A Few Interactions Improve Distributed Nonparametric Estimation, Optimally

Consider the problem of nonparametric estimation of an unknown β -Hölder smooth density pXY at a given point, where X and Y are both d dimensional. An infinite sequence of i.i.d. samples (Xi,Yi) are generated according to this distribution, and two terminals observe (Xi) and (Yi) , 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 square risk is order (k k )-2β d+2β for one-way protocols and k-2β d+2β for interactive protocols. The logarithmic improvement is nonexistent in the parametric counterparts, and therefore can be regarded as a consequence of nonparametric nature of the problem. Moreover, a few rounds of interactions achieve the interactive minimax rate: the number of rounds can grow as slowly as the super-logarithm (i.e., inverse tetration) of k. The proof of the upper bound is based on a novel multi-round scheme for estimating the joint distribution of a pair of biased Bernoulli variables, and the lower bound is built on a sharp estimate of a symmetric strong data processing constant for biased Bernoulli variables.