This is a corrigendum for the article “An Unbiased Risk Estimator for Partial Label Learning with Augmented Classes” published in ACM Trans. Intell. Syst. Technol. 15(6): 131:1-131:22 (2024).

Weakly supervised learning aims to construct effective predictive models from imperfectly labeled data. The recent trend of weakly supervised learning has focused on how to learn an accurate classifier from completely unlabeled data, given little supervised information such as class priors. In this paper, we consider a newly proposed weakly supervised learning problem called multi-class classification from multiple unlabeled datasets, where only multiple sets of unlabeled data and their class priors (i.e., the proportions of each class) are provided for training the classifier. To solve this problem, we first propose a classifier-consistent method (CCM) based on a probability transition matrix. However, CCM cannot guarantee risk consistency and lacks of purified supervision information during training. Therefore, we further propose a risk-consistent method (RCM) that progressively purifies supervision information during training by importance weighting. We provide comprehensive theoretical analyses for our methods to demonstrate the statistical consistency. Experimental results on multiple benchmark datasets and various prior matrices demonstrate the superiority of our proposed methods.

Partial Label Learning (PLL) is a typical weakly supervised learning task, which assumes each training instance is annotated with a set of candidate labels containing the ground-truth label. Recent PLL methods adopt identification-based disambiguation to alleviate the influence of false positive labels and achieve promising performance. However, they require all classes in the test set to have appeared in the training set, ignoring the fact that new classes will keep emerging in real applications. To address this issue, in this article, we focus on the problem of Partial Label Learning with Augmented Class (PLLAC), where one or more augmented classes are not visible in the training stage but appear in the inference stage. Specifically, we propose an unbiased risk estimator with theoretical guarantees for PLLAC, which estimates the distribution of augmented classes by differentiating the distribution of known classes from unlabeled data and can be equipped with arbitrary PLL loss functions. Besides, we provide a theoretical analysis of the estimation error bound of the estimator, which guarantees the convergence of the empirical risk minimizer to the true risk minimizer as the number of training data tends to infinity. Furthermore, we add a risk-penalty regularization term in the optimization objective to alleviate the influence of the over-fitting issue caused by negative empirical risk. Extensive experiments on benchmark, UCI, and real-world datasets demonstrate the effectiveness of the proposed approach.

Multiple-instance learning (MIL) is a significant weakly supervised learning problem, where the training data consist of bags containing multiple instances and bag-level labels. Most previous MIL research required fully labeled bags. However, collecting such data is challenging due to the labeling costs or privacy concerns. Fortunately, we can easily collect pairwise comparison information, indicating one bag is more likely to be positive than the other. Therefore, we investigate a novel MIL problem about learning a bag-level binary classifier only from pairwise comparison bags. To solve this problem, we display the data generation process and provide a baseline method to train an instance-level classifier based on unlabeled-unlabeled learning. To achieve better performance, we propose a convex formulation to train a bag-level classifier and give a generalization error bound. Comprehensive experiments show that both the baseline method and the convex formulation achieve satisfactory performance, while the convex formulation performs better. 1

Multiple-instance learning (MIL) solves the problem where training instances are grouped in bags, and a binary (positive or negative) label is provided for each bag. Most of the existing MIL studies need fully labeled bags for training an effective classifier, while it could be quite hard to collect such data in many real-world scenarios, due to the high cost of data labeling process. Fortunately, unlike fully labeled data, triplet comparison data can be collected in a more accurate and human-friendly way. Therefore, in this article, we for the first time investigate MIL from only triplet comparison bags , where a triplet (X a , X b , X c ) contains the weak supervision information that bag X a is more similar to X b than to X c . To solve this problem, we propose to train a bag-level classifier by the empirical risk minimization framework and theoretically provide a generalization error bound. We also show that a convex formulation can be obtained only when specific convex binary losses such as the square loss and the double hinge loss are used. Extensive experiments validate that our proposed method significantly outperforms other baselines.

In contrast to the standard learning paradigm where all classes can be observed in training data, learning with augmented classes (LAC) tackles the problem where augmented classes unobserved in the training data may emerge in the test phase. Previous research showed that given unlabeled data, an unbiased risk estimator (URE) can be derived, which can be minimized for LAC with theoretical guarantees. However, this URE is only restricted to the specific type of one-versus-rest loss functions for multi-class classification, making it not flexible enough when the loss needs to be changed with the dataset in practice. In this paper, we propose a generalized URE that can be equipped with arbitrary loss functions while maintaining the theoretical guarantees, given unlabeled data for LAC. To alleviate the issue of negative empirical risk commonly encountered by previous studies, we further propose a novel risk-penalty regularization term. Experiments demonstrate the effectiveness of our proposed method.

Inmultiple-instance learning(MIL), each training example is represented by a bag of instances. A training bag is either negative if it contains no positive instances or positive if it has at least one positive instance. Previous MIL methods generally assume that training bags are fully labeled. However, the exact labels of training examples may not be accessible, due to security, confidentiality, and privacy concerns. Fortunately, it could be easier for us to access the pairwise similarity between two bags (indicating whether two bags share the same label or not) and unlabeled bags, as we do not need to know the underlying label of each bag. In this paper, we provide the first attempt to investigate MIL from only similar-dissimilar-unlabeled bags. To solve this new MIL problem, we first propose a strong baseline method that trains an instance-level classifier by employing an unlabeled-unlabeled learning strategy. Then, we also propose to train a bag-level classifier based on a convex formulation and theoretically derive a generalization error bound for this method. Comprehensive experimental results show that our instance-level classifier works well, while our bag-level classifier even has better performance.

