Credal Deep Ensembles for Uncertainty Quantification
Authors: Kaizheng Wang
Fabio Cuzzolin
Shireen Kudukkil Manchingal
Keivan Shariatmadar
David Moens
Hans Hallez
Venue: NeurIPS 2024 poster (2024)
Keywords: neural networks
credal inference
uncertainty estimation and quantification
classification
Primary area: probabilistic_methods

Abstract: This paper introduces an innovative approach to classification called Credal Deep Ensembles (CreDEs), namely, ensembles of novel Credal-Set Neural Networks (CreNets). CreNets are trained to predict a lower and an upper probability bound for each class, which, in turn, determine a convex set of probabilities (credal set) on the class set. The training employs a loss inspired by distributionally robust optimization which simulates the potential divergence of the test distribution from the training distribution, in such a way that the width of the predicted probability interval reflects the epistemic uncertainty about the future data distribution. Ensembles can be constructed by training multiple CreNets, each associated with a different random seed, and averaging the outputted intervals. Extensive experiments are conducted on various out-of-distributions (OOD) detection benchmarks (CIFAR10/100 vs SVHN/Tiny-ImageNet, CIFAR10 vs CIFAR10-C, ImageNet vs ImageNet-O) and using different network architectures (ResNet50, VGG16, and ViT Base). Compared to Deep Ensemble baselines, CreDEs demonstrate higher test accuracy, lower expected calibration error, and significantly improved epistemic uncertainty estimation.

Credal Deep Ensembles for Uncertainty Quantification
Kaizheng Wang Fabio Cuzzolin Shireen Kudukkil Manchingal
Keivan Shariatmadar David Moens Hans Hallez'
KKU Leuven. Department of Computer Science, DistriNet
KU Leuven, Department of Mechanical Engineering, LMSD
Oxford Brookes University, Visual Artificial Intelligence Laboratory "Flanders Make @ KU Leuven
(kaizheng. wang, keivan. shariatmadar, david mavid.moens, moens, hans (fabio. cuzzolin, 19185895) ackuk
Abstract
This paper introduces innovative approach to classification called Credal Deep Ensembles (CreDEs), namely, ensembles of novel Credal-Ser Neural Networks (CreNets). CreNets are trained to predict lower and upper probability bound for each class, which, in turn, determine a convex set of probabilities (credal set) the class set. The training employs loss inspired by distributionally robust optimization which simulates the potential divergence of the test distribution from the training distribution, in such way that the width of the predicted probabil- ity interval reflects the "epistemic' uncertainty about the future data distribution. Ensembles can be constructed by training multiple CreNets, each associated with a different random seed, and averaging the outputted intervals. Extensive experi- ments conducted on various out-of-distributions (OOD) detection benchmarks (CIFAR 10/100 vs SVHN/Tiny- Imaggee,, CIFARIO vs CIFAR10-C, ImageNet vs ImageNet-O) and using different network architectures (ResNet50, VGG16, and ViT Base). Compared to Deep Ensemble baselines, CreDEs demonstrate higher test accuracy, lower expected calibration error, and significantly improved epistemic uncertainty estimation.
Introduction
The quantification of the uncertainty associated with neural network predictions has recently attracted increasing attention, enhance the reliability and robustness of neural networks. Researchers agree to distinguish aleatory uncertainty (AU) from epistemic uncertainty (EU): the former arises from the inherent randomness, e.g., data noise, and irreducible. The latter caused by lack knowledge about the process which generates the data, due to the limited availability of training data, and is reducible [ 36). Effective EU quantification is beneficial for -ittt--distribution (OOD) detection 30 [5] and can contribute to variety of safety-critical applications, including autonomous driving
21, medical diagnosis
44 flood uncertainty estimation (10), structural health monitoring 72
In classification, standard neural networks (SNNs) whose predictions amount to single probability distributions are unable to account for epistemic uncertainty, because single distribution assumes precise knowledge about the dependency between inputs and outputs. To properly capture the EU, the network's outcome needs to express the uncertainty about prediction's uncertainty itself 35 62
The most well-known approach to EU quantification in deep learning leverages Bayesian neural networks (BNNs) 7 22 38). BNNs model network parameters distributions and thus predict "second-order" distribution (i.c., distribution of distributions) [36], although in practice predictions
"Corresponding author.
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

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are often generated by running the network on sample parameters extracted from a posterior. While
efficient training techniques (such as sampling [3 57 and variational inference [2] have been
developed to mitigate their complexity, practical challenges persist for BNNs, including the scaling to large datasets, handling complex network architectures, and real-time applicability (54)
An alternative approach, called Deep Ensembles (DEs), handles uncertainty quantification by aggre- gating multiple individually-trained SNNs [43], so that predictions amount finite sets of probability distributions over the classes. DEs, often viewed as an approximation of Bayesian model averaging (BMA) [38, have become a powerful baseline for uncertainty estimation [2 23 54 60 However, empirical evidence [2] suggests that DEs could yield relatively low- low-quality estimates of epistemic uncertainty. Further, DEs lack a sound theoretical justification 47).
(which predicts convex sets of probability distributions or credal sets [46] on the target space) can provide an alternative way of quantifying epistemic uncertainty. Credal representations have been extensively studied within the broader field of machine learning, including, e.g., the naive credal classifier (14), the notion of credal network [13] or credal random forest classification [65]. 'Imprecise' BNNs have been recently introduced which model both network weights and predictions as credal sets 8 While these models demonstrate robustness in Bayesian sensitivity analysis, their computational complexity is on a par with that of ensembles of BNNs, significantly limiting their practical applicability.
Credal inference 13
36
62
Novelty and Main Contributions. This paper presents an innovative approach to classification tasks called Credal Deep Ensembles (CreDEs), ensembles of novel Credal-Set Neural Networks (CreNets), aiming to improve EU quantification in the framework of credal inference. At the cost of merely doubling the number of output nodes compared to classical SNNs, CreNets are trained to predict a lower and an upper probability bound for each class rather than a single probability value. Such probability intervals over classes thus efficiently determine prediction in the form of a credal set. The training strategy is inspired by Distributionally Robust Optimization [42 55 61, which simulates the potential divergence of the test distribution from the training distribution. As result, the width of the predicted probability interval reflects the "epistemic" uncertainty about the future data distribution. Adopting an ensemble strategy, CreDEs derive the final prediction by averaging the probability intervals outputted by the members of the ensemble. A conceptual comparison between CreDEs and DEs is illustrated in Figure 0
Extensive experimental validation conducted on several OOD detection benchmarks, including CI- FAR10/100 (ID)
(OOD), CIFARIO (ID) vs CIFARIO-C (OOD), ImageNet (ID) vs ImageNet-O (OOD), and across different network architectures: ResNet50, VGG 16 and Visual Transformer Base (ViT Base). Compared to traditional Deep Ensembles, our CreDEs achieve higher test accuracy and lower expected calibration error (ECE) on ID samples, and significantly improve the quality of EU estimation.
Deep Ensembles
Credal Deep Ensembles
Average
Aggregate
SAN-11
-pp
CreNer-1
q
SNN-3
Lighet
D (Crodal Sets Formed by Probabilily Intervals)
SNN-2
(Single Probability Distributions )
CreNeet2
Model Space
Target Spuce Y
Model Spuce
Target Space ?
Figure 1: Comparison between the proposed Credal Deep Ensembles and traditional Deep Ensem- bles. The former aggregate collection of credal set predictions from CreNets as the final (credal) prediction, whereas the latter average set of single probability distributions from standard SNNs as the outcome, F.g.. in the probability simplex [16] associated with the target space Y = { B, D) (the triangle in the figure), a probability vector (qA, (D) is represented as single point. For each CreNet, the predicted lower and upper probabilities of each class act as constraints (parallel lines) which determine a credal prediction (in gray). Single credal predictions are aggregated in Sec. 2.4
Related Work Besides BNNs, DEs, and credal inference, other 'second-order" uncertainty estimation approaches exist, such as Dirichlet-based methods (9 48 (50 63], in which predictions are represented as Dirichlet distributions. One significant challenge for the latter is the absence of ground

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truth labels. Although various loss functions have been proposed, these models' performance often diverges from theoretical EU assumptions 7T. Another rationale for the exclusion of Dirichlet-basse based approaches as baselines for our CreDE work is that such models often necessitate the inclusion of OOD data during training 48
49 56]. This challenges their practical adaptability, as it cannot guarantee their robustness against other forms of 'unseen' OOD data [71]. Moreover, recent study [39] has shown that these methods often fail to capture the EU properly, making the resulting measures difficult to interpret quantitatively.
Paper Outline The remainder of this paper is structured follows. Sec. 2 presents our CreNets and CreDEs in full detail. Sec. [ddecribes the experimental validations and results. Sec. summarizes our conclusions and future work. Appendices report mathematical proofs in { additional experiments in SB] implementation details in O the analysis of alternative ensemble strategies for CreDEs in D and further discussion on
future work in S (including achieving statistical guarantees using conformal learning and the framework's 's extension to regression), respectively.
Approach
The proposed Credal-Set Neural Network architecture and forward propagation are introduced in Sec. 1 CreNets' training procedure is discussed in Sec. 2.2 The class prediction and uncertainty Credal Deep Ensembles are presented in Sec. 2.4
quantification are discussed in Sec. 2.3
2.1 Credal-Set Neural Networks
Architecturally, our CreNet design focuses only on the final classification layers, and can therefore be applied on top of any representation layers of neural network models. The final layers of a CreNet (Figure 2 first output a deterministic interval for each class, using for each class output node associated with the interval midpoint and one associated with its half-length h, respectively (a total 20 nodes for classes). Let be the input vector to the final layer. CreNets compute m and (the vectors col- lecting interval midpoints and half-lengths for all classes) as:
Daltor wentormenggg wedie Interal
Interval MoltMox
m m=g(Wio.z+bic)
(1)
h h=((Wonac 2-oon22--b++200)
where W W1.0g b Wc+1:20, 1 are the weights and biases associated with the first C' and the remaining C' nodes, respectively. Here g(-) is arbitrary activation function and ((-) denotes the Softplus function [79 that ensures the non-negativity of h.
Criddl Probebilty Simplex
Figure 2: CreNet final layer structure for three classes.
The deterministic intervals associated with all classes, denoted [a,, an] == (lar,> au, 1] can then be obtained as [ar, arl (m-h, m-).
proper mapping from such deterministic intervals (ar., atd] a collection of probability intervals [qL, qu:]= (192., qu:1).- for each class needs to ensure that [qL, qu. satisfies the conditions:
QLi qu, Vi=1.... C and Σ OL. S SI<EE qU
(2)
The former condition guarantees a proper [qL, qu.] for each class. The latter enables the resulting collection of probability intervals to determine a non-empty credal set, Q, as follows [52]:
= {q 9: E(g,,, -u. Ec= ==
The probability vectors in Q meet the normalization condition, and their probability value per class constrained by the probability intervals (Eq. (2).
Traditional SoftMax activation cannot ensure that the convexity conditions in Eq. (2) are met when
computing [qL. qu. using qu = =ooftMa((ar) and qu = (oftMax(ar)) respectively. A toy example
3

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is given in Appendix { Therefore, we employ Interval SoftMax activation proposed in 75 to compute qr. qul from ar. arl, as follows:
exp(ar.)
exp(av.)
qLi exp(al,)+ Exx, deg 18L2
qu exp(au, ) E exp( Qtg taLk
(4)
exp(
where qr, and qu; are the lower and upper probability bound for the 7th class, respectively. As proven in Appendix A the probability intervals generated by the Interval SoftMax duly satisfy Eq. (2)
2.2 Training Procedure
The rationale for the training of a CreNet is for the predicted lower and upper bounds (Eq. 4 to the probability of the classes, qr and qu. to express the epistemic uncertainty (induced by the limited size and variability of the training set) about how different the distribution of future test data may be from that the training data.
To this extent, we designed composite loss function with two components: one, which applies classical cross entropy to the upper probability vector, encourages the latter optimistically assume that test data distribution will be similar. The other, inspired by Distributionally Robust Optimization
(DRO) [42 55 61,, pushes the lower probability to reflect 'pessimistic" stance on future distri- butional divergence. The width of the resulting interval will thus reflect the epistemic uncertainty associated with the prediction.
We first contrast the classical training strategy with that of Distributionally Robust Optimization in Sec. 2.2.1
We then delve into the design and implementation of our CreNet loss in Sec. 2.2.2
2.2.1
Classical and DRO Training Strategy
Vanilla Strategy Given a training set D= far,, bn N 1= the conventional neural network training process aims to solve the following optimization problem
Σ n=1 [((En, tn).
minimize BE
(5)
where 0 denotes the model's trainable parameters in the space O and L denotes an arbitrary loss function. The underlying assumption that the training and distributions are identical. As result, the trained network serves as an empirical risk minimizer [36]. However, this ideal assumption often results in optimoptiii predictions because the test observations may, practice, significantly differ from the training data [34].
DRO Strategy In contrast to the vanilla strategy, the objective of DRO 6 20 is to minimize the worst-case expected risk (R(0)) over an uncertain set of distributions 4, as follows:
minimize OEO
R(0)
sup (( VEll
(6)
in which E is the expectation operation. In practice, a group DRO setting [59 is adopted in which the training distribution assumed be a mixture of groups Pg. indexed by g G {1,2, .... m. Because the optimum of linear program is attained at a vertex, the worst-case risk in Eq. (6) is equivalent to maximum over the expected loss each group, as follows:
R(0) maximize P,,,t)pP, C(xx, t). gEG
In practice, the group DRO model minimizes the empirical worst-group risk R(0), namely:
[x(0) = maximiie E (x,t)) C((x ((t,t))))))) 1).01
minimize de
(8)
gEg
where Ps the empirical distribution of the g-th group of training points. Therefore, group DRO learns models with good worst-group training loss across groups [59] One special form of group DRO adversarially reweighted learning (42), which consists of minimax game between a learner and adversary. The learner optimizes for the main classification task and aims to learn the best

