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(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 α=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 λα(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 ↓
GSC ↑
TopK
90.6 ± 0.1
16.00 ± 0.00
58.2 ± 1.3
ICP
90.2 ± 0.2
16.71 ± 0.23
57.4 ± 1.4
Learnt CP
90.2 ± 0.3
15.72 ± 0.15
84.3 ± 0.5
CFC (ours)
90.3 ± 0.5
15.53 ± 0.12
88.7 ± 0.7
CFC-P (ours)
90.8 ± 0.6
15.87 ± 0.16
89.1 ± 0.5
Table 1: Synthetic-data results at α=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 α=0.25 , GSM8K α=0.10 , and Flickr8k α=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 α . Across all three datasets, the conditional methods flatten the miscoverage profile relative to marginal baselines, especially on the hardest groups.
Methods
α=0.20
α=0.25
ECR
GSC ↑
APSS ↓
ECR
GSC ↑
APSS ↓
TopK
81.6 ± 0.3
69.7 ± 1.5
1.75 ± 0.00
73.4 ± 0.3
55.9 ± 1.6
1.00 ± 0.00
ICP
80.0 ± 0.5
65.1 ± 1.5
1.43 ± 0.02
74.9 ± 0.3
56.7 ± 1.9
1.08 ± 0.01
Learnt CP
79.6 ± 0.5
76.3 ± 1.5
1.69 ± 0.04
74.7 ± 0.4
74.0 ± 1.1
1.22 ± 0.03
CFC (ours)
76.2 ± 0.5
65.4 ± 1.7
1.21 ± 0.01
72.7 ± 0.4
65.2 ± 1.8
1.03 ± 0.03
CFC-P-F (ours)
80.1 ± 0.5
76.3 ± 1.6
1.72 ± 0.04
75.3 ± 0.4
74.6 ± 1.0
1.32 ± 0.10
Table 2: TriviaQA results for α∈{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−α are preferred. Appendix C reports the full experiments with additional details.
Setting
CFC
CFC-P-F
Entropy-linear Φ
66.9 / 45.1 / 1.02
70.6 / 53.2 / 1.32
Max-loss-linear Φ
67.6 / 56.5 / 0.99
70.7 / 57.4 / 1.26
Chosen Φ
68.4 / 62.8 / 0.88
70.0 / 69.2 / 0.99
Table 3: TriviaQA feature-map ablation at α=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 ↑
APSS ↓
TopK
96.42 ± 0.39
86.52 ± 1.11
1.00 ± 0.00
ICP
95.09 ± 1.42
79.85 ± 6.53
4.73 ± 0.09
Learnt CP
94.91 ± 1.03
88.48 ± 3.05
4.01 ± 0.98
CFC (ours)
94.82 ± 0.97
88.48 ± 2.32
2.35 ± 0.43
CFC-P-F (ours)
95.24 ± 1.40
88.79 ± 3.01
4.59 ± 0.62
Table 4: GSM8K results at α=0.05 using the first five sampled candidates per prompt and the quadratic basis Φ(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−α are preferred. Appendix C reports the full sweep.
Method
N=5
N=20
ECR
GSC ↑
APSS ↓
ECR
GSC ↑
APSS ↓
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 α=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 ↓
GSC ↑
TopK
96.37 ± 0.17
1.00 ± 0.00
93.23 ± 0.47
ICP
95.58 ± 0.54
1.84 ± 0.01
85.21 ± 3.14
Learnt CP
96.16 ± 0.17
1.22 ± 0.07
94.48 ± 0.26
CFC (ours)
95.81 ± 0.38
0.99 ± 0.00
93.23 ± 0.47
CFC-P-F (ours)
97.27 ± 0.21
1.42 ± 0.07
95.21 ± 0.77
Table 6: VLM experiment on Flickr8k with Qwen2-VL-7B-Instruct at α=0.03 using up to two cached candidates per image and the quadratic basis Φ(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−α=97% are preferred.
Appendix figures & tables9 assets
Supplementary material from the paper’s appendix.
Appendix
Figure 5: Learned threshold λα(X) versus prompt difficulty in the updated synthetic run ( α=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
α=0.10
α=0.15
ECR
GSC ↑
APSS ↓
ECR
GSC ↑
APSS ↓
TopK
90.6 ± 0.1
58.2 ± 1.3
16.00 ± 0.00
85.0 ± 1.6
44.9 ± 2.3
10.80 ± 0.98
ICP
90.2 ± 0.2
57.4 ± 1.4
16.71 ± 0.23
85.3 ± 0.6
46.5 ± 1.4
12.11 ± 0.28
Learnt CP
90.2 ± 0.3
84.3 ± 0.5
15.72 ± 0.15
85.2 ± 0.6
77.4 ± 0.6
12.44 ± 0.17
CFC (ours)
90.3 ± 0.5
88.7 ± 0.7
15.53 ± 0.12
85.2 ± 0.6
82.7 ± 0.8
12.42 ± 0.09
CFC-PAC (ours)
90.8 ± 0.6
89.1 ± 0.5
15.87 ± 0.16
85.6 ± 0.6
83.4 ± 1.1
12.66 ± 0.07
Appendix
Table 7: Results at different target error rates α .
