Noise-Insensitive Boolean-Functions are Juntas

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Noise-Insensitive Boolean-Functions are Juntas

• Guy Kindler Muli Safra Slides prepared
  with help of Adi Akavia

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Influential People

• The theory of the influence of variables on
  Boolean functions BL, KKL and related issues,
  has been introduced to tackle social choice
  problems, furthermore has motivated a magnificent
  sequence of works, related to economics K,
  percolation BKS, Hardness of approximation
  DSRevolving around the Fourier/Walsh analysis
  of Boolean functions
• And the real important question

3
Where to go for Dinner?

• Who has suggestions
• Each cast their vote in an (electronic) envelope,
  and have the system decided, not necessarily
  according to majority
• It turns out someone in the Florida wing- has
  the power to flip some votes

Power
influence
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Voting Systems

• n agents, each voting either for (T) or
  against (F) a Boolean function over n
  variables f is the outcome
• The values of the agents (variables) may each,
  independently, flip with probability ?
• It turns out one cannot design an f that would
  be robust to such noise -that is, would, on
  average, change value w.p. lt ?O(1)- unless taking
  into account only very few of the votes

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Dictatorship

• Def a Boolean function P(n)?-1,1 is a
  monotone e-dictatorships --denoted fe--if

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Juntas

• Def a Boolean function fP(n)?-1,1 is a
  j-Junta if ?J?n where J j, s.t. for every
  x?n f(x) f(x ? J)
• Def f is an ?, j-Junta if ? j-Junta f s.t.
• Def f is an ?, j, p-Junta if ? j-Junta f
  s.t.

We would tend to omit p
p-biased, product distribution
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Long-Code

• In the long-code Ln? 0,12n each element is
  encoded by an 2n-bits
• This is the most extensive binary code, having
  one bit for every subset in P(n)

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Long-Code

• Encoding an element e?n
• Ee legally-encodes an element e if Ee fe

T
F
F
T
T
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Long-Code ? Monotone-Dictatorship

• The truth-table of a Boolean function over n
  elements, can be considered as a 2n bits long
  string (each corresponding to one input setting
  or a subset of n)For a long-code, the legal
  code-words are all monotone dictatorshipsHow
  about the Hadamard code?

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Long-code Tests

• Def (a long-code test) given a code-word w,
  probe it in a constant number of entries, and
• accept w.h.p if w is a monotone dictatorship
• reject w.h.p if w is not close to any monotone
  dictatorship

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Efficient Long-code Tests

• For some applications, it suffices if the test
  may accept illegal code-words, nevertheless, ones
  which have short list-decoding
• Def(a long-code list-test) given a code-word w,
  probe it in 2/3 places, and
• accept w.h.p if w is a monotone dictatorship,
• reject w.h.p if w is not even approximately
  determined by a short list of domain elements,
  that is, if ?? a Junta J?n s.t. f is close to
  f and f(x)f(x?J) for all x
• Note a long-code list-test, distinguishes
  between the case w is a dictatorship, to the case
  w is far from a junta.

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General Direction

• These tests may vary
• The long-code list-test a, in particular the
  biased case version, seem essential in proving
  improved hardness results for approximation
  problems
• Other interesting applications
• Hence finding simple, weak as possible,
  sufficient-conditions for a function to be a
  junta is important.

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Background

• Thm (Friedgut) a Boolean function f with small
  average-sensitivity is an ?,j-junta
• Thm (Bourgain) a Boolean function f with small
  high-frequency weight is an ?,j-junta
• Thm (KindlerSafra) a Boolean function f with
  small high-frequency weight in a p-biased measure
  is an ?,j-junta
• Corollary a Boolean function f with small
  noise-sensitivity is an ?,j-junta
• Parameters average-sensitivity BL,KKL,F
  high-frequency weight KKL,B noise-sensiti
  vity BKS

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Noise-Sensitivity

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

n
n
I
I
z
x
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Noise-Sensitivity

• Def(??,p,xn ) Let 0lt?lt1, and x?P(n). Then
  y??,p,x, if y (x\I)?? z where
• I??n is a noise subset, and
• z ?pI is a replacement.
• Def(?-noise-sensitivity) let 0lt?lt1, then
• When p½ equivalent to flipping each coordinate
  in x w.p. ?/2.

