Source-linked AI summary
Uncertainty Quantification in Pairwise Difference Learning for Classification
Mohamed Karim Belaid, Maximilian Rabus, Eyke Hüllermeier
TL;DR
Reliable predictions require uncertainty estimates, yet uncertainty quantification has been sparsely explored for pairwise difference learning. This paper develops a multiclass pairwise difference classifier and uncertainty framework suited to different base learners; empirical studies report improved classification accuracy and recommend approaches for uncertainty estimation.
Problem
Method
The paper converts multiclass classification into binary pairwise class-equality prediction and adapts uncertainty representations and measures to the base learner’s uncertainty-awareness.
Source: §1 · p. 2, §7 · p. 17Results
The PDL classifier shows significantly improved accuracy over conventional classification, while empirical studies recommend ULP, or UP for probabilistic non-Bayesian base learners, and find variance-based measures consistently outperform information-theoretic measures.
Source: §7 · p. 17Takeaways & Limitations
The authors recommend ULP, or UP for probabilistic non-Bayesian base learners, and variance-based uncertainty measures for PDL classifiers.
Source: §7 · p. 17The classification setup assumes data points are independently and identically distributed according to an unknown joint probability distribution.
Source: §3 · p. 4
Abstract
from arXiv · showhide
Instead of learning a mapping from instances to outcomes in the standard way, the key idea of pairwise difference learning (PDL) is to learn a function that takes two instances as input and predicts the difference between the respective outcomes. Given a function of this kind, predictions for a query instance are derived from every training example and then averaged. We propose a version of PDL for the task of classification and propose a suitable meta-learning technique for this purpose. Moreover, we address the problem of uncertainty quantification, i.e., of representing and quantifying the uncertainty of a PDL classifier's predictions in a faithful manner. Building on recent work on the decomposition of total uncertainty into aleatoric and epistemic components, we propose different ways of extending corresponding measures to the setting of PDL, depending on how and to what extent uncertainty is represented by the underlying base learner. In large-scale empirical studies, we analyze both the predictive accuracy and the uncertainty awareness of our PDL classifier.
1 Introduction
The paper extends pairwise difference learning to multiclass classification and focuses on representing and quantifying uncertainty in its predictions. It proposes a classification meta-learning technique, uncertainty measures, and empirical evaluation of accuracy and uncertainty awareness.
1 Introduction
- PDL predicts differences between paired inputs, then averages predictions derived from training examples; this can improve accuracy, especially with small datasets.Pairing expands the training examples to the square of the original count, at the cost of losing statistical independence.Source: §1 · p. 2
- The paper shifts PDL classification research toward faithful uncertainty quantification, covering aleatoric and epistemic uncertainty alongside predictive accuracy.The authors motivate reliability as important for high-stakes decisions, including healthcare, autonomous systems, and risk management.Source: §1 · p. 2
- The proposed meta-learning technique turns any multiclass classification problem into a single binary problem.The paper also establishes a taxonomy of total, aleatoric, and epistemic uncertainty representations and evaluates accuracy and uncertainty awareness on real-world and synthetic datasets.Source: §1 · p. 2
2 Related Work
Prior work has extended PDL from regression to classification and applied it across varied domains. However, uncertainty quantification within PDL remains sparsely explored, particularly the decomposition of total uncertainty into aleatoric and epistemic components.
2 Related Work
- Earlier PDL work spans neural-network and classical regression, then classification applications in natural language processing and tabular data.Benchmark evaluations covered more than 400 supervised-learning datasets, and reported applications include chemistry, image processing, quantum reactions, and drug activity ranking.Source: §2 · p. 3
- Prior uncertainty work uses model outputs, perturbations, ensembles, or intermediate PDL predictions, while combined perturbation approaches remain insufficiently mature for real-world applications.Source: §2 · p. 3
- Aleatoric and epistemic uncertainty have distinct sources, and decomposing total uncertainty complicates selection of effective measures.Prior work reported that epistemic-uncertainty methods can also estimate aleatoric uncertainty in large language models.Source: §2 · p. 3
- PDL uncertainty studies include pair-input methods and variance over pair predictions, but the latter measured total uncertainty without decomposing aleatoric and epistemic components.Related work also considered reducing aleatoric uncertainty caused by experimental measures.Source: §2 · p. 3
3 Pairwise Difference Classifier
The Pairwise Difference Classifier learns whether two inputs share a class label, then uses each training example as evidence to estimate a query’s class probabilities. It combines a global learned pair model with averaged, example-specific predictions.
