Ordinal Cross-Entropy Loss

• Ordinal cross-entropy loss is a loss function that incorporates the inherent ordering of classes by penalizing errors in proportion to their distance from the true label.
• It uses techniques such as distance-weighting, unimodal constraints, and soft encoding to align outputs with ordinal relationships in applications like medical imaging and age estimation.
• Empirical evaluations show that ordinal-aware loss methods improve model calibration, interpretability, and reduce severe misclassifications compared to standard cross-entropy loss.

Ordinal cross-entropy loss refers to a class of loss functions designed to accommodate the inherent ordering of categories in an ordinal classification problem, remedying the limitations of traditional categorical cross-entropy loss which treats all class mispredictions equivalently. Unlike the standard cross-entropy loss, ordinal cross-entropy losses impose penalties that reflect the magnitude of deviation from the ground-truth class, typically employing distance-based weighting, unimodal constraints, or structured regularization to enhance prediction quality and interpretability under ordinal settings.

1. Ordinal Cross-Entropy: Conceptual Foundations

In standard categorical classification, the cross-entropy loss function optimizes the predicted label probability vector against a one-hot target that indicates the true class. This classic treatment is suboptimal for ordinal regression tasks, where misclassifying an instance as a class adjacent to the true category should incur a lower penalty than predicting a distant class. The need for ordinal-aware loss is particularly acute in fields such as medical imaging, risk scoring, and age estimation, where class ordering is critical. Recent literature addresses this gap by introducing ordinal cross-entropy losses that integrate class distance or ordering directly into the loss computation, penalizing misclassifications according to their ordinal divergence (Beckham et al., 2017, Polat et al., 2022, Polat et al., 2 Dec 2024).

2. Methods for Ordinal Cross-Entropy Loss Construction

Multiple approaches to constructing ordinal cross-entropy losses have emerged:

• Distance-weighted cross-entropy: CDW-CE (Class Distance Weighted Cross-Entropy) modifies the standard cross-entropy by scaling the penalty for each class by an order-based factor $|i - c|^\alpha$, where $i$ is the predicted class index, $c$ is the true class, and $\alpha$ is a hyperparameter controlling sensitivity to class distance (Polat et al., 2022, Polat et al., 2 Dec 2024).

$$\text{CDW-CE} = -\sum_{i=0}^{N-1} \log(1 - \hat{y}_i) \cdot |i - c|^\alpha$$

• Probability distribution shaping: Probability outputs from a deep network are shaped using unimodal distributions parameterized by Poisson or binomial PMFs to produce a single-peaked distribution over the ordinal classes, directly enforcing unimodality and ensuring that neighboring classes have similar probabilities (Beckham et al., 2017).

For example, the binomial variant:

$p(k; n, p) = \binom{n}{k} p^k (1-p)^{n-k}$

• Soft ordinal encoding and regularization: Instead of one-hot encoding, targets are soft-encoded as distributions with mass concentrated around the true label, and regularization terms enforce unimodal structure in the output. The ORCU loss introduces a soft-encoded cross-entropy plus an order-aware regularization term, directly improving calibration and ordinal consistency (Kim et al., 21 Oct 2024).

3. Geometric and Divergence-Based Extensions

Research leveraging entropy-regularized optimal transport (Fenchel-Young losses) and $f$-divergence generalizations proposes embedding inter-class costs directly into the loss function. Geometric losses enable attaching a cost matrix $C(y, y')$ representing the penalty for predicting label $y'$ instead of true label $y$, and these costs can naturally reflect ordinal structure via $|y - y'|$ or $|y - y'|^2$ metrics (Mensch et al., 2019, Roulet et al., 30 Jan 2025).

The Fenchel-Young construction for an $f$-divergence is (using reference measure $q$):

$$\ell_f(\theta, y; q) = \text{softmax}_f(\theta; q) + D_f(y, q) - \langle y, \theta \rangle$$

where $D_f(y, q) = \sum_j f(y_j / q_j) q_j$ and $\text{softmax}_f$ is obtained through the maximization $$\max_{p \in \Delta^k} \langle p, \theta \rangle - D_f(p, q)$$.

This framework generalizes KL-based cross-entropy to any convex $f$-divergence, with tuning of reference distributions and generator functions to tailor loss behavior for ordinal class structures.

4. Ordinality-Aware Regularization and Calibration

Overconfidence and non-unimodal predictions are typical failure modes of conventional cross-entropy for ordinal regression. Recent methods introduce soft encoding and explicit regularization:

• Soft ordinal encoding: Labels are encoded as a probability distribution over classes using a similarity-based function, usually via exponential decay with respect to label distance. This avoids single-class spikes and aligns model outputs with ordinal relationships (Kim et al., 21 Oct 2024).
• Unimodality constraints: Regularization terms on the logits enforce monotonic increase then decrease in output probabilities around the correct label, ensuring distributions peak near the true class and decay as ordinal distance grows (Beckham et al., 2017, Kim et al., 21 Oct 2024).
• Calibration metrics: Ordinal cross-entropy formulations are quantitatively evaluated on calibration errors (SCE, ACE, ECE) and unimodality, showing improved trustworthiness of prediction probabilities in ordinal settings (Kim et al., 21 Oct 2024).

5. Empirical Evaluation and Performance Metrics

Ordinal cross-entropy losses are benchmarked across multiple domains:

Dataset/Task	Loss Type	Key Metrics	Findings
Ulcerative Colitis Severity	CDW-CE	QWK, F1, MAE, CAMs	Higher QWK, improved interpretability, superior CAM
Diabetic Retinopathy	Unimodal Binomial	Top-k, QWK	Smoothed probability mass, less severe penalties
Age Estimation (Adience)	ORCU, Binomial	SCE, ACE, Unimodality	SOTA calibration, unimodal outputs

Across studies, ordinal cross-entropy losses demonstrate tangible improvements over vanilla cross-entropy, including:

• Reduction in large-distance misclassifications
• Better clustering of latent representations (Silhouette scores)
• Alignment of attention maps with domain expert expectations
• Enhanced calibration with unimodal probability distributions centered on true labels

6. Implementation and Adaptability

Ordinal cross-entropy loss formulations are computationally straightforward to integrate into existing deep learning pipelines. Distance-weighted schemes and soft-encoding approaches require minimal changes to loss computation logic. Parametric regularization (e.g., the margin $m$ in CDW-CE) or temperature parameters ($\tau$ for tuning unimodality) can be learned or set during hyperparameter optimization (Polat et al., 2 Dec 2024, Beckham et al., 2017). Alternatives such as $f$-divergence-based losses require solving one-dimensional root-finding problems (e.g., by bisection), but parallelizable algorithms have been established to maintain practical efficiency (Roulet et al., 30 Jan 2025).

Ordinal extensions have also been applied to binary settings, e.g., solar flare prediction using proximity-weighted BCE, indicating generalizability to diverse problem types (Pandey et al., 5 Oct 2025).

7. Significance and Outlook

Ordinal cross-entropy loss methods constitute a principled advancement for tasks where class order is semantically meaningful. These methods provide improved model calibration, interpretability, and robustness by incorporating ordinal relationships directly into the optimization objective. Experimental evidence across medical image analysis, computer vision, and risk assessment validates their utility. Ongoing research explores structured entropy, cost-sensitive losses, and geometric optimal transport as promising avenues for further enhancing ordinal loss design (Lucena, 2022, Mensch et al., 2019). A plausible implication is increased adoption of ordinal-aware cross-entropy variants in safety-critical and clinical applications, where not only prediction but also probability interpretation is essential.