We consider the noise complexity of differentially private mechanisms in the setting where the user asks d linear queries f : R n → R non-adaptively. Here, the database is represented by a vector in R n and proximity between databases is measured in the 1-metric. We show that the noise complexity is determined by two geometric parameters associated with the set of queries. We use this connection to give tight upper and lower bounds on the noise complexity for any d ≤ n. We show that for d random linear queries of sensitivity 1, it is necessary and sufficient to add 2-error
Hardt and Talwar REF give tight upper and lower bounds on the amount of noise needed to ensure differential privacy for d non-adaptive linear queries, where the database is a vector in R n .
On the geometry of differential privacy
STOC '10
Computer Science Mathematics