Communication Complexity with Small Advantage Thomas Watson - - PowerPoint PPT Presentation

Communication Complexity with Small Advantage

Thomas Watson

University of Memphis

Communication complexity

F : {0, 1}n × {0, 1}n → {0, 1}

Communication complexity

F : {0, 1}n × {0, 1}n → {0, 1} (Alice: x) (Bob: y)

Communication complexity

F : {0, 1}n × {0, 1}n → {0, 1} (Alice: x) − − − − − − − − − → (Bob: y)

Communication complexity

F : {0, 1}n × {0, 1}n → {0, 1} (Alice: x) − − − − − − − − − → (Bob: y) ← − − − − − − − − −

Communication complexity

F : {0, 1}n × {0, 1}n → {0, 1} (Alice: x) − − − − − − − − − → (Bob: y) ← − − − − − − − − − − − − − − − − − − →

Communication complexity

F : {0, 1}n × {0, 1}n → {0, 1} (Alice: x) − − − − − − − − − → (Bob: y) ← − − − − − − − − − − − − − − − − − − → ← − − − − − − − − −

Communication complexity

F : {0, 1}n × {0, 1}n → {0, 1} (Alice: x) − − − − − − − − − → (Bob: y) ← − − − − − − − − − − − − − − − − − − → ← − − − − − − − − − − − − − − − − − − →

Communication complexity

F : {0, 1}n × {0, 1}n → {0, 1} (Alice: x) − − − − − − − − − → (Bob: y) ← − − − − − − − − − − − − − − − − − − → ← − − − − − − − − − − − − − − − − − − → F(x, y) F(x, y)

Communication complexity

F : {0, 1}n × {0, 1}n → {0, 1} (Alice: x) − − − − − − − − − → (Bob: y) ← − − − − − − − − − − − − − − − − − − → ← − − − − − − − − − − − − − − − − − − → F(x, y) F(x, y) Randomized protocols:

Communication complexity

F : {0, 1}n × {0, 1}n → {0, 1} (Alice: x) − − − − − − − − − → (Bob: y) ← − − − − − − − − − − − − − − − − − − → ← − − − − − − − − − − − − − − − − − − → F(x, y) F(x, y) Randomized protocols:

◮ Correctness:

∀(x, y) : P[output is F(x, y)] ≥ 3/4

Communication complexity

F : {0, 1}n × {0, 1}n → {0, 1} (Alice: x) − − − − − − − − − → (Bob: y) ← − − − − − − − − − − − − − − − − − − → ← − − − − − − − − − − − − − − − − − − → F(x, y) F(x, y) Randomized protocols:

◮ Correctness:

∀(x, y) : P[output is F(x, y)] ≥ 3/4

◮ Cost:

max(x,y), random outcomes(# bits communicated)

Communication complexity

F : {0, 1}n × {0, 1}n → {0, 1} (Alice: x) − − − − − − − − − → (Bob: y) ← − − − − − − − − − − − − − − − − − − → ← − − − − − − − − − − − − − − − − − − → F(x, y) F(x, y) Randomized protocols:

◮ Correctness:

∀(x, y) : P[output is F(x, y)] ≥ 3/4

◮ Cost:

max(x,y), random outcomes(# bits communicated)

◮ Complexity:

R(F) = mincorrect protocols(cost)

Classic results

Classic results

R(Inner-Product) = Θ(n) R(Set-Intersection) = Θ(n) R(Gap-Hamming) = Θ(n)

Classic results

R(Inner-Product) = Θ(n) R(Set-Intersection) = Θ(n) R(Gap-Hamming) = Θ(n) R : success probability ≥ 3/4

Classic results

R(Inner-Product) = Θ(n) R(Set-Intersection) = Θ(n) R(Gap-Hamming) = Θ(n) R : success probability ≥ 3/4 Small advantage: R1/2+ǫ : success probability ≥ 1/2 + ǫ

Classic results — revisited

R1/2+ǫ(Inner-Product) = Θ(n) (unless ǫ ≤ 2−Ω(n)) R1/2+ǫ(Set-Intersection) = Θ(ǫ · n) R1/2+ǫ(Gap-Hamming) = Θ(ǫ2 · n)

Classic results — revisited

R1/2+ǫ(Inner-Product) = Θ(n) (unless ǫ ≤ 2−Ω(n)) R1/2+ǫ(Set-Intersection) = Θ(ǫ · n) R1/2+ǫ(Gap-Hamming) = Θ(ǫ2 · n) . . .

