Adaptive Refinement Protocols for Distributed Distribution Estimation under $\ell^p$-Losses

Consider the communication-constrained estimation of discrete distributions under $\ell^p$ losses, where each distributed terminal holds multiple independent samples and uses limited number of bits to describe the samples. We obtain the minimax optimal rates of the problem in most parameter regimes. An elbow effect of the optimal rates at $p=2$ is clearly identified. To show the optimal rates, we first design estimation protocols to achieve them. The key ingredient of these protocols is to introduce adaptive refinement mechanisms, which first generate rough estimate by partial information and then establish refined estimate in subsequent steps guided by the rough estimate. The protocols leverage successive refinement, sample compression, thresholding and random hashing methods to achieve the optimal rates in different parameter regimes. The optimality of the protocols is shown by deriving compatible minimax lower bounds.