Differentially Private Hamiltonian Monte Carlo

We present DP-HMC, a variant of Hamiltonian Monte Carlo (HMC) that is differentially private (DP). We use the penalty algorithm of Yildirim et al. to make the acceptance test private, and add Gaussian noise to the gradients of the target distribution to make the HMC proposal private. Our main contribution is showing that DP-HMC has the correct invariant distribution, and is ergodic. We also compare DP-HMC with the existing penalty algorithm, as well as DP-SGLD and DP-SGNHT.