CFC
CFC-PAC
Ncal
M
True cov.
Mean set
True cov.
Mean set
2000
50
0.799 ± 0.006
10.60 ± 0.35
0.808 ± 0.005
10.92 ± 0.30
2000
100
0.809 ± 0.012
9.77 ± 0.31
0.819 ± 0.012
10.15 ± 0.29
2000
150
0.796 ± 0.006
9.03 ± 0.43
0.807 ± 0.006
9.37 ± 0.43
5000
50
0.801 ± 0.004
10.45 ± 0.12
0.808 ± 0.004
10.69 ± 0.16
5000
100
0.803 ± 0.008
9.52 ± 0.19
0.809 ± 0.007
9.74 ± 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
α=0.20
α=0.25
α=0.30
ECR
GSC ↑
APSS ↓
ECR
GSC ↑
APSS ↓
ECR
GSC ↑
APSS ↓
TopK
81.6 ± 0.3
69.7 ± 1.5
1.75 ± 0.00
73.4 ± 0.3
55.9 ± 1.6
1.00 ± 0.00
73.4 ± 0.3
55.9 ± 1.6
1.00 ± 0.00
ICP
80.0 ± 0.5
65.1 ± 1.5
1.43 ± 0.02
74.9 ± 0.3
56.7 ± 1.9
1.08 ± 0.01
69.5 ± 0.8
49.3 ± 2.7
0.90 ± 0.02
Learnt CP
79.6 ± 0.5
76.3 ± 1.6
1.69 ± 0.04
74.7 ± 0.4
74.0 ± 1.1
1.22 ± 0.03
69.5 ± 0.7
68.7 ± 1.3
0.97 ± 0.02
CFC (ours)
76.2 ± 0.5
65.4 ± 1.7
1.21 ± 0.01
72.7 ± 0.4
65.2 ± 1.8
1.03 ± 0.03
68.4 ± 0.7
62.8 ± 2.0
0.88 ± 0.01
CFC-PAC (ours)
76.4 ± 0.5
65.4 ± 1.7
1.22 ± 0.01
73.1 ± 0.5
65.3 ± 1.8
1.05 ± 0.04
68.8 ± 0.7
62.9 ± 2.1
0.89 ± 0.02
Appendix
Table 9: Full TriviaQA sweep across target error rates α under the calibration-defined split max{Tent(X),Tloss(X)}≥q0.925 . 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−α are preferred.
Methods
α=0.05
α=0.10
α=0.15
ECR
GSC ↑
APSS ↓
ECR
GSC ↑
APSS ↓
ECR
GSC ↑
APSS ↓
TopK
96.42 ± 0.39
86.52 ± 1.11
1.00 ± 0.00
96.42 ± 0.39
86.52 ± 1.11
1.00 ± 0.00
96.42 ± 0.39
86.52 ± 1.11
1.00 ± 0.00
ICP
95.09 ± 1.42
79.85 ± 6.53
4.73 ± 0.09
90.39 ± 1.44
56.36 ± 6.20
4.36 ± 0.08
83.91 ± 2.15
27.73 ± 8.14
3.97 ± 0.11
Learnt CP
94.91 ± 1.03
88.48 ± 3.05
4.01 ± 0.98
90.09 ± 1.38
86.06 ± 0.77
2.22 ± 0.07
84.70 ± 1.11
79.09 ± 1.56
1.92 ± 0.06
CFC (ours)
94.82 ± 0.97
88.48 ± 2.32
2.35 ± 0.43
90.18 ± 1.41
86.36 ± 1.07
1.49 ± 0.04
84.97 ± 1.12
79.55 ± 1.52
1.34 ± 0.04
CFC-FULL (ours)
95.03 ± 0.97
88.94 ± 2.74
4.08 ± 0.93
90.30 ± 1.40
86.36 ± 1.07
2.23 ± 0.08
85.09 ± 1.03
79.55 ± 1.52
1.96 ± 0.06
Appendix
Table 10: Full GSM8K sweep across target error rates α using the first five sampled candidates per prompt, T(X) equal to mean verifier loss, and the quadratic basis Φ(X)=[1,T(X),T(X)2] . ECR and GSC are reported in percent; for ECR, values closest to the target coverage 1−α are preferred.
