$\Delta$-UQ: Accurate Uncertainty Quantification via Anchor Marginalization

• Existing uncertainty methods are often restricted to classification or require specialized supervised training, leaving no accurate, model-agnostic single-model estimator for both regression and classification.Source: §1 · p. 1
• Δ-UQ combines a random anchor R with a recoverable pretext encoding Δ(X, R), allowing a deterministic predictive model to estimate uncertainty by marginalizing over anchors.Source: §1 · p. 1
• The method links anchor-induced uncertainties to improper input sampling and inherent data noise, supporting estimation of total uncertainty comprising aleatoric and epistemic components.Source: §1 · p. 1
• The framework makes no assumptions about the model or task and is demonstrated across regression applications including model fitting, sequential optimization, and model-based inversion.Source: §1 · p. 1
• Across these use cases and classification tasks involving out-of-distribution detection and calibration under distribution shifts, Δ-UQ consistently outperforms many benchmark uncertainty estimators.Source: §1 · pp. 1–2

2.1 Overview

• Uncertainty arises when different anchors produce inconsistent decoding or predictions, with poorly sampled regions yielding larger discrepancies and prediction uncertainty.Exact decoding removes this uncertainty, whereas insufficient sampling can increase it.Source: §2.1 · p. 2

2.2 Anchoring-based Uncertainty Estimation

• Δ-UQ reparameterizes each input as an anchor R plus an encoding Δ(X,R) from which X remains recoverable, allowing deterministic models to estimate uncertainty through anchor variation.The anchor prior need not equal the input distribution, although the paper uses P(R)=P(X) for simplicity.Source: §2.2 · p. 2, §2.1 · p. 2
• At inference, the model is evaluated with K different anchors and the resulting predictions are marginalized to obtain a mean and uncertainty estimate.Training exposes each sample to randomly chosen anchors across multiple epochs.Source: §2.2 · pp. 2–3, §2.1 · p. 2
• The encoding choice controls the fidelity–calibration trade-off: Δ(X,R)=X gives unreliable uncertainty, two anchors improve calibration but reduce fidelity, and Δ(X,R)=X−R performs best on both metrics.The comparison uses R2 prediction fidelity and Spearman correlation between uncertainty and generalization error.Source: §2.2 · p. 3, Table 1 · p. 3
• In a toy classification demonstration, uncertainty concentrates near decision boundaries when classes are separable and reflects aleatoric ambiguity when classes are nearly inseparable.Total uncertainty is measured as the sum of prediction-probability variances across the three classes.Source: §2.2 · p. 3, Figure 2 · p. 3

3 Related Work

• Bayesian methods can be difficult to scale, while deep ensembles require training several models and can create a computational bottleneck.Deep ensembles are described as accurate and task-agnostic, but their multiple independently initialized models increase training cost.Source: §3 · pp. 3–4
• The related methods differ in how they obtain uncertainty, including posterior weight sampling, feature-space distances, and explicit uncertainty estimators.These approaches motivate a general-purpose alternative that does not depend on the same model-specific mechanisms.Source: §3 · pp. 3–4
• Single-model methods include DUQ and DEUP, whereas Δ-UQ estimates inherent uncertainty through anchor marginalization without a post-hoc step.The paper positions Δ-UQ as a single-model alternative to methods requiring specialized or additional uncertainty estimation procedures.Source: §3 · p. 4

4 Experiments

• The experiments cover model fitting, sequential optimization, model-based inversion, OOD detection, and shifted-distribution calibration to assess ∆-UQ across diverse regression and classification settings.The benchmark scope is motivated by the method’s general-purpose design for different models and prediction tasks.Source: §4 · p. 4

Uncertainty Calibration on OpenML Benchmarks

• Across the evaluated datasets, ∆-UQ consistently produces higher-quality uncertainty estimates than ensembling and DEUP, as shown by higher Spearman rank-correlation values in five independent trials.The experiments use random-forest regressors with 200 training samples, five independent trials, and 100 anchors for ∆-UQ.Source: Uncertainty Calibration on OpenML… · p. 4
• ∆-UQ reaches higher function values with fewer iterations than competing uncertainty estimators in sequential model optimization.Table 3 compares maximum values after fixed iteration budgets across five independent trials, while the Multi-Optima experiment reports that DEUP requires twice as many iterations to reach approximately 0.951.Source: Uncertainty Calibration on OpenML… · pp. 4–5, Table 3 · p. 5
• In model-based optimization, ∆-UQ consistently meets target digit thicknesses and outperforms MC Dropout and deep ensembles, including targets beyond the training regime.Models are trained on images with thickness below 200, then evaluated at target thicknesses from 150 to 250.Source: Uncertainty Calibration on OpenML… · pp. 5–6, Table 4 · p. 6

