Approximate Algorithms for Verifying Differential Privacy with Gaussian Distributions

2025-09-10

AI Summary

This work addresses the verification challenge of differential privacy (DP) algorithms under the Gaussian mechanism, focusing on probabilistic distribution approximation for loop-free programs that sample from discrete and continuous distributions (Gaussian and Laplace). We propose a verification method combining high-precision numerical integration, adaptive probability density function (PDF) approximation, and tight tail-probability bounds. For the first time, we theoretically establish near-decidability of $(varepsilon,delta)$-DP for continuous distributions—including Gaussian—thereby overcoming a fundamental limitation of existing tools that struggle with continuous sampling. We implement our approach in DipApprox, a verification tool built upon the FLINT library. Experiments successfully verify canonical DP algorithms—including the Sparse Vector Technique and Noisy Max—confirming their privacy guarantees and detecting subtle violations. Our method significantly improves both accuracy and practicality of DP verification in continuous-distribution settings.

Technology Category

• Machine Learning: Privacy
• Reasoning under Uncertainty: Probabilistic Programming
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Security and Privacy: Data transparency and provenance
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Data and user privacy-enhancing technologies for the Web

Abstract

The verification of differential privacy algorithms that employ Gaussian distributions is little understood. This paper tackles the challenge of verifying such programs by introducing a novel approach to approximating probability distributions of loop-free programs that sample from both discrete and continuous distributions with computable probability density functions, including Gaussian and Laplace. We establish that verifying $(ε,δ)$-differential privacy for these programs is emph{almost decidable}, meaning the problem is decidable for all values of $δ$ except those in a finite set. Our verification algorithm is based on computing probabilities to any desired precision by combining integral approximations, and tail probability bounds. The proposed methods are implemented in the tool, DipApprox, using the FLINT library for high-precision integral computations, and incorporate optimizations to enhance scalability. We validate {ourtool} on fundamental privacy-preserving algorithms, such as Gaussian variants of the Sparse Vector Technique and Noisy Max, demonstrating its effectiveness in both confirming privacy guarantees and detecting violations.

Problem

• Verifying differential privacy for Gaussian-based algorithms
• Approximating probability distributions in loop-free programs
• Establishing almost decidability for (ε,δ)-differential privacy verification

Innovation

• Approximates probability distributions for loop-free programs
• Uses integral approximations and tail bounds
• Implements high-precision computations with FLINT library