We study the problem of sampling from a distribution under local differential privacy (LDP). Given a private distribution $P \in \mathcal{P}$, the goal is to generate a single sample from a distribution that remains close to $P$ in $f$-divergence while satisfying the constraints of LDP. This task captures the fundamental challenge of producing realistic-looking data under strong privacy guarantees. While prior work by Park et al. (NeurIPS'24) focuses on global minimax-optimality across a class of distributions, we take a local perspective. Specifically, we examine the minimax risk in a neighborhood around a fixed distribution $P_0$, and characterize its exact value, which depends on both $P_0$ and the privacy level. Our main result shows that the local minimax risk is determined by the global minimax risk when the distribution class $\mathcal{P}$ is restricted to a neighborhood around $P_0$. To establish this, we (1) extend previous work from pure LDP to the more general functional LDP framework, and (2) prove that the globally optimal functional LDP sampler yields the optimal local sampler when constrained to distributions near $P_0$. Building on this, we also derive a simple closed-form expression for the locally minimax-optimal samplers which does not depend on the choice of $f$-divergence. We further argue that this local framework naturally models private sampling with public data, where the public data distribution is represented by $P_0$. In this setting, we empirically compare our locally optimal sampler to existing global methods, and demonstrate that it consistently outperforms global minimax samplers.

翻译：我们研究在局部差分隐私（LDP）约束下从分布中采样的问题。给定一个私有分布 $P \in \mathcal{P}$，目标是在满足 LDP 约束的同时，生成一个来自与 $P$ 在 $f$-散度意义上保持接近的分布的样本。这一任务捕捉了在强隐私保证下生成逼真数据的基本挑战。尽管 Park 等人（NeurIPS'24）的先前工作聚焦于一类分布上的全局极小极大最优性，我们则采取局部视角。具体而言，我们考察固定分布 $P_0$ 邻域内的极小极大风险，并刻画其精确值，该值同时依赖于 $P_0$ 和隐私水平。我们的主要结果表明，当分布类 $\mathcal{P}$ 被限制在 $P_0$ 的邻域内时，局部极小极大风险由全局极小极大风险决定。为证明这一点，我们（1）将先前工作从纯 LDP 扩展到更一般的泛函 LDP 框架，以及（2）证明当约束于 $P_0$ 附近的分布时，全局最优的泛函 LDP 采样器可产生最优的局部采样器。在此基础上，我们还推导了局部极小极大最优采样器的一个简单闭式表达式，该表达式不依赖于 $f$-散度的选择。我们进一步论证，该局部框架自然地建模了带有公开数据的隐私采样，其中公开数据分布由 $P_0$ 表示。在此设定下，我们通过实验将我们的局部最优采样器与现有全局方法进行比较，并证明其始终优于全局极小极大采样器。