Making Polynomials Robust to Noise Alexander Sherstov U C L A Noise - - PowerPoint PPT Presentation

Making Polynomials Robust to Noise

Alexander Sherstov

U C L A

Noise in computation

2

Noise in computation

2

human error

Noise in computation

2

human error malicious third party

Noise in computation

2

human error malicious third party randomness

Decision trees

3

x1 x1 x2 x7 1 1

Decision trees

3

Feige, Peleg, Raghavan, Upfal (STOC ’90):

PARITYn , MAJORITYn require depth

Ω(n log n)

x1 x1 x2 x7 1 1

Broadcast networks

4

x1 x2 x3 xn ... f (x1, x2, . . . , xn)

Broadcast networks

4

x1 x2 x3 xn ... f (x1, x2, . . . , xn)

broadcasts enough for any

Gallager (1984):

O(n log log n) f

Broadcast networks

4

x1 x2 x3 xn ... f (x1, x2, . . . , xn)

Goyal, Kindler, Saks (FOCS ’05):

broadcasts necessary for

Ω(n log log n) f = id

broadcasts enough for any

Gallager (1984):

O(n log log n) f

Polynomials

5

Polynomials

5

f : {−1, +1}n → {−1, +1}

Minimum degree of a real polynomial s.t. Approximate degree

max

x∈{−1,+1}n |f (x) − ˜

f (x)| ≤ 1 3 ˜ f

g deg(f )

Polynomials

5

[Minsky & Papert 1969]

f : {−1, +1}n → {−1, +1}

Minimum degree of a real polynomial s.t. Approximate degree

max

x∈{−1,+1}n |f (x) − ˜

f (x)| ≤ 1 3 ˜ f

g deg(f )

Motivation

6

• Circuit complexity
• Communication complexity
• Quantum query complexity
• Complexity of learning
• Proof complexity

Motivation

6

• Circuit complexity
• Communication complexity
• Quantum query complexity
• Complexity of learning
• Proof complexity

via lower bounds on

h

}

g deg(f )

Motivation

7

• Learning algorithms
• Approximate inclusion-exclusion

via upper bounds on

h

}

g deg(f )

A difficulty

8

f ≈ ˜ f , g ≈ ˜ g

A difficulty

8

f ≈ ˜ f , g ≈ ˜ g = ⇒ f (g, g, . . . , g) ≈ ˜ f (˜ g, ˜ g, . . . , ˜ g)

A difficulty

8

Reason:

˜ f (˜ g, ˜ g, . . . , ˜ g)

cannot handle noisy input

f ≈ ˜ f , g ≈ ˜ g = ⇒ f (g, g, . . . , g) ≈ ˜ f (˜ g, ˜ g, . . . , ˜ g)

Robust approximation

9

Buhrman, Newman, Röhrig, de Wolf (2003)

|f (x) − ˜ f (x + ✏)| ≤ 1 3 ∀✏ = (✏1, . . . , ✏n) ∈ [− 1

3, 1 3]n

Robust approximation

9

Buhrman, Newman, Röhrig, de Wolf (2003)

|f (x) − ˜ f (x + ✏)| ≤ 1 3 ∀✏ = (✏1, . . . , ✏n) ∈ [− 1

3, 1 3]n

Robust approximation

9

Buhrman, Newman, Röhrig, de Wolf (2003)

|f (x) − ˜ f (x + ✏)| ≤ 1 3 ∀✏ = (✏1, . . . , ✏n) ∈ [− 1

3, 1 3]n

✔

f ≈ ˜ f , g ≈ ˜ g = ⇒ f (g, g, . . . , g) ≈ ˜ f (˜ g, ˜ g, . . . , ˜ g)

Robust polynomials

10

Problem (Buhrman, Newman, Röhrig, de Wolf ’03)

Does every have a robust approximating polynomial

• f degree O(g

deg(f ))? f

Robust polynomials

10

Problem (Buhrman, Newman, Röhrig, de Wolf ’03)

Does every have a robust approximating polynomial

• f degree O(g

deg(f ))?

