Non Linear Invariance Principles Non Linear Invariance Principles with Applicationswith Applications

Elchanan MosselElchanan MosselU.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=1

n 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) Ni

2 S2/2 + F(3)(R) Ni

3 S3/6 + F(4)(*) Ni4 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?