Monotonicity.ppsx

On

Monotonicity

Testing

Boolean

Isoperimetric

type

Theorems

Subhash

Khot

Dor

Minzer

Property Testing

•

subset

P

of universe

(graphs, function, etc.)

Property:

•

accepts

an object in

P

;

•

rejects

if



-

far

from

P

.

(one-sided)

tester

•

Specific to each property.

Hamming

Distance

P

P



far

close

BF Monotonicity Testing

Universe: Boolean Functions

partial order on Boolean cube

P={f: f Monotone}

• P monotone functions.

Distance from monotonicity

Normalized hamming:

fraction of inputs

f,g differ on

Normalized hamming:

fraction of inputs

f,g differ on

}

1

,

0

{

}

1

,

0

{

:



f

i

i

y

x

i

y

x









)

(

)

(

x

f

y

f

y

x









)

,

(

min

)

(

f

f

g

P

g





Partial Timeline

1998 2002 2013 2014 2015

Introduced

by GGLRS

Introduced

by GGLRS

Lower bound

FLNRRS

Lower bound

FLNRRS

o(n)

algorithm CS

o(n)

algorithm CS

Improved

algorithm

CST

Improved

algorithm

CST

Tight

algorithm

Tight

algorithm

All proposed algorithms

•

Pair Testers

•

Non Adaptive

All proposed algorithms

One-sided Pair Tester

PickQuery xf(x)y ,

f(y)

Reject if non-mono. detected

One-Sided, Non-adaptive Pair-tester

•

Accept

Reject

w.h.p

f



P

( f) >

Edge Tester[GGLRS]

Pick xR{0, 1}nand iR[n] Query f(x), f(xi)

Reject if non-mono. detected

Thm[GGLRS]:

•

detection probability of edge-tester





(f)/n

Denote x

for x with

coordinate i flipped

Denote

x

for

x

with

Coming Up

Lower Bound

•

in the middle Section

•

0/1 outside.

random anti-dictator

•

Picks

x<y

points

•



(x, y)



O(



n)

non-adaptive pair-tester

•

w/probability



O(1/



n)

Detects violation



n

Lower Bound[FLNRRS]

negligible

negligible

negligible

negligible

1

0

Hamming

Weight

Hamming

Weight

0n 1n

Simplistic

:

detection prob. for



n

-

distance is



n



edge-tester’s?

NO

!?

Simplistic

:

detection prob. for



n

-

distance is



n



edge-tester’s?

NO

!?

Pick

x<y

,



(x, y) =



n

Query

f(x)

,

f(y)

Reject

if non-mono. detected

n

c

x





2

i

x

x

k even.

majority on a

unless a is balanced: i anti dictator on b.

k even.

majority on a

unless a is balanced:

i anti dictator on b.

Tiebreaker function

detection prob



n



detection of edge

tester?

No!

Detection occurs only if

both points balanced in

a



distance



n/k

k/2 > 0 k/2 < 1 k/2 = 1-x_i

set k=n/2

gives



(f)/n

or



(f)

/



n

set

k=n/2

gives



(f)/n

or



(f)

/



n

Pick

x



y

,



(x, y) =



n

Query

f(x)

,

f(y)

Reject

if non-mono. detected

n k

k

x

x

x

x

,...,

,

,...,

Previous Work

•

Testing is possible in

O(n/



)

queries.

•

Can one do better?

GGLRS

:

•

An affirmative answer by [CS 13],

possible in

O(n



)

•

Later improved by [CST14] O(n



).

Improve

:

•

with

O(n



)

queries.

•

Optimal up to logarithmic factors by

the lower bound of [FLNRRS].

:

Our Tester

Pick lR [½log(n)] Pick xR {0, 1}n

w/prob. ½ pick S{i| xi=0} of size 2l

o/w S{i|xi=1} of same size

Query f(x),

f(xS)

Reject if a non-mono. detected.

