11

Noise-Insensitive Noise-Insensitive Boolean-Functions Boolean-Functions

are Juntasare Juntas

Guy Kindler & Muli SafraGuy Kindler & Muli Safra

Slides prepared with help of: Adi AkaviaSlides prepared with help of: Adi Akavia

22

InfluentialInfluential People People The theory of the The theory of the InfluenceInfluence of of

Variables on Boolean FunctionsVariables on Boolean Functions [KKL,BL,R,M][KKL,BL,R,M] and related issues, has and related issues, has been introduced to tackle been introduced to tackle social choicesocial choice problems. This area has motivated a problems. This area has motivated a magnificent sequence of works, magnificent sequence of works, related to Erelated to Economicsconomics [K],[K], percolationpercolation [BKS],[BKS], Hardness of ApproximationHardness of Approximation [DS][DS]Revolving around the Revolving around the Fourier/Walsh Fourier/Walsh analysis of Boolean functionsanalysis of Boolean functions… …

And the real important question:And the real important question:

33

Where to go for Dinner?Where to go for Dinner?

The The alternativesalternatives

Diners would cast their vote in Diners would cast their vote in an (electronic) envelope.an (electronic) envelope.The system would decide –The system would decide –not necessarily by majority…not necessarily by majority…

It turns out someone –in the It turns out someone –in the Florida wing- has the ability Florida wing- has the ability to flip some votesto flip some votes

PowerPower

influenceinfluence

44

Voting SystemsVoting Systems nn agents, each voting either “for” ( agents, each voting either “for” (TT) )

or “against” (or “against” (FF) – a Boolean function ) – a Boolean function over over nn variables variables ff is the outcome is the outcome

The values of the agents (variables) The values of the agents (variables) may each, independently, flip with may each, independently, flip with probability probability

Bottom LineBottom Line: one cannot design an : one cannot design an ff that would be robust to such noise --that would be robust to such noise --that is, would, on average, change that is, would, on average, change value w.p.value w.p.< < O(1)O(1)-- unless taking into account -- unless taking into account only very few of the votesonly very few of the votes

55

DictatorshipDictatorship

DefDef: a Boolean function : a Boolean function P([n])P([n]){-1,1}{-1,1} is is a a monotone monotone ee--dictatorshipsdictatorships --denoted --denoted ffee--if:--if:

e

T e xf x

F e x

e

T e xf x

F e x

66

JuntasJuntas

DefDef: a Boolean function : a Boolean function f:P([n])f:P([n]){-1,1}{-1,1} is a is a jj--JuntaJunta if if JJ[n] [n] wherewhere |J|≤ j|J|≤ j, , s.t. for every s.t. for every xxP([n])P([n]), , f(x) = f(x f(x) = f(x J) J)

DefDef: : ff is an is an [[, j], j]--JuntaJunta if if j- j-Junta Junta f’f’ s.t. s.t.

DefDef: : ff is an is an [[, j, p], j, p]--JuntaJunta if if j- j-Junta Junta f’f’ s.t. s.t.

nx~U

f x f ' xPr nx~U

f x f ' xPr

px~

f x f ' xPr

px~

f x f ' xPr

We would tend to omit p

p-biased, product distribution

77

Long-CodeLong-Code In the long-code In the long-code L:[n]L:[n] {0,1} {0,1}22nn

each each element is encoded by an element is encoded by an 22nn-bits-bits

This is the most extensive binary code, This is the most extensive binary code, having one bit for every subset in having one bit for every subset in P([n])P([n])

88

Long-CodeLong-Code Encoding an element Encoding an element ee[n][n]:: EEee legally-encodeslegally-encodes an element an element ee if if EEee = f = fee

FF FF TT TT TT

99

Long-Code Long-Code Monotone- Monotone-DictatorshipDictatorship

The truth-table of a Boolean function The truth-table of a Boolean function over over nn elements, can be considered elements, can be considered as a as a 22nn bits long string (each bits long string (each corresponding to one input setting – corresponding to one input setting – or a subset of or a subset of [n][n]))

For a long-code, the legal code-words For a long-code, the legal code-words are all monotone dictatorshipsare all monotone dictatorships

How about the Hadamard code?How about the Hadamard code?

