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

Transcription

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

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

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

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

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

6 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

7 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 x 2 x x 3 x x 1 x 4 S 1 = {x 1,x 3 },S 2 = {x 2,x 4 }

8 Block-mul8linearity Theorem in [AA15] Let f :{ 1,1} n [ 1,1] be any bounded block-mul8linear Boolean func8on with degree k. Then there exists a randomized algorithm that, on input x { 1,1} n, 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. Conjecture: This theorem works for arbitrary polynomials n 1 1/k Yes, via decoupling!

9 Block-mul8linearity Theorem in [AA15] Let f :{ 1,1} n [ 1,1] be any bounded block-mul8linear Boolean func8on with degree k. Then there exists a randomized algorithm that, on input x { 1,1} n, 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. Quantum algorithm makes t queries to x { 1,1} n The probability that the algorithm accepts can be expressed as a Boolean func8on with degree at most 2t. The algorithm can be simulated by a classical algorithm with O(n 1-1/(2t) ) queries.

10 Block-mul8linearity Theorem in [AA15] Let f :{ 1,1} n [ 1,1] be any bounded block-mul8linear Boolean func8on with degree k. Then there exists a randomized algorithm that, on input x { 1,1} n, 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. Can we extend this algorithm to arbitrary Boolean func8ons? Yes, via decoupling!

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

12 Examples of decoupling f(x 1,x 2,x 3 ) = x 1 x 2 x 3! f (y1,y 2,y 3,z 1,z 2,z 3,w 1,w 2,w 3 ) = 1 6 y 1 z 2 w y 1 w 2 z z 1 y 2 w z 1 w 2 y w 1 y 2 z w 1 z 2 y 3

13 Examples of decoupling Maj 3 (x 1,x 2,x 3 ) = 1 2 x x x x 1 x 2 x 3 Maj! (y1 3,y 2,y 3,z 1,z 2,z 3,w 1,w 2,w 3 ) = 1 6 y z w y z w y z w y 1 z 2 w y 1 w 2 z z 1 y 2 w z 1 w 2 y w 1 y 2 z w 1 z 2 y 3

14 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]

15 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]

16 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]

17 Block-mul8linearity Theorem in [AA15] Let f :{ 1,1} n [ 1,1] be any bounded block-mul8linear Boolean func8on with degree k. Then there exists a randomized algorithm that, on input x { 1,1} n, 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. f(x) = f! (x,...,x)

18 Block-mul8linearity Theorem in [AA15] Let f :{ 1,1} n [ 1,1] be any bounded block-mul8linear Boolean func8on with degree k. Then there exists a randomized algorithm that, on input x { 1,1} n, 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. f(x) = f! (x,...,x) f :{ 1,1} n [ 1,1] f! :{ 1,1} kn [ C,C]?

19 Theorem 1. Let Theorem 2. For all t > 0, Decoupling inequality Φ :! 0! 0 (k is the degree of f) be convex and non-decreasing. E[Φ( f! (x (1),...,x (k) ) )] E[Φ(C k f(x) )] Pr[ f! (x (1),...,x (k) ) > C k t] D k Pr[ f(x) > t] Comments: 1. C k,d k = k O(k) 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 [de la Peña 92] [Peña Montgomery-Smith 95, Giné 98]

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

21 Block-mul8linearity Theorem in [AA15] Let f :{ 1,1} n [ 1,1] be any bounded block-mul8linear Boolean func8on with degree k. Then there exists a randomized algorithm that, on input x { 1,1} n, 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. f(x) = f! (x,...,x) ε ' = ε /C k f f! f! /C k [ 1,1] [ C k,c k ] C k = 2 O(k) [ 1,1]

22 Block-mul8linearity Theorem in [AA15] Let f :{ 1,1} n [ 1,1] be any bounded block-mul8linear Boolean func8on with degree k. Then there exists a randomized algorithm that, on input x { 1,1} n, 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. f(x) = f! (x,...,x) ε ' = ε /C k f f! f! /C k [ 1,1] [ C k,c k ] C k = 2 O(k) [ 1,1]

23 Applica8on 2: AA Conjecture Let f :{ 1,1} n [ 1,1] be a Boolean func8on with degree at most k. Then MaxInf[f] poly(var[f]/k). Def: S [n] S i S f(x) = f! (S) x i Var[ f ]= f! (S) 2 S i Inf i [ f ]= f! (S) 2 MaxInf[ f ]= max{inf i [n] i [ f ]} Maj 3 (x 1,x 2,x 3 ) = 1 x + 1 x + 1 x 1 x x x Var[Maj 3 ]= 1 Inf i [Maj 3 ]= 1 2 MaxInf[Maj 3 ]= 1 2

