[PPT] - Hardness vs Randomness for Bounded Depth Arithmetic Circuits PowerPoint Presentation

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Hardness vs Randomness for Bounded Depth Arithmetic Circuits

Chi-Ning Chou Mrinal Kumar Noam Solomon Harvard University

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Outline

Arithmetic circuits and algebraic complexity classes
Polynomial identity testing (PIT)
Hardness vs Randomness for arithmetic circuits
Polynomial factorization
Open problems

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SLIDE 3

Outline

Arithmetic circuits and algebraic complexity classes
Polynomial identity testing (PIT)
Hardness vs Randomness for arithmetic circuits
Polynomial factorization
Open problems

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Arithmetic circuits

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× × × + + + + +

x1 x2 x3 x4

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Arithmetic circuits

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× × × + + + + + Multivariate polynomial P ∈ F[x1, x2, . . . , xn]

x1 x2 x3 x4

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Arithmetic circuits

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× × × + + + + +

Σ Σ

Y

Multivariate polynomial P ∈ F[x1, x2, . . . , xn]

x1 x2 x3 x4

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Arithmetic circuits

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× × × + + + + +

Σ Σ

Y

Depth - 3 Multivariate polynomial P ∈ F[x1, x2, . . . , xn]

x1 x2 x3 x4

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Arithmetic circuits

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× × × + + + + +

Σ Σ

Y

Depth - 3 Size - 8 Multivariate polynomial P ∈ F[x1, x2, . . . , xn]

x1 x2 x3 x4

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Arithmetic circuits

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× × × + + + + +

Σ Σ

Y

Depth - 3 Size - 8

*Assume F = Q

Multivariate polynomial P ∈ F[x1, x2, . . . , xn]

x1 x2 x3 x4

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Algebraic complexity classes

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Algebraic complexity classes

C = n {f1, f2, . . . }

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Algebraic complexity classes

C = n {f1, f2, . . . }

For simplicity, denote .

f = fn

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Algebraic complexity classes

: Polynomials computed by poly(n) size, poly(n)

degree arithmetic circuits (e.g Determinant).

C = n {f1, f2, . . . }

VP

For simplicity, denote . f = fn

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Algebraic complexity classes

: Polynomials computed by poly(n) size, poly(n)

degree arithmetic circuits (e.g Determinant).

: Polynomials computed by poly(n) size,

poly(n) degree, and depth- arithmetic circuits.

C = n {f1, f2, . . . }

Depth-∆

VP

For simplicity, denote . f = fn

∆

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Algebraic complexity classes

: Polynomials computed by poly(n) size, poly(n)

degree arithmetic circuits (e.g Determinant).

: Polynomials computed by poly(n) size,

poly(n) degree, and depth- arithmetic circuits.

Many more such as VF, VBP, VNP…

C = n {f1, f2, . . . }

Depth-∆

VP

For simplicity, denote . f = fn

∆

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Hardness - Lower bounds

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Hardness - Lower bounds

Goal: Find an explicit such that .

{fn} {fn} 62 C

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Hardness - Lower bounds

Goal: Find an explicit such that .

[Strassen 73, Baur & Strassen 83] An n log n lower

bound for general arithmetic circuits.

{fn} {fn} 62 C

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Hardness - Lower bounds

Goal: Find an explicit such that .

[Strassen 73, Baur & Strassen 83] An n log n lower

bound for general arithmetic circuits.

[Kalorkoti 87] A quadratic lower bound for arithmetic

formula.

{fn} {fn} 62 C

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Hardness - Lower bounds

Goal: Find an explicit such that .

[Strassen 73, Baur & Strassen 83] An n log n lower

bound for general arithmetic circuits.

[Kalorkoti 87] A quadratic lower bound for arithmetic

formula.

[Kumar 17] A quadratic lower bound for homogeneous

algebraic branching programs.

{fn} {fn} 62 C

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Hardness - Lower bounds

Goal: Find an explicit such that .

[Strassen 73, Baur & Strassen 83] An n log n lower

bound for general arithmetic circuits.

[Kalorkoti 87] A quadratic lower bound for arithmetic

formula.

[Kumar 17] A quadratic lower bound for homogeneous

algebraic branching programs.

[NW’95, GKKS’14, FLMS’14, KS’14] Exponential lower

bounds for depth-3 and depth-4 circuits.