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parameters 0 that minimize the expected loss. In contrast, the adversary maximizes the expected loss by making an adversarial assignment of weights wn. collected in a vector w. Consequently, the training optimization problem assumes the form
minimize A
{
maximize WES
1 N
Σ
N n=1 Wi ((( (x)=
where the set S of weight vectors varies across different implementations
42 55 61
2.2.2 CreNet Loss Design and Implementation
Design As anticipated, the CreNet training process applies the vanilla training strategy to the upper probability prediction vector qu (Eq. ) and the DRO strategy to the lower probability prediction qL (Eq. (9)). The resulting overall loss function has composite structure
CereNet:
CE(qU,, tn)
maximize WEB
Wn CE(qLa tn),
(10)
N
Vanilla Component
DRO Component
where CE denotes the classical cross-entropy loss function used in classification. Given a predicted discrete probability vector and the ground-truth label t, CE is defined as: CE:=- > 4a logg Qt- The vanilla component applied to the upper probability vector Qu because such a loss takes the training distribution at face value and is thus more likely to encourage 'optimistic" (overconfident) or "upper bound' predictions for the class scores. The DRO component is computed on the lower probability vectors qL, it weighs training outliers to simulate future differrnnces in data distribution at test time, encouraging 'pessimistic" or 'lower bound' class score predictions. Thus, the width of the resulting probability interval will reflect the uncertainty associated with the model's 's ignorance of how much the future test distribution will differ from the train distribution, using the boundary/outleer cases observed at training time to guess what the uncertainty on future test cases will be.
Cross-Entroon of Lower/Upper Probability Vectors Please note that in Eq. T0 the CE is applied to lower/upper probability vectors, which are not (normalized) probability vectors. However, as the ground truth (label) vector t equals for the true class j and 0 for all the other elements, calculating CE(q, t) for any predicted probability vector q reduces to log, q(j*). Consequently, all probability vectors with the same component for the true class will generate the same CE for that sample.
The consequence for CreNet training is that feeding lower (upper) probability vector qu (qu) to Eq. (TO) is equivalent to computing the CE with any one of the probability vectors the credal prediction (Eq. (3) whose probability for the true class j" equals the lower (upper) probability value there. It can be shown that these form one of the 'faces' of the boundary of the credal set.
Importantly, because of the functional structure of Interval SoftMax activation (Eq. (4)), upper and lower probability vectors are not computed independently, but are correlated. Thus, they minimized together via the total loss (Eq. (IO)), with the DRO component also influencing the upper probability qu, driving the solution away from the trivial one (all-ones upper probability vectors).
Implementation As the S of weight vectors in Eq. (0)) varies across different implementations in Eq. (T0p is not straightforward when using batch-wise training 34], we resort to a simpler heuristic proposed by [34]. For each training batch, only the 5 [0.5, 1) portion of samples with the highest cross- cross-entropy with the lower probability vector (CE(qr... tx))) are selected to compute the DRO component of the loss. As a result, Wn, > is implicitly set in Eq. (10) for selected samples while w., = for deselected samples.
42 (55 61 and estimating
The underlying rationale is the following. Within a batch of samples, those instances that demonstrate high losses are identified 'hard-to-learn' samples, essentially representing the 'minority group' within a training dataset [3]] Setting a value å thus identifies what fraction of the training points is chosen to represent potential future domain shifts at test time. A smaller 8 signifies more cautious approach, in which even a few training outliers can indicate future challenges.
The lower bound to the design range for 8 is 0.5 because we empirically observed that values of 5 < 0.5 may destabilize the training process, as a too-large averaged loss is returned for backpropagation. When 8 approaches 1, the data distribution of the samples considered by the vanilla and the DRO components of the loss becomes similar, implicitly assuming a less pronounced divergence between train and test distributions. The corresponding predicted probability intervals become narrower. If 8
5

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were theoretically set to 1, all samples would be selected for backpropagation, implying that w, for any n in Eq. (TO. Consequently, the loss Eq. (TO) would be the sum of the vanilla component on qu and the vanilla component on qr.- Empirically, we observed that this leads to collapse of the upper and lower probability bounds to single values.
The implementation of the CreNet training procedure is shown in Algorithm
Algorithm 1 CreNet Training Procedure
Training dataset D (mo; In kn)=-n Portion of samples per batch de [0.5, 1); Batch size ! while enable training do
Input:
1. Compute CE(qU,, tn)) and CE(qL,, tu) for each sample
2. Sort the sample indices (m1,..,,M.,) in descending order of CE(qLn. bx)
3. Define γ) = [ôn]
4. Minimize LCieNet 7 E CE(qU,, tn)+ do E CE(QLmy tmyy
end while
2.3 Class Prediction and Uncertainty Quantification
Class Prediction For the class prediction we employ the 'maximin' and 'maximax" criteria 666
min =- arginxx qL.5 imax :- argmax q0
(11)
which output (respectively) the class indices with the highest lower and upper reachable probability (4i, and qū: within the same credal set induced by the predicted lower and upper probabilities qr, QL,-UU,- Figure illlsttrats how the lower and upper probabilities qI, Qt, that determine the credal
set Q may differ Trom the probabilities qi., and qú, actually reachable for each class within Q. The reachable lower and upper probabilities for class can be easily obtained as follows [17
.i. Wü=min qUi 1 -E 91. max QL.: 1
- qUi j
(12)
Uncertainty Quantification Given credal set prediction, upper and lower entropies generalizing Shannon's entropy, denoted as ((Q) and H(Q), be defined which may serve measures for TU and AU, respectively S 36).
Computing (() boils down solving the following optimization problem:
Σe
C カ=
=(Q))maximixe
1=1 -a- logl qi L.. S. S nú, Vi and
(13)
This seeks the highest entropy value of probability distribution within the predicted credal set Q H(Q), for which maximize is replaced by minimize, searches for the minimal such entropy. Such optimization problems can be addressed using standard solver, e.g., the SciPy optimization package
73
Epistemic uncertainty can then be quantified as ((Q) HQQ) 36)
Computational Complexity Reduction To reduce the computational complexity of Eq. (3) for large value of C' (e.g.. C²= 1000), we propose original approach called Probability Interval Dimension Reduction (PIDR) in Algorithm [ This method first identifies the K-1 classes with the highest lower probability values, then merges the remaining elements into a single class with the associated upper and lower probability calculated using Eq. (12). Consequently, the dimension the probability interval is reduced from C' to K.
2.4 Credal Deep Ensembles
Inspired by conventional DEs 43 the final step of our approach to introduce Credal Deep Ensembles (CreDEs). CreDEs aggregate M individually trained CreNets and predict the aggregated probability intervals, denoted as [ 97 as follows:
e M Σ
Li..
m Pum.
(14)
where (q..., , Ti is the set of reachable probability intervals predicted by the m-th CreNet. Eq. (20 in Appendix D proves that [ q satisfies the convexity condition in Eq. (2) for constructing

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Algorithm 2 Probability Interval Dimension Reduction Algorithm
Input: qi. qú-: Chosen number of classes K
probability intervals [rr, rvll 1. Index vector of q.. descending order: t L-aggsort (
Output:
2. Define the upper and lower probability per selected class:
L, + - -u, Eqi, for jo =.,,.., K -
Figure 3: intervals are redun- dant, some of the (e.g.) upper probabilities qUA may not be ac- tually reachable in the credal set that results from the intersection of all interval constraints.
3. Define upper and lower probability for deselected classes:
TLK f max (1- -ie Li=lee Tú, - TL,);
T0x min(minn1))) le Ei AL, E5=1 TU,)
non-empty credal set. Therefore, class prediction and uncertainty estimation as described in Sec. 2.3 apply to CreDEs. We discuss the rationale for averaging strategy and the alternative ensemble approaches for CreDEs in Appendix $D
3 Experimental Validation
Setup We assessed CreDEs through OOD) detection benchmarks across various dataset pairings (ID OOD samples). including CIFARIO /TTCIIARI00 40] vs SVHN Bi//iinyImageeer
45). CIFARIO vs CIFARIO-C [29]. ImageNet 18 vs ImageNet-O [3]] We trained 15 CreNets (using 8 0.5) and SNNs on the ResNet50 architecture 28 starting from different random seeds, using the training set as per the ID dataset in the pair. Following this, we constructed 15 dif- ferent CreDEs and DEs, respectively, by randomly selecting five members from the pool of 15 trained models. The same ensemble member lists are used for both DEs and CreDEs, with each ensemble strictly guaranteed to be distinct. More details are given in Appendix $C Codes are available at https://gitlabb htt.ttall..uulleuuu kuleuven be/m- group- cacnppus-buggee wtwpss/bbbbebeeeseeeteee.oee http//irittettuuiiiiieee -redal--eep--nnsebll
git
Uncertainty Quantification in DEs Total uncertainty (TU) can be quantified in DEs via the Shannon entropy (H)) of the averaged predicted distribution. The AU, on the other hand, can be obtained by averaging the entropies of the predictions of each ensemble member [2 33). Namely,
M TU: (q) EM_ qm), AU:=ii(q)= EM ((g)),
(15)
where M is the number of networks, and qm denote the average probability vector and the single probability vector of the m-th SNN model, respectively. The level of epistemic uncertainty,
representing an approximation of mutual information (36), can be obtained as EU: (
Test Accuracy and ECE on ID Samples We evaluated the test accuracy and expected calibration error (ECE) 24 58 of CreDEs-5 and DEs-5 on the test set of each ID dataset. lower ECE value signifies a closer alignment between the model's confidence scores and the true probabilities of the events. Since ECE is designed for a singular probability vector, we implemented compromise calculation as follows. Suppose our model predicts the class indices k and j when using the imin and immx criteria, respectively, ECE values are then computed based on the associated lower qt, and upper 98, reachable probabilities in the credal set.
Table rrports the test accuracy and ECE for DEs-5 and CreDEs-5 on the various datasets, indicating that our CreDEs-5 achieved higher test accuracy and lower ECE on ID samples. Note that employing the min prediction showed higher ECE on the challenging ImageNet dataset. This is likely because the strategy, selecting the class with the highest lower reachable probability, is conservative one.
Table I: Test accuracy (%. 1) and ECE (4) of DEs-5 and CreDEs-5 using CIFARI0, CIFARI00, and ImageNet as ID datasets over 15 runs. The better performance marked in bold.
CIFARIO
CIFAR100
ImageNet
Test Accuracy
ECF
Test Accuracy
ECF
Test Accuracy
ECE
DEs-5
93. 13
100000000000
80
0.039240.0027
77.9250.02
0.241 540.0009
CreDEs-5
93.7520.11
0.0092:0.0016
79.542021
0.0366:90.025
78.41+9.02
0.593020.0006
(Ours)
93,7420.11
0.010820.0017
79.6520.19
0.026829.023
78.5129.02
0.168520.0004

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Table 2: OOD detection AUROC and AUPRC performance (%, 1) between CreDEs-5 and DEs-5 based on ResNet50 using EU uncertainty metrics CIFARI0/100 vs. SVHN/Tiny-Imagee ImageNet and ImageNet vs. ImageNet-O. Results are averaged over 15 runs. Best results in bold.
ID Samples
CIFARIO
CIFAR100
ImageNet
OOD Samples
SVHN
Tiny-ImageNet
SVHN
Tiny-ImageNer
ImageNot-O
Performance Indicator
AUROC
AUPRC
AUROC
AUPRC
AUROC
AUPRC
AUROC
AUPRC
AUROC
AUPRC
DEs-5
(())-B(() 99.588093 92.29=1.00 oo 86.87a0.20 20 83 .2.0220... 1e (73883) 997 84, 25 a5 20000.02 74, 4468860 27 (65 03 62.77a0.38
CreDEs-5 (Q) 96.55t0.25 98.1720.17 88.1020.26 87.85+00.5 778.55e1.15 86.5779.65 (2.55222.. 77.60200.4 67.8220.06 62.80:0.12
EU Quantification for OOD Detection It is our hypothesis that OOD data express a higher EU. Hence, we can use better EU quantification as the means to improve the OOD detec-
tion [54]. Thus, superior OOD detection performance provides compelling evidence of en- hanced uncertainty estimation quality. For the OOD detection performance assessment, we employed AUROC (Area Under the Re- ceiver Operating Characteristic curve) and AUPRC (Area Under the Precision-Recal curve) scores. AUROC captures true and false positive rates, while AUPRC assesses precision and recall trade-offs, offering valu- able insights into model effectiveness across various confidence levels. When calculat- ing H(Q) and H(Q) in the ImageNet vs ImageNet-O experiment, we employed our PIDR Algorithm 2 with K 20. Table
AURAC
BURRO
0.99
0.90
20.89
0.80
Overall Compption Intensity
Overal Cormuptin intersity
H(0) -R())
Figure 4: OOD detection (CIFAR10 vS
CIFARIO-C) over increased corruption intensity.
reports the OOD detection performance of
WVHN/TinyIImage/e and ImageNet vs ImageNet-O settings. As the CIFARIO-C dataset contemplates data from CIFAR 10 corrupted in 15 distinet ways, each with 5 different intensities, Figure presents averaged AUROC and AUPRC scores for OOD detection on CIFARIO vs CIFAR10-C across types of corruption, against the intensity of corruption. Table and Figure 4 confirm CreDEs-5's superior OOD detection performance over DEs-5. This indicates the effectiveness of CreDEs in improving the EU quantification quality, using ((Q)= - (Q) as the uncertainty measures.
CreDEs-5 and DEs-5 in the CIFARIO/CIFARIOS
Qualitative Evaluation Due to the high dimensionality, visualizing directly computing the size of the credal set becomes challenging as C' increases. Consequently, we indirectly evaluate whether our CreDEs consistently generate nearly Dirac credal sets predictions through the maximum attainable upper bound probability of the prediction. The closer this probability is to 1, the more it approximates Dirac credal set. Figure 5 shows the results of ResNet50-basee CreDEs-5 for the CIFARIO, SVHN, and
datasets. It verifies that our method does not consistently generate nearly Dirac credal sets, especially for OOD samples. For CIFARIO. substantial proportion of (but not all) the credal sets are quasi-Dirae. This observation is reasonable as it is consistent with the high test accuracy of CreDEs and the low ECE reported in Table 1
6000
CIFARIO
750
10000
ImapeNet
4000
500
2000
250
Probability
it value
Probabiitty
Figure Maximum reachable upper probability max (qū, l.a) per sample from 15 runs.
Additionally, Figure shows the reliability diagram of the ResNet50-based DEs-5 and CreDEs-5 on the CIFARIO dataset, demonstrating better calibration performance of our CreDEs. Figure showcases the EU estimation plots for these models. Although the EU estimates for DEs-5 and CreDEs-5 are not directly comparable due to differing representations, CreDEs-5 demonstrates significantly higher EU estimates for OOD samples, observed qualitatively.
Ablation Study on Various Network Architectures We also performed an ablation study on network backbones different from ResNet50, including VGG16 (67] and Vision Transformer Base (ViT Base)
8

