Distributed Statistical Estimation of High-Dimensional and Nonparametric Distributions

We consider the problem of estimating high-dimensional and nonparametric distributions in distributed networks, where each sensor in the network observes an independent sample from the underlying distribution and can communicate it to a central processor by writing at most k bits on a public blackboard. We obtain matching upper and lower bounds for the minimax risk of estimating the underlying distribution under L
₁
loss. Our results reveal that the minimax risk reduces exponentially in k. Instead of relying on strong data processing inequalities for the converse as commonly done in the literature, we build on a new representation of the communication constraint, which leads to a tight characterization of the problem.