{fn} {fn} 62 C

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Outline

Arithmetic circuits and algebraic complexity classes
Polynomial identity testing (PIT)
Hardness vs Randomness for arithmetic circuits
Polynomial factorization
Open problems

7

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Randomness - Polynomial identity testing (PIT)

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Randomness - Polynomial identity testing (PIT)

Goal: Given , determine whether . f ∈ C f ≡ 0

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Randomness - Polynomial identity testing (PIT)

Goal: Given , determine whether . f ∈ C f ≡ 0

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Randomness - Polynomial identity testing (PIT)

Goal: Given , determine whether .

Easy when using randomness: Schwartz-Zippel.

f ∈ C f ≡ 0

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Randomness - Polynomial identity testing (PIT)

Goal: Given , determine whether .

Easy when using randomness: Schwartz-Zippel.
No non-trivial deterministic PIT for and .

f ∈ C f ≡ 0 Depth-∆ VP

sub-exponential time

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Randomness - Polynomial identity testing (PIT)

Goal: Given , determine whether .

Easy when using randomness: Schwartz-Zippel.
No non-trivial deterministic PIT for and .

f ∈ C f ≡ 0

PIT = Hitting Set

Depth-∆ VP

sub-exponential time

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Randomness - Polynomial identity testing (PIT)

Goal: Given , determine whether .

Easy when using randomness: Schwartz-Zippel.
No non-trivial deterministic PIT for and .

f ∈ C f ≡ 0

PIT = Hitting Set

is a hitting set for if for any non-zero P C f ∈ C Depth-∆ VP 9a 2 P, f(a) 6= 0.

sub-exponential time

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Randomness - Polynomial identity testing (PIT)

PIT = Hitting Set

is a hitting set for if for any non-zero P C f ∈ C 9a 2 P, f(a) 6= 0. Goal: Explicitly construct a hitting set for . C P

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Randomness - Polynomial identity testing (PIT)

PIT = Hitting Set

is a hitting set for if for any non-zero P C f ∈ C 9a 2 P, f(a) 6= 0. Goal: Explicitly construct a hitting set for .

Running time is .

C P poly(n, |P|)

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Outline

Arithmetic circuits and algebraic complexity classes
Polynomial identity testing (PIT)
Hardness vs Randomness for arithmetic circuits
Polynomial factorization
Open problems

9

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Hardness vs Randomness

Randomness Hardness

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Hardness vs Randomness

Lower Bound PIT

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Hardness vs Randomness

Lower Bound PIT

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Hardness vs Randomness

[KI’04]: Permanent not in => PIT for

Lower Bound PIT VP VP

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Hardness vs Randomness

[KI’04]: Permanent not in => PIT for
[DSY’09]: for => PIT for

Lower Bound PIT ω(poly(n)) Depth-∆ Depth-∆ − 5 VP VP

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Hardness vs Randomness

[KI’04]: Permanent not in => PIT for
[DSY’09]: for => PIT for

Lower Bound PIT ω(poly(n))

with bounded individual degree

Depth-∆ Depth-∆ − 5 VP VP

multilinear

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Our result

Theorem: For any , ∆ ≥ 6

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Our result

Theorem: For any , ∆ ≥ 6

multilinear and with degree O(log2 n/ log2 log n)

ω(poly(n))lower bound for Depth-∆

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Our result

Theorem: For any , ∆ ≥ 6

multilinear and with degree O(log2 n/ log2 log n)

ω(poly(n))lower bound for Depth-∆ Sub-exponential time PIT for Depth-∆ − 5

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Our result

Theorem: For any , ∆ ≥ 6

multilinear and with degree O(log2 n/ log2 log n)

ω(poly(n))lower bound for Depth-∆ Sub-exponential time PIT for Depth-∆ − 5

∆ − 2 k − 2

O ( l

gk

n / l

gk

l

g

n )

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Our result

Theorem: For any , ∆ ≥ 6

multilinear and with degree O(log2 n/ log2 log n)

ω(poly(n))lower bound for Depth-∆ Sub-exponential time PIT for Depth-∆ − 5

∆ − 2 k − 2

O ( l

gk

n / l

gk

l

g

n )

Don’t be bothered by the constant in depth!

