Laplace Noise Mechanism Reminder: Counting queries Given a predicate - - PowerPoint PPT Presentation

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Jan 28, 2024 133 likes •252 views

Laplace Noise Mechanism Reminder: Counting queries Given a predicate q : X ! { 0 , 1 } (e.g., smoker?), we define the corresponding (normalized) counting query n q ( X ) = 1 X q ( x i ) . n i =1 A workload Q of counting queries is given by

SLIDE 1

Laplace Noise Mechanism

SLIDE 2

Reminder: Counting queries

Given a predicate q : X ! {0, 1} (e.g., “smoker?”), we define the corresponding (normalized) counting query q(X) = 1 n

i=1

q(xi). A workload Q of counting queries is given by predicates q1, . . . , qk. Q(X) = B @ q1(X) . . . qk(X) 1 C A 2 [0, 1]k. E.g., “smoker?”, “smoker and over 30?”, “smoker and heart disease?”, etc.

" smoker , heart disease ,

? "

SLIDE 3

Answering counting queries with Randomized Response

How

can I answer

counting

queries

e .

using

instances

( one

per

query )

with (q)

)

whole mechanism

E- DP by

composition

than

All

queries

get

answers

c- IL

error

with prob

. I - of

long

as n →kIe8l

Exercise :

Do the

£42 details !

SLIDE 4

Sensitivity

The `1 sensitivity of f : X n ! Rk is ∆1f = max

X⇠X 0 kf (X) f (X 0)k1 = max X⇠X 0 k

i=1

|f (X)i f (X 0)i| Measure of how much a person can influence f .

e.g. text QQ

% neighbouring

e.g

, if

f :D

→ IR

then ns.f-zinqylfkl

f 't'll

SLIDE 5

Sensitivity of a workload of counting queries

Suppose

flx )

alt)
(

g! It, )

for

workload

f k counting queries

Give

upper bound

, Q

gilt) - t

gil

qilxj

)

c- soil's

D , q .

. = In

Ei !9i"?ia%""

' a.

SLIDE 6

Laplace noise mechanism

The Laplace noise mechanism MLap (for a function f : X n ! Rk) outputs MLap(X) = f (X) + Z, where Z 2 Rk is sampled from Lap(0, ∆1f

ε ).

Lap(µ, b) is the Laplace distribution on Rk with expectation µ 2 Rk and scale b > 0, and has pdf p(z) = 1 (2b)k ekzµk1/b = 1 (2b)k exp 1 b

i=1

|zi µi| !

. . . .. . Z ,

are independent

from

ne
dimensional

Laplo,

µwpise-DpTy

SLIDE 7

Privacy of the Laplace noise mechanism

For any f , MLap is ε-DP. Let X ∼ X 0, and let p be the pdf of M(X), and p0 the pdf of M(X 0). p(z) = ✓ 1 2∆1f ◆k eεkzf (X)k1/∆1f p0(z) = ✓ 1 2∆1f ◆k eεkzf (X 0)k1/∆1f Claim: enough to show maxz2Rk p(z)

p0(z) ≤ eε 14

THE;fCXnhapHH.FI/=flHltZ

Lap

lap

~ Laplftx't . At)

E E

Henk

→ Ethel

IPCMLAPIXIES )

dZ Else ' p' czidz

ee fgptztdz

= e' IPCMCX ' ) c- S ) .

SLIDE 8

Privacy of the Laplace noise mechanism

p(z) = ✓ 1 2∆1f ◆k eεkzf (X)k1/∆1f p0(z) = ✓ 1 2∆1f ◆k eεkzf (X 0)k1/∆1f

Need

;

Kzn

¥7,

Y

E

= exp (Ff ( UZ

') " ,

, ))

e ee

' IZ)

my ←

EHFIX)

fit 'm ,

fl X' Ill

E HZ

fl till,

k fit)

fix'll!

Triangle inequality

¥i

SLIDE 9

Histograms

Suppose the query workload Q = {q1, . . . , qk} “partitions” the data.

∀x ∈ X : at most one of q1(x), . . . , qk(x) equals 1.
e.g., “votes for the Liberal Party per riding”

What is the sensitivity?

"' mi.

X of 4 .

. . . .

Xu 's

' =

. . . . . ti ' , . . . .tn }

fray

"QIN

QH'll!
finger

, ÷! Iq .

K)
gilt'll

E E

because

E 2

query

values

change by

c- In

each .

SLIDE 10

Accuracy of the Laplace noise mechanism

If Z ∈ Rk is a Laplace random variable from Lap(µ, b), then, for every i P(|Zi − µi| ≥ t) = et/b.

Generalizes

" K

norm

mechanism

Hardt

, Talwar

Map ( H
Lap ( fit )

, diet )

Rl ?

at llk.pk/i-fCXIilza1z?zlPllkaplxt.-fctHzH

E k

. e

debit

, then

is ,ft kn

Minar

error z d) else-1¥

E.g

. if

f-(x)

Q (H

E B

i.e.

answers to

k counting

n z Hulking

queries