  We study the mixing time of the Metropolis-adjusted Langevin algorithm (MALA)
for sampling from a log-smooth and strongly log-concave distribution. We
establish its optimal minimax mixing time under a warm start. Our main
contribution is two-fold. First, for a $d$-dimensional log-concave density with
condition number $\kappa$, we show that MALA with a warm start mixes in $\tilde
O(\kappa \sqrt{d})$ iterations up to logarithmic factors. This improves upon
the previous work on the dependency of either the condition number $\kappa$ or
the dimension $d$. Our proof relies on comparing the leapfrog integrator with
the continuous Hamiltonian dynamics, where we establish a new concentration
bound for the acceptance rate. Second, we prove a spectral gap based mixing
time lower bound for reversible MCMC algorithms on general state spaces. We
apply this lower bound result to construct a hard distribution for which MALA
requires at least $\tilde \Omega (\kappa \sqrt{d})$ steps to mix. The lower
bound for MALA matches our upper bound in terms of condition number and
dimension. Finally, numerical experiments are included to validate our
theoretical results.
