Settling the Query Complexity of Non-Adaptive Junta Testing Erik - PowerPoint PPT Presentation

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

How Adaptivity Helps: Binary Search f 7 / 29

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

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

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

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

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

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

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

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

Upper bounds Can we test k -juntas with query complexity independent of n ? Yes! 8 / 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. 8 / 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 � O ( k 2 /ε ) in the non-adaptive model. 8 / 29

Two Algorithms Theorem (Blais 08) There exists a non-adaptive algorithm for testing k-juntas making � O ( 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

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

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 � O ( k 3 / 2 ) /ε Non-adaptive Ω( k log k ) Adaptive O ( k /ε + k log k ) Ω( k ) 10 / 29

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 . 11 / 29

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

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 � k log 1+ c ( k ) � Ω ≫ O ( k log k ) log log k 11 / 29

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

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

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

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

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

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

Overview of the proof ✬ ✩ ✬ ✩ ✲ Alg ′ for SSSQ Alg for D yes , D no ✫ ✪ ✫ ✪ ✻ ✬ ✩ ✬ ✩ ❄ D yes and D no Lower bound for SSSQ ✫ ✪ ✫ ✪ 14 / 29

Overview of the proof ✬ ✩ ✬ ✩ ✲ Alg ′ for SSSQ Alg for D yes , D no ✫ ✪ ✫ ✪ ✻ ✬ ✩ ✬ ✩ ❄ D yes and D no Lower bound for SSSQ ✫ ✪ ✫ ✪ 14 / 29

Overview of the proof ✬ ✩ ✬ ✩ ✲ Alg ′ for SSSQ Alg for D yes , D no ✫ ✪ ✫ ✪ ✻ ✬ ✩ ✬ ✩ ❄ 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

Overview of the proof ✬ ✩ ✬ ✩ “must work a certain way” ✲ Alg ′ for SSSQ Alg for D yes , D no ✫ ✪ ✫ ✪ ✻ ✬ ✩ ✬ ✩ ❄ 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

Overview of the proof ✬ ✩ ✬ ✩ “must work a certain way” Set-Size-Set-Queries( p y , p n ) ✲ Alg ′ for SSSQ Alg for D yes , D no ✫ ✪ ✫ ✪ ✻ ✬ ✩ ✬ ✩ ❄ 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