1. Seminar in Foundations of Privacy 1.Adding Consistency to Differential Privacy 2.Attacks on Anonymized Social Networks Inbal Talgam March 2008

2. 1. Adding Consistency to Differential Privacy

3. 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.

4. 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). ##… 2 k attribute settings 0…00…1… Contingency Table 83… 2 j attribute settings 09… 2 i attribute settings Marginals j << k

5. Main Contribution Solve following consistency problem: At low accuracy cost 20… Marginals Noise NaN-0.5… Contingency Table +

6. Outline Discussion of: 1.Privacy 2.Accuracy & Consistency Key method - Fourier basis The algorithm –Part I –Part II

7. Privacy – Definition Intuition: The risk is the same with or without participating in the DB Definition: DB 1 DB 2 Differing on 1 element A randomized function K gives ε -differential privacy if for all DB 1, DB 2 differing on at most 1 element

8. Privacy - Mechanism Noise Pls let me know f ( DB ) DB Goal: Noise K ( DB ) = f ( DB )+ Noise Laplace noise: Pr[ K ( DB )= a ] exp (|| f ( DB ) - a|| 1 / σ)

9. The Calibrated Noise Mechanism for 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 → R d, the L 1-sensitivity of f is for all DB 1, DB 2 differing on at most 1 element

10. 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)

11. Accuracy & Consistency 83… Contingency Table 20… 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? 83… Contingency Table + Noise 32… Marginals Compromise accuracy Non-calibrated, binomial noise Var=Θ(2 k )

12. Key Approach Non-redundant representation Specific for required marginals 83… Contingency Table 20… 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

13. 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. DB

14. Outline Discussion of: 1.Privacy 2.Accuracy & Consistency Key method - Fourier basis The algorithm –Part I –Part II

15. 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 C β β determines attributes Remember: x α, α ≤ β, C β (x), C β (x) γ ##… Contingency Table x 0…0 x 0…1 x α where 20… Marginal C β (x) :

16. The Fourier Basis –Orthonormal basis for space of contingency tables x (R 2 k ). Motivation: Any marginal C β (x) can be written as a combination of few f α ’s. –How few? Depends on order of marginal. f α : …

17. Writing marginals in Fourier Basis Theorem: Marginal of x with attributes β Write x in Fourier basis Linearity Proof. For any coordinate By definition of marginal operator and Fourier vector

18. 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

19. Algorithm – Part I INPUT: Required marginals {C β } {f α } = Fourier vectors needed to write marginals Releasing marginals {C β (x)} = releasing coeffs OUTPUT: Noisy coeffs {Φ α } METHOD: Add calibrated noise Sensitivity depends on |{α}| on order of C β ’s

20. Part II – Non-negativity by LP INPUT: Noisy coeffs {Φ α } 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 | Φ α - | ≤ b

21. Algorithm Summary Input: Contingency table x, required marginals {C β } Output: Marginals {C β } of new contingency table x'' {f α } = Fourier vectors needed to write marginals Compute noisy Fourier coefficients {Φ α } Find non-negative x' with nearly the correct Fourier coefficients Round to x'' Part I Part II

22. Accuracy Guarantee - Revisited With probability 1-δ, #Coefficients

23. 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

24. Any Questions?