We propose a new method called the Metropolis-adjusted Mirror Langevin algorithm for approximate sampling from distributions whose support is a compact and convex set. This algorithm adds an accept-reject filter to the Markov chain induced by a single step of the mirror Langevin algorithm (Zhang et al., 2020), which is a basic discretisation of the mirror Langevin dynamics. Due to the inclusion of this filter, our method is unbiased relative to the target, while known discretisations of the mirror Langevin dynamics including the mirror Langevin algorithm have an asymptotic bias. We give upper bounds for the mixing time of the proposed algorithm when the potential is relatively smooth, convex, and Lipschitz with respect to a self-concordant mirror function. As a consequence of the reversibility of the Markov chain induced by the algorithm, we obtain an exponentially better dependence on the error tolerance for approximate sampling. We also present numerical experiments that corroborate our theoretical findings.

翻译：我们提出了一种新方法，称为Metropolis调整的Mirror Langevin算法，用于从支集为紧凸集的分布中进行近似采样。该算法在Mirror Langevin动力学一步离散化所诱导的马尔可夫链（Zhang等人, 2020）基础上添加了接受-拒绝滤波器。由于包含该滤波器，我们的方法相对于目标分布是无偏的，而已知的Mirror Langevin动力学离散化方法（包括Mirror Langevin算法）存在渐近偏差。我们给出了当势函数相对光滑、凸且关于自协调镜像函数是Lipschitz时，所提算法混合时间的上界。利用算法所诱导马尔可夫链的可逆性，我们在近似采样的误差容忍度上获得了指数级更好的依赖性。我们还呈现了数值实验，以佐证我们的理论发现。