Credal Machine Learning for Risk-Averse Decision Making

1 Introduction

• Existing risk-averse learning methods do not account for uncertainty about the true data-generating distribution, which can matter most where the model knows least.The true CVaR is uncertain when training data are scarce or deployment conditions differ from training.Source: §1 · p. 1
• The medical example illustrates lower CVaR and avoidance of catastrophic decisions with the proposed method.The introduction frames this as a need to control worst outcomes, not merely optimize average performance.Source: §1 · p. 1, Figure 1 · p. 2
• The paper combines a credal predictor that represents epistemic uncertainty with a decision rule mapping credal sets to distributions for CVaR minimization.The predictor is designed to require only a single forward pass.Source: §1 · p. 2
• The paper reports that its approach avoids catastrophic decisions while sacrificing little expected performance across classification, distribution shift, and reinforcement learning.It presents the combination of credal prediction and CVaR minimization as essential for risk-averse decisions under known and unknown risk.Source: §1 · p. 2

2 Risk-Averse ML

• A low mean error can still be unsafe in safety-critical settings when the loss distribution has a heavy upper tail.The setup considers probabilistic predictors trained from i.i.d. labeled data.Source: §2 · p. 2

2.1 Conditional-Value-at-Risk

• CVaR at level τ is the mean of the upper τ-tail of the loss distribution, with the essential supremum and expected risk as limiting cases.Its minimum-threshold representation is attained at v*=VaR_τ(L).Source: §2.1 · pp. 2–3
• Empirically, CVaR can be estimated by averaging the ⌈τN⌉ largest sample losses, equivalent to top-k loss when τ=k/N.The section also situates CVaR among risk-averse learning objectives and related guarantees.Source: §2.1 · p. 3

2.2 Minimizing the CVaR

• The variational form of CVaR makes prediction separable by instance at a fixed threshold, with instances coupled only through that threshold.Unlike expected loss, CVaR itself does not decompose additively over instance-wise problems.Source: §2.2 · p. 3
• The CVaR-optimal prediction requires the unavailable ground-truth conditional and joint distributions, and errors matter especially in the loss tail.The hinge makes losses below the threshold irrelevant and shifts probability toward classes in the tail set; if all class losses are below threshold, the ground-truth distribution remains optimal.Source: §2.2 · p. 4

3 Credal Prediction

• Credal sets are sets of probability distributions that represent epistemic uncertainty and partial knowledge about a prediction.A single estimate of the conditional distribution cannot express that uncertainty.Source: §3.1 · p. 4, §3 · p. 4

3.1 Credal Sets from Relative Likelihood

• Relative likelihood defines plausible models by how well they explain the training data compared with the maximum-likelihood estimate, yielding nested likelihood regions as α varies.Under stated regularity conditions, the likelihood region can be viewed as a confidence region for the ground-truth model.Source: §3.1 · p. 4

3.2 Learning and Approximation

• The method assesses plausibility locally at each test instance, using how well candidate conditional distributions explain nearby training data.This construction adapts to shifted or out-of-distribution instances.Source: §3.2 · p. 5
• CreWare uses kernel-weighted training embeddings to form local class evidence and a KL-divergence ball whose radius depends on that evidence.Little nearby evidence produces a larger ball, while more evidence produces a smaller one; class-wise extremes over the ball define the credal set.Source: §3.2 · p. 5

4 From Credal Predictions to Risk-Averse Decisions

• The rule applies the worst-case principle to uncertainty over the ground-truth distribution by considering the worst case over each credal set.It is Γ-maximin applied to the hinged loss.Source: §4 · p. 6
• At α=1 the credal set collapses to the maximum-likelihood estimate, recovering precise CVaR-optimal prediction; at τ=1 the rule becomes Γ-maximin expected loss.As τ approaches zero, it tends toward Wald’s minimax act over outcomes deemed plausible by the credal set.Source: §4 · p. 6
• When α approaches zero, the credal set becomes the whole simplex and the rule reduces to Wald’s minimax act over all outcomes, independently of τ.The parameters interpolate among expected-risk minimization, precise CVaR prediction, robust Bayes, and Wald’s minimax.Source: §4 · p. 6
• The threshold v is a population-level scalar jointly determined with predictions, since each threshold induces predictions and their resulting loss distribution.The procedure evaluates candidate thresholds on hold-out data and selects the one minimizing empirical CVaR.Source: §4 · pp. 6–7

