Conditional Factuality Controlled LLMs with Generalization Certificates via Conformal Sampling

Kai Ye; Qingtao Pan; Shuo Li

cs.LGMar 28, 2026

Conditional Factuality Controlled LLMs with Generalization Certificates via Conformal Sampling

Authors: Kai Ye, Qingtao Pan, Shuo Li

Organizations: Case Western Reserve University

Abstract

Large language models (LLMs) need reliable test-time control of hallucinations. Existing conformal methods for LLMs typically provide only marginal guarantees and rely on a single global threshold, which can under-cover hard prompts, over-cover easy ones, and produce oversized prediction sets. We propose Conditional Factuality Control (CFC), a post-hoc conformal framework that returns set-valued outputs with conditional coverage guarantees. CFC defines a continuous, feature-conditional acceptance threshold through augmented quantile regression on a latent ``success'' score, and deploys it through a fixed-point threshold rule at inference time. Theoretically, we show that CFC satisfies a conditional coverage guarantee under exchangeability and analyze its efficiency, proving that, under mild assumptions on the score distributions, the conditional rule is strictly more sample-efficient than marginal conformal prediction at the same target coverage. We further derive a PAC-style variant, CFC-PAC, which shrinks the nominal risk level based on a stability bound, yielding a finite-sample certificate that the conditional miscoverage deviates from the target by at most $O(\sqrt{\log(1/δ)/N})$ . Empirically, on synthetic data, real-world reasoning and QA benchmarks, and a Flickr8k VLM setting, CFC and CFC-PAC consistently attain near-target coverage across difficulty groups while using smaller prediction sets than CP and non-CP baselines.

Figures & tables

Figure 1: Limitation of marginal CP and advantage of proposed CFC. Left: A single global threshold learned from the marginal score mixture yields only marginal coverage and can under‑cover hard prompts while over‑covering easy ones. Right: Our CFC learns a data‑dependent threshold via conformal quantile regression, adapting the acceptance level to input features and achieving conditional coverage across subgroups.

Figure 2: Groupwise miscoverage on synthetic data across 10 difficulty bins. The dashed line marks the target miscoverage

\alpha=0.10

. Learnt CP improves over marginal baselines, but CFC and CFC-P remain closest to the target across all bins, especially on hard prompts.

Figure 3: Learned threshold $\widehat{\lambda}_{\alpha}(X)$ versus prompt difficulty. Easy prompts receive stricter thresholds, while harder prompts receive looser thresholds, which is the mechanism behind the improved group-wise reliability of CFC.

Method	ECR	APSS $\downarrow$	GSC $\uparrow$
TopK	90.6 $\pm$ 0.1	16.00 $\pm$ 0.00	58.2 $\pm$ 1.3
ICP	90.2 $\pm$ 0.2	16.71 $\pm$ 0.23	57.4 $\pm$ 1.4
Learnt CP	90.2 $\pm$ 0.3	15.72 $\pm$ 0.15	84.3 $\pm$ 0.5
CFC (ours)	90.3 $\pm$ 0.5	15.53 $\pm$ 0.12	88.7 $\pm$ 0.7
CFC-P (ours)	90.8 $\pm$ 0.6	15.87 $\pm$ 0.16	89.1 $\pm$ 0.5

Table 1: Synthetic-data results at

\alpha=0.10

. Learnt CP uses a learned difficulty-conditional threshold but no exact conformal correction, isolating learning from feature-conditional conformalization.

Figure 4: Groupwise miscoverage on real-world datasets at representative target errors: TriviaQA

\alpha=0.25

, GSM8K

\alpha=0.10

, and Flickr8k

\alpha=0.03

. Bars show mean miscoverage over split seeds and the upper error bars show one standard deviation. The TriviaQA panel uses the chosen two-group feature map; Appendix C gives its exact construction. The GSM8K and Flickr8k panels use five equal-frequency difficulty groups ordered from easy to hard. The dashed line marks the target miscoverage

\alpha

. Across all three datasets, the conditional methods flatten the miscoverage profile relative to marginal baselines, especially on the hardest groups.

