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Settling the Query Complexity of Non-Adaptive Junta Testing Erik - - PowerPoint PPT Presentation

Jul 02, 2023 244 likes •1.09k views

Settling the Query Complexity of Non-Adaptive Junta Testing Erik Waingarten, Columbia University Based on joint work with Xi Chen (Columbia University) Rocco Servedio (Columbia University) Li-Yang Tan (Toyota Technological Institute) Jinyu

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SLIDE 1

Settling the Query Complexity of Non-Adaptive Junta Testing

Erik Waingarten, Columbia University Based on joint work with Xi Chen (Columbia University) Rocco Servedio (Columbia University) Li-Yang Tan (Toyota Technological Institute) Jinyu Xie (Columbia University)

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SLIDE 2

Boolean Function Property Testing

Given query (black-box) access to an unknown Boolean function f : {0, 1}n → {0, 1}, does it have some property P? With as few queries as possible, a randomized tester to tell if f has property P vs. f is far from having property P

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SLIDE 3

Boolean Function Property Testing: FAQ

What does far from P mean?

◮ Distance between two functions f and g:

dist(f , g) = Pr

x∈{0,1}n

  • f (x) = g(x)
  • ◮ dist(f , P) = ming∈P dist(f , g) ≥ ε.

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SLIDE 4

Rules of the Game

Given query access to an unknown f : {0, 1}n → {0, 1} and a parameter ε > 0: If f has property P, accept w.p. > 2/3; If f is ε-far from having property P, reject w.p. > 2/3; Otherwise: doesn’t matter what we do. Given P, number of queries needed in terms of n and ε?

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SLIDE 5

This talk: P = k-juntas

Definition

A Boolean function f : {0, 1}n → {0, 1} is a k-junta if it depends on at most k variables.

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SLIDE 6

Non-Adaptive vs. Adaptive Algorithms

f x1, . . . , xq f (x1), . . . , f (xq)

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SLIDE 7

Non-Adaptive vs. Adaptive Algorithms

f x1, . . . , xq f (x1), . . . , f (xq) f x1

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SLIDE 8

Non-Adaptive vs. Adaptive Algorithms

f x1, . . . , xq f (x1), . . . , f (xq) f f (x1)

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SLIDE 9

Non-Adaptive vs. Adaptive Algorithms

f x1, . . . , xq f (x1), . . . , f (xq) f x2

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SLIDE 10

Non-Adaptive vs. Adaptive Algorithms

f x1, . . . , xq f (x1), . . . , f (xq) f f (x2)

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SLIDE 11

Non-Adaptive vs. Adaptive Algorithms

f x1, . . . , xq f (x1), . . . , f (xq) f . . .

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SLIDE 12

Non-Adaptive vs. Adaptive Algorithms

f x1, . . . , xq f (x1), . . . , f (xq) f f (. . . )

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SLIDE 13

Non-Adaptive vs. Adaptive Algorithms

f x1, . . . , xq f (x1), . . . , f (xq) f xq

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SLIDE 14

Non-Adaptive vs. Adaptive Algorithms

f x1, . . . , xq f (x1), . . . , f (xq) f f (xq)

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SLIDE 15

Non-Adaptive vs. Adaptive Algorithms

f x1, . . . , xq f (x1), . . . , f (xq) f f (xq) Adaptive algorithms with q queries can simulate non-adaptive algorithms with q queries.

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SLIDE 16

Non-Adaptive vs. Adaptive Algorithms

f x1, . . . , xq f (x1), . . . , f (xq) f f (xq) Adaptive algorithms with q queries can simulate non-adaptive algorithms with q queries. Non-adaptive algorithms with 2q queries can simulate adaptive algorithms with q queries.

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SLIDE 17

Non-Adaptive vs. Adaptive Algorithms

f x1, . . . , xq f (x1), . . . , f (xq) f f (xq) Adaptive algorithms with q queries can simulate non-adaptive algorithms with q queries. Non-adaptive algorithms with 2q queries can simulate adaptive algorithms with q queries. Exponential gaps are known:

◮ Signed majorities [Matulef, O’Donnell, Rubinfeld, Servedio 09], [Ron,

Servedio 13].

