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 𝑇1, … , 𝑇𝑙

Block-multilinearity

A homogeneous polynomial function 𝑔 with degree 𝑙 is Block-multilinear if we can partition the input variables into 𝑙 blocks 𝑇1, … , 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- 𝑇* = 𝑦*, 𝑦, , 𝑇+ = 𝑦+, 𝑦-

[Khot Naor 08, Lovett 10, Kane Meka13, Aaronson Ambainis15]

Max-E3-Lin-2 Anti-concentrations of Gaussian polynomial PRG for Liptschitz functions of polynomials Classical simulation for 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). Def:

𝑔 𝑦*,𝑦+,𝑦,,𝑦- = 1 2 𝑦*𝑦++ 1 2 𝑦+𝑦, + 1 2 𝑦,𝑦- − 1 2 𝑦*𝑦-

𝑔 𝑦 = I 𝑔 J(𝑇) K𝑦L

L∈N N⊆[5]

Var[𝑔] = I 𝑔 J(𝑇)+

NP∅

InfL[𝑔] = I 𝑔 J(𝑇)+

N∋L

MaxInf 𝑔 = maxL∈[5]{InfL 𝑔 } Var[𝑔] = 1 Inf*[𝑔] = 1 2 MaxInf 𝑔 = 1 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 BQP𝐵 ⊂ AvgP𝐵 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!!

First block Rest variables Then use hypercontractivity and Cauchy-Schwartz

𝑔 𝑧,𝑨 = I𝑧L𝑕L(𝑨)

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!!

Yes, via decoupling!

Can we extend this proof to arbitrary Boolean polynomials?

Decoupling

general polynomial block-multilinear degree 𝑙 degree 𝑙 𝑜 variables 𝑙𝑜 variables (𝑙 blocks of 𝑜 variables)

• 1. 𝑔 𝑦 = 𝑔

c(𝑦, 𝑦, … ,𝑦)

• 2. 𝑔

c and 𝑔 has similar properties

decoupling

𝑙 copies of 𝑦

𝑔 𝑔 c

Examples of decoupling

𝑔 𝑦*, 𝑦+,𝑦, = 𝑦*𝑦+𝑦, 𝑔 c 𝑧*, 𝑧+, 𝑧,, 𝑨*, 𝑨+, 𝑨,, 𝑥*, 𝑥+,𝑥, = *

e𝑧*𝑨+𝑥,

Examples of decoupling

𝑔 𝑦*, 𝑦+,𝑦, = 𝑦*𝑦+𝑦, 𝑔 c 𝑧*, 𝑧+, 𝑧,, 𝑨*, 𝑨+, 𝑨,, 𝑥*, 𝑥+,𝑥, = *

e𝑧*𝑨+𝑥, + * e𝑧*𝑥+𝑨,

Examples of decoupling

𝑔 𝑦*, 𝑦+,𝑦, = 𝑦*𝑦+𝑦, 𝑔 c 𝑧*, 𝑧+, 𝑧,, 𝑨*, 𝑨+, 𝑨,, 𝑥*, 𝑥+,𝑥, = *

e𝑧*𝑨+𝑥, + * e𝑧*𝑥+𝑨, + * e𝑨*𝑧+𝑥, + * e𝑨*𝑥+𝑧, + * e𝑥*𝑧+𝑨, + * e𝑥*𝑨+𝑧,

Var 𝑔 c = 1 𝑙! Var 𝑔 Infgh 𝑔 c = 1 𝑙! i 𝑙 Infjh 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). 𝑔 𝑔 c

Var 𝑔 c = 1 𝑙! Var 𝑔 MaxInf 𝑔 c = 1 𝑙! i 𝑙 MaxInf 𝑔

decoupling

Decoupling inequality

Theorem 1. Let Φ: ℝmn → ℝmn be convex and non-decreasing. Theorem 2. For all 𝑢 > 0, 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

[de la Peña 92] [Peña Montgomery-Smith 95, Giné 98]

(𝑙 is the degree of 𝑔)

E Φ 𝑔 c 𝑦 * , …, 𝑦 ' ≤ E[Φ(𝐷'|𝑔(𝑦)|)] Pr 𝑔 c 𝑦 * , … , 𝑦 ' > 𝐷'𝑢 ≤ 𝐸'Pr 𝑔(𝑦) > 𝑢

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). 𝑔 𝑔 c 𝑔 c/𝐷'

Var 𝑔 c = 1 𝑙! Var 𝑔 MaxInf 𝑔 c = 1 𝑙! i 𝑙 MaxInf 𝑔 Var 𝑔 c/𝐷' = 1 𝐷'

+ Var 𝑔

c MaxInf 𝑔 c/𝐷' = 1 𝐷'

+ MaxInf 𝑔

c [−1,1] [−𝐷', 𝐷'] [−1,1] 𝐷' = exp 𝑙 log𝑙 from decoupling inequality

decoupling

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.

First block Rest variables Then use hypercontractivity and Cauchy-Schwartz

We only need 𝑔 to be a linear map on 𝑧.

