Communication with Imperfect Shared Randomness

Transcription

1 Communication with Imperfect Shared Randomness (Joint work with Venkatesan Guruswami (CMU), Raghu Meka (?) and Madhu Sudan (MSR)) Who? Clément Canonne (Columbia University) When? November 19, / 17

2 Communication & Complexity There is a world outside of n Context There is Alice, Bob, what they communicate and what they don t have to. 2 / 17

3 Communication & Complexity There is a world outside of n Context There is Alice, Bob, what they communicate and what they don t have to. The f : {0, 1} n {0, 1} n {0, 1}, they compute; the protocol Π they use; from which D x, D y their inputs come; what is blue and what red means. 2 / 17

4 Communication & Complexity But context is not perfect... Noise, misunderstandings, false assumptions Context is almost never perfectly shared. 3 / 17

5 Communication & Complexity But context is not perfect... Noise, misunderstandings, false assumptions Context is almost never perfectly shared. My periwinkle is your orchid. 3 / 17

6 Communication & Complexity But context is not perfect... Noise, misunderstandings, false assumptions Context is almost never perfectly shared. My periwinkle is your orchid. the printer on the 5 th floor of Columbia is not exactly the model my laptop has a driver for. 3 / 17

7 Communication & Complexity But context is not perfect... Noise, misunderstandings, false assumptions Context is almost never perfectly shared. My periwinkle is your orchid. the printer on the 5 th floor of Columbia is not exactly the model my laptop has a driver for. what precisely is a French baguette around here? 3 / 17

8 Communication & Complexity What about randomness? Equality testing Complexity? I have x {0, 1} n, you have y {0, 1} n, are they equal? Deterministic det-cc(eq) = Θ(n) Private randomness private-cc(eq) = Θ(log n) Shared randomness psr-cc(eq) = O(1) (Recall Newman s Theorem: private-cc(p) psr-cc(p) + O(log n).) 4 / 17

9 This work Randomness and uncertainty ISR (Imperfectly Shared Randomness) What if the randomness ( context ) was not perfectly in sync? To compute f (x, y): Alice: has access to r {±1}, gets input x {0, 1} n Bob: has access to s {±1}, gets input y {0, 1} n w/ r ρ s: Er i = Es i = 0, Er i s i = ρ, (r i, s i ) (r j, s j ). 5 / 17

10 This work Randomness and uncertainty ISR (Imperfectly Shared Randomness) What if the randomness ( context ) was not perfectly in sync? To compute f (x, y): Alice: has access to r {±1}, gets input x {0, 1} n Bob: has access to s {±1}, gets input y {0, 1} n w/ r ρ s: Er i = Es i = 0, Er i s i = ρ, (r i, s i ) (r j, s j ). Studied (independently) by [BGI14] (different focus: referee model ; more general correlations). 5 / 17

11 ISR: general relations For every P with x, y {0, 1} n and 0 ρ ρ 1, psr-cc(p) isr-cc ρ (P) isr-cc ρ (P) private-cc(p) psr-cc(p) + O(log n). (also true for one-way: psr-cc ow, isr-cc ow ρ, private-cc ow ) but for many problems, log n is already huge. 6 / 17

12 Rest of the talk 1 A first example: the Compression problem 2 General upperbound on ISR in terms of PSR 3 Strong lower bound: Alice, Bob, Charlie and Dana. 7 / 17

13 First result: Compression Compression with uncertain priors Previous work P = Q P Q P Q Alice has P, gets m P; Bob knows Q P, wants m. H(P) (Huffman coding) H(P) + 2 [JKKS11] (w/ shared randomness) O(H(P) + + log log N) [HS14] (deterministic) 8 / 17

14 First result: Compression Compression with uncertain priors Previous work P = Q P Q P Q Alice has P, gets m P; Bob knows Q P, wants m. H(P) (Huffman coding) H(P) + 2 [JKKS11] (w/ shared randomness) O(H(P) + + log log N) [HS14] (deterministic) This work For all ɛ > 0, isr-cc ow ρ (Compress ) 1+ɛ 1 h( 1 ρ 2 )(H(P) O(1)) natural protocol 8 / 17

15 General upperbound It s inner products all the way down! Theorem ρ > 0, c < such that k, we have PSR-CC(k) ISR-CC ow ρ (c k ). Proof. (Outline) Define GapInnerProduct, complete for PSR-CC(k) (see strategies as X R, Y R {0, 1} 2k ; use Newman s Theorem to bound # R s); Show there exists a (Gaussian-based) isr protocol for GapInnerProduct, with O ρ (4 k ) bits of comm. 9 / 17

16 General upperbound Can we do better? For problems in PSR-CC ow (k)? PSR-CC ow (k) ISR-CC ow ρ (c o(k) )? For ISR-CC ρ? PSR-CC(ω(k)) ISR-CC ρ (c k )? 10 / 17

17 General upperbound Can we do better? For problems in PSR-CC ow (k)? PSR-CC ow (k) ISR-CC ow ρ (c o(k) )? For ISR-CC ρ? PSR-CC(ω(k)) ISR-CC ρ (c k )? Answer No. 10 / 17

18 Strong converse: lower bound It s as good as it gets. Theorem k, P = (P n ) n N s.t. psr-cc ow (P) k, yet ρ < 1 isr-cc ρ (P) = 2 Ωρ(k). Proof. (High-level) Define SparseGapInnerProduct, relaxation of GapInnerProduct. Show it has as O(log q)-bit one-way psr protocol (Alice uses the shared randomness to send one coordinate to Bob) isr lower bound: argue that for any (fixed) * strategy of Alice and Bob using less that q bits, either (a) something impossible happens in the Boolean world, or (b) something impossible happens in the Gaussian world. 11 / 17

