CSC2412: Adaptive Data Analysis via Di ff erential Privacy Sasho - - PowerPoint PPT Presentation

SLIDE 1

CSC2412: Adaptive Data Analysis via Differential Privacy

Sasho Nikolov

1

SLIDE 2

The adaptive data analysis problem

SLIDE 3

Estimating population counts

• Unknown distribution D on X
• Predicates q1, . . . , qk : X → {0, 1}

Want to estimate, for all i = 1 . . k: qi(D) = Ex∼D[qi(x)].

2

juni

verse of possible

data points

• → models

the

population

• E. g

.

if ,

i ? smoker

qz

• ? smoker and male

ch

.

• ? smoker

.any

PhD

• fraction of

the

population

satisfying

9 i

SLIDE 4

The classical solution

Draw a sample X = {x1, . . . , xn} iid from D. Hope that ∀i : qi(X) ≈ qi(D)

3

gilt )

• I

,

qidxj )

Effi (x))

• qi

( D)

Is independent , info

,

I}

Hotting :

"

R ( I qilx)

• gil DH H)
• Pll qilxl
• Eq

.

I > d )

E L

e

• 2h22

Blt

i

: lqiltl

• qi CDH > a)

c- 2k . e

• 2nd Ep

if fnzhg.la?kh#-/

=L

SLIDE 5

Adaptive queries?

What if qi depends on q1, . . . , qi−1?

4

• the

estimates

for

• g. ( D)

,

. . ., qi , CD)

• E. g

.

qi

is

chosen

based

• n

q , ( H

, . . .

, qi, (X )

E.g

.

g.

= ? smokers

and

male

%

'

?

smokers

and female }

→ it

even

split

ask

• g. so ? smokers

and

235 yrs

ebesto#

Suppose

we

ask

• 9. ( X ) , g. CH

.

• -

n , gut )

for Kan

,

q

. .

random

and

we

• invert
• to

learn

X

predicates

9k .# HI

• f ! I:X ⇒ qµ , # =L

. But if

D

is uniform

• n

R

then Kiki , (D) =D

SLIDE 6

A simple solution

5

Break

X =L x

. . . . . . rn}

into X

'

• htt

.. . Kyl

X? { tinges

.

• -1×2%1

Answer

g.

CD )

by gilt

' )

:

%

by qdx

')

this

*

. . . .

• in }

ya by

quark)#

get

error

2

W

prob

I - p

I

need

n

z ful2%1

Can

we

do better ?

I ¥kln!Yf)

• -

SLIDE 7

Transfer theorem

Theorem Suppose M takes a dataset X and answers k adaptive queries q1, . . . , qk. If

• 1. ∀X ∈ X n, P(∃i : |qi(X) − M(X)i| > ↵) < ↵,
• 2. M is (↵, ↵)-DP,

then P(∃i : |M(X)i − qi(D)| > C↵) < C.

6

µ

answers

%

w/

UK)±

q, determined

from

tf!!

) ,

→ U answers willHk

by analysts

• →

U accurate

• n

the

dataset for

a

constant

C

µ

KD

"

X - D

"

⇐ X

• Six ,

. . - , rut

x ;

a D independently

SLIDE 8

Improving on the simple solution

Can get error ↵ with ≈

√k log k α2

samples.

7

Simple

solution ;

error

d with

a k¥431 22

Gaussian

noise t

advanced

composition

answer

q

.

ur

giant Zi

Zi

• N

'ns!¥iE

and

we

get

( e

, 81

• DP

for

any

d and

e=fi

Transfer

Him :

we

need

( d. did

• DP

g

= t g)

Std

dev per q,

is a F

=

sad

if

n

→ kTH

I

SLIDE 9

Key Lemma

Lemma Suppose W is (", )-DP, and on input X outputs a counting query q. Let X ∼ Dn. Then |E[q(D) | q = W(X)] − E[q(X) | q = W(X)]| ≤ eε − 1 + .

8

tell

"

q:&-3,4?!! distr

.

