Enhancing Image-Conditional Coverage in Segmentation: Adaptive Thresholding via Differentiable Miscoverage Loss
Authors: Rui Luo
Jie Bao
Xiaoyi Su
Wen Jung Li
Suqun Cao
Venue: ICLR 2026 Poster (2026)
Keywords: image segmentation
conditional coverage
conformal prediction
conformal risk control
Primary area: alignment, fairness, safety, privacy, and societal considerations

Abstract: Current deep learning models for image segmentation often lack reliable uncertainty quantification, particularly at the image-specific level. While Conformal Risk Control (CRC) offers marginal statistical guarantees, achieving image-conditional coverage, which ensures prediction sets reliably capture ground truth for individual images, remains a significant challenge. This paper introduces a novel approach to address this gap by learning image-adaptive thresholds for conformal image segmentation. We first propose AT (Adaptive Thresholding), which frames threshold prediction as a supervised regression task. Building upon the insights from AT, we then introduce COAT (Conditional Optimization for Adaptive Thresholding), an innovative end-to-end differentiable framework. COAT directly optimizes image-conditional coverage by using a soft approximation of the True Positive Rate (TPR) as its loss function, enabling direct gradient-based learning of optimal image-specific thresholds. This novel differentiable miscoverage loss is key to enhancing conditional coverage. Our methods provide a robust pathway towards more trustworthy and interpretable uncertainty estimates in image segmentation, offering improved conditional guarantees crucial for safety-critical applications. The code is available at https://github.com/bjbbbb/Conditional-Optimization-for-Adaptive-Thresholding.

Published as a conference paper at ICLR 2026
ENHANCING IMAGE-CONDITIONAL COVERAGE IN SEGMENTATION: ADAPTIVE THRESHOLDING VIA DIF- FERENTIABLE MISCOVERAGE Loss
Rui Luo' Jie Bao", Xiaoyi Su², Wen Jung Li', Suqun Cao² City University of Hong Kong, Huaiyyn Institute of Technology
ABSTRACT
Current deep learning models for image segmentation often lack reliable un- certainty quantification, particularly at the image-specific level. While Confor- mal Risk Control (CRC) offers marginal statistical guarantees, achieving image- conditional coverage, which ensures prediction sets reliably capture ground truth for individual images, remains significant challenge. This paper introduces novel approach to address this gap by learning image-adaptive thresholds for con- formal image segmentation. We first propose AT (Adaptive Thresholding), which frames threshold prediction as a supervised regression task. Building upon the in- sights from AT, then introduce COAT (Conditional Optimization for Adaptive Thresholding), an innovative end-to-end differentiable framework. COAT directly optimizes Iageecconiiiioal coverage by using soft approximation of the True Positive Rate (TPR) as its loss function, enabling direct gradient- based learning of optimal image-specific thresholds. This novel differentiable miscoverage loss is key to enhancing conditional coverage. Our methods provide robust path- way towards more trustworthy and interpretable uncertainty estimates image segmentation, offering improved conditional guarantees crucial for safety-critical applications. The code is available at htttt:: // com/b iwttaaal-ppiiiizttiiiiii for woroonn-ooorroodoroooeeeeeaeeoee ive Teehhhhhhllleee
INTRODUCTION
Image segmentation a fundamental computer vision task with critical applications medical di- agnostics, autonomous driving, and remote sensing. While deep learning has significantly advanced segmentation performance, reliable uncertainty quantification remains challenging but essential for carety-critical applications. Traditional evaluation metries like Dice IoU provide overall perfor- mance measures but fail to offer anstance-wise reliability guarantees.
Conformal prediction (CP) has emerged as powerful framework for providing distribution-fie uncertainty quantification with finite-sample guarantees. It constructs prediction regions that contain the true label with user-speciffed probability, regardless of the underlying data distribution. Recent work on Conformal Risk Control (CRC) (Angelopoulos et al.) 2024) has extended this framework handle more complex performance metries beyond simple miscoverage, such as controlling the false negative rate in segmentation tasks.
However, key limitation of standard CRC, particularly in image-level tasks like segmentation. is its focus marginal guarantees. While CRC ensures that the average risk across dataset is controlled. the risk for individual images (i.e., the conditional risk) can vary substantially. safety- critical domains, ensuring that each image's prediction is reliable, rather than just the average over many images, is paramount. This image-specific variability risk is a significant challenge that current approaches struggle to address effectively.
This paper proposes novel approach to achieve image-conditional coverage in conformal image segmentation by learning image-adaptive thresholds. Our core idea is to train a model that predicts
"Corresponding author. E-mail: mullu@@citu..cu..hk

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Published as a conference paper at ICLR 2026
Senuattic segrediation
Calivatiin:: mininal
Peolents
in2(t]
1-0}
COAT
Lss Rexsstuull "XX"")"
Figure 1: Schematic Overview of the COAT Framework Pipeline.
a unique threshold for each input image, aiming to satisfy the desired coverage level for that specific image. We introduce two distinct methods to realize this:
AT (Adaptive Thresholding): As an initial step, this method treats the problem of threshold prediction supervised regression task. We pre-compute optimal hard thresholds for training images that achieve the target coverage, and then train a neural network to predict these thresholds given the image and its base segmentation model outputs.
2. COAT (Conditional Optimization for Adaptive Thresholding): Building upon AT's con- cept, we propose innovative end-to-end differentiable framework, which we name COAT (as illustrated Figure DD. Instead of relying on pre-computed hard thresholds, COAT di- rectly optimizes for mage-conditional coverage. It achieves this by utilizing soft, differ- entiable approximation of the True Positive Rate (TPR) to define its loss function, enabling direct gradient-basee based learning of optimal image-specific thresholds. This novel differen- tiable miscoverage loss is key contribution for enhancing conditional coverage.
By learning image- adaptive thresholds and, particularly through the end-to-end differentiable op- timization of COAT, our methods provide a robust pathway towards more trustworthy and inter- pretable uncertainty estimates in image segmentation, offering significantly improved conditional guarantees, which are crucial for the deployment Al systems in high-stakes applications.
PRELIMINARIES AND PROBLEM SETUP
2.1 PROBLEM SETUP
For image segmentation, we consider an input image X; with its ground truth binary mask Y; E {0, 13 which delineates region of interest. Our primary objective is to construct prediction C(X,) C 21,2, ..., N that controls the false negative rate (FNR) in expectation. The FNR quantifies the proportion of true positive pixels that are incorrectly excluded from the prediction set, critical metric in applications such as medical imaging where missing regions of interest can have severe consequences. Specifically, we aim to ensure:
Σ J= C(X) Yb]
E
< a,
(1)
where E (0, is user-specified risk level. Here, En YbJi denotes the number of positive pixels in the ground truth mask. We also define as small positive constant (e.g.. 10-9) used to prevent division by zero in certain calculations. The expectation in equation i taken over random draws of the test data, reflecting the average performance of the prediction set ((x)).
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However, this marginal guarantee, while ensuring that the average FNR across the entire dataset is controlled, does not guarantee consistent performance for individual images. Due to the inherent variability in image "difficulty" or characteristies, applying a single threshold to all images can lead to over-coverage for "easy" images and severe under-coverage for "difficult" ones. This implies that, while the FNR might be met on average, the conditional coverage (i.e., FNR for single image) for specific images can deviate significantly from the target level 1 - a. For safety-critical applications, such image-to-image variability is unacceptable, necessitating stronger guarantee: not only must the marginal FNR be controlled, but the prediction reliability for each image should also be close as possible the target level.
That is, for an input image X; Diest and its corresponding ground-trun -rutt label Y E Diest, we should ensure the fulfillment of the following conditions:
E C(X) -YbI
Coverage
(2)
Drese
ΣN Y.
X,EDiest
Subsequently, we should strive to narrow the coverage gap (Kaur et al. 2025):
Σラ- N O(X)L) Y!
Coverage Gap = [Deest] X.EDiere
Σ
(1 01)
(1-a))
(3)
Σラ YM
2.2 CONFORMAL RISK CONTROL (CRC)
Conformal Risk Control (CRC) (Angelopoulos et al. 2024) extends the principles of conformal prediction offer distribution-free guarantees on the expected value of any monotone loss function. For image segmentation, CRC achieves control over the false negative rate by applying a specific threshold to the pixel-wise probabilities generated by base segmentation model. This optimal threshold is determined through a calibration procedure performed on a dedicated held-out dataset.
Given base segmentation model that outputs probability map p(X,) (f1 (X.),...,,j* pN (Xx)) for an input image X,, where p, (X.) estimates P(j E Yi(X;) for pixel j. the CRC approach proceeds as follows:
Prediction Set Definition: For a given threshold т, the prediction set Ö(2,, r) is defined by including all pixels j whose predicted probability (X;)) is greater than equal to
C(x,,T)= = p 2 2
(4)
2. Calibrated Threshold Computation: The calibrated threshold 7' is determined using a calibration dataset Deat. is the largest threshold that satisfies the empirical risk constraint on the calibration set:
1 Ee 7(x -.D)
7' sup
T
1
Σ IED..
B n+1
saj.
(S)
n
En Ybl
where n is the size of the calibration set Deat, is an upper bound of the loss function (typically B = 1 for the FNR).
3. Final Prediction Set: The final prediction set for new test image X, is then constructed using the calibrated threshold
(xx.)=((X..))
(6)
This methodology guarantees that E MN O(X)LI-LL ΣM rib
< the data distribution, pro-
viding a distribution-fres control of the false negative rate, as established by Theorem in An- gelopoulos et al. (Angelopoulos al. 2024).
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1.0
0.8
Coverage
0.6
0.2790
0.2
0..337
=00.1107
Target Coverage (0.9)
0.0.0
0.6
Threshold (T)
Figure 2: A figure presenting the relationship between the coverage rates of various images and the variable T.
3 METHODOLOGY
We introduce two novel methods for learning an image-adaptive threshold f(X) for conformal risk control in image segmentation. The first, AT (Adaptive Thresholding), serves supervised base- line, while the second, COAT (Conditional Optimization for Adaptive Thresholding), is our more advanced, end-to-end differentiable approach.
3.1
AT: SUPERVISED THRESHOLD PREDICTION
The AT approach frames the problem as learning direct mapping from an image to its optimal segmentation threshold.
3.1.1 PREDICTION
The threshold predictor fo takes the input image X and the corresponding probability map ((X) from the base model to predict single scalar threshold f(X).
f(X)= = (x(,
The notation ((X) implies a combination of these inputs, typically through channel-wise con- catenation and any necessary spatial alignment, to form the input tensor for fD.
3.1.2 TRAINING AND LOSS FUNCTION
This method requires a pre-computation step to generate a "ground-truth" threshold 7 (x Y) for each image in the training set. This is determined via numerical search (e.g.. binary search) as the value that makes the TPR of the resulting hard segmentation mask equal to the target coverage level a.
