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Selecting the step size for the Metropolis-adjusted Langevin algorithm (MALA)is necessary in order to obtain satisfactory performance. However, finding anadequate step size for an arbitrary target distribution can be a difficult taskand even the best step size can perform poorly in specific regions of the spacewhen the target distribution is sufficiently complex. To resolve this issue weintroduce autoMALA, a new Markov chain Monte Carlo algorithm based on MALA thatautomatically sets its step size at each iteration based on the local geometryof the target distribution. We prove that autoMALA has the correct invariantdistribution, despite continual automatic adjustments of the step size. Ourexperiments demonstrate that autoMALA is competitive with relatedstate-of-the-art MCMC methods, in terms of the number of log densityevaluations per effective sample, and it outperforms state-of-the-art samplerson targets with varying geometries. Furthermore, we find that autoMALA tends tofind step sizes comparable to optimally-tuned MALA when a fixed step sizesuffices for the whole domain.

autoMALA: Locally adaptive Metropolis-adjusted Langevin algorithm