Privacy Preserving Mining of Association Rules Alexandre Evfimievski, Ramakrishnan Srikant, Rakesh...

Privacy Preserving Mining of Association Rules

Alexandre Evfimievski,

Ramakrishnan Srikant,

Rakesh Agrawal,

Johannes Gehrke

IBM Almaden Research CenterCornell University

Data Mining and Privacy

• The primary task in data mining: development of models about aggregated data.

• Can we develop accurate models without access to precise information in individual data records?

Data Mining and Privacy

• The primary task in data mining: development of models about aggregated data.

• Can we develop accurate models without access to precise information in individual data records?

• Answer: yes, by randomization.– R. Agrawal, R. Srikant “Privacy Preserving Data Mining,”

SIGMOD 2000– for numerical attributes, classification

• How about association rules?

Randomization Overview

Recommendation Service

B. Spears,baseball,cnn.com,…

J.S. Bach,painting,nasa.gov,…

B. Marley,camping,linux.org,…

Associations

Recommendations

Associations

Recommendations

Metallica,painting,nasa.gov,…

B. Spears,soccer,bbc.co.uk,…

B. Marley,camping,microsoft.com…

Support Recovery

Associations Recap

• A transaction t is a set of items (e.g. books)• All transactions form a set T of transactions• Any itemset A has support s in T if

• Itemset A is frequent if s smin

• If A B , then supp (A) supp (B).

tATtAs

|#supp

Associations Recap

• A transaction t is a set of items (e.g. books)• All transactions form a set T of transactions• Any itemset A has support s in T if

• Itemset A is frequent if s smin

• If A B , then supp (A) supp (B).

• Example:– 20% transactions contain beer,– 5% transactions contain beer and diapers;– Then: confidence of “beer diapers” is 5/20 = 0.25 = 25%.

tATtAs

|#supp

The Problem

• How to randomize transactions so that– we can find frequent itemsets– while preserving privacy at transaction level?

Talk Outline

• Introduction• Privacy Breaches• Our Solution• Experiments• Conclusion

Uniform Randomization

• Given a transaction,– keep item with 20% probability,– replace with a new random item with 80%

probability.

Example: {x, y, z}

1% have

{x, y, z}

5% have{x, y}, {x, z},or {y, z} only

10 M transactions of size 10 with 10 K items:

94%have one or zeroitems of {x, y, z}

Example: {x, y, z}

1% have

{x, y, z}

Uniform randomization: How many have {x, y, z} ?

Example: {x, y, z}

1% have

{x, y, z}

0.008%800 ts.

0.00016%16 trans.

less than 0.00002%2 transactions

• 0.22 • 8/10,000• 0.23at most

• 0.2 • (9/10,000)2

Example: {x, y, z}

1% have

{x, y, z}

0.008%800 ts.97.8%

0.00016%16 trans.

less than 0.00002%2 transactions

• 0.22 • 8/10,000• 0.23at most

• 0.2 • (9/10,000)2

Example: {x, y, z}• Given nothing, we have only 1% probability that {x, y, z}

occurs in the original transaction

• Given {x, y, z} in the randomized transaction, we have about 98% certainty of {x, y, z} in the original one.

• This is what we call a privacy breach.

• Uniform randomization preserves privacy “on average,” but not “in the worst case.”

Privacy Breaches• Suppose:

– t is an original transaction;– t’ is the corresponding randomized transaction;– A is a (frequent) itemset.

• Definition: Itemset A causes a privacy breach of level (e.g. 50%) if, for some item z A,

– Assumption: no external information besides t’.

tAtz |Pr

Talk Outline

Our Solution

• Insert many false items into each transaction• Hide true itemsets among false ones

“Where does a wise man hide a leaf? In the forest. But what does he do if there is no forest?”

“He grows a forest to hide it in.”

G.K. Chesterton

Our Solution

• Insert many false items into each transaction• Hide true itemsets among false ones

Can we still find frequent itemsets while having sufficient privacy?

“Where does a wise man hide a leaf? In the forest. But what does he do if there is no forest?”

“He grows a forest to hide it in.”

G.K. Chesterton

Definition of cut-and-paste• Given transaction t of size m, construct t’:

a, b, c, u, v, w, x, y, zt =

t’ =

– Choose a number j between 0 and Km (cutoff);

a, b, c, u, v, w, x, y, zt =

t’ =j = 4

– Include j items of t into t’;

a, b, c, u, v, w, x, y, zt =

b, v, x, zt’ =j = 4

– Include j items of t into t’;

– Each other item is included into t’ with probability pm .

