  Selecting the step size for the Metropolis-adjusted Langevin algorithm (MALA)
is necessary in order to obtain satisfactory performance. However, finding an
adequate step size for an arbitrary target distribution can be a difficult task
and even the best step size can perform poorly in specific regions of the space
when the target distribution is sufficiently complex. To resolve this issue we
introduce autoMALA, a new Markov chain Monte Carlo algorithm based on MALA that
automatically sets its step size at each iteration based on the local geometry
of the target distribution. We prove that autoMALA has the correct invariant
distribution, despite continual automatic adjustments of the step size. Our
experiments demonstrate that autoMALA is competitive with related
state-of-the-art MCMC methods, in terms of the number of log density
evaluations per effective sample, and it outperforms state-of-the-art samplers
on targets with varying geometries. Furthermore, we find that autoMALA tends to
find step sizes comparable to optimally-tuned MALA when a fixed step size
suffices for the whole domain.