The model fD is then trained using a standard Mean Squared Error (MSE) loss between the predicted threshold F(X) and the ground-trut -ruth threshold -" The loss function over the training distribution is:
LAT EXXYY Dew (F(P)//M_F((FFFFFFFF) - -xxx))7((X,))
(8)
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3.2 COAT: END-TO-END DIFFERENTIABLE MISCOVERAGE LOSS
As shown in Figure 2 due to the non-continuous and non-increasing relationship between and the target coverage across different images, directly training т may not necessarily yield satisfac- tory coverage performance and requires pre-caleulating the relationship between and coverage. Moreover, the non-continuous and non-increasing nature also renders direct training of coverage infeasible. To circumvent the need for pre-calculating ground-truth thresholds, COAT enables end- to-end training by defining a fully differentiable loss function that directly optimizes for the target coverage.
3.2.1 PREDICTION
The prediction model fD has the same architecture in AT, and it also takes the probability map from the base model as input.
=( = fxx,xxxXX)))
Algorithm COAT: Conditional Optimization for Adaptive Thresholding
1: Input: labeled data Dusin, unlabel test data Dest. target coverage 1 - temperature T, small
constant
2: Randomly split Duain into Dj D2. and Deat-
3: Train a semantic segmentation model using D1.
4: Initialize parameters of threshold predictor fD.
5: for epoch in training epochs do
6: for each (X,, Yt) E D2 do
7: Obtain probability map p; + (())).
8: Predict threshold: f; + (D ((X,))
Compute soft mask: Moott (X,) = o ((Xx))
9:
T
Compute differentiable TPR: TPR X Yi, ((X))) Σ Man MX. (X.I6)-X
10:
Σラ 4.bl+
11:
Compute loss: LCOAT + (TPR(X, Y, (1-)))
Update parameters of fn by descending the gradient V LCOAT.
13: end for
14: end for
15: Compute base thresholds: R + (x(,,, for (X,, ((X)) E Deat-
16: Define coverage function: R(t)= + PD.i Σ. E (x,(2:))
Σn YGG
17: Find minimal correction: / inf (t R(t)) 2 ((Dtall 1)(1 (eull
18: for each (X,, ((X,)) E Diest do
Compute the base threshold: 7 + fD p(x:))
20: Calculate the adjusted threshold after calibration: + clip(, - -t,0,1).
21: Generate the prediction set: C(X,) (p(X,) 2 +/.
22: end for
Output: ((X;) for Drest-
3.2.2 TRAINING AND LOSS FUNCTION
The of this method is differentiable approximation of the TPR. Instead of applying hard threshold, we use the predicted threshold F(X) ) to generate soft, probabilistic segmentation mask Munn- The loss is then the MSE between the TPR calculated from this soft mask and the target coverage level a.
The full loss function is defined follows:
LCOAT
E
IPR(X,Y,77XX)) - (1 a))
(X. (D
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where the predicted TPR, TPR, is computed via: TPR(X, Σn= Moon (X) Y
Σ/= YD] +e
Here, [4, w| denotes the pixel value at (h, w) for the ground truth mask Y. The soft mask Moot is defined using the sigmoid function o (.) and a temperature parameter T >
Mon (X) = (((X) (718)~7(44 )
T
This formulation allows gradients to flow from the final loss back to the parameters of the threshold predictor fD. enabling direct optimization towards the desired conditional coverage without inter- mediate supervision. As detailed in Algorithm | we present a comprehensive description of the COAT framework for image segmentation, covering all critical implementation components.
The primary objective of COAT is to learn the intricate relationship between an image's characteristics and its target conditional coverage. COAT achieves this through an innovative end-to- end differentiable miscoverage loss, which directly optimizes for the desired conditional coverage. This direct optimization circumvents the need to explicitly pre-calculate the complex, non-linear relationship between individual and coverage for each image. Following this learning phase, calibration set is used to apply a global adjustment, t', to the predicted image-specific thresholds. This final calibration step, which involve either positive the marginal coverage rate, as defined by Equation 2 across the entire dataset. This two-stage pro- cess directly optimizing for mmage-conditional reliability and then performing marginal calibra- tion provides robust pathway towards more trustworthy and interpretable uncertainty estimates.
Remark:
negative t', statistically ensures
3.3 THEORETICAL GUARANTEES
Theorem (Coverage Guarantees). Let Deal ((x,, Y.) be the calibration set and (Xn41, Yn+1) be a test sample. Suppose {(X,, Y,));= are exchangeable. Then the final prediction sel C(Xa11) given by AT or COAT satisfies:
ΣE ((Xm+))B)-Y+t1 E
21-a. a.
(10)
Proof. For each sample (X,, Yi), we define parameterized loss function (t(t) for t [0, 1]. where t global correction parameter. The loss is the false negative rate (FNR) for the prediction set formed by the adjusted threshold:
N (t = 1 E (x, clip(f. - f, 0,] .Yb
(11)
E
Here, न. = fx(X,, p(),) is the base threshold predicted by AT or COAT. This loss function L,(t) a non-increasing and right-continuous function of t. It satisfies
(,(1) 0 < and sup (1) < 1.
(12)
We compute the empirical risk on the calibration set:
(t) Σ
(13)
1=1
where R(1) is the empirical coverage defined in Algorithm
D
Define
inf
n n n [..
{a inf { R(I) 2 -+ (1 a)
(14)
n
n
n
By Theorem in Angelopoulos et al.
(2024), have E [1 (1') S a, which implies
Σ
C(X Y Yntild
E
2 1 a
(15)
Σn Yn+1
((x (8,(()))1) clip(fat1 1', 1)].

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This theorem shows that both AT and COAT provide powerful finite-sample guarantee for marginal coverage. iee. the mean TPR. We also discuss in Appendix A.1 the assumptions under which they can achieve conditional validity.
3.4
RELATED WORK
3.4.1
CONFORMAL RISK CONTROL
Unlike other uncertainty quantification methods (Chen al. 2025a/c|bb conformal prediction (Vovk) al., 2005) provides distribution-fra uncertainty quantification with finite-sample guarantees. Its split conformal variant (Lei et al.) 2018) is widely used for its computational efficiency. CP has been successfully applied to both classification (Luo & Colombo [024] Luo & Zhouu 2024 and regression tasks (Luo & Zhou 2025c d Bao et al. 20025a|
et al. (026). Its flexibility allows adaptation to various real-world scenarios, including games (Bao et al., 20055 Luo et al. 2026).
time-series forecasting (Su et al. (2024). and graph-based applications (Luo et al. 2023
Tang et al.,
2025| Luo & Zhou| [025b] Wang et al., 2025a Luo & Colombo| 2025|Zhang Zhang et al. 2025). Re-
cent extensions, notably Conformal Risk Control (CRC) (Angelopoulos et al. 20244] Bates et al.) 2021] allow for controlling the expected value of various monotone loss functions, such as the false negative rate in segmentation.
While initial CRC applications often used a single global threshold, recent works have explored more nuanced control. Teneggi et al. (Teneggi et al. 2023) proposed grouping pixels to control risk through a convex surrogate loss, and further extended this to semantic-speciffi control for medical imaging (Teneggi al. 2025). Bereska et al. (Bereska et al., 2025) introduced Spatially- Conformal Prediction (SACP) to adapt uncertainty estimates based on proximity to critical struc- tures. He el al. (He et al. 2025) integrated CRC into model training for quality assurance in radi- ation therapy. These advancements highlight growing need for more adaptive and context-aware risk control beyond global guarantees, which our image-conditional approach directly addresses.
3.4.2 CONDITIONAL CONFORMAL PREDICTION
Achieving conditional coverage ensuring prediction sets attain the desired coverage for every possible covariate value is generally impossible without strong distributional assumptions (Vovk) 2012 Foygel Barber et al. 202T). However, for high-stakes applications, marginal guarantees are often insufficient due potential disparities in coverage across subpopulations.
Many works aim to improve conditional validity by modifying the calibration step Lei & Wasser- man 2014: Guan 2023 Barber al. 2023 Luo & Zhou 2025a) altering the initial prediction rule
[Romano et al. 2019: Sesia & Romano| [202 |] Chernozhukov et al. 2021). Some research focuses
202I|Chernozhukor
on coverage under covariate shift (Lei & Wasserman) 2014 Tibshirani al. 2019 Izbicki al.
2022], with frameworks like that by Gibbs et al. (Gibbs et al. 2025) aiming for exact finite-sample
coverage across shifts.
Group conditional guarantees have also been explored (Toccaceli & Gammerman 2019 Gupta et al. 2020 Ding et al. 2023). Mondrian conformal prediction (Vovk et al.] (0003) provides exact cover- age for disjoint groups. More flexible methods for overlapping groups exist (Foygel Barber et al.)
2021 Jung et al., 2023), though they can be computationally intensive or rely on distributional as- sumptions. Other approaches learn features to improve conditional coverage, such density-based atypicality (Yuksekgonul et al. 2023) or learning covariate space partitions (Kiyani et al. 2024)..
Our method contributes to this line by providing novel way to achieve image-conditional coverage in segmentation, moving beyond group-basee approaches to an instance-specific adaptive threshold- ing mechanism, particularly through the end-to-end differentiable optimization.
EXPERIMENTS
This section presents the empirical evaluation of our proposed AT and COAT methods. We assess their performance in controlling the False Negative Rate (FNR) on diverse image segmentation tasks. Our experiments aim to demonstrate the effectiveness of amage-adaptive thresholding compared to approaches like CRC Angelopoulos al. (2024) and AA-CRC (Blot al. 2025), and to validate

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the robustness of our methods across different base segmentation models and datasets. Detailed experimental settings can found in Appendixs A.2 and
Desired FNR=0.1
Original
1708-0613
JNHHBAAS
CRC
JNitoo 101
Jhit-a
Jhib-o 135
UNi-a167
EMDi- 100
FiMbo 112
COAY
Figure 3: Qualitative comparison of CRC and COAT prediction sets a significance level 0.1. The top row shows original polyp images, the middle row displays CRC prediction sets, and the bottom row presents COAT prediction sets. White pixels represent true positives, red false negatives, and cyan false positives. FNR values are given for each prediction. COAT demonstrates more consistent coverage and false negatives across images compared to CRC, highlighting the merit of our conditional risk control approach.
As shown in Figure 3 and Figure we conducted a qualitative analysis on the polyp dataset (with alpha 0.1) and the skin dataset (with alpha 0.2) respectively. The base segmentation models presented in both cases are PSPNet. It can be observed that COAT is capable of better maintaining the given target coverage rate for each image. On the polyp dataset, COAT maintains a more stable FNR while also achieving lower false positive rate. On the skin dataset, COAT consistently keeps the images at the given coverage rate. COAT adaptively adjusts thresholds achieve the target False Negative Rate (FNR); while this may increase false positíves, such instances often stem from the base model's inherent uncertainties. safety-critical scenarios where false negatives are more detrimental than false positives, COAT's precise FNR control proves a significant advantage for robust risk management.
As shown in Table D and Figure 6 randomly partitioned the dataset 20 times and then tested the mean and standard deviation Marginal Coverage and Coverage Gap. Across all base models and datasets, COAT consistently outperformed CRC and AA-CRC in terms of Coverage Gap. with COAT consistently achieving the best Coverage Gap.
