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Communication complexity of estimating correlations
Name
1901.09100.pdf
Description
Submitted version
Size
333.63 KB
Format
Adobe PDF
Checksum (MD5)
34f6f6e794957e20ba64b6c1a03ffb8a
Date Issued
June 2019
Journal
Proceedings of the Annual ACM Symposium on Theory of Computing
Publisher
Association for Computing Machinery (ACM)
Citation
2019. "Communication complexity of estimating correlations." Proceedings of the Annual ACM Symposium on Theory of Computing.
Version
Original manuscript
Abstract
© 2019 Association for Computing Machinery. 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.
Terms of Use
Creative Commons Attribution-Noncommercial-Share Alike
Persistent DSpace Link
DOI of Published Version
10.1145/3313276.3316332
