-Testing Sofya Raskhodnikova Penn State University, visiting - PowerPoint PPT Presentation

𝑀 𝑞 -Testing Sofya Raskhodnikova Penn State University, visiting Boston University and Harvard Joint work with Piotr Berman (Penn State), Grigory Yaroslavtsev (Penn State → Brown) 1

Property Testing Models Tolerant Property Tester Property Tester [Rubinfeld Sudan 96, Goldreich Goldwasser Ron 98] [Parnas Ron Rubinfeld 06] YES YES Accept with Accept with probability ≥ 𝟑/𝟒 probability ≥ 𝟑/𝟒 𝜁 1 𝜁 Don’t care 𝜁 2 Close to YES Don’t care Far from Far from Reject with Reject with   YES YES probability 2/3 probability 2/3 Equivalent to tolerant testing: estimating distance to the property. Two objects are at distance 𝜁 = they differ in an 𝜁 fraction of places 2

Why Hamming Distance? • Nice probabilistic interpretation – probability that two functions differ on a random point in the domain • Natural measure for – algebraic properties (linearity, low degree) – properties of graphs and other combinatorial objects • Motivated by applications to probabilistically checkable proofs (PCPs) • It is equivalent to other natural distances for – properties of Boolean functions 3

Which stocks grew steadily? Data from http://finance.google.com

𝑀 𝑞 -Testing for properties of real-valued data

Use 𝑀 𝑞 -metrics to Measure Distances • Functions 𝑔, 𝑕: 𝐸 → 0,1 over (finite) domain 𝐸 • For 𝑞 ≥ 1 1/𝑞 𝑞 𝑀 𝑞 𝑔, 𝑕 = 𝑔 − 𝑕 𝑞 = 𝑔 𝑦 − 𝑕 𝑦 𝑦∈𝐸 𝑀 0 𝑔, 𝑕 = 𝑔 − 𝑕 0 = 𝑦 ∈ 𝐸: 𝑔 𝑦 ≠ 𝑕 𝑦 𝒈 −𝑕 𝑞 • 𝑒 𝑞 𝑔, 𝑕 = 1 𝒒 6

𝑴 𝒒 -Testing and Tolerant 𝑴 𝒒 -Testing Tolerant Property Tester Property Tester YES YES Accept with Accept with probability ≥ 𝟑/𝟒 probability ≥ 𝟑/𝟒 𝜁 1 𝜁 Don’t care 𝜁 2 Close to YES Don’t care Far from Far from Reject with Reject with   YES YES probability 2/3 probability 2/3 𝑔−𝑕 𝑞 Functions 𝑔, 𝑕: 𝐸 → [0,1] are at distance 𝜁 if 𝑒 𝑞 = = 𝜁 . 𝟐 𝑞 7

New 𝑀 𝑞 -Testing Model for Real-Valued Data • Generalizes standard 𝑀 0 -testing • For 𝑞 > 0 still have a nice probabilistic interpretation: distance 𝑒 𝑞 𝑔, 𝑕 = 𝐅 𝒈 − 𝒉 𝒒 1/𝑞 • Compatible with existing PAC-style learning models (preprocessing for model selection) • For Boolean functions, 𝑒 0 𝑔, 𝑕 = 𝑒 𝑞 𝑔, 𝑕 𝑞 . 8

Our Contributions 1. Relationships between 𝑀 𝑞 -testing models 2. Algorithms – 𝑀 𝑞 -testers for 𝑞 ≥ 1 • monotonicity, Lipschitz, convexity – Tolerant 𝑀 𝑞 -tester for 𝑞 ≥ 1 • monotonicity in 1D (aka sortedness)  Our 𝑀 𝑞 -testers beat lower bounds for 𝑀 0 -testers  Simple algorithms backed up by involved analysis  Uniformly sampled (or easy to sample) data suffices 3. Nearly tight lower bounds 9

Implications for 𝑴 𝟏 -Testing Some techniques/observations/results carry over to 𝑀 0 -testing – Improvement on Levin’s work investment strategy Gives improvements in run time of testers for • Connectivity of bounded-degree graphs [Goldreich Ron 02] • Properties of images [R 03] • Multiple-input problems [Goldreich 13] – First example of monotonicity testing problem where adaptivity helps – Improvements to 𝑀 0 -testers for Boolean functions 10