19 Strong converse: lower bound Two-pronged impossibility, first prong. Case (a) Definition (Agreement distillation) The strategies (f r, g s ) r,s have common high-influence variable (recall the one-way psr protocol). But then, two players Charlie and Dana can * leverage this strategies to win an agreement distillation game: Charlie and Dana have no inputs. Their goal is to output w C and w D satisfying: (i) Pr[ w C = w D ] γ; (ii) H (w C ), H (w D ) κ. But this requires Ω(κ) log(1/γ) bits of communication (via [BM10, Theorem 1]). 12 / 17

20 Strong converse: lower bound Two-pronged impossibility, second prong. Case (b) f r : {0, 1} n K A [0, 1] 2k, g s : {0, 1} n K B [0, 1] 2k have no common high-influence variable. We then show that this implies k = 2 Ω( q), by using an Invariance Principle (in the spirit of [Mos10]) to go to the Gaussian world : if f, g are low-degree polynomials with no common influential variable, then E (x,y) N n [ f (x), g(y) ] E (X,Y ) G n [ F (X ), G(Y ) ] and Charlie and Dana can use this solve (yet another) problem, the Gaussian Inner Product (GaussianCorrelation ξ ). But... a reduction to Disjointness shows that (even with psr) this requires Ω1/ξ bits of communication. 13 / 17

21 Conclusions Summary Dealing with more realistic situations: Alice, Bob, and what they do not know about each other; comes into play when n is huge (Newman s Theorem becomes loose); show general and tight relations and reductions in this model, with both upper and lower bounds. a new invariance theorem, and use in comm. complexity. 14 / 17

22 Conclusions Summary What about... Dealing with more realistic situations: Alice, Bob, and what they do not know about each other; comes into play when n is huge (Newman s Theorem becomes loose); show general and tight relations and reductions in this model, with both upper and lower bounds. a new invariance theorem, and use in comm. complexity. more general forms of correlations? cases where even randomness is expensive? (minimize its use) one-sided error? 14 / 17

23 Thank you. (Questions?) 15 / 17

24 Theorem (Our Invariance Principle) Fix any two parameters p 1, p 2 ( 1, 1). For all ε (0, 1], l N, θ 0 [0, 1), and closed convex sets K 1, K 2 [0, 1] l there exist τ > 0 and mappings T 1 : {f : {+1, 1} n K 1 } {F : R n K 1 } T 2 : {g : {+1, 1} n K 2 } {G : R n K 2 } such that for all θ [ θ 0, θ 0 ], if f, g satisfy ( ) max min max Inf i(d)f j, max Inf i(d)g j i [n] j [l] j [l] τ then, for F = T 1 (f ) and G = T 2 (g), we have where N = N p1,p 2,θ and G is the Gaussian distribution which matches the first and second-order moments of N. 16 / 17

25 Theorem (Invariance Theorem of [GHM + 11]) Let (Ω, µ) be a finite prob. space with each prob. at least α 1/2. Let b = Ω and L = {χ 0 = 1, χ 1, χ 2,..., χ b 1 } be a basis for r.v. s over Ω. Let Υ = {ξ 0 = 1, ξ 1,..., ξ b 1 } be an ensemble of real-valued Gaussian r.v. s with 1 st and 2 nd moments matching those of the χ i s; and h = (h 1, h 2,..., h t ): Ω n R t s.t. Inf i (h l ) τ, Var(h l ) 1 for all i [n] and l [t]. For η (0, 1), let H l (l [t]) be the multilinear polynomial associated with T 1 η h l w.r.t. L. If Ψ: R t R is Λ-Lipschitz (w.r.t. the L 2 -norm), then [Ψ E ( H 1 (L n ),, H t (L n ) )] [ E Ψ ( H 1 (Υ n ),, H t (Υ n ) )] for some constant C = C(t). C(t) Λ τ η 18 log 1 α = oτ (1) 17 / 17

26 Mohammad Bavarian, Dmitry Gavinsky, and Tsuyoshi Ito. On the role of shared randomness in simultaneous communication. In Javier Esparza, Pierre Fraigniaud, Thore Husfeldt, and Elias Koutsoupias, editors, ICALP (1), volume 8572 of Lecture Notes in Computer Science, pages Springer, Andrej Bogdanov and Elchanan Mossel. On extracting common random bits from correlated sources. CoRR, abs/ , Venkatesan Guruswami, Johan Håstad, Rajsekar Manokaran, Prasad Raghavendra, and Moses Charikar. Beating the random ordering is hard: Every ordering CSP is approximation resistant. SIAM J. Comput., 40(3): , Elad Haramaty and Madhu Sudan. Deterministic compression with uncertain priors. In Proceedings of the 5th Conference on Innovations in Theoretical Computer Science, ITCS 14, pages , New York, NY, USA, ACM. Brendan Juba, Adam Tauman Kalai, Sanjeev Khanna, and Madhu Sudan. Compression without a common prior: an information-theoretic justification for ambiguity in language. In Bernard Chazelle, editor, ICS, pages Tsinghua University Press, Elchanan Mossel. Gaussian bounds for noise correlation of functions. Geometric and Functional Analysis, 19(6): , / 17