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POOL: Propagated Uncertainty Over Lookalikes
Rounak Sharma, Ananya B. Sai, Soumyabrata Pal
TL;DR
Black-box confidence estimation must distinguish likely-correct from likely-incorrect answers while balancing accuracy and generation cost. POOL clusters semantically related queries, evaluates representative medoids, propagates scores, and rescues high-disagreement cases; with HY@5, it preserves most AUROC while reducing generations.
Problem
Black-box confidence estimation must separate likely-correct from likely-incorrect outputs while remaining cheap enough for large benchmarks and production query streams.
Method
POOL shares a black-box estimator across overlapping semantic query clusters by evaluating medoids, softly propagating scores, and selectively evaluating high-disagreement queries.
Results
Across six domains, three datasets, and five LLMs, POOL-HY@5 retains 93.5–97.9% of unpooled HY@5 AUROC while saving 19.3–39.3% of generations.
Takeaways & Limitations
Semantic redundancy in paraphrase-dense workloads can reduce confidence-estimation generations, with savings reaching 73-76%.
Takeaways & Limitations
POOL requires a batch available upfront and its savings accrue only after a sufficient query cache has been built.
Abstract
from arXiv · showhide
Black-box large language models need confidence scores that can separate likely-correct from likely-incorrect outputs, enabling systems to prioritize human review, route uncertain cases to stronger models, or choose abstention thresholds on development data. Yet existing confidence estimators face a cost-quality trade-off: verbal confidence is cheap but is often overconfident, while sampling-based uncertainty is more informative but scales linearly with the number of samples per query. We propose \textsc{POOL} (\emph{Propagated Uncertainty Over Lookalikes}),a cost-efficient framework that addresses this trade-off taking inspiration from group-testing.\textsc{POOL} clusters query stems with overlaps, evaluates a base estimator on representative medoids, softly propagates confidence scores to nearby queries, and selectively evaluates high-disagreement cases. We instantiate this framework with \textsc{Hy@}$p$, a hybrid estimator that combines verbal confidence with spectral answer diversity computed from the negative von Neumann entropy of sampled answer embeddings.Across six domains from three datasets and five black-box LLMs, \textsc{Hy@}5 achieves higher average AUROC than verbal confidence and \textsc{Vn@}10 sampling while using half as many samples as \textsc{Vn@}10. \textsc{POOL}-\textsc{Hy@}5 retains 93.5--97.9\% of its AUROC while saving 19.3--39.3\% of generations. On paraphrase-dense workloads, generation savings rise to 73-76\%, showing that semantic redundancy can be leveraged to lower confidence-estimation costs.
1 Introduction
POOL addresses the accuracy–cost trade-off in black-box confidence estimation by sharing estimator calls across semantically related queries. Its hybrid estimator and pooled evaluation preserve ranking quality while reducing generations.
- Black-box confidence scores must separate likely-correct from likely-incorrect outputs while remaining cheap for large-scale use.They support abstention, stronger-model or human routing, inference-budget allocation, and review prioritization.
- Verbal confidence is inexpensive but often overconfident, whereas sampling-based uncertainty provides a different accuracy–cost trade-off.
- POOL clusters overlapping query stems, evaluates representative medoids, propagates scores to member queries, and selectively rescues high-disagreement cases.The framework is inspired by group testing and treats the base estimator as a black box.
- HY@5 combines standard black-box signals as a low-cost query-specific estimator and achieves the best average AUROC among the reported per-query baselines.The evaluation covers multiple base estimators and black-box LLMs.
- 93.5–97.9% of unpooled AUROC is retained while confidence-estimation generations fall by 19.3–39.3%; paraphrase-dense workloads reach 73-76% savings.These results show that semantic redundancy can be exploited for cheaper confidence estimation.
2 Related Works
The related work spans black-box confidence elicitation, sampling-based uncertainty estimation, and classical group testing. POOL combines these lines by applying pooled evaluation to semantically similar LLM queries.
- Black-box confidence research includes verbal elicitation, sampling-based uncertainty estimation, and classical group testing.
