Ryan O’Donnell Carnegie Mellon University

Part 1: A. Fourier expansion basics B. Concepts: Bias, Influences, Noise Sensitivity C. Kalai’s proof of Arrow’s Theorem

10 Minute Break

Part 2: A. The Hypercontractive Inequality B. Algorithmic Gaps

Sadly no time for: Learning theory Pseudorandomness Arithmetic combinatorics Random graphs / percolation Communication complexity Metric / Banach spaces Coding theory etc.

1A. Fourier expansion basics

f : {0,1} n  {0,1}

f : {−1,+1} n  {−1,+1}

ℝ3ℝ3 (+1,+1,+1) (−1,−1,−1) (+1,+1,−1) (+1,−1,+1) (−1,+1,+1) −1 +1 −1

ℝ3ℝ3 (+1,+1,+1) (−1,−1,−1) +1 −1

ℝ3ℝ3 (+1,+1,+1) (−1,−1,−1) −1 +1 −1

ℝ3ℝ3 (+1,+1,+1) (−1,−1,−1) +1 −1

ℝ3ℝ3 (+1,+1,+1) (−1,−1,−1) +1

ℝ3ℝ3 (+1,+1,+1) (−1,−1,−1) −1

ℝ3ℝ3 (+1,+1,+1) (−1,−1,−1) −1 +1 −1 +1 −1

ℝ3ℝ3 (+1,+1,+1) (−1,−1,−1) −1 +1 −1 +1 −1 +1 −1

ℝ3ℝ3 (+1,+1,+1) (−1,−1,−1) +1 −1 +1 −1

ℝ3ℝ3 (+1,+1,+1) (−1,−1,−1) −1 +1 −1

(+1,+1,+1) +1 −1 +1 −1 (+1,+1,−1) (+1,−1,−1)

=

=

Proposition: Every f : {−1,+1} n  {−1,+1} can be expressed as a multilinear polynomial, That’s it. That’s the “Fourier expansion” of f. (uniquely) (indeed, → ℝ )

Proposition: Every f : {−1,+1} n  {−1,+1} can be expressed as a multilinear polynomial, That’s it. That’s the “Fourier expansion” of f. (uniquely) (indeed, → ℝ )

⇓ Rest: 0

Why? Coefficients encode useful information. When? 1. Uniform probability involved 2. Hamming distances relevant

Parseval’s Theorem: Let f : {−1,+1} n  {−1,+1}. Then avg { f(x) 2 }

“Weight” of f on S ⊆ [n] =

{2} {1} ∅ {3} {1,3}{1,2}{2,3} {1,2,3}

{2} {1} ∅ {3} {1,3}{1,2}{2,3} {1,2,3}

{2} {1} ∅ {3} {1,3}{1,2}{2,3} {1,2,3}

{2} {1} ∅ {3} {1,3}{1,2}{2,3} {1,2,3}

{2} {1} ∅ {3} {1,3}{1,2}{2,3} {1,2,3}

{2} {1} ∅ {3} {1,3}{1,2}{2,3} {1,2,3}

1B. Concepts: Bias, Influences, Noise Sensitivity

Social Choice: Candidates ±1 n voters Votes are random f : {−1,+1} n  {−1,+1} is the “voting rule”

Bias of f: avg f(x) = Pr[+1 wins] − Pr[−1 wins] Fact: Weight on ∅ = measures “imbalance”.

Influence of i on f: Pr[ f(x) ≠ f(x ( ⊕ i) ) ] = Pr[voter i is a swing voter] Fact:

{2} {1} ∅ {3} {1,3}{1,2}{2,3} {1,2,3} Maj(x 1, x 2, x 3 )

+1 −1 Inf i (f) = Pr[ f(x) ≠ f(x ( ⊕ i) ) ]

+1 −1 Inf i (f) = Pr[ f(x) ≠ f(x ( ⊕ i) ) ]

avg Inf i (f) = frac. of edges which are cut edges

LMN Theorem: If f is in AC 0 then avg Inf i (f)

⇒ avg Inf i (Parity n ) = 1 ⇒ Parity ∉ AC 0 ⇒ avg Inf i (Maj n ) = ⇒ Majority ∉ AC 0

KKL Theorem: If Bias(f) = 0, then Corollary: Assuming f monotone, −1 or +1 can bribe o(n) voters and win w.p. 1−o(1).

