1. Noise stability of functions with low influences:
Invariance & Optimality
Elchanan Mossel (Statistics, Berkeley)
Ryan O'Donnell (Microsoft Research)
Krzysztof Oleszkiewicz (Mathematics, Warsaw)
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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 ] ]
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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.
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4. 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 )/
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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).
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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
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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.
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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-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.
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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! +
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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 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
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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.
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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.
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