  Conventional wisdom in the sampling literature, backed by a popular diffusion
scaling limit, suggests that the mixing time of the Metropolis-Adjusted
Langevin Algorithm (MALA) scales as $O(d^{1/3})$, where $d$ is the dimension.
However, the diffusion scaling limit requires stringent assumptions on the
target distribution and is asymptotic in nature. In contrast, the best known
non-asymptotic mixing time bound for MALA on the class of log-smooth and
strongly log-concave distributions is $O(d)$. In this work, we establish that
the mixing time of MALA on this class of target distributions is
$\widetilde\Theta(d^{1/2})$ under a warm start. Our upper bound proof
introduces a new technique based on a projection characterization of the
Metropolis adjustment which reduces the study of MALA to the well-studied
discretization analysis of the Langevin SDE and bypasses direct computation of
the acceptance probability.