In addition, have plotted the training progress of COAT. As can be seen from the loss function in Figure 5) regardless of the segmentation model employed and the corresponding dataset used, the training of COAT's loss function is highly stable, with rapid decline that eventually approaches zero. The qualitative results for the Fire dataset are in Appendix A.4 We also analyzed the reference temperature T' in the COAT method, examining how different temperatures affect the coverage gap (see Appendix A.5). We also discussed other aspects such as the time consumption of these methods (see Appendices A.6 A.10
Desired FNR=0,2 (a=0.2)
Originl
JNCE--AA1
INR--150
IXB-00206
1N0--0204
INRc0 20P
EXI--0300
ENR-0.110 150
ENR-00201
ENR--1000
ENR--1100
CRC
EXC 0208
IXRL-0225
FXR--O115
JXR8-0.101
INR-0.200 2011
EXR-0221
EXB-- 2018
ENR-00197
ENR--1.00
ENR--1202
COAT
Figure 4: Qualitative comparison of CRC and COAT prediction sets at a significance level 0.2.
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0.1
= 0.2
Dataset
Model
Method
Marginal Coverage
Coverage Gap
Marginal Coverage
Coverage Gap
CRC
0.907 (0.015)
0.145 (0.010)
0.796 (0.028)
0.232 (0.007)
Deeplab v3+
AA-CRC
0,900 (0.017)
0.122 (0.015)
0.797 (0.025)
0.167 (0.020)
AT
0.899 (0.024)
0.115 (0.014)
0.802 (0.047)
0.174 (0.019)
COAT
0.899 (0.016)
0.106 (0.010)
0.802 (0.020)
0.162 (0.011)
CRC
0.901 (0.018)
0.135 (0.013)
0.797 (0.025)
0.256 (0.016)
AA-CRC
0.908 (0.017)
0.131 (0.012)
0.793 (0.024)
0,217 (0.017)
UNet
AT
0.903 (0.013)
0.127 (0.009)
0.805 (0.022)
0,209 (0.013)
Polyp
COAT
0.904 (0.022)
0.122 (0.012)
0.803 (0.018)
0.199 (0.014)
CRC
0.906 (0.019)
0.150 (0.015)
0.804 (0.026)
0,249 (0.008)
AA-CRC
0,908 (0.018)
0.119 (0.016)
0.796 (0.022)
0.162 (0.025)
PSPNet
AT
0.899
0.119 (0.014)
0.796 (0.020)
0.166 (0.014)
COAT
(0.016)
0.110 (0.015)
0.796 (0.021)
0.144 (0.013)
CRC
(0,088)
0.149 (0.014)
0.803 (0.026)
0,255 (0.009)
AA-CRC
0,906 (0.026)
0.126 (0.014)
0.799 (0.022)
0,182 (0.014)
SINet
AT
0.899 (0.024)
0.119 (0.013)
0.809 (0.051)
0.170 (0.014)
COAT
0.896 (0.016)
0.102 (0.010)
0.800 (0.021)
0.148 (0.014)
CRC
0.901 (0.003)
0.067 (0.001)
0,803 (0.005)
0.092 (0.001)
AA-CRC
0.903 (0.004)
0.062 (0.002)
0.804 (0.005)
0.083 (0.006)
Deeplab v3+
AT
0.901 (0.003)
0.061 (0.003)
0,806 (0.031)
0.086 (0.015)
CONT
0.900 (0.002)
0.058 (0.001)
0.799 (0.003)
0.076 (0.002)
CRC
0.899 (0.005)
0.077 (0:001)
0,802 (0.006)
0.103 (0.001)
AA-CRC
0.902 (0.005)
0.068 (0.004)
0.803 (0.005)
0.088 (0.006)
UNet
AT
0.900 (0.003)
0.063 (0.003)
0,800 (0.004)
0.085 (0.006)
Fire
CONT
0.900 (0.003)
0.061 (0.001)
0.800 (0.003)
0.079 (0.002)
CRC
0.901 (0.003)
0.065 (0.001)
0,801 (0.005)
0.091 (0.001)
AA-CRC
0.904 (0.005)
0.063 (0.003)
0.808 (0.008)
0.079 (0.010)
PSPNet
AT
(0.0119))
0.065 (0.003)
0.799 (0.004)
0.089 (0.005)
COAT
0.900 (0.003)
0.060 (0.001)
0.800 (0.004)
0,077 (0.002)
CRC
0.901 (0.004)
0.071 (0.001)
0,800 (0.006)
0.101 (0.001)
AA-CRC
0.903 (0.008)
0.063 (0.002)
0.803 (0.009)
0.082 (0.006)
SINet
AT
0.900 (0.003)
0.065 (0.004)
0.799 (0.00S)
0.090 (0.007)
COAT
0.900 (0.002)
0.059 (0.002)
0.799 (0.004)
0 (0.003)
CRC
0.900 (0.003)
0.072 (0.001)
0,802 (0.006)
0.107 (0.002)
AA-CRC
0.905 (0.004)
0.057 (0.003)
0,806 (0.005)
0.079 (0.010)
Deeplab v3+
AT
(0.016)
0.061 (0.009)
0.809 (0.039)
0.090 (0.023)
COAT
0.899 (0.003)
0.054 (0.001)
0,800 (0.005)
0.073 (0.002)
CRC
0.900 (0.003)
0.062 (0.001)
0,800 (0.006)
0.097 (0.002)
AA-CRC
0.908 (0.003)
0.056 (0.003)
0,807 (0.005)
0.081 (0.004)
UNet
AT
0.899 (0.003)
0.059 (0.002)
0.800 (0.004)
0.090 (0.006)
Skin
COAT
0.899 (0.003)
0.054 (0.001)
0,800 (0.004)
0.079 (0.002)
CRC
0.902 (0.003)
0.069 (0.001)
0.804 (0.006)
0,103 (0.001)
AA-CRC
0.906 (0.005)
0.057 (0.005)
0.806 (0.005)
0.071 (0.011) 1)
PSPNet
AT
0.903 (0.015)
0.061 (0.008)
0.809 (0.039)
0.076 (0.025)
COAT
0.899 (0.003)
0.050 (0.002)
0.799 (0.004)
0.064 (0.002)
CRC
0.905 (0.004)
0.078 (0:001)
0.806 (0.004)
0,113 (0.001)
AA-CRC
0.905 (0.005)
0.063 (0.006)
0.805 (0.005)
0.075 (0.010)
SINet
AT
0.906 (0.022)
0.065 (0.010)
0.800 (0.003)
0.074 (0.005)
COAT
0.899 (0.003)
0.055 (0.001)
0.800 (0.004)
0.071 (0.002)
Table 1: Marginal Coverage and Coverage Gap Results at a 0.1 and 0.2 Across Different
Models and Conformal Methods. Each dataset result is the mean and standard deviation of 20 random splits.
Remark on Coverage Gap Limitations: It is important to note that the coverage gap cannot be reduced to zero in practice due to several fundamental limitations. First, finite sample effects mean that with limited calibration data, perfect estimation of image-specific coverage is statistically im- possible. Second, there cxists inherent birs-variance trade-off in threshold prediction while our adaptive methods reduce bias by learning image-specins patterns, they introduce variance through the learned predictor. Third, model capacity constraints limit how well our threshold predictor can capture the complex relationship between image characteristics and optimal thresholds.
Despite these theoretical limitations, our COAT method consistently achieves the smallest cover- age gap across all experimental settings, demonstrating its effectiveness in learning meaningful image-adaptive patterns. The end-to-end differentiable optimization in COAT provides a princi-

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Figure 5: Loss function graphs for different segmentation models trained using the COAT method, with datasets shown from left right being polyp, fire, and skin respectively.
AM--0
A--RC XI
Deeplabbv va-
17NH
P5PNH
SDH
CIC M-3IC
AT
COAT
CBC
M-3CC
AT
COAT
C3 M-33C
AT
COAT
CRC
AV-RC
COAT
Deeplab
T
A6-aC
Figure 6: The box plot results for the Coverage Gap obtained from different datasets and various base segmentation models are presented. For each experimental setup, 20 random splits were conducted. The three rows of plots, from top to bottom, display the results for the polyp, fire, and skin datasets, respectively.
pled approach that directly optimizes for the target coverage, leading to more reliable conditional guarantees compared to both global thresholding (CRC) and supervised approaches (AA-CRC and AT). This improvement is particularly valuable for safety-critical applications where consistent per- image reliability paramount.
5 CONCLUSION
In this paper, we address the limitation of existing segmentation methods that provide only marginal guarantees and lack image-level conditional coverage safety-critical applications by proposing two novel methods for learning adaptive thresholds: AT and COAT. As our core contribution, COAT introduces an innovative end-to-end differentiable miscoverage loss, enabling the precise learning of an optimal threshold for each image by directly optimizing the conditional coverage target. Ex- tensive experiments across multiple datasets demonstrate that our methods, particularly COAT, sig- nificantly reduce the Coverage Gap while maintaining the target marginal coverage rate, thereby exhibiting stronger consistency and reliability across different images. However, COAT exhibits limitations in few-shot samples, multi-class segmentation, and False Discovery Rate (FDR) risk control. This work provides a robust pathway toward building more trustworthy and interpretable AI systems for critical applications.
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ACKNOWLEDGMENTS
This work was supported by the National Natural Science Foundation of China (Grant 62506315) and City University of Hong Kong (Grants 9610639, 7020161).
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A APPENDIX
LARGE LANGUAGE MODEL (LLM) USAGE
During the preparation of this manuscript, large language model (LLM), specifically [Gemini-2.5- Flash]. was employed as a general-purpose assist tool. The LLM's contributions were primarily in the following areas:
Code Optimization for Experimental Visualization: The LLM assisted in optimizing and refining Python code snippets used for experimental visualization and data processing routines. This collaboration led to more efficient and readable implementations, partieu- larly for generating the figures (e.g., Figure 3, Figure 4, Figure 7) and tables (e.g., Table 1, Table 2) presented in the paper.
Writing Assistance and Refinement: The LLM was utilized for drafting and refining cer- tain sections of the paper, including improving clarity, grammar, and stylistic coherence. This involved generating initial textual descriptions and polishing existing content to en- hance its overall quality and readability.
The authors maintained full responsibility for reviewing, editing, and validating all content generated or optimized with the assistance of the LLM, ensuring its accuracy, originality, and adherence to scientific standards. The LLM was not involved in the core research ideation, experimental design, data collection, or primary analysis leading to the scientific conclusions. The scientific content, conclusions, and any potential errors remain solely the responsibility of the authors.
