ROC-n-reroll: How verifier imperfection affects test-time scaling
This work provides theoretical insights for researchers and practitioners in language model optimization, addressing a gap in understanding verifier effects on test-time scaling, though it is incremental as it builds on existing empirical studies.
The paper tackles the lack of theoretical understanding of how verifier imperfection affects test-time scaling methods like Best-of-N and Rejection Sampling, proving that instance-level accuracy is characterized by the verifier's ROC curve, with experiments showing RS outperforms BoN for fixed compute and both converge in the infinite-compute limit.
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.