◮ Read-once width-2 OBDD [Ron, Tsur 12] [Brody, Matulef, Wu 11]. 6 / 29

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SLIDE 18

How Adaptivity Helps: Binary Search

f

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SLIDE 19

How Adaptivity Helps: Binary Search

f x, y

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How Adaptivity Helps: Binary Search

f f (x), f (y)

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SLIDE 21

How Adaptivity Helps: Binary Search

f f (x), f (y)

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SLIDE 22

How Adaptivity Helps: Binary Search

f

  • z

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SLIDE 23

How Adaptivity Helps: Binary Search

f

  • z

Recurse on path for O(log n) steps.

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SLIDE 24

How Adaptivity Helps: Binary Search

f

  • z

Recurse on path for O(log n) steps. Will find some edge (x, y) in direction i f (x) = f (y).

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SLIDE 25

How Adaptivity Helps: Binary Search

f

  • z

Recurse on path for O(log n) steps. Will find some edge (x, y) in direction i f (x) = f (y). With O(log n) many queries, can find one important direction.

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SLIDE 26

Upper bounds

Can we test k-juntas with query complexity independent of n?

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Upper bounds

Can we test k-juntas with query complexity independent of n? Yes!

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SLIDE 28

Upper bounds

Can we test k-juntas with query complexity independent of n? Yes!

Theorem (Fisher, Kindler, Ron, Safra, Samorodnitsky 04)

One can ε-test k-juntas for any k with poly(k, ε−1) queries.

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Upper bounds

Can we test k-juntas with query complexity independent of n? Yes!

Theorem (Fisher, Kindler, Ron, Safra, Samorodnitsky 04)

One can ε-test k-juntas for any k with poly(k, ε−1) queries. Additionally, one can achieve O(k2/ε) in the non-adaptive model.

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Two Algorithms

Theorem (Blais 08)

There exists a non-adaptive algorithm for testing k-juntas making

  • O(k3/2)/ε many queries.

Theorem (Blais 09)

There exists an adaptive algorithm for testing k-juntas making O(k/ε + k log k) queries. Non-adaptive: estimate variation of blocks of coordinates. Adaptive: use binary search on blocks of coordinates.

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SLIDE 31

Two Lower Bounds

Theorem (Chockler and Gutfreund 04)

Testing juntas adaptively requires Ω(k) queries for some ε = Ω(1).

Theorem (Buhrman, Garcia-Soriano, Matsliah, de Wolf 13)

Testing juntas non-adaptively requires Ω(k log k) queries for some ε = Ω(1).

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SLIDE 32

Two Lower Bounds

Theorem (Chockler and Gutfreund 04)

Testing juntas adaptively requires Ω(k) queries for some ε = Ω(1).

Theorem (Buhrman, Garcia-Soriano, Matsliah, de Wolf 13)

Testing juntas non-adaptively requires Ω(k log k) queries for some ε = Ω(1). Model Upper bound Lower bound Non-adaptive

  • O(k3/2)/ε

Ω(k log k) Adaptive O(k/ε + k log k) Ω(k)

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SLIDE 33

Adaptivity can help

Theorem (Servedio, Tan, Wright 15)

Testing k-juntas non-adaptively requires Ω

  • k log k

εc log(log k/εc)

  • for any c < 1.

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SLIDE 34

Adaptivity can help

Theorem (Servedio, Tan, Wright 15)

Testing k-juntas non-adaptively requires Ω

  • k log k

εc log(log k/εc)

  • for any c < 1.

When ε = Θ(1), lower bound is Ω(k log k/ log(log k)).

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SLIDE 35

Adaptivity can help

Theorem (Servedio, Tan, Wright 15)

Testing k-juntas non-adaptively requires Ω

  • k log k

εc log(log k/εc)

  • for any c < 1.

When ε = Θ(1), lower bound is Ω(k log k/ log(log k)). When ε = 1/ log k, lower bound is Ω k log1+c(k) log log k

  • ≫ O(k log k)

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SLIDE 36

Questions

Model Upper bound Lower bound Non-adaptive

  • O(k3/2)/ε

Ω(k log k/(εc log(log(k)/εc))) Adaptive O(k/ε + k log k) Ω(k)

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SLIDE 37

Questions

Model Upper bound Lower bound Non-adaptive

  • O(k3/2)/ε

Ω(k log k/(εc log(log(k)/εc))) Adaptive O(k/ε + k log k) Ω(k) When does adaptivity help?

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SLIDE 38

Questions

Model Upper bound Lower bound Non-adaptive

  • O(k3/2)/ε

Ω(k log k/(εc log(log(k)/εc))) Adaptive O(k/ε + k log k) Ω(k) When does adaptivity help? Can the adaptive algorithm be made non-adaptive?

