Rényi Differential Privacy of the Sampled Gaussian Mechanism (1908.10530v1)

Abstract: The Sampled Gaussian Mechanism (SGM)---a composition of subsampling and the additive Gaussian noise---has been successfully used in a number of machine learning applications. The mechanism's unexpected power is derived from privacy amplification by sampling where the privacy cost of a single evaluation diminishes quadratically, rather than linearly, with the sampling rate. Characterizing the precise privacy properties of SGM motivated development of several relaxations of the notion of differential privacy. This work unifies and fills in gaps in published results on SGM. We describe a numerically stable procedure for precise computation of SGM's R\'enyi Differential Privacy and prove a nearly tight (within a small constant factor) closed-form bound.

• The paper unifies disparate analyses by integrating subsampling with Gaussian noise addition under a stable Rényi differential privacy framework.
• It derives a near-optimal closed-form upper bound and introduces a numerically robust technique for precise privacy loss computation.
• The comparative study demonstrates that Rényi differential privacy provides tighter, reliable bounds over conventional methods for iterative data access.

R{ényi Differential Privacy of the Sampled Gaussian Mechanism

The paper "R{ényi Differential Privacy of the Sampled Gaussian Mechanism" by Ilya Mironov, Kunal Talwar, and Li Zhang, offers an in-depth exploration into the privacy characteristics of the Sampled Gaussian Mechanism (SGM) within the framework of Rényi Differential Privacy (RDP). This paper is motivated by the need for rigorously quantifying the privacy guarantees provided by mechanisms in machine learning that employ subsampling combined with additive Gaussian noise.

Summary and Contributions

The foundation of the research presented in this paper revolves around differential privacy, particularly focusing on the SGM's privacy amplification by sampling. The SGM integrates two principal components—random dataset subsampling followed by the addition of Gaussian noise—both of which are standard techniques in privacy-preserving data analysis.

The paper makes several notable contributions:

1. Unified Treatment of SGM Results: The authors collate and comprehensively address gaps in the current understanding of SGM, drawing connections between disparate analyses.
2. Numerically Stable Procedure: A significant portion of the paper is devoted to developing and detailing a numerically stable method for precisely calculating the RDP of SGM. This approach allows for exact computation of privacy bounds, which is essential for both theoretical and practical applications.
3. Closed-form Bound: The authors derive a near-optimal closed-form upper bound for the privacy loss under SGM, helping facilitate easier adoption without resorting to computational methods for obtaining privacy parameters under specific conditions.
4. Comparative Analysis with CDP and Other Postulates: Through comparative paper, the paper illustrates relationships with other privacy concepts, such as concentrated differential privacy (CDP) and truncated CDP, underlining similarities and differences specifically in application to the Gaussian Mechanism.

Key Results and Implications

The paper reports numerical results affirming that the RDP offers a slightly conservative but robust estimate of privacy loss in comparison to existing methods like the moments accountant. The closed form and numerical stability discussed allow for effective tracking of privacy loss across compositions of SGM over repetitive data accesses.

By tightening bounds on privacy loss using RDP theory, the manuscript provides important implications for the design of privacy-preserving algorithms in scenarios with frequent adaptive queries, such as in training machine learning models. Despite this rigor, future work may focus on broadening the applicability of these bounds to cover more general settings, such as non-spherical noise distributions or varying subsampling strategies.

Speculation on Future Developments

The insights and methodologies developed in this paper are poised to influence several aspects of machine learning and data privacy research. As AI models become increasingly sophisticated and privacy concerns grow, understanding the nuanced privacy dynamics using robust measures like RDP becomes indispensable. We anticipate further research into extending this framework to cover various data modalities, interactions with other types of uncertainties, and implications on model utility, particularly in federated learning and edge computing contexts.

In summary, while the paper provides robust bounds and an effective computational paradigm for SGM privacy analysis, the evolving landscape of data privacy will continuously present challenges and opportunities for careful reevaluation and sophisticated application of such fundamental theoretical insights.