Private frequency estimation via projective geometry

In this work, we propose a new algorithm ProjectiveGeometryResponse (PGR) for locally differentially private (LDP) frequency estimation. For universe size of k and with n users, our eps-LDP algorithm has communication cost ceil(log_2 k) and computation cost O(n + k\exp(eps) log k) for the server to approximately reconstruct the frequency histogram, while achieve optimal privacy-utility tradeoff. In many practical settings this is a significant improvement over the O~(n+k^2) computation cost that is achieved by the recent PI-RAPPOR algorithm (Feldman and Talwar; 2021). Our empirical evaluation shows a speedup of over 50x over PI-RAPPOR while using approximately 75x less memory. In addition, the running time of our algorithm is comparable to that of HadamardResponse (Acharya, Sun, and Zhang; 2019) and RecursiveHadamardResponse (Chen, Kairouz, and Ozgur; 2020) which have significantly worse reconstruction error. The error of our algorithm essentially matches that of the communication- and time-inefficient but utility-optimal SubsetSelection (SS) algorithm (Ye and Barg; 2017). Our new algorithm is based on using Projective Planes over a finite field to define a small collection of sets that are close to being pairwise independent and a dynamic programming algorithm for approximate histogram reconstruction for the server.