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Compare with [Dvir-Shpilka-Yehudayoff’09]

[DSY’09] This work

Lower bound for With degree PIT for With bounded individual degree Depth-∆ Depth-∆ − 5 O(log2 n/ log2 log n)

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Hardness vs Randomness framework [KI’04, DSY’09]

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Hardness vs Randomness framework [KI’04, DSY’09]

Nisan-Wigderson generator Reduce #variables from n → `

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Hardness vs Randomness framework [KI’04, DSY’09]

Nisan-Wigderson generator Schwartz-Zippel lemma Reduce #variables from n → ` Brute-force to find hitting set in time dO(`)

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Hardness vs Randomness framework [KI’04, DSY’09]

Nisan-Wigderson generator Schwartz-Zippel lemma Reduce #variables from n → ` Brute-force to find hitting set in time dO(`)

Reduce to factoring problem!

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NW generator - reducing #variables

q ∈ C

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NW generator - reducing #variables

q ∈ C n

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NW generator - reducing #variables

q ∈ C n

P ⊆ Fn Goal: Hitting set

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NW generator - reducing #variables

q ∈ C n y `

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NW generator - reducing #variables

q ∈ C n y S1 Sn S2 `

Nisan- Wigderson Design

m

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NW generator - reducing #variables

q ∈ C n y S1 y|S1 y|Sn Sn y|S2 S2 `

Nisan- Wigderson Design

m

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NW generator - reducing #variables

q ∈ C n y f S1 y|S1 y|Sn Sn f f y|S2 S2 ` m

Nisan- Wigderson Design

… m

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NW generator - reducing #variables

q ∈ C n y `

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NW generator - reducing #variables

q ∈ C n y ` Q(y) = q

f(y|S1), f(y|S2), . . . , f(y|Sn)

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NW generator - reducing #variables

q ∈ C n y ` Q(y) = q

f(y|S1), f(y|S2), . . . , f(y|Sn)
Want: If then .

q 6⌘ 0 Q 6⌘ 0

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Key lemma

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Key lemma

q ∈ C y Q(y) = q

f(y|S1), f(y|S2), . . . , f(y|Sn)
Goal: If , then .

q 6⌘ 0 Q 6⌘ 0

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Key lemma

Lemma: Let non-zero and a m-variate multilinear polynomial of degree . If q ∈ Depth-∆ f Q(y) = q

f(y|S1), f(y|S2), . . . , f(y|Sn)
≡ 0

d

q ∈ C y Q(y) = q

f(y|S1), f(y|S2), . . . , f(y|Sn)

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Key lemma

Lemma: Let non-zero and a m-variate multilinear polynomial of degree . If Then, f can be computed by a size and depth circuit. q ∈ Depth-∆ f Q(y) = q

f(y|S1), f(y|S2), . . . , f(y|Sn)
≡ 0

∆ + 5 poly(n, d

√ d)

d

q ∈ C y Q(y) = q

f(y|S1), f(y|S2), . . . , f(y|Sn)

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Key lemma

Lemma: Let non-zero and a m-variate multilinear polynomial of degree . If Then, f can be computed by a size and depth circuit. q ∈ Depth-∆ f Q(y) = q

f(y|S1), f(y|S2), . . . , f(y|Sn)
≡ 0

∆ + 5 poly(n, d

√ d)

d f / ∈ Depth-∆ + 5 Q(y) 6⌘ 0, 8q 2 Depth-∆

q ∈ C y Q(y) = q

f(y|S1), f(y|S2), . . . , f(y|Sn)

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Key lemma

Lemma: Let non-zero and a m-variate multilinear polynomial of degree . If Then, f can be computed by a size and depth circuit. q ∈ Depth-∆ f Q(y) = q

f(y|S1), f(y|S2), . . . , f(y|Sn)
≡ 0

∆ + 5 poly(n, d

√ d)

d f / ∈ Depth-∆ + 5 Q(y) 6⌘ 0, 8q 2 Depth-∆

q ∈ C y Q(y) = q

f(y|S1), f(y|S2), . . . , f(y|Sn)
S

c h w a r t z

Z

i p p e l

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Proof sketch of the key lemma

∃q ∈ Depth-∆, Q(y) ≡ 0 f ∈ Depth-∆ + 5

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Proof sketch of the key lemma

