# Differential privacy guarantees of Markov chain Monte Carlo algorithms

arXiv: https://arxiv.org/abs/2502.17150 · HTML: https://mlsanity.com/paper/2502.17150/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 18 of its 39 references that are in the map.
- First posted on arXiv 24 Feb 2025.

## Builds on (most cited first, 18 of 18)

- [Deep Learning with Differential Privacy](https://mlsanity.com/paper/1607.00133/) (2016)
- [Renyi Differential Privacy](https://mlsanity.com/paper/1702.07476/) (2017)
- [The Composition Theorem for Differential Privacy](https://mlsanity.com/paper/1311.0776/) (2013)
- [Non-asymptotic convergence analysis for the Unadjusted Langevin Algorithm](https://mlsanity.com/paper/1507.05021/) (2015)
- [High-dimensional Bayesian inference via the Unadjusted Langevin Algorithm](https://mlsanity.com/paper/1605.01559/) (2016)
- [Differential Privacy Dynamics of Langevin Diffusion and Noisy Gradient Descent](https://mlsanity.com/paper/2102.05855/) (2021)
- [Privacy for Free: Posterior Sampling and Stochastic Gradient Monte Carlo](https://mlsanity.com/paper/1502.07645/) (2015)
- [Privacy of Noisy Stochastic Gradient Descent: More Iterations without More Privacy Loss](https://mlsanity.com/paper/2205.13710/) (2022)
- [Differentially Private Learning Needs Hidden State (Or Much Faster Convergence)](https://mlsanity.com/paper/2203.05363/) (2022)
- [Convergence of Langevin Monte Carlo in Chi-Squared and Renyi Divergence](https://mlsanity.com/paper/2007.11612/) (2020)
- [Differential Privacy Guarantees for Stochastic Gradient Langevin Dynamics](https://mlsanity.com/paper/2201.11980/) (2022)
- [Differentially Private Markov Chain Monte Carlo](https://mlsanity.com/paper/1901.10275/) (2019)
- [R'enyi Differential Privacy Mechanisms for Posterior Sampling](https://mlsanity.com/paper/1710.00892/) (2017)
- [On Connecting Stochastic Gradient MCMC and Differential Privacy](https://mlsanity.com/paper/1712.09097/) (2017)
- [Privacy Loss of Noisy Stochastic Gradient Descent Might Converge Even for Non-Convex Losses](https://mlsanity.com/paper/2305.09903/) (2023)
- [Rosenthal-type inequalities for linear statistics of Markov chains](https://mlsanity.com/paper/2303.05838/) (2023)
- [DP-Fast MH: Private, Fast, and Accurate Metropolis-Hastings for Large-Scale Bayesian Inference](https://mlsanity.com/paper/2303.06171/) (2023)
- [Federated Bayesian Computation via Piecewise Deterministic Markov Processes](https://mlsanity.com/paper/2210.13816/) (2022)

## Abstract (arXiv)

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.