The Optimal Noise-Adding Mechanism in Differential Privacy

Differential privacy is a framework to quantify to what extent individual privacy in a statistical database is preserved while releasing useful aggregate information about the database. In this paper, within the classes of mechanisms oblivious of the database and the queries beyond the global sensitivity, we characterize the fundamental tradeoff between privacy and utility in differential privacy, and derive the optimal epsilon-differentially private mechanism for a single real-valued query function under a very general utility-maximization (or cost-minimization) framework. The class of noise probability distributions in the optimal mechanism has staircase-shaped probability density functions which are symmetric (around the origin), monotonically decreasing and geometrically decaying. The staircase mechanism can be viewed as a geometric mixture of uniform probability distributions, providing a simple algorithmic description for the mechanism. Furthermore, the staircase mechanism naturally generalizes to discrete query output settings as well as more abstract settings. We explicitly derive the parameter of the optimal staircase mechanism for l(1) and l(2) cost functions. Comparing the optimal performances with those of the usual Laplacian mechanism, we show that in the high privacy regime (epsilon is small), the Laplacian mechanism is asymptotically optimal as epsilon -> 0; in the low privacy regime (epsilon is large), the minimum magnitude and second moment of noise are Theta (Delta e((-epsilon/2))) and Theta(Delta(2)e((-2 epsilon/3))) as epsilon -> +infinity, respectively, while the corresponding figures when using the Laplacian mechanism are Delta/epsilon and 2 Delta(2)/epsilon(2), where Delta is the sensitivity of the query function. We conclude that the gains of the staircase mechanism are more pronounced in the moderate-low privacy regime.