If Q(y) = q

f(y|S1), f(y|S2), . . . , f(y|Sn)
≡ 0

∃q ∈ Depth-∆, Q(y) ≡ 0 f ∈ Depth-∆ + 5

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Proof sketch of the key lemma

If Q(y) = q

f(y|S1), f(y|S2), . . . , f(y|Sn)
≡ 0

q ∈ C x1 x2 xn

6⌘ 0

∃q ∈ Depth-∆, Q(y) ≡ 0 f ∈ Depth-∆ + 5

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Proof sketch of the key lemma

If Q(y) = q

f(y|S1), f(y|S2), . . . , f(y|Sn)
≡ 0

q ∈ C x1 x2 xn y q ∈ C

6⌘ 0 ≡ 0

∃q ∈ Depth-∆, Q(y) ≡ 0 f ∈ Depth-∆ + 5

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Proof sketch of the key lemma

∃q ∈ Depth-∆, Q(y) ≡ 0 f ∈ Depth-∆ + 5

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By hybrid argument, there exists f y|S1 f y|S2 xn q ∈ C

Proof sketch of the key lemma

xi

∃q ∈ Depth-∆, Q(y) ≡ 0 f ∈ Depth-∆ + 5

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By hybrid argument, there exists f y|S1 f y|S2 xn q ∈ C

Proof sketch of the key lemma

xi

z = {x1, . . . , xi−1, xi+1, . . . , xn, y}

˜ Q(z, xi)

∃q ∈ Depth-∆, Q(y) ≡ 0 f ∈ Depth-∆ + 5

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By hybrid argument, there exists f y|S1 f y|S2 xn q ∈ C

Proof sketch of the key lemma

xi

z = {x1, . . . , xi−1, xi+1, . . . , xn, y}

˜

Q(z, xi) ˜ Q(z, xi) 6⌘ 0

∃q ∈ Depth-∆, Q(y) ≡ 0 f ∈ Depth-∆ + 5

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By hybrid argument, there exists f y|S1 f y|S2 xn q ∈ C

Proof sketch of the key lemma

z = {x1, . . . , xi−1, xi+1, . . . , xn, y}

f

y|Si ˜ Q(z, xi) ˜ Q(z, xi) 6⌘ 0 ˜ Q(z, f(z)) ≡ 0

∃q ∈ Depth-∆, Q(y) ≡ 0 f ∈ Depth-∆ + 5

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By hybrid argument, there exists xn q ∈ C

Proof sketch of the key lemma

f

y|Si ˜ Q(z, xi) ˜ Q(z, xi) 6⌘ 0 ˜ Q(z, f(z)) ≡ 0

∃q ∈ Depth-∆, Q(y) ≡ 0 f ∈ Depth-∆ + 5

Fixed Fixed

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By hybrid argument, there exists xn q ∈ C

Proof sketch of the key lemma

f

y|Si ˜ Q(z, xi) ˜ Q(z, xi) 6⌘ 0 ˜ Q(z, f(z)) ≡ 0

∃q ∈ Depth-∆, Q(y) ≡ 0 f ∈ Depth-∆ + 5

Fixed Fixed

* ˜

Q(z, xi) ∈ Depth-∆ + 1

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Proof sketch of the key lemma

˜

Q(z, xi) 6⌘ 0 ˜ Q(z, f(z)) ≡ 0 ˜ Q(z, xi) ∈ Depth-∆ + 1

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Proof sketch of the key lemma

˜

Q(z, xi) 6⌘ 0 ˜ Q(z, f(z)) ≡ 0

xi − f(z) ˜ Q(z, xi) divides

˜ Q(z, xi) ∈ Depth-∆ + 1

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Proof sketch of the key lemma

˜

Q(z, xi) 6⌘ 0 ˜ Q(z, f(z)) ≡ 0

xi − f(z) ˜ Q(z, xi) divides Reducing to polynomial factorization!

