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