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 conditions for \emph{exact} optimality. We prove that one of the 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 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：我们研究了在通信和局部差分隐私约束下的均值估计问题。尽管已有工作针对该问题提出了\emph{阶}-最优算法（即随着比特数增加而渐近最优的算法），但在非渐近设定下实现\emph{精确}最优性仍未达成。本文在共享随机性（服务器与用户之间共享的随机变量）存在的情况下，向刻画\emph{精确}最优方法迈进一步，并确定了实现\emph{精确}最优的若干条件。我们证明其中一个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，其中码本为随机旋转单纯形——满足\emph{精确}最优码本的性质。该机制基于$k$-最近邻编码，我们证明该编码对于随机旋转单纯形码本而言是\emph{精确}最优的。