Folklore

O(g deg(f ) log n) f

Robust polynomials

10

Problem (Buhrman, Newman, Röhrig, de Wolf ’03)

Does every have a robust approximating polynomial

• f degree O(g

deg(f ))?

Folklore

O(g deg(f ) log n)

Buhrman et al. (2003)

O(g deg(f ) log g deg(f )) f

Robust polynomials

10

Problem (Buhrman, Newman, Röhrig, de Wolf ’03)

Does every have a robust approximating polynomial

• f degree O(g

deg(f ))?

Folklore

O(g deg(f ) log n)

Buhrman et al. (2003)

O(g deg(f ) log g deg(f ))

Buhrman et al. (2003)

O(n) f

Our result

11

Problem (Buhrman, Newman, Röhrig, de Wolf ’03)

Does every have a robust approximating polynomial

• f degree O(g

deg(f ))? f

Yes.

✔

Our result

11

Every polynomial can be made robust with constant overhead in degree.

p: {−1, +1}n → [−1, +1]

Main result.

Problem (Buhrman, Newman, Röhrig, de Wolf ’03)

Does every have a robust approximating polynomial

• f degree O(g

deg(f ))? f

Yes.

✔

Our result

12

Main result. For any , there is s.t.

probust : Rn → R δ > 0, p: {−1, +1}n → [−1, +1]

Our result

12

Main result. For any , there is s.t.

probust : Rn → R δ > 0, |p(x) − probust(x + ✏)| ≤ ∀x ∈ {−1, +1}n, ✏ ∈ [− 1

3, 1 3]n

.

p: {−1, +1}n → [−1, +1]

Our result

12

Main result. For any , there is s.t.

probust : Rn → R δ > 0, |p(x) − probust(x + ✏)| ≤ ∀x ∈ {−1, +1}n, ✏ ∈ [− 1

3, 1 3]n

.

deg probust = O ✓ deg p + log 1 δ ◆

.

p: {−1, +1}n → [−1, +1]

Our result

12

Main result. For any , there is s.t.

probust : Rn → R δ > 0, |p(x) − probust(x + ✏)| ≤ ∀x ∈ {−1, +1}n, ✏ ∈ [− 1

3, 1 3]n

.

deg probust = O ✓ deg p + log 1 δ ◆

.

p: {−1, +1}n → [−1, +1]

• ptimal

✔

Our result

12

Main result. For any , there is s.t.

probust : Rn → R δ > 0, |p(x) − probust(x + ✏)| ≤ ∀x ∈ {−1, +1}n, ✏ ∈ [− 1

3, 1 3]n

.

deg probust = O ✓ deg p + log 1 δ ◆

.

p: {−1, +1}n → [−1, +1]

• ptimal

✔

explicit

Our solution

13

• 3. Robust approximation of arbitrary p
• 1. Robust approximation of a monomial

p(x) =

n

Y

i=1

xi

• 2. Robust approximation of homogeneous p

p(x) = X

|S|=d

aS Y

i∈S

xi

Our solution

13

• 3. Robust approximation of arbitrary p
• 1. Robust approximation of a monomial

p(x) =

n

Y

i=1

xi

• 2. Robust approximation of homogeneous p

p(x) = X

|S|=d

aS Y

i∈S

xi

hardest part

h

}

Our solution

13

• 3. Robust approximation of arbitrary p
• 1. Robust approximation of a monomial

p(x) =

n

Y

i=1

xi

• 2. Robust approximation of homogeneous p

p(x) = X

|S|=d

aS Y

i∈S

xi

hardest part

h

}

Robust approximation of ∏xi

14

(. . . , xi + ✏i, . . . ) 7!

n

Y

i=1

xi ± 2−Ω(n) ∀x1, . . . , xn = ±1

Robust approximation of ∏xi

14

(. . . , xi + ✏i, . . . ) 7!

