# Majority of the Bests: Improving Best-of-N via Bootstrapping

arXiv: https://arxiv.org/abs/2511.18630 · HTML: https://mlsanity.com/paper/2511.18630/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Placed in Test-time compute and efficient thinking, one of 17 directions in LLM reasoning.
- Builds on 21 of its 40 references that are in the map.
- First posted on arXiv 23 Nov 2025.

## Placed in

- LLM reasoning / [Test-time compute and efficient thinking](https://mlsanity.com/lr/test-time-compute-and-efficient-thinking/)
- Synthetic data and self-improvement for LLMs / [Self-rewarding alignment and self-training](https://mlsanity.com/synth/self-rewarding-alignment-and-self-training/)

## Builds on (most cited first, 21 of 21)

- [The Llama 3 Herd of Models](https://mlsanity.com/paper/2407.21783/) (2024)
- [Chain-of-Thought Prompting Elicits Reasoning in Large Language Models](https://mlsanity.com/paper/2201.11903/) (2022)
- [Training Verifiers to Solve Math Word Problems](https://mlsanity.com/paper/2110.14168/) (2021)
- [Self-Consistency Improves Chain of Thought Reasoning in Language Models](https://mlsanity.com/paper/2203.11171/) (2022)
- [Measuring Mathematical Problem Solving With the MATH Dataset](https://mlsanity.com/paper/2103.03874/) (2021)
- [Let's Verify Step by Step](https://mlsanity.com/paper/2305.20050/) (2023)
- [Self-Refine: Iterative Refinement with Self-Feedback](https://mlsanity.com/paper/2303.17651/) (2023)
- [Tree of Thoughts: Deliberate Problem Solving with Large Language Models](https://mlsanity.com/paper/2305.10601/) (2023)
- [MMLU-Pro: A More Robust and Challenging Multi-Task Language Understanding Benchmark](https://mlsanity.com/paper/2406.01574/) (2024)
- [CommonsenseQA: A Question Answering Challenge Targeting Commonsense Knowledge](https://mlsanity.com/paper/1811.00937/) (2018)
- [s1: Simple test-time scaling](https://mlsanity.com/paper/2501.19393/) (2025)
- [Large Language Monkeys: Scaling Inference Compute with Repeated Sampling](https://mlsanity.com/paper/2407.21787/) (2024)
- [Gemma: Open Models Based on Gemini Research and Technology](https://mlsanity.com/paper/2403.08295/) (2024)
- [Solving math word problems with process- and outcome-based feedback](https://mlsanity.com/paper/2211.14275/) (2022)
- [Reasoning with Language Model is Planning with World Model](https://mlsanity.com/paper/2305.14992/) (2023)
- [Interpretable Preferences via Multi-Objective Reward Modeling and Mixture-of-Experts](https://mlsanity.com/paper/2406.12845/) (2024)
- [ReST-MCTS*: LLM Self-Training via Process Reward Guided Tree Search](https://mlsanity.com/paper/2406.03816/) (2024)
- [Skywork-Reward-V2: Scaling Preference Data Curation via Human-AI Synergy](https://mlsanity.com/paper/2507.01352/) (2025)
- [Regularizing Hidden States Enables Learning Generalizable Reward Model for LLMs](https://mlsanity.com/paper/2406.10216/) (2024)
- [Consistency of random forests](https://mlsanity.com/paper/1405.2881/) (2014)
- [Evaluation of Best-of-N Sampling Strategies for Language Model Alignment](https://mlsanity.com/paper/2502.12668/) (2025)

## Built on by (most cited first, 7 of 7)

- [What If We Allocate Test-Time Compute Adaptively?](https://mlsanity.com/paper/2602.01070/) (2026)
- [FUSE: Ensembling Verifiers with Zero Labeled Data](https://mlsanity.com/paper/2604.18547/) (2026)
- [AVIS: Adaptive Test-Time Scaling for Vision-Language Models](https://mlsanity.com/paper/2606.11576/) (2026)
- [Believe Your Model: Distribution-Guided Confidence Calibration](https://mlsanity.com/paper/2603.03872/) (2026)
- [From the Inside Out: Progressive Distribution Refinement for Confidence Calibration](https://mlsanity.com/paper/2603.16500/) (2026)
- [LACE: Lattice Attention for Cross-thread Exploration](https://mlsanity.com/paper/2604.15529/) (2026)
- [Tail Competition Explains Scaling in Best-of-N Sampling for Verifiable Problems](https://mlsanity.com/paper/2610.19362/) (2026)

## Abstract (arXiv)

Sampling multiple outputs from a Large Language Model (LLM) and selecting the most frequent (Self-consistency) or highest-scoring (Best-of-N) candidate is a popular approach to achieve higher accuracy in tasks with discrete final answers. Best-of-N (BoN) selects the output with the highest reward, and with perfect rewards, it often achieves near-perfect accuracy. With imperfect rewards from reward models, however, BoN fails to reliably find the correct answer and its performance degrades drastically. We consider the distribution of BoN's outputs and highlight that, although the correct answer does not usually have a probability close to one under imperfect rewards, it is often the most likely outcome. This suggests that the mode of this distribution can be more reliably correct than a sample from it. Based on this idea, we propose Majority-of-the-Bests (MoB), a novel selection mechanism that estimates the output distribution of BoN via bootstrapping and selects its mode. Experimental results across five benchmarks, three different base LLMs, and two reward models demonstrate consistent improvements over BoN in 25 out of 30 setups. We also provide theoretical results for the consistency of the bootstrapping. MoB serves as a simple, yet strong alternative to BoN and self-consistency, and more broadly, motivates further research in more nuanced selection mechanisms.