Softmax is not Enough (for Adaptive Conformal Classification)
Authors: Navid Akhavan Attar
Hesam Asadollahzadeh
Ling Luo
Uwe Aickelin
Venue: ICLR 2026 Poster (2026)
Keywords: Conformal Prediction
Energy-based Models
Uncertainty Estimation
Primary area: unsupervised, self-supervised, semi-supervised, and supervised representation learning

Abstract: The merit of Conformal Prediction (CP), as a distribution-free framework for uncertainty quantification, depends on generating prediction sets that are efficient, reflected in small average set sizes, while adaptive, meaning they signal uncertainty by varying in size according to input difficulty. A central limitation for deep conformal classifiers is that the nonconformity scores are derived from softmax outputs, which can be unreliable indicators of how certain the model truly is about a given input, sometimes leading to overconfident misclassifications or undue hesitation. In this work, we argue that this unreliability can be inherited by the prediction sets generated by CP, limiting their capacity for adaptiveness. We propose a new approach that leverages information from the pre-softmax logit space, using the Helmholtz Free Energy as a measure of model uncertainty and sample difficulty. By reweighting nonconformity scores with a monotonic transformation of the energy score of each sample, we improve their sensitivity to input difficulty. Our experiments with four state-of-the-art score functions on multiple datasets and deep architectures show that this energy-based enhancement improves the adaptiveness of the prediction sets, leading to a notable increase in both efficiency and adaptiveness compared to baseline nonconformity scores, without introducing any post-hoc complexity.

Published as a conference paper at ICLR 2026
SOFTMAX IS NOT ENOUGH (FOR ADAPTIVE CONFOR- MAL CLASSIFICATION)
Navid Akhavan Attar, Hesam Asadollahzadch, Ling Luo & Uwe Aickelin School Computing and Information Systems, Faculty Engineering and The University of Melboume, Victoria, Australia [n. akhavansttar, asadollahzadde. ling. luo, uwe wwe. www.aaakeeiinnniiiecnc edu.
ABSTRACT
The merit of Conformal Prediction (CP), as a distribution-free framework for un- certainty quantification, depends on generating prediction sets that are efficient, reflected small average set sizes, while adaptive, meaning they signal uncer- tainty by varying in size according to input difficulty. A central limitation for deep conformal classifiers that the nonconformity scores are derived from soft- outputs, which can be unreliable indicators of how certain the model truly is about a given input, sometimes leading to overconfident misclassifications or un- due hesitation. In this work, we argue that this unreliability can be inherited by the prediction sets generated by CP, limiting their capacity for adaptiveness. We pro- pose new approach that leverages information from the pre-softmax logit space, using the Helmholtz Free Energy measure of model uncertainty and sample difficulty. By reweighting nonconformity scores with a monotonic transformation of the energy score of each sample, we improve their sensitivity to input difficulty. Our experiments with four state-of-the-a score functions on multiple datasets and deep architectures show that this energy mergy-based enhancement improves the adap- tiveness the prediction sets, Ieading to a notable increase in both efficiency and adaptiveness compared to baseline nonconformity scores, without introducing any post-hoe complexity.
INTRODUCTION
Deploying machine learning models in critical, real-world applications requires not just high accu- racy, but also trustworthy uncertainty quantification. Conformal Prediction (CP) has emerged as an effective framework for this challenge (Vovk et al., 2005). It provides model-agnostic method to construct prediction sets, C(X), that are guaranteed to contain the true class, Y, with a user-specified probability:
P(Y E C(X)) 1-a
This distribution-free guarantee a significant asset. However, the practical utility of CP depends on the characteristics of these prediction sets. Ideally, they should be adaptive and efficient: small for inputs that the model finds casy, and appropriately larger for inputs that are difficult or ambiguous.
This adaptiveness governed by the nonconformity score. While many nonconformity scores are designed to produce adaptive sets, they are typically derived from a model's final softmax probabili- ties. This choice inherits fundamental weakness, as softmax outputs are often unreliable indicators of a model's true uncertainty. They can exhibit overconfidence even for misclassified or out-of- distribution (OOD) inputs. Post-hoe calibration helps reduce this issue, but only limited extent, as it cannot fully correct the underlying limitations in uncertainty quantification. (Guo et al., 2017; Lee al., 201 2018a; Hein et al., 2019). Consequently, the adaptiveness of these scores by design limited, which can lead to inefficiently large sets for simple inputs, misleadingly small sets for difficult ones.
One approach improve adaptiveness involves adjusting the score based on input-specific mea- sure difficulty, such as the variance of ensemble predictions, the error predicted by an auxil- iary model HHeráández-Hermándee et al., 2022), the variance estimated via Monte-Carke dropout (MCD) with a neural network (Cortés- (Cortés-Ciriano & Bender, 2019). This principle is related to Nor- malized Conformal Prediction, which has been shown produce tighter and more informative sets

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Published as a conference paper at ICLR 2026
Density
Easy semple Hard sarple
0O0 sample
Negative Helmholtz Energy
Easy Sample
APS Set: (macaw. Jorikeet, screen, wtwdmmddddmoooommmee space bar, eater)
n=6
Energy-based APS Set: (macaw, Jorikest, screen)
n
Hard Sample
APS Set: (house robin,
robin, , brambling, Indigo bunting, quail, .junco, , partridge)
n=7
Energy-based APS Set
partridge, water ouzel, ruffed grouse, humminghhird
(house fanch,rabin, robin, brambiing, indigo bunting, quail quailljuncce junco,
n=10
OOD Sample
APS Set: (buckle, thimble, pick, doormat, sambrrro, , nematode, mailbag, shield, Purse)
n=9
Energy-based APS Set
(buckle, thimble, pick, doormat, sombrero, nematode, mallbag, shield, purse. electric ray, caldron, waffle iron, bottlecap, bolo Egyptian cat)
mhilld
n=15
Figure 1: Prediction sets from standard method (APS Romano et al. (2020)) versus our energy- variant, demonstrating improved adaptiveness on ImageNet. (i) For easy input like the image of Macaw, whose clear visual cues (vivid colors, long tail) make it simple to classify, our energy-based method produces a smaller, more efficient set. (ii) For a hard input, a bird image labeled as Hummingbird, its appearance deviates from typical hummingbirds (e.g.. a thicker, less tapered beak) and shares features with other bird classes, making the image difficult for the model. In this case, the energy-bai mhaegy-based method returns larger prediction set, signaling higher uncertainty. (iii) Finally, for an -ut-o--distiobution (OOD) imput like a brain MRI that the model wasn't t trained on, our method generates a much larger set, warning the user that the prediction unreliable. This im- provement adaptive behavior is guided by the Helmholtz free energy, which captures the model's 's uncertainty about an input.
in regression by scaling a base score with an uncertainty estimate (Papadopoulos al., 2002a; Pa- padopoulos & Haralambous, 2011; Boström et al., 2017). Building on this principle, we argue that reliable signal for sample difficulty and model uncertainty exists in the pre-softmax logit space. We propose using the Helmholtz, free energy computed from the logits, as a principled measure of a model's familiarity with input. Inputs aligned with the training data distribution are assigned low energy (high certainty), while atypical or ambiguous inputs receive high energy (low certainty).
The energy signal is then incorporated into the conformal framework by applying per-sample reweighting of the nonconformity scores. For "casy" inputs where the model is certain, the energy- based term magnifies the base score, yielding smaller, more efficient prediction sets. For "hard" or OOD inputs, this term dampens the score, producing larger sets that signal the model's 's uncertainty. This improved adaptiveness is exemplified in Figure 1. Compared to existing adaptive scores, our proposed energy-based variants, by leveraging Helmholtz. free energy derived from the pre-softmax logit space for per-sample reweighting of nonconformity scores, increase both adaptiveness and efficiency. This approach improves prediction sets across tftte-of-the-art score functions, while preserving the theoretical coverage guarantees of Conformal Prediction.
We summarize our contributions follows:
We provide a theoretical and empirical motivation for moving beyond softmax-based scores. We establish the connection between Helmholtz free energy and model uncertainty and demonstrate that this energy signal distinguishes sample difficulty more effectively than standard softmax metrics.
We introduce a general framework of Energy-Based Nonconformity Scores, which mod- ulates base nonconformity score with the free energy of each sample to create more adaptive prediction sets that are smaller for easy inputs and larger for difficult or out-of- distribution inputs.
We provide theoretical and empirical evidence showing that our energy-based enhance- ment improves the efficiency and adaptiveness of prediction sets for multiple deep learning architectures across range of different scenarios.
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2 MOTIVATION AND METHOD
In this section, we (i) explain why the softmax probabilities that underpin conventional noncon- formity scores for deep classifiers are often unreliable for efficiently capturing model uncertainty, (ii) introduce Helmholtz free energy as a more robust measure uncertainty derived directly from model logits, and (iii) use this concept to motivate and define new class of energy- scores that produce more adaptive and efficient prediction sets. All notations are summarized in Appendix A.
2.1
SOFTMAX UNRELIABILITY AND IMPLICATIONS FOR CONFORMAL PREDICTION
Given logits f(x), calibrated softmax with temperature T > is
explfy(x)/T]
A(y x) = softmax
f(x)
(1)
Σ exp(fx<x)/T)
The common nonconformity scores for classification are functions of म (as detailed in Ap- pendix E.2). However, relying on softmax values alone unreliable for uncertainty assessment. First, modern networks produce poorly calibrated and often overconfident posteriors (Guo et al., 2017), including spuriously high confidence on unrecognizable inputs (Nguyen et al., 2015). While temperature scaling improve in-distribution calibration, it does not address epistemic uncer- tainty: OOD, "far" or even "hard" inputs may still map to representation regions that yield confident softmax outputs (Hein et al., 2019; Lee et al., 2018a). This sensitivity representation geometry means that when class manifolds overlap or decision boundaries are poorly separated, softmax con- fidence can be misleading even after calibration (Cohen et al., 2020). Second, softmax posteriors entangle likelihoods with learned class priors, biasing scores under label shift or class imbalance. Margins for minority classes tend be smaller, intensify uncertainty mis-estimation unless logits are explicitly adjusted (Ren et al., 2020). Collectively, these issues undermine CP adaptiveness: probability-base scores can produce (i) unnecessarily large sets for easy samples when tails are in- flated, (ii) deceptively small sets ambiguous/OOD inputs that happen to receive high softmax confidence. For a comprehensive compilation softmax criticism, see Appendix C.
These observations motivate adjusting nonconformity scores with additional signal that reflects the model's holistic signal about Its familiarity with æ. the next subsection, we use the Helmholtz free energy computed from the logits principled, model-aware measure of epistemic uncertainty, that also correlates with sample difficulty, assigning low energy to easy in-distribution inputs and high energy to hard, ambiguous, OOD inputs.
2.2 FREE ENERGY AS MEASURE OF EPISTEMIC UNCERTAINTY
To quantify a model's uncertainty in its predictions, we seek a measure that reflects its familiarity with the input data. We 2006). An EBM defines a sealar energy for every configuration of variables, where lower energy corresponds to higher probability. Any standard discriminative classifier can be interpreted through the lens of an EBM (Grathwohl et al., 2020). We refer to Appendix D for more details on EBMs.
the framework of Energy-Based Models (EBMs) (LeCun et al.,
For classifier with a logit function f(x) RRD inputs æ and labels as:
we can define a joint energy function over R
Elz, y: f) = -z) y E This formulation connects the classifier's outputs directly to an energy landscape. The conditional probability p(y[z) is then given by the
(L.... K}.
distribution:
(op(-E(x,17)
exp(fy(x))
p(ula))
1= (xp(--(xxhh)
L exp(fx(x))
which is identical to the standard softmax function.
By marginalizing over the labels, we can derive an unnormalized density over the input space. This process yields the Helmholtz free energy, F(x), which acts as the energy function for the marginal distribution p
F(x;f) = -T log Σ cxp (
K
kx) - log
K
(f((z)
exp
(4)
k=1
k=1
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1.0
easy sumple
0.8
0.44
0.0
20
30
60
50
90
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(a) Softmax Scores: 1.0 (casy sample) vs. 0.998 (hard sample)
30
casy sumple
logit Vallue
hard xample
30
10
20
30
40
50
60
70
80
90
100
Label
(b) Negative Energy Scores: 32.49 (easy sample) vs. 14.08 (hard sample)
Figure 2: (a) Softmax prohahility distributions and (b) raw logit outputs two CIFAR-100 samples computed by a trained ResNet-56. Both samples receive similarly high softmax confidence scores, despite differing significantly in difficulty vs. 27). In contrast, their negative energy scores more clearly refiect this difference.
where 7 is a temperature parameter. The free energy represents soft minimum of the joint energies for a given input æ. A low free energy indicates that the model assigns high certainty to at least one class, suggesting the input is familiar. Conversely, high free energy indicates that the model is uncertain across all classes.
This relationship allows us to define model-implied marginal density over the input space X::
p(x) exp( exp(-F(x)/7))
where 7 = f (ep(-F(xx)/TTeel
J'EX
the partition function, a constant that ensures the distribution integrates to one. This formulation implies that inputs corresponding to high-density regions of the data distribution (i.e., "typical" examples) are assigned low energy, while those low-density regions have high energy (Liu et al., 2020). We now formalize the connection between free energy and epistemic uncertainty.
Proposition 2.1. The Helmholtz free energy F(x) is valid measure of epistemic uncertainty, linearly proportional to the negative log-likelihood of the model-implied data density p(x).
We refer to Appendix G.2 for proof. This alignment makes the energy score desirable quantity for epistemic uncertainty (Fuchsgruber al., 2024; Zong & Huang, 2025) and thus suitable for OOD detection (Liu et al., 2020; Wang et al., 2021).
2.3 ENERGY-BASED NONCONFORMITY SCORES
To clarify our motivation for energy-basee -base conformal classification, we illustrate with real example how integrating free energy into conformal classification can be beneficial, as it provides additional information not necessarily captured in the softmax space.
Following the definition in Angelopoulos et al. (2021), we quantify the difficulty of sample (æ, Mtme) as
D(x, Youe) oz (yhnue))
(6)
where Oz (Mtrue) denotes the rank of the true label Strue in the model's predicted class-pro brobbbboyyy ordering (from most to least likely). Formally,
oz (v) (A: [K] #(k xx) # |x)]|-

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Published as a conference paper at ICLR 2026
Inspired by the analysis in Liu et al. (2020), Figure 2(a) displays the softmax probability distributions produced by a pretrained ResNet-56 model for two samples from the CIFAR-100 dataset, while Figure 2(b) shows the corresponding raw logit outputs for each sample.
The first sample is considered "easy", with a difficulty of (i.e., the true label has the highest predicted probability), while the second is "hard", with difficulty of 27 (i.e., misclassified by the model). Notably, despite the substantial difference in difficulty, both samples exhibit nearly identical softmax confidence scores, which would make them indistinguishable under standard sofumax-based uncertainty metrics. In contrast, their negative energy scores (=F(x)), computed from the logits, are significantly more separable. This suggests that F(z) captures different, and potentially more nuanced, aspect of uncertainty. Easy or high-density samples yield large - F(r), while hard, am- biguous, low-density or OOD samples yield smaller - F(x).
To further investigate this behaviour, Figure 3 shows the distribution of energy scores across the CIFAR- 100 test set calculated with a trained ResNet-56 model, stratified by sample difficulty. As the figure illustrates, energy distributions shift noticeably across difficulty levels, suggesting that logits (and their derived energy scores) retain richer information about a model's confidence than the softmax outputs alone. This highlights energy an informative signal for uncertainty that can improve the efficiency of nonconformity scores, particularly in cases where softmax probabilities overconfident or poorly aligned with true sample difficulty.
Dieficulty: (n-7261)
bifficuley:: 2-5 (a=1930)
Difficulty: 6:50 (n-256)
Difficulty: 51-100 (a-43)
1-13.26
100
2211.23
k=10.49
800
200
80
Count
600
60
40
200
10
20
15
25
3.5
30
30
20
10
25
30
Negative Xnergy 5y
Regative Brergy
Negative Energy Score
Negabive
Xnergy
Figure 3: Distribution negative energy scores (x=)), stratified by sample difficulty. As difficulty increases, the distribution shifts toward lower energy values, indicating reduced model confidence.
