# Risk bounds for PU learning under Selected At Random assumption

arXiv: https://arxiv.org/abs/2201.06277 · HTML: https://mlsanity.com/paper/2201.06277/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 6 of its 31 references that are in the map.
- First posted on arXiv 17 Jan 2022.

## Builds on (most cited first, 6 of 6)

- [Learning from positive and unlabeled data: a survey](https://mlsanity.com/paper/1811.04820/) (2018)
- [Mixture Proportion Estimation via Kernel Embedding of Distributions](https://mlsanity.com/paper/1603.02501/) (2016)
- [Self-PU: Self Boosted and Calibrated Positive-Unlabeled Training](https://mlsanity.com/paper/2006.11280/) (2020)
- [Beyond the Selected Completely At Random Assumption for Learning from Positive and Unlabeled Data](https://mlsanity.com/paper/1809.03207/) (2018)
- [Estimating the class prior and posterior from noisy positives and unlabeled data](https://mlsanity.com/paper/1606.08561/) (2016)
- [Instance-Dependent PU Learning by Bayesian Optimal Relabeling](https://mlsanity.com/paper/1808.02180/) (2018)

## Abstract (arXiv)

Positive-unlabeled learning (PU learning) is known as a special case of semi-supervised binary classification where only a fraction of positive examples are labeled. The challenge is then to find the correct classifier despite this lack of information. Recently, new methodologies have been introduced to address the case where the probability of being labeled may depend on the covariates. In this paper, we are interested in establishing risk bounds for PU learning under this general assumption. In addition, we quantify the impact of label noise on PU learning compared to standard classification setting. Finally, we provide a lower bound on minimax risk proving that the upper bound is almost optimal.