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Frequency
0.8
1,4582-10"
0.3-
0.8-
20
0.6
0.6
Accuracc
0.6-
0.0
0.2
0.4
estimates of Cre
Frequency
0.2
0.2
0.2-
Theperage (D0A)
0.0
0.0
0.6
Confidence
Cont
Confidence
estimates DES:5
Figure 6: Reliability diagram of ResNet50-based DEs-5 and
Figure 7: EU estimates compari- son of ResNee50-bbsed models,
CreDEs-5 (using Emin and Emaz. respectively) CIFARIO.
77. Table 3 reports the test accuracy and ECE of CreDEs-5 and DEs-5 on the CIFARIO test set (representing ID samples) and the OOD detection performance on CIFARIO vs SVHNTTnn--mageeN Figure X compares OOD detection performance in the CIFARIO vs CIFARIO-C setting against the intensity of corruption, using both AUPRC and AUROC as metrics. The results consistently demonstrate that CreDEs achieve higher test accuracy. lower ECE. and significantly improved epistemic uncertainty estimation, leading to enhanced OOD detection performance.
Table 3: Test accuracy (%, 1) and ECE (L) of DEs-5 and CreDEs-5 CIFARIO ID dataset (left).
AUROC and AUPRC scores (%, T) for OOD detection on CIFARIO vs SVHN/Tiny-Ima -mage-et (right).
Results averaged over 15 runs. The Best results are in bold.
CIFARIO (T))
CIFARIO SVHN
CIPAR vs Tiny-Imagehe
Test Accuracy
ECE
AUROC
AUPRC
AUROC
AUPRC
DEs-5
85,5340.10
0,0815200011
((())
82,19
87,524081
78. 78
73.2840.23
VGG16
CreDFs.5
Juin
87.94H0.11
0.0203H0.014
(7(0)-1(1)
87.6840.73
93.47
82.56+0.28
80.81
(Ours)
87.9240.11
0.061110.0012
DE
90.414097
0.0181
(1 (())
77.71
88.7360
82.27700.9
78.854081
Base
CreDEs
Amin
93.6011040
0.010718.0014
((Q)-
88.57=2.08
93.24
88.73
87.84+9.52
(Ours)
93.59t0039
0.0104
Rugraiee AUPRC
Bvereggu AUPAC
0.95
Batrigged Autce
0,95
3.80
84
82
0,90
0.70
0.80
Overal Corruption Intensity
Overal Corruption Intensity
Overall Comuption Intensity
Overall Comugttinn Intensity
VGG16
VIT Base
(00)(0m))
Figure 8: OOD detection on CIFARIO vs CIFARIO-C against increased corruption intensity, using VGG16 and VIT Base as backbones.
Ablation Study on Hyperparameter for CreNet Training In our main evaluation, we set by default 5 = 0.5 to reflect a balanced assessment of the train-test divergence and show how such a value allows our model to outperform the baselines. Table T reports the test accuracy and OOD detection performance (using EU estimates) of CreDEs-5 under various values of 6. The
ablation findings verify the robustness of Cre- DEs across hyperparameter setups and indicate the 0.5 might be too pessimistic choice in CIFAR10 settings. Performance peaks at == 0.875 most cases, implying that 5 5= 0.875 may provide the 'optimal' estimate of how test and train sets diverge for CIFAR1O. One pos- sible way find the 'best' 8 in practice is to conduct standard cross-validation on specific test scenarios. However, the method is not par- ticularly sensitive to this hyperparameter. Per- spectively, interesting option, in the presence of multiple datasets (e.g.acquired over time in continual learning setting), could be applying the DRO loss component to different components of the training set, and assessing the results robustly select 8.
Table 4: Test accuracy (%, 1) and OOD detection performance (%, 1) of CreDEs-5 using various Results are averaged over 15 runs.
0.5 0.625 0.75 0.875 0.9375 0.96875
Accuracy (CIFARIO)
frne
93.74
94.54
93.88
93.99
fuis
93.75
94.55
94.47
94.56
93.87
93.99
AUROC
97.44 97.92
97.42
(OOD Detection) AUPRC
93.07 96.34 97.00 96.92
98.79
98.82
Tlay-InageNet
AUROC 88 89.01 89.10 89.18
89.85
89.24
(OOD Detection) AUPRC 88.13 89.81 89.76 89.72 89.18
89.26
We also report the average EU estimation values of CreDEs-5 for each dataset in Table 5 In- creasing the value of d (i.c., giving less importance to the divergence between test and training

---

distributions) leads to decreasing trend in the average EU estimates per dataset (particularly for ID CIFARIO samples). This aligns with the intuition that, if the model is more uncertain about the divergence of the distributions (smaller 8), it should express a larger FU. Despite smaller uncertainty values at high 8's, the difference between ID and OOD samples remains notice- able. This explains why a 8 closer to 1 does not always lead to low-performance OOD de- tection and why model's OOD detection performance robust against the choice 8.
Table 5: Averaged EU estimates of CreDEs-5 using
various 8.
5
0.5
0.625
0.75
0.875 0.9375 0.96875
CIFARIO
|0.3557 0.0611 0.0586
0,0572 0.0222 0.0215
SVHN Tiny-ImageNed
1.6093
0.2450
0,2553
0.2593
0. 0.1612
0.1574
1 1.49%6 0.2030 0.1957 0.1970 0.1025 0.1005
Model Inference Complexity rrepors the parameter count and inference cost on one NVIDIA 100-5XM4440GB GPU for single SNN and CreNet on ImageNet. CreNets show a marginal increase in complexity due to its minor architectural modifications. More discussions on the inference and training complexity are presented in Appendix $
Additional Experiments Appendix $B.1 discusses the implementation and performance of EU quantification in CreDEs when using a different uncertainty measure for credal sets (the generalized Hartley measure 4 36)). The results demonstrate that our CreDEs consistently enhance the quality of EU quantification, Appendix re.2ott an ablation study for the hyperparameter K of our PIDR Algorithm 2 which shows the effect of K on CreDFi''s uncer- tainty quantification and time cost. Appendix
exhibiting robustness against different measures. Table Model complexity of ResNee55-bbasee SNN and CreNet on ImageNet dataset.
Model
Parameters (million) Inference time per sample (ms)
SB.3 assesses the ability of CreDEs to evaluate total uncertainty (TU) (as opposed to EU) in OOD detection, suggesting that our CreDEs also achieve an improved TU estimation, compared to DEs. Appendices $B.4, B.5 and BB.6 compare the uncertainty quantification abilities of CreDEs versus those of traditional DEs that also apply the DRO strategy, DEs that apply the 'product of experts' (32) ensemble setting, several BNN baselines, respectively. CreDEs continue to demonstrate superior performance in uncertainty estimation. Ap- pendix $B.7 assess CreDEs in case study involving active learning [23] [54]. All these additional experiments demonstrate that our CreDEs deliver improved uncertainty quantification.
vs CreNer 25.557 27.606
5.5 vs 5.740.3
4 Conclusion
Conclusion In this paper, we introduced novel Credal-Set Neural Network (CreNet) for classi- fication tasks. Given any given input instance, CreNet is designed to predict lower and an upper probability for each class, rather than single probability value, thus providing an efficient and effective implementation of credal inference. We also proposed Credal Deep Ensembles (CreDEs), ensembles of CreNets, which extend the traditional deep ensemble idea the credal domain. Ex- tensive experimental validation was conducted on several OOD detection benchmarks, and across different network architectures and uncertainty measures. Compared to traditional Deep Ensembles, our CreDEs achieve higher test accuracy and lower ECE on ID samples, while significantly improving the quality of EU and TU estimation, leading in turn to strongly enhanced OOD detection perfor- mance. Hence, we believe our work can potentially improve neural network safety and reliability, and have wide applicability to real-world scenarios such as medical image analysis.
Limitation Despite the superior performance of CreDEs, neither traditional DEs nor CreDEs may be desirable when memory usage stringent and computational resources are limited.
Future Work Three essential objectives of our future research include elaborating on statistical coverage guarantees of our CreDEs (outlined in Appendix $E.2). extending our framework to regression tasks (a roadmap is provided in Appendix }E.3), and assessing our CreDEs alongside other uncertainty-aware models in real-world applications comprehensively. like medical image analysis.
Acknowledgement
We thank the anonymous reviewers for their valuable feedback. This work has received funding from the European Union's Horizon 2020 research and innovation program under grant agreement No. 964505 (E-pi).
10

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A Mathematical Proof
Toy Problem for Unavailability of Traditional SoftMax The traditional SoftMax activation function cannot be used to define the credal set, it cannot ensure the condition in Eq. (2) when
computing (qL, Qu- as qT. - SootSoooMaaaee (ar.) and qU = SoftMax (ar)) respectively.
For example, assuming that we have ap :- (-1, 0, 1) and au := (0, 1, from CreNet, the qu and qu computed using SoftMax are:
qr So = (0.0900, 0.2447, 0.6653), qu SSooMMax(((a)) (0.0420, 0. 0.1142, 0.8438).
The resulting 'probability intervals' are not properly defined and appear unreasonable, as some lower does satisfy
bounds are considerably higher than the upper bounds.
Mathematical Proof for Interval SoftMax The proof that Interval SoftMax in Eq. the conditions in Eq. 22 is straightforward:
exp(ar,)
exp
au, au,tol; 2
t=1
q
i=1
exp( 2u to + exp(a2,)
1=1
k
exp
RU to
) + expl 2
a2i tal,
(16)
exp(av;)
974
exp(
on toL2 2
+ explav,)
1=1
1=1
B Additional Experiments
In this section, Appendix BB.1 discusses the implementation and performance of EU quantification of CreDEs using another uncertainty measure for credal sets: the generalized Hartley measure. Appendix B.2 performs an ablation study on the value of the hyperparameter K of the PIDR Algorithm Appendix $B.3 assessed CreDEs' performance in the OOD detection task when quantifying uncertainty using TU instead of EU. Appendices $B.4 |B.5] and comprre CreDEs versus traditional DEs that also apply the DRO strategy, DEs that apply the 'product of experts' ensemble setting [32], and several BNN baselines, respectively. Appendix $B.7 assessed CreDEs in an active learning case study.
B.1 Generalized Hartley Measure for EU Quantification of CreDEs
Uncertainty quantification in credal sets merits further investigation. For instance, recent research 35 has explored, e.g., the use of probability interval length measure of epistemic uncertainty, in the special case of binary classification. However, these measures cannot be readily extended to multi-class cases. Recently, the most established methods for decomposing the total uncertainty of credal sets are generalized Entropy
B 36 and the generalized Hartley Measure 36).
The generalized Hartley measure [4] GH(Q). measures the non-specificity across the distributions in the credal set, and can be seen as a proxy for volume [36). Mathematically. GH(Q) calculates the expectation of the Hartley measure 26 over all possible subsets B on the target space
Definition
as follows [4]
GH (2) = E mQ (BB) (og) (B)),
(17)
in which me denotes the mass assignment function associated to Q and [B] indicates the cardinality of B. MB) can be computed using the Möbius inverse of the capacity function 12 (36), as follows:
mQ (B) E-)
(18)
where B\ = (vly E and y & A} and vi describes the lower probability of all possible subsets A C B.
Efficient Implementation in CreDEs One of the reasons that hinder the application of the Gen- eralized Hartley Measure is its computational complexity. In our work, we proposed an efficient implementation and an approximate approach (the PIDR Algorithm 2 for computing it.
In classification, target space Y comprises finite of class labels, namely Y = (B1, yc).
16

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In our case, the lower probability (A(A) associated with the predicted credal set can be readily computed as follows:
( (Euza - vyeer 9ü,
(19)
where q7. and qi, are the reachable lower and upper probability values per class in the defined credal set. They can be easily obtained from Eq. (12). Figure [ illustrates how the lower and upper probabilities qU, that determine the credal set Q may differ from the probabilities qi, and qü, actually reachable for each class within Q
The full (H( calculation process is presented in Algorithm
Algorithm 3 (H) Calculation
Input: [9/. Tú (122, &. Target space W
Output: GH(
Initialize: GH(Q)=1 =0
for all B S Y and [B] 2 do
Initialize: mg(B) for all A C B do
Compute (A) using Eq. 19 ma (IB) (-) (A) (Eq. 18)
end for (Q( ( + log. (B)) (Eq. 17
end for
Although the use of probability intervals simplifies the calculation of GH(Q) ( in general, significant challenge arises for large values of (c.g. C'= 100) due to the complexity of involving subsets C.
However, when applying proposed PIDR Algorithm 2 the dimension of the probability interval reduced from C to K; therefore, calculating GH(Q) requires only 2 subsets.
Experimental Validation of GH(Q) for OOD Detection The OOD detection results for CreDEs-5 using GH(Q) are shown in Tables 7 and in Figure Probability Interval Dimension Reduction (PIDR) (Algorithm 2 is utilized with settings K 4 and ==10 when computing GH(Q) for dataset pairs containing CIFARIO and CIFARI10//mageen
respectively. The results verify that
* Our CreDEs consistently enhance the quality of EU quantification, exhibiting robustness across different uncertainty measures, i.e., the generalized Shannon entropy and the gen- eralized Hartley measure. This improved EU quantification leads to better OOD detection performance compared to Deep Ensemble baselines.
The proposed PIDR algorithm ensures an efficient implementation of the generalized Hartley measure in our framework. An ablation study on PIDR's 's hyperparameter is conducted in Appendix B.22
Note, however, that applying K= 10 for the setting ImageNet vs ImageNet-O does not yield better result, due to the coarseness of approximating 1000 classes using only 10. This suggests that computing GH(Q) is still challenging for tasks involving 1000 or more classes.
B.2 Ablation Study on Hyperparameter of PIDR Algorithm
Effect on ( Quantification Figure 10 illustrates the influence of various settings of K on
GH(() quantification. The average value GH (Q) suggests that the use of the PIDR Algorithm 2
results in an underestimated (H) value, compared to the result without using PIDR (K = 10).
Consequently, increasing the value of K enhances OOD detection performance. However, as K
grows execution time increases exponentially, due to the iterative calculations of m (B) and (A(A) in
Algorithm across 2 subsets. The time cost is measured on a single Intel Xeon Gold 8358 CPU@2.6 GHz. While higher than the time cost for EU calculation of DEs (4.1e-4 ms), this figure shows that calculating GH for 10 classes (K = for 17 ms) remains practical without actual computational constraints. Besides, the numbers reported are for GH calculation without any optimization: more
17