n

Y

i=1

xi ± 2−Ω(n) ∀x1, . . . , xn = ±1 xi = xi + ✏i p (xi + ✏i)2

Robust approximation of ∏xi

14

(. . . , xi + ✏i, . . . ) 7!

n

Y

i=1

xi ± 2−Ω(n) ∀x1, . . . , xn = ±1 xi = xi + ✏i p (xi + ✏i)2 = (xi + ✏i) ·

∞

X

d=0

✓−1/2 d ◆ ((xi + ✏i)2 − 1)d

(Maclaurin)

Robust approximation of ∏xi

15

n

Y

i=1

xi = X

d1,...,dn n

Y

i=1

(xi + ✏i) ✓−1/2 di ◆ ((xi + ✏i)2 − 1)di

Robust approximation of ∏xi

15

n

Y

i=1

xi = X

d1,...,dn n

Y

i=1

(xi + ✏i) ✓−1/2 di ◆ ((xi + ✏i)2 − 1)di = 2xi✏i + ✏2

i

Ÿ

Robust approximation of ∏xi

15

n

Y

i=1

xi = X

d1,...,dn n

Y

i=1

(xi + ✏i) ✓−1/2 di ◆ ((xi + ✏i)2 − 1)di = 2xi✏i + ✏2

i

Ÿ

< 2−Ω(d1+d2+···+dn)

Ÿ

2O(n)

Robust approximation of ∏xi

15

n

Y

i=1

xi = X

d1,...,dn n

Y

i=1

(xi + ✏i) ✓−1/2 di ◆ ((xi + ✏i)2 − 1)di = 2xi✏i + ✏2

i

Ÿ

d1 + d2 + · · · + dn ≥ 10n

Can discard terms with .

⇤

< 2−Ω(d1+d2+···+dn)

Ÿ

2O(n)

Our solution

16

• 3. Robust approximation of arbitrary p
• 1. Robust approximation of a monomial

p(x) =

n

Y

i=1

xi

• 2. Robust approximation of homogeneous p

p(x) = X

|S|=d

aS Y

i∈S

xi

✔

Our solution

16

• 3. Robust approximation of arbitrary p
• 1. Robust approximation of a monomial

p(x) =

n

Y

i=1

xi

• 2. Robust approximation of homogeneous p

p(x) = X

|S|=d

aS Y

i∈S

xi

✔

Reduction to homogeneous case

17

Reduction to homogeneous case

18

Fact (V.A. Markov, 1893). The coefficients of every degree-d polynomial are bounded by

[1, +1] 7! [1, +1] (1 + √ 2)d.

Reduction to homogeneous case

18

Fact (V.A. Markov, 1893). The coefficients of every degree-d polynomial are bounded by

[1, +1] 7! [1, +1] (1 + √ 2)d.

best possible

Reduction to homogeneous case

18

Fact (V.A. Markov, 1893). The coefficients of every degree-d polynomial are bounded by

[1, +1] 7! [1, +1] (1 + √ 2)d.

best possible We prove: 4d

Reduction to homogeneous case

19

p: {−1, +1}n → [−1, +1]

Reduction to homogeneous case

19

p: {−1, +1}n → [−1, +1] p = p0 + p1 + · · · + pd

• Decompose into homogeneous parts:

Reduction to homogeneous case

19

p: {−1, +1}n → [−1, +1] p = p0 + p1 + · · · + pd

• Decompose into homogeneous parts:

|pi(x) ˜ pi(x + ✏)| < c−dkpik∞ deg ˜ pi = O(d)

• Robustly approximate each :

pi

Reduction to homogeneous case

19

p: {−1, +1}n → [−1, +1] p = p0 + p1 + · · · + pd

• Decompose into homogeneous parts:

˜ p = ˜ p0 + ˜ p1 + · · · + ˜ pd

• Set

|pi(x) ˜ pi(x + ✏)| < c−dkpik∞ deg ˜ pi = O(d)

