Settling the Query Complexity of Non-Adaptive Junta Testing

Transcription

1 Settling the Query Complexity of Non-Adaptive Junta Testing Erik Waingarten, Columbia University Based on joint work with Xi Chen (Columbia University) Rocco Servedio (Columbia University) Li-Yang Tan (Toyota Technological Institute) Jinyu Xie (Columbia University) 1 / 29

2 Boolean Function Property Testing Given query (black-box) access to an unknown Boolean function f : {0, 1} n {0, 1}, does it have some property P? With as few queries as possible, a randomized tester to tell if f has property P vs. f is far from having property P 2 / 29

3 Boolean Function Property Testing: FAQ What does far from P mean? Distance between two functions f and g: dist(f, g) = dist(f, P) = ming P dist(f, g) ε. [ ] Pr f (x) g(x) x {0,1} n 3 / 29

4 Rules of the Game Given query access to an unknown f : {0, 1} n {0, 1} and a parameter ε > 0: If f has property P, accept w.p. > 2/3; If f is ε-far from having property P, reject w.p. > 2/3; Otherwise: doesn t matter what we do. Given P, number of queries needed in terms of n and ε? 4 / 29

5 This talk: P = k-juntas Definition A Boolean function f : {0, 1} n {0, 1} is a k-junta if it depends on at most k variables. 5 / 29

6 Non-Adaptive vs. Adaptive Algorithms x 1,..., x q f f (x 1 ),..., f (x q ) 6 / 29

7 Non-Adaptive vs. Adaptive Algorithms x 1,..., x q x 1 f f f (x 1 ),..., f (x q ) 6 / 29

8 Non-Adaptive vs. Adaptive Algorithms x 1,..., x q f f f (x 1 ),..., f (x q ) f (x 1 ) 6 / 29

9 Non-Adaptive vs. Adaptive Algorithms x 1,..., x q x 2 f f f (x 1 ),..., f (x q ) 6 / 29

10 Non-Adaptive vs. Adaptive Algorithms x 1,..., x q f f f (x 1 ),..., f (x q ) f (x 2 ) 6 / 29

11 Non-Adaptive vs. Adaptive Algorithms x 1,..., x q... f f f (x 1 ),..., f (x q ) 6 / 29

12 Non-Adaptive vs. Adaptive Algorithms x 1,..., x q f f f (x 1 ),..., f (x q ) f (... ) 6 / 29

13 Non-Adaptive vs. Adaptive Algorithms x 1,..., x q x q f f f (x 1 ),..., f (x q ) 6 / 29

14 Non-Adaptive vs. Adaptive Algorithms x 1,..., x q f f f (x 1 ),..., f (x q ) f (x q ) 6 / 29

15 Non-Adaptive vs. Adaptive Algorithms x 1,..., x q f f f (x 1 ),..., f (x q ) f (x q ) Adaptive algorithms with q queries can simulate non-adaptive algorithms with q queries. 6 / 29

16 Non-Adaptive vs. Adaptive Algorithms x 1,..., x q f f f (x 1 ),..., f (x q ) f (x q ) Adaptive algorithms with q queries can simulate non-adaptive algorithms with q queries. Non-adaptive algorithms with 2 q queries can simulate adaptive algorithms with q queries. 6 / 29

17 Non-Adaptive vs. Adaptive Algorithms x 1,..., x q f f f (x 1 ),..., f (x q ) f (x q ) Adaptive algorithms with q queries can simulate non-adaptive algorithms with q queries. Non-adaptive algorithms with 2 q queries can simulate adaptive algorithms with q queries. Exponential gaps are known: Signed majorities [Matulef, O Donnell, Rubinfeld, Servedio 09], [Ron, Servedio 13]. Read-once width-2 OBDD [Ron, Tsur 12] [Brody, Matulef, Wu 11]. 6 / 29

18 How Adaptivity Helps: Binary Search f 7 / 29

19 How Adaptivity Helps: Binary Search x, y f 7 / 29

20 How Adaptivity Helps: Binary Search f f (x), f (y) 7 / 29

21 How Adaptivity Helps: Binary Search f f (x), f (y) 7 / 29

22 How Adaptivity Helps: Binary Search z f 7 / 29

23 How Adaptivity Helps: Binary Search z f Recurse on path for O(log n) steps. 7 / 29

24 How Adaptivity Helps: Binary Search z f Recurse on path for O(log n) steps. Will find some edge (x, y) in direction i f (x) f (y). 7 / 29