1010

Long-code Tests Long-code Tests Def Def (a (a long-code testlong-code test): given a code-): given a code-

word word ww, probe it in a constant number , probe it in a constant number of entries, andof entries, and accept w.h.p if accept w.h.p if ww is a monotone is a monotone

dictatorshipdictatorship reject w.h.p if reject w.h.p if ww is not close to any is not close to any

monotone dictatorshipmonotone dictatorship

1111

Efficient Long-code TestsEfficient Long-code Tests

For some applications, it suffices if the test For some applications, it suffices if the test may accept illegal code-words, nevertheless, may accept illegal code-words, nevertheless, ones which have short ones which have short list-decodinglist-decoding::

DefDef(a (a long-code list-testlong-code list-test): given a code-word ): given a code-word ww, probe it in 2 or 3 places, and, probe it in 2 or 3 places, and accept w.h.p if accept w.h.p if w w is a monotone dictatorship,is a monotone dictatorship, reject w.h.p if reject w.h.p if ww is not evenis not even approximately approximately

determined by a short list of domain elementsdetermined by a short list of domain elements

that is, if that is, if a a JuntaJunta JJ[n][n] s.t. s.t. f f is close to is close to f’ f’ and and f’(x)=f’(xf’(x)=f’(xJ) J) for allfor all x x

NoteNote: a long-code list-test, distinguishes between the case : a long-code list-test, distinguishes between the case ww is a is a dictatorshipdictatorship, to the case , to the case ww is far from a is far from a juntajunta..

1313

BackgroundBackground Thm (Friedgut)Thm (Friedgut): a Boolean function : a Boolean function ff with small with small

average-sensitivityaverage-sensitivity is an is an [[,j]-,j]-juntajunta

Thm (Bourgain)Thm (Bourgain): a Boolean function : a Boolean function f f with small with small high-high-frequency weight frequency weight is an is an [[,j]-,j]-juntajunta

ThmThm: a Boolean function : a Boolean function f f with small with small high-frequency high-frequency weight weight in a in a pp--biased biased measure is an measure is an [[,j]-,j]-juntajunta

CorollaryCorollary: a Boolean function : a Boolean function f f with with smallsmall noise-noise-sensitivitysensitivity is an is an [[,j]-,j]-juntajunta

ParametersParameters: : average-sensitivity average-sensitivity [M,R,BL,KKL,F] [M,R,BL,KKL,F] high-frequency weight high-frequency weight [KKL,B][KKL,B]noise-sensitivity noise-sensitivity [BKS][BKS]

1414

[n][n]x

IIz

[n][n]

Noise-SensitivityNoise-Sensitivity

How often does the value of How often does the value of ff changes changes when the input is perturbed?when the input is perturbed?

x

IIz

1515

DefDef((,p,x,p,x[n] [n] ): Let ): Let 0<0<<1<1, and , and xxP([n])P([n]). .

Then Then y~y~,p,x,p,x, if , if y = (x\I)y = (x\I) z z where where I~I~

[n][n] is a is a noise subsetnoise subset, and, and z~ z~ pp

II is a is a replacementreplacement..

DefDef((-noise-sensitivity-noise-sensitivity): let ): let 0<0<<1<1, then, then

[ When [ When p=½p=½ equivalent to flipping each equivalent to flipping each coordinate in coordinate in xx w.p. w.p. /2/2.].]

[n] [n]p ,p,xx~ ,y~

ns f = Pr f x f y

[n] [n]p ,p,xx~ ,y~

ns f = Pr f x f y

[n] [n] xIIz

Noise-SensitivityNoise-Sensitivity

1616

Fourier/Walsh TransformFourier/Walsh Transform

Write Write f:{-1, 1}f:{-1, 1}nn{-1, 1}{-1, 1} as a polynomial as a polynomial

What would be the monomials?What would be the monomials?

For every set For every set SS[n][n] we have a monomial which we have a monomial which is the product of all variables in is the product of all variables in SS (the only (the only relevant powers are either relevant powers are either 00 or or 11))

It now makes sense to consider the degree of It now makes sense to consider the degree of ff or to break it or to break it according to the various degrees of the monomials..according to the various degrees of the monomials..

( )[ ]

S

S n

ff S cÍ

= å ( )[ ]

S

S n

ff S cÍ

= å

1717

High/Low FrequenciesHigh/Low Frequencies

DefDef: the : the high-frequencyhigh-frequency portion of portion of ff::

DefDef: the : the low-frequency low-frequency portion of portion of ff: :

DefDef: the : the high-frequency-weighthigh-frequency-weight is: is:

DefDef: the : the low-frequency-weightlow-frequency-weight is: is:

kS

S k

ff S

kS

S k

ff S

kS

S k

ff S

kS

S k

ff S

22k

2S k

ff S

22k

2S k

ff S

22k

2S k

ff S

22k

2S k

ff S

1818

Low High-Frequency WeightLow High-Frequency WeightPropProp: the : the -noise-sensitivity can be expressed in -noise-sensitivity can be expressed in

Fourier transform terms asFourier transform terms as

PropProp: Low : Low nsns Low Low high-freq weighthigh-freq weightProofProof: By the above proposition, low noise-sensitivity : By the above proposition, low noise-sensitivity

impliesimplies

nevertheless, nevertheless, ff being being {-1, 1}{-1, 1} function, by Parseval function, by Parseval formula (that the formula (that the norm 2norm 2 of the function and its of the function and its Fourier transform are equal) impliesFourier transform are equal) implies

