Recent work has suggested using Monte Carlo methods based on piecewise deterministic Markov processes (PDMPs) to sample from target distributions of interest. PDMPs are non-reversible continuous-time processes endowed with momentum, and hence can mix better than standard reversible MCMC samplers. Furthermore, they can incorporate exact sub-sampling schemes which only require access to a single (randomly selected) data point at each iteration, yet without introducing bias to the algorithm's stationary distribution. However, the range of models for which PDMPs can be used, particularly with sub-sampling, is limited. We propose approximate simulation of PDMPs with sub-sampling for scalable sampling from posterior distributions. The approximation takes the form of an Euler approximation to the true PDMP dynamics, and involves using an estimate of the gradient of the log-posterior based on a data sub-sample. We thus call this class of algorithms stochastic-gradient PDMPs. Importantly, the trajectories of stochastic-gradient PDMPs are continuous and can leverage recent ideas for sampling from measures with continuous and atomic components. We show these methods are easy to implement, present results on their approximation error and demonstrate numerically that this class of algorithms has similar efficiency to, but is more robust than, stochastic gradient Langevin dynamics.

翻译：近期研究提出使用基于分段确定性马尔可夫过程的蒙特卡洛方法对目标分布进行采样。PDMP是一类具有动量的非可逆连续时间过程，因此比标准的可逆MCMC采样器具有更好的混合特性。此外，它们可以结合精确的子采样方案，该方案在每次迭代中仅需访问单个（随机选择的）数据点，且不会对算法的平稳分布引入偏差。然而，PDMP（特别是结合子采样时）的适用模型范围有限。本文提出采用子采样的PDMP近似模拟方法，用于后验分布的可扩展采样。该近似采用对真实PDMP动力学的欧拉近似形式，并基于数据子样本的对数后验梯度估计值进行计算。因此我们将此类算法称为随机梯度PDMP。值得注意的是，随机梯度PDMP的轨迹是连续的，且能够利用从具有连续与原子分量的测度中采样的最新思想。我们证明这些方法易于实现，给出了其近似误差的理论结果，并通过数值实验表明此类算法与随机梯度朗之万动力学具有相近的效率，且具有更强的鲁棒性。