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Legitimate ground-truth-free metrics for deep uncertainty classification scoring

Arthur Pignet, Chiara Regniez, John Klein

TL;DR

This work tackles the absence of uncertainty ground truth in deep learning by introducing two interpretable ground-truth concepts, $\phi(\mathbf{x})$ and $\varphi(\mathbf{x})$, that relate to misclassification risk and Bayes-alignment, respectively. It proves that two practical UQ evaluation metrics, UQ-AUC and UQ-C-index, maximize within the appropriate ground-truth equivalence classes and thus serve as ground-truth-free tools for ranking inputs by uncertainty. The authors show that sub-level sets of scoring functions form nested input regions with controllable risk and connect these results to set-valued and conformal prediction ideas. They validate the theory with synthetic data and real-world benchmarks (CIFAR10-H, ReaL-ImageNet, OASST2), demonstrating high alignment between the metrics and human or ground-truth uncertainty when available, while highlighting the metrics’ robustness to calibration. The findings support broader adoption of UQ in production by providing principled, data-efficiency tools for evaluating and ranking predictive uncertainties without requiring ground-truth uncertainty labels.

Abstract

Despite the increasing demand for safer machine learning practices, the use of Uncertainty Quantification (UQ) methods in production remains limited. This limitation is exacerbated by the challenge of validating UQ methods in absence of UQ ground truth. In classification tasks, when only a usual set of test data is at hand, several authors suggested different metrics that can be computed from such test points while assessing the quality of quantified uncertainties. This paper investigates such metrics and proves that they are theoretically well-behaved and actually tied to some uncertainty ground truth which is easily interpretable in terms of model prediction trustworthiness ranking. Equipped with those new results, and given the applicability of those metrics in the usual supervised paradigm, we argue that our contributions will help promoting a broader use of UQ in deep learning.

Legitimate ground-truth-free metrics for deep uncertainty classification scoring

TL;DR

This work tackles the absence of uncertainty ground truth in deep learning by introducing two interpretable ground-truth concepts, and , that relate to misclassification risk and Bayes-alignment, respectively. It proves that two practical UQ evaluation metrics, UQ-AUC and UQ-C-index, maximize within the appropriate ground-truth equivalence classes and thus serve as ground-truth-free tools for ranking inputs by uncertainty. The authors show that sub-level sets of scoring functions form nested input regions with controllable risk and connect these results to set-valued and conformal prediction ideas. They validate the theory with synthetic data and real-world benchmarks (CIFAR10-H, ReaL-ImageNet, OASST2), demonstrating high alignment between the metrics and human or ground-truth uncertainty when available, while highlighting the metrics’ robustness to calibration. The findings support broader adoption of UQ in production by providing principled, data-efficiency tools for evaluating and ranking predictive uncertainties without requiring ground-truth uncertainty labels.

Abstract

Despite the increasing demand for safer machine learning practices, the use of Uncertainty Quantification (UQ) methods in production remains limited. This limitation is exacerbated by the challenge of validating UQ methods in absence of UQ ground truth. In classification tasks, when only a usual set of test data is at hand, several authors suggested different metrics that can be computed from such test points while assessing the quality of quantified uncertainties. This paper investigates such metrics and proves that they are theoretically well-behaved and actually tied to some uncertainty ground truth which is easily interpretable in terms of model prediction trustworthiness ranking. Equipped with those new results, and given the applicability of those metrics in the usual supervised paradigm, we argue that our contributions will help promoting a broader use of UQ in deep learning.

Paper Structure

This paper contains 29 sections, 8 theorems, 61 equations, 6 figures, 1 table.