3 Pairwise Difference Classifier
- The setup assumes feature vectors with discrete class labels drawn i.i.d. from an unknown joint distribution.The objective is to obtain accurate predictions by minimizing a prespecified expected loss.Source: §3 · p. 4
- The classifier learns a binary decision about whether paired inputs share the same class label.Its probabilistic predictor γ(x, x′) models the probability that the two labels coincide, using a binary classifier trained on transformed pairs.Source: §3 · p. 4
- For a query, each training example supplies evidence: γ gives the probability that the query has the example’s label, and the prior is conditioned on that uncertain event.The resulting posterior assigns the example’s label probability γ and proportionally rescales other class probabilities; evidence from all training examples is then averaged.Source: §3 · pp. 4–5
- PDL averages predictions from one learned difference model anchored on different training examples, combining global model learning with local example-based predictions.This resembles ensemble averaging, but aggregates predictions from the same model rather than multiple randomized models; the individual predictions also provide a basis for uncertainty quantification.Source: §3 · p. 5
4 Uncertainty Quantification for PDL Classifiers
The section organizes PDL uncertainty by the uncertainty information available from its base learner and by whether predictions are averaged or integrated. It extends first- and second-order predictions with measures of total, aleatoric, and epistemic uncertainty.
4 Uncertainty Quantification for PDL Classifiers
- PDL distinguishes deterministic predictors, first-order probabilistic predictors, and second-order Bayesian predictors according to the uncertainty their outputs can represent.Second-order predictions are distributions over first-order probability distributions; integrating second-order predictions would produce a third-order prediction that is difficult to interpret and handle computationally.Source: §4 · pp. 5–6
4.1 Deterministic Predictors
- Deterministic base learners yield probability distributions under PDL by aggregating their hard predictions across input pairs.The pdc1-NN variant produces full probability distributions, unlike 1-NN, which provides no uncertainty information.Source: §4.1 · pp. 6–7, Figure 1 · p. 7
4.2 Probabilistic Predictors
- Averaging probabilistic base-learner predictions gives a first-order distribution that captures aleatoric uncertainty, while integration yields a second-order distribution that captures epistemic uncertainty.For the Decision Tree example, both PDL variants provide smoother, more precise first-order uncertainty estimates; integration also distinguishes second-order distributions for inputs with similar first-order predictions.Source: §4.2 · p. 7
4.3 Second-order Predictors
- Integrating pairwise predictions from a second-order base learner creates nested uncertainty information, which can be condensed into second-order distributions using UL, UP, or ULP.The example uses a ten-tree Random Forest and 300 anchor points; the three approaches yield distributions ranging from nearly Gaussian to almost uniform.Source: §4.3 · pp. 8–9
4.4 Uncertainty Measures
- Total uncertainty decomposes into aleatoric uncertainty, reflecting data randomness, and epistemic uncertainty, reflecting the learner’s lack of knowledge.The section notes that additional data will not reduce aleatoric uncertainty, whereas epistemic uncertainty typically diminishes with more or better data.Source: §4.4 · pp. 9–10
4.4.1 Entropy-based Information Measures
- Entropy-based measures define total uncertainty as entropy of the expected first-order prediction and decompose it into conditional entropy and mutual information.For UL, UP, and ULP, total uncertainty coincides because all three approaches produce the same average prediction.Source: §4.4.1 · p. 10
4.4.2 Variance-Based Measures
- Variance-based measures represent categorical outcomes as one-hot Bernoulli variables and sum their uncertainty contributions across classes.The decomposition uses the law of total variance to obtain aleatoric and epistemic components.Source: §4.4.2 · pp. 10–11
5 Evaluation
The evaluation tests PDC on 99 OpenML datasets, using macro F1 for predictive performance and rank-correlation or calibration-error analyses for uncertainty quality. It compares PDC with its base learners and finds predictive and uncertainty-quantification benefits, with outcomes varying by uncertainty type and base-learner setting.
