Non Linear Invariance Principles Non Linear Invariance Principles with Applicationswith Applications

Elchanan MosselElchanan Mossel

U.C. Berkeley U.C. Berkeley http://stat.berkeley.edu/~mosselhttp://stat.berkeley.edu/~mossel

Lecture Plan• Background: Noise Stability in Gaussian

Spaces– Noise := Ornstein-Uhlenbeck process. – Bubbles and half-spaces. – Double Bubbles and the “Peace Sign” Conjecture.

• An invariance principle – Half-Spaces = Majorities are stablest– Peace-signs = Pluralities are stablest?– Voting schemes. – Computational hardness of graph coloring.

Gaussian Noise

• Let 0 1 and f, g : Rn Rm.

•Define <f, g> := E[<f(N) , g(M) >], where

N,M ~ Normal(0,I) with E[Ni Mj] = (i,j).

• For sets A,B let: <A,B> := <1A,1B>

• Let n := standard Gaussian volume

• Let n := Lebsauge measure.

• Let n-1, n-1 := corresponding (n-1)-dims areas.

Some isoperimetric results

• I. Ancient: Among all sets with

n(A) = 1 the minimizer of n-1(

A) is A = Ball.

• II. Recent (Borell, Sudakov-Tsierlson

70’s) Among all sets with n(A) =

a the minimizer of n-1( A) is A =

Half-Space.

• III. More recent (Borell 85): For all , among all sets with (A) = a

the maximizer of <A,A> is

given by A = Half-Space.

Double bubbles•Thm1 (“Double-Bubble”):

•Among all pairs of disjoint sets A,B with n(A) = n(B) = a, the

minimizer of -1( A B) is a

“Double Bubble”

•Thm2 (“Peace Sign”):

•Among all partitions A,B,C of Rn with (A) = (B) = (C) = 1/3 , the minimum of ( A B C) is obtained for the “Peace Sign”

• 1. Hutchings, Morgan, Ritore, Ros. +

Reichardt, Heilmann, Lai, Spielman 2. Corneli, Corwin, Hurder, Sesum, Xu, Adams, Dvais, Lee, Vissochi

The Peace-Sign Conjecture•Conj:

• For all 0 1,

•all n 2

•The maximum of

<A, A> + <B, B> + <C, C>

among all partitions (A,B,C) of

Rn with n(A) = n(B) = n(C) =

1/3 is obtained for

(A,B,C) = “Peace Sign”

Lecture Plan• Background: Noise Stability in Gaussian

Spaces– Noise := Ornstein-Uhlenbeck operator. – Bubbles and half-spaces. – Double Bubbles and the “Peace Sign” Conjecture.

• An invariance principle – Half-Spaces = Majorities are stablest– Peace-signs = Pluralities are stablest?– Voting schemes. – Computational hardness of graph coloring.

Influences and Noise in product Spaces

• Let X be a probability space.

• Let f L2(Xn,R). The i’th influence of f is given by:

Ii(f) := E[ Var[f | x1,…,xi-1,xi+1,…,xn] ]

(Ben-Or,Kalai,Linial, Efron-Stein 80s)

•Given a reversible Markov operator T on X and

f, g: Xn R define the T - noise form by

<f, g>T := E[f T n g]

•The 2nd eigen-value (T) of T is defined by

(T) := max {|| : spec(T), < 1}

• Let X = {-1, 1} with the uniform measure.

• For the dictator function xj: Ii(xj) = (i,j).

• For the majority m(x) = sgn(1 i n xi) function:

Ii(m) (2 n)-1/2.

• Let T be the “Beckner Operator” on X:

Ti,j = (i,j) + (1-)/2.

•T xi = xi and <xi, xi>T = .

•<m, m>T ~ 2 arcsin() /

• (T) = .

Influences and Noise in product Spaces – Example 1

Definition of Voting Schemes

• A population of size n is to choose between two options / candidates.

• A voting scheme is a function that associates to each configuration of votes an outcome.

• Formally, a voting scheme is a function f : {-1,1}n ! {-1,1}.

• Assume below that f(-x1,…,-xn) = -f(x1,…,xn)

• Two prime examples: – Majority vote,– Electoral college.

• At the morning of the vote:

• Each voter tosses a coin.

• The voters vote according to the outcome of the coin.

A voting model

• Which voting schemes are more robust against noise? • Simplest model of noise: The voting machine flips each

vote independently with probability .• <f, f> = correlation of intended vote with actual

outcome.

A model of voting machines

Intended vote Registered vote

1

prob1-

prob 1

prob1-

prob

-1

-1

1

-1

• <m, m> 2 arcsin / [n ] 1 – c(1-)1/2 [ 1]

for m(x) = majority(x) = sgn(i=1n xi)

• Result is essentially due to Sheppard (1899): “On the application of the theory of error to cases of normal distribution and normal correlation”.

