1. Dana Ron Tel Aviv University
Exponentially improved algorithms and lower bounds for testing signed majorities
Dana Ron
Tel Aviv University
Rocco A. Servedio Columbia University

2. Testing properties of Boolean functions: the basics
Let C be a class of Boolean functions mapping
e.g. C = {all monotone functions}, C = {all GF(2)-linear functions}, C = {all conjunctions}, etc.
Testing algorithm has black-box access to unknown and arbitrary

3. Testing properties of Boolean functions, cont.
Success criterion for testing algorithm:
output “accept” with prob. > 2/3 if C;
outputs “reject” with prob. > 2/3 if is e-far from every C.
Measure performance of algorithm by # of oracle calls to that it makes.
Several natural properties are testable with query complexity independent of n:

4. Some Boolean function testing results
Class of functions # of queries
parity functions [BLR93]
degree-d GF(2) polynomials [AKK+03]
literals [PRS02]
conjunctions [PRS02]
J-juntas [FKRSS04, B08, B09]
s-term monotone DNF [PRS02]
halfspaces [MORS09]

5. Different flavors of testing problems
Logical/combinatorial properties: conjunctions, juntas, size-s decision trees, DNF formulas, etc. Algebraic properties: low-degree GF(2) polynomials, low-degree/sparse Fourier representations, etc. Geometric properties: halfspaces

6. Halfspaces A halfspace is a function
[MORS09] gave poly(1/e)-query algorithm for testing halfspaces.

7. This work: testing signed majorities
A signed majority: a halfspace such that
, each
Highly symmetrical class of halfspaces
Correspond to fair voting schemes where each voter has one of two opposing orientations
For rest of talk, C = {signed majorities}

8. Testing signed majorities: previous results
Perhaps surprisingly, testing signed majorities is provably harder than testing halfspaces.
[MORS09b] gave
-query nonadaptive algorithm for testing C;
-query lower bound for nonadaptive algorithms that test C.

9. This paper’s results
Exponentially improved bounds (both upper and lower bounds) for testing signed majorities.
Theorem 1: A query adaptive algorithm for testing signed majorities.
Computationally efficient – time
Previous algorithm (nonadaptive) used queries

10. Our results, continued Exponentially improved lower bound:
Theorem 2: Any non-adaptive algorithm for testing signed majorities must make queries, even for testing when
Implies lower bound for adaptive testing algorithms
Previous lower bound was for non-adaptive testing algorithms

11. The lower bound -- sketch
Standard approach for nonadaptive lower bounds [Yao77]:
1) Define
Yes-distribution – distribution DYes over functions f in the class
No-distribution – distribution DNo over functions g far from the class
2) Show that for any fixed vector of q inputs (x1,…, xq) from {+1,-1}n, the two distributions over response vectors
(f(x1),…,f(xq)) and (g(x1),…,g(xq))
are statistically close to each other.
This gives a lower bound of q queries for nonadaptive testers.
where f~DYes
where g~DNo

12. Lower bound sketch, cont.
Our Dyes distribution:
, each
Our Dno distribution:
, each
Same distributions as in earlier [MORS09b] lower bound.
Proof uses multidimensional invariance tools [BO10,GOWZ10,M08].
Rest of talk: the algorithmic results

13. Fourier basics
Recall
For monotone/unate f, these are the influences of variables (up to +- sign)

14. The [MORS09b] algorithm
In signed majority function have Not hard to show that signed majorities are precisely the functions that maximize

15. The [MORS09b] algorithm, cont
In signed majority function have
[MORS09b]: If is -far from every signed majority, then fraction of all coordinates have
 natural nonadaptive algorithm: sample coordinates , estimate for each.

16. This work: How to avoid poly(n) query complexity?
First intuition: use degree-1 Fourier coefficients “collectively” rather than individually (a la [MORS09] algorithm for testing general halfspaces)
Look at sum of squares of degree-1 Fourier coefficients
Second intuition: algorithm had better exploit adaptiveness somewhere (recall lower bound for nonadaptive algorithms…)
Sequence of restrictions fixing more and more variables – use adaptiveness to confirm that have “right” restriction before extending it

17. High-level idea behind algorithm
Let be any function that is e-far from every signed majority. Then either
Degree-1 Fourier weight is far from “right” value ; or
Some individual degree-1 Fourier coefficient is large; or
Can find a restriction of such that is defined over variables and is far from every signed majority over variables.
If neither (1) nor (2) is detected, iterate on
Can check this efficiently [MORS09]
“large” ~ e/log n; can
check this efficiently
This is the hard part of the analysis…
Uses adaptiveness!
After iterations, reach function on variables which can be tested easily with queries.

18. Sketch of the algorithm
Estimate , reject if too far from
Check if any ; if yes then reject, otherwise continue to next step
Pick random subset of variables. Try random restrictions fixing until get one such that resulting is roughly balanced. (Reject if too many failures.)
Check that degree-1 Fourier coefficients of restriction, ,
are “compatible” with corresponding degree-1 Fourier coefficients of original function,
(5) Recurse on .
“compatible”: the two vectors are close – roughly same length, point in roughly same direction
Do this until defined on variables; then use naïve method to test that is close to signed majority on variables.

19. Completeness: f is a signed MAJ
Estimate , reject if too far from
Check if any ; if yes then reject, otherwise continue to next step
Pick random subset of variables. Try random restrictions fixing until get one such that resulting is roughly balanced. (Reject if too many failures.)
Check that degree-1 Fourier coefficients of restriction, ,
are “compatible” with corresponding degree-1 Fourier coefficients of original function,
(5) Recurse on .
Do this until defined on variables; then use naïve method to test that is close to signed majority on variables.

20. Two key lemmas for soundness
Lemma 1: (roughly stated): if is far from every signed MAJ, then level-1 Fourier coefficients of are far from those of any signed MAJ.
Gets us “off the ground” in working with level-1 Fourier coefficients
Lemma 2: (roughly stated): if degree-1 Fourier coefficients of are far from those of any signed MAJ and gets to step (4), then whp over choice of , the degree-1 Fourier coefficients of a compatible are also far from those of any signed MAJ.

21. Soundness: f far from every signed MAJ
Lemma 1  level-1 Fourier coeff of f are far from every signed MAJ
Estimate , reject if too far from
Check if any ; if yes then reject, otherwise continue to next step
Pick random subset of variables. Try random restrictions fixing until get one such that resulting is roughly balanced. (Reject if too many failures.)
Check that degree-1 Fourier coefficients of restriction, ,
are “compatible” with corresponding degree-1 Fourier coefficients of original function,
(5) Recurse on .
If f passes this step, Lemma 2  level-1 Fourier coefficients of f’ are also far from every signed MAJ
If test doesn’t reject earlier, it will reject here!
Do this until defined on variables; then exhaustively check that level-1 Fourier coefficients of match those of some signed majority on variables.

22. Summary
Exponentially improved results (both upper and lower bounds) for testing signed majorities. Theorem 1: A -query adaptive algorithm for testing signed majorities.
Theorem 2: Any non-adaptive algorithm for testing signed majorities must make queries, even for testing when

23. Future work
Apply ingredients in our adaptive algorithm to get better adaptive testers for monotonicity?

24. THANK YOU