Theorem:

• detection prob. of our tester (f)2/_n

Denote x

for x with

coordinates S flipped

Denote

x

for

x

with

coordinates

S

flipped

Proof’s

Road Map

G

f

-

negative edges

Event

R

Subgraph

G

’

Event

R

’

Then:

Estimate

R

’

Issuess:

-

Persistency

-

overcounting

Issuess:

-

Persistency

-

overcounting

U’ regular

D’ bounded degree

Large

U’

regular

D’

bounded degree

Analysis

Hypercube graph

• Vertices: {0,1}n

• Edges x_↔xi

A Boolean function

• 0/1 labeling of vertexes

Subgraph

G

-• _{only monotonicity violating edges (red)} • _{remove isolated vertices.} • _{is a bipartite graph}

1

1

0

0

0

0

0

0

1

1

0

0

1

1

1

1

1

1

Pick lR

[½log(n/log n)] Pick xR {0, 1}n

w/prob. xi=0} of size ½ pick S2l{i| o/w S{i|xi=1} of same size

Query f(x),

f(xS)

Event R

•

in

D

labeled

1

;

in

U

labeled

0

G

Bipartite

•

x<y, (x, y)



G

-•

y<z

Consider

(x, y, z)

•

tester picks

x, z

•

f(z)=0

Event

R

:

1 1 0 0 0 0 0 0 1 1 1 1 0 0 0 0 1 1 0 0 D U

x

y

z

Pick lR [½log(n/log n)]

Pick x_R {0, 1}n

w/prob. ½ pick S{i|x_i=0} of size 2l

o/w S{i|x_i=1} of same size

Query f(x), f(xS)

Reject if a violation is found.

} : ) ( , 0 ) ( , : ) ,

{(x z x D f z y x x y z

R

vs



•

Thm

•

detection charged to

(x, y)

times

#edges

in

G

-Proof

•

f(z)



f(y)

? (

persistency

)

•

z

might be counted

multiple times

(for many

y

’s)

Issues!

x

y

z

Pick lR [½log(n/log n)]

Pick x_R {0, 1}n

w/prob. ½ pick S{i|x_i=0} of size 2l

o/w S{i|x_i=1} of same size

Query f(x), f(xS)

Reject if a violation is found.

Solution

:

Move to Sub-Graph G’ s.t

these issues are negligible

.

Solution

:

Move to Sub-Graph

G’

s.t

these issues are negligible

.

)

/

)

(

(

~

)

Sub-graph

•



subgraph

G’=



D’



U’, E’



of

G

and

d

s.t.

•

U’

-regular:

deg(y)=d

for every

y



U’

•

D’

-bound:

deg(x)



2d

for every

x



D’

•

Size: let



=

|U’|/2

Lemma(main):

•

main technical difficulty of this work

Proof is the

•

R’

•

Let us prove that

Assuming this

•

Set



=



n/log n

and estimate detection

probability conditioned on distance =



Proof:

How?

Consider all possible d’s

(powers of 2)

How?

Consider all possible

d

’s

(powers of 2)

) ) ( (

~ 2

2_{d  f}

   ) / ) ( ( ~ ) '

Pr(_{R    f} 2 _n

} : ) ( , 0 ) ( , : ) ,

{(x z x D f z y x x y z

R'{(x,z):xD' ,f(z) 0 , y (x) :x_ y_ z}

A Bound On Total Influence

Let

• _I-=n . Fraction negative edges

• I+=n _. Fraction positive edges

• I=n . Fraction influential edges

Lemma

• If I-/I+ < 1-δ then I=O_δ(n) • One line proof.

Corollary

• _{negative and the positive influence must be}

very close for the total influence to be large.

We may therefore

assume

I<O(



n)

.

O/w edge-tester

detects

non-monotonicity w/prob.



(



n)

!

We may therefore

assume

I<O(



n)

.

O/w edge-tester

detects

non-monotonicity w/prob.



(



n)

Non-Persistency a Non-Issue

•

y

is called (

τ-

1

)-non-persistent if

where

z

is a random point

τ-

1

steps above

y

Definition

•

Let



be fraction of (

τ-

1

)-non-persistent

inputs. Then

Lemma

•

Most vertices in

U’

are presistent!

Corrolary

Due to regularity:

Almost all edges!

Due to regularity:

Almost all edges!

Previous lemma

Previous lemma

1

)) ( )

( (

Pr_z f z  f y 

)

(

)

(

)

(









o

n

I

n

o

n

I











n n /log 

•

Is a non-Issue

•

Probability

z

is above

two

y

’s is negligible

Over-counting

•

Simple calculation

(inclusion-exclusion).

The rest

Overcoming Over-counting

•subgraph G’=D’U’, E’ ofG- _{and d s.t.}

•U’-regular: Deg(y)=d for every yU’

•D’-bound: Deg(x)2d for every xD’

•Size: let  = |U’|/2n

Lemma(main):

• _{main technical difficulty of this work}

Proof is the

•_R’

•Let us prove that

Assuming this

•Set =n/log n and estimate detection probability conditioned on distance =

Proof:

)) ( (

~ 2

Boolean

Isoperimetric

type

Isoperimetric Inequalities

(

Directed

)

Sensitivity )Directed( Sensitivity

•

For x such that f(x)=1

•

If(x) – The number of sensitive (influential) edges touching x

.