• For n1/2 £ n1/2 electoral college f <f,f> 1- c (1-)1/4 [n , 1]

• <f,f>-1/2 determined prob.

of Condorcet Paradox (Kalai)

Majority and Electoral College

• Noise Theorem (folklore): Dictatorship, f(x) = xi is the

most stable balanced voting scheme.

• In other words, for all schemes, for all f : {-1,1}n {-1,1} with E[f] = 0 it holds that <f, f> = <x1, x1>

• Harder question: What is the “stablest” voting scheme not allowing dictatorships or Juntas?

• For example, consider only symmetric monotone f.

• More generally: What is the “stablest” voting scheme f satisfying for all voters i: Ii(f) = P[f(x1,…,xi,…,xn) f(x1,

…,-xi,…,xn)] < where n and 0.

An easy answer and a hard question

X X

• Let X = {0,1,2} with the uniform measure.

• Let Ti,j = ½ (i j)

•Then (T) = ½ and

•Claim (Colouring Graph): Consider Xn as a

graph where (x,y) Edges(Xn) iff xi yi for all i.

Let A,B Xn. Then <A, B>T = 0 iff there are no

edges between A and B. In particular, A is an

independent set iff <A, A>T = 0.

•Q: How do “large” independent sets look like?

Influences and Noise in product Spaces – Example 2

Graph Colouring – An Algorithmic Problem

• Let (G) := min # of colours needed

to colour the vertices of a graph G so that no edge is monochramatic.

• ApxCol(q,Q):

Given a graph G, is (G) q or (G) Q ?

• This is an algorithmic problem. How hard is it?

• For q=2 easy: simply check bipartiteness

• For q=3, no efficient algorithms are known unless Q >|G|0.1

• Efficient := Running time that is polynomial in |G|.

• Also known that (3,4) and (3,5) are NP-hard.

• NP-hard := “Nobody believes polynomial time algorithms exist”.

• What about (3,6) ?????

Graph Colouring – An Algorithmic Problem

• In 2002, Khot introduced a family of algorithmic problems called “games”. He speculated that these problems are NP-hard.

• These problems resisted multiple algorithmic attacks.

• Subhash “games conjecture”

• Claim: Consider {0,1,2}n as a graph G where (x,y) Edges(G) iff xi yi for all i.

• Let Q > 3. Suppose that such that for all n if there are no edges between A and B {0,1,2}n (<A,B>T =

0) and |A|,|B| > 3n/Q then there exists an i such that Ii(A) > and Ii(B) >

• Then ApxColor(3,Q) is NP hard.

Graph Colouring – An Algorithmic Problem

u

Graph Colouring – An Algorithmic Problem

u[u]

Influences and Noise in product Spaces – Example 3

• Let X = {0,1,2} with the uniform measure.

• Let 0, 1, 2 = (1,0,0), (0,1,0),(0,0,1) R3.

• Let d : Xn R3 defined by d(x) := x(1)

• Let p : Xn R defined by p(x) = y

where y is the most frequent value among the

xi.

• Ii(d) = 2/3 (i,1); Ii(p) c n-1/2.

• For 0 1, let T be the Markov operator on X

defined by Ti,j = (i,j) + (1-)/3.

•<d, d>T = Var(d).

Gaussian Noise Bounds •Def: For a, b, [0,1] , let

(a, b, := sup {< F,G > | F,G R, [F] =

a, [G] = b}

a, b, := inf {< F,G > | F,G R, [F] = a,

[G] = b}

•Thm: Let X be a finite space. Let T be a reversible Markov operator on X with = (T) < 1.

•Then > 0 > 0 such that for all n and all

f,g : Xn [0,1] satisfying maxi min(Ii(f),

Ii(g)) <

• It holds that <f, g>T (E[f], E[g], ) + and

• <f, g>T (E[f], E[g], ) -

• M-O’Donnell-Oleskiewicz-05 + Dinur-M-Regev-06

Example 1•Taking T on {-1,1} defined by Ti,j = (i,j) + (1-

)/2

•Thm : Claim: f : {-1,1}n {-1,1} with Ii(f) <

for all i and E[f] = 0 it holds that:

•<f, f>T <F, F> + where F(x) = sgn(x)

• <F, F> = 2 arcsin()/ (F is known by Borell-

85)

•So “Majority is Stablest”: Most Stable “Voting Scheme” among low influence ones.

•Weaker results obtained by Bourgain 2001.

• “” tight in-approximation result for MAX-CUT.

