In this paper, we examine the computational complexity of sampling from a Bayesian posterior (or pseudo-posterior) using the Metropolis-adjusted Langevin algorithm (MALA). MALA first employs a discrete-time Langevin SDE to propose a new state, and then adjusts the proposed state using Metropolis-Hastings rejection. Most existing theoretical analyses of MALA rely on the smoothness and strong log-concavity properties of the target distribution, which are often lacking in practical Bayesian problems. Our analysis hinges on statistical large sample theory, which constrains the deviation of the Bayesian posterior from being smooth and log-concave in a very specific way. In particular, we introduce a new technique for bounding the mixing time of a Markov chain with a continuous state space via the $s$-conductance profile, offering improvements over existing techniques in several aspects. By employing this new technique, we establish the optimal parameter dimension dependence of $d^{1/3}$ and condition number dependence of $\kappa$ in the non-asymptotic mixing time upper bound for MALA after the burn-in period, under a standard Bayesian setting where the target posterior distribution is close to a $d$-dimensional Gaussian distribution with a covariance matrix having a condition number $\kappa$. We also prove a matching mixing time lower bound for sampling from a multivariate Gaussian via MALA to complement the upper bound.

翻译：本文研究了使用Metropolis调整Langevin算法（MALA）从贝叶斯后验（或伪后验）分布中采样的计算复杂度。MALA首先采用离散时间的Langevin随机微分方程提出新状态，然后通过Metropolis-Hastings拒绝机制对提议状态进行调整。现有大多数MALA的理论分析依赖于目标分布的平滑性和强对数凹性，而这些性质在实际贝叶斯问题中往往缺失。我们的分析基于统计大样本理论，该理论以特定方式约束了贝叶斯后验偏离平滑和强对数凹性的程度。特别地，我们引入了一种通过$s$-电导率配置文件对连续状态空间马尔可夫链混合时间进行界定的新技术，该技术在多个方面改进了现有方法。通过应用这一新技术，我们在标准贝叶斯场景下（目标后验分布接近一个$d$维高斯分布，其协方差矩阵的条件数为$\kappa$）建立了MALA在预烧期后的非渐近混合时间上界的最优参数维度依赖性$d^{1/3}$和条件数依赖性$\kappa$。同时，我们证明了从多元高斯分布通过MALA采样的匹配混合时间下界，以补充上界结果。