25 How Adaptivity Helps: Binary Search z f Recurse on path for O(log n) steps. Will find some edge (x, y) in direction i f (x) f (y). With O(log n) many queries, can find one important direction. 7 / 29

26 Upper bounds Can we test k-juntas with query complexity independent of n? 8 / 29

27 Upper bounds Can we test k-juntas with query complexity independent of n? Yes! 8 / 29

28 Upper bounds Can we test k-juntas with query complexity independent of n? Yes! Theorem (Fisher, Kindler, Ron, Safra, Samorodnitsky 04) One can ε-test k-juntas for any k with poly(k, ε 1 ) queries. 8 / 29

29 Upper bounds Can we test k-juntas with query complexity independent of n? Yes! Theorem (Fisher, Kindler, Ron, Safra, Samorodnitsky 04) One can ε-test k-juntas for any k with poly(k, ε 1 ) queries. Additionally, one can achieve Õ(k2 /ε) in the non-adaptive model. 8 / 29

30 Two Algorithms Theorem (Blais 08) There exists a non-adaptive algorithm for testing k-juntas making Õ(k 3/2 )/ε many queries. Theorem (Blais 09) There exists an adaptive algorithm for testing k-juntas making O(k/ε + k log k) queries. Non-adaptive: estimate variation of blocks of coordinates. Adaptive: use binary search on blocks of coordinates. 9 / 29

31 Two Lower Bounds Theorem (Chockler and Gutfreund 04) Testing juntas adaptively requires Ω(k) queries for some ε = Ω(1). Theorem (Buhrman, Garcia-Soriano, Matsliah, de Wolf 13) Testing juntas non-adaptively requires Ω(k log k) queries for some ε = Ω(1). 10 / 29

32 Two Lower Bounds Theorem (Chockler and Gutfreund 04) Testing juntas adaptively requires Ω(k) queries for some ε = Ω(1). Theorem (Buhrman, Garcia-Soriano, Matsliah, de Wolf 13) Testing juntas non-adaptively requires Ω(k log k) queries for some ε = Ω(1). Model Upper bound Lower bound Non-adaptive Õ(k 3/2 )/ε Ω(k log k) Adaptive O(k/ε + k log k) Ω(k) 10 / 29

33 Adaptivity can help Theorem (Servedio, Tan, Wright 15) Testing k-juntas non-adaptively requires ( ) k log k Ω ε c log(log k/ε c ) for any c < / 29

34 Adaptivity can help Theorem (Servedio, Tan, Wright 15) Testing k-juntas non-adaptively requires ( ) k log k Ω ε c log(log k/ε c ) for any c < 1. When ε = Θ(1), lower bound is Ω(k log k/ log(log k)). 11 / 29

35 Adaptivity can help Theorem (Servedio, Tan, Wright 15) Testing k-juntas non-adaptively requires ( ) k log k Ω ε c log(log k/ε c ) for any c < 1. When ε = Θ(1), lower bound is Ω(k log k/ log(log k)). When ε = 1/ log k, lower bound is Ω ( k log 1+c ) (k) O(k log k) log log k 11 / 29

36 Questions Model Upper bound Lower bound Non-adaptive Õ(k 3/2 )/ε Ω(k log k/(ε c log(log(k)/ε c ))) Adaptive O(k/ε + k log k) Ω(k) 12 / 29

37 Questions Model Upper bound Lower bound Non-adaptive Õ(k 3/2 )/ε Ω(k log k/(ε c log(log(k)/ε c ))) Adaptive O(k/ε + k log k) Ω(k) When does adaptivity help? 12 / 29

38 Questions Model Upper bound Lower bound Non-adaptive Õ(k 3/2 )/ε Ω(k log k/(ε c log(log(k)/ε c ))) Adaptive O(k/ε + k log k) Ω(k) When does adaptivity help? Can the adaptive algorithm be made non-adaptive? 12 / 29

39 Main Result Theorem Testing juntas non-adaptively requires Ω(k 3/2 /ε) queries. Model Upper bound Lower bound Non-adaptive Õ(k 3/2 )/ε Ω(k 3/2 /ε) Adaptive O(k/ε + k log k) Ω(k) 13 / 29

40 Main Result Theorem Testing juntas non-adaptively requires Ω(k 3/2 /ε) queries. Model Upper bound Lower bound Non-adaptive Õ(k 3/2 )/ε Ω(k 3/2 /ε) Adaptive O(k/ε + k log k) Ω(k) Goal for this talk: Ω(n 3/2 ) for 3n 4 -junta testing with ε = Ω(1). 13 / 29

41 Main Result Theorem Testing juntas non-adaptively requires Ω(k 3/2 /ε) queries. Model Upper bound Lower bound Non-adaptive Õ(k 3/2 )/ε Ω(k 3/2 /ε) Adaptive O(k/ε + k log k) Ω(k) Goal for this talk: Ω(n 3/2 ) for 3n 4 -junta testing with ε = Ω(1). For general k, we use a padding argument. 13 / 29

42 Overview of the proof Alg for D yes, D no Alg for SSSQ D yes and D no Lower bound for SSSQ 14 / 29

43 Overview of the proof Alg for D yes, D no Alg for SSSQ D yes and D no Lower bound for SSSQ 14 / 29

44 Overview of the proof Alg for D yes, D no Alg for SSSQ D yes and D no Lower bound for SSSQ One class: parameter p p y for D yes and p n for D no 14 / 29

45 Overview of the proof must work a certain way Alg for D yes, D no Alg for SSSQ D yes and D no Lower bound for SSSQ One class: parameter p p y for D yes and p n for D no 14 / 29

46 Overview of the proof must work a certain way Set-Size-Set-Queries(p y, p n ) Alg for D yes, D no Alg for SSSQ D yes and D no Lower bound for SSSQ One class: parameter p p y for D yes and p n for D no 14 / 29

47 Overview of the proof must work a certain way Set-Size-Set-Queries(p y, p n ) Alg for D yes, D no Alg for SSSQ D yes and D no One class: parameter p p y for D yes and p n for D no Lower bound for SSSQ Bin(n/2, p y ) vs. Bin(n/2, p n ) 14 / 29

48 Overview of the proof must work a certain way Set-Size-Set-Queries(p y, p n ) Alg for D yes, D no Alg for SSSQ D yes and D no One class: parameter p p y for D yes and p n for D no Lower bound for SSSQ Bin(n/2, p y ) vs. Bin(n/2, p n ) 14 / 29

49 D yes : 3n/4-juntas M = n 2. A = n 4. i [2 n/2 ], h i is a random function depending on S i A. S i includes k A with probability 1 n. f (x) = h x M (x). 15 / 29

50 D yes : 3n/4-juntas x M = 2 M = n 2. A = n 4. i [2 n/2 ], h i is a random function depending on S i A. S i includes k A with probability 1 n. f (x) = h x M (x). 15 / 29

51 D yes : 3n/4-juntas M = n 2. A = n 4. x M = 2 Ex: influence of M and A? i [2 n/2 ], h i is a random function depending on S i A. S i includes k A with probability 1 n. f (x) = h x M (x). 15 / 29

52 D no : 3n/4 + n-juntas M = n 2. A = n 4 + n. i [2 n/2 ], h i is a random function depending on S i A. S i includes k A with probability 1 n. f (x) = h x M (x). 16 / 29

53 D yes vs. D no D yes : M is random set of size n 2. A = {k M : w.p 1 2 }. S i = {k A : w.p 1 n }. D no : M is random set of size n 2. A = {k M : w.p n }. S i = {k A : w.p 1 n }. 17 / 29

54 D yes vs. D no D yes : M is random set of size n 2. A = {k M : w.p 1 2 }. S i = {k A : w.p 1 n }. p y = 1 2. D no : M is random set of size n 2. A = {k M : w.p n }. S i = {k A : w.p 1 n }. p n = n. 17 / 29

55 D yes vs. D no D yes : M is random set of size n 2. A = {k M : w.p 1 2 }. S i = {k A : w.p 1 n }. D no : M is random set of size n 2. A = {k M : w.p n }. S i = {k A : w.p 1 n }. Lemma (D yes are k-juntas) With probability 1 o(1), f D yes is a k-junta. Lemma (D no are far from k-juntas) With probability 1 o(1), f D no is Ω(1)-far from being a k-junta. 18 / 29

56 D yes vs. D no D yes : M is random set of size n 2. A = {k M : w.p 1 2 }. S i = {k A : w.p 1 n }. D no : M is random set of size n 2. A = {k M : w.p n }. S i = {k A : w.p 1 n }. Lemma (D yes are k-juntas) With probability 1 o(1), f D yes is a k-junta. Lemma (D no are far from k-juntas) With probability 1 o(1), f D no is Ω(1)-far from being a k-junta. Hint: f D no has 3n 4 + n 4 relevant variables! 18 / 29

57 Overview of the proof must work a certain way Set-Size-Set-Queries(p y, p n ) Alg for D yes, D no Alg for SSSQ D yes and D no One class: parameter p p y for D yes and p n for D no Lower bound for SSSQ Bin(n/2, p y ) vs Bin(n/2, p n ) 19 / 29

58 Overview of the proof must work a certain way Set-Size-Set-Queries(p y, p n ) Alg for D yes, D no Alg for SSSQ D yes and D no One class: parameter p p y for D yes and p n for D no Lower bound for SSSQ Bin(n/2, p y ) vs Bin(n/2, p n ) 19 / 29

59 Structure of functions Both D yes and D no follow the structure: 20 / 29

60 Structure of functions Both D yes and D no follow the structure: Indexing (M), and Evaluation (h 1,..., h 2 n/2). 20 / 29

61 Structure of functions Both D yes and D no follow the structure: Indexing (M), and Evaluation (h 1,..., h 2 n/2). Indexing restricts the algorithms we look at, evaluation gives the lower bound. 20 / 29

62 Structure of functions Both D yes and D no follow the structure: What happens if algorithm has queries x 1,..., x q indexed into different sub-functions h i? 21 / 29

63 Structure of functions Both D yes and D no follow the structure: What happens if algorithm has queries x 1,..., x q indexed into different sub-functions h i? f (x 1 ), f (x 2 ),..., f (x q ) look like random bits. 21 / 29

64 Structure of functions Both D yes and D no follow the structure: What happens if algorithm has queries x 1,..., x q indexed into different sub-functions h i? f (x 1 ), f (x 2 ),..., f (x q ) look like random bits. Alice cannot distinguish between D yes and D no. 21 / 29

65 Structure of functions Both D yes and D no follow the structure: What happens if algorithm has queries x 1,..., x q indexed into different sub-functions h i? f (x 1 ), f (x 2 ),..., f (x q ) look like random bits. Alice cannot distinguish between D yes and D no. Alice wants multiple queries indexed to same sub-function. 21 / 29

66 Structure of functions x 1 x 2 x 3 x l Alice wants to query x 1,..., x l sub-function h / 29

67 Structure of functions x 1 x 2 M x 3 x l Alice wants to query x 1,..., x l sub-function h / 29

68 Structure of functions x 1 x 2 M x 3 x l Alice wants to query x 1,..., x l sub-function h 2. How different can M be on x 1,..., x l? 22 / 29

69 Structure of functions x 1 x 2 M x 3 x l Alice wants to query x 1,..., x l sub-function h 2. How different can M be on x 1,..., x l? If dist(x i, x j ) is large, then very likely x i and x j disagree on M! 22 / 29

70 Structure of functions x 1 x 2 M x 3 x l Alice wants to query x 1,..., x l sub-function h 2. How different can M be on x 1,..., x l? If dist(x i, x j ) is large, then very likely x i and x j disagree on M! Condition: when multiple queries indexed to same function, queries are close. 22 / 29

71 Overview of the proof must work a certain way Set-Size-Set-Queries(p y, p n ) Alg for D yes, D no Alg for SSSQ D yes and D no One class: parameter p p y for D yes and p n for D no Lower bound for SSSQ Bin(n/2, p y ) vs Bin(n/2, p n ) 23 / 29

72 Overview of the proof must work a certain way Set-Size-Set-Queries(p y, p n ) Alg for D yes, D no Alg for SSSQ D yes and D no One class: parameter p p y for D yes and p n for D no Lower bound for SSSQ Bin(n/2, p y ) vs Bin(n/2, p n ) 23 / 29

73 Set-Size-Set-Queries Problem (SSSQ) Definition The SSSQ(p y, p n, m) problem seeks to distinguish the following two cases: Yes: A [m] is a random set includes each element i.i.d w.p p y. No: A [m] is a random set includes each element i.i.d w.p p n. Query access: Set queries are sets T [m] Responses are sets V T where V includes i T A w.p 1 2m. Complexity: for queries T 1,..., T q, cost = q j=1 T j. 24 / 29

74 Intuition: Alg SSSQ D yes : M is random set of size n 2. A = {k M : w.p 1 2 }. S i = {k A : w.p 1 n }. D no : M is random set of size n 2. A = {k M : w.p n }. S i = {k A : w.p 1 n }. Junta Testing SSSQ D yes and D no SSSQ( 1 2, n, n 2 ) x 1,..., x l indexed to h i Influential directions seen in S i Query complexity Sets T i Response set V i T i A Cost 25 / 29

75 Intuition: Alg SSSQ D yes : M is random set of size n 2. A = {k M : w.p 1 2 }. S i = {k A : w.p 1 n }. D no : M is random set of size n 2. A = {k M : w.p n }. S i = {k A : w.p 1 n }. Junta Testing SSSQ D yes and D no SSSQ( 1 2, n, n 2 ) x 1,..., x l indexed to h i Influential directions seen in S i Query complexity Sets T i Response set V i T i A Cost Theorem If there is a q-query non-adaptive tester Alg for D yes vs D no, then there is an algorithm for SSSQ( 1 2, n, n 2 ) with cost O(q log(n)). 25 / 29

76 Overview of the proof must work a certain way Set-Size-Set-Queries(p y, p n ) Alg for D yes, D no Alg for SSSQ D yes and D no One class: parameter p p y for D yes and p n for D no Lower bound for SSSQ Bin(n/2, p y ) vs Bin(n/2, p n ) 26 / 29

77 Overview of the proof must work a certain way Set-Size-Set-Queries(p y, p n ) Alg for D yes, D no Alg for SSSQ D yes and D no One class: parameter p p y for D yes and p n for D no Lower bound for SSSQ Bin(n/2, p y ) vs Bin(n/2, p n ) 26 / 29

78 Lower bound for SSSQ Lemma (hope to prove) SSSQ( 1 2, n, n 2 ) requires cost Ω(n 3/2 ). Main Ideas: A Bin( n 2, 1 2 ) (in D yes) vs A Bin( n 2, n ) (in D no ). Each sample costs Ω( n), since we observe k V when k A w.p 1 n. Ω(n) Ω( n) = Ω(n 3/2 ) }{{}}{{} distinguish 1 2 vs seeing one example n 27 / 29

79 Putting things together Theorem If non-adaptive tester Alg for D yes vs D no with q queries, then SSSQ( 1 2, n, n 2 ) has cost O(q log(n)). 28 / 29

80 Putting things together Theorem If non-adaptive tester Alg for D yes vs D no with q queries, then SSSQ( 1 2, n, n 2 ) has cost O(q log(n)). Lemma SSSQ( 1 2, n, n 2 ) requires cost Ω(n 3/2 ). 28 / 29

81 Putting things together Theorem If non-adaptive tester Alg for D yes vs D no with q queries, then SSSQ( 1 2, n, n 2 ) has cost O(q log(n)). Lemma SSSQ( 1 2, n, n 2 ) requires cost Ω(n 3/2 ). Theorem Testing 3n 4 -juntas non-adaptively requires Ω(n 3/2 ) queries. 28 / 29

82 Open Problems Tradeoffs in query complexity and adaptivity? See next! [Canonne and Gur 17]. Other natural properties of Boolean functions with gaps in adaptive and non-adaptive query complexity? More applications of this class of random functions? 29 / 29

83 Open Problems Tradeoffs in query complexity and adaptivity? See next! [Canonne and Gur 17]. Other natural properties of Boolean functions with gaps in adaptive and non-adaptive query complexity? More applications of this class of random functions? Thanks! 29 / 29