Other Perturbation Techniques

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Other Perturbation Techniques
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Outline

• Randomized Responses
• Sketch
• Project ideas

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Randomized Responses

• Problem description
• A provides the answer to Bs question
• A wants to preserve his/her privacy
• Question/answer can be sensitive
• The method
• Assume the answer can be yes or no
• A has a probability ? to be honest, and the
  probability 1- ? to give a random response
• We can estimate the real probability of yes and
  no from the randomized responses

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• Notations
• O(yes) observed probability of yes from the
  randomized responses
• of yes/total of responses
• P(yes) real probability of yes
• Inference
• O(yes) P(yes) ? P(no)(1-?)
• P(yes) ? (1-P(yes))(1-?)
• ? P(yes) (O(yes)?-1)/(2?-1)

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• Extend to multiple categories
• The answer ci has a prob ?ij changed to cj
• O((c1,c2,,cn)) observed prob of ci
• P((c1,c2,,cn)) real prob of ci
• The relationship between O and P

Note When ? is invertible, use matrix inversion
to solve P. Otherwise, use iterative
methods similar to that in Rakeshs paper
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• Different perturbation matrices can be used.
  Which one is the best?
• Balance between privacy and utility?

Zero privacy is preserved, while full data
utility is preserved
Uniform randomization, privacy is fully
preserved, while no data utility is left
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Optimizing both privacyutility

• Read paper 33
• Privacy similar to previous discussion
• Based on accuracy of estimation
• A Bayes method
• C c1,c2,,cn)
• Y is the perturbed value, X is the original
  value, and X is the estimated value

Accuracy of estimation
It can be calculated by checking the original
data, the perturbed data and the estimated data
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• Privacy
• Average 1- (accuracy of estimation)
• Worst case
• Utility
• P(ci) the original prob, O(ci) the prob on
  perturbed data, P(ci) is the estimated prob
• Utility depends on the difference between the
  original prob and the estimated prob

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Optimization algorithm

• Find the perturbation that balance the two
  metrics
• The evolutionary algorithm
• Start with a set of initial RR matrices
• Repeat the following steps in each iteration
• Mating selecting two RR matrices in the pool
• Crossover exchanging several columns between the
  two RR matrices
• Mutation change some values in a RR matrix
• Meet the privacy bound filtering the resultant
  matrices
• Evaluate the fitness value for the new RR
  matrices.
• Note the fitness values is defined in terms of
  privacy and utility metrics

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summary

• Randomized response is the basic technique for
  perturbing categorical data
• Boolean
• Multi-category

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Sketch

• Address the problem of high-dimensional sparse
  data
• Multiplicative perturbation
• Randomized responses
• Market basket data
• Bag of words

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Definition of sketch

• Similar to projection perturbation
• Map d dimensional data ? r dimensional data, rltltd
• Difference for each record the mapping matrix is
  different
• Definition
• X (x1,xd), S(s1,,sr)

is randomly drawn from -1, 1
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property

• Dot product of the original data X and Y can be
  approximated with their sketches
• Dot product is important in calculating Euclidean
  distances!

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• Accuracy of the dot product estimation

Large r ? smaller variance ? better quality
however, ? lower privacy
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Privacy

• Original data value can be estimated
• Sparse data
• Most are canceled in sketch
• Estimate of xk

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privacy

• ? - anonimity

Suppress the record if this condition is not
satisfied
Another concept K-variance paper 29 for more
details.
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• Applications
• Dot product estimation
• Determine the length of sparse transaction ( of
  non-zero items in boolean vector)
• Determine Euclidean distance
• Average of a set of records (centroid of a
  cluster)