1. Ryan Donnell Carnegie Mellon University O

2. 1. Describe some TCS results requiring variants of the Central Limit Theorem. Talk Outline 2. Show a flexible proof of the CLT with error bounds. 3. Open problems and an advertisement.

3. 1. Describe some TCS results requiring variants of the Central Limit Theorem. Talk Outline 2. Show a flexible proof of the CLT with error bounds. 3. Open problems and an advertisement.

4. Linear Threshold Functions

6. Learning Theory [O-Servedio08] Thm: Can learn LTFs f in poly(n) time, just from correlations E[f(x)x i ]. Key: G ~ N(0,1) when all |c i |.

7. Property Testing [Matulef-O-Rubinfeld-Servedio09] Thm: Can test if is -close to an LTF with poly(1/) queries. Key: when all |c i |.

8. Derandomization [Meka-Zuckerman10] Thm: PRG for LTFs with seed length O(log(n) log(1/)). Key: even when x i s not fully independent.

9. Multidimensional CLT? when all small compared to For

10. Derandomization+ [Gopalan-O-Wu-Zuckerman10] Thm: PRG for functions of O(1) LTFs with seed length O(log(n) log(1/)). Key: Derandomized multidimensional CLT.

11. Property Testing+ [Blais-O10] Thm: Testing if is a Majority of k bits needs k Ω(1) queries. Key: assuming E[X i ] = E[Y i ], Var[X i ] = Var[Y i ], and some other conditions. (actually, a multidimensional version)

12. Social Choice, Inapproximability [Mossel-O-Oleszkiewicz05] Thm: a) Among voting schemes where no voter has unduly large influence, Majority is most robust to noise. b) Max-Cut is UG-hard to.878-approx. Key: If P is a low-deg. multilin. polynomial, assuming P has small coeffs. on each coord.

13. 1. Describe some TCS results requiring variants of the Central Limit Theorem. Talk Outline 2. Show a flexible proof of the CLT with error bounds. 3. Open problems and an advertisement.

14. Gaussians Standard Gaussian: G ~ N(0,1). Mean 0, Var 1. a + bG also a Gaussian: N(a,b 2 ) Sum of independent Gaussians is Gaussian: If G ~ N(a,b 2 ), H ~ N(c,d 2 ) are independent, then G + H ~ N(a+c,b 2 +d 2 ). Anti-concentration: Pr[ G [u, u+] ] O().

15. X 1, X 2, X 3, … independent, ident. distrib., mean 0, variance σ 2, Central Limit Theorem (CLT)

16. CLT with error bounds X 1 + · · · + X n is close toN(0,1), assuming X i is not too wacky. X 1, X 2, …, X n independent, ident. distrib., mean 0, variance 1/n, wacky:

17. Niceness of random variables Say E[X] = 0, stddev[X] = σ. eg: ±1. N(0,1). Unif on [-a,a]. not nice: def: ( σ). def: X is nice if

18. Niceness of random variables Say E[X] = 0, stddev[X] = σ. eg: ±1. N(0,1). Unif on [-a,a]. not nice: def: ( σ). def: X is C-nice if

19. Y -close to Z: Berry-Esseen Theorem X 1, X 2, …, X n independent, ident. distrib., mean 0, variance 1/n, X 1 + · · · + X n is -close toN(0,1), assuming X i is C-nice, where [Shevtsova07]:.7056

20. General Case X 1, X 2, …, X n independent, ident. distrib., mean 0, X 1 + · · · + X n is -close toN(0,1), assuming X i is C-nice, [Shiganov86]:.7915

21. Berry-Esseen: How to prove? 1. Characteristic functions 2.Steins method 3.Replacement = think like a cryptographer X 1, X 2, …, X n indep., mean 0, S = X 1 + · · · + X n G ~ N(0,1).-close to

22. Indistinguishability of random variables S -close to G:

23. Indistinguishability of random variables S -close to G: u

24. Indistinguishability of random variables S -close to G: u t

25. Indistinguishability of random variables S -close to G:

26. Replacement method S -close to G: u δ

27. Replacement method X 1, X 2, …, X n indep., mean 0, S = X 1 + · · · + X n G ~ N(0,1) For smooth

28. Replacement method X 1, X 2, …, X n indep., mean 0, G = G 1 + · · · + G n For smooth S = X 1 + · · · + X n Hybrid argument

29. X 1, X 2, …, X n indep., mean 0, S Y = Y 1 + · · · + Y n For smooth S X = X 1 + · · · + X n Invariance principle Y 1, Y 2, …, Y n Var[X i ] = Var[Y i ] =

30. Hybrid argument Def: Z i = Y 1 + · · · + Y i + X i+1 + · · · + X n S X = Z 0, S Y = Z n X 1, X 2, …, X n, Y 1, Y 2, …, Y n, independent, matching means and variances. S X = X 1 + · · · + X n S Y = Y 1 + · · · + Y n vs.

31. Hybrid argument Z i = Y 1 + · · · + Y i + X i+1 + · · · + X n Goal: X 1, X 2, …, X n, Y 1, Y 2, …, Y n, independent, matching means and variances.

32. Z i = Y 1 + · · · + Y i + X i+1 + · · · + X n

33. where U = Y 1 + · · · + Y i1 + X i+1 + · · · + X n. Note: U, X i, Y i independent. Goal: U T

34. = by indep. and matching means/variances!

35. Variant Berry-Esseen: Say If X 1, X 2, …, X n & Y 1, Y 2, …, Y n indep. and have matching means/variances, then

36. Usual Berry-Esseen: If X 1, X 2, …, X n indep., mean 0, u δ Hack

37. Usual Berry-Esseen: If X 1, X 2, …, X n indep., mean 0, Variant Berry-Esseen + Hack Usual Berry-Esseen except with error O( 1/4 )

38. Extensions are easy! Vector-valued version: Use multidimensional Taylor theorem. Derandomized version: If X 1, …, X m C-nice, 3-wise indep., then X 1 +···+ X m is O(C)-nice. Higher-degree version: X 1, …, X m C-nice, indep., Q is a deg.-d poly., then Q(X 1, …, X m ) is O(C) d -nice.

39. 1. Describe some TCS results requiring variants of the Central Limit Theorem. Talk Outline 2. Show a flexible proof of the CLT with error bounds. 3. Open problems, advertisement, anecdote?

40. Open problems 1.Recover usual Berry-Esseen via the Replacement method. 2.Vector-valued: Get correct dependence on test sets K. (Gaussian surface area?) 3.Higher-degree: improve (?) the exponential dependence on degree d. 4.Find more applications in TCS.

41. Do you like LTFs and PTFs? Do you like probability and geometry?