💬 NLP

Agreement in Representation Space for Open-Ended Self-Consistency

This paper introduces Embedding-Based Agreement (EBA), a training-free method that extends self-consistency to open-ended generation tasks by leveraging geometric clustering in representation space to identify high-quality outputs, demonstrating that semantic concentration in embedding space is a more robust and scalable signal for reliability than exact symbolic matching or recent LLM-based selection approaches.

Original authors: Paula Ontalvilla, Gorka Azkune, Aitor Ormazabal

Published 2026-06-11

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Paula Ontalvilla, Gorka Azkune, Aitor Ormazabal

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Problem: Asking for a Second Opinion

Imagine you ask a very smart but sometimes chatty AI to solve a math problem or write a piece of code. To get the best answer, you ask it the same question 100 times. This is called "sampling."

In simple tasks with one right answer (like "What is 2+2?"), you can just count the votes. If 90 times it says "4" and 10 times it says "5," you pick "4." This is the old way of doing things, called Self-Consistency.

But what if the task is open-ended?

Code: You ask for a function to sort a list. One answer uses Python, another uses JavaScript, and a third uses a different sorting algorithm. They all work, but they look totally different. You can't just "count votes" because no two answers are exactly the same word-for-word.
Summarization: You ask to summarize a news article. One summary is short and punchy; another is long and detailed. Both are good, but they don't match.

The old method fails here because it looks for exact matches (like finding identical twins). The authors of this paper wanted to find a way to measure agreement even when the answers look different on the surface.

The New Idea: The "Crowded Room" Analogy

The authors propose a new way to think about agreement. Instead of looking at the words the AI generates, they look at the meaning behind them.

Imagine the AI's "brain" (its internal representation space) is a giant, invisible room.

The Hypothesis: When the AI generates many different answers that are actually correct and similar in meaning, they don't scatter randomly. Instead, they tend to huddle together in a specific, crowded corner of the room.
The Noise: When the AI generates wrong or weird answers, they tend to wander off into the empty, isolated corners of the room, far away from the crowd.

So, "agreement" isn't about everyone saying the exact same sentence. It's about everyone standing in the same geometric neighborhood.

The Solution: EBA (Embedding-Based Agreement)

To test this, the authors created a tool called EBA. Here is how it works, step-by-step:

Ask the AI: Generate 100 different answers to the same question.
Map the Answers: Turn every single answer into a coordinate point in that giant invisible room (using something called an "embedding").
Find the Crowd: Look for the biggest cluster of points. Where are most of the answers gathering?
Pick the Leader: Choose the answer that is closest to the center of that biggest crowd.

The Analogy: Imagine you are at a party and you want to know what the "main topic" of conversation is.

Old Method: You listen for people shouting the exact same phrase. If no one shouts the exact same phrase, you give up.
EBA Method: You look at the room. You see a big group of people standing in a circle, laughing and talking. Even if they are saying different words, their body language and location show they are all agreeing on the same topic. You pick the person standing right in the middle of that circle as the "best" answer.

What They Found

The authors tested this on three types of tasks: Math, Coding, and Summarization.

It Works Better Than Random: Just picking a random answer from the 100 generated is like picking a random person in the whole building. EBA picks someone from the "smart crowd," and it consistently gets better results.
More Samples = Better Results: The more answers you ask the AI to generate, the clearer the "crowd" becomes. It's like if you only have 5 people in a room, it's hard to tell where the group is. If you have 200 people, the crowd is obvious. EBA gets smarter as you give it more data.
It's Stable: Other methods that try to solve this problem (like asking a second AI to judge the answers) often get confused or run out of memory if you ask for too many answers. EBA stays steady and reliable even with huge numbers of samples.
The "Center" is King: They discovered something fascinating: The answers that are located near the center of the geometric cluster are almost always the best, most reliable answers. The answers that are far out on the edges (the "peripheral" ones) are usually the bad or inaccurate ones.

Why This Matters

This paper changes how we view AI consistency.

Before: We thought consistency meant "saying the same thing."
Now: We know consistency is a geometric property. It means "standing in the same neighborhood of meaning."

This allows us to use the "Self-Consistency" trick (asking many questions and picking the best one) for complex tasks like writing code or summarizing books, not just simple math problems. It proves that even when AI answers look different on the outside, their "inner geometry" reveals which ones are truly in agreement.

Technical Summary: Agreement in Representation Space for Open-Ended Self-Consistency

Problem Statement

Self-consistency has emerged as a primary inference-time strategy for enhancing the reasoning capabilities of Large Language Models (LLMs). Classical methods, such as those proposed by Wang et al. (2022), operate by sampling multiple reasoning paths and selecting the most frequent final answer via majority voting. While highly effective for tasks with categorical outputs (e.g., mathematical reasoning, multiple-choice questions), this approach relies on exact symbolic matching.

