# A Unified Empirical Risk Minimization Framework for Flexible N-Tuples Weak Supervision

arXiv: https://arxiv.org/abs/2507.07771 · HTML: https://mlsanity.com/paper/2507.07771/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 15 of its 32 references that are in the map.
- First posted on arXiv 10 Jul 2025.

## Builds on (most cited first, 15 of 15)

- [Positive-Unlabeled Learning with Non-Negative Risk Estimator](https://mlsanity.com/paper/1703.00593/) (2017)
- [Learning from Complementary Labels](https://mlsanity.com/paper/1705.07541/) (2017)
- [Complementary-Label Learning for Arbitrary Losses and Models](https://mlsanity.com/paper/1810.04327/) (2018)
- [Learning with Multiple Complementary Labels](https://mlsanity.com/paper/1912.12927/) (2019)
- [Partial Label Learning with Self-Guided Retraining](https://mlsanity.com/paper/1902.03045/) (2019)
- [On the Minimal Supervision for Training Any Binary Classifier from Only Unlabeled Data](https://mlsanity.com/paper/1808.10585/) (2018)
- [Classification from Pairwise Similarity and Unlabeled Data](https://mlsanity.com/paper/1802.04381/) (2018)
- [Mitigating Overfitting in Supervised Classification from Two Unlabeled Datasets: A Consistent Risk Correction Approach](https://mlsanity.com/paper/1910.08974/) (2019)
- [Binary Classification from Positive-Confidence Data](https://mlsanity.com/paper/1710.07138/) (2017)
- [Classification from Pairwise Similarities/Dissimilarities and Unlabeled Data via Empirical Risk Minimization](https://mlsanity.com/paper/1904.11717/) (2019)
- [Semi-Supervised AUC Optimization based on Positive-Unlabeled Learning](https://mlsanity.com/paper/1705.01708/) (2017)
- [Pointwise Binary Classification with Pairwise Confidence Comparisons](https://mlsanity.com/paper/2010.01875/) (2020)
- [Learning from Similarity-Confidence Data](https://mlsanity.com/paper/2102.06879/) (2021)
- [Binary Classification from Multiple Unlabeled Datasets via Surrogate Set Classification](https://mlsanity.com/paper/2102.00678/) (2021)
- [Classification from Triplet Comparison Data](https://mlsanity.com/paper/1907.10225/) (2019)

## Abstract (arXiv)

To alleviate the annotation burden in supervised learning, N-tuples learning has recently emerged as a powerful weakly-supervised method. While existing N-tuples learning approaches extend pairwise learning to higher-order comparisons and accommodate various real-world scenarios, they often rely on task-specific designs and lack a unified theoretical foundation. In this paper, we propose a general N-tuples learning framework based on empirical risk minimization, which systematically integrates pointwise unlabeled data to enhance learning performance. This paper first unifies the data generation processes of N-tuples and pointwise unlabeled data under a shared probabilistic formulation. Based on this unified view, we derive an unbiased empirical risk estimator that generalizes a broad class of existing N-tuples models. We further establish a generalization error bound for theoretical support. To demonstrate the flexibility of the framework, we instantiate it in four representative weakly supervised scenarios, each recoverable as a special case of our general model. Additionally, to address overfitting issues arising from negative risk terms, we adopt correction functions to adjust the empirical risk. Extensive experiments on benchmark datasets validate the effectiveness of the proposed framework and demonstrate that leveraging pointwise unlabeled data consistently improves generalization across various N-tuples learning tasks.