- Verbal-confidence studies examine whether language models can assess uncertainty without access to internal logits.Prior work reports partial self-knowledge, improved calibration from prompting, and confidence collapse in instruction-tuned models.
- Sampling-based methods use agreement among stochastic generations as a black-box uncertainty signal.Examples include majority-vote self-consistency, hallucination detection through inter-sample consistency, and semantic entropy.
- Classical group testing identifies defective items by testing pools rather than individuals, reducing the need for individual tests.
- POOL applies a group-testing reduction to LLM confidence estimation by clustering semantically similar questions and evaluating only cluster representatives.
3 Method
POOL wraps a per-query confidence estimator in a two-tier pipeline: representative medoid evaluation and soft propagation, followed by targeted rescue of disagreement cases. Its clustering and rescue controls trade generation cost against ranking quality.
- POOL produces scores that rank likely-correct outputs above likely-incorrect ones rather than calibrated probabilities.
- The base method M is treated as a black box, allowing VC, VN@p, HY@p, MAJ@p, and SE@p to plug into the framework unchanged.
- Tier-1: Representative Selection and Soft Propagation: Agglomerative clustering selects medoids that cover the batch in stem-embedding space, while the cutoff τ controls the compression ratio c = K/N.Smaller τ yields fewer, larger clusters.
- Tier-1: Representative Selection and Soft Propagation: POOL evaluates M once per medoid, then combines sufficiently similar medoid scores with soft attention so boundary queries can blend evidence from multiple medoids.Queries fall back to the nearest medoid when no medoid meets the similarity threshold.
- Tier-1: Representative Selection and Soft Propagation: The same attention weights provide a disagreement-based reliability signal without extra computation.Small disagreement indicates neighbour agreement; large disagreement indicates that the soft average may hide a conflict.
- Tier-2: Disagreement-Triggered Rescue: POOL directly reevaluates the top-⌈qN⌉ non-medoid queries by disagreement and overwrites their propagated scores.This query-level rescue targets uncertain cases rather than spending extra budget uniformly across clusters.
- Tier-2: Disagreement-Triggered Rescue: The total cost is K + ⌈qN⌉ base-method calls, with q controlling the rescue budget from the Tier-1 floor to the unpooled estimator.τ and q are deployment-level cost–quality knobs, while embeddings and clustering are computed once per batch.
4 Experiments
Across six domains, three datasets, and five black-box LLMs, POOL evaluates confidence estimators on representative query clusters, propagates scores, and rescues high-disagreement queries. HY@5 is the strongest base estimator, while POOL preserves most AUROC and reduces generations, especially for redundant workloads.
- Experimental setup: Experiments cover six domains across MMLU-Pro, MATH-500, and TriviaQA using five black-box LLMs.The evaluation spans multiple question formats and model families, including reasoning and non-reasoning configurations.
- Method: POOL wraps a black-box per-query estimator with clustered medoid evaluation, soft similarity-weighted propagation, and disagreement-triggered rescue.Queries are clustered using embeddings; medoids receive base-estimator calls, while selected non-medoids are directly reevaluated.
- Main results: HY@5 has the highest average AUROC at 0.752, ahead of VN@10 at 0.737 and verbal confidence at 0.689.HY@5 matches or beats VN@10 in 19 of 30 cells, with mean +0.014 AUROC, while using five rather than ten samples.
- Main results: POOL-HY@5 retains 93.4%/95.9%/97.9% of unpooled AUROC at 39.3%/29.3%/19.3% generation savings as q increases from 0 to 0.20.The rescue fraction q provides a cost-quality trade-off, while τ controls the number of directly evaluated queries.
- Workload structure: Dense MMLU-Pro and MATH-500 splits yield 20–54% savings at τ =0.55, with the largest savings on Law and MATH-500 and the smallest on Psychology.Savings track stem redundancy and are determined by workload structure rather than model correctness.