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A.1 CONDITIONAL VALIDITY UNDER CONSERVATIVE DOMINANCE
In general, distribution-Fie conditional validity is impossible without additional assumptions. Here we show that, under a conservative dominance condition (relative to an oracle conditional thresh- old), the adjusted thresholding rules of the proposed AT and COAT enjoy conditional validity. Let (X,Y Y ~ P denote new test image-mask pair drawn from an unknown distribution on x V and let Px be the marginal distribution of X. Let 7()) be the fixed base threshold estimated by AT or COAT and ((-) be the fixed base segmentation model. For a threshold E [O, 1J, we define the prediction set T( (j (X) 2 7) and the FNR loss for (X,Y):
2 Yb
L(X, YiT =
E 1],
(16)
Σ Yb]
where we assume C(x,r'), we have
P(E N Yb] 0)
For any fixed (x, and any T', since C(a, 7) 2
N
N
Σ
(z)()) 2 7] M Σ (t) 27 sbi.,
(17)
which implies that L(x, < L( y: T'. Consequently, the conditional risk
[(x,T) ( x 2) j= 1 (x) YD]
X x
(18)
N Σ Yb]
also non-decreasing in
Assumption 1 (Right-comtinuity). For Px-almost every x, the map + L(x, is right-cootiiusss
on [0, 1].
For any fixed a E (0, and X = x, we define the oracle threshold as
-( (x) (x) sup {r [0, (a,r)
(19)
Under Assumption 1 we have Ü(,, ((a) (())))) S for Px-almost every x.
Assumption (Conservative dominance). Fix global correction parameter E [0, 1] and define the adjusted threshold Fy (z) == clip(f(x)) - t', 1). Assume that
(x( S ((0) for Px-almost every T.
(20)
Proposition 1 (Conditional Validity). Under Assumptions 7 and 2 for Px-almost every x.
Σ N
Σ/= (x) 2 (X)] -Yb]
E
x = 2 1 a.
(21)
Σ YbJ
Proof. Fix x E X. By monotonicity i(x 7) and Assumptions
and
((,,Fe(x)) < M(x, ( (x) aa
(22)
Equivalently,
N (x) 2 (x) I 1 1(2, (x))
(23)
En Ybl
for Px-almost every x.
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A.2 DATASETS
We utilized three distinct image segmentation datasets to evaluate the robustness of our proposed algorithms: polyp segmentation, skin lesion segmentation, and flame segmentation. These tasks particularly critical for FNR control, as missing parts of the region of interest can lead to severe consequences.
For each dataset, we followed a specific data partitioning strategy. The initial training set was used for training the base segmentation models (UNet (Ronneberger et al. 2013). DeepLab v3+ (Chen et al. 2018), PSPNet (Zhao et al., 2017), SINet (Fan et al.] (220aa). The remaining data was desig- nated as the test set. This test set was then further partitioned. One half of the test set was reserved for final performance evaluation. The other half was designated as calibration set. For AA-CRC, AT and COAT methods, this calibration set was further equally divided into a training subset (for training the adaptive threshold prediction model fo) and calibration subset (for determining the final calibrated threshold). For the standard Conformal Risk Control (CRC) method, the entire cali- bration set utilized for its calibration procedure.
Polyp Dataset: Following similar setups as Angelopoulos et al. (Angelopoulos et al., 2024 blot et al. (Blot et al. (025)) and Fan ef al. (Fan et al.) 2020b), this dataset (Jha et al.
2019; [ernal et al. [2017] Väzquez et al. 2017: Tajbakhsh et al. 2015 Silva et al.
2014 comprised 1450 images for training the base segmentation models and 798 images
for the test set.
Skin Lesion Dataset: We employed the HAM10000 skin image dataset (Tschandl al., 2018). This dataset was split with 50% (5007 images) allocated for training the base models and the remaining 50% (5008 images) for the test set.
Fire Dataset: For image fire segmentation experiments, used the dataset provided by Aktaş (Aktas) 2023). This dataset was partitioned with 80% (21968 images) for training the base models and 20% (5492 images) for the test set.
Additionally, for both the AT and COAT methods, the threshold predictor fo was implemented using a ResNet-50 (He et al. 2016) architecture. The training epochs for AT were set to 30, while for COAT, they were set to 60. For the COAT method, the temperature parameter T used in the sigmoid function for the soft mask calculation was set to 0.05.
A.3 EXPERIMENTAL DETAILS
The source code and experiment scripts used to generate the results in this paper will be made publicly available upon publication of the paper.
Reproducibility Statement:
Implementation Details: All experiments were conducted on a server equipped with an NVIDIA RTX 4090 GPU (24GB of RAM), running Ubuntu 24.04. Our models were implemented using Python 301 PyTorch 2.3.0, and the system was configured with CUDA 12.6. The learning rate of the COAT method is consistently set 5e -4, and the batch size is 64. All the basic segmentation models underwent 20 epochs of training. Additionally, a unified learning rate of le-4 and a batch size of 24 were employed for all models. The random seed to 42 for all experiments.
A.4 4
FIRE RESULTS
Due to space constraints, we have included the qualitative analysis of the fire dataset in the appendix. is evident that COAT consistently maintains target coverage, albeit with a certain degree of false positive rate. However, when compared to CRC, COAT demonstrates lower and more stable false negative rate.
A.5 SENSITIVITY ANALYSIS
In this section, we employed the SINet basic segmentation model on various datasets and conducted grid search sensitivity analysis with parameters set 0.1 and T [0.001, 0.01, 0.05, 0.1, 1, 10, 100). Similarly, each experiment was carried out with 20 random splits, and the results are presented as the mean values and standard deviations.
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Desired FNR=0.2 (a=0.2)
Original
FNB--0.00
ENR-00779
FND-00514
FNIR-0.500
FNR-0.30
FNR--2301
FNR-00.00
FNR--0199
FNR--0
CRC
FNR-0332
ENR-00197
FNR-00.18
COAT
Figure 7: Qualitative comparison of CRC and COAT prediction sets at a significance level of
0.2.
0.1
Dataset
Marginal Coverage
Coverage Gap
100
0.901 (0.016)
0.151 (0.012)
10
0.899 (0.023)
0.153 (0.018)
0.899 (0.023)
0.156 (0.017)
Polyp
0.1
0.899 (0.012)
0.114 (0.015)
0.05
0.896 (0.016)
0.102 (0.010)
0.01
0.900 (0.020)
0.148 (0.013)
0.001
0.901 (0.026)
0.147 (0.018)
100
0.900 (0.003)
0.071 (0.001)
10
0.901 (0.003)
0.070 (0.001)
1
0.900 (0.003)
0.070 (0.001)
Fire
0.1
0.900 (0.003)
0.062 (0.001)
0.05
0.900 (0.003)
0.059 (0.001)
0.01
0,900 (0.004)
(0.001)
0.001
0.900 (0.003)
0.066 (0.014)
100
0.900 (0.001)
0.081 (0.001)
10
0.901 (0.001)
0.081 (0.001)
0.900 (0.002)
0.081 (0.001)
Skin
0.1
0.901 (0.004)
0.056 (0.001)
0.05
0.899 (0.003)
0.055 (0.001)
0.01
0.900 (0.004)
0.056 (0.002)
0.001
0.899 (0.003)
0.058 (0.009)
Table 2: Marginal Coverage and Coverage Gap Results at 0.1 Across Different Models and T. Each dataset result the mean and standard deviation of 20 random splits.
A.6 TIME METRICS
In this section, we calculated the training, calibration, and testing time for each method. Specifi- cally, we ran each method 10 times using the PSPNet segmentation model with 0.2, and then computed the mean and standard deviation of the results for all methods.
A.7 DISTRIBUTION OF CONDITIONAL COVERAGE
To further elucidate COAT's advantages, we also visualize the distribution of individual image cov- erage for each dataset.
Figures g to 10 illustrate the density distribution coverage, for individual test images across CRC, CRC. AT, and COAT. These results are obtained using the PSPNet base model various
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Number of sct samples
Dataset
Method
training
calibration
testing
Training time (s)
Calibration time
Testing time
CRC
399
399
0.0000 (0.0000)
0.1961 (0.0021)
0,5414 (0.0092)
AA-CRC
200
198
399
38.2870 (0.3807)
120.8917 (60.4003)
0.4720 (0.0699)
Polyp
200
198
399
151.7819 (0.6079)
2.1570 (0.0140)
3,0021 (0.0631)
COAT
200
198
399
24.3318 (0.5924)
0.3764 (0.0821)
0.4807 (0.0607)
2746
2746
0.0000 (0.0000)
0.1961 (0.0021)
0.5413 (0.0092)
AA-RCC
1373
1373
2746
260.0201 (0.4249)
647.3359 (213.7751)
3.0068 (0.0583)
1373
1373
2746
547.3396 (0.4609)
6.6958 (0.2020)
9.5174 (0. (0.1679)
COAT
1373
1373
2746
166 8045 (0.3589)
2.3913 (0.0361)
3.3785 (0.2682)
CRC
2504
2504
0,0000 (0,0000)
1,3292 (0.0315)
0.4895 (0.0061)
Skin
AA CRC
1252
1252
2504
7810 (0.2400)
8790 (214.2507)
2.7394 (0.03683)
AT 1252
1252
2504
598955699/ (1 L0257)
6.0459 (0.0214)
8.6309 (0.0247)
COAT
1252
1252
2504
131.7819 (0.6079)
2.1570 (0.0140)
3.0021 (0.0631)
Table The time consumed by various methods during training, calibration, and testing on different datasets was recorded, utilizing the PSPNet base model with 0.2 and 10 random runs.
7.5
Target (0.90)
7.5
Target
(0.80)
Densily
CRC
Density
CRC
5.0
AA-CRC
AA-CCC
AT
AT
2.5
COAT
2.5
COAT
0.0
0.0
0.4
1.0
0.2
0.4
1.0
Coverage
Coverage
Figure 8: Density distribution of image-level coverage for CRC, AA-CRC, AT, and COAT using the
Deeplab v3+ model on Polyp.
7.5
Target (0.90)
7.5
Target (0.60)
Donsily
CRC
Densily
CRC
5.0
AA-CRC
5.0
AA-CRC
AT
2.5
COAT
2.5
COAT
0.0
0.0
0.1
0.2
0.3
0.4 0.5 Coverage
0.6
0.7
0.8
1.0
0.0 0.1 0.2
0.3 0.4 0.6
0.8
1.0
Coverage
Figure 9: Density distribution of image-level coverage for CRC, AA-CRC, AT, and COAT using the Deeplab v3+ model on Fire.
7.5
Target (0.90)
Target (0.80)
Densily
CRC
Densily
CRC
5.0
AA-CRC
5.0
AA-CRC
2.5
COAT
2.5
COAT
0.0
0.0
0.1
0.2
0.3
0.4
0.5 Coverage
0.6
0.7
0.8
0.9
1.0
0.0
0.1
0.2 0.3 0.4 0.5 0.6
0.7
0.8
0.9
1.0
Coverage
Figure 10: Density distribution of image-level coverage for CRC, AA-CRC, AT, and COAT using the Deeplab v3+ model on Skin.
datasets. The x-axis denotes the coverage per image, discretized into 0.1 intervals, while the y-axis indicates the density of images falling within each respective coverage interval. vertical dashed line marks the target coverage of
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A clear observation from these distributions is that COAT consistently exhibits the highest density of samples precisely at the target coverage, with a near-absence of samples in the low-coverage regions. This demonstrates that COAT provides the most consistent and reliable image-wiss condi- tional coverage across all evaluated methods. Empirical evidence further suggests that its end-to-end differentiable optimization framework is highly effective in minimizing the diserepancy between the realized and target coverage for individual images. This, in turn, significantly enhances the trust- worthiness and interpretability of uncertainty estimation in critical segmentation tasks.
