Noise stability of functions with low influences: Invariance & Optimality Elchanan Mossel (Statistics, Berkeley) Ryan O'Donnell (Microsoft Research) Krzysztof Oleszkiewicz (Mathematics, Warsaw) 11/14/2018

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 ] ] 11/14/2018

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. 11/14/2018

Majority is Stablest Conj (Khot-Kindler-M-O’Donnell-04) Thm(M-O’Donnell-Oleskiewicz-05): “Majority is Stablest”: For all  ¸ 0, lim ! 0 sup[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 )/ 11/14/2018

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). 11/14/2018

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 11/14/2018

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. 11/14/2018

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-linear polynomial, 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. 11/14/2018

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! + 11/14/2018

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 8 smooth F ( |F’’’| · C ), E[F(p(x1,…,xn)] is close to E[F(p(N1,…,Nn))]. Proof similar to Lindberg proof of CLT Uses Hypercontractivity 11/14/2018

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. 11/14/2018

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. 11/14/2018

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