New Algorithms and Lower Bounds for Monotonicity Testing of Boolean Functions
Rocco Servedio
Joint work with Xi Chen and Li-Yang TanColumbia University
Institute for Advanced StudyMarch 2014
Property Testing
All Boolean functions
Property P
Query access to unknown f on any input x
With as few queries as possible, decide if
Far from P
Simplest question about a Boolean function: Does it have some property P?
f has Property P vs. f does not have Property P
Unreasonable: Easy lower bound of
f is far from having Property P
Rules of the game
1. If f has Property P, accept w.p. > 2/3.
2. If f is ε-far from having Property P, reject w.p. > 2/3.
3. Otherwise: doesn’t matter what we do.
All Boolean functions
Property P Far from P
Query access to unknown f on any input x
This work: P = monotonicity
Monotone Far from monotone
Well-studied problem:
[GGR98, GGL+98, DGL+99, FLN+02, HK08, BCGM12, RRS+12, BBM12, BRY13, CS13, …]
but still significant gaps in our understanding.
This work: P = monotonicity
Background and prior results
For all monotone functions g: A monotone function is one that satisfies:
Monotone Far from monotone
Quick reminder about monotonicity
“Flipping an input bit from 0 to 1 cannot make f go from 1 to 0”
Monotone
1
01
1
01
Far from monotone
0n 0n
1n 1n
First test that comes to mind
Pick random edge
y = 1 1 1 1 0 0 1 0 1 0 1 0 0 1
x = 1 1 1 1 0 0 1 1 1 0 1 0 0 1
Reject!
f(x) = 0
f(y) = 1
f(x) = 1
f(y) = 1
f(x) = 0
f(y) = 0
f(x) = 1
f(y) = 0
Repeat
?x
y
Call such an edge a “violating edge”
Theorem [Goldreich et al. 1998] If f is ε-far from monotone, Ω(ε/n) fraction of edges are violations.
Therefore tester will reject within O(n/ε) queries.
An observation and a theorem
Reject!
f(x) = 0
f(y) = 1
Call such an edge a “violating edge”
Simple observation: If f is monotone, tester never rejects.
Question: If f is ε-far from monotone, how likely to catch violating edge?
?x
y
Tester = Sample random edges, check for violations.
An exponential gap between upper and lower bounds
Goldreich et al. [FOCS 1998]
Introduced problem, gave tester with O(n) query complexity.
Fischer et al. [STOC 2002]
Any non-adaptive tester must make Ω(log n) queries. Therefore, any adaptive tester must make Ω(log log n) queries.
Chakrabarty-Seshadhri [STOC 2013]
O(n7/8)-query non-adaptive tester!
[GGR98, DGL+99, HK08, BCGM12, RRS+12, BRY13, CS13, …]
Lower bound:
Any non-adaptive tester must make queries.Therefore, any adaptive tester must make queries.
Exponential improvements over Ω(log n) and Ω(log log n) lower bounds of Fischer et al. (2002)
Upper bound:There is a non-adaptive tester that make
queries.
Polynomial improvement over O(n7/8) upper bound of Chakrabarty and Seshadhri (2013)
Our results
Rest of talk
The lower bound
• Main idea of the argument in a simpler setting
• Key technical ingredient: Multidimensional Central LimitTheorems
• Generalizing the lower bound: Testing monotonicity on hypergrids
Yao’s minimax principle
Lower bound against randomized algorithms
Tricky distribution over inputs to deterministic algorithms
implies
Yao’s principle in our setting
Monotone Far from Monotone
Distribution Dno supported on far from monotone functions
Indistinguishability. For all T = deterministic tester that makes o(n1/5) queries,
Distribution Dyes supported
on monotone functions
Dno : = −1 with prob 0.1, 7/3 with prob 0.9
Main Structural Result: Indistinguishability
Any deterministic tester that makes few queries cannot tell Dyes from Dno
Key property: , .
Dyes : = uniform from {1,3}
Our Dyes and Dno distributions
Both supported on Linear Threshold Functions (LTFs) over {−1,1}n:
Verify: Dyes LTFs are monotone, Dno LTFs far from monotone w.h.p.
Claim. For all T = deterministic tester that makes q = o(n1/5) queries,
Indistinguishability: starting small
q = 1 query
Non-trivial proof of a triviality
Claim. Let T = deterministic tester that makes 1 query, on string z. Then:
(*)
vs.
Tester sees:
Central Limit Theorems. Sum of many independent “reasonable” random variables converges to Gaussian of same mean and variance.
Main analytic tool (Baby version):
Goal: Upper bound
Recall key property:
Claim. Let T = deterministic tester that makes 1 query. Then:
We just proved:
Plenty of room to spare!Would be happy with < 0.1
q queries instead of 1
Multidimensional CLTs. Sum of many independent “reasonable” q-dimensional random variables converge to q-dimensional
Gaussian of same mean and variance.
