Imperfectly Shared Randomness in Communication

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1 Imperfectly Shared Randomness in Communication Madhu Sudan Harvard Joint work with Clément Canonne (Columbia), Venkatesan Guruswami (CMU) and Raghu Meka (UCLA). 11/16/2016 UofT: ISR in Communication 1 of 22

xx xx Alice Encoder Decoder Bob Remarkable mathematical discoveries: Prob.

2 Classical theory of communication Shannon (1948) Clean architecture for reliable communication. xx xx Alice Encoder Decoder Bob Remarkable mathematical discoveries: Prob. Method, Entropy, (Mutual) Information, Compression schemes, Error-correcting codes. But does Bob really need all of xx? 11/16/2016 UofT: ISR in Communication 2 of 22

3 [Yao 80] Communication Complexity The model xx (with shared randomness) ff: xx, yy Σ RR = $$$ yy Alice Bob CCCC ff = # bits exchanged by best protocol ff(xx, yy) w.p. 2/3 11/16/2016 UofT: ISR in Communication 3 of 22

4 Communication Complexity - Motivation Standard: Lower bounds Circuit complexity, Streaming, Data Structures, extended formulations This talk: Communication complexity as model for communication Food for thought: Shannon vs. Yao? Which is the right model. Typical (human-human/computer-computer) communication involves large context and brief communication. Contexts imperfectly shared. 11/16/2016 UofT: ISR in Communication 4 of 22

5 This talk Example problems with low communication complexity. New form of uncertainty and overcoming it. 11/16/2016 UofT: ISR in Communication 5 of 22

6 Aside: Easy CC Problems Equality testing: Protocol: EEEE xx, yy = 1 xx = yy; CCCC EEEE Fix = OO ECC 1 EE: 0,1 nn 0,1 NN ; Shared randomness: ii NN ; xx = (xx HH kk xx, yy = 1 Δ xx, yy kk; CCCC HH kk Exchange pppppppp = OO(kk kk Protocol: 1,, xx nn ) logeekk) xx[huang ii yy, EE= yyyy ii ; etal.] 1,, yy nn Accept Use common iff EE xx randomness ii = EE yy ii. kk xx, yy = 1 wt xx, wt yy kk & to ii hash ss. tt. xx nn ii = yy [kk xx, ii = 2 1. ] yy xx ii yy ii Hamming distance: Small set intersection: CCCC kk = OO(kk) [Håstad Wigderson] Gap (Real) Inner Product: Main Insight: xx, yy R nn ; xx 2, yy 2 = 1; If GG NN 0,1 nn, then GGGGPP cc,ss xx, yy = 1 if xx, yy εε; = 0 if xx, yy 0; EE GG, xx GG, yy = xx, yy Summary CCCC GGGGPP cc,ss from = OO 1 εε2 ; [Alon, Matias, Szegedy] [Ghazi, Kamath, S. 16] ii 11/16/2016 UofT: ISR in Communication 6 of 22

7 Uncertainty in Communication Overarching question: Are there communication mechanisms that can overcome uncertainty? What is uncertainty? A f R B This talk: Alice, Bob don t share randomness perfectly; only approximately. 11/16/2016 UofT: ISR in Communication 7 of 22

8 Rest of this talk Model: Imperfectly Shared Randomness Positive results: Coping with imperfectly shared randomness. Negative results: Analyzing weakness of imperfectly shared randomness. 11/16/2016 UofT: ISR in Communication 8 of 22

9 Model: Imperfectly Shared Randomness Alice rr ; and Bob ss where rr, ss = i.i.d. sequence of correlated pairs rr ii, ss ii ii ; rr ii, ss ii { 1, +1}; EE rr ii = EE ss ii = 0; EE rr ii ss ii = ρρ 0. Notation: iiiirr ρρ (ff) = cc of ff with ρρ-correlated bits. cccc(ff): Perfectly Shared Randomness cc. pppppppp ff : cc with PRIVate randomness Starting point: for Boolean functions ff cccc ff iiiirr ρρ ff pppppppp ff cccc ff + log nn What if cccc ff log nn? E.g. cccc ff = OO(1) = iiiirr 1 ff = iiiirr 0 ff ρρ ττ iiiirr ρρ ff iiiirr ττ ff 11/16/2016 UofT: ISR in Communication 9 of 22

