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.

翻译：考虑在$\ell^p$损失下对离散分布进行通信受限的估计问题，其中每个分布式终端持有多个独立样本，并使用有限比特数来描述这些样本。我们在大多数参数范围内获得了该问题的极小极大最优率。最优率在$p=2$处表现出明显的“拐点效应”。为证明这些最优率，我们首先设计了能够达到这些速率的估计协议。这些协议的关键在于引入自适应优化机制：该机制首先利用部分信息生成粗略估计，随后在粗略估计的引导下，通过后续步骤建立精细估计。协议结合了逐次优化、样本压缩、阈值处理以及随机哈希方法，以在不同参数范围内实现最优率。通过推导相容的极小极大下界，证明了这些协议的最优性。