• Robustly approximate each :

pi

20

Reduction to homogeneous case

|p(x) − ˜ p(x + ✏)| ≤

d

X

i=0

|pi(x) − ˜ pi(x + ✏)|

20

Reduction to homogeneous case

|p(x) − ˜ p(x + ✏)| ≤

d

X

i=0

|pi(x) − ˜ pi(x + ✏)| 

d

X

i=0

c−dkpik∞

20

Reduction to homogeneous case

|p(x) − ˜ p(x + ✏)| ≤

d

X

i=0

|pi(x) − ˜ pi(x + ✏)| 

d

X

i=0

c−dkpik∞

h

}

≤ 4d

20

Reduction to homogeneous case

|p(x) − ˜ p(x + ✏)| ≤

d

X

i=0

|pi(x) − ˜ pi(x + ✏)| ≤ 2−Ω(d) 

d

X

i=0

c−dkpik∞

h

}

≤ 4d

20

Reduction to homogeneous case

|p(x) − ˜ p(x + ✏)| ≤

d

X

i=0

|pi(x) − ˜ pi(x + ✏)| ≤ 2−Ω(d) 

d

X

i=0

c−dkpik∞

h

}

≤ 4d

Reduction to homogeneous case

21

For any

• d

X

i=0

pi(x)

• ≤ 1

x ∈ {−1, +1}n,

Reduction to homogeneous case

21

For any

• d

X

i=0

pi(x)

• ≤ 1

x ∈ {−1, +1}n, max

−1≤t≤1

• d

X

i=0

pi(x)ti

• ≤ 1

= ⇒

Reduction to homogeneous case

21

For any

• d

X

i=0

pi(x)

• ≤ 1

x ∈ {−1, +1}n, max

−1≤t≤1

• d

X

i=0

pi(x)ti

• ≤ 1

= ⇒

= E " d X

i=0

pi(x) #

Reduction to homogeneous case

21

For any

• d

X

i=0

pi(x)

• ≤ 1

x ∈ {−1, +1}n, max

−1≤t≤1

• d

X

i=0

pi(x)ti

• ≤ 1

= ⇒

are coefficients of a polynomial

p0(x), . . . , pd(x) [1, +1] 7! [1, +1]

= ⇒

= E " d X

i=0

pi(x) #

Our solution

22

• 3. Robust approximation of arbitrary p
• 1. Robust approximation of a monomial

p(x) =

n

Y

i=1

xi

• 2. Robust approximation of homogeneous p

p(x) = X

|S|=d

aS Y

i∈S

xi

✔ ✔

Our solution

22

• 3. Robust approximation of arbitrary p
• 1. Robust approximation of a monomial

p(x) =

n

Y

i=1

xi

• 2. Robust approximation of homogeneous p

p(x) = X

|S|=d

aS Y

i∈S

xi

✔ ✔

Homogeneous case

23

max

x∈{−1,+1}n |p(x)| ≤ 1

Given: s.t.

p = X

|S|=d

aSχS

Homogeneous case

23

max

x∈{−1,+1}n |p(x)| ≤ 1

Given: s.t.

p = X

|S|=d

aSχS

Define

˜ p = X

|S|=d

aS ˜ χS

Homogeneous case

23

max

x∈{−1,+1}n |p(x)| ≤ 1

Given: s.t.

p = X

|S|=d

aSχS

Define

˜ p = X

|S|=d

aS ˜ χS

Seems crazy!

|p(x) − ˜ p(x + ✏)| ≤ X

|S|=d

|aS||S(x) − ˜ S(x + ✏)|

Homogeneous case

23

max

x∈{−1,+1}n |p(x)| ≤ 1

≤ X

|S|=d

|aS| · c−d

Given: s.t.

p = X

|S|=d

aSχS

Define

˜ p = X

|S|=d

aS ˜ χS

Seems crazy!