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SLIDE 39

Main Result

Theorem

Testing juntas non-adaptively requires Ω(k3/2/ε) queries. Model Upper bound Lower bound Non-adaptive

  • O(k3/2)/ε
  • Ω(k3/2/ε)

Adaptive O(k/ε + k log k) Ω(k)

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SLIDE 40

Main Result

Theorem

Testing juntas non-adaptively requires Ω(k3/2/ε) queries. Model Upper bound Lower bound Non-adaptive

  • O(k3/2)/ε
  • Ω(k3/2/ε)

Adaptive O(k/ε + k log k) Ω(k) Goal for this talk: Ω(n3/2) for 3n

4 -junta testing with ε = Ω(1).

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SLIDE 41

Main Result

Theorem

Testing juntas non-adaptively requires Ω(k3/2/ε) queries. Model Upper bound Lower bound Non-adaptive

  • O(k3/2)/ε
  • Ω(k3/2/ε)

Adaptive O(k/ε + k log k) Ω(k) Goal for this talk: Ω(n3/2) for 3n

4 -junta testing with ε = Ω(1).

For general k, we use a “padding” argument.

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SLIDE 42

Overview of the proof

✬ ✫ ✩ ✪ ✻ ❄ ✲

Dyes and Dno

✬ ✫ ✩ ✪

Alg for Dyes, Dno

✬ ✫ ✩ ✪

Alg′ for SSSQ

✬ ✫ ✩ ✪

Lower bound for SSSQ

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SLIDE 43

Overview of the proof

✬ ✫ ✩ ✪ ✻ ❄ ✲

Dyes and Dno

✬ ✫ ✩ ✪

Alg for Dyes, Dno

✬ ✫ ✩ ✪

Alg′ for SSSQ

✬ ✫ ✩ ✪

Lower bound for SSSQ

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SLIDE 44

Overview of the proof

One class: parameter p py for Dyes and pn for Dno

✬ ✫ ✩ ✪ ✻ ❄ ✲

Dyes and Dno

✬ ✫ ✩ ✪

Alg for Dyes, Dno

✬ ✫ ✩ ✪

Alg′ for SSSQ

✬ ✫ ✩ ✪

Lower bound for SSSQ

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SLIDE 45

Overview of the proof

One class: parameter p py for Dyes and pn for Dno “must work a certain way”

✬ ✫ ✩ ✪ ✻ ❄ ✲

Dyes and Dno

✬ ✫ ✩ ✪

Alg for Dyes, Dno

✬ ✫ ✩ ✪

Alg′ for SSSQ

✬ ✫ ✩ ✪

Lower bound for SSSQ

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SLIDE 46

Overview of the proof

One class: parameter p py for Dyes and pn for Dno “must work a certain way” Set-Size-Set-Queries(py, pn)

✬ ✫ ✩ ✪ ✻ ❄ ✲

Dyes and Dno

✬ ✫ ✩ ✪

Alg for Dyes, Dno

✬ ✫ ✩ ✪

Alg′ for SSSQ

✬ ✫ ✩ ✪

Lower bound for SSSQ

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SLIDE 47

Overview of the proof

One class: parameter p py for Dyes and pn for Dno “must work a certain way” Set-Size-Set-Queries(py, pn) Bin(n/2, py) vs. Bin(n/2, pn)

✬ ✫ ✩ ✪ ✻ ❄ ✲

Dyes and Dno

✬ ✫ ✩ ✪

Alg for Dyes, Dno

✬ ✫ ✩ ✪

Alg′ for SSSQ

✬ ✫ ✩ ✪

Lower bound for SSSQ

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SLIDE 48

Overview of the proof

One class: parameter p py for Dyes and pn for Dno “must work a certain way” Set-Size-Set-Queries(py, pn) Bin(n/2, py) vs. Bin(n/2, pn)

✬ ✫ ✩ ✪ ✬ ✫ ✩ ✪ ✻ ❄ ✲

Dyes and Dno

✬ ✫ ✩ ✪

Alg for Dyes, Dno

✬ ✫ ✩ ✪

Alg′ for SSSQ

✬ ✫ ✩ ✪

Lower bound for SSSQ

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SLIDE 49

Dyes: 3n/4-juntas

|M| = n

2.

|A| = n

4.

i ∈ [2n/2], hi is a random function depending on Si ⊂ A. Si includes k ∈ A with probability

1 √n.

f (x) = hx|M(x).