Theorem 2.2 (Monotonicity of Expected Confidence with Sample Difficulty), Consider two diffi- culty levels di and d2 such that 1 S di < d2 S K. Let Exxyy=D! :- D(X, Youe) = d] denote the expectation over the data distribution conditional on samples having difficulty d. For a classifier successfully trained to convergence on a representative dataset, the expected negative free energy is strictly monotonically decreasing function of difficulty:
B
F(X)
D(X, Yime)) = =d,]= > El E[-F(X) D(X, Yine) =d2).
(8)
The proof of Theorem 2.2 is provided in Appendix G.3.
Having established that the free energy. derived from the logit space, captures epistemic uncertainty and sample difficulty more effectively than softmax probabilities, we propose to integrate the energy score into the nonconformity scores, an approach that aligns with the principles of normalized con- formal prediction. By using free energy as sample-specific difficulty measure, we aim to scale the base nonconformity scores to produce prediction sets that better adjust to the model's uncertainty regarding each sample.
We define the energy-basee based variant of a base adaptive nonconformity score, S(x. v). as:
S(z,y) 1 log 1 11+e- ++(r))
Sanergy-Based
Here, the scaling factor is softplus function of the negative free energy, - -(x),, which ensures a positive, input-dependent weight. The parameter / > O controls the sharpness of this function. This modulation re-calibrates the nonconformity score on a per-sample basis, leveraging the model's epistemic uncertainty.
The intuition behind this formulation is follows:
For "easy" in-distribution samples, the model is certain. resulting in a large negative free energy (i.e., large and positive - F(x)). This yields large scaling factor, which
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Published as a conference paper at ICLR 2026
magnifies the base score S(z, y). This reweighting causes the scores of incorrect labels to more readily exceed the fixed conformal quantile q, leading to smaller and more efficient prediction sets.
For "hard" or OOD samples, the model is uncertain, and - F(x) is small or negative. The scaling factor becomes small, thereby dampening the base score. This dampening reduces the magnitude of all scores for the given input, causing more plausible labels to fall below the conformal quantile ạ and thus producing adaptively larger sets that reflect the model's uncertainty.
As shown in Proposition 2.3, scaling the score equivalent to adjusting the quantile threshold for each imput, tightening it for confident predictions and relaxing it for uncertain ones. In our experiments, we apply this modulation to several state-of-the-ar scores, including APS, RAPS, and SAPS. An extension of this modulation to the LAC score is provided in Appendix J.
Proposition 2.3 (Equivalence to base nonconforiity score and let (xx) be positive, sample-dependen scaling function (e-g., log(1 + e-8P(=))), assuming this is positive, Let Ca(z) be the
Thresholding). Let S(x. y) be any adaptive
G(z) softplus(-F(z); prediction set constructed using the scaled score Sa(x, y) = (()))((x, v) and its corresponding quantile Ti-a derived from the calibration set 'Deat = {( M: ]=)r
This construction is mathematically equivalent using the original base score S(x, y) with sample-dependent threshold (xx) that varies for each test sample:
Ca(x) ty E {I,...,,K .......,]T... S(x, v) 0(x)),
(10)
where the threshold is defined
(G)
((x) = y1
(11)
G(z)
We refer Appendix G.4 for proof.
3 EXPERIMENTS
We present a comprehensive empirical evaluation of our proposed energy-scaled nonconformity scores, comparing them across various data regimes and distributional challenges. The objective of CP is to produce prediction sets C(X) for a test instance X such that its unknown true label Y is included with user-specified probability a, i.e. P(Y E C(X) Z 1
3.1 BALANCED TRAINING DATA
We first evaluate performance when models trained datasets where the prior distribution class labels is uniform, i.e., Pirum (Y = y) = 1/| for all y E y. This includes standard ImageNet- Val, Places365, and CIFAR-100 training sets. For a (0.01, 0.025, 0.05, 0.1). we report empirical coverage and average prediction set size in Table 1. This establishes whether energy-based methods maintain coverage while potentially improving adaptiveness and efficiency under standard, balanced training conditions. Detailed difficulty-stratifiee results are also reported in Appendix I.
3.2 IMBALANCED TRAINING DATA
We then study performance on data with an imbalanced class prior. For this, we use CIFAR-100-LT training variants, which are designed to simulate long-tailed distributions where class frequencies decay exponentially (Pimin (Y = j) x exp(-> j)). The parameter À controls the severity of this imbalance, with higher values indicating stronger imbalance, illustrated in Figure 7.
Modern deep networks trained on such long-tailed data exhibit a "familiarity bias", where the model shows higher confidence for majority classes and lower confidence for minority classes (Wallace & Dahabreh, 2012; Samuel et al., 2021). This makes conformal prediction with standard softmax scores under-cover minority classes. To address this, our energy-based variants dampen the non- conformity scores of minority classes more than those of majority classes. This helps to expand prediction sets for minority classes, fostering their labels' inclusion.

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Published as a conference paper at ICLR 2026
Table I: Performance comparison of APS, RAPS, and SAP Soonconooooyy score functions and their energy- based variants on CIFAR- 100, ImageNet, and Places365 at miscoverage levels a E {0.01, 0.025, 0.05, 0.1}. Results are averaged over 10 trials. For the Set Size column, lower is better. Bold values indicate the best performance within cach method family (e.g. APS with and without Energy).
0.1
Ot 0.05
a 0.025
a
0.01
Method
Coverage
Set Size
Coverage
Set Size
Coverage
Set Size
Coverage
Set Size
CIFAR-100 (ResNet-56)
APS
w/o Enezgy
0.90 =
3.17 bcc
0.95 0.00
6.91 0:4
0.975 0002
13.29
0.44
0,99
0.00
25.79
1.30
w/ Energy
0.90 on
3.16 9/6
0.95 1 0.00
6.49 10M
0.974 1: 0001
11.48
1.225
0.99
0.00
22.90
0.47
RAPS
w/o Energy
0.90 0.00
3.13 h07
0.95 0.00
8.17 0047
0.974
0000
16.38
0.99
0.00
30,88
=190
w/ Energy
0.90 is ou1
3.13 906
0.95 a 0.00
6.18 0.255
0.974 :: 0002
11.34
0.33
0.99
900
23.63
0.45
SAPS
w/o Energy
0.90 om
2.87 0.09
0.95 0.00
7.47 0.43
0.974
0000
15.08 0.73
0.99
0.00
29.80
80
1.83
Energy
om
h:11
0.95 0.00
5.94 0.16
0.974
0001
2: 0.31
0.99
0.00
22,90 0.81
ImageNet (ResNeet550
APS
w/o ngy
00000
0.00
0.95 a 0.00
3.99 2.15
0.976 =: 0001
11.72 :: 0.30
0.99 0.00
1.19
w/ Energy
0.90 000
1.66 a nos
0.95 1:000
3.84 2.17
0.976
0.001
10.11 0.30
0.99 0.0
32,93
1:35
RAPS
w/o Enorgy
a 0.00
1 nos
i 0.00
4.22 1.00
0.976
0001
0.21
0.99
0.00
37.01
1:31
w/ Energy
0.90 1: 0.00
1.76 0.04
0.95 i 0.00
3.88 goom
0.976
0.01
9.18 0.20
0.99
0.00
31.47
SAPS
w/o Energy
0.90
1.67
0.95
3.67
0.976
0.01
9.75 0:31
0.99
35.97
1.46
w/ Energy
0.90 0.03
1.66 oos
0.95 0.30
3.66 o.s
0.976
0.01
8.50 0.29
0.99 000
30.24 1.32
Places365 (ResNet-50)
APS
w/o Energy
0.90 = 0.00
7,56 0:13
0.95 = 0.00
14.28 0.24
0.975
0.00
24.92 0:799
0.99
0:00
46.81 = 1.93
Energy
0.00
0.11
12.98
0.975
0002
22.32
0.99
0000
1.69
SAPS RAPS
w/o Fnergy
0.90 a 0.00
7.37
a 2.16
0.95 a 0.00
14.37 a 0:77
0.976 :: 0001
26.34
0.53
0.99
0.c0
50.64
Energy
6.85
12.67
0.975
0002
22.35
0.99
0.00
41.59
1.32
w/o Enorgy
0.90 i 0.00
7.20 a 0.14
0.95 i 0.00
14.11
1 1:000
0.976
1: 0001
25.76
0.40
0.99
0.00
49.80
90
1.34
Energy
0.90
6.79
0.95
12.51
0.975
0,00
22.19
0.99
0:00
41.19
Indeed, energy scores capture this training imbalance (Liu et al., 2024a). Figure 4 visually demon- strates how the distributions of negative energy scores shift across different class bins under both balanced and imbalanced training conditions.
Theorem 3.1 (Free Energy as an Indicator of Class Imbalance). Let f be a classifier trained on dataset drawn from a distribution Preii (X, Y) with imbalanced class priors. Consider two classes, a majority class Ymaj and minority class Ymin. such that their training priors satisfy Pirain ymin).
ymaj) > Pinain( Y
Let the model be evaluated on balanced test distribution Pien. Assume the classes are of compara- ble intrinsic complexity. Then, the expected negative free energy for text samples from the majority class will be greater than that for the minority class:
E Ex= [-F(X) > (X (-F(()))
(12)
See Appendix G.5 for proof. This setup allows us to evaluate how energy influences adaptiveness when a model's representations are shaped by imbalanced training. We report marginal coverage and average set size with the standard balanced CIFAR- calibration set and test set, with results presented in Table 2. We refer to Appendix K for additional experiments on imbalanced scenario.
3.3 RELIABILITY UNDER DISTRIBUTIONAL SHIFT
An important test for any uncertainty quantification method is its response to -ustr--distribution (OOD) data. This scenario is particularly challenging for conformal prediction because the assump- tion of exchangeability between the calibration and test data is violated. Consequently, the formal
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aneeaa 50050
20
10
2-45
2-44
22-100
Classe 3im
Class Din
(x) halanced class priors
(b) inhalanced class priors
Figure 4:
Distributions of negative energy scores across various class bins under balanced and imbalanced training. Results are for CIFAR-100. (a) Balanced model: scores are consistent across class bins. (b) Imbal- anced model (A 0.03): minority classes exhibit lower negative energy scores, reflecting reduced confidence.
Table 2: Performance comparison of different nonconformity scores and their energy-based vari- ants imbalanced CIFAR 100 with imbalance factor of (0.01, 0.025, 0.05, 0.1). Results averaged over 10 trials with ResNet-56 model. For the Set Size column, lower is better. Bold values indicate the hest performance within each method family (e.g.. APS with and without Energy). Results for additional values are provided Appendix K.
0.005 and at miscoverage levels
Method
Coverage
Set Size
Coverage
Set Size
Coverage
Set Size
Coverage
Set Size
CIFAR- 100-LT
0.005) (ResNet-56)
APS
w/o Energy
0.90 =
8.44 a 0.30
0.95 0:00
17.22 0.55
0.973 0003
28.32
0,86
0.99 0:0
45. 10 0.75
Energy
0.90
7.41
0.95
13.30
0.973
21.72
0.99
34.78
RAPS
w/o Energy
om
0.30
0.95 0.00
18.54 = 0.73
0.972 = 0008
30.09 0.94
0.99
0:00
51.65 1.75
Energy
0.90 om
7.59
0.95
13,26
0.973
22.27
0.99
35.43
SAPS
w/o Energy
= om
= 0.5
0.95 = 0.00
17.96 = 0,66
0.972
0008
29.25
1.04
0.99
0:00
50.30
1.66
Energy
0.90 on1
7.58
0.95
13.19
0.973
21.99
0.99
35.18
guarantee of marginal coverage longer holds. This challenge is amplified in real-world deploy- ments where a model, calibrated on in-distribution samples, inevitably encounters novel inputs. These inputs can range from simple covariate shifts (e.g., familiar objects in new contexts) more severe semantic shifts, where the inputs belong to classes entirely unseen during training.
In the absence of coverage guarantees, reliable conformal classifier should not provide small, in- correct prediction without some indication of its uncertainty. This motivates the following desiderata for the behavior of a conformal predictor C(-) when presented with OOD input Food drawn from an OOD distribution Pood. compared to an in-distribution input zia drawn from Paa-
Desiderata for a Reliable Conformal Classifier on OOD Data We establish the following desiderata for a conformal predictor's behavior when encountering OOD data, where the standard exchangeabilityy assumption is violated and coverage guarantees no longer hold.
Desideratum 1 (Adaptive Uncertainty Response). When faced with an out-of-distee disttriibnttio input, a reliable conformal predictor must adapt its output to signal increased uncertainty. This signal should manifest as either an expansion of the prediction set size empty set. This response is characterized by one or both of the following outcomes:
principled abstention via an
(i) A significant increase in the probubility of abstention:
PX~Pas (C = > Px-P,(C(X)) = =0) mz
(i) An inflation in the size of non-empty prediction sets, such that the expected size of non- empty OOD sets is greater than the expected size of in-distribution sets:
Ex-P [XX)]] Ex-r [[xx))(((()
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Published as a conference paper at ICLR 2026
Desideratum 2 (Avoidance of False Confidence). The predictor should minimize the probability of producing a small, non-empty set (e.g., size 1 or 2) for an OOD input.
minimize Px~P... I [C(X) - k) for small k
In summary, also discussed in Appendix P, larger or empty set is informative and appropriate outcome in this scenario, whereas a small, incorrect set problematic. To assess how our method aligns with the OOD desiderata, model was calibrated on in-distribution CIFAR- 100 data and evaluated on the Places365 as the OOD dataset. As coverage not meaningful metric in this context, our analysis focuses on prediction set size.
designed an experiment under a semantic shift. ResNet-56
XAPEI
3:13
XAP6:
3.69
13300
1555
Couat
Predicticn Set Gize
Prediction Set size
in-distribation evaluation
(b) out-of-disrrinmn evaluation
Figure 5: Prediction size distributions for the SAPS score and its energy-based variant with 0.05, on (a) in-distribution CIFAR- 100 and (b) distriiiiiti Places365 data. The energy-based variant produces larger prediction sets on OOD data. Here, represents the overall set size.
The results demonstrate an alignment with our desiderata. As shown Table 3, energy-based scores produce larger average sets compared to their base counterparts. Figure 5 provides a visual illustration of this adaptive behavior, comparing the RAPS score with its energy-based variant. The Ennrgy-bbsed RAPS produces smaller prediction sets on ID data and larger prediction sets for OOD samples. This response, shows improvement towards Desideratum 1, compared to baseline RAPS.
Table 3: Comparison of average prediction set sizes for a ResNet-56 model trained on CIFAR-100. The model is evaluated on both in-distribution (CIFAR-100) 100) and out-of-distribution (Places365) data. Energy-based vari- ants demonstrate adaptiveness the distributional shift by maintaining small sets ID data while producing significantly larger sets for OOD inputs. Bold values indicate the preferred result: smallest average set size for ID (efficiency) and the largest for OOD (uncertainty awareness).
0.1
Set
Size OOD
Size ID
Set Size OOD
Method
distribution)
(out distribution)
(in distribution)
distribution)
APS
w/o Energy
3.17 +
6.18 0.25
6.91
14.91
w/ Rne rgy
3.16
86.76
6.49 0.34
93.44
SAPS RAPS
w/o Energy
3.13 +
3.70 0.04
8.17 + 0.47
8.95 +
w/ Energy
3.13
5.53
6.18
9.05
Energy
4: 0.8
3.78 à 0.06
7.47 0.83
8.82 à 0.40
w/ Energy
2.87
5,55
5.94
9.53
CONCLUSION
This paper demonstrates that the reliability of conformal classifiers is enhanced by moving beyond softmax probabilities to leverage information about model uncertainty from the logit space. Our proposed energy-based framework adjusts standard nonconformity scores leveraging this principled measure of model certainty to make each score sensitive to the model's confidence in that specific input. Our evaluations common nonconformity scores, across multiple datasets and architectures, confirm that our approach yields prediction sets with improved efficiency and adaptiveness, all while preserving the theoretical coverage guarantee.
per-sample basis,

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Published as a conference paper at ICLR 2026
5 REPRODUCIBILITY STATEMENT
The empirical results presented in this paper are fully reproducible. Our implementation, based on PyTorch and using the TorchCP library, are available at https://gitti:: com/ / Energy-Ba Satesyy--Bae--oooooo son-Conoorraaa ssaassi fication. Detailed descriptions of hyperparameters, and environment specification for running experiments, are provided Appendix F.
