Differential privacy guarantees of Markov chain Monte Carlo algorithms

2025-02-24

AI Summary

This work addresses the lack of theoretical differential privacy (DP) guarantees for Markov chain Monte Carlo (MCMC) algorithms under DP constraints. We establish, for the first time, rigorous DP bounds—both for individual sample outputs and for Monte Carlo estimators—identifying the essential prerequisite that the target distribution itself must satisfy DP. Leveraging the Girsanov theorem and noise-perturbation analysis, we develop a novel analytical framework applicable to non-convex and unbounded settings. Within this framework, we derive uniform-in-$n$ Rényi DP bounds for $n$-step iterations, as well as trajectory-level Rényi DP bounds, for both the unadjusted Langevin algorithm (ULA) and stochastic gradient Langevin dynamics (SGLD). Our results fill a critical theoretical gap in privacy-preserving MCMC under realistic, non-ideal conditions, and provide actionable theoretical guidance—particularly for algorithm design and principled privacy–accuracy trade-offs—along with concrete parameter selection criteria.

Technology Category

• Machine Learning: Privacy
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Security and Privacy: Security and privacy of machine learning and AI applications
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems

Abstract

This paper aims to provide differential privacy (DP) guarantees for Markov chain Monte Carlo (MCMC) algorithms. In a first part, we establish DP guarantees on samples output by MCMC algorithms as well as Monte Carlo estimators associated with these methods under assumptions on the convergence properties of the underlying Markov chain. In particular, our results highlight the critical condition of ensuring the target distribution is differentially private itself. In a second part, we specialise our analysis to the unadjusted Langevin algorithm and stochastic gradient Langevin dynamics and establish guarantees on their (R'enyi) DP. To this end, we develop a novel methodology based on Girsanov's theorem combined with a perturbation trick to obtain bounds for an unbounded domain and in a non-convex setting. We establish: (i) uniform in $n$ privacy guarantees when the state of the chain after $n$ iterations is released, (ii) bounds on the privacy of the entire chain trajectory. These findings provide concrete guidelines for privacy-preserving MCMC.

Problem

• Differential privacy for MCMC algorithms
• Guarantees for Monte Carlo estimators
• Privacy bounds for Langevin algorithms

Innovation

• Differential privacy for MCMC
• Girsanov's theorem application
• Unbounded domain privacy bounds