Noise Sensitivity of f at : NS(f) = Pr[wrong winner wins], when each vote misrecorded w/prob f( ) ) +−++−−+−− −−+++++−−

Learning Theory principle: [LMN’93, …, KKMS’05] If all f ∈ C have small NS(f) then C is efficiently learnable.

{2} {1} ∅ {3} {1,3}{1,2}{2,3} [3]

Proposition: for small, with Electoral College: 10 1

1C. Kalai’s proof of Arrow’s Theorem

Ranking 3 candidates Condorcet [1775] Election: => (x_i, y_i, z_i) are Not All Equal (no ) Condorcet: Try f = Maj. Outcome can be “irrational” A > B > C > A. [easy eg] Maybe some other f? A > B? B > C? C > A?

Ranking 3 candidates Condorcet [1775] Election: => (x_i, y_i, z_i) are Not All Equal (no ) Condorcet: Try f = Maj. Outcome can be “irrational” A > B > C > A. [easy eg] Maybe some other f? A > B? B > C? C > A? “ C > A > B ” “ A > B > C ” “ B > C > A ” + − − + + − + − − − + − − − + − + − + − + − + +

Ranking 3 candidates Condorcet [1775] Election: => (x_i, y_i, z_i) are Not All Equal (no ) Condorcet: Try f = Maj. Outcome can be “irrational” A > B > C > A. [easy eg] Maybe some other f? A > B? B > C? C > A? “ C > A > B ” “ A > B > C ” “ B > C > A ” f() ) ) = + = + = − Society: “A > B > C” + − − + + − + − − − + − − − + − + − + − + − + +

Ranking 3 candidates Condorcet [1775] Election: => (x_i, y_i, z_i) are Not All Equal (no ) Condorcet: Try f = Maj. Outcome can be “irrational” A > B > C > A. [easy eg] Maybe some other f? A > B? B > C? C > A? “ C > A > B ” “ A > B > C ” “ B > C > A ” f() ) ) = + = + = − Society: “A > B > C” + − − + + − + − + − + − − − + − + − + − + − + +

Ranking 3 candidates Condorcet [1775] Election: => (x_i, y_i, z_i) are Not All Equal (no ) Condorcet: Try f = Maj. Outcome can be “irrational” A > B > C > A. [easy eg] Maybe some other f? “ C > A > B ” “ A > B > C ” “ B > C > A ” Society: “A > B > C” A > B? B > C? C > A? f() ) ) = + = + = + + − − + + − + − + − + − − − + − + − + − + − + +

Ranking 3 candidates Condorcet [1775] Election: => (x_i, y_i, z_i) are Not All Equal (no ) Condorcet: Try f = Maj. Outcome can be “irrational” A > B > C > A. [easy eg] Maybe some other f? “ C > A > B ” “ A > B > C ” “ B > C > A ” Society: “A > B > C > A” ? A > B? B > C? C > A? f() ) ) = + = + = + + − − + + − + − + − + − − − + − + − + − + − + +

Arrow’s Impossibility Theorem [1950]: If f : {−1,+1} n  {−1,+1} never gives irrational outcome in Condorcet elections, then f is a Dictator or a negated-Dictator.

Gil Kalai’s Proof [2002]:

“ C > A > B ” “ A > B > C ” “ B > C > A ” A > B? B > C? C > A? f() ) ) = + = + = − + − − + + − + − − − + − − − + − + − + − + − + +

“ C > A > B ” “ A > B > C ” “ B > C > A ” A > B? B > C? C > A? f() ) ) = + = + = − + − − + + − + − − − + − − − + − + − + − + − + +

Gil Kalai’s Proof:

Gil Kalai’s Proof, concluded: f never gives irrational outcomes ⇒ equality ⇒ all Fourier weight “at level 1” ⇒ f(x) = ±x j for some j (exercise).

⇓ Guilbaud’s Theorem [1952] Guilbaud’s Number ≈.912

Corollary of “Majority Is Stablest” [MOO05]: If Inf i (f) ≤ o(1) for all i, then Pr[rational outcome with f]

Part 2: A. The Hypercontractive Inequality B. Algorithmic Gaps

2A. The Hypercontractive Inequality AKA Bonami-Beckner Inequality

KKL TheoremFriedgut’s TheoremTalagrand’s TheoremEvery monotone graph property has a sharp thresholdFKN TheoremBourgain’s Junta TheoremMajority Is Stablest Theorem all use “Hypercontractive Inequality”

Hoeffding Inequality: Let F = c 0 + c 1 x 1 + c 2 x 2 + ··· + c n x n, where x i ’s are indep., unif. random ±1.

