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Computational Applications of Noise Sensitivity

- Ryan ODonnell

Includes joint work with Elchanan Mossel Rocco

Servedio Adam Klivans Nader Bshouty Oded

Regev Benny Sudakov

Intro to Noise Sensitivity

Election schemes

- suppose there is an election between two parties,

called 0 and 1 - assume unrealistically that n voters cast votes

independently and unif. randomly - an election scheme is a boolean function f

0,1n ? 0,1 mapping votes to winner - what if there are errors in recording of votes?

suppose each vote is misrecorded independently

with prob. e.

(No Transcript)

Election schemes

- suppose there is an election between two parties,

called 0 and 1 - assume unrealistically that n voters cast votes

independently and unif. randomly - an election scheme is a boolean function f

0,1n ? 0,1 mapping votes to winner - what if there are errors in recording of votes?

suppose each vote is misrecorded independently

with prob. e. - what is the prob. this affects elec.s outcome?

Definition

- Let f 0,1n ? 0,1 be any boolean function.
- Let 0 e ½, the noise rate.
- Let x be a uniformly randomly chosen string in

0,1n, and let y be an e-noisy version of x. - Then the noise sensitivity of f at e is
- NSe(f) Pr f(x) ? f(y).

x,y

Examples

- Suppose f is the constant function f(x) 1.
- Then NSe(f) 0.
- Suppose f is the dictator function f(x) x1.
- Then NSe(f) e.
- In general, for fixed f, NSe(f) is a function of

e.

Examples parity

- The parity (xor) function on n bits 1 iff there

are an odd number of 1s in the input. - In calculating Prf(x) ? f(y), it doesnt matter

what x is, just how many flips there are. - NSe(PARITYn) Prodd number of heads in n

e-biased coin flips - ½ ½(1 2e)n.

NSe(PARITY10) ½ ½(1 2e)10

Basic facts about NS

- NSe(f) is an increasing, (log-)concave function

of e which is 0 at 0 and 2p(1-p) at ½ (where

pPrf 1). - this follows from a formula for NSe(f) in terms

of Fourier coefficients - NSe(f) 2f(Ø) 2 S (1-2e)S f (S)2.

S µ n

PARITY, MAJORITY, dictator, and AND on 5 bits

PARITY, MAJORITY, dictator, and AND on 15 bits

PARITY, MAJORITY, dictator, and AND on 45 bits

History of Noise Sensitivity (in computer science)

History of Noise Sensitivity

- Kahn-Kalai-Linial 88
- The Influence of Variables on Boolean Functions

Kahn-Kalai-Linial 88

- implicitly studied noise sensitivity
- motivation study of random walks on the

hypercube where the initial distribution is

uniform over a subset - the question, What is the prob. that a random

walk of length en, starting uniformly in f-1(1),

ends up outside f-1(1)? is essentially asking

about NSe(f) - famous for using Fourier analysis and

Bonami-Beckner inequality in TCS

History of Noise Sensitivity

- Håstad 97
- Some Optimal Inapproximability Results

Håstad 97

- breakthrough hardness of approximation results
- decoding the Long Code given access to the

truth-table of a function, want to test that it

is significantly determined by a junta (very

small number of variables) - roughly, does a noise sensitivity test picks x

and y as in n.s., tests f(x)f(y)

History of Noise Sensitivity

- Benjamini-Kalai-Schramm 98
- Noise Sensitivity of Boolean Functions and

Applications to Percolation

Benjamini-Kalai-Schramm 98 Noise Sensitivity of

Boolean Functions and Applications to

Percolation

Benjamini-Kalai-Schramm 98

- intensive study of noise sensitivity of boolean

functions - introduced asymptotic notions of noise

sensitivity/stability, related them to Fourier

coefficients - studied noise sensitivity of percolation

functions, threshold functions - made conjectures connecting noise sensitivity to

circuit complexity - and more

This thesis

- New noise sensitivity results and applications
- tight noise sensitivity estimates for boolean

halfspaces, monotone functions - hardness amplification thms. (for NP)
- learning algorithms for halfspaces, DNF (from

random walks), juntas - new coin-flipping problem, and use of reverse

Bonami-Beckner inequality

Hardness Amplification

Hardness on average

- def We say f 0,1n ? 0,1 is (1-e)-hard for

circuits of size s if there is no circuit of size

s which computes f correctly on more than (1-e)2n

inputs. - def A complexity class is (1-e)-hard for

polynomial circuits if there is a function family

(fn) in the class such that for suff. large n, fn

is (1-e)-hard for circuits of size poly(n).

Hardness of EXP, NP

- Of course we cant show NP is even (1-2-n)-hard

for poly ckts, since this is NPµP/poly. - But lets assume EXP, NP µ P/poly. Then just how

hard are these for poly circuits? - For EXP, extremely strong results known

BFNW93,Imp95,IW97,KvM99,STV99 if EXP is

(1-2-n)-hard for poly circuits, then it is (½

1/poly(n))-hard for poly circuits. - What about NP?

Yaos XOR Lemma

- Some of the hardness amplification results for

EXP use Yaos XOR Lemma - Thm If f is (1-e)-hard for poly circuits,

then PARITYk f is (½½(1-2e)k)-hard for poly

circuits. - Here, if f is a boolean fcn on n inputs and g is

a boolean fcn on k inputs, g f is the function

on kn inputs given by g(f(x1), , f(xk)). - No coincidence that the hardness bound for

PARITYk f is 1-NSe(PARITYk).

A general direct product thm.

