On Sensitivity and Chaos Elchanan Mossel U C

On Sensitivity and Chaos Elchanan Mossel, U. C. Berkeley mossel@stat. berkeley. edu, http: //www. stat. berkeley. edu/~mossel/ 1/6/2022 1

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, 1}n ! {-1, 1}. • Assume below that f(-x 1, …, -xn) = -f(x 1, …, xn) • Two prime examples: – Majority vote, – Electoral college. 1/6/2022 2

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. 1/6/2022 3

Ranking 3 candidates • Each voter tosses a dice. • Vote according to the corresponding order on A, B and C. 1/6/2022 4

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 -1 prob 1 - -1 prob -1 1 prob 1 - 1/6/2022 1 5

Formal model • Assume x is chosen uniformly in {-1, 1}n. • 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) = E[f(x) f(N (x))]? • Which is the most sensitive / stable f? • Is there an f which is both stable and sensible? • Maj? Electoral college? 1/6/2022 6

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[ E[f(x) |y] > 1 - ] be for small ? 1/6/2022 7

Arrow’s Paradox for ranking • Assume x is chosen uniformly in S 3 n • Let x. ABi = 1 if voter i prefers A to B. • Suppose we declare A preferable to B if f(x. AB) = 1 and similarly for the preference of A to C and B to C. • What is the probability of a paradox: • PDX(f) = P[f(x. AB) = f(x. BC) = f(x. CA)]? • Arrow’s: If f is non-trivial – probability is non-zero. • Kalai 02: PDX(f) = ¼ - ¾ S 1/3(f) B A 1/6/2022 C 8

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 theory of error to cases of normal distribution and normal correlation” • An n 1/2 £ n 1/2 electoral college gives Er (f) = ( 1/4). • Machine error ~ 0. 01% ) voting error ~ 10%!. 1/6/2022 9

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 (Arrow’s): • f(x) = xi minimizes the probability of a non-rational outcome. • Exit poll calculations: • For dictator we know the outcome with probability 1 -. 1/6/2022 10

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 i’th variable of f : {-1, 1}n ! {-1, 1} is Ii(f) = P[f(x 1, …, xi, …, xn) f(x 1, …, -xi, …, xn)] • More generally for f : {-1, 1}n ! R let Ii(f) = i 2 S f. S 2. • Examples: Ii(f(x) = xj) = i, j, Ii(Maj) ~ n-1/2 • Notion introduced by Ben. Or-Linial, Later KKL … • From now on look at function with low influences – these are not determined by small # of coordinates. 1/6/2022 X X 11

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 E[f(x) f(y) : 8 i, Ii(f) · , E[f] = a} • s = supn, f E[f(x) f(y) : E[f] = 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. 1/6/2022 12

Some Conjectures – now theorems • • Let I(f) = max Ii(f). Conj (Kalai-01) Thm: (M-O’Donnell-Oleskiewicz-05): For f with low influences – “it ain’t over until it’s over: ” As ! 1, ( ! 0 and ! 0): sup[V (f, ) : I(f) · , E[f] = 0. • Conj (Kalai-02) Thm: (M-O’Donnell-Oleskiewicz-05): • “The probability of an Arrow Paradox” • As ! 0 sup[PDX(f) : I(f) · , E[f] = 0] is minimized by the Majority function • Recall: PDX(f) = ¼ - ¾ S 1/3(f) 1/6/2022 13

Some Conjectures – now theorems • Conj (Khot-Kindler-M-O’Donnell-04) • Thm(M-O’Donnell-Oleskiewicz-05): “Majority is Stablest” • As ! 0 the quantity sup[S (f) : I(f) · , E[f] = 0] = (2 arcsin )/ is maximized by the majority function for all > 0. • Motivated by “MAX-CUT” (more later). • Thm (Dinur-M-Regev): “Large independent set” • Look at {0, 1, 2}n 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. 1/6/2022 14

What is MAX-CUT? • • G = (V, E) C = (Sc, S), partition of V w(C) = |(Sx. Sc) E| w : E ―> R+ weighted unweighted 1/6/2022 15

What is MAX-CUT? • OPT = OPT(G) = maxc {|C|} • MAX-CUT problem: find C with w(C)= OPT • -approximation: find C with w(C) ≥ ·OPT In Khot. Kindler. MO’Donnel, given Unique Games and Maj is Stablest we obtain hardness of MAX-CUT that matches GW factor ~. 878567. • 1/6/2022 1 st time: Hardness result matches semi-definite alg. 16

The Fourier connection • Look at {-1, 1}n. • Define (T f)(x) = E[f(N x) | x] and S (f) = E[f(x) f(N x)] = E[f T f]. • T has the eigenvectors u. S(x) = i 2 S xi, corresponding to the eigenvalues |S|. • Proof: Txi = E[N xi | xi] = (1 - ) xi - xi = xi • {Us} is an orthogonal basis. • f(x) = S fs u. S(x) : = Fourier expansion of f. • fs = E[f*u. S] : = Fourier coefficient at S. • E[f*g] = S fs gs ) S f. S 2 = 1 • Conclusion: S (f) = S f. S 2 |S|. • Thm (Kalai): PDX(f) = ¼ - ¾ S 1/3(f) 1/6/2022 U 1, 2, 3 U 1 17