Mean: μ = c 0 Variance: Hoeffding Inequality: Let F = c 0 + c 1 x 1 + c 2 x 2 + ··· + c n x n,

Mean: μ = Variance: Hypercontractive Inequality*: Let

Then for all q ≥ 2, Hypercontractive Inequality: Let

Then F is a “reasonable d ” random variable. Hypercontractive Inequality: Let

Then for all q ≥ 2, Hypercontractive Inequality: Let

Then “q = 4” Hypercontractive Inequality: Let

Then “q = 4” Hypercontractive Inequality: Let

KKL TheoremFriedgut’s TheoremTalagrand’s TheoremEvery monotone graph property has a sharp thresholdFKN TheoremBourgain’s Junta TheoremMajority Is Stablest Theorem all use Hypercontractive Inequality

KKL TheoremFriedgut’s TheoremTalagrand’s TheoremEvery monotone graph property has a sharp thresholdFKN TheoremBourgain’s Junta TheoremMajority Is Stablest Theorem just use “q = 4” Hypercontractive Inequality

“q = 4” Hypercontractive Inequality: Let F be degree d over n i.i.d. ±1 r.v.’s. Then Proof [MOO’05] : Induction on n. Obvious step. Use induction hypothesis. Use Cauchy-Schwarz on the obvious thing. Use induction hypothesis. Obvious step.

2B. Algorithmic Gaps

Opt best poly-time guarantee ln(N) “Set-Cover is NP-hard to approximate to factor ln(N)”

Opt LP-Rand-Rounding guarantee ln(N) “Factor ln(N) Algorithmic Gap for LP-Rand-Rounding”

Opt( S ) LP-Rand-Rounding( S ) ln(N) “Algorithmic Gap Instance S for LP-Rand-Rounding”

Algorithmic Gap instances are often “based on” {−1,+1} n.

Sparsest-Cut: Algorithm: Arora-Rao-Vazirani SDP. Guarantee:Factor

Opt = 1/n

f(x) = sgn( )

Opt = 1/n f(x) = sgn(r 1 x r n x n ) ARV gets

Opt = 1/n ARV gets gap:

Algorithmic Gaps → Hardness-of-Approx LP / SDP-rounding Alg. Gap instance n optimal “Dictator” solutions “generic mixture of Dictators” much worse + PCP technology = same-gap hardness-of-approximation

Algorithmic Gaps → Hardness-of-Approx LP / SDP-rounding Alg. Gap instance n optimal “Dictator” solutions “generic mixture of Dictators” much worse + PCP technology = same-gap hardness-of-approximation

KKL / Talagrand Theorem: If f is balanced, Inf i (f) ≤ 1/n.01 for all i, then avg Inf i (f) ≥ Gap: Θ(log n) = Θ(log log N).

[CKKRS05]: KKL + Unique Games Conjecture ⇒ Ω(log log log N) hardness-of-approx.

2-Colorable 3-Uniform hypergraphs: Input:2-colorable, 3-unif. hypergraph Output:2-coloring Obj:Max. fraction of legally colored hyperedges

2-Colorable 3-Uniform hypergraphs: Algorithm: SDP [KLP96]. Guarantee: [Zwick99]

Algorithmic Gap Instance Vertices:{−1,+1} n 6 n hyperedges:{ (x,y,z) : poss. prefs in a Condorcet election} (i.e., triples s.t. (x i,y i,z i ) NAE for all i)

Elts: {−1,+1} n Edges: Condorcet votes (x,y,z) 2-coloring = f : {−1,+1} n → {−1,+1} frac. legally colored hyperedges = Pr[“rational” outcome with f] Instance 2-colorable? ✔ (2n optimal solutions: ±Dictators)

Elts: {−1,+1} n Edges: Condorcet votes (x,y,z) SDP rounding alg. may output Random weighted majority also rational-with-prob.-.912! [same CLT arg.] f(x) = sgn(r 1 x r n x n )

Algorithmic Gaps → Hardness-of-Approx LP / SDP-rounding Alg. Gap instance n optimal “Dictator” solutions “generic mixture of Dictators” much worse + PCP technology = same-gap hardness-of-approximation

Corollary of Majority Is Stablest: If Inf i (f) ≤ o(1) for all i, then Pr[rational outcome with f] Cor: this + Unique Games Conjecture ⇒.912 hardness-of-approx*

2C. Future Directions

Develop the “structure vs. pseudorandomness” theory for Boolean functions.