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SLIDE 50

Dyes: 3n/4-juntas

x|M = 2 |M| = n

2.

|A| = n

4.

i ∈ [2n/2], hi is a random function depending on Si ⊂ A. Si includes k ∈ A with probability

1 √n.

f (x) = hx|M(x).

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SLIDE 51

Dyes: 3n/4-juntas

x|M = 2

✛ ✚ ✘ ✙

Ex: influence of M and A? |M| = n

2.

|A| = n

4.

i ∈ [2n/2], hi is a random function depending on Si ⊂ A. Si includes k ∈ A with probability

1 √n.

f (x) = hx|M(x).

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SLIDE 52

Dno: 3n/4 + √n-juntas

|M| = n

2.

|A| = n

4 + √n.

i ∈ [2n/2], hi is a random function depending on Si ⊂ A. Si includes k ∈ A with probability

1 √n.

f (x) = hx|M(x).

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SLIDE 53

Dyes vs. Dno

Dyes: M is random set of size n

2.

A = {k ∈ M : w.p 1

2}.

Si = {k ∈ A : w.p

1 √n}.

Dno: M is random set of size n

2.

A = {k ∈ M : w.p 1

2 + 1 √n}.

Si = {k ∈ A : w.p

1 √n}.

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SLIDE 54

Dyes vs. Dno

Dyes: M is random set of size n

2.

A = {k ∈ M : w.p 1

2}.

Si = {k ∈ A : w.p

1 √n}.

py = 1

2.

Dno: M is random set of size n

2.

A = {k ∈ M : w.p 1

2 + 1 √n}.

Si = {k ∈ A : w.p

1 √n}.

pn = 1

2 + 1 √n.

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SLIDE 55

Dyes vs. Dno

Dyes: M is random set of size n

2.

A = {k ∈ M : w.p 1

2}.

Si = {k ∈ A : w.p

1 √n}.

Dno: M is random set of size n

2.

A = {k ∈ M : w.p 1

2 + 1 √n}.

Si = {k ∈ A : w.p

1 √n}.

Lemma (Dyes are k-juntas)

With probability 1 − o(1), f ∼ Dyes is a k-junta.

Lemma (Dno are far from k-juntas)

With probability 1 − o(1), f ∼ Dno is Ω(1)-far from being a k-junta.

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SLIDE 56

Dyes vs. Dno

Dyes: M is random set of size n

2.

A = {k ∈ M : w.p 1

2}.

Si = {k ∈ A : w.p

1 √n}.

Dno: M is random set of size n

2.

A = {k ∈ M : w.p 1

2 + 1 √n}.

Si = {k ∈ A : w.p

1 √n}.

Lemma (Dyes are k-juntas)

With probability 1 − o(1), f ∼ Dyes is a k-junta.

Lemma (Dno are far from k-juntas)

With probability 1 − o(1), f ∼ Dno is Ω(1)-far from being a k-junta. Hint: f ∼ Dno has 3n

4 + √n 4 relevant variables!

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SLIDE 57

Overview of the proof

One class: parameter p py for Dyes and pn for Dno “must work a certain way” Set-Size-Set-Queries(py, pn) Bin(n/2, py) vs Bin(n/2, pn)

✬ ✫ ✩ ✪ ✻ ❄ ✲

Dyes and Dno

✬ ✫ ✩ ✪

Alg for Dyes, Dno

✬ ✫ ✩ ✪

Alg′ for SSSQ

✬ ✫ ✩ ✪

Lower bound for SSSQ

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SLIDE 58

Overview of the proof

One class: parameter p py for Dyes and pn for Dno “must work a certain way” Set-Size-Set-Queries(py, pn) Bin(n/2, py) vs Bin(n/2, pn)

✬ ✫ ✩ ✪ ✻ ❄ ✲

Dyes and Dno

✬ ✫ ✩ ✪ ✬ ✫ ✩ ✪

Alg for Dyes, Dno

✬ ✫ ✩ ✪

Alg′ for SSSQ

✬ ✫ ✩ ✪

Lower bound for SSSQ

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SLIDE 59

Structure of functions

Both Dyes and Dno follow the structure:

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SLIDE 60

Structure of functions

Both Dyes and Dno follow the structure: Indexing (M), and Evaluation (h1, . . . , h2n/2).

