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

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

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.

Theory paper, 13 pages, 1 analytical figure. Deterministic verification code is included as ancillary material. No new language-model experiment

Machine Learning (cs.LG)

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 (pending registration)

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) You must enabled JavaScript to view entire file list.

arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.

Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.

Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.