Seminar in Foundations of Privacy

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Seminar in Foundations of Privacy

• Adding Consistency to Differential Privacy
• Attacks on Anonymized Social Networks
• Inbal Talgam
• March 2008

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1. Adding Consistency to Differential Privacy
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Differential Privacy

• 1977 Dalenius - The risk to ones 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
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).
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Main Contribution

• Solve following consistency problem
• At low accuracy cost

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Outline

• Discussion of
• Privacy
• 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

A randomized function K gives e-differential
privacy if for all DB1, DB2 differing on at most
1 element
DB1
DB2
Differing on 1 element
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Privacy - Mechanism
K(DB) f(DB)
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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 ?f.
• Definition
• All useful functions should be insensitive
• (e.g. marginals)

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The Calibrated Noise Mechanism How Much Noise

• Main result To ensure e-differential privacy for
  a query of sensitivity ?f, add Laplace noise with
  s ?f/e.
• Why does it work? Remember
• Laplace Definition
• PrK(DB)a
• exp (f(DB) - a1 / s)

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Accuracy Consistency
So smoking is one of the leading causes of
statistics?

• Compromise consistency
• May lead to technical problems and confusion

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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.
Accuracy Few Fourier coefficients are needed for
low-order marginals, so low sensitivity and small
error.
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-d,
• Remark Advantage of working in the interactive
  model.

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Outline

• Discussion of
• Privacy
• Accuracy Consistency
• Key method - Fourier basis
• The algorithm
• Part I
• Part II

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Notation Preliminaries
x00
x01

• x1 ?
• We say a ß if ß has all as attributes (and
  more)
• e.g. 0110 0111 but not 0110 0101
• Introduce the linear marginal operator Cß
• ß determines attributes
• Remember xa, a ß, Cß(x), Cß(x)?

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The Fourier Basis

• Orthonormal basis for space of contingency tables
  x (R2k).
• Motivation Any marginal Cß(x) can be written as
  a combination of few fas.
• How few? Depends on order of marginal.
• fa

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Writing marginals in Fourier Basis

• Theorem

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Outline

• Discussion of
• Privacy
• 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 Cß
• fa Fourier vectors needed to write marginals
• Releasing marginals Cß(x) releasing coeffs
  ltfa,xgt
• OUTPUT Noisy coeffs Fa
• METHOD Add calibrated noise
• Sensitivity depends on a on order of
  Cßs

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Part II Non-negativity by LP

• INPUT Noisy coeffs Fa
• OUTPUT Non-negative contingency table x'
• METHOD Minimize difference between Fourier
  coefficients
• Most entries x'? in a vertex solution are 0
• Rounding adds small error

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Algorithm Summary

• Input Contingency table x, required marginals
  Cß
• Output Marginals Cß of new contingency table
  x''
• fa Fourier vectors needed to write marginals
• Compute noisy Fourier coefficients Fa
• Find non-negative x' with nearly the correct
  Fourier coefficients
• Round to x''

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Accuracy Guarantee - Revisited

• With probability 1-d,

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?