( ) ( )2S

S

1 f S ~1l-å ( ) ( )2S

S

1 f S ~1l-å

2

S

f S 1 2

S

f S 1

2S

S

2 ns f =1 1 f S 2S

S

2 ns f =1 1 f S

1919

Average and RestrictionAverage and Restriction

DefDef: Let : Let II[n],[n], xxP([n]\I)P([n]\I), , the the restriction function restriction function isis

DefDef: the : the average function average function isis

NoteNote::

I

Iy P I

A f : P I

A f x E f x y

I

Iy P I

A f : P I

A f x E f x y

I

I

f x : P I 1,1

f x y f x y

I

I

f x : P I 1,1

f x y f x y

I Iy P I

A f x E f x y

I Iy P I

A f x E f x y

[n]I

x

y

[n]I

x

y y

y yy

2020

Fourier ExpansionFourier Expansion PropProp: :

PropProp:: I SS I

A ff (S)

I S

S I

A ff (S)

I STS I T I S

f x f T x

I STS I T I S

f x f T x

2121

InfluenceInfluence/Variation/Variation

DefDef: the : the variation variation of of II on on ff::

PropProp: the following are equivalent : the following are equivalent definitions to the definitions to the variation variation of of II on on ff::

22

I I 2S I

ff A ff S

variation 22

I I 2S I

ff A ff S

variation

I Iy P Ix P I

f E var f x y

variation

I I

y P Ix P If E var f x y

variation

Influencei(f) = variationi(f) = variation{i}(f)

2525

Low-frequencies Variation and Low-frequencies Variation and a.s.a.s.

DefDef: the : the low-frequency variationlow-frequency variation is: is:

DefDef:: the the average-sensitivity average-sensitivity isis

And in Fourier representation:And in Fourier representation:

DefDef: the : the low-frequency average-sensitivity low-frequency average-sensitivity is:is:

ii [n]

ff

as variation ii [n]

ff

as variation

2

S

ff (S) S as 2

S

ff (S) S as

2

k kI I

S IS k

ff f S

variation variation 2

k kI I

S IS k

ff f S

variation variation

2

i [n] S k

ff f (S) S

k kias variation

2

i [n] S k

ff f (S) S

k kias variation

2626

Biased Walsh Product Biased Walsh Product [Talagrand][Talagrand]

DefDef: In the : In the pp-biased product distribution -biased product distribution pp, , the probability of a subset the probability of a subset x x isis

The usual The usual Fourier basis Fourier basis is not orthogonal is not orthogonal with respect to the with respect to the biased inner-product,biased inner-product,

Hence, we use the Hence, we use the Biased Walsh ProductBiased Walsh Product::

x n xnp x p (1 p) x n xnp x p (1 p)

p1 p i x

xi x1 p

p

i

p1 p i x

xi x1 p

p

i

ii S

x x

S ii S

x x

S

2727

Main ResultMain Result

TheoremTheorem: :

constant constant >0>0 s.t. any Boolean function s.t. any Boolean function

f:P([n])f:P([n]){-1,1}{-1,1} satisfying satisfying

is an is an [[,j]-junta ,j]-junta for for j=O(j=O(-2-2kk332k2k))..

CorollaryCorollary: :

fix a fix a pp-biased distribution -biased distribution pp over over P([n])P([n]). .

Let Let >0>0 be any parameter. be any parameter.

Set Set k=logk=log1-1-(1/2)(1/2). .

Then Then constant constant >0>0 s.t. any Boolean function s.t. any Boolean function

f:P([n])f:P([n]){-1,1}{-1,1} satisfying satisfying

is an is an [[,j]-junta ,j]-junta for for j=O(j=O(-2-2kk332k2k))..

2k22

f Ok

2k22

f Ok

2ns f O k 2ns f O k

2828

The The KKL/FreidgutKKL/Freidgut Framework Framework

ThmThm: any Boolean function : any Boolean function ff is an is an [[,j]-,j]-junta junta for for

ProofProof::1.1. Specify the juntaSpecify the junta

where, let where, let k=O(as(f)/k=O(as(f)/)) and fix and fix =2=2-O(k)-O(k)

2.2. Show the complement of Show the complement of JJ has small variation has small variation

f / O asj = 2 f / O asj = 2

iJ i | f variation iJ i | f variation

[n]

J

3030

KKL/FreidgutKKL/FreidgutLemmaLemma::

ProofProof: :

Now, lets bound each argument:Now, lets bound each argument:

PropProp[KKL][KKL]: :

ProofProof: characters of size : characters of size kk contribute to contribute to the the average-sensitivityaverage-sensitivity at least at least (since(since ))