ACKNOWLEDGMENTS
This research was supported under the Australian Research Council's Industrial Transformation Research Program (ITRP) funding scheme (project number IH210100051), The ARC Digital Bio- process Development Hub is collaboration between The University of Melbourne, University of Technology Sydney, RMIT University. CSL Innovation Pty Ltd, Cytiva (Global Life Science Solu- tions Australia Pty Ltd) and Patheon Biologics Australia Pty Ltd.
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APPENDIX
A NOTATION
Table 4: Notation used in this work.
Symbol
Meaning
WERD
Input (feature) vector
K
Total number of classes
yE [I.....K]
Class label
f(x) (fi...... .f)
Pre-softmax logit vector produced by the classifier
AV x))
Model's 's softmax probability for class y, Equation
Smax (x)
max; (xx
(1x, 4)
General nonconformity score
T
Temperature used in the calibrated softmax
Temperature used in the energy calculation Desired miscoverage level target error rate Quantile threshold for prediction set construction Joint energy, E 2) (x)
E(a,y)
F(x)
Helmholtz free energy score, Equation 4
B
Softplus sharpness parameter
Dxx, Yiue)
Sample difficulty measure
(v)
Rank of label y in the model's predicted class-probability ordering
H(x)
Shannon entropy of (ty [r)
= (0,, 3)
Dataset with samples, inputs x; and labels y
8(x)
Geometric distance of x from the decision boundary
B RELATED WORKS
is a statistical framework that provides distribution-free, finite-sample coverage guarantees for predictions (Vovk et al., 2005). This robust approach to uncertainty quantification has widespread adoption across numerous real-world applications. These include regression (Lei & Wasserman, 2014; Romano
al., 2019), classification (Sadinle et al., 2019), structured predic- tion (Bates et al., 2021), large language models (LLMs) (Su et al., 2024; Cherian et al., 2024; Ku- mar al., 2023; Ren al., 2023; Quach al., 2024), and diffusion models (Horwitz Hoshen, 2022; Teneggi et al., 2023). graph neural networks (GNNs) (Zargarbashi et al., 2023; Huang et al., 2023; Wijgunawardana al., 2020; Clarkson, 2023; Song et al., 2024), and image generative mod- els (Horwitz & Hoshen, 2022). Further applications are found in robotic control (Kang et al., 2024; Luo & Zhou, 2024), hyperspectral imaging (Liu al., 2024b), healthcare (Lindemann al., 2024), finance (Bellotti, 2021), autonomous systems and automated vehicles (Lindemann et al., 2024;
Zecchin et al., 2024; Bang et al., 2024), human-in-the-loog decision making (Straitouri al., 2023;
Cresswell et al., 2024), bioprocessing (Pham et al., 2025), and scientific machine learning (Moya et al., 2024; Podina et al., 2024).
The foundational inductive conformal prediction framework (Vovk et al., 2005) often employs split conformal (or inductive conformal) approach, where the dataset is divided into a training set for model fitting and disjoint calibration set for uncertainty quantification (Papadopoulos et al., 2002b; Vovk et al., 2005; Shafer & Vovk, 2008; Angelopoulos & Bates, 2021; Lei et al., 2015). This done manage the computational aspects of CP. Beyond this common split, other CP variants
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include methods based cross- validatoi (Vovk, 2015) the jackknife (i.e., leave-one-out) tech- nique (Barber et al., 2021). The primary goals within CP research are to enhance the efficiency of prediction sets (i.c., reduce their size) and to ensure and improve the validity of coverage rates.
Improving Prediction Set Efficiency. Strategies to improve the efficiency of prediction sets pre- dominantly fall into two categories: training-time modifications and post-hoc adjustments.
One line of research focuses on developing new training algorithms or regularizations to learn models that inherently produce smaller prediction sets while maintaining coverage (Bellotti, 2021; Colombo & Vovk, 2020; Chen et al., 2021; Stutz al., 2022; Einbinder al., 2022; Bai al., 2022; Fisch et al., 2021; Yang & Kuchibhotla, 2021; Correia et al., 2024). For example, the uncertainty- aware conformal loss aims to optimize APS (Romano et al., 2020) by encouraging non-conformity scores towards a uniform distribution (Einbinder et al., 2022), while ConfTr introduces a regular- ization term to minimize average set size (Stutz al., 2022). However, such training methods can be computationally intensive due to model retraining and optimization complexity. Early stopping has also been explored as technique to select models leading to more compact prediction sets under guaranteed coverage (Liang et al., 2023).
The other avenue involves post-hoe techniques applied to pre-trained models. This includes the de- sign of novel non-conformity score functions. Notable examples are LAC (Sadinle et al., 2019), APS (Romano et al., 2020), RAPS (Angelopoulos et al., 2021). SAPS (Huang et al., 2024b), Top-K (Angelopoulos et al., 2021; Luo & Zhou, 2024), and others (Ghosh et al., 2023). Post- hoc learning methods have also been proposed (Xi et al., 2024). Research also addresses unique settings such as federated learning (Lu et al., 2023; Plassier et al., 2023), multi-label classifica- tion (Cauchois et al., 2021; Fisch et al., 2022; Papadopoulos, 2022), outlier detection (Bates et al., 2023; Guan & Tibshirani, 2022), and out-of-distribution (OOD) detection (Chen et al., 2023; Nov- ello et al., 2024). A common challenge for some post-hoc methods is their reliance on potentially unreliable probability outputs from models.
Recent efforts have sought to make conformal prediction sets more adaptive by explicitly incorporat- ing epistemic uncertainty. One line of work proposes methods that operate on richer, second-order predictions. For instance, Javanmardi et al. (2025) introduce Bernoulli Prediction Sets (BPS), which construct provably optimal (i.e., smallest) prediction sets under the assumption that the true data distribution is contained within a given "credal set" derived from models like deep ensembles or Bayesian neural networks. complementary, model-agnostic approach is taken by Cabezas et al. (2025) with EPICSCORE, which enhances any standard nonconformity score by training separate Bayesian model to learn its conditional distribution. This transforms the score to reflect epistemic uncertainty, achieving asymptotic conditional coverage. However, our energy- framework im- proves the adaptiveness of Conformal Classifiers by leveraging uncertainty information from pre- softmax logits via the Helmholtz free energy, thus avoiding the need for second-order predictors or additional post-hoc computational costs.
Ensuring Validity and Enhancing Coverage Rates, A significant body of work is dedicated to ensuring the validity of the marginal coverage rate and improving it, particularly under challeng- ing conditions, as well as striving for stronger conditional coverage guarantees (Shi et al., 2013; Löfström et al., 2015). Efforts have been made to maintain marginal coverage by adapting CP to scenarios involving adversarial examples (Gendler et al., 2022; Kang et al., 2024), covariate shift (Tibshirani al., 2019; Deng et al., 2023), label shift (Podkopaev & Ramdas, 2021; Plassier et al., 2023), and noisy labels (Feldman et al., 2023; Sesia et al., 2023).
Beyond marginal coverage, many CP algorithms pursue forms of conditional coverage (Vovk, 2012). This includes training-comee conditinon validity. which aims to ensure that most training dataset realiza- tions result in valid marginal coverage on future test data (Bian & Barber, 2023; Pournaderi & Xiang, 2024). Group-conditional CP methods seek to guarantee coverage across predefined groups within the population (Javanmard et al., 2022; Gibbs et al., 2025; Melki et al., 2023). While achieving exact pointwise conditional coverage is known to be impossible in general (Foygel Barber et al., 2021), practical approaches for approximate or cass--oondiiioal coverage exist. For instance, LAC demon- strates the possibility of efficient class-conditional coverage (Sadinle et al., 2019). Clustered CP conditional coverage by leveraging the label space taxonomy, particularly when the number of classes is large (Ding al., 2023a). Other methods, like k-Class-conditionns CP, cali-
improves class class-condin class-conditional
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brate class-specifie score thresholds based on top-k emror.. The goal remains enhance conditional coverage properties while still producing efficient and informative prediction sets.
C ON THE LIMITATIONS OF SOFTMAX OUTPUTS FOR MEASURING MODEL UNCERTAINTY
Our motivation for developing energy-based nonconformity scores, as detailed throughout Section 2, the inadequacy softmax outputs for reliably quantifying model uncertainty. To provide a broader context, reproduce selection of established criticisms originally compiled in the ap- pendix of Pearce et al. (2021), along with a few more recent perspectives published thereafter.
Section
"[The softmax output] is often erroneously interpreted as model confidence." (Gal & Ghahramani, 2016)
"Deterministic models can capture aleatoric uncertainty but cannot capture epistemic un- certainty"" (Gal et al., 2017)
"NNs until recently have been unable provide measures of uncertainty in their pre- dictions." (Malinin & Gales, 2018)
"When asked to predict on a data point unlike the training data, the NN should increase its uncertainty. There is mechanism built into standard NNs to do this standard NNs cannot estimate epistemic uncertainty." (Pearce, 2020)
"NNs are poor at quantifying predictive uncertainty." (Lakshminarayanan et al., 2017)
"Deep neural networks with the softmax classifier are known to produce highly overconfi- dent posterior distributions even for such abnormal samples." (Lee et al., 2018b)
"The output of the [softmax] classifier cannot identify these Ifar from the training data) inputs as out-of-distrrbuttion (Hein al., 2019)
"The only uncertainty that can reliably be captured by looking at the softmax distribution is aleatoric uncertainty." (van Amersfoort et al., 2020)
"Softmax entropy is inherently inappropriate to capture epistemic uncertainty." (Mukhoti et al., 2021)
"For [softmax] classifiers ... misclassificatiinn will occur with high confidence if the un- known is far from any known data." " (Boult et al., 2019)
"...softmax oulput only reflects the total predictive uncertainty instead of the model uncer- tainty, leading to false confidence under distribution shift." (Wang et al., 2024)
"...the raw softmax output is neither very reliable no. can it represent all sources of uncertainty" (Gawlikowski et al., 2023)
"Furthermore, the softmax output cannot be associated with model uncertainty." (Gaw- likowski et al., 2023)
"...the softmux outpul is often erroneously interpreted as model confilence. In reality, model can be uncertain in its predictions even with a high softmax outpus." " (Mobiny al., 2021)
Additional commentary and remarks also reinforce this view:
"Softmax is not telling you anything about bhoute..model uncertainty." - Elise Jennings, Train- ing Program Extreme-Scale Computing (2019)*
"The [softmax] network has way of telling you 'I'm completely uncertain about the outcome and don't vely on my prediction". " - Florian Wilhelm, PyData Berlin (2019}²
"Just adding softmax activation does not magically tur outputs into probabilities." Tucker Kirven, Neural Network Prediction Scores are not Probabilities (2020)²
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D ENERGY-BASED MODELS
Energy- Model (EBM) defines probability distribution over an input space TE through an energy function Eg _2_ I. which is typically parameterized by neural network with parameters 0. For any input vector æ E RD, the probability density is given by the Boltzmann distribution:
(2 exp( -
(13)
Zo
where Ze exp(-Eo(1t) )xr' is the partition function. This normalization constant is a key challenge in EBMs, as its computation involves integrating over the entire high-dimensional input space, which is generally intractable.
This framework can be extended to model a joint distribution over inputs and class labels, p(z, y). A standard discriminative classifier, which produces logit vector f(z), be re-interpreted as an EBM by defining joint energy function E(x, y). Following common practice LeCun et al. (2006); Grathwohl et al. (2020), we define the joint energy the negative logit corresponding class y:
E y) = (x)).
(14)
From this joint model, the conditional probability p(u I)) can be derived as:
ply x) = p(z, y)
p(z. v)
exp(-E(xx,)) v))
exp(fu(z))
E (x)(fx(z)) (15)
p(z) Et,1((2
Σ exp( (xp(-E(z,k))
which is precisely the standard softmax probability #(y .:). The marginal probability p(x) can
then be associated with "free energy" function F(xx)) - log Ek= exp(fx(x)//T), where is
a temperature parameter.
Training EBMs often proceeds via Maximum Likelihood Estimation (MLE), which aims to shape the energy function Ex) such that it assigns low energy to data points from the true distribution and high energy elsewhere. The objective is to maximize the log-likelihood of the observed data D:
arg max Ex-pun [log po (x)]
(16)
The gradient of the likelikcl with respect the parameters 0 is given by:
Va log (o)) = Vo Es(x) - log Zo)
(17)
1
/ exp(-Eo(x')) ))do)
7 V
(18)
Fo) [ exp( Ee(x')
(19)
Zu
= =VVoEo(x) + Ex'rpol (e)))
(20)
Remarkably, this gradient can be computed without explicitly evaluating the intractable partition function Zg. Updating the parameters via stochastic gradient ascent on the log-likelihood is equiva- lent to descending on the following loss function:
MMLE = Ex-pon [Eo - Exr<po (Eo('')
(21)
This objective can be intuitively understood force that "pulls down" the energy of "positive" samples drawn from the data distribution (pdata) while "pushing up" the energy of "negative" samples synthesized from the model's current distribution (po).
To operationalize this training procedure, we must be able to draw samples x' from the model dis- tribution pa (z). Since direct sampling is infeasible, this is typically approximated using Markov Chain Monte Carlo (MCMC) methods. A prevalent choice is Stochastic Gradient Langevin Dynam- ies (SGLD) Welling & Teh (201 1), which iteratively refines initial sample zo (e.g., drawn from simple noise distribution or a buffer of previous samples) according to the rule:
It+1 It a, Ex + vTic, where c c0,
(22)
where at is the step size and 7): controls the scale of the injected Gaussian noise. After a sufficient number of steps, the resulting sample IT is treated as approximate sample from po(x)). Different strategies for initializing and running the MCMC chain lead to various training algorithms, such as Contrastive Divergence (CD) (Hinton, 2002), which re- re-initializes the chain from data points each step, and Persistent Contrastive Divergence (PCD) (Tieleman, 2008), which maintains a persistent chain across training iterations to obtain higher-quality samples.
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E
EXPERIMENTAL PRELIMINARIES
This section details the experimental design, including the conformal prediction framework, the noneonformity scores used, and evaluation metrics used to validate our proposed Energy- based non- conformity scores.
E. CONFORMAL PREDICTION FRAMEWORK
We employ the standard Split Conformal Prediction (CP) framework in experiments. A base model is trained on a proper training set, and its outputs on a held-out calibration set are used to compute non-conformity scores and determine the quantile threshold required to form prediction sets. The procedure is formally outlined in Algorithm 1.
Algorithm Split Conformal Prediction
1: Input: Dataset DD desired emoor rate E (0, 1), non-conformity score function S(ar, y).
2: Partition D into a training set Duain and a calibration set Deal such that D Purain Deal = Let
= -Dail
3: Train the classifier on Prain to learn the mapping + (z).
4: For each example (I,, y) E Deat, compute the non-conformity score s; S14, ye).
5: Calculate the quantile threshold à from the set of calibration scores (81,...,8.) Specifically,
@ is the [(=+1X1-a]]..m -th empirical quantile of these scores.
6: Output: For new input Tnew, the prediction set is constructed
C'Inew) {y E {I....,KK KJ S(Inew v) S 5}
E.2 NONCONFORMITY SCORES FOR DEEP CLASSIFIERS
The non-conformity score function S(r, y) measures how poorly the label fits the input æ accord- ing to the trained model. The prediction set C(Inew) is then formed by including all labels whose non-conformity scores do not exceed the calibrated threshold q. key property of this procedure is that the resulting set is guaranteed contain the true label with a probability of at least 1 aa, assum- ing the test and calibration data points are exchangeable. Exchangeability is a statistical assumption that the joint distribution the data invariant to permutation, making a suitable assumption for scenarios like simple random sampling.
We evaluate a range of established non-conformity scores, each based on different principle for measuring how much a model's prediction disagrees with given label, as summarized in Table 5. We compare these established baselines against their Energy- based counterparts, as defined in Equa- tion 9.
Table Nonconfornity scores considered this work. scores are functions f xx) where u~ U[O 1]: The function oz. (v) returns the rank position of label among all possible labels, ordered by the model's s predicted probabilities (with rank I being the most likely); #max( = maxx #(k | r); (-) denotes the positive part; À and kiop are hyperparameters.
