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Non Linear Invariance Principles with Applications. Elchanan Mossel U.C. Berkeley http://stat.berkeley.edu/~mossel. Lecture Plan. Background: Noise Stability in Gaussian Spaces Noise := Ornstein-Uhlenbeck process. Bubbles and half-spaces.
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Non Linear Invariance Principles with Applications Elchanan Mossel U.C. Berkeley http://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 withn(A) = 1the minimizer ofn-1( A)is A = Ball. • II. Recent (Borell, Sudakov-Tsierlson 70’s) Among all sets withn(A) = athe minimizer ofn-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.
Thm1 (“Double-Bubble”): • Among all pairs of disjoint sets A,Bwith n(A) = n(B) = a, the minimizer of n-1( A B) is a “Double Bubble” • Thm2(“Peace Sign”): • Among all partitions A,B,C ofRnwith (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 Double bubbles
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 Rnwith 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). Thei’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}
Influences and Noise in product Spaces – Example 1 • Let X = {-1, 1} with the uniform measure. • For the dictator function xj: Ii(xj) = (i,j). • For the majoritym(x) = sgn(1 i n xi) function: Ii(m) (2 n)-1/2. • Let Tbe the “Beckner Operator” on X: • Ti,j = (i,j) + (1-)/2. • T xi = xiand <xi, xi>T = . • <m, m>T ~ 2 arcsin() / • (T) = .
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.
A voting model • At the morning of the vote: • Each voter tosses a coin. • The voters vote according • to the outcome of the coin.
A model of voting machines • Which voting schemes are more robust against noise? • Simplest model of noise: The voting machine flips each vote independently with probability . • <f, f>1-2 = correlation of intended vote with actual outcome. Registered vote Intended vote 1 prob e -1 -1 prob 1 - e -1 prob e 1 1 prob 1 - e
Majority and Electoral College • <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)
An easy answer and a hard question • 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 monotonef. • 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. X
Influences and Noise in product Spaces – Example 2 • Let X = {0,1,2} with the uniform measure. • Let Ti,j = ½ (i j) • Then (T) = ½ and • Claim (Colouring Graph): ConsiderXn 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?
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 ani such that Ii(A) > and Ii(B) > . • Then ApxColor(3,Q) is NP hard.
[u] Graph Colouring – An Algorithmic Problem 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 yis 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, r) := sup {< F,G > | F,G R, [F] = a, [G] = b} • (a, b, r) := 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 Xwith = (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>T 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, r) := sup {< F,G > | F,G R, [F] = a, [G] = b} • (a, b, r) := 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 Xwith = (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() = L2(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 aSi 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)(*) Ni4 S4/24 • E[F(R+ Ni S)] = E[F(R)] + E[F’’(R)] E[Ni2] /2 + E[F(4)(*)Ni4S4]/24 • E[F(R + Xi S)] = E[F(R)] + E[F’’(R)] E[Xi2] /2 + E[F(4)(*)Xi4 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?