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.

翻译：本地差分隐私（Local Differential Privacy, LDP）已成为隐私保护数据分析的金标准框架。然而，刻画最优隐私-效用权衡（Privacy-Utility Trade-off, PUT）及对应的最优LDP信道仍高度碎片化，依赖于特定问题的逐例分析。在本工作中，我们发展了一个统一的理论框架，系统地刻画了针对一般隐私保护统计决策问题的最优PUT与最优LDP信道。首先，我们识别了贝叶斯风险与极小极大风险作为LDP信道函数的关键函数性质，包括数据处理不等式（Data Processing Inequality, DPI）、直和拟凸性（或可加性）、凹性以及对称不变性。利用这些性质，我们简化了计算最优PUT所需的优化域。此外，基于凸几何的洞见，我们在Blackwell序下的极大LDP信道与一个有限维多面体之间建立了一一对应，从而给出了精确的几何刻画。这一结果通过顶点枚举或线性规划使得最优PUT的计算变得可行。进一步地，当底层问题呈现由传递群作用刻画的对称性时，我们推导出了最优PUT的精确解析表达式，从而无需数值优化即可得到闭式解。我们的框架广泛适用于超越风险最小化的场景，涵盖了在LDP信道上最大化互信息、f-散度以及Fisher信息等信息论度量。通过恢复或加强若干已知结果，并推导出此前未解决的具体任务中最优PUT的精确解析表达式，我们展示了这一理论框架的有效性。