Polynomial bounds for decoupling, with applica8ons Ryan ODonnell, - PowerPoint PPT Presentation

Polynomial bounds for decoupling, with applica8ons Ryan O’Donnell, Yu Zhao Carnegie Mellon University

Boolean func8ons f :{ − 1,1} n → ! ⎧ if at least two inputs are 1 = 1 Maj 3 ( x 1 , x 2 , x 3 ) ⎨ if at least two inputs are -1 − 1 ⎩ x 2 x 3 x 1 Output 1 1 1 1 Maj 3 ( x 1 , x 2 , x 3 ) = 1 2 x 1 + 1 2 x 2 + 1 2 x 3 − 1 1 1 -1 1 2 x 1 x 2 x 3 1 -1 1 1 1 -1 -1 -1 -1 1 1 1 -1 1 -1 -1 -1 -1 1 -1 -1 -1 -1 -1

Boolean func8ons 2 = 1 x i Fourier expansion: the unique mul8linear polynomial representa8on of a Boolean func8on ! ( S ) ∑ ∏ f ( x ) = f x i i ∈ S ⊆ [ n ] S Maj 3 ( x 1 , x 2 , x 3 ) = 1 2 x 1 + 1 2 x 2 + 1 2 x 3 − 1 2 x 1 x 2 x 3

Proper8es of Boolean func8ons Low circuit complexity Monotonicity Linear threshold Bounded Block-mul8linearity Low degree Small influence Small variance Homogeneity

Block-mul8linearity A homogeneous Boolean func8on f with degree k is Block-mul*linear

Block-mul8linearity A homogeneous Boolean func8on f with degree k is Block-mul*linear if we can par88on the input variables into k blocks S 1 , …, S k

Block-mul8linearity A homogeneous Boolean func8on f with degree k is Block-mul*linear if we can par88on the input variables into k blocks S 1 , …, S k such that each monomial in the Fourier expansion of f contains exactly 1 variable in each block. [Khot Naor 08, LoveW 10, Kane Meka13, Aaronson Ambainis15] Sort( x 1 , x 2 , x 3 , x 4 ) = 1 2 x 1 x 2 + 1 2 x 2 x 3 + 1 2 x 3 x 4 − 1 2 x 1 x 4 S 1 = { x 1 , x 3 }, S 2 = { x 2 , x 4 }

Block-mul8linearity f :{ − 1,1} n → [ − 1,1] Theorem in [AA15] Let be any bounded block-mul8linear Boolean func8on with degree k . Then there exists a randomized algorithm that, on input , non-adap8vely queries 2 O ( k ) ( n / ε 2 ) 1-1/ k bits x ∈ { − 1,1} n of x , and then es8mate the output of f within error ε with high probability. Conjecture: This theorem works for n 1 − 1/ k arbitrary polynomials Yes, via decoupling!

Block-mul8linearity f :{ − 1,1} n → [ − 1,1] Theorem in [AA15] Let be any bounded block-mul8linear Boolean func8on with degree k . Then there exists a randomized algorithm that, on input , non-adap8vely queries 2 O ( k ) ( n / ε 2 ) 1-1/ k bits x ∈ { − 1,1} n of x , and then es8mate the output of f within error ε with high probability. x ∈ { − 1,1} n Quantum algorithm makes t queries to The probability that the algorithm accepts can be expressed as a Boolean func8on with degree at most 2 t . The algorithm can be simulated by a classical algorithm with O ( n 1-1/(2 t ) ) queries.

Block-mul8linearity f :{ − 1,1} n → [ − 1,1] Theorem in [AA15] Let be any bounded block-mul8linear Boolean func8on with degree k . Then there exists a randomized algorithm that, on input , non-adap8vely queries 2 O ( k ) ( n / ε 2 ) 1-1/ k bits x ∈ { − 1,1} n of x , and then es8mate the output of f within error ε with high probability. Can we extend this algorithm to arbitrary Boolean func8ons? Yes, via decoupling!

