We study two basic statistical tasks in non-interactive local differential privacy (LDP): learning and refutation.
In comparison, with only private samples, this problem cannot be solved even for simple query classes with VC-dimension one, and without any private samples, a larger public sample of size $d/\alpha^2$ is needed.
Our main result is a new approximation algorithm with an approximation guarantee that depends only on the dimension $d$ of the vectors and not on the size $k$ of the output set.
Absent privacy constraints, this problem requires $O(\log k)$ samples from $p$, and it was recently shown that the same complexity is achievable under (central) differential privacy.
We give new characterizations of the sample complexity of answering linear queries (statistical queries) in the local and central models of differential privacy: *In the non-interactive local model, we give the first approximate characterization of the sample complexity.
Our main result is to obtain improved approximation algorithms for the $A$-optimal design problem by introducing the proportional volume sampling algorithm.
We show that every symmetric normed space admits an efficient nearest neighbor search data structure with doubly-logarithmic approximation.