Extending self-consistency to open-ended generation tasks—such as code synthesis, text summarization, and free-form reasoning—presents a significant challenge. In these settings, semantically equivalent responses often differ substantially at the lexical or structural level, rendering exact matching ineffective. Recent attempts to address this, such as Universal Self-Consistency (USC) (Chen et al., 2023) which uses LLM-based evaluation, or Self-Certainty (SCe) (Kang et al., 2026) which relies on uncertainty signals, introduce new limitations regarding scalability, computational cost, and stability.

Methodology: Embedding-Based Agreement (EBA)

The authors hypothesize that self-consistency in open-ended generation is not merely a property of symbolic overlap but a geometric property of the generation space. Specifically, they posit that semantically compatible generations concentrate in similar, dense regions of the representation space, while inconsistent or low-quality generations appear isolated or sparsely distributed.

To operationalize this hypothesis, the paper introduces Embedding-Based Agreement (EBA), a training-free framework that estimates consistency through the following steps:

Sampling: Given an input prompt $x$ , $N$ candidate generations $\{y_1, \dots, y_N\}$ are sampled from a language model.
Embedding: Each generation is mapped into a continuous representation space using an embedding function $\phi(\cdot)$ , producing vectors $\{e_1, \dots, e_N\}$ .
Clustering: The embeddings are clustered using agglomerative clustering with cosine distance. The number of clusters is determined automatically via silhouette analysis.
Selection: Clusters containing a higher number of generations are interpreted as regions of strong agreement. The final prediction is selected as the sample closest to the centroid of the dominant cluster.

The framework is designed to be model-agnostic and does not require additional training or external LLM evaluators for the selection process.

Key Contributions

Geometric Interpretation of Self-Consistency: The paper reframes self-consistency from a discrete voting mechanism to a geometric phenomenon where agreement emerges from the collective organization of sampled generations in representation space.
EBA Framework: The introduction of a simple, training-free method (EBA) that leverages clustering in embedding space to select high-quality outputs for open-ended tasks.
Empirical Validation: Comprehensive experiments across three distinct domains: mathematical reasoning (MATH500), code generation (HumanEval), and summarization (CNN/DM).
Analysis of Scaling and Robustness: The study demonstrates that agreement signals strengthen with increased sampling budgets and remain robust across different model families (Llama, Qwen), model scales (8B to 70B), and embedding spaces (pretrained models vs. native hidden representations).
Geometric Quality Correlation: The analysis reveals a strong correlation between the geometric location of a generation and its quality. Generations concentrated near the central regions of the representation space yield more reliable outputs, whereas peripheral generations are significantly less accurate.

Experimental Results

The authors evaluated EBA against random selection, USC, and SCe using Llama 3.1-8B/70B and Qwen 3-8B/32B models.

Performance Improvement: EBA consistently outperforms random selection. Improvements are most pronounced in code generation (up to +4 points on HumanEval) and mathematical reasoning (up to +17 points on MATH500). Gains in summarization are smaller but consistent (+1 ROUGE-1 point).
Scaling Behavior: Unlike USC, which often degrades as the number of candidates increases due to context limits or evaluation noise, and SCe, which shows unstable scaling, EBA exhibits stable, near-monotonic improvement as the sampling budget ( $n$ ) increases from 1 to 256.
Robustness: The effectiveness of EBA holds across different embedding models (Qwen, Gemma) and even when using native hidden representations extracted directly from the generation model, suggesting the signal is intrinsic to the generation distribution.
Connection to Classical SC: On MATH500, when applied to extracted final answers, EBA closely mirrors the scaling behavior of classical majority voting, suggesting it is a continuous extension of the original self-consistency principle.

Significance and Claims

The paper claims that viewing self-consistency as a geometric property of sampled generations offers a unified perspective for both categorical and open-ended tasks. The primary significance of this work lies in:

Providing a robust, scalable, and computationally efficient alternative to LLM-based evaluation for selecting consistent outputs in open-ended generation.
Demonstrating that consistency signals are stable across model scales and embedding spaces, including native representations.
Establishing that the "center" of the representation space occupied by a distribution of generations is a strong indicator of output reliability.

The authors conclude that their findings support a shift in understanding self-consistency away from exact symbolic overlap toward the geometric organization of the generation space, opening new directions for inference-time selection based on representation-space structure.

Limitations

The paper acknowledges several limitations:

Representation Geometry: Current embedding spaces may exhibit low-separation geometric structures (e.g., concentration in narrow high-dimensional cones), meaning proximity does not always perfectly equate to semantic equivalence.
Clustering Heuristics: The reliance on agglomerative clustering and silhouette analysis may not fully capture fine-grained local structures in the generation space.
Computational Overhead: While cheaper than LLM evaluation, EBA still requires generating multiple samples and computing embeddings, which can be costly for very large sampling budgets or long generations.
Scope: Evaluation is limited to summarization, code, and math; further assessment in domains like dialogue or agentic reasoning is needed.