Fast-Mixing Markov Chains without Gradients

2026-06-25

AI Summary

Existing methods for accelerating Markov chain mixing often rely on costly geometric information