Ryan O’Donnell

Carnegie Mellon University

Karl Wimmer

CMU & Duquesne

Amir Shpilka

Technion

Rocco Servedio

Columbia

Parikshit Gopalan

UW & Microsoft SVC

joint with

Outline

1. Testing boolean functions overview

2. Statement & proof sketch of Result 2

3. Statements of other results

Property Testing Boolean Functions

• Query access to

• Say YES whp if f has property P

• Say NO whp if f is ϵ-far from all g w/ ppty P

• P “testable” if doable with # queries

depending only on ϵ

Property Testing

Characterizing testability fairly

well-understood for

graphs [AS05a,AS05b,AFNS06,AT08,

…]

& codes [KS07,KS08,…].

But wide open for boolean functions.

Testable boolean function properties

• Linearity (f(x) = λ•x) [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/ϵ) q’s

Characterization??

Fischer’s survey [Fis01]

suggests Fourier may be key.

This paper:

Having concise Fourier representation

is testable.

Fourier analysis of boolean functions

Fourier analysis of boolean functions

Fourier analysis of boolean functions

There are 2n linear functions,

Every uniquely expressible as

a real #

Fourier sparsity

Def: f is s-sparse

⇔ # of nonzero is ≤ s

Eg: Linear functions are 1-sparse.

k-juntas are 2k-sparse.

Result #2

Thm: “Is s-sparse?”

is ϵ-testable with poly(s/ϵ)

nonadaptive queries.

Proof ingredients

• Hashing Fourier coefficients

to affine subspaces [FGKP06]

• New structural facts about

s-sparse boolean functions

Physical space Fourier space

Physical space Fourier space

+ + − − −

− + − − +

− − + + +

− + + + −

− − − − +

0 0 0 0 0

0 0 0 ⅝ 0

0 –⅜ 0 0 0

0 0 0 0 0

0 0 0 ¾ 0

Fourier space

Hashing Fourier coefficients idea

s2

buckets

Hashing Fourier coefficients idea

• Birthday Paradox ⇒ s Fourier coeffs split

• Test that at most s buckets are nonzero?

s2

buckets

Hashing to affine subspaces

Pick α1, …, αd ∈ at random, d = 2 log s.

α1 • λ = 0

α2 • λ = 0

αd • λ = 0

λ :subpace of

codimension d

Hashing to affine subspaces

Pick α1, …, αd ∈ at random, d = 2 log s.

α1 • λ = b1

α2 • λ = b2

αd • λ = bd

λ :subpace of

codimension d

affine

b ∈

Hashing Fourier coefficients idea

Birthday Paradox? OK, by pairwise independence.

2d = s2

buckets(aff. subsps.)

indexed by F2

Projection functions

Projection functions

Projection functions

b

Physical to Fourier link

b

ex:

Physical to Fourier link

b

ex: Pbf(x) = avg. of  ±f on 2d = s2 strings rel’d to x

Projection functions

• We can compute Pbf(x) exactly,

for any bucket b, with s2 queries

• Pbf(x) is always

The Test

1. Hash to a random 2d = s2 buckets b.

2. For each b,

3. ϵ’-test if Pbf(x) ≡ 0, for ϵ’ =

poly(ϵ/s)

4. Let B = {b : Pbf wasn’t ≡ 0}

5. Say NO if |B| > s

The Test

1. Hash to a random 2d = s2 buckets b.

2. For each b,

3. ϵ’-test if Pbf(x) ≡ 0, for ϵ’ = poly(ϵ/s)

4. Let B = {b : Pbf wasn’t ≡ 0}

5. Say NO if |B| > s

6. Say YES

The Test

1. Hash to a random 2d = s2 buckets b.

2. For each b,

3. ϵ’-test if Pbf(x) ≡ 0, for ϵ’ = poly(ϵ/s)

4. Let B = {b : Pbf wasn’t ≡ 0}

5. Say NO if |B| > s

6. Say YES poly(s/ϵ) nonadapt. queries

The Test

Recall that for b ∈ B we “should” have Pbf =

Analysis easier if you also do:

6. For each b ∈ B,

7. Test |Pbf| constant

8. Test sgn(Pbf) is linear, using [BLR90]

The Test

Recall that for b ∈ B we “should” have Pbf =

Analysis easier if you also do:

6. For each b ∈ B,

7. Test |Pbf| constant

8. Test sgn(Pbf) is linear, using [BLR90]

Key for analysis

Granularity Theorem:

If is s-sparse, then

for all λ.

(hence 0 or > 1/s)

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 ϵ-testable

with 2O(k)/ϵ nonadaptive queries.

Other results

Result 3: Lower bounds.

Even for nonadaptive testers, ϵ = .49,

• queries needed for s-sparsity

• 2Ω(k) queries need for k-dimensionality

Improves on [AKKLR03, BFNR08].

Other results

Result 4: Exact, proper learning algorithm

for s-sparse functions under unif.

(Easy consequence of Granularity Theorem.)

Other results

Result 5: “Every subclass of k-dimensional

functions is testable with 2O(k)/ϵ

nonadaptive queries.”

(Uses “Testing by implicit learning” [DLMORSW07])

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)?