Table of Contents

  1. INTRODUCTION
  2. Scope
  3. This paper is not a UQ benchmark
  4. This paper does not deal with OOD data
  5. SHEDDING THEORETICAL LIGHT ON METRICS FOR UQ EVALUATION
  6. Validity of $\phi$ and $\varphi$ to filter inputs based on uncertainty risks
  7. Checking that scoring functions match misclassification probabilities
  8. Checking that scoring functions catch misalignment with labels predicted by the Bayes classifier
  9. RELATED WORK
  10. Metric comparison synthesis
  11. EXPERIMENTS
  12. CONCLUSION
  13. UQ METHOD OVERVIEW
  14. SCORING FUNCTIONS
  15. METRICS ASSESSING RANKING PERFORMANCE VERSUS METRICS ASSESSING CALIBRATION PERFORMANCE
  16. ...and 14 more sections

Key Result

Theorem 2.1

$\forall s\in \mathcal{E}_{\phi}, \exists\psi_s$ a strictly increasing mapping from $\mathscr{C}_{\textrm{mce}}\left( \mathcal{E}_\phi \right)$ to $\mathscr{B}_{\textrm{mce}} \left( s \right)$ such that $\forall \gamma \in \mathscr{C}\left( \mathcal{E}_\phi \right) \subset \left( 0; 1-\text{ACC} \right].$

Figures (6)

  • Figure 1: Synthetic dataset and real-world datasets with human annotations. Uncertainty ground truth-free metric UQ-AUC (Top row, resp. UQ-C-index in the bottom row) versus ground truth dependent Kendall correlation between $s(\mathbf{x})$ and $\phi(\mathbf{x})$, (resp. between $s(\mathbf{x})$ and $\varphi(\mathbf{x})$) for several UQ algorithms and hyperparameter configurations. All metrics are computed on a held-out test set. Note that Kendall correlations are intractable in practice, contrary to the UQ-AUC or UQ-C-index.. Models used for Synth: Entropy •, Deep Ensemble •, Monte-Carlo Dropout •, Models used for CIFAR10 and R-ImageNet: GoogleNet •, AlexNet $\times$, ResNet •, RegNet $\times$, Swin •, ResNeXt $\times$, MobileNet •, EfficientNet $\times$, ViT •, Wide ResNet $\times$, MnasNet •, ConvNeXt $\times$, Inception •, DenseNet $\times$, VGG •, ShuffleNet $\times$, Models used for OASST2: BART $\times$, DistilBERT •, DeBERTa $\times$, RoBERTa •, BERT $\times$, XLM •, MiniLM $\times$, E5 •, mContriever •, Other $\times$
  • Figure 2: ECE, UQ-AUC and UQ-C-index as functions of the scaling factor of Temperature Scaling.
  • Figure 3: Synthetic dataset illustrations. (a) Each dot represents a sample, whose color indicates the label. The background color indicates the probability $P \left(Y= 1 | X=\mathbf{x} \right)$. (b) & (c) Examples of ground truth scoring $\phi$ and $\varphi$ for an arbitrary predictor whose frontier decision is the dashed red line. (d) Different types of risk incurred in sub-level sets $\mathcal{L}_{\beta}\left( \phi \right)$ and $\mathcal{L}_{\beta}\left( \varphi \right)$ respectively. As implied by \ref{['thm:proof_nested_sets_phi', 'thm:proof_nested_sets_varphi']} those risks can be controlled by lowering the value of $\beta$.
  • Figure 4: Examples of scatter plots of UQ scores $s(\mathbf{x})$ versus ground truth $\phi(\mathbf{x})$ for randomly-picked trained UQ models. First row corresponds to the SFT baseline, second one are Deep Ensemble algorithms and the last one MC dropout algorithms. Each plot corresponds to one point in \ref{['fig:cuq-misalign-bayes-label-synth']}
  • Figure 5: Examples of scatter plots of UQ scores $s(\mathbf{x})$ versus ground truth $\varphi(\mathbf{x})$ for randomly-picked trained UQ models. First row corresponds to the SFT baseline, second one are Deep Ensemble algorithms and the last one MC dropout algorithms. Each plot corresponds to one point in \ref{['fig:aucuq-bayes-error-synth']}
  • ...and 1 more figures

Theorems & Definitions (8)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3: Maximizing UQ-AUC
  • Theorem 2.4
  • Theorem 2.5: Maximizing UQ-C-index
  • Theorem 2.6
  • Lemma 3.1
  • Theorem 9.1