AutoStep: Locally adaptive involutive MCMC

2024-10-24

arXiv.org

AI Summary

To address the challenge of selecting a globally appropriate step size for Markov chain Monte Carlo (MCMC) methods under complex, multi-scale target distributions, this paper proposes AutoStep MCMC—a novel method embedding a local geometry-aware adaptive step-size mechanism within the involutive MCMC framework. Its key contribution is the first integration of gradient- and curvature-informed dynamic step-size scheduling with rigorous π-invariance preservation: at each iteration, the step size is automatically tuned based on local geometric structure (e.g., curvature), complemented by acceptance-rate–driven adaptation. We provide theoretical guarantees of both invariance and stability. Empirical evaluations demonstrate that AutoStep significantly improves effective sample size per unit computational cost on multi-scale tasks, achieving performance competitive with state-of-the-art adaptive MCMC methods.

Technology Category

• Search and Optimization: Sampling/Simulation-based Search
• Machine Learning: Learning with Manifolds
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

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 AutoStep MCMC is $pi$-invariant and has other desirable properties under mild assumptions on the target distribution $pi$ and involutive proposal. Empirical results examine the effect of various step size selection design choices, 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.

Problem

• Adapts step size locally for MCMC efficiency
• Addresses multiscale target distribution challenges
• Ensures π-invariance with adaptive proposals

Innovation

• Locally adaptive step size selection
• Involutive MCMC with geometry adaptation
• Competitive effective sample size efficiency