- Yao doesnt help for NP if you have a hard

function fn in NP, PARITYk fn probably isnt in

NP. - We generalize Yao and determine the hardness of g

fn for any g in terms of the noise

sensitivity of g - Thm If f (balanced) is (1-e)-hard for poly

circuits, then g fn is roughly (1-NSe(g))-hard

for poly circuits.

Why noise sensitivity?

- Suppose f is balanced and (1-e)-hard for poly

circuits. x1, , xk are chosen uniformly at

random, and you, a poly circuit, have to guess

g(f(x1), , f(xk)). - Natural strategy is to try to compute each yi

f(xi) and then guess g(y1,,yk). - But f is (1-e)-hard for you! So Prf(xi)?yi

e. - Success prob.
- Prg(f(x1)f(xk))g(y1yk) 1-NSe(g).

Hardness of NP

- If (fn) is a (hard) function family in NP, and

(gk) is a monotone function family, then (gk

fn) is in NP. - We give constructions and prove tight bounds for

the problem of finding monotone g such that

NSe(g) is very large (close to ½) for e very

small. - Thm If NP is (1-1/poly(n))-hard for poly ckts,

then NP is (½ 1/vn)-hard for poly ckts.

Learning algorithms

Learning theory

- Learning theory (Valiant84) deals with the

following scenario - someone holds an n-bit boolean function f
- you know f belongs to some class of fcns (eg,

parities of subsets, poly size DNF) - you are given a bunch of uniformly random labeled

examples, (x, f(x)) - you must efficiently come up with a hypothesis

function h that predicts f well

Learning noise-stable functions

- We introduce a new idea for showing function

classes are learnable - Noise-stable classes are efficiently learnable
- Thm Suppose C is a class of boolean fcns on n

bits, and for all f ? C, NSe(f) ß(e). Then

there is an alg. for learning C to within

accuracy e in time - nO(1)/ß (e).

-1

Example halfspaces

- E.g., using Peres98, every boolean function f

which is the intersection of two halfspaces has

NSe(f) O(ve). - Cor The class of intersections of two

halfspaces can be learned in time nO(1/e²). - No previously known subexponential alg.
- We also analyze the noise sensitivity of some

more complicated classes based on halfspaces and

get learning algs. for them.

Why noise stability?

- Suppose a function is fairly noise stable. In

some sense this means if you know f(x), you have

a good guess for f(y) for ys which are somewhat

close to x in Hamming distance. - Idea Draw a net of examples (x1, f(x1)),

(xM, f(xM)). To hypothesize about y, compute a

weighted average of known labels, based on dist.

to y hypothesis - sgn w(?(y,x1))f(x1) w(?(y,xM))f(xM) .

Learning from random walks

- Holy grail of learning Learn poly size DNF

formulas in polynomial time. - Consider natural weakening of learning examples

not iid, come from random walk. - We show DNF poly-time learnable in this model.

Indeed, also in a harder model NS-model

examples are (x,f(x),y,f(y)) - Proof estimate NS on subsets of input bits ?

find large Fourier coefficients.

Learning juntas

- The essential blocking issue for learning poly

size DNF formulas is that they can be O(log

n)-juntas. - Previously, no known algorithm for learning

k-juntas in time better than the trivial nk. - We give the first improvement algorithm runs in

time n.704k. - Can the strong relationship between juntas and

noise sensitivity improve this?

Coin flipping

The T1-2e operator

- T1-2e operates on the space of functions 0,1n

? R - T1-2e(f) (x) E f(y) ( Prf(y) 1).
- Notable fact about T1-2e the Bonami-Beckner

Bon68 hypercontractive inequality T?(f)2

f1?²

y noisee(x)

Bonami, Beckner

The T1-2e operator

- It follows easily that
- NSe(f) ½ - ½ Tv1-2e(f)2.
- Thus studying noise sensitivity is equivalent to

studying the (2-)norm of the T1-2e operator. - We consider studying higher norms of the T1-2e

operator. The problem can be phrased

combinatorially, in terms of a natural coin

flipping problem.

Cosmic coin flipping

- n random votes cast in an election
- we use a balanced election scheme, f
- k different auditors get copies of the votes

however, each gets an e-noisy copy - what is the probability all k auditors agree on

the winner of the election? - Equivalently, k distributed parties want to flip

a shared random coin given noisy access to a

cosmic random string.

Relevance of the problem

- Application of this scenario Everlasting

security of DingRabin01 a cryptographic

protocol assuming that many distributed parties

have access to a satellite broadcasting stream of

random bits. - Also a natural error-correction problem without

encoding, can parties attain some shared entropy?

Success as function of k

- Most interesting asymptotic case e a small

constant, n unbounded, k ? 8. What is the maximum

success probability? - Surprisingly, goes to 0 only polynomially
- Thm The best success probability of k players

is Õ(1/k4e), with the majority function being

essentially optimal.

Reverse Bonami-Beckner

- To prove that no protocol can do better than

k-O(1), we need to use a reverse Bonami-Beckner

inequality Bor82 for f 0, t 0, - T?(f)1-t/? f1-t?
- Concentration of measure interpretation Let A

be a reasonably large subset of the cube. Then

almost all x have Pry ? A somewhat large.

Conclusions

Open directions

- estimate the noise sensitivity of various classes

of functions general intersections of threshold

functions, percolation functions, - new hardness of approx. results using NS-junta

connection DS02,Kho02,DF03? - find a substantially better algorithm for

learning juntas - explore applications of reverse Bonami-Beckner

coding theory, e.g.?