grigory yaroslavtsev joint work with piotr berman and sofya raskhodnikova
TRANSCRIPT
-Testing
Grigory Yaroslavtsev
Joint work with Piotr Berman and Sofya Raskhodnikova
⇒ ⇒
Testing Big Data
• Q: How to understand properties of large data looking only at a small sample?
• Q: How to ignore noise and outliers?• Q: How to minimize assumptions about the
sample generation process?• Q: How to optimize running time?
Property Testing [Goldreich, Goldwasser, Ron; Rubinfeld, Sudan]
NO
YES
Randomized Algorithm
Accept with probability
Reject with probability
⇒
⇒⇒
YES
NO
Property Tester
-close
Accept with probability
Reject with probability
⇒
⇒Don’t care
-close : fraction has to be changed to become YES
Tolerant Property Testing [Parnas, Ron, Rubinfeld]
-close : fraction has to be changed to become YES
⇒
YES
NO
Property Tester
-close
Accept with probability
Reject with probability
⇒
⇒Don’t care
⇒
Tolerant Property Tester
Accept with probability
Reject with probability
⇒
⇒Don’t care
NO
-close-close
YES
• = class of monotone functions
• -close:
Tolerant “Property Testing”
⇒
Tolerant “ Property Tester”
Accept with probability
Reject with probability
⇒
⇒Don’t care
NO
-close-close
YES
8
New -Testing Model for Real-Valued Data
• Generalizes standard Hamming testing
• For still have a probabilistic interpretation:
• Compatible with existing PAC-style learning models (preprocessing for model selection)
• For Boolean functions, .
9
Our Contributions
1. Relationships between -testing models2. Algorithms
– -testers for • monotonicity, Lipschitz, convexity
– Tolerant -tester for • monotonicity in 1D (sublinear algorithm for isotonic regression)
Our -testers beat lower bounds for Hamming testersSimple algorithms backed up by involved analysisUniformly sampled (or easy to sample) data suffices
3. Nearly tight lower bounds
10
Implications for Hamming Testing
Some techniques/results carry over to Hamming testing
– Improvement on Levin’s work investment strategy• Connectivity of bounded-degree graphs [Goldreich, Ron ‘02]
• Properties of images [Raskhodnikova ‘03]
• Multiple-input problems [Goldreich ‘13]
– First example of monotonicity testing problem where adaptivity helps
– Improvements to Hamming testers for Boolean functions
Definitions
• (D = finite domain/poset)• , for • Hamming weight (# of non-zero values)• Property = class of functions (monotone,
convex, linear, Lipschitz, …)
Relationships: -Testing
(,) = query complexity of -testing property at distance
• (Cauchy-Shwarz)
Boolean functions
Relationships: Tolerant -Testing(,) = query complexity of tolerant -testing property with distance parameters
• No general relationship between tolerant -testing and tolerant Hamming testing
• -testing for is close in complexity to -testing
For Boolean functions =
Our Results: Testing Monotonicity• Hypergrid ()
• adaptive tester for Boolean functions
Upper bound
[Dodis et al. ’99,…, Chakrabarti, Seshadhri ’13]
Lower bound
[Dodis et al.’99…, Chakrabarti, Seshadhri ’13]
Non-adaptive 1-sided error
Monotonicity: Key Lemma
• M = class of monotone functions• Boolean slicing operator
if otherwise.
• Theorem:
Proof sketch: slice and conquer
1) Closest monotone function with minimal -norm is unique (can be denoted as an operator ).
2) and commute:
=
2) 3)1)
-Testers from Boolean Testers Thm: A nonadaptive, 1-sided error -test for monotonicity of is also an -test for monotonicity of .Proof:• A violation :• A nonadaptive, 1-sided error test queries a random set and
rejects iff contains a violation.• If is monotone, will not contain a violation.• If then • W.p., set contains a violation for
12
𝒇 (𝒙) 𝒇 (𝒚 )¿
Distance Approximation and Tolerant Testing
[Saks Seshadhri 10]
Approximating -distance to monotonicity
• Time complexity of tolerant -testing for monotonicity is
– Better dependence than what follows from distance appoximation for
– Improves adaptive distance approximation of [Fattal,Ron’10] for Boolean functions
-Testers for Other PropertiesVia combinatorial characterization of -distance to the property• Lipschitz property :
Via (implicit) proper learning: approximate in up to error , test approximation on a random -sample• Convexity :
(tight for ) • Submodularity
[Feldman, Vondrak 13]
Open Problems• All our algorithms for for were obtained directly from -testers.Can one design better algorithms by working directly with -distances?• Our complexity for -testing convexity grows exponentially with d
Is there an -testing algorithm for convexity with subexponential dependence on the dimension?
• Our -tester for monotonicity is nonadaptive, but we show that adaptivity helps for Boolean range.
Is there a better adaptive tester?• We designed tolerant tester only for monotonicity (d=1,2).
Tolerant testers for higher dimensions? Other properties?