Optimal Privacy-Utility Trade-Offs in LDP: Functional and Geometric Perspectives

Local differential privacy (LDP) has emerged as a gold-standard framework for privacy-preserving data analysis. However, characterizing the optimal privacy-utility trade-off (PUT) and the corresponding optimal LDP channels remains largely fragmented, relying on problem-specific, case-by-case analyses. In this work, we develop a unified theoretical framework that systematically characterizes the optimal PUT and optimal LDP channels for general privacy-preserving statistical decision-making problems. We first identify key functional properties of Bayesian and minimax risks as functions of the LDP channel, including the data processing inequality (DPI), direct-sum quasi-convexity (or additivity), concavity, and symmetry invariance. Leveraging these properties, we reduce the optimization domain required to compute the optimal PUT. Additionally, building on convex geometric insights, we establish a one-to-one correspondence between maximal LDP channels under the Blackwell order and a finite-dimensional polytope, yielding an exact geometric characterization. This result renders the optimal PUT computationally tractable via vertex enumeration or linear programming. Furthermore, when the underlying problem exhibits symmetries characterized by a transitive group action, we derive an exact analytic expression for the optimal PUT, leading to closed-form solutions without numerical optimization. Our framework applies broadly beyond risk minimization, encompassing the maximization of information-theoretic measures such as mutual information, $f$-divergences, and Fisher information over LDP channels. We demonstrate the efficacy of our theoretical framework by recovering or strengthening several known results, and deriving exact analytic expressions for the optimal PUTs in specific tasks that were previously unaddressed.