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Differential privacy guarantees of Markov chain Monte Carlo algorithms

Andrea Bertazzi, Tim Johnston, Gareth O. Roberts, Alain Durmus

TL;DR

A novel methodology based on Girsanov's theorem combined with a perturbation trick is developed to obtain bounds for an unbounded domain and in a non-convex setting and provides concrete guidelines for privacy-preserving MCMC.

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 (Rényi) 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.

Differential privacy guarantees of Markov chain Monte Carlo algorithms

TL;DR

A novel methodology based on Girsanov's theorem combined with a perturbation trick is developed to obtain bounds for an unbounded domain and in a non-convex setting and provides concrete guidelines for privacy-preserving MCMC.

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 (Rényi) 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 privacy guarantees when the state of the chain after iterations is released, (ii) bounds on the privacy of the entire chain trajectory. These findings provide concrete guidelines for privacy-preserving MCMC.

Paper Structure

This paper contains 32 sections, 21 theorems, 129 equations.

Table of Contents

  1. Introduction
  2. Contributions of this work
  3. Preliminaries
  4. From convergence to differential privacy
  5. Differential privacy of a Markov chain after $n$ steps
  6. Differential privacy of Monte Carlo estimators
  7. Characterising the Privacy of Diffusions
  8. Differential Privacy with Radon-Nikodym Derivatives
  9. Differential privacy of stochastic differential equations
  10. Privacy of Langevin-based algorithms in the non-convex setting
  11. Sampling from Bayesian posteriors with ULA
  12. Noisy stochastic gradient descent
  13. Commentary on Bounds Presented
  14. Conclusions
  15. Proofs and general results for \ref{['sec:convergence_and_DP']}
  16. ...and 17 more sections

Key Result

Proposition 3.2

Consider the randomised algorithm $\mathcal{A}_s({\mathcal{D}}) \sim \nu_{\mathcal{D}} P_{\mathcal{D}}^n$ and suppose ass:convergence_markovchain is verified. The following statements hold:

Theorems & Definitions (47)

  • Definition 2.1
  • Definition 2.2
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • proof
  • Proposition 3.4
  • proof
  • Proposition 3.7
  • proof
  • ...and 37 more