A.8 FNR AND FPR CONTROL PERFORMANCE
A.8.1 FPR CONTROL PERFORMANCE
In this section, evaluate the performance of COAT (Conditional Optimization for Adaptive Thresholding) method against the conventional Conformal Risk Control (CRC) approach in achiev- ing False Positive Rate (FPR) control for image segmentation. Controlling FPR is crucial in many applications, especially when minimizing false alarms or over-segmentation is critical.
We assess performance using two key metries:
Marginal Coverage: Defined 1 FPR, this metric quantifies the overall proportion of true negative pixels (background) that are correctly classified as background across the entire test set. The goal to achieve marginal coverage close to 1 a. For instance, the target FPR = 0.1, the desired marginal coverage is 0.9. This means we aim for the overall FPR be at most a.
This metric measures the deviation of the conditional FPR from the target a for individual images. A smaller coverage gap indicates better conditional coverage, meaning the method can adaptively adjust its threshold to maintain the FPR close to a for each specific input image, rather than just on average. Specifically, it often reflects how well FPR FPR(X;) adheres to for any given image X, A common definition for coverage gap is B [|]FPR(X (X;) - a.], or similar measure of conditional FPR violation, where a smaller value is highly desirable.
Coverage Gap:
The results in Table demonstrate the superior performance of COAT in achieving robust FPR control, particularly in terms of conditional coverage. While both CRC and COAT can provide marginal FPR coverage guarantees, COAT significantly outperforms CRC by providing much tighter conditional coverage, as evidenced by the consistently smaller coverage gaps. This ability per-image basis makes COAT
to adapt thresholds requiring precise FPR control and trustworthy uncertainty estimates, especially where minimizing false positives paramount.
reliable method for applications
A.8.2 WEIGHTED COAT TRAINING LOSS CONTROL
In this section, we employ PSPNet as the base model. We prioritize the FNR loss and conduct weighted training on the FPR loss, while maintaining the calibration steps for FNR. By comparing different loss weights, we observe how the FPR changes under the condition that COAT ensures marginal FNR.
As shown in Table 5 and Table 6 when we employ the weighted loss, we find that 0.1 par- ticularly good performance in the polyp dataset. There is solid reason for this. other datasets, there is an overlap between the False Negative Rate (FNR) and the False Positive Rate (FPR), which causes this part the loss to be almost zero. However, the polyp dataset with a = 0.1, there is no such overlap, resulting in relatively large loss for the FPR, which makes the training process easier. For detailed reasons, please refer to Appendix A.
A.9 EXTENDING TO MULTI-TARGET IMAGE SEGMENTATION
This section details the extension of our COAT (Conditional Optimization for Adaptive Threshold- ing) framework to address multi-target image segmentation tasks. such applications, the goal is often accurately identify and delineate multiple distinct object classes within an image. Unlike
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ar 0.1
a= 0.2
Dutaset
Model
Method
Marginal Coverage
Coverage Gap
Marginal Civerage
Coverage Gap
Deeplab v3+
CRC
0.903 (0.008)
0.072 (0.004)
0.795 (0.011)
0.133 (0.005)
COAT
0.901 (0.004)
0.048 (0.005)
0.797 (0.012)
0.081 (0.012)
CRC
0.899 (0.006)
0.055 (0.002)
0.804 (0.009)
0.082 (0.003)
UNet
Polyp
COAT
0.901 (0.006)
0.051 (0.006)
0,800 (0.011)
0.101 (0.013)
CRC
0.902 (0:010)
0.065 (0.004)
0.797 (0.009)
0,107 (0.003)
PSPNet
COAT
0.899 (0.005)
0.049 (0.004)
0.800 (0.010)
0.091 (0.008)
CRC
0.899 (0.006)
0.044 (0.002)
0.797 (0.011)
0.076 (0.002)
SINet
COAT
0.900 (0.006)
0.039 (0.002)
0.803 (0.008)
(0.006)
CRC
0,900 (0.004)
0.076 (0.002)
0.801 (0.006)
0.133 (0.002)
Deeplab v3+
COAT
0,900 (0:001)
0.030 (0.003)
0,801 (0.002)
0.047 (0.005)
CRC
0,900 (0.003)
0.073 (0.001)
0.799 (0.007)
0.127 (0.001)
UNet
Fire
COAT
0.900 (0.001)
0.028 (0.002)
0.801 (0.001)
0.033 (0.003)
CRC
(0.000)
0.062 (0.001)
0,800 (0.003)
0.087 (0.001)
PSPNet
COAT
0.900 (0.001)
0.026 (0.002)
0.800 (0.001)
0.033 (0.003)
SINet
CRC
0.900 (0.000)
0.030 (0.000)
0.800 (0.002)
0.038 (0.000)
COAT
0,900 (0.001)
0.020 (0.001)
0.801 (0.002)
0.032 (0.002)
CRC
0.899 (0.004)
0.091 (0.002)
0.799 (0.007)
0.176 (0.004)
Deeplab v3+
COAT
0.901 (0.002)
0.054 (0.003)
0.800 (0.003)
0.082 (0.003)
CRC
0.900 (0.005)
0.106 (0.002)
0.801 (0.007)
0.174 (0.003)
UNer
Skin
COAT
0.900 (0.002)
0.051 (0.002)
0.801 (0.003)
0.073 (0.003)
CRC
0.899 (0.003)
0.081 (0.002)
0.800 (0.007)
0.147 (0.002)
PSPNet
COAT
0.900 (0.002)
0,040 (0.001)
0.801 (0.002)
0.055 (0.003)
CRC
0.899 (0.004)
0.087 (0.002)
0.801 (0.006)
0.151 (0.002)
SINet
COAT
0.900 (0.003)
0.060 (0.002)
0.799 (0.004)
0.081 (0.005)
Table 4: Marginal Coverage and Coverage Gap Results for FPR Control at = 0.1 and a = 0.2
Across Different Models and Conformal Methods. Each dataset result is the mean and standard deviation of 20 random splits.
Dutsset
FNR loss weight
FPR loss weight
Marginal Coverage (1 (NR)
Covenge Cap (1 FNR) Marginal Coverage (1 (FR)
er === 0.1
Coverage Gup EPR)
0
0.396 (0,016)
0.302 (0.110)
0.696 (0.091)
0.315 (0.072)
0:02
0.894 (0:012)
0,303 (0.815)
0.819 (0:047)
(D.040))
Polyp
0.2
0.897 (0.018)
0.107 (0.015)
0.815 (0:057)
0,211 (0.054)
(0.011)
0.340 (0.012)
0.105 (0.0144
(0.022)
(0.016)
(0,024)
0,00 (0:002)
0.987 (0.018)
0.088
(0.002)
(0.001)
(0.080)
0.2
0.900 (0:0003)
0.000
(0.090)
0.088
0 399 (0.000)
0.064 (0.005)
0.957 (0.001)
0.087 (0.001)
2
0.398 (0.0015)
0.139 (0.0844)
0.984 (0:001)
0.084 (0.003)
0
01399 (0.0LE)
0,050 (0.002)
0.985
0,994 (1.001)
0.02
0 3.000 (0:003)
0.051 (0.001)
0.985 (0:001)
0.094 (0.001)
Skin
0.2
0,900 (0:002)
0,051 (0.001)
0.983 (0:001)
0.093 (0.001)
0.561 (C.0EE)
03057 (0.612)
0.985 (0:002)
0,090 (1.003)
0.399 (0.006)
0,109 (0.003)
0.990 (0.001)
0.090 (0.001)
Table 5: Marginal Coverage and Coverage Gap Results for FNR and FPR weighted Control at Each dataset result is the mean and standard deviation of 20 random splits.
single-class segmentation, multi-target scenarios introduce additional complexities, especially when aiming for reliable uncertainty quantification across different categories simultancously.
A.9.1 PROBLEM FORMULATION FOR MULTI-TARGET SEGMENTATION
For multi-target segmentation, we are interested in controlling the False Negative Rate (FNR) for a union of specified target classes. Given an input image X; and its ground truth mask Y,, which now contains multiple class labels, we define set of target class indices Kungen- Our objective is to construct a prediction set C(X,) that covers any pixel belonging to any of these target classes with a user-specifted probability 1 a. That is, we aim control the FNR for the combined ground truth
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Dunaset
FNR weight
weight
Margatal Coverage (I-FNR)
Covenge Stop (I-FNR)
Marginal Coverage (I-FPR)
Coverage Eap (I-FPR)
(0:021)
(0.014)
(0.019)
0.02
0.797 (0:015)
0.146 (0.013)
0.895 (0.024)
0.238 (0.015)
Polyp
0.2
0.799 (0.0144)
0.147 (0.115)
0.114 (0.4120)
0.224 (0.0133
0 1.501 (0.025)
0.240 (0.009)
0.982 (0.006)
0.185(0.004
2
0.307 (0,029)
0.242 (0.007)
0,982 (0.004)
0.182 (0.084)
01
0.799 (0.0044)
0,080 (Q,003)
0.993 (0,000)
0.193 (0,000)
0.02
0.798 (0.0013)
0,877 (0J002)
0.995 (0:000)
0.19%
Fire
0.2
0 0.500 (0.004)
0,077 (0.002)
0.943 (0.000)
0.195 (0.3000000))
(0.0014)
(0,0000
(0.000)
(0.008)
0.194 (0.001)
799
0.195
(0:002)
(0.001)
(0:901)
0.2
0 (0-0002)
(0.001)
(0:001)
0.300 (0. (0.003)
0.067 (0.012)
(0,002)
0.192 (0.001)
0.300 (0.087)
0.367 (0.005)
0.997 (0.000)
0.197 (0,000)
Table 6: Marginal Coverage and Coverage Gap Results for FNR and FPR weighted Control at a = 0.2. Each dataset result is the mean and standard deviation of 20 random splits.
mask Yrunion = Kkee fj:je E Y, and YM = k3. The FNR is then calculated as:
FNR(X;) = Σ3= O(X)Li) Y, yiuion [i]
Σ Y, uunion b]
And we seek to achieve E[FNR(X,)) < a (marginal coverage) while minimizing the conditional coverage gap, i.e., making (NR(X) as close to as possible for individual images X;
A.9.2 MULTI-CLASS COAT STRATEGY: PER-CLASS ADAPTIVE THRESHOLDING
To extend COAT to multi target segmentation, we adopt "per-class adaptive thresholding" strategy- Instead of training a single COAT model to predict a global threshold for the union of classes, we train separate COAT model for each individual target class k E Kurrget Each class-specifie COAT model fD is trained to predict an optimal image-adaptive threshold (X) for its corresponding class's probability map pk (X).
The process is as follows:
Base Model Predictions: A pre-trained base segmentation model (e.g.. PSPNet) outputs multi-channel probability map (XX) (X). ... (X)))), where P& (X) is the pixel- wise probability of class
Per-Class COAT Training: For each target class E Kunrget
We extract the probability map (X) for class k and the binary ground truth mask (x) (where (X) if pixel belongs to class and 0 otherwise).
