Lower Bounds for Testing Properties of Functions on

Hypergrids

Grigory Yaroslavtsevhttp://grigory.us

Joint with: Eric Blais (MIT)

Sofya Raskhodnikova (PSU)

⇒ ⇒

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

Ultra-fast Approximate Decision Making

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]

Communication Complexity [Yao’79]

Alice: Bob:

𝒇 (𝒙 ,𝒚 )=?

Shared randomness

…

𝒇 (𝒙 ,𝒚 )• = min. communication (error ) • min. -round communication (error )

• -linear function: where • -Disjointness: ,

, iff .

Alice: Bob:

0?

/2-disjointness -linearity [Blais, Brody,Matulef’11]

/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)

-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

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

Functions [Blais, Raskhodnikova, Y.]

monotone functions over

Previous for monotonicity on the line ():• [Ergun, Kannan, Kumar, Rubinfeld, Viswanathan’00]• [Fischer’04]

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

Functions [Blais, Raskhodnikova, Y.]

• Augmented Index: S; ()

• [Miltersen, Nisan, Safra, Wigderson, 98]

𝑺⊆ [ 𝑙𝑜𝑔𝒎 ] 𝒊∈ [ 𝑙𝑜𝑔𝒎 ] ,𝑺∩ [𝒊−1]

?

Walsh functions: For :,

where is the -th bit of

Functions [Blais, Raskhodnikova, Y.]

𝒙𝒘 {𝟒 }=¿

𝒙𝒘 {𝟏}=¿

𝒙𝒘 {𝟐}=¿

…

Step functions. For :

Functions [Blais, Raskhodnikova, Y.]

𝒙𝑠𝑡𝑒𝑝2=¿

• 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.]

Functions [Blais, Raskhodnikova, Y.]

𝒊∈ [𝒏𝑙𝑜𝑔𝒎 ] ,𝑺∩ [𝒊−1 ]

Embed into -th coordiante using -dimensional Walsh and step functions:• Walsh functions: • Step functions:

⇒

…, ,

⇒

Functions [Blais, Raskhodnikova, Y.]

𝑺𝟏 ,…,𝑺𝒏⊆ [ log𝒎 ]…, ,

• Walsh functions: • Step functions:

+

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

+

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]

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. ?