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.

翻译：在Gold[1967]的经典极限识别模型中，学习器逐轮接收正例流，并最终必须恢复目标假设。近期，Kleinberg和Mullainathan[2024]引入极限生成范式，要求学习器最终输出目标假设支撑集的新元素。这两条研究路线均聚焦于纯正例或全标注数据。然而，许多自然监督信号本质上是关系性的而非单例性的——它们编码的是样例间的关系而非单个样例的标签。我们率先开展极限对比识别与生成的研究，在此框架下学习器观察对比式数据呈现：流式输入无序对$\{x,y\}$，满足未知目标二值假设$h$的$h(x)\ne h(y)$，但正例元素对学习器隐藏。首先，我们在无噪声环境下给出三个结果：对比可识别类的精确刻画（Angluin[1980]特征条件的一维几何精炼）、称为对比闭包维度的组合维度（Raman等[2025]闭包维度的对比类比，具有紧样本复杂度的均匀对比生成精确刻画），以及对比生成与文本识别互不可比的严格层次关系。随后我们证明有限对抗污染下的尖锐反转：存在某些类，可通过单一与预算无关的算法在任何有限污染预算下从对比对中识别，但在仅受到一个污染观测时无法从正例中识别。统一的技术对象是公共交叉图，它通过单一覆盖与入射语言编码了成对歧义性、族级生成障碍与污染缺陷。