Minimax Risks and Optimal Procedures for Estimation under Functional Local Differential Privacy

As concerns about data privacy continue to grow, differential privacy (DP) has emerged as a fundamental concept that aims to guarantee privacy by ensuring individuals' indistinguishability in data analysis. Local differential privacy (LDP) is a rigorous type of DP that requires individual data to be privatized before being sent to the collector, thus removing the need for a trusted third party to collect data. Among the numerous (L)DP-based approaches, functional DP has gained considerable attention in the DP community because it connects DP to statistical decision-making by formulating it as a hypothesis-testing problem and also exhibits Gaussian-related properties. However, the utility of privatized data is generally lower than that of non-private data, prompting research into optimal mechanisms that maximize the statistical utility for given privacy constraints. In this study, we investigate how functional LDP preserves the statistical utility by analyzing minimax risks of univariate mean estimation as well as nonparametric density estimation. We leverage the contraction property of functional LDP mechanisms and classical information-theoretical bounds to derive private minimax lower bounds. Our theoretical study reveals that it is possible to establish an interpretable, continuous balance between the statistical utility and privacy level, which has not been achieved under the $\epsilon$-LDP framework. Furthermore, we suggest minimax optimal mechanisms based on Gaussian LDP (a type of functional LDP) that achieve the minimax upper bounds and show via a numerical study that they are superior to the counterparts derived under $\epsilon$-LDP. The theoretical and empirical findings of this work suggest that Gaussian LDP should be considered a reliable standard for LDP.