Non Linear Invariance Principles with Applications Elchanan Mossel U.C. Berkeley

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 : R n  R m. Define  := E[ ], where N,M ~ Normal(0,I) with E[N i M j ] =   (i,j). For sets A,B let:  :=  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  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 R n 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  +  +  among all partitions (A,B,C) of R n 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  L 2 (X n,R). The i’th influence of f is given by: I i (f) := E[ Var[f | x 1,…,x i-1,x i+1,…,x n ] ] (Ben-Or,Kalai,Linial, Efron-Stein 80s) Given a reversible Markov operator T on X and f, g: X n  R define the T - noise form by T := E[f T  n g] The 2 nd 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 x j : I i (x j ) =  (i,j). For the majority m(x) = sgn(  1  i  n x i ) function: I i (m)  (2  n) -1/2. Let T be the “Beckner Operator” on X: T i,j =   (i,j) + (1-  )/2. T x i =  x i and T = . 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(-x 1,…,-x n ) = -f(x 1,…,x n ) 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 .   = correlation of intended vote with actual outcome. A model of voting machines Intended vote Registered vote 1 prob1-   1 1-   1

  2 arcsin  /  [n   ]  1 – c(1-  ) 1/2 [   1] for m(x) = majority(x) = sgn(  i=1 n x i ) Result is essentially due to Sheppard (1899): “On the application of the theory of error to cases of normal distribution and normal correlation”. For n 1/2 £ n 1/2 electoral college f   1- c (1-  ) 1/4 [n  ,   1] -1/2 determined prob. of Condorcet Paradox (Kalai) Majority and Electoral College

Noise Theorem (folklore): Dictatorship, f(x) = x i 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    =  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: I i (f) = P[f(x 1,…,x i,…,x n )  f(x 1,…,- x i,…,x n )] <  where n   and   0. An easy answer and a hard question X

Let X = {0,1,2} with the uniform measure. Let T i,j = ½  (i  j) Then  (T) = ½ and Claim (Colouring Graph): Consider X n as a graph where (x,y)  Edges(X n ) iff x i  y i for all i. Let A,B  X n. Then T = 0 iff there are no edges between A and B. In particular, A is an independent set iff 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 x i  y i 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 ( T = 0) and |A|,|B| > 3 n /Q then there exists an i such that I i (A) >  and I i (B) >  Then ApxColor(3,Q) is NP hard.

Graph Colouring – An Algorithmic Problem u

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)  R 3. Let d : X n  R 3 defined by d(x) :=  x(1) Let p : X n  R defined by p(x) =  y where y is the most frequent value among the x i. I i (d) = 2/3  (i,1); I i (p)  c n -1/2. For 0    1, let T be the Markov operator on X defined by T i,j =   (i,j) + (1-  )/3. T =  Var(d).

Gaussian Noise Bounds Def: For a, b,   [0,1], let  (a, b,  := sup {  | F,G  R,  [F] = a,  [G] = b}  a, b,  := inf {  | 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 : X n  [0,1] satisfying max i min(I i (f), I i (g)) <  It holds that T   (E[f], E[g],  ) +  and T   (E[f], E[g],  ) -  M-O’Donnell-Oleskiewicz-05 + Dinur-M-Regev-06

Example 1 Taking T on {-1,1} defined by T i,j =   (i,j) + (1-  )/2 Thm  : Claim:  f : {-1,1} n  {-1,1} with I i (f) <  for all i and E[f] = 0 it holds that: T   +  where F(x) = sgn(x)  = 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 “  ” tight in-approximation result for MAX-CUT. Khot-Kindler-M-O’Donnell-05

Example 2 Taking T on {0,1,2} defined by T i,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. I i (A), I i (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 T i,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 I i (f) <  for all i, it holds that    lim n   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  R,  [F] = a,  [G] = b}  a, b,  := inf {  | 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 : X n  [0,1] satisfying max i min(I i (f), I i (g)) <  It holds that T   (E[f], E[g],  ) +  and 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 L 2 (  ) = L  (N 1,N 2,…). Let H  [a,b] be:  n { f : X n  [a, b] |  i: I i (f) < , E[f] = 0, E[f 2 ] = 1} Then: H  “  “ {f  L 2 (  ) : E[f] =0, E[f 2 ] = 1, a  f  b}  noise forms in H  [a,b] ~ noise forms of [a, b] bounded functions in L 2 (  )

An Invariance Principle For example, we prove: Invariance Principle [ M+O’Donnell+Oleszkiewicz(05)]: Let p(x) =  0 < |S| · k a S  i 2 S x i be a degree k multi- linear polynomial with |p| 2 = 1 and I i (p)   for all i. Let X = (X 1,…,X n ) be i.i.d. P[X i =  1] = 1/2. N = (N 1,…,N n ) 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(X 1,…,X n )] is close to E[F(p(N 1,…,N n ))].

Invariance Principle – Proof Sketch Write: p(X 1,…,X i-1, N i, N i+1,…,N n ) = R + N i S p(X 1,…,X i-1, X i, N i+1,…,N n ) = R + X i S F(R+N i S) = F(R) + F’(R) N i S + F’’(R) N i 2 S 2 /2 + F (3) (R) N i 3 S 3 /6 + F (4) (*) N i 4 S 4 /24 E[F(R+ N i S)] = E[F(R)] + E[F’’(R)] E[N i 2 ] /2 + E[F (4) (*)N i 4 S 4 ]/24 E[F(R + X i S)] = E[F(R)] + E[F’’(R)] E[X i 2 ] /2 + E[F (4) (*)X i 4 S 4 ]/24 |E[F(R + N i S) – E[F(R + X i S)|  C E[S 4 ] But, E[S 2 ] = I i (p). And by Hyper-Contractivity, E[S 4 ]  9 k-1 E[S 2 ] So: |E[F(R + N i S) – E[F(R + X i S)  C 9 k I i 2

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?