•

I

-f(x) - The number of negatively sensitive (influential) edges touching x

•

For f(x)=0, both are defined to be 0

.

Well known

Well known

•

I  var(f)

GGLRS: directed parallel

GGLRS: directed parallel

•

I- _{ }(f)

1

1

1

1

1

1

0

0

0

0 0₀

0

In CS13

In CS13

• I

-f Γ-f,matching =Ω(ε(f)2)

A Directed Version

A Directed Version

•

Margulis inequality

I_f Γ_f =Ω(var(f)2)

Directed version

Directed version

•

Talagrand generalization

of Margulis

?

1

1

0

0

0

0

0

0

1

1

1

1

0

0

0

0

1

1 _D

U

Directed – Undirected Analogy

Maximal matching in G

-Maximal matching in

G

-Fraction of vertices

with non-zero

sensitivity

Fraction of vertices

with non-zero

Isoperimetric Inequalities

•

For every Boolean function

f

Thm[Talagrand 93’]

Thm[Talagrand 93’]

•

var(f)

for Boolean

f

is roughly the size of

the minority value

•

Inductive step: split according to a

variable, account for the difference

between those functions

.

proof

proof

•

we prove a directed parallel

For our purposes

For our p

urposes

I_f(x) – The number of sensitive(influential)

edges touching x

.

I_f(x) – The number of sensitive(influential)

edges touching x . 1 1 1 1 1 1 0 0 0

0 0₀

0 0

))

(var(

)

)

(

(

I

x

f

E





))

(

(

)

)

(

(

I

x

f

Isoperimetric Inequalities

Remarks

•

proof is very different from Talagrand’s

•

Why: (f) misbehaves in induction

•

Switching

•

Results in a function on

the same number of bits

•

Monotonize on

i

Switch(i)

[

GGLRS

]

•

Monotonize on a set of

indices

•

Not commutative

Sequential

applicatio

n

•

Ρ

a permutation,

S

applies

switches in the order of

ρ

•

γ(f) = E

(Δ(f,S

f))

Distance

by

Switches

i

i

)}

(

),

(

max{

)

1

,

)(

(

x

f

i

x

f

x

x

f

S







Distance By Switches

Lemma(Fatal, Ron)

•

For every

f

:

ε(f) ≤ γ(f) ≤

2

ε(f)

Proof

• Left side clear. For the right side observe that for every f,g & i: Δ(S_i°g,S_i°f)≤ Δ(g,f)

• Let g be the closest monotone, then

• Δ(g,S_ρ°f) = Δ(S_ρ°g,S_ρ°f) ≤ Δ(g,f) = ε(f)

• By the triangle inequality

Overview of Proof

•

Analyze the simpler case in

which every variable has only

negative or positive influence

Pure

Functions

•

Reduce the general case to

pure functions using an

operator called

split

.

Reduce

•

On the splitting order

.

Throw

Pure (Unate) functions

•

f

is

pure

if each variable has either positive or

negative (not both) influence on

f

Definition

•

Let

A

be the set of positively influential variables:

p

=E

(f(y,x

=z))

•

Define

g

to be constant

0/1

on restriction

z

according to

p

Why is it simple for pure functions?

•

g

is monotone (active only on positive variables)

•

f

is

O(ε(f))

close to

g

(proof uses FKG inequality)

Claims

} 2 / 1 {

1

)

,

Pure functions

•

ε(f)≈Δ(g,f)=E

z

(Δ(g(y,x

=z),f(y,x

=z)))

•

The last term is

≈E

z

(var(f(y,x

=z)))

Observe

•

last inequality is Talagrand’s!

Finishing the proof

))

(

(

))]

,

(

(var(

[

]

)

(

[

]

)

(

[

]

)

(

[

f

z

x

f

E

y

I

E

y

I

E

x

I

E



















1

)

,

Open Questions

Adaptivity

•

Chen et al showed a matching

lower bound for two sided testers

•

Can adaptivity help?(current

lower bound logarithmic)

Isoperime

tric

•

No dependence on the dimension

in Talagrand’s theorem

.

•

Is the dependency on n necessary

in the directed version