• ther functions?

Climbing the polynomial hierarchy

NP: R1/2+ǫ(Set-Intersection) = Θ(ǫ · n)

Climbing the polynomial hierarchy

NP: R1/2+ǫ(Set-Intersection) = Θ(ǫ · n) [Braverman–Moitra STOC’13, G¨

• ¨
• s–Watson RANDOM’14]

(information complexity) (corruption)

Climbing the polynomial hierarchy

NP: R1/2+ǫ(Set-Intersection) = Θ(ǫ · n) [Braverman–Moitra STOC’13, G¨

• ¨
• s–Watson RANDOM’14]

(information complexity) (corruption) Σ2P, Π2P:

Climbing the polynomial hierarchy

NP: R1/2+ǫ(Set-Intersection) = Θ(ǫ · n) [Braverman–Moitra STOC’13, G¨

• ¨
• s–Watson RANDOM’14]

(information complexity) (corruption) Σ2P, Π2P: R1/2+ǫ(Tribes) = Θ(ǫ · n)

Climbing the polynomial hierarchy

NP: R1/2+ǫ(Set-Intersection) = Θ(ǫ · n) [Braverman–Moitra STOC’13, G¨

• ¨
• s–Watson RANDOM’14]

(information complexity) (corruption) Σ2P, Π2P: R1/2+ǫ(Tribes) = Θ(ǫ · n) Higher levels? (read-once AC0 formulas)

Climbing the polynomial hierarchy

NP: R1/2+ǫ(Set-Intersection) = Θ(ǫ · n) [Braverman–Moitra STOC’13, G¨

• ¨
• s–Watson RANDOM’14]

(information complexity) (corruption) Σ2P, Π2P: R1/2+ǫ(Tribes) = Θ(ǫ · n) Higher levels? (read-once AC0 formulas) Constant advantage: well-understood [Jayram–Kopparty–Raghavendra/Leonardos–Saks CCC’09]

Climbing the polynomial hierarchy

NP: R1/2+ǫ(Set-Intersection) = Θ(ǫ · n) [Braverman–Moitra STOC’13, G¨

• ¨
• s–Watson RANDOM’14]

(information complexity) (corruption) Σ2P, Π2P: R1/2+ǫ(Tribes) = Θ(ǫ · n) Higher levels? (read-once AC0 formulas) Constant advantage: well-understood [Jayram–Kopparty–Raghavendra/Leonardos–Saks CCC’09] Small advantage: open

Function definitions

Set-Intersection:

x x x x x y y

1 1 2 2 3 3 4 4 5 5

y y y

Function definitions

Set-Intersection:

x x x x x y y

1 1 2 2 3 3 4 4 5 5

y y y

Tribes:

x x x x y

2 2 3 3 4 4 5 5

y y y x

6 6

x y

1 1

x

7 7 x 8

x

9 9

y y y n n y

8

What’s known about Tribes?

R(Tribes) = Θ(n)

What’s known about Tribes?

R(Tribes) = Θ(n) [Jayram–Kumar–Sivakumar STOC’03, Harsha–Jain FSTTCS’13] (information complexity) (smooth rectangle bound)

What’s known about Tribes?

R(Tribes) = Θ(n) [Jayram–Kumar–Sivakumar STOC’03, Harsha–Jain FSTTCS’13] (information complexity) (smooth rectangle bound) R1/2+ǫ(Tribes) = ??

What’s known about Tribes?

R(Tribes) = Θ(n) [Jayram–Kumar–Sivakumar STOC’03, Harsha–Jain FSTTCS’13] (information complexity) (smooth rectangle bound) R1/2+ǫ(Tribes) = ?? [G¨

• ¨
• s–Watson] trick:

R1/2+ǫ ≥ Ω(ǫ · corruption bound)

What’s known about Tribes?

