some topics in analysis of boolean functions
TRANSCRIPT
R ya n O’Donne llC a rne g ie Me llon Unive rs ity
P a rt 1:
A. F ourie r e xpa ns ion ba s ic s
B . Conc e pts :
B ia s , In flue nc e s , Nois e S e ns itiv ity
C. Ka la i’s proof o f Arrow’s Th e ore m
10 Minute B re a k
P a rt 2:
A. Th e Hype rc ontra c tive Ine qua lity
B . Alg orith m ic G a ps
S a dly no tim e for:
Le a rn ing th e ory
P s e udora ndomne s s
Arith m e tic c ombina toric s
R a ndom g ra ph s / pe rc ola tion
Communic a tion c omple xity
Me tric / B a na c h s pa c e s
Coding th e ory
e tc .
1A. F ourie r e xpa ns ion ba s ic s
f : {0,1}n {0,1}
f : {−1,+1}n {−1,+1}
ℝ3(+1,+1,+1)
(−1,−1,−1)
(+1,+1,−1)
(+1,−1,+1)
(−1,+1,+1)
−1
−1
−1
+1+1
+1
+1
−1
ℝ3(+1,+1,+1)
(−1,−1,−1)
+1
+1
+1
+1−1
−1
−1
−1
ℝ3(+1,+1,+1)
(−1,−1,−1)
−1
−1
−1
+1−1
−1
−1
−1
ℝ3(+1,+1,+1)
(−1,−1,−1)
+1
+1
+1
+1+1
+1
+1
−1
ℝ3(+1,+1,+1)
(−1,−1,−1)
+1
+1
+1
+1+1
+1
+1
+1
ℝ3(+1,+1,+1)
(−1,−1,−1)
−1
−1
−1
−1−1
−1
−1
−1
ℝ3(+1,+1,+1)
(−1,−1,−1)
−1
−1
+1
+1−1
+1
+1
−1
ℝ3(+1,+1,+1)
(−1,−1,−1)
−1
+1
−1
+1+1
−1
+1
−1
ℝ3(+1,+1,+1)
(−1,−1,−1)
+1
−1
−1
+1+1
+1
−1
−1
ℝ3(+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)
=
=
P ropos ition:
E ve ry f : {−1,+1}n {−1,+1} c a n be
e xpre s s e d a s a multilinear po lynom ia l,
Th a t’s it. Th a t’s th e “F ourie r e xpa ns ion” o f f.
(un ique ly)
(inde e d, )→ ℝ
P ropos ition:
E ve ry f : {−1,+1}n {−1,+1} c a n be
e xpre s s e d a s a multilinear po lynom ia l,
Th a t’s it. Th a t’s th e “F ourie r e xpa ns ion” o f f.
(un ique ly)
(inde e d, )→ ℝ
⇓
R e s t: 0
Why?
Coe ffic ie nts e nc ode us e fu l in form a tion.
When?
1. Uniform proba b ility involve d
2. Ha m m ing d is ta nc e s re le va nt
P a rs e va l’s Th e ore m :
Le t f : {−1,+1}n {−1,+1}.
Th e n
a vg { f(x)2 }
“We ig h t” o f f on S [n ]⊆
=
{2}{1}
∅
{3}
{1,3}{1,2} {2,3}
{1,2,3}
{2}{1}
∅
{3}
{1,3}{1,2} {2,3}
{1,2,3}
{2}{1}
∅
{3}
{1,3}{1,2} {2,3}
{1,2,3}
{2}{1}
∅
{3}
{1,3}{1,2} {2,3}
{1,2,3}
{2}{1}
∅
{3}
{1,3}{1,2} {2,3}
{1,2,3}
{2}{1}
∅
{3}
{1,3}{1,2} {2,3}
{1,2,3}
1B . Conc e pts :
B ia s , In flue nc e s , Nois e S e ns itiv ity
S oc ia l Ch oic e :
Ca ndida te s ±1
n vote rs
Vote s a re ra ndom
f : {−1,+1}n {−1,+1}
is th e “voting ru le ”
B ia s o f f:
a vg f(x) = Pr[+1 wins ] − Pr[−1 wins ]
F a c t:
We ig h t on = m e a s ure s “im ba la nc e ”.∅
In flue nc e o f i on f:
Pr[ f(x) ≠ f(x( i)⊕ ) ]
= Pr[vote r i is a s wing vote r]
F a c t:
{2}{1}
∅
{3}
{1,3}{1,2} {2,3}
{1,2,3}
Maj(x1,x2,x3)
+1
+1
+1+1
−1
−1
−1−1
In fi(f) = Pr[ f(x) ≠ f(x( i)⊕ ) ]
+1
+1
+1+1
−1
−1
−1−1
In fi(f) = Pr[ f(x) ≠ f(x( i)⊕ ) ]
a vg In fi(f) = fra c . o f e dg e s wh ic h
a re c ut e dg e s
LMN Th e ore m :
If f is in AC 0
th e n a vg In fi(f)
⇒ a vg In fi(P a rityn) = 1
⇒ P a rity AC∉ 0
⇒ a vg In fi(Ma jn) =
⇒ Ma jority AC∉ 0
KKL Th e ore m :
If B ia s (f) = 0,
th e n
Corolla ry:
As s uming f m onotone ,
−1 or +1 c a n b ribe o(n) vote rs
a nd win w.p. 1−o(1).
