On Sensitivity and Chaos

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On Sensitivity and Chaos
Elchanan Mossel, U.C. Berkeley mossel_at_stat.berkel
ey.edu, http//www.stat.berkeley.edu/mossel/
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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 which option to
  choose.
• Formally, a voting scheme is a function f
  -1,1n ! -1,1.
• Assume below that
• f(-x1,,-xn) -f(x1,,xn)
• Two prime examples
• Majority vote,
• Electoral college.

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A mathematical model of voting

• At the morning of the vote
• Each voter tosses a coin.
• The voters want to vote according
• to the outcome of the coin.

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Ranking 3 candidates

• Each voter tosses a dice.
• Vote according to the corresponding
• order on A,B and C.

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A mathematical model of voting machines

• Which voting schemes are more robust against
  noise?
• Simplest model of noise The voting machine flips
  each vote independently with probability ?.

Registered vote
Intended vote
prob ?
1
-1
prob 1-?
-1
prob ?
-1
1
prob 1-?
1
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Formal model

• Assume x is chosen uniformly in -1,1n.
• Let ? 1-2?.
• Let y N?(x) is obtained from x by flipping each
  of x coordinates with probability ?.
• Equivalently yi xi with probability ?
  otherwise an
• independent coin-toss.
• Question What is the probability that the
  population voted for who they meant to vote for?
• What is S?(f) Ef(x) f(N?(x))?
• Which is the most sensitive / stable f?
• Is there an f which is both stable and sensible?
• Maj? Electoral college?

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Predicting the outcome of the elections

• How many of the votes do we need to see to
  predict the outcome of the elections?
• Let yi 0 (unknown) with probability ? and
• yi xi 1 with probability 1-?.
• How large can
• V?(f,?) Py Ef(x) y gt 1 - ?
• be for small ??

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Arrows Paradox for ranking

• Assume x is chosen uniformly in S3n
• Let xABi 1 if voter i prefers A to B.
• Suppose we declare A preferable to B if
• f(xAB) 1 and similarly for the preference of A
  to C and B to C.
• What is the probability of a paradox
• PDX(f) Pf(xAB) f(xBC) f(xCA)?
• Arrows If f is non-trivial probability is
  non-zero.
• Kalai 02 PDX(f) ¼ - ¾ S1/3(f)

B
C
A
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Examples majority elec. college

• It is easy to calculate that
• For f Majority on n voters
• limn ! 1 Er?(f) ½ - arcsin(1 2 ?)/?
• When ? is small Er?(f) 2 ?1/2/?.
• Machine error 0.01 ) voting error 1.

• Result is essentially due to Sheppard (1899!)
  On the application of the theory of error to
  cases of normal distribution and normal
  correlation
• An n1/2 n1/2 electoral college gives Er?(f)
  ?(?1/4).
• Machine error 0.01 ) voting error 10!.

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Some easy answers

• Noise Theorem (folklore) Dictatorship, f(x) xi
  is the most stable balanced voting scheme.
• In other words, for all schemes
• (error in voting) (error in machines)
• Paradox Theorem (Arrows)
• f(x) xi minimizes the probability of a
  non-rational outcome.
• Exit poll calculations
• For dictator we know the outcome with
• probability 1-?.

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Low influences and democracy

• But in fact, we do not care about Dictators or
  Juntas
• Want functions that depend on all (many)
  coordinates.
• The influence of ith variable of f -1,1n !
  -1,1 is
• Ii(f) Pf(x1,,xi,,xn) ? f(x1,,-xi,,xn)
• More generally for f -1,1n ! R let Ii(f) ?i
  2 S fS2.
• Examples Ii(f(x) xj) ?i,j, Ii(Maj) n-1/2
• Notion introduced by BenOr-Linial, Later KKL
• From now on look at function with low influences
  
• these are not determined by small of
  coordinates.