R(Tribes) = Θ(n) [Jayram–Kumar–Sivakumar STOC’03, Harsha–Jain FSTTCS’13] (information complexity) (smooth rectangle bound) R1/2+ǫ(Tribes) = ?? [G¨

• ¨
• s–Watson] trick:

R1/2+ǫ ≥ Ω(ǫ · corruption bound)

◮ Doesn’t work for Tribes: corruption bound ≈ √n

What’s known about Tribes?

R(Tribes) = Θ(n) [Jayram–Kumar–Sivakumar STOC’03, Harsha–Jain FSTTCS’13] (information complexity) (smooth rectangle bound) R1/2+ǫ(Tribes) = ?? [G¨

• ¨
• s–Watson] trick:

R1/2+ǫ ≥ Ω(ǫ · corruption bound)

◮ Doesn’t work for Tribes: corruption bound ≈ √n

?? Similar trick: R1/2+ǫ ≥ Ω(ǫ · smooth rectangle bound) ??

What’s known about Tribes?

R(Tribes) = Θ(n) [Jayram–Kumar–Sivakumar STOC’03, Harsha–Jain FSTTCS’13] (information complexity) (smooth rectangle bound) R1/2+ǫ(Tribes) = ?? [G¨

• ¨
• s–Watson] trick:

R1/2+ǫ ≥ Ω(ǫ · corruption bound)

◮ Doesn’t work for Tribes: corruption bound ≈ √n

?? Similar trick: R1/2+ǫ ≥ Ω(ǫ · smooth rectangle bound) ??

◮ Not true in general

Our approach for Tribes

Information complexity:

Our approach for Tribes

Information complexity:

◮ Ω(1)-advantage for Tribes [JKS’03]

Our approach for Tribes

Information complexity:

◮ Ω(1)-advantage for Tribes [JKS’03] ◮ ǫ-advantage for Set-Inter [BM’13]

Our approach for Tribes

Information complexity:

◮ Ω(1)-advantage for Tribes [JKS’03] ◮ ǫ-advantage for Set-Inter [BM’13] ◮ Combine?

Our approach for Tribes

Information complexity:

◮ Ω(1)-advantage for Tribes [JKS’03] ◮ ǫ-advantage for Set-Inter [BM’13] ◮ Combine?

4-step approach:

Our approach for Tribes

Information complexity:

◮ Ω(1)-advantage for Tribes [JKS’03] ◮ ǫ-advantage for Set-Inter [BM’13] ◮ Combine?

4-step approach:

• 1. Conditioning and direct sum
• 2. Uniformly covering a pair of gadgets
• 3. Relating information and probabilities for inputs
• 4. Relating information and probabilities for transcripts

Preliminaries

Idea from [BM’13]:

Preliminaries

Idea from [BM’13]: Suffices to use 3Eq gadget instead of And gadget   1 1 1   1

• 1. Conditioning and direct sum

info cost o(ǫ · n)

• info cost o(ǫ)

x x x x y

2 2 3 3 4 4 5 5

y y y x

6 6

x y

1 1

x

7 7 x 8

x

9 9

y y y n n y

8

3Eq 3Eq 3Eq 3Eq 3Eq 3Eq 3Eq 3Eq 3Eq

• x

2 2

y x y

1 1

3Eq 3Eq

• 1. Conditioning and direct sum

info cost o(ǫ · n)

• info cost o(ǫ)

x x x x y

2 2 3 3 4 4 5 5

y y y x

6 6

x y

1 1

x

7 7 x 8

x

9 9

y y y n n y

8

3Eq 3Eq 3Eq 3Eq 3Eq 3Eq 3Eq 3Eq 3Eq

• x

2 2

y x y

1 1

3Eq 3Eq

Want to show: advantage ≤ O(info cost)

• 1. Conditioning and direct sum

x

2 2

y x y

1 1

3Eq 3Eq

Want to show: advantage ≤ O(info cost)