Nois e S e ns itivity o f f a t :
NS (f) = Pr[wrong winne r wins ],
wh e n e a c h vote mis re c orde d w/prob
f(
f(
)
)
+ − + + − − + − −
− − + + + + + − −
Le a rn ing Th e ory princ ip le :
[LMN’93, …, KKMS ’05]
If a ll f ∈ C h a ve s m a ll NS (f)
th e n C is e ffic ie ntly le a rna b le .
{2}{1}
∅
{3}
{1,3}{1,2} {2,3}
[3]
P ropos ition:
for s ma ll ,
with Electoral College:
10
1
1C. Ka la i’s proof o f Arrow’s Th e ore m
R a nking 3 c a ndida te s
Condorc e t [1775] E le c tion:
=> (x_i, y_i, z _i) a re Not All E qua l (no 111 -1-1-1)
Condorc e t: Try f = Ma j. Outc om e c a n be “irra tiona l” A
> B > C > A. [e a s y e g ]
Ma ybe s om e oth e r f?
A > B ?
B > C?
C > A?
R a nking 3 c a ndida te s
Condorc e t [1775] E le c tion:
=> (x_i, y_i, z _i) a re Not All E qua l (no 111 -1-1-1)
Condorc e t: Try f = Ma j. Outc om e c a n be “irra tiona l” A
> B > C > A. [e a s y e g ]
Ma ybe s om e oth e r f?
• • • • • •
A > B ?
B > C?
C > A?
“C >
A >
B”
“A >
B >
C”
“B >
C >
A”
+
−
+
+
+
−
+
+
−
+
−
−
−
+
−
−
−
+
−
+
−
+
−
+
−
+
+
R a nking 3 c a ndida te s
Condorc e t [1775] E le c tion:
=> (x_i, y_i, z _i) a re Not All E qua l (no 111 -1-1-1)
Condorc e t: Try f = Ma j. Outc om e c a n be “irra tiona l” A
> B > C > A. [e a s y e g ]
Ma ybe s om e oth e r f?
• • • • • •
A > B ?
B > C?
C > A?
“C >
A >
B”
“A >
B >
C”
“B >
C >
A”
f( )
f( )
f( )
= +
= +
= −
S oc ie ty: “A > B > C”
+
−
+
+
+
−
+
+
−
+
−
−
−
+
−
−
−
+
−
+
−
+
−
+
−
+
+
R a nking 3 c a ndida te s
Condorc e t [1775] E le c tion:
=> (x_i, y_i, z _i) a re Not All E qua l (no 111 -1-1-1)
Condorc e t: Try f = Ma j. Outc om e c a n be “irra tiona l” A
> B > C > A. [e a s y e g ]
Ma ybe s om e oth e r f?
• • • • • •
A > B ?
B > C?
C > A?
“C >
A >
B”
“A >
B >
C”
“B >
C >
A”
f( )
f( )
f( )
= +
= +
= −
S oc ie ty: “A > B > C”
+
−
+
+
+
−
+
+
−
+
−
+
−
+
−
−
−
+
−
+
−
+
−
+
−
+
+
R a nking 3 c a ndida te s
Condorc e t [1775] E le c tion:
=> (x_i, y_i, z _i) a re Not All E qua l (no 111 -1-1-1)
Condorc e t: Try f = Ma j. Outc om e c a n be “irra tiona l” A
> B > C > A. [e a s y e g ]
Ma ybe s om e oth e r f?