|p(x) − ˜ p(x + ✏)| ≤ X

|S|=d

|aS||S(x) − ˜ S(x + ✏)|

Homogeneous case

23

max

x∈{−1,+1}n |p(x)| ≤ 1

≤ X

|S|=d

|aS| · c−d

Given: s.t.

h

}

≤ ✓n d ◆1/2 · c−d 1 p = X

|S|=d

aSχS

Define

˜ p = X

|S|=d

aS ˜ χS

Seems crazy!

|p(x) − ˜ p(x + ✏)| ≤ X

|S|=d

|aS||S(x) − ˜ S(x + ✏)|

Homogeneous case

23

max

x∈{−1,+1}n |p(x)| ≤ 1

Given: s.t.

p = X

|S|=d

aSχS

Define

˜ p = X

|S|=d

aS ˜ χS

Seems crazy!

|p(x) − ˜ p(x + ✏)| ≤ X

|S|=d

|aS||S(x) − ˜ S(x + ✏)|

Homogeneous case

24

max

x∈{−1,+1}n |p(x)| ≤ 1

s.t. Given:

p = X

|S|=d

aSχS

Define

˜ p = X

|S|=d

aS ˜ χS

Homogeneous case

24

max

x∈{−1,+1}n |p(x)| ≤ 1

s.t. use this directly Given:

p = X

|S|=d

aSχS

Define

˜ p = X

|S|=d

aS ˜ χS

Homogeneous case

24

max

x∈{−1,+1}n |p(x)| ≤ 1

s.t. use this directly Find s.t.

z1, z2, . . . , zi, . . . ∈ [0, 1]n p(x) − ˜ p(x + ✏) =

∞

X

i=1

⇠ip(zi),

∞

X

i=1

|ξi| < 2−Ω(d)

Given:

p = X

|S|=d

aSχS

Define

˜ p = X

|S|=d

aS ˜ χS

Homogeneous case

24

max

x∈{−1,+1}n |p(x)| ≤ 1

s.t.

h

}

inverting infinite matrix use this directly Find s.t.

z1, z2, . . . , zi, . . . ∈ [0, 1]n p(x) − ˜ p(x + ✏) =

∞

X

i=1

⇠ip(zi),

∞

X

i=1

|ξi| < 2−Ω(d)

Given:

p = X

|S|=d

aSχS

Define

˜ p = X

|S|=d

aS ˜ χS

Warmup: Boolean inputs

25

Theorem.

φ: {−1, +1}n → R δ: {−1, +1}d → R

homogeneous of degree d symmetric Then:

max

x∈{−1,+1}n

• X

|S|=d

ˆ φ(S)δ(x|S)

•  dd

d! kφk∞kˆ δk1

Warmup: Boolean inputs

25

Theorem.

φ: {−1, +1}n → R δ: {−1, +1}d → R

homogeneous of degree d symmetric

what we want to robustly approximate

Then:

max

x∈{−1,+1}n

• X

|S|=d

ˆ φ(S)δ(x|S)

•  dd

d! kφk∞kˆ δk1

Warmup: Boolean inputs

25

Theorem.

φ: {−1, +1}n → R δ: {−1, +1}d → R

homogeneous of degree d symmetric

what we want to robustly approximate error for a single monomial

Then:

max

x∈{−1,+1}n

• X

|S|=d

ˆ φ(S)δ(x|S)

•  dd

d! kφk∞kˆ δk1

Warmup: Boolean inputs

25

Theorem.

φ: {−1, +1}n → R δ: {−1, +1}d → R

homogeneous of degree d symmetric

what we want to robustly approximate error for a single monomial

Then:

cumulative error

max

x∈{−1,+1}n

• X

|S|=d

ˆ φ(S)δ(x|S)

•  dd

d! kφk∞kˆ δk1

h

}

independent of n

Warmup: Boolean inputs

25

Theorem.

