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

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

Block-multilinearity A homogeneous polynomial function 𝑔 with degree 𝑙 is Block-multilinear

Block-multilinearity A homogeneous polynomial function 𝑔 with degree 𝑙 is Block-multilinear if we can partition the input variables into 𝑙 blocks 𝑇 1 , … , 𝑇 𝑙

Block-multilinearity A homogeneous polynomial function 𝑔 with degree 𝑙 is Block-multilinear if we can partition the input variables into 𝑙 blocks 𝑇 1 , … , 𝑇 ' such that each monomial in 𝑔 contains exactly 1 variable in each block. 𝑔 𝑦 * , 𝑦 + ,𝑦 , , 𝑦 - = 1 2 𝑦 * 𝑦 + + 1 2 𝑦 + 𝑦 , + 1 2 𝑦 , 𝑦 - − 1 2 𝑦 * 𝑦 - Anti-concentrations of 𝑇 * = 𝑦 * , 𝑦 , , 𝑇 + = 𝑦 + , 𝑦 - Gaussian polynomial Max-E3-Lin-2 [Khot Naor 08, Lovett 10, Kane Meka13, Aaronson Ambainis15] PRG for Liptschitz Classical simulation for functions of polynomials quantum query algorithm

AA Conjecture Let 𝑔: {−1,1} 5 → [−1,1] be a bounded Boolean polynomial with degree at most 𝑙 . Then MaxInf[𝑔] ≥ poly(Var[𝑔]/𝑙) . 𝑔 𝑦 * ,𝑦 + ,𝑦 , ,𝑦 - = 1 2 𝑦 * 𝑦 + + 1 2 𝑦 + 𝑦 , + 1 2 𝑦 , 𝑦 - − 1 Def: J(𝑇) K𝑦 L 𝑔 𝑦 = I 𝑔 2 𝑦 * 𝑦 - N⊆[5] L∈N J(𝑇) + Var[𝑔] = 1 Var[𝑔] = I 𝑔 NP∅ Inf * [𝑔] = 1 J(𝑇) + Inf L [𝑔] = I 𝑔 2 N∋L MaxInf 𝑔 = 1 MaxInf 𝑔 = max L∈[5] {Inf L 𝑔 } 2

AA Conjecture Let 𝑔: {−1,1} 5 → [−1,1] be a bounded Boolean polynomial with degree at most 𝑙 . Then MaxInf[𝑔] ≥ poly(Var[𝑔]/𝑙) . Suppose AA Conjecture holds: 1. There exists some deterministic simulation of a quantum algorithm; 2. P = P#P implies BQP 𝐵 ⊂ AvgP 𝐵 with probability 1 for a random oracle 𝐵 .

AA Conjecture, weak version Let 𝑔: {−1,1} 5 → [−1,1] be a Boolean polynomial with degree at most 𝑙 . Then MaxInf[𝑔] ≥ Var[𝑔] + /exp (𝑙) .

AA Conjecture, weak version Let 𝑔: {−1,1} 5 → [−1,1] be a Boolean polynomial with degree at most 𝑙 . Then MaxInf[𝑔] ≥ Var[𝑔] + /exp (𝑙) . There exists an easy proof for block-multilinear function!! 𝑔 𝑧,𝑨 = I𝑧 L 𝑕 L (𝑨) L First block Then use hypercontractivity and Cauchy-Schwartz Rest variables

AA Conjecture, weak version Let 𝑔: {−1,1} 5 → [−1,1] be a Boolean polynomial with degree at most 𝑙 . Then MaxInf[𝑔] ≥ Var[𝑔] + /exp (𝑙) . There exists an easy proof for block-multilinear function!! Can we extend this proof to arbitrary Boolean polynomials? Yes, via decoupling!

Decoupling decoupling c 𝑔 𝑔 general polynomial block-multilinear degree 𝑙 degree 𝑙 𝑜 variables 𝑙𝑜 variables ( 𝑙 blocks of 𝑜 variables) 𝑙 copies of 𝑦 c(𝑦, 𝑦, … ,𝑦) 1. 𝑔 𝑦 = 𝑔 c and 𝑔 has similar properties 2. 𝑔

Examples of decoupling 𝑔 𝑦 * , 𝑦 + ,𝑦 , = 𝑦 * 𝑦 + 𝑦 , c 𝑧 * , 𝑧 + , 𝑧 , , 𝑨 * , 𝑨 + , 𝑨 , , 𝑥 * , 𝑥 + ,𝑥 , 𝑔 = * e 𝑧 * 𝑨 + 𝑥 ,

