# Cost-Sensitive Unbiased Risk Estimation for Multi-Class Positive-Unlabeled Learning

arXiv: https://arxiv.org/abs/2510.25226 · HTML: https://mlsanity.com/paper/2510.25226/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 17 of its 42 references that are in the map.
- First posted on arXiv 29 Oct 2025.

## Builds on (most cited first, 17 of 17)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift](https://mlsanity.com/paper/1502.03167/) (2015)
- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification](https://mlsanity.com/paper/1502.01852/) (2015)
- [Deep Learning for Classical Japanese Literature](https://mlsanity.com/paper/1812.01718/) (2018)
- [Positive-Unlabeled Learning with Non-Negative Risk Estimator](https://mlsanity.com/paper/1703.00593/) (2017)
- [Learning from positive and unlabeled data: a survey](https://mlsanity.com/paper/1811.04820/) (2018)
- [Cost Sensitive Learning of Deep Feature Representations from Imbalanced Data](https://mlsanity.com/paper/1508.03422/) (2015)
- [Class-prior Estimation for Learning from Positive and Unlabeled Data](https://mlsanity.com/paper/1611.01586/) (2016)
- [On the Minimal Supervision for Training Any Binary Classifier from Only Unlabeled Data](https://mlsanity.com/paper/1808.10585/) (2018)
- [Dist-PU: Positive-Unlabeled Learning from a Label Distribution Perspective](https://mlsanity.com/paper/2212.02801/) (2022)
- [Rademacher Complexity Bounds for a Penalized Multiclass Semi-Supervised Algorithm](https://mlsanity.com/paper/1607.00567/) (2016)
- [Rademacher Complexity and Generalization Performance of Multi-category Margin Classifiers](https://mlsanity.com/paper/1812.00584/) (2018)
- [Auxiliary Tasks Enhanced Dual-affinity Learning for Weakly Supervised Semantic Segmentation](https://mlsanity.com/paper/2403.01156/) (2024)
- [Identifying leading indicators of product recalls from online reviews using positive unlabeled learning and domain adaptation](https://mlsanity.com/paper/1703.00518/) (2017)
- [A review of unsupervised learning in astronomy](https://mlsanity.com/paper/2406.17316/) (2024)

## Abstract (arXiv)

Positive--Unlabeled (PU) learning considers settings in which only positive and unlabeled data are available, while negatives are missing or left unlabeled. This situation is common in real applications where annotating reliable negatives is difficult or costly. Despite substantial progress in PU learning, the multi-class case (MPU) remains challenging: many existing approaches do not ensure unbiased risk estimation, which limits performance and stability. We propose a cost-sensitive multi-class PU method based on adaptive loss weighting. Within the empirical risk minimization framework, we assign distinct, data-dependent weights to the positive and inferred-negative (from the unlabeled mixture) loss components so that the resulting empirical objective is an unbiased estimator of the target risk. We formalize the MPU data-generating process and establish a generalization error bound for the proposed estimator. Extensive experiments on eight public datasets, spanning varying class priors and numbers of classes, show consistent gains over strong baselines in both accuracy and stability.