φ: {−1, +1}n → R δ: {−1, +1}d → R

homogeneous of degree d symmetric

what we want to robustly approximate error for a single monomial

Then:

cumulative error

max

x∈{−1,+1}n

• X

|S|=d

ˆ φ(S)δ(x|S)

•  dd

d! kφk∞kˆ δk1

h

}

independent of n

Warmup: Boolean inputs

25

Theorem.

φ: {−1, +1}n → R δ: {−1, +1}d → R

homogeneous of degree d symmetric

what we want to robustly approximate error for a single monomial

Then:

cumulative error

max

x∈{−1,+1}n

• X

|S|=d

ˆ φ(S)δ(x|S)

•  dd

d! kφk∞kˆ δk1

Warmup: Boolean inputs

26

Idea: express error as linear combination of

φ(x), x ∈ {−1, +1}n

Warmup: Boolean inputs

26

Idea: express error as linear combination of

φ(x), x ∈ {−1, +1}n

Key: operator Av : R{+1,−1}n → R{−1,+1}n

h

}

(Avf )(x) =

j-th coordinate E

z∈{−1,+1}d

z1z2 . . . zd f . . . , z1xv1

j

+ z2xv2

j

+ · · · + zdxvd

j

d , . . . !

Warmup: Boolean inputs

26

Idea: express error as linear combination of

φ(x), x ∈ {−1, +1}n

Key: operator Av : R{+1,−1}n → R{−1,+1}n

h

}

(Avf )(x) =

j-th coordinate E

z∈{−1,+1}d

z1z2 . . . zd f . . . , z1xv1

j

+ z2xv2

j

+ · · · + zdxvd

j

d , . . . !

Warmup: Boolean inputs

26

Idea: express error as linear combination of

φ(x), x ∈ {−1, +1}n

Key: operator Av : R{+1,−1}n → R{−1,+1}n

evaluate on non-Boolean inputs by identifying f with its multilinear extension to Rn

Warmup: Boolean inputs

27

✔ linear

Warmup: Boolean inputs

27

✔ linear ✔ bounded: kAvk∞→∞ = 1

Warmup: Boolean inputs

27

✔ linear ✔ bounded: kAvk∞→∞ = 1

symmetric

✔

Warmup: Boolean inputs

27

✔ linear ✔ bounded: kAvk∞→∞ = 1

symmetric

✔ ✔ Avχ{1,2,...,d} = d!

dd E

T ∈(

{1,2,...,d} v1+···+vd)

χT

Warmup: Boolean inputs

28

AvχS = d! dd E

T ∈(

S v1+···+vd)

χT (|S| = d)

Warmup: Boolean inputs

28

AvχS = d! dd E

T ∈(

S v1+···+vd)

χT (|S| = d) X

T ∈(

S k)

χT ✓d k ◆dd d! A1k0d−kχS = = ⇒

Warmup: Boolean inputs

29

X

T ∈(

S k)

χT ✓d k ◆dd d! A1k0d−kχS =

Warmup: Boolean inputs

30

X

T ∈(

S k)

χT ✓d k ◆dd d! A1k0d−kχS =

d

X

k=0

ˆ δ({1, . . . , k})

d

X

k=0

ˆ δ({1, . . . , k})

Warmup: Boolean inputs

30

X

T ∈(

S k)

χT = δ(x|S) ✓d k ◆dd d! A1k0d−kχS =

d

X

k=0

ˆ δ({1, . . . , k})

d

X

k=0

ˆ δ({1, . . . , k})

Warmup: Boolean inputs

31

X

T ∈(

S k)

χT = δ(x|S) X

|S|=d

ˆ φ(S) ✓d k ◆dd d! A1k0d−kχS = X

|S|=d

ˆ φ(S)

d

X

k=0

ˆ δ({1, . . . , k})

d

X

k=0

ˆ δ({1, . . . , k})

Warmup: Boolean inputs

31

X

T ∈(

S k)