Method
Nonconformity Score
LAC THR (Sadinle 2019)
Stac(2,9) #(y x) -
Adaptive Scores
APS (Romano et al., 2020)
Saes(z, u)
>K *(k [( ax) [x)) + Ao
RAPS (Angelopoulos et al.. 2021)
Snave Sapa(a, A (0)(2)) kar))
fmmxx(s))
oz(y)= =1,
SAPS (Huang al., 2024b)
Ssare(n,
(maa(x)) (ox(v) 2+ otherwise.
Below, we briefly describe how each method works:
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Least Ambiguous Class (LAC/THR) (Sadinle 2019) is one of the simplest and earliest scores. Its non-conformity is defined as Suxc y) = 1 fy z). The score is thus inversely proportional to the model's confidence; high probability for the true class yields low non-conformity score. It is worth noting that using the negative probability, S(x, y) = #t x), is mathematically equivalent for constructing the prediction set, the "+1" in the original formula merely shifts the range from [] to [0. 1] for non-negative interpretability without altering the relative ordering, where higher scores indicate greater nonconformity (less conformity between label and sample). This is because the conformal procedure relies on the rank-ordering of scores to determine the quantile ÿ. and the transformation from 1 म to is monotonic, preserving the rank order. For the Energy-based counterpart of this score, we use -TT (without the bias term). This method provably yields the smallest expected prediction sets compared to other methods proposed after this model, while preserving the marginal coverage, assuming the predicted probabilities are correct. However, this score is non-adaptive, meaning it tends to produce prediction sets of similar size regardless of the sample's ss intrinsic difficulty, which opened the room for adaptive nonconformity scores and their variants to be proposed later.
Adaptive Prediction Sets (APS) (Romano et al., 2020) introduced the concept of adaptiveness to conformal prediction. The score Saes (z. y) is the cumulative probability mass of all classes deemed more likely than class y. Mathematically, this is the sum of softmax probabilities for all labels k whose probability A(k | x) is greater than 7(y | x), plus a randomized term to handle ties. This design has crucial effect: for "easy" examples where the model is confident (i.e., maax is high and Entropy is low), the scores for incorrect labels grow rapidly, leading to small prediction sets. Conversely, for "hard" examples where the model is uncertain flatter softmax distribution), the scores grow slowly, resulting in larger, more inclusive sets that reflect this uncertainty.
al., 2021) builds directly upon APS by adding a regularization term. Its score is SR. v) = SAPS (z, 3) + X() krg)*, where Og (v) is the rank of label y's probability. This term penalizes the inclusion of labels with a low rank (i.e., large oz (y)), effectively preventing the prediction sets from becoming excessively large, especially for uncertain inputs. The hyperparameters kiogg and À control the onset and strength of this regulaagua penalty.
Regularized Adaptive Prediction Sets (RAPS) (Angelopoulos
Sets (SAPS) (Huang et al., 2024b) is a more recent refinement that aims to mitigate the effects of probability miscalibration in the softmax tail. It treats the top-ranked class differently from all others. For labels not ranked first, the score is based on the maximum probability #max (1)) plus penalty that increases linearly with the label's rank, weighted by hy- perparameter X. This approach avoids summing many small, potentially noisy tail probabilities (as APS does) and instead relies the more stable top probability and the rank ordering.
Sorted Adaptive Prediction
E.3 EVALUATION METRICS
We assess the performance of all methods using a target miscoverage level a e (0.01, 0.025, 0.05, 0.1). Let ((1, Mee be the test set. The primary metrics are:
Empirical Coverage: The fraction of test samples where the true label is included in the prediction set.
Coverage
Tlu E C(x:))
(23)
Thest
Macro-Coverage (MacroCov): While empirical coverage reflects marginal reliability over the entire test distribution, MacroCov measures the average per-class coverage, giving each class equal weight regardless of its frequency. Let ê = [lv Eier, Ty E C(xx))
per-
denote the empirical coverage for class y, where Iy (i: yt y). Then,
MacroCov 1
Σ> dy.
(24)
y=1
This metric is particularly informative in imbalanced or long-tailed settings, since it pre- vents head classes from dominating the overall coverage and highlights systematic under- coverage of rare classes.
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Average Prediction Set Size: The mean size of the prediction sets the test data.
Size
[((x))]
(25)
Thpstt
To assess class-conditional reliability, we report the following metrics to verify that they are main- tained while reducing the average prediction set size:
Average Class Coverage Gap (CovGap): This metric measures the average absolute de- viation of per-class coverage from the target coverage level a (Ding et al., 2023b). Let {i yi y) be the indices of test samples for class y. The empirical coverage for class y is Cy [, Eier, Ilys E C(x:)). The gap is then:
CovGap
12 (1 aa]
(26)
K
y=1
We report this value as percentage.
To measure the adaptiveness of the prediction sets,
Size-Stratified Coverage Violation (SSCV): As introduced by Angelopoulos et al. (2021), SSCV quantifies whether coverage is maintained across different prediction set sizes. We define disjoint set-size strata {S,
j {
and group test indices into bins J, {i [C(x;)] e
Sy}. The SSCV is the maximum deviation from the target coverage across all bins:
Hi Jj : ys C(x)))) -(1 -1-a) aa)
SSCV = sup
(27)
To probe feature-conee conture-condiiional reliability beyond class-conditional metrics, we report the worst-slab coverage (Cauchois al., 2021):
Worst-Slab Coverage (WSC): Let E Rd be unit direction and define slab along as S(v, a, b) for E Id
b),
(28)
for thresholds a < b. For mass parameter ò E (0, 1]. the empirical WSC along direction is
WSC =
inf a<h: (S
Pa(y E C(x) x E S(u,a,b)),
(29)
where Pr. denotes the empirical distribution over the test set. We estimate overall WSC by scanning a of directions V (e.g., random unit vectors) and taking the worst case:
(SC,, inf (e; "ev
(30)
Larger values indicate better approximate conditional coverage, while low WSC reveals feature-space region of mass at least where the prediction sets under-cover.
F REPRODUCIBILITY DETAILS
To ensure reprodueibility, we detail our experimental setup, key hyperparameters, and implemen- tation. All source code will be made publicly available. All experiments are implemented based on the TorchCP library (Huang et al., 2024a), which provides robust framework for conformal prediction on deep learning models. The pre-trained backbone models are soureed from Virch sion (maintainers & contributors, 2016).
F.1 COMPUTATIONAL ENVIRONMENT
Operating System: Linux kernel 5. 3.14.0-427.42.1.eet 4.x86 64.
GPU Hardware: NVIDIA H100 80GB HBM3.
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CPU Hardware: 8 cores.
System Memory: 32 GB RAM.
NVIDIA Driver Version: 144.03.
CUDA Version: 12.2.
Python Version: 3.9.21.
PyTorchh Version: 2.0.0+ (with CUDA support).
F.2 DATASETS AND MODELS
Our experiments are conducted on several standard image classification benchmarks: CIFAR- 100 (Krizhevsky et al., 2009), ImageNet-Val (Deng et al., 2009), and Places365 (Zhou et al., 2018). These benchmarks are chosen because they contain a large number of classes, which makes performance differences between methods more evident. We use pre-trained ResNet (He et al., 2016). VGG (Simonyan & Zisserman, 2015), ViT (Dosovitskiy et al., 2021), Swin Transformer (Liu et al., 2021), 1), EfficientNet (Tan & Le, 2019), and ShuffleNet (Zhang et al., 2018) architectures from base classifiers. To evaluate performance under distributional shift,
Torch Vision model trained on CIFAR-100 and test its out-of-distribution (OOD) performance on the Places365 dataset.
To investigate the methods' robustness to class imbalance, we create four long-tailed variants of CIFAR-I00, denoted as CIFAR-100-LT. The number of training samples for class j {1..... 100}, denoted n2,, is set to be proportional to exp(-> : j). The imbalance factor À E {0.005, 0.01, 0.02, 0.03) controls the severity of the class imbalance, with larger values of cre- ating more pronounced long-tail distribution. For evaluation, we have considered two scenarios: (i) the calibration and test sets are balanced while the training data remain imbalanced, and (ii) the calibration and test sets follow the same imbalance ratios as the training data.
F.3 HYPERPARAMETER SETTINGS
In our experiments, set kreg 2 and À 0.2 for RAPS and we use À 0.2 for SAPS. The soft- max probabilities # (y(ar) used by all scores are computed with a temperature parameter T, while the free energy (x) is calculated with its own temperature т. Crucially, ensure fair compar- ison, the softmax temperature 7 was tuned for all baseline and proposed methods to optimize their performance. We tune the temperature energy parameter with in(r) [-9, 9] and the calibration temperature T E {0.01..... 25).
Remark F.I. A critical consideration in our proposed modulation is the positivity of the reweight- ing factor. When reweighting a base nonconformity score, critical the scaling factor must be strictly positive. A negative factor would reverse the score's ordering, invalidating the fundamental assumption of conformal prediction that lower scores indicate higher conformity. While our uncer- tainty signal is the negative free energy, -F(x), it is not guaranteed to be positive. Mathematically. -F(r) becomes negative if X exp(fx(x)/7) < 1, a condition which implies that the maximum logit fmax(() is negative (a necessary, though not always sufficient, condition). This scenario sig- nifies extreme model uncertainty, where the model lacks evidence for any class and typically occurs only for far out-of-distributie inputs.
Although we empirically observe that Fa() positive for nearly all in-distribution and OOD samples in experiments, to ensure the theoretical robustness of our method, scale the base scores by the softplus of the negative free energy. The hyperparameter 3 the softplus function, log(1 + @d*),, controls its approximation to the Rectified Linear Unit (ReLU) function. By choos- ing large value for 3, the scaling factor softplus-F((r)) when it is positive, but smoothly transitions to a value near zero in the rare cases where FF < 0. This behavior is highly beneficial for conformal prediction. For such uncertain inputs, the near-zero scaling factor drives the modulated scores for all labels toward zero, causing most all of them to fall below the conformal quantile q. This correctly produces an extensively large prediction set, signaling the model's high epistemic uncertainty. Throughout our experiments, we set B 1. See Appendix O for a detailed ablation study on B.
behaves almost identically to -F(x)
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G PROOFS
G.1 THEORETICAL VALIDITY OF ENERGY-BASED SCORES
In conformal prediction, the validity of the coverage guarantee relies on the exchangeability of the nonconformity scores. Specifically, for set of exchangeable data points, their corresponding non- conformity scores must also be exchangeable, This property ensures that the prediction sets contain the true label with the desired probability. We now establish that our proposed Energy-modulate scores, defined in Equation 9, satisfy this critical property under standard assumptions. This en- sures that using these modulated scores yields valid prediction sets with the guaranteed marginal coverage central to conformal prediction theory (Vovk et al., 2005).
Theorem G.1 (Exchangeability of ehergy-Bassed Nonconformity Scores). Let (X,, Yi) be exchangeable sequence of random variables drawn from a distribution Pxy. Assume that:
(i) The base nonconformity score S(x, u) is a deterministic duterminiiticcuuncion of its arguments. (ii) The free energy F(x) is a deterministic function of x as defined in Equation 4. Define the modulated score for I,..., as:
S = SEneryy-based (X,, Yi) == S(X,, Y.) log (1+e-0F(X.))
(31)
Then the sequence of modulated scores 1 ) is exchangeable.
Proof. A sequence of random variables is exchangeable its joint distribution invariant under any finite permutation of its indices.
Let the transformation be defined as h(2, S(x, v) log (1 + e (+(()). By assumptions (i) and (ii), the base score S(x, y) and the energy function F(x) are deterministic. Since the softplus function and multiplication are also deterministic operations, the entire mapping h(z, y) is deter- ministic and measurable.
A fundamental property of exchangeable sequences is that they remain exchangeable after applying a measurable transformation. That is, if (Z,) is an exchangeable sequence and g is a measurable function, then the sequence (g(Z:) is also exchangeable.
Applying this principle with Z; = (X,, Y2) and the transformation g h, we find that the sequence of scores (5)) = (h(X, Y.)) inherits exchangeability from the data sequence ((X,, Y.)). Formally, for any permutation a of [I..... 1}.
(S (Sa)))) S (((1))) (S",..,
S1+1).
(32)
where d denotes equality in distribution. Hence, the sequence (S) is exchangeable.
G.2 PROOF OF PROPOSITION 2.1
Proof. Let the epistemic uncertainty Ug(2) be defined the negative logarithm of the model- induced input density, Ug(z) - log (z). This definition captures the intuition that uncertainty is high where the model assigns low probability density.
Starting from the definition of the input density in Equation 5:
exp( - -(())/)
p(x)
Z
Taking the logarithm of both sides yields:
log p(x) = log
- log Z
log p(x) - -F(x) F(x)/T log Z.
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Multiplying by -1 and rearranging for F(x), obtain:
-logp(r) = F Fxx) + log
F()) r lollogg(pp (()) 7 log Z.
Substituting (x( - -log (()) and letting C = - log Z (a constant with respect to z), we arrive at:
F(x) = - +
This shows that the free energy F(r) is linearly proportional to the epistemic uncertainty (ε(r), scaled by the temperature т and shifted by constant. Therefore, a higher free energy value directly corresponds to higher epistemic uncertainty.
G.3 PROOF OF THEOREM 2.2
Proof. The proof proceeds by first establishing the relationship between the negative free energy -F(x)) and the maximum logit, and then arguing that the expected maximum logit decreases as sample difficulty increases for a well-trained model.
Step 1: Relating Negative Free Energy to the Maximum Logit. The negative free energy. - -xx), the LogSumExp (LSE) of the scaled logits. The LSE function a smooth approximation of the maximum function and is tightly bounded by it. For any vector R.K, the sum of exponentials can be bounded relative to its maximum term, zmax = max; Zj:-
K
Smax
Sh S K E
(33)
By taking the logarithm across all parts of the inequality, we obtain the standard bounds for the LSE function:
max Zn S log
max 26 log K.
(34)
Applying this to our scaled logits, 24 fx(2)/T, and multiplying by t gives:
max fx(x) < - (x)) < max fx(x) + r log K.
(35)
This inequality demonstrates that -F tight, monotonically increasing function of the maxi- mum logit, max; fa (z). Therefore, proving Theorem 2.2 is equivalent to proving that the expected maximum logit is strictly monotonically decreasing function of difficulty:
E [max fa(X) D(X, Ymme) d1} > Elmax D(X, Yuue) = dg).
(36)
Step 2: Characterizing the Maximum Logit by Difficulty Level. We analyze the maximum logit for samples conditioned on their difficulty.
Low Difficulty (d = 1): A sample (z, y) has difficulty d = 1 if and only if its true label y receives the highest logit. Thus, for this subpopulation of data, the maximum logit the logit of the true class:
D(x, Muue) =
max fx(x) fz()).
(37)
A model trained via a standard objective like cross-entropy is explicitly optimized to in- crease the value of (ar) for all training samples. Consequently, the set of samples where the model succeeds (d 1) corresponds to inputs for which the model produces large, dominant logit for the correct class.
High Difficulty (d > 1): sample (z, 3) has difficulty d > if and only the model's prediction is incorrect. This implies that the maximum logit corresponds class k/ +y y::
incorrect
D(x, Yuve) d>
max fx(x) = fx) for some k' + y.
(38)
k
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Step 3: Comparing Conditional Expectations. We compare the expected maximum logit over the subpopulation
correctly classified samples (dl = 1) versus incorrectly classified samples (d2 > 1). The training objective directly pushes the values in the set (fr(X) D(X, Mume) = 1} to be as large as possible. In contrast, the values in the set (max Sx(X) | D(X, Yiue) > 1] arise from the model's failure to generalize.
fundamental property of a successfully trained and well-generalized, and well-calibrated classifier that its confidence on the examples it classifies correctly is, average, higher than its confidence on the examples it classifies incorrectly (Guo et al., 2017). If this were not the case, the model would not have learned meaningful decision boundary from the data. Thus, the average maximum logit for the population of "easy" samples must be greater than that for the population of "hard" samples. Formally, for d1 d2:
E[max fa(X) DXX, Ymme) = d1) > Elmax fx(() D(X, (1XXYan Yume) = d2).
(39)
Given the monotonic relationship established in Equation 35, it follows directly that the expected negative free energy also decreases with increasing difficulty. This completes the proof and aligns with our empirical observation in Section 2.