Uncertainty based OOD Detection

• For CIFAR-10 versus SVHN, ∆-UQ estimates uncertainty with a single ResNet-18 model and compares against predictive entropy, DUQ, and temperature scaling for OOD detection.The evaluation uses AUROC on 10,000 CIFAR-10 and 26,032 SVHN test images; uncertainty-based scaling further improves over the mean-only model and other baselines.Source: Uncertainty based OOD Detection · p. 6, Table 5 · p. 6

Calibration Under Distribution Shifts

• ∆-UQ competitively calibrates classifiers under CIFAR-10-C distribution shifts across 16 corruptions and five intensity levels.Figure 4 summarizes calibration error with boxplots across corruption types, comparing ∆-UQ against MC-Dropout, deep ensembles, temperature scaling, and SVI.Source: Calibration Under Distribution Shifts · pp. 6–7, Figure 4 · p. 7

• Δ-UQ estimates both aleatoric and epistemic uncertainties using a single model, without additional training modules or specific regularization strategies.The method re-parameterizes each input as an anchor plus a pretext encoding and marginalizes over anchor choices.Source: §5 · p. 7
• Δ-UQ produces meaningful uncertainties for regression and classification and is consistently superior to several existing uncertainty estimators.Source: §5 · p. 7
• Fixing one anchor during training limits optimization to point estimates, while larger training-time anchor sets might make mean predictions more robust.Source: §5 · p. 7
• Using P(R) = P(X) is a design choice whose effect on prediction fidelity and uncertainty calibration remains important for future anchor-prior design.Source: §5 · p. 7

• For randomly chosen 2D test samples, five anchors produce prediction probabilities from the Δ-model, while marginalization yields mean predictions and uncertainty estimates.Training data are shown as circles and anchors as stars.Source: Appendix A · p. 9

• The evaluation uses Sinusoid and Multi Optima in 1D, plus Booth, Levi N.13, and Ackley in 2D.Source: §B.1 · p. 9
• Neural-network methods use fully connected networks with three 128-unit ReLU hidden layers, while deep ensembles contain three independently seeded models.Source: §B.1 · p. 9
• MC Dropout uses a dropout rate of 0.3, DEUP uses a matching three-layer error predictor, and the Gaussian-process baseline uses BoTorch’s native functions.Source: §B.1 · p. 9
• Each optimization run starts with 6 samples and adds one sample per iteration using expected improvement.Source: §B.1 · p. 9
• Predictive models are trained with Adam at learning rate 1e-3 for 200 epochs after each sample addition.Source: §B.1 · p. 9

B.2 Model-based Optimization

• In a model-based optimization experiment on MNIST digits, thickness was the target attribute, and models were trained only on images with thickness ≤200.Thickness is defined as the total number of pixels with intensity >0.2.Source: §B.2 · p. 9
• Uncertainty-aware latent optimization seeks solutions that match the desired thickness and have low uncertainty under the forward model.The inverse model is optimized through predictions from the forward model in cyclical training.Source: §B.2 · p. 9
• The inverse model is fixed across methods and maps 100-dimensional latent noise plus target thickness through a five-layer fully connected network to digit images.Source: §B.2 · p. 9
• Testing runs cyclical optimization for 1000 iterations from 50 randomly initialized latent points for each desired thickness.Figure 7a shows example Δ-UQ realizations for different target values.Source: §B.2 · p. 9

Anchor count

• Increasing the number of test-time anchors improves uncertainty performance until a certain value on CIFAR-10-C.The experiment fixes K = 1 during training and evaluates correlation with true generalization error under impulse noise at intensity 5.Source: §C.1 · p. 9

Anchor distribution

• Anchor-prior performance is evaluated by varying distributions in the CIFAR-SVHN separation experiment with K = 10.The priors include CIFAR-10-C at different corruption intensities, a standard normal distribution, SVHN, and the CIFAR training distribution.Source: §C.1 · pp. 9–10
• Using CIFAR-10-C as anchors degrades performance proportionally, relative to training-distribution anchors (“CIFAR”) and competitive baselines DUQ and Deep Ensembles.Source: §C.1 · p. 10
• A simple random prior performs better than DUQ, indicating that carefully chosen random priors may further improve performance.This result is reported for the CIFAR-SVHN separation experiment.Source: §C.1 · p. 10
• SVHN anchors perform similarly to CIFAR anchors, suggesting anchors may be usable at test time without access to the training distribution.Both conditions are evaluated with the same distributions for the CIFAR and SVHN datasets.Source: §C.1 · p. 10

Metrics and distribution shifts

• Brier Score and Negative Log Likelihood supplement expected calibration error for evaluating calibration under distribution shift.The additional metrics use the same training setup, model, and hyperparameters as the main text.Source: §C.2 · p. 10, Figure 7 · p. 11
• Across CIFAR-10-C intensities, the proposed method has higher accuracy and lower calibration error than most single-model methods, approaching deep ensembles.The calibration-error trend observed with ECE also holds for Brier Score and NLL, with results averaged across 5 random seeds.Source: §C.2 · pp. 10–11, Figure 8 · p. 12