Achieving the Exactly Optimal Privacy-Utility Trade-Off with Low Communication Cost via Shared Randomness

We consider a discrete distribution estimation problem under a local differential privacy (LDP) constraint in the presence of shared randomness. For this problem, we propose a new class of LDP schemes achieving the exactly optimal privacy-utility trade-off (PUT), with the communication cost less than or equal to the size of the input data. Moreover, it is shown as a simple corollary that one-bit communication is sufficient for achieving the exactly optimal PUT for a high privacy regime if the size of the input data is an even number. The main idea is to decompose a block design scheme proposed by Park et al. (2023), based on the combinatorial concept called resolution. We call the resultant decomposed LDP scheme with shared randomness as a resolution of the original block design scheme. A resolution of a block design scheme has a communication cost less than or equal to that of the original block design scheme. Also, the resolution of a block design scheme is exactly optimal whenever the original block design scheme is exactly optimal. Accordingly, we provide a resolution of the exactly optimal subset selection scheme proposed by Ye and Barg (2018), called the Baranyai's resolution. The Baranyai's resolution is not only exactly optimal, but also it achieves the minimum communication cost among all exactly optimal resolutions of block design schemes.