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.

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.

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)

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).

• 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

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

• 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