The choice of Km and pm is based on the desired level of privacy.

a, b, c, u, v, w, x, y, zt =

b, v, x, zt’ = œ, å, ß, ξ, ψ, €, א, ъ, ђ, …j = 4

Partial SupportsTo recover original support of an itemset, we need

randomized supports of its subsets.• Given an itemset A of size k and transaction size m,• A vector of partial supports of A is

– Here sk is the same as the support of A.

– Randomized partial supports are denoted by

lAtTtT

,,...,, 10 where

Transition Matrix• Let k = |A|, m = |t|. • Transition matrix P = P (k, m) connects randomized

partial supports with original ones:

• Randomized supports are distributed as a sum of multinomial distributions.

lAtlAtP

The Unbiased Estimators

• Given randomized partial supports, we can estimate original partial supports:

1, PQsQs whereest

The Unbiased Estimators

• Given randomized partial supports, we can estimate original partial supports:

• Covariance matrix for this estimator:

• To estimate it, substitute sl with (sest)l .– Special case: estimators for support and its variance

1, PQsQs whereest

ljlijiliji

QlDQsT

Class of Randomizations

• Our analysis works for any randomization that satisfies two properties:– A per-transaction randomization applies the same

procedure to each transaction, using no information about other transactions;

– An item-invariant randomization does not depend on any ordering or naming of items.

Class of Randomizations

• Our analysis works for any randomization that satisfies two properties:– A per-transaction randomization applies the same

procedure to each transaction, using no information about other transactions;

– An item-invariant randomization does not depend on any ordering or naming of items.

• Both uniform and cut-and-paste randomizations satisfy these two properties.

AprioriLet k = 1, candidate sets = all 1-itemsets.

Repeat:1. Count support for all candidate sets

2. Output the candidate sets with support smin

3. New candidate sets = all (k + 1)-itemsets s.t. all their k-subsets are candidate sets with support smin

4. Let k = k + 1

Stop when there are no more candidate sets.

The Modified AprioriLet k = 1, candidate sets = all 1-itemsets.

Repeat:1. Estimate support and variance (σ2) for all candidate sets

2. Output the candidate sets with support smin

3. New candidate sets = all (k + 1)-itemsets s.t. all their k-subsets are candidate sets with support smin - σ

4. Let k = k + 1

Stop when there are no more candidate sets, or the estimator’s precision becomes unsatisfactory.

Privacy Breach Analysis• How many added items are enough to protect

privacy?– Have to satisfy Pr [z t | A t’] < ( no privacy breaches)– Select parameters so that it holds for all itemsets.– Use formula ( ):

llkl PsPstAtz

0,,#Pr 0 stzlAtsl

Privacy Breach Analysis• How many added items are enough to protect

privacy?– Have to satisfy Pr [z t | A t’] < ( no privacy breaches)– Select parameters so that it holds for all itemsets.– Use formula ( ):

• Parameters are to be selected in advance!– Construct a privacy-challenging test: an itemset whose all

subsets have maximum possible support.– Enough to know maximal support of an itemset for each size.

llkl PsPstAtz

0,,#Pr 0 stzlAtsl

Graceful Tradeoff

• Want more precision or more privacy?– Adjust privacy breach level– A small relaxation of privacy restrictions results in

a small increase in precision of estimators.

Talk Outline

• Introduction• Privacy Breaches• Our Solution• Experiments

– Support recovery vs. parameters– Real-life data

• Conclusion

Lowest Discoverable Support• LDS is s.t., when predicted, is 4 away from zero.• Roughly, LDS is proportional to

LDS vs. number of transactions

1 10 100Number of transactions, millions

1-itemsets 2-itemsets 3-itemsets

|t| = 5, = 50%

LDS vs. Breach Level

30 40 50 60 70 80 90

Privacy Breach Level, %

1-itemsets

2-itemsets

3-itemsets

|t| = 5, |T| = 5 M

• Reminder: breach level is the limit on Pr [z t | A t’]

Talk Outline

• Introduction• Privacy Breaches• Our Solution• Experiments

– Support recovery vs. parameters– Real-life data

• Conclusion

Real datasets: soccer, mailorder• Soccer is the clickstream log of WorldCup’98 web site,

split into sessions of HTML requests.– 11 K items (HTMLs), 6.5 M transactions– Available at http://www.acm.org/sigcomm/ITA/

• Mailorder is a purchase dataset from a certain on-line store– Products are replaced with their categories– 96 items (categories), 2.9 M transactions

A small fraction of transactions are discarded as too long.

