Optimal Noise Adding Mechanisms for Approximate Differential Privacy

We study the (nearly) optimal mechanisms in (epsilon, delta)-differential privacy for integer-valued query functions and vector-valued (histogram-like) query functions under a utility-maximization/cost-minimization framework. Within the classes of mechanisms oblivious of the database and the queries beyond the global sensitivity, we characterize the tradeoff between epsilon and delta in utility and privacy analysis for histogram-like query functions, and show that the (epsilon, delta)-differential privacy is a framework not much more general than the (epsilon, 0)-differential privacy and (0, delta)-differential privacy in the context of l(1) and l(2) cost functions, i.e., minimum expected noise magnitude and noise power. In the same context of l(1) and l(2) cost functions, we show the near-optimality of uniform noise mechanism and discrete Laplacian mechanism in the high privacy regime (as (epsilon, delta) -> (0, 0)). We conclude that in (epsilon, delta)-differential privacy, the optimal noise magnitude and the noise power are Theta(min((1/epsilon), (1/delta))) and Theta(min((1/epsilon(2)), (1/delta(2)))), respectively, in the high privacy regime.