The main goal of this article is to convince you, the reader, that supervised learning in the Probably Approximately Correct (PAC) model is closely related to -- of all things -- bipartite matching! En-route from PAC learning to bipartite matching, I will overview a particular transductive model of learning, and associated one-inclusion graphs, which can be viewed as a generalization of some of the hat puzzles that are popular in recreational mathematics. Whereas this transductive model is far from new, it has recently seen a resurgence of interest as a tool for tackling deep questions in learning theory. A secondary purpose of this article could be as a (biased) tutorial on the connections between the PAC and transductive models of learning.

Most work in the area of learning theory has focused on designing effective Probably Approximately Correct (PAC) learners. Recently, other models of learning such as transductive error have seen more scrutiny. We move toward showing that these problems are equivalent by reducing agnostic learning with a PAC guarantee to agnostic learning with a transductive guarantee by adding a small number of samples to the dataset. We first rederive the result of Aden-Ali et al. arXiv:2304.09167 reducing PAC learning to transductive learning in the realizable setting using simpler techniques and at more generality as background for our main positive result. Our agnostic transductive to PAC conversion technique extends the aforementioned argument to the agnostic case, showing that an agnostic transductive learner can be efficiently converted to an agnostic PAC learner. Finally, we characterize the performance of the agnostic one inclusion graph algorithm of Asilis et al. arXiv:2309.13692 for binary classification, and show that plugging it into our reduction leads to an agnostic PAC learner that is essentially optimal. Our results imply that transductive and PAC learning are essentially equivalent for supervised learning with pseudometric losses in the realizable setting, and for binary classification in the agnostic setting. We conjecture this is true more generally for the agnostic setting.

We study the problem of adversarially robust learning in the transductive setting. For classes $\mathcal{H}$ of bounded VC dimension, we propose a simple transductive learner that when presented with a set of labeled training examples and a set of unlabeled test examples (both sets possibly adversarially perturbed), it correctly labels the test examples with a robust error rate that is linear in the VC dimension and is adaptive to the complexity of the perturbation set. This result provides an exponential improvement in dependence on VC dimension over the best known upper bound on the robust error in the inductive setting, at the expense of competing with a more restrictive notion of optimal robust error.