- Paraphrase-augmented datasets: On paraphrase-augmented workloads, POOL-HY@5 saves 73–76% of generations while retaining approximately 96% of unpooled AUROC.Answer-preserving near-duplicates allow representative selection to convert semantic density into fewer LLM calls.
5 Conclusion
POOL reduces black-box confidence-estimation cost by amortizing estimator calls across semantic neighborhoods while retaining most ranking quality. Its HY@5 instantiation preserves 93.5–97.9% of unpooled AUROC with 19.3–39.3% fewer generations, and savings reach 73–76% on paraphrase-dense workloads.
- Conclusion: POOL amortizes confidence-estimator calls across semantic neighborhoods and can wrap any base estimator treated as a black box.Queries are clustered by semantic similarity, with representative scores propagated to related queries.
- Conclusion: 93.5–97.9% of unpooled HY@5 AUROC is retained while saving 19.3–39.3% of generations, depending on rescue budget.HY@5 combines verbal confidence with spectral sample diversity and serves as POOL’s supporting instantiation.
- Conclusion: 73–76% generation savings are achieved on paraphrase-dense workloads.The reported savings indicate that compression increases with workload redundancy.
- Conclusion: Adaptive medoid selection, alternative rescue triggers, and extension to open-ended generation remain open directions.These directions target local embedding density, rescue selection, and broader generation settings.
Limitations
POOL’s savings depend on having a sufficiently populated query batch or cache and on the assumption that similar query embeddings imply similar confidence profiles. These constraints are weaker for dense, redundant workloads and may fail in adversarial or sparse settings.
- Limitations: POOL requires the full query batch upfront for clustering, medoid selection, and score propagation.Online one-at-a-time query arrival does not directly provide the batch structure required by the method.
- Limitations: Cold-start savings accrue only after a sufficient query cache has been built; initially, POOL may evaluate every query individually.The paper recommends a warm-up phase using the unpooled estimator before activation.
- Limitations: The required warm-up size is workload-dependent, with dense topical workloads converging faster than sparse open-domain benchmarks.Customer-support FAQs are given as an example of a denser workload.
- Limitations: Input-space propagation assumes that cosine-similar query stems have similar confidence profiles, but this may break down in extreme adversarial scenarios.The limitation is inherent to similarity-based approaches and is stress-tested in the paper.
A Ethics Statement
The work targets lower-cost confidence estimation for black-box LLMs to support selective prediction and abstention. Its evaluation uses public academic benchmarks without collecting private or personally identifiable data or involving human subjects.
- A Ethics Statement: POOL aims to reduce black-box LLM confidence-estimation cost and make selective prediction and abstention mechanisms more accessible.
- A Ethics Statement: The experiments use publicly available MMLU-Pro, MATH-500, and TriviaQA benchmarks, with no private or personally identifiable data collected.
- A Ethics Statement: No human subjects were involved in the experiments.
- A Ethics Statement: No new models were trained; the method uses inference calls to existing commercial and open-weight APIs.
B Additional Results
Additional analyses support HY’s complementary signal fusion and characterize POOL’s cost overhead and cost-savings behavior. Z-scored fusion improves AUROC over individual channels, while embedding and clustering add little to total LLM cost.
- B Additional Results: Approximately 0.06 AUROC is gained by fusing verbal confidence and VN@5, whose errors are complementary.The two signals are weakly correlated, with Spearman ρ = 0.24.
- B Additional Results: Replacing the z-scored sum with max loses 0.04 AUROC, while dropping z-scoring costs 0.017 AUROC.The max rule can allow one over-confident channel to promote wrong answers, and the channels otherwise operate on incompatible scales.
- B Additional Results: HY@5 and HY@10 have Spearman correlation ρ = 0.89, indicating that increasing the sample count refines rather than qualitatively changes the ranking.VC is only weakly correlated with VN@5 (ρ = 0.24) and VN@10 (ρ = 0.28).
- B Additional Results: At q = 0.20, embedding dollars account for 0.7–2.3% of total per-query cost, averaging 1.18%.The reported operating point incurs approximately 3.3–5.0 LLM generations and 2.2–3.0 embedding calls per query.
