Putting a Junta to the Test

Putting a Junta to the Test

Joint work with Eldar Fischer, Dana Ron, Shmuel Safra, and Alex Samorodnitsky

Guy KindlerGuy Kindler

Property TestingProperty Testing

o P – propertyo f – input

o Goal: Distinguish, using the fewest possible queries, between• f has P• f is -far from having P

d(f,g) = Prx[f(x)≠g(x)]

d(f,g) = Prx[f(x)≠g(x)]

HistoryHistory

o Testing Proofs (PCP): BLR

o Combinatorial properties: GGR

o PRS: Logic AND, monotonous DNF.

JuntasJuntas

Boolean Functions:

nf : { 1,1} { 1,1}

f( )=

n entries

JuntasJuntas

Boolean Functions:

1 -1 1 1 1 1 -1 -1 1 -1 -1 1 1 -1 -1 -1f( )=-1

nf : { 1,1} { 1,1}

Juntas

1 -1 1 1 1 1 -1 -1 1 -1 -1 1 1 -1 -1 -1f( )=-1

j-junta: depends on at most j coordinates.

1 -1 1 1 1 1 -1 -1 1 -1 -1 1 1 -1 -1 -1

-1

JuntasJuntas

1 -1 1 1 1 1 -1 -1 1 -1 -1 1 1 -1 -1 -1f( )=-1-1 1 1 1 -1 1 1

j-junta: depends on at most j coordinates.

-1

1 -1 1 1 1 1 -1 -1 -1-1 1 1 1 -1 1 1f( )=

Definition of j-Junta TestDefinition of j-Junta Test

1 -1 1 1 1 1 -1 -1 1 -1 -1 1 1 -1 -1 -1

1 1 1 -1 -1 1 -1 1 1 1 -1 1 1 1 1 1

-1 1 1 -1 -1 -1 1 -1 1 -1 1 -1 1 -1 -1 1

1 -1 -1 -1 1 -1 1 -1 1 1 1 -1 1 -1 1 -1

-1 -1 -1 1 -1 1 1 -1 1 1 -1 1 -1 1 -1 -1

f( )=

f( )=

f( )=

f( )=

-1

1

1

1

-1

1 -1 1 1 1 1 -1 -1 -1-1 1 1 1 -1 1 1f( )=1 -1 1 1 1 1 -1 -1 1 -1 -1 1 1 -1 -1 -1

1 1 1 -1 -1 1 -1 1 1 1 -1 1 1 1 1 1

-1 1 1 -1 -1 -1 1 -1 1 -1 1 -1 1 -1 -1 1

1 -1 -1 -1 1 -1 1 -1 1 1 1 -1 1 -1 1 -1

-1 -1 -1 1 -1 1 1 -1 1 1 -1 1 -1 1 -1 -1

f( )=

f( )=

f( )=

f( )=

-1

1

1

1

-1

Accept?Accept?

Reject?Reject?

Definition of j-Junta TestDefinition of j-Junta Test

Before we test juntas…

Given a set I of coordinates, can we verify that f does not depend on it?

Given a set I of coordinates, can we verify that f does not depend on it?

I-independence testI-independence test

I-independence testI-independence test

I

-1 1 -1 -1 1 -1 1 1 1 -1 1

-1 1 -1 -1 1 -1 1 1 1 -1 1

I-independence testI-independence test

I

w

f( )=

f( )=

-1 1 -1 -1 1 -1 1 1 1 -1 1

-1 1 -1 -1 1 -1 1 1 1 -1 1 1 1 -1 -1 1

-1 1 1 -1 -1

-1

1

I-independence testI-independence test

I

f( )=

wf( )=

-1 1 -1 -1 1 -1 1 1 1 -1 1

-1 1 -1 -1 1 -1 1 1 1 -1 1 1 1 -1 -1 1

-1 1 1 -1 -1

-1

1

Claim: If Pr[I is detected] ≤ then f is at most ”-dependent on I” g independent of I, d(f,g)≤

variationf(I)

Claim: If Pr[I is detected] ≤ then f is at most ”-dependent on I”

I-independence testI-independence test

I

f( )=

wf( )=

-1 1 -1 -1 1 -1 1 1 1 -1 1

-1 1 -1 -1 1 -1 1 1 1 -1 1 1 1 -1 -1 1

-1 1 1 -1 -1

-1

1

g independent of I, d(f,g)≤

variationf(I)

Claim: If Pr[I is detected] ≤ then f is at most ”-dependent on I”

I-independence testI-independence test

Proof: Let g(w z0)=Majz{f(wz)}

Define p(w)=Prz[f(wz)≠g(w z)]

variationf(I)

dist(f,g)

wE p(w)

1 2w,z ,z 1 2 Pr [f(w z ) f(w z )]

wE 2p(w)(1 p(w))

I

f( )=

wf( )=

-1 1 -1 -1 1 -1 1 1 1 -1 1

-1 1 -1 -1 1 -1 1 1 1 -1 1 1 1 -1 -1 1

-1 1 1 -1 -1

-1

1

g independent of I, d(f,g)≤

variationf(I)

1 2w z ,z 1 2E Pr [f(w z ) f(w z )]

w,zPr f(w z) g(w z)

1. Partition the coordinates into r subsets.

The j-Junta TestThe j-Junta Test

I1 Ir

r=10j2r=10j2

A j-junta is independent of all but j subsets !

A j-junta is independent of all but j subsets !

1. Partition the coordinates into r subsets.

2. Run the independence-test r/ times on each subset.

The j-Junta TestThe j-Junta Test

I1 Ir

If f has variation “/r” on a subset, it is almost surely detected!

