Parallel computations for Metropolis Markov chains with Picard maps

TL;DR

This paper introduces parallel algorithms for gradient-free Metropolis Markov chains using Picard maps, achieving faster convergence in high-dimensional sampling tasks with minimal gradient information.

Contribution

The authors develop novel parallel algorithms for zeroth-order Metropolis chains that significantly improve sampling efficiency in high dimensions.

Findings

Achieves $ ilde{O}(\sqrt{d})$ parallel iterations for sampling in $\mathbb{R}^d$

Reduces convergence time by a factor of $\sqrt{d}$ compared to sequential methods

Demonstrates effectiveness in high-dimensional regression, epidemic modeling, and precision medicine applications.

Abstract

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 $\pi$ on $\mathbb{R}^d$, our algorithm generates samples close to $\pi$ in $\mathcal{O}(\sqrt{d})$ parallel iterations with $\mathcal{O}(\sqrt{d})$ processors, therefore speeding up the convergence of the corresponding sequential implementation by a factor $\sqrt{d}$. Furthermore, a modification of our algorithm generates samples from an approximate measure $ \pi_r$ in $\mathcal{O}(1)$ parallel iterations and $\mathcal{O}(d)$ 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 $\pi$ using only point-wise evaluations of $\log\pi$ and parallel computing.