5 Empirical Results

• Across the evaluations, the credal predictor adapts to shift and OoD inputs, the full pipeline reduces CVaR under shift, and the approach applies to offline reinforcement learning.Source: §5 · p. 7

5.1 Credal Sets under Distribution Shift

• CreWare’s sets grow substantially as BloodMNIST corruption severity rises and accuracy falls, while baseline set sizes increase only slightly.Source: §5.1 · p. 7, Figure 2 · p. 8

5.2 Out-of-Distribution Detection

• CreWare achieves the highest AUROC in both near- and far-OoD settings, with Table 1 reporting results for CIFAR-10 as the in-distribution dataset.Source: §5.2 · p. 8, Table 1 · p. 8
• The OoD evaluation does not resolve the concern that supervised OoD detection is misspecified because epistemic uncertainty may concern labels rather than distribution membership.Source: §5.2 · p. 8

5.3 Risk-Aversion under Distribution Shift

• On PathMNIST, CreWare plus the rule matches or outperforms risk-averse baselines and CreWra on CVaR across τ levels and corruption severities, while sacrificing little mean-loss performance.Source: §5.3 · p. 9, Figure 3 · p. 9

5.4 Risk-Averse Reinforcement Learning

• In offline driving, Credal + rule maintains a 0% crash rate across observation counts; MLE + rule remains unsafe while hazards are under-observed.Source: §5.4 · pp. 9–10, Figure 4 · p. 10

• Existing credal decision rules hedge against ignorance about the ground-truth distribution but aggregate outcomes by expectation, so they can accept severe risks that are favorable on average.Source: §6 · p. 10
• CVaR and distributionally robust optimization address outcome-tail risk by reweighting observed losses, whereas credal sets range over plausible models; their parameters separately govern tail risk and epistemic uncertainty.Source: §6 · p. 10
• In sequential settings, safety under epistemic uncertainty requires both tail-risk control and uncertainty over models; Rigter et al. (2023) addresses both through a second-order posterior rather than prior-free credal sets.Source: §6 · p. 10

• The approach combines CVaR minimization over a credal set with a predictor based on local relative likelihood to support risk-averse decisions under epistemic uncertainty.Source: §7 · p. 10
• CVaR is challenging to use for learning because it does not decompose over training instances, while credal prediction is proposed to improve reliability.Source: §7 · p. 10
• The sensitivity of the approach to embedding quality needs further investigation, and whether its credal sets can provide guarantees like conformal prediction remains open.Source: §7 · p. 11

A Variations on the CVaR

• Marginal CVaR takes the loss tail under the joint distribution over instances and outcomes, avoiding compensation between high-risk instances that can occur when instance-wise CVaRs are averaged.Source: A Variations on the CVaR · p. 17
• Estimating instance-wise aleatoric tail risk requires ground-truth conditional distributions, which are usually unavailable in machine learning.Source: A Variations on the CVaR · p. 17
• The paper deliberately uses marginal CVaR because safety-critical applications concern rare large errors in the population as a whole, regardless of their origin.Source: A Variations on the CVaR · p. 17

B Extended Related Work

• Unlike prior credal predictors whose set width is governed by training procedures, this construction fits the set post hoc for each test instance.Source: B Extended Related Work · p. 18
• Existing credal decision rules hedge against ignorance but aggregate outcomes by expectation, while CVaR-based methods target outcome tails; keeping these roles distinct lets τ and α govern them separately.Source: B Extended Related Work · p. 18

Regularity conditions

• The proof identifies the tail classes using the threshold and shows that moving toward their conditional distribution decreases the expected loss under that tail distribution.Strict propriety makes the tail-conditional distribution differ from the ground truth when the tail is non-degenerate.Source: §C.1 · p. 19, Regularity conditions · p. 19