Methods	$\alpha=0.20$			$\alpha=0.25$
	ECR	GSC $\uparrow$	APSS $\downarrow$	ECR	GSC $\uparrow$	APSS $\downarrow$
TopK	81.6 $\pm$ 0.3	69.7 $\pm$ 1.5	1.75 $\pm$ 0.00	73.4 $\pm$ 0.3	55.9 $\pm$ 1.6	1.00 $\pm$ 0.00
ICP	80.0 $\pm$ 0.5	65.1 $\pm$ 1.5	1.43 $\pm$ 0.02	74.9 $\pm$ 0.3	56.7 $\pm$ 1.9	1.08 $\pm$ 0.01
Learnt CP	79.6 $\pm$ 0.5	76.3 $\pm$ 1.5	1.69 $\pm$ 0.04	74.7 $\pm$ 0.4	74.0 $\pm$ 1.1	1.22 $\pm$ 0.03
CFC (ours)	76.2 $\pm$ 0.5	65.4 $\pm$ 1.7	1.21 $\pm$ 0.01	72.7 $\pm$ 0.4	65.2 $\pm$ 1.8	1.03 $\pm$ 0.03
CFC-P-F (ours)	80.1 $\pm$ 0.5	76.3 $\pm$ 1.6	1.72 $\pm$ 0.04	75.3 $\pm$ 0.4	74.6 $\pm$ 1.0	1.32 $\pm$ 0.10

Table 2: TriviaQA results for

\alpha\in\{0.20,0.25,0.30,0.35\}

under the chosen calibration-defined feature map. APSS below

1

indicates abstention on some prompts. ECR and GSC are reported in percent; for ECR, values closest to the target coverage

1-\alpha

are preferred. Appendix C reports the full experiments with additional details.

Setting	CFC	CFC-P-F
Entropy-linear $\Phi$	66.9 / 45.1 / 1.02	70.6 / 53.2 / 1.32
Max-loss-linear $\Phi$	67.6 / 56.5 / 0.99	70.7 / 57.4 / 1.26
Chosen $\Phi$	68.4 / 62.8 / 0.88	70.0 / 69.2 / 0.99

Table 3: TriviaQA feature-map ablation at

\alpha=0.30

(target coverage

70\%

). Each cell reports ECR/GSC/APSS. TopK and ICP are unchanged across feature maps and remain fixed at

73.4/55.9/1.00

and

69.5/49.3/0.90

, respectively.

Methods	ECR	GSC $\uparrow$	APSS $\downarrow$
TopK	96.42 $\pm$ 0.39	86.52 $\pm$ 1.11	1.00 $\pm$ 0.00
ICP	95.09 $\pm$ 1.42	79.85 $\pm$ 6.53	4.73 $\pm$ 0.09
Learnt CP	94.91 $\pm$ 1.03	88.48 $\pm$ 3.05	4.01 $\pm$ 0.98
CFC (ours)	94.82 $\pm$ 0.97	88.48 $\pm$ 2.32	2.35 $\pm$ 0.43
CFC-P-F (ours)	95.24 $\pm$ 1.40	88.79 $\pm$ 3.01	4.59 $\pm$ 0.62

Table 4: GSM8K results at

\alpha=0.05

using the first five sampled candidates per prompt and the quadratic basis

\Phi(X)=[1,T(X),T(X)^{2}]

with

T(X)

equal to mean verifier loss. ECR and GSC are reported in percent; for ECR, values closest to the target coverage

1-\alpha

are preferred. Appendix C reports the full sweep.

Method	$N=5$			$N=20$
	ECR	GSC $\uparrow$	APSS $\downarrow$	ECR	GSC $\uparrow$	APSS $\downarrow$
CFC (ours)	90.18	86.36	1.49	90.30	87.73	3.79
CFC-P-F (ours)	91.91	88.48	2.34	92.06	88.79	7.97

Table 5: GSM8K sample-budget ablation at

\alpha=0.10

under the same mean-loss quadratic rule. The larger budget gives only a small subgroup-coverage gain, but it substantially inflates APSS.

