Download Problem Set 5: Analysis of Boolean Functions and more Exercises Computer Architecture and Organization in PDF only on Docsity!
Analysis of Boolean Functions CMU 18-859S, Spring 2007
PROBLEM SET 5
Due: Tuesday, April 24, beginning of class
Homework policy: I encourage you to try to solve the problems by yourself. However, you may collaborate
as long as you do the writeup yourself and list the people you talked with.
Do at least 4 out of 7.
- Poincar´e Inequality III. For this problem, please assume the setup and results of Problem 1 on Home-
work #4. Given f : X
n → R, we will write “
f (S)
2 ” for E x
[(f
S (x))
2 ].
(a) Show that Var[f ] =
S 6 =∅
f (S)
2 .
(b) Given I ⊆ [n], let y denote a random draw from X
¯ I , where
I = [n] \ I, as usual. Show that
E
y
[E[f y→
¯ I
]
2 ] =
S⊆
¯ I
f (S)
2 .
(c) For i ∈ [n], define
Inf i
(f ) = E
y
[Var[f y→
¯ I
]],
where we are writing I = {i}. Show that Inf i
(f ) =
S 3 i
f (S)
2 , and conclude that
Var[f ] ≤
n ∑
i=
Infi(f ).
Remark: This fact (actually, a slightly different equivalent fact) is sometimes known as the Efron-Stein
Inequality.
- Talagrand’s Lemma. Let f : {− 1 , 1 }
n → { 0 , 1 } and write p = E[f ]. (Think of p as small.) Ta-
lagrand’s Lemma states that W 1
(f ) =
|S|=
f (S)
2 ≤ O(p
2 log(1/p)). In this problem we will show a
slight generalization: the result holds for any f : {− 1 , 1 }
n → [− 1 , 1], when p = E[|f |].
(a) Let
` : {− 1 , 1 }
n → R be given by
`(x) =
n
i=
ˆ f (i)
σ
xi, where σ =
W 1 (f ).
1 Show that for any
t ≥ 0 ,
σ = E[ 1 {|
˜ `|≤t}
· f ·
`] + E[ 1
{|
˜ `|>t}
· f ·
`].
(b) Upper-bound the above by pt + O(exp(−t
2 /2)). (Hint: Use Hoeffding.)
(c) Deduce W 1
(f ) ≤ O(p
2 log(1/p)).
1 If σ = 0 then we’re already done.
- Degree- 1 versus influence. Let f : {− 1 , 1 }
n → {− 1 , 1 }. Show that |
f (i)| ≤ Inf i
(f ). Show that this
is false in general for f : {− 1 , 1 }
n → [− 1 , 1].
- Majority Is Stablest for small ρ. Suppose f : {− 1 , 1 }
n → [− 1 , 1] is “(, 1)-quasirandom”; i.e.,
f (i)
2 ≤ for all i.
(a) Show that W 1
(f ) ≤
2
π
+ O(
). (Hint: Study E[f ·
`] as in Problem 2 and use Berry-Esseen.)
(b) Show that if in addition E[f ] = 0, then S ρ
(f ) ≤
2
π
arcsin ρ + O(ρ
2
) for ρ ≥ 0.
(Thus the Majority Is Stablest Theorem holds for “small” ρ.)
- Reverse Majority Is Stablest. Recall that the Majority Is Stablest Theorem is the following:
Fix 0 ≤ ρ ≤ 1. Then if f : {− 1 , 1 }
n → [− 1 , 1] is (, 1 / log(1/))-quasirandom and satisfies E[f ] = 0,
then Sρ(f ) ≤
2
π
arcsin ρ + O(
log log(1/)
log(1/)
Use this to deduce the following “reversed” version:
Fix − 1 ≤ ρ ≤ 0. Then if f : {− 1 , 1 }
n → [− 1 , 1] is (, 1 / log(1/))-quasirandom (we don’t assume
E[f ] = 0), then S ρ
(f ) ≥
2
π
arcsin ρ − O(
log log(1/)
log(1/)
(Hint: Odd-ize.)
- Attenuated influences vs. influences, and the noise sensitivity of Tribes.
(a) Let f : {− 1 , 1 }
n → {− 1 , 1 }. Show that Inf
(ρ)
i
(f ) ≤ (Infi(f ))
2 /(1+ρ) .
(b) Let T denote the Tribes function on n bits. Show that for 0 ≤ γ ≤ 1 / 2 ,
|S|≥ 1
|S|γ
|S|− 1 ˆ T (S)
2
≤
O(log
2
n)
n
1 − 2 γ
(c) Assume 1 /n ≤ γ ≤ 1 / log n. Show that NS 1
2
−γ
(T ) ≥
1
2
− γ · O(
log
2 n
n
(This implies that Tribes is an excellent combining function for hardness amplification — if the hardness is
already near
1
2
to start.)
- Minimum balanced k-cuts in Kneser-like graphs.
(a) Suppose f : {− 1 , 1 }
n → { 0 , 1 } has E[f ] = p. Show that S ρ
(f ) ≤ p
2 /(1+ρ) .
(b) Let > 0 and let k ∈ N be a power of 2. Consider the weighted complete graph on the vertex set
n in which the weight on the edge (u, v) is equal to
Pr
x,y∼ 1 −
x
[(x, y) = (u, v)].
A balanced k-cut in this graph is a partition of the vertices into k equal-sized parts. The value of the cut is
equal to the total weight of edges that have endpoints in different parts. Since the total weight in the graph
is 1 , the value of a cut is in the range [0, 1]. Show that in fact the minimum value among balanced k-cuts in
this graph is at least 1 − o k→∞
