Locally Differentially Private Sparse Vector Aggregation

Vector mean estimation is a central primitive in federated analytics. In vector mean estimation, each user $i \in[n]$ holds a real-valued vector $v_{i} \in[-1,1]^{d}$, and a server wants to estimate the mean of all n vectors; we would additionally like to protect each user’s privacy. In this paper, we consider the k-sparse version of the vector mean estimation problem. That is, suppose each user’s vector has at most k non-zero coordinates in its d-dimensional vector, and moreover, $k \ll d$. In practice, since the universe size d can be very large (e.g., the space of all possible URLs), we would like the per-user communication to be succinct, i.e., independent of or (poly-)logarithmic in the universe size.In this paper, we show matching upper- and lower-bounds for the k-sparse vector mean estimation problem under local differential privacy (LDP). Specifically, we construct new mechanisms that achieve asymptotically optimal error as well as succinct communication, either under user-level-LDP or event-level-LDP. We implement our algorithms and evaluate them on synthetic and real-world datasets. Our experiments show that we can often achieve one or two orders of magnitude reduction in error compared with prior work under typical choices of parameters, while incurring insignificant communication cost.