Non-Interactive Simulation and Dimension Reduction for Polynomials - PowerPoint PPT Presentation

Non-Interactive Simulation and Dimension Reduction for Polynomials Pritish Kamath joint work with Badih Ghazi Prasad Raghavendra CCC UCSD June 24, 2018 1 / 9

2 / 9 Talk outline... • Motivation • Motivation • Motivation • “ Dimension Reduction for Polynomials ” lemma • Summary & Open Directions!

Talk outline... 2 / 9 • Motivation • Motivation • Motivation • “ Dimension Reduction for Polynomials ” lemma • Summary & Open Directions! •

Randomness Models in Distributed Tasks randomness randomness randomness randomness randomness In Information Theory Common Information [Gács-Körner ’73, Wyner ’75] Distributed Source Coding [Slepian-Wolf ’73] In Computer Science Information Tieoretic Crypto! Key Agreement, Secure Computation, … ? 3 / 9

Randomness Models in Distributed Tasks randomness randomness randomness randomness randomness In Information Theory Common Information [Gács-Körner ’73, Wyner ’75] Distributed Source Coding [Slepian-Wolf ’73] In Computer Science Information Tieoretic Crypto! Key Agreement, Secure Computation, … ? 3 / 9

Randomness Models in Distributed Tasks randomness randomness randomness randomness randomness In Information Theory Common Information [Gács-Körner ’73, Wyner ’75] Distributed Source Coding [Slepian-Wolf ’73] In Computer Science Information Tieoretic Crypto! Key Agreement, Secure Computation, … ? 3 / 9

Randomness Models in Distributed Tasks randomness randomness randomness randomness randomness In Information Theory Common Information [Gács-Körner ’73, Wyner ’75] Distributed Source Coding [Slepian-Wolf ’73] In Computer Science Information Tieoretic Crypto! Key Agreement, Secure Computation, … ? 3 / 9

Randomness Models in Distributed Tasks randomness Key Agreement, Secure Computation, … ? Information Tieoretic Crypto! In Computer Science [Slepian-Wolf ’73] Distributed Source Coding [Gács-Körner ’73, Wyner ’75] Common Information In Information Theory 3 / 9 randomness randomness randomness randomness ( 1 − ε ) 2 0 0 ε / 2 ε / 2 1 1 ( 1 − ε ) 2

Randomness Models in Distributed Tasks Common Information Key Agreement, Secure Computation, … ? Information Tieoretic Crypto! In Computer Science [Slepian-Wolf ’73] Distributed Source Coding [Gács-Körner ’73, Wyner ’75] In Information Theory randomness randomness randomness randomness randomness 3 / 9 P ( X , Y ) X Y

Randomness Models in Distributed Tasks randomness Key Agreement, Secure Computation, … ? Information Tieoretic Crypto! In Computer Science [Slepian-Wolf ’73] [Gács-Körner ’73, Wyner ’75] 3 / 9 randomness randomness randomness randomness In Information Theory . . . P ( X , Y ) ▶ Common Information ▶ Distributed Source Coding X Y ▶ · · ·

Randomness Models in Distributed Tasks randomness randomness randomness randomness randomness [Gács-Körner ’73, Wyner ’75] [Slepian-Wolf ’73] Key Agreement, Secure Computation, … ? 3 / 9 In Information Theory . . . ▶ Common Information ▶ Distributed Source Coding ▶ · · · In Computer Science . . . ▶ Information Tieoretic Crypto!

Randomness Models in Distributed Tasks randomness [Canonne-Guruswami-Meka-Sudan ’15] [Bavarian-Gavinsky-Ito ’14] Key Agreement, Secure Computation, … ? [Slepian-Wolf ’73] [Gács-Körner ’73, Wyner ’75] randomness randomness randomness randomness 3 / 9 In Information Theory . . . P ( X , Y ) ▶ Common Information ▶ Distributed Source Coding X Y ▶ · · · In Computer Science . . . ▶ Information Tieoretic Crypto! ▶ Communication Complexity

Randomness Models in Distributed Tasks [Gács-Körner ’73, Wyner ’75] difgerent joint distributions! Understand the power of Abstract Goal: [Canonne-Guruswami-Meka-Sudan ’15] [Bavarian-Gavinsky-Ito ’14] Key Agreement, Secure Computation, … ? [Slepian-Wolf ’73] randomness 3 / 9 randomness randomness randomness randomness In Information Theory . . . P ( X , Y ) ▶ Common Information ▶ Distributed Source Coding X Y ▶ · · · In Computer Science . . . ▶ Information Tieoretic Crypto! ▶ Communication Complexity

Non-Interactive Simulation of Joint Distributions When can simulate ? Main Qvestion How can a “constant-sized” problem be HARD? 4 / 9

Non-Interactive Simulation of Joint Distributions When can simulate problem be HARD? How can a “constant-sized” Main Qvestion ? 4 / 9 ( 1 − ε ) P ( X , Y ) 0 0 2 1 0 0 1 0 1 0 0 1 0 1 1 · · · X Y ε / 2 ε / 2 ( 1 − ε ) 1 1 1 0 1 1 0 1 1 0 0 0 1 1 · · · 2 BSS ε

Non-Interactive Simulation of Joint Distributions Main Qvestion problem be HARD? How can a “constant-sized” 4 / 9 ( 1 − ε ) P ( X , Y ) 0 0 2 1 0 0 1 0 1 0 0 1 0 1 1 · · · X Y ε / 2 ε / 2 ( 1 − ε ) 1 1 1 0 1 1 0 1 1 0 0 0 1 1 · · · 2 BSS ε When can BSS ε simulate BSS δ ? a , b ∈ { 0, 1 } a b ( a , b ) ∼ BSS δ

