Instance-Level Costs for Nuanced Classifier Evaluation

We propose a unified algebraic framework for classification performance evaluation that encompasses binary, multiclass, multilabel, ordinal, hierarchical, cost-sensitive, and soft-label settings within a single formalism. The foundation is a representation of actual and predicted labels as binary indicator matrices, combined with three aggregation operators -- global, column-wise, and row-wise -- that correspond exactly to micro, macro/weighted, and exemplar averaging. Any binary performance measure expressed in terms of true/positive/negative counts extends automatically to all settings by substituting these operators, generating multiclass and multilabel versions without measure-specific derivations. The framework further accommodates soft classifier outputs via argmax or thresholding, soft ground truth via triangular norms, ordinal classification via membership functions or cumulative encodings, and cost-sensitive evaluation via a cost matrix that subsumes MAE and MSE as special cases. We establish several theoretical results: micro-averaging equals denominator-weighted macro-averaging; the product $t$-norm is the unique one preserving the confusion-matrix partition; skew-invariant measures are characterised as functions of recall and specificity; and micro-precision, micro-recall, and micro-$F_1$ are all equal to accuracy in multiclass settings. Empirical illustrations on synthetic and real data confirm the theoretical findings.

High-stakes decision systems in credit scoring, fraud detection, healthcare, and industrial safety require reliable uncertainty quantification under severe class imbalance and asymmetric error costs. Standard marginal conformal prediction (CP) provides valid overall coverage guarantees; however, we show that it severely under-covers rare, costly minority classes, with minority-class coverage dropping to as low as 0.5% on certain datasets. To characterize and address this limitation, we conduct a comprehensive benchmark comparing marginal CP, class-conditional (Mondrian) CP, and cost-controlled abstention mechanisms across 15 real-world imbalanced tabular datasets, 7 classification models, 3 probability calibration techniques, and 10 random seeds, resulting in 3,150 experimental runs. Our results show that Mondrian CP restores valid minority-class coverage, achieving an average minority-coverage improvement of 61.7 percentage points over marginal CP (p < 1e-80). Furthermore, combining Mondrian CP with cost-controlled abstention significantly reduces expected decision cost compared with standard decision boundaries, confidence-based rejectors, and risk-controlled rejectors under realistic human review budgets. We further quantify dataset-specific break-even thresholds at which deferring ambiguous instances to human experts becomes cost-effective. These findings provide practical guidance for deploying distribution-free, cost-aware uncertainty quantification in high-stakes decision support systems.

Class imbalance poses a significant challenge in classification tasks, often causing standard learning algorithms to become biased toward the majority class. Cost-sensitive learning (CSL) addresses this issue by assigning higher penalties to minority-class misclassifications. However, conventional CSL typically applies a uniform penalty to all minority-class instances, ignoring the fact that minority samples may differ substantially in terms of local safety, overlap, boundary ambiguity, and outlier-like behavior. Uniform penalization can therefore introduce undue bias, increasing the number of misclassifications. In this study, we propose iCost, an instance-complexity-aware CSL framework that assigns adaptive penalties to minority-class samples according to their estimated learning difficulty. This fine-grained penalization strategy ensures fairer weighting, reduces unwarranted bias, and improves overall classification performance. Two complementary complexity estimation strategies are introduced: Neighbor-iCost, based on local neighborhood composition, and Gini-iCost, based on Gini-impurity-based feature-space partitioning. Extensive experiments on 65 binary and 10 multiclass imbalanced datasets show that iCost outperforms conventional CSL by a clear margin and remains highly competitive with widely used resampling methods. To support reproducibility and practical adoption, the proposed algorithm has been released as a scikit-learn-compatible Python package through PyPI. This work offers a fresh perspective on imbalanced learning by integrating instance-level data complexity into the learning process, opening new avenues for developing adaptive, complexity-aware strategies for imbalanced classification.