Differential Privacy for Relational Algebra: improving the - - PowerPoint PPT Presentation

Differential Privacy for Relational Algebra: improving the sensitivity bounds via constraint systems

Marco Stronati

Universit` a di Pisa, Italy marco@stronati.org

Catuscia Palamidessi

INRIA and LIX, Ecole Polytechnique, France catuscia@lix.polytechnique.fr

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Introduction

Statistical Disclosure Control

Revealing accurate statistics vs Preserving the privacy of individuals.

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Introduction

Statistical Disclosure Control

Revealing accurate statistics vs Preserving the privacy of individuals.

How many people have cancer?

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Introduction

Statistical Disclosure Control

Revealing accurate statistics vs Preserving the privacy of individuals.

How many people have cancer? Does John Doe have cancer?

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Background Quantitative approach

Information Hiding

Dalenius’ ad omnia privacy desideratum (’77): nothing about an individual should be learnable from the database that could not be learned without access to the database.

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Background Quantitative approach

Information Hiding

Dalenius’ ad omnia privacy desideratum (’77): nothing about an individual should be learnable from the database that could not be learned without access to the database.

Trade off between privacy and utility

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Background Quantitative approach

Information Hiding

Dalenius’ ad omnia privacy desideratum (’77): nothing about an individual should be learnable from the database that could not be learned without access to the database.

Trade off between privacy and utility Quantitative Approach

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Differential Privacy

Differential Privacy - Dwork, McSherry, Smith, Nissim

A randomized function H : R → R satisfies ǫ-differential privacy if for all pairs R, R′ ∈ R, with R ∼ R′, and all X ⊆ R: Pr[H(R) ∈ X] ≤ Pr[H(R′) ∈ X] · eǫ

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Differential Privacy

Differential Privacy - Dwork, McSherry, Smith, Nissim

A randomized function H : R → R satisfies ǫ-differential privacy if for all pairs R, R′ ∈ R, with R ∼ R′, and all X ⊆ R: e−ǫ ≤ Pr[H(R) ∈ X] Pr[H(R′) ∈ X] ≤ eǫ

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Differential Privacy

Differential Privacy - Dwork, McSherry, Smith, Nissim

A randomized function H : R → R satisfies ǫ-differential privacy if for all pairs R, R′ ∈ R, with R ∼ R′, and all X ⊆ R: e−ǫ ≤ Pr[H(R) ∈ X] Pr[H(R′) ∈ X] ≤ eǫ ǫ-indistinguishability

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Differential Privacy

Overview

(oblivious case)

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Differential Privacy Noise addition

Noise addition

Laplacian distribution Lap(x | b) = 1 2b exp

• −|x|

b

• Marco Stronati (APVP’12)

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Differential Privacy Noise addition

Noise addition

Laplacian distribution Lap(x | b) = 1 2b exp

• −|x|

b

• Theorem (Dwork06)

For Q : R → R, the randomized mechanism H that adds noise with distribution Lap(∆Q/ǫ) enjoys ǫ-differential privacy.

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Differential Privacy Noise addition

Noise addition

Laplacian distribution Lap(x | b) = 1 2b exp

• −|x|

b

• Theorem (Dwork06)

For Q : R → R, the randomized mechanism H that adds noise with distribution Lap(∆Q/ǫ) enjoys ǫ-differential privacy. ↑ ∆Q ↓ ǫ

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Differential Privacy Sensitivity

Sensitivity

Definition (Sensitivity Dwork06)

Given a query Q : R → R, the sensitivity of Q, denoted by ∆Q, is defined as: ∆Q = sup

R∼R′ | Q(R) − Q(R′) |

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Contribution

Contribution

◮ a compositional method to compute a bound on the sensitivity of a query

expressed in relational algebra

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Contribution

Contribution

◮ a compositional method to compute a bound on the sensitivity of a query

expressed in relational algebra

◮ constraints used to obtain the exact sensitivity

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Differential Privacy for Relational Algebra

Differential Privacy for Relational Algebra

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Differential Privacy for Relational Algebra Relational Algebra

