Distributed Nonparametric Estimation: from Sparse to Dense Samples per Terminal

Consider the communication-constrained problem of nonpar ametric function estimation, in which each distributed terminal holds multiple i.i.d. Under certain regu larity assumptions, we characterize the minimax optimal rates for all regimes, and identify phase transitions of the optimal rates as the samples per terminal vary from sparse to dense. This fully solves the problem left open by previous works, whose scopes are limited to regimes with either dense samples or a single sample per terminal. To achieve the optimal rates, we design a layered estimation protocol by exploiting protocols for the parametric density estimat ion problem. We show the optimality of the protocol using information-theoretic methods and strong data processing inequalities, and incorporating the classic balls and bins model. The optimal rates are immediate for various special c ases such as density estimation, Gaussian, binary, Poisson and heteroskedastic regression models. Distributed nonparametric estimation problems have attra cted wide attention, and related theoretical studies can shed light on the understanding of modern applications such as federated learning [1], [2], [3].