Download presentation

Presentation is loading. Please wait.

Published byCornelius Ridout Modified over 2 years ago

1
Kevin Matulef MIT Ryan O’Donnell CMU Ronitt Rubinfeld MIT Rocco Servedio Columbia

2
= Property Testing Linear Threshold Functions =

3
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.

4
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.”

5
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.

6
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

7
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]

8
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

9
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)

10
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]

11
Intuition for the test Q: Which subset of half of the [sphere/Gaussian space] maximizes probability of vectors landing in same side? the test

12
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

13
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.

14
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] ?

15
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:

16
The “NO” case the test

17
Analyzing the pass probability Fact: Cor: the test for all i. The tail part,

18
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

19
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

20
Analyzing the pass probability But: (since f is § 1-valued) the test Gaussian facts

21
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) )

22
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]

23
Boolean version The “NO” case Let’s PgUp and see what needs to change!

24
Analyzing the pass probability But: (since f is § 1-valued) the test ??? False: is possible: f (x) = x 1.

25
Idea But: (since f is § 1-valued) False: is possible: f (x) = x 1. ???

26
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

27
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

Similar presentations

OK

Review of Probability. Important Topics 1 Random Variables and Probability Distributions 2 Expected Values, Mean, and Variance 3 Two Random Variables.

Review of Probability. Important Topics 1 Random Variables and Probability Distributions 2 Expected Values, Mean, and Variance 3 Two Random Variables.

© 2018 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on history of internet free download Ppt on reuse of waste materials Ppt on happy diwali Ppt on new delhi tourism Ppt on security guard training Cell surface display ppt on ipad Ppt on companies act 2013 vs companies act 1956 Download ppt on my role model Download ppt on water scarcity Ppt on earth movements and major landforms in europe