Examples of decoupling 𝑔 𝑦 * , 𝑦 + ,𝑦 , = 𝑦 * 𝑦 + 𝑦 , c 𝑧 * , 𝑧 + , 𝑧 , , 𝑨 * , 𝑨 + , 𝑨 , , 𝑥 * , 𝑥 + ,𝑥 , 𝑔 = * e 𝑧 * 𝑨 + 𝑥 , + * e 𝑧 * 𝑥 + 𝑨 ,

Examples of decoupling 𝑔 𝑦 * , 𝑦 + ,𝑦 , = 𝑦 * 𝑦 + 𝑦 , c 𝑧 * , 𝑧 + , 𝑧 , , 𝑨 * , 𝑨 + , 𝑨 , , 𝑥 * , 𝑥 + ,𝑥 , 𝑔 = * e 𝑧 * 𝑨 + 𝑥 , + * e 𝑧 * 𝑥 + 𝑨 , + * e 𝑨 * 𝑧 + 𝑥 , + * e 𝑨 * 𝑥 + 𝑧 , + * e 𝑥 * 𝑧 + 𝑨 , + * e 𝑥 * 𝑨 + 𝑧 , c = 1 1 c = Var 𝑔 𝑙! Var 𝑔 Inf g h 𝑔 𝑙! i 𝑙 Inf j h 𝑔

AA Conjecture, weak version Let 𝑔: {−1,1} 5 → [−1,1] be a Boolean function with degree at most 𝑙 . Then MaxInf[𝑔] ≥ Var[𝑔] 2 /exp (𝑙). decoupling c 𝑔 𝑔 c = 1 Var 𝑔 𝑙! Var 𝑔 1 c = MaxInf 𝑔 𝑙! i 𝑙 MaxInf 𝑔

Decoupling inequality ( 𝑙 is the degree of 𝑔 ) Theorem 1. Let Φ: ℝ mn → ℝ mn be convex and non-decreasing. c 𝑦 * , …, 𝑦 ' E Φ 𝑔 ≤ E[Φ(𝐷 ' |𝑔(𝑦)|)] [de la Peña 92] Theorem 2. For all 𝑢 > 0 , c 𝑦 * , … , 𝑦 ' Pr 𝑔 > 𝐷 ' 𝑢 ≤ 𝐸 ' Pr 𝑔(𝑦) > 𝑢 [Peña Montgomery-Smith 95, Giné 98] Comments: 1. 𝐷 ' , 𝐸 ' = exp (𝑙 log𝑙) 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 multilinear neccesarily

AA Conjecture, weak version Let 𝑔: {−1,1} 5 → [−1,1] be a Boolean function with degree at most 𝑙 . Then MaxInf[𝑔] ≥ Var[𝑔] + /exp (𝑙). decoupling c c/𝐷 ' 𝑔 𝑔 𝑔 [−1,1] [−𝐷 ' , 𝐷 ' ] [−1,1] c/𝐷 ' = 1 c = 1 c Var 𝑔 + Var 𝑔 𝐷 ' = exp 𝑙 log𝑙 Var 𝑔 𝑙! Var 𝑔 𝐷 ' from decoupling inequality 1 c/𝐷 ' = 1 c = c MaxInf 𝑔 𝑙! i 𝑙 MaxInf 𝑔 MaxInf 𝑔 + MaxInf 𝑔 𝐷 '

Summary of classical decoupling Advantage: Transfer a general function 𝑔 to a block- multilinear function. Disadvantage: Introduce an exponential factor on 𝑙 in decoupling inequality. L

Summary of classical decoupling Sometimes we don’t need the function to be all- blocks-multilinear. We only need 𝑔 to be a linear map on 𝑧 . 𝑔 𝑧,𝑨 = I𝑧 L 𝑕 L (𝑨) L First block Then use hypercontractivity and Cauchy-Schwartz Rest variables

One-block-multilinear A polynomial function 𝑔 with degree 𝑙 is one- block-multilinear if there exists a subset of the input variables 𝑇 such that each monomial (except the constant term) in 𝑔 contains exactly 1 variable in 𝑇 . 𝑔 𝑧,𝑨 = I𝑧 L 𝑕 L (𝑨) L

Partial decoupling, with polynomial bounds Our result: Partial decoupling w 𝑔 𝑔 general function One-block-multilinear function degree 𝑙 degree 𝑙 𝑜 variables 2𝑜 variables ( 2 blocks of 𝑜 variables)

