Learning Distributions from Multiple Data Providers

Motivated by learning from heterogeneous and overlapping data providers, we study a stylized model of distribution learning from restricted conditional samples. The goal is to learn an unknown distribution p on a finite domain [n]. The learner is given a fixed family of queryable sets \mathscr{S} \subseteq 2^{[n]}, and each query to S \in \mathscr{S} returns an independent sample from the conditional distribution p(\cdot \mid S). Learnability is governed by the co-occurrence graph associated with \mathscr{S}: two domain elements are adjacent if they appear together in some queryable set. Pointwise consistency is achievable when this graph is connected on the target support. PAC learning requires more: it is possible when the co-occurrence graph is complete. The optimal sample complexity of PAC learning ranges from nearly linear to quadratic. Every query family with complete co-occurrence graph admits sample complexity \widetilde O(n^2/ε^2), and this bound is tight in the worst case. On the other hand, if [n] is queryable then ordinary sampling improves the bound to Θ(n/ε^2), and this cannot be improved further even if every set is queryable. More generally, we identify hierarchical comparabilityas a sufficient structural condition on \mathscr S under which the optimal complexity is nearly linear, \widetilde Θ(n/ε^2), with pairwise query families as a canonical example. Finally, the full range of polynomial rates between linear and quadratic is attainable: for every α\in (1,2), there exists a query family with optimal PAC rate \widetilde Θ(n^α/ε^2).