Structured Proper Loss Geometries for Multiclass Classification: Theory and Controlled Empirical Evaluation
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Abstract (arXiv)
Strictly proper scoring rules identify the true conditional class distribution at population level, but their curvature can alter optimization and finite-sample behavior. We study three multiclass objectives: a class-aware quadratic Bregman score (CAPM), a strongly convex generator with constrained log-cosh ridges (HPG), and an HPG objective with an annealed probability-margin penalty (APMS). CAPM is treated as a structured instance of established quadratic scoring-rule theory. We derive conditional-regret, curvature, range, and logit-gradient bounds for CAPM and HPG, and prove exact penalty-range and conditional-target displacement bounds for APMS. Controlled five-seed experiments use Digits, Wisconsin breast cancer, and synthetic confusion and long-tail problems under clean labels, symmetric and pair-flip corruption, class imbalance, calibration evaluation, input corruption, and first-order adversarial perturbations. The candidates are close to cross-entropy on clean data and show descriptive gains in some noisy-label cells, but the five-seed comparisons are interpreted descriptively rather than as significance evidence. The selected noisy-label baselines perform better on Digits with 40% symmetric label noise, and explicit prior-adjustment methods perform better in the 30:1 synthetic long-tail experiment. Ablations do not show a consistent benefit from the candidate-specific graph, ridge, or margin components. The mathematical analysis establishes the stated properties, and the experiments delimit the empirical evidence; together they do not support a claim of general superiority.
Key facts. No citing papers in the map yet (data through 14 Oct 2026). Builds on 24 of its 34 references that are in the map. First posted on arXiv 28 Jun 2026.
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References · 24
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12 more
- 2019Symmetric Cross Entropy for Robust Learning with Noisy Labelscited by 500
- 2017Robust Loss Functions under Label Noise for Deep Neural Networkscited by 443
- 2020Balanced Meta-Softmax for Long-Tailed Visual Recognitioncited by 400
- 2020Normalized Loss Functions for Deep Learning with Noisy Labelscited by 257
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- 2022PolyLoss: A Polynomial Expansion Perspective of Classification Loss Functionscited by 63
- 2019Robust Bi-Tempered Logistic Loss Based on Bregman Divergencescited by 61
- 2020On Focal Loss for Class-Posterior Probability Estimation: A Theoretical Perspectivecited by 21
- 2024Language Generation with Strictly Proper Scoring Rulescited by 8
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- 2023LegendreTron: Uprising Proper Multiclass Loss Learningcited by 2
- 2026Tailoring Strictly Proper Scoring Rules for Downstream Tasks: An Application to Causal Inferencecited by 1
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