AutoStep: Locally adaptive involutive MCMC (2410.18929v3)

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. We prove 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.

• The paper introduces AutoStep MCMC that autonomously adjusts step sizes based on local geometry for optimal sampling in involutive MCMC methods.
• The method preserves π-invariance, irreducibility, and aperiodicity while enhancing effective sample size per computational cost.
• Empirical results demonstrate that AutoStep MCMC outperforms state-of-the-art techniques in Bayesian inference on multiscale models.

Overview of AutoStep: Locally Adaptive Involutive MCMC

The paper introduces AutoStep MCMC, a novel class of Markov chain Monte Carlo (MCMC) methodologies that optimize step size selection on a local scale for involutive MCMC methods. This approach is designed to address the difficulties associated with choosing a suitable global step size for complex, multiscale target distributions, offering adaptability by tuning the step size at each iteration in accordance with the local geometry of the target distribution.

Core Contributions and Methodology

AutoStep MCMC operates within the domain of involutive MCMC, leveraging deterministic involutive proposals characterized by a step size parameter. The main contribution of AutoStep is its ability to autonomously adjust this step size during runtime, guided by the local behavior of the target distribution, denoted as $\pi$.

Theoretically, the authors establish that AutoStep MCMC maintains $\pi$-invariance, irreducibility, and aperiodicity under mild conditions. The method utilizes an augmented space approach incorporating both state and step size as part of the proposal distribution, enabling dynamic adaptation while preserving necessary MCMC properties.

Empirical Results

The empirical assessments conducted demonstrate the competitiveness of AutoStep MCMC against state-of-the-art alternatives, particularly noted in terms of effective sample size per computational cost across various challenging distributions. The flexibility of the proposed method is evident as it adapts effectively to different local structures of the target distribution.

Technical Observations

AutoStep MCMC integrates a round-based procedure for parameter tuning, optimizing the initial step size, jitter, and mass matrix over a series of iterations. The paper details a comprehensive step size selection strategy grounded in both operational efficiency and theoretical soundness, avoiding excessive computational burdens associated with exact calculations by implementing a heuristic range-based strategy for acceptance.

Implications and Future Directions

The AutoStep framework presents a robust mechanism for locally adaptive sampling, promising enhancements in exploring target distributions with intricate structures. Theoretically, this approach could inspire further investigations into MCMC sampling efficiency metrics, potentially exploring asymptotic properties and variance analysis.

In terms of practical application, AutoStep MCMC could influence a range of domains requiring Bayesian inference over multiscale models, offering enhanced sampling efficiency and reduced human intervention in tuning parameters.

Overall, this work enriches the toolbox for adaptive MCMC methods, underscoring the impact of local adaptivity and theoretical integration in developing efficient sampling algorithms. Future work could explore broader classes of targets, extend theoretical insights, and harness AutoStep MCMC's adaptability to complex, high-dimensional problems.