# Communication Complexity of Randomness of 17 Randomness Processing Industry Dispersers, Extractors,

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• of 17May 29, 2019 TIFR: Comm. Comp. Randomness Manipulation 1

Communication Complexity of Randomness Manipulation

Madhu Sudan Harvard University

Joint work with Mitali Bafna (Harvard), Badih Ghazi (Google), and Noah Golowich (Harvard)

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Randomness Processing Industry

 Dispersers, Extractors, Merges, Condensers, PRGs …

 Interest not in the exact function. (E=RSA most boring.)

 Rather in its manipulation of distributions …  Distribution of 𝑋𝑋 vs. that of 𝐸𝐸(𝑋𝑋)

 Long history … omitted. Key ingredients  single processor “𝐸𝐸”  unknown source 𝑋𝑋 ∈ 𝒳𝒳

May 29, 2019 TIFR: Comm. Comp. Randomness Manipulation 2

𝐸𝐸𝑋𝑋 𝐸𝐸(𝑋𝑋)

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 2-player setting:

 Is 𝛿𝛿 = 0 possible? If not minimize 𝛿𝛿! Etc.  Alice + Bob can use private randomness  Zero communication version: “Non-Interactive

Simulation” (NIS).

Distributed Randomness Processing

May 29, 2019 TIFR: Comm. Comp. Randomness Manipulation 3

Alice 𝑈𝑈

Bob𝑌𝑌 𝑉𝑉

≤ 𝐶𝐶 bits, ≤ 𝑟𝑟 Rounds

𝑋𝑋𝑋𝑋1,𝑋𝑋2, …

𝑌𝑌1,𝑌𝑌2, …

𝑋𝑋,𝑌𝑌 ∼ 𝑃𝑃 Goal: 𝑈𝑈,𝑉𝑉 ≈𝛿𝛿 𝑄𝑄

Infinite sequence

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A classical example

 Zero communication  𝑋𝑋,𝑌𝑌 ∼ Unif 0,0 , 0,1 , 1,0  𝑈𝑈,𝑉𝑉 ∼ 𝜌𝜌-correlated bits.

 𝑈𝑈,𝑉𝑉 ∼ Unif 0,1 , Pr 𝑈𝑈 = 𝑉𝑉 = 1+𝜌𝜌 2

 [Witsenhausen ‘70s]:  Can’t achieve 𝜌𝜌 = 1. Not even 𝜌𝜌 = .51

 (Naïve strategy achieves 𝜌𝜌 = 1 4 )

 Can achieve 𝜌𝜌 = 1 3

by a general method.

 Best 𝜌𝜌? Open!!

May 29, 2019 TIFR: Comm. Comp. Randomness Manipulation 4

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Witsenhausen …

 Intermediate Problem: Output (𝐴𝐴,𝐵𝐵) w. 𝐴𝐴,𝐵𝐵 ∼ 𝑁𝑁 0,1 with maximum correlation.

 Definition: Max-Corr 𝜌𝜌 𝑋𝑋,𝑌𝑌 ≜ max 𝑓𝑓,𝑔𝑔

𝔼𝔼𝑋𝑋,𝑌𝑌 𝑓𝑓 𝑋𝑋 𝑔𝑔(𝑌𝑌)

 Where 𝑓𝑓,𝑔𝑔:Ω → ℝ, 𝔼𝔼𝑋𝑋 𝑓𝑓 𝑋𝑋 = 𝔼𝔼𝑌𝑌 𝑔𝑔 𝑌𝑌 = 0 𝔼𝔼𝑋𝑋 𝑓𝑓 𝑋𝑋 2 = 𝔼𝔼𝑌𝑌 𝑔𝑔 𝑌𝑌 2 = 1

 Thm 1: For any 𝑈𝑈,𝑉𝑉 output, 𝜌𝜌 𝑈𝑈,𝑉𝑉 ≤ 𝜌𝜌(𝑋𝑋,𝑌𝑌)  Thm 2: For Gaussian output (𝐴𝐴,𝐵𝐵), can achieve

𝜌𝜌 𝐴𝐴,𝐵𝐵 = 𝜌𝜌(𝑋𝑋,𝑌𝑌)  Thm 3: For binary 𝑈𝑈,𝑉𝑉 , can achieve

1 − 2 ⋅ cos−1 𝜌𝜌 𝑋𝑋,𝑌𝑌

𝜋𝜋 ≤ 𝜌𝜌 𝑈𝑈,𝑉𝑉 ≤ 𝜌𝜌 𝑋𝑋,𝑌𝑌

May 29, 2019 TIFR: Comm. Comp. Randomness Manipulation 5

For Bernoulli/Gaussians Max-Correlation = Correlation

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“Tensorization”