• 1. Conditioning and direct sum

x

2 2

y x y

1 1

3Eq 3Eq

Want to show: advantage ≤ O(info cost) Usual info complexity proofs use Pinsker: statistical distance ≤ O( √ mutual info)

• 1. Conditioning and direct sum

x

2 2

y x y

1 1

3Eq 3Eq

Want to show: advantage ≤ O(info cost) Usual info complexity proofs use Pinsker: statistical distance ≤ O( √ mutual info) Instead: exploit symmetry properties of 3Eq to get higher-order terms to “cancel out”

• 2. Uniformly covering a pair of gadgets

• 2. Uniformly covering a pair of gadgets

Uniformly cover with

1 1 − 1 − 1

x

2 2

y x y

1 1

3Eq 3Eq

• 2. Uniformly covering a pair of gadgets

Uniformly cover with

1 1 − 1 − 1

x

2 2

y x y

1 1

3Eq 3Eq

Lemma: Linear combination of acceptance probabilities ≤ O( four contributions to info cost)

• 2. Uniformly covering a pair of gadgets

Uniformly cover with

1 1 − 1 − 1

x

2 2

y x y

1 1

3Eq 3Eq

Lemma: Linear combination of acceptance probabilities ≤ O( four contributions to info cost) Uniform covering + Lemma: advantage ≤ O(info cost)

• 3. Relating information and probabilities for inputs

1 1 − 1 − 1

“Input Lemma”: Linear combination of acceptance probabilities ≤ O( four contributions to info cost) Prove analogous “Transcript Lemma”? (then could sum over transcripts)

• 3. Relating information and probabilities for inputs

[BM’13] transcript lemma: Our transcript lemma:

• 3. Relating information and probabilities for inputs

[BM’13] transcript lemma: Our transcript lemma:

− 1 1 1 − 1

• 3. Relating information and probabilities for inputs

[BM’13] transcript lemma: Our transcript lemma:

− 1 1 1 − 1 − 1 2 2 − 1 2 2 − 1 − 1 −4

• 3. Relating information and probabilities for inputs

[BM’13] transcript lemma: Our transcript lemma:

− 1 1 1 − 1 − 1 2 2 − 1 2 2 − 1 − 1 −4

∀ transcript: contribution to lin comb of prob ≤ O(contribution to info costs)

• 3. Relating information and probabilities for inputs

• 3. Relating information and probabilities for inputs

accepting rejecting rejecting

− 1 2 2 − 1 2 2 − 1 − 1 −4

+ 2 ·

1 − 1 − 1 1

+ 2 ·

1 − 1 − 1 1

• 3. Relating information and probabilities for inputs

accepting rejecting rejecting

− 1 2 2 − 1 2 2 − 1 − 1 −4

+ 2 ·

1 − 1 − 1 1

+ 2 ·

1 − 1 − 1 1

=

1 1 − 1 − 1

• 4. Relating information and probabilities for transcripts

− 1 2 − 1 2 −4 2 − 1 2 − 1

1 2 + δ 1 2 1 2 1 2 1 2 + γ 1 2

• 4. Relating information and probabilities for transcripts

− 1 2 − 1 2 −4 2 − 1 2 − 1

1 2 + δ 1 2 1 2 1 2 1 2 + γ 1 2

lin comb of probabilities = 2 · green area = Θ(δγ)

• 4. Relating information and probabilities for transcripts

− 1 2 − 1 2 −4 2 − 1 2 − 1

1 2 + δ 1 2 1 2 1 2 1 2 + γ 1 2

lin comb of probabilities = 2 · green area = Θ(δγ) ≤ contribution to info costs = Θ(δ2 + γ2)

Generalized Tribes

x x x x y

2 2 3 3 4 4 5 5

y y y x

6 6

x y

1 1

x

7 7 x 8

x

9 9

y y y m l y

8

Generalized Tribes

x x x x y

2 2 3 3 4 4 5 5

y y y x

6 6

x y

1 1

x

7 7 x 8

x

9 9

y y y m l y

8

Open: Small-advantage complexity when ℓ is small

The end