2k

2

as ff k 2k

2

as ff k

Jf 2

variation Jf 2

variation

2k kJ J 2

ff f variation variation 2k kJ J 2

ff f variation variation

[n]

J

2k

2k f

2k

2k f

2

S

as ff S S 2

S

as ff S S

3232

Beckner/Nelson/BonamiBeckner/Nelson/Bonami InequalityInequality

DefDef: let : let TT be the following operator on be the following operator on ff

ThmThm: for any : for any p≥rp≥r andand ≤((r-1)/(p-1))≤((r-1)/(p-1))½½

CorollaryCorollary: for : for gg of degree of degree kk

1 ,p,xy

f x f yE

T 1 ,p,xy

f x f yE

T

rpff T

rpff T

4 44k

4 2g g

4 44k

4 2g g

3434

k

i J

2 2

O(k)S S

i S,S k i S,S ki J i J2 r

2 4/ r

O(k) O(k)S S

i S i Si J i Jr 2

22/ rO(k)

kJ

O(k) r

i J

f

f (S) 2 f(S)

2 f(S) 2 f(S)

as f2 f 2

f

i

iinfluenc

variation vari on

e

ati

k

i J

2 2

O(k)S S

i S,S k i S,S ki J i J2 r

2 4/ r

O(k) O(k)S S

i S i Si J i Jr 2

22/ rO(k)

kJ

O(k) r

i J

f

f (S) 2 f(S)

2 f(S) 2 f(S)

as f2 f 2

f

i

iinfluenc

variation vari on

e

ati

FreidgutFreidgut’s Proof’s ProofPropProp::

ProofProof:: k

Jf 4

variation kJ

f 4 variation

we do not know whether as(f) is

small!

True only since this is a {-1,0,1} function.

So we cannot proceed this way with only this way with only asaskk!!

3535

If If kk were 1 were 1

Easy caseEasy case (!?!): If we’d have a bound on the (!?!): If we’d have a bound on the non-linear weight, we should be done.non-linear weight, we should be done.

The linear part is a set of independent The linear part is a set of independent characters (the singletons)characters (the singletons)

Concentration of measureConcentration of measure: In order for those : In order for those to hit close to to hit close to 11 or or -1-1 most of the time, they most of the time, they must avoid the law of large numbers, must avoid the law of large numbers, namely be almost entirely placed on one namely be almost entirely placed on one singleton [by Chernoff like bound]singleton [by Chernoff like bound]

(!) (!) [FKN, ext.] [FKN, ext.] if if ff is close to is close to linear linear then then ff is is close to close to shallowshallow ( ( a constant function or a a constant function or a dictatorship) dictatorship)

3636

Almost Linear Almost Linear Almost Almost ShallowShallow

Thm(Thm([FKN][FKN])): : global constant global constant MM, , s.t. s.t. Boolean function Boolean function ff, , shallow shallow Boolean function Boolean function gg, s.t. , s.t.

Hence, Hence, ||f||fII[x][x]>1>1||||22 is small is small ffII[x][x] is is close to close to shallowshallow!!

22 1

2 2f g M f

22 1

2 2f g M f

3737

How to Deal with Dependency How to Deal with Dependency between Characters?between Characters?

RecallRecall

(theorem’s premise)(theorem’s premise)

IdeaIdea: Let: Let Partition Partition [n]\J[n]\J into into II11,…,I,…,Irr, for , for r >> kr >> k w.h.p w.h.p ffII[x][x] is close to is close to linearlinear (low freq (low freq

characters intersect characters intersect II expectedly by expectedly by 11 element, while high-frequency weight is low).element, while high-frequency weight is low).

2k kJ J2

ff f variation + variation 2k kJ J2

ff f variation + variation

2k22

1f Ok

2k22

1f Ok

J i | f kivariation J i | f kivariation

[n]

J

I1

I2IrI

3838

So what?So what?

ffII[x][x] is close to is close to linearlinear

By By [FKN][FKN],, ffII[x][x] is shallow for any is shallow for any xx

Still, Still, ffII[x][x] could be a different could be a different dictatorship for different dictatorship for different xx’s, hence ’s, hence the variation of each the variation of each iiII might be might be low!!low!!

P([n])

J

I1

I2IrI

3939

Dictatorship and its Dictatorship and its SingletonSingleton

PropProp: for a dictatorship : for a dictatorship hh, , coordinate coordinate ii s.t.s.t. (where (where pp is the bias). is the bias).