˜ Q(z, xi) ∈ Depth-∆ + 1

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Outline

Arithmetic circuits and algebraic complexity classes
Polynomial identity testing (PIT)
Hardness vs Randomness for arithmetic circuits
Polynomial factorization
Open problems

19

SLIDE 80

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Polynomial factorization (Simplified setting)

Goal: For any such that . Show that . P(z, y) ∈ C P(z, f(z)) = 0 f ∈ C0

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Polynomial factorization (Simplified setting)

Goal: For any such that . Show that . P(z, y) ∈ C P(z, f(z)) = 0 f ∈ C0 [Kal89]

C C0 VP VP

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Polynomial factorization (Simplified setting)

Goal: For any such that . Show that . P(z, y) ∈ C P(z, f(z)) = 0 f ∈ C0 [Kal89] [DSY09]

with bounded individual degree

C C0 VP VP

Depth-∆ Depth-∆ + 3

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Polynomial factorization (Simplified setting)

Goal: For any such that . Show that . P(z, y) ∈ C P(z, f(z)) = 0 f ∈ C0 [Kal89] [DSY09]

with bounded individual degree

[DSS18]

C C0 VP VP

Depth-∆ Depth-∆ + 3

(resp. VBP(nlog n), VNP(nlog n))

VF(nlog n))

(resp. VBP(nlog n), VNP(nlog n))

VF(nlog n))

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Polynomial factorization (Simplified setting)

Goal: For any such that . Show that . P(z, y) ∈ C P(z, f(z)) = 0 f ∈ C0 [Kal89] [DSY09]

with bounded individual degree

[DSS18] Our result

with degree

C C0 VP VP

Depth-∆ Depth-∆ + 3

(resp. VBP(nlog n), VNP(nlog n))

VF(nlog n))

(resp. VBP(nlog n), VNP(nlog n))

VF(nlog n)) Depth-∆ Depth-∆ + 3

O(log2 n/ log2 log n)

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Polynomial factorization (Simplified setting)

Goal: For any such that . Show that . P(z, y) ∈ C P(z, f(z)) = 0 f ∈ C0 [Kal89] [DSY09]

with bounded individual degree

[DSS18] Our result

with degree

C C0 VP VP

Depth-∆ Depth-∆ + 3

(resp. VBP(nlog n), VNP(nlog n))

VF(nlog n))

(resp. VBP(nlog n), VNP(nlog n))

VF(nlog n))

non-deterministic (existential)

Depth-∆ Depth-∆ + 3

O(log2 n/ log2 log n)

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Factorization for bounded depth circuits

Goal: For any s.t. . Show that . P(z, y) ∈ Depth-∆ P(z, f(z)) = 0 f ∈ Depth-∆ + O(1)

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Factorization for bounded depth circuits

Goal: For any s.t. . Show that . P(z, y) ∈ Depth-∆ P(z, f(z)) = 0 f ∈ Depth-∆ + O(1) Newton iteration

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Factorization for bounded depth circuits

Goal: For any s.t. . Show that . P(z, y) ∈ Depth-∆ P(z, f(z)) = 0 f ∈ Depth-∆ + O(1) Newton iteration Structure lemma

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Factorization for bounded depth circuits

Goal: For any s.t. . Show that . P(z, y) ∈ Depth-∆ P(z, f(z)) = 0 f ∈ Depth-∆ + O(1) Newton iteration Depth reduction Structure lemma

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Newton iteration (Sloppy Hensel Lifting)

Goal: H≤i[hi] = H≤i[f].

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Newton iteration (Sloppy Hensel Lifting)

Goal: H≤i[hi] = H≤i[f].

Def: (Homogeneous components)

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Newton iteration (Sloppy Hensel Lifting)

Goal: H≤i[hi] = H≤i[f].

Def: (Homogeneous components) The degree i homogeneous component is the collection

f monomials of degree i.

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Newton iteration (Sloppy Hensel Lifting)

Goal: H≤i[hi] = H≤i[f].

Def: (Homogeneous components) The degree i homogeneous component is the collection

f monomials of degree i.

Example: f(x1, x2, x3) = x3

1x2 + x1x2x3 + x2 2 + x1x3 + x4 3

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Newton iteration (Sloppy Hensel Lifting)

Goal: H≤i[hi] = H≤i[f].

Def: (Homogeneous components) The degree i homogeneous component is the collection

f monomials of degree i.

Example:

f(x1, x2, x3) = x3

1x2 + x1x2x3 + x2 2 + x1x3 + x4 3

H0[f] = 0

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Newton iteration (Sloppy Hensel Lifting)

Goal: H≤i[hi] = H≤i[f].

Def: (Homogeneous components) The degree i homogeneous component is the collection

f monomials of degree i.

