A Metropolis-Hastings step is widely used for gradient-based Markov chain Monte Carlo methods in uncertainty quantification. By calculating acceptance probabilities on batches, a stochastic Metropolis-Hastings step saves computational costs, but reduces the effective sample size. We show that this obstacle can be avoided by a simple correction term. We study statistical properties of the resulting stationary distribution of the chain if the corrected stochastic Metropolis-Hastings approach is applied to sample from a Gibbs posterior distribution in a nonparametric regression setting. Focusing on deep neural network regression, we prove a PAC-Bayes oracle inequality which yields optimal contraction rates and we analyze the diameter and show high coverage probability of the resulting credible sets. With a numerical example in a high-dimensional parameter space, we illustrate that credible sets and contraction rates of the stochastic Metropolis-Hastings algorithm indeed behave similar to those obtained from the classical Metropolis-adjusted Langevin algorithm.

翻译：在不确定性量化中，Metropolis-Hastings步骤被广泛用于基于梯度的马尔可夫链蒙特卡洛方法。通过基于批次计算接受概率，随机Metropolis-Hastings步骤节省了计算成本，但降低了有效样本量。我们表明，这种障碍可以通过一个简单的修正项来避免。我们研究了将修正后的随机Metropolis-Hastings方法应用于非参数回归设定下吉布斯后验分布采样时，所得链的平稳分布的统计性质。聚焦于深度神经网络回归，我们证明了一个PAC-Bayes oracle不等式，该不等式给出了最优收缩率，并分析了所得置信集的直径及其高覆盖概率。通过高维参数空间中的数值示例，我们阐明了随机Metropolis-Hastings算法的置信集和收缩率确实与经典Metropolis调整后的Langevin算法所获得的结果表现相似。