Local Privacy and Statistical Minimax Rates

Working under a model of privacy in which data remains private even from the statistician, we study the tradeoff between privacy guarantees and the utility of the resulting statistical estimators. We prove bounds on information-theoretic quantities, including mutual information and Kullback-Leibler divergence, that influence estimation rates as a function of the amount of privacy preserved. When combined with standard minimax techniques such as Le Cam's and Fano's methods, these inequalities allow for a precise characterization of statistical rates under local privacy constraints. In this paper, we provide a complete treatment of three canonical problem families: mean estimation in location family models, parameter estimation in fixed-design regression, and convex risk minimization. For all of these families, we provide lower and upper bounds that match up to constant factors, giving privacy-preserving mechanisms and computationally efficient estimators that achieve the bounds.