# Mixture Proportion Estimation and PU Learning: A Modern Approach

arXiv: https://arxiv.org/abs/2111.00980 · HTML: https://mlsanity.com/paper/2111.00980/

## Key facts

- Cited by 52 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 45 references that are in the map.
- First posted on arXiv 1 Nov 2021.

## Builds on (most cited first, 16 of 16)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding](https://mlsanity.com/paper/1810.04805/) (2018)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Understanding deep learning requires rethinking generalization](https://mlsanity.com/paper/1611.03530/) (2016)
- [Striving for Simplicity: The All Convolutional Net](https://mlsanity.com/paper/1412.6806/) (2014)
- [Detecting and Correcting for Label Shift with Black Box Predictors](https://mlsanity.com/paper/1802.03916/) (2018)
- [What is the Effect of Importance Weighting in Deep Learning?](https://mlsanity.com/paper/1812.03372/) (2018)
- [Failing Loudly: An Empirical Study of Methods for Detecting Dataset Shift](https://mlsanity.com/paper/1810.11953/) (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)
- [Regularized Learning for Domain Adaptation under Label Shifts](https://mlsanity.com/paper/1903.09734/) (2019)
- [A Unified View of Label Shift Estimation](https://mlsanity.com/paper/2003.07554/) (2020)
- [Mixture Proportion Estimation via Kernel Embedding of Distributions](https://mlsanity.com/paper/1603.02501/) (2016)
- [Nonparametric semi-supervised learning of class proportions](https://mlsanity.com/paper/1601.01944/) (2016)
- [Class Proportion Estimation with Application to Multiclass Anomaly Rejection](https://mlsanity.com/paper/1306.5056/) (2013)

## Built on by (most cited first, 30 of 52)

- [Dist-PU: Positive-Unlabeled Learning from a Label Distribution Perspective](https://mlsanity.com/paper/2212.02801/) (2022)
- [How Does Unlabeled Data Provably Help Out-of-Distribution Detection?](https://mlsanity.com/paper/2402.03502/) (2024)
- [Domain Adaptation under Open Set Label Shift](https://mlsanity.com/paper/2207.13048/) (2022)
- [RLSbench: Domain Adaptation Under Relaxed Label Shift](https://mlsanity.com/paper/2302.03020/) (2023)
- [ShapePU: A New PU Learning Framework Regularized by Global Consistency for Scribble Supervised Cardiac Segmentation](https://mlsanity.com/paper/2206.02118/) (2022)
- [Anatomy-Guided Weakly-Supervised Abnormality Localization in Chest X-rays](https://mlsanity.com/paper/2206.12704/) (2022)
- [Plugin estimators for selective classification with out-of-distribution detection](https://mlsanity.com/paper/2301.12386/) (2023)
- [Domain Adaptation under Missingness Shift](https://mlsanity.com/paper/2211.02093/) (2022)
- [Online Label Shift: Optimal Dynamic Regret meets Practical Algorithms](https://mlsanity.com/paper/2305.19570/) (2023)
- [Beyond Myopia: Learning from Positive and Unlabeled Data through Holistic Predictive Trends](https://mlsanity.com/paper/2310.04078/) (2023)
- [Learning with Complementary Labels Revisited: The Selected-Completely-at-Random Setting Is More Practical](https://mlsanity.com/paper/2311.15502/) (2023)
- [Unsupervised Learning under Latent Label Shift](https://mlsanity.com/paper/2207.13179/) (2022)
- [A Unified Positive-Unlabeled Learning Framework for Document-Level Relation Extraction with Different Levels of Labeling](https://mlsanity.com/paper/2210.08709/) (2022)
- [Imprecise Label Learning: A Unified Framework for Learning with Various Imprecise Label Configurations](https://mlsanity.com/paper/2305.12715/) (2023)
- [Mixture Proportion Estimation Beyond Irreducibility](https://mlsanity.com/paper/2306.01253/) (2023)
- [A Unified Approach to Count-Based Weakly-Supervised Learning](https://mlsanity.com/paper/2311.13718/) (2023)
- [Positive Unlabeled Contrastive Learning](https://mlsanity.com/paper/2206.01206/) (2022)
- [ZScribbleSeg: Zen and the Art of Scribble Supervised Medical Image Segmentation](https://mlsanity.com/paper/2301.04882/) (2023)
- [A General Framework for Learning from Weak Supervision](https://mlsanity.com/paper/2402.01922/) (2024)
- [Complementary Benefits of Contrastive Learning and Self-Training Under Distribution Shift](https://mlsanity.com/paper/2312.03318/) (2023)
- [Making Binary Classification from Multiple Unlabeled Datasets Almost Free of Supervision](https://mlsanity.com/paper/2306.07036/) (2023)
- [A Positive-Unlabeled Metric Learning Framework for Document-Level Relation Extraction with Incomplete Labeling](https://mlsanity.com/paper/2306.14806/) (2023)
- [Out-of-Distribution Detection using Synthetic Data Generation](https://mlsanity.com/paper/2502.03323/) (2025)
- [Positive-Unlabeled Reinforcement Learning Distillation for On-Premise Small Models](https://mlsanity.com/paper/2601.20687/) (2026)
- [Uncertainty-aware Pseudo-label Selection for Positive-Unlabeled Learning](https://mlsanity.com/paper/2201.13192/) (2022)
- [Offline Reinforcement Learning with Domain-Unlabeled Data](https://mlsanity.com/paper/2404.07465/) (2024)
- [ModelMix: A New Model-Mixup Strategy to Minimize Vicinal Risk across Tasks for Few-scribble based Cardiac Segmentation](https://mlsanity.com/paper/2406.13237/) (2024)
- [Positive and Unlabeled Data: Model, Estimation, Inference, and Classification](https://mlsanity.com/paper/2407.09735/) (2024)
- [Rethinking Consistent Multi-Label Classification Under Inexact Supervision](https://mlsanity.com/paper/2510.04091/) (2025)
- [Learning From Positive and Unlabeled Data Using Observer-GAN](https://mlsanity.com/paper/2208.12477/) (2022)

## Abstract (arXiv)

Given only positive examples and unlabeled examples (from both positive and negative classes), we might hope nevertheless to estimate an accurate positive-versus-negative classifier. Formally, this task is broken down into two subtasks: (i) Mixture Proportion Estimation (MPE) -- determining the fraction of positive examples in the unlabeled data; and (ii) PU-learning -- given such an estimate, learning the desired positive-versus-negative classifier. Unfortunately, classical methods for both problems break down in high-dimensional settings. Meanwhile, recently proposed heuristics lack theoretical coherence and depend precariously on hyperparameter tuning. In this paper, we propose two simple techniques: Best Bin Estimation (BBE) (for MPE); and Conditional Value Ignoring Risk (CVIR), a simple objective for PU-learning. Both methods dominate previous approaches empirically, and for BBE, we establish formal guarantees that hold whenever we can train a model to cleanly separate out a small subset of positive examples. Our final algorithm (TED)n, alternates between the two procedures, significantly improving both our mixture proportion estimator and classifier