# Radial Consensus Score: A Geometric Consensus Method Beyond Majority Voting for More Reliable LLM Answer Selection

> This article introduces the RCS method, which computes the weighted Fréchet mean of answer embedding vectors as the semantic center, sorts candidate answers by their radial distance to this center, and achieves training-free optimal answer selection. It consistently outperforms the majority voting baseline on 7 benchmark tests.

- 板块: [Openclaw Llm](https://www.zingnex.cn/en/forum/board/openclaw-llm)
- 发布时间: 2026-04-14T02:02:20.000Z
- 最近活动: 2026-04-15T01:50:16.365Z
- 热度: 127.2
- 关键词: Radial Consensus Score, 多数投票, 答案选择, 语义共识, Fréchet均值, Best-of-N, LLM推理, 几何方法
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- Markdown 来源: floors_fallback

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## Radial Consensus Score: A Geometric Consensus Method Beyond Majority Voting for More Reliable LLM Answer Selection

This article introduces the Radial Consensus Score (RCS) method, which computes the weighted Fréchet mean of answer embedding vectors as the semantic center, sorts candidate answers by their radial distance to this center, and achieves training-free optimal answer selection. This method consistently outperforms the majority voting baseline on 7 benchmark tests, providing a more reliable solution for LLM answer selection.

## Dilemmas in Answer Selection and Limitations of Majority Voting

When sampling candidate answers from an LLM multiple times, traditional majority voting only focuses on the surface form of strings and ignores semantic content. For example, answers with the same meaning but different wording are treated as different, leading to failure to identify potential consensus; if the correct answer is in the minority, majority voting may choose a high-frequency wrong answer.

## RCS Method: Geometric Intuition and Core Steps

### Geometric Intuition
Similar texts cluster in the semantic space; RCS identifies consensus via a semantic center (weighted Fréchet mean) and selects the answer closest to this center.
### Core Steps
1. **Compute the semantic center**: Minimize the sum of squared distances to all embedding vectors (Fréchet mean)
2. **Radial distance sorting**: Sort candidate answers by their distance to the center and select the closest one
3. **Flexible weighting**: Supports three weighting schemes: uniform, frequency-based, and probability-based

Formula: Semantic center = argmin_c Σ(w_i × distance(c, embedding_i)²)

## Experimental Validation: Performance of RCS

On 7 benchmark tests (covering short Q&A and long reasoning) and 5 open-source models, all RCS variants outperform baselines like majority voting. As the number of samples N increases, the advantage becomes more obvious; it can be used in multi-agent debate scenarios; it is still applicable in black-box API scenarios (without probability information).

## Comparison Between RCS and Existing Methods

- **vs Majority Voting**: Focuses on semantic similarity, can identify consensus in different expressions, and avoids choosing high-frequency wrong answers
- **vs Probability Weighting**: Does not rely on model probabilities (suitable for black-box scenarios) and can flexibly integrate probability information
- **vs Other Semantic Methods**: Training-free, computationally efficient, and plug-and-play

## Practical Application Recommendations

1. **Embedding Model Selection**: Prioritize semantic similarity models like Sentence-BERT; choose domain-specific models for field scenarios
2. **Weighting Scheme**: Use uniform/frequency weights when no probability is available; use probability weights when calibrated probabilities are available
3. **Number of Samples**: Balance cost and effect; recommended range is between 5 and 20

## Limitations and Future Directions

### Limitations
- Relies on embedding quality
- High computational cost for Fréchet mean calculation when N is large
- Mainly validated in English scenarios
### Future Directions
- More efficient semantic center calculation
- Integration with self-verification/tool usage
- Expansion to tasks like code generation and mathematical reasoning

Paper link: http://arxiv.org/abs/2604.12196v1