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. For this algorithm, we also give upper bounds for the number of iterations taken to mix to a constrained distribution whose potential is relatively smooth, convex, and Lipschitz continuous with respect to a self-concordant mirror function. As a consequence of the reversibility of the Markov chain induced by the inclusion of the Metropolis-Hastings filter, we obtain an exponentially better dependence on the error tolerance for approximate constrained sampling. We also present numerical experiments that corroborate our theoretical findings.

翻译：我们提出一种名为Metropolis调整的镜像Langevin算法的新方法，用于从支撑集为紧凸集的分布中近似采样。该算法向单步镜像Langevin算法（Zhang等人，2020）生成的马尔可夫链中添加接受-拒绝滤波器，其中镜像Langevin算法是镜像Langevin动力学的一种基本离散化形式。由于引入了该滤波器，我们的方法相对于目标分布是无偏的，而已知的镜像Langevin动力学离散化（包括镜像Langevin算法）存在渐近偏差。针对该算法，我们还给出了混合到约束分布所需迭代次数的上界，该分布关于自协调镜像函数的势函数相对光滑、凸且利普希茨连续。由于Metropolis-Hastings滤波器的引入使得马尔可夫链具有可逆性，我们在近似约束采样的误差容限上获得了指数级更好的依赖关系。我们同时呈现了数值实验，验证了我们的理论发现。