Method	ECR	APSS $\downarrow$	GSC $\uparrow$
TopK	96.37 $\pm$ 0.17	1.00 $\pm$ 0.00	93.23 $\pm$ 0.47
ICP	95.58 $\pm$ 0.54	1.84 $\pm$ 0.01	85.21 $\pm$ 3.14
Learnt CP	96.16 $\pm$ 0.17	1.22 $\pm$ 0.07	94.48 $\pm$ 0.26
CFC (ours)	95.81 $\pm$ 0.38	0.99 $\pm$ 0.00	93.23 $\pm$ 0.47
CFC-P-F (ours)	97.27 $\pm$ 0.21	1.42 $\pm$ 0.07	95.21 $\pm$ 0.77

Table 6: VLM experiment on Flickr8k with Qwen2-VL-7B-Instruct at

\alpha=0.03

using up to two cached candidates per image and the quadratic basis

\Phi(X)=[1,T(X),T(X)^{2}]

with

T(X)

equal to mean verifier loss. ECR and GSC are reported in percent; for ECR, values closest to the target coverage

1-\alpha=97\%

are preferred.

Appendix figures & tables9 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 5: Learned threshold $\widehat{\lambda}_{\alpha}(X)$ versus prompt difficulty in the updated synthetic run ( $\alpha=0.10$ , 5 bins). Easy prompts receive stricter thresholds, while harder prompts receive looser thresholds, explaining the improved group-wise coverage of CFC relative to global-threshold baselines.

Methods	$\alpha=0.10$			$\alpha=0.15$
	ECR	GSC $\uparrow$	APSS $\downarrow$	ECR	GSC $\uparrow$	APSS $\downarrow$
TopK	90.6 $\pm$ 0.1	58.2 $\pm$ 1.3	16.00 $\pm$ 0.00	85.0 $\pm$ 1.6	44.9 $\pm$ 2.3	10.80 $\pm$ 0.98
ICP	90.2 $\pm$ 0.2	57.4 $\pm$ 1.4	16.71 $\pm$ 0.23	85.3 $\pm$ 0.6	46.5 $\pm$ 1.4	12.11 $\pm$ 0.28
Learnt CP	90.2 $\pm$ 0.3	84.3 $\pm$ 0.5	15.72 $\pm$ 0.15	85.2 $\pm$ 0.6	77.4 $\pm$ 0.6	12.44 $\pm$ 0.17
CFC (ours)	90.3 $\pm$ 0.5	88.7 $\pm$ 0.7	15.53 $\pm$ 0.12	85.2 $\pm$ 0.6	82.7 $\pm$ 0.8	12.42 $\pm$ 0.09
CFC-PAC (ours)	90.8 $\pm$ 0.6	89.1 $\pm$ 0.5	15.87 $\pm$ 0.16	85.6 $\pm$ 0.6	83.4 $\pm$ 1.1	12.66 $\pm$ 0.07

Appendix

Table 7: Results at different target error rates

\alpha

		CFC		CFC-PAC
$N_{\text{cal}}$	$M$	True cov.	Mean set	True cov.	Mean set
2000	50	0.799 $\pm$ 0.006	10.60 $\pm$ 0.35	0.808 $\pm$ 0.005	10.92 $\pm$ 0.30
2000	100	0.809 $\pm$ 0.012	9.77 $\pm$ 0.31	0.819 $\pm$ 0.012	10.15 $\pm$ 0.29
2000	150	0.796 $\pm$ 0.006	9.03 $\pm$ 0.43	0.807 $\pm$ 0.006	9.37 $\pm$ 0.43
5000	50	0.801 $\pm$ 0.004	10.45 $\pm$ 0.12	0.808 $\pm$ 0.004	10.69 $\pm$ 0.16
5000	100	0.803 $\pm$ 0.008	9.52 $\pm$ 0.19	0.809 $\pm$ 0.007	9.74 $\pm$ 0.15

Appendix

Table 8: Ablation of CFC vs. CFC-PAC on synthetic data, with 10 bins.

Figure 6: Group miscoverage for CFC and CFC-PAC, with 10 bins.