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Fourier/Walsh Transform

• Write f-1, 1n?-1, 1 as a polynomial
• What would be the monomials?
• For every set S?n we have a monomial which is
  the product of all variables in S (the only
  relevant powers are either 0 or 1)?????
• Make sense now to consider the degree of f or to
  break it according to the various degrees of the
  monomials..

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High/Low Frequencies and their Weights

• Def the high-frequency portion of f
• Def the low-frequency portion of f
• Def the high-frequency-weight is
• Def the low-frequency-weight is

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Low High-Frequency Weight

• Prop the ?-noise-sensitivity can be expressed in
  Fourier transform terms as
• Prop Low ns?? Low high-freq weight
• Proof By the above proposition, low
  noise-sensitivity implies nevertheless, f
  being -1, 1 function, by Parseval formula (that
  the norm 2 of the function and its Fourier
  transform are equal) implies

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Average and Restriction
n

• Def Let I?n, x?P(n\I), the restriction
  function is
• Def the average function is
• Note

I
y
x
n
I
y
y
y
y
y
x
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Fourier Expansion

• Prop
• Prop????
• Corollary

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Variation

• Def the variation of f
• Prop the following are equivalent definitions to
  the variation of f

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Proof

• Recall
• Therefore

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Proof Cont.

• Recall
• Therefore (by Parseval)

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Proof

• First, lets show

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Low-freq Variation and Low-freq
Average-Sensitivity

• Def the low-frequency variation is
• Def the average sensitivity is
• And in Fourier representation
• Def the low-frequency average sensitivity is

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Biased Walsh Product Talagrand

• Def In the p-biased product distribution ?p, the
  probability of a subset x is
• The usual Fourier basis ?is not orthogonal with
  respect to the biased inner-product,
• Hence, we use the Biased Walsh Product

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Main Result

• Theorem ? constant ?gt0 s.t. any Boolean
  function fP(n)?-1,1 satisfying is an
  ?,j-junta for jO(?-2k3?2k).
• Corollary fix a p-biased distribution ?p over
  P(n). Let ?gt0 be any parameter. Set
  klog1-?(1/2). Then ? constant ?gt0 s.t. any
  Boolean function fP(n)?-1,1 satisfying is
  an ?,j-junta for jO(?-2k3?2k).

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Where to go for Dinner?
Of course theyll have to discuss it over dinner.

• Who has suggestions
• Each cast their vote in an (electronic) envelope,
  and have the system decided, not necessarily
  according to majority
• It turns out someone in the Florida wing- has
  the power to flip some votes

Form a Committee
Power
influence
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First Attempt Following Freidguts Proof

• Thm any Boolean function f is an ?,j-junta for
  
• Proof
• Specify the juntawhere, let kO(as(f)/?) and fix
  ?2-O(k)
• Show the complement of J has small variation

P(n)
J
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Proving n\J has small variation

• Prop Let f be a Boolean function, s.t.
  variationJ(f)?? ?/2, then f is an ?,J-junta.
• Proof define a junta f as follows
  f(x)f(x?J)???????? then f is a J-junta,
  andhence

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Following Freidgut - Cont

• Lemma
• Proof
• Now, lets bound each argument
• Prop
• Proof characters of size ?k contribute to the
  average-sensitivity at least (since )

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Following Freidgut - Cont

• Lemma
• Proof
• Now, lets bound each argument
• Prop
• Proof characters of size??k contribute to the
  average-sensitivity at least (since )

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Following Freidgut - Cont
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 as?k! ?

• Prop
• Proof

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If k were 1

• Easy case (!?!) If wed have a bound on the
  non-linear weight, we should be done.
• The linear part is a set of independent
  characters (the singletons)
• In order for those to hit close to 1 or -1 most
  of the time, they must avoid the law of large
  numbers, namely be almost entirely placed on one
  singleton by Chernoff like boundThmFKN,
  ext. Assume f is close to linear, then f is
  close to shallow (? a constant function or a
  dictatorship)

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How to Deal with Dependency between Characters

• Recall
• (theorems premise)
• Idea Let
• Partition n\J into I1,,Ir, for r gtgt k
• w.h.p fIx is close to linear (low freq
  characters intersect I expectedly by ?1 element,
  while high-frequency weight is low).

P(n)
I2
Ir
I
I1
J
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So what?