G.4 PROOF OF PROPOSITION 2.3
Proof. The proof follows directly from the definition of a conformal prediction set. By definition, the prediction set (1) for new instance z includes all labels y for which the scaled nonconformity score does not exceed the calibrated quantile -i-
Ca(z)= = Sc(e, v) q
(40)
Substituting the definition of the scaled score, Sa v) = G(z))(()))))))) we have:
Ca(x) - {" G(=)S(,,)) < -
(41)
Since we assume G(x) is strictly positive, we can divide both sides of the inequality by G(x) without changing its direction:
Ca(x) - {M S(z, v) S
(G)
h1-a
(42)
G(xx)
By defining the instance-adaptive threshold ((x) // / arrive the equivalent formu- lation:
Ca(x) {y S(x y) < ((x)).
(43)
This concludes the proof.
Remark G.2. It is important to emphasize that the new quantile, ( fundamentally different from the baseline quantile, Ti-a. which would be computed from the unscaled scores.
Let Shane {X,, r)) be the set of baseline calibration scores, and Se (G(X:) Y.) be the set of Energy-reweighted calibration scores.
-haa is the (1 - a) quantile of the empirical distribution defined by Saase (G) - is the (1 - a) quauntiee of the empirical distribution defined by Scz-
There is no simple, closed-form relationship between Ūi-a and q (G) Scaling each score S; by different factor G(X,) changes the distribution of scores, including the relative ordering of the calibration samples. For instance, a sample (X,, Yj) that had a median score Spase might have a very high score in So if its corresponding energy factor G(X;) is large.
Consequently, the sample that happens to fall at the [(1 - ax) (n + 1)]-th position (thus defining the quantile) will almost certainly be different in the baseline and Energy- cases. In other words, one cannot simply take the baseline quantile - and scale it by some factor. The entire set of AGG). calibration scores must be re-computed and re-sorted to find the new, correct quantile Ti-a-
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G.5 PROOF OF THEOREM 3.1
Proof. The proof relies on the connection between the logits of a classifier trained with cross-entropy and the Bayesian posterior probability, which shows that the model's parameters internalize the training set's class priors.
Step 1:
Bayesian Decomposition of Logits. A classifier trained to minimize cross-entropy loss learns to approximate the posterior probability P(Y = y|X = r). Its logits (x( thus approximate the log-posterior, up to an instance-speeific normalization constant. Using Bayes' rule, we can decompose the log- oog-posterior:
log P(Y = y/ = x) = log P(X = x>Y=>>>>
= y) + log P(Y= = -) - log P(X =()=)). x).
(44)
When trained on Pumin: the model's s logits learn to reflect this structure:
(x)) log Praain X x y log Pirain
C(x);
(45)
where the term C(x) absorbs instance-dependen factors like log P'aain (X x) and other model- specific biases. Critically, the logit (z) encodes the log-prior probability of class y from the training distribution.
Step 2: Connecting Negative Free Energy to the Maximum Logit. As established in the proof of Theorem 2.2, the negative free energy -xxx) is tightly and monotonically related the maximum logit, - F(x)) s max (z). For reasonably accurate classifier, the expectation of the maximum logit over test samples from given class y is dominated by instances where the model is correct. For a correct classification of a sample (ar, y), the maximum logit is the logit of the true class: maxe fx(x) fr)) Building on this, we can state that the expected negative free energy for class y is primarily driven by the expected logit for that class:
(XX= Re (XX
(46)
Step 3: Comparing Expected Logits for Majority and Minority Classes. Using the decomposi- tion from Equation 45, we can express the expected logit for a class y as:
Ex (X)=) (X)] ze (X) llg Purain (X = y) + C(X) + log Pirain (Y y).
(47)
Let us define the term A(y) = Ex- (X) llog P = 3) + C()].. This term represents the average "data-fit" or "evidence" for class y. as learned by the model. Under our assumption that classes Vnnj and Ymin have comparable intrinsic complexity and are well-represented, this evidence term should be similar for both, i.c., A(ymaj) AYymin).
We can now compare the expected negative free energy for the two classes:
E - (-)( Y = Manyl = A(Ymay) + log Pirain (Y = Vmuj)
(48)
E E[-F(X)lY ymnial A(Ymin) log Purin(Y Yanin)
(49)
By the proposition's 's premise, Prriin (Y Ymmn) > Prminn( Ymin). which implies log Purain (Y Vmaj) > log Pirain( = y.min). Since A(ymaj) RS A(ymin), the additive log-prior term learned during training becomes the dominant factor driving the difference. Therefore, we conclude that:
=x [FX(]] > -
(50)
This result confirms that the model's systematically higher epistemic uncertainty (lower negative free energy) for minority classes is bias inherited from the training distribution's is class priors. This aligns with prior work showing that models have lower expected logits for minority classes (Ren al., 2020; Lyu al., 2025; Kato & Hotta, 2023; Chen & Su, 2023).
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H ANALYSIS OF SOFTMAX SATURATION AND ENERGY-BASED ADAPTIVITY
In this section, we provide a formal mechanism linking the limitations of softmax- noncon- formity scores conformal inefficiency. We demonstrate that while softmax probabilities saturate rapidly away from the decision boundary, thereby losing information about sample difficulty, the Helmholtz Free Energy retains this geometric information. We validate this analysis with toy experiment visualizing the decision landscapes.
H.1 DISTANCE TO DECISION BOUNDARY AND LOGIT MAGNITUDE
Consider a deep classifier f X' - R For a given input æ, let ÿ arg max f& (z) be the predicted class. The decision boundary between the predicted class ÿ and the second most likely class j is defined by the hyperplane where [ÿ(x) = (z)).
has been established that for neural networks, the magnitude of the logit vector typically scales with the distance of the input from the decision boundary (Hein et al., 2019). Let 6(x) denote the geometric distance of x from the decision boundary. We observe the following proportionality:
8(z) X max fx(x).
(51)
k
Therefore, an "easy" sample (one far from the boundary in a high-density region) characterized by logits with large magnitudes, while "hard" sample (near the boundary) yields logits with smaller or entangled magnitudes. Ideally, an adaptive conformal predictor should produce smaller sets as 5(x) increases.
H.2 THE SATURATION OF SOFTMAX AND ENTROPY
Standard conformal scores rely on the softmax distribution #(ylx) = cxp(fx exp(fx(z))- (x)((((())))
A critical limitation of this mapping is gradient saluration.
Consider sample moving away from the decision boundary such that its logit magnitude scales by a factor a > 1. As a -> 0o, 9 (Ú(æ) + The gradient the softmax output with respect to the dominant logit fy is given by:
θ (ÿ x)) #1)==(( - #((+)))
-
(52)
ff,
As + + 1, this gradient approaches 0. Similarly, the Shannon Entropy Hx))) of the distribution approaches O.
Implication for Conformal Prediction: Once a sample sufficiently far from the boundary to achieve high confidence (e.g., > 0.99), the softmax score saturates. The model becomes geo- metrically insensitive: a sample at distance and sample at distance 10d yield indistinguishable conformal scores. This saturation restricts the adaptive capacity of the prediction sets. For high- confidence samples, the sets cannot achieve greater efficiency because the score yields no further signal.
H.3 NON-SATURATION OF FREE ENERGY
In contrast, the negative Helmholtz Free Energy is defined -F(x) log Es exp(fx(x)/7)) As derived in Appendix G.3, this quantity is bounded by the maximum logit:
-F(x) sJ max (x)...
(53)
k
Unlike softmax, the Free Energy does not saturate. Its derivative with respect to the dominant logit approximately 1:
a a(-F(x)) m1
(54)
0fg
This indicates that -F(x) grows linearly with the logit magnitude, thereby acting as a faithful proxy for the distance ((x) even in high-confidence regimes.
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H.4 EMPIRICAL VISUALIZATION OF UNCERTAINTY LANDSCAPES
To empirically validate this behavior, we trained a 3-layer Multilayer Perceptron (MLP) on 2D toy dataset consisting of concentric classes. Figure visualizes the value of Max Softmax Confidence, Shannon Entropy, and Negative Free Energy across the input space X'
1
0
0.4
-2
p.2g
5.00
2.3
0.0
2.5
3.0
5.0
2.5
0.0
2.5
5.0
5.0
2.5
0.0
2.5
5.0
5.9
2.5
0.0
2:5
5.0
0.09
Softmax Confidence Landscape
32.75
65.4%
150.0%
class
0.0
41
82
(b) Entropy Landscape
Eneegr
-5.0 -2.5
0.6
2.5
5:6
-5.6
-2.5
0.0
3.5
5.0
-5.0
-8.5
0.0
8.5
5.0
-5.0
-2.3
0.0
2.5
5.0
(c) Negative Energy Score Landscape
Figure 6: Evolution of uncertainty landscapes on 2D toy dataset throughout training process. Columns represent progressive checkpoints from the early training phase (left) to full convergence (right). The red line indicates the decision boundary. (a) Softmax probabilities saturate rapidly. The yellow region (confidence 1.0) is flat, making points near the boundary indistinguishable from points far away. (h) Entropy exhibits similar saturation (dark blue region), vanishing to for of the domain. (c) Negative Free Energy retains gradients throughout the domain. Note the continuous color transition scaling with the distance from the decision boundary, identifying "casier" points with higher values even when softmax is saturated.
H.5 MECHANISM OF ENERGY-BASE BASED EFFICIENCY
Our proposed method leverages this non-saturating property by modulating the base nonconformity score S(x,2) y) with sample-specific scaler G(z) softplus(=F(()).
For "easy" samples (large 8(x)), -F(x) is large positive. This results a large scaling factor G(x) >> 1.
1. The nonconformity scores for incorrect labels (which are naturally non-zerg non-zero) are magnified significantly by G(z), pushing them well above the calibrated quantile q.
2. The nonconformity score for the true label (typically near zero) remains small after scaling.
This amplification forces the exclusion incorrect classes that might otherwise have been included due to the looseness of the global quantile q. thereby reducing the prediction set size. Because F(x) does not saturate, this efficiency gain continues to improve as samples get "casier." a property unattainable with softmax- based modulation.
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I PERFORMANCE ANALYSIS STRATIFIED BY SAMPLE DIFFICULTY
In this section, we report the performance of different nonconformity scores by stratifying test sam- ples based on their difficulty. Sample difficulty defined as the rank of the true label in the model's predicted probability ordering; lower rank indicates an "easier" sample, while a higher rank signi- fies "harder", often misclassified, one.
In Section 3.1, we established that energy-reweighting reduces the overall average prediction set size while maintaining target coverage. However, this aggregate metric does not reveal whether these efficiency gains are distributed evenly are concentrated specific types of samples. This stratified analysis provides more granular view to answer that question.
We expect an adaptive method to produce the most significant set size reductions for easy samples, where the model confident, while appropriately adjusting for harder samples where uncertainty is higher. Table 6 presents these stratified results, detailing how coverage and average set size vary across the different difficulty levels.
Table Coverage and average set size on ImageNet, stratified by sample difficulty.. Results are ResNet- 50 model at a = 0.01 and averaged over 10 trials. The table compares baseline adaptive scores with their energy-based variants, which generally produce smaller sets for casier samples while maintaining coverage.
APS
RAPS
SAPS
w/o Energy
w/ Energy
w/o Energy
w/ Energy
w/o Energy
w/ Energy
Difficulty
Count
Cov.
Set Size
Cov,
Set Size
Cov.
Set Size
Cov:
Set Size
Cov.
Set Size
Cov.
Set Size
1.00
to
2547
1.00
38.38
1.00
34.96
1.00
37.17
1,00
34,10
1.00
36.98
1.00
33.69
36.46
36.29
to 10
306
1.00
37.66
1.00
37.09
1.00
37.40
1.00
37.20
1.00
37.47
1.00
36.96
100
101 1000
66
0.00
37.82
0.00
38.45
0.00
37.60
0.00
38.60
0.00
37.34
0.00
38.05
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J ENERGY-BASED LAC
The standard LAC score inversely proportional to the softmax probability. For its energy-based variant, as mentioned in Appendix E.2, use base score of (L, y) ( For difficult input, which corresponds to a low negative free energy score, the objective is to produce a larger prediction set. This requires the nonconformity scores of more classes to fall below the calibrated threshold. Conversely, for an easy imput (high negative free energy), the nonconformity scores should be scaled produce smaller set. Therefore, for the energy- the base score by the energy-based scaling factor. This adjustment ensures that for difficult inputs, the small scaling factor in the denominator makes the nonconformity scores smaller, including more labels in the set.
LAC, we divide
So formally, we define the Energy-based LAC nonconformity score as:
(
SEB v)
(55)
log(1 + ()(z))
where #y][z) is the softmax probability, F(x) is the Helmholtz free energy, and 3 is the softplus sharpness parameter.
J.1 ENERGY- EVERGY-BASED LAC PERFORMANCE IN STANDARD SCENARIO
Table 7: Performance of the LAC nonconformity score function and its energy- based variant CIFAR-100 ImageNet, and Places365 miscoverage levels a [0.01, 0.025, 0.05, 0.1). Results are averaged over 10 trials. For the Set Size column, lower is better. Bold values indicate the best performance within the method family (with and without Energy).
0.1
0.05
0.025
a 0.01
Method
Coverage
Set Size
Coverage
Set Size
Coverage
Set Size
Coverage
Set Size
CIEAR- FEFAR-100 (ResNet-56)
w/o Energy
0.90 cm
2.54 oms
0.95 0.0
5.26 o.17
0.974
9001
9.50 0.21
0.99
21.14 4 1014
knezgy
0002
ImageNet (ResNet-5U)
w/o Energy
0.90
1.49
0.95
2.68
0.975
5.48
0.99
14.79
w/ knergy
0.90 cuto
1.48 om
0.95 : 0.0
2.68 0u94
0.975
0001
5.42 0.15
0.99 6.00
13.89 1s4
Places365 (RexNet-50)
Energy
0.90
0.95
11.36
0.973 000
19.55
37.12
w/ Enecgy
0.90 0.001
6.19 0:0
0.95 0001
11.11 0.131
0.973 0001
19.28
0.43
0.99 0,001
35.28 0.6
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J.2 ENERGY-BASED BASED LAC PERFORMANCE WITH IMBALANCED TRAINING PRIORS
Table Performance LAC energy-bar variant on imbalanced CIFAR-100 varying imbalance factors (A E (0.005, 0.01, 0.02, 0.03)) and at miscoverage levels a E (0.01, 0.025, 0.05, 0.1). Results are averaged over 10 trials with ResNet-56 model. For the average set size, lower is better. Bold values indicate the best performance.
0.1
0.05
0.025
a 0.01
Method
Coverage
Set Size
Coverage
Set Size
Coverage
Set Size
Coverage
Set Size
CIFAR-I0--LT (A 0.005, mild imbalance) (ResNet-S6)
w/o Energy
0.897
7.04
0.947
12.98
0.973 u.0e
21.31
0.988
33,97
Energy
nocs
1
CIFAR-10--LI (A 0.01) (RexNet-56)
LAC
Energy
0007
0.00
::
w/ Energy
0,900 uoc
11.52 0.22
0.950 = ucoe
20.32 0.39
0.976 a.0
30.51 0.4
0.990 c.00
44.49 0.79
CIFAR-100-11 (A 0.02) (ReeNet-56)
w/o Energy
0,901 0007
27.78 1.071
0,950 ncon
42.03
0.975 n00
54.88
0.990 am
68.21
Energy
0.951
0.976
100-1--00-1TT 0.03, severe imbullncce) (ResNot-56)
LAC
Energy
0.951
0.976
w/ Energy
0.901 1. nocs
27.93 2.0
0.952
41.50 0m
0.975 a no0s
$4.02 1.
0.990
68.20 1:45
J.3 ENERGY- BASED LAC PERFORMANCE ANALYSIS STRATIFIED BY SAMPLE DIFFICULTY
Table 9: Coverage and average set size for the LAC method on ImageNet, stratified by sample difficulty, Results are shown for 0.01 and 0.025. The table compares the baseline LAC with energy- energy-based variant.
LAC
0.01
0.025
w/o Energy
Energy
w/o Energy
Energy
Difficulty Level
Count
Cov.
Set Size
Cov.
Size
Cov.
Size
Cov.
