Learning Credal Ensembles via Distributionally Robust Optimization

David Moens; Fabio Cuzzolin; Ghifari Adam Faza; Hans Hallez; Kaizheng Wang; Siu Lun Chau

REVIEW 2 major objections 4 minor 2 cited by

Learning Credal Ensembles via Distributionally Robust Optimization

T0 review · 2 major / 4 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read CreDRO trains an ensemble with varied degrees of i.i.d. relaxation so epistemic uncertainty reflects potential train–test shifts, not just random initialization.

desk verdict A useful empirical recipe for credal ensembles, but the central mechanism — the spread of δ_i — is never isolated by a fixed-δ control experiment. read the letter →

arxiv 2602.08470 v3 pith:SOE6I2WR submitted 2026-02-09 cs.LG stat.ML

Kaizheng Wang , Ghifari Adam Faza , Fabio Cuzzolin , Siu Lun Chau , David Moens , Hans Hallez This is my paper · ORCID

classification cs.LGstat.ML

keywords epistemicuncertaintycredalsetsdistributionallyrobustoptimizationdeepensemblesout-of-distributiondetectionselectiveclassificationprobabilityintervalsensembledisagreement

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper argues that epistemic uncertainty in deep learning should come from disagreement about how much the training and test distributions may differ, not mainly from which random initialization a model happened to get. To do this it proposes CreDRO, an ensemble in which each member is trained with a different degree of distributionally robust optimization: member i keeps only the top δ_i highest-loss samples in each batch, interpolating from a worst-case δ_G to full empirical risk minimization. At inference, the members' softmax outputs define a box credal set—a convex set of probability vectors bounded by per-class min/max probabilities—and epistemic uncertainty is measured by the entropy gap across that set. On out-of-distribution detection and selective classification in histopathology, the paper reports that CreDRO outperforms existing credal classifiers and deep ensembles.

What carries the argument

The load-bearing object is the truncation schedule δ_i = (1−δ_G)/(M−1)·(i−1)+δ_G applied inside the adversarially reweighted learning objective. For each batch, samples are sorted by loss and only the top δ_i fraction is backpropagated; this is the paper's pragmatic stand-in for group DRO, with δ parameterizing the degree of assumed train–test divergence. The second ingredient is the box credal set K_B = {p : p_k ∈ [min_i p_{i,k}, max_i p_{i,k}], Σ p_k = 1}, built from the ensemble's softmax outputs. Epistemic uncertainty is then the difference H(K_B) − H(K_B), computable by two constrained entropy optimizations whose cost depends on the number of classes, not the ensemble size.

What would settle it

Run CreDRO with the δ_i assignment shuffled across members (or reversed) while keeping the same multiset of δ values; if OOD detection AUROC is unchanged, the explanatory story—ordered degrees of i.i.d. relaxation—is not what drives the gains. Additionally, train a variant that replaces top-δ truncation with exact sample reweighting from an explicit divergence ball; large performance differences would show the heuristic, not DRO, is responsible.

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Extended reading notes

Core claim

The central claim is that disagreement between models trained under different relaxations of the i.i.d. assumption is a more meaningful source of epistemic uncertainty than disagreement from random initialization. The paper formalizes this by assigning each ensemble member a different truncation fraction δ_i in an adversarially reweighted DRO objective: during each batch, only the top δ_i fraction of samples by loss are used for backpropagation, which implicitly upweights hard-to-learn examples and simulates a hypothesized distribution shift. The δ_i are spread uniformly from a user-chosen worst-case δ_G (default 0.5) to 1 (vanilla ERM). The resulting softmax probabilities are converted to a

Load-bearing premise

The whole construction rests on the assumption that keeping the top-δ highest-loss samples in a batch behaves like a true DRO reweighting and that the uniform grid of δ values covers the plausible train–test shifts; if either fails, the ensemble diversity is arbitrary and the epistemic uncertainty is not really about distribution shift.

