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 mathscrS subseteq 2[n], and each query to S ∈ mathscrS returns an independent sample from the conditional distribution p(· mid S). Learnability is governed by the co-occurrence graph associated with mathscrS: 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(n2/ε2), and this bound is tight in the worst case. On the other hand, if [n] is queryable then ordinary sampling improves the bound to Theta(n/ε2), 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 Theta(n/ε2), with pairwise query families as a canonical example. Finally, the full range of polynomial rates between linear and quadratic is attainable: for every α ∈ (1,2), there exists a query family with optimal PAC rate widetilde Theta(nα/ε2).