χT = δ(x|S) X

|S|=d

ˆ φ(S)

= cumulative error

✓d k ◆dd d! A1k0d−kχS = X

|S|=d

ˆ φ(S)

d

X

k=0

ˆ δ({1, . . . , k})

d

X

k=0

ˆ δ({1, . . . , k})

Warmup: Boolean inputs

32

X

T ∈(

S k)

χT = = δ(x|S) X

|S|=d

ˆ φ(S) ✓d k ◆dd d! A1k0d−kφ

d

X

k=0

ˆ δ({1, . . . , k})

d

X

k=0

ˆ δ({1, . . . , k})

= cumulative error

h

}

bounded by 1

Warmup: Boolean inputs

32

X

T ∈(

S k)

χT = = δ(x|S) X

|S|=d

ˆ φ(S) ✓d k ◆dd d! A1k0d−kφ

d

X

k=0

ˆ δ({1, . . . , k})

d

X

k=0

ˆ δ({1, . . . , k})

= cumulative error

h

}

kˆ δk1

bounded by

h

}

bounded by 1

Warmup: Boolean inputs

32

X

T ∈(

S k)

χT = = δ(x|S) X

|S|=d

ˆ φ(S) ✓d k ◆dd d! A1k0d−kφ

d

X

k=0

ˆ δ({1, . . . , k})

d

X

k=0

ˆ δ({1, . . . , k})

= cumulative error

h

}

kˆ δk1

bounded by

h

}

bounded by 1

Warmup: Boolean inputs

32

X

T ∈(

S k)

χT = = δ(x|S) X

|S|=d

ˆ φ(S) ✓d k ◆dd d! A1k0d−kφ ⇤

d

X

k=0

ˆ δ({1, . . . , k})

d

X

k=0

ˆ δ({1, . . . , k})

= cumulative error

Just proved:

33

Theorem.

φ: {−1, +1}n → R δ: {−1, +1}d → R

homogeneous of degree d symmetric Then:

✔

max

x∈{−1,+1}n

• X

|S|=d

ˆ φ(S)δ(x|S)

•  dd

d! kφk∞kˆ δk1

Real inputs

34

Theorem.

φ: {−1, +1}n → R

homogeneous of degree d

X = [−1.1, −0.9] ∪ [0.9, 1.1]

Real inputs

34

Theorem.

φ: {−1, +1}n → R

homogeneous of degree d

X = [−1.1, −0.9] ∪ [0.9, 1.1]

Then can be approximated on within by a polynomial of degree .

Xn 100−n O(d) φ

Real inputs

34

Theorem.

φ: {−1, +1}n → R

homogeneous of degree d

X = [−1.1, −0.9] ∪ [0.9, 1.1]

Then can be approximated on within by a polynomial of degree .

Xn 100−n O(d) φ

Real inputs

35

P(x) = X

|S|=d

ˆ φ(S)p(x|S)

Approximant:

Real inputs

35

P(x) = X

|S|=d

ˆ φ(S)p(x|S)

Approximant: degree-O(d) robust approximant for a single monomial

Real inputs

35

P(x) = X

|S|=d

ˆ φ(S)p(x|S)

Approximant: degree-O(d) robust approximant for a single monomial Idea: express error as linear combination of

φ(x), x ∈ {−1, +1}n

Real inputs

36

v ∈ Nd

Key: operator Av : R{+1,−1}n → RXn

Real inputs

36

v ∈ Nd

Key: operator Av : R{+1,−1}n → RXn

h

}

j-th coordinate

(Avf )(x) = E

z∈{−1,+1}d

z1z2 . . . zd f . . . , z1xj · 4v1(x2

j − 1)v1 + · · · + zdxj · 4vd(x2 j − 1)vd

d , . . . !

