TY - GEN

T1 - Communication complexity of estimating correlations

AU - Hadar, Uri

AU - Liu, Jingbo

AU - Polyanskiy, Yury

AU - Shayevitz, Ofer

N1 - Publisher Copyright:
© 2019 Association for Computing Machinery.

PY - 2019/6/23

Y1 - 2019/6/23

N2 - We characterize the communication complexity of the following distributed estimation problem. Alice and Bob observe infinitely many iid copies of ρ-correlated unit-variance (Gaussian or ±1 binary) random variables, with unknown ρ ∈ [−1, 1]. By interactively exchanging k bits, Bob wants to produce an estimate ρ of ρ. We show that the best possible performance (optimized over interaction protocol Π and estimator ρ) satisfies infΠρ supρ E[|ρ − ρ|2] = k−1(2 ln12 +o(1)). Curiously, the number of samples in our achievability scheme is exponential in k; by contrast, a naive scheme exchanging k samples achieves the same Ω(1/k) rate but with a suboptimal prefactor. Our protocol achieving optimal performance is one-way (non-interactive). We also prove the Ω(1/k) bound even when ρ is restricted to any small open sub-interval of [−1, 1] (i.e. a local minimax lower bound). Our proof techniques rely on symmetric strong data-processing inequalities and various tensorization techniques from information-theoretic interactive common-randomness extraction. Our results also imply an Ω(n) lower bound on the information complexity of the Gap-Hamming problem, for which we show a direct information-theoretic proof.

AB - We characterize the communication complexity of the following distributed estimation problem. Alice and Bob observe infinitely many iid copies of ρ-correlated unit-variance (Gaussian or ±1 binary) random variables, with unknown ρ ∈ [−1, 1]. By interactively exchanging k bits, Bob wants to produce an estimate ρ of ρ. We show that the best possible performance (optimized over interaction protocol Π and estimator ρ) satisfies infΠρ supρ E[|ρ − ρ|2] = k−1(2 ln12 +o(1)). Curiously, the number of samples in our achievability scheme is exponential in k; by contrast, a naive scheme exchanging k samples achieves the same Ω(1/k) rate but with a suboptimal prefactor. Our protocol achieving optimal performance is one-way (non-interactive). We also prove the Ω(1/k) bound even when ρ is restricted to any small open sub-interval of [−1, 1] (i.e. a local minimax lower bound). Our proof techniques rely on symmetric strong data-processing inequalities and various tensorization techniques from information-theoretic interactive common-randomness extraction. Our results also imply an Ω(n) lower bound on the information complexity of the Gap-Hamming problem, for which we show a direct information-theoretic proof.

KW - Communication complexity

KW - Correlation estimation

KW - Distributed estimation

UR - http://www.scopus.com/inward/record.url?scp=85068767533&partnerID=8YFLogxK

U2 - 10.1145/3313276.3316332

DO - 10.1145/3313276.3316332

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AN - SCOPUS:85068767533

T3 - Proceedings of the Annual ACM Symposium on Theory of Computing

SP - 792

EP - 803

BT - STOC 2019 - Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing

A2 - Charikar, Moses

A2 - Cohen, Edith

PB - Association for Computing Machinery

T2 - 51st Annual ACM SIGACT Symposium on Theory of Computing, STOC 2019

Y2 - 23 June 2019 through 26 June 2019

ER -