Parallel computations for Metropolis Markov chains with Picard maps

Abstract

We develop parallel algorithms for simulating zeroth-order (also known as 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 using O(d) processors, thereby 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 pointwise evaluations of π and parallel computing.