• • • • • •
“C >
A >
B”
“A >
B >
C”
“B >
C >
A”
S oc ie ty: “A > B > C”
A > B ?
B > C?
C > A?
f( )
f( )
f( )
= +
= +
= +
+
−
+
+
+
−
+
+
−
+
−
+
−
+
−
−
−
+
−
+
−
+
−
+
−
+
+
R a nking 3 c a ndida te s
Condorc e t [1775] E le c tion:
=> (x_i, y_i, z _i) a re Not All E qua l (no 111 -1-1-1)
Condorc e t: Try f = Ma j. Outc om e c a n be “irra tiona l” A
> B > C > A. [e a s y e g ]
Ma ybe s om e oth e r f?
• • • • • •
“C >
A >
B”
“A >
B >
C”
“B >
C >
A”
S oc ie ty: “A > B > C > A”?A > B ?
B > C?
C > A?
f( )
f( )
f( )
= +
= +
= +
+
−
+
+
+
−
+
+
−
+
−
+
−
+
−
−
−
+
−
+
−
+
−
+
−
+
+
Arrow’s Impos s ib ility Th e ore m [1950]:
If
f : {−1,+1}n {−1,+1} never g ive s
irra tiona l outc ome in Condorc e t e le c tions ,
th e n
f is a Dictator or a negated-Dic tator.
G il Ka la i’s P roof [2002]:
• • • • • •
“C >
A >
B”
“A >
B >
C”
“B >
C >
A”
A > B ?
B > C?
C > A?
f( )
f( )
f( )
= +
= +
= −
+
−
+
+
+
−
+
+
−
+
−
−
−
+
−
−
−
+
−
+
−
+
−
+
−
+
+
• • • • • •
“C >
A >
B”
“A >
B >
C”
“B >
C >
A”
A > B ?
B > C?
C > A?
f( )
f( )
f( )
= +
= +
= −
+
−
+
+
+
−
+
+
−
+
−
−
−
+
−
−
−
+
−
+
−
+
−
+
−
+
+
G il Ka la i’s P roof:
G il Ka la i’s P roof:
G il Ka la i’s P roof, c onc lude d:
f never g ive s irra tiona l outc ome s ⇒ e qua lity
⇒ a ll F ourie r we ig h t “a t le ve l 1”
⇒ f(x) = ±xj for s om e j (e xe rc is e ).
⇓
G uilba ud’s Th e ore m [1952]
Guilba ud’s Numbe r ≈ .912
Corolla ry o f “Ma jority Is S ta b le s t” [MOO05]:
If In fi(f) ≤ o (1) for a ll i,
th e n
Pr[ra tiona l outc ome with f]
P a rt 2:
A. Th e Hype rc ontra c tive Ine qua lity
B . Alg orith m ic G a ps
2A. Th e Hype rc ontra c tive Ine qua lity
AKA B ona m i-B e c kne r Ine qua lity
a ll us e “Hype rc ontra c tive Ine qua lity”
Hoe ffd ing Ine qua lity:
Le t
F = c 0 + c 1 x1 + c 2 x2 + ··· + c n xn,
wh e re x i’s a re inde p., un if. ra ndom ±1.
Me a n: μ = c 0 Va ria nc e :
Hoe ffd ing Ine qua lity:
Le t
F = c 0 + c 1 x1 + c 2 x2 + ··· + c n xn,
Me a n: μ = Va ria nc e :
Hype rc ontra c tive Ine qua lity*:
Le t
Th e n for a ll q ≥ 2,
Hype rc ontra c tive Ine qua lity:
Le t
Th e n F is a “re a s ona b le d” ra ndom va ria b le .
Hype rc ontra c tive Ine qua lity:
Le t
Th e n for a ll q ≥ 2,
Hype rc ontra c tive Ine qua lity:
Le t
Th e n
“q = 4” Hype rc ontra c tive Ine qua lity:
Le t
Th e n
“q = 4” Hype rc ontra c tive Ine qua lity:
Le t
a ll us e Hype rc ontra c tive Ine qua lity
ju s t us e “q = 4” Hype rc ontra c tive Ine qua lity
“q = 4” Hype rc ontra c tive Ine qua lity:
Le t F be de g re e d ove r n i.i.d. ±1 r.v.’s .
Th e n
P roof [MOO’05]: Induc tion on n.
Obvious s te p.
Us e induc tion h ypoth e s is .
Us e Ca uc h y-S c h wa rz on th e obvious th ing .
