Kevin MatulefMIT

Ryan O’DonnellCMU

Ronitt RubinfeldMIT

Rocco ServedioColumbia

= Property Testing

Linear Threshold Functions =

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.

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.”

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

−1

−1

1

i.e.

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] ·

uniformGaussian

The truth about the Boolean test

2-queryGaussian

test

non-balancedcase

Boolean,

“low-influences”

version

testingfor low

influences

“cross-testing” two

low-influence halfspaces

stitching together halfspaces,

LP bounds

junta-testing[FKRSS’02]

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

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)

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]

Intuition for the test

Q: Which subset of half of the [sphere/Gaussian space]

maximizes probability of vectors landing in same side?

the test

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

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.

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] ?

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:

The “NO” case the test

Analyzing the pass probability

Fact:

Cor:

the test

for all i.

The tail part,

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

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

Analyzing the pass probability

But:

(since f is §1-valued)

the test

Gaussian facts

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)

)

The truth about the Boolean test

2-queryGaussian

test

non-balancedcase

Boolean,

“low-influences”

version

testingfor low

influences

“cross-testing” two

low-influence halfspaces

stitching together halfspaces,

LP bounds

junta-testing[FKRSS’02]

Boolean version

The “NO” case

Let’s PgUp and see what needs to change!

Analyzing the pass probability

But:

(since f is §1-valued)

the test

???False: is possible: f (x) = x1.

Idea

But:

(since f is §1-valued)

False: is possible: f (x) = x1.

???

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

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