A dedicated COAT model fD.: is trained using the differentiable miscoverage loss, optimizing (X) to achieve 1 - coverage for class k.
Per-Class Calibration: Each trained fD. is then calibrated using class class-specific calibra-
tion set to find a correction term '/-
For a new test image X, the final prediction set Ō(X) for the union of target classes is formed by taking the logical OR (union) of the individual, calibrated prediction masks from each target class:
Combined Prediction Set:
Ō(x) = U ( (A clip(fx(X) - t/6, 0,1)
A.9.3 COMPARISON WITH MULTI-CLASS CRC
Our multi-class COAT method compared with a conformal prediction baseline method. To ensure fair comparison with class-wise COAT strategy, adapt this baseline method to compute an independent threshold for each target class. Specifically, this class-wise conformal prediction method computes a unique threshold X for each target class & Kunges by calibrating specifically
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for that class's False Negative Rate (FNR) the calibration set. Subsequently, these class-specife thresholds A are applied to the probability maps of their respective target classes.
The final prediction set for this bascline method is formed by combining the predictions of individual classes:
CCRC(()
U PA }
is important to note that while this class-wise thresholding approach can provide class-marginal coverage guarantees (i.e., independently controlling each class's ' FNR on the calibration set), it does not inherently guarantee mmage-conditions coverage or clss--coniitinal coverage.
Therefore, the key distinction lies in the adaptability and scope of guarantees. While this class- wise baseline method provides marginal guarantees for each class's FNR, its reliance on static, class-specific thresholds inherently limits its ability to control image- and class-specific conditional coverage. Our class-wise COAT strategy, by training an adaptive threshold predictor for each class, aims provide more fine-grained image-conditionaal FNR control for each target class. When these individually optimized class predictions are combined, we expect to achieve tighter control over the class-union conditional FNR, thereby closing the coverage gap.
A.9.4 MULTI-CLASS IMAGE SEGMENTATION RESULTS: CRC vs. COAT
We evaluate our multi-class COAT on the Cityscapes dataset (Cordts et al. 2016 challenging benchmark for urban scene understanding. We selected three diverse target classes:
Road (index 0): A common and extensive background class.
Person (index 11): A critical and often smaller foreground class.
Car (index 13): Another important foreground class, varying in size and shape.
Our experiments are conducted with a target FNR level of a = 0.1. To ensure robustness, we perform 20 random splits of the dataset for training, calibration, and testing, and report mean and standard deviation for performance metrics. The base segmentation model used is PSPNet, pre- trained on Cityscapes.
Metric
CRC
COAT
Coverage
0.9152 (0.0091)
0.9048 (0.0107)
Coverage Gap
0.0844 (0.0048)
0.0811 (0.0150)
Coverage (road)
0.9031 (0.0113)
0.8978 (0.0127)
Coverage gap (road)
0.0978 (0.0068)
0.0938 (0.0188)
Coverage (person)
0.9044 (0.0139)
0.9017 (0.0155)
Coverage gap (person)
1069 (0.0075)
0.0997 (0.0088)
Coverage (car)
0.9035 (0.0092)
0.9013 (0.0113)
Coverage gap (car)
0.0827 (0.0029)
0.0670 (0.0036)
Table 7: Performance Comparison of CRC and COAT on Cityscapes Multi- Target Segmentation
As shown Table 7 compared with CRC, COAT not only achieves smaller conditional coverage and more stable marginal coverage across categories, but also attains smaller coverage.
Remark. As shown in Figure 11 our multi-class COAT strategy demonstrates promising results extending adaptive thresholding to multi-target image segmentation, achieving better class- conditional coverage compared to the baseline. However, it is important to acknowledge certain considerations for its broader application. The current approach involves training a separate COAT model for each target class. While effective, this can become computationally intensive and time- consuming when dealing with a very large number of target classes. Future work will explore the design of a joint loss function that enables simultaneous adaptive thresholding for multiple classes within a single optimization framework, aiming to improve efficiency without compromising perfor- mance. Additionally, the effectiveness of COAT, like other conformal prediction methods, relies on
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Desired FNR-0.1 (a 0.1) for classes: road, person, car
Original
CRC
PND-0.200
PNDD-0,115
TAdteo
INID-1:200
PXD--6.110
COAT
False Fositive (FP)
False Negalive (FNN
Figure 11: Qualitative comparison of CRC and COAT prediction sets at significance level of 0.1.
the assumption of exchangeability. scenarios with extremely limited datasets, this assumption may be compromised, potentially leading degradation in the method's performance and reliability.
A.10 DUAL COAT FOR SYNCHRONOUS FNR AND FPR CONTROL
Simultaneously controlling False Negative Rate (FNR) and False Positive Rate (FPR) is crucial in safety-critical applications. While our COAT framework primarily targets FNR by minimizing coverage gaps, its differentiable nature allows direct extension to FPR control, enabling a flexible approach balance both risks with single threshold.
A.10.1 PROBLEM FORMULATION FOR DUAL CONTROL
In image segmentation, a predicted set C(X, r) is defined by a base model's probability map ((X)
and threshold
C(X, r) = (j By (X) 2
False Negative Rate (FNR): Proportion of true positive pixels incorrectly excluded.
FNR(X,T)= = E T(X,T) Yb
Σ YbJ
Objective: FNR(X,7) S OFNN. Lower reduces FNR. FNR is non-increasing with
False Positive Rate (FPR): Proportion of true negative pixels incorrectly included. For the binary mask representation where Y E {0, 1).N, the complement is 1 Y[j] for each pixel j.
EN O(X, (1-Yb))
FPR(X,
(1-
Objective: FPR(X, S OFFR. Higher reduces FPR. FPR is non-decreasing with 7.
The challenge lies in finding single that satisfies both FNR(X, 7) < apna and FPR(X, 7) < OPPR, due to their opposing dependencies on т.
A.1 INDIVIDUAL COAT MODELS FOR FNR AND FPR
We train two independent COAT models, each optimized for a specific risk by learning image- adaptive thresholds:
1. COATENR: Learns FFNR(X) aiming to achieve FNR(X, (xNFF(((XX) aFNR.
2. COATFPR! Learns Frrn(X) aiming achieve FPR X, FFrn(X)) (X)) OFPR
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A.10.3 INTERPOLATION FOR JOINT MARGINAL RISK CONTROL
To balance overall (marginal) FNR and FPR across the dataset, we propose an interpolation strategy:
1. Obtain Image-Adaptive Thresholds: Train COAT and COAT independently. then obtain calibrated thresholds Tp and (X) for all test images using the calibra- tion procedure from Algorithm
2. Linear Interpolation: For each image X, define an interpolated threshold (x(X) using
À E 1]:
Tx(X) = A (X)( + (1 - X) (X)
This transitions between FNR FNR-priority (A = 0) and FPR- (A = 1) thresholds.
0.23
PMR
Anerae E93
Anarage 7FB.
Sarge
Terpit
3.8
3:23
3.00
3579
1129
Landba
Lambid
Figure 12: Left: afNr 0.1, aPPR 0.1; Right: apNa = 0.1, aFPR 0.2. On the fire dataset, when À takes any value within the interval [0, 1|, we can simultancously control both the FNR and FPR.
3320
Jeerae
Dripd ENR (S-1)
3:335
Torpe FSR
Sarget
Target 111113.23
3:23
222
3.00
3.75
124
1123
3180
05
0.
1.2
Landbds
Lambid
Figure 13: Left: aPNa 0.1, aFPR = 0.1; Right: OFNR = 0.1, aPpR 0.2. On the skin dataset, when À takes any value within the interval [0, 1J. we can simultaneously control both the FNR and FPR.
3. Marginal Risk Curve Generation: Iterate À from to 1. For each X, compute (x) for all test images. Then, for each image X, construct the prediction set C(X, (x))) (j (x( 2 (X)). Finally, calculate the average (marginal) FNR and FPR across the test set using these prediction sets. Plot these average FNR and FPR values.
4. Interpretation and User Guidance: The resulting curve visualizes the FNR-FPR trade- off:
A = 0: (X) = (X).(X). Expected marginal FNR near aFNR. FPR potentially higher.
= (xX) TFPR(X) Expected marginal FPR near aPPR FNR potentially
higher.
As À increases, (X) generally increases, causing marginal FNR to typically increase and marginal FPR to typically decrease.
This curve allows users to select optimal À to balance marginal FNR and FPR according to ipplicatiionspeciiii safety and performance needs, aiming for a region where both risks are at or below their target levels.
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Anarce
Awares PTP
(NA.OL)
323
Sagad P7A (L1)
27. 0022
0.6
0.3
Lantidda
Lantidd
Figure 14: Left: arnn = 0.1, orrn 0.1; Right: aFNR = 0.2, orra = 0.2. On the polyp dataset, due its greater difficulty and the extremely small dataset sample size, the basic model yields poor prediction performance. This makes it impossible for us to control both the FNR and FPR simultaneously. As we increase our a, as shown the right panel, we become able control both FNR and FPR concurrently.
A.10.4 EXPERIMENTAL ANALYSIS
Remark. Note that Figures T2 13 and 14 which plot average FNR/FPR versus X, intended to serve as empirical guidance for applying this method to new sample X. For any such sample, our method computes calibrated thresholds: TENR designed satisfy E(FNR) S OFNN. and TFPR TPRR designed to satisfy E(FPR) I OPPR. This leads to two scenarios:
Feasible Solution (Tern S TEND): A feasible interval exits. Any threshold T selected from this interval guarantees that both risk constraints (E(FNR) OFNR and E(FPR) s OFFR are satisfied. In this case, is a preference parameter to choose a point within this guaranteed-sod safe region, guided by the average performance trends shown in the plots.
Desired FNR=FPR=002 (alpha =0.2)
Original Imzgee
(tye + Dwwl/2
Ground Turh
176--1,86, 17R-6.200
PN8I-1.26, 1188-d 011
1N00-6.29, R7B-0.09 834
I76--1.AA 17R-6.200
INI--1288 1950-2029
INI0-6.00 1788-0.004
FNB-U0000
Figure 15: Visualization of results for different values on the polyp dataset when aFNR aFPR 0.2.
No Feasible Solution (TEPR > TPxR): No single threshold can satisfy both constraints, necessitating a trade-off. Here, X directly governs this choice: (1) X selects TFN guaranteeing average FNR control. The plot provides empirical estimate of the resulting (violated) FPR; and (2) selects FPPR guaranteeing average FPR control. The plot estimates the resulting FNR. Values of X between O and I represent smooth interpolation
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between these two strategies. However, any choice of X E (0, does not guarantee control over either the FNR or the FPR in this case.
In addition, two key points deserve emphasis. First, the risk control discussed here refers to marginal (average) risk control over the data distribution, conditional guarantee for individual samples. Second, the curves show the expected trend; the actual FNR/FPR achieved by applying a X-chosen threshold may lie exactly on the curve. Consequently, while this method provides a flexible decision-supper tool, the above considerations should be carefully weighed, particularly in the infeasible regime. When simultaneous risk control is possible, however, does enable empirical performance optimization within guaranteed sufe region.