10 Distill Perfect Randomness from ISR? Agreement Distillation: Alice rr; Bob ss; rr, ss ρρ-corr. unbiased bits Outputs: Alice uu; Bob vv; HH uu, HH vv kk Communication = cc bits; What is max. prob. ττ of agreement (uu = vv)? Well-studied in the literature! [Ahlswede-Csiszar 70s]: ττ 1 cc = Ω kk (one-way) [Bogdanov-Mossel 2000s]: cc = 0 ττ exp( kk) [Our work]: ττ = exp( kk + OO cc ) (two-way) Summary: Can t distill randomness! 11/16/2016 UofT: ISR in Communication 10 of 22

11 Results Model first studied by [Bavarian,Gavinsky,Ito 14] ( Independently and earlier ). Their focus: Simultaneous Communication; general models of correlation. They show iiiiii Equality = OO 1 (among other things) Our Results: Generally: cccc ff kk iiiiii ff 2 kk Converse: ff with cccc ff kk & iiiiii ff 2 kk 11/16/2016 UofT: ISR in Communication 11 of 22

12 Equality Testing (our proof) Key idea: Think inner products. Encode xx XX = EE(xx);yy YY = EE yy ;XX, YY 1, +1 NN xx = yy XX, YY = NN xx yy XX, YY NN/2 Estimating inner products: Building on sketching protocols Alice: Picks Gaussians GG 1, GG tt R NN, Sends ii tt maximizing GG ii, XX to Bob. Bob: Accepts iff GG ii, YY 0 Analysis: OO ρρ (1) bits suffice if GG ρρ GG Gaussian Protocol 11/16/2016 UofT: ISR in Communication 12 of 22

13 General One-Way Communication Idea: All communication Inner Products (For now: Assume one-way-cc ff kk) For each random string RR Alice s message = ii RR 2 kk Bob s output = ff RR (ii RR ) where ff RR : 2 kk 0,1 W.p. 2 3 over RR, ff RR ii RR is the right answer. 11/16/2016 UofT: ISR in Communication 13 of 22

14 General One-Way Communication For each random string RR Alice s message = ii RR 2 kk Bob s output = ff RR (ii RR ) where ff RR : 2 kk 0,1 W.p. 2 3, ff RR ii RR is the right answer. Vector representation: ii RR xx RR 0,1 2kk (unit coordinate vector) ff RR yy RR 0,1 2kk (truth table of ff RR ). ff RR ii RR = xx RR, yy RR ; Acc. Prob. XX, YY ; XX = xx RR RR ; YY = yy RR RR Gaussian protocol estimates inner products of unit vectors to within ±εε with OO ρρ 1 εε 2 communication. 11/16/2016 UofT: ISR in Communication 14 of 22

Two-way communication Still decided by inner products. Simple lemma: KK AA kk, KK BB kk R 2kk convex, that describe private coin k-bit comm. strategies for Alice, Bob s.t. accept prob.

15 Two-way communication Still decided by inner products. Simple lemma: KK AA kk, KK BB kk R 2kk convex, that describe private coin k-bit comm. strategies for Alice, Bob s.t. accept prob. of ππ AA KK AA kk, ππ BB KK BB kk equals ππ AA, ππ BB Putting things together: Theorem: cccc ff kk iiiiii ff OO ρρ (2 kk ) 11/16/2016 UofT: ISR in Communication 15 of 22

