Preserving Randomness for Adaptive Algorithms Adam R. Klivans - - PowerPoint PPT Presentation

Preserving Randomness for Adaptive Algorithms

William M. Hoza Adam R. Klivans May 25, 2017 Caltech Theory of Computing Seminar

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Randomized estimation algorithms

◮ Algorithm Est(C) estimates some value µ(C) ∈ Rd

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Randomized estimation algorithms

◮ Algorithm Est(C) estimates some value µ(C) ∈ Rd

Pr[Est(C) − µ(C)∞ > ε] ≤ δ

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Randomized estimation algorithms

◮ Algorithm Est(C) estimates some value µ(C) ∈ Rd

Pr[Est(C) − µ(C)∞ > ε] ≤ δ

◮ Canonical example:

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Randomized estimation algorithms

◮ Algorithm Est(C) estimates some value µ(C) ∈ Rd

Pr[Est(C) − µ(C)∞ > ε] ≤ δ

◮ Canonical example:

◮ C is a Boolean circuit 2 / 20

Randomized estimation algorithms

◮ Algorithm Est(C) estimates some value µ(C) ∈ Rd

Pr[Est(C) − µ(C)∞ > ε] ≤ δ

◮ Canonical example:

◮ C is a Boolean circuit ◮ µ(C)

def

= Prx[C(x) = 1] (d = 1)

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Randomized estimation algorithms

◮ Algorithm Est(C) estimates some value µ(C) ∈ Rd

Pr[Est(C) − µ(C)∞ > ε] ≤ δ

◮ Canonical example:

◮ C is a Boolean circuit ◮ µ(C)

def

= Prx[C(x) = 1] (d = 1)

◮ Est(C) evaluates C at several randomly chosen points 2 / 20

Executing Est many times

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Executing Est many times

◮ Goal: Execute Est(C1), Est(C2), . . . , Est(Ck)

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Executing Est many times

◮ Goal: Execute Est(C1), Est(C2), . . . , Est(Ck) ◮ Say Est uses n random bits

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Executing Est many times

◮ Goal: Execute Est(C1), Est(C2), . . . , Est(Ck) ◮ Say Est uses n random bits ◮ Na¨

ıve implementation: nk random bits

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Executing Est many times

◮ Goal: Execute Est(C1), Est(C2), . . . , Est(Ck) ◮ Say Est uses n random bits ◮ Na¨

ıve implementation: nk random bits

◮ Can we do better?

3 / 20

Executing Est many times

◮ Goal: Execute Est(C1), Est(C2), . . . , Est(Ck) ◮ Say Est uses n random bits ◮ Na¨

ıve implementation: nk random bits

◮ Can we do better? ◮ Theorem: Can use just n + O(k log(d + 1)) random bits!

3 / 20

Executing Est many times

◮ Goal: Execute Est(C1), Est(C2), . . . , Est(Ck) ◮ Say Est uses n random bits ◮ Na¨

ıve implementation: nk random bits

◮ Can we do better? ◮ Theorem: Can use just n + O(k log(d + 1)) random bits!

◮ Slight increases in error, failure probability 3 / 20

Nonadaptive setting

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Nonadaptive setting

C1, . . . , Ck

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Nonadaptive setting

C1, . . . , Ck Est(C1), . . . , Est(Ck)

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Nonadaptive setting

C1, . . . , Ck Est(C1), . . . , Est(Ck)

◮ Algorithm that uses just n random bits:

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Nonadaptive setting

C1, . . . , Ck Est(C1), . . . , Est(Ck)

◮ Algorithm that uses just n random bits:

• 1. Pick X ∈ {0, 1}n uniformly at random once

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Nonadaptive setting

C1, . . . , Ck Est(C1), . . . , Est(Ck)

◮ Algorithm that uses just n random bits:

• 1. Pick X ∈ {0, 1}n uniformly at random once
• 2. Execute Est(C1, X), Est(C2, X), . . . , Est(Ck, X)

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Nonadaptive setting

C1, . . . , Ck Est(C1), . . . , Est(Ck)

◮ Algorithm that uses just n random bits:

• 1. Pick X ∈ {0, 1}n uniformly at random once
• 2. Execute Est(C1, X), Est(C2, X), . . . , Est(Ck, X)