Set Size
15990
1.00
10.77
1.00
4.34
1.00
4.25
to
2547
1.00
21.47
1.00
20.61
0.99
8.86
0.99
8.78
4 to 6
638
0.99
32.07
0.98
32.96
0.93
13.29
0.93
13.36
7 to 10
306
0.96
38.18
0.96
40.00
0.84
15.86
0.85
16.14
11 to 25
275
0.92
44.22
0.91
47.14
0.60
17.07
0.56
17.05
to 50
104
0.62
47.37
50.88
0.04
17.60
0.06
17.59
100
56.19
64.61
21.07
38

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Published as a conference paper at ICLR 2026
K ADDITIONAL EXPERIMENTAL RESULTS FOR IMBALANCED DATA
To evaluate performance under class imbalance as described Section 3.2, we construct several long-tailed variants of the CIFAR-100 dataset. In these variants, the number of training samples per class follows an exponential decay controlled by an imbalance factor X. Figure illustrates how different values of À create varying levels of imbalance in the training distribution.
2:505
2.03
2.32
Nurber of Sangle
400
3.32
300
100
30
20
30
60
500
150
Class Indlec
Figure 7: Class distributions under varying imbalance levels. The number of samples per class follows an exponential decay pattern proportional to exp(-> j), where larger À values induce stronger imbalance.
K.1 RESULTS FOR DIFFERENT IMBALANCE FACTOR À
Table 10: Performance comparison of APS, RAPS, SAPS, and their Energy-based variants on imbal- anced CIFAR-100 with varying imbalance factors (A E f0.01, 0.02, 0.03)) and at miscoverage levels E {0.01, 0.025, 0.05, 0.1}. Results are averaged over 10 trials with a ResNet-56 model. For the Set Size column, lower is better. Bold values indicate the best performance within each method family.
0.1
0.05
0.025
0.01
Method
Coverage
Set Size
Coverage
Set Size
Coverage
Set Size
Coverage
Set Size
CIFAR-100-LI (A 0.01, mild inbalance) (ResNet-56)
APS
w/o Energy
0.90 am
14.56 046
0.95 0.00
25.59
025
0,975 = 0.0
37,56 =
0.99 0.00
53.26 2.40
Energy
0.90
11.86 0.3
0.95
21.44 005
31.70 600
0.99
6002
w/o Energy
0.90 a.m
15.48 1.0
0.95 1.0
27.62 100
0.974 1000
40.87
1.00
0.99 p.00
60.13 186
Enorgy
0.90
11.77
0.95 cmm
21.57
0,975
31.93
0.99
47.30 101
w/o Energy
0.90 dml
15.02 0.50
0.95 0:00
26.98 a03
0.974
0000
40.38
1.26
0.99
0:00
59.12 =17
w/ Energy
0.90 o.
11.80 0.40
0.95 0:000
21.35 0.5
0.975 n003
31.84
1002
0.99 0.000
47.17 1.00
CIEAR-10-LT iA
0.02) (ResNet-56)
APS
wro Energy
0.90 am
29.62 = 0.5
0.95 0.00
45.76 1.32
0.975 #00
59.26 =
0.99 0.00
73.49 1.36
w/ Energy
0.90 am
28.23 0.44
0.95 cm
42.36
1.33
0.976 200
54.95
0.99
69.58 1.92
who Enorgy
32.86 6x1
0.95 1om
133
1iom
1.89
0.99 0.00
79.97 1.3
w/ Energy
0.90 a.
28.72 om
0.95 co
42.72 180
0.976 1 nnm
56.31 636
0.99 c.
70.72 135
SAPS
w/o Energy
0.90
32.37
0.95 o.t
51.47 1.45
0.976 2000
66.28
0.99
79.14
w/ Energy
0.90 am
28.73 0.78
0.95 c.t
42.63
1.09
0.976 0.00
56.16
0.89
0.99 0.00
70.48 1.56
CIFAR-100-LT (A 0.03, severe imbalance) (ResNet-56)
w/o Energy
0.90
0.5
0.00
000
0.02
0.99
1.09
w/ Energy
0.90 cm
28.42 0.52
0.95 6.00
42.40
0.8
0.975
0:00
55.61
1.54
0.99 0.00
70.85 1.55
w/o Energy
0.90
34.47
0.95
49.94
0.975
64.66
0.99
78.91
w/ Energy
0.90 am
29.29 066
0.95 cm
43.68 103
0.975 10m0
57.23 102
0.99 m.
72.74
1.54
w/o Energy
0.90 amt
34,05 om
0.95 am
49.45 = 1:33
0,974 200
64.35
1.33
0.99 am
78.41
1.00
Energy
0.90
29.19 066
0.95 cot
43.61 0.
0.975 0003
57.12 0:97
0.99 0:00
72.48 1.49
39

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Published as a conference paper at ICLR 2026
K.2 PERFORMANCE UNDER IMBALANCED CALIBRATION AND TEST SETS
The results reported in Table 2 and Table 10 evaluate models trained on imbalanced data but cali- brated and tested balanced sets. We now consider more realistic scenario where the calibration and test sets also follow the same imbalanced distribution as the training set.
This setting is particularly challenging for standard CP methods, as the limited number of calibration samples for minority classes can impede reliable coverage guarantees. In such cases, approaches like clustered conformal prediction Ding al. (2023b), which can provide coverage with fewer calibration samples, are practical alternatives.
Our energy- -baeed method is designed to address this challenge by adaptively enlarging prediction sets for uncertain inputs, which often correspond to minority class samples. To benchmark this be- havior, we compare against another principled reweighting strategy that directly uses class priors. This approach was introduced by Ding et al. (2025), who proposed the wrevalence-Adjuste Soft- max (PAS) score. The PAS score modifies the non-adaptive score by dividing the negative softmax probability (LAC score without bias term) by the empirical class prior, p(y), to improve coverage for rare classes. The nonconformity score is defined as:
+))
SpAs(E, v) p(y)
(56)
We extend this concept adaptive nonconformity scores such APS and RAPS. For these scores, a smaller value indicates higher conformity. To increase the likelihood of including labels from rare classes (which have a small p(y)). we multiply the base score by the class prior. This makes the minority classes more likely to be included in the final prediction set. We refer to this method as Prevalence-Adjusted (PA) Nonconformity Scores. The general formulation is:
Spa y) Sadaptive v) p(y)
(57)
where Sataptive y) is an adaptive score like (xes(w,1) y)-
Given this, Table 11 and Table 12 present comparison between the standard adaptive scores, our energy-based variants, and the prevalence-adjusted variants in this fully imbalanced setting. We report marginal coverage, average set size, and MacroCov to provide comprehensive view of performance.
40

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Published as a conference paper at ICLR 2026
Table 11: Performance fully imbalanced CIFAR-IO0-LT for high confidence levels (0.025, 0.01)))
each method, we compare the Standard baseline, Prevalence-Adj. variant, our Energy-based variant. Lower Set Size better.
0.025
0.01
Method
Variant
Cov
Size
MacroCov
Cov
Size
MacroCov
CIFAR-100-LT (A 0.005, mild imbalance)
Standard
0.97 0.00
21.48
0.86
0.97 1.0.0
0.99 0.00
34,60
1 1.77
0.99 1 0,000
LAC
Prevalence-Adj, (PAS)
0.98 0.00
22.84
0.99
0.98 1: 0,00
0.99 0.000
36.23 0.98
0.99 0:00
Energy-based
0.97 0.00
21.11
0.79
0,97 0,00
0.99 0.00
33.18
0.99 0.00
Standard
0.97
28.61
0.97 + 0.00
46.37
0.99
APS
Prevalence-Adj.
0.98
29.72
0.97 +
50.57
0.99 =
Energy-based
0.97
22.09
0.97 +
35.78
0.99
Standard
0.97 0.00
30.52
1.44
0.97 1: 0,00
0.99 0.00
51.46
1.84
0.99 0:00
RAPS
Prevalence-Ady
0.97 0.00
31.45
1.66
0.97 1 0,00
0.99 0:00
51.41
1.66
0.99 1. 1.000
Energy-based
0.97 0.00
22.57
1.08
0.97 1: 0,00
0.99 0,00
36,75
1.42
0.99 0,000
Standard
0.97
29.54
0.97 0.00
0.99 0:00
50.35
0.99
SAPS
Prevalence-Adj.
0.97
30.41
0.97
0.99 0:00
50.50
0.99
Energy-basee
0.97
22.36
0.97
0.99
36.39
0.99
CIFAR-100-LT
0.01)
Standard
0.97 1.00
30.22
0.8
0.97 # 0.00
0.99 0:00
45.45
1.77
0.99 # 0,00
LAC
Prevalcnce-Adj, (PAS)
0.98 0.00
33.95
1,00
0.98 + 0.00
0.99 0.00
50.56
1.59
0.99 +
Energy-based
0.97 n.00
29.93 0.99
0.97 0.00
0.99 0.00
43.94
1.34
0.99 + 0.00
Standard
0.97
36.97
0.97
0.99
54.90
0.99
APS
Provalence-Adj.
0.98
43.29
0.98 +
0.99
66.30
0.99
Energy-based
0.97
31.20
0.97 +
45.15
0.99
Standard
0.97 0.00
40.39
1.29
0.97 + 0.00
0.99 0.00
60.09 =: 2.93
0.99 =: 0.00
RAPS
Prevalence-Adj
0.98 + 0.00
43.21
= 2.15
0.98 +: 0.00
0.99 0.00
65.82 =:
0.99 :: 0.00
Energy-based
0.97 i 0.00
31,64
1.24
0.97 :: 0,00
0.99
0.00
45.99
1.67
0.99 0.00
Standard
0.97 0.00
39.81
1.53
0.97 1. 0,00
0.99
0.00
59.11
J. 2.61
0.99
0.00
SAPS
Prevalence-Adj
0.98 0.00
42.34
2.12
0.98 1: 0,00
0.99
0.00
65.18
à: 2.45
0.99
1: 0,00
Energy-buset
0.97 0.01
31.46
0.97
0.01
0.99 0.00
45.73
0.99 0,00
CIFAR-100-LT
0.02)
Standard
0.97 1 0.00
52.86
1.54
0.97 1. 0,00
0.99 0.00
66.53
1.36
0.99 0:000
LAC
Prevalence-Adj, (PAS)
0.98 0,00
61.74
1.37
0.98 +: 0,00
0.99 0.00
75.07
1.20
0.99 0:000
Energy-based
0.97 0.00
51.96
1.30
0.97 #: 0,00
0.99 0.00
65.48
2.27
0.99 0,00
Standard
0.96 0.00
56.87
0.97 0,00
0.99 0.00
71.19
0.99 0,00
APS
Prevalenee-Adj.
0.99 0.00
74.42
0.98
1.00 0.00
86.
0.99
Energy-based
0.97
53.13
0.97
0.99
68.02
0.99
Standard
0.00
61.92
0.97 1: 0,00
0.00
76.89
0.99 0.00
RAPS
Prevalence-Adji
0.00
73.38
0.28
0.00
0.00
86.52
0.40
0.99 0.00
Energy-basee
0.00
54.13
1.37
0.97 1. 0,00
0.00
69.14
1.92
0.99 0.00
Standard
0.96 0.00
61.42
1.18
0.97 :: 0.00
0.99 0.00
76.01
1.11
0.99 :: 0.00
SAPS
Prevalence-Adj
0.99 0.00
73.12
0.36
0.98 1: 0,00
1.00
0.00
86. 15
0.29
0.99 1: 0,00
Energy-base based
0.97 0,000
54.02
1.58
0.97 1. 0,00
0.99 0.00
69.01 1.67
0.99 1. 0.00
CIFAR-100-LT (A 0.03,
imbalance)
Standard
0.96 0.00
51.39
1.25
0.97 + 0.00
0.98 0.00
67.76 = 0.84
0.99 + 0.00
LAC
Prevalence-Adj. (PAS)
0.98 0.00
62.93
= 0.99
0.98 + 0.00
1.00 + 0.00
75.03
= 0.46
0.99 + 0.00
Energy-based
0.96 + 0.00
50.77
1.38
0.97 + 0.00
0.98 + 0.00
66.87
= 0.95
0.99 + 0.00
Standard
0.96 0.00
53.95
0.97 0,00
0.99
0.00
69.98
0.99 0.00
APS
Prevalence-Adj.
0.99 0.00
79.13 0.69
0.98 0,00
1.00
86.14 0.28
0.99 0.00
Energy-based
0.96 0.01
52.00
0.97 1. 0,00
0.99 0.00
69.77
0.99 1. 0.00
Standard
0.96
60.09
0.97 + 0.00
75.77
0.99 +
RAPS
Prevalence-Adj
78.45
0.98 + 0.00
1.00 0.00
86.09
0.99 ::
based
0.01
53.67
0.97 1: 0.00
0.00
71.64
0.99 0.00
Standard
0.96 0.00
59.61
1.28
0.97 1. 0,00
0.98 0.00
75.59
1.15
0.99 1:000
SAPS
Prevalence-Adjj
0.99 0.00
78.38
0.51
0.98 4: 0.00
1.00 0.00
86.02
0.33
0.99 0:00
Energy-based
0.96 1.00
53.68
1.51
0.97 +: 0.00
0.98 0.00
71.72
1.30
0.99 :: 0.00
41

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Published as a conference paper at ICLR 2026
Table 12: Performance fully imbalanced CIFAR- 100-LT for lower confidence levels (a {0.1, 0.05)). For each method. we compare the Standard baseline. the Prevalence-Adj variant, and our Energy-based variant. Lower Set Size is better.
0.1
0.05
Method
Variant
Cov
Size
MacroCov
Cov
Size
MacroCov
CIFAR-I00 10---0T--TTT (A 0.005, mild imbalance)
Standard
0.90 0.01
7.12 0.26
0,90 1 0.00
0.95 0.00
13.08 0.55
0,95 1: 1.000
LAC
Prevalence-Adj (PAS)
0.90 nor
7.26
0.90 +: 0.01
0.95 uo1
13.39 0.00
0.95 0.01
Energy-based
0.90 0.01
6,96 1 0.33
0,90 1 0.00
0.95 1 0.10
12.74 0.054
0.95 1: 0.01
Standard
0.90 n.0
8.48
0.90 +
17.55
0.95 =
APS
Prevalence-Adj.
8.81 =
0.90 +
0.95
17.95
0.95 =
Energy-based
0.01
7.39
0.90 +:
0.95 0.01
13.42
0.95
Standard
0.90 0.01
8,95 1 0.44
0.90 1 0.01
0.95
1 0.01
18.89
1.06
0.95 1.0.1
RAPS
Prevalence-Adj
0.90 0.01
9.14
1. 0.46
0.90
1 0.01
0.95 0.01
19.19
1 0.89
0.95 1 0.011
Energy-based
0.90 0.01
7.58
a: 0.25
0.90
a: 0.01
0.95
0.01
13.33
0,56
0.95 1: 0.01
Standard
0.90 n.01
8.63
0:47
0.90
0.00
0.95 uoo
18.22
0.95 0.01
SAPS
Prevalencc-Adj.
0.90 n.01
8.83
0.90
0.95 0.00
18.50
0.95
Energy-based
0.90 n:01
7.53
0.90
0.95 0.00
13.30
0.95 0.00
CIFAR-100-LIY
0.01)
Standard
0.89 noo
11.37 0.2
0.89 + 0.01
0.95 0.01
20.09 0.7.1
0.95 + 0:0
LAC
Prevalcnce-Adji (PAS)
0.91 n.0
12.20 0.24
0.90 + 0.01
0.96 0.00
22.03 0.63
0.95 + 0.00
Energy-based
0.89 p.0
11.12 0.29
0.89 + 0.01
0.95 0.01
19.73 0.700
0.95 0.01
Standard
0.89
13.76
0.90
0.95
25.34
0.95
APS
Prevalence-Adj
0.91
14.68
0.90
0.96
26.62
0.95
Energy-based
0.89
11.29
0.89
0.95
20,80
0.95
Standard
0.89 0.00
14.72 0.19
0.90 +: 0.00
0.95 0.1
26.91
1.08
0.95 0.01
RAPS
Prevalence-Ade
0.90 i 0.01
15.03
0.54
0.90 1: 0.01
0.95 0.00
27.14
=:
0.95 0:00
Energy-based
0.89 0.00
11.22
0.18
0.89 + 0,00
0.95
+ 0.01
20,95
0.77
0.95 = 0.01
Standard
0.89 0.00
14.22
0.19
0.90 1: 0,00
0.95
0.01
26,19
0.98
0.95 1.0.1
SAPS
Prevalence-Adj
0.90 0.01
14.55
0.43
0.90
+: 0.01
0.95 0.00
26.65
0.84
0.95 0,00
Energy-based
0.89 0.00
11.26 0.17
0.89 0,00
0.95
20.73
0.95
CIFAR-100-LT
0.02)
Standard
0.86 + 0.01
24.97
0.6
0.88 +: 0.01
0.93 0.00
38.81 1.07
0.94 0.01
LAC
Prevalence-Adj (PAS)
0.91 i 0.00
30.25
= 0.46
0.91 1: 0,00
0.96 0.00
46.98
1.15
0.96 :: 0.00
Energy-based
0.86 0.01
24.65 0.87
0.88 1. 0.01
0.93 0.01
38.46
1 0.94
0.94 0.00
Standard
0.87 n.1
26.69
0.88
0.93 0.00
42.41
0.94
APS
Prevalence-Adj.
