We introduce a Probably Approximately Correct (PAC) learning framework where each hypothesis is represented by a graph, with edges indicating positive interactions, such as between users and items. This framework subsumes the classical binary and multi-class PAC learning models as well as multi-label learning with partial feedback, where only a single random correct label per example is observed, rather than all correct labels.
Our work uncovers a rich statistical and algorithmic landscape, with nuanced boundaries on what can and cannot be learned. Notably, classical methods like Empirical Risk Minimization fail in this setting, even for simple hypothesis classes with only two hypotheses. To address these challenges, we develop novel algorithms that learn exclusively from positive data, effectively minimizing both precision and recall losses. Specifically, in the realizable setting, we design algorithms that achieve optimal sample complexity guarantees. In the agnostic case, we show that it is impossible to achieve additive error guarantees (i.e., additive regret)—as is standard in PAC learning—and instead obtain meaningful multiplicative approximations.