This article tackles the problem of multilabel learning with missing labels. For this problem, it is widely accepted that label correlations can be used to recover the ground-truth label matrix. Most of the existing approaches impose the low-rank assumption on the observed label matrix to exploit label correlations by decomposing it into two matrices, which describe the latent factors of instances and labels, respectively. The quality of these latent factors highly influences the recovery of ground-truth labels and the construction of the multilabel classification model. In this article, we propose recovering the ground-truth label matrix by regularized matrix factorization. Specifically, the latent factors of instances are regularized by the local topological structure derived from the feature space, which can be further used to induce an effective multilabel model. Moreover, the latent factors of labels and the label correlations are mutually adapted via label manifold regularization. In this way, the recovery of the ground-truth label matrix and the construction of the multilabel classification model are optimized jointly and can benefit from the regularized matrix factorization. Extensive experimental studies show that the proposed approach significantly outperforms the state-of-the-art algorithms on both full-label and missing-label data.

To alleviate the data requirement for training effective binary classifiers in binary classification, many weakly supervised learning settings have been proposed. Among them, some consider using pairwise but not pointwise labels, when pointwise labels are not accessible due to privacy, confidentiality, or security reasons. However, as a pairwise label denotes whether or not two data points share a pointwise label, it cannot be easily collected if either point is equally likely to be positive or negative. Thus, in this paper, we propose a novel setting called pairwise comparison (Pcomp) classification, where we have only pairs of unlabeled data that we know one is more likely to be positive than the other. Firstly, we give a Pcomp data generation process, derive an unbiased risk estimator (URE) with theoretical guarantee, and further improve URE using correction functions. Secondly, we link Pcomp classification to noisy-label learning to develop a progressive URE and improve it by imposing consistency regularization. Finally, we demonstrate by experiments the effectiveness of our methods, which suggests Pcomp is a valuable and practically useful type of pairwise supervision besides the pairwise label.

Multiple-instance learning (MIL) is an important weakly supervised binary classification problem, where training instances are arranged in bags, and each bag is assigned a positive or negative label. Most of the previous studies for MIL assume that training bags are fully labeled. However, in some real-world scenarios, it could be difficult to collect fully labeled bags, due to the expensive time and labor consumption of the labeling task. Fortunately, it could be much easier for us to collect similar and dissimilar bags (indicating whether two bags share the same label or not), because we do not need to figure out the underlying label of each bag in this case. Therefore, in this paper, we for the first time investigate MIL from only similar and dissimilar bags. To solve this new MIL problem, we propose a convex formulation to train a bag-level classifier based on empirical risk minimization and theoretically derive a generalization error bound. In addition, we also propose a strong baseline for this new MIL problem, which aims to train an instance-level classifier by minimizing the instance-level empirical risk. Extensive experimental results clearly demonstrate that our proposed baseline works well, while our proposed convex formulation is even better.

Positive-unlabeled (PU) learning handles the problem of learning a predictive model from PU data. Past few years have witnessed the boom of PU learning, while the existing learning algorithms are limited to binary classification and cannot be directly applied to multi-class PU data. In this paper, we present an unbiased estimator of the original classification risk for multi-class PU learning, and show that the direct empirical risk minimization suffers from the severe overfitting problem because the risk is unbounded below. To address this problem, we propose an alternative risk estimator, and theoretically establish an estimation error bound. We show that the estimation error of its empirical risk minimizer achieves the optimal parametric convergence rate. Extensive experiments on multiple datasets demonstrate the effectiveness of the proposed approach for multi-class PU learning.

Trained with the standard cross entropy loss, deep neural networks can achieve great performance on correctly labeled data. However, if the training data is corrupted with label noise, deep models tend to overfit the noisy labels, thereby achieving poor generation performance. To remedy this issue, several loss functions have been proposed and demonstrated to be robust to label noise. Although most of the robust loss functions stem from Categorical Cross Entropy (CCE) loss, they fail to embody the intrinsic relationships between CCE and other loss functions. In this paper, we propose a general framework dubbed Taylor cross entropy loss to train deep models in the presence of label noise. Specifically, our framework enables to weight the extent of fitting the training labels by controlling the order of Taylor Series for CCE, hence it can be robust to label noise. In addition, our framework clearly reveals the intrinsic relationships between CCE and other loss functions, such as Mean Absolute Error (MAE) and Mean Squared Error (MSE). Moreover, we present a detailed theoretical analysis to certify the robustness of this framework. Extensive experimental results on benchmark datasets demonstrate that our proposed approach significantly outperforms the state-of-the-art counterparts.

Partial multi-label learning (PML) deals with the problem where each training example is assigned multiple candidate labels, only a part of which are correct. To learn from such PML examples, the straightforward model training tends to be misled by the noise candidate label set. To alleviate this problem, a coupled framework is established in this paper to learn the desired model and perform the relabeling procedure alternatively. In the relabeling procedure, instead of simply extracting relative label confidences, or deterministically eliminating low confidence labels and preserving high confidence labels as ground-truth ones, we introduce a soft sign thresholding operator to adaptively strengthen candidate labels with high confidence and weaken candidate labels with low confidence, which enlarges the difference of confidences of candidate labels within allowable range. We further show that the resulting nonconvex quadratic programming (QP) optimization problem can be relaxed into a convex QP problem with proper conditions. Extensive experiments on synthesized and real-world data sets demonstrate the effectiveness of our proposed approach.