  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.