Decoupling ! decoupling f f general func8on block-mul8linear func8on degree k degree k n variables kn variables ( k blocks of n variables) k copies of x ! ( x ,..., x ) f ( x ) = f 1. ! f 2. and f has similar proper8es

Examples of decoupling f ( x 1 , x 2 , x 3 ) = x 1 x 2 x 3 ! ( y 1 , y 2 , y 3 , z 1 , z 2 , z 3 , w 1 , w 2 , w 3 ) f = 1 6 y 1 z 2 w 3 + 1 6 y 1 w 2 z 3 + 1 6 z 1 y 2 w 3 + 1 6 z 1 w 2 y 3 + 1 6 w 1 y 2 z 3 + 1 6 w 1 z 2 y 3

Examples of decoupling Maj 3 ( x 1 , x 2 , x 3 ) = 1 2 x 1 + 1 2 x 2 + 1 2 x 3 − 1 2 x 1 x 2 x 3 ! ( y 1 , y 2 , y 3 , z 1 , z 2 , z 3 , w 1 , w 2 , w 3 ) Maj 3 = 1 6 y 1 + 1 6 z 1 + 1 6 w 1 + 1 6 y 2 + 1 6 z 2 + 1 6 w 2 + 1 6 y 3 + 1 6 z 3 + 1 6 w 3 − 1 12 y 1 z 2 w 3 − 1 12 y 1 w 2 z 3 − 1 12 z 1 y 2 w 3 − 1 12 z 1 w 2 y 3 − 1 12 w 1 y 2 z 3 − 1 12 w 1 z 2 y 3

Block-mul8linearity A homogeneous Boolean func8on f with degree k is Block-mul*linear if we can par88on the input variables into k blocks S 1 , …, S k such that each monomial in the Fourier expansion of f contains exactly 1 variable in each block. [KN08, Lov10, KM13, AA15]

Block-mul8linearity A homogeneous Boolean func8on f with degree k is Block-mul*linear if we can par88on the input variables into k blocks S 1 , …, S k such that each monomial in the Fourier expansion of f contains exactly 1 variable in each block. [KN08, Lov10, KM13, AA15]

Block-mul8linearity A homogeneous Boolean func8on f with degree k is Block-mul*linear if we can par88on the input variables into k blocks S 1 , …, S k such that each monomial in the Fourier expansion of f contains at most 1 variable in each block. [KN08, Lov10, KM13, AA15]

Block-mul8linearity f :{ − 1,1} n → [ − 1,1] Theorem in [AA15] Let be any bounded block-mul8linear Boolean func8on with degree k . Then there exists a randomized algorithm that, on x ∈ { − 1,1} n input , non-adap8vely queries 2 O ( k ) ( n / ε 2 ) 1-1/ k bits of x , and then es8mate the output of f within error ε with high probability. ! ( x ,..., x ) f ( x ) = f

Block-mul8linearity f :{ − 1,1} n → [ − 1,1] Theorem in [AA15] Let be any bounded block-mul8linear Boolean func8on with degree k . Then there exists a randomized algorithm that, on x ∈ { − 1,1} n input , non-adap8vely queries 2 O ( k ) ( n / ε 2 ) 1-1/ k bits of x , and then es8mate the output of f within error ε with high probability. ! ( x ,..., x ) f ( x ) = f f :{ − 1,1} n → [ − 1,1] ! :{ − 1,1} kn → [ − C , C ]? f

Decoupling inequality ( k is the degree of f ) Φ : ! ≥ 0 → ! ≥ 0 Theorem 1. Let be convex and non-decreasing. ! ( x (1) ,..., x ( k ) )|)] ≤ E[ Φ ( C k | f ( x )|)] E[ Φ (| f [de la Peña 92] Theorem 2. For all t > 0, ! ( x (1) ,..., x ( k ) )| > C k t ] ≤ D k Pr[| f ( x )| > t ] Pr[| f [Peña Montgomery-Smith 95, Giné 98] Comments: C k , D k = k O ( k ) 1. 2. The inputs can be any independent random variables with all moments finite. 3. The reverse inequality also holds with some worse constants. 4. f does not need to be mul8linear neccesarily

