# Beyond Majority Voting: Efficient Best-Of-N with Radial Consensus Score

arXiv: https://arxiv.org/abs/2604.12196 · HTML: https://mlsanity.com/paper/2604.12196/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 38 of its 43 references that are in the map.
- First posted on arXiv 14 Apr 2026.

## Builds on (most cited first, 30 of 38)

- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [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)
- [Sentence-BERT: Sentence Embeddings using Siamese BERT-Networks](https://mlsanity.com/paper/1908.10084/) (2019)
- [Qwen3 Technical Report](https://mlsanity.com/paper/2505.09388/) (2025)
- [Efficient Memory Management for Large Language Model Serving with PagedAttention](https://mlsanity.com/paper/2309.06180/) (2023)
- [Self-Consistency Improves Chain of Thought Reasoning in Language Models](https://mlsanity.com/paper/2203.11171/) (2022)
- [Let's Verify Step by Step](https://mlsanity.com/paper/2305.20050/) (2023)
- [GPQA: A Graduate-Level Google-Proof Q&A Benchmark](https://mlsanity.com/paper/2311.12022/) (2023)
- [Gemma 2: Improving Open Language Models at a Practical Size](https://mlsanity.com/paper/2408.00118/) (2024)
- [MMLU-Pro: A More Robust and Challenging Multi-Task Language Understanding Benchmark](https://mlsanity.com/paper/2406.01574/) (2024)
- [Improving Factuality and Reasoning in Language Models through Multiagent Debate](https://mlsanity.com/paper/2305.14325/) (2023)
- [A Survey on Hallucination in Large Language Models: Principles, Taxonomy, Challenges, and Open Questions](https://mlsanity.com/paper/2311.05232/) (2023)
- [Math-Shepherd: Verify and Reinforce LLMs Step-by-step without Human Annotations](https://mlsanity.com/paper/2312.08935/) (2023)
- [Semantic Uncertainty: Linguistic Invariances for Uncertainty Estimation in Natural Language Generation](https://mlsanity.com/paper/2302.09664/) (2023)
- [SelfCheckGPT: Zero-Resource Black-Box Hallucination Detection for Generative Large Language Models](https://mlsanity.com/paper/2303.08896/) (2023)
- [Crowdsourcing Multiple Choice Science Questions](https://mlsanity.com/paper/1707.06209/) (2017)
- [Aligning AI With Shared Human Values](https://mlsanity.com/paper/2008.02275/) (2020)
- [Generating with Confidence: Uncertainty Quantification for Black-box Large Language Models](https://mlsanity.com/paper/2305.19187/) (2023)
- [Deep Think with Confidence](https://mlsanity.com/paper/2508.15260/) (2025)
- [Universal Self-Consistency for Large Language Model Generation](https://mlsanity.com/paper/2311.17311/) (2023)
- [Scalable Best-of-N Selection for Large Language Models via Self-Certainty](https://mlsanity.com/paper/2502.18581/) (2025)
- [Kernel Language Entropy: Fine-grained Uncertainty Quantification for LLMs from Semantic Similarities](https://mlsanity.com/paper/2405.20003/) (2024)
- [BoNBoN Alignment for Large Language Models and the Sweetness of Best-of-n Sampling](https://mlsanity.com/paper/2406.00832/) (2024)
- [Debate or Vote: Which Yields Better Decisions in Multi-Agent Large Language Models?](https://mlsanity.com/paper/2508.17536/) (2025)
- [Reasoning Aware Self-Consistency: Leveraging Reasoning Paths for Efficient LLM Sampling](https://mlsanity.com/paper/2408.17017/) (2024)
- [Looking for a Needle in a Haystack: A Comprehensive Study of Hallucinations in Neural Machine Translation](https://mlsanity.com/paper/2208.05309/) (2022)
- [TokUR: Token-Level Uncertainty Estimation for Large Language Model Reasoning](https://mlsanity.com/paper/2505.11737/) (2025)
- [Integrate the Essence and Eliminate the Dross: Fine-Grained Self-Consistency for Free-Form Language Generation](https://mlsanity.com/paper/2407.02056/) (2024)

## Built on by (most cited first, 2 of 2)

- [Reachability Is Not Realization: Tracing the Sources of LLM Benchmark Gains](https://mlsanity.com/paper/2608.03219/) (2026)
- [Uncertainty Quantification for LLM Agents: A Taxonomy, an Evaluation Protocol, and an Empirical Study](https://mlsanity.com/paper/2609.07395/) (2026)

## Abstract (arXiv)

Large language models (LLMs) frequently generate multiple candidate responses for a given prompt, yet selecting the most reliable one remains challenging, especially when correctness diverges from surface-level majority agreement. Existing approaches, such as self-consistency, rely on discrete voting, while probability-based methods often fail to capture relationships among candidate answers or tend to underweight high-quality but less frequent responses, and do not fully leverage the geometric structure of answer representations. To address these limitations, we introduce Radial Consensus Score (RCS), a simple, efficient, and training-free method for best-of-N selection. RCS models semantic consensus by computing a weighted Fr'echet mean (semantic center) of answer embeddings and ranking candidates by their radial distance to this center. Importantly, RCS provides a general framework that supports multiple weighting schemes, including uniform, frequency-based, and probability-based variants, enabling flexible integration of agreement signals and model confidence while remaining fully applicable in black-box settings. Extensive experiments across seven benchmarks covering short-form QA and long-form reasoning tasks, and five open-weight models, demonstrate that RCS variants consistently outperform strong baselines, with gains becoming more pronounced as the sampling budget increases. RCS also serves as an effective drop-in replacement for majority voting in multi-agent debate and exhibits strong robustness in black-box scenarios. Overall, these results highlight geometric consensus as a scalable and broadly applicable principle for reliable answer selection, extending beyond majority voting to more expressive and robust aggregation in LLM inference.