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Kevin Matulef MIT Ryan O’Donnell CMU Ronitt Rubinfeld MIT Rocco Servedio Columbia

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= Property Testing Linear Threshold Functions =

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Main Theorem: There is a poly(1/ ) query, nonadaptive, two-sided error property testing algorithm for being a halfspace. Given black-box access to, f a halfspace ) alg. says YES with prob. ¸ 2/3; f -far from all halfspaces ) alg. says YES with prob. · 1/3. Halfspaces are testable.

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Motivation “Usual” property testing motivation…? ‘precursor to learning’ motivation makes some sense Not many poly(1/ )-testable classes known. Core test is 2-query: f a halfspace ) Pr[ f passes] ¼ c f -far from all halfspaces ) Pr[ f passes] · c − poly( ) Local tests really characterize the class: “Halfspaces maximize this quadratic form, and anything close to maximizing is close to a halfspace.”

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2-query test Promise: f is balanced 1. Pick to be -correlated inputs. are such that 2. Test if Thm: f a halfspace ) Pr[ f passes] ¸ f -far from all halfspaces ) Pr[ f passes] · uniform 1 −1 1 −1 1 i.e.

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2-query test Promise: f is balanced 1. Pick to be -correlated inputs. 2. Test if Thm: f a halfspace ) Pr[ f passes] ¸ f -far from all halfspaces ) Pr[ f passes] · uniform Gaussian

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The truth about the Boolean test 2-query Gaussian test non-balanced case Boolean, “low-influences” version testing for low influences “cross-testing” two low-influence halfspaces stitching together halfspaces, LP bounds junta-testing [FKRSS’02]

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Gaussian testing setting Domain: Class to be tested is Halfspaces: thought of as having Gaussian distribution: Each coord 1,…, n distributed as a standard N(0,1) Gaussian Unknown Gaussian

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Facts about Gaussian space Rotationally invariant The r.v. has distribution N(0, ). With overwhelming probability, Hence essentially same as uniform distribution on the sphere. “ are -correlated n-dim. Gaussians:” are i.i.d. “ -correlated 1-dim. Gaussians:” – draw, set (proof: =, which has same distribution as by rotational symmetry)

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Why Gaussian space? You:“Ryan, why are you hassling us with all this Gaussian stuff? I only care about testing on {−1, 1} n.” Me:“Sorry, you have to be able to solve this problem first.” But also: Much nicer setting because of rotational invariance. might really be a function in disguise. [class of halfspaces $ class of halfspaces]

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Intuition for the test Q: Which subset of half of the [sphere/Gaussian space] maximizes probability of vectors landing in same side? the test

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Intuition for the test A: Halfspace, for each value of 2 [0,1]. (And each value of ½.) (Gaussian: [Borell’85] ; Sphere: [Feige-Schechtman’99], others? ) the test

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But does this characterize halfspaces? Q: If a set passes the test with probability close to that of a halfspace, is it itself close to a halfspace? A: Not known, in general. But: We will show that this is true when is close to 0.

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The “YES” case Suppose f is a balanced halfspace. 1.By spherical symmetry, we can assume 2.Thus iff. 3.This probability is [Sheppard’99] the test Pr[ f passes] ?

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The “NO” case Suppose is any balanced function. Def: Given, define their “correlation” to be “Usual Fourier analysis thing”: where f = 0 is the “constant part” of f, f = 1 is the “linear part” of f, etc. the test Pr[ f passes] ? Def: any expressible as:

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The “NO” case the test

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Analyzing the pass probability Fact: Cor: the test for all i. The tail part,

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Analyzing the pass probability What is the “constant part” of f ? Prototypical constant function is Fact: = 0 in our case, since I promised f balanced. the test

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Analyzing the pass probability What is the “linear part” of f ? A linear function looks like Fact: Cor: the test Let’s write in place of

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Analyzing the pass probability But: (since f is § 1-valued) the test Gaussian facts

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The “NO” case completed with equality iff I.e., for any f : if is close to, then f is close to being a halfspace. In particular, with a little more analytic care, one concludes: the test (in fact, the sgn of its linear part) )

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The truth about the Boolean test 2-query Gaussian test non-balanced case Boolean, “low-influences” version testing for low influences “cross-testing” two low-influence halfspaces stitching together halfspaces, LP bounds junta-testing [FKRSS’02]

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Boolean version The “NO” case Let’s PgUp and see what needs to change!

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Analyzing the pass probability But: (since f is § 1-valued) the test ??? False: is possible: f (x) = x 1.

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Idea But: (since f is § 1-valued) False: is possible: f (x) = x 1. ???

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Idea ??? Central Limit Theorem:If each is “small”, say, (with error bounds) then is “close” in distribution to. “ i th influence ” Germ of remainder of proof: 1.Possible to test if all i ’s small 2. for at most i’s

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Open directions 1. this result + “Every lin. thresh. fcn. has a low-weight approximator” [Servedio ’06] = we understand Boolean halfspaces somewhat thoroughly. Can we use this to solve some more open problems? 2.Which classes of functions testable? Consider the class “isomorphic to Majority;” i.e., Another chunk of the paper shows an lower bound! (# queries depends only on

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