If f has variation “/r” on a subset, it is almost surely detected!

1. Partition the coordinates into r subsets.

2. Run the independence-test r/ times on each subset.

3. Accept if ≤j of the subsets are detected.

The j-Junta TestThe j-Junta Test

I1 Ir

Completeness:Completeness:

Soundness:

If f is far from being a junta,

then the test rejects with

probability ½.

Soundness:

If f is far from being a junta,

then the test rejects with

probability ½.

Lemma: For every Boolean f, unless

f is –close to a j-junta,

w.h.p.,

the test rejects.

SoundnessSoundness

at least j+1 subsets have variation /r

[n], ||≤j, variationf([n]\)<

over the partitions of [n],

VariationsVariations

21f̂ (S )

22f̂ (S )

23f̂ (S )

24f̂ (S )

26f̂ (S )

27f̂ (S )

11

VariationsVariations

I

21f̂ (S )

22f̂ (S )

23f̂ (S )

24f̂ (S )

26f̂ (S )

27f̂ (S )

21f̂ (S )

22f̂ (S )

26f̂ (S )

VariationsVariations

I

21f̂ (S )

22f̂ (S )

23f̂ (S )

24f̂ (S )

26f̂ (S )

27f̂ (S )

f Ivariation

We’ll prove that unless

f is –close to a j-junta,

w.h.p.

the test rejects.at least j+1 subsets have variation

/r

||≤j, and variationf([n]\)<

over the partitions of [n],

For t /r, let fi | i tvariationJ

I1 Ir

If ||>j, easy !!If ||>j, easy !!

We’ll prove that unless

f is –close to a j-junta,

w.h.p.

the test rejects.at least j+1 subsets have variation

/r

||≤j, and variationf([n]\)<

over the partitions of [n],

Fix t /r and let ii | f t variationJ

I1 Ir

Assume variationf([n]\)>.

Then !!!

Assume variationf([n]\)>.

Then !!!

mf[ (I )] / rE variation

Claim: w.h.p. mf (I \ ) / 2rvariation J

mi I ]=1/ r=J , Pr[o Recall: For each i in

Claim: w.h.p. mf (I \ ) / 2rvariation J

I

21f̂ (S )

22f̂ (S )

23f̂ (S )

24f̂ (S )

26f̂ (S )

27f̂ (S )

mi I ]=1/ r=J , Pr[o Recall: For each i in

Claim: w.h.p. mf (I \ ) / 2rvariation J

23f̂ (S )

I

The Unique-VariationThe Unique-Variation

21f̂ (S )

22f̂ (S )

23f̂ (S )

24f̂ (S )

26f̂ (S )

27f̂ (S )

I

The Unique-VariationThe Unique-Variation

21f̂ (S )

22f̂ (S )

23f̂ (S )

24f̂ (S )

26f̂ (S )

27f̂ (S )

I

The Unique-VariationThe Unique-Variation

21f̂ (S )

22f̂ (S )

23f̂ (S )

24f̂ (S )

26f̂ (S )

27f̂ (S )

21f̂ (S )

22f̂ (S )

23f̂ (S )

24f̂ (S )

26f̂ (S )

27f̂ (S )

22f̂ (S )

I

The Unique-VariationThe Unique-Variation

21f̂ (S )

22f̂ (S )

23f̂ (S )

24f̂ (S )

26f̂ (S )

27f̂ (S )

21f̂ (S )

22f̂ (S )

23f̂ (S )

24f̂ (S )

26f̂ (S )

27f̂ (S )

22f̂ (S )

I fu_variation

I

The Unique-VariationThe Unique-Variation

21f̂ (S )

22f̂ (S )

23f̂ (S )

24f̂ (S )

26f̂ (S )

27f̂ (S )

21f̂ (S )

22f̂ (S )

23f̂ (S )

24f̂ (S )

26f̂ (S )

27f̂ (S )

22f̂ (S )

I fu_variation

ff u_variation variationJ J

I

The Unique-VariationThe Unique-Variation

21f̂ (S )

22f̂ (S )

23f̂ (S )

24f̂ (S )

26f̂ (S )

27f̂ (S )

21f̂ (S )

22f̂ (S )

23f̂ (S )

24f̂ (S )

26f̂ (S )

27f̂ (S )

22f̂ (S )

I fu_variation

mI f / r E u_variation

I

The Unique-VariationThe Unique-Variation

21f̂ (S )

22f̂ (S )

23f̂ (S )

24f̂ (S )

26f̂ (S )

27f̂ (S )

21f̂ (S )

22f̂ (S )

23f̂ (S )

24f̂ (S )

26f̂ (S )

27f̂ (S )

22f̂ (S )

I fu_variation

mI with high probability !2r !f /variation

I

The Unique-VariationThe Unique-Variation

21f̂ (S )

22f̂ (S )

23f̂ (S )

24f̂ (S )

26f̂ (S )

27f̂ (S )

21f̂ (S )

22f̂ (S )

23f̂ (S )

24f̂ (S )

26f̂ (S )

27f̂ (S )

22f̂ (S )

I fu_variation

mI with high probability !2r !f /variation

Q.E.DQ.E.D

Other ResultsOther Results

o Shown number of queries: j4/o Using adaptivity: j3/o Using two-sidedness: j2/ o Allowing (2j)-juntas: j2/ o Variables in General probability

spaces.o “f” is “g” test, where g is a j-junta.o Lower Bound: at least (j)1/2 queries are

needed

Open ProblemsOpen Problems

o Improve lower bound to j2/(perhaps via random-walk convergence on Z2)

o “f is g” for non-juntas?

o Characterize efficiently testable properties via Fourier transform??