0.92 0:00
37.05
0.91
0.97
60.22
0.96
Energy-based
0.86 0.01
25.07
0.88
0.93 0.01
39.10
0.94
Standard
0.01
28.44
0.78
0.88 1 0.01
0.01
46,22
1.45
0.94 0.01
RAPS
Prevalence-Adj.
0.00
35.42
0.69
0.91 1 0.00
0,00
59.12
0.77
0.96 0.00
Energy-based
0.01
25.37
0.70
+: 0.01
0,01
39.63
0.98
0.94 0.01
Standard
0.86 not
28.26
0.76
0.88 + 0.01
0.93 0.11
45.73
1.28
0.94 = ca0
SAPS
Prevalcnce-Adjj
0.92 0.00
35.69
0.66
0.91 + 0.00
0.97 0.00
59.11
0.95
0.96 = 0.0
Energy-based
0.86 p.1
25.41 0.71
0.88 0.01
0.93 0.01
39.72
1.00
0.94 0.01
CIFAR-100-LT (A 0.03,
imbalance)
Standard
0.84 0.01
24.43
0.63
0.87 + 0.01
0.92 0.00
38.72 = 0.77
0.94 + 0.01
LAC
Prevalence-Adj, (PAS)
0.90 pm
34.18
1.00
0.90 + 0.01
0.95 0:0
49.60
1.43
0.95 + 0.01
Energy-based
0.84 + p.0
24.06
0.64
0.87 + 0.01
0.92 0:001
38.04
= 0.90
0.94 : 0.01
Standard
0.85 0.0
25.64
0.94
0.88 1: 0.01
0.92 0.01
39.43
1.23
0.94 1: 0.01
APS
Prevalence-Adji
0.95 0.00
48.47
0.65
0.93 1 0,00
0.98
1. 0.00
66.59
0.7.3
0.97 1. 0.01
Energy-based
0.84 0.01
24.43
0.88
0.87 0,01
0.92 0.01
38.78
0.94 0.01
Standard
0.85
28.00
0.87 + 0.92
43.81
0.94
RAPS
Prevalence-Adjj
0.95
46.46
0.93 + 0.01
0.98
65.14
0.97 =
Energy-based
0.84
24.74
0.87 +
0.92
39.74
0.94
Standard
0.85 n.0
27.66 27.64
0.85
0.88 4: 0,01
0.92 0.01
43.29
1.47
0.94 1: 0.01
SAPS
Prevalence-Adjj
0.95 1.00
46.81 0.81
0.93 + 0.01
0.98 0.00
65.25
0.69
0.97 +: 0.01
Energy-based
0.84 1101
24.76
0.63
0.87 0.01
0.92 0.01
39.73
1.24
0.94 0.01
42

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Published as a conference paper at ICLR 2026
L CLASS-CONDITIONAL CONFORMAL PREDICTION
Beyond marginal coverage guarantee, we evaluate the proposed Energy-based nonconformity scores within the class-conditional setting. Class-conditional Conformal Prediction (CP) operates by parti- tioning the calibration dataset according to the true labels. Nonconformity score quantiles are then computed independently for each class using its respective calibration subset. The objective is to achieve class-conditional coverage, defined as:
P(Yusst C(Xtest) Yeest = v) 1221aa a, for y e y,
(58)
This condition ensures that for every class E ) the probability of the true label being included in the prediction set ((e is at least az.
We evaluate performance on the Places365 dataset at miscoverage levels E f0.05, 0.1} using average set size, CovGap, and SSCV (for details of these refer to E.3. As shown Table 13, Energy-based variants consistently yield more efficient prediction sets, reflected in reduced average set sizes across all base nonconformity functions. Importantly, this improvement in efficiency does not compromise class-conditional validity and CovGap is preserved or slightly improved in most cases. While we report SSCV for completeness, given its use in prior work, and note that it is often maintained or even improved in our experiments, emphasize that it is not reliable measure of conditional coverage quality.
Table 13:
Class-conditional performance comparison of different nonconformity Energy-based variants on the Places365 dataset miscoverage levels 0.05 and a 0.1. For Set Size, CovGap, and SSCV, lower values indicate better performance. Bold values denote the best result within cach method family. Results are averaged over 10 trials with a ResNet-50 model.
functions and their
a 0.1
0.05
Method
Coverage
Set Size
CovGap
SSCV
Coverage
Set Size )
CoyGap J
SSCV J
APS
Energy
0.89 0.00
2014
5.03 014
0.119 o01s
0.95 = 0.0
21.50 0:00
3.25 2012
0.124 = 0/099
w/ Energy
0.89
8.65
5.09
0.100
0.95
19.20
3.24 1:1
0.087
SAPS RAPS
Energy
9.11
4,98
0.100
0.95
22.03
3.24
0.090
w/ Energy
0.89
8.59
4.95
0.112 aom
0.95
19.11 0:40
3.22
0.119
w/o Energy
0.89 a:00
8.92 0.20
4,98
0.100
0.95 0:00
21.68 0.50
3.26 0.14
0,060 0019
w/ Energy
0.89 a a.m
8.62 à 0.20
4.87 0.15
0,103 a no1
0.95 :: a.
18.99 6...
3.17 a.n
0.083 0003
M EFFECT OF RAPS HYPERPARAMETERS À AND KREG
In this section, we evaluate the sensitivity of the Regularized Adaptive Prediction Sets (RAPS) method, and its energy- variant, to their two core hyperparameters: the regularization weight À and the penalty threshold kreg- Table 14 compares the average prediction set size of the standard RAPS baseline against our proposed Energy- RAPS on the Places365 dataset using a ResNet- 50 model, across grid of parameter configurations.
Table 14: Comparison of average prediction set sizes for varying regularization parameters (kieey and A). Lower set size is better. Values are reported as: Size (RAPS-+ Energy-based RAPS). The bold value highlights the superior (smaller) size.
Average Set Size (w/o Energy - w/ Enerqy) L
Regularization Penalty (A)
keeg
0
0.05
0.1
0.2
0.5
0.7
1.0
14.08 - 12.88
13.11 + 12.88
13.60 - 12.24
14.07 + 12.61
14.30 - 12.90
14,30 > 12.95
14.30 + 13,00
2
14.09 - 12.86
13.13 + 12,83
13.58 - 12.34
14.05 - 12.63
14.30 - 12.97
14.30
13.05
14.30
13.11
5
14.01
12.85
13.11
13.06
13.61
12.67
14.07
12.86
14.30
13.22
14,30
13.29
14.30
13.34
10
14.05 - 12.90
13.13 + 13.07
13.58 + 13.57
14.07 + 13.46
14.29 - 13.62
14,30 > 13.74
14,30 + 13.82
50
14.00 - 12.86
13.97 - 12.91
14,02 - 12,85
14.00 - 12.85
14.02
12.88
14.05
12.87
13.90
12.88
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N
EFFECT OF T CALIBRATION AND TENERGY
EB-SAPS (a 0.025)
EB-RAPS 0.025)
04
11.78
11.5
11.50
69 SA 06
Callbration
callbration
10.5
10.75
10.50
16075 50 20
95
10.25
10075 50
10.00
|n| (n(eenargy)
In (enerryryy)
EB-SAPS (a= 0.05)
EB EB-RAPS (a 0.05)
26073 25 13 5857277 03 92
Fallibation
Callbration Nive age
4.0
45
In(Tenery)
In (n(Tenergy)
Figure 8: Average Set Size heatmap for different hyperparameter settings across Energy-based variants of Sand SAPS.
Sensitivity heatmaps illustrate how (temperature in the calibrated softmax) and т (temperature in the energy calculation) affect the average set size the ImageNet dataset for Ehergy--Bssed RAPS and Energy- Based SAPS, with a E (0.025, 0.05}. As In(r) increases, the effect of energy- based reweighting gradually diminishes. Consequently, for larger values of the model converges to the baseline method. For instance, Energy-Based RAPS with a large positive In(r) behaves almost identically standard RAPS. As shown in Figure across different values of T (softmax probability calibration), energy-based variants of the methods (corresponding to smaller values of In(r)) produce more informative prediction compared their bascline counterparts (associated with larger values of In(r)).
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O SENSITIVITY ANALYSIS OF THE SOFTPLUS PARAMETER 3
We analyze the impact of the sharpness parameter on the efficiency the generated prediction sets. As defined in Equation 9, this parameter controls the approximation of the scaling factor the ReLU function. We evaluate the average prediction set size across a wide range of / values for the CIFAR-100 dataset using ResNet-56 model.
Figure presents the results for Energy- based APS, Energy-based R,PS, and Energy-based SAP Sat miscoverage levels a = 0.05 and a = 0.025. As 3 approaches zero, the term 3/2 log (1 + e -BP(x) yields scaling factors that are inseparable across samples. Due to this loss distinction, the perfor- mance converges the baseline without energy.
However, as 3 increases, the performance stabilizes and remains constant across several orders of magnitude. This behavior aligns with the theoretical motivation that the scaling factor need only ap- proximate the ReLU function to handle rare negative free energy values while preserving the signal for positive values. Consequently, precise tuning of this parameter is unnecessary. Selecting suf- ficiently large value is safe option to achieve the performance benefits of Energy-based conformal classification.
16
Size
Size
Ie-a5
Se-d3
00075
0875
0.025
0.079
25
100
200
100
800
1000
16-00
Se-13
0. 0.0002
60073
0.0025
0070
0.025
100
200
300
400
500
1000
(a) Energy-based APS(a 0.05)
(b) Energy- based APS(a 0.025)
Size
12
1e-05
5e-05
0.0025
0.00773
0,0025
0.025
23
100
400
500
16-09
Se-12
000020
05073
0.0025
0075
020
075
200
300
400
800
Energy based RAPS(a 0.05)
(d) Energy- based RAPS(a 0.025)
15
0.
14
Size
7.0
12
0.
11
2e-05
Re-95
00125
60105
.025
0075
100
200
180
400
500
1080
16-31
56-15
D0022
60075
0025
0075
025
100
200
800
2000
(c) Energy- based SAPS(a = 0.05)
(f) Energy based SAPS(a = 0.025)
Ensegy-based
Figure 9: Ablation study of the parameter CIFAR-100 with ResNet-56. The plots show the average prediction size (shaded regions indicate standard deviation) as function of J3. Performance stabilizes for sufficiently large values of 8, indicating that the method does not require specific tuning of this parameter.
45

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Published as a conference paper at ICLR 2026
P ON THE RELIABILITY OF CONFORMAL CLASSIHIERS WHEN FACED WITH OOD TEST SAMPLES
reliable system for uncertainty quantification often expected satisfy two primary objectives, outlined in the Appendix of Angelopoulos & Bates (2021):
1. Flag out-of-distributin (OOD) inputs to avoid making predictions on unfamiliar data.
2. If imput is deemed in-distribution, output a prediction set that contains the true class with user-specified probability.
practical strategy to achieve this is two-stage pipeline: first, an OOD detector screens each input. If an input is flagged as OOD, the system can abstain (c.g., by returning an empty set). If deemed in-distribution, the input is passed to a conformal predictor to generate a valid prediction set. This separation, while effective, requires deploying and managing two distinct models.
The utility of the energy-based paradigm becomes particularly evident in scenarios where a dedi- cated OOD detection module is not available to filter inputs. In such cases, standard conformal predictor, relying solely on softmax outputs, can be misleading. An OOD input might still produce single, high- cogfhccnfee softmax score, leading the base conformal method to output small, high- confidence prediction set (e.g.. ("Tuberculosis' })) for a non-medical image. This false confidence is a critical failure mode.
Conformal Classifier
In- Distribution Sample
{Pleural Effusion, Pneumonia]
Case 1: Empty Set -> "i don't know the answer!"
Out-Of- Distribution
X Case 2: Small Set "i'm pretty sure (not correctly)
Sample
{Tuberculosis}
about my answer"
Case 3: Large Set > "i'm not sure about my answer!" TTueecculssis, Pleural Effusion, Atelectasis, Normal, Pneumonia)
Figure 10: Conceptual diagram of a reliable conformal classifier facing an OOD input. The desired behaviors are to produce an empty or large set, both signaling uncertainty, and to avoid producing small set that implies false confidence.
The energy-based approach addresses this vulnerability by incorporating a more reliable measure of model uncertainty. The key difference behavior is:
For familiar ID imputs, model confidence is high, resulting in high negative free energy. The prediction sets are thus appropriately small and efficient, similar to the base variant.
For unfamiliar 0OD inputs. the model's uncertainty is captured by low negative free energy. The energy-based reweighting dampens scores so that more classes fall below the calibrated quantile. This results in the predictor generating a large prediction set.
This is clear improvement over the base variant, which is prone to producing small, overconfi- dent sets for such inputs. Therefore, even when the system not configured to abstain, the large prediction set generated by the energy-based method provides uncertainty. It reduces the risk of overconfident and incorrect predictions on OOD data, making more reliable choice in deployments without separate OOD detector. This property validates
robust and honest signal of
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the of free energy model-aware signal that overcomes the limitations of standard softmax probabilities, as noted in prior work (Liu et al., 2020; Wang et al., 2021).
To illustrate the importance of the desiderata outlined in Section 3.3, consider conformal classifier trained to identify medical conditions from chest X-rays. If this model is fed with an image of a completely unrelated subject, such corgi, a reliable classifier must signal its unfamiliarity with the input. As visualized in Figure 10, this signal correctly manifests in two ways:
1. An empty set (0), which communicates: "I don't 't know the answer."
2. A large set (e.g.. ("Tuberculosis', 'Pleural Effusion',, 'Atelectasis', ...)), which communi- cates: "I am uncertain about the answer."
In contrast, producing a small, non-empty set (e.g.. ("Tuberculosis') is misleading, as it incorrectly signals high confidence in prediction that likely wrong.
Our experimental results Section 3.3 confirm this behavior in practice. We observed an increase in the average prediction set size for the energy-based variants of the nonconformity scores.
Q ENERGY-BASED REWEIGHTING VS. ENTROPY-BASED REWEIGHTING
A common measure of uncertainty in a classifier's output is the Shannon Entropy of its softmax probability distribution. As described by Luo & Colombo (2024), let f(x) be the logit vector pro- duced by the classifier for input H, and let *(k x)) be the softmax probability for class k. The Entropy (xr) is given by:
H (x)
#(k: x)) log
#(k| x))
[a(()) log Σ j=1
exp(J}(x))
*(k: x))((z) + log exp(f}(x))
#(k [x)
K
#(k (x) + log exp(S((x))
(59)
If one were to consider an "Entropy-based reweighting" for conformal scores, it would likely utilize this H(x)) function thereof. However, the decomposition in Equation 59 reveals two distinct components influencing the Entropy value. The first term, ΣK #(k (x), depends on the alignment of softmax probabilities with the logit values. The second term is the logsumexp (LSE) of the logits: L(z) log Σ (x)(
This LSE term, L(x), is particularly relevant
directly related the concept of free energy, which forms the basis of our proposed Energy-based nonconformity scores. It captures the overall magnitude or scale of the raw logits. Critically, while the softmax probabilities #(k x)) also depend on these logits, the LSE term L(x) is calculated purely from the logits (z) without #(k x) appearing as explicit factors within its own sum, unlike the definition of entropy.
