# Multi-Complementary and Unlabeled Learning for Arbitrary Losses and Models

arXiv: https://arxiv.org/abs/2001.04243 · HTML: https://mlsanity.com/paper/2001.04243/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 13 Jan 2020.

## Built on by (most cited first, 5 of 5)

- [Reduction from Complementary-Label Learning to Probability Estimates](https://mlsanity.com/paper/2209.09500/) (2022)
- [Intra-Cluster Mixup: An Effective Data Augmentation Technique for Complementary-Label Learning](https://mlsanity.com/paper/2509.17971/) (2025)
- [libcll: an Extendable Python Toolkit for Complementary-Label Learning](https://mlsanity.com/paper/2411.12276/) (2024)
- [Q&A Label Learning](https://mlsanity.com/paper/2305.04539/) (2023)
- [Learning from N-Tuple Data with M Positive Instances: Unbiased Risk Estimation and Theoretical Guarantees](https://mlsanity.com/paper/2510.18406/) (2025)

## Abstract (arXiv)

A weakly-supervised learning framework named as complementary-label learning has been proposed recently, where each sample is equipped with a single complementary label that denotes one of the classes the sample does not belong to. However, the existing complementary-label learning methods cannot learn from the easily accessible unlabeled samples and samples with multiple complementary labels, which are more informative. In this paper, to remove these limitations, we propose the novel multi-complementary and unlabeled learning framework that allows unbiased estimation of classification risk from samples with any number of complementary labels and unlabeled samples, for arbitrary loss functions and models. We first give an unbiased estimator of the classification risk from samples with multiple complementary labels, and then further improve the estimator by incorporating unlabeled samples into the risk formulation. The estimation error bounds show that the proposed methods are in the optimal parametric convergence rate. Finally, the experiments on both linear and deep models show the effectiveness of our methods.