Real inputs

37

✔ linear ✔ bounded: kAvk∞→∞ = 1

symmetric

✔

Real inputs

37

✔ linear ✔ bounded: kAvk∞→∞ = 1

symmetric

✔ ✔ Avχ{1,...,d} = d!

dd · 4v1+···+vd E

σ∈Sd d

Y

j=1

xj(x2

j − 1)vσ(j)

Real inputs

38

E

σ∈Sd d

Y

j=1

xj(x2

j − 1)vσ(j)

1 4v1+···+vd · dd d! Avχ{1,...,d} =

Rewriting:

Real inputs

39

E

σ∈Sd d

Y

j=1

xj(x2

j − 1)vσ(j)

1 4v1+···+vd · dd d! Avχ{1,...,d} = X

|v|≥D

✓−1/2 v1 ◆ · · · ✓−1/2 vd ◆ X

|v|≥D

✓−1/2 v1 ◆ · · · ✓−1/2 vd ◆

Real inputs

39

E

σ∈Sd d

Y

j=1

xj(x2

j − 1)vσ(j)

1 4v1+···+vd · dd d! Avχ{1,...,d} = X

|v|≥D

✓−1/2 v1 ◆ · · · ✓−1/2 vd ◆ X

|v|≥D

✓−1/2 v1 ◆ · · · ✓−1/2 vd ◆ = δ(x1, . . . , xd),

error for a single monomial

Real inputs

40

1 4v1+···+vd · dd d! Avχ{1,...,d} X

|v|≥D

✓−1/2 v1 ◆ · · · ✓−1/2 vd ◆ δ(x1, . . . , xd) =

Real inputs

41

X

|v|≥D

✓−1/2 v1 ◆ · · · ✓−1/2 vd ◆ δ(x|S) = 1 4v1+···+vd · dd d! AvχS

Real inputs

41

X

|v|≥D

✓−1/2 v1 ◆ · · · ✓−1/2 vd ◆ δ(x|S) = 1 4v1+···+vd · dd d! AvχS X

|S|=d

ˆ φ(S) X

|S|=d

ˆ φ(S)

Real inputs

41

X

|v|≥D

✓−1/2 v1 ◆ · · · ✓−1/2 vd ◆ δ(x|S) = 1 4v1+···+vd · dd d! AvχS

cumulative error

X

|S|=d

ˆ φ(S) X

|S|=d

ˆ φ(S)

Real inputs

42

X

|v|≥D

✓−1/2 v1 ◆ · · · ✓−1/2 vd ◆ δ(x|S) = X

|S|=d

ˆ φ(S)

cumulative error

1 4v1+···+vd · dd d! Avφ

h

}

bounded by 1

Real inputs

42

X

|v|≥D

✓−1/2 v1 ◆ · · · ✓−1/2 vd ◆ δ(x|S) = X

|S|=d

ˆ φ(S)

cumulative error

1 4v1+···+vd · dd d! Avφ

h

}

bounded by

100−d

h

}

bounded by 1

⇤

Real inputs

42

X

|v|≥D

✓−1/2 v1 ◆ · · · ✓−1/2 vd ◆ δ(x|S) = X

|S|=d

ˆ φ(S)

cumulative error

1 4v1+···+vd · dd d! Avφ

Theorem.

φ: {−1, +1}n → R

homogeneous of degree d

X = [−1.1, −0.9] ∪ [0.9, 1.1]

Then can be approximated on within by a polynomial of degree .

Xn 100−n O(d) φ

Just proved:

43

✔

Our solution

44

• 3. Robust approximation of arbitrary p
• 1. Robust approximation of a monomial

p(x) =

n

Y

i=1

xi

• 2. Robust approximation of homogeneous p

p(x) = X

|S|=d

aS Y

i∈S

xi

✔ ✔ ✔

Open problems

45

• Prove that g

deg(f (g, g, . . . , g)) = Θ(g deg(f ) g deg(g)) g deg(f ) f

• Prove that is polynomially related to

sensitivity of

• (Fortnow 1996) Prove a polynomial relationship

between the degree of f as a polynomial and a rational function.

Questions?