Testing

1. Kevin MatulefMIT Ryan O’DonnellCMU Ronitt RubinfeldMIT Rocco ServedioColumbia Testing Halfspaces

2. Testing = Property Testing Halfspaces Linear Threshold Functions =

3. Main Theorem: Halfspaces are testable. 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.

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] ¼c 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. 1 1 −1 −1 1 −1 −1 1 2-query test Promise: f is balanced uniform i.e. 1. Pick to be -correlated inputs. 2. Test if are such that Thm: f a halfspace )Pr[ f passes] ¸ f -far from all halfspaces )Pr[ f passes] ·

6. 2-query test Promise: f is balanced Gaussian uniform 1. Pick to be -correlated inputs. 2. Test if Thm: f a halfspace )Pr[ f passes] ¸ f -far from all halfspaces )Pr[ f passes] ·

7. 2-queryGaussiantest The truth about the Boolean test junta-testing[FKRSS’02] “cross-testing” twolow-influence halfspaces testingfor lowinfluences Boolean, “low-influences”version non-balancedcase stitching together halfspaces,LP bounds

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

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

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

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 the test Suppose f is a balanced halfspace. • By spherical symmetry, we can assume • Thus iff . • This probability is Pr[ f passes] ? [Sheppard’99]

15. The “NO” case the test 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. Pr[ f passes] ? Def: any expressible as:

16. The “NO” case the test

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

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

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

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

21. The “NO” case completed the test 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: (in fact, the sgn of its linear part) )

22. 2-queryGaussiantest The truth about the Boolean test junta-testing[FKRSS’02] “cross-testing” twolow-influence halfspaces testingfor lowinfluences Boolean, “low-influences”version non-balancedcase stitching together halfspaces,LP bounds

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

24. Analyzing the pass probability the test But: (since f is §1-valued) False: is possible: f (x) = x1. ???

25. ??? Idea But: (since f is §1-valued) False: is possible: f (x) = x1.

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

27. Open directions • 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? • 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 )