𝑔 𝑧,𝑨 = I𝑧L𝑕L(𝑨)

L

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 = I𝑧L𝑕L(𝑨)

L

Partial decoupling, with polynomial bounds

Our result:

Partial decoupling general function One-block-multilinear function degree 𝑙 degree 𝑙 𝑜 variables 2𝑜 variables (2 blocks of 𝑜 variables)

𝑔 𝑔 w

Examples of partial decoupling

𝑔 𝑦*, 𝑦+,𝑦, = 𝑦*𝑦+𝑦, 𝑔 w 𝑧*,𝑧+, 𝑧,, 𝑨*, 𝑨+, 𝑨, = *

,𝑧*𝑨+𝑨, + * ,𝑨*𝑧+𝑨, + * ,𝑨*𝑨+𝑧,

Var 𝑔 w = 1 𝑙 Var 𝑔 Infgh 𝑔 w = 1 𝑙+ Infjh 𝑔 Infxh 𝑔 w = 𝑙 − 1 𝑙+ Infjh 𝑔

Partial decoupling, with polynomial bounds

Our result: Theorem 1. Let Φ:ℝmn → ℝmn be convex and non-decreasing. Theorem 2. For all 𝑢 > 0, With constants: .

Boolean Boolean, homogeneous standard Gaussian poly(𝑙) E Φ 𝑔 y 𝑧, 𝑨 ≤ E[Φ(𝐷'|𝑔(𝑦)|)] Pr 𝑔 w 𝑧, 𝑨 > 𝐷'𝑢 ≤ 𝐸'Pr 𝑔(𝑦) > 𝑢

𝑃(𝑙+) 𝑃(𝑙,/+) 𝑃(𝑙) 𝐷' = { 𝐸' = exp (𝑙 log𝑙)

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). 𝑔 𝑔 w 𝑔 w/𝐷'

Var 𝑔 w = 1 𝑙 Var 𝑔 MaxInf 𝑔 w ≥ 1 𝑙+ MaxInf 𝑔 Var 𝑔 w/𝐷' = 1 𝐷'

+ Var 𝑔

w MaxInf 𝑔 w/𝐷' = 1 𝐷'

+ MaxInf 𝑔

w [−1,1] [−𝐷', 𝐷'] [−1,1] 𝐷' = poly 𝑙 from new decoupling inequality

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). 𝑔 𝑔 w 𝑔 w/𝐷'

Var 𝑔 w = 1 𝑙 Var 𝑔 MaxInf 𝑔 w ≥ 1 𝑙+ MaxInf 𝑔 Var 𝑔 w/𝐷' = 1 𝐷'

+ Var 𝑔

w MaxInf 𝑔 w/𝐷' = 1 𝐷'

+ MaxInf 𝑔

w [−1,1] [−𝐷', 𝐷'] [−1,1] 𝐷' = poly 𝑙 from new decoupling inequality

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 = I𝑧L𝑕L(𝑨)

L

Comparisons

Full decoupling Partial decoupling Block-multilinear One-block-multilinear

General inputs with all finite moments Boolean or Gaussian 𝐷' = exp (𝑙 log𝑙) 𝐷' = poly (𝑙)

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

Theorem in [AA15] Let be any bounded block-multilinear Boolean function with degree k. Then there exists a randomized algorithm that,

• n input , non-adaptively queries 2O(k)(n/ε2)1-1/k

bits of x, and then estimate the output of f within error ε with high probability.

f :{−1,1}n →[−1,1] x ∈{−1,1}n

f(x)= f

!(x,...,x)

[−1,1] [−Ck,Ck]

f

f

!

f

! /Ck

[−1,1]

ε'= ε /Ck Ck = 2O(k)

Application 2

Theorem in [AA15] Let be any bounded block-multilinear Boolean function with degree k. Then there exists a randomized algorithm that,

• n input , non-adaptively queries 2O(k)(n/ε2)1-1/k

bits of x, and then estimate the output of f within error ε with high probability.

f :{−1,1}n →[−1,1] x ∈{−1,1}n

f(x)= f

!(x,...,x)

[−1,1] [−Ck,Ck]

f

f

!

f

! /Ck

[−1,1]

ε'= ε /Ck Ck = 2O(k)

Application 3: Tight bounds for DFKO Theorems

DFKO Inequality:

f :Rn → R a polynomial with degree k

Standard Gaussian/Boolean inputs (for Boolean, is small)

MaxInf[f] Var[f]≥1

Pr[|f|>t]≥ exp(−O(t2k2logk))

There exists some function f such that

Pr[|f|>t]≤ exp(−O(t2k2))

A gap of log k

Pr[|f|>t]≤ exp(−O(t2))

[Dinur Friedgut Kindler O’Donnell 07]

Future direction

Our result: Theorem 1. Let Φ:ℝmn → ℝmn be convex and non-decreasing. Theorem 2. For all 𝑢 > 0, With constants: .

Boolean Boolean, homogeneous standard Gaussian poly(𝑙) E Φ 𝑔 y 𝑧, 𝑨 ≤ E[Φ(𝐷'|𝑔(𝑦)|)] Pr 𝑔 w 𝑧, 𝑨 > 𝐷'𝑢 ≤ 𝐸'Pr 𝑔(𝑦) > 𝑢

𝑃(𝑙+) 𝑃(𝑙,/+) 𝑃(𝑙) 𝐷' = { 𝐸' = exp (𝑙 log𝑙)

Future direction

• 1. One-block decoupling inequalities are tight

with Gaussian inputs. What about Boolean case?

• 2. Can we generalize them to arbitrary inputs

with all moments finite?

• 3. Do the reverse inequalities hold?
• 4. Prove (or disprove) AA Conjecture for one-

block-multilinear functions.

Thank you!