Editorial extensions

If this is right

Epistemic uncertainty now reflects potential distribution shift, so OOD and shifted-domain samples receive higher uncertainty, improving out-of-distribution detection when uncertainty is used as the score.
Selective classification in a realistic medical domain shift (histopathology from unseen scanners) improves, enabling safer abstention based on shift-induced disagreement.
The method requires no change to network architecture—only per-member top-δ loss truncation during training—so it can wrap existing deep ensembles and loss functions.
Performance is stable across the choice of δ_G because the ensemble spans a range of δ values; smaller δ_G produces wider probability intervals, consistent with more pessimistic shift assumptions.
DRO-trained ensembles alone, without credal post-processing, also outperform standard deep ensembles, locating part of the benefit in the training procedure rather than in the credal representation.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the mechanism is really the ordering of members by shift sensitivity, then a direct test is to shuffle or reverse the δ_i assignment across members: the OOD detection gain should degrade, since the ensemble would no longer span meaningful shift levels.
The less competitive performance of a closely related method that uses a single fixed DRO parameter suggests that the diversity of δ values, not DRO alone, is what matters; a reader could isolate this by training a single DRO model and perturbing it with different top-δ schedules.
A natural extension the paper only sketches is regression, where Gaussian predictions would form a credal set of mixtures; one could test whether the same shift-derived disagreement transfers to continuous targets.
Because the paper offers no theoretical link between δ and a conventional divergence measure, the interpretation of δ as a 'degree of distribution shift' remains a heuristic; connecting δ to, say, a chi-square or KL ball would make the method's semantics testable.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

CreDRO is a simple, well-engineered recipe: train each ensemble member with a different top-δ truncation of high-loss samples, then take min/max softmax bounds into a box credal set. The OOD detection and selective classification results are consistently better than strong baselines across several benchmarks, and the ablations (ensemble size, δG, box vs convex hull) are reasonably thorough. The idea that EU should reflect disagreement under varying relaxations of the i.i.d. assumption is a genuinely useful reframing, and the per-member δ grid is new relative to CreDE's single fixed DRO level.

The soft spot is the one the stress-test identifies, and I think it lands. There is no control experiment where every member is trained with the same δ but otherwise identical setup (same architecture, loss, credal construction). EN-DRO also uses the δ grid, CreDE changes the architecture and loss, and Table 4 varies δG while keeping the spread to 1.0. So we don't actually know whether the spread of δ_i is what drives the gains, or whether it's just that each member is trained with hard-example reweighting, which individually improves accuracy and calibration (Table 7 shows DRO members at ~95.4% vs ~94.3% for CreWra). If a fixed-δ ensemble performs the same, the paper's central claim—that EU captures shift-meaningful disagreement—is not supported; it would reduce to 'hard-example reweighting improves members.' The lack of any theoretical link between top-δ truncation and a divergence measure makes this gap load-bearing, though not fatal.

Minor issues: no code in the preprint (promised on publication), no significance tests, CreRL/CreEns reported at their best α, and CreDE results missing on CIFAR100-C with a somewhat hand-wavy explanation. The self-citations are heavy but not abusive; they're mostly to the authors' own credal framework, and CreDRO doesn't reduce to those prior results.

Who this is for: people working on epistemic uncertainty, selective classification, OOD detection. It's an empirical paper with a clean recipe and consistently favorable results; the central mechanism just isn't isolated yet. A serious referee should engage; the fixed-δ control is the one experiment that would clarify the claim. I'd send it to review, and I'd expect revision rather than rejection.

Referee Report

2 major / 4 minor

Summary. The paper proposes CreDRO, a deep-ensemble method in which each member is trained under a different degree of i.i.d.-assumption relaxation, implemented via batch-wise truncation to the top-δ high-loss samples. The resulting softmax predictions are converted to a box credal set, and epistemic uncertainty is quantified as the upper-minus-lower entropy of that set. The authors claim that CreDRO captures distribution-shift-related epistemic uncertainty, and they report consistent gains over deep-ensemble and state-of-the-art credal baselines across OOD detection benchmarks and a medical selective-classification task.