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Table 7: OOD detection AUROC and AUPRC performance (%, 1) between CreDEs-5 and DEs-5 based on ResNet50 using EU as uncertainty metrics on CIFARI0/100 vs. SVHN/Tiny-Imagee and ImageNet vs. ImageNet-O. Results are averaged over 15 runs. Best results are in bold.
ID Samples
CI
CIFAR100
ImageNer
OOD Samples
SVHN
Tiny-ImageNet
SVHN
Tiny-ImageNet
ImageNet-O
Performance Indicator
AUROC
AUPRC
AUROC
AUPRC:
AUROC
AUPRC
AUROC
AUPRO,
AUROC
AUPRC
DEs-5
(q)
5840099 92.29H1.00 oo) 86.87 20 83 0260 1e 73. 8365) 997 84 26 8000.00 74 14.6840.22 27
65 03:0.53
62. 02.7740.35
CreDEs-5
QQ) ( 96.55+0.25
98. 17.117777788. 88.1020.26 87.85+0.35
788.551115 86.57#8.65 (2254420.26 77.69=0.44
67.82:0.06
62.80
(ours)
(H
96.7220.24 98.2520.17 89.5420.16 88.7420.24
79.23e1.19 87.1729.66 N3.0120.24 78.95e0.44
63.462205
58. 13
Table 8: AUROC and AUPRC scores (%, 1) for OOD detection on CIFARIO SVHN/Tiny- ImageNet. Results averaged over 15 runs. The Best results are in bold.
CIPARIO SVHN
CIFARIO Tiny-ImageNer
AUROC
AUPRC
AUROC
AUPRC
DEs-5
(())
1920.2
87.524081
78.5880.15
73.2840.23
VGG16
CreDEs-5
H(Q)-H(()
87.68+0.73
93.4750.57
82.5660.28
(Ours)
GH(Q)
86.99=9.72
93.18t9.41
82.2330.18
80.83:9924
DEs-5
H(4)
77.71#1.67
88.7340033
82.27:0079
78.8510.8
VTT Base
CrDDss-5 (Ours)
((Q)-H(0)
88.57a2.06
93.2441.25
88.73.0.32
87.8429.52
GII(Q)
89.07711.66
93.32+1106
89.199t0.4
88.21
Averaged AUPRC
0,95
Averaged AUPRC
0.85
Averaged AUPRC
0.95
0.90
0.80
0.90
0.75
0.85
0.85
0.70
0.80
0.80
0.65
4
Overall Corruption Intensity
Overall Corruption Intensity
Overal I Corruption Intensity
0.96
0.86
@veraged AUROC
Averaged AUROC
Averaged AUROC
0.94
0.84
0.92
0.82
0.90
0.90
0.80
0.78
50.85
Overall Corruption Intensity
Overall Corruption Intensity
Overal 1 Corruption Intensity
ResNet50
VGG16
VIT Base
H - ((())
GH(Q)(0um))
H(@)
86g)
Figure 9: OOD detection CIFAR10 CIFARIO-C against increased corruption intensity, using ResNet50, VGG16, and ViT Base as backbones.
18

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17
Averaged Time Cost (ms/sample)
0.07
0% MO: CIFARIO
InD: CIFRRO)
0.06
Averanees CH C Der sant
12
CIFARIO
SVHN
0.05
8
Tiny imgeNNet
0.04
8 8
90
1%
SVHN
AUROC
Averagen AUPRC
SVHN
Tiny-ImageNet
0.03
Tiny-ImageNe
0.02
Aver Averages seee
83
0.02
8
Applied
[K]
Figure 10: Average time cost of (H(Q) (black dotted line) and GH( value per sample across
various datasets (blue lines), along with the AUROC/AUPRC scores (green/purple lines) for OOD detection versus increasing values of K.
efficient code implementation could significantly reduce the cost. The effect of various settings of K on H (QQ) shown as follows.
Effect on TU Quantification In this experiment, we examine the effect of K on TU estimation.
Figure 11 shows the average values of TU estimates (H()) sample, together with the AUROC and The results indicate that applying PIDR (Algorithm{) tends to underestimate TU values. Consequently, increasing the value of K improves the OOD detection performance, but it also leads an increase in execution time. This is because solving the constrained optimization problem in Eq. (3) involves more variables and constraints.
AUPRC scores for CIFAR 100 SVHN/Tiny-Iageeh
The reported time cost is measured single Intel Xeon Gold 8358 CPU@2.6 GHz, without optimization in the calculation process. We believe a more efficient code implementation could significantly mitigate this.
Time Cost (melsample)
100
107
Tora uncertainty Estimation
AUROC/AUPRC Scarms 1%6)
88
Tiny- imageNet
SVHNAURDC
AUROC
SVHNAUURC
Tiny-mageNNtt
AUPRC
3-
72
100
Applicd
Applied
Applicd
Figure 11: AverageH (Q) time cost, average TQQQ) value per sample, and OOD performance on the OOD detection benchmark (CIFARI00 SVHN/Tny-ImageNe)) for increasing values of K.
B.3 Total Uncertainty Estimation Evaluation via OOD Detection
In further additional experiments, we also assess the quality of the total uncertainty (TU) estimates produced by CreDEs-5 on the various OOD detection benchmarks (4,554) The results Tables 9 and 10 consistently demonstrate CreDEs' improved OOD detection performance using TU as metric.
B.4 Comparison between CreDEs and DEs with DRO Strategy
In this experiment, we additionally train 15 R based SNNs on the CIFAR1O and CIFAR100 datasets using the DRO loss component For fair comparison, we set 8 0.5 (just as for CreNets), using the same random seeds and training epochs. Other training configurations are described Appendix C We named the resulting deep ensembles with 5 ensemble members D-5
same training strategy as in Algorithm | respectively.
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Table 9: OOD detection AUROC and AUPRC performance (%, 1) between CreDEs-5 and DEs-5 based on ResNet50 using TU as uncertainty metrics on CIFARI0/100 vs. SVHN/Tiny -eeaeee and ImageNet vs. ImageNet-O. Results are averaged over 15 runs. Best results in bold.
Samples
CIFARIO
CIFAR100
ImageNet
OOD Samples
SVHN
Tiny-ImageNer
SVHN
Tiny-ImageNe
ImageNet-O
Performance Indicator]
AUROC
AUPRC
AUROC
AUPRC:
AUROC
AUPRC
AUROC
AUPRC
AUROC
AUPRC
DEs-5
(())
94. 83 97, 27.46600.29 88, 8040 19 87. $7.21 78.53e1 88.83+101 80, 75 66.6.65... 19 50.20=0007 50,4340.06
CreDEs-5
(
95.71=0.22 97.73+9.27 89.02+0.10 88,02+0.15
79,44 88. 1050.79 83,49H0.17 80.61#0333 67.55 62.79+9.188
Table 10: OOD detection AUROC and AUPRC performance (%, f) between CreDEs-55 and DEs-5 based on VGG16 and ViT Base using TU as uncertainty metrics on CIFAR10 vs. SVHN/Tiny- ImageNet. Results are averaged over 15 runs. Best results in bold.
VGG16
VIT Base
SVIN (OOD)
Tiny ImageNet (ODD)
SVHI (OOD)
Tiny Mmyy-ImaeNNet (OOD)
AUROC
AUPRC
AUROC
AUPRC
AUROC
AUPRC
AUROC
AUPRC
H(q)
84.5010.99
90.78a0.35
79
75.91
79 79.80=1.75
87.97.1.17
83.81a081
6840.9
CreDFs-5
((
87.05+a.00
93.3622042
14.144
80.81
87.30+1.77
92.24
88.1740.44
86.9420.60
We compare test accuracy and ECE for DEs*-5 and CreDEs-5 in Table IT and their OOD detection performance on the CIFARION 10/100 (ID) vs WeAlN/Tiny-ImagNNN (OOD) benchmark in Table 12
Table 11:
Test accuracy and ECE of D and CreDEs-5 on the CIFARIO and CIFAR 100 datasets.
Best results in bold.
CIFARIO
CIFAR
Accuracy
Accuracy
DEs
91.53+0.22
0.01 890.99.00019
68. 34
0.0372+00033
CreDEs-5
Rasi
93.75a0.11
0.0092x0.0016
79.54.0.21
0.036620.0025
93.74:0.11
0.0108:0.0017
79.65=0.19
0.0268:0.0033
Table 12: OOD detection performance comparison of DEs*-5 and CreDEs-5 using the dataset pairs CIFAR10/100 (ID) vs SVHN/Tiny-lmageNet (OOD).
CI (ID)
CIFAR 100 (ID)
SVHN (OOD)
Tiny NmtgeNet (OOD)
SVHN (OOD)
Tiny ImageNet (OOD)
AUROC AUPRC
AUROC
AUPRC
AUROC
AUPRC
AUROC
AUPRC
TU
(E(())
91.82=096 95. 25.1130.700
86. 36.2660.30
84.09H0.22
78.70=1.1
88.20+0.91
76.99H0.28
73.03+0.37
(Q)
95.71:0.42 97.7320.27
89.0240.10
88.02.0.15
79.44.144
88. 10
83.4920.17
80.6120.33
Dbst-::H/())-i00
87.21=149 31.0911.99
$4.5840.30
80.80 80e0..4
74.3821.39
84.6740.66
75.27a01.88
70.8020.45
EU CreDEs CreDEss (()
96.55en.25
98.17=1.17
88.10e0.26
87.85e0.35
78,55=1.15
86.5740.65
82.54401.6 77.60=n.44
CrDDEs-S:CH(C)
96.7220.24 98.2520.17 89.5440.16 88.7420.24
79.23s1.19 87.1720.66 83,0120.44 78.9520.44
The reported results demonstrate that CreDEs-5 outperforms DEs* ensembles by achieving higher test accuracy and lower ECE values. Concerning OOD detection tasks, it can be found that CreDEs in general improve the AUPRC and AUROC scores using either the TU or the EU metric, pretty much across the board. These results suggest that CreDEs provide higher-quality EU and TU estimation.
In Table 12 a 0.1%% %% drop in AUPRC using the TU metric can be observed. However, remember that CreDEs calculate TU (using the upper entropy) by solving a constrained optimization problem Eq. (13) using a numerical solver from SciPy. The slight performance decrease is likely due to numerical errors during the optimization process.
B.5 Comparison between CreDEs and DEs with POE Ensemble Setting
In this experiment, we conduct a comparison between our CreDEs and DEs with the 'product of experts' (POE) 32 ensemble setting, opposed to the more commonly employed 'mixture of
20