Methods	$\alpha=0.20$			$\alpha=0.25$			$\alpha=0.30$
	ECR	GSC $\uparrow$	APSS $\downarrow$	ECR	GSC $\uparrow$	APSS $\downarrow$	ECR	GSC $\uparrow$	APSS $\downarrow$
TopK	81.6 $\pm$ 0.3	69.7 $\pm$ 1.5	1.75 $\pm$ 0.00	73.4 $\pm$ 0.3	55.9 $\pm$ 1.6	1.00 $\pm$ 0.00	73.4 $\pm$ 0.3	55.9 $\pm$ 1.6	1.00 $\pm$ 0.00
ICP	80.0 $\pm$ 0.5	65.1 $\pm$ 1.5	1.43 $\pm$ 0.02	74.9 $\pm$ 0.3	56.7 $\pm$ 1.9	1.08 $\pm$ 0.01	69.5 $\pm$ 0.8	49.3 $\pm$ 2.7	0.90 $\pm$ 0.02
Learnt CP	79.6 $\pm$ 0.5	76.3 $\pm$ 1.6	1.69 $\pm$ 0.04	74.7 $\pm$ 0.4	74.0 $\pm$ 1.1	1.22 $\pm$ 0.03	69.5 $\pm$ 0.7	68.7 $\pm$ 1.3	0.97 $\pm$ 0.02
CFC (ours)	76.2 $\pm$ 0.5	65.4 $\pm$ 1.7	1.21 $\pm$ 0.01	72.7 $\pm$ 0.4	65.2 $\pm$ 1.8	1.03 $\pm$ 0.03	68.4 $\pm$ 0.7	62.8 $\pm$ 2.0	0.88 $\pm$ 0.01
CFC-PAC (ours)	76.4 $\pm$ 0.5	65.4 $\pm$ 1.7	1.22 $\pm$ 0.01	73.1 $\pm$ 0.5	65.3 $\pm$ 1.8	1.05 $\pm$ 0.04	68.8 $\pm$ 0.7	62.9 $\pm$ 2.1	0.89 $\pm$ 0.02

Appendix

Table 9: Full TriviaQA sweep across target error rates

\alpha

under the calibration-defined split

\max\{T_{\mathrm{ent}}(X),T_{\mathrm{loss}}(X)\}\geq q_{0.925}

. APSS below

1

indicates that the method abstains and returns the empty set on some prompts. ECR and GSC are reported in percent; for ECR, values closest to the target coverage

1-\alpha

are preferred.

Methods	$\alpha=0.05$			$\alpha=0.10$			$\alpha=0.15$
	ECR	GSC $\uparrow$	APSS $\downarrow$	ECR	GSC $\uparrow$	APSS $\downarrow$	ECR	GSC $\uparrow$	APSS $\downarrow$
TopK	96.42 $\pm$ 0.39	86.52 $\pm$ 1.11	1.00 $\pm$ 0.00	96.42 $\pm$ 0.39	86.52 $\pm$ 1.11	1.00 $\pm$ 0.00	96.42 $\pm$ 0.39	86.52 $\pm$ 1.11	1.00 $\pm$ 0.00
ICP	95.09 $\pm$ 1.42	79.85 $\pm$ 6.53	4.73 $\pm$ 0.09	90.39 $\pm$ 1.44	56.36 $\pm$ 6.20	4.36 $\pm$ 0.08	83.91 $\pm$ 2.15	27.73 $\pm$ 8.14	3.97 $\pm$ 0.11
Learnt CP	94.91 $\pm$ 1.03	88.48 $\pm$ 3.05	4.01 $\pm$ 0.98	90.09 $\pm$ 1.38	86.06 $\pm$ 0.77	2.22 $\pm$ 0.07	84.70 $\pm$ 1.11	79.09 $\pm$ 1.56	1.92 $\pm$ 0.06
CFC (ours)	94.82 $\pm$ 0.97	88.48 $\pm$ 2.32	2.35 $\pm$ 0.43	90.18 $\pm$ 1.41	86.36 $\pm$ 1.07	1.49 $\pm$ 0.04	84.97 $\pm$ 1.12	79.55 $\pm$ 1.52	1.34 $\pm$ 0.04
CFC-FULL (ours)	95.03 $\pm$ 0.97	88.94 $\pm$ 2.74	4.08 $\pm$ 0.93	90.30 $\pm$ 1.40	86.36 $\pm$ 1.07	2.23 $\pm$ 0.08	85.09 $\pm$ 1.03	79.55 $\pm$ 1.52	1.96 $\pm$ 0.06

