Learning Distributions from Multiple Data Providers

arXiv:2607.24732v1 Announce Type: cross Abstract: 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(dot 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/epsilon^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 Theta(n/epsilon^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 Theta(n/epsilon^2), with pairwise query families as a canonical example. Finally, the full range of polynomial rates between linear and quadratic is attainable: for every alpha in (1,2), there exists a query family with optimal PAC rate widetilde Theta(n^alpha/epsilon^2).

Source: arXiv cs.LG | 2026-07-28