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experts' approach in our primary analysis. Here, DEs"-S denotes the deep ensembles that process the final predictions from the ensemble members using the POE strategy. The experimental setup mirrors that used by OOD detection benchmarks, involving data pairs CIFARIO (ID) vs SVHN/Tiny- ImageNet (OOD).
Table 13 shows that DEsP-5 could improve test accuracy but significantly reduce the calibration performance of DEs-5 (larger ECE values). Among these comparisons, CreDEs-5 emerged as the most superior method. Furthermore, we evaluate the uncertainty estimation through the OOD detection benchmark. Specifically, =the entropy of the final prediction of DEs is calculated to quantify the total uncertainty. For CreDEs-5 and DEs-5, use the upper entropy, (Q)), and H respectively. The results in jable[[4]cnssisentty demonstrate the superior performance of our method. Although the POE strategy improves the test. aceuracy of classical DEs, it significantly degrades calibration performance and leads to inferior OOD detection performance.
Table 13: Test accuracy (ACC) (%) and ECE comparison on the CIFAR10 10 dataset, using the ResNet50, VGG16, and ViT Base architectures.
ResNer50
VGG16
VIT
ACC
ECB
ACC
ECE
ACC
ECE
CreDEs (
93.74Hl.11
0.0109:0.0017
87.92e0.11
0.0611+0.0012
93.59=00399
0.0104#0.0012
CreDEs-5 (lai))
93.75+0.11
0.0092:0.0016
87.9449011
0.0203a0.0014
93.60=0.40
0.0107:8.0014
DEs-5
93.32+0.13
0.0131=00010
85.53+0.10
0.081 140.0001
90.43=0097
0.0181#0.0019
D5
93.4740.11
0.0610-00011
85.55+0:08
0.1368H0.0008
90.56-0:90
0.0894+0.007
Table 14: OOD detection performance comparison (%) on CIFAR10 vs SVHN/Tnn-lmageNee, using the ResNet50, VGG16, and ViT Base architectures.
ResNet50
VGG16
VIT Base
SVHN
Tiny ImageNet
SVIIN
Tiny ImageNet
SVIN
ImageNet
AUROC
AUPRC
AUROC
AUPRC
AUROC
AUPRC
AUROC
AUPRC
AUROC
AUPRC
AUROC
AUPRC
CreDEs.595.71:0..5 955.5557.....977735337 89, 220.0222. 85.02....7... (7.56870000000000 93.36
17:0.4486
DEs-5 94.80+0.43 97, 26-0 88, 80+0 19 20.29 84,5000-00000000 50+0.40 90.78.0. 78-0 79.40Ho 90...1100.14770 (tt....77.977111111 97 -08 81.7
DEs 93.90s0.24 96. 10.021 88.00 84.1 120.32 14.1000..2200223
89.83 2.0.
B.6 Comparison between CreDEs and Bayesian Neural Networks
As discussed in the main body, the main reason for excluding Bayesian neural network (BNN) approaches in our main evaluation is that they generally have difficulty scaling to large datasets and complex network architectures [54]. In this section, we conducted additional comparison between Cre and DEs, MCDropout [22), and two maasssollowstanddrdizeee BNNs (BNN-R (51) and BNN-F 776) All the models are trained on the ResNet50 for the CIFAR10 dataset from scratch. The input data shape is (32, 32, 3). The Adam optimizer is applied with a learning rate scheduler, initialized at 0.001. The learning rate is subject to reduction of 0.1 at epochs 80 and 120. For BNNs, 10 forward passes are used for uncertainty estimation.
The uncertainty evaluation via OOD detection on the CIFARI0 vs SVHN/Tiyy dataset is reported in Table 15 The results consistently demonstrate the significant improvements of our CreDEs.
B.7 Case Study on Active Learning Settings
Active learning (AL) aims to efficiently train models with minimal data by acquiring additional samples from vast pool of unlabeled data, which are then labeled by experts [II]. After each acquisition step, the model is retrained using the expanded training set. The iterative process continues until either the desired accuracy the maximum allowable acquired samples reached. Efficient data acquisition can be a reliable estimate of the uncertainty of models [23 54]
Setup We deploy 5 (8 0.5) and DEs-5 (baseline) using the ResNet18 18 architecture and utiliz- ing clean MNIST samples in the pool set. TU and EU estimations from each approach for the acquisi- tion functions are utilized. We begin with initial training set of 20 randomly selected MNIST points.
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Table 15: OOD detection AUROC and AUPRC performance (%, 1) between CreDEs-5 and Bayesian models based on ResNet50 using EU and TU as uncertainty metries on CIFARIO vs. SVHN/Tiny- ImageNet. Results are averaged over 15 runs. The best results are in bold. The 'drop' denotes the dropout rate applied MCDropout.
Model
Epistemic Uncertainty Mensure as Metric
Total Uncertainty Measure as Metric
SVHN (OOD)
TTny-ImageNet (OOD)
SVHN (OOD)
Tiny-ImageNNt (OOD)
AUROC
AUPRC
AUROC
ALPRC
AUROC
AUPRC
AUROC
CreDFs-5
79.14
86.84+1.18
82.85a0.29
80.71
81,0020.75
88.66120.46
84.0620.11
82, 16
DEs-5
73.53&165
83.81+142
76.13
70.86:0477
77.93
84.92s0.39
80.22ah.26
76.94a0.30
BNN-R
70.30+3.55
42.41+245
72.91=201
07.82+2.10
73.37=2.00
82.69 .8
73.98
70.52+1.80
BNN-F
70.1564.38
02.04
73.6621446
68.52+1.53
73.77=2.62
12.90
74.57=1.30
MCDropout
0.1 drop
74.19+1.55
82.93+101
75.04+0.77
68.25+1.31
76.92+1.85
85. 15.93e1.2 1.22
77.48+0.56
73.63+0.62
0.4 drop
61.661 11.6611.89
73.47+1.27
67.24a1.36
59.55
79.25a0.96
86.04420.77
76.0420057
72.73a006
In each iteration, we acquire the 5 samples with the highest reported uncertainty estimates (EU or TU per model). After each step, we train models using the Adam optimizer for 20 epochs and select the with the best accuracy from the validation set. AL process stops when the training set size reaches 150.
Test Accuracy
80
EUcre:N(()) H())
Results Figure 12 shows the result comparison between CreDEs-5 and DES-5 using TU and EU estimates the acquisition functions per model. In the evaluation using MÑIST, aiming for a 90% accuracy maximum sample count of 150, CreDEs-5 employing acquisition functions TU (H)) and EU (G)(Q)) demonstrates superior per- formance compared to DEs-5 using TU (q)). In addition, CreDEs-5 with EU (H) H(Q)) outperforms DEs-5 with EU (H) H (q)). The additional evidence verifies the improved quality EU and TU estimation of CreDEs, compared to DEs. In future work, we aim to explore the potential integration of our methods into other active learn- ing benchmarks (70) and real-world applications further improve on them.
EUcre:GH(()
T(cre:((())
60
Euge ((4) ((())-((0))
Tuge: H(g)
20
50
100
150
Acquired Dataset Size
MNIST
Figure 12: AL experiments us-
ing different acquisition functions. Achieved test accuracy vs. acquired training set size.
C Experiment Implementation Details
For the main experiments on the ResNet50 backbone, we used two Tesla P100-SXM2-16GBB GPUs devices to independently train 15 SNNs and CreNets using CIFARIO and CIFAR100 datasets. The input shape of both networks was (224, 224, 3). We employed the Adam optimizer, with a learning rate scheduler set at 0.001 and reduced to 0.0001 during the last five training epochs.
Figure T3 shows the averaged training and validation accuracy for training process monitoring.
In the ImageNet experiments, we employed three NVIDIA A A100-SXM4-80GH GPUs. To create deep ensembles, we independently retrained 15 deep SNNs based on pre-trained ResNet50 model for 3 epochs, using the Adam optimizer with an initialized learning rate of 1e For CreDEs, we initialized CreNet weights using pre-trained ResNet50 model and independently retrained 15 CreNet models for 5 epochs, using the Adam optimizer with an initialized learning rate of 16-5, The choice of larger learning rate value and epoch count for CreNets is consequence of their modified final later compared to SNNs. For the ablation study on various network architectures, we again utilized two Tesla 21000-XX2--12
GPUs and one NVIDIA A A100-XXM4-00GB GPU as devices to independently train 15 SNNs and CreNets, based on VGG16 and ViT Base architectures, respectively, and using the CIFARIO dataset. VGGl6-based SNNs and CreNets were trained for 20 epochs. SNNs and CreNets using the VIT Base backbone were trained for 25 and 40 epochs, respectively. The input
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100
100
80
Accuracy (%6
Accuracy (96)
60
60
40
40
CreNetU
CreNetL-Val
CreNetU
CreNetL-Val
CreNett
SNN SNN-Val
CreNett
SNN
CreNert--aal
Va
SNN-Val
0
10
20
25
1
15
20
25
30
35 40
Training Epoch
Training Epoch
(a) CIFARIO
(b) CIFAR100
Figure 13: Averaged training and validation (Val) accuracy (%) for CreNets and SNNs over 15 runs. The U and L in the labels of CreNets represent accuracies associated with upper and lower probability bounds, namely imx and mmnn. respectively.
shape for both networks was set to (224, 224, 3). For optimization, we employed the Adam optimizer with learning rate scheduler initialized at 0.001 and reduced to 0.0001 during the final training epochs.
Training Complexity We did not include the report of training time complexity in the main paper as CreNets use custom training loop, unlike the TennsoFloow-standdarieedd training of standard neural networks (SNNs), precluding fair comparison.
Nevertheless, we did train single CreNet and single SNN based on the ResNet50 architecture on the CIFARI0 dataset, from scratch and on single A100 GPU. The training time per epoch 16.36s for the SNN and 73.77s for CreNet, respectively. Given the evidence that CreNets only marginally increases the inference time (single forward pass), we are optimistic that by standardizing and optimizing the customized training loop and adopting a more efficient code implementation of Algorithm D we could significantly reduce the training load.
Further Discussion Inference Complexity As discussed previously, regarding inference time, doubling the final layer nodes would slightly increase the inference time. For instance, the inference time per sample for ResNet50 architecture single standard neural network, vs 5.7 ms for single CreNet (a marginal increase). The inference cost the CIFARIO/IN dataset reported in Table 16 further demonstrates slight increase in inference complexity in our method. Morcover, Table presents the inference cost, evaluated on single AMD EPYC 7643 48-core CPU. The results indicate no significant overhead of our CreDEs and also demonstrate that employing VGG16, a lighter model architecture compared to ResNet50, substantially reduces the inference cost for both DEs and CreDEs.
the ImageNet dataset is 5.5 ms for
Table 16: Complexity comparison between ResNet50-based SNNs and CreNets using CIFARIOW 10/100 datasets. The inference cost per dataset measured by single NVIDIA P100 SXM2-16GB GPU for both models.
Dataset
Model
Parameters (million)
Inference time per sample (ms)
CIFARIO
SNN's CreNets
26.216 26.221
60.6el17 63.081111 62.5e0.5 140.7
CIFAR100
SNN' vs CreNets
26.262 vs 26.314
Table 17: Inference cost comparison on CPU between SNNs and CreNets per single CIFARIO input of different architectures.
VGG16 (ms)
ResNe150 (ms)
SNNs vs CreNets
19.2=3. 23,145.2
148.2+49 163.3-39.4
Regarding uncertainty estimation cost, we report the cost of calculating the Generalized Hartley (GH) measure and the upper entropy in Figures 10 and IT] respectively. For example, the time cost for GH calculation for CIFARTO without approximation is ms (0.02 ms in the reduced case considering out of 10 classes) while calculating the EU in deep ensembles for CIFAR10 takes
23

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1x 1x00.'ms mss measured on the same single CPU. Though higher CreDEs remain practical without actual computational constraints. addition, the reported numbers are without code efficiency optimization: more efficient code implementation could significantly reduce the cost.
The practical takeaway here is that, as demonstrated by extensive experimental variation, exhibit strong potential to enhance the uncertainty quantification performance of DEs in real-world applications, with only modest increase computational complexity. However, if DES are already deemed impractical due to computational limitations, our CreDEs would not be a suitable alternative.
CreDEs
D
Discussions on Ensemble Approaches
D.I Rationale for Averaging Ensemble Strategy
The randomness of parameter initialization in neural networks is one of the reasons leading to (epistemic) uncertainty about the "ground-truth' model. As we gather more information, both epistemic and total uncertainty should decrease. For example, if we assume that we can train an infinite number of standard neural networks, then the Deep Ensembles would eliminate the source of ignorance caused by the randomness of parameter initialization.
Our proposed averaging approach to creating an ensemble of CreNets follows a similar rationale. Specifically, if aggregate infinite number of ensemble members, the uncertainty caused by the randomness would vanish. The outputted probability interval of CreDEs, primarily acknowledges the lack of precise insights into the divergence between the training and test distributions.
D.2 Possible Alternative Ensemble Approaches
CreDEs aggregate predictions from multiple individually trained CreNets, producing credal sets based on probability intervals. addition averaging, two alternative approaches, namely union (disjunctive combination) and intersection (conjunctive combination) [17], can be envisaged.
These alternative methods are illustrated in Figure
(a)
(b)
(c)
Figure 14: Representation of three ensemble approaches: averaging (a), union (b), and intersection (c). In each subfigure, the ultimate credal set (highlighted in dark red) is formed by aggregating two individual credal sets, each constrained by probability intervals indicated in light green and blue, respectively.
Averaging CreDEs average the upper and lower probabilities per class from M CreNets and predict the aggregated probability intervals, denoted as [qř, 9. discussed in Eq. (14. can be proved that 1q2 Lū is guaranteed to generate a non-empty credal set, as follows:
trained
E LL. = DM
MM DC qim. 515 M
GÜm, - T..-
(20)
The semantic behind averaging is that equally trust all pieces of information (individual credal sets) without judging the authenticity the information. Similar to traditional deep ensembles (DEs), the averaging ensemble approach alleviate the influence of training process randomness.
Union Given collection of convex probability intervals, denoted as (( Fú MM M.
De Campos
al. have proposed the computationally efficient way calculating the union of credal sets [17] as

---

follows:
TZ.
min m
9im..
9ú
max
(21)
[E......M
The union ensemble method implies that at least one piece information is considered to true.
The union operation in Eq. (1 has a significant limitation, it results in an expanded credal set and
introduces an overestimation effect on the precise union credal sets, as shown Figure 14
Intersection A collection of convex probability intervals {lqi... Lu, De can formulate an intersection
L.. (M)
9im,' qu.
min (E()....M))
Um..
(22)
However, the obtained [q7 Qi does not inherently satisfy the condition outlined in Eq. 2 for constructing a credal set [17]. Therefore, the intersection approach is not applicable in CreDEs.
Empirical evaluation In this experiment, we mainly evaluate the impact of averaging and union ensemble approaches on the EU estimation (GH(Q)) of CreDEs. Utilizing 15 individually trained ResNet50-based CreNets on CIFARII 10 dataset, we formulate 15 CreDEs-M by varying the ensemble number from 2 to through averaging and union ensemble methodologies. Each kind of CreDEs- M is assessed for the averaged GH(Q) concerning samples and the quantity of CreDEs-M, and the averaged standard deviation (STD) of GH(Q) related to samples and the quantity of CreDEs-M The results are plotted in Figure 15 (b) and (a), respectively. Besides, we also present the AUPRC and AUROC scores of OOD detection using GH(Q) QQ) as the uncertainty metric Figure 15 accordingly.
c and (d),
0.25
0..20
0.115
0.75
Averaged
0.10
0.50
0.05
0.25
40.00
0.00
6
10
5
8
9
10
Number of Ensemble Members
Number of Ensemble Members
(a)
(b)
AUPRC of #F00 Detaction
AUROC of OCD Detecctiionn
0.95
0.95
0.90
0.85
1.
10
10
Number of Ensemble Members
Number of Ensemble Members
(c)
(d)
CIFAR10 (Avg)
CIFAR10 (Union) --t-- SVHN (Avg)
SVHN (Union) ...- Tinylmage (Avg)
Tinylmage (Union)
Figure 15: Impact of averaging (Avg) and union on the EU estimation of CreDEs on OOD detection benchmark involving CIFAR10 EVHN/Tiny-mageehe (Tinylmage), implemented ResNet50 architecture.
Figure 15 (a) illustrates reduction in the averaged standard deviation (STD) of GH(Q) estimates as the number of ensemble members increases. This suggests that averaging the ensemble helps mitigate the uncertainty resulting from the randomness in the CreNet training process. Consequently, the AUROC and AUPRC scores, shown in Figure [5 (c) and (d), exhibit enhancement through the utilization of the averaging ensemble approach, accompanied by concurrent reduction in the STDs of the scores as the number of ensemble members increases. In contrast, Figure [5 (b) highlights overestimation of EU across various datasets when employing the union ensemble method.
While the average EU estimates for ID samples are overall lower than those for OOD instances, the overestimation may lead OOD mis-detection in some specific samples. This explains the
"fluctuatoons" in the associated AUPRC and AUROC score curves in Figure 15
25