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SLIDE 61

Structure of functions

Both Dyes and Dno follow the structure: Indexing (M), and Evaluation (h1, . . . , h2n/2). Indexing restricts the algorithms we look at, evaluation gives the lower bound.

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SLIDE 62

Structure of functions

Both Dyes and Dno follow the structure: What happens if algorithm has queries x1, . . . , xq indexed into different sub-functions hi?

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SLIDE 63

Structure of functions

Both Dyes and Dno follow the structure: What happens if algorithm has queries x1, . . . , xq indexed into different sub-functions hi? f (x1), f (x2), . . . , f (xq) look like random bits.

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SLIDE 64

Structure of functions

Both Dyes and Dno follow the structure: What happens if algorithm has queries x1, . . . , xq indexed into different sub-functions hi? f (x1), f (x2), . . . , f (xq) look like random bits. Alice cannot distinguish between Dyes and Dno.

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SLIDE 65

Structure of functions

Both Dyes and Dno follow the structure: What happens if algorithm has queries x1, . . . , xq indexed into different sub-functions hi? f (x1), f (x2), . . . , f (xq) look like random bits. Alice cannot distinguish between Dyes and Dno. Alice wants multiple queries indexed to same sub-function.

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SLIDE 66

Structure of functions

x1 x2 x3 xℓ Alice wants to query x1, . . . , xℓ sub-function h2.

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SLIDE 67

Structure of functions

x1 x2 x3 xℓ

✬ ✫ ✩ ✪

M Alice wants to query x1, . . . , xℓ sub-function h2.

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SLIDE 68

Structure of functions

x1 x2 x3 xℓ

✬ ✫ ✩ ✪

M Alice wants to query x1, . . . , xℓ sub-function h2. How different can M be on x1, . . . , xℓ?

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SLIDE 69

Structure of functions

x1 x2 x3 xℓ

✬ ✫ ✩ ✪

M Alice wants to query x1, . . . , xℓ sub-function h2. How different can M be on x1, . . . , xℓ? If dist(xi, xj) is large, then very likely xi and xj disagree on M!

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SLIDE 70

Structure of functions

x1 x2 x3 xℓ

✬ ✫ ✩ ✪

M Alice wants to query x1, . . . , xℓ sub-function h2. How different can M be on x1, . . . , xℓ? If dist(xi, xj) is large, then very likely xi and xj disagree on M! Condition: when multiple queries indexed to same function, queries are close.

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SLIDE 71

Overview of the proof

One class: parameter p py for Dyes and pn for Dno “must work a certain way” Set-Size-Set-Queries(py, pn) Bin(n/2, py) vs Bin(n/2, pn)

✬ ✫ ✩ ✪ ✻ ❄ ✲

Dyes and Dno

✬ ✫ ✩ ✪ ✬ ✫ ✩ ✪

Alg for Dyes, Dno

✬ ✫ ✩ ✪

Alg′ for SSSQ

✬ ✫ ✩ ✪

Lower bound for SSSQ

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SLIDE 72

Overview of the proof

One class: parameter p py for Dyes and pn for Dno “must work a certain way” Set-Size-Set-Queries(py, pn) Bin(n/2, py) vs Bin(n/2, pn)

✬ ✫ ✩ ✪ ✻ ❄ ✲

Dyes and Dno

✬ ✫ ✩ ✪ ✬ ✫ ✩ ✪

Alg for Dyes, Dno

✬ ✫ ✩ ✪ ✬ ✫ ✩ ✪

Alg′ for SSSQ

✬ ✫ ✩ ✪

Lower bound for SSSQ

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SLIDE 73

Set-Size-Set-Queries Problem (SSSQ)

Definition

The SSSQ(py, pn, m) problem seeks to distinguish the following two cases: Yes: A ⊂ [m] is a random set includes each element i.i.d w.p py. No: A ⊂ [m] is a random set includes each element i.i.d w.p pn. Query access: Set queries are sets T ⊂ [m] Responses are sets V ⊂ T where V includes i ∈ T ∩ A w.p 1 √ 2m . Complexity: for queries T1, . . . , Tq, cost = q

j=1 |Tj|.

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SLIDE 74

Intuition: Alg → SSSQ

Dyes: M is random set of size n

2.

A = {k ∈ M : w.p 1

2}.

Si = {k ∈ A : w.p

1 √n}.

Dno: M is random set of size n

2.

A = {k ∈ M : w.p 1

2 + 1 √n}.