X
X
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Low influences and PCPs

• Khot 02 suggested a paradigm for proving that
  problems are hard to approximate.
• (Very) Roughly speaking the hardness of
  approximation factor is given by c/s where
• c lim? ! 0 supn,f Ef(x) f(y) 8 i, Ii(f) ?,
  Ef a
• s supn,f Ef(x) f(y) Ef a
• x and y are correlated inputs. The correlation
  between them is related to the problem for which
  one wants to
• prove hardness.
• It is not known if Khots paradigm is equivalent
  to classical NP hardness.
• But the paradigm have given sharp hardness
  factors for many problems.

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Some Conjectures now theorems

• Let I(f) max Ii(f).
• Conj (Kalai-01) Thm (M-ODonnell-Oleskiewicz-05)
  
• For f with low influences it aint over until
  its over
• As ? ! 1, (? ! 0 and ? ! 0)
• supV?(f,?) I(f) ?, Ef 0 0.
• Conj (Kalai-02) Thm (M-ODonnell-Oleskiewicz-05)
  
• The probability of an Arrow Paradox
• As ? ! 0
• supPDX(f) I(f) ?, Ef 0
• is minimized by the Majority function
• Recall PDX(f) ¼ - ¾ S1/3(f)

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Some Conjectures now theorems

• Conj (Khot-Kindler-M-ODonnell-04)
• Thm(M-ODonnell-Oleskiewicz-05) Majority is
  Stablest
• As ? ! 0 the quantity
• supS?(f) I(f) ?, Ef 0 (2 arcsin ?)/
  p
• is maximized by the majority function for all
  ? gt 0.
• Motivated by MAX-CUT (more later).
• Thm (Dinur-M-Regev) Large independent set
• Look at 0,1,2n with edges between x and y if xi
  ? yi for all i. Then an independent set of size
  10-6 has an influencial variable.
• Motivated by hardness of coloring.

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What is MAX-CUT?

• G (V,E)
• C (Sc,S), partition of V
• w(C) (SxSc) ? E
• w E ?gt R

weighted ? unweighted
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What is MAX-CUT?

• OPT OPT(G) maxc C
• MAX-CUT problem find C with w(C) OPT
• ?-approximationfind C with w(C) ?OPT

• In KhotKindlerMODonnel, given Unique Games and
  Maj is Stablest we obtain hardness of MAX-CUT
  that matches GW factor .878567.
• 1st time Hardness result matches semi-definite
  alg.

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The Fourier connection

• Look at -1,1n.
• Define (T?f)(x) Ef(N?x) x and
• S?(f) Ef(x) f(N? x) Ef T? f.
• T? has the eigenvectors uS(x) ?i 2 S xi,
  corresponding to the eigenvalues ?S.
• Proof Txi EN? xi xi (1-?) xi - ? xi ?
  xi
• Us is an orthogonal basis.
• f(x) ?S fs uS(x) Fourier expansion of f.
• fs EfuS Fourier coefficient at S.
• Efg ?S fs gs ) ?S fS2 1
• Conclusion S?(f) ?S fS2 ?S .
• Thm (Kalai) PDX(f) ¼ - ¾ S1/3(f)

U1
U1,2,3
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The Fourier connection

• For it aint over until its over.
• Look at Zn where Z -?-1/2,0,?-1/2 with
  probabilities (?/2,1-?,?/2).
• Writing f(x) ?S fs uS(x) we show that
• Ef(x) y has the same distribution as
• T?1/2 ?S fs vS(z) where
• vs(z) ?i 2 S zi
• T?1/2 vs(z) ?S/2 vs(z).
• Bottom line Have to understand maximizers of
  norms and tail probabilities of general T?
  operators.

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Fourier proof dictator is stablest

• Write f(x) ?S fS uS(x).
• Ef1 0 ) f ?S ? fSUS ) Tf ?S ?
  fS ?S US
• S?(f) EfTf ?S ? fS2 ?S ?
• Dictatorship, f(x) xi is the only optimal
  function.

• S?(f) ?S ? fS2 ?S ) in general stability
  is determined by how much of the Fourier mass
  lies on low degree coefficients.

