On the Stability of Spherical Hellinger-Kantorovich Flows and Their Implications for Differential Privacy
Aratrika Mustafi, Soumya Mukherjee
Key claim
Perturbation theory improves SHK gradient flow sampling accuracy.
This paper introduces a perturbation theory for spherical Hellinger-Kantorovich gradient flows, allowing for the comparison of flows from different potentials. A key result is the establishment of uniform bounds for log-likelihood ratios and divergences, which can be applied to enhance sampling methods in differential privacy.
The paper develops a new perturbation theory for SHK gradient flows, extending existing methodologies.
The methodology is solid, with clear bounds and applications to differential privacy.
Deep reliability assessment
The methodology supports the stability analysis of SHK gradient flows under perturbations of the potential, providing dimension-free bounds on divergences. However, the practical application to differential privacy is contingent on assumptions like the log-Sobolev inequality, which may not hold in all real-world scenarios.
Reproducibility
No open source code or dataset is provided, making it challenging to reproduce the results without implementing the numerical methods and experiments described.
Discussion questions
- How does the assumption of a common log-Sobolev constant across different datasets affect the generality of the results?
- What are the practical implications of these stability bounds for builders implementing differential privacy in real-world systems?
- What experimental results or scenarios would falsify the claimed stability of SHK gradient flows under potential perturbations?
Key figure
Figure 1 likely illustrates the comparison between the empirical pointwise log-ratio and the theoretical bounds on divergences over time, demonstrating the stability of SHK flows.