– longer than 10 (for soccer) or 7 (for mailorder)

Modified Apriori on Real Data

Itemset Size

True Itemsets

True Positives

False Drops

False Positives

1 266 254 12 31

2 217 195 22 45

3 48 43 5 26

Itemset Size

True Itemsets

True Positives

False Drops

False Positives

1 65 65 0 0

2 228 212 16 28

3 22 18 4 5

Soccer:

smin = 0.2%

0.07% for 3-itemsets

Mailorder:

smin = 0.2%

0.05% for 3-itemsets

Breach level = 50%. Inserted 20-50% items to each transaction.

False Drops False Positives

Size < 0.1 0.1-0.15 0.15-0.2 0.2

1 0 2 10 254

2 0 5 17 195

3 0 1 4 43

Size < 0.1 0.1-0.15 0.15-0.2 0.2

1 0 7 24 254

2 7 10 28 195

3 5 13 8 43

Size < 0.1 0.1-0.15 0.15-0.2 0.2

1 0 0 0 65

2 0 1 15 212

3 0 1 3 18

Size < 0.1 0.1-0.15 0.15-0.2 0.2

1 0 0 0 65

2 0 0 28 212

3 1 2 2 18

Soccer

Mailorder

Pred. supp%, when true supp 0.2%

True supp%, when pred. supp 0.2%

Actual Privacy Breaches

• Verified actual privacy breach levels• The breach probabilities are counted in the datasets

for frequent and near-frequent itemsets.• If maximum supports were estimated correctly, even

worst-case breach levels fluctuated around 50%– At most 53.2% for soccer,– At most 55.4% for mailorder.

Talk Outline

Summary

• Privacy breaches: identified problem and provided a solution for controlling breaches

• Derived estimators of support and variance for a class of randomization operators

• Algorithm for discovering associations in randomized data

• Validated on real-life datasets• Can find associations while preserving privacy at the

level of individual transactions

Future Work

• Control of more general privacy breaches– What about other properties of transactions, for example

item z t breach caused by A t’ = ?– What about external information?

• Theoretical limits of discoverability for a given privacy breach level– How to compute theoretical limits?– How to attain them by an algorithm?

Thank You!

BACK-UPS

Our Solution: Example• Old set-up:

– Given 10,000 items, 10 M transactions of size 10– 100,000 transactions (1%) contain A = {x, y, z}

• In addition to uniform randomization with p = 80%, insert 500 new random items to each transaction.– ~ 800 transactions contain {x, y, z} before and after;– Roughly (10 M) • (500 / 10,000)3 = 1250 transactions contain

none before and full {x, y, z} after.

• Presence of {x, y, z} in a randomized transaction now says little about the original transaction.

Privacy Breach Analysis

• GIVEN: itemset A, and item z A • WANTED:• Assume that partial supports are probabilities:

• Define: • Then we have:

lAtAs ll #Prsupp

0,,#Pr 0 stzlAtsl

tAtz |Pr

llkl PsPstAtz

Limiting Privacy Breaches

• We want to make sure that always

• But we do not know supports in advance.• Solution: For each itemset size k, give “privacy-

challenging” test values to .– It is an itemset whose subsets have maximum supports – We need to estimate maximum support values prior to

randomization

llkl PsPs

ll ss ,

LDS vs. Transaction Size = 50%, |T| = 5 M

• Too long transactions cannot be used for prediction

1 2 3 4 5 6 7 8 9 10Transaction Size

1-itemsets

2-itemsets3-itemsets

Related Work

R. Agrawal, R. Srikant “Privacy Preserving Data Mining,” SIGMOD 2000:

• Each client has a numerical attribute xi

• Client i sends xi + yi , where yi = random offset, with known distribution

• Server reconstructs the distribution of original attributes (~ EM algorithm)

• The distribution is then used for classification– Numerical attributes only

Related Work

• Y. Lindell and B. Pinkas “Privacy Preserving Data Mining,” Crypto 2000

• J. Vaidya and C. Clifton “Privacy Preserving Association Rule Mining in Vertically Partitioned Data”

• …

Privacy Concern

• Popular press: – “The End of Privacy”, “The Death of Privacy”

• Government directives:– European directive on privacy protection (Oct 98)– Canadian Personal Information Protection Act (Jan 2001)

• Surveys of Web users:– 17% fundamentalists, 56% pragmatic majority, 27%

marginally concerned (April 99)– 82% said having privacy would matter (July 99)

Privacy Preserving Mining of Association Rules Alexandre Evfimievski, Ramakrishnan Srikant, Rakesh...

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