Programmatic Weak Supervision (PWS) aims to reduce the cost of constructing large high quality labeled datasets often used in training modern machine learning models. A major component of the PWS pipeline is the label model, which amalgamates predictions from multiple noisy weak supervision sources, i.e. labeling functions (LFs), to label datapoints. While most label models are either probabilistic or adversarial, a recently proposed label model achieves strong empirical performance without falling into either camp. That label model constructs a polytope of plausible labelings using the LF predictions and outputs the "center" of that polytope as its proposed labeling. In this paper, we attempt to theoretically study that strategy by proposing Bayesian Balsubramani-Freund (BBF), a label model that implicitly constructs a polytope of plausible labelings and selects a labeling in its interior. We show an assortment of statistical results for BBF: log-concavity of its posterior, its form of solution, consistency, and rates of convergence. Extensive experiments compare our proposed method against twelve baseline label models over eleven datasets. BBF compares favorably to other Bayesian label models and label models that don't use datapoint features -- matching or exceeding their performance on eight out of eleven datasets.

The accurate labeling of datasets is often both costly and time-consuming. Given an unlabeled dataset, programmatic weak supervision obtains probabilistic predictions for the labels by leveraging multiple weak labeling functions (LFs) that provide rough guesses for labels. Weak LFs commonly provide guesses with assorted types and unknown interdependences that can result in unreliable predictions. Furthermore, existing techniques for programmatic weak supervision cannot provide assessments for the reliability of the probabilistic predictions for labels. This paper presents a methodology for programmatic weak supervision that can provide confidence intervals for label probabilities and obtain more reliable predictions. In particular, the methods proposed use uncertainty sets of distributions that encapsulate the information provided by LFs with unrestricted behavior and typology. Experiments on multiple benchmark datasets show the improvement of the presented methods over the state-of-the-art and the practicality of the confidence intervals presented.

Labeling data via rules-of-thumb and minimal label supervision is central to Weak Supervision, a paradigm subsuming subareas of machine learning such as crowdsourced learning and semi-supervised ensemble learning. By using this labeled data to train modern machine learning methods, the cost of acquiring large amounts of hand labeled data can be ameliorated. Approaches to combining the rules-of-thumb falls into two camps, reflecting different ideologies of statistical estimation. The most common approach, exemplified by the Dawid-Skene model, is based on probabilistic modeling. The other, developed in the work of Balsubramani-Freund and others, is adversarial and game-theoretic. We provide a variety of statistical results for the adversarial approach under log-loss: we characterize the form of the solution, relate it to logistic regression, demonstrate consistency, and give rates of convergence. On the other hand, we find that probabilistic approaches for the same model class can fail to be consistent. Experimental results are provided to corroborate the theoretical results.