C.1 Minimizer of Expected Risk versus CVaR

• For sufficiently small perturbations toward the tail-conditional distribution, the CVaR objective falls, so the ground-truth distribution is not even a local minimizer under the stated regularity conditions.The argument assumes full support, finite continuous losses, and no class loss exactly at the CVaR threshold.Source: Regularity conditions · p. 19

C.2.1 Log-loss

• Under log-loss with p∗ = (0.8, 0.2) and v = 1, the minimizer is (e^-1/2, 1 − e^-1/2), reducing the likely class probability to the hinge threshold.The minimizer remains unchanged for every p∗(y1) > e^-1/2.Source: §C.2.1 · p. 19

C.2 Derivations of Examples

• For log-loss, the objective is convex and piecewise smooth, with hinge activation changing at the two stated probability breakpoints.The minimum follows where the likely-class probability reaches the breakpoint at which its hinge becomes active.Source: §C.2.1 · p. 19

C.2.2 Brier score

• For the binary Brier-score example with p∗ = (0.8, 0.2) and v = 1, the minimizer is approximately (0.65, 0.35), pushing down the likely-class probability to the hinge threshold.For the one-coordinate binary Brier score with threshold v = 1/4, the uniform distribution is the unique minimizer and has J = 0.Source: §C.2.2 · p. 20

Fitting

• CreWare fits post hoc by embedding training examples, applying L2 normalization and selected whitening, then storing whitened embeddings, labels, the whitening map, and kernel bandwidth.Whitening strength is selected using leave-one-out accuracy and a predicted-class-share constraint; the Gaussian-kernel bandwidth is set from nearest-reference distances.Source: D Algorithmic Details · p. 21

Prediction

• Prediction compares each whitened test embedding with 200 reference points, constructs class-wise evidence and credal bounds, and uses 50-step bisections to obtain bounds at a cost of K bisections per instance.The described prediction cost also includes one forward pass and one N × d distance computation.Source: D Algorithmic Details · p. 21

Efficiency and embedding compression

• Compressing N reference embeddings to M landmarks makes distance computation cost M · d per instance, removing its linear dependence on training-set size.Landmarks store cluster centroids and class histograms.Source: D Algorithmic Details · p. 21
• As landmark count M falls, fitting gets faster while inference time does not decrease significantly and AUROC remains largely the same on CIFAR-10.The figure reports AUROC, fitting time, and inference time across six landmark counts, with means and standard deviations over three seeds.Source: D Algorithmic Details · p. 21, Figure 5 · p. 22

Finding the CVaR-minimax distribution

• The CVaR-minimax rule solves the inner worst-case distribution exactly—by water-filling for interval sets or vertex enumeration for convex hulls—then minimizes over predictions with 300 Adam iterations.The predictor is parameterized with a softmax, and gradients use the exact inner solution.Source: D Algorithmic Details · p. 21

Calibration

• Calibration uses 100 thresholds up to the largest hold-out loss of the maximum-entropy prediction, and one sweep serves all risk levels because the calibrated prediction does not depend on τ.Beyond that maximum, the hinge vanishes and C(v) can only increase.Source: D Algorithmic Details · p. 22

E.1 Compute Resources

• Jobs used a single GPU on the compute node and ran under Slurm in a Singularity container with PyTorch NGC 26.03.The compute-resource table summarizes the node used for experiments.Source: Table 2 · p. 23, §E.1 · p. 23

E.2 Data; E.3.1 ResNet18

• The paper overviews datasets, references, and licenses in Table 3, and adapts ResNet18 to CIFAR-10 by changing the first convolution from 7 × 7 to 3 × 3 and removing the first max-pooling layer.Models were trained without early stopping, and the checkpoint with the lowest validation loss was used.Source: §E.2 · p. 23, §E.3.1 · p. 23

E.4.1 Credal Ensembling; E.4.2 Credal Bayesian Neural Networks; E.4.3 Credal Wrapper