• u 't
• a Etf

t f

n

• ver

random

choice

• f

X - D

" "

E

and

randomness of

N

A

DP algorithm

cannot

find

a query

that

distinguishes

X

from

D

.

SLIDE 10

Proof of Key Lemma

E[q(X) | q = W(X)] = 1 n

n

X

i=1

E[q(xi) | q = W(X)] = 1 n

n

X

i=1

P(q(xi) | q = W(X))

9

quit

.

• th E. qui

'

q :&

→ so

,

I}

h

.

Take

4.

n D

independently

from everything

else .

X

'

• Sir.

. . . . , ri-nxi.xi.ie .

• yxn}

trudging

plqcx.it/q=WKHEfeElPlqcriiitlq--WK' 1) to

( E.tl

• DP
• f

W

SLIDE 11

Proof part 2

10

X - th

.

• - in }

qEon

:

( xi

, X '

)

has

the

so:L

.fm

.

• u

X 's 4h

. .

• - Hi , tie '

'

• - 'i'n }

as

( x !

, X )

plqcx.it/q=WlXl)EeElPlqcxiiitlq--WK' 1) to

quit

91¥49

""" =

e' Plgirittlq

• WH) ) + of

=

EE EIQCD

) Iq

.

• with

+ T

IE Iqlxllq

• WHY

E

e ' 14¥?

Efqctllq

• NIH) - IEIqcdllq-WIXHL.ee
• I

c- T

Z

• Cee- ly f )

analogous

SLIDE 12

SLIDE 13

Aside: Generalization from DP

Theorem For any non-negative loss `(✓, (x, y)), X = {(x1, y1), . . . , (xn, yn)} ∼ Dn, and LX(✓) = 1 n

n

X

i=1

`(✓, (xi, yi)) LD(✓) = E(x,y)∼D[`(✓, (x, y))], if ✓ is computed by an (", )-DP algorithm, then E[LD(✓)] ≤ eεE[LX(✓)] + max

θ,x,y `(✓, (x, y)). 11

→ Almost

the

same proof as

the

lemma

( exercise ) DP

pine

:*.

Population

loss

is

not

much

more

than

empirical

loss

for

DD algo

.

SLIDE 14

A simpler transference theorem

Theorem If the mechanism M satisfies that

• 1. ∀X ∈ X n, and all sequence of adaptive queries q1, . . . , qk,

E[maxi |qi(X) − M(X)i|] ≤ ↵

• 2. M is (", )-DP,

then E[max

i

|qi(D) − M(X)i|] ≤ ↵ + eε − 1 +

12

n

n

tf ,

t - Ik

• I

ate + of

q ,

,

. . . . %

are

adaptively

chosen

based

• n MHI . . -MAI

X-

D

"

SLIDE 15

Proof

Trick: Suppose that if qi is asked, so is 1 − qi, and is answered by 1 − M(X)i. Then maxk

i=1 |qi(D) − M(X)i| = maxk i=1 qi(D) − M(X)i. 13

qilx)

• I

⇐

I - qilxl

• O

q ok)

• O

H l

• gild

.

• I

lqil Dl

• lllxlil

=

max

9 gild

)

• Mahi, Mtk
• gild )}
• 11

I

• gild )
• ( t - Murli )

Define

w

set

.

it

x) simulates

it

• n

the

adaptive

M is c.ft

• Dp
• f
• prod

""'t queries

• h

, . . . , q,

I

r

,

⇒ w

is 1481

• Dp

H Outputs qi

sat . gild

) - NIH,

• - junk,atqjlD

)

Yi has

mat

error f

• UCH;

SLIDE 16

Proof pt 2

14

IE Hia qi

CD )

• U Chi
• E Iq

. ID)

• MIX ) ;

I qi

.

• WH))

= IE I q .

. ID )

• qi CHI qi

.

• WH))

N

e

'

• I to

1- IE E q .

• H )
• MkIi l qi
• NCH )

N

l

by lemma

⇐ enjoy qjkl

• Umd

:

E

e

E

• I tft L

.