24 Applica8on 2: AA Conjecture Let f :{ 1,1} n [ 1,1] 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.

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

26 Applica8on 2: AA Conjecture, weak version Let f :{ 1,1} n [ 1,1] 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!! f(y,z) = g i (z) i y i First block Rest variables Then use hypercontrac8vity and Cauchy-Schwartz

27 Examples of decoupling f(x 1,x 2,x 3 ) = x 1 x 2 x 3! f (y1,y 2,y 3,z 1,z 2,z 3,w 1,w 2,w 3 ) = 1 6 y 1 z 2 w y 1 w 2 z z 1 y 2 w z 1 w 2 y w 1 y 2 z w 1 z 2 y 3 Var[ f! ]= 1 k! Var[ f ] Inf yi [ f! ]= 1 k! k Inf x i [ f ]

28 Applica8on 2: AA Conjecture, weak version Let f :{ 1,1} n [ 1,1] be a Boolean func8on with degree at most k. Then MaxInf[f] Var[f] 2 /exp(k). f f! f! /C k [ 1,1] [ C k,c k ] Var[ f! ]= 1 k! Var[ f ] Inf i [ f! ]= 1 k! k Inf i [ f ] [ 1,1] Var[ f! /C k ]= 1 C k 2 Var[ f! ] Inf i [ f! /C k ]= 1 C k 2 Inf i [ f! ]

29 Applica8on 2: AA Conjecture, weak version Let f :{ 1,1} n [ 1,1] be a Boolean func8on with degree at most k. Then MaxInf[f] Var[f] 2 /exp(k log k). f f! f! /C k [ 1,1] [ C k,c k ] Var[ f! ]= 1 k! Var[ f ] Inf i [ f! ]= 1 k! k Inf i [ f ] [ 1,1] Var[ f! /C k ]= 1 C k 2 Var[ f! ] Inf i [ f! /C k ]= 1 C k 2 Inf i [ f! ]

30 Summary of classical decoupling Advantage: Transfer a general func8on f to a blockmul8linear func8on. Disadvantage: Introduce an exponen8al factor on k in decoupling inequality. L

31 Summary of classical decoupling Some8mes we don t need the func8on to be allblocks-mul8linear. We only need f to be a linear map on y. f(y,z) = g i (z) i y i First block Rest variables Then use hypercontrac8vity and Cauchy-Schwartz

32 One-block-mul8linear A Boolean func8on f with degree k is one-blockmul*linear if there exists a subset of the input variables S such that each monomial (except the constant term) in the Fourier expansion of f contains exactly 1 variable in S. f(y,z) = g i (z) Sort(x 1,x 2,x 3,x 4 ) = 1 2 x 1 x x 2 x x 3 x x 1 x 4 i y i Maj 3 (x 1,x 2,x 3 ) = 1 2 x x x x 1 x 2 x 3

33 Par8al decoupling, with polynomial bounds Our result: f Par8al decoupling f general func8on degree k n variables One-block-mul8linear func8on degree k 2n variables (2 blocks of n variables)

34 Examples of par8al decoupling f(x 1,x 2,x 3 ) = x 1 x 2 x 3 f(y 1,y 2,y 3,z 1,z 2,z 3 ) = y 1 z 2 z 3 + z 1 y 2 z 3 + z 1 z 2 y 3

35 Examples of par8al decoupling Maj 3 (x 1,x 2,x 3 ) = 1 2 x x x x 1 x 2 x 3! Maj (y1 3,y 2,y 3,z 1,z 2,z 3 ) = 1 2 y y y 3

36 Examples of par8al decoupling Maj 3 (x 1,x 2,x 3 ) = 1 2 x x x x 1 x 2 x 3! Maj (y1 3,y 2,y 3,z 1,z 2,z 3 ) = 1 y + 1 y + 1 y 1 y z z 1 z y z 1 z z y kf(x) = f(x,x) Var[ f ] Var[ f ] kvar[ f ] Inf [ f ]= Inf [ f ] yi xi for homogeneous case only Inf [ f ] (k 1)Inf [ f ] zi xi

37 Our result: Theorem 1. Let Theorem 2. For all t > 0, With constants:. Par8al decoupling, with polynomial bounds Φ :! 0! 0 be convex and non-decreasing. E[Φ( f(y,z) )] E[Φ(C k f(x) )] Pr[ f(y,z) > C k t] D k Pr[ f(x) > t] poly(k) O(k 2 ) Boolean D k = k O(k) C k = O(k 3/2 ) Boolean, homogeneous O(k) standard Gaussian

38 Applica8on 2: AA Conjecture, weak version Let f :{ 1,1} n [ 1,1] be a Boolean func8on with degree at most k. Then MaxInf[f] Var[f] 2 /exp(k). f f [ 1,1] [ C k,c k ] Var[ f ] Var[ f ] MaxInf[ f ] kmaxinf[ f ] Var[ f Inf i [ f f /C k [ 1,1] /C k ]= 1 C k 2 Var[ f ] /C k ]= 1 C k 2 Inf i [ f ]

