Exact Optimality of Communication-Privacy-Utility Tradeoffs in Distributed Mean Estimation

Berivan Isik, Wei-Ning Chen, Ayfer Ozgur, Tsachy Weissman, Albert No

Keywords: distributed mean estimation, privacy, compression, communication, federated analytics.

TL;DR: We study the exact optimality conditions, together with an exact optimal mechanism for a family of schemes, for distributed mean estimation problem under communication and privacy constraints.

Abstract: 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. In this work, we take a step towards characterizing the exact-optimal approach in the presence of shared randomness (a random variable shared between the server and the user) and identify several conditions for 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 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.

Submission Number: 1977