Laplace Noise Mechanism

Reminder: Counting queries

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

(normalized) counting query

q(X ) =1

n

nX

i=1

q(xi ).

A workload Q of counting queries is given by predicates q1, . . . , qk .

Q(X ) =

0

B@q1(X )

...

qk(X )

1

CA 2 [0, 1]k .

E.g., “smoker?”, “smoker and over 30?”, “smoker and heart disease?”, etc.

9

" smoker,

heart disease,over 30 ? "

Answering counting queries with Randomized Response

10

How can I answer k counting queries w/ e . Dp

using RR ?

K instances of RR ( one per query )with (q) - Dp ) whole mechanismis E- DP by

composition than

All queries get answers we c- IL error

with prob . I - of

as long as n→kIe8l Exercise : Do the

£42 details !

Sensitivity

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

�1f = maxX⇠X 0

kf (X )� f (X 0)k1 = maxX⇠X 0

kX

i=1

|f (X )i � f (X 0)i |

Measure of how much a person can influence f .

11

e.g. textQQ - f }

% neighbouringe.g , if f :D

"→ IR

then ns.f-zinqylfkl- f 't'll

Sensitivity of a workload of counting queries

12

Suppose flx ) -

- alt) -- (g! It, ) for a workload

of k counting queries

Give an upper bound on D,Q

gilt) - t . gil qilxj) c- soil 's D, q .

.= In ki

1a --

em. Ei !9i"?ia%"" ' a.

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" ).

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)ke�kz�µk1/b =

1

(2b)kexp

�1

b

kX

i=1

|zi � µi |!

13

Z. .

. .. .

Z,are independent

Zi is from a one - dimensional Laplo,

µwpise-DpTy

Privacy of the Laplace noise mechanism

For any f , MLap is "-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

e�"kz�f (X )k1/�1f p0(z) =

✓1

2�1f

◆k

e�"kz�f (X 0)k1/�1f

Claim: enough to show maxz2Rkp(z)p0(z) e"

14

THE;fCXnhapHH.FI/=flHltZLap lap

~ Laplftx't . At)E E

Henk → Ethel'IPCMLAPIXIES ) - 1PM dZ Else' p' czidz

= ee fgptztdz = e' IPCMCX

' ) c- S ).

Privacy of the Laplace noise mechanism

p(z) =

✓1

2�1f

◆k

e�"kz�f (X )k1/�1f p0(z) =

✓1

2�1f

◆k

e�"kz�f (X 0)k1/�1f

15

Need ;Kzn ¥7, ee

'

Y E

FI= exp (Ff ( UZ - TH

'

) ",- Hz- HNK

, )) e eep'

IZ) my

←EHFIX ) - fit 'm ,

HZ - fl X' Ill , E HZ - fl till, t k fit ) - fix 'll! Triangle inequality

¥i

Histograms

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

• 8x 2 X : at most one of q1(x), . . . , qk(x) equals 1.

• e.g., “votes for the Liberal Party per riding”

What is the sensitivity?

16

"' mi.

a

X of 4 . . ..

.Xu 's

- F '= 94 . .

. .

. ti'

,. . . .tn }

fray,"QIN - QH 'll! -

-

finger, ÷! Iq .

-

K ) - gilt 'll E E

because E 2 queryvalues

change by c- In each.

Accuracy of the Laplace noise mechanism

If Z 2 Rk is a Laplace random variable from Lap(µ, b), then, for every i

P(|Zi � µi | � t) = e�t/b.

17

Generalizes to " 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 if n zHulkingqueries he