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Computational Applications of Noise Sensitivity Ryan O’Donnell

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Includes joint work with : Elchanan Mossel Rocco Servedio Adam Klivans Nader Bshouty Oded Regev Benny Sudakov

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Intro to Noise Sensitivity

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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 : {0,1} n → {0,1} mapping votes to winner what if there are errors in recording of votes? suppose each vote is misrecorded independently with prob. ε.

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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 : {0,1} n → {0,1} mapping votes to winner what if there are errors in recording of votes? suppose each vote is misrecorded independently with prob. ε. what is the prob. this affects elec.’s outcome?

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Definition Let f : {0,1} n → {0,1} be any boolean function. Let 0 ≤ ε ≤ ½, the noise rate. Let x be a uniformly randomly chosen string in {0,1} n, and let y be an ε-noisy version of x. Then the noise sensitivity of f at ε is: NS ε (f) = Pr [f(x) ≠ f(y)]. x,y

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Examples Suppose f is the constant function f(x) = 1. Then NS ε (f) = 0. Suppose f is the “dictator” function f(x) = x 1. Then NS ε (f) = ε. In general, for fixed f, NS ε (f) is a function of ε.

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Examples – parity The parity (xor) function on n bits 1 iff there are an odd number of 1’s in the input. In calculating Pr[f(x) ≠ f(y)], it doesn’t matter what x is, just how many flips there are. NS ε ( PARITY n ) = Pr[odd number of heads in n ε-biased coin flips] = ½ – ½(1 – 2ε) n.

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NS ε ( PARITY 10 ) = ½ – ½(1 – 2ε) 10

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Basic facts about NS NS ε (f) is an increasing, (log-)concave function of ε which is 0 at 0 and 2p(1-p) at ½ (where p=Pr[f = 1]). this follows from a formula for NS ε (f) in terms of Fourier coefficients: NS ε (f) = 2f(Ø) – 2 Σ (1-2ε) |S| f (S) 2. S µ [n]

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PARITY, MAJORITY, dictator, and AND on 5 bits

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PARITY, MAJORITY, dictator, and AND on 15 bits

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PARITY, MAJORITY, dictator, and AND on 45 bits

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History of Noise Sensitivity (in computer science)

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History of Noise Sensitivity Kahn-Kalai-Linial ’88 The Influence of Variables on Boolean Functions

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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 εn, starting uniformly in f -1 (1), ends up outside f -1 (1)?” is essentially asking about NS ε (f) famous for using Fourier analysis and “Bonami-Beckner inequality” in TCS

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History of Noise Sensitivity Håstad ’97 Some Optimal Inapproximability Results

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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)

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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

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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…

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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

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Hardness Amplification

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Hardness on average def: We say f : {0,1} n → {0,1} is (1-ε)-hard for circuits of size s if there is no circuit of size s which computes f correctly on more than (1-ε)2 n inputs. def: A complexity class is (1-ε)-hard for polynomial circuits if there is a function family (f n ) in the class such that for suff. large n, f n is (1-ε)-hard for circuits of size poly(n).

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Hardness of EXP, NP Of course we can’t show NP is even (1-2 -n )- hard for poly ckts, since this is NP µ P/poly. But let’s assume EXP, NP µ P/poly. Then just how hard are these for poly circuits? For EXP, extremely strong results known – [ BFNW 93,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?

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Yao’s XOR Lemma Some of the hardness amplification results for EXP use Yao’s XOR Lemma: Thm: If f is (1-ε)-hard for poly circuits, then PARITY k f is (½+½(1-2ε) 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(x 1 ), …, f(x k )). No coincidence that the hardness bound for PARITY k f is 1-NS ε ( PARITY k ).

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A general direct product thm. Yao doesn’t help for NP – if you have a hard function f n in NP, PARITY k f n probably isn’t in NP. We generalize Yao and determine the hardness of g f n for any g – in terms of the noise sensitivity of g: Thm: If f (balanced) is (1-ε)-hard for poly circuits, then g f n is roughly (1-NS ε (g))- hard for poly circuits.

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Why noise sensitivity? Suppose f is balanced and (1-ε)-hard for poly circuits. x 1, …, x k are chosen uniformly at random, and you, a poly circuit, have to guess g(f(x 1 ), …, f(x k )). Natural strategy is to try to compute each y i = f(x i ) and then guess g(y 1,…,y k ). But f is (1-ε)-hard for you! So Pr[f(x i )≠y i ] = ε. Success prob.: Pr[g(f(x 1 )…f(x k ))=g(y 1 …y k )] = 1-NS ε (g).

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Hardness of NP If (f n ) is a (hard) function family in NP, and (g k ) is a monotone function family, then (g k f n ) is in NP. We give constructions and prove tight bounds for the problem of finding monotone g such that NS ε (g) is very large (close to ½) for ε very small. Thm: If NP is (1-1/poly(n))-hard for poly ckts, then NP is (½ + 1/√n)-hard for poly ckts.

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Learning algorithms

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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

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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, NS ε (f) ≤ β(ε). Then there is an alg. for learning C to within accuracy ε in time: n O(1)/β (ε).

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Example – halfspaces E.g., using [Peres98], every boolean function f which is the “intersection of two halfspaces” has NS ε (f) ≤ O(√ε). Cor: The class of “intersections of two halfspaces” can be learned in time n O(1/ε²). 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.

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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 y’s which are somewhat close to x in Hamming distance. Idea: Draw a “net” of examples: (x 1, f(x 1 )), … (x M, f(x M )). To hypothesize about y, compute a weighted average of known labels, based on dist. to y: hypothesis =… sgn[ w(Δ(y,x 1 ))f(x 1 ) + ··· + w(Δ(y,x M ))f(x M ) ].

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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.

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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 n k. We give the first improvement: algorithm runs in time n.704k. Can the strong relationship between juntas and noise sensitivity improve this?

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Coin flipping

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The T 1-2ε operator T 1-2ε operates on the space of functions {0,1} n → R: T 1-2ε (f) (x) = E [f(y)] (= Pr[f(y) = 1]). Notable fact about T 1-2ε : the Bonami- Beckner [Bon68] “hypercontractive” inequality:||T λ (f)|| 2 ≤ ||f|| 1+λ² y = noise ε (x)

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Bonami, Beckner

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The T 1-2ε operator It follows easily that: NS ε (f) = ½ - ½ ||T √1-2ε (f)|| 2. Thus studying noise sensitivity is equivalent to studying the (2-)norm of the T 1-2ε operator. We consider studying higher norms of the T 1-2ε operator. The problem can be phrased combinatorially, in terms of a natural coin flipping problem.

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“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 ε-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.

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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?

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Success as function of k Most interesting asymptotic case: ε a small constant, n unbounded, k → ∞. What is the maximum success probability? Surprisingly, goes to 0 only polynomially: Thm: The best success probability of k players is Õ( 1/k 4ε ), with the majority function being essentially optimal.

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Reverse Bonami-Beckner To prove that no protocol can do better than k -Ω(1), we need to use a reverse Bonami- Beckner inequality [Bor82]: for f ≥ 0, t ≥ 0, ||T λ (f)|| 1-t/λ ≥ ||f|| 1-tλ Concentration of measure interpretation: Let A be a reasonably large subset of the cube. Then almost all x have Pr[y ∈ A] somewhat large.

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Conclusions

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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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