16 Main Technical Result: Matching lower bound Theorem: There exists a (promise) problem ff s.t. cccc ff kk iiiirr ρρ ff exp (kk) The Problem: Gap Sparse Inner Product (G-Sparse-IP). Alice gets sparse xx 0,1 nn ; wt xx 2 kk nn Bob gets yy 0,1 nn Promise: xx, yy.9 2 kk nn or xx, yy.6 2 kk nn. Decide which. G-Sparse-IP: xx, yy 00, 11 nn ; wwww xx 22 kk nn Decide xx, yy (. 99) 22 kk nn or xx, yy (. 66) 22 kk nn? 11/16/2016 UofT: ISR in Communication 16 of 22

pppppp Protocol for G-Sparse-IP Note: Gaussian protocol takes OO 2 kk bits. Need to get exponentially better. Idea: xx ii 0 yy ii correlated with answer.

17 pppppp Protocol for G-Sparse-IP Note: Gaussian protocol takes OO 2 kk bits. Need to get exponentially better. Idea: xx ii 0 yy ii correlated with answer. Use (perfectly) shared randomness to find random index ii s.t. xx ii 0. Shared randomness: ii 1, ii 2, ii 3, uniform over [nn] Alice Bob: smallest index jj s.t. xx iijj 0. Bob: Accept if yy iijj = 1 Expect jj 2 kk ; cccc kk. G-Sparse-IP: xx, yy 00, 11 nn ; wwww xx 22 kk nn Decide xx, yy (. 99) 22 kk nn or xx, yy (. 66) 22 kk nn? 11/16/2016 UofT: ISR in Communication 17 of 22

Lower Bound Idea Catch: Dist., Det. Protocol w. comm. kk.

Main Idea: xx Protocol can look like G-Sparse-Inner-Product But

hard Protocol can ignore sparsity Requires 2 kk bits Protocol can

18 Lower Bound Idea Catch: Dist., Det. Protocol w. comm. kk. Need to fix strategy first, construct dist. later. Main Idea: xx Protocol can look like G-Sparse-Inner-Product But implies players can agree on common index to focus on Agreement is hard Protocol can ignore sparsity Requires 2 kk bits Protocol can look like anything! Invariance Principle [MOO, Mossel ] 11/16/2016 UofT: ISR in Communication 18 of 22

19 Invariance Principle [MOO 08] Informally: Analysis of prob. processes with bits often hard. Analysis of related processes with Gaussian variables easy. Invariance Principle under sufficiently general conditions prob. of events invariant when switching from bits to Gaussian Theorem: For every convex KK 1, KK 2 1,1 l transformations TT 1, TT 2 s.t. if ff: 0,1 nn KK 1 and gg: 0,1 nn KK 2 have no common influential variable, then FF = TT 1 ff: R nn KK 1 and GG = TT 2 gg: R nn KK 2 satisfy Exp xx,yy ff xx, gg yy Exp XX,YY FF XX, GG YY 11/16/2016 UofT: ISR in Communication 19 of 22

20 Summarizing kk bits of comm. with perfect sharing 2 kk bits with imperfect sharing. This is tight Invariance principle for communication Agreement distillation Low-influence strategies G-Sparse-IP: xx, yy 00, 11 nn ; wwww xx 22 kk nn Decide xx, yy (. 99) 22 kk nn or xx, yy (. 66) 22 kk nn? 11/16/2016 UofT: ISR in Communication 20 of 22

21 Conclusions Imperfect agreement of context important. Dealing with new layer of uncertainty. Notion of scale (context LARGE) Many open directions+questions: Imperfectly shared randomness: One-sided error? Does interaction ever help? How much randomness? More general forms of correlation? 11/16/2016 UofT: ISR in Communication 21 of 22

22 Thank You! 11/16/2016 UofT: ISR in Communication 22 of 22

Communication Amid Uncertainty

Communication Amid Uncertainty Madhu Sudan Harvard University Based on joint works with Brendan Juba, Oded Goldreich, Adam Kalai, Sanjeev Khanna, Elad Haramaty, Jacob Leshno, Clement Canonne, Venkatesan