When À equals in Figures 12
13 and 14 the corresponding scenarios are equivalent to the results in Tables 5 and 6 where the ratio of FNR loss FPR loss is O. In the polyp dataset (Figure [4). there exists no value of À that can simultancously ensure the marginal coverage of both FNR and FPR. When employ the weighted loss, significantly enhances the performance of the COAT model. Conversely, in Figures 12 and 13 where FNR and FPR are guaranteed to be less than or equal a, there no substantial improvement in performance with the weighted loss.
Desired ENR=FPR=0.2 0.2 (a=0.2)
Original Imagge
tyM
120,-126,, 19K-0.414
PNB--1156, 1196-1566
INII-6.544 R71:-0.200
IN00-1110 198-0-2
INI--1300, 1156-1785
UNN0-1,000 (PR-0.230
Figure 16: Visualization of results for different À values the polyp dataset when OFNe OPPR 0.2.
we have visualized the results for different values of lambda on the polyp dataset when apna OFPR = 0.2. The 2nd to 4th columns represent the results for == 1, 1,0.5 and 0, respectively. All of them simultaneously control both the FNR and FPR below 0.2. Users select different values of lambda within the range based their preference for other metrics. As shown Figure [6] it presents an opposite scenario on the polyp dataset, that is, for certain images, there is feasible solution, and we have make choice between controlling the FNR or the FPR.
As shown in Figure 15
A.11 COAT CORE CODE IMPLEMENTATION
To enhance the reproducibility our work, this appendix provides the core PyTorch implementation details for our COAT (Conditional Optimization for Adaptive Thresholding) method. We have split the implementation into two parts, mirroring the two main stages of the framework: the end-to-end training of the threshold predictor, and the subsequent post-hoc calibration and inference process.
26

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Published as a conference paper at ICLR 2026
A.11.1 ALGORITHM FOR COAT TRAINING PHASE
This first algorithm details the training procedure for the adaptive threshold predictor, fD. The core components are the network architecture itself and the novel differentiable miscoverage loss func- tion, LCoaT, which enables direct, gradient- pradient-based optimization towards the conditional coverage target.
Training Phase. This code outlines the end-to-end training of the thrrsholdPredicce network. It includes the network architecture, the differ- entiable TPR approximation ccalculate, b computation within single training step function cactiioonncottoo (coat_trraiingg_tee)
Algorithm Python Pseudo-Code for COAT:
and the main training loss
import torch
import torch. nn
models import resmet50, ResNei50_ Weights
class ThresholdPredictor (nn., Modulc): """ adaptive threshold predictor network, I_D. the paper. "M
def __init__ (self, pretrained: bool = True):
super.)_.inntt..... self. resnel
Weights DEFAULT)
Modify input layer for channels (Image original_convi
Probability Map)
11
self.resnett convi
12
self. resnet..eee.....e....e coavl nn. Conv2d(4, 64, kernel_size=7, stride=2. padding=3.bias=Falte bias=False with torch ():
14
self resnet. convl. weight [:. conv....hht...
:3, :. :l:1= original_convi clone
15
res.eet.coovl weight 3, keepdim=True)
original_convl weight. clone (0.ee))meee mean (dmmn,
16.
17
Modify output layer for single threshold self resaet. fe an. Lincar resner. fesnet.fe.lie in_ 1)
18
19
def forward image: torch Tensor, prob_map: torch. Tensor) torch. Tensor:
21
input_tensor torch. cat c prob_map. (l)) dim=l)
return lorch sigmoid (elff resnet resnet
23
24
calculate difflieetddiiffeeetaabb.ee _
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prob_map: Tensor. true_mask: torch. Tensor, pred_tau: torch Tensor.
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T: float. epsilon: ]loat] torch. Tensor:
""Coomputes the differentiable TPR "
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soft_mask = torch. sigmoid (( prob_map pred_tau) T) intersection
soft, mask true stu__mas))
30
true maskmaakaaaa wask_size torch socch.sum((i strue carmtttt_mssss mask)
31
return intersection (truc_mask_st) + opsilon)
32
35 def pout_traainng_
model: ThresholdPredictor optimizer: torch, optim Optimizer.
35
bntch_batch: tarch, Tensor, bat_map_bbeee torch. alpha: fleat, float, epsilan: float):
torch.Tensor. Tensor rrue_mask. batch: batch
turch. Tensor
37
.. Performs single training step for the COAT model. ...
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model. train ()
aptimizer. zero_grad
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Predict image- thresholds
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pred_taus = model
prob_map_batch
46
total_loss 0.0
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targel_coverage aipha Compute loss for each sample in the batch for i in mange(image hatch size
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tpr_prime calculate caleulate_diiferreme differentiablee
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prob_map trob_map_baatee bathhbtch tre sue__aak___ mash_batchhiil...... pred_taus[i], T. epsilon)
50
1_OOAT (TPR (1 a
loss (tpr_prime- target_coverage)==2
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tolal_loss #= loss
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Update parameters
avg_loss tosal_loss image_batch, (0)
56
backward (ag_woss.bbckwwrr
57
optimizer, (tep))
return avg_loss. item
27

---

Published as a conference paper at ICLR 2026
A.11.2 ALGORITHM FOR COAT CALIBRATION AND INFERENCE PHASE
Once the "ThresooldPredictoor trained, this second algorithm details the procedure for calibration and inference. A held-out calibration set, Doat, is used to compute single, global correction term t'. This correction is then applied the predicted thresholds for all test images to generate the final prediction sets, C(X), which are guaranteed to satisfy the marginal coverage property.
Algorithm 3 Python Pseudo-Code for COAT: Calibration and Inference Phase. This code shows the post-training procedure. First, cal ibrate e_cott the trained model and calibration to find the optimal correction term t t_prime. Then, apply_ca wiibarrerroooooo uses this term generate final prediction sets the test data.
import import numpy
def calibrate_coate
model: torch nn Madule, col_er torch. utils data. DataLoader, alpha: float) -> float: *" Performs calihration find the correction term tene model. eval
gase_taus_com
10
Compute base thresholds for the calibration set D_eal with torch no_grad
12
for (images, prob_ maps. _) in cal, loader:
base_taus_call
(images, prob_maps) cpu numpy ())
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15
the empirical coverage function R(t) def srs_empprical__ovrrgge
16
float) float:
17
coverages |1
cal_datasei- cal_loader, dataset
19
for in range range(len (len (aaaeeeaaaasssae
_datasett): [al___aaatttstt]]
=* prob_map, true_ mask_makk cal_datasct[i]]
21
Apply correction: tau_i -
22
adjusted_ = пр.
(basc_taus_eaa
t. 0,
25
pred_set prob_map >= adjusted_tau if soue_mask.sun >
24
coverage (pred_set & j...mmakk)mmmm / true_mask, up.
26
appendd coverages
item
28
29
Find minimal correction t' via search (c.g., binary search)
30
Target R(U) (ID (1__calltt//IDeee len laal__laderrddee cal_loader.datasett) dataset)
(1-alpha)
31
target_eal aal__aa___o_____ coverage
(n
-alpha
33
A placeholder for binary search function to find (argel_cal_.overage)) [prime binary_search_ peg.rrah_.or..ggg..mmmiittt.... eep.eeppcrrcal_ceereeeeeee
35
36
return (_rrime
38
def apply_calibeee calibrated coat
model: torch nn. Module, lest_loader: torch, utils, data. DataLoader, t_prime: float) list: Applics the calibrated model to the test set."" model..evai()
40
42
predictian_sets
43
with torch. mo_grad)
44
for (images. prob_maps, in text_loader:
45
Compute hase thresholds for test images
midages,, prob_maps) base_taus_ted (imagees)) for i in (ange(len Calculate adjusted_tan tau torch torch.clamp( clamp haamppphaaaaas__a___ee [ -_prime, 0, 1)
47
adjusted threshold
49
50
Generate the final prediction set pred_set prob_maps[i] >= adjusted_tn prediction_sete sets append pred_set) set)
52
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return prediction_sett Line 23: Output C(X.))
