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：为获得满意性能，必须为Metropolis调整Langevin算法（MALA）选取步长。然而，针对任意目标分布寻找合适的步长可能十分困难，且当目标分布足够复杂时，即使最优步长在空间的特定区域表现也可能不佳。为解决此问题，我们提出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结构，在每次迭代中自动设置步长。我们证明了autoMALA在持续自动调整步长的条件下仍具有正确的不变分布。实验表明，就每有效样本的对数密度评估次数而言，autoMALA与相关最先进MCMC方法性能相当，并且在几何结构变化的目标上优于最先进的采样器。此外，我们发现当固定步长足以覆盖整个定义域时，autoMALA倾向于找到与优化调优MALA相当的步长。