◮ Overall failure probability is still kδ (union bound)

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Adaptive setting

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Adaptive setting

C1

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Adaptive setting

C1 Est(C1)

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Adaptive setting

C1 Est(C1) C2

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Adaptive setting

C1 Est(C1) C2 Est(C2)

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Adaptive setting

C1 Est(C1) C2 Est(C2) . . . Ck Est(Ck)

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Adaptive setting

C1 Est(C1) C2 Est(C2) . . . Ck Est(Ck)

◮ Let X be the randomness used for Est(C1)

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Adaptive setting

C1 Est(C1) C2 Est(C2) . . . Ck Est(Ck)

◮ Let X be the randomness used for Est(C1) ◮ C2 is stochastically dependent on X

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Adaptive setting

C1 Est(C1) C2 Est(C2) . . . Ck Est(Ck)

◮ Let X be the randomness used for Est(C1) ◮ C2 is stochastically dependent on X ◮ Failure probability of Est(C2, X) is ???

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Concentrated functions

◮ f : {0, 1}n → Rd

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Concentrated functions

◮ f : {0, 1}n → Rd ◮ Definition: f is (ε, δ)-concentrated at µ ∈ Rd if

Pr

X [f (X) − µ∞ > ε] ≤ δ.

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Concentrated functions

◮ f : {0, 1}n → Rd ◮ Definition: f is (ε, δ)-concentrated at µ ∈ Rd if

Pr

X [f (X) − µ∞ > ε] ≤ δ. ◮ Example: f (X) def

= Est(C, X)

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Randomness steward model

Owner Steward

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Randomness steward model

Owner Steward f1

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Randomness steward model

Owner Steward f1 Y1

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Randomness steward model

Owner Steward f1 Y1 f2

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Randomness steward model

Owner Steward f1 Y1 f2 Y2

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Randomness steward model

Owner Steward f1 Y1 f2 Y2 . . . fk Yk

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Randomness steward model

Owner Steward f1 Y1 f2 Y2 . . . fk Yk

◮ Each fi is (ε, δ)-concentrated at some µi

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Randomness steward model

Owner Steward f1 Y1 f2 Y2 . . . fk Yk

◮ Each fi is (ε, δ)-concentrated at some µi ◮ Steward requirement: For any owner,

Pr

• max

i

Yi − µi∞ > ε′

• ≤ δ′

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One-query stewards

◮ Definition: One-query steward: Only accesses each fi by

querying a single point fi(Xi)

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One-query stewards

◮ Definition: One-query steward: Only accesses each fi by

querying a single point fi(Xi)

◮ Querying fi corresponds to executing Est 8 / 20

One-query stewards

◮ Definition: One-query steward: Only accesses each fi by

querying a single point fi(Xi)

◮ Querying fi corresponds to executing Est ◮ The owner does not see Xi 8 / 20

Warm-up steward

◮ Theorem: For any n, k, ε, δ, there exists a one-query steward

for d = 1 with

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Warm-up steward

◮ Theorem: For any n, k, ε, δ, there exists a one-query steward

for d = 1 with

◮ Error ε′ ≤ O(ε)

(vs. na¨ ıve ε)

9 / 20

Warm-up steward

◮ Theorem: For any n, k, ε, δ, there exists a one-query steward

for d = 1 with

◮ Error ε′ ≤ O(ε)

(vs. na¨ ıve ε)

◮ Failure probability δ′ ≤ 2k · δ

(vs. na¨ ıve kδ)

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Warm-up steward

◮ Theorem: For any n, k, ε, δ, there exists a one-query steward

for d = 1 with

◮ Error ε′ ≤ O(ε)

(vs. na¨ ıve ε)

◮ Failure probability δ′ ≤ 2k · δ

(vs. na¨ ıve kδ)

◮ Randomness n

(vs. na¨ ıve nk)

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Warm-up steward

◮ Theorem: For any n, k, ε, δ, there exists a one-query steward

for d = 1 with

◮ Error ε′ ≤ O(ε)

(vs. na¨ ıve ε)

◮ Failure probability δ′ ≤ 2k · δ

(vs. na¨ ıve kδ)

◮ Randomness n

(vs. na¨ ıve nk)

◮ The steward: “Reuse randomness and round”