Examples of partial decoupling 𝑔 𝑦 * , 𝑦 + ,𝑦 , = 𝑦 * 𝑦 + 𝑦 , w 𝑧 * ,𝑧 + , 𝑧 , , 𝑨 * , 𝑨 + , 𝑨 , 𝑔 = * , 𝑧 * 𝑨 + 𝑨 , + * , 𝑨 * 𝑧 + 𝑨 , + * , 𝑨 * 𝑨 + 𝑧 , w = 1 w = 1 Var 𝑔 𝑙 Var 𝑔 Inf g h 𝑔 𝑙 + Inf j h 𝑔 w = 𝑙 − 1 Inf x h 𝑔 Inf j h 𝑔 𝑙 +

Partial decoupling, with polynomial bounds Our result: Theorem 1. Let Φ:ℝ mn → ℝ mn be convex and non-decreasing. y 𝑧, 𝑨 E Φ 𝑔 ≤ E[Φ(𝐷 ' |𝑔(𝑦)|)] Theorem 2. For all 𝑢 > 0 , w 𝑧, 𝑨 Pr 𝑔 > 𝐷 ' 𝑢 ≤ 𝐸 ' Pr 𝑔(𝑦) > 𝑢 poly(𝑙) With constants: 𝑃(𝑙 + ) Boolean 𝑃(𝑙 ,/+ ) Boolean, homogeneous 𝐷 ' = { 𝐸 ' = exp (𝑙 log𝑙) . 𝑃(𝑙) standard Gaussian

AA Conjecture, weak version Let 𝑔: {−1,1} 5 → [−1,1] be a Boolean function with degree at most 𝑙 . Then MaxInf[𝑔] ≥ Var[𝑔] + /exp (𝑙). w w/𝐷 ' 𝑔 𝑔 𝑔 [−1,1] [−𝐷 ' , 𝐷 ' ] [−1,1] w/𝐷 ' = 1 w = 1 w Var 𝑔 + Var 𝑔 𝐷 ' = poly 𝑙 Var 𝑔 𝑙 Var 𝑔 𝐷 ' from new decoupling inequality w ≥ 1 w/𝐷 ' = 1 w MaxInf 𝑔 𝑙 + MaxInf 𝑔 MaxInf 𝑔 + MaxInf 𝑔 𝐷 '

AA Conjecture Let 𝑔: {−1,1} 5 → [−1,1] be a Boolean function with degree at most 𝑙 . Then MaxInf[𝑔] ≥ Var[𝑔] + /poly (𝑙). w w/𝐷 ' 𝑔 𝑔 𝑔 [−1,1] [−𝐷 ' , 𝐷 ' ] [−1,1] w/𝐷 ' = 1 w = 1 w Var 𝑔 + Var 𝑔 𝐷 ' = poly 𝑙 Var 𝑔 𝑙 Var 𝑔 𝐷 ' from new decoupling inequality w ≥ 1 w/𝐷 ' = 1 w MaxInf 𝑔 𝑙 + MaxInf 𝑔 MaxInf 𝑔 + MaxInf 𝑔 𝐷 '

AA Conjecture Let 𝑔: {−1,1} 5 → [−1,1] be a Boolean function with degree at most 𝑙 . Then MaxInf[𝑔] ≥ Var[𝑔] + /poly (𝑙). The conjecture holds for one-block-multilinear functions. 𝑔 𝑧,𝑨 = I𝑧 L 𝑕 L (𝑨) L

Comparisons Full decoupling Partial decoupling One-block-multilinear Block-multilinear 𝐷 ' = exp (𝑙 log𝑙) 𝐷 ' = poly (𝑙) General inputs Boolean or Gaussian with all finite moments

Decoupling with polynomial bounds Main result: Prove the decoupling inequalities for one-block decoupling with polynomial bounds. Applications: 1. Give an easy proof for the weak version of AA Conjecture. Show that AA Conjecture holds iff it holds for all one-block-multilinear functions; 2. Generalize a randomized algorithm to arbitrary Boolean functions with the same query complexity; 3. Prove the tight bounds for DFKO Theorems.

Application 2 f :{ − 1,1} n → [ − 1,1] Theorem in [AA15] Let be any bounded block-multilinear Boolean function with degree k . Then there exists a randomized algorithm that, on input , non-adaptively queries 2 O ( k ) ( n / ε 2 ) 1-1/ k x ∈ { − 1,1} n bits of x , and then estimate 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]