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We identify fundamental tradeoffs between statistical utility and privacyunder local models of privacy in which data is kept private even from thestatistician, providing instance-specific bounds for private estimation andlearning problems by developing the \emph{local minimax risk}. In contrast toapproaches based on worst-case (minimax) error, which are conservative, thisallows us to evaluate the difficulty of individual problem instances anddelineate the possibilities for adaptation in private estimation and inference.Our main results show that the local modulus of continuity of the estimand withrespect to the variation distance---as opposed to the Hellinger distancecentral to classical statistics---characterizes rates of convergence underlocally private estimation for many notions of privacy, including differentialprivacy and its relaxations. As consequences of these results, we identify analternative to the Fisher information for private estimation, giving a morenuanced understanding of the challenges of adaptivity and optimality.

The Right Complexity Measure in Locally Private Estimation: It is not the Fisher Information