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Warm-up steward

◮ Theorem: For any n, k, ε, δ, there exists a one-query steward

for d = 1 with

◮ Error ε′ ≤ O(ε)

(vs. na¨ ıve ε)

◮ Failure probability δ′ ≤ 2k · δ

(vs. na¨ ıve kδ)

◮ Randomness n

(vs. na¨ ıve nk)

◮ The steward: “Reuse randomness and round”

◮ Pick X ∈ {0, 1}n uniformly at random once 9 / 20

Warm-up steward

◮ Theorem: For any n, k, ε, δ, there exists a one-query steward

for d = 1 with

◮ Error ε′ ≤ O(ε)

(vs. na¨ ıve ε)

◮ Failure probability δ′ ≤ 2k · δ

(vs. na¨ ıve kδ)

◮ Randomness n

(vs. na¨ ıve nk)

◮ The steward: “Reuse randomness and round”

◮ Pick X ∈ {0, 1}n uniformly at random once ◮ For i = 1 to k: Return fi(X), rounded to multiple of 2ε 9 / 20

Analysis of warm-up steward

2ε

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Analysis of warm-up steward

2ε µ

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Analysis of warm-up steward

2ε µ A(µ)

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Analysis of warm-up steward

2ε µ A(µ) B(µ)

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Analysis of warm-up steward

2ε µ A(µ) B(µ)

◮ Imagine if the steward always returns A(µ) or B(µ)...

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Analysis of warm-up steward

2ε µ A(µ) B(µ)

◮ Imagine if the steward always returns A(µ) or B(µ)...

f1

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Analysis of warm-up steward

2ε µ A(µ) B(µ)

◮ Imagine if the steward always returns A(µ) or B(µ)...

f1 f A

2

f B

2

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Analysis of warm-up steward

2ε µ A(µ) B(µ)

◮ Imagine if the steward always returns A(µ) or B(µ)...

f1 f A

2

f B

2

f AA

3

f AB

3

f BA

3

f BB

3

10 / 20

Analysis of warm-up steward

2ε µ A(µ) B(µ)

◮ Imagine if the steward always returns A(µ) or B(µ)...

f1 f A

2

f B

2

f AA

3

f AB

3

f BA

3

f BB

3

f AAA

4

f AAB

4

f ABA

4

f ABB

4

f BAA

4

f BAB

4

f BBA

4

f BBB

4

10 / 20

Analysis of warm-up steward

2ε µ A(µ) B(µ)

◮ Imagine if the steward always returns A(µ) or B(µ)...

f1 f A

2

f B

2

f AA

3

f AB

3

f BA

3

f BB

3

f AAA

4

f AAB

4

f ABA

4

f ABB

4

f BAA

4

f BAB

4

f BBA

4

f BBB

4 ◮ Union bound: Pr[X good for every function in tree] ≥ 1 − 2kδ

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Analysis of warm-up steward

2ε µ A(µ) B(µ)

◮ Imagine if the steward always returns A(µ) or B(µ)...

f1 f A

2

f B

2

f AA

3

f AB

3

f BA

3

f BB

3

f AAA

4

f AAB

4

f ABA

4

f ABB

4

f BAA

4

f BAB

4

f BBA

4

f BBB

4 ◮ Union bound: Pr[X good for every function in tree] ≥ 1 − 2kδ ◮ If so, inductively, every fi is in the tree!

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Main result

◮ Theorem: For all n, k, d, ε, δ, γ, there is an efficient one-query

steward with

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Main result

◮ Theorem: For all n, k, d, ε, δ, γ, there is an efficient one-query

steward with

◮ Error ε′ ≤ O(εd) 11 / 20

Main result

◮ Theorem: For all n, k, d, ε, δ, γ, there is an efficient one-query

steward with

◮ Error ε′ ≤ O(εd) ◮ Failure probability δ′ ≤ kδ + γ 11 / 20

Main result

◮ Theorem: For all n, k, d, ε, δ, γ, there is an efficient one-query

steward with

◮ Error ε′ ≤ O(εd) ◮ Failure probability δ′ ≤ kδ + γ ◮ # random bits n + O(k log(d + 1) + log k log(1/γ)) 11 / 20

Main steward

◮ Pick random seed X, compute (X1, . . . , Xk) = Gen(X)

