提出球面赫林格-坎托罗维奇流的稳定性理论，用于提升差分隐私采样精度。
On the Stability of Spherical Hellinger-Kantorovich Flows and Their Implications for Differential Privacy
- 基于球面赫林格-坎托罗维奇几何构建梯度流，融合传输与反应机制。
- 给出潜在函数差异对采样轨迹影响的统一控制，实现维度无关的对数似然比界。
- 为差分隐私中的指数机制提供纯差分隐私和近似差分隐私的动态保障。
梯度流采样将吉布斯分布视为概率测度上能量泛函的极小化器，并生成收敛于目标的动态过程。在球面赫林格-坎托罗维奇（SHK）几何下，该流耦合了传输与反应，等价于出生-死亡朗之万动力学。本文发展了SHK梯度流的扰动理论：对于两个势函数 $V$ 与 $V^{ ext{′}}$，从相同初始状态出发，量化其对应流之间的偏差随时间传播规律。统一的扰动界给出了对数似然比和瑞尼散度的维度无关、逐点控制；额外结构下还可导出KL散度界。我们将这些结果应用于差分隐私中的指数机制近似采样：对数似然比控制提供了基于SHK采样的时变纯差分隐私保证，而KL界通过曲棍球棒散度给出了近似差分隐私证书。同时，我们推导出一个效用界，分离了指数机制固有的次优性与有限时间采样误差。
原文摘要 · Abstract (English)
Gradient-flow sampling interprets a Gibbs distribution as the minimizer of an energy functional over probability measures and generates dynamics converging to this target. Under spherical Hellinger-Kantorovich (SHK) geometry, the flow couples transport and reaction and coincides with birth-death Langevin dynamics. In this work, we develop a perturbation theory for SHK gradient flows. For two potentials $V$ and $V^{\prime}$, we compare the associated flows from a common initialization and quantify how potential discrepancies propagate over time. A uniform perturbation bound yields dimension-free, pointwise control of the log-likelihood ratio and Rényi divergence, while additional structure allows us to derive bounds for the KL divergence as well. We apply these results to approximate sampling for the exponential mechanism in differential privacy. The likelihood-ratio control provides explicit time-dependent Pure-DP guarantees for SHK-based samplers, while the KL bound yields Approximate-DP certificates via hockey-stick divergence. We also derive a utility bound separating intrinsic exponential-mechanism suboptimality from finite-time sampling error.
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