• Credal Ensembling, Credal Bayesian Neural Networks, and Credal Wrapper construct credal sets from ensemble distributions, Bayesian-network outputs, or class-wise probability intervals, respectively.The first two use convex hulls; the wrapper uses intervals and an intersection probability when a single distribution is needed, except that the proposed decision rule is used in risk-averse experiments.Source: §E.4.1 · p. 24, §E.4.2 · p. 24, §E.4.3 · p. 24

E.4.4 Credal Distributionally Robust Optimization; E.4.5 Credal Relative Likelihood; E.4.6 Efficient Credal Prediction

• CreDro and Credal Relative Likelihood use ensembles to construct interval-based credal sets, while Efficient Credal Prediction converts bounded-logit perturbations into probability intervals.The stated configurations include 10 CreDro members with δG = 0.5 and α = 0.95 for the two likelihood-based methods.Source: §E.4.4 · p. 24, §E.4.5 · p. 24, §E.4.6 · pp. 24–25

E.4.7 SQwash; E.4.8 AdaCVaR

• SQwash trains on the worst τ-fraction of batch losses, whereas AdaCVaR uses an adversarial sampler to focus training batches on high-loss instances.Both methods produce a single network with precise conditional distributions.Source: §E.4.7 · p. 25, §E.4.8 · p. 25

F.1 Distribution Shift

• Distribution-shift experiments measure credal-set size and regular prediction accuracy on BloodMNIST and CIFAR-10 corruptions across severity levels, averaging corruption types and reporting three-seed variation.The shifted benchmarks are MedMNIST-C and CIFAR-10-C, with 11 and 15 corruption types respectively, each at five severity levels.Source: §F.1 · p. 26

F.2 Out-of-Distribution Detection; F.2.1 MedMNIST

• OOD detection is framed as balanced binary iD-versus-OoD classification, with AUROC reported over three runs; MedMNIST uses BloodMNIST as iD, PathMNIST and TissueMNIST as near-OoD, and four other datasets as far-OoD.The far-OoD datasets are DermaMNIST, RetinaMNIST, OCTMNIST, and BreastMNIST.Source: §F.2 · p. 26, §F.2.1 · p. 26

F.3 Coverage versus Efficiency; F.4 Risk-Aversion under Distribution Shift

• Coverage is the fraction of CIFAR-10-H human soft-label distributions contained in predicted credal sets, while efficiency is one minus their mean width; risk-averse shift tests evaluate CVaR of log-loss and Brier loss on MedMNIST-C.Risk-averse baselines use the same τ-level as evaluation, and losses are averaged across corruption types with means and standard deviations over three seeds.Source: §F.3 · p. 26, §F.4 · pp. 26–27

F.5 Cost-Sensitive Decision Making

• The triage experiment makes sending a septic patient home the uniquely worst action, while precaution costs 6–8 for healthy patients.It uses scarce training data for atypically presenting septic patients and compares calibrated point predictors and CreWare over 10 seeds.Source: §F.5 · p. 27, Table 5 · p. 27, Table 6 · p. 27

F.6 Risk-Averse Reinforcement Learning

• Across 150 seeds, the MLE route planner averaged 19.1 reward with a 0.27 crash rate; these results were evaluated over 1,000 drives per planned route.The CVaR reward is the population CVaR0.1 of pooled returns, while crash rate and mean reward are averaged over seeds.Source: §F.6 · p. 29

G.1 Credal Sets under Distribution Shifts

• On CIFAR-10-C, CreWare increases credal-set size as accuracy diminishes across corruption severities.The figure reports mean and standard deviation of set size and accuracy; competing sets increase less in size.Source: §G.1 · p. 30, Figure 6 · p. 30

G.2 Coverage-Efficiency of Credal Predictor

• On CIFAR-10-H, CreWare stays in the high-coverage region while maintaining good efficiency.The evaluation uses human annotations as approximations of ground-truth distributions.Source: Figure 7 · p. 30, §G.2 · p. 30

G.3 Out-of-distribution Detection; G.3.1 Entropy-Based versus Set-Size-Based Measures; G.3.2 MedMNIST

• CreWare outperforms other methods across near- and far-OoD datasets using set size and on MedMNIST; many methods perform worse than random on the medical suite.The additional CIFAR-10 evaluation includes MNIST and Textures; tables report mean and standard deviation of AUROC over three runs.Source: §G.3 · p. 30, §G.3.2 · p. 31, §G.3.1 · p. 31

