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Adaptive Margin Ordinal Loss: Penalizing Center-Class Hedging in Ordinal Classification

Manisha Kandel

arXiv:2609.10752v1cs.LGcs.AI

TL;DR

Ordinal classifiers trained with cross-entropy can hedge toward center classes, while existing losses do not directly encode this prediction-space failure. The paper proposes AMOL and CHR to target and measure center hedging, achieving the best QWK across four datasets and complete reproducible elimination of hedging with AMOL-asym on Abalone.

  • Problem

    Cross-entropy encourages center-class hedging in ordinal classification, and existing ordinal losses do not directly suppress this prediction-space failure as a function of true-label extremeness.

  • Method

    AMOL applies a multiplicative per-class weight based jointly on candidate proximity to center and true-label extremeness, while CHR directly measures center-class hedging.

  • Results

    Across four datasets and five random seeds, AMOL achieves the best QWK of compared methods on all four datasets, while AMOL-asym reaches CHR = 0.000 ± 0.000 on Abalone.

  • Takeaways & Limitations

    AMOL is recommended for general ordinal performance, whereas AMOL-asym is recommended when eliminating center-class hedging is critical.

  • Takeaways & Limitations

    The paper identifies future work in theoretical analysis, medical ordinal applications, and extension to larger K and non-symmetric ordinal scales.

Abstract

from arXiv · show

Standard cross-entropy loss causes neural networks trained on ordinal classification tasks to hedge predictions toward center classes, a failure mode we term \emph{center-class hedging}. This occurs because predicting the middle class minimizes expected symmetric loss, making it the path of least resistance regardless of the true label. Existing ordinal losses address related problems such as large-error penalization and rank consistency, but none directly suppresses center-class hedging as a function of where the true label lies relative to the ordinal center. We propose the Adaptive Margin Ordinal Loss (AMOL), a multiplicative weight applied to per-class loss terms of the form $m(k,y) = 1 + α\cdot (1 - |k-c|/c) \cdot (|y-c|/c)$, where $c$ is the center class, $k$ is the candidate class, and $y$ is the true label. The weight encodes a joint condition: it is large only when the candidate class is near center and the true label is far from center, collapsing to standard behavior otherwise. We further introduce the Center-Hedging Rate (CHR) as a diagnostic metric that directly quantifies this failure mode. Across four ordinal classification benchmarks and five random seeds, AMOL achieves the best or tied-best Quadratic Weighted Kappa (QWK) on all four datasets compared to cross-entropy, OLL, and SORD baselines. An asymmetric variant (AMOL-asym) eliminates center-class hedging entirely on the Abalone dataset ($\text{CHR} = 0.000 \pm 0.000$ across all five seeds, $n \approx 266$ extreme-class test samples per run), compared to $0.074 \pm 0.005$ for standard cross-entropy.

1 Introduction

Ordinal classifiers trained with cross-entropy exhibit center-class hedging, while existing losses do not directly target this prediction-space failure. AMOL introduces a center-aware loss weight and CHR diagnostic, improving QWK across four benchmarks.

  • Center-class hedging arises because the middle class minimizes expected loss under symmetric distance penalties, biasing predictions toward the ordinal center.This produces excess center probability and underconfident predictions for extreme classes.
  • Existing ordinal losses address large errors, rank consistency, or class-frequency imbalance but do not explicitly target center-class probability mass for extreme labels.These approaches operate on axes distinct from prediction-space center bias.
  • CHR directly measures center-class hedging, providing an interpretable diagnostic for this failure mode.
  • AMOL applies an adaptive multiplicative weight that jointly targets center-proximate candidate classes and extreme true labels.The proposed condition is designed to tax center-adjacent probability mass specifically when the true label is far from the ordinal center.
  • Across four benchmarks and five random seeds, AMOL improves QWK consistently, with AMOL-asym reaching CHR = 0.000 ± 0.000 on Abalone versus 0.074 ± 0.005 for cross-entropy.

2 Background and Related Work

Ordinal labels encode unequal error magnitudes, but standard cross-entropy treats classes nominally and can encourage center hedging. Existing ordinal methods improve distance or rank handling without encoding the joint condition of candidate center proximity and true-label extremeness.

  • Cross-entropy is blind to where incorrect probability mass goes, so center and opposite-extreme hedging receive identical treatment.
  • SORD uses Gaussian soft targets centered on the true label but does not adapt penalties to the true label’s position relative to center.
  • SLACE conditions balance sensitivity on training-set class frequency rather than prediction-space center bias.
  • OLL penalizes prediction distance from the true class, whereas AMOL targets candidate distance from the ordinal center conditioned on an extreme true label.
  • AMOL addresses a joint prediction-space condition not encoded by the existing losses described here.

3 Method

The method measures center prediction on extreme-label examples with CHR and weights per-class loss terms according to candidate proximity to center and true-label extremeness. Its asymmetric variant penalizes only inward hedging toward center.

  • CHR measures the probability that the predicted class is the center class given an extreme true label, with lower values indicating less center bias.CHR_ext pools the two most extreme classes on each side when single-extreme samples are too sparse for stable estimation.
  • The AMOL weight is largest only when a candidate class is near center and the true label is far from center.The design leaves correct extreme predictions unpenalized and recovers standard loss when α = 0 or the true label is central.
  • AMOL multiplies per-class loss terms by m(k, y) while using a Gaussian soft reference distribution centered on the true label.When α = 0, the formulation reduces to KL divergence from the model output to that soft reference distribution.
  • AMOL-asym restricts extra penalties to classes between the true label and the center, allowing mass toward the opposite extreme under uncertainty.