Main analytic tool (Grown-up version):
Multidimensional CLTs
[Mossel 08, Gopalan-O’Donnell-Wu-Zuckerman 10]
Lower bound
[Valiant-Valiant 11]
Main technical work:
Adapting multidimensional CLT for Earth Mover Distance to get
Claim. For all T = deterministic tester that makes q = o(n1/5) queries,
Let’s prove the real thing:
Setting things up
Arrange the q queries of tester T in a q x n matrix Q
i-th query string
Recall: Tester’s goal is to distinguish
versus
What does the tester see?
Goal: Upper bound
Tester sees:
(coordinate-wise sign)
Goal: Upper bound
Random variables supported on 2q orthants of Rq
union of orthants“roughly equal weight on any union of orthants”
Fixed
from product distribution over Rn sum of n independent vectors in Rq
Ditto:
Goal: Two sums of n independent vectors in Rq are “close”
where closeness = roughly equal weight on any union of orthants.
The final setup
Furthermore, since
suffices to show each are close to q-dimensional Gaussian with matching mean and covariance matrix.
Valiant-Valiant Multidimensional CLT
Sum of many independent “reasonable” q-dimensional random variables is close to q-dimensional Gaussian of same mean and variance.
with respect to Earth Mover Distance:
Minimum amount of work necessary to “change one PDF into the other one”,
where work := mass x distance
Technical lemma:Closeness in EMD roughly equal
weight on any union of orthants
Valiant-Valiant Multidimensional CLT
Key technical lemma
Closeness in EMD Roughly equal weight on any union of orthants
Let’s consider the contrapositive:for all unions of orthants
≥ 0.1 unit of mass must be moved out of
Recall: work = mass x distance
Slight technical obstacle: What if this 0.1 unit of mass only moves a tiny distance?
Solution: Then G has ≥ 0.1 weight on red boundary.
Not possible thanks to anti-concentration of G
If we want to make S look like G,
In slightly more detail
For all r > 0, define Br := radius r boundary around
Therefore:
, has to be moved out of
Br
We just proved
Gaussian anti-concentration
Valiant-ValiantCLT
The details
t=O(1) in our setting
= in our setting
Optimal choice of makes RHS
Pick , and RHS is o(1) as desired.
Indistinguishability. For all T = deterministic tester that makes q = o(n1/5) queries,
Recap
Lower bound:
Any non-adaptive tester must make queries.Therefore, any adaptive tester must make queries.
Testing monotonicity of Boolean functions over general hypergrid domains
Boolean functions over hypergrids
Monotone Far from monotone
All Boolean functionsAll Boolean-valued functions over hypergrid
Testing monotonicity of Boolean functions over hypergrids
Lower bound: *Any non-adaptive tester for testing monotonicity of
.must make queries.
* only for even m
Proof by reduction to m=2 case (Boolean hypercube)
A useful characterization
Theorem. [Fischer et al. 2002]
There exists ε .
mn vertex-disjoint pairs such that and .
is ε-far from monotone
“violating pair”
(Upward direction is easy)
0
1
0
1
0
1
0
1
0
1
0
10
1
Given any , define as follows:
Reducing to m = 2
numbers in [m] bits in {0,1}
Easy: If f is monotone then so is F.
Remains to argue:If f is ε-far from monotone then so is F.
Exists ε2n vertex-disjoint pairs in {0,1}n that are violations w.r.t. f
Exists ε .
mn vertex-disjoint pairs in [m]n that are violations w.r.t. F
Useful characterization
Useful characterization
Each violating pair (m/2)n vertex-disjoint violating pairs
suffices to show:
f(x) = 0
f(y) = 1
1
0 S(x)
S(y){0,1}n
[m]n
|S(x)| = |S(y)| = (m/2)n
where
Easy end game: exhibit order-preserving bijection between S(x) and S(y)
Conclusion
A polynomial lower bound for testing monotonicity of Boolean functions
Main technical ingredient: multidimensional central limit theorems Proof extends to testing monotonicity over general hypergrid domains
(when m is even)
Open Problem: Polynomial lower bounds against adaptive testers?
Lower bound:
Any non-adaptive tester must make queries.Therefore, any adaptive tester must make queries.
What I didn’t talk about
An improved one-sided, non-adaptive tester
Builds on ingredients from [Chakrabarty and Seshadhri 2013]: trade off edge tester versus “path tester”
We give a different “path tester” with an improved analysis
Open Problems: can we exploit adaptivity or two-sided error to obtain more efficient testers?
Upper bound:There is a non-adaptive tester that make
queries.
Thank you!