Appendix

Table 10: Full GSM8K sweep across target error rates

\alpha

using the first five sampled candidates per prompt,

T(X)

equal to mean verifier loss, and the quadratic basis

\Phi(X)=[1,T(X),T(X)^{2}]

. ECR and GSC are reported in percent; for ECR, values closest to the target coverage

1-\alpha

are preferred.

Methods	$N=5$			$N=20$
	ECR	GSC $\uparrow$	APSS $\downarrow$	ECR	GSC $\uparrow$	APSS $\downarrow$
TopK	96.42 $\pm$ 0.39	86.52 $\pm$ 1.11	1.00 $\pm$ 0.00	96.70 $\pm$ 0.36	88.48 $\pm$ 1.11	1.00 $\pm$ 0.00
ICP	90.39 $\pm$ 1.44	56.36 $\pm$ 6.20	4.36 $\pm$ 0.08	90.15 $\pm$ 1.37	55.76 $\pm$ 5.52	16.75 $\pm$ 0.35
Learnt CP	90.09 $\pm$ 1.38	86.06 $\pm$ 0.77	2.22 $\pm$ 0.07	90.15 $\pm$ 0.81	87.42 $\pm$ 1.41	7.24 $\pm$ 0.33
CFC (ours)	90.18 $\pm$ 1.41	86.36 $\pm$ 1.07	1.49 $\pm$ 0.04	90.30 $\pm$ 0.63	87.73 $\pm$ 1.47	3.79 $\pm$ 0.12
CFC-FULL (ours)	90.30 $\pm$ 1.40	86.36 $\pm$ 1.07	2.23 $\pm$ 0.08	90.45 $\pm$ 0.70	87.73 $\pm$ 1.47	7.50 $\pm$ 0.18

Appendix

Table 11: GSM8K sample-budget ablation at

\alpha=0.10

under the mean-loss quadratic rule. Each budget uses the same basis

\Phi(X)=[1,T(X),T(X)^{2}]

, differing only in the number of retained sampled candidates per prompt.

Methods	$\alpha=0.01$			$\alpha=0.02$			$\alpha=0.03$
	ECR	GSC $\uparrow$	APSS $\downarrow$	ECR	GSC $\uparrow$	APSS $\downarrow$	ECR	GSC $\uparrow$	APSS $\downarrow$
TopK	97.75 $\pm$ 0.19	95.62 $\pm$ 0.71	2.00 $\pm$ 0.00	97.75 $\pm$ 0.19	95.62 $\pm$ 0.71	2.00 $\pm$ 0.00	96.37 $\pm$ 0.17	93.23 $\pm$ 0.47	1.00 $\pm$ 0.00
ICP	97.29 $\pm$ 0.29	93.33 $\pm$ 1.45	1.93 $\pm$ 0.01	96.25 $\pm$ 0.66	88.23 $\pm$ 3.66	1.87 $\pm$ 0.02	95.58 $\pm$ 0.54	85.21 $\pm$ 3.14	1.84 $\pm$ 0.01
Learnt CP	97.39 $\pm$ 0.33	95.52 $\pm$ 0.85	1.74 $\pm$ 0.06	97.06 $\pm$ 0.31	95.10 $\pm$ 0.63	1.34 $\pm$ 0.07	96.16 $\pm$ 0.17	94.48 $\pm$ 0.26	1.22 $\pm$ 0.07
CFC (ours)	96.37 $\pm$ 0.17	93.23 $\pm$ 0.47	1.00 $\pm$ 0.00	96.37 $\pm$ 0.17	93.23 $\pm$ 0.47	1.00 $\pm$ 0.00	95.81 $\pm$ 0.38	93.23 $\pm$ 0.47	0.99 $\pm$ 0.00
CFC-FULL (ours)	97.66 $\pm$ 0.16	95.62 $\pm$ 0.71	1.86 $\pm$ 0.04	97.27 $\pm$ 0.21	95.21 $\pm$ 0.77	1.40 $\pm$ 0.06	96.27 $\pm$ 0.24	94.58 $\pm$ 0.26	1.25 $\pm$ 0.08