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Reminder Majority is Stablest

• From now on we will try to prove it

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From discrete to Gaussian stability

• Consider the Gaussian measure on Rk.
• Suppose f Rk ! -1,1 is smooth and EGf
  0.
• Let fn -1,1kn ! -1,1 be defined by
• fn(x) f((x1xn)/n1/2,,(x(k-1)n1xkn)/n1/2
  )
• By smoothness limn ! 1 max1 i kn Ii(fn) 0.
• By the Central Limit Theorem limn ! 1 EUfn(x)
  0
• limn ! 1 EUfnT?fn EGf(X) f(Y), where
• X Gaussian vector and Y U? X.
• U? X ? X (1-?2)1/2 Z, where Z independent
  Gaussian.
• (X,Y) (X1,,XK,Y1,,Yk) is a normal vector with
• EXi Xj EYi Yj ?i,j and EXi Yj ? ?i,j

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From discrete to Gaussian stability 2

• Majority is Stablest )
• for all smooth f Rk ! -1,1 with EGf 0
• EGf U? f EGf(X) f(Y) 1
  (2/?) arcos ?
• By density the same should hold for all f 2 L2.
• Note that we obtain equality when m(x) sgn(x1)
  
• m(x) is the limit of the majority functions.
• So Majority is Stablest implies Gaussian
  results that may be easier to verify.
• Indeed was proved by Borell85 using Erhard
  symmetrization.
• Easiest to see via 2pt symmetrization on the
  sphere.

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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 theorem again
• CLT If a2 1 and supi ai ? then
• ? ai xi N where means
• supx P?i ai xi x PN x ? and ? ! 0
  as ? ! 0
• Different formulation
• Let f -1,1n ! R be a linear function f(x)
  ? ai xi and
• f2 1.
• Ii(f) ? for all i.
• Then f ? ai Ni where
• Ni are i.i.d. Gaussians.

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From Gaussian to discrete stability

• A new limit theorem MODonnellOleszkiewicz
• Let f ?0 lt S k aS ?i 2 S xi be a degree k
  polynomial such that
• f2 1
• Ii(f) ? for all i.
• Then f ?0 lt S k aS ?i 2 S Ni
• 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).
• Step 1 (classical) Suffices to show that for
  every smooth F --F C, it holds that
• EF(p(x1,,xn) is close to EF(p(N1,,Nn)).

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Sketch of proof of Lemma

• p(,xi-1,Ni,) Ri Ni Si and p(,xi,Ni1,)
  Ri xi Si
• By Calculus and independence
• EF(Ri Ni Si) EF(Ri xi Si)
• sup F (Exi3 ENi3) ESi3 /6
  C ESi3
• If we could say ESi3 C ESi23/2
• then were done since ESi23/2 Ii3/2.
• This is a hyper-contractive inequality.
• So all that is left is to prove

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Sketch of proof of Lemma

• This is a standard argument.
• Somewhat similar results (same proof idea)
• Rotar (75) Slightly different setting. No
  Berry-Essen bounds. Lindenberg conditions instead
  of hypercontractivity.
• Chaterjee (04) Elegant but conditions are too
  strong uses worst case influences instead of
  average case.

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Conclusion

• Weve seen how Gaussian techniques can help solve
  discrete stability problems.
• Future work
• Better understanding of the dependency on all
  parameters for general prob. spaces.
• Applications to
• Social choice.
• PCPs
• Learning.
• Sometimes need better Gaussian understanding.
• Example
• Suppose we want to partition Gaussian space to 3
  parts of equal measure what is the most stable
  way?

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Properties of voting schemes

• Some properties of voting schemes
• Some properties we may require from voting
  schemes
• The function f is anti-symmetric f(x) f(x)
  where (z1,,zn) (1 z1,,1-zn).
• The function f is balanced EUniff 0.
• stronger support in a candidate shouldnt hurt
  her
• The function f is monotone x y ) f(x) f(y),
  where x y if xi yi for all i.
• Note that both majority and the electoral college
  are anti-symmetric and monotone.

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Stability of voting schemes

• Which voting schemes are more robust against
  noise?
• Simplest model of noise The voting machine flips
  each vote independently with probability ? (not
  realistic).
• Simplest model of voter distribution i.i.d.
  distribution where each voter votes 0/1 with
  probability ½.
• Very far from reality
• Buy maybe good model for critical voting.