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：本文提出一种名为 Metropolis-adjusted Mirror Langevin 算法的新方法，用于从支撑集为紧凸集的分布中进行近似采样。该算法在 Mirror Langevin 算法（Zhang 等人，2020）单步诱导的马尔可夫链上增加了接受-拒绝过滤器，而 Mirror Langevin 算法本身是 Mirror Langevin 动力学的基本离散化形式。由于该过滤器的引入，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）均存在渐近偏差。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，其中该约束分布的势函数相对于自协调镜像函数具有相对光滑性、凸性及 Lipschitz 连续性。通过引入 Metropolis-Hastings 过滤器所诱导的马尔可夫链的可逆性，我们在近似约束采样问题上获得了关于误差容忍度的指数级更优依赖关系。本文还提供了数值实验以验证理论结果。