Abstract: Many common Markov chain Monte Carlo (MCMC) kernels can be formulated using a deterministic involutive proposal with a step size parameter. Selecting an appropriate step size is often a challenging task in practice; and for complex multiscale targets, there may not be one choice of step size that works well globally. In this work, we address this problem with a novel class of involutive MCMC methods---AutoStep MCMC---that selects an appropriate step size at each iteration adapted to the local geometry of the target distribution. Weprove that under mild conditions AutoStep MCMC is $\pi$-invariant, irreducible, and aperiodic, and obtain bounds on expected energy jump distance and cost per iteration. Empirical results examine the robustness and efficacy of our proposed step size selection procedure, and show that AutoStep MCMC is competitive with state-of-the-art methods in terms of effective sample size per unit cost on a range of challenging target distributions.

Lay Summary:

Markov Chain Monte Carlo (MCMC) methods are powerful tools used to sample from complex probability distributions. A key challenge in using these methods is choosing the right “step size,” which controls how far the algorithm moves at each step. If the step size is too small, the algorithm moves slowly; if it is too large, the algorithm frequently gets rejected. In many real-world problems, there is no single step size that works well everywhere.Our work introduces AutoStep MCMC, a method that automatically selects a “good” step size at each iteration based on the local shape of the target distribution. This makes the algorithm more reliable, especially for challenging models that vary widely across different regions. We prove that our method explores properly and does not get stuck, and we show it performs competitively with the best existing samplers on a range of problems.By making MCMC more adaptive and easier to use, AutoStep helps researchers get better results with less manual tuning, opening the door to more robust and automated statistical inference.