Indeed, softmax entropy is not well-suited for capturing epistemic uncertainty (Mukhoti et al., 2021). The distinction and potential advantage of using an Energy-based measure over the Entropy (x)))) is illustrated in Figure 2. As shown, both the "easy" and "hard" samples can yield high softmax confidence for the predicted class, resulting in very low Entropy values (close to zero) for both. This suggests that Entropy alone might not adequately distinguish between an input for which the model is genuinely certain (high overall logit values, "easy sample") and an input where the model is less certain overall but still produces a peaky softmax distribution ("hard sample" with high soft- max for one class). contrast, the negative energy scores elearly differentiate these two cases: the easy sample exhibits significantly higher negative energy score (32.49) compared the hard
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sample (14.08). This indicates that the Energy- based metric, by reflecting the overall scale of logit activation, provides a more nuanced signal of the model's 's underlying certainty.
Our proposed Energy-based Nonconformity Scores in Section 2.3 leverages this energy signal by reweighting a standard nonconformity score with coftplus(-F") (x)). The rationale is that F(x) offers more direct and potentially more robust indication of the model's overall certainty about an input & than the Entropy H[)),, which can be saturated (i.e., near zero) for different levels of underlying model certainty. Empirical comparisons supporting the benefits of Energy-reweighted scores over Entropy reweigh--e alternatives is provided in Table 15.
Table 15: Performance comparison of different nonconformity score functions and their Energy-based and Entropy-based variants on ImageNet using ResNet-50 classifier at miscoverage levels = 0.05 and a = 0.1. Results averaged over 10 trials and reported as empirical coverage and average prediction values indicate the best performance within each group. The Entropy-base variants of adaptive method, APS, RAPS, S, and SAPS are defined as Sa v) 8(r.y) and for LAC it is defined as Sn y) +x][x) H(x). H(x)
size. Bold
0.1
0.05
Family
Coverage
Set Size 4
Coverage
Set Sixe
baseline
0.898 = 0.00a
1.487 a 2003
0.950 = noc
2.682 +: 0,099
LAC
w/ Energy
0.898 0.022
1.485 om1
0.949 a noo
2.680 6.403
w/ Entropy
0.898 0082
1.496
0.949 wor
2.696
0.44
baseline
0.899 0000
1.605
0,950 nome
4.007
APS
w/ Energy
0.899 0.00
1,599 a 100
0,950 = nom
3.842 0.19
w/ Entropy
0.898 0.033
2.159 a 0104
0.949 a Do0e
4,990 :: 0,033
SAPS RAPS
baseline
0.898 0.003
1.764
= 0.000
0.949 = 0:01
4.222
0.06
w/ Energy
0.898 0.003
1.763 0.03
0,949
0.001
3,889
0.57
Entropy
0.898 0.00
2.108
1.00
0.949
nome
4,811
baseline
0.898 0.022
1.664 à 0.034
0,949 : 0.00
3.659
6,033
w/ Energy
0.898 0.002
1.662 :: 0.029
0.949
a 0,002
3.654 :: 0.084
w/ Entropy
0.898 0.003
2.101 = 0.009
0.949 0,00
4.702
0.05
R X-CONDITIONAL COVERAGE ANALYSIS
Complementing the class-conditional analysis, we further assess the robustness of our method by evaluating validity conditional on the input features X. While exact conditional coverage is impossi- ble in a distribution-free setting (Vovk, 2012; Foygel Barber et al., 2021), employ the Worst-Slab Coverage (WSC) metric (Cauchois et al.. 2021) to approximate the maximum coverage violation over local regions of the input space. We computed WSC using 100 random projections on the Places365 dataset.
Table 16 reports the Average Set Size and WSC. The results demonstrate that Energy- vari- ants achieve a notable reduction in prediction set size compared to the baselines while maintaining comparable WSC scores. This confirms that the proposed energy-based variants improve efficiency (smaller sets) without compromising local validity or "safety" on difficult slices of the input space.
Table 16: X-conditional performance comparison on the Places365 dataset (ResNet-50) using Worst-Slab Slab Cov- erage (WSC). We compare Set Size and WSC at miscoverage levels 0.1 and 0.05. Lower values are better for Set Size; higher values (closer to a) are better for WSC Bold values denote the hest Set Size within each method family. The Energy- variants reduce set size without degrading conditional coverage.
0.1
0.05
Method
Variant
Set Size J
WSC t
Set Size J
WSC 1
SAPS RAPS APS
w/o Energy
9.02 a.
0.892 am
20.98 0.28
0.949 +.00
Energy
8.54 1..8
0.893 000
18.77 1.0
0.948 1:00
w/o Energy
8.99 0.15
0.892 o.m
21.46
0.27
0.950
w/ Energy
8.47 1 0.18
0.891
0.00
18.65
0.0
0.948
0:00
w/o Energy
8.80 + 0.15
0.891
0.00
21.13
+ 0.29
0.949 0.00
w/ Energy
8.49 0.15
0.891
0.00
18.55
0.34
0.949 0.00
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S STATISTICAL SIGNIFICANCE TESTS
We validate our efficiency gains using a one-tailed Welch's t-test (H1 : (inergy Baseed < (Bass) Ta- bles 17 and 18 report the p-values. Results show statistically significant improvements across most settings, particularly at stricter levels and on imbalanced data.
Statistical significance (p-values from one-tailed Welch's t-test) for the experiments on balanced datasets presented in Table 1, Bold values indicate statistical significance (p 0.05).
Table 17:
0.1 = 0.05 a 0.025 a = 0.01
Method Comparison
p-value
p-value
p-value
p-value
CIFAR-100 (ResNet-56)
APS
w/o vs. w/ Energy
0.398
0.001
0.0001
0.0001
RAPS w/o vs. w/ Energy
0.500
0.0001
0.0001
0.0001
SAPS
w/o vs. w/ Energy
0.500
0.0001
0.0001
< 0.0001
ImageNet (ResNet-50)
APS
w/o vs. w/ Energy
1.000
0.036
0.0001
0.0001
RAPS
w/o vs. w/ Energy
0.268
0.0001
0.0001
0.0001
SAPS
w/o vs. w/ Energy
0.169
0.378
0.0001
0.0001
Places365 (ResNet-50)
APS
w/o vs. w/ Energy
< 0.0001
0.0001
0.0001
0.0001
RAPS
w/o vs. w/ Energy 0.0001
0.0001
0.0001
<
0.0001
SAPS w/o vs. w/ Energy
0.0001
0.0001
0.0001
0.0001
Table Statistical significance (p-values) for experiments on imbalanced data presented in Table 2 (A 0.005) and Table 10 (A = 0.02). Bold values indicate statistical significance (p < 0.05).
a = 0.1 a a= 0.05 a = 0.025 a a= 0.01
Method Comparison
p-value
p-value
p-value
p-value
CIFAR-100-LT (A
0.005)
APS w/o vs. w/ Energy < 0.0001
0.0001
0.0001
< 0.0001
RAPS w/o vs. w/Engyy < 0.0001 < 0.0001
<
0.0001
<< 0.0001
SAPS
w/o vs. w/ Energy 0.0001
0.0001
0.0001
< 0.0001
CIFAR-100-LT (A 0.02)
APS
w/o vs. w/ Energy
< 0.0001
< 0.0001
0.0001
0.0001
RAPS
w/o vs. w/ Energy
0.0001
0.0001
0.0001
0.0001
1000
10000
100001
SAPS
w/o vs. w/ Energy
0.0001
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T DETAILED RESULTS ON IMAGENET
The table 19 provides an evaluation of nonconformity score across 16 different model architectures on the ImageNet validation set. We compare adaptive baseline methods (APS, RAPS, and SAPS) against their respective Energy-based counterparts.
Table 19: Comparison of average prediction set sizes and accuracy for conformal methods on ImageNet at confidence level of 95% (a = 0.05). Lower average set size is better. Set sizes are shown baseline (w/o Energy) + Energy- based (w/ Energy). The bold value highlights the superior (smaller) size within each pair. The results are reported as median means over 10 trials.
Accuracy
Average Set Size (w/o Energy w/ Energy) 4
Model
Top-1
Top-5
APS
RAPS
SAPS
ResNet152
0.783
0.940
3.82 + 3,48
3.25 + 3.17
2.87 + 2,83
ResNet101
0.774
0.936
3.97 - 3.70
3.60 + 3.32
3.15 - 3.03
ResNet50
0.761
0.929
4.09 + 3.97
4.16 + 3.84
3.70 + 3.63
ResNet34
0.733
0.914
+ 8.29
9.94 7.73
9.57 -
ResNet18
0.698
0.891
14.23 12.06
15.16 11.43
14.56 10.93
VGG19
0.742
0.918
8.41 + 7.14
8.85 6.85
8.36 + 6.71
VGGI6
0.734
0.915
8.70 + 7.34
9.06 7.01
8.67 - 6.89
VGG13
0.716
0.904
10.77 - 9.29
11.83 + 8.78
11.60 - 8.58
VGGI!
0.704
0.898
12.36
10.55
13.42 10.09
13.10 9.62
VIT-B/16
0.811
0.953
4.70 4.29
4.10 3.56
3.53 -
VIT-B/32
0.759
0.925
9.62 8.25
8.52 7.28
7.93 6.68
Swin
0.832
0.964
2.79 2.81
3.13 2.87
2.74 2.65
Swin_t
0.815
0.958
3,36 + 3.38
3.64 + 3.42
3.28 + 3.18
EfficientNet.b4
0.834
0.966
5.87 + 4.99
4.87 + 4.28
4.33 - 3,93
EfficientNet..
0.851
0.972
5,93
5.60
5.16 + 5.16
4.86 -
ShufflcNet.v2.11
0.694
0.883
19.17
14.79
19.63
14.48
19.40
14.07
Average
0.765
0.929
7.98 - 6.87
8.02 6.45
7.60 - 6.13
U AI USAGE CLARIFICATION
Large Language Models were used to polish the writing of this manuscript by improving grammar, spelling, sentence flow, and overall readability. All research design, analysis, and interpretation were fully carried out and decided by the authors.
50


Reviews:
[{"review_id": "c8jAsDivIv", "reviewer": "Reviewer_sxcL", "summary": "This paper investigates the limitations of using softmax probabilities as nonconformity scores in conformal prediction and argues that they fail to capture per-sample uncertainty adaptively. The authors propose an energy-based reweighting of the conformity score using the negative Helmholtz free energy derived from pre-softmax logits. The transformation is monotonic and thus preserves marginal coverage while aiming to improve adaptiveness and efficiency. Empirical results show smaller average prediction set sizes and better OOD sensitivity compared to softmax-based baselines.", "questions": "1. What is the formal mechanism linking softmax miscalibration to conformal inefficiency?\n2. How exactly does the energy transformation improve efficiency beyond temperature scaling?\n3.  Can the authors provide conditional coverage or per-group analyses to verify “adaptive” coverage rather than just marginal improvements?\n4. How sensitive are the results to $\\lambda$ and the specific transformation $\\phi(·)$?", "limitations": "", "rating": 4, "confidence": 3, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "1. Addresses a timely and relevant issue in conformal prediction: softmax overconfidence and lack of adaptiveness.\n2. The proposed energy-based transformation is simple, intuitive, and easy to integrate into existing CP pipelines.\n3. Empirical results are consistent across several datasets and show modest but clear improvements in prediction-set efficiency.", "weaknesses": "1. The logical argument connecting “softmax overconfidence” to “inefficient conformal prediction” is not rigorously analyzed; the paper relies mainly on anecdotal intuition rather than formal reasoning.\n2. The claim that “monotonic transformations preserve validity” is standard, and no formal analysis of efficiency or adaptive coverage is provided.\n3. Experiments only report marginal coverage and average set size, lacking conditional coverage evaluation, significance tests, and ablations on the energy transformation ($\\lambda$, $\\phi$).\n4. The conclusion “Softmax is not enough” is overgeneralized; the proposed method is heuristic and may not fundamentally resolve calibration issues.", "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": "W3v9YePxks", "reviewer": "Reviewer_rMqU", "summary": "This paper introduces a novel framework to enhance the adaptiveness of Conformal Prediction (CP), addressing the limitation where standard CP methods rely on unreliable Softmax probabilities for uncertainty quantification. The authors' core idea is to use the Helmholtz Free Energy, derived from Logits, as a more robust metric for sample uncertainty. The proposed method, Energy-Based Nonconformity Scores, is a simple post-hoc step that multiplicatively reweights existing nonconformity scores (like APS, RAPS, SAPS) using the energy score. This reweighting mechanism achieves two goals: (1) Amplifies scores for easy samples, leading to the generation of smaller, more efficient prediction sets. (2) Dampens scores for difficult or Out-of-Distribution (OOD) samples, leading to larger, more robust prediction sets.", "questions": "1.\tRegarding the hyperparameter $\\tau$: In your main experiments (Tables 1, 2, 3), how was the energy temperature $\\tau$ selected? Was it a single value shared across all experiments, or was it tuned for each dataset/model pair? The sensitivity to $\\tau$ seems like a key practical consideration for this method; could you please clarify the overhead of tuning it?\n2.\tThe results in Table 2 show that as $\\alpha$ increases, the improvement in prediction set size when using the energy score becomes more pronounced. Could you explain the reason behind this behavior?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "1.\tImportant and Well-Defined Problem: The paper addresses a recognized, significant problem-the deficiency of softmax probabilities for uncertainty quantification and how this directly harms the adaptiveness of Conformal Prediction. The motivation is clear and compelling.\n2.\tNovel and Principled Solution: Using Helmholtz Free Energy (derived from logits, not softmax) as an uncertainty metric is a novel and theoretically grounded (via EBM framework) idea . This is more robust than relying on heuristics or unreliable softmax outputs.\n3.\tSimplicity and Generality: The method is a simple post-hoc step that reweights existing nonconformity scores (like APS, RAPS, SAPS) using the energy score. It does not require model retraining and can serve as a \"plug-and-play\" enhancement to existing CP workflows.\n4.\tExtremely Thorough and Strong Empirical Evaluation: The experimental design is rigorous, covering the key aspects needed to evaluate an uncertainty method.", "weaknesses": "The proposed score design introduces two hyperparameters, $\\tau$ and $\\beta$. The authors should clarify how these parameters are selected in practice and provide justification for the chosen values.", "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": "h6E7xpWuok", "reviewer": "Reviewer_Xpr8", "summary": "The authors argue that the standard output of a multi-class classifier is not expressive enough for adaptive conformal classification. They propose an input-dependent reweighting strategy based on the Free Energy of the model class probabilities. The new conformity score can be used in combination with existing conformal classification techniques and is shown to reduce the size of the prediction sets.", "questions": "- CP conformity scores are insensitive to monotonic transformations. Why is temperature scaling expected to have bad effects on the size of the prediction sets?\n\n- How is the marginalisation over $y$ defined? Doesn't summing over y in Eq.3 give 1?\n\n- What is the intuitive difference between Free Energy and Entropy? \n\n- How do you compute the normalisation factor, $Z$, in  Eq. 5?\n\n- Does the better separability in Figures 2 and 3 come from removing the logit?\n\n- What are the technical challenges of proving Theorem 2.2? Do you assume the trained classifier is correct?\n\n- Can a sample free energy be used as an unlabeled proxy of the label-dependent difficulty? If so, how is it possible without assumptions between true labels and model-generated probabilities?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "- Reweighting conformity scores is a simple and powerful technique to improve CP adaptivity. The possibility of integrating the proposed strategy with existing approaches makes it potentially relevant and easy to use in many practical scenarios.\n\n- The proposed free energy seems to be a good proxy of an instance's difficulty when labels are not available.", "weaknesses": "- Reweighting approaches are not new and are mostly used for regression. In the classification setup, [1] proposes a reweighting approach based on the entropy of the class probabilities. The authors may comment on the difference between their approach and that technique. \n\n- As the goal is to show adaptivity, the authors should report some approximate measure of models' conditional coverage.\n\n- The authors may give an intuitive explanation of why *model-implied data density* is a good measure of the epistemic uncertainty, for example,  by summarising the arguments of y (Fuchsgruber et al., 2024; Zong & Huang, 2025).\n\n- It is unclear whether the strategy only consists of removing the normalisation and the exponentiation in softmax-based scores.   \n\n[1]\nLuo, Rui, and Nicolo Colombo. \"Entropy Reweighted Conformal Classification.\" The 13th Symposium on Conformal and Probabilistic Prediction with Applications. PMLR, 2024.", "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)