for testing signed majorities Dana Ron Tel Aviv University Rocco A Servedio Columbia University Testing properties of Boolean functions the basics Let C be a class of Boolean functions mapping ID: 631528
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Slide1
Exponentially improved algorithms and lower boundsfor testing signed majorities
Dana RonTel Aviv University
Rocco A. Servedio
Columbia UniversitySlide2
Testing properties of Boolean functions: the basicsLet 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
tounknown and arbitrarySlide3
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:Slide4
Some Boolean function testing results
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]
Class of functions # of queries Slide5
Different flavors of testing problemsLogical/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: halfspacesSlide6
HalfspacesA halfspace is a function
[MORS09]
gave poly(1/
e
)-query algorithm for testing
halfspaces
.Slide7
This work: testing signed majoritiesA 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}Slide8
Testing signed majorities:previous resultsPerhaps 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.Slide9
This paper’s resultsExponentially 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 queriesSlide10
Our results, continuedExponentially 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 algorithmsSlide11
The lower bound -- sketchStandard approach for nonadaptive lower bounds [Yao77]: 1) DefineYes-distribution – distribution
DYes over functions f in the class
No-distribution – distribution
D
No 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(x
1
),…,
f(x
q
)
) and (g(x1),…,g(xq))are statistically close to each other. This gives a lower bound of q queries for nonadaptive testers.where f~DYeswhere
g~DNoSlide12
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 resultsSlide13
Fourier basicsRecall
For monotone/
unate
f
, these are the
influences
of variables (up to +- sign)Slide14
The [MORS09b] algorithmIn signed majority function have
Not hard to show that signed majorities are precisely the functions that maximize
Slide15
The [MORS09b] algorithm, cont natural
nonadaptive algorithm: sample coordinates , estimate for each.
In
signed majority function
have
[MORS09b]
: If is -far from every signed majority, then fraction of all coordinates have Slide16
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)
Second intuition: algorithm had better exploit
adaptiveness
somewhere (recall lower bound for
nonadaptive
algorithms…)
Look at
sum of squares of degree-1 Fourier coefficients
Sequence of
restrictions
fixing more and more variables – use
adaptiveness
to confirm that have “right” restriction before extending itSlide17
High-level idea behind algorithmLet 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 .
After iterations, reach function on variables which can be tested easily with queries.
Can check this efficiently
[MORS09]
“large” ~
e
/log
n
; can
check this efficiently
This is the hard part of the analysis…
Uses
adaptiveness
! Slide18
Sketch of the algorithmEstimate , reject if too far fromCheck 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 coefficientsof 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.
“compatible”: the two vectors are close – roughly same length, point in roughly same directionSlide19
Completeness: f is a signed MAJEstimate , reject if too far from
Check if any ; if yes then reject, otherwise continue to next stepPick 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. Slide20
Two key lemmas for soundnessLemma 1: (roughly stated): if is far from every signed MAJ, then level-1 Fourier coefficients of are far from those of any signed MAJ.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.
Gets us “off the ground” in working with level-1 Fourier coefficientsSlide21
Soundness: f far from every signed MAJEstimate , reject if too far from
Check if any ; if yes then reject, otherwise continue to next stepPick 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 exhaustively check that level-1 Fourier coefficients of match those of some signed majority on variables.
Lemma 1
level-1 Fourier
coeff
of
f
are far from every signed MAJ
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! Slide22
SummaryExponentially 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 whenSlide23
Future workApply ingredients in our adaptive algorithm to getbetter adaptive testers for monotonicity?Slide24
THANK YOU