4 Experiments

The experiments compare cross-entropy, OLL, SORD, and AMOL across four datasets using QWK and center-hedging diagnostics. The study also evaluates AMOL variants and reports results across five random seeds.

  • Table 1 summarizes the datasets used in the experimental evaluation.
  • The real-data experiments use stratified 60/20/20 splits, standardized features, and a two-layer MLP, with results averaged across five seeds.
  • Figure 1 visualizes AMOL’s per-class weight as candidate class varies across true labels and contrasts symmetric with asymmetric weighting.
  • AMOL is compared with CE, OLL, and SORD, while ablations test asymmetric, exponential, OLL-based, and CE-based variants.
  • QWK is the primary performance metric, while CHR_ext is the primary center-hedging diagnostic alongside Accuracy, MAE, and AMAE.

5 Results

Across four datasets, AMOL improves QWK and reduces center-class hedging relative to cross-entropy, while AMOL-asym produces the strongest anti-hedging results on Abalone and Wine White. Wine Red results require caution because its extreme-class sample is very small.

  • 5.1 Baseline Comparison: AMOL achieves the best QWK on all four datasets, with gains over CE from 0.011 on Synthetic to 0.064 on Wine White.Abalone improves from 0.693 to 0.735, while Wine White improves from 0.530 to 0.594.
  • 5.1 Baseline Comparison: AMOL reduces CHR_ext on all four datasets, whereas OLL can worsen center-class hedging relative to CE.On Wine Red, OLL has CHR_ext 0.250 versus CE’s 0.229.
  • 5.3 Abalone: 0.000 ± 0.000 CHR on Abalone is achieved by AMOL-asym across five seeds, compared with 0.074 ± 0.005 for CE.The Abalone evaluation includes approximately 266 extreme-class test samples per seed.
  • 5.2 Wine White: 73% lower CHR_ext is obtained by AMOL-asym than CE on Wine White, alongside QWK of 0.599 ± 0.005 versus 0.530 ± 0.013.The probability-distribution figure shows reduced center-class mass, although predictions are not necessarily concentrated at the correct extreme classes.
  • 5.4 Wine Red: Wine Red’s extreme-class test set contains approximately 5 samples, so its CHR results are interpreted cautiously despite AMOL’s best reported CHR_ext and QWK.The paper explicitly avoids strong CHR claims for this dataset.

6 Ablation Study

The Abalone ablations show that directional weighting is critical for eliminating hedging, while the linear functional form is stable and the KL-to-Gaussian-target base loss is essential for performance.

  • Directionality: AMOL-asym reaches CHR = 0 while AMOL reaches 0.038, but AMOL achieves higher QWK at 0.735 versus 0.726.The directional restriction trades some general accuracy for stronger anti-hedging performance.
  • Functional Form: AMOL-exp performs nearly identically to AMOL on QWK and CHR, supporting the linear form for interpretability and stability.The ablation indicates that the choice of functional form has little practical effect in this setting.
  • Base Loss: Multiplying the margin onto OLL reduces QWK to 0.510 versus 0.735 and accuracy to 0.19 versus 0.38 on Abalone.The KL-to-Gaussian-target base supplies correctness information that the margin alone does not provide.

7 Discussion

AMOL is most useful when center-heavy labels create a strong incentive to hedge and enough extreme-class samples support reliable CHR measurement. Its symmetric and asymmetric variants serve different priorities, while the method has tuning, center-definition, and calibration limitations.

  • 7 Discussion: Gains are most pronounced on center-heavy datasets with sufficient extreme-class test samples, especially Abalone and Wine White.
  • 7 Discussion: AMOL is recommended for general ordinal performance, whereas AMOL-asym is recommended when completely eliminating center-class hedging is critical.The variants are presented as complementary rather than competing.
  • 7 Discussion: AMOL requires tuning α, and AMOL-asym assumes a well-defined ordinal center that may be ambiguous for multimodal distributions.
  • 7 Discussion: AMOL sacrifices calibration guarantees because it is not a strictly proper scoring rule, leaving calibration analysis for future work.

8 Conclusion

The paper introduces AMOL to penalize center-class probability mass for extreme-label samples and CHR to measure center-class hedging. Across four datasets and five seeds, AMOL leads QWK on every dataset, while future work targets theoretical, clinical, and broader ordinal extensions.

  • 8 Conclusion: AMOL adaptively penalizes center-class probability mass for extreme-label samples, while CHR directly measures center-class hedging.
  • 8 Conclusion: Across four datasets and five random seeds, AMOL achieves the best QWK of any compared method on all four datasets.AMOL-asym reaches CHR = 0.000 ± 0.000 on Abalone across five seeds, with approximately 266 extreme-class test samples per run.
  • 8 Conclusion: Future work includes theoretical analysis, medical ordinal grading applications, and extension to larger K and non-symmetric ordinal scales.

NeurIPS Paper Checklist

The checklist states that the paper’s main claims accurately reflect its contributions and scope.

  • NeurIPS Paper Checklist: The checklist records that the paper’s abstract and introduction accurately reflect its contributions and scope.

2. Experiments

The checklist reports that the experiments have empirical support and a complete setup description, while theoretical justification is partial and no foreseeable societal impact is identified.

  • 2. Experiments: Experimental results are reported as mean ± std across five random seeds on four datasets.
  • 2. Experiments: The experimental setup fully specifies architecture, optimizer, hyperparameters, and dataset splits, with code planned for release upon acceptance.
  • 2. Experiments: Theoretical claims are only partially justified because full properness analysis is left for future work.
  • 2. Experiments: The checklist identifies no foreseeable negative societal impact for this methods paper proposing an ordinal-classification loss.
  • 2. Experiments: The paper reports that its limitations are discussed in Section 7.
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