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 inO(d)parallel iterations usingO(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 πrinO(1)parallel iterations andO(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.