Corollary (from [FKN])Corollary (from [FKN]): : global constant global constant MM, s.t. , s.t. Boolean function Boolean function hh, either, eitheror or

{}( )h i p> {}( )h i p>

h i p h i p

21n 2

h M hvariation 21n 2

h M hvariation

{1} {2} {i} {n} {1,2} {1,3} {n-1,n} S {1,..,n}

weight

Characters

Total weight of no more than Total weight of no more than 1-p1-p

4040

Main LemmaMain Lemma LemmaLemma: : >0>0, s.t. for any , s.t. for any and any and any

function function g:P([m])g:P([m]) , the following , the following holds: holds:

m

4 24k k k

2 2x~Pr g x O g g

m

4 24k k k

2 2x~Pr g x O g g

Low-freq high-freq

4141

Probability ConcentrationProbability Concentration Simple BoundSimple Bound:: ProofProof::

Low-freq BoundLow-freq Bound: Let : Let g:P([m])g:P([m]) be be of degree of degree kk and and >0>0, then , then >0>0 s.t. s.t.

ProofProof: recall the corollary:: recall the corollary:

m

tt

tx~Pr g x g

m

tt

tx~Pr g x g

m

44 4k

2x~Pr g x g

m

44 4k

2x~Pr g x g

4 44k

4 2g g

4 44k

4 2g g

4242

Lemma’s ProofLemma’s Proof Now, let’s prove the lemma: Now, let’s prove the lemma: Bounding low and high freq Bounding low and high freq

separately:separately:,,

simple bound

4 224 4k k k

2 2g g

4 224 4k k k

2 2g g

m

m m

x~

k k

x~ x~

Pr g x

Pr g x Pr g x

m

m m

x~

k k

x~ x~

Pr g x

Pr g x Pr g x

Low-freq bound

4343

ffII[x][x] Mostly Constant Mostly Constant

LemmaLemma: : >0>0, s.t. for any , s.t. for any and any and any function function g:P([m])g:P([m])

DefDef: Let : Let DDII be the set of be the set of xxP(P(II)), s.t. , s.t. ffII[x][x] is a dictatorship is a dictatorship

Next we show, that Next we show, that |D|DII|| must be small, must be small, hence for most hence for most xx, , ffII[x][x] is constant. is constant.

I ID x P I : i I ,s.t. f x i p I ID x P I : i I ,s.t. f x i p

m

4 24k k k1 22 2x~

Pr g x M g M g

m

4 24k k k1 22 2x~

Pr g x M g M g

4444

LemmaLemma:: ProofProof: denote: denote , then, then

|D|DII|| must be small must be small

[n]

24k kI 1 2 2x~

Pr x D M M f

[n]

24k kI 1 2 2x~

Pr x D M M f

i Ig x f x i i Ig x f x i

[n]

i i

I ix~ x P Ii I

4 24k k k1 22 2

i I i I

4 2

4k1 S 2 S

i I S [n],S k,S I i i I S [n],S k,S I i2 2

22 24k k

1 2 2i I S [n],S k,S I i i I

Pr x D Pr g x p

M g M g

M f S M f S

M f S M f

[n]

i i

I ix~ x P Ii I

4 24k k k1 22 2

i I i I

4 2

4k1 S 2 S

i I S [n],S k,S I i i I S [n],S k,S I i2 2

22 24k k

1 2 2i I S [n],S k,S I i i I

Pr x D Pr g x p

M g M g

M f S M f S

M f S M f

Prev lemma

I TT [n],T I S

f x S f T

I TT [n],T I S

f x S f T

Each S is counted only for one index iI.

(Otherwise, if S was counted for both i and j

in I, then |SI|>1!)

Parseval

4545

Simple PropSimple Prop PropProp: let : let {a{aii}}iiII be be sub-distributionsub-distribution, ,

that is, that is, iiIIaaii11, , 00aaii, then , then iiIIaaii

22maxmaxiiII{a{aii}}.. ProofProof::

2 2i max max

maxi I

1a a aa

2 2i max max

maxi I

1a a aa

1 2 3 max n

ai no more than no more than 11

1

1 2 3 n

ai

1/a1/amaxmax

1

4646

|D|DII|| must be small - Cont must be small - Cont ThereforeTherefore

(since(since ), ),

HenceHence

2

2 2k

i I i Ii I S [n],S k, S [n],S k,S I i S I i

f S max f S max f

ivariation

2

2 2k

i I i Ii I S [n],S k, S [n],S k,S I i S I i

f S max f S max f

ivariation

2

S [n],S k,S I i

f S 1

2

S [n],S k,S I i

f S 1

[n]

24k kI 1 2 2x~

Pr x D M M f

[n]

24k kI 1 2 2x~

Pr x D M M f

5151

Where to go for Dinner?Where to go for Dinner?

The The alternativesalternatives

Diners would cast their vote in Diners would cast their vote in an (electronic) envelope.an (electronic) envelope.The system would decide –The system would decide –not necessarily by majority…not necessarily by majority…

It turns out someone –in the It turns out someone –in the Florida wing- has the ability Florida wing- has the ability to flip some votesto flip some votes

PowerPower

influenceinfluence

Of course they’ll have to discuss it

over dinner….

