CSC2412: Properties of Di ff erential Privacy & More Mechanisms - - PowerPoint PPT Presentation

CSC2412: Properties of Differential Privacy & More Mechanisms

Sasho Nikolov

1

Review

Data model

Data set: (multi-)set X of n data points X = {x1, . . . , xn}.

• each data point (or row) xi is the data of one person
• each data point comes from a universe X

We call two data sets X and X 0 neighbouring if

• 1. (variable n) we can get X 0 from X by adding or removing an element
• 2. (fixed n) we can get X 0 from X by replacing an element with another

2

e.g

se

• so , Bd

( →

we will mostly

x~x

'

⇐

• X. X

'

neighbouring

use

this

Differential Privacy

Definition A mechanism M is ε-differentially private if, for any two neighbouring datasets X, X 0, and any set of outputs S P(M(X) ∈ S) ≤ eεP(M(X 0) ∈ S).

3

C- Range ( M )

Basic Properties

Composition motivation

It would be nice if we can:

• Post-process outputs of DP algorithms without losing privacy.
• Build complex DP algorithms from simple ones.
• Allow an analyst to adaptively choose queries to ask

4

• E. g

average

Hete

'

for

the output ( T

, ,

E.

g

use

RR

to

answer

many

counting

queries

• E. g

" smokers ?

"

" smokers

are

under 25 yrs

• ld ? '

+

<259

.

Composition theorem

Suppose

• M1(·) is ε1-DP
• M2(·, y) is ε2-DP for any y in the range of M1

Then M(·) given by M(X) = M2(X, M1(X)) is (ε1 + ε2)-DP.

5

M ,

takes

X

M ,

takes

X

and

the

• utput of ll

,

Epsilon addupost.pro#gy

Ust

, t)

Es

• f) p

tf

2- c-

Range

( Nz) If

U,

is

O - DP

i.e

µ, is only

a

air , nine" 'M

is

an:{

"Itsy

. .?

" / no:¥ Iheu ,

then Mill , 1H)

is

e ,

• DP

Proof of the composition theorem

6

Take

some

X

• X

'

Ito

prove

:

MIX ) -

• Milt ,MINI

S

E

Range Ilk)

is

CE

, -14) - DP

lP( MIX)

c- S)

=

E

Plucky

) es)

• PIM , KI
• y )

ye Range Ill

, )

E E

e' a plucky )

c- S)

• d

' PIM , IX ')

• y )

yethauglll

, )

=

elite

2-

BLUE Kyles )

. RCM

, K's

.

• y)

ye Rangel U. )

=

ee

. -19

PCU (t't

c- S )

Group Privacy

What protection is offered to small groups rather than individuals?

• E.g., what can an adversary find out about my immediate family?

Definition Two data sets X, X 0 are t-neighbours if they differ in the data of ≤ t individuals. For any ε-DP mechanism M, any t-neighbours X, X 0, and any set S of outputs P(M(X) ∈ S) ≤ etεP(M(X 0) ∈ S).

7

X = { X

, , Xz ,

• -

i Xj

• -

, Xu }

X

'

• ha .
• , ti

'

,

, xj .

Proof of group privacy property

8

• X. t

'

t - neighbouring

⇒ tht

, Nix?

. . XIX

'

• tht

't

,

X

'

• t}

• 'nxt

F-Yi

x

. . K ,

• - Xi

.

• - Yu 's

xk={ ×

, ,x

"

,

• -i'ji

X

'

• ha . . .

, ti

'

,

, rj

's

"

x2

AS

set of

• utputs

PLUNKS)EeEp( MIX's c-Sleek plumes)

• e etc IPIMLX

' )

c- S)