Parikshit Gopalan Microsoft Research Rocco Servedio Columbia Univ Avi Wigderson IAS Princeton and Avishay Tal IAS Princeton see ECCC version Real degree of Boolean functions ID: 633390
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Slide1
Degree and Sensitivity: tails of two distributions
Parikshit
Gopalan
Microsoft Research
Rocco
Servedio
Columbia Univ.
Avi Wigderson
IAS
,
Princeton
and
Avishay
Tal
IAS
,
Princeton
*
(*see ECCC version)Slide2
(Real) degree of Boolean functions
f
: {-1,1
}
n
{-1,1
} in R
[
x
1
,
x
2
,…,
x
n
]
deg
(
f
) = min
d
:
∃
Real polynomial
p
of degree
d
such that
p
(
x
)=
f
(
x
)
x
{
-1,1
}
n
Ex1
:
Maj
(
x
,
y
,
z
) =
½
(
x
+
y
+
z
–
xyz
)
deg
=3
Ex2
: NAE(
x
,
y
,
z
) =
½
(
xy
+
yz
+
xz
–1
)
deg
=2
p
f
=
unique
multilinear
p
s.t
.
p
(
x
)=
f
(
x
)
x
{-1,1
}
n
p
f
=
T
[
n
]
f’
(
T
)
∏
i
T
x
i
f’
(
T
)
=
Fourier
coefficients
deg
(
f
)
=
deg
(
p
f
) = max |
T
|:
f’
(
T
)
≠
0Slide3
Complexity measures
f
: {-1,1
}
n
{-1,1}D(f) – Deterministic decision tree complexityR(f) – Probabilistic decision tree complexityQ(f) – Quantum decision tree complexityN(f) – Certificate complexitydeg(f) – Real degreedeg∞(f) – L∞ approximate degreebs(f) – Block sensitivity……[Nisan,…] All parameters are polynomially relatedsen(f) – Sensitivity ??
independent of nSlide4
Sensitivity
f
: {-1,1
}
n
{-1,1}1-11-111-1-1Gfs(x) = sensitivity of x = vertex degree of x in G
f sen(
f)
=
max
x
s
(
x
)
[Nisan-
Szegedy
]
sen
(
f
) ≤
deg(f)2[Sens-Conjecture] deg(f) ≤ sen(f)c
Understand
G
f
!
!
New parameters
low sensitivity = smooth
Smooth
=
SimpleSlide5
Fourier dist. &
Real approx.
f
: {-1,1
}
n {-1,1} L2-approximationdegε (f) = min d: ∃ Real polynomial q of degree d such that Ex {-1,1}n [|q(x)-f(x)|2] ≤ ε T f’(T)2 =1 Fourier dist: T [n] with prob
f’(T)2ε
t
=
|
T
|>
t
f’
(
T
)
2
:
Tails
of the Fourier
distribution
deg
ε
(f) = min d: εd ≤ ε - Best approximator q is a truncation of pf - deg0(f) = deg
(
f) Slide6
Main result
f
: {-1,1
}
n
{-1,1}deg0(f) = deg(f) degε(f) = min d: εd ≤ ε (best approximation in L2)[Sens-Conj] deg0(f) ≤ sen(f)c[Thm1] ε>0 degε (f) ≤ sen(f) log(1/ε)
[Thm2] This is “optimal”:∃c>0,δ
<1 deg
ε
(
f
) ≤
sen
(
f
)
c
log
(1/
ε
)
δ
[Sens-Conj] Slide7
1
-1
1
-1
1
1-1-1
Sensitive trees
f
: {-1,1
}
n
{-1,1
}
A sensitive tree in
G
f
is a
subgraph
H
of
G
f
H
is a tree
dimensions of edges(H) distinct ts(f) = max {dim H: H
sens
tree}
sen(f) = max
{dim H:
H sens
star}
[Thm3]
deg(f
) ≤ ts(
f)2
[TS-Conj]
deg(f
) ≤ ts
(f)
1
32
1
3
2
3
2Slide8
Moments: Fourier vs. Sensitivity
f
: {-1,1
}
n
{-1,1}, pf = T f’(T)XT D: draw T [n] with probability f’(T)2Dk=ED[|T|k] : Fourier momentsS: draw x {-1,1}n uniformly.Sk=Ex[s(x)k]:
Sensitivity moments[Moment-Conj]
k
D
k
≤
a
k
S
k
(independent of
f
,
n
)
[Fact]
D
1
=
S
1 (Total influence), [Kalai] D2 = S2[Thm4] deg(f) ≤ ts(f) [Moment-Conj
]
Average-case
variants of deg(
f) & sens(
f)Slide9
Proof of the main result
f
: {-1,1
}
n
{-1,1}degε(f) = min d: εd ≤ ε (best approximation in L2)[Thm1] ε>0 degε (f) ≤100 sen(f) log(1/ε) s k (t=100sk) εt = |T|>t f’(T)2 = Pr
D[|T|>t] =
PrD[
|
T
|
k
>
t
k
] ≤
≤ E
[
|
T
|
k
]/
t
k
≤
………… ≤ exp(-k) ≤(n/t)k Pr[deg(fρ)
=
k] ≤
(1) (2)
Random restriction
Leaving k var
aliveSlide10
Random restrictions
ρ
:
{
x1,x2,…,xn} {-1,1,*} at random fromRk= {ρ = (K,y) : K=ρ-1(*), |K|=k, y {-1,1}n-K }(1) ED[|T|k] ≤ nk Prρ[deg(fρ) = k]Proof:
Prρ [deg(f
ρ) =
k
] =
Pr
[
f’
ρ
(
K
)
≠0]
≥ 2
-2
k
E[
f’
ρ
(
K
)2] Granularity of Fourier = T Prρ [KT] fρ(T)2 Heredity
of
Fourier = (
k/n
)k E
D[|T|
k]Slide11
A switching lemma
(2)
Pr
ρ
[
deg (fρ) = k] ≤ (10sk/n)k(2’) Prρ[ts (fρ) = k] ≤ (10sk/n)kBad = { ρ : ts (fρ) = k } RkBad [2n]×[s]k×[2]k ρ A DFS path in the sensitive tree
|Bad|/|Rk| < (10
sk/n
)
k
[
Moment-
Conj
]
Hastad’s
switching lemma
Has max degree ≤
s
In paper we use
“proper walks”Slide12
Applications
Learning algorithm for low-sensitivity functions in time
(1/
ε
)
poly(s) Under uniform dist: Uses [LMN] Exact learning alg: Uses [GNSTW] New proof of the switching lemmaBetter bounds on Entropy-Influence conj: [EntInf-Conj] Ent(f) ≤ c.Inf(f) [Fact] Ent(f) ≤ c.Inf(f)
.log n
[EntInf-Conj
]
Ent
(
f
) ≤
c
.
Inf
(
f
)
.
log
sen
(
f
)Slide13
Conclusions & open problems
Prove consequences of
[
Sens-
Conj
] [GSTW] degε (f) ≤ sen(f) log(1/ε) [GNSTW] depth(f) ≤ poly(sen(f))Prove [Sens-Conj] !!!If not…deg(f) ≤ ts(f)? Relate new parametersk ED[|T|k] ≤ ak
Ex[s(x)
k] ?
k
E
x
[
s
(
x
)
k
] ≤
b
k
E
D
[|
T
|k] ?