Exact Optimality in Communication-Privacy-Utility Tradeoffs

We study the mean estimation problem under communication and local differential privacy constraints. While previous work has proposed order-optimal algorithms for the same problem (i.e., asymptotically optimal as we spend more bits), exact optimality (in the non-asymptotic setting) still has not been achieved. We take a step towards characterizing the exact-optimal approach in the presence of shared randomness and identify several necessary conditions for 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 exact-optimal codebook. The proposed mechanism is based on a $k$-closest encoding which we prove to be exact-optimal for the randomly rotated simplex codebook.