28


Reviews:
[{"review_id": "tYXO9MhS2B", "reviewer": "Reviewer_2Y8X", "summary": "This paper addresses the challenge of achieving image-conditional coverage in conformal image segmentation, where existing Conformal Risk Control (CRC) methods only provide marginal guarantees. The authors propose two novel methods: Adaptive Thresholding (AT), which formulates threshold prediction as a supervised regression task, and COAT (Conditional Optimization for Adaptive Thresholding), an end-to-end differentiable framework that directly optimizes for image-conditional coverage using a soft approximation of the True Positive Rate (TPR) as its loss. The approach enables image-specific threshold learning, offering more reliable and interpretable uncertainty estimates for segmentation models, particularly in safety-critical domains.", "questions": "N/A", "limitations": "", "rating": 4, "confidence": 3, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "The paper tackles an important and underexplored problem in conformal prediction—extending coverage guarantees from marginal to image-conditional levels. The proposed COAT framework is technically sound and conceptually novel, offering a differentiable formulation for conditional risk optimization. The introduction of a soft TPR-based loss is elegant and allows gradient-based training, which is rarely explored in this context. The work is clearly written, well-motivated, and highly relevant to safety-critical applications such as medical imaging. Experimental results (if included) would likely demonstrate meaningful improvements in both conditional reliability and interpretability over standard CRC baselines.", "weaknesses": "While the proposed idea is novel and well-motivated, several limitations remain.\n\nLack of empirical depth: The paper would benefit from more extensive experiments across diverse segmentation datasets and backbone architectures to demonstrate the generality of AT and COAT beyond the presented setting.\n\nLimited comparison scope: The baseline comparisons appear restricted to CRC-based approaches; including Bayesian or ensemble-based uncertainty methods could strengthen the evaluation.\n\nTheoretical guarantees: Although COAT introduces a differentiable loss to approximate conditional coverage, the paper lacks a clear theoretical justification of when or why this approximation yields valid conditional guarantees.\n\nAblation clarity: It is not entirely clear how much of the observed improvement stems from the adaptive thresholding itself versus the differentiable optimization scheme; more detailed ablation or visualization would help clarify this.\n\nPractical considerations: The additional computational cost and training stability of COAT are not well quantified, which could limit adoption in large-scale or real-time applications.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "PE8oRYmTV9", "reviewer": "Reviewer_hqSc", "summary": "This paper addresses a critical limitation in deep learning-based image segmentation: the lack of reliable image-conditional uncertainty quantification. While methods like Conformal Risk Control (CRC) provide marginal guarantees on performance metrics (e.g., False Negative Rate) across a dataset, they fail to ensure reliability for individual images, a crucial requirement for safety-critical applications. To bridge this gap, the authors propose learning image-adaptive thresholds to achieve image-conditional coverage.\nThe paper introduces two methods: first, a supervised baseline named AT (Adaptive Thresholding), which treats threshold prediction as a regression task on pre-computed optimal thresholds. The primary contribution is COAT (Conditional Optimization for Adaptive Thresholding), a novel end-to-end differentiable framework. COAT introduces a differentiable miscoverage loss, which uses a soft, sigmoid-based approximation of the True Positive Rate (TPR). This allows the model to directly optimize for the desired conditional coverage via gradient-based learning, circumventing the need for pre-computed targets.\nExperimental results across multiple datasets and base models demonstrate that COAT significantly reduces the \"Coverage Gap\"—the average deviation from the target coverage per image—compared to existing methods like CRC, while still upholding the marginal coverage guarantee. The work presents a robust pathway towards more trustworthy and reliable instance-wise uncertainty estimates in image segmentation.", "questions": "1. The differentiable loss is a core contribution of this work. Its soft mask is defined as Msoft(X)=σ( (pb(X) - τb(X)) / T ). Could the authors please provide the rationale behind this formulation?\n2. The paper presents COAT as an advancement over the supervised AT baseline. While the results in Table 1 are strong, AT and COAT are compared within a group of other methods. To better isolate the specific contribution of the end-to-end differentiable framework, could the authors provide a more direct, head-to-head comparison between AT and COAT?\n3. For safety-critical applications, the computational cost of a method is a crucial factor. While the appendix notes the hardware used, a comparative analysis of performance is missing. Could the authors please provide the average inference latency (e.g., in milliseconds per image) for the CRC, AT, and COAT methods? This would be essential for understanding the practical trade-offs involved in adopting the proposed adaptive frameworks.\n4. The qualitative results effectively showcase instances where COAT succeeds. To provide a more complete picture of the method's capabilities, could the authors include or discuss any failure cases or types of images where COAT still struggles to achieve the target coverage? An analysis of such challenging cases would offer deeper insights into the method's limitations and highlight promising directions for future research.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. Originality\nThe paper introduces a novel formulation of conditional coverage for semantic segmentation, addressing a key limitation of traditional conformal prediction methods that only ensure marginal coverage. The proposed COAT framework is conceptually original in two ways:\n(1) It reframes adaptive threshold selection as a differentiable optimization problem using image-specific TPR-based objectives.\n(2) It presents a new end-to-end surrogate loss that approximates conditional miscoverage, enabling training of a threshold predictor that generalizes across diverse image characteristics.\nThis combination of conditional risk control and differentiable threshold learning is innovative and extends conformal prediction into a domain where such guarantees are rare and previously unattainable.\n2. Quality\nThe methodological quality is strong. \nThe algorithmic design is principled, with clear motivation, precise mathematical formulation, and a calibration step that preserves marginal coverage guarantees. Experiments are extensive, evaluating multiple backbone architectures and several conformal baselines. Results are reported across 20 random data splits, with means and standard deviations, showcasing statistical rigor.\nThe paper also includes a theoretical justification proving asymptotic conditional coverage under reasonable assumptions.\n3. Clarity\nThe paper is generally well written and easy to follow. Algorithm 1 is detailed, explicit, and matches the narrative description. Figures provide intuitive interpretations of coverage behavior and prediction stability. The appendix further improves clarity by discussing theoretical assumptions, asymptotic properties, and the role of LLM-assisted content generation.\n4. Significance\nThe paper makes a substantial contribution to reliable machine learning for safety-critical applications. Ensuring per-image conditional coverage is significantly more relevant than marginal guarantees in medical settings, where rare under-covered samples can lead to severe consequences. The proposed COAT method improves reliability not only statistically but also visually, producing prediction sets with fewer false negatives and more consistent behavior across images.", "weaknesses": "While the paper presents a strong and compelling contribution, there are several areas where it could be improved to enhance its clarity, reproducibility, and impact.\n1. The definition of the soft mask Msoft(X), using temperature parameter T to control the sharpness of the approximation. The paper provides no rationale for this specific design choice. To solidify this core component of the methodology, the authors should clarify if this formulation offers specific theoretical or empirical advantages, or provide an ablation study comparing it to the more conventional approach.\n2. The paper positions COAT as a more advanced, end-to-end extension of the supervised AT baseline. However, the experimental results group the comparison of AT and COAT together with external baselines. A more focused, head-to-head comparison is needed to clearly demonstrate the value added by COAT's end-to-end differentiable framework. A direct analysis isolating the performance difference (especially in Coverage Gap) between AT and COAT would be crucial to empirically justify the increased complexity and novelty of the COAT approach.\n3. The paper does not provide a quantitative analysis of the computational overhead. While the appendix mentions the hardware used for training, it omits a comparison of training times and, more critically, inference latency against the other methods. For safety-critical domains like autonomous driving, the practicality of a method depends heavily on its speed. A comparison of the inference time per image for CRC, AT, and COAT would be a vital addition for assessing the method's real-world viability.\n4. The qualitative analysis focuses primarily on successful examples where COAT performs well. While useful, this presents an incomplete picture of the method's capabilities. To provide a more nuanced understanding of its limitations, the authors are encouraged to include and discuss some failure cases or instances where COAT still struggles to close the coverage gap. This would offer valuable insights into the types of images for which adaptive thresholding remains a challenge.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "m0l6OcqjMp", "reviewer": "Reviewer_ch3e", "summary": "This paper addresses the challenge of achieving image-conditional coverage in image segmentation tasks, a crucial problem for safety-critical applications. The authors introduce two novel methods for adaptive thresholding: Adaptive Thresholding and Conditional Optimization for Adaptive Thresholding. These methods aim to improve uncertainty quantification in segmentation tasks by learning image-specific thresholds that control the FNR at an image level, rather than relying on marginal guarantees that apply to the entire dataset. The COAT method, in particular, introduces a differentiable loss function that allows end-to-end optimization for the target conditional coverage, enabling more reliable and interpretable uncertainty estimates. Through extensive experiments, the authors show that COAT consistently outperforms other methods, such as CRC and AA-CRC, in controlling the FNR and achieving more consistent and accurate conditional coverage across different images and datasets.", "questions": "If the goal is to control the FDR (not FPR), the monotonicity between FDR and the threshold can be discussed. Additionally, it would be beneficial to explore how to optimize other performance metrics while controlling the FNR.\n\n\nI will consider adjusting the score based on the response.", "limitations": "", "rating": 6, "confidence": 5, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The paper tackles the significant challenge of achieving image-specific uncertainty quantification in segmentation tasks. I think conditional guarantees are critical.  \n2. The COAT method introduces a novel end-to-end differentiable miscoverage loss, allowing for direct optimization of image-conditional coverage without the need for pre-calculated thresholds, this enhances the flexibility and accuracy of uncertainty estimates", "weaknesses": "In some applications, such as medical diagnostics, controlling both false positives and false negatives is essential, and the method could benefit from a more balanced approach to risk control. For example, is it possible to simultaneously derive threshold intervals that control both the FNR and FPR? An analysis of the balance between the two would be valuable. \n\nIt would be valuable to extend the evaluation to multi-class segmentation problems, where each pixel could belong to more than one class. \n\nthe paper focuses on image level, in some cases, different regions within an image may have varying degrees of importance. For example, in medical imaging, certain organs or lesions may require stricter coverage guarantees than others. Introducing class-specific or region-specific thresholds could enhance the reliability", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "clyoDUFps2", "reviewer": "Reviewer_PHB7", "summary": "The paper targets uncertainty quantification for high-risk image segmentation and argues that standard Conformal Risk Control (CRC) only offers marginal (overall) risk control, leaving large per-image variability. It proposes Conformal Risk Adaptation (CRA): recast threshold selection as a weighted quantile problem (eliminating grid search), and use a two-stage calibration—(i) monotone non-parametric pixel-probability recalibration (e.g., isotonic regression), then (ii) stratified calibration at the image level based on the total predicted probability mass to improve conditional risk behavior.", "questions": "1. Stratification guarantees (Eq. (8)): Must g be fixed before training? If Kand strat thresholds are chosen from calibration data, do you require data splitting or multiple-testing correction to maintain finite-sample guarantees?\n2. Ties & discretization: CRA scores rely on cumulative probability mass. How do you handle ties (e.g., quantized/duplicate probabilities)? Do ties induce threshold jitter or systematic conservativeness in coverage?\n3. Stratifier robustness: Why choose $\\sum_j \\hat{p}_j$ specifically? Have you evaluated alternatives (e.g., entropy, peakiness, predicted foreground fraction), and their effects on conditional risk and normalized size?\n4. Multi-class extension: For multi-class segmentation, do you compute per-class quantiles or a shared joint threshold? How do you preserve both overall and class-conditional risk guarantees?\n5. Fairness & small targets: On small-volume/rare-object subgroups, does CRA reduce Coverage Gap and FN hotspots (false-negative heatmaps)? Please report size-stratified results to substantiate the “per-case stability” claim.\n6. Comparisons & ablations: Could you add results combining CRA with spatial priors/shape regularization (e.g., CRFs/morphology) and an ablation of CRA components (isotonic recalibration / stratification / weighted quantile) to show the accuracy–size–risk trade-offs?\n7. Constants & upper bound B(Eq. (3)): What is the source and magnitude of B? If different loss bounds or surrogate risks (e.g., smoothed FNR) are used, how do the threshold solution and guarantees change?", "limitations": "", "rating": 4, "confidence": 3, "soundness": 2, "presentation": 3, "contribution": 3, "strengths": "1.\tRecasting risk control as weighted quantile estimation yields a closed-form threshold solution, avoiding discretization error and the computational overhead of grid search.\n2.\tRanking by normalized cumulative probability mass makes the method self-adaptive to images with different “brightness/confidence” profiles—directly addressing the source of conditional-risk heterogeneity.\n3.\tThe calibration design is sensible: pixel-level monotone recalibration improves the fidelity of total mass estimates; stratified, image-level calibration provides group-conditional control that improves fairness and stability.\n4.\tCode repository provided; good reproducibility.", "weaknesses": "1. Theory scope/assumptions insufficiently specified: In Theorem 1, the “arbitrary scoring function” premise should clarify what monotonicity/thresholdability conditions are required so that CRC-style isotonicity and risk upper bounds hold. For stratified calibration (Eq. (7)–(8)), assumptions on the stratification function g(fixed a priori vs. data-dependent) are unclear; if Kand stratification thresholds are tuned on the calibration set, the finite-sample guarantees may be compromised without data splitting or multiplicity control. Connections to group-conditional CP and finite-sample smoothing/quotas need tightening.\n\n2. Using $\\sum_j \\hat{p}_j$ as the stratifier could correlate strongly with target area/prevalence, potentially systematically treating small objects or low-contrast images differently. Analyze fairness/bias impacts.\n\n3. Missing comparisons to spatially aware/topology-aware conformal methods or pixel-grouping approaches; Only reporting t-tests is weak—please add effect sizes and confidence intervals; The Coverage Gap (mean absolute deviation) may be high-variance for small ∣Y∣; consider tail-risk metrics (CVaR, quantile gaps).\n\t\n4. Multi-label extension is under-specified (class coupling, probability-sum constraints, shared vs. per-class CRA) and lacks guarantees.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)