5252

DiscussionDiscussion Tests that look at only 2 or 3 places Tests that look at only 2 or 3 places

cannot produce a large gap between cannot produce a large gap between probability of acceptance of a probability of acceptance of a dictatorship and that of a function not dictatorship and that of a function not so close to a juntaso close to a junta

Nevertheless, if requiring the function Nevertheless, if requiring the function to have additional properties, such as to have additional properties, such as local-maximality, one may be able to local-maximality, one may be able to design a test with a large gapdesign a test with a large gap

5353

Shallow FunctionShallow Function DefDef: a function : a function ff is is linearlinear, if only singletons , if only singletons

have non-zero weight have non-zero weight DefDef: a function : a function ff is is shallowshallow, if , if ff is either a is either a

constant or a dictatorship.constant or a dictatorship. ClaimClaim: Boolean linear functions are shallow.: Boolean linear functions are shallow.

0 1 2 3 k n

weight

Charactersize

5454

Boolean Linear Boolean Linear Shallow Shallow ClaimClaim: Boolean linear functions are : Boolean linear functions are

shallow.shallow. ProofProof: let : let ff be Boolean linear be Boolean linear

function, we next show:function, we next show:1.1. {i{ioo}} s.t. s.t.

((i.e.i.e. ))

2.2. And conclude, that eitherAnd conclude, that either or or i.e.i.e. ff is shallow is shallow

0S , i ,f S 0 0S , i ,f S 0 00 iff fi 00 iff fi

ff ff 00 iffi 00 iffi

5555

Claim 1Claim 1 Claim 1Claim 1: let : let ff be boolean linear be boolean linear

function, then function, then {i{ioo}} s.t. s.t. ProofProof: w.l.o.g assume: w.l.o.g assume

for any for any zz{3,…,n}{3,…,n}, consider , consider xx0000=z=z, , xx1010=z=z{1}{1}, , xx0101=z=z{2}{2}, , xx1111=z=z{1,2}{1,2}

thenthen .. Next value must be far from Next value must be far from {-1,1}{-1,1},, A contradiction! (boolean function) A contradiction! (boolean function) ThereforeTherefore

00 iff fi 00 iff fi

f 1 f 2 0 f 1 f 2 0

ab a'b'a,b a' ,b' : f x f x min f 1 , f 2 ab a'b'a,b a' ,b' : f x f x min f 1 , f 2

ab a'b'

ab a'b'1 1

ab a'b'2 2

f x f x

f 1 x x

f 2 x x

ab a'b'

ab a'b'1 1

ab a'b'2 2

f x f x

f 1 x x

f 2 x x

f 2 0 f 2 01

-1

?

5757

Claim 2Claim 2 Claim 2Claim 2: let : let ff be boolean function, s.t. be boolean function, s.t.

Then eitherThen either or or ProofProof: consider : consider f(f()) and and f(if(i00))::

ThenThen but but ff is boolean, hence is boolean, hence thereforetherefore

00 iff fi 00 iff fi

ff ff 00 iffi 00 iffi

0

0 0

ff fi

fi ffi

0

0 0

ff fi

fi ffi

0 0fi f 2 fi 0 0fi f 2 fi

0fi 0,1 0fi 0,1 0fi f 0,2 0fi f 0,2

1

-1

0 f f

0fi 0fi

0fi 0fi

5959

Proving FKN: Proving FKN: almost-linear almost-linear close to close to

shallowshallow TheoremTheorem: Let : Let f:P([n])f:P([n]) be be linearlinear, ,

LetLet let let ii00 be the index s.t. is maximal be the index s.t. is maximal

then then

NoteNote: : ff is is linearlinear, hence, hencew.l.o.g., assume w.l.o.g., assume ii00=1=1, then all we need to , then all we need to

show is:show is:

We show that in the following claim and We show that in the following claim and lemma.lemma.

0fi 0fi

2

2f 1

2

2f 1

0

2

0 i2

ff fi 1 o 1

0

2

0 i2

ff fi 1 o 1

n

ii 1

ff fi

n

ii 1

ff fi

n 2

i 2

fi 1 o 1

n 2

i 2

fi 1 o 1

6060

CorollaryCorollary CorollaryCorollary: Let : Let ff be linear, and be linear, and

then then a a shallow booleanshallow boolean function function gg s.t.s.t.

