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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| Published in: | IEEE transactions on information theory Vol. 62; no. 2; pp. 925 - 951 |
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| Main Authors: | , |
| Format: | Journal Article |
| Language: | English |
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New York
IEEE
01.02.2016
The Institute of Electrical and Electronics Engineers, Inc. (IEEE) |
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| ISSN: | 0018-9448, 1557-9654 |
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| Abstract | 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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| AbstractList | 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. 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 $\ell _{1}$ and $\ell _{2}$ 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 $\epsilon \to 0$ ; 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 \to +\infty $ , 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. 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 $ell _...1...$ and $ell _...2...$ 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 $epsilon to 0$ ; in the low privacy regime ( $epsilon $ is large), the minimum magnitude and second moment of noise are $Theta (Delta e^...(-...epsilon .../...2...)...)$ and $Theta (Delta ^...2... e^...(-...2epsilon .../...3...)...)$ as $epsilon to +infty $ , respectively, while the corresponding figures when using the Laplacian mechanism are $...Delta .../...epsilon ...$ and $...2Delta ^...2.../...epsilon ^...2...$ , 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. (ProQuest: ... denotes formulae/symbols omitted.) |
| Author | Quan Geng Viswanath, Pramod |
| Author_xml | – sequence: 1 surname: Quan Geng fullname: Quan Geng email: gengquanshine@gmail.com organization: Coordinated Sci. Lab., Univ. of Illinois at Urbana-Champaign, Urbana, IL, USA – sequence: 2 givenname: Pramod surname: Viswanath fullname: Viswanath, Pramod email: pramodv@illinois.edu organization: Coordinated Sci. Lab., Univ. of Illinois at Urbana-Champaign, Urbana, IL, USA |
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| SubjectTerms | Asymptotic properties Context Data privacy Density Information theory Laplace equations Noise Optimization Privacy Probability Probability distribution Queries randomized algorithm Sensitivity Staircases |
| Title | The Optimal Noise-Adding Mechanism in Differential Privacy |
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