seminar in foundations of privacy 1.adding consistency to differential privacy 2.attacks on...
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![Page 1: Seminar in Foundations of Privacy 1.Adding Consistency to Differential Privacy 2.Attacks on Anonymized Social Networks Inbal Talgam March 2008](https://reader030.vdocuments.us/reader030/viewer/2022033107/56649d2b5503460f94a00119/html5/thumbnails/1.jpg)
Seminar in Foundations of Privacy
1. Adding Consistency to Differential Privacy2. Attacks on Anonymized Social Networks
Inbal TalgamMarch 2008
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1. Adding Consistency to Differential Privacy
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Differential Privacy
• 1977 Dalenius - The risk to one’s privacy is the same with or without access to the DB.
• 2006 Dwork & Naor – Impossibe (auxiliary info).• 2006 Dwork et al – The risk is the same with or
without participating in the DB.
Plus: Strong mechanism of Calibrated Noise to achieve DP while maintaining accuracy.
• 2007 Barak et al - Adding consistency.
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Setting – Contingency Table and Marginals
k binary attributes
n participants DB
0 1 0 0 1 1 1 0
0 0 1 0 1 0 …
Terminology: Contingency table (private), marginals (public).
# # …
2k attribute settings
0…0 0…1 …
Contingency Table
8 3 …
2j attribute settings
0 9 …
2i attribute settings
Marginals
j << k
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Main Contribution
• Solve following consistency problem:
• At low accuracy cost
2 0 …
Marginals
Noise NaN -0.5 …
Contingency Table
+
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Outline
• Discussion of:1. Privacy
2. Accuracy & Consistency
• Key method - Fourier basis
• The algorithm– Part I– Part II
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Privacy – Definition• Intuition: The risk is the same with or without
participating in the DB• Definition:
DB1 DB2Differing on 1 element
A randomized function K gives ε-differential privacy if
for all DB1, DB2 differing on at most 1 element
)exp(
)(
)(
2
1 SDBKPR
SDBKPR
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Privacy - Mechanism
Noise
Pls let me know f(DB)
DB
Goal: Noise
K(DB) = f(DB)+
NoiseLaplace noise:
Pr[K(DB)=a]
exp (||f(DB) - a||1 / σ)
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The Calibrated Noise Mechanismfor DP
• Main idea: Amount of noise to add to f(DB) is calibrated according to the sensitivity of f, denoted Δf.
• Definition:
• All useful functions should be insensitive…
(e.g. marginals)
For f : D → Rd, the L1-sensitivity of f is
for all DB1, DB2 differing on at most 1 element121
,)()(max
21
DBfDBffDBDB
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The Calibrated Noise Mechanism – How Much Noise
• Main result: To ensure ε-differential privacy for a query of sensitivity Δf, add Laplace noise with σ = Δf/ε.
• Why does it work? Remember: Laplace: Definition:
Pr[K(DB)=a]exp (||f(DB) - a||1 / σ)
)exp(
)(
)(
2
1 SDBKPR
SDBKPR
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Accuracy & Consistency
8 3 …
Contingency Table
2 0 …
Marginals
Noise+
NaN -0.5 …
New Table
• Compromise consistency
• May lead to technical problems and confusion
So smoking is one of the
leading causes of statistics?
8 3 …
Contingency Table
+
Noise
3 2 …
Marginals
• Compromise accuracy
• Non-calibrated, binomial noise Var=Θ(2k)
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Key Approach
• Non-redundant representation
• Specific for required marginals
8 3 …
Contingency Table
2 0 …
Marginals
+
Small number of coefficients of the Fourier
basis
Consistency:
Any set of Fourier coefficients correspond
to a (fractional and possibly negative) contingency table.
Accuracy:
Few Fourier coefficients are needed for low-
order marginals, so low sensitivity and small
error.
Noise
+
Linear Programming +
Rounding
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Accuracy – What is Guaranteed
• Let C be a set of original marginals, each on ≤ j attributes.
• Let C’ be the result marginals.
• With probability 1-δ, :
• Remark: Advantage of working in the interactive model.
Cc
DB
CCCcc j /)/log(2' 3
1
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Outline
• Discussion of:1. Privacy
2. Accuracy & Consistency
• Key method - Fourier basis
• The algorithm– Part I– Part II
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Notation & Preliminaries
• ||x||1 = ?
• We say α ≤ β if β has all α’s attributes (and more) e.g. 0110 ≤ 0111 but not 0110 ≤ 0101
• Introduce the linear marginal operator Cβ
β determines attributes
• Remember: xα, α ≤ β, Cβ(x), Cβ(x)γ
# # …
Contingency Table
x0…0 x0…1 xα where k}1,0{
:2kRx
:
))(( xxC2 0 …
Marginal
Cβ(x) :
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The Fourier Basis
• – Orthonormal basis for space of contingency
tables x (R2k).
• Motivation: Any marginal Cβ(x) can be written as a combination of few fα’s.– How few? Depends on order of marginal.
• fα:
}}1,0{|{ kf
2/, 2/)1( kf …2/2/1 k
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Writing marginals in Fourier Basis
• Theorem: 0fC
fCxffxfCxC ,,Marginal of x with
attributes β
Write x in Fourier basis
Linearity
fCxf ,
Proof. For any coordinate
:
2/, 2/)1( kfC
By definition of marginal operator and Fourier vector
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Outline
• Discussion of:1. Privacy
2. Accuracy & Consistency
• Key method - Fourier basis
• The algorithm– Part I – adding calibrated noise– Part II – non-negativity by linear
programming
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Algorithm – Part I
INPUT: Required marginals {Cβ}• {fα} = Fourier vectors needed to write marginals• Releasing marginals {Cβ(x)} = releasing coeffs <fα,x>
OUTPUT: Noisy coeffs {Φα}
METHOD: Add calibrated noise• Sensitivity depends on |{α}| on order of Cβ’s
8 3 …
Contingency Table
2 0 …
Marginals
+
Small number of coefficients of the Fourier
basis
Noise
+
8 3 …
Contingency Table
8 3 …
Contingency Table
2 0 …
Marginals
2 0 …
Marginals
+
Small number of coefficients of the Fourier
basis
NoiseNoise
+
fCxf ,
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Part II – Non-negativity by LPINPUT: Noisy coeffs {Φα} OUTPUT: Non-negative contingency table x'METHOD: Minimize difference between Fourier coefficients
• Most entries x'γ in a vertex solution are 0 Rounding adds small error
minimize b
subject to:
x'γ ≥ 0
|Φα - <fα,x'>| ≤ b
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Algorithm Summary
Input: Contingency table x, required marginals {Cβ} Output: Marginals {Cβ} of new contingency table x''
• {fα} = Fourier vectors needed to write marginals• Compute noisy Fourier coefficients {Φα}
• Find non-negative x' with nearly the correct Fourier coefficients
• Round to x''
)/(, Lapxf
',min xf
Part I
Part II
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}{/)/}{log(}{22 jRounding
LP
Bound on Laplace noise per coefficient
Accuracy Guarantee - Revisited
• With probability 1-δ, 1'cc
#Coefficients
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Summary & Open Questions
• Algorithm for marginals release• Guarantees privacy, accuracy & consistency
– Consistency: can reconstruct a synthetic, consistent table
– Accuracy: error increases smoothly with order of marginals
• Open questions: – Improving efficiency – Effect of noise on marginals’ statistical properties
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Any Questions?