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Main steward

◮ Pick random seed X, compute (X1, . . . , Xk) = Gen(X) ◮ For i = 1 to k:

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Main steward

◮ Pick random seed X, compute (X1, . . . , Xk) = Gen(X) ◮ For i = 1 to k:

◮ Obtain Wi = fi(Xi) 12 / 20

Main steward

◮ Pick random seed X, compute (X1, . . . , Xk) = Gen(X) ◮ For i = 1 to k:

◮ Obtain Wi = fi(Xi) ◮ Shift and round Wi to determine output Yi 12 / 20

Main steward

◮ Pick random seed X, compute (X1, . . . , Xk) = Gen(X) ◮ For i = 1 to k:

◮ Obtain Wi = fi(Xi) ◮ Shift and round Wi to determine output Yi

◮ Ingredient 1: Gen: PRG for block decision trees

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Main steward

◮ Pick random seed X, compute (X1, . . . , Xk) = Gen(X) ◮ For i = 1 to k:

◮ Obtain Wi = fi(Xi) ◮ Shift and round Wi to determine output Yi

◮ Ingredient 1: Gen: PRG for block decision trees ◮ Ingredient 2: Deterministic shifting and rounding algorithm

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Shifting and rounding algorithm

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(d + 1) · 2ε Wi1 Wi2 Wi3 Wi4 Wi5

Shifting and rounding algorithm

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(d + 1) · 2ε Wi1 Wi2 Wi3 Wi4 Wi5

Shifting and rounding algorithm

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(d + 1) · 2ε Wi1 Wi2 Wi3 Wi4 Wi5

Shifting and rounding algorithm

13 / 20

(d + 1) · 2ε Wi1 Wi2 Wi3 Wi4 Wi5

Shifting and rounding algorithm

13 / 20

(d + 1) · 2ε Wi1 Wi2 Wi3 Wi4 Wi5

Shifting and rounding algorithm

13 / 20

(d + 1) · 2ε Wi1 Wi2 Wi3 Wi4 Wi5

Shifting and rounding algorithm

13 / 20

(d + 1) · 2ε Wi1 Wi2 Wi3 Wi4 Wi5

Shifting and rounding algorithm

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(d + 1) · 2ε Wi1 Wi2 Wi3 Wi4 Wi5

Analysis of shifting and rounding algorithm

◮ For W ∈ Rd and ∆ ∈ [d + 1], define R∆(W ) ∈ Rd by shifting

W according to ∆, then rounding

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Analysis of shifting and rounding algorithm

◮ For W ∈ Rd and ∆ ∈ [d + 1], define R∆(W ) ∈ Rd by shifting

W according to ∆, then rounding

◮ By construction, Yi = R∆(Wi) for some ∆

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Analysis of shifting and rounding algorithm

◮ For W ∈ Rd and ∆ ∈ [d + 1], define R∆(W ) ∈ Rd by shifting

W according to ∆, then rounding

◮ By construction, Yi = R∆(Wi) for some ∆ ◮ Imagine if Yi = R∆(µi) for some ∆...

14 / 20

Analysis of shifting and rounding algorithm

◮ For W ∈ Rd and ∆ ∈ [d + 1], define R∆(W ) ∈ Rd by shifting

W according to ∆, then rounding

◮ By construction, Yi = R∆(Wi) for some ∆ ◮ Imagine if Yi = R∆(µi) for some ∆...

f1

14 / 20

Analysis of shifting and rounding algorithm

◮ For W ∈ Rd and ∆ ∈ [d + 1], define R∆(W ) ∈ Rd by shifting

W according to ∆, then rounding

◮ By construction, Yi = R∆(Wi) for some ∆ ◮ Imagine if Yi = R∆(µi) for some ∆...

f1 f R1(µ1)

2

f R2(µ1)

2

f R3(µ1)

2

⊥

14 / 20

Analysis of shifting and rounding algorithm

◮ For W ∈ Rd and ∆ ∈ [d + 1], define R∆(W ) ∈ Rd by shifting

W according to ∆, then rounding

◮ By construction, Yi = R∆(Wi) for some ∆ ◮ Imagine if Yi = R∆(µi) for some ∆...

f1 f R1(µ1)

2

f R2(µ1)

2

f R3(µ1)

2

⊥ f R2(µ1),R1(µ2)