 “Single-letter” Problems: Given (1) structure 𝑆𝑆 (graph, distribution, game) (2) product operation 𝑆𝑆⊗𝑡𝑡, (3) measure 𝑀𝑀, compute lim

𝑡𝑡→∞ 𝑀𝑀 𝑆𝑆⊗𝑡𝑡

 Shannon capacity, Compression length, Channel capacity, parallel repetition value of 2-prover game, Direct sum complexity, NIS

 “Tensorizing bound”:  Typically: 𝑀𝑀 𝑆𝑆⊗𝑡𝑡 ≥ 𝑀𝑀(𝑆𝑆).  Find: ℳ(⋅) s.t. 𝑀𝑀 𝑆𝑆 ≤ ℳ(𝑆𝑆) and ℳ 𝑆𝑆⊗𝑡𝑡 = ℳ(𝑆𝑆)

 Max-Correlation “Tensorizes”  If 𝑋𝑋,𝑌𝑌 ∼ 𝜇𝜇 & 𝑋𝑋𝑡𝑡,𝑌𝑌𝑡𝑡 ∼ 𝜇𝜇⊗𝑡𝑡 then 𝜌𝜌 𝑋𝑋𝑡𝑡 ,𝑌𝑌𝑡𝑡 = 𝜌𝜌(𝑋𝑋,𝑌𝑌)  Proof idea: 𝜇𝜇 described by matrix 𝑃𝑃; 𝜌𝜌 related to its

singular values, 𝜇𝜇⊗𝑡𝑡 described by 𝑃𝑃⊗𝑡𝑡

May 29, 2019 TIFR: Comm. Comp. Randomness Manipulation 6

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Single-letter Embarassment Problem Base Single-letter

version Compressibility P P

Channel capacity NP P

May 29, 2019 TIFR: Comm. Comp. Randomness Manipulation 7

Glossary: 0 ≤ P ≤ 𝑁𝑁𝑃𝑃 ≤ EXP ≤ Computable ≤ CA Computably Approximable ≤ ∞

Problem Base Single-letter version

Compressibility P P

Channel capacity NP P

Independent set/Shannon capacity

NP [0,∞]

Value of 2-prover game NP [NP,∞]

Problem Base Single-letter version

Compressibility P P

Channel capacity NP P

Independent set/Shannon capacity

NP [0,∞]

Value of 2-prover game NP [NP,∞]

Non-interactive-simulation (“correlation”)

NP [0,𝐶𝐶𝐴𝐴]

Communication Complexity [NP,Exp?] 0,𝐶𝐶𝐴𝐴

Problem Base Single-letter version

Compressibility P P

Channel capacity NP P

Independent set/Shannon capacity

NP [0,∞]

Value of 2-prover game NP [NP,∞]

Non-interactive-simulation (“correlation”)

NP [0,𝐶𝐶𝐴𝐴]

Communication Complexity [NP,Exp?] 0,𝐶𝐶𝐴𝐴

? P [NP,∞]

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Complexity of NIS

 “Computably approximable”  [Ghazi,Kamath,S.], [De,Mossel,Neeman]2,

[Ghazi,Kamath,Raghavendra]  There exists a finite time algorithm determining if we can

get 𝜖𝜖-close to (𝑈𝑈,𝑉𝑉) from i.i.d. samples of (𝑋𝑋,𝑌𝑌)

 Idea: “Invariance Priniciple” [Mossel]  Either there is a method using few samples of (𝑋𝑋,𝑌𝑌) or

many samples 𝑋𝑋1,𝑌𝑌1 , … , 𝑋𝑋𝑛𝑛,𝑌𝑌𝑛𝑛 each with low influence on 𝑈𝑈,𝑉𝑉 .

 If latter, can replace 𝑋𝑋,𝑌𝑌 by finite # Gaussians.  Gives computable upper bound on #samples needed to

get close to optimal strategy.

May 29, 2019 TIFR: Comm. Comp. Randomness Manipulation 8

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Communication?

 Restrict to “Common Randomness Generation” (𝑘𝑘-CRG): Output (𝑈𝑈 = 𝑉𝑉), with 𝑈𝑈 ∼ Unif( 0,1 𝑘𝑘)

 First considered ~70 [Ahlswede-Csizar]:  “Characterization”: Can communicate 𝑅𝑅 ⋅ 𝑘𝑘 + 𝑜𝑜 𝑘𝑘 bits, where 𝑅𝑅 = min

Π ICint(Π) ICext Π

 Computable? Computably approximable?  For one-way communication: 𝑅𝑅 = 1 − 𝜌𝜌 𝑋𝑋,𝑌𝑌 2

[Zhao-Chia] (see also [Guruswami-Radhakrishnan, Ghazi-Jayram, S.-Tyagi-Watanabe])

May 29, 2019 TIFR: Comm. Comp. Randomness Manipulation 9

Alice 𝑈𝑈

Bob 𝑉𝑉

≤ 𝐶𝐶 bits, ≤ 𝑟𝑟 Rounds

𝑋𝑋1,𝑋𝑋2, …

𝑌𝑌1,𝑌𝑌2, …

𝑋𝑋,𝑌𝑌 ∼ 𝑃𝑃 Goal: 𝑈𝑈,𝑉𝑉 ≈𝛿𝛿 𝑄𝑄

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This talk: Rounds in CRG

 Does increasing #rounds ease CRG? (For some (𝑋𝑋,𝑌𝑌)?) … Is the following true?

