We introduce the problem of learning conditional averages in the PAC framework. The learner receives a sample labeled by an unknown target concept from a known concept class, as in standard PAC learning. However, instead of learning the target concept itself, the goal is to predict, for each instance, the average label over its neighborhood -- an arbitrary subset of points that contains the instance. In the degenerate case where all neighborhoods are singletons, the problem reduces exactly to classic PAC learning. More generally, it extends PAC learning to a setting that captures learning tasks arising in several domains, including explainability, fairness, and recommendation systems. Our main contribution is a complete characterization of when conditional averages are learnable, together with sample complexity bounds that are tight up to logarithmic factors. The characterization hinges on the joint finiteness of two novel combinatorial parameters, which depend on both the concept class and the neighborhood system, and are closely related to the independence number of the associated neighborhood graph.

翻译：我们在PAC框架中引入了学习条件期望的问题。学习者接收由已知概念类中未知目标概念标记的样本，这与标准PAC学习相同。然而，目标并非学习目标概念本身，而是对每个实例预测其邻域（包含该实例的任意点集）上的平均标记。当所有邻域均为单点集时，该问题完全退化为经典PAC学习。更一般地，它将PAC学习扩展至一个能够涵盖多个领域（包括可解释性、公平性和推荐系统）中学习任务的设定。我们的主要贡献在于完整刻画了条件期望何时可学习，并给出了在至多对数因子范围内紧致的样本复杂度界。该刻画依赖于两个新颖组合参数的联合有限性，这些参数同时取决于概念类和邻域系统，并与相关邻域图的独立数密切相关。