On geometric convergence for the Metropolis-adjusted Langevin algorithm under simple conditions
研究了Metropolis调整Langevin算法在温和假设下的几何收敛性，仅需目标分布的尾部和平滑性条件，并分析了步长对收敛界的影响。
Summary While the Metropolis-adjusted Langevin algorithm is a popular and widely used Markov chain Monte Carlo method, very few papers derive conditions that ensure its convergence. In particular, to the authors’ knowledge, assumptions that are both easy to verify and guarantee geometric convergence, are still missing. In this work, we establish V-uniformly geometric convergence for the Metropolis-adjusted Langevin algorithm under mild assumptions about the target distribution. Unlike previous work, we only consider tail and smoothness conditions for the potential associated with the target distribution. These conditions are quite common in the Markov chain Monte Carlo literature. Finally, we pay special attention to the dependence of the bounds we derive on the step size of the Euler–Maruyama discretization, which corresponds to the proposed Markov kernel of the Metropolis-adjusted Langevin algorithm.