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
The lower bound given in REF depends on the volume of a convex body associated with the linear query functions, and the lower bound works for arbitrary linear query functions.
On the geometry of differential privacy
STOC '10
Computer Science Mathematics