Example:

f(x1, x2, x3) = x3

1x2 + x1x2x3 + x2 2 + x1x3 + x4 3

H0[f] = 0 H1[f] = 0

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Newton iteration (Sloppy Hensel Lifting)

Goal: H≤i[hi] = H≤i[f].

Def: (Homogeneous components) The degree i homogeneous component is the collection

f monomials of degree i.

Example:

f(x1, x2, x3) = x3

1x2 + x1x2x3 + x2 2 + x1x3 + x4 3

H2[f] = x2

2 + x1x3

H0[f] = 0 H1[f] = 0

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22

Newton iteration (Sloppy Hensel Lifting)

Goal: H≤i[hi] = H≤i[f].

Def: (Homogeneous components) The degree i homogeneous component is the collection

f monomials of degree i.

Example:

f(x1, x2, x3) = x3

1x2 + x1x2x3 + x2 2 + x1x3 + x4 3

H2[f] = x2

2 + x1x3

H0[f] = 0 H1[f] = 0 H3[f] = x1x2x3

SLIDE 98

22

Newton iteration (Sloppy Hensel Lifting)

Goal: H≤i[hi] = H≤i[f].

Def: (Homogeneous components) The degree i homogeneous component is the collection

f monomials of degree i.

Example:

f(x1, x2, x3) = x3

1x2 + x1x2x3 + x2 2 + x1x3 + x4 3

H2[f] = x2

2 + x1x3

H0[f] = 0 H1[f] = 0 H3[f] = x1x2x3 H4[f] = x3

1x2 + x4 3

SLIDE 99

22

Newton iteration (Sloppy Hensel Lifting)

Goal: H≤i[hi] = H≤i[f].

SLIDE 100

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Newton iteration (Sloppy Hensel Lifting)

Goal: Update: H≤i[hi] = H≤i[f]. hi = hi−1 − Hi[P(z, hi−1(z))] δ .

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Newton iteration (Sloppy Hensel Lifting)

Goal: Update: H≤i[hi] = H≤i[f]. hi = hi−1 − Hi[P(z, hi−1(z))] δ .

* Homogenization & partial derivative preserve depth

SLIDE 102

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Newton iteration (Sloppy Hensel Lifting)

Goal: Update: Intuition: Taylor’s expansion. H≤i[hi] = H≤i[f]. hi = hi−1 − Hi[P(z, hi−1(z))] δ .

* Homogenization & partial derivative preserve depth

SLIDE 103

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Newton iteration (Sloppy Hensel Lifting)

Goal: Update: Intuition: Taylor’s expansion. H≤i[hi] = H≤i[f]. hi = hi−1 − Hi[P(z, hi−1(z))] δ . Q: How to efficiently update?

* Homogenization & partial derivative preserve depth

SLIDE 104

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Structure lemma

SLIDE 105

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Structure lemma

Goal: For any s.t. . Show that . P(z, y) ∈ Depth-∆ P(z, f(z)) = 0 f ∈ Depth-∆ + O(1)

SLIDE 106

23

Structure lemma

Goal: For any s.t. . Show that .

P as an univariate polynomial:

P(z, y) ∈ Depth-∆ P(z, f(z)) = 0 f ∈ Depth-∆ + O(1) P(z, y) =

k

X

i=0

Ci(z)yi.

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Structure lemma

Goal: For any s.t. . Show that .

P as an univariate polynomial:

Lemma [DSY’09]: For each , there exists polynomial such that P(z, y) ∈ Depth-∆ P(z, f(z)) = 0 f ∈ Depth-∆ + O(1) P(z, y) =

k

X

i=0

Ci(z)yi. i = 1, 2, . . . , d = deg(f) Ai H≤i[f] = H≤i[Ai(C0, C1, . . . , Ck)].

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23

Structure lemma

Goal: For any s.t. . Show that .

P as an univariate polynomial:

Lemma [DSY’09]: For each , there exists polynomial such that P(z, y) ∈ Depth-∆ P(z, f(z)) = 0 f ∈ Depth-∆ + O(1) P(z, y) =

k

X

i=0

Ci(z)yi. i = 1, 2, . . . , d = deg(f) Ai H≤i[f] = H≤i[Ai(C0, C1, . . . , Ck)].

