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Random Data Perturbation Techniques and Privacy Preserving Data Mining

Gunjan Gupta

(Authors: H. Kargupta, S. Datta, Q. Wang & K. Sivakumar)

April 26, 2005

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Privacy & Good Service: Often Conflicting Goals

• Privacy– Customer: I don’t want you to share my personal information with anyone.– Business: I don’t want to share my data with a competitor.

• Quantity, Cost & Quality of Service– Customer: I want you to provide me lower cost of service– and good quality.– and at lower cost.

• Paradox: lower cost often comes from being able to use/share sensitive data that can be used or misused:– Provide better service by predicting consumer needs better, or sell information

to marketers.– Optimize load sharing between competing utilities or preempting competition.– Doctor saving patient by knowing patient history or insurance companies

declining coverage to individuals with preexisting conditions.

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Can we use privacy sensitive data to optimize cost and quality of a service without compromising any privacy?

Central Question:

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Short Answer:

No!

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Long Answer:

Maybe compromise a small amount of privacy (low cost increase) to improve quality and cost of service (high cost

savings) substantially.

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Why anonymous exact records not so secure?

• Example : medical insurance premium estimation based on patient history– Predictive fields often generic: age, sex, disease history, first two digits of zip

code (not allowed in Germany). no. of kids etc.– Specifics such as record id (key), name, address omitted.

• This could be easily broken by matching non-secure records with secure anonymous records:

Susan Calvin, 121 Norwood Cr. Austin, TX-78753

Hi, I am Susan, and here are pictures of me, my husband, and my 3 wonderful kids from my 43rd birthday party!

Female, 43, 3 kids, 78---,married, anonymous medical record 1

Female, 43, 2 kids, 78---, single anonymous medical record 2

Yellowpages

Personal website

Anonymous “privacy preserving records”

Susan Calvin, 43, 3 kids, Address, 78733, now labeled med. Records!

Internal Human + Automated hacker

Broken Exact record

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Two approaches to Privacy Preserving

• Distributed:– Suitable for multi-party platforms. Share sub-models.– Unsupervised: Ensemble Clustering, Privacy Preserving Clustering etc.– Supervised: Meta-learners, Fourier Spectrum Decision Trees, Collective

Hierarchical Clustering and so on..– Secure communication based: Secure sum, secure scalar product

• Random Data Perturbation: Our focus– Perturb data by small amounts to protect privacy of individual records.– Preserve intrinsic distributions necessary for modeling.

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Recovering approximately correct anonymous features also breaks privacy

• Somewhat inexactly recovered anonymous record values might also be sufficient:

Susan Calvin, 121 Norwood Cr. Austin, TX-78753

Hi, I am Susan, and here are pictures of me, my husband, and my 3 wonderful kids from my 43rd birthday party!

yellowpages

Personal website

“Denoised” privacy preserving records

Susan Calvin, 43, 3 kids, Address, 78733, now labeled med. Records!

Internal Human + Automated hacker

Broken Exact record

Female, 44.5, 3.2 kids, 78---,married, anonymous medical record 1

Female, 42.2, 2.1 kids, 78---, single, anonymous medical record 2

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Anonymous records (with or without) small perturbations not secure: not a recently noticed phenomena

• 1979, Denning & Denning: The Tracker: A Threat to Statistical Database Security– Show why anonymous records are not secure.– Show example of recovering exact salary of a professor from anonymous

records.– Present a general algorithm for an Individual Tracker.– A formal probabilistic model and set of conditions that make a dataset support

such a tracker.

• 1984, Traub & Yemin: The Statistical Security of a Statistical Database:– No free lunch: perturbations cause irrecoverable loss in model accuracy.– However, the holy grail of random perturbation:

We can try to find a perturbation algorithm that best trades off between loss of privacy vs. model accuracy.

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Recovering perturbed distributions: Earlier work

• Reconstructing Original Distribution from Perturbed Ones. Setup:– N samples U1, U2, U3.. Xn– N noise values V1, V2, V3.. Vn all taken from a public(known) distribution

V.– Visible noisy data: W1=U1+V1, W2=U2+V2 . .– Assumption: Such noise can allow you to recover the distribution of

X1,X2,X3 ..Xn, but not the individual record’s.