39 Applica8on 2: AA Conjecture Let f :{ 1,1} n [ 1,1] be a Boolean func8on with degree at most k. Then MaxInf[f] Var[f] 2 /poly(k). f f [ 1,1] [ C k,c k ] f /C k [ 1,1] Var[ f ] Var[ f ] Var[ f /C k ]= 1 ] 2 C Var[ f k Inf i [ f /C k ]= 1 ] 2 C Inf [ f i k MaxInf[ f ] kmaxinf[ f ]

40 Applica8on 2: AA Conjecture Let f :{ 1,1} n [ 1,1] be a Boolean func8on with degree at most k. Then MaxInf[f] Var[f] 2 /poly(k). The conjecture holds for one-block-mul8linear func8ons. f(y,z) = g i (z) i y i

41 Comparisons Full decoupling Block-mul8linear C k = exp(k) Par8al decoupling One-block-mul8linear C k = poly(k) Var[ f! ] exp( O(k))Var[ f ] Var[ f ] Var[ f ] kvar[ f ] f(x) = f! (x,...,x) kf(x) = f (x,x) for homogeneous case only General inputs with all finite moments Boolean or Gaussian

42 The rest of my talk 1. Applica8on 3: Tight bounds for DFKO Theorems 2. Proof sketch for our decoupling inequali8es

43 Applica8on 3: Tight bounds for DFKO Theorems DFKO Inequality: f :R n R a polynomial with degree k Standard Gaussian/Boolean inputs (for Boolean, is small) Var[ f ] 1 [Dinur Friedgut Kindler O Donnell 07] Pr[ f > t] exp( O(t 2 k 2 logk)) MaxInf[ f ] Pr[ f > t] exp( O(t 2 )) A gap of log k There exists some func8on f such that Pr[ f > t] exp( O(t 2 k 2 ))

44 Applica8on 3: Tight bounds for DFKO Theorems Pr[ f (y,z) > t] exp( O(k +t 2 )) (by hypercontrac8vity) Pr[ f(y,z) > C k t] D k Pr[ f(x) > t] Gaussian case: C k = O(k),D k = k O(k) = exp(o(klogk)) Pr[ f > t] exp( O(t 2 k 2 ))

45 Proof sketch for Gaussian case Theorem 1. Let Φ :! 0! 0 be convex and non-decreasing. E[Φ( f(y,z) )] E[Φ(C k f(x) )] f(y,z) = c f(a y + b z) i i i a i 2 + b i 2 = 1 i i a i y + b i z N(0,1) n c i = C k = O(k)

46 Proof sketch for Gaussian case Theorem 1. Let Φ :! 0! 0 be convex and non-decreasing. E[Φ( f(y,z) )] E[Φ(C k f(x) )] E[Φ( f(y,z) )]= E Φ c i f(a i y + b i z) i c i E Φ C C k k f(a i y + b i z) i c = i E Φ( C C k k f(x) ) i ( ) = E Φ C k f(x) ( )

47 Proof sketch for Gaussian case Theorem 1. Let Φ :! 0! 0 f(y,z) = c i f(a i y + b i z) i a i 2 + b i 2 = 1 i c i = C k = O(k) be convex and non-decreasing. E[Φ( f(y,z) )] E[Φ(C k f(x) )] f(x) = x 1 x 2 f(y,z) = y 1 z 2 + y 2 z 1 f(a i y + b i z) = (a i y 1 + b i z 1 )(a i y 2 + b i z 2 ) y 1 z 2 + y 2 z 1 = c i a 2 i y 1 y 2 + c i a i b i (y 1 z 2 + y 2 z 1 ) + c i b 2 i z 1 z 2 i i 2 2 c i a i = 0 c i a i b i = 1 c i b i = 0 i i i i Best choice we got: a i b i = k i

48 Summary Main result: Prove the decoupling inequali8es for one-block decoupling with polynomial bounds. Applica8ons: 1. Generalize a randomized algorithm to arbitrary Boolean func8ons with the same query complexity; 2. Give an easy proof for the weak version of AA Conjecture. Show that AA Conjecture holds iff it holds for all one-block-mul8linear func8ons; 3. Prove the 8ght bounds for DFKO Theorems.

49 Future direc8on 1. One-block decoupling inequali8es are 8ght with Gaussian inputs. What about Boolean case? 2. Can we generalize them to arbitrary inputs with all moments finite? 3. Do the reverse inequali8es hold? 4. Prove (or disprove) AA Conjecture for oneblock-mul8linear func8ons.

50 Thank you!