5 Evaluation
- PDC was evaluated using five Scikit-learn base learners: probabilistic DecisionTree and ExtraTree, and Bayesian RandomForest, ExtraTrees, and Bagging of MLP.The evaluation considered interactions among architecture, base learner, uncertainty representation, and entropy- versus variance-based metrics.Source: §5 · p. 11
5.1 Data
- The study selected 99 OpenML datasets, focusing on smaller datasets; pairwise comparisons increased the effective sample size to approximately 360,000 data points.The selection followed criteria similar to OpenML-CC18 and previous PDL research.Source: §5.1 · p. 11
5.2 Metrics
- Predictive performance used macro F1, while uncertainty quality used accuracy-rejection analysis summarized by Goodman and Kruskal’s Gamma and EU-to-calibration-error comparisons.ECE approximated true calibration error; reported performance was averaged over 25 runs per dataset, with significance assessed by Student’s t-test (α = 0.05).Source: §5.2 · pp. 12–13
5.3 Baseline
- The study compared each PDC model with its corresponding underlying base learner and did not directly compare against Tynes et al.’s regression-specific method.The authors note distinct computational formulations for that method and their variance-based TU estimation using UP.Source: §5.3 · p. 13
5.4.1 Predictive Performance
- Ensemble-backed PDC achieved the highest average Macro F1 across 99 datasets, surpassing all baselines; significant wins favored PDC in direct comparisons.The comparison is summarized in Table 2.Source: Table 2 · p. 14, §5.4.1 · p. 14
5.4.2 UQ of PDC Using a Probabilistic Base Classifier
- With probabilistic base learners, PDC improved TU rank correlation to a mean of 0.56, with only 5% negative correlations, from a baseline median of 0.23.PDC also enabled EU assessment, with predominantly positive ECE–EU rank correlations.Source: §5.4.2 · p. 14, Figure 4 · p. 15
- For probabilistic classifiers, entropy- and variance-based measures both quantified uncertainty meaningfully, and AU and TU had high Pearson correlation (ρ = 0.98).The strong correlation indicates TU was predominantly driven by aleatoric noise in these scenarios.Source: §5.4.2 · p. 14
5.4.3 UQ of PDC Using a Second-order Base Classifier
- With Bayesian base learners, TU and AU were highly correlated and both guided selective prediction, while EU quality depended on the uncertainty-representation strategy.ULP produced the highest EU rank correlations, with a slight, non-significant improvement over a baseline Bayesian classifier.Source: §5.4.3 · p. 14, Figure 5 · p. 15
6 Discussion
The discussion attributes PDL’s predictive performance to local and global information, binary task reduction, and averaging across anchors. It also reports improved uncertainty quantification, including separation of aleatoric and epistemic uncertainty in synthetic examples.
6.1 Effect of PDC on the predictive performance
- PDL combines anchor-based local information with a global classifier, reduces multiclass classification to a binary same-label task, and averages predictions across anchors.The authors describe anchor averaging as an ensemble effect that reduces prediction-error variance.Source: §6.1 · pp. 15–16
6.2 Effect of PDC on the Uncertainty Quantification
- Variance-based measures matched or outperformed Shannon entropy-based measures, while probabilistic PDC reduced negative TU–misclassification correlations from 29% to 5%.On a synthetic dataset, PDC(DT) produced TU that increased with sparsity, and PDC(Bagging(MLP)) separated noisy from sparse regions.Source: §6.2 · p. 16, Figure 6 · p. 16
- A probabilistic-base PDC model delivered EU estimates comparable to a baseline second-order ensemble despite using only a single trained tree.The authors identify reduced computational complexity as a potential benefit.Source: §6.2 · p. 16
- For second-order base learners, ULP gave the best EU quantification, apparently by combining variability from base learners and anchor-point predictions.UL and UP each average out one of these sources of variability.Source: §6.2 · p. 16
- PDC(DT) and PDC(RandomForest) assigned highest AU to the noisy center and highest EU to the sparse region, unlike RandomForest’s allocation of excessive AU to an empty region.The comparison is shown in Fig. 7.Source: §6.2 · p. 16, Figure 7 · p. 17
7 Conclusion
The paper extends pairwise difference learning to multiclass classification by recasting outcome differences as class equality or inequality, and reports improved accuracy over conventional classification. It also develops uncertainty quantification that adapts to the base learner and recommends approaches that performed best most of the time in its evaluations.
7 Conclusion
- PDC extends PDL to multiclass classification by defining outcome differences through class (in)equality, reducing the task to binary classification.Empirical studies report significantly improved accuracy compared with conventional classification.Source: §7 · p. 17
- PDC maps deterministic base learners to probabilistic uncertainty, and probabilistic learners to second-order representations of aleatoric and epistemic uncertainty.For Bayesian base learners, the framework keeps uncertainty second-order to avoid overly complex interpretation.Source: §7 · p. 17
- ULP is recommended overall, while UP coincides with ULP for probabilistic, non-Bayesian base learners; variance-based measures consistently outperform information-theoretic measures.The recommended approaches yield the best results most of the time, although alternatives can perform better on specific datasets.Source: §7 · p. 17
Appendix A Choice of Base Learners
PDL can wrap binary classifiers for binary and multiclass tasks, but paired-data dependence and the joint feature representation create model-specific suitability concerns. Pair construction also expands the training and prediction workload.
