Best-of-Majority: Minimax-Optimal Strategy for Pass@$k$ Inference Scaling
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915532035325952 |
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| author | Di, Qiwei Ji, Kaixuan Li, Xuheng Zhao, Heyang Gu, Quanquan |
| author_facet | Di, Qiwei Ji, Kaixuan Li, Xuheng Zhao, Heyang Gu, Quanquan |
| contents | LLM inference often generates a batch of candidates for a prompt and selects one via strategies like majority voting or Best-of- N (BoN). For difficult tasks, this single-shot selection often underperforms. Consequently, evaluations commonly report Pass@$k$: the agent may submit up to $k$ responses, and only the best of them is used when computing regret. Motivated by this, we study inference scaling in the more general Pass@$k$ inference setting, and prove that neither majority voting nor BoN exhibits the desirable scaling with $k$ and the sampling budget $N$. Combining the advantages of majority voting and BoN, we propose a new inference strategy called Best-of-Majority (BoM), with a pivotal step that restricts the candidates to the responses with high frequency in the $N$ samples before selecting the top-$k$ rewards. We prove that when the sampling budget is $N=\tildeΩ(C^*)$, the regret of BoM is $O(ε_{\mathrm{opt}}+\sqrt{ε_{\mathrm{RM}}^2C^*/k})$, where $C^*$ is the coverage coefficient, $ε_{\mathrm{RM}}$ is the estimation error of the reward model, and $ε_{\mathrm{opt}}$ is the estimation error of reward at the optimal response. We further establish a matching lower bound, certifying that our algorithm is minimax optimal. Beyond optimality, BoM has a key advantage: unlike majority voting and BoN, its performance does not degrade when increasing $N$. Experimental results of inference on math problems show BoM outperforming both majority voting and BoN. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_03199 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Best-of-Majority: Minimax-Optimal Strategy for Pass@$k$ Inference Scaling Di, Qiwei Ji, Kaixuan Li, Xuheng Zhao, Heyang Gu, Quanquan Machine Learning LLM inference often generates a batch of candidates for a prompt and selects one via strategies like majority voting or Best-of- N (BoN). For difficult tasks, this single-shot selection often underperforms. Consequently, evaluations commonly report Pass@$k$: the agent may submit up to $k$ responses, and only the best of them is used when computing regret. Motivated by this, we study inference scaling in the more general Pass@$k$ inference setting, and prove that neither majority voting nor BoN exhibits the desirable scaling with $k$ and the sampling budget $N$. Combining the advantages of majority voting and BoN, we propose a new inference strategy called Best-of-Majority (BoM), with a pivotal step that restricts the candidates to the responses with high frequency in the $N$ samples before selecting the top-$k$ rewards. We prove that when the sampling budget is $N=\tildeΩ(C^*)$, the regret of BoM is $O(ε_{\mathrm{opt}}+\sqrt{ε_{\mathrm{RM}}^2C^*/k})$, where $C^*$ is the coverage coefficient, $ε_{\mathrm{RM}}$ is the estimation error of the reward model, and $ε_{\mathrm{opt}}$ is the estimation error of reward at the optimal response. We further establish a matching lower bound, certifying that our algorithm is minimax optimal. Beyond optimality, BoM has a key advantage: unlike majority voting and BoN, its performance does not degrade when increasing $N$. Experimental results of inference on math problems show BoM outperforming both majority voting and BoN. |
| title | Best-of-Majority: Minimax-Optimal Strategy for Pass@$k$ Inference Scaling |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2510.03199 |