Relational Algebra - A Formal SQL

◮ T : universe of tuples ◮ Relation R: a set of tuples ◮ R: universe of relations

Definition (Relation Schema)

name(a1 : D1, a2 : D2, . . . , an : Dn)

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Differential Privacy for Relational Algebra Relational Algebra

Relational Algebra - A Formal SQL

◮ T : universe of tuples ◮ Relation R: a set of tuples ◮ R: universe of relations

Definition (Relation Schema)

name(a1 : D1, a2 : D2, . . . , an : Dn)

Example

Items { Item : String, Price : Int, Cost : Int } Item Price Cost Oil 100 10 Salt 50 11

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Differential Privacy for Relational Algebra Constraints

Constraints

CREATE TABLE products ( product no integer, name text, price numeric CHECK (price > 0) );

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Differential Privacy for Relational Algebra Constraints

Constrained Schema

T (C) R(C) = 2T (C)

Items { Item : String, Price : Int, Cost : Int } { < Cost ≤ 1000 Cost ≤ Price ≤ 1000 }

c-schema: schema + set of constraints C

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Differential Privacy for Relational Algebra Constraints

Constrained Schema

T (C) R(C) = 2T (C)

Items { Item : String, Price : Int, Cost : Int } { < Cost ≤ 1000 Cost ≤ Price ≤ 1000 }

c-schema: schema + set of constraints C Transformation from c-schema to c-schema

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Differential Privacy for Relational Algebra Sensitivity Constrained

Sensitivity Constrained

Definition (Sensitivity constrained)

Given f : (X, dX) → (Y , dY ), set of constraints C on X ∆f (C) = sup

x,x′∈ sol(C) x=x′

dY (f (x), f (x′)) dX(x, x′)

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Differential Privacy for Relational Algebra Metric Spaces

Metric Spaces

Adjacency relation (R, ∼)

− − − − − − − − − − − − − − ∼ − − − − − − − − − − − − − − − − − − − − −

Hamming Graph

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Differential Privacy for Relational Algebra Metric Spaces

Metric Spaces

Adjacency relation (R, ∼)

− − − − − − − − − − − − − − ∼ − − − − − − − − − − − − − − − − − − − − −

Hamming Graph

Definition (Hamming distance dH)

Given R, R′ ∈ R dH(R, R′) = |R ⊖ R′| = |(R \ R′) ∪ (R′ \ R)|

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Differential Privacy for Relational Algebra Metric Spaces

Metric Spaces

Definition (n-Hamming Distance dnH)

Given R, R

′ ∈ Rn:

dnH(R, R

′) = max (dH(R1, R′ 1), . . . , dH(Rn, R′ n))

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Differential Privacy for Relational Algebra Metric Spaces

Metric Spaces

Definition (n-Hamming Distance dnH)

Given R, R

′ ∈ Rn:

dnH(R, R

′) = max (dH(R1, R′ 1), . . . , dH(Rn, R′ n))

Definition (Euclidean Distance dE)

Given x, x′ ∈ R dE(x, x′) = |x − x′|

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Operators

Structure of a Query

(Rn, dnH)

Op

− → . . .

Op

− → (Rn, dnH)

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Operators

Structure of a Query

(Rn, dnH)

Op

− → . . .

Op

− → (Rn, dnH)

AγF

− − → (R, dE)

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Operators

Structure of a Query

(Rn, dnH)

Op

− → . . .

Op

− → (Rn, dnH)

AγF

− − → (R, dE)

• p

∈ ∪, ∩, \, σϕ π, ×, ⊲ ⊳

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Operators Operators Sensitivity

Operators Sensitivity

∆op(C) = sup R,R′∈ R(C)

R=R′ dH(op(R),op(R′)) dH(R,R′)

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Operators Operators Sensitivity

Operators Sensitivity

∆op(C) = sup R,R′∈ R(C)

R=R′ dH(op(R),op(R′)) dH(R,R′)

= min

• ∆op(∅), diam(C ⊗ Cop)
• Marco Stronati (APVP’12)

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Operators Operators Sensitivity

Operators Sensitivity

∆op(C) = sup R,R′∈ R(C)