Individual degree

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24

Structure lemma

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24

Structure lemma

Lemma (This work): For each , there exists polynomial such that Ai i = 1, 2, . . . , d = deg(f) H≤i[f] = H≤i[Ai(g0, g1, . . . , gd)]

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24

Structure lemma

Lemma (This work): For each , there exists polynomial such that where Ai i = 1, 2, . . . , d = deg(f) H≤i[f] = H≤i[Ai(g0, g1, . . . , gd)] gi = H≤d  ∂i ∂yi P(z, H[f])

− H0

 ∂i ∂yi P(z, H[f])

.

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24

Structure lemma

Lemma (This work): For each , there exists polynomial such that where

size degree at most O(d6) d

Ai i = 1, 2, . . . , d = deg(f) H≤i[f] = H≤i[Ai(g0, g1, . . . , gd)] gi = H≤d  ∂i ∂yi P(z, H[f])

− H0

 ∂i ∂yi P(z, H[f])

.

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24

Structure lemma

Lemma (This work): For each , there exists polynomial such that where

size degree at most O(d6) d

Ai i = 1, 2, . . . , d = deg(f) H≤i[f] = H≤i[Ai(g0, g1, . . . , gd)] gi = H≤d  ∂i ∂yi P(z, H[f])

− H0

 ∂i ∂yi P(z, H[f])

.

Depth with top layer Σ ∆ + 1

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24

Structure lemma

Lemma (This work): For each , there exists polynomial such that where

size degree at most O(d6) d

Ai i = 1, 2, . . . , d = deg(f) H≤i[f] = H≤i[Ai(g0, g1, . . . , gd)] gi = H≤d  ∂i ∂yi P(z, H[f])

− H0

 ∂i ∂yi P(z, H[f])

.

Depth with top layer Σ ∆ + 1 * Homogenization & partial derivative preserve depth

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25

Structure lemma

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25

Structure lemma

P(z, y) ∈ Depth-∆ P(z, f(z)) = 0

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25

Structure lemma

f = hd z P(z, y) ∈ Depth-∆ P(z, f(z)) = 0

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25

Structure lemma

f = hd g0 g1 gd … z P(z, y) ∈ Depth-∆ P(z, f(z)) = 0

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25

Structure lemma

f = hd A0 A1 Ad g0 g1 gd … … z P(z, y) ∈ Depth-∆ P(z, f(z)) = 0

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25

Structure lemma

f = hd + A0 A1 Ad g0 g1 gd … … z P(z, y) ∈ Depth-∆ P(z, f(z)) = 0

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25

Structure lemma

f = hd + A0 A1 Ad g0 g1 gd … … z

Depth ∆ + 1

P(z, y) ∈ Depth-∆ P(z, f(z)) = 0

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25

Structure lemma

f = hd + A0 A1 Ad g0 g1 gd … … z

Depth ∆ + 1 Depth?

P(z, y) ∈ Depth-∆ P(z, f(z)) = 0

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26

Depth reduction [Gupta-Kamath-Kayal-Saptharishi’13]

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26

Depth reduction [Gupta-Kamath-Kayal-Saptharishi’13]

x1 x2 xn

size
degree d

s

x1 x2 xn

size
depth 3, i.e.,

(snd)O(

√ d)

ΣΠΣ

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26

x1 x2 xn

size
degree d

s

x1 x2 xn

size
depth

(snd)O(d)1/k

Depth reduction [Agrawal-Vinay’08, Koiran’12, Tavenas’13]

2k

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27

Factorization for bounded depth circuits (Wrap up)

Goal: For any s.t. . Show that . P(z, y) ∈ Depth-∆ P(z, f(z)) = 0 f ∈ Depth-∆ + O(1) Newton iteration Depth reduction Structure lemma

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27

Factorization for bounded depth circuits (Wrap up)

Goal: For any s.t. . Show that . P(z, y) ∈ Depth-∆ P(z, f(z)) = 0 f ∈ Depth-∆ + O(1) Newton iteration Depth reduction Structure lemma H≤i[hi] = H≤i[f].