Non-Interactive Simulation of Joint Distributions Answer: problem be HARD? How can a “constant-sized” Main Qvestion NO YES 4 / 9 ( 1 − ε ) P ( X , Y ) 0 0 2 1 0 0 1 0 1 0 0 1 0 1 1 · · · X Y ε / 2 ε / 2 ( 1 − ε ) 1 1 1 0 1 1 0 1 1 0 0 0 1 1 · · · 2 BSS ε When can BSS ε simulate BSS δ ? δ ≥ ε a , b ∈ { 0, 1 } δ < ε a b ( a , b ) ∼ BSS δ

Non-Interactive Simulation of Joint Distributions Main Qvestion problem be HARD? How can a “constant-sized” 4 / 9 P ( X , Y ) 1 / 3 0 0 1 0 0 0 0 0 0 0 1 0 1 1 · · · X Y 1 / 3 1 / 3 1 1 0 1 0 0 0 1 0 1 0 1 0 0 · · · DISJ When can DISJ simulate BSS δ ? a , b ∈ { 0, 1 } a b ( a , b ) ∼ BSS δ

Non-Interactive Simulation of Joint Distributions (Partial) problem be HARD? How can a “constant-sized” Main Qvestion NO OPEN YES Answer: 4 / 9 P ( X , Y ) 1 / 3 0 0 1 0 0 0 0 0 0 0 1 0 1 1 · · · X Y 1 / 3 1 / 3 1 1 0 1 0 0 0 1 0 1 0 1 0 0 · · · DISJ When can DISJ simulate BSS δ ? δ ≥ 3 8 [ ) a , b ∈ { 0, 1 } 4 , 3 1 δ ∈ 8 a b δ < 1 ( a , b ) ∼ BSS δ 4

Non-Interactive Simulation of Joint Distributions Main Qvestion How can a “constant-sized” problem be HARD? 4 / 9 P ( X , Y ) X 1 , X 2 , X 3 , X 4 , X 5 , . . . X Y Y 1 , Y 2 , Y 3 , Y 4 , Y 5 , . . . When can P simulate BSS δ ? a , b ∈ { 0, 1 } a b ( a , b ) ∼ BSS δ

Non-Interactive Simulation of Joint Distributions Main Qvestion How can a “constant-sized” problem be HARD? 4 / 9 P ( X , Y ) X 1 , X 2 , X 3 , X 4 , X 5 , . . . X Y Y 1 , Y 2 , Y 3 , Y 4 , Y 5 , . . . When can P simulate Q ? a , b ∈ [ k ] a b ( a , b ) ∼ Q

Non-Interactive Simulation of Joint Distributions Main Qvestion problem be HARD? How can a “constant-sized” Not obvious! Algorithmically decidable? OPEN in most cases! Analytically? 4 / 9 P ( X , Y ) X 1 , X 2 , X 3 , X 4 , X 5 , . . . X Y Y 1 , Y 2 , Y 3 , Y 4 , Y 5 , . . . When can P simulate Q ? a , b ∈ [ k ] a b ( a , b ) ∼ Q

Non-Interactive Simulation of Joint Distributions Main Qvestion problem be HARD? How can a “constant-sized” Not obvious! Algorithmically decidable? OPEN in most cases! Analytically? 4 / 9 P ( X , Y ) X 1 , X 2 , X 3 , X 4 , X 5 , . . . X Y Y 1 , Y 2 , Y 3 , Y 4 , Y 5 , . . . When can P simulate Q ? a , b ∈ [ k ] a b ( a , b ) ∼ Q

Tensor Power Problems Non-interactive Simulation falls under the category of “Tensor Power” problems. 5 / 9

Tensor Power Problems Non-interactive Simulation falls under the category of “Tensor Power” problems. In Information Theory, 5 / 9 ▶ Zero-error Shannon capacity ▶ Zero-error Witsenhausen rate

Tensor Power Problems Non-interactive Simulation falls under the category of “Tensor Power” problems. In Information Theory, In Computer Science, 5 / 9 ▶ Zero-error Shannon capacity ▶ Zero-error Witsenhausen rate ▶ (Classical) Amortized value of 2-prover 1-round games ▶ (Qvantum) Entangled value of 2-prover 1-round games ▶ (Qvantum) Local State Transformation ▶ Computing SDP integrality gaps for CSPs ▶ Amortized communication complexity

Tensor Power Problems Non-interactive Simulation falls under the category of “Tensor Power” problems. In Information Theory, In Computer Science, [Braverman-Rao ’11], [Braverman-Schneider ’15] 5 / 9 ▶ Zero-error Shannon capacity ▶ Zero-error Witsenhausen rate ▶ (Classical) Amortized value of 2-prover 1-round games ▶ (Qvantum) Entangled value of 2-prover 1-round games ▶ (Qvantum) Local State Transformation ▶ Computing SDP integrality gaps for CSPs [Raghavendra-Steurer ’09] ▶ Amortized communication complexity = Information complexity

Tensor Power Problems Non-interactive Simulation falls under the category of “Tensor Power” problems. In Information Theory, In Computer Science, [Braverman-Rao ’11], [Braverman-Schneider ’15] 5 / 9 ▶ Zero-error Shannon capacity [Open] ▶ Zero-error Witsenhausen rate [Open] ▶ (Classical) Amortized value of 2-prover 1-round games [Open] ▶ (Qvantum) Entangled value of 2-prover 1-round games [Open] ▶ (Qvantum) Local State Transformation [Open] ▶ Computing SDP integrality gaps for CSPs [Raghavendra-Steurer ’09] ▶ Amortized communication complexity = Information complexity

Decidability via “Dimension Reduction” Main Qvestion 6 / 9 Can P simulate Q ?