R=R′ dH(op(R),op(R′)) dH(R,R′)

= min

• ∆op(∅), diam(C ⊗ Cop)
• Marco Stronati (APVP’12)

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Operators Operators Sensitivity

Operators Sensitivity

∆op(C) = sup R,R′∈ R(C)

R=R′ dH(op(R),op(R′)) dH(R,R′)

= min

• ∆op(∅), diam(C ⊗ Cop)
• Marco Stronati (APVP’12)

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Operators Operators Sensitivity

Other operators

• p

∆ Schema ∪ 2 (A, C1 ∨ C2) ∩ 2 (A, C1 ∧ C2) \ 2 (A, C1 ∧ (¬C2)) σϕ 1 (A, C ∧ ϕ) πA′ 1 (A′, C)

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Operators Cartesian product

Cartesian product

× : (R2, d2H) → (R, dH)

Name Age Height John 30 180 Alice 45 160 × Car Owner Fiat Alice Ford Alice = Name Age Height Car Owner John 30 180 Fiat Alice John 30 180 Ford Alice Alice 45 160 Fiat Alice Alice 45 160 Ford Alice

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Operators Cartesian product

Cartesian product

× : (R2, d2H) → (R, dH)

Name Age Height John 30 180 Alice 45 160 × Car Owner Fiat Alice Ford Alice = Name Age Height Car Owner John 30 180 Fiat Alice John 30 180 Ford Alice Alice 45 160 Fiat Alice Alice 45 160 Ford Alice

The (unrestricted) cartesian product has unbounded sensitivity.

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Operators Cartesian product

Cartesian product

× : (R2, d2H) → (R, dH)

Name Age Height John 30 180 Alice 45 160 × Car Owner Fiat Alice Ford Alice = Name Age Height Car Owner John 30 180 Fiat Alice John 30 180 Ford Alice Alice 45 160 Fiat Alice Alice 45 160 Ford Alice

The (unrestricted) cartesian product has unbounded sensitivity.

Join ⊲ ⊳

R ⊲ ⊳

R.ai=T.ai T = σ(R.ai=T.ai)(R×T)

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Operators Query Sensitivity

Intermediate query sensitivity

◮ op any of ∪, ∩, \, σ, π, ×, ×1 ◮ op : (Rn, dnH) → (R, dH) ◮ Cop constraint obtained after op

S(Id) = min(1, diam(CId)) base case S(op ◦ Q) = min(∆op · S(Q), diam(Cop◦Q)) if n = 1 S(op ◦ (Q1, Q2)) = min(∆op · max(S(Q1), S(Q2)), diam(Cop◦(Q1,Q2))) if n = 2

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Operators Query Sensitivity

Intermediate query sensitivity

◮ op any of ∪, ∩, \, σ, π, ×, ×1 ◮ op : (Rn, dnH) → (R, dH) ◮ Cop constraint obtained after op

S(Id) = min(1, diam(CId)) base case S(op ◦ Q) = min

• ∆op · S(Q), diam(Cop◦Q)
• if n = 1

S(op ◦ (Q1, Q2)) = min

• ∆op · max(S(Q1), S(Q2)), diam(Cop◦(Q1,Q2))
• if n = 2

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Aggregation γ

Aggregation γ

{a1,...,am} γ {f1,...,fk} : (R, dH) → (R, dH) ◮ groups tuples with the same values of ai ◮ computes fj for each group (count,max,min,avg,sum) ◮ returns a single tuple for each group, with ai and fj.

SELECT Car, Count(*), Avg(Height) FROM R GROUPBY Car

{Car}γ{Count,Avg(Height)}

  

Name Age Height Car Alice 45 160 Ford John 30 180 Fiat Frank 45 165 Bmw Eve 20 170 Ford

   =

Car Count Avg(Height) Ford 2 165 Fiat 1 180 Bmw 1 165

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Aggregation γ Functions

Functions

Assume ∅γf

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Aggregation γ Functions

Functions

Assume ∅γf R ∼ R′

∆count(C) = 1 ×n ∆sumai (C) = max{| sup(C, ai)|, | inf(C, ai)|} ×n ∆avgai (C) = | sup(C, ai) − inf(C, ai)| × 1