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E Further Discussion on Future work
E.1
Generalizing Cross-Entropy for Lower/Upper Probability
As stated in Sec. 2.2.2. calculating the cross-entropy (CE) loss for the lower and upper probability works due to the 'one-hot" labeling nature of the ground truth vector t. However, generalizing the CE. which corresponds to the Kulback Leibler (KL) divergence
((F) 4
(xl())a)= = Σ (0) log j=I,..,C
between predicted probability vector and the general ground truth vector t, to lower/upper probabilities still an open research subject [68 69].
In our case, the credal set Q is defined by the outputted proper probability intervals [qL. qul- Therefore, the KL divergence for lower probability inducing a credal set may be calculated by:
Finding the probability vector that best approximates it. For probability intervals, there are two established such ways: normalizing either the lower or the upper probability (see 774 or computing the so-called 'intersection probability' (see (55)-
Computing the KI. divergence between the ground truth vector and the approximation obtained.
In future work, we aim to investigate the approach and compare those other well-founded methods for calculating the cross-entropy loss with the one used in the paper.
E.2 Theoretical Coverage Guarantees
In the current stage, our CreNets do not provide coverage guarantees, e.g., how likely it is for the divergence of future data distributions to be within the modcled bounds. Nevertheless, various approaches to incorporating statistical guarantees in our framework can be envisaged.
In particular, CreDE, being a classifier, can be employed as the 'underlying model' in an inductive conformal learning framework [64), which builds an empirical cumulative distribution of the 'non- conformity' scores set of calibration samples and test time outputs the set of labels whose empirical CDF is above desired significance level (e.g., 90%).
Namely, given a test input & and the associated predictive system of probability intervals [gt.,, q...,c c 1,.... C' (the output of CreDE), a sensible choice, for instance, is to set as non- conformity score of pair (test input, class), (2r, c), the complement of the upper probability for that class, given input æ:
s(x, c) = qu.
(i.c., label would be considered non-conformal if its predicted upper probability, for that input low), and compute predictive regions as standard in conformal learning:
r(x)= -CEC C >E
where
pe = (x) s(t),)))))))))))))))))(( :
s(x,
(()))))) s(); (z) = s(x,
q
a+1
(zy,e;) is the j-th calibration point, q is the number of calibration points, and u ~ u(0, 1) (the uniform distribution on the interval (0, 1)).
We plan to explore this integration as the next step of our future work.
E.3 Extension for Regression Framework
The vast majority of papers using credal sets in machine learning focus classification 12 5 78 or, more recently, on self-supervised learning (but still classification setting (37]). Nevertheless, recent study [19] has shown that the formalism of belief functions (a special class of credal sets) can
26

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be extended to regression, by leveraging random fuzzy sets. It might thus be possible to explore such connections between probability intervals and random fuzzy sets and devise suitable regression framework based on CreDEs.
The following section outlines a future research direction.
direct possible extension of CreDEs regression problems as
Remember that CreNet outputs credal set the simplex of probability distributions over the classes. Each vertex of this credal set is therefore a probability distribution over the target space (the set of classes C' for classification).
On the other hand, Bayesian regressor network (trained to learn distribution of its weights) would output (continuous) probability density over the target space (for the sake of simplicity, assume Y IR).
One could then train an ensemble Bayesian regressor networks to predict a credal set with fixed number vertices (one network outputting one vertex probability) that the final predicted credal set is the convex closure of those. Figure illlssttrte the concept briefly.
Bayesian regression
network
Y=R
1... G
Credal prediction on the target space Y
Y=R
Network G
Figure 16: Concept of credal regressor.
The Distributionally Robust Optimization (DRO) framework employed for CreDE training is model the divergence between the data distribution of samples belonging to different groups within the training set. A full DRO formulation with G loss components, in combination with Bayesian deep learning techniques such as variational inference, could then be employed to drive the training of the G credal vertex networks, encouraged generate diverse (probabilistic) predictions model different possible data distributions, in generalization of the two-component loss used here.
Broader Impacts
The main objective of this paper is to advance the field of Machine Learning by improving the quality of uncertainty quantification. There are many potential societal consequences of our work, none of which we feel must specifically highlighted here. The proposed method demonstrates superior performance in detecting out-of-distributie (OOD) samples. Such capability can potentially safeguard end users from misguided decisions that stem from the incorrect predictions of neural networks OOD instances. Therefore, our approach potentially improve the safety, reliability, and trustworthiness of machine learning systems for classification tasks and be applied mission- critical domains, such as autonomous driving and medical sciences.
27

