sampling Temperature Rise with the Budget? Sufficient Conditions for Pass@k Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Computer Science > Machine Learning arXiv:2608.14665 (cs) [Submitted on 1 Aug 2026] Title: When Does the Best Sampling Temperature Rise with the Budget? Sufficient Conditions for Pass@k Authors: Changsu Jeong (Independent Researcher) View a PDF of the paper titled When Does the Best Sampling Temperature Rise with the Budget? Sufficient Conditions for Pass@k, by Changsu Jeong (Independent Researcher) View PDF HTML (experimental) Abstract: The temperature that maximizes pass@$k$ is often low for a small sampling budget and higher for a large budget. This pattern has been reported from Codex through recent multi-sample inference studies. It is not an algebraic property of pass@$k$: as Slocum et al. (ICLR 2025) observe, for one fixed task the maximizing temperature is independent of $k$. Building on that fixed-task observation and the hard/easy-task explanation, we give a formal population-level sufficient condition for the aggregate pattern. For task $X$, let $p_t(X)$ be one-sample success probability at temperature $t$, and define the conditional log-success response $m_t(u)=\mathbb{E}[\dot p_t(X)\mid p_t(X)=u]/u$. If $m_t(u)$ is nonincreasing in current success probability, then the normalized temperature derivative of aggregate pass@$k$ is nondecreasing in $k$. Consequently, derivative signs are nested across budgets; if each temperature-performance curve is strictly single-peaked, its unique maximizer is nondecreasing in $k$. The proof identifies the mechanism as a monotone-likelihood-ratio power tilt toward lower-success tasks. We derive a closed-form two-stratum phase diagram, including upward and downward regimes, and show that the marginal temperature derivative admits an exact $\mathrm{Beta}(2,k)$ kernel representation whose kernel concentrates at one-sample success of order $1/k$. Interpreting that scale as task-level localization additionally requires a regular, nonvanishing density-response factor near zero. A signed-moment representation yields diagnostic shape restrictions, while a short appendix records exact discrete refinements of the existing multi-configuration allocation formulation. No language model is trained, and no model query is used as an experimental measurement: the contribution is a conditional theory of an established empirical phenomenon, with assumptions that can be tested in future work. Comments: Theory paper, 13 pages, 1 analytical figure. Deterministic verification code is included as ancillary material. No new language-model experiment Subjects: Machine Learning (cs.LG) Cite as: arXiv:2608.14665 [cs.LG] (or arXiv:2608.14665v1 [cs.LG] for this version) https://doi.org/10.48550/arXiv.2608.14665 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Changsu Jeong [ view email ] [v1] Sat, 1 Aug 2026 08:43:36 UTC (48 KB) Full-text links: Access Paper: View a PDF of the paper titled When Does the Best Sampling Temperature Rise with the Budget? Sufficient Conditions for Pass@k, by Changsu Jeong (Independent Researcher) View PDF HTML (experimental) TeX Source view license Ancillary-file links: Ancillary files ( details ) : REPRODUCIBILITY.txt THEORY_NOTES.txt passk_temperature_theory/__init__.py passk_temperature_theory/make_figure.py passk_temperature_theory/test_extended_theory.py passk_temperature_theory/test_verify_theory.py passk_temperature_theory/verify_theory.py (2 additional files not shown) Current browse context: cs.LG < prev | next > new | recent | 2026-08 Change to browse by: cs References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... 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