Lower Bounds for Testing Properties of Functions on Hypergrids

1. Lower Bounds for Testing Properties of Functions on Hypergrids GrigoryYaroslavtsev http://grigory.us Joint with: Eric Blais (MIT) SofyaRaskhodnikova (PSU)

2. Property Testing [Goldreich, Goldwasser, Ron, Rubinfeld, Sudan] Randomized algorithm Property tester YES YES Accept with probability Accept with probability -close Don’t care No No Reject with probability Reject with probability -close: fraction can bechanged to become YES

3. Ultra-fast Approximate Decision Making

4. Property Testing [Goldreich, Goldwasser, Ron, Rubinfeld, Sudan] Property = set of YES instances Query complexity of testing • = Adaptive queries • = Non-adaptive (all queries at once) • = Queries in rounds ()

5. Communication Complexity [Yao’79] Shared randomness Bob: Alice: … • = min. communication (error ) • min. -round communication (error )

6. /2-disjointness -linearity [Blais, Brody,Matulef’11] • -linear function: where • -Disjointness: , , iff. Alice: Bob: 0?

7. /2-disjointness -linearity [Blais, Brody,Matulef’11] • is -linear • is -linear, ½-far from -linear • Test for -linearity using shared randomness • To evaluate exchange and (2 bits)

8. -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

9. 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

10. Functions[Blais, Raskhodnikova, Y.] monotone functions over Previous for monotonicity on the line (): • [Ergun, Kannan, Kumar, Rubinfeld, Viswanathan’00] • [Fischer’04]

11. 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 • CAny non-adaptive tester for monotonicity of has complexity

12. Functions[Blais, Raskhodnikova, Y.] • Augmented Index: S; () • [Miltersen, Nisan, Safra, Wigderson, 98] ?

13. Functions[Blais, Raskhodnikova, Y.] Walsh functions: For : , where is the -th bit of …

14. Functions[Blais, Raskhodnikova, Y.] Step functions. For :

15. Functions[Blais, Raskhodnikova, Y.] • 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,

16. Functions[Blais, Raskhodnikova, Y.] Embed into -thcoordiante using -dimensional Walsh and step functions: • Walsh functions: • Step functions: …,,

17. Functions[Blais, Raskhodnikova, Y.] • Walsh functions: • Step functions: …,, +

18. Functions[Blais, Raskhodnikova, Y.] • Only coordinate matters: • Coordinates < cancelled by Bob’s Walsh terms • Coordinates > cancelled by Bob’s Step terms • Coordinate behaves as in the case …,, +

19. 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]

20. 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. ?