3

f R2(µ1),R2(µ2)

3

f R2(µ1),R3(µ2)

3

⊥

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Certification tree

f1 f R1(µ1)

2

f R2(µ1)

2

f R3(µ1)

2

⊥ f R2(µ1),R1(µ2)

3

f R2(µ1),R2(µ2)

3

f R2(µ1),R3(µ2)

3

⊥

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Certification tree

f1 f R1(µ1)

2

f R2(µ1)

2

f R3(µ1)

2

⊥ f R2(µ1),R1(µ2)

3

f R2(µ1),R2(µ2)

3

f R2(µ1),R3(µ2)

3

⊥

◮ A sequence (X1, . . . , Xk) of query points determines:

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Certification tree

f1 f R1(µ1)

2

f R2(µ1)

2

f R3(µ1)

2

⊥ f R2(µ1),R1(µ2)

3

f R2(µ1),R2(µ2)

3

f R2(µ1),R3(µ2)

3

⊥

◮ A sequence (X1, . . . , Xk) of query points determines:

◮ A transcript (f1, Y1, f2, Y2, . . . , fk, Yk) 15 / 20

Certification tree

f1 f R1(µ1)

2

f R2(µ1)

2

f R3(µ1)

2

⊥ f R2(µ1),R1(µ2)

3

f R2(µ1),R2(µ2)

3

f R2(µ1),R3(µ2)

3

⊥

◮ A sequence (X1, . . . , Xk) of query points determines:

◮ A transcript (f1, Y1, f2, Y2, . . . , fk, Yk) ◮ A path P through tree 15 / 20

Certification tree

f1 f R1(µ1)

2

f R2(µ1)

2

f R3(µ1)

2

⊥ f R2(µ1),R1(µ2)

3

f R2(µ1),R2(µ2)

3

f R2(µ1),R3(µ2)

3

⊥

◮ A sequence (X1, . . . , Xk) of query points determines:

◮ A transcript (f1, Y1, f2, Y2, . . . , fk, Yk) ◮ A path P through tree

◮ If we pick X1, . . . , Xk independently and u.a.r.,

Pr

(X1,...,Xk)[P has a ⊥ node] ≤ kδ

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Certification tree

f1 f R1(µ1)

2

f R2(µ1)

2

f R3(µ1)

2

⊥ f R2(µ1),R1(µ2)

3

f R2(µ1),R2(µ2)

3

f R2(µ1),R3(µ2)

3

⊥

◮ A sequence (X1, . . . , Xk) of query points determines:

◮ A transcript (f1, Y1, f2, Y2, . . . , fk, Yk) ◮ A path P through tree

◮ If we pick X1, . . . , Xk independently and u.a.r.,

Pr

(X1,...,Xk)[P has a ⊥ node] ≤ kδ ◮ (Certification) No ⊥ nodes in P =

⇒ every Yi has error O(εd)

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Block decision trees

◮ (k, n, q) block decision tree: Full q-ary tree of height k

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Block decision trees

◮ (k, n, q) block decision tree: Full q-ary tree of height k

v va vb vc vaa vab vac vba vbb vbc vca vcb vcc

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Block decision trees

◮ (k, n, q) block decision tree: Full q-ary tree of height k ◮ Each internal node vs has a function vs : {0, 1}n → [q]

v va vb vc vaa vab vac vba vbb vbc vca vcb vcc

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Block decision trees

◮ (k, n, q) block decision tree: Full q-ary tree of height k ◮ Each internal node vs has a function vs : {0, 1}n → [q]

v va vb vc vaa vab vac vba vbb vbc vca vcb vcc 00, 11 01 10

16 / 20

Block decision trees

◮ (k, n, q) block decision tree: Full q-ary tree of height k ◮ Each internal node vs has a function vs : {0, 1}n → [q] ◮ Tree reads nk bits and outputs a leaf

v va vb vc vaa vab vac vba vbb vbc vca vcb vcc 00, 11 01 10

16 / 20

PRG for block decision trees

◮ Theorem: There is an efficient γ-PRG for block decision trees

with seed length n + O(k log q + log k log(1/γ))

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PRG for block decision trees

◮ Theorem: There is an efficient γ-PRG for block decision trees

with seed length n + O(k log q + log k log(1/γ))

◮ Proof idea: Modify parameters of INW generator

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PRG for block decision trees

◮ Theorem: There is an efficient γ-PRG for block decision trees

with seed length n + O(k log q + log k log(1/γ))

◮ Proof idea: Modify parameters of INW generator ◮ This generator fools the certification tree

17 / 20

PRG for block decision trees

◮ Theorem: There is an efficient γ-PRG for block decision trees

with seed length n + O(k log q + log k log(1/γ))

◮ Proof idea: Modify parameters of INW generator ◮ This generator fools the certification tree ◮ No need to fool steward/owner protocol!

