We study the mean estimation problem under communication and local differential privacy constraints. While previous work has proposed \emph{order}-optimal algorithms for the same problem (i.e., asymptotically optimal as we spend more bits), \emph{exact} optimality (in the non-asymptotic setting) still has not been achieved. In this work, we take a step towards characterizing the \emph{exact}-optimal approach in the presence of shared randomness (a random variable shared between the server and the user) and identify several necessary conditions for \emph{exact} optimality. We prove that one of the necessary conditions is to utilize a rotationally symmetric shared random codebook. Based on this, we propose a randomization mechanism where the codebook is a randomly rotated simplex -- satisfying the necessary properties of the \emph{exact}-optimal codebook. The proposed mechanism is based on a $k$-closest encoding which we prove to be \emph{exact}-optimal for the randomly rotated simplex codebook.

翻译：我们研究在通信和本地差分隐私约束下的均值估计问题。虽然已有工作提出了该问题的阶最优算法（即随着比特数增加渐进最优），但精确最优性（在非渐进设定下）仍未实现。本文在存在共享随机性（服务器与用户共享的随机变量）的情况下，向刻画精确最优方法迈出了一步，并确定了精确最优性的若干必要条件。我们证明其中一个必要条件是使用旋转对称的共享随机码本。基于此，我们提出一种随机化机制，其码本为随机旋转的单纯形——满足精确最优码本的必要性质。所提机制基于k近邻编码，我们证明该编码对随机旋转单纯形码本具有精确最优性。