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- W2912332216 abstract "We characterize the communication complexity of the following distributed estimation problem. Alice and Bob observe infinitely many iid copies of $rho$-correlated unit-variance (Gaussian or $pm1$ binary) random variables, with unknown $rhoin[-1,1]$. By interactively exchanging $k$ bits, Bob wants to produce an estimate $hatrho$ of $rho$. We show that the best possible performance (optimized over interaction protocol $Pi$ and estimator $hat rho$) satisfies $inf_{Pi hatrho}sup_rho mathbb{E} [|rho-hatrho|^2] = tfrac{1}{k} (frac{1}{2 ln 2} + o(1))$. Curiously, the number of samples in our achievability scheme is exponential in $k$; by contrast, a naive scheme exchanging $k$ samples achieves the same $Omega(1/k)$ rate but with a suboptimal prefactor. Our protocol achieving optimal performance is one-way (non-interactive). We also prove the $Omega(1/k)$ bound even when $rho$ 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 $Omega(n)$ lower bound on the information complexity of the Gap-Hamming problem, for which we show a direct information-theoretic proof." @default.
- W2912332216 created "2019-02-21" @default.
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- W2912332216 date "2019-01-25" @default.
- W2912332216 modified "2023-09-27" @default.
- W2912332216 title "Communication Complexity of Estimating Correlations" @default.
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