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Application: Randomness-efficient Goldreich-Levin

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Application: Randomness-efficient Goldreich-Levin

◮ Oracle access to x ∈ {0, 1}2n

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Application: Randomness-efficient Goldreich-Levin

◮ Oracle access to x ∈ {0, 1}2n ◮ Theorem: Can find all Hadamard codewords that agree with x

in ( 1

2 + θ)-fraction of positions

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Application: Randomness-efficient Goldreich-Levin

◮ Oracle access to x ∈ {0, 1}2n ◮ Theorem: Can find all Hadamard codewords that agree with x

in ( 1

2 + θ)-fraction of positions

◮ Runtime poly(n, 1/θ, log(1/δ))

(δ = failure prob)

18 / 20

Application: Randomness-efficient Goldreich-Levin

◮ Oracle access to x ∈ {0, 1}2n ◮ Theorem: Can find all Hadamard codewords that agree with x

in ( 1

2 + θ)-fraction of positions

◮ Runtime poly(n, 1/θ, log(1/δ))

(δ = failure prob)

◮ O(n + log n log(1/δ)) random bits (independent of θ!) 18 / 20

Application: Randomness-efficient Goldreich-Levin

◮ Oracle access to x ∈ {0, 1}2n ◮ Theorem: Can find all Hadamard codewords that agree with x

in ( 1

2 + θ)-fraction of positions

◮ Runtime poly(n, 1/θ, log(1/δ))

(δ = failure prob)

◮ O(n + log n log(1/δ)) random bits (independent of θ!)

◮ Previous best: O(n log(n/θ) log(1/(δθ))) random bits

(Bshouty et al. ’04)

18 / 20

Application: Randomness-efficient Goldreich-Levin

◮ Oracle access to x ∈ {0, 1}2n ◮ Theorem: Can find all Hadamard codewords that agree with x

in ( 1

2 + θ)-fraction of positions

◮ Runtime poly(n, 1/θ, log(1/δ))

(δ = failure prob)

◮ O(n + log n log(1/δ)) random bits (independent of θ!)

◮ Previous best: O(n log(n/θ) log(1/(δθ))) random bits

(Bshouty et al. ’04)

◮ Proof ingredients:

18 / 20

Application: Randomness-efficient Goldreich-Levin

◮ Oracle access to x ∈ {0, 1}2n ◮ Theorem: Can find all Hadamard codewords that agree with x

in ( 1

2 + θ)-fraction of positions

◮ Runtime poly(n, 1/θ, log(1/δ))

(δ = failure prob)

◮ O(n + log n log(1/δ)) random bits (independent of θ!)

◮ Previous best: O(n log(n/θ) log(1/(δθ))) random bits

(Bshouty et al. ’04)

◮ Proof ingredients:

◮ Standard Goldreich-Levin algorithm 18 / 20

Application: Randomness-efficient Goldreich-Levin

◮ Oracle access to x ∈ {0, 1}2n ◮ Theorem: Can find all Hadamard codewords that agree with x

in ( 1

2 + θ)-fraction of positions

◮ Runtime poly(n, 1/θ, log(1/δ))

(δ = failure prob)

◮ O(n + log n log(1/δ)) random bits (independent of θ!)

◮ Previous best: O(n log(n/θ) log(1/(δθ))) random bits

(Bshouty et al. ’04)

◮ Proof ingredients:

◮ Standard Goldreich-Levin algorithm ◮ Our steward with d = poly(1/θ) 18 / 20

Application: Randomness-efficient Goldreich-Levin

◮ Oracle access to x ∈ {0, 1}2n ◮ Theorem: Can find all Hadamard codewords that agree with x

in ( 1

2 + θ)-fraction of positions

◮ Runtime poly(n, 1/θ, log(1/δ))

(δ = failure prob)

◮ O(n + log n log(1/δ)) random bits (independent of θ!)

