Communication Complexity of Estimating Correlations

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 $\rho\in[-1,1]$. By interactively exchanging $k$ bits, Bob wants to produce an estimate $\hat\rho$ of $\rho$. We show that the best possible performance (optimized over interaction protocol $\Pi$ and estimator $\hat \rho$) satisfies $\inf_{\Pi,\hat\rho}\sup_\rho \mathbb{E} [|\rho-\hat\rho|^2] = \Theta(\tfrac{1}{k})$. Furthermore, we show that the best possible unbiased estimator achieves performance of $1+o(1)\over {2k\ln 2}$. Curiously, thus, restricting communication to $k$ bits results in (order-wise) similar minimax estimation error as restricting to $k$ samples. 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. Notably, the protocol achieving (almost) optimal performance is one-way (non-interactive). For one-way protocols we also prove the $\Omega(\tfrac{1}{k})$ bound even when $\rho$ is restricted to any small open sub-interval of $[-1,1]$ (i.e. a local minimax lower bound). %We do not know if this local behavior remains true in the interactive setting. Our proof techniques rely on symmetric strong data-processing inequalities, various tensorization techniques from information-theoretic interactive common-randomness extraction, and (for the local lower bound) on the Otto-Villani estimate for the Wasserstein-continuity of trajectories of the Ornstein-Uhlenbeck semigroup.