ProofProof: let: let , let , let gg be the be the boolean function closest to boolean function closest to ll. . Then,Then,this is true, as this is true, as is small (by theorem),is small (by theorem), and additionallyand additionally is small, sinceis small, since

2f g 3 o 1 2f g 3 o 1

0ffi 0ffi

2

2f g 9 o 1 2

2f g 9 o 1

2

2f 1

2

2f 1

2l g

2l g

2fl

2fl

2

2f 1

2

2f 1

6161

Claim 1Claim 1 Claim 1Claim 1: Let : Let f f be linear. be linear.

w.l.o.g., assumew.l.o.g., assumethen then global constant global constant c=min{p,1-p}c=min{p,1-p} s.t. s.t. i 2,...,n: fi c i 2,...,n: fi c

f 1 f 2 ... f n f 1 f 2 ... f n

{} {1} {2} {i} {n} {1,2} {1,3} {n-1,n} S {1,..,n}

weight

Characters

Each of weight no more than Each of weight no more than cc

6262

Proof of Claim1Proof of Claim1 ProofProof: assume: assume

for any for any zz{3,…,n}{3,…,n}, consider , consider xx0000=z=z, , xx1010=z=z{1}{1}, , xx0101=z=z{2}{2}, , xx1111=z=z{1,2}{1,2}

thenthen Next value must be far from Next value must be far from {-1,1} {-1,1} !! A contradiction! (toA contradiction! (to ))

2

2f 1

2

2f 1

f 2 c f 2 c

ab a'b'a,b a' ,b' : f x f x min f 1 , f 2 c ab a'b'a,b a' ,b' : f x f x min f 1 , f 2 c

ab a'b'

ab a'b'1 1

ab a'b'2 2

f x f x

f 1 x x

f 2 x x

ab a'b'

ab a'b'1 1

ab a'b'2 2

f x f x

f 1 x x

f 2 x x

1

-1

?

6464

LemmaLemma LemmaLemma: Let : Let g g be linear, let be linear, let

assumeassume , then , then CorrolaryCorrolary: The theorem follows from : The theorem follows from

the combination of the combination of claim1claim1 and the and the lemmalemma:: Let Let mm be the minimal index s.t. be the minimal index s.t. ConsiderConsider If If m=2m=2: the theorem is obtained (by : the theorem is obtained (by

lemma)lemma) Otherwise -- a contradiction to Otherwise -- a contradiction to

minimality of minimality of m m ::

20

2g c

20

2g c 20

2g 1 o 1 20

2g 1 o 1

n 2

i m

fi c 2

n 2

i m

fi c 2

n 2

i m

g b fi

n 2

i m

g b fi

note n 220

2i m

g fi

n 220

2i m

g fi

1

n 2

mi

fi c 1 o 1 c o 1

1

n 2

mi

fi c 1 o 1 c o 1

2

2g 1

2

2g 1

6565

Lemma’s ProofLemma’s Proof Lemma’s ProofLemma’s Proof: Note: Note

Hence, all we need to show is that Hence, all we need to show is that

Intuition:Intuition: Note that Note that |g||g| and and |b||b| are far from are far from 00

(since (since |g||g| is is -close to -close to 11, and , and cc-close to -close to bb).). AssumeAssume b>0b>0, then for almost all inputs , then for almost all inputs xx, ,

g(x)=|g(x)| g(x)=|g(x)| (as(as ) ) Hence Hence E[g] E[g] E[|g(x)|] E[|g(x)|], and , and therefore therefore var(g) var(g) var(|g|) var(|g|)

20

2g var g 20

2g var g

2

2var g g 1 2

2var g g 1

var g var g var g var g

20

2g c

20

2g c

6666

EE22[g] - E[g] - E22[|g|] = 2E[|g|] = 2E22[|g|1[|g|1{f<0}{f<0}] ] o( o() ) (by Azuma’s inequality)(by Azuma’s inequality)

We next show We next show var(g) var(g) var(|g|) var(|g|):: By the premiseBy the premise however however

thereforetherefore

Proof-mapProof-map::

|g|,|b| are far from 0|g|,|b| are far from 0

g(x)=|g(x)| for almost all xg(x)=|g(x)| for almost all x

E[g] E[g] E[|g|] E[|g|]

var(g) var(g) var(|g|) var(|g|)

var g var g

var g var g o var g var g o

2 2 2 2

2var g g E g var g E g E g 2 2 2 2

2var g g E g var g E g E g

6767

Variation LemmaVariation Lemma

LemmaLemma(variation): (variation): >0>0, and , and r>>k r>>k s.t.s.t.

CorollaryCorollary: for most : for most II and and xx, , ffII[x][x] is almost is almost constantconstant

J1r

24k kII ~ 2

E f O f

variation J1r

24k kII ~ 2

E f O f

variation

P([n])

J

I1

I2IrI

I Iy Ix P I

f E var f x y

variation

I I

y Ix P If E var f x y

variation

6868

By union bound on By union bound on II11,…,I,…,Irr: :

(set(set ))

Let Let f’(x) = sign( Af’(x) = sign( AJJ[f](x[f](xJ) )J) ). . f’f’ is the boolean function closest to is the boolean function closest to AAJJ[f][f], , thereforetherefore

Hence Hence ff is an is an [[,j],j]-junta.-junta.