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27

Factorization for bounded depth circuits (Wrap up)

Goal: For any s.t. . Show that . P(z, y) ∈ Depth-∆ P(z, f(z)) = 0 f ∈ Depth-∆ + O(1) Newton iteration Depth reduction Structure lemma H≤i[hi] = H≤i[f]. hi − hi−1 = Ai(g0, g1, . . . , gd)

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27

Factorization for bounded depth circuits (Wrap up)

Goal: For any s.t. . Show that . P(z, y) ∈ Depth-∆ P(z, f(z)) = 0 f ∈ Depth-∆ + O(1) Newton iteration Depth reduction Structure lemma H≤i[hi] = H≤i[f]. hi − hi−1 = Ai(g0, g1, . . . , gd)

Depth with top layer Σ ∆ + 1 size degree at most O(d6) d

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27

Factorization for bounded depth circuits (Wrap up)

Goal: For any s.t. . Show that . P(z, y) ∈ Depth-∆ P(z, f(z)) = 0 f ∈ Depth-∆ + O(1) Newton iteration Depth reduction Structure lemma H≤i[hi] = H≤i[f]. hi − hi−1 = Ai(g0, g1, . . . , gd)

Depth with top layer Σ ∆ + 1 size degree at most O(d6) d

hi − hi−1 = ˜ Ai(g0, g1, . . . , gd)

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27

Factorization for bounded depth circuits (Wrap up)

Goal: For any s.t. . Show that . P(z, y) ∈ Depth-∆ P(z, f(z)) = 0 f ∈ Depth-∆ + O(1) Newton iteration Depth reduction Structure lemma H≤i[hi] = H≤i[f]. hi − hi−1 = Ai(g0, g1, . . . , gd)

Depth with top layer Σ ∆ + 1 size degree at most O(d6) d

hi − hi−1 = ˜ Ai(g0, g1, . . . , gd)

Depth size with top layer Σ ∆ + 3 poly(n, d

√ d)

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28

Conclusion

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28

Conclusion

Theorem: For any s.t. . If , then . P(z, y) ∈ Depth-∆ P(z, f(z)) = 0 d

√ d = poly(n)

f ∈ Depth-∆ + 3

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28

Conclusion

Theorem: For any s.t. . If , then . Theorem: For any . If there’s a lower bound for with degree , then there’s a sub-exponential time PIT for . P(z, y) ∈ Depth-∆ P(z, f(z)) = 0 d

√ d = poly(n)

f ∈ Depth-∆ + 3 ∆ ≥ 6 ω(poly(n)) Depth-∆ Depth-∆ − 5 O(log2 n/ log2 log n)

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28

Conclusion

Theorem: For any s.t. . If , then . Theorem: For any . If there’s a lower bound for with degree , then there’s a sub-exponential time PIT for . P(z, y) ∈ Depth-∆ P(z, f(z)) = 0 d

√ d = poly(n)

f ∈ Depth-∆ + 3 ∆ ≥ 6 ω(poly(n)) Depth-∆ Depth-∆ − 5 O(log2 n/ log2 log n)

[DSY’09] This work Lower bound for With degree PIT for With bounded individual degree

Depth-∆ Depth-∆ − 5 O(log2 n/ log2 log n)

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Outline

Arithmetic circuits and algebraic complexity classes
Polynomial identity testing (PIT)
Hardness vs Randomness for arithmetic circuits
Polynomial factorization
Open problems

29

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30

Open problems

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30

Open problems

Hardness vs Randomness

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30

Open problems

Hardness vs Randomness

✦ Remove the degree condition(s)?

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30

Open problems

Hardness vs Randomness

✦ Remove the degree condition(s)? ✦ More circuit classes?

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30

Open problems

Hardness vs Randomness

✦ Remove the degree condition(s)? ✦ More circuit classes?

Polynomial factorization

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30

Open problems

Hardness vs Randomness

✦ Remove the degree condition(s)? ✦ More circuit classes?

Polynomial factorization

✦ Remove the degree condition(s)?

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30

Open problems

Hardness vs Randomness

✦ Remove the degree condition(s)? ✦ More circuit classes?

Polynomial factorization

✦ Remove the degree condition(s)? ✦ Sparse polynomials?

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30

Open problems

Hardness vs Randomness

✦ Remove the degree condition(s)? ✦ More circuit classes?

Polynomial factorization

✦ Remove the degree condition(s)? ✦ Sparse polynomials? ✦ Closure results for VF, VBP?

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30

Open problems

Hardness vs Randomness

✦ Remove the degree condition(s)? ✦ More circuit classes?

Polynomial factorization

✦ Remove the degree condition(s)? ✦ Sparse polynomials? ✦ Closure results for VF, VBP?