 Question: ∀𝜖𝜖 > 0 , 𝑟𝑟, ∃(𝑋𝑋,𝑌𝑌) s.t. ∀𝑘𝑘  ∃ 𝑟𝑟, 𝜖𝜖𝑘𝑘 -protocol for 𝑘𝑘-CRG 𝑋𝑋,𝑌𝑌  No 𝑟𝑟 − 1, 1 − 𝜖𝜖 .𝑘𝑘 -protocol for 𝑘𝑘-CRG(𝑋𝑋,𝑌𝑌)

 Still don’t know. Partial progress.  Thm. [BGGS’19] : ∀𝑛𝑛,𝑘𝑘, 𝑟𝑟 there exists 𝑋𝑋,𝑌𝑌 s.t.

 ∃ 𝑟𝑟,𝑂𝑂(𝑟𝑟 log 𝑛𝑛 -protocol for 𝑘𝑘-CRG(𝑋𝑋,𝑌𝑌).

 No 𝑟𝑟 2 − 3, min 𝑘𝑘, 𝑛𝑛

polylog 𝑛𝑛 -protocol for 𝑘𝑘-CRG(𝑋𝑋,𝑌𝑌)

May 29, 2019 TIFR: Comm. Comp. Randomness Manipulation 10

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Our Source:

May 29, 2019 TIFR: Comm. Comp. Randomness Manipulation 11

𝑖𝑖0

𝜋𝜋1

𝜋𝜋2

𝜋𝜋3

𝜋𝜋𝑟𝑟

𝐴𝐴1 𝐴𝐴2 𝐴𝐴3 𝐴𝐴𝑛𝑛 𝐵𝐵1 𝐵𝐵2 𝐵𝐵3 𝐵𝐵𝑛𝑛… …

Alice’s Inputs 𝑋𝑋 = (𝜋𝜋1,𝜋𝜋3, … ;𝐴𝐴1, … ,𝐴𝐴𝑛𝑛)

Bob’s Inputs 𝑌𝑌 = (𝑖𝑖0;𝜋𝜋2,𝜋𝜋4, … ;𝐵𝐵1, … ,𝐵𝐵𝑛𝑛)

𝑖𝑖0 ∼ [𝑛𝑛]

𝜋𝜋1, … ,𝜋𝜋𝑛𝑛 ∼ 𝑆𝑆𝑛𝑛 𝐴𝐴1, … ,𝐴𝐴𝑛𝑛 ∼ 0,1 𝑘𝑘

𝐵𝐵1, … ,𝐵𝐵𝑛𝑛 ∼ 0,1 𝑘𝑘

𝐴𝐴𝑖𝑖𝑟𝑟 = 𝐵𝐵𝑖𝑖𝑟𝑟 where 𝑖𝑖𝑗𝑗 = 𝜋𝜋𝑗𝑗 𝑖𝑖𝑗𝑗−1

“Pointer Chasing Source” (PCS)

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CRG from Pointer Chasing Source

 Easy direction easy: (𝑟𝑟, 𝑟𝑟 log𝑛𝑛)-protocol for 𝑘𝑘-CRG  Hardness?

 Pointer chasing is hard: [Duris-Galil-Schnitger, Nisan-Wigderson,…].

 CRG from PCS requires pointer chasing?  No! Can solve problem without pointers!  Small non-deterministic complexity!  Hardness needs to use hardness of

disjointness? And of pointer-chasing?  How to combine modularly?

May 29, 2019 TIFR: Comm. Comp. Randomness Manipulation 12

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Our solution: Pointer Verification Problem

 PVP: Distinguish 𝑋𝑋,𝑌𝑌 ∼ 𝑃𝑃𝑉𝑉YES from 𝑋𝑋,𝑌𝑌 ∼ 𝑃𝑃𝑉𝑉No  𝑃𝑃𝑉𝑉YES:𝑋𝑋 = 𝜋𝜋1,𝜋𝜋3, …𝜋𝜋𝑟𝑟−1 ; 𝑌𝑌 = 𝑖𝑖0, 𝑖𝑖𝑟𝑟; 𝜋𝜋2,𝜋𝜋4, …𝜋

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