Appendix

Table 12: Full Flickr8k sweep with Qwen2-VL-7B-Instruct using up to two cached candidates per image,

T(X)

equal to mean verifier loss, and the quadratic basis

\Phi(X)=[1,T(X),T(X)^{2}]

. ECR and GSC are reported in percent; for ECR, values closest to the target coverage

1-\alpha

are preferred.

Setting	CFC	CFC-PAC-FULL
$N=2$ , max-loss, poly2	96.02 / 1.00 / 93.02	97.62 / 1.89 / 95.93
Chosen: $N=2$ , mean-loss, poly2	95.81 / 0.99 / 93.23	97.27 / 1.42 / 95.21
$N=3$ , mean-loss, poly2	96.00 / 1.00 / 93.02	97.81 / 2.16 / 96.35

Appendix

Table 13: Flickr8k setting ablation at

\alpha=0.03

(target coverage

97\%

). Each cell reports ECR/APSS/GSC for the corresponding method.

Explore similar work

Inference-Time Conformal Reasoning with Valid Factuality Control for Large Language Models
Jun 7, 2026Ting Wang, Yuanjie Shi, Yan Yan +1LLM Inference LLM Reasoning with Graphs
University of Illinois Urbana-Champaign · School of EECS, Washington State University.
Differentiable Conformal Training for LLM Reasoning Factuality
Apr 22, 2026Nathan Hittesdorf, Marco Salzetta, Lu ChengLLM Hallucination Mitigation Conformal Prediction
Department of Computer Science, University of Illinois at Chicago, Chicago, United States · Department of Physics, University of Illinois at Urbana-Champaign, Urbana, United States
Adaptive Conformal Prediction for Improving Factuality of Generations by Large Language Models
Apr 15, 2026Aleksandr Rubashevskii, Dzianis Piatrashyn, Preslav Nakov +1Conformal Prediction Language Model Calibration
Mohamed bin Zayed University of Artificial Intelligence (MBZUAI), Abu Dhabi, UAE

Jun 7, 2026J/K move · Enter open · S save

Inference-Time Conformal Reasoning with Valid Factuality Control for Large Language Models

Ting Wang, Yuanjie Shi, Yan Yan, Huan Zhang

University of Illinois Urbana-Champaign · School of EECS, Washington State University.

LLM Inference LLM Reasoning with Graphs Conformal Prediction LLM Uncertainty Estimation LLM Reasoning

Open paper AskPDF arXiv

Large language models (LLMs) increasingly perform multi-step reasoning, where intermediate claims form implicit directed acyclic graphs whose node correctness is structurally conditioned on their ancestors. This makes factuality uncertainty structural, rather than a trivial accumulation of node-wise errors, and necessitates inference-time uncertainty quantification over the reasoning structure. While conformal prediction (CP) offers flexible user-specified factuality control, existing work remains post-hoc and cannot intervene during generation. To fill the gap between CP's flexibility and its post-hoc limitation, we propose an Inference-Time Conformal Reasoning (ITCR) framework that integrates CP directly into reasoning graph generation. ITCR learns a structure-level factuality uncertainty function that aggregates claim-level factuality signals over reasoning graphs without complex modeling assumptions. We then design the non-conformity score based on graph-level factuality uncertainty and calibrate the conformal threshold to decide when to stop generation. We theoretically show such generation is nested, yielding valid coverage guarantees for factuality control. Experiments over multiple datasets and coverage objectives demonstrate empirically valid coverage. In downstream reasoning tasks, inference-time calibrated graphs yield more accurate generation than post-hoc pruned graphs.

Abstract

Explore similar work

Inference-Time Conformal Reasoning with Valid Factuality Control for Large Language Models

Differentiable Conformal Training for LLM Reasoning Factuality

Adaptive Conformal Prediction for Improving Factuality of Generations by Large Language Models