Si = {k ∈ A : w.p

1 √n}.

Junta Testing SSSQ Dyes and Dno SSSQ( 1

2, 1 2 + 1 √n, n 2)

x1, . . . , xℓ indexed to hi Sets Ti Influential directions seen in Si Response set Vi ⊂ Ti ∩ A Query complexity Cost

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SLIDE 75

Intuition: Alg → SSSQ

Dyes: M is random set of size n

2.

A = {k ∈ M : w.p 1

2}.

Si = {k ∈ A : w.p

1 √n}.

Dno: M is random set of size n

2.

A = {k ∈ M : w.p 1

2 + 1 √n}.

Si = {k ∈ A : w.p

1 √n}.

Junta Testing SSSQ Dyes and Dno SSSQ( 1

2, 1 2 + 1 √n, n 2)

x1, . . . , xℓ indexed to hi Sets Ti Influential directions seen in Si Response set Vi ⊂ Ti ∩ A Query complexity Cost

Theorem

If there is a q-query non-adaptive tester Alg for Dyes vs Dno, then there is an algorithm for SSSQ( 1

2, 1 2 + 1 √n, n 2) with cost O(q log(n)).

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SLIDE 76

Overview of the proof

One class: parameter p py for Dyes and pn for Dno “must work a certain way” Set-Size-Set-Queries(py, pn) Bin(n/2, py) vs Bin(n/2, pn)

✬ ✫ ✩ ✪ ✻ ❄ ✲

Dyes and Dno

✬ ✫ ✩ ✪ ✬ ✫ ✩ ✪

Alg for Dyes, Dno

✬ ✫ ✩ ✪

Alg′ for SSSQ

✬ ✫ ✩ ✪

Lower bound for SSSQ

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SLIDE 77

Overview of the proof

One class: parameter p py for Dyes and pn for Dno “must work a certain way” Set-Size-Set-Queries(py, pn) Bin(n/2, py) vs Bin(n/2, pn)

✬ ✫ ✩ ✪ ✻ ❄ ✲

Dyes and Dno

✬ ✫ ✩ ✪ ✬ ✫ ✩ ✪

Alg for Dyes, Dno

✬ ✫ ✩ ✪

Alg′ for SSSQ

✬ ✫ ✩ ✪ ✬ ✫ ✩ ✪

Lower bound for SSSQ

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SLIDE 78

Lower bound for SSSQ

Lemma (hope to prove)

SSSQ( 1

2, 1 2 + 1 √n, n 2) requires cost

Ω(n3/2). Main Ideas: |A| ∼ Bin( n

2, 1 2) (in Dyes) vs |A| ∼ Bin( n 2, 1 2 + 1 √n) (in Dno).

Each sample costs Ω(√n), since we observe k ∈ V when k ∈ A w.p

1 √n.

Ω(n)

  • distinguish 1

2 vs 1 2 + 1 √n

× Ω(√n)

seeing one example

= Ω(n3/2)

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SLIDE 79

Putting things together

Theorem

If non-adaptive tester Alg for Dyes vs Dno with q queries, then SSSQ( 1

2, 1 2 + 1 √n, n 2) has cost O(q log(n)).

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SLIDE 80

Putting things together

Theorem

If non-adaptive tester Alg for Dyes vs Dno with q queries, then SSSQ( 1

2, 1 2 + 1 √n, n 2) has cost O(q log(n)).

Lemma

SSSQ( 1

2, 1 2 + 1 √n, n 2) requires cost

Ω(n3/2).

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SLIDE 81

Putting things together

Theorem

If non-adaptive tester Alg for Dyes vs Dno with q queries, then SSSQ( 1

2, 1 2 + 1 √n, n 2) has cost O(q log(n)).

Lemma

SSSQ( 1

2, 1 2 + 1 √n, n 2) requires cost

Ω(n3/2).

Theorem

Testing 3n

4 -juntas non-adaptively requires

Ω(n3/2) queries.

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SLIDE 82

Open Problems

Tradeoffs in query complexity and adaptivity? See next! [Canonne and Gur 17]. Other “natural” properties of Boolean functions with gaps in adaptive and non-adaptive query complexity? More applications of this class of random functions?

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SLIDE 83

Open Problems

Tradeoffs in query complexity and adaptivity? See next! [Canonne and Gur 17]. Other “natural” properties of Boolean functions with gaps in adaptive and non-adaptive query complexity? More applications of this class of random functions?

Thanks!

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