The Fourier connection • For “it ain’t over until it’s over”. • Look at Zn where Z = {- -1/2, 0, -1/2} with probabilities ( /2, 1 - , /2). • Writing f(x) = S fs u. S(x) we show that • E[f(x) | y] has the same distribution as T 1/2 S fs v. S(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. 1/6/2022 18

Fourier proof – dictator is stablest • • Write f(x) = S f. S u. S(x). E[f*1] = 0 ) f = S ; f. SUS ) Tf = S ; f. S |S| US S (f) = E[f*Tf] = S ; f. S 2 |S| · Dictatorship, f(x) = xi is the only optimal function. = • S (f) = S ; f. S 2 |S| ) in general: stability is determined by how much of the Fourier mass lies on “low degree” coefficients. 1/6/2022 19

Reminder: Majority is Stablest • From now on we will try to prove it … 1/6/2022 20

From discrete to Gaussian stability • Consider the Gaussian measure on Rk. • Suppose f : Rk ! [-1, 1] is “smooth” and EG[f] = 0. • Let fn : {-1, 1}kn ! [-1, 1] be defined by fn(x) = f((x 1+…+xn)/n 1/2, …, (x(k-1)n+1+…+xkn)/n 1/2) • By smoothness: limn ! 1 max 1 · i · kn Ii(fn) = 0. • By the Central Limit Theorem: limn ! 1 EU[fn(x)] = 0 & • limn ! 1 EU[fn*T fn] = EG[f(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) = (X 1, …, XK, Y 1, …, Yk) is a normal vector with • E[Xi Xj] = E[Yi Yj] = i, j and E[Xi Yj] = i, j 1/6/2022 21

From discrete to Gaussian stability 2 • “Majority is Stablest” ) • for all smooth f: Rk ! [-1, 1] with EG[f] = 0: EG[f U f] = EG[f(X) f(Y)] · 1 – (2/ ) arcos • By “density” the same should hold for all f 2 L 2. • Note that we obtain equality when m(x) = sgn(x 1) – 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 Borell 85 using Erhard symmetrization. • Easiest to see via 2 pt symmetrization on the sphere. 1/6/2022 22

From Gaussian to discrete stability • Is there a way to deduce the discrete results from the Gaussian result? • Let’s look at the CLT theorem again: • CLT: If |a|2 = 1 and supi |ai| · then • ai xi ~ N where ~ means – supx |P[ i ai xi · x] – P[N · x]| · and ! 0 as ! 0 • Different formulation: • Let f : {-1, 1}n ! R be a linear function: f(x) = ai xi and – |f|2 = 1. – Ii(f) · for all i. • Then f ~ ai Ni where • Ni are i. i. d. Gaussians. 1/6/2022 23

From Gaussian to discrete stability • A new limit theorem [M+O’Donnell+Oleszkiewicz]: • Let f = 0 < |S| · k a. S i 2 S xi be a degree k polynomial such that – |f|2 = 1 – Ii(f) · for all i. • Then f ~ 0 < |S| · k a. S i 2 S Ni • Similar result for other discrete spaces. • Generalizes: – CLT – Gaussian chaos results for U and V statistics. 1/6/2022 24

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 g. S u. S then |h-g|2 is small. • Let h’ = |S| · k g. S i 2 S Ni • Then: <h, T h> = <h’, U h’> is large and by the new limit theorem: • h’ is close in L 2 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! + 1/6/2022 25

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(x 1, …, xn) ~ p(N 1, …, Nn). Step 1 (classical): Suffices to show that for every smooth F --|F’’’| · C, it holds that • E[F(p(x 1, …, xn)] is close to E[F(p(N 1, …, Nn))]. 1/6/2022 26

Sketch of proof of Lemma • p(…, xi-1, Ni, …) = Ri + Ni Si and p(…, xi, Ni+1, …) = Ri + xi Si • By Calculus and independence: • |E[F(Ri + Ni Si)] – E[F(Ri + xi Si)] |· sup |F’’’| (E[|xi|3] + E[|Ni|3]) E[|Si|3] /6 · C E[|Si|3] • If we could say E[|Si|3] · C’ E[Si 2]3/2 then we’re done since E[Si 2]3/2 = Ii 3/2. • This is a “hyper-contractive” inequality. • So all that is left is to prove: 1/6/2022 27

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. 1/6/2022 28

Conclusion • We’ve 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. – PCP’s – 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? 1/6/2022 29

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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 ~(z 1, …, zn) = (1 – z 1, …, 1 -zn). • The function f is balanced: EUnif[f] = 0. • stronger support in a candidate shouldn’t 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. 1/6/2022 31

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”. 1/6/2022 32