Ryan Donnell Carnegie Mellon University O

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.

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.

Linear Threshold Functions

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 |.

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

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

Multidimensional CLT? when all small compared to For

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.

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)

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.

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.

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().

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

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:

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

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

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

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

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

Indistinguishability of random variables S -close to G:

Indistinguishability of random variables S -close to G: u

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

Indistinguishability of random variables S -close to G:

Replacement method S -close to G: u δ

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

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

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 ] =

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.

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.

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

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

= by indep. and matching means/variances!

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

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

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 )

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.

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?

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.

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