ROC-n-reroll: How verifier imperfection affects test-time scaling

Test-time scaling aims to improve language model performance by leveraging additional compute during inference. While many works have empirically studied techniques like Best-of-N (BoN) and rejection sampling that make use of a verifier to enable test-time scaling, there is little theoretical understanding of how verifier imperfection affects performance. In this work, we address this gap. Specifically, we prove how instance-level accuracy of these methods is precisely characterized by the geometry of the verifier's ROC curve. Interestingly, while scaling is determined by the local geometry of the ROC curve for rejection sampling, it depends on global properties of the ROC curve for BoN. As a consequence when the ROC curve is unknown, it is impossible to extrapolate the performance of rejection sampling based on the low-compute regime. Furthermore, while rejection sampling outperforms BoN for fixed compute, in the infinite-compute limit both methods converge to the same level of accuracy, determined by the slope of the ROC curve near the origin. Our theoretical results are confirmed by experiments on GSM8K using different versions of Llama and Qwen to generate and verify solutions.