Contrastive Identification and Generation in the Limit
In the classical identification in the limit model of Gold [1967], a stream of positive examples is presented round by round, and the learner must eventually recover the target hypothesis. Recently, Kleinberg and Mullainathan [2024] introduced generation in the limit, where the learner instead must eventually output novel elements of the target's support. Both lines of work focus on positive-only or fully labeled data. Yet many natural supervision signals are inherently relational rather than singleton, which encode relationships between examples rather than labels of individual ones. We initiate the study of contrastive identification and generation in the limit, where the learner observes a contrastive presentation of data: a stream of unordered pairs $\{x,y\}$ satisfying $h(x)\ne h(y)$ for an unknown target binary hypothesis $h$, but which element is positive is hidden from the learner. We first present three results in the noiseless setting: an exact characterization of contrastive identifiable classes (a one-line geometric refinement of Angluin [1980]'s tell-tale condition), a combinatorial dimension called contrastive closure dimension (a contrasitive analogue of the closure dimension in Raman et al. [2025]) and exactly characterizing uniform contrastive generation with tight sample complexity, and a strict hierarchy in which contrastive generation and text identification are mutually incomparable. We then prove a sharp reversal under finite adversarial corruption: there exist classes identifiable from contrastive pairs under any finite corruption budget by a single budget-independent algorithm, yet not identifiable from positive examples under even one corrupted observation. The unifying technical object is the common crossing graph, which encodes pairwise ambiguity, family-level generation obstructions, and corruption defects in a single coverage-and-incidence language.