2

∆maxai (C) = | sup(C, ai) − inf(C, ai)| ∆minai (C) = | sup(C, ai) − inf(C, ai)|

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Aggregation γ Functions

Functions

Assume ∅γf dH(R, R′) = n

∆count(C) = 1 ×n ∆sumai (C) = max{| sup(C, ai)|, | inf(C, ai)|} ×n ∆avgai (C) = | sup(C, ai) − inf(C, ai)| ×

n 1+n

∆maxai (C) = | sup(C, ai) − inf(C, ai)| ∆minai (C) = | sup(C, ai) − inf(C, ai)|

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Global Sensitivity

Global Sensitivity

Definition (global sensitivity)

The global sensitivity GS of a query γf (Q) is defined as: GS(γf (Q)) = ∆f (CQ) · S(Q) if f = count, sum, avg ∆f (CQ) if f = max, min

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Global Sensitivity

Global Sensitivity

Definition (global sensitivity)

The global sensitivity GS of a query γf (Q) is defined as: GS(γf (Q)) = ∆f (CQ) · S(Q) if f = count, sum, avg ∆f (CQ) if f = max, min

Theorem (Soundness and strictness)

The sensitivity bound computed by GS(·) is sound and strict. Namely: GS(γf (Q)) = ∆γf (Q)

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Global Sensitivity Example

Example

c-schema ({Weight, Height}, CI) CI = {Weight ∈ [0, 150] ∧ Height ∈ [0, 200]}

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Global Sensitivity Example

Example

c-schema ({Weight, Height}, CI) CI = {Weight ∈ [0, 150] ∧ Height ∈ [0, 200]} γavg(Weight)(σWeight≤Height−100(R))

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Global Sensitivity Example

Example

c-schema ({Weight, Height}, CI) CI = {Weight ∈ [0, 150] ∧ Height ∈ [0, 200]} γavg(Weight)(σWeight≤Height−100(R)) CQ = {Weight ∈ [0, 150] ∧ Height ∈ [0, 200] ∧ Weight ≤ Height − 100}

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Global Sensitivity Example

Example

c-schema ({Weight, Height}, CI) CI = {Weight ∈ [0, 150] ∧ Height ∈ [0, 200]} γavg(Weight)(σWeight≤Height−100(R)) CQ = {Weight ∈ [0, 150] ∧ Height ∈ [0, 200] ∧ Weight ≤ Height − 100} ∆(CI, γavg(Weight)) = |max(CI ,Weight)−min(CI ,Weight)|

2

= 75 ∆(CQ, γavg(Weight)) = |max(CQ,Weight)−min(CQ,Weight)|

2

= 50

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Future work

Future Work

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Future work Join datasets

Join datasets

Join ⊲ ⊳

◮ ×n: product with blocks of a fixed n size , to obtain n sensitivity. policies to

pick these representative elements

◮ ×γ: single record is built as an aggregation of the relation, thus falling in the

case of ×1 sensitivity

◮ a mix the two approaches could be considered, building n aggregations,

possibly using the operator {ai}γf

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Future work Compositional ǫ analysis

Compositional ǫ analysis

Sequential composition: Qi queries each providing ǫi differential privacy, Qn ◦ . . . ◦ Q1 provides (

i ǫi)-differential privacy.

Parallel composition: Qi queries each providing ǫ differential privacy, parallel application to disjoint subsets of the input provides again ǫ-differential privacy. Extend static analysis to compute ǫ and optimize to parallelize

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Future work Comparison of metric spaces

Comparison of metric spaces

Distance dnH dnH((R1, . . . , Rn), (R′

1, . . . , R′ n)) = max(dH(R1, R′ 1), . . . , dH(Rn, R′ n))

At first was the Manhattan distance d2H((R1, R2), (R3, R4)) = dH((R1, R2)) + dH(R3, R4)

◮ lower sensibilities on many operators, namely ∪, ∩, \ all had sensitivity 1 ◮ did not allow us to compute the real sensitivity but just a bound

Explore further the comparison

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Future work Comparison of metric spaces

Thanks

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