Significance. The central idea—defining epistemic uncertainty as disagreement among models trained under different plausible train–test shift scenarios rather than only under random initialization—is timely and conceptually appealing. If the mechanism is validated, the paper would offer a practical way to inject distribution-shift awareness into credal ensembles with no architectural change. The experimental effort is extensive: multiple OOD benchmarks, corrupted-data tests, selective classification on Camelyon17, and ablations for ensemble size, δG, credal-set construction, and member accuracy. The plan to release code is a positive for reproducibility. However, the paper's central claim about the source of the EU quality is not yet isolated experimentally, and the interpretation of δ as a level of distribution shift is heuristic rather than derived.

major comments (2)

[§3.1, Eq. (6); Table 1; Table 4; Table 7] The paper attributes CreDRO's EU quality to 'disagreement among models trained with varying relaxations of the i.i.d. assumption' (Abstract; §3.1). No experiment isolates the variation mechanism. EN-DRO also uses the δ_i grid of Eq. (6), so the DE-vs-EN-DRO comparison does not control for the spread; CreDE uses a fixed DRO level but changes the architecture and loss. The δG ablation (Table 4) always keeps the spread to 1.0, so it cannot reveal whether the spread matters. Table 7 shows per-member accuracy improves from ≈94.3% (CreWra) to ≈95.4% (CreDRO), so the AUROC gains could plausibly be explained by better per-member predictions. I request an ablation in which all M members train with the same δ (e.g., δ=0.5, 0.75, 0.9), while keeping architecture, schedule, and credal construction identical. If a fixed-δ ensemble matches CreDRO, the central claim reduces to 'hard-example reweighting
[§3.1, Eq. (4)–(6)] The top-δ truncation is presented as a 'flexible approximation' of adversarially reweighted learning, but no derivation connects the truncation to a distribution-shift neighborhood in Eq. (2) or (4). The interpretation of δ_i as a 'degree of relaxation of the i.i.d. assumption' is therefore a heuristic. This matters because the conceptual contribution is precisely that the ensemble disagreement reflects distribution-shift uncertainty. Please either (a) provide a formal characterization of the induced reweighting (e.g., as a CVaR/divergence-ball maximization with a known δ), or (b) soften the causal claim to describe the method as an empirically useful heuristic. The fixed-δ control from the previous comment would also help distinguish the effect of varying δ from the effect of DRO training per se.

minor comments (4)

[§4.1] Clarify the hyperparameter selection for CreRL and CreEns. 'Report their best results' is too vague; state whether α was selected on a validation split and provide the sensitivity of the baselines to α in an appendix or table.
[Figure 4 / §4.5] CreDE is missing for CIFAR100 vs CIFAR100-C; the explanation is in Appendix C but should be summarized in the caption or main text so the reader is not left puzzled by the absent curve.
[Algorithm 1] The last step says 'Minimize' but does not explicitly state that this defines the per-batch loss for an optimizer update; please clarify notation and connect Step 5 to the standard SGD/backprop update.
[Table 1] Some advantages are within 1–2 standard deviations (e.g., Places: CreDRO 92.7±0.1 vs CreRL 91.8±0.2); consider reporting paired significance tests or effect sizes, and specify whether runs share seeds across methods for fairer comparisons.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: CreDRO's central claim is evaluated against external benchmarks, and its δ-grid is a design choice rather than a fitted or self-referential prediction.

full rationale

CreDRO's derivation chain is self-contained in the relevant sense. The training procedure (Algorithm 1; Eq. 6) fixes member-specific DRO levels via a user-chosen δ_G and a uniform interpolation formula; no parameter is fitted to the OOD/selective-classification targets and then renamed as a prediction. The resulting box credal set (Eqs. 7–8) and its EU measure (upper-lower entropy, Eq. 10) are applied to these trained members, and performance is scored on external benchmarks (SVHN, Places365, CIFAR100, FMNIST, ImageNet, CIFAR-C, Camelyon17). This is an empirical evaluation of an explicit design, not a reduction of the output to the input by construction. The paper cites the authors' prior credal frameworks (CreDE, CreWra) as baselines and background, but the central mechanism—varying δ_i to produce distribution-shift-related disagreement—does not reduce to those prior results. The paper itself flags the subjective nature of the weight set and the absence of a rigorous theoretical link ('The predefined set of weight vectors W is a subjective design choice...'; 'a rigorous theoretical analysis... left for future work'), and the lack of a fixed-δ control is a potential confound; these are correctness/experimental-design concerns rather than circularity. No specific equation or fitted quantity is identical to another by construction, so no circular step is established.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

