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.

翻译：为获得令人满意的性能，需为Metropolis-adjusted Langevin算法（MALA）选取步长。然而，对于任意目标分布，寻找合适的步长是一项困难的任务，且即使是最佳步长，在目标分布足够复杂时，也可能在空间的特定区域表现不佳。为解决此问题，我们提出了autoMALA，一种基于MALA的新型马尔可夫链蒙特卡洛算法，该算法能根据目标分布的局部几何形状，在每次迭代中自动设定步长。我们证明，尽管持续进行步长的自动调整，autoMALA仍具有正确的不变分布。实验表明，autoMALA在每次有效样本的对数密度评估次数方面，与相关的最先进MCMC方法相当，并且在几何形状变化的目标分布上优于最先进的采样器。此外，我们发现在固定步长足以覆盖整个域的情况下，autoMALA倾向于找到与最优调优MALA相当的步长。