Anomaly detection is the task of identifying examples that do not behave as expected. Because anomalies are rare and unexpected events, collecting real anomalous examples is often challenging in several applications. In addition, learning an anomaly detector with limited (or no) anomalies often yields poor prediction performance. One option is to employ auxiliary synthetic anomalies to improve the model training. However, synthetic anomalies may be of poor quality: anomalies that are unrealistic or indistinguishable from normal samples may deteriorate the detector's performance. Unfortunately, no existing methods quantify the quality of auxiliary anomalies. We fill in this gap and propose the expected anomaly posterior (EAP), an uncertainty-based score function that measures the quality of auxiliary anomalies by quantifying the total uncertainty of an anomaly detector. Experimentally on 40 benchmark datasets of images and tabular data, we show that EAP outperforms 12 adapted data quality estimators in the majority of cases.

With the increasing deployment of machine learning models in many socially-sensitive tasks, there is a growing demand for reliable and trustworthy predictions. One way to accomplish these requirements is to allow a model to abstain from making a prediction when there is a high risk of making an error. This requires adding a selection mechanism to the model, which selects those examples for which the model will provide a prediction. The selective classification framework aims to design a mechanism that balances the fraction of rejected predictions (i.e., the proportion of examples for which the model does not make a prediction) versus the improvement in predictive performance on the selected predictions. Multiple selective classification frameworks exist, most of which rely on deep neural network architectures. However, the empirical evaluation of the existing approaches is still limited to partial comparisons among methods and settings, providing practitioners with little insight into their relative merits. We fill this gap by benchmarking 18 baselines on a diverse set of 44 datasets that includes both image and tabular data. Moreover, there is a mix of binary and multiclass tasks. We evaluate these approaches using several criteria, including selective error rate, empirical coverage, distribution of rejected instance's classes, and performance on out-of-distribution instances. The results indicate that there is not a single clear winner among the surveyed baselines, and the best method depends on the users' objectives.

Anomaly detection methods identify examples that do not follow the expected behaviour, typically in an unsupervised fashion, by assigning real-valued anomaly scores to the examples based on various heuristics. These scores need to be transformed into actual predictions by thresholding, so that the proportion of examples marked as anomalies equals the expected proportion of anomalies, called contamination factor. Unfortunately, there are no good methods for estimating the contamination factor itself. We address this need from a Bayesian perspective, introducing a method for estimating the posterior distribution of the contamination factor of a given unlabeled dataset. We leverage on outputs of several anomaly detectors as a representation that already captures the basic notion of anomalousness and estimate the contamination using a specific mixture formulation. Empirically on 22 datasets, we show that the estimated distribution is well-calibrated and that setting the threshold using the posterior mean improves the anomaly detectors' performance over several alternative methods. All code is publicly available for full reproducibility.

Anomaly detection attempts at finding examples that deviate from the expected behaviour. Usually, anomaly detection is tackled from an unsupervised perspective because anomalous labels are rare and difficult to acquire. However, the lack of labels makes the anomaly detector have high uncertainty in some regions, which usually results in poor predictive performance or low user trust in the predictions. One can reduce such uncertainty by collecting specific labels using Active Learning (AL), which targets examples close to the detector's decision boundary. Alternatively, one can increase the user trust by allowing the detector to abstain from making highly uncertain predictions, which is called Learning to Reject (LR). One way to do this is by thresholding the detector's uncertainty based on where its performance is low, which requires labels to be evaluated. Although both AL and LR need labels, they work with different types of labels: AL seeks strategic labels, which are evidently biased, while LR requires i.i.d. labels to evaluate the detector's performance and set the rejection threshold. Because one usually has a unique label budget, deciding how to optimally allocate it is challenging. In this paper, we propose a mixed strategy that, given a budget of labels, decides in multiple rounds whether to use the budget to collect AL labels or LR labels. The strategy is based on a reward function that measures the expected gain when allocating the budget to either side. We evaluate our strategy on 18 benchmark datasets and compare it to some baselines.

Machine learning models always make a prediction, even when it is likely to be inaccurate. This behavior should be avoided in many decision support applications, where mistakes can have severe consequences. Albeit already studied in 1970, machine learning with a reject option recently gained interest. This machine learning subfield enables machine learning models to abstain from making a prediction when likely to make a mistake. This survey aims to provide an overview on machine learning with a reject option. We introduce the conditions leading to two types of rejection, ambiguity and novelty rejection. Moreover, we define the existing architectures for models with a reject option, describe the standard learning strategies to train such models and relate traditional machine learning techniques to rejection. Additionally, we review strategies to evaluate a model's predictive and rejective quality. Finally, we provide examples of relevant application domains and show how machine learning with rejection relates to other machine learning research areas.