CreDRO introduces no new particles, forces, or latent variables. Its free parameters are limited to the hand-set δG and the design choices of ensemble size and credal-set type. The method rests primarily on domain assumptions connecting DRO truncation to distribution shift and box credal sets to epistemic uncertainty.

free parameters (1)

δG (global DRO truncation hyperparameter) = 0.5 (default; searched over {0.5,0.6,0.7,0.8,0.9} in ablation)
Chosen by hand to represent the assumed worst-case train–test divergence. It fully determines the per-member schedule δi via Eq. (6). The ablation shows robustness, but no principled way to set it is given.

assumptions (5)

domain assumption Top-δ high-loss sample truncation approximates adversarially reweighted group DRO and simulates potential domain shifts.
Section 3.1 and Algorithm 1: only the top δ fraction of each batch is backpropagated, with the rationale that hard-to-learn instances correspond to minority groups. This link is asserted, not validated.
domain assumption Uniform interpolation of δi between δG and 1 spans the range of plausible train–test distribution shifts.
Eq. (6) defines δi = (1−δG)/(M−1)·(i−1)+δG. The paper assumes this schedule produces 'plausible models' reflecting different degrees of i.i.d. relaxation.
domain assumption The box credal set built from per-class min/max softmax probabilities is a valid representation of predictive epistemic uncertainty.
Section 3.2, Eqs. (7)–(8), following De Campos et al. (1994). The choice of the box over the convex hull is justified empirically in Section 4.4, not theoretically.
domain assumption Upper–lower entropy difference H(KB)−H(KB) is an appropriate measure of epistemic uncertainty for credal predictions.
Section 3.2, based on Abellán et al. (2006). Alternative measures are acknowledged in the related-work appendix, and the binary-case width measure is used for Camelyon17.
domain assumption OOD-detection AUROC and selective-classification AUC are valid proxies for epistemic-uncertainty quality.
Section 4.1: since ground-truth EU does not exist, OOD detection is used as a practical benchmark. The paper relies on this external proxy rather than a direct EU ground truth.

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Cite this review

Pith. "Pith review of Learning Credal Ensembles via Distributionally Robust Optimization." pith.science (2026). https://pith.science/paper/SOE6I2WR

@misc{pith2026260208470,
  author       = {Pith},
  title        = {Pith review of: Learning Credal Ensembles via Distributionally Robust Optimization},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SOE6I2WR}},
  note         = {Machine review of arXiv:2602.08470}
}

read the original abstract

Credal predictors are models that are aware of epistemic uncertainty and produce a convex set of probabilistic predictions. They offer a principled way to quantify predictive epistemic uncertainty (EU) and have been shown to improve model robustness in various settings. However, most state-of-the-art methods mainly define EU as disagreement caused by random training initializations, which mostly reflects sensitivity to optimization randomness rather than uncertainty from deeper sources. To address this, we define EU as disagreement among models trained with varying relaxations of the i.i.d. assumption between training and test data. Based on this idea, we propose CreDRO, which learns an ensemble of plausible models through distributionally robust optimization. As a result, CreDRO captures EU not only from training randomness but also from meaningful disagreement due to potential distribution shifts between training and test data. Empirical results show that CreDRO consistently outperforms existing credal methods on tasks such as out-of-distribution detection across multiple benchmarks and selective classification in medical applications.

Figures

Figures reproduced from arXiv: 2602.08470 by the authors.