• fIx is close to linear
• By FKN fIx is either a constant-function or a
  dictatorship, for any x
• Still, fIx could be a different dictatorship
  for every x, hence the variation of each i?I
  might be low

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almost linear ? almost shallow

• Theorem(FKN) ?global constant M, s.t.
  ?Boolean function f, ?shallow Boolean function
  g, s.t.
• Hence, fIxgt12 is small ?? fIx is close to
  shallow!

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Dictatorship and its Singleton

• Prop if fIx is a dictatorship, then
  ?coordinate i s.t. (where p is the
  bias).
• Corollary (from FKN) ?global constant M, s.t.
  ?Boolean function h, eitheror

weight
Total weight of no more than 1-p
Characters
1 2 i n 1,2 1,3 n-1,n S 1,..,n
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fIx Mostly Constant

• Lemma ??gt0, s.t. for any ? and any function
  gP(m)?? ?
• Def Let DI be the set of x?P(I), s.t. fIx is
  a dictatorship
• Next we show, that DI must be small, hence for
  most x, fIx is constant.

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DI must be small
Parseval
Prev lemma

• Lemma
• Proof let , then

Each S is counted only for one index i?I.
(Otherwise, if S was counted for both i and j in
I, then S?Igt1!)
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Simple Prop

• Prop let aii?I be sub-distribution, that is,
  ?i?Iai?1, 0?ai, then ?i?Iai2?maxi?Iai.
• Proof

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DI must be small - Cont

• Therefore(since ),
• Hence

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Obtaining the Lemma

• It remains to show that indeed
• Prop1
• Prop2

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Obtaining the Lemma Cont.

• Prop3
• Proof separate by freq
• Small freq
• Large freq
• Corollary(from props 2,3)

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Obtaining the Lemma Cont.

• Recall by corollary from FKN, Either or
• Hence
• By Corollary
• Combined with Prop1 we obtain

DI is small
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prop1
DI must be small
prop2
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Important Lemma

• Lemma ??gt0, s.t. for any ? and any function
  gP(m)?? ?, the following holds

high-freq
Low-freq
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Beckner/Nelson/Bonami Inequality

• Def let T? be the following operator on f
• Thm for any pr and ?((r-1)/(p-1))½
• Corollary for f s.t. fgtk0

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Beckner/Nelson/Bonami Corollary

• Proof

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Probability Concentration

• Simple Bound
• Proof
• Low-freq Bound Let gP(m)?? ? be of degree k
  and ?gt0, then ??gt0 s.t.
• Proof recall the corollary

?
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Lemmas Proof

• Now, lets prove the lemma
• Bounding low and high freq separately???,

simple bound
Low-freq bound
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Shallow Function

• Def a function f is linear, if only singletons
  have non-zero weight
• Def a function f is shallow, if f is either a
  constant or a dictatorship.
• Claim Boolean linear functions are shallow.

weight
Charactersize
0 1 2 3 k n
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Boolean Linear ?? Shallow

• Claim Boolean linear functions are shallow.
• Proof let f be Boolean linear function, we next
  show
• ?io s.t. (i.e. )
• And conclude, that either or i.e. f is shallow

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Claim 1

• Claim 1 let f be boolean linear function, then
  ?io s.t.
• Proof w.l.o.g assume
• for any z?3,,n, consider x00z, x10z?1,
  x01z?2, x11z?1,2
• then .
• Next value must be far from -1,1,
• A contradiction! (boolean function)
• Therefore

?
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Claim 1

• Claim 1 let f be boolean linear function, then
  ?io s.t.
• Proof w.l.o.g assume
• for any z?3,,n, consider x00z, x10z?1,
  x01z?2, x11z?1,2
• then .
• But this is impossible as f(x00),f(x10) ,f(x01),
  f(x11) ? -1,1, hence their distances cannot all
  be gt0 !
• Therefore .

?
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Claim 2

• Claim 2 let f be boolean function,
  s.t. Then either or
• Proof consider f(?) and f(i0)
• Then
• but f is boolean, hence
• therefore

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Linearity and Dictatorship

• Prop Let f be a balanced linear boolean function
  then f is a dictatorship.
• Prooff(?),f(i0)??-1,1, hence
• Prop Let f be a balanced boolean function s.t.
  as(f)1, then f is a dictatorship.
• Proof , but f is balanced, (i.e. ),
  therefore f is also linear.

