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 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 inf_Πρ sup_ρ E[|ρ − ρ|²] = k−¹(_{2 ln}¹₂ +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.