Ryan O’Donnell Microsoft Theory Group

1. Stability & Chaos in Boolean Functions Ryan O’Donnell Microsoft Theory Group

2. 2004 Governor’s Election in WA vs. Ruth Bennett (L) – about 2% Christine Gregoire (D) Dino Rossi (R) Votes: 1,371,153 1,371,414 (49.995%) (50.005%) 1,372,442 1,372,484 (49.9992%) (50.0008%) 1,373,361 1,373,232 (50.002%) (49.998%) Rossi lead by 10−5 fraction! Governor

3. A simple model for 2-party elections • n voters, each i.i.d. voting for +or − • , an election “function” + − + + − − + − + + + + − n f ( ) = + (winner)

4. Noise sensitivity and stability • n voters, each i.i.d. voting for +or − • , an election “function” + − + + − − + − + + + + − n f ( ) = + (true winner) (winner) (errors introduced in counting) f ( ) + − − + − − −+ + + + − − = − (declared winner)

5. Noise sensitivity of Majority Fact: • error in the WA state recount seemed to be on the order of about 1..5  10−4 • (counting one ballot per sec, that’s about one mistake per hour) • (much greater than Rossi’s/Gregoire’s margins of victory, about 1..5  10−5!) • probability counting errors affect a typical Maj-based vote: about 10−2 = 1%

6. Electoral College EC: Break voters into m blocks of n/m voters, take recursive Maj. ≈ 5% if ε≈1..5  10−4

7. 1996 General Election in BC vs. Glen Clark (NDP) Gordon Campbell (Lib) Votes: 625,395 661,929 (48.5%) (51.5%) Seats: 39 33 Premier

8. Noise sensitivity applications • voting systems (“social choice” in economics) computational • learning theory • hardness amplification (complexity theory) • combinatorics (set systems, independent sets) • embeddability of metric spaces • hardness of approximation & “Long Code tests”

9. Plan for the rest of the talk “Which function is least noise-sensitive?” • formulate the question in a workable way • mention why it’s useful to know the answer • say a little about the proof

10. Which function is least noise sensitive? Fix 0 < ε < ½ and n >> 1/ε. Question: Which hassmallest NoiseSensε( f ) ? Answer: Constant functions (e.g., f (x) = +1) have NoiseSensε( f ) = 0. So let’s assume f is “balanced”, Pr[f = −1] = Pr[f = +1] = ½.

11. Which function is least noise sensitive? Question: Which balanced function is least noise sensitive? Answer: “Dictator functions” f (x) = ± xj NoiseSensε( f ) = ε Unsatisfying – these aren’t “really” functions on n bits. (Very unsatisfying from the p.o.v. of elections.) Want to exclude them, but a bit hard to quantify…

12. Which function is leastnoise sensitive? Idea: each voter should be “rarely relevant” – i.e., each voter i should have “small influence” on f. • Infi( j-Dictator ) = δij Infi(Maj) = • Infi(EC) = Infi(Parity) = 1

13. Which function is leastnoise sensitive? Khot-Kindler-Mossel-O ’04, in a paper about hardness of approximation for MAX-CUT: “Conjecture:  0 < ε < ½, if f balanced and Infi(f) = o(1) for all i, then NoiseSensε( f ) > NoiseSensε(Maj) − o(1).”

14. Majority Is Stablest [Mossel-O-Oleszkiewicz-’05] proves this conjecture, along with some generalizations. Consequences: • .878 hardness for MAX-CUT subject to UGC, other tight UGC-hardness results, equivalence of UGC and MAX-2LIN(q) hardness [KKMO’04] • resolution of some social choice questions of Kalai (“It Ain’t Over Till It’s Over”, “most rational” social choice function) • resolves combinatorics questions re. indep. sets in product graphs [ADFS’04] • UGC-hardness of C-colouring 3-colourable graphs, for all const. C [DMR’05] • stronger non-embeddability results for l1[KN’05, KV’05]

15. A bit about the proof • View boolean cube as {−1, +1}n  ( -radius sphere in Rn) • Make a graph on it, connecting x and yif their dot product is about (1−2ε)n. • balanced boolean function f ½-cut in the graph • noise sensitivity  cut value • “halfspace cut”: f = sgn( ci xi). E.g., dictator, Maj. • “Except for cuts unfairly correlated with dictators, min-bisection is achieved by Maj (or any random halfspace).”

16. A bit about the proof • This is true with no caveats on the n-dim. sphere! [follows fairly easily from Baernstein-Taylor ’76, folklorically known in the ’80s, written explicitly in Feige-Schechtman ’01] • Also true in “Gaussian space” (Rn with the Gaussian probability measure).[Borell ’85] • Proof idea: pass from ±1 setting to Gaussian setting. • Inspiration: Central Limit Theorem – sum of n i.i.d. ±1random variables is distributed like a Gaussian.

17. Central Limit Theorem Let Let G= [a standard Gaussian] If ci2 = o(1) for all i, then X and G are o(1)-close. [ for all t, |Pr[X < t] – Pr[G < t]| = o(1). ]

18. Every function can be expressed as a multilinear polynomial in n real variables: (“Fourier expansion”) Some neat formulas hold:

19. Central Limit Theorem Let Let G= [a standard Gaussian] If ci2 = o(1) for all i, then X and G are o(1)-close. [ for all t, |Pr[X < t] – Pr[G < t]| = o(1). ]

20. Central Limit Theorem – Revisited Let Let If ci2 = o(1) for all i, then X and G are o(1)-close. [ for all t, |Pr[X < t] – Pr[G < t]| = o(1). ]

21. Central Limit Theorem – Revisited Let Q be an n-variate multilinear polynomial of degree 1. Let X = Q(x1,…xn) Let G = Q(g1,…gn) If = o(1) for all i, then X and G are o(1)-close. [ for all t, |Pr[X < t] – Pr[G < t]| = o(1). ]

22. Central Limit Theorem – Revisited Let Q be an n-variate multilinear polynomial of degree 1. Let X = Q(x1,…xn) Let G = Q(g1,…gn) If “Infi(Q)” = = o(1) for all i, then X and G are o(1)-close. [ for all t, |Pr[X < t] – Pr[G < t]| = o(1). ]

23. An extension [Rotar' ’79, MOO’05] Let Q be an n-variate multilinear polynomial of degree d = O(1). Let X = Q(x1,…xn) Let G = Q(g1,…gn) If “Infi(Q)” = = o(1) for all i, then X and G are o(1)-close. [ for all t, |Pr[X < t] – Pr[G < t]| = o(1). ]

24. Majority Is Stablest “ 0 < ε < ½, if f balanced and Infi(f) = o(1) for all i,NoiseSensε( f ) > NoiseSensε(Maj) − o(1).” Given f with small influences, express it as a multilinear polynomial.  “high-degree” part doesn’t matter too much. f is close to its “Gaussian version”. Now use Borell’s result.