Noise stability of functions with low influences: Invariance & Optimality

1. Noise stability of functions with low influences: Invariance & Optimality Elchanan Mossel (Statistics, Berkeley) Ryan O'Donnell (Microsoft Research) Krzysztof Oleszkiewicz(Mathematics, Warsaw)

2. Influences and Noise-Stability • The Influence of the i’th variable on f : {-1,1}n! {-1,1} measures how much f depends on the i’th coordinate: Ii(f) := P[f(x1,…,xi-1,-1,xi+1,…,xn)  f(x1,…xi-1,1,xi+1,…,xn)] • Let I(f) := maxi I(f) . • The -Noise-Stability of f : {-1,1}n! R is the correlation between the values of f on two inputs that are -correlated: • S(f) := E[f(x) f(y)] where zi = (xi,yi) are independent with E[xi] = E[yi] = 0 and E[xi yi] = (P[xi = yi] = (1+)/2) • Definition of Ii and I extends to f : {-1,1}n! R by: Ii(f) := E[Vari[f]] = E[ Var[ f | x1,…xi-1,xi+1,…,xn ] ]

3. Low influences, Stability and UGC • Often, using unique-games-conjecture (Khot 2002), after constructing the outer-verifier, • (Very) roughly speaking the hardness of approximation factor is given by c/s where • c = lim! 0 supn,f {S(f) : I(f) ·, E[f] = 0} • s = supn,f E[S(f) : E[f] = 0} for an appropriate value of  (sometimes need variant of S) • s is typically easy to analyze – it is maximized by a dictator. • It is harder to find c.

4. Majority is Stablest • Conj (Khot-Kindler-M-O’Donnell-04) • Thm(M-O’Donnell-Oleskiewicz-05): • “Majority is Stablest”: • For all ¸ 0, lim! 0sup[S(f) : f: {-1,1}n! [-1,1], I(f) ·, E[f] = 0] = (2 arcsin )/ p • Majn(x) := sgn(i=1n xi) has • I(Majn) ! 0 and S(Majn) ! (2 arcsin )/

5. Tight hardness factors assuming UCG • Maj-Stablest + UGC implies: • (From Khot 2002): (1-,1-O(1/2)) hardness for MAX-2-LIN (mod 2) and MAX-2-SAT. • From (Khot-Kindler-M-O’Donnell 2005): • Goemans-Williamson algorithm achieves tight approximation factor (0.878…) for MAX-CUT. • 8 > 0 9 q() such that MAX-2-LIN(mod q) has (1-,) hardness. • MAX-q-CUT has (1-1/q+o(1/q)) hardness factor (matches Frieze & Jerrum semi-definite algorithm).

6. First attempt at Maj-Stablest • Instead of proving it – assume it and let • f : Rk! [-1,1]. • N,M = standard normal vectors & E[Ni Mj] = (i = j). • Define S(f) = E[f(N) f(M)]. • “Majority is Stablest” ) • Thm B:sup { S(f) : E[f] = 0} = 2 arcsin  / . • Pf: Approximate f by fn : {-1,1}k n! {-1,1} with low influences and use Majority is Stablest and the Central Limit Theorem. • Thm B was proven by Borell 85. • The optimizer f is the indicator of a half space. N M

7. From Gaussian to discrete stability • Is there a way to deduce the discrete results from the Gaussian result? • Let’s look at the CLT: • CLT: If |a|2 = 1 and supi |ai| · then • supx |P[i ai xi· x] – P[N · x]| · O() • Different formulation: • Let f : {-1,1}n! R be a linear function: f(x) =  ai xi and • |f|2 = 1. • I(f) ·. • Then supt |P[i ai xi· t] – P[i ai Ni· t]| · O(), where • Ni are i.i.d. Gaussians.

8. From Gaussian to discrete stability • A new limit theorem [M+O’Donnell+Oleszkiewicz(05)]: • Let f : {-1,1}n! R be a degree k multi-linearpolynomial, f(x) = 0 < |S| · k aSi 2 S xi such that • |f|2 = 1 • I(f) ·. • Then for all t: |P[f · t] - P[0 < |S| · k aSi 2 S Ni · t]| · O(k 1/(4k)) • We prove similar result for other discrete spaces. • Generalizes: • CLT • Gaussian chaos results for U and V statistics.

9. A proof sketch : maj is stablest • Idea: Truncate and follow your nose. • Suppose f : {-1,1}n! [-1,1] has small influences but E[f T f] = is large. • Then the same is true for g = T f (() < 1). • Let h= |S| · k gS uS then |h-g|2 is small. • Let h’ = |S| · k gSi 2 S Ni • Then: <h,T h> = <h’, U h’> is large and by the new limit theorem: • h’ is close in L2 to a [-1,1] R.V. • Take g’(x) = h’(x) if |h’(x)| · 1 and g’(x) = sgn(h’(x)). • E[g’ U g’] is too large – contradiction! +

10. A proof sketch : new limit theorem • Recall: p a degree k multi-linear polynomial with: • |p|2 = 1 and Ii(p) · for all i. • Want to show p(x1,…,xn) ~ p(N1,…,Nn). • Suffices to show that 8smoothF ( |F’’’| · C ), E[F(p(x1,…,xn)] is close to E[F(p(N1,…,Nn))]. • Proof similar to Lindberg proof of CLT • Uses Hypercontractivity

11. Other results in the paper • Conj (Kalai-02) Thm: (M-O’Donnell-Oleskiewicz-05): • Majority is Stablest ) “The probability of an Arrow Paradox” among all low influence function is minimized by the majority function. • It is assumed that voters rank • 3 candidates uniformly in S3n • A “paradox” is the event that the • overall preference is A over B over • C over A using an aggregation • function f : {-1,1}n! {-1,1} B A C • Conj (Kalai-01) Thm: (M-O’Donnell-Oleskiewicz-05): • For f with low influences – “it ain’t over until it’s over.” • This means that for every , the probability to be (1-)-“certain” of value of the function given a random fraction  of the inputs goes to 0 as ! 0.

12. Conclusion • We prove new invariance principle that allows to translate stability problems between different settings: • Discrete spaces • Gaussian measures • Spherical measures • For Maj-Is-Stablest Gaussian analogue is known. • Connections suggest interesting future work. • In recent work (Dinur+M+Regev) same philosophy applied to show UGC ) hardness of coloring.