Methods
N=5
N=20
ECR
GSC ↑
APSS ↓
ECR
GSC ↑
APSS ↓
TopK
96.42 ± 0.39
86.52 ± 1.11
1.00 ± 0.00
96.70 ± 0.36
88.48 ± 1.11
1.00 ± 0.00
ICP
90.39 ± 1.44
56.36 ± 6.20
4.36 ± 0.08
90.15 ± 1.37
55.76 ± 5.52
16.75 ± 0.35
Learnt CP
90.09 ± 1.38
86.06 ± 0.77
2.22 ± 0.07
90.15 ± 0.81
87.42 ± 1.41
7.24 ± 0.33
CFC (ours)
90.18 ± 1.41
86.36 ± 1.07
1.49 ± 0.04
90.30 ± 0.63
87.73 ± 1.47
3.79 ± 0.12
CFC-FULL (ours)
90.30 ± 1.40
86.36 ± 1.07
2.23 ± 0.08
90.45 ± 0.70
87.73 ± 1.47
7.50 ± 0.18
Appendix
Table 11: GSM8K sample-budget ablation at α=0.10 under the mean-loss quadratic rule. Each budget uses the same basis Φ(X)=[1,T(X),T(X)2] , differing only in the number of retained sampled candidates per prompt.
Methods
α=0.01
α=0.02
α=0.03
ECR
GSC ↑
APSS ↓
ECR
GSC ↑
APSS ↓
ECR
GSC ↑
APSS ↓
TopK
97.75 ± 0.19
95.62 ± 0.71
2.00 ± 0.00
97.75 ± 0.19
95.62 ± 0.71
2.00 ± 0.00
96.37 ± 0.17
93.23 ± 0.47
1.00 ± 0.00
ICP
97.29 ± 0.29
93.33 ± 1.45
1.93 ± 0.01
96.25 ± 0.66
88.23 ± 3.66
1.87 ± 0.02
95.58 ± 0.54
85.21 ± 3.14
1.84 ± 0.01
Learnt CP
97.39 ± 0.33
95.52 ± 0.85
1.74 ± 0.06
97.06 ± 0.31
95.10 ± 0.63
1.34 ± 0.07
96.16 ± 0.17
94.48 ± 0.26
1.22 ± 0.07
CFC (ours)
96.37 ± 0.17
93.23 ± 0.47
1.00 ± 0.00
96.37 ± 0.17
93.23 ± 0.47
1.00 ± 0.00
95.81 ± 0.38
93.23 ± 0.47
0.99 ± 0.00
CFC-FULL (ours)
97.66 ± 0.16
95.62 ± 0.71
1.86 ± 0.04
97.27 ± 0.21
95.21 ± 0.77
1.40 ± 0.06
96.27 ± 0.24
94.58 ± 0.26
1.25 ± 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 Φ(X)=[1,T(X),T(X)2] . ECR and GSC are reported in percent; for ECR, values closest to the target coverage 1−α 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 α=0.03 (target coverage 97% ). Each cell reports ECR/APSS/GSC for the corresponding method.
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.
Ting Wang, Yuanjie Shi, Yan Yan +1
University of Illinois Urbana-Champaign · School of EECS, Washington State University.
Large Language Models (LLMs) frequently hallucinate, limiting their reliability in critical applications. Conformal Prediction (CP) addresses this by calibrating error rates on held-out data to provide statistically valid confidence guarantees. Recent work extends CP to LLM factuality to filter out risky claims, ensuring that hallucination rates remain below a user-specified level (e.g., 10%). While prior methods treat claims independently, Coherent Factuality extends to multi-step reasoning by representing outputs as dependency graphs and jointly validating claims with their logical ancestors. A key limitation is that Coherent Factuality is not differentiable, requiring hand-crafted scorers that at high reliability levels remove nearly 60% of true claims. We introduce Differentiable Coherent Factuality (DCF), a fully differentiable relaxation that enables learning improved scorers while provably recovering the original algorithm's guarantees. Experiments on two benchmark reasoning datasets demonstrate DCF achieves up to 141% improvement in claim retention while maintaining reliability guarantees, representing a significant step towards reliable conformal LLM systems.
Nathan Hittesdorf, Marco Salzetta, Lu Cheng
Department of Computer Science, University of Illinois at Chicago, Chicago, United States · Department of Physics, University of Illinois at Urbana-Champaign, Urbana, United States
Large language models (LLMs) are prone to generating factually incorrect outputs. Recent work has applied conformal prediction to provide uncertainty estimates and statistical guarantees for the factuality of LLM generations. However, existing approaches are typically not prompt-adaptive, limiting their ability to capture input-dependent variability. As a result, they may filter out too few items (leading to over-coverage) or too many (under-coverage) for a given task or prompt. We propose an adaptive conformal prediction approach that extends conformal score transformation methods to LLMs, with applications to long-form generation and multiple-choice question answering. This enables prompt-dependent calibration, retaining marginal coverage guarantees while improving conditional coverage. In addition, the approach naturally supports selective prediction, allowing unreliable claims or answer choices to be filtered out in downstream applications. We evaluate our approach on multiple white-box models across diverse domains and show that it significantly outperforms existing baselines in terms of conditional coverage.
Aleksandr Rubashevskii, Dzianis Piatrashyn, Preslav Nakov +1
Mohamed bin Zayed University of Artificial Intelligence (MBZUAI), Abu Dhabi, UAE