Noise stability of functions with low influences:

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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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Influences and Noise-Stability

• The Influence of the ith variable on f -1,1n
  ! -1,1 measures how much f depends on the ith
  coordinate
• Ii(f) Pf(x1,,xi-1,-1,xi1,,xn) ?
  f(x1,xi-1,1,xi1,,xn)
• Let I(f) maxi I(f) .
• The ?-Noise-Stability of f -1,1n ! R is the
  correlation between the values of f on two inputs
  that are ?-correlated
• S?(f) Ef(x) f(y) where zi (xi,yi) are
  independent with Exi Eyi 0 and Exi yi
  ? (Pxi yi (1?)/2)
• Definition of Ii and I extends to f -1,1n ! R
  by
• Ii(f) EVarif E Var f
  x1,xi-1,xi1,,xn

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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) ?, Ef 0
• s supn,f ES?(f) Ef 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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Majority is Stablest

• Conj (Khot-Kindler-M-ODonnell-04)
• Thm(M-ODonnell-Oleskiewicz-05)
• Majority is Stablest
• For all ? 0,
• lim? ! 0 supS?(f) f -1,1n ! -1,1, I(f)
  ?, Ef 0
• (2 arcsin ?)/ p
• Majn(x) sgn(?i1n xi) has
• I(Majn) ! 0 and S?(Majn) ! (2 arcsin ?)/?

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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-ODonnell 2005)
• Goemans-Williamson algorithm achieves tight
  approximation factor (0.878) for MAX-CUT.
• 8 ? gt 0 9 q(?) such that
• MAX-2-LIN(mod q) has (1-?,?) hardness.
• MAX-q-CUT has (1-1/qo(1/q)) hardness factor
  (matches Frieze Jerrum semi-definite
  algorithm).

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First attempt at Maj-Stablest

• Instead of proving it assume it and let
• f Rk ! -1,1.
• N,M standard normal vectors ENi Mj ? ?(i
  j).
• Define S?(f) Ef(N) f(M).
• Majority is Stablest )
• Thm B sup S?(f) Ef 0 2 arcsin ? / ?.
• Pf Approximate f by fn -1,1k 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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From Gaussian to discrete stability

• Is there a way to deduce the discrete results
  from the Gaussian result?
• Lets look at the CLT
• CLT If a2 1 and supi ai ? then
• supx P?i ai xi x PN x O(?)
• Different formulation
• Let f -1,1n ! R be a linear function f(x)
  ? ai xi and
• f2 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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From Gaussian to discrete stability

• A new limit theorem MODonnellOleszkiewicz(05)
  
• Let f -1,1n ! R be a degree k multi-linear
  polynomial,
• f(x) ?0 lt S k aS ?i 2 S xi such that
• f2 1
• I (f) ?.
• Then for all t
• Pf t - P?0 lt 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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A proof sketch maj is stablest

• Idea Truncate and follow your nose.
• Suppose f -1,1n ! -1,1 has small influences
  but Ef T? f is large.
• Then the same is true for g T? f (?(?) lt 1).
• Let h ?S k gS uS then h-g2 is small.
• Let h ?S k gS ?i 2 S Ni
• Then lth,T? hgt lth, U? hgt 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)).
• Eg U? g is too large contradiction!

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A proof sketch new limit theorem

• Recall p a degree k multi-linear polynomial
  with
• p2 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 ),
  EF(p(x1,,xn) is close to EF(p(N1,,Nn)).

• Proof similar to Lindberg proof of CLT
• Uses Hypercontractivity

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Other results in the paper

• Conj (Kalai-02) Thm (M-ODonnell-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 -1,1n ! -1,1

B
C
A

• Conj (Kalai-01) Thm (M-ODonnell-Oleskiewicz-05)
  
• For f with low influences it aint over until
  its 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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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 (DinurMRegev) same philosophy
  applied to show UGC ) hardness of coloring.

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