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Computational Applications of Noise Sensitivity

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Title: Computational Applications of Noise Sensitivity Author: Ryan Last modified by: Ryan Created Date: 4/1/2003 10:12:00 PM Document presentation format – PowerPoint PPT presentation

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Title: Computational Applications of Noise Sensitivity


1
Computational Applications of Noise Sensitivity
  • Ryan ODonnell

2
Includes joint work with Elchanan Mossel Rocco
Servedio Adam Klivans Nader Bshouty Oded
Regev Benny Sudakov
3
Intro to Noise Sensitivity
4
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.

5
(No Transcript)
6
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?

7
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
8
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.

9
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.

10
NSe(PARITY10) ½ ½(1 2e)10

11
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
12
PARITY, MAJORITY, dictator, and AND on 5 bits
13
PARITY, MAJORITY, dictator, and AND on 15 bits
14
PARITY, MAJORITY, dictator, and AND on 45 bits
15
History of Noise Sensitivity (in computer science)
16
History of Noise Sensitivity
  • Kahn-Kalai-Linial 88
  • The Influence of Variables on Boolean Functions

17
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

18
History of Noise Sensitivity
  • Håstad 97
  • Some Optimal Inapproximability Results

19
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)

20
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
21
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

22
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

23
Hardness Amplification
24
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).

25
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?

26
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).

27
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.

28
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).

29
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.

30
Learning algorithms
31
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

32
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
33
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.

34
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) .

35
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.

36
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?

37
Coin flipping
38
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)
39
Bonami, Beckner
40
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.

41
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.

42
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?

43
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.

44
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.

45
Conclusions
46
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.?
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