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 queriesqueries beyond the global...

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Veröffentlicht in:IEEE transactions on information theory Jg. 62; H. 2; S. 925 - 951
Hauptverfasser: Quan Geng, Viswanath, Pramod
Format: Journal Article
Sprache:Englisch
Veröffentlicht: New York IEEE 01.02.2016
The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Schlagworte:
Asymptotic properties
Context
Data privacy
Density
Information theory
Laplace equations
Noise
Optimization
Privacy
Probability
Probability distribution
Queries
randomized algorithm
Sensitivity
Staircases
Data privacy
randomized algorithm
ISSN:0018-9448, 1557-9654
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Zusammenfassung: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 queriesqueries beyond the global sensitivity, we characterize the fundamental tradeoff between privacy and utility in differential privacy, and derive the optimal ϵ-differentially private mechanism for a single realvalued 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 (ϵ is small), the Laplacian mechanism is asymptotically optimal as ϵ → 0; in the low privacy regime (ϵ is large), the minimum magnitude and second moment of noise are Θ(Δe (-ϵ/2) ) and Θ(Δ 2 e (-2ϵ/3) ) as ϵ → +∞, respectively, while the corresponding figures when using the Laplacian mechanism are Δ/ϵ and 2Δ 2 /ϵ 2 , where Δ 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.
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ISSN:0018-9448
1557-9654
DOI:10.1109/TIT.2015.2504967