  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.
Not only so, we would like to protect each individual user's privacy. In this
paper, we consider the $k$-sparse version of the vector mean estimation
problem, that is, suppose that 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 are the first to show matching upper- and lower-bounds for
the $k$-sparse vector mean estimation problem under local differential privacy.
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 as
well as real-world datasets. Our experiments show that we can often achieve one
or two orders of magnitude reduction in error in comparison with prior works
under typical choices of parameters, while incurring insignificant
communication cost.
