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Ryan ODonnell Carnegie Mellon University Karl Wimmer CMU & Duquesne Amir Shpilka Technion Rocco Servedio Columbia Parikshit Gopalan UW & Microsoft SVC joint with

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Outline 1. Testing boolean functions overview 2. Statement & proof sketch of Result 2 3. Statements of other results

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Property Testing Boolean Functions Query access to Say YES whp if f has property P Say NO whp if f is -far from all g w/ ppty P P testable if doable with # queries depending only on

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Property Testing Characterizing testability fairly well-understood for graphs [AS05a,AS05b,AFNS06,AT08,…] &codes [KS07,KS08,…]. But wide open for boolean functions.

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Testable boolean function properties Linearity (f(x) = λx) [BLR90] Degree k [AKKLR03] Dicatators [BGS95] Conjunctions, size-s mono. DNF [PRS01] s-juntas ( s relevant vbls) [FKKRS03,B09] size-s DNFs, DTs, BPs, formulas, ckts, polynomials [DLMORSW07] Halfspaces [MORS09] poly(s/) qs

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Characterization?? Fischers survey [Fis01] suggests Fourier may be key. This paper: Having concise Fourier representation is testable.

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Fourier analysis of boolean functions

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There are 2 n linear functions, Every uniquely expressible as a real #

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Fourier sparsity Def: f is s-sparse # of nonzero is s Eg:Linear functions are 1-sparse. k-juntas are 2 k -sparse.

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Result #2 Thm: Is s-sparse? is -testable with poly(s/) nonadaptive queries.

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Proof ingredients Hashing Fourier coefficients to affine subspaces [FGKP06] New structural facts about s-sparse boolean functions

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Physical spaceFourier space

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Physical spaceFourier space ++ ++ +++ +++ + 00000 0000 0–000 00000 000¾0

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Fourier space

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Hashing Fourier coefficients idea s 2 buckets

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Hashing Fourier coefficients idea Birthday Paradox s Fourier coeffs split Test that at most s buckets are nonzero? s 2 buckets

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Hashing to affine subspaces Pick α 1, …, α d at random, d = 2 log s. α 1 λ = 0 α 2 λ = 0 α d λ = 0 λ : subpace of codimension d

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Hashing to affine subspaces Pick α 1, …, α d at random, d = 2 log s. α 1 λ = b 1 α 2 λ = b 2 α d λ = b d λ : subpace of codimension d affine b

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Hashing Fourier coefficients idea Birthday Paradox? OK, by pairwise independence. 2 d = s 2 buckets (aff. subsps.) indexed by F2

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Projection functions

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b

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Physical to Fourier link b ex:

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Physical to Fourier link b ex: P b f(x) = avg. of ±f on 2 d = s 2 strings reld to x

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Projection functions We can compute P b f(x) exactly, for any bucket b, with s 2 queries P b f(x) is always

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The Test 1.Hash to a random 2 d = s 2 buckets b. 2.For each b, 3.-test if P b f(x) 0, for = poly(/s) 4.Let B = {b : P b f wasnt 0} 5.Say NO if |B| > s

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The Test 1.Hash to a random 2 d = s 2 buckets b. 2.For each b, 3.-test if P b f(x) 0, for = poly(/s) 4.Let B = {b : P b f wasnt 0} 5.Say NO if |B| > s 6.Say YES

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The Test 1.Hash to a random 2 d = s 2 buckets b. 2.For each b, 3.-test if P b f(x) 0, for = poly(/s) 4.Let B = {b : P b f wasnt 0} 5.Say NO if |B| > s 6.Say YES poly(s/) nonadapt. queries

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The Test Recall that for b B we should have P b f = Analysis easier if you also do: 6.For each b B, 7.Test |P b f| constant 8.Test sgn(P b f) is linear, using [BLR90]

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The Test Recall that for b B we should have P b f = Analysis easier if you also do: 6.For each b B, 7.Test |P b f| constant 8.Test sgn(P b f) is linear, using [BLR90]

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Key for analysis Granularity Theorem: If is s-sparse, then for all λ. (hence 0 or > 1/s)

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Other results Def: f is k-dimensional if { } lies in a k-dimensional subspace. f is a k-junta of parities Result 1: Is f k-dimensional? is -testable with 2 O(k) / nonadaptive queries.

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Other results Result 3: Lower bounds. Even for nonadaptive testers, =.49, queries needed for s-sparsity 2 Ω(k) queries need for k-dimensionality Improves on [AKKLR03, BFNR08].

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Other results Result 4: Exact, proper learning algorithm for s-sparse functions under unif. (Easy consequence of Granularity Theorem.)

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Other results Result 5: Every subclass of k-dimensional functions is testable with 2 O(k) / nonadaptive queries. (Uses Testing by implicit learning [DLMORSW07] )

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Open problems More on characterizing testability of boolean function properties. Test functions with (cf [GS06] ) Characterize s-sparse boolean functions. Are they disj. unions of poly(s) many affine subspaces of codimension O(log s)?

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