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This research establishes a theoretical framework linking verifier imperfections to the geometry of ROC curves in test-time scaling methods. The study demonstrates that Rejection Sampling outperforms Best-of-N under fixed compute constraints while proving that low-compute observations cannot reliably predict high-compute performance limits for these techniques. These findings provide critical guidance for optimizing inference strategies as models scale toward infinite-compute regimes.
ROC-n-reroll: How verifier imperfection affects test-time scaling
arXiv:2507.12399v3 Announce Type: replace Abstract: Test-time scaling aims to improve language model performance by leveraging additional compute during inference. Many works have empirically studied techniques such as Best-of-N (BoN) and Rejection Sampling (RS) that make use of a verifier to enable test-time scaling. However, to date there is little theoretical understanding of how verifier imperfection affects performance -- a gap we address in this work. Specifically, we prove that the instance-level accuracy of these methods is precisely characterized by the geometry of the verifier's ROC curve. Our theory has two important takeaways, confirmed by experiments with Qwen and LLama models on GSM8K and MATH500. First, RS outperforms BoN for fixed compute, while both methods converge to the same accuracy in the infinite-compute limit. Second, it is generally impossible to predict the high-compute performance of either method based on observations in the low-compute regime.
Abstract:Test-time scaling aims to improve language model performance by leveraging additional compute during inference. Many works have empirically studied techniques such as Best-of-N (BoN) and Rejection Sampling (RS) that make use of a verifier to enable test-time scaling. However, to date there is little theoretical understanding of how verifier imperfection affects performance -- a gap we address in this work. Specifically, we prove that the instance-level accuracy of these methods is precisely characterized by the geometry of the verifier's ROC curve. Our theory has two important takeaways, confirmed by experiments with Qwen and LLama models on GSM8K and MATH500. First, RS outperforms BoN for fixed compute, while both methods converge to the same accuracy in the infinite-compute limit. Second, it is generally impossible to predict the high-compute performance of either method based on observations in the low-compute regime.
Machine Learning (cs.LG); Machine Learning (stat.ML)
arXiv:2507.12399 [cs.LG]
(or arXiv:2507.12399v3 [cs.LG] for this version)
From: Florian E. Dorner [view email] [v1] Wed, 16 Jul 2025 16:44:29 UTC (3,950 KB) [v2] Fri, 10 Oct 2025 07:34:34 UTC (3,868 KB) [v3] Mon, 17 Aug 2026 15:56:15 UTC (3,936 KB)
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