Us e induc tion h ypoth e s is .
Obvious s te p.
2B . Alg orith m ic G a ps
Opt
be s t po ly-tim eg ua ra nte e
ln(N)
“S e t-Cove r is NP -h a rd to
a pproxim a te to fa c tor ln(N)”
Opt
LP -R a nd-R oundingg ua ra nte e
ln(N)
“F a c tor ln(N) Algorithmic Gap
for LP -R a nd-R ounding ”
Opt(S )
LP -R a nd-R ounding (S )
ln(N)
“Algorithmic Gap Ins tance S
for LP -R a nd-R ounding ”
Alg orith mic G a p ins ta nc e s
a re o fte n “ba s e d on” {−1,+1}n.
S pa rs e s t-Cut:
Alg orith m : Arora -R a o-Va z ira n i S DP .
G ua ra nte e : F a c tor
Opt = 1/n
Opt = 1/n
Opt = 1/n
Opt = 1/n
f(x) = s g n( )
Opt = 1/n
f(x) = s g n(r1x1 + ••• + rnxn)
AR V g e ts
Opt = 1/n
AR V g e ts
g a p:
Alg orith mic G a ps Ha rdne s s -o f-Approx→
LP / S DP -rounding Alg . G a p ins ta nc e
• n optim a l “Dic ta tor” s o lutions
• “g e ne ric m ixture o f Dic ta tors ” muc h wors e
+ P CP te c h nolog y
= s a me -g a p h a rdne s s -o f-a pproxim a tion
Alg orith mic G a ps Ha rdne s s -o f-Approx→
LP / S DP -rounding Alg . G a p ins ta nc e
• n optim a l “Dic ta tor” s o lutions
• “g e ne ric m ixture o f Dic ta tors ” muc h wors e
+ P CP te c h nolog y
= s a me -g a p h a rdne s s -o f-a pproxim a tion
KKL / Ta la g ra nd Th e ore m :
If f is b a la nc e d,
In fi(f) ≤ 1/n .01 for a ll i,
th e n
a vg In fi(f) ≥
G a p: Θ(log n) = Θ(log log N).
[CKKR S 05]: KKL + Unique G a me s Conje c ture
⇒ Ω(log log log N) h a rdne s s -o f-a pprox.
2-Colora b le 3-Uniform h ype rg ra ph s :
Input: 2-c o lora b le , 3-un if. h ype rg ra ph
Output: 2-c o loring
Ob j: Ma x. fra c tion o f le g a lly
c o lore d h ype re dg e s
2-Colora b le 3-Uniform h ype rg ra ph s :
Alg orith m : S DP [KLP 96].
G ua ra nte e :
[Zwic k99]
Alg orith m ic G a p Ins ta nc e
Ve rtic e s : {−1,+1}n
6 n h ype re dg e s :{ (x,y,z ) : pos s . pre fs in
a Condorc e t e le c tion}
(i.e., triples s.t. (xi,y i,z i) NAE for all i)
E lts : {−1,+1}n E dg e s : Condorc e t vote s (x,y,z )
2-c o loring = f : {−1,+1}n {−1,+1}→
fra c . le g a lly c o lore d h ype re dg e s
= Pr[“ra tiona l” outc ome with f]
Ins ta nc e 2-c o lora b le ? ✔
(2n optima l s o lutions : ±Dic ta tors )
E lts : {−1,+1}n E dg e s : Condorc e t vote s (x,y,z )
S DP rounding a lg . m a y output
R a ndom we ig h te d m a jority a ls o
ra tiona l-with -prob .-.912! [s a m e CLT a rg .]
f(x) = s g n(r1x1 + ••• + rnxn)
Alg orith mic G a ps Ha rdne s s -o f-Approx→
LP / S DP -rounding Alg . G a p ins ta nc e
• n optim a l “Dic ta tor” s o lutions
• “g e ne ric m ixture o f Dic ta tors ” muc h wors e
+ P CP te c h nolog y
= s a me -g a p h a rdne s s -o f-a pproxim a tion
Corolla ry o f Ma jority Is S ta b le s t:
If In fi(f) ≤ o (1) for a ll i,
th e n
Pr[ra tiona l outc ome with f]
Cor: th is + Unique Ga me s Conje c ture
⇒ .912 ha rdne s s -of-a pprox*
2C. F uture Dire c tions
De ve lop th e “s truc ture vs . ps e udora ndom ne s s ”
th e ory for B oole a n func tions .