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Proving FKN almost-linear ? close to shallow

• Theorem Let fP(n)?? ? be linear,
• Let
• let i0 be the index s.t. is maximal
• then
• Note f is linear, hence w.l.o.g., assume
  i01, then all we need to show is We show
  that in the following claim and lemma.

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Corollary

• Corollary Let f be linear, andthen ? a shallow
  boolean function g s.t.
• Proof let , let g be the boolean function
  closest to l. Then,this is true, as
• is small (by theorem),
• and additionally is small, since

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Claim 1

• Claim 1 Let f be linear. w.l.o.g., assumethen
  ?global constant cminp,1-p s.t.

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Proof of Claim1

• Proof assume
• for any z?3,,n, consider x00z, x10z?1,
  x01z?2, x11z?1,2
• then
• Next value must be far from -1,1 !
• A contradiction! (to )

?
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Proof of Claim1
they cannot all be near -1,1!

• Proof assume .
• for any z?3,,n, consider x00z, x10z?1,
  x01z?2, x11z?1,2
• then .
• Hence
• Therefore, for a random x this holds w.p. at
  least c, and therefore -- a contradiction.

?
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Lemma

• Lemma Let g be linear, let assume , then
• Corrolary The theorem follows from the
  combination of claim1 and the lemma
• Let m be the minimal index s.t.
• Consider
• If m2 the theorem is obtained (by lemma)
• Otherwise -- a contradiction to minimality of m

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Lemmas Proof

• Lemmas Proof Note
• Hence, all we need to show is that
• Intuition
• Note that g and b are far from 0(since g
  is ?-close to 1, and c?-close to b).
• Assume bgt0, then for almost all inputs x,
  g(x)g(x) (as )
• Hence Eg ? Eg(x), and
• therefore var(g) ? var(g)

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Proof-map g,b are far from 0 g(x)g(x) for
almost all x Eg ? Eg var(g) ? var(g)
?
?
?

• E2g - E2g 2E2g1flt0 ? o(?) (by
  Azumas inequality)
• We next show var(g) ? var(g)
• By the premise
• however
• therefore

?
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Variation Lemma
P(n)
I2
Ir
I
I1

• Lemma(variation) ??gt0, and rgtgtk s.t.
• Corollary for most I and x, fIx is almost
  constant

J
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Using Idea2
P(n)
I2
Ir
I
I1

• By union bound on I1,,Ir
• 
  (set )
• Let f(x) sign( AJf(x?J) ). f is the
  boolean function closest to AJf, therefore
• Hence f is an ?,j-junta.

J
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variation-Lemma - Proof Plan

• Lemma(variation) ??gt0, and rgtgtk s.t.
• Sketch for proving the variation lemma
• w.h.p fIx is almost linear
• w.h.p fIx is close to shallow
• fIx cannot be close to dictatorship too often.

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• The End

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XOR Test

• Let ? be a random procedure for choosing two
  disjoint subsets x,y s.t.?i?n, i?x\y w.p
  1/3, i?y\x w.p 1/3, andi?x?y w.p 1/3.
• Def(XOR-Test) Pick ltx,ygt?,
• Accept if f(x)??f(y),
• Reject otherwise.

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Example

• Claim Let f be a dictatorship, then f passes the
  XOR-test w.p. 2/3.
• Proof Let i be the dictator, then
  Prltx,ygt?f(x)??f(y)Prltx,ygt? i?x?y2/3
• Claim Let f be a ??-close to a dictatorship f,
  then f passes the XOR-test w.p. 2/3
  2/3??(?-?2).
• Proof see next slide

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Local Maximality

• Def f is locally maximal with respect to a test,
  if ??f obtained from f by a change on one input
  x0, that is, Prltx,ygt?f(x)??f(y) ?
  Prltx,ygt?f(x)??f(y)
• Def Let ?x be the distribution of all y such
  that ltx,ygt?.
• Claim if f is locally maximal then f(x)
  -sign(Ey?(x)f(y)).

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• Claim
• Proof immediate from the Fourier-expansion, and
  the fact that y?x?

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• Conjecture Let f be locally maximal (with
  respect to the XOR-test), assume f passes the
  XOR-test w.p ? 1/2 ?, for some constant ?gt0,
  then f is close to a junta.