◮ Previous best: O(n log(n/θ) log(1/(δθ))) random bits

(Bshouty et al. ’04)

◮ Proof ingredients:

◮ Standard Goldreich-Levin algorithm ◮ Our steward with d = poly(1/θ) ◮ Goldreich-Wigderson sampler 18 / 20

Landscape of stewards

ε′ δ′ Randomness complexity Reference

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Landscape of stewards

ε′ δ′ Randomness complexity Reference ε kδ nk Na¨ ıve

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Landscape of stewards

ε′ δ′ Randomness complexity Reference ε kδ nk Na¨ ıve O(ε) 2kδ n

(works for d = 1 only)

This work

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Landscape of stewards

ε′ δ′ Randomness complexity Reference ε kδ nk Na¨ ıve O(ε) 2kδ n

(works for d = 1 only)

This work O(εd) kδ + γ n + O(k log(d + 1) + log k log(1/γ)) This work

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Landscape of stewards

◮ Steward model captures derandomization constructions in

literature ε′ δ′ Randomness complexity Reference ε kδ nk Na¨ ıve O(ε) 2kδ n

(works for d = 1 only)

This work O(εd) kδ + γ n + O(k log(d + 1) + log k log(1/γ)) This work

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Landscape of stewards

◮ Steward model captures derandomization constructions in

literature ε′ δ′ Randomness complexity Reference ε kδ nk Na¨ ıve O(ε) 2kδ n

(works for d = 1 only)

This work O(εd) kδ + γ n + O(k log(d + 1) + log k log(1/γ)) This work O(εkd/γ) kδ + γ n + O(k log k + k log d + k log(1/γ)) ≈ SZ ’99

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Landscape of stewards

◮ Steward model captures derandomization constructions in

literature ε′ δ′ Randomness complexity Reference ε kδ nk Na¨ ıve O(ε) 2kδ n

(works for d = 1 only)

This work O(εd) kδ + γ n + O(k log(d + 1) + log k log(1/γ)) This work O(εkd/γ) kδ + γ n + O(k log k + k log d + k log(1/γ)) ≈ SZ ’99 O(ε) kδ + k/2nΩ(1) O(n6 + kd) ≈ IZ ’89

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Landscape of stewards

◮ Steward model captures derandomization constructions in

literature ε′ δ′ Randomness complexity Reference ε kδ nk Na¨ ıve O(ε) 2kδ n

(works for d = 1 only)

This work O(εd) kδ + γ n + O(k log(d + 1) + log k log(1/γ)) This work O(εkd/γ) kδ + γ n + O(k log k + k log d + k log(1/γ)) ≈ SZ ’99 O(ε) kδ + k/2nΩ(1) O(n6 + kd) ≈ IZ ’89 O(ε) kδ + γ n + O(kd + log k log(1/γ)) This work

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Landscape of stewards

◮ Steward model captures derandomization constructions in

literature ε′ δ′ Randomness complexity Reference ε kδ nk Na¨ ıve O(ε) 2kδ n

(works for d = 1 only)

This work O(εd) kδ + γ n + O(k log(d + 1) + log k log(1/γ)) This work O(εkd/γ) kδ + γ n + O(k log k + k log d + k log(1/γ)) ≈ SZ ’99 O(ε) kδ + k/2nΩ(1) O(n6 + kd) ≈ IZ ’89 O(ε) kδ + γ n + O(kd + log k log(1/γ)) This work Any Any ≤ 0.2 n + Ω(k) − log(δ′/δ) This work

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Open questions

◮ Optimal randomness complexity when d is large?

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Open questions

◮ Optimal randomness complexity when d is large? ◮ Simultaneously achieve error ε′ ≤ O(ε) and randomness

complexity n + O(k log(d + 1))?

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Open questions

◮ Optimal randomness complexity when d is large? ◮ Simultaneously achieve error ε′ ≤ O(ε) and randomness

complexity n + O(k log(d + 1))?

◮ Thanks! Questions? ◮ This material is based upon work supported by the National Science

Foundation Graduate Research Fellowship under Grant No. DGE-1610403.

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