On the Stability of Spherical Hellinger-Kantorovich Flows and Their Implications for Differential Privacy
2026-05-22 • Cryptography and Security
Cryptography and SecurityMachine Learning
AI summaryⓘ
The authors study a way to sample from complex probability distributions by viewing the problem as an evolving flow guided by energy. They focus on a special geometry called spherical Hellinger-Kantorovich, which combines spreading and changing of mass, corresponding to a kind of birth-death process with noise. They develop a theory to understand how small changes in the energy function affect the sampling flows over time, providing precise bounds on differences in likelihood and divergence measures. Their results apply to differential privacy, giving clear guarantees about privacy loss and utility when using these sampling methods.
gradient flowGibbs distributionspherical Hellinger-Kantorovich geometrybirth-death Langevin dynamicsRényi divergenceKL divergencedifferential privacyexponential mechanismlikelihood ratiopure-DP
Authors
Aratrika Mustafi, Soumya Mukherjee
Abstract
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.