Communication Complexity of Two-party Nonparametric Global Density Estimation

Consider the problem of nonparametric estimation of an unknown $\beta$ -Hölder smooth density function $p_{XY}$ with compact support, where $X$ and $Y$ are both $d$ dimensional. An infinite sequence of i.i.d. samples $(X_{i}, \ Y_{i})$ are generated according to this distribution, and two terminals observe $(X_{i})$ and $(Y_{i})$, respectively. They are allowed to exchange $k$ bits either in oneway or interactively in order for Bob to estimate the unknown density. We show that the minimax mean integrated square risk is order $(\frac{k}{\log k})^{-\frac{\beta}{d+\beta}}$ for one-way protocols, and between $(\frac{k}{\log k})^{-\frac{\beta}{d+\beta}}$ and $k^{-\frac{\beta}{d+\beta}}$ for interactive protocols. These rates are different from the case of pointwise density estimation which we recently determined in another work. The interactive lower bound in this work used, among other things, a recent result of Ordentlich and Polyanskiy regarding the optimality of binary inputs in certain optimizations related to the strong data processing constant.