Relationships between 𝑀 𝑞 -Testing Models

Relationships Between 𝑀 𝑞 -Testing Models 𝐷 𝒒 ( 𝑸 , 𝜻 ) = complexity of 𝑀 𝒒 -testing property 𝑸 with distance parameter 𝜻 • e.g., query or time complexity • for general or restricted (e.g., nonadaptive) tests For all properties 𝑸 • 𝑀 𝟐 -testing is no harder than Hamming testing 𝐷 𝟐 ( 𝑸 , 𝜻 ) ≤ 𝐷 𝟏 ( 𝑸 , 𝜻 ) • 𝑀 𝒒 -testing for 𝒒 > 1 is close in complexity to 𝑀 𝟐 -testing 𝐷 𝟐 ( 𝑸 , 𝜻 ) ≤ 𝐷 𝒒 ( 𝑸 , 𝜻 ) ≤ 𝐷 𝟐 ( 𝑸 , 𝜻 𝒒 ) 12

Relationships Between 𝑀 𝑞 -Testing Models 𝐷 𝒒 ( 𝑸 , 𝜻 ) = complexity of 𝑀 𝒒 -testing property 𝑸 with distance parameter 𝜻 • e.g., query or time complexity • for general or restricted (e.g., nonadaptive) tests For properties of Boolean functions 𝒈: 𝐸 → 0,1 • 𝑀 𝟐 -testing is equivalent to Hamming testing 𝐷 𝟐 ( 𝑸 , 𝜻 ) = 𝐷 𝟏 ( 𝑸 , ε ) • 𝑀 𝒒 -testing for 𝒒 > 1 is equivalent to 𝑀 𝟐 -testing with appropriate distance parameter 𝐷 𝒒 ( 𝑸 , 𝜻 ) = 𝐷 𝟐 ( 𝑸 , 𝜻 𝒒 ) 13

Relationships: Tolerant 𝑀 𝑞 -Testing Models 𝐷 𝒒 ( 𝑸 , 𝜻 𝟐 , 𝜻 𝟑 ) = complexity of tolerant 𝑀 𝒒 -testing property 𝑸 with distance parameters 𝜁 1 , 𝜁 2 • E.g., query or time complexity • for general or restricted (e.g., nonadaptive) tests For all properties 𝑸 • No obvious relationship between tolerant 𝑀 𝟐 -testing and tolerant Hamming testing • 𝑀 𝒒 -testing for 𝒒 > 1 is close in complexity to 𝑀 𝟐 -testing 𝒒 , ε 2 ) ≤ 𝐷 𝒒 ( 𝑸 , 𝜻 𝟐 , 𝜻 𝟑 ) ≤ 𝐷 𝟐 ( 𝑸 , 𝜻 𝟐 , 𝜻 𝟑 𝒒 ) 𝐷 𝟐 ( 𝑸 , 𝜻 𝟐 14

Relationships: Tolerant 𝑀 𝑞 -Testing Models 𝐷 𝒒 ( 𝑸 , 𝜻 𝟐 , 𝜻 𝟑 ) = complexity of tolerant 𝑀 𝒒 -testing property 𝑸 with distance parameters 𝜁 1 , 𝜁 2 • E.g., query or time complexity • for general or restricted (e.g., nonadaptive) tests For properties of Boolean functions 𝒈: 𝐸 → 0,1 • 𝑀 𝟐 -testing is equivalent to Hamming testing 𝐷 𝟐 ( 𝑸 , 𝜻 𝟐 , 𝜻 𝟑 ) = 𝐷 𝟏 ( 𝑸 , 𝜻 𝟐 , 𝜻 𝟑 ) • 𝑀 𝒒 -testing for 𝒒 > 1 is equivalent to 𝑀 𝟐 -testing with appropriate distance parameters d 𝒒 , 𝜻 𝟑 𝒒 ) 𝐷 𝒒 ( 𝑸 , 𝜻 𝟐 , 𝜻 𝟑 ) = 𝐷 𝟐 ( 𝑸 , 𝜻 𝟐 15

𝑃𝑣𝑠 𝑆𝑓𝑡𝑣𝑚𝑢𝑡 Property: Monotonicity

Monotonicity • Domain D= [𝑜] 𝑒 (vertices of 𝑒 -dim hypercube) (𝑒, 𝑒, 𝑒) • A function 𝑔: 𝐸 → R is monotone if increasing a coordinate of 𝑦 does not decrease 𝑔 𝑦 . • Special case 𝑒 = 1 (1,1,1) 𝑔: [𝑜] → R is monotone ⇔ 𝑔 1 , … 𝑔(𝑜) is sorted. One of the most studied properties in property testing [Ergün Kannan Kumar Rubinfeld Viswanathan , Goldreich Goldwasser Lehman Ron, Dodis Goldreich Lehman R Ron Samorodnitsky, Batu Rubinfeld White, Fischer Lehman Newman R Rubinfeld Samorodnitsky, Fischer, Halevy Kushilevitz, Bhattacharyya Grigorescu Jung R Woodruff, ..., Chakrabarty Seshadhri, Blais R Yaroslavtsev, Chakrabarty Dixit Jha Seshadhri] 17

Monotonicity Testers: Running Time 𝑀 𝑞 𝑔 𝑀 0 𝑜 Θ log 𝑜 1 Θ → [0,1] 𝜻 𝑞 𝜻 [Ergün Kannan Kumar Rubinfeld Viswanathan 00, Fischer 04] 𝑜 𝑒 𝑒 𝑒 Θ 𝑒 ⋅ log 𝑜 O 𝜻 𝑞 log 𝜻 𝑞 → [0,1] 𝜻 1 1 Ω 𝜻 𝑞 log 𝜻 𝑞 for 𝑒 = 2 [Chakrabarty Seshadhri 13] nonadaptive 1-sided error 18

Monotonicity Testers: Running Time 𝑀 𝑞 𝑔 𝑀 0 𝑜 Θ 1 1 Θ → {0,1} 𝜻 𝑞 𝜻 𝑒 𝑒 Θ 𝑒 𝜻 ⋅ log 3 𝑒 O 𝜻 𝑞 log 𝜻 𝑞 𝑜 𝑒 𝜻 1 1 Ω 𝜻 𝑞 log 𝜻 𝑞 for 𝑒 = 2 → {0,1} [Dodis Goldreich Lehman R Samorodnitsky 99] nonadaptive 1-sided error 1 Θ for constant 𝑒 𝜻 𝑞 adaptive 1-sided error 19

𝑀 1 -Testing of Monotonicity 20

Monotonicity: Reduction to Boolean Functions Given 𝒈: 𝐸 → [0,1], a Boolean threshold function 𝒈 (𝒖) : 𝐸 → {0,1} 𝒈 (𝒖) 𝑦 = 1 if 𝒈 𝑦 ≥ 𝑢 0 otherwise 𝒈 (𝒖) 𝑦 1 𝒈 (𝒖) 𝑦 𝑒𝒖 • Decomposition: 𝑔 𝑦 = 0 1 𝒖 • M = class of monotone functions 0 𝒈 𝑦 Characterization Theorem 1 𝑀 1 𝒈 (𝒖) , 𝑁 𝑒𝒖 𝑀 1 𝒈, 𝑁 = 0 21

Characterization Theorem: One Direction 1 𝑀 1 𝒈 (𝒖) , 𝑁 𝑒𝒖 𝑀 1 𝒈, 𝑁 ≤ 0 • ∀𝑢 ∈ 0,1 , let 𝑕 𝑢 =closest monotone (Boolean) function to 𝒈 (𝒖) . 1 𝑕 𝑢 𝑒𝒖 . Then 𝒉 is monotone, since 𝑕 𝑢 are monotone. • Let 𝒉 = 0 𝑀 1 𝒈, 𝑁 ≤ 𝑔 − 𝑕 1 Because 𝒉 is monotone 1 𝒈 (𝒖) 𝑒𝒖 − 0 1 𝑕 𝑢 𝑒𝒖 = 0 Decomposition & definition of 𝒉 1 1 (𝒈 (𝒖) −𝑕 𝑢 )𝑒𝒖 = 0 1 1 𝒈 (𝒖) − 𝑕 𝑢 1 𝑒𝒖 ≤ 0 Triangle inequality 1 𝑀 1 𝒈 (𝒖) , 𝑁 𝑒𝒖 = 0 Definition of 𝑕 𝑢 22

Monotonicity: Using Characterization Theorem Characterization Theorem 1 𝑒 1 𝒈 (𝒖) , 𝑁 𝑒𝒖 𝑒 1 𝒈, 𝑁 = 0 We use Characterization Theorem to get monotonicity 𝑀 1 -testers and tolerant testers from standard property testers for Boolean functions. 23

𝑀 1 -Testers from Testers for Boolean Ranges A nonadaptive, 1-sided error 𝑀 0 -test for monotonicity of 𝑔: 𝐸 → {0,1} is also an 𝑀 1 -test for monotonicity of 𝑔: 𝐸 → [0,1] . Proof: > 𝒈(𝒛) 𝒈(𝒚) • A violation (𝑦, 𝑧) : • A nonadaptive, 1-sided error test queries a random set 𝑅 ⊆ 𝐸 and rejects iff 𝑅 contains a violation. • If 𝑔: 𝐸 → [0,1] is monotone, 𝑅 will not contain a violation. • If 𝑒 1 𝑔, 𝑁 ≥ 𝜁 then ∃𝒖 ∗ : 𝑒 0 𝒈 (𝒖 ∗ ) , 𝑁 ≥ 𝜻 • W.p. ≥ 2/3 , set 𝑅 contains a violation (𝑦, 𝑧) for 𝒈 (𝒖 ∗ ) 𝒈 (𝒖 ∗ ) 𝑦 = 1, 𝒈 (𝒖 ∗ ) 𝑧 = 0 ⇓ 𝒈 𝑦 > 𝒈 𝑧 24