**Figure 1.** CreDRO Concept. ① Training: An ensemble is trained using distributionally robust optimization with members weighted differently to simulate varying degrees of train-test distribution shifts (see Section 3.1). ② Inference: Softmax probabilities are converted into class-wise probability intervals, creating a box credal set (see Section 3.2). Contributions. In response, we formulate EU as disagreement among ensemble me… view at source ↗

**Figure 2.** AUROC (%) for OOD detection using EU, across methods and ensemble sizes. samples assigned to classes 0 and 1, respectively. The model’s uncertainty estimate is then used as the prediction score, and the performance is evaluated via the area under the receiver operating characteristic curve (AUROC). Setup. We benchmark our CreDRO against several SOTA credal classifiers considered in a recent study [Löhr et al., 2025]… view at source ↗

**Figure 3.** Kernel density plots of EU estimates on ID and OOD data from distinct methods. (ensemble size: M = 5; first-run results) 4.3 Ablation study on training hyperparameter δG Setup. In the main experiment, we set δG to 0.5 by default to reflect a balanced assessment of train-test divergence and to evaluate how this value helps our model outperform the baselines. This experiment conducts an ablation study on different val… view at source ↗

Figures from the paper (9 more)

**Figure 4.** Figure 4: OOD detection score comparison across increasing levels of corruption. Results are averaged from 15 runs. Note that the results of CreDE on CIFAR100 vs CIFAR100-C are missing because its generated probability intervals are too tight (see [PITH_FULL_IMAGE:figures/full_…

**Figure 5.** Figure 5: AR (top) and normalized AR (bottom) curves, along with the average AUC values. Note that the accuracy is set to 100% when the rejection rate reaches 1.0. To decouple selective performance from the model’s base classification accuracy, we further report normalized AR cu…

**Figure 3.** Figure 3: WSI (patient_017_node_2) with ground-truth annotations and model predictions. Predictions have been generated from a ResNet Ensemble. The tumor confidence decreases in areas near the border of the annotation, while the uncertainty thereby increases. This result is cons…

**Figure 7.** Figure 7: Kernel density plots of the averaged probability interval length (PIL) over classes on CIFAR100-C datasets with 19 types of corruptions: CreDE (top) and CreDRO (bottom). The averaged PIL for an input instance is computed as PIL = 1 100 P100 k=1 pk − p k . The results s…

**Figure 8.** Figure 8: Kernel density plots of EU estimates on ID and OOD data from distinct methods. (ensemble size: M = 10; first-run results) 0.0 0.5 1.0 1.5 EU Estimates 0 5 10 Density DE 0.0 0.5 1.0 1.5 EU Estimates 0 5 10 EN-DRO 0 1 2 3 EU Estimates 0 1 2 3 CreDRO 0 1 2 3 EU Estimates …

**Figure 9.** Figure 9: Kernel density plots of EU estimates on ID and OOD data from distinct methods. (ensemble size: M = 15; first-run results) [PITH_FULL_IMAGE:figures/full_fig_p027_9.png]

**Figure 10.** Figure 10: Kernel density plots of EU estimates on correctly and incorrectly classified medical test samples, across different classifiers. The plots show that CreDRO produces noticeably higher EU values for incorrectly classified samples than for correctly classified ones, qual…

**Figure 11.** Figure 11: Kernel density plots of the averaged probability interval length (PIL) over classes on different datasets. The results empirically show that larger values of δG generally lead to larger box credal sets. 28 [PITH_FULL_IMAGE:figures/full_fig_p028_11.png]

**Figure 12.** Figure 12: Code implementation for computing the entropy difference of the box credal set (left) and the convex hull credal set (right), following Löhr et al. [2025]. 30 [PITH_FULL_IMAGE:figures/full_fig_p030_12.png]

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Reference graph

Works this paper leans on

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Additionally, Wang et al

CreBNN represents weights and predictions as credal sets and has demonstrated robustness; however, the setting of credal set prior still relies on random initialization and its computational cost is comparable to that of Bayesian neural network ensembles, significantly limiting practical applicability. Additionally, Wang et al. [2025c] introduces credal-s...

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!') #' %(

δG Individuals’{δ i}5 i=1 per Member Number of Selected TopδSamples per Batch 0.5 0.50, 0.625, 0.75, 0.875, 1.00 64, 80, 96, 112, 128 0.7 0.70, 0.775, 0.85, 0.925, 1.00 89, 99, 108, 118, 128 0.9 0.90, 0.925, 0.95, 0.975, 1.00 115, 118, 121, 124, 128 A smaller value ofδG, which reflects a more pessimistic belief about potential distribution divergence, may...