Appendix A Choice of Base Learners
- PDL can use any probabilistic binary classifier as a base learner for binary or multiclass settings, though some classifiers are more appropriate in practice.Non-parametric methods are described as generally more robust and superior-performing; experiments predominantly use tree-based methods, which also train quickly.Source: Appendix A · pp. 20–21
- Pairing makes the paired dataset dependent even when the original data are i.i.d., and some methods may be more sensitive to these violations than others.An instance can recur across multiple pairs, as in metric-learning scenarios.Source: Appendix A · p. 21
- The joint representation of each pair and its feature differences introduces perfect multicollinearity, which can hinder some models, notably linear ones.Other models handle this feature dependence more effectively.Source: Appendix A · p. 21
Appendix B Complexity
- Pair expansion yields N^2 training pairs and 3F joint features; test prediction uses 2MA predictions, with each test point paired to A anchors and evaluated twice for symmetry.The classification task is reduced to a binary decision.Source: Appendix B · p. 21
C.1 Analyzing the Unique Contribution
PDC raises the best average Macro F1 across 99 datasets, extending the previous record through improvements on 75 datasets.
C.1 Analyzing the Unique Contribution
- Across 99 datasets, PDC raised the highest average Macro F1 from 0.8112 ± 0.0035 to 0.8243 ± 0.0031, an incremental gain of 0.0131, with improvements on 75 datasets.Source: §C.1 · p. 21
- 75 datasets improved with PDC, whose 0.0131 incremental gain is reported as 32 times greater than that of the best baseline.The comparison concerns each dataset’s best performance, rather than only head-to-head baseline wins.Source: §C.1 · p. 21
- The paper frames the primary objective as achieving the best performance on each dataset, even when PDL does not win every head-to-head baseline comparison..Source: §C.1 · p. 21
C.2 Analyzing Overfitting
The appendix reports reduced overfitting across dataset sizes and describes a synthetic dataset designed to visualize aleatoric and epistemic uncertainty patterns.
C.2 Analyzing Overfitting
- Across 199 cross-validation runs with non-significantly different training Macro F1 scores, PDL consistently had a smaller train-test gap.The trend also appears without restricting the analysis to similar training scores; the test Macro F1 comparison is scoped to runs with non-significantly different training scores.Source: §C.2 · p. 21, Table C2 · p. 22
- PDL is reported to enhance test performance and mitigate overfitting regardless of dataset size.The authors characterize the approach as robust across diverse data scales, particularly for small datasets.Source: §C.2 · pp. 21–22
Appendix D Visualizing the Uncertainty
- The synthetic X dataset places high aleatoric uncertainty in the central class-overlap region and high epistemic uncertainty in four peripheral low-density regions.Its nonlinearly separable classes are mixtures of Gaussians arranged along diagonal and anti-diagonal alignments.Source: Appendix D · p. 22
Appendix E Extended Benchmark Results
Appendix E assesses uncertainty informativeness using accuracy- and ECE-rejection curves. The curves examine how retained-set accuracy and calibration error change as increasingly uncertain predictions are rejected.
E.1 Accuracy-Rejection Curves
- Accuracy-Rejection Curves assess uncertainty informativeness by removing the most uncertain test predictions at each rejection percentage and measuring remaining accuracy; effective estimates should yield monotonically rising accuracy, while non-informative estimates remain nearly flat.The appendix shows 5 example datasets.Source: §E.1 · p. 23
- The appendix presents five accuracy rejection curves, one for each Dataset 772, Dataset 991, Dataset 1462, Dataset 40704, and Dataset 44490, labeled Figures E1 through E5 respectively.Source: Figure E1 · p. 23, Figure E2 · p. 24, Figure E3 · p. 25
E.2 ECE-Rejection Curves
- ECE-rejection curves sort test instances by descending epistemic uncertainty and examine whether the remaining instances’ average ECE decreases quasi-monotonically as rejection increases.Per-instance ECE approximates unobservable true calibration error using bins of predictions and average true labels.Source: §E.2 · p. 28