- B Additional Results: POOL’s relative cost reduction depends only on clustering compression and rescue fraction, not on the base estimator’s generation cost.The per-query cost formula causes the base-method factor to cancel in the savings ratio.
- B Additional Results: Adding embedding, clustering, and similarity-search overhead multiplies the generation-based cost by at most 1.02–1.03× in the worst case.CPU operations complete in less than one second per split, compared with approximately 1–3 seconds per LLM generation.
C Per-Query Base Estimator
The paper compares verbal, sample-diversity, hybrid, majority-agreement, and semantic-entropy estimators for black-box confidence. The hybrid combines verbal confidence with spectral sample diversity, while POOL-specific normalization uses evaluated calls only.
- Verbal confidence is cheap but prone to confidence collapse, whereas sample-based estimators cannot recover the model’s own commitment.
- VN@p draws p answers, embeds them, and forms a Gram matrix that is PSD-projected and trace-normalized into a spectral representation.
- Higher negative von Neumann entropy values, closer to zero, indicate spectrally concentrated samples and therefore higher confidence.
- HY@p fuses verbal confidence with VN@p because verbal confidence can be confidently wrong and VN@p can be misled by paraphrases of one wrong answer.
- For POOL-wrapped HY@p, z-score statistics come only from evaluated medoid and rescue calls, while verbal confidence is elicited alongside one sampled answer.
- MAJ@p measures the fraction of samples matching the modal answer, while SE@p uses hard semantic equivalence classes and Shannon entropy over cluster sizes.
D Baseline Details
The baseline analysis compares complete per-query estimators, POOL rescue rules, and supervised input-based probes. Supervised probes rank poorly despite label access, while POOL reads signal directly from sampled model behavior.
- End-to-end baselines produce full per-query confidence estimates, whereas rescue baselines share POOL’s propagation and vary only in Tier-2 selection.
- rand+kNN matches POOL’s medoid budget, evaluates randomly chosen queries, and predicts remaining scores with similarity-weighted k-nearest-neighbor regression.
- P(IK)-LR, P(IK)-MLP, and APRICOT are adapted into black-box probes using question or answer embeddings and supervised correctness-related targets.
- Despite label access, supervised probes rank poorly because stem-only features weakly predict instance correctness, the MLP overfits, and APRICOT captures coarse topic difficulty.
- Table 12 evaluates nine per-query estimators across five models and six datasets, including VC, VN@p, HY@p, MAJ@p, and SE@p.
- The rescue rules select random queries, median-boundary scores, confidence tails, disagreement, or an oracle based on true propagation error.
E Metric Details
The evaluation separates ranking quality from calibration quality. AUROC is the primary metric for separating correct from incorrect outputs, while ECE assesses whether scores behave like probabilities.
- The paper evaluates confidence estimates with AUROC for ranking quality and Expected Calibration Error for calibration quality.
- AUROC measures the probability that a randomly chosen correct answer receives a higher score than a randomly chosen incorrect answer.
- AUROC uses 0.5 for chance and 1.0 for perfect separation, and its invariance to monotone rescaling permits comparison across score scales.
- ECE partitions predictions into 15 equal-width confidence bins and compares empirical accuracy with mean predicted confidence, with lower values better and 0 indicating perfect calibration.
- Because VN@p and HY@p are unbounded z-scored values, the paper maps them to probabilities with one-dimensional Platt scaling before computing ECE.
F Additional Embedding Models
The evaluation checks whether POOL’s ranking and cost-quality trade-off depend on the embedding encoder. It also documents implementation prompts and encoder-specific threshold recalibration.
- Repeating the evaluation with bge-large-en-v1.5 and e5-large-v2 preserves POOL’s ranking and cost-quality trade-off alongside text-embedding-3-small.
- Because cosine-similarity scales differ across encoders, τ is recalibrated per encoder to match mean compression K/N while keeping θ = τ − 0.15.
- The appendix reports average AUROC across embedding encoders with this matched-compression recalibration.