• Two well known methods and definitions:– Agrawal & Srikant:

Interval based: Privacy(X) at Confidence 0.95= X2-X1

– Agrawal & Aggarwal:

Distributional Privacy(X)=2h(x)

X1 X2

f(x) f’(x)

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Interval Based Method: Agrawal & Srikant in more detail

• N samples U1, U2, U3.. Xn• N noise values V1, V2, V3.. Vn all taken from a public(known) distribution V.• W1=U1+V1, W2=U2+V2 . .• Visible noisy data: W1, W2, W3 ..

Given: noise function fV , using Bayes’ Rule, we can show that the cumulative posterior distribution function of u in terms of w (visible) and fV , and unknown desired function fu ,

Differentiating w.r.t. u we get an important recursive definition:

Notation issue (in paper): f‘ simply means approximation of true f, not derivative of f !

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Interval Based Method: Agrawal & Srikant in more detail

Seed with a uniform distribution for J=0

sum over discrete z intervals instead of integral for speed

Algorithm in practice:

replaced integration with summation over i.i.d samples

STEP J+1

STEP J

• Converges to a local minima? Different than uniform initialization might give a different result. Not explored by authors.

• For large enough samples, hope to get close to true distribution.

• Stop when fU(J+1) – fU(J) becomes small.

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Interval Based Method: Good Results for a variety of noises

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Revisiting an Essential Assumption in the Random Perturbation

Assumption: Such noise can allow you to recover the distribution of X1,X2,X3 ..Xn, but not the individual record’s.

• The Authors in this paper challenge this assumption.

• Claim randomness addition can be mostly visual and not real:

• Many simple forms of random perturbations are “breakable”.

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Exploit predictable properties of Random data to design a filter to break the perturbation encryption?

Spiral data Random data

All eigen-values close to 1!

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Spectral Filtering: Main Idea: Use eigen-values properties of noise to filter

• U+V data

• Decomposition of eeigen-values of noise and original data

• Recovered data

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Decomposing eigen-values: separating data from noise

Let –

U and V be the m x n data and noise matrices

P the perturbed matrix UP= U+V

Covariance matrix of UP = UP T UP = (U+V) T (U+V) = UTU + VTU + UTV + UTU

Since signal and noise are uncorrelated in random perturbation, for large no. of observations: VTU ~ 0 and UTV ~ 0, therefore

UP T UP = UTU + VTV

Since the above 3 matrices are correlation matrices, they are symmetric and positive semi-definite, therefore, we can perform eigen decomposition:

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With bunch of algebra and theorems from Matrix Perturbation theory, authors show that in the limit (lots of data)..

Giving us the following algorithm:

1. Find a large no. of eigen values of the perturbed data P.

2. Separate all eigen values inside λmin and λmax and save row indices IV

3. Take the remaining eigen indices to get the “peturbed” but not noise eigens coming from true data U: save their row indices IU

4. Break perturbed eigenvector matrix QP into AU = QP (IU), AV = QP (IV).

5. Estimate true data as projection :

Wigner’s law: Describes distribution of eigen values for normal random matrices:

• eigen values for noise component V stick in a thin range given by λmin and λmax (show example next page) with high probability.

• Allows us to compute λmin and λmax.Solution!

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Exploit predictable properties of Random data to design a filter to break the perturbation encryption?

Spiral data Random data

All eigen-values close to 1!

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Results: Quality of Eeigen values recovery

Only the real eigen’s got captured, because of the nice automatic thresholding !

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Results: Comparison with Aggarwal’s reproduction

Agrawal & Srikant (no breakingof encryption) Agrawal & Srikant (estimated from broken

encryption)

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Discussion

• Amazing amount of experimental results and comparisons presented by authors in the Journal version.

• Extension to a situation where perturbing distribution form is known but exact first , second or higher order statistics not known: discussed but not presented.

• Comparison of performance with other obvious techniques for noise reduction in signal processing community: – Moving Averages and Weiner Filtering.– PCA Based filtering.

• Pros and Cons of the perturbation analysis by authors (and in general):– Effect of more and more noise: rapid degradation of results.– Problem in dealing with inherent noise in original data.– Technique fails when features independent of each other because of

Covariance matrix exploitation: Points to a major improvement possibility in encryption: perform ICA/PCA and then randomize?

– Results suggest that more complex noise models might be harder to break. Not clear if this improves privacy-model quality tradeoff?

– eigen decomposition has an inherent metric assumption?

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A not-so-ominous* application of noise filtering: Nulling Interferometer on Terrestrial Planet Finder-I

*but maybe not if you believe Hollywood movies such as Independence Day!

alien ship