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 $\mathbb{R}^d$, our algorithm generates samples close to $π$ 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 $ π_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 $π$ using only point-wise evaluations of $\logπ$ and parallel computing.

翻译：我们针对基于Picard映射的零阶（即无梯度）Metropolis Markov链模拟开发了并行算法。对于以$\mathbb{R}^d$上对数凹分布$π$为目标的随机游走Metropolis Markov链，我们的算法通过$\mathcal{O}(\sqrt{d})$个处理器在$\mathcal{O}(\sqrt{d})$次并行迭代中生成接近$π$的样本，从而将对应顺序实现的收敛速度提升$\sqrt{d}$倍。此外，算法的改进版本可通过$\mathcal{O}(d)$个处理器在$\mathcal{O}(1)$次并行迭代中从近似测度$π_r$生成样本。我们通过高维回归问题、梯度不可用的流行病模型以及精准医学中的实际应用，对所提算法的性能进行了实证评估。这些算法易于实现，可为仅依赖$\logπ$的点值评估和并行计算、需要从指定分布$π$采样的实践者提供有效工具。