Using Idea2Using Idea2

24k kJ 2

f r O f variation 24k kJ 2

f r O f variation

24k k

2O f 24k k

2O f

22 4k k

1 2ff ' O f 22 4k k

1 2ff ' O f

P([n])

J

I1

I2IrI

4kr 4kr

2

2

2 2

J J2 2

24k k

2

ff '

f A f A ff '

2 O f

2

2

2 2

J J2 2

24k k

2

ff '

f A f A ff '

2 O f

6969

variation-Lemma - Proof variation-Lemma - Proof PlanPlan

LemmaLemma(variation): (variation): >0>0, and , and r>>k r>>k s.t.s.t.

Sketch for proving the variation lemmaSketch for proving the variation lemma::

1.1. w.h.p w.h.p ffII[x][x] is almost linear is almost linear

2.2. w.h.p w.h.p ffII[x][x] is close to shallow is close to shallow

3.3. ffII[x] [x] cannot be close to cannot be close to dictatorship dictatorship too often.too often.

J1r

2 24k k 4k kII ~ 2 2

E f O f O f

variation J1r

2 24k k 4k kII ~ 2 2

E f O f O f

variation

O k2 O k2

7070

The EndThe End

7171

XOR TestXOR Test Let Let be a random procedure for be a random procedure for

choosing two disjoint subsets choosing two disjoint subsets xx,,y y s.t.:s.t.:ii[n][n], , iix\y x\y w.p w.p 1/31/3, , iiy\x y\x w.p w.p 1/31/3, and, andiixxy y w.pw.p 1/3 1/3..

DefDef((XOR-TestXOR-Test): Pick ): Pick <x,y>~<x,y>~,, Accept if Accept if f(x)f(x)f(y)f(y),, Reject otherwise.Reject otherwise.

7272

ExampleExample ClaimClaim: Let : Let ff be a dictatorship, then be a dictatorship, then ff

passes the XOR-test w.p. passes the XOR-test w.p. 2/32/3. . ProofProof: Let : Let ii be the dictator, then be the dictator, then

PrPr<x,y>~<x,y>~[f(x)[f(x)f(y)]=Prf(y)]=Pr<x,y>~<x,y>~

[i[ixxy]=2/3 y]=2/3

ClaimClaim: Let : Let f’f’ be a be a -close to a -close to a dictatorship dictatorship ff, then , then f’f’ passes the XOR- passes the XOR-test w.p. test w.p. 2/3 – 2/32/3 – 2/3((--22)). .

ProofProof: see next slide…: see next slide…

7373

x,y ~

x,y ~

x,y ~

x,y ~

2 2

2 2

2

f ' x f ' y

f x f y f ' x f x f ' y f y

f x f y f ' x f x f ' y f y

f x f y f ' x f x f ' y f y f ' x f x f ' y f y

2 2 11 2 13 3 32 1 132 2

3 3

Pr

Pr

Pr

Pr

x,y ~

x,y ~

x,y ~

x,y ~

2 2

2 2

2

f ' x f ' y

f x f y f ' x f x f ' y f y

f x f y f ' x f x f ' y f y

f x f y f ' x f x f ' y f y f ' x f x f ' y f y

2 2 11 2 13 3 32 1 132 2

3 3

Pr

Pr

Pr

Pr

7474

Local MaximalityLocal Maximality DefDef: : ff is is locally maximallocally maximal with respect to a with respect to a

test, test, if if f’f’ obtained from obtained from ff by a change on one by a change on one input input xx00, that is,, that is,

PrPr<x,y>~<x,y>~[f(x)[f(x)f(y)] f(y)] Pr Pr<x,y>~<x,y>~[f’(x)[f’(x)f’(y)]f’(y)] DefDef: Let : Let xx be the distribution of all be the distribution of all y y such such

that that <x,y>~<x,y>~.. ClaimClaim: if : if ff is is locally maximallocally maximal then then

f(x) = -sign(Ef(x) = -sign(Ey~y~(x)(x)[f(y)])[f(y)])..

0

0 0

f x x xf ' x

f x x x

0

0 0

f x x xf ' x

f x x x

7575

ClaimClaim::

ProofProof: immediate from the Fourier-: immediate from the Fourier-expansion, and the fact that expansion, and the fact that yyx=x=

y~ (x) S x

E f y f S

y~ (x) S x

E f y f S

7676

ConjectureConjecture: Let : Let ff be be locally maximallocally maximal (with respect to the XOR-test), (with respect to the XOR-test), assume assume ff passes the XOR-test w.p passes the XOR-test w.p 1/2 + 1/2 + , for some constant , for some constant >0>0, , then then ff is close to a junta. is close to a junta.