Parallel computations for Metropolis Markov chains with Picard maps

We develop parallel algorithms for simulating zeroth-order (aka gradient-free) Metropolis Markov chains based on the Picard map. For Random Walk Metropolis Markov chains targeting log-concave distributions π on ℝd, our algorithm generates samples close to π in O(√d) parallel iterations with O(√d) processors, therefore speeding up the convergence of the corresponding sequential implementation by a factor √d. Furthermore, a modification of our algorithm generates samples from an approximate measure πr in O(1) parallel iterations and O(d) processors. We empirically assess the performance of the proposed algorithms in high-dimensional regression problems, an epidemic model where the gradient is unavailable and a real-word application in precision medicine. Our algorithms are straightforward to implement and may constitute a useful tool for practitioners seeking to sample from a prescribed distribution π using only point-wise evaluations of logπ and parallel computing.