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Shuchi Chawla, Cynthia Dwork, Frank McSherry, Adam Smith, Larry Stockmeyer, Hoeteck Wee From Idiosyncratic to Stereotypical: Toward Privacy in Public Databases
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Shuchi Chawla 2 Database Privacy Census data – a prototypical example Individuals provide information Census bureau publishes sanitized records Privacy is legally mandated; what utility can we achieve? Our Goal: What do we mean by preservation of privacy? Characterize the trade-off between privacy and utility – disguise individual identifying information – preserve macroscopic properties Develop a “good” sanitizing procedure with theoretical guarantees
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Shuchi Chawla 3 An outline of this talk A mathematical formalism What do we mean by privacy? Prior work An abstract model of datasets Isolation; Good sanitizations A candidate sanitization A brief overview of results General argument for privacy of n-point datasets Open issues and concluding remarks
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Shuchi Chawla 4 Privacy… a philosophical view-point [Ruth Gavison] … includes protection from being brought to the attention of others … Matches intuition; inherently desirable Attention invites further loss of privacy Privacy is assured to the extent that one blends in with the crowd Appealing definition; can be converted into a precise mathematical statement!
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Shuchi Chawla 5 Database Privacy Statistical approaches Alter the frequency ( PRAN/DS/PERT ) of particular features, while preserving means. Additionally, erase values that reveal too much Query-based approaches involve a permanent trusted third party Query monitoring: dissallow queries that breach privacy Perturbation: Add noise to the query output [Dinur Nissim’03, Dwork Nissim’04] Statistical perturbation + adversarial analysis [Evfimievsky et al ’03] combine statistical techniques with analysis similar to query-based approaches
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Shuchi Chawla 6 Everybody’s First Suggestion Learn the distribution, then output: A description of the distribution, or, Samples from the learned distribution Want to reflect facts on the ground Statistically insignificant facts can be important for allocating resources
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Shuchi Chawla 7 A geometric view Abstraction : Points in a high dimensional metric space – say R d ; drawn i.i.d. from some distribution Points are unlabeled; you are your collection of attributes Distance is everything Real Database (RDB) – private n unlabeled points in d-dimensional space. Sanitized Database (SDB) – public n’ new points possibly in a different space.
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Shuchi Chawla 8 The adversary or Isolator Using SDB and auxiliary information (AUX), outputs a point q q “isolates” a real point x, if it is much closer to x than to x’s neighbors, T-radius of x – distance to its T-nearest neighbor x is “safe” if x > (T-radius of x)/(c-1) B(q, c x ) contains x’s entire T-neighborhood (c-1) c – privacy parameter; eg. 4 q x cc large T and small c is good i.e., if B(q,c ) contains less than T RDB points
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Shuchi Chawla 9 A good sanitization Sanitizing algorithm compromises privacy if the adversary is able to considerably increase his probability of isolating a point by looking at its output A rigorous (and too ideal) definition D I I ’ w.o.p RDB 2 R D n aux z x 2 RDB : | Pr[ I (SDB,z) isolates x] – Pr[ I ’ (z) isolates x] | · /n Definition of can be forgiving, say, 2 - (d) or (1 in a 1000) Quantification over x : If aux reveals info about some x, the privacy of some other y should still be preserved Provides a framework for describing the power of a sanitization method, and hence for comparisons
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Shuchi Chawla 10 The Sanitizer The privacy of x is linked to its T-radius Randomly perturb it in proportion to its T-radius x’ = San(x) R S(x,T-rad(x)) Intuition: We are blending x in with its crowd If the number of dimensions (d) is large, there are “many” pre-images for x’. The adversary cannot conclusively pick any one. We are adding random noise with mean zero to x, so several macroscopic properties should be preserved.
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Shuchi Chawla 11 Results on privacy.. An overview DistributionNum. of points Revealed to adversaryAuxiliary information Uniform on surface of sphere 2Both sanitized pointsDistribution, 1-radius Uniform over a bounding box or surface of sphere nOne sanitized point, all other real points Distribution, all real points Gaussian2 o(d) n sanitized pointsDistribution Gaussian2 (d) Work under progress
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Shuchi Chawla 12 Results on utility… An overview Distributional/ Worst-case ObjectiveAssumptionsResult Worst-caseFind K clusters minimizing largest diameter - Optimal diameter as well as approximations increase by at most a factor of 3 DistributionalFind k maximum likelihood clusters Mixture of k Gaussians Correct clustering with high probability as long as means are pairwise sufficiently far
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Shuchi Chawla 13 A special case - one sanitized point RDB = {x 1,…,x n } The adversary is given n-1 real points x 2,…,x n and one sanitized point x’ 1 ; T = 1; c=4; “flat” prior Recall: x’ 1 2 R S(x 1,|x 1 -y|) where y is the nearest neighbor of x 1 Main idea: Consider the posterior distribution on x 1 Show that the adversary cannot isolate a large probability mass under this distribution
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Shuchi Chawla 14 Let Z = { p R d | p is a legal pre-image for x’ 1 } Q = { p | if x 1 =p then x 1 is isolated by q } We show that Pr[ Q ∩ Z | x’ 1 ] ≤ 2 - (d) Pr[ Z | x’ 1 ] Pr[x 1 in Q ∩ Z | x’ 1 ] = prob mass contribution from Q ∩ Z / contribution from Z = 2 1-d /(1/4) A special case - one sanitized point Q q x’ 1 x2x2 x3x3 x4x4 x5x5 Z Q∩ZQ∩Z x6x6 |p-q| · 1/3 |p-x’ 1 |
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Shuchi Chawla 15 Contribution from Z Pr[x 1 =p | x’ 1 ] Pr[x’ 1 | x 1 =p] 1/r d (r = |x’ 1 -p|) Increase in r x’ 1 gets randomized over a larger area – proportional to r d. Hence the inverse dependence. Pr[x’ 1 | x 1 2 S] s S 1/r d solid angle subtended at x’ 1 Z subtends a solid angle equal to at least half a sphere at x’ 1 x’ 1 x2x2 x3x3 x4x4 x5x5 Z x6x6 S r p
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Shuchi Chawla 16 Contribution from Q Å Z The ellipsoid is roughly as far from x’ 1 as its longest radius Contribution from ellipsoid is 2 -d x total solid angle Therefore, Pr[ x 1 2 Q Å Z ] / Pr[ x 1 2 Z ] 2 -d Q q x’ 1 x2x2 x3x3 x4x4 x5x5 Z Q∩ZQ∩Z x6x6 rr
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Shuchi Chawla 17 The general case… n sanitized points Initial intuition is wrong: Privacy of x 1 given x 1 ’ and all the other points in the clear does not imply privacy of x 1 given x 1 ’ and sanitizations of others! Sanitization is non-oblivious – Other sanitized points reveal information about x, if x is their nearest neighbor Where we are now Consider some example of safe sanitization (not necessarily using perturbations) Density regions? Histograms? Relate perturbations to the safe sanitization Uniform distribution; histogram over fixed-size cells exponentially low probability of isolation
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Shuchi Chawla 18 Future directions Extend the privacy argument to other “nice” distributions For what distributions is there no meaningful privacy— utility trade-off? Characterize acceptable auxiliary information Think of auxiliary information as an a priori distribution The low-dimensional case – Is it inherently impossible? Discrete-valued attributes Our proofs require a “spread” in all attributes Extend the utility argument to other interesting macroscopic properties – e.g. correlations
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Shuchi Chawla 19 Conclusions A first step towards understanding the privacy- utility trade-off A general and rigorous definition of privacy A work in progress!
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Shuchi Chawla 20 Questions?
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