Differentially Private Hamiltonian Monte Carlo
This work addresses privacy-preserving Bayesian inference for data-sensitive applications, representing an incremental improvement over prior DP MCMC methods.
The paper tackles the challenge of ensuring differential privacy in Hamiltonian Monte Carlo (MCMC) methods, presenting a DP-HMC algorithm that adds noise to gradients and uses the Metropolis-Hastings test, proving convergence and showing it outperforms or matches existing DP MCMC algorithms in performance and consistency.
Markov chain Monte Carlo (MCMC) algorithms have long been the main workhorses of Bayesian inference. Among them, Hamiltonian Monte Carlo (HMC) has recently become very popular due to its efficiency resulting from effective use of the gradients of the target distribution. In privacy-preserving machine learning, differential privacy (DP) has become the gold standard in ensuring that the privacy of data subjects is not violated. Existing DP MCMC algorithms either use random-walk proposals, or do not use the Metropolis--Hastings (MH) acceptance test to ensure convergence without decreasing their step size to zero. We present a DP variant of HMC using the MH acceptance test that builds on a recently proposed DP MCMC algorithm called the penalty algorithm, and adds noise to the gradient evaluations of HMC. We prove that the resulting algorithm converges to the correct distribution, and is ergodic. We compare DP-HMC with the existing penalty, DP-SGLD and DP-SGNHT algorithms, and find that DP-HMC has better or equal performance than the penalty algorithm, and performs more consistently than DP-SGLD or DP-SGNHT.