Decoupling inequality ( k is the degree of f ) Φ : ! ≥ 0 → ! ≥ 0 Theorem 1. Let be convex and non-decreasing. ! ( x (1) ,..., x ( k ) )|)] ≤ E[ Φ ( C k | f ( x )|)] E[ Φ (| f [de la Peña 92] Theorem 2. For all t > 0, ! ( x (1) ,..., x ( k ) )| > C k t ] ≤ D k Pr[| f ( x )| > t ] Pr[| f [Peña Montgomery-Smith 95, Giné 98] Comments: 5. If f is a homogeneous func8on with Boolean input, C k can be improved to 2 O ( k ) . [Kwapień 87] " ! p ≤ C k ! f ! p Φ = | ⋅ | p ! f 6. " ! ∞ ≤ C k ! f ! ∞ . p → ∞ ! f

Block-mul8linearity f :{ − 1,1} n → [ − 1,1] Theorem in [AA15] Let be any bounded block-mul8linear Boolean func8on with degree k . Then there exists a randomized algorithm that, on input , non-adap8vely queries 2 O ( k ) ( n / ε 2 ) 1-1/ k bits x ∈ { − 1,1} n of x , and then es8mate the output of f within error ε with high probability. ! / C k ! ! ( x ,..., x ) ε ' = ε / C k f ( x ) = f f f f C k = 2 O ( k ) [ − 1,1] [ − C k , C k ] [ − 1,1]

Block-mul8linearity f :{ − 1,1} n → [ − 1,1] Theorem in [AA15] Let be any bounded block-mul8linear Boolean func8on with degree k . Then there exists a randomized algorithm that, on input , non-adap8vely queries 2 O ( k ) ( n / ε 2 ) 1-1/ k bits x ∈ { − 1,1} n of x , and then es8mate the output of f within error ε with high probability. ! / C k ! ! ( x ,..., x ) ε ' = ε / C k f ( x ) = f f f f C k = 2 O ( k ) [ − 1,1] [ − C k , C k ] [ − 1,1]

Applica8on 2: AA Conjecture f :{ − 1,1} n → [ − 1,1] Let be a Boolean func8on with degree at most k . Then MaxInf[ f ] ≥ poly(Var[ f ]/ k ). ! ( S ) ∑ ∏ Def: f ( x ) = Maj 3 ( x 1 , x 2 , x 3 ) = 1 2 x 1 + 1 2 x 2 + 1 2 x 3 − 1 f x i 2 x 1 x 2 x 3 i ∈ S ⊆ [ n ] S ! ( S ) 2 ∑ Var[Maj 3 ] = 1 Var[ f ] = f S ≠∅ ! ( S ) 2 ∑ Inf i [Maj 3 ] = 1 Inf i [ f ] = f 2 S ∍ i MaxInf[Maj 3 ] = 1 MaxInf[ f ] = max [ n ] {Inf i [ f ]} 2 i ∈

Applica8on 2: AA Conjecture f :{ − 1,1} n → [ − 1,1] Let be a Boolean func8on with degree at most k . Then MaxInf[ f ] ≥ poly(Var[ f ]/ k ). Suppose AA Conjecture holds: 1. There exists some determinis8c simula8on of a quantum algorithm; ⊂ 2. P = P #P implies BQP A AvgP A with probability 1 for a random oracle A .

Applica8on 2: AA Conjecture, weak version f :{ − 1,1} n → [ − 1,1] Let be a Boolean func8on with degree at most k . Then MaxInf[ f ] ≥ Var[ f ] 2 /exp( k ).

Applica8on 2: AA Conjecture, weak version f :{ − 1,1} n → [ − 1,1] Let be a Boolean func8on with degree at most k . Then MaxInf[ f ] ≥ Var[ f ] 2 /exp( k ). There exists an easy proof for block-mul8linear func8on!! ∑ g i ( z ) f ( y , z ) = y i i First block Then use hypercontrac8vity and Cauchy-Schwartz Rest variables

Examples of decoupling f ( x 1 , x 2 , x 3 ) = x 1 x 2 x 3 ! ( y 1 , y 2 , y 3 , z 1 , z 2 , z 3 , w 1 , w 2 , w 3 ) f = 1 6 y 1 z 2 w 3 + 1 6 y 1 w 2 z 3 + 1 6 z 1 y 2 w 3 + 1 6 z 1 w 2 y 3 + 1 6 w 1 y 2 z 3 + 1 6 w 1 z 2 y 3 ! ] = 1 ! ] = 1 Var[ f k !Var[ f ] Inf y i [ f k ! ⋅ k Inf x i [ f ]