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，Metropolis调整的Langevin算法（MALA）需选择合适的步长。然而，针对任意目标分布寻找合适的步长是一项困难的任务，且当目标分布足够复杂时，即使最优步长也可能在空间特定区域表现不佳。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，该算法根据目标分布的局部几何特征在每个迭代步骤自动设定步长。我们证明，尽管步长持续自动调整，autoMALA仍具有正确的稳态分布。实验表明，在有效样本的对数密度评估次数方面，autoMALA与相关最新MCMC方法具有竞争力，并在几何特征变化的目标分布上优于当前最优采样器。此外，我们发现当固定步长足以覆盖整个定义域时，autoMALA倾向于找到与最优调谐MALA相当的步长。