G.4 Computation Time Comparison

• CreWare’s training cost is an order of magnitude below ensemble-based methods and more than 20 times below CreBNN; only EffCre is comparable.Every method except CreBNN produces credal sets within a few seconds on the full test split.Source: §G.4 · pp. 31–32, Figure 8 · p. 32

G.5.1 Log-loss

• Under log-loss, CreWare matches or exceeds risk-averse and credal baselines on CVaR with little mean-loss sacrifice, and its advantage grows with distribution shift.As τ increases, its CVaR advantage diminishes while its mean-loss advantage increases; the supplied figures cover PathMNIST and BloodMNIST.Source: §G.5.1 · pp. 33–34, Figure 9 · p. 33, Figure 10 · p. 33

G.5.2 Brier-loss

• Under Brier-loss, bounded losses make results less extreme, so increasing distribution shift does not create the large baseline gap seen with log-loss.Brier-loss never exceeds 2; the supplied results cover PathMNIST and BloodMNIST.Source: §G.5.2 · p. 34, Figure 11 · p. 34, Figure 12 · p. 34

G.5.3 Ablation on the effect of α

• As α approaches 0, pipeline predictions tend toward uniform, performing well on CVaR but sacrificing much in mean loss.The ablation uses α values from 0.0 to 1.0 and evaluates CVaRτ of log-loss on PathMNIST and BloodMNIST.Source: §G.5.3 · p. 35, Figure 13 · p. 35, Figure 14 · p. 35
• At α = 1, predictions reduce to precise CVaR-optimal predictions under the maximum-likelihood estimate, with best mean loss on clean data but higher CVaR.With stronger distribution shift, these predictions incur increasingly higher losses because the MLE cannot express epistemic uncertainty.Source: §G.5.3 · p. 35
• CreWare shows lower CVaR and avoids catastrophic decisions in the cost-sensitive comparison.This supplied figure-level result complements the α ablation but does not specify an α value.Source: Figure 15 · p. 36

G.6 Cost-Sensitive Decision Making

• The triage scenario compares actions across healthy, flu, and sepsis outcomes, with sending a septic patient home costing 20; septic patients are underrepresented and can resemble healthy patients.The approach is compared with MLE, SQwash, AdaCVaR, and rule-based ablations.Source: §G.6 · p. 36
• CVaR falls from 10.2 to 6.7 at mean cost 1.97, versus the MLE’s mean cost of 1.63; the point-estimate and CreWra rule ablations incur roughly three times the mean cost.Figure 15’s mean-cost decomposition shows catastrophic decisions, such as septic patients sent home, dominate the risk-neutral baseline’s cost.Source: §G.6 · p. 36
• Across 0–120 septic training cases, the approach’s CVaR stays nearly flat, from 6.9 to 6.4; the credal set avoids catastrophic decisions even when sepsis evidence is absent.SQwash’s protection collapses at zero coverage, while the point-estimate baselines can hedge only once some sepsis cases are represented.Source: §G.6 · p. 36, Figure 16 · p. 37
• With 30 subgroup sepsis cases, the MLE recovers to CVaR 7.4 and the arms converge: CreWra+rule reaches 6.36 and the approach reaches 6.49.The credal advantage is largest when evidence is scarcest and diminishes as evidence accumulates.Source: §G.6 · p. 36, Figure 17 · p. 37

G.7 Risk-Averse Reinforcement Learning

• The larger-map driving task pits a 91-step shortcut with an approximately 13.5% crash probability against a 136-step safe detour; the shortcut has higher expected return, 169.4 versus 164.The map has 329 hazard cells, and the comparison varies the number of hazard observations k.Source: §G.7 · p. 38
• On the larger driving map, Credal + rule detours from the first observation and has a 0% crash rate at every k, while MLE crashes in approximately 13% of episodes.MLE + rule remains unsafe through k = 90 and is safe only from k = 167 onward.Source: §G.7 · p. 38, Figure 18 · p. 38