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.

我们提出了一种名为 Metropolis-adjusted Mirror Langevin 算法的新方法，用于从支持为紧凸集的分布中进行近似抽样。该算法在镜像 Langevin 算法（Zhang 等人，2020）的单步离散化所产生的马尔可夫链中添加了接受 - 拒绝过滤器，而已知的镜像 Langevin 算法的离散化具有渐近偏差。当势函数相对平滑、凸且在自共轭镜像函数上满足 Lipschitz 条件时，我们给出了所提算法的混合时间的上界。作为算法产生的马尔可夫链可逆性的结果，我们对近似抽样的误差容限得到了指数级的改善依赖。我们还进行了数值实验以验证我们的理论发现。