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Reviews:
[{"review_id": "lB1ctewDI2", "reviewer": "Reviewer_HFiw", "summary": "In this paper, the authors propose a novel method for uncertainty estimation in deep networks called Credal Deep Ensembles, which combines credal inference and ensembling approaches. During inference, the model predicts intervals (lower and upper probability values) for each class, resulting in a final output that is a simplex subset defined by these intervals. The ensembling technique predicts N such intervals and averages them out. Based on this final prediction, the authors derive several uncertainty measures (both aleatoric and epistemic). The authors validate their method through extensive experiments on several image classification tasks (CIFARs and ImageNets) and various model architectures (ResNets, VGGs, ViTs), demonstrating significant improvements over ensembles in terms of model accuracy, quality of uncertainty estimations, OOD detection, and model calibration.", "questions": "* Given the increased complexity and number of parameters in the proposed approach compared to other methods like ensembles, how do the authors justify its practicality and ease of implementation? Are there any strategies to mitigate these complexities (for example, how to chose $\\delta$ parameters for training or $K$ parameter for the optimized inference)?\n\n* I believe that comparisons with and discussions of second-order methods such as evidential learning in the context of credal inference are important. How does the proposed approach perform relative to these techniques?\n\n* The paper mentions significant improvements over ensembles but does not clearly explain why credal inference brings additional benefits. Can the authors provide a more detailed explanation of the source of these improvements?", "limitations": "The authors provide an adequate discussion of limitations in the last section.", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "* The paper is clearly written and easy to follow. The idea is intuitive and easy to grasp. The related work section provides an adequate discussion of existing approaches to credal learning and nicely describes the proposed method in detail.\n\n* The proposed methodology is interesting and provides a novel perspective on second-order approaches as evidential learning by combining credal inference with ensembling techniques.\n\n* The authors provided an extensive evaluation of their approach, showing improved performance, uncertainty quality, OOD detection, and model's calibration in several popular and important setups.", "weaknesses": "* The approach involves too many parameters and procedures for training and inference, making it more complex compared to other methods such as ensembles. This added complexity could limit its practicality (and applicability) and ease of implementation.\n\n* The paper does not provide a comparison with second-order methods such as evidential learning, which leaves a gap in the evaluation of the proposed approach. Including such comparisons could be important since the proposed credal set approach is deeply connected to second-order distribution prediction methods. Including such comparisons would strengthen the validation of the proposed approach and offer a more comprehensive assessment of its performance against established techniques.\n\n* The output size is doubled, which, for tasks with a high number of classes, could significantly increase the number of weights. This could result in higher computational costs and memory requirements, potentially limiting the scalability and efficiency of the approach.\n\n* It is not entirely clear where the improvements over ensembles are coming from or why credal inference brings additional benefits. More explanation is needed to understand the source of these enhancements.\n\n* From the perspective of the experimental evaluation, I would be curious to see evidence that the behavior demonstrated in the paper would hold in other domains, such as texts, graphs, more complicated vision tasks (e.g. segmentation), not limiting to image classification task.\n\nI tend to assess the paper positively and would be happy to increase my score after the discussion.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Dear Reviewer,\n\nWe express our sincere gratitude for acknowledging our work and your valuable feedback. In the following, we address your concerns.\n\n**Regarding W4 and Q3: A more detailed explanation of the source of the improvements**\n\n**Response:** We would like to make a clearer explanation below.\n\nEpistemic uncertainty in predictions is induced by the probability distribution(s) that generated the training data may be very different from the distribution that generates a test point. It is well known that using CE as a loss leads to the maximum likelihood estimator of the model parameters, under the assumption that all training points are drawn i.i.d. from a single, unknown distribution $P$:\n$$\n\\hat{\\theta} = \\arg \\min_\\theta\\text{CE} = \\theta_{MLE},\n$$\nwhere $\\hat{\\theta}$ denotes the learned parameters.\n\nWhat does Distributionally Robust Optimization (DRO) do? It takes a cautious stance, and assumes training points are generated by one of many potential distributions $\\mathcal{U}$, and minimizes the upper bound of the expected risk (Eq. (6) of the paper). In practice, this is approximated using Adversarially Reweighted Learning as in Eq. (9). Therefore, it learns using the following equation:\n\t\t\t$$\\hat{\\theta} = \\arg \\min_{P \\in \\mathcal{U}} E_P (Loss)$$\n\nHowever, if we apply DRO to a traditional network, the network will learn a single parameter value (Eq. (9) of the paper); for a new test data point $x_t$ it will then produce a single predicted probability vector, the one produced by the model that, in the long run, minimizes the worst-case risk.\n\nThis, though, does not express at all the (epistemic) uncertainty about *which* of the distributions in $\\mathcal{U}$ has generated the particular test datapoint $x_t$; as a result, it does not express how uncertain the prediction may be for that particular data point.\n\nAllowing a network to output lower and upper probability vectors for each input, instead, allows this epistemic uncertainty to be explicitly modelled.\n\nWhat is the best-case scenario? That in which all data (including the test point $x_t$) are indeed generated by the same distribution – in this case, $\\theta_{MLE}$ is the best choice, in a maximum likelihood sense.\n\nWhat is the worst case? That in which $x_t$ is generated by a distribution that is far away from the groups of distributions $P_g$ (see lines 140-141) that generated the training data. In this case, the DRO solution will still be far from perfect, but still the best one can do, given the evidence in the training data.\n\nTherefore, if I minimize the CE versus $\\boldsymbol{q}_U$ (first component of Eq. (10)), I get a credal set Cre of predictions whose upper probability is such that, for each class $i = 1,…,C$, there exists a probability vector $\\boldsymbol{q}\\in \\mathbb{Q}$ such that $q(i)$ is the MLE prediction for that class (please remember the discussion of lines 172-175).\n\nSimilarly, if I minimize the DRO loss versus $\\boldsymbol{q}_L$ (the second component of Eq. (10)), for each class $i = 1,…,C$, there exists a probability vector $\\boldsymbol{q} \\in \\mathbb{Q}$ such that $q(i)$ is the DRO prediction for that class. \n\nMinimizing the two components together, as in Eq. (10), thus encourages the predicted credal set to extend from the best-case to the worst-case prediction.\n\n**Regarding Q1: any strategies to mitigate these complexities ($\\delta$ parameters for training or $K$ parameter for the optimized inference)**\n\n**Response:** In our main evaluation, we set by default $\\delta=0.5$ to reflect a balanced assessment of the train-test divergence and show how such a value allows our model to outperform the baselines. Most importantly, the ablation study on $\\delta$ (Lines 287-312) demonstrates that our CreDEs are not particularly sensitive to $\\delta$.\n\nOne possible way to find the best $\\delta$ in practice is to conduct standard cross-validation on specific test scenarios. Perspectively, an interesting option, in the presence of multiple datasets (e.g.acquired over time in a continual learning setting), could be applying the DRO loss component to different components of the training set, and assessing the results to robustly select $\\delta$.\n\n$K$ is a trade-off parameter between uncertainty estimation and computational efficiency. The ablation study in Appendix B.2 shows that increasing the value of $K$ improves the OOD detection performance, but leads to an increase in execution time. Applying  $K<=10$ to the Generalized Hartly (GH) measure for EU estimation is more practical. Choosing $K$ for upper and lower entropy measures is more flexible. For instance, in Table 2 of the paper,  $K=20$ for the ImageNet dataset (original 1k classes) shows an improved OOD detection performance compared to DEs.\n\n**Regarding concerns about the complexity (W1 and W3) and the comparison baselines (W2 and Q2)**\n\n**Response:** We kindly invite you to follow the detailed response in our global rebuttal part 1 and part 2, respectively.\n\n**Regarding W5: Expecting evidence in other domains**\n\n**Response:** We fully agree with your comments that showing more evidence of performance improvement of our CreDEs in other domains can further strengthen our work. Given the consistent performance improvements of CreDEs on a wide range of image classification tasks (which you acknowledged as our strength), we are optimistic about successfully extending our work to other domains. We will include your suggestion in the discussion of our future work and find its proper place within the expanded scope of that work.", "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": "aV08P8xsks", "reviewer": "Reviewer_AC8P", "summary": "This paper introduces Credal Deep Ensembles, which are ensembles of Credal-Set Neural Networks designed to predict lower and upper probability bounds for each class, representing epistemic uncertainty.", "questions": "Why were Dirichlet-based approaches and other recent uncertainty quantification methods excluded from the comparison? \n(See also comment on baseline comparisons).\n\nPlease elaborate on the loss function (10). Why is this a sensible choice? It is not immediately clear to me from the corresponding explanations in the paper. \n\nHow does the Interval SoftMax function handle cases where the intervals for different classes overlap significantly? What impact does this have on the final predictions and uncertainty estimates?\n\nInstead of learning upper/lower probabilities, what is the difference to get the upper/lower probabilities e.g. from an ensemble of NNs? \n\nHow do you guarantee that the computation of the lower and upper probabilities within the credal set is scalable for high-dimensional classification tasks?", "limitations": "There is a very short discussion of potential limitations at the end of the paper.", "rating": 5, "confidence": 3, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "Novel approach to uncertainty quantification in deep learning. Empirically it seems that the proposed approach is performing not too bad, while I am not entirely sure about the interpretation of the results (a lot of the results seem to indicate only marginal improvement).", "weaknesses": "Empirical results are not very convincing, since the improvements (if there are any) seem to be of marginal nature.\n\nWhile the paper introduces novel methods and validates them empirically, it could benefit from a deeper theoretical analysis of why the proposed methods improve uncertainty quantification. E.g. the paper introduces novel components like the Interval SoftMax function and the application of DRO, but it lacks a detailed theoretical analysis that justifies their effectiveness.\n\nLack of baseline comparisons (some model classes were excluded).", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Dear Reviewer,\n\nWe appreciate your efforts in reviewing our work and your recognition of our novelty. Below we address your concerns.\n\n**Response to W1:** We believe that we have provided a comprehensive evaluation of our approach, which consistently shows significantly improved performance, uncertainty quality, OOD detection, and model calibration in several popular and important setups. For instance, in Table 3 of our paper, we can observe the OOD detection improvements of about 5% and 11% for CIFAR 10 vs SVHN (AUROC) on VGG16 and ViT base architectures, respectively. Such visible improvements can be consistently observed in, for example, Table 2, Figures 5, 6, and Tables 8-12 of the paper.\n\n**Response to W2 and Q2:** We value the opportunity to provide a more rigorous argument of the theoretic analysis and loss design. \n\nEpistemic uncertainty in predictions is induced by the probability distribution(s) that generated the training data may be very different from the distribution that generates a test point. It is well known that using CE as a loss leads to the maximum likelihood estimator of the model parameters, under the assumption that all training points are drawn i.i.d. from a single, unknown distribution $P$:\n$$\n\\hat{\\theta} = \\arg \\min_\\theta\\text{CE} = \\theta_{MLE}.\n$$\n\nWhat does Distributionally Robust Optimization (DRO) do? It takes a cautious stance, and assumes training points are generated by one of many potential distributions $\\mathcal{U}$, and minimizes the upper bound of the expected risk (Eq. (6) of the paper). In practice, this is approximated using Adversarially Reweighted Learning as in Eq. (9). Therefore, it learns\n\t\t\t$\\hat{\\theta} = \\arg \\min_{P \\in \\mathcal{U}} E_P (Loss)$.\nHowever, if we apply DRO to a traditional network, the network will learn a single parameter value (Eq. (9) of the paper); for a new test data point $x_t$ it will then produce a single predicted probability vector, the one produced by the model that, in the long run, minimizes the worst-case risk.\n\nThis, though, does not express at all the (epistemic) uncertainty about *which* of the distributions in $\\mathcal{U}$ has generated the particular test datapoint $x_t$; as a result, it does not express how uncertain the prediction may be for that particular data point.\n\nAllowing a network to output lower and upper probability vectors for each input, instead, allows this epistemic uncertainty to be explicitly modeled.\n\nWhat is the best-case scenario? That in which all data (including the test point $x_t$) are indeed generated by the same distribution – in this case, $\\theta_{MLE}$ is the best choice, in a maximum likelihood sense.\n\nWhat is the worst case? That in which $x_t$ is generated by a distribution that is far away from the groups of distributions $P_g$ (see lines 140-141) that generated the training data. In this case, the DRO solution will still be far from perfect, but still the best one can do, given the evidence in the training data.\n\nTherefore, if I minimize the CE versus $\\boldsymbol{q}_U$ (first component of Eq. (10)), I get a credal set Cre of predictions whose upper probability is such that, for each class $i = 1,…,C$, there exists a probability vector $\\boldsymbol{q}\\in \\mathbb{Q}$ such that $q(i)$ is the MLE prediction for that class (please remember the discussion of lines 172-175).\n\nSimilarly, if I minimize the DRO loss versus $\\boldsymbol{q}_L$ (the second component of Eq. (10)), for each class $i = 1,…,C$, there exists a probability vector $\\boldsymbol{q} \\in \\mathbb{Q}$ such that $q(i)$ is the DRO prediction for that class. \n\nMinimizing the two components together, as in Eq. (10), thus encourages the predicted credal set to extend from the best-case to the worst-case prediction.\n\n**Response to W3 and Q1:** We kindly invite you to follow the detailed response in our global rebuttal (Part 2) for the comparison baselines.\n\n**Response to Q3:** Let us consider a numerical example in a three-element case. \nIf we have three intervals such as $[1.1, 3.1]$, $[1, 3]$, and $[1, 2]$ as the inputs of the Interval SoftMax, the Interval SoftMax returns the following probability intervals as the outcomes: $[0.15612615, 0.71893494]$, $[0.12732467, 0.70053512]$, $[0.09810377, 0.43282862]$. We can observe that the Interval SoftMax also generates overlapped probability intervals to correctly reflect the overlap in the input. In extreme cases, If all the probability intervals over all classes are significantly overleaped, the resulting credal set could include the uniform probability. However, this does not prevent us from using the maximin and maximax criteria in Eq. (11) to derive a class prediction according to the conservativeness of the decision maker. Uncertainty estimation methods for the credal sets are also valid.\n\n**Response to Q4:** The main difference is that getting upper/lower probabilities e.g. from an ensemble of NNs would be a post-processing procedure only in the prediction space, no training process is involved. In our CreDEs, the model learns the probability interval from data and could encourage the predicted credal set to extend from the best-case to the worst-case prediction.\n\n**Response to Q5:** The Interval SoftMax Eq.(4) can ensure a valid credal set regardless of the dimension of the tasks. To reduce the computational complexity for high-dimensional classes, we performed Alg. 2 to reduce the dimension from high $C$ to $K$. Alg. 2 has a strong mathematical background in probability intervals [17] (Eq. (12) of the paper) and can rigorously guarantee a valid credal set. In addition, the ablation study in Appendix B.2 and the OOD detection cases for ImageNet in Table 1 show the effectiveness of the reduction in uncertainty quantification of credal sets.\n\n**Response to L1:** Following your comments, we will duly extend the discussion on the computational complexity and comparison baselines in the revised version, as we discussed in the global 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": "sv4iUDwUmX", "reviewer": "Reviewer_7GAw", "summary": "The paper proposes credal neural networks, which output probability intervals for each class as opposed to a single probability estimate. They also propose to use ensembles of these models, averaging the outputs of the members. Their ensembles of credal neural networks show a higher performance than traditional ensembles in various experiments, from epistemic uncertainty (OOD detection), to the normal accuracy on ImageNet.", "questions": "To calculate accuracy, did you use i_max or i_min?\n\nDo you have some insights into how big the credal intervals are usually, etc? I’m asking because I want to rule out the alternate hypothesis that your method is producing almost-Dirac credal sets.\n\nI understand your motivation of why you apply the vanilla cross entropy loss to the upper bound estimate of the credal set $q_U$ as opposed to the midpoint $m$, but do you have ablations quantitatively supporting this choice?\n\nDo the computational complexity studies in Table 6 include the constrained optimization search for maximum / minimum entropy? In Appendix C it seems like this could be a burden, where training a single ensemble credal ensemble member takes five times as long as training a standard cross-entropy loss member.\n\nThere’s a small typo in line 284, “AUORC”", "limitations": "The authors acknowledge that the runtime and RAM costs for ensembles are a hindering aspect of their work.", "rating": 7, "confidence": 4, "soundness": 4, "presentation": 4, "contribution": 4, "strengths": "The method is benchmarked across an impressive range of tasks and datasets in both main paper and appendix and shows consistent improvements over traditional deep ensembles.\n\nThe method has many hidden details, e.g., trivial solutions it could collapse to. The paper uncovers many of these potential problems and explains how they are solved by neat implementation tweaks. These tweaks can be relevant to other credal approaches.\n\nReproducible code is provided, along with pseudo code in the paper.", "weaknesses": "In all experiments, there is only one baseline, namely traditional deep ensembles. I would encourage to pull results for the standard ensemble with DRO loss (Called DEs*-5) from the appendix to the main paper and to also implement other methods, like Mahalanobis (https://arxiv.org/abs/1807.03888), which recently showed SOTA performance on EU estimation (https://arxiv.org/pdf/2402.19460)\n\nThe resampling setup is internally correlated as the authors construct their 15 ensembles by drawing 5 members out of 15 trained models repeatedly. The 15 ensembles thus overlap which will likely lead to underestimating the error bars in the results", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Dear Reviewer,\n\nWe sincerely appreciate your high recognition of our work and valuable feedback and suggestions. Below we address your concerns.\n\n**Regarding W1**\n\n**Response:** We fully agree that moving the additional results of DEs$^*$-5 (standard ensemble with DRO loss) from the appendix to the main body is a better presentation. Following your suggestion, we include the MC Dropout model in Table 2 of the rebuttal file, which is claimed to be a good choice for uncertainty quantification across the board in the evaluation of [A]. The additional comparison further demonstrates the outperformance of our CreDEs.  We did not report the result in the rebuttal because the Mahalanobis method is a distance-based method for uncertainty quantification and was outside the original scope, which was to address uncertainty quantification using ''second-order'' representations. However, we will properly extend the above discussion in the limitations section of the revised paper. For a more detailed response to the baseline choice, we invite you to read our global rebuttal (part 2).\n\n**Regarding W2**\n\n**Response:** In our experiments, we randomly selected five members from the 15 trained models to construct the ensemble. The ensemble member lists of each ensemble were strictly guaranteed to be different. Therefore, we believe that this setting could reflect the error bars of the results. Based on your comments, we will highlight the setting to improve clarity.\n\n**Regarding Q1**\n\n**Response:** Yes, we reported the accuracy of the models using $i_{max}$ and $i_{min}$ in Table 1 and Table 3 of the paper. They both show improvement in test accuracy.\n\n**Regarding Q2** \n\n**Response:** Due to the high dimensionality, it is hard to visualize and compute the size of the credal set when there are $C>3$ elements. However, we can indirectly judge whether the generated credal set resembles a Dirac credal set by calculating the maximum reachable upper bound probability of each sample, namely $\\max{(q^*_{U_1}, ..., q^*_{U_C})}$. The closer the probability value is to 1, the closer it is to an almost Dirac credal set. We show the results of ResNet50-based CreDEs-5 on the CIFAR10, SVHN, and Tiny-ImageNet datasets in *Figure 1 of the rebuttal file*. We can conclude that our method does not always generate almost Dirac credal sets, especially on OOD samples. For CIFAR10 samples, there are a large number of quasi-Dirac credal sets (but not all), which is reasonable considering the high test accuracy of CreDEs and the low ECE (shown in Table 1 of the paper).\n\n**Regarding Q3** \n\n**Response:** If we first calculate the probability vector based on the midpoint $\\boldsymbol{m}$ and then calculate the cross-entropy loss, we observed that we could not make the model reasonably learn to generate half-length $\\boldsymbol{h}$ to generate the probability interval from the training process, as the $\\boldsymbol{h}$ is not incorporated in the training. In another case, we could not calculate the midpoint of the generated probability intervals as $\\boldsymbol{q}_U-\\boldsymbol{q}_L$ for cross-entropy calculation, it is not normalized. In Appendix E.1, we discuss another sensible way of generalizing cross-entropy for lower/upper probability as our future exploration.\n\n**Regarding Q4** \n\n**Response:** No, the inference cost in Table 6 does not include the uncertainty calculation cost. We reported the cost of calculating the generalized Hartley (GH) measure and the upper entropy in Figures 7 and 8 of the paper. For instance, the time cost for GH calculation for CIFAR10 without approximation is 17 ms (0.02 ms in the reduced case considering 4 out of 10 classes) while calculating EU in deep ensembles for CIFAR10 takes $4.1\\times 10^{-4}$ ms, measured on the same single CPU. Though higher CreDEs remain practical without actual computational constraints. Besides, the numbers reported are without code efficiency optimization: a more efficient code implementation could significantly reduce the cost.\n\nRegarding the training complexity, the CreNet uses a custom training loop, unlike the TensorFlow-standardized training standard neural networks, precluding a fair comparison. Given the evidence that CreNets only marginally increases the inference time (single forward pass) in Table 6, we are optimistic that by standardizing and optimizing the custom training loop and adopting a more efficient code implementation of Algorithm 1, the training effort could be significantly reduced. \n\n**Regarding Q5**\n\n**Response:** Thank you for pointing the typo out. We will revise it accordingly.\n\n[A]  Benchmarking uncertainty disentanglement: Specialized uncertainties for specialized tasks. 2024", "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": "hzVC009JIw", "reviewer": "Reviewer_YsPp", "summary": "This paper presents Credal Deep Ensembles (CreDEs), an ensemble framework of Credal-Set Neural Networks (CreNets) to produce a high-quality epistemic uncertainty by using a credal set with the Distributionally Robust Optimization (DRO) technique. \nContributions are:\n- A rigorous CreNet final layer in Section 2.1, where the number of nodes is doubled to present the confidence intervals with lower and upper probability bounds for each class. The final set of probability intervals is shown to be proper by satisfying convexity conditions.\n- A training procedure with CreNet loss in Section 2.2, where the standard cross-entropy loss is trained with the upper-class probability bounds, and adversarially reweighted DRO is trained with the lower ones.\n- Experimental results show CreDEs are better than Deep Ensembles (DEs) by having a higher test accuracy, lower ECE in IID classification, AUROC/AUPRC in OOD detection tasks.", "questions": "1. Can you qualitatively evaluate the uncertainty quality by using the reliability diagrams [8] and the PDF plot of predictive entropy $\\mathrm{H}(p(y|x))$ like Fig. 3 in [9]?\n2. Regarding the quantitative evaluation of the epistemic uncertainty in Table 5: How about the performance of DEs? Can you add an evaluation by calculating $Var_i[\\mu_i(x)]$ in [10]?\n3. Regarding the robustness performance, can you compare your method with DEs regarding the Negative log-likelihood, test accuracy, and ECE on OOD data (e.g., train on CIFAR-10, test on CIFAR-10-C)?\n4. Regarding Table 1: What is the number of $K$ used for this setting? Could you please specify why there are significant differences in your reported ECE with [1] on ImageNet about DEs?\n5. I am also curious about the results of different settings of DEs. For instance, the DEs used in your paper is the mixture of experts (or operation), how about the product of experts (and operation)?\n\nReferences:\n\n[1] Nado et al., Uncertainty Baselines: Benchmarks for Uncertainty & Robustness in Deep Learning, arXiv preprint arXiv:2106.04015, 2021.\n\n[2] Gal et al., Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learning, ICML, 2016.\n\n[3] Wen et al., J. Batchensemble: an alternative approach to efficient ensemble and lifelong learning, ICLR, 2020.\n\n[4] Dusenberry et al., Efficient and scalable Bayesian neural nets with rank-1 factors, ICML, 2020.\n\n[5] Havasi et al., Training independent subnetworks for robust prediction, ICLR, 2021.\n\n[6] Liu et al., Simple and principled uncertainty estimation with deterministic deep learning via distance awareness, NeurIPS, 2020.\n\n[7] Charpentier et al., Natural posterior network: Deep bayesian predictive uncertainty for exponential family distributions, ICLR, 2022.\n\n[8] Guo et al., On Calibration of Modern Neural Networks, ICML, 2017.\n\n[9] Lakshminarayanan et al., Simple and scalable predictive uncertainty estimation using deep ensembles, NeurIPS, 2017.\n\n[10] Valdenegro-Toro et al., A Deeper Look into Aleatoric and Epistemic Uncertainty Disentanglement, CVPR Workshop, 2022.\n\n---\n\n**After rebuttal.** The authors provide detailed additional results, and I believe that they are valuable to support the claim of the main paper. So, I increased my initial rating for this paper.", "limitations": "Please see my comments on the Weaknesses.", "rating": 5, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "- The paper is well-written and clear to understand. Sections are well organized, and the proposed method and its components are clear.\n- I like the proposed CreNet final layer. I also like the motivation of this paper, the topic of improving uncertainty and robustness performance in Neural Networks (NNs) is important and has broad applications in the AI safety field.\n- The empirical evaluation covers different task performances in multiple experimental settings.", "weaknesses": "1. The theoretical contributions of this paper are weak. There are no theories to show the proposed method and the loss function can improve the uncertainty or robustness quality. The only theoretically sound result is showing the results in Eq. 4 satisfy convexity conditions in Eq. 2, but this can be derived straightforwardly by using the sum of standard exponential functions for the output vector.\n2. Regarding the practical novelty, I also raise a concern about the computational limitation of the proposed method. Because DEs has been raised as computationally inefficient in estimating the NN uncertainty field [1, 3, 4, 5], CreDEs is even less efficient than DEs. In particular, for each ensemble member, in training time, the complexity of Alg. 1 with sorting and the training objective function is more complex than Standard Neural Network (SNN). In test time, the model also requires doubling the number of output nodes than SNNs and needs to solve two optimization problems in Eq. 13, causing a higher latency.\n3. The proposed method only works for classification tasks. Additionally, it is also a heuristic and depends on hyperparameter tuning. In particular, choosing the adversarial weights $\\displaystyle w$ for DRO is non-trivial and can significantly change the training objective function. Furthermore, choosing the number of class $K$ in Alg. 2 is also heuristic, causing a trade-off between uncertainty estimation quality and computational efficiency. \n4. Lack of empirical comparison with many related works in uncertainty baselines [1]. I would be happy to raise my initial score if important baselines will be added, including MC Dropout [2], BatchEnsemble [3], Rank-1 BNN [4], MIMO [5], SNGP [6], and NatPN [7].\n5. Lack of uncertainty evaluations and comparisons (see Questions). Better performance in OOD detection or active learning tasks is only a necessary condition, it is not a sufficient condition for a high-quality epistemic uncertainty algorithm.\n6. The training complexity of CreNet is considerably higher than SNN in L-712, causing CreDEs's complexity to be even much higher than DEs by linearly increasing the number of ensemble sizes. The model inference complexity in Tab. 6 is not evaluated in different model architectures across different hardware settings, e.g., GPU, CPU, etc.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Dear Reviewer,\n\nWe thank you for your valuable feedback. \n\n**Response to W1:** We believe that lines 172-175 in the original paper are also an important result, which shows that cross-entropy loss (CE) can be applied to upper probability vectors as representatives of part of the boundary of the predicted credal set.\n\nHowever, in the original paper, indeed, the only validation of our hypothesis that credal prediction leads to better accuracy, calibration, and OOD detection is empirical. Below we outline a more rigorous argument.\n\nEpistemic uncertainty in predictions is induced by the probability distribution(s) that generated the training data may be very different from the distribution that generates a test point. It is well known that using CE as a loss leads to the maximum likelihood estimator of the model parameters, under the assumption that all training points are drawn i.i.d. from a single, unknown distribution $P$:\n$$\n\\hat{\\theta} = \\arg \\min_\\theta\\text{CE} = \\theta_{MLE}.\n$$\n\nWhat does Distributionally Robust Optimization (DRO) do? It takes a cautious stance, and assumes training points are generated by one of many potential distributions $\\mathcal{U}$, and minimizes the upper bound of the expected risk (Eq. (6) of the paper). In practice, this is approximated using Adversarially Reweighted Learning as in Eq. (9). Therefore, it learns\n\t\t\t$\\hat{\\theta} = \\arg \\min_{P \\in \\mathcal{U}} E_P (Loss)$.\nHowever, if we apply DRO to a traditional network, the network will learn a single parameter value (Eq. (9) of the paper); for a new test data point $x_t$ it will then produce a single predicted probability vector, the one produced by the model that, in the long run, minimizes the worst-case risk.\n\nThis, though, does not express at all the (epistemic) uncertainty about *which* of the distributions in $\\mathcal{U}$ has generated the particular test datapoint $x_t$; as a result, it does not express how uncertain the prediction may be for that particular data point.\n\nAllowing a network to output lower and upper probability vectors for each input, instead, allows this epistemic uncertainty to be explicitly modeled.\n\nWhat is the best-case scenario? That in which all data (including the test point $x_t$) are indeed generated by the same distribution – in this case, $\\theta_{MLE}$ is the best choice, in a maximum likelihood sense.\n\nWhat is the worst case? That in which $x_t$ is generated by a distribution that is far away from the groups of distributions $P_g$ (see lines 140-141) that generated the training data. In this case, the DRO solution will still be far from perfect, but still the best one can do, given the evidence in the training data.\n\nTherefore, if I minimize the CE versus $\\boldsymbol{q}_U$ (first component of Eq. (10)), I get a credal set Cre of predictions whose upper probability is such that, for each class $i = 1,…,C$, there exists a probability vector $\\boldsymbol{q}\\in \\mathbb{Q}$ such that $q(i)$ is the MLE prediction for that class (please remember the discussion of lines 172-175).\n\nSimilarly, if I minimize the DRO loss versus $\\boldsymbol{q}_L$ (the second component of Eq. (10)), for each class $i = 1,…,C$, there exists a probability vector $\\boldsymbol{q} \\in \\mathbb{Q}$ such that $q(i)$ is the DRO prediction for that class. \n\nMinimizing the two components together, as in Eq. (10), thus encourages the predicted credal set to extend from the best-case to the worst-case prediction.\n\n**Response to W2, W4, and W6**: We kindly invite you to follow the detailed response in our global rebuttal. Based on your suggestions, we further reported additional inference complexity comparison in Table 1 of the rebuttal file and included MC Dropout and two TensorFlow-standardized BNNs.\n\n**Response to W3**: Yes, our method is for classification and we provide a roadmap for extending it to regression tasks in Appendix E.3. Our training includes a hyperparameter $\\delta$. We performed the ablation study (lines 287-312), showing that our CreDEs are not particularly sensitive to $\\delta$. $K$ is a design choice by the end user. The ablation study in Appendix B.2 shows that increasing the value of $K$ improves performance and increases cost. In addition, the reduction algorithm has a strong mathematical background in probability intervals [17], which can rigorously guarantee a valid credal set.\n\n**Response to W5, Q1, Q2**: Based on your advice, we reported the entropy plots and reliability diagrams in the rebuttal file. Regarding Table 5, the EU estimates of DEs-5 are 0.0996 (CIFAR10), 0.4758 (SVHN), and 0.4753 (Tiny-ImageNet). Although the EU estimates of DEs-5 and CreDEs-5 are not comparable in values due to different representations, CreDEs-5 exhibits much higher EU estimates of OOD samples than those of ID data and more reliable predictions on ID samples, qualitatively. We could not report $Var_i[\\mu_i(x)]$ in our cases because the calculation is for models with a custom softmax layer, according to Eq. (6-8) in [10].\n\n**Response to Q3**: In our settings, the corrupted CIFAR10 samples will be detected as OOD data and be to abstain. Therefore, we did not report the accuracy and ECE values on CIFAR10-C. For completeness, we reported the numbers in Table 3 of the rebuttal file, showing the visibly improved performance of CreDEs compared to DEs.\n\n**Response to Q4**: $K$ is used to reduce the complexity of uncertainty estimation. We do not employ Alg. 2 for accuracy and ECE calculation. The difference is most likely because the training settings are significantly different. As the ImageNet experiments are conducted fairly for DEs and CreDEs (shown in Appendix C), we believe the results can properly reflect the performance improvements.\n\n**Response to Q5**: We randomly selected five members from the 15 models to construct the ensemble. The member list of each ensemble was strictly different. Could you please specify the settings that you are curious about? We are eager to continue the discussion.", "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:
Dear Reviewers,

