autoMALA: Locally adaptive Metropolis-adjusted Langevin algorithm

TL;DR

autoMALA is a novel MCMC algorithm that adaptively adjusts its step size based on local geometry, improving sampling efficiency for complex target distributions.

Contribution

It introduces autoMALA, which automatically tunes step sizes during sampling, maintaining correctness and outperforming existing methods on complex targets.

Findings

autoMALA maintains correct invariant distribution despite automatic step size adjustments

autoMALA outperforms state-of-the-art samplers on targets with varying geometries

autoMALA finds step sizes similar to optimally-tuned MALA in simpler regions

Abstract

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.