• Khot-Kindler-M-O’Donnell-05

Example 2

•Taking T on {0,1,2} defined by Ti,j = ½ (i j)

•Thm Claim: > 0 > 0 s.t. if A,B {0,1,2}n have no edges between them and P[A], P[B] then

•There exists an i s.t. Ii(A), Ii(B) .

•Proof follows from Borell-85 showing (,,1/2) > 0.

•Claim Hardness of approximation result for graph-colouring:

• “For any constant K, it is NP hard to

colour 3-colorable graphs using K colours”.

Dinur-M-Regev-06

Example 3

•Taking T on {0,1,2} defined by Ti,j = (i,j) +

(1-)/3

•Recall: 0,1,2 = (1,0,0),(0,1,0),(0,0,1)

•Thm + “Peace Sign Conjecture”

•Claim: (“Plurality is Stablest”):

f : {0,1,2}n {0,1,2} with E[f] =

(1/3,1/3,1/3) and Ii(f) < for all i, it holds that

<f, f> limn <p , p>T + , where

•p is the plurality function on n inputs (“Plurality is Stablest”)

•Claim “Optimal Hardness of approximation result” for MAX-3-CUT.

More results

•More applied results use Noise-Stability bounds:

•Social choice: Kalai (Paradoxes).

•Hardness of approximation:

•Dinur-Safra, Khot, Khot-Regev, Khot-Vishnoy etc.

Gaussian Noise Bounds •Def: For a, b, [0,1] , let

(a, b, := sup {< F,G > | F,G R, [F] =

a, [G] = b}

a, b, := inf {< F,G > | F,G R, [F] = a,

[G] = b}

•Thm: Let X be a finite space. Let T be a reversible Markov operator on X with = (T) < 1.

•Then > 0 > 0 such that for all n and all

f,g : Xn [0,1] satisfying maxi min(Ii(f),

Ii(g)) <

• It holds that <f, g>T (E[f], E[g], ) + and

• <f, g>T (E[f], E[g], ) -

• M-O’Donnell-Oleskiewicz-05 + Dinur-M-Regev-06

Gaussian Noise Bounds •Proof Idea:

• Low influence functions are close to functions in

L2() = L(N1,N2,…).

• Let H[a,b] be:

n{ f : Xn [a, b] | i: Ii(f) < , E[f] = 0, E[f2]

= 1}

•Then: H ““ {f L2() : E[f] =0, E[f2] = 1, a f

b}

• noise forms in H [a,b] ~ noise forms of [a,

b] bounded functions in L2()

An Invariance Principle

• For example, we prove:• Invariance Principle

[M+O’Donnell+Oleszkiewicz(05)]:

• Let p(x) = 0 < |S| · k aS i 2 S xi be a degree k multi-linear polynomial with |p|2 = 1 and Ii(p) for all i.

• Let X = (X1,…,Xn) be i.i.d. P[Xi = 1] = 1/2 .

N = (N1,…,Nn) be i.i.d. Normal(0,1).

• Then for all t:|P[p(X) · t] - P[p(N) · t]| · O(k 1/(4k))

•Note: Noise form “kills” high order monomials.

•Proof works for any hyper-contractive random vars.

Invariance Principle – Proof Sketch

•Suffices to show that 8 smooth F (sup |F(4)| ·

C ), E[F(p(X1,…,Xn)] is close to E[F(p(N1,…,Nn))].

Invariance Principle – Proof Sketch

•Write: p(X1,…,Xi-1, Ni, Ni+1,…,Nn) = R + Ni S

• p(X1,…,Xi-1, Xi, Ni+1,…,Nn) = R + Xi S

• F(R+Ni S) = F(R) + F’(R) Ni S + F’’(R) Ni2 S2/2 +

F(3)(R) Ni3 S3/6 + F(4)(*) Ni

4 S4/24

• E[F(R+ Ni S)] = E[F(R)] + E[F’’(R)] E[Ni2] /2 + E[F(4)(*)Ni

4S4]/24

• E[F(R + Xi S)] = E[F(R)] + E[F’’(R)] E[Xi2] /2 + E[F(4)(*)Xi

4 S4]/24

• |E[F(R + Ni S) – E[F(R + Xi S)| C E[S4]

• But, E[S2] = Ii(p).

• And by Hyper-Contractivity, E[S4] 9k-1 E[S2]

• So: |E[F(R + Ni S) – E[F(R + Xi S) C 9k Ii2

Summary

•Prove the “Peace Sign Conjecture” (Isoperimetry)

• “Plurality is Stablest” (Low Inf Bounds)

• MAX-3-CUT hardness (CS) and voting.

•Other possible application of invariance principle:

•To Convex Geometry?

•To Additive Number Theory?