We appreciate your efforts in reviewing our paper and recognizing the novelty and strengths of our work. In the following, we would like to address your concerns regarding the complexity of our Credal Deep Ensembles (CreDEs) method and the comparison baselines.

**Part 1: added complexity compared to classical Deep Ensembles**

**Response:** We fully acknowledge the practical limitations of our method when there are strict computational resources (Lines 347-348). However, the extensive experiments show the scalability of our CreDEs on larger datasets (e.g., ImageNet) and larger network architectures (e.g., vision transformer). In addition, CreDEs consistently demonstrate visible improvement in a wide range of tasks and datasets, compared to deep ensembles (DEs). Therefore, CreDEs have great practical potential to enhance the performance of classical DEs in diverse real-world applications, such as medical image analysis [42], flood uncertainty estimation [10], structural health monitoring [70], or uncertainty-aware human motion forecasting [E].

Regarding inference time, doubling the final layer nodes would slightly increase the inference time. For instance, the inference time per sample for a ResNet50 architecture on the ImageNet dataset is 5.5 ms for a single standard neural network, vs 5.7 ms for a single CreNet (a marginal increase). The inference cost on the CIFAR10/100 dataset reported in *Table 1 of the rebuttal file* further demonstrates a slight complexity increase in our method.

Regarding the uncertainty estimation cost, we reported the cost of calculating the Generalized Hartley (GH) measure and the upper entropy in Figures 7 and 8 of the paper, respectively. For instance, the time cost for GH calculation for CIFAR10 without approximation is 17 ms (0.02 ms in the reduced case considering 4 out of 10 classes) while calculating the Epistemic Uncertainty (EU) in deep ensembles for CIFAR10 takes $4.1\times 10^{-4}$ ms, measured on the same single CPU. Though higher CreDEs remain practical without actual computational constraints. Besides, the numbers reported are without code efficiency optimization: a more efficient code implementation could significantly reduce the cost.

Regarding the training complexity, the CreNet uses a custom training loop, unlike the TensorFlow-standardized training standard neural networks, precluding a fair comparison. Given the evidence that CreNets only marginally increases the inference time (single forward pass) in Table 6, we are optimistic that by standardizing and optimizing the custom training loop and adopting a more efficient code implementation of Algorithm 1, the training effort could be significantly reduced. 

**Part 2: Comparison limited to DEs**

**Response:** We chose DEs for comparison, as they serve as a strong baseline for uncertainty estimation, e.g., shown in studies [A, B, C].  More recently, study [A] shows that DEs are good choices across the board for uncertainty estimation in their evaluation. Through extensive evaluation, we draw our main conclusion: CreDEs show significantly and consistently improved performance, uncertainty quality, OOD detection, and model calibration in several popular and important setups. We much appreciate the reviewers' acknowledgment of the extensive evaluation as one of the strengths of our paper.

The main reason for excluding Bayesian neural network (BNN) approaches in our original paper is that they generally have difficulty scaling to large datasets and complex network architectures [C]. Inspired by the suggestions, we conducted an additional comparison between CreDEs and DEs, MCDropout [F], and two TensorFlow-standardized BNNs (BNN-R [G]  and BNN-F [H]). All the models are trained on the ResNet50 for the CIFAR10 dataset from scratch. The Adam optimizer is applied with a learning rate scheduler, initialized at 0.001. The learning rate is subject to a reduction of 0.1 at epochs 80 and 120. For BNNs, 10 forward passes are used for uncertainty estimation. The uncertainty evaluation via OOD detection on the CIFAR10 vs SVHN/Tiny-ImageNet dataset is reported in *Table 2 of the rebuttal file*. The results consistently demonstrate the significant improvements of our methods.

We excluded the Dirichlet-based approaches (lines 69-76) (other branches of second-order models) mainly because, they are generally trained to predict Dirichlet distributions from only one-hot label data. A criticism of these methods is the lack of Dirichlet distribution labels in the training process, and the performance of the models often deviates from theoretical EU assumptions [69]. In addition, a recent study [D] showed that epistemic uncertainty is generally not faithfully represented in these methods and the resulting measures of epistemic uncertainty cannot be interpreted quantitatively.

We will duly extend the above discussion in the Limitation and Future work in the revised version. One of the essential objectives of our future work is to build a universal uncertainty evaluation framework, and comprehensively assess our methods together with various other models. We believe that the kind of broader comparison you suggest will find its right place within the expanded scope of that work.

[A] Benchmarking uncertainty disentanglement: Specialized uncertainties for specialized tasks. 2024.

[B] Deep ensembles work, but are they necessary? NeurIPS 2022

[C] Deep deterministic uncertainty: A new simple baseline. CVPR 2023

[D] Is Epistemic Uncertainty Faithfully Represented by Evidential Deep Learning Methods? ICML 2024

[E] De-tgn: Uncertainty-aware human motion forecasting using deep ensembles. IEEE Robotics and Automation Letters 2024

[F] Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learning, ICML 2016

[G] Variational dropout sparsifies deep neural network. ICML 2017

[H] Flipout: Efficient pseudo-independent weight perturbations on mini-batches. iCLR 2018

Meta-review: Dear reviewers, thank you for all the hard work you put in your reviews and in engaging with the authors and their responses.
To the Authors, thank you for giving such a detailed rebuttal.

After reading all of the reviews and comments, I am very happy to recommend this paper for acceptance. Especially after the authors in depth rebuttal.  

Best,

    AC
Decision: Accept (poster)