2025
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Specifically, WSIs from centers 0, 1, and 3, which use 3DHistech scanners, make up the ID training and validation set

To convey a strong domain shift scenario [Mehrtens et al., 2023], the dataset is split so that the test set only contains images from scanners not present in the in-distribution (ID) training data, introducing an additional technological variation. Specifically, WSIs from centers 0, 1, and 3, which use 3DHistech scanners, make up the ID training and valid...

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Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.

[1] [4]

Additionally, Wang et al

CreBNN represents weights and predictions as credal sets and has demonstrated robustness; however, the setting of credal set prior still relies on random initialization and its computational cost is comparable to that of Bayesian neural network ensembles, significantly limiting practical applicability. Additionally, Wang et al. [2025c] introduces credal-s...

2021

[2] [5]

!') #' %(

δG Individuals’{δ i}5 i=1 per Member Number of Selected TopδSamples per Batch 0.5 0.50, 0.625, 0.75, 0.875, 1.00 64, 80, 96, 112, 128 0.7 0.70, 0.775, 0.85, 0.925, 1.00 89, 99, 108, 118, 128 0.9 0.90, 0.925, 0.95, 0.975, 1.00 115, 118, 121, 124, 128 A smaller value ofδG, which reflects a more pessimistic belief about potential distribution divergence, may...

2025

[3] [6]

Specifically, WSIs from centers 0, 1, and 3, which use 3DHistech scanners, make up the ID training and validation set

To convey a strong domain shift scenario [Mehrtens et al., 2023], the dataset is split so that the test set only contains images from scanners not present in the in-distribution (ID) training data, introducing an additional technological variation. Specifically, WSIs from centers 0, 1, and 3, which use 3DHistech scanners, make up the ID training and valid...

2023

[4] [7]

Distributionally robust neural networks for group shifts: On the importance of regularization for worst-case generalization

Shiori Sagawa, Pang Wei Koh, Tatsunori B Hashimoto, and Percy Liang. Distributionally robust neural networks for group shifts: On the importance of regularization for worst-case generalization. arXiv preprint arXiv:1911.08731,

arXiv 1911

[5] [8]

doi: 10.1016/j

ISSN 0925-2312. doi: 10.1016/j. neucom.2024.127929. URLhttp://dx.doi.org/10.1016/j.neucom.2024.127929. Rui Tuo and Wenjia Wang. Uncertainty quantification for Bayesian optimization. In Gustau Camps- Valls, Francisco J. R. Ruiz, and Isabel Valera, editors,Proceedings of The 25th International Conference on Artificial Intelligence and Statistics, volume 151...

arXiv 2024

[6] [9]

Credal ensemble distillation for uncertainty quantification.arXiv preprint arXiv:2511.13766, 2025a

Kaizheng Wang, Fabio Cuzzolin, David Moens, and Hans Hallez. Credal ensemble distillation for uncertainty quantification.arXiv preprint arXiv:2511.13766, 2025a. Kaizheng Wang, Fabio Cuzzolin, Keivan Shariatmadar, David Moens, and Hans Hallez. Credal wrapper of model averaging for uncertainty estimation in classification. InThe Thirteenth International Con...

arXiv

[7] [10]

""deffun(x: np.ndarray) -> np.ndarray:return-entropy(x, base=base)x0=probs.mean(axis=1)constraints={

Table 10:Summary of computational complexities for entropy optimization.TB,TC denote the number of iterations of the numerical optimizer. Quantity Problem Size Complexity per Instance BoxK B upper entropyCvariablesO(T B·C) BoxK B lower entropyCvariablesO(T B·C) Convex hullK C upper entropyMvariablesO(T C·M·C) Convex hullK C lower entropy trivialO(M·C) imp...

arXiv 2025

[8] [17]

For the training configurations, we follow the official GitHub repositoryhttps://github.com/ DBO-DKFZ/uncertainty-benchmark to train the deep ensemble (DE) model (M = 5), using the approach outlined in Mehrtens et al. [2023]. We then adjust the training of CreDRO and CreDE by setting the training hyperparameter to0.5, as suggested in the original work, to...

arXiv 2023