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Seminar in Foundations of Privacy. Adding Consistency to Differential Privacy Attacks on Anonymized Social Networks Inbal Talgam March 2008. 1. Adding Consistency to Differential Privacy. Differential Privacy.

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seminar in foundations of privacy

Seminar in Foundations of Privacy

Adding Consistency to Differential Privacy

Attacks on Anonymized Social Networks

Inbal Talgam

March 2008

differential privacy
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 Noiseto achieve DP while maintaining accuracy.

  • 2007 Barak et al - Adding consistency.
setting contingency table and marginals

Contingency Table

#

#

0…0

0…1

2k attribute settings

Marginals

8

3

0

9

j << k

2j attribute settings

2i attribute settings

Setting – Contingency Table and Marginals

0 1 0 0 1 1 1 0

0 0 1 0 1 0 …

n participants

DB

k binary attributes

Terminology: Contingency table (private), marginals (public).

main contribution

Contingency Table

NaN

-0.5

Marginals

2

0

Noise

Main Contribution
  • Solve following consistency problem:
  • At low accuracy cost

+

outline
Outline
  • Discussion of:
    • Privacy
    • Accuracy & Consistency
  • Key method - Fourier basis
  • The algorithm
    • Part I
    • Part II
privacy definition
Privacy – Definition
  • Intuition: The risk is the same with or without participating in the DB
  • Definition:

A randomized function K gives ε-differential privacy if

for all DB1, DB2 differing on at most 1 element

DB1

DB2

Differing on 1 element

privacy mechanism

Pls let me know f(DB)

DB

Laplace noise:

Pr[K(DB)=a]

exp (||f(DB) - a||1 / σ)

Noise

Noise

Noise

Goal:

Privacy - Mechanism

K(DB) = f(DB)+

the calibrated noise mechanism for dp

For f : D→ Rd, the L1-sensitivity of f is

for all DB1, DB2 differing on at most 1 element

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)

the calibrated noise mechanism how much noise
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)

accuracy consistency

Contingency Table

Contingency Table

New Table

+

NaN

8

8

3

3

-0.5

Marginals

Marginals

3

2

2

0

Noise

Noise

  • Compromise accuracy
  • Non-calibrated, binomial noise Var=Θ(2k)
Accuracy & Consistency

So smoking is one of the leading causes of statistics?

+

  • Compromise consistency
  • May lead to technical problems and confusion
key approach

Contingency Table

8

3

Marginals

2

0

Noise

Key Approach

Small number of coefficients of the Fourier basis

  • Non-redundant representation
  • Specific for required marginals

+

+

Consistency:

Any set of Fourier coefficients correspond to a (fractional and possibly negative) contingency table.

Linear Programming + Rounding

Accuracy:

Few Fourier coefficients are needed for low-order marginals, so low sensitivity and small error.

accuracy what is guaranteed

DB

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.
outline14
Outline
  • Discussion of:
    • Privacy
    • Accuracy & Consistency
  • Key method - Fourier basis
  • The algorithm
    • Part I
    • Part II
notation preliminaries

Contingency Table

xα where

#

#

Marginal

2

0

Cβ(x) :

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 linearmarginal operatorCβ

β determines attributes

  • Remember: xα, α ≤ β, Cβ(x), Cβ(x)γ

x0…0

x0…1

the fourier basis

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:
writing marginals in fourier basis

Write x in

Fourier basis

Marginal of x with

attributes β

Linearity

Proof. For any coordinate

By definition of marginal operator and Fourier vector

Writing marginals in Fourier Basis
  • Theorem:
outline18
Outline
  • Discussion of:
    • Privacy
    • Accuracy & Consistency
  • Key method - Fourier basis
  • The algorithm
    • Part I – adding calibrated noise
    • Part II – non-negativity by linear programming
algorithm part i
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
part ii non negativity by lp

minimize b

subject to:

x'γ ≥ 0

|Φα - <fα,x'>| ≤ b

Part II – Non-negativity by LP

INPUT: 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
algorithm summary

Part I

Part II

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''
accuracy guarantee revisited
Accuracy Guarantee - Revisited
  • With probability 1-δ,

#Coefficients

summary open questions
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