Lower Bounds for Testing Properties of Functions on Hypergrids
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Lower Bounds for Testing Properties of Functions on
Hypergrids
Grigory Yaroslavtsevhttp://grigory.us
Joint with: Eric Blais (MIT)
Sofya Raskhodnikova (PSU)
⇒ ⇒
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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 can be changed to become YES
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Ultra-fast Approximate Decision Making
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Property = set of YES instances
Query complexity of testing • = Adaptive queries• = Non-adaptive (all queries at once)• = Queries in rounds ()
Property Testing [Goldreich, Goldwasser, Ron, Rubinfeld, Sudan]
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Communication Complexity [Yao’79]
Alice: Bob:
𝒇 (𝒙 ,𝒚 )=?
Shared randomness
…𝒇 (𝒙 ,𝒚 )
• = min. communication (error ) • min. -round communication (error )
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• -linear function: where • -Disjointness: ,
, iff .
Alice: Bob:
0?
/2-disjointness -linearity [Blais, Brody,Matulef’11]
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/2-disjointness -linearity [Blais, Brody,Matulef’11]
• is -linear• is -linear, ½-far from -linear
𝑺⊆ [𝒏 ] ,|𝑺|=𝒌 /𝟐 𝐓⊆ [𝒏 ] ,|𝑻|=𝒌/𝟐𝝌𝑺=⊕𝑖 ∈𝑺 𝑥𝑖 𝝌𝑻=⊕𝑖 ∈𝑻 𝑥 𝑖
𝝌=𝜒 𝑺⊕ 𝜒𝑻
• Test for -linearity using shared randomness• To evaluate exchange and (2 bits)
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-Disjointness• [Razborov, Hastad-Wigderson] • [Folklore + Dasgupta, Kumar, Sivakumar’12; Buhrman, Garcia-Soriano, Matsliah, De Wolf’12]
where [Saglam, Tardos’13]
• [Braverman, Garg, Pankratov, Weinstein’13]
{ times
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Property testing lower bounds via CC
• Monotonicity, Juntas, Low Fourier degree, Small Decision Trees [Blais, Brody, Matulef’11]
• Small-width OBDD properties [Brody, Matulef, Wu’11]
• Lipschitz property [Jha, Raskhodnikova’11]• Codes [Goldreich’13, Gur, Rothblum’13]• Number of relevant variables [Ron, Tsur’13]
(Almost) all: Boolean functions over Boolean hypercube
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Functions [Blais, Raskhodnikova, Y.]
monotone functions over
Previous for monotonicity on the line ():• [Ergun, Kannan, Kumar, Rubinfeld, Viswanathan’00]• [Fischer’04]
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Functions [Blais, Raskhodnikova, Y.]
• Proof ideas: – Reduction from Augmented Index (widely used in
streaming, e.g [Jayram, Woodruff’11; Molinaro, Woodruff, Y.’13])
– Fourier analysis over basis of characters => Fourier analysis over : basis of Walsh functions
• C Any non-adaptive tester for monotonicity of has complexity
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Functions [Blais, Raskhodnikova, Y.]
• Augmented Index: S; ()
• [Miltersen, Nisan, Safra, Wigderson, 98]
𝑺⊆ [ 𝑙𝑜𝑔𝒎 ] 𝒊∈ [ 𝑙𝑜𝑔𝒎 ] ,𝑺∩ [𝒊−1]
?
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Walsh functions: For :,
where is the -th bit of
Functions [Blais, Raskhodnikova, Y.]
𝒙𝒘 {𝟒 }=¿
𝒙𝒘 {𝟏}=¿
𝒙𝒘 {𝟐}=¿
…
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Step functions. For :
Functions [Blais, Raskhodnikova, Y.]
𝒙𝑠𝑡𝑒𝑝2=¿
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• Augmented Index Monotonicity Testing
• is monotone• is ¼ -far from monotone• Only -th frequency matters: higher frequencies are
cancelled, lower don’t affect monotonicity• Thus,
𝑺⊆ [ 𝑙𝑜𝑔𝒎 ]
𝒊∈ [ 𝑙𝑜𝑔𝒎 ] ,𝑺∩ [𝒊−1]
Functions [Blais, Raskhodnikova, Y.]
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Functions [Blais, Raskhodnikova, Y.]
𝒊∈ [𝒏𝑙𝑜𝑔𝒎 ] ,𝑺∩ [𝒊−1 ]
Embed into -th coordiante using -dimensional Walsh and step functions:• Walsh functions: • Step functions:
⇒
…, ,
⇒
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Functions [Blais, Raskhodnikova, Y.]
𝑺𝟏 ,…,𝑺𝒏⊆ [ log𝒎 ]…, ,
• Walsh functions: • Step functions:
+
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Functions [Blais, Raskhodnikova, Y.]
𝑺𝟏 ,…,𝑺𝒏⊆ [ log𝒎 ]…, ,
• Only coordinate matters:– Coordinates < cancelled by Bob’s Walsh terms– Coordinates > cancelled by Bob’s Step terms– Coordinate behaves as in the case
+
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Functions [Blais, Raskhodnikova, Y.]
• monotone functions over
• -Lipschitz functions over • separately convex functions over • monotone axis-parallel -th derivative over • convex functions over
– Can’t be expressed as a property of axis-parallel derivatives!
Thm. [BRY] For all these properties These bounds are optimal for and [Chakrabarty, Seshadhri, ‘13]
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Open Problems
• Adaptive bounds and round vs. query complexity tradeoffs for functions – Only known: [Fischer’04; Chakrabarty Seshadhri’13]
• Inspired by connections of CC and Information Complexity– Direct information-theoretic proofs?– Round vs. query complexity tradeoffs in property testing?
• Testing functions – -testing model [Berman, Raskhodnikova, Y. ‘14]– Testing convexity: vs. ?