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

arXiv: https://arxiv.org/abs/2608.14665 · HTML: https://mlsanity.com/paper/2608.14665/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 10 of its 17 references that are in the map.
- First posted on arXiv 1 Aug 2026.

## Builds on (most cited first, 10 of 10)

- [Evaluating Large Language Models Trained on Code](https://mlsanity.com/paper/2107.03374/) (2021)
- [Inference-Aware Fine-Tuning for Best-of-N Sampling in Large Language Models](https://mlsanity.com/paper/2412.15287/) (2024)
- [Pass@K Policy Optimization: Solving Harder Reinforcement Learning Problems](https://mlsanity.com/paper/2505.15201/) (2025)
- [How Do Large Language Monkeys Get Their Power (Laws)?](https://mlsanity.com/paper/2502.17578/) (2025)
- [Weight Ensembling Improves Reasoning in Language Models](https://mlsanity.com/paper/2504.10478/) (2025)
- [Optimizing Temperature for Language Models with Multi-Sample Inference](https://mlsanity.com/paper/2502.05234/) (2025)
- [Diverse Preference Learning for Capabilities and Alignment](https://mlsanity.com/paper/2511.08594/) (2025)
- [On the Role of Temperature Sampling in Test-Time Scaling](https://mlsanity.com/paper/2510.02611/) (2025)
- [Why Pass@k Optimization Can Degrade Pass@1: Prompt Interference in LLM Post-training](https://mlsanity.com/paper/2602.21189/) (2026)
- [Learning Adaptive LLM Decoding](https://mlsanity.com/paper/2603.09065/) (2026)

## Abstract (arXiv)

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 pt(X) be one-sample success probability at temperature t, and define the conditional log-success response mt(u)=mathbbE[dot pt(X)mid pt(X)=u]/u. If mt(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 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.