Quadratically Tight Relations for Randomized Query Complexity Rahul - PowerPoint PPT Presentation

Quadratically Tight Relations for Randomized Query Complexity Quadratically Tight Relations for Randomized Query Complexity Rahul Jain Hartmut Klauck Srijita Kundu Troy Lee Miklos Santha Swagato Sanyal Jevg¯ enijs Vihrovs Centre for Quantum Technologies, National University of Singapore, Centre for Quantum Computer Science, University of Latvia. 8th June, 2018 The 13th International Computer Science Symposium in Russia, CSR 2018

Quadratically Tight Relations for Randomized Query Complexity Query Complexity Outline 1 Query Complexity 2 Expectational Certificate Complexity 3 Partition Bound

Quadratically Tight Relations for Randomized Query Complexity Query Complexity Query Complexity We want to compute some Boolean function f : { 0 , 1 } n → { 0 , 1 } . The input is x = ( x 1 , . . . , x n ). With a single query we can ask the value of any x i . The cost of the computation is the number of queries made.

Quadratically Tight Relations for Randomized Query Complexity Query Complexity Query Complexity Determistic query complexity D( f ) (minimum worst-case number of queries). Randomized query complexity R( f ) (correct with probability ≥ 2 / 3). Exact randomized query complexity R 0 ( f ) (minimum worst-case expected number of queries). R( f ) ≤ R 0 ( f ) ≤ D( f ) .

Quadratically Tight Relations for Randomized Query Complexity Query Complexity Example: Recursive Majority 3-Maj ( x 1 , x 2 , x 3 ) = 1 ⇐ ⇒ x 1 + x 2 + x 3 ≥ 2. D( 3-Maj h ) = 3 h . R 0 ( 3-Maj h ) ≤ (8 / 3) h . Maj Maj Maj Maj x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9

Quadratically Tight Relations for Randomized Query Complexity Query Complexity Query Complexity In this work, we study which measures M( f ) can characterize R 0 ( f ) or R( f ) quadratically: M( f ) ≤ R( f ) ≤ M( f ) 2 ? We show two results: 1 The expectational certificate complexity bounds R 0 ( f ) quadratically: EC( f ) ≤ R 0 ( f ) ≤ O (EC( f ) 2 ) . 2 The partition bound bounds R( f ) quadratically for product distributions µ : D µ 1 / 3 ( f ) ≤ O (prt 1 / 3 ( f ) 2 ) .

Quadratically Tight Relations for Randomized Query Complexity Expectational Certificate Complexity Outline 1 Query Complexity 2 Expectational Certificate Complexity 3 Partition Bound

Quadratically Tight Relations for Randomized Query Complexity Expectational Certificate Complexity Certificate Complexity A certificate for an input x is a set of positions of x that have to be revealed to know the value of f ( x ) with certainty. The length of a certificate is the number of positions revealed. A minimal certificate of x is a certificate of smallest length C( f , x ). The certificate complexity of f is C( f ) = max x C( f , x ). It is known that C( f ) ≤ R 0 ( f ) ≤ C( f ) 2 .

Quadratically Tight Relations for Randomized Query Complexity Expectational Certificate Complexity Example: AND-OR C( And-Or n ) = √ n . And Or Or Or x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 0 0 0 1 1 1

Quadratically Tight Relations for Randomized Query Complexity Expectational Certificate Complexity Fractional Certificate Complexity Fractional certificate complexity FC( f ) is given by the optimal value of the following LP: [Tal / Gilmer, Saks, Srinivasan] � minimize max w x ( i ) x i ∈ [ n ] � subject to ∀ x , y s.t. f ( x ) � = f ( y ) : w x ( i ) ≥ 1 i : x i � = y i ∀ x , i : 0 ≤ w x ( i ) ≤ 1 . FC( f ) ≤ C( f ). It is known that FC( f ) ≤ R( f ) ≤ R 0 ( f ) ≤ FC( f ) 3 .

Quadratically Tight Relations for Randomized Query Complexity Expectational Certificate Complexity Example: Majority 3-Maj ( x 1 , x 2 , x 3 ) = 1 ⇐ ⇒ x 1 + x 2 + x 3 ≥ 2. C( f , 000) = 2. FC( f , 000) = 3 / 2. Weights w 1 = w 2 = w 3 = 1 / 2. 000 110 101 011

Quadratically Tight Relations for Randomized Query Complexity Expectational Certificate Complexity Fractional Certificate Complexity Hypothesis: R 0 ( f ) ≤ FC( f ) 2 . If that is true, then R 0 ( f ) ≤ Q( f ) 4 . (Quantum query complexity.) Currently the best upper bound is R 0 ( f ) ≤ Q( f ) 6 . A quadratic separation is known, R( And-Or n ) = Ω( n ), FC( And-Or n ) = √ n .

Quadratically Tight Relations for Randomized Query Complexity Expectational Certificate Complexity Expectational Certificate Complexity Expectational certificate complexity EC( f ) is given by the optimal value of the following program: � minimize max w x ( i ) x i ∈ [ n ] � subject to ∀ y s.t. f ( x ) � = f ( y ) : w x ( i ) w y ( i ) ≥ 1 , i : x i � = y i ∀ x , i : 0 ≤ w x ( i ) ≤ 1 . Not a linear program anymore!

Quadratically Tight Relations for Randomized Query Complexity Expectational Certificate Complexity Expectational Certificate Complexity R( f ) = O (EC( f ) 2 ) algorithm: Repeat O (EC( f )) times: Pick any consistent (with previous queries) input z s.t. f ( z ) = 1; If there is no such z , return 0 . Independently query each x i with probability w z ( i ); Return 1 . Each round takes � n i =1 w z ( i ) ≤ EC( f ) queries on expectation; hence query complexity is O (EC( f ) 2 ). Expected amount of weight removed from w x each round is � i : x i � = z i w x ( i ) w z ( i ) ≥ 1; hence, O (EC( f )) many rounds is enough.

Quadratically Tight Relations for Randomized Query Complexity Expectational Certificate Complexity Expectational Certificate Complexity Properties: FC( f ) ≤ EC( f ) ≤ C( f ). EC( f ) ≤ C( f ) 2 , tight! EC( f ) ≤ R 0 ( f ) ≤ O (EC( f ) 2 ). EC( f ) ≤ O (FC( f ) 3 / 2 ) . EC( f ) 2 / 3 ≤ R( f ) ≤ O (EC( f ) 2 ).

Quadratically Tight Relations for Randomized Query Complexity Partition Bound Outline 1 Query Complexity 2 Expectational Certificate Complexity 3 Partition Bound

Quadratically Tight Relations for Randomized Query Complexity Partition Bound Partition Bound The ǫ -partition bound of f (denoted by prt ǫ ( f )), is given by the log 2 of the optimal value of the following LP: [Jain, Klauck] � � w z , A · 2 | A | minimize subject to ∀ x : w f ( x ) , A ≥ 1 − ǫ, z , A A ∋ x � ∀ x : w z , A = 1 , z , A ∋ x ∀ z , A : w z , A ≥ 0 . Lower bound, 1 2 prt ǫ ( f ) ≤ R ǫ ( f ).

Quadratically Tight Relations for Randomized Query Complexity Partition Bound Partition Bound Example: prt( And-Or n ) = Ω( n ). Known that R( f ) = O (prt( f )) 3 . Best separation is quadratic, R( f ) = Ω(prt( f ) 2 ). [Ambainis, Kokainis, Kothari] Is prt( f ) quadratically tight for R( f )?

Quadratically Tight Relations for Randomized Query Complexity Partition Bound Distributional Query Complexity Let µ be a probability distribution over inputs { 0 , 1 } n . Distributional query complexity D µ ǫ ( f ) is the minimum worst-case cost of a deterministic algorithm A such that x ∼ µ [ A ( x ) = f ( x )] ≥ 1 − ǫ. Pr Yao’s theorem: D µ R ǫ ( f ) = max ǫ ( f ) . µ

Quadratically Tight Relations for Randomized Query Complexity Partition Bound Block Sensitivity An input x is sensitive on a subset of positions B ⊆ [ n ], if f ( x ) � = f ( x B ). The block sensitivity of x , denoted by bs( f , x ), is the maximum number of disjoint sensitive blocks. The block sensitivity of f is max x bs( f , x ).

Quadratically Tight Relations for Randomized Query Complexity Partition Bound Corruption Bound Let µ be a probability distribution over the inputs. Let A be an ǫ -error b -certificate under µ , if x ∼ µ [ f ( x ) � = b | x ∈ A ] ≤ ǫ. Pr Query corruption bound: corr b ,µ ( f ) = min {| A | | A is an ǫ -error b -certificate under µ } . ǫ Query corruption bound: corr b ,µ corr ǫ ( f ) = max max ( f ) . ǫ µ b

Quadratically Tight Relations for Randomized Query Complexity Partition Bound Corruption Bound Minimum query corruption bound over product distributions: corr × b corr b ,µ min ,ǫ ( f ) = max min ( f ) , ǫ µ where µ is a product distribution. µ is a bit-wise product distribution if for all x , n � µ ( x ) = µ i ( x i ) . i =1

Quadratically Tight Relations for Randomized Query Complexity Partition Bound Corruption Bound We adapt the proof of D( f ) ≤ C( f ) bs( f ) to prove that D µ 4 ǫ ( f ) = O (corr × min ,ǫ ( f ) · bs( f )) for product distributions. Since corr × min ,ǫ ( f ) ≤ corr ǫ ( f ) and bs( f ) ≤ corr ǫ ( f ), we get D µ 4 ǫ ( f ) = O (corr ǫ ( f ) 2 ) .

Quadratically Tight Relations for Randomized Query Complexity Partition Bound Partition Bound Since bs( f ) = O ( 1 ǫ prt ǫ ( f )) and corr × min , 2 ǫ ( f ) ≤ prt ǫ ( f ), we get � 1 � D µ ǫ prt ǫ ( f ) 2 8 ǫ ( f ) = O . A polylogarithmic improvement over previous best upper bound; constant error instead of inverse polynomial error. [Harsha, Jain, Radhakrishnan]

Quadratically Tight Relations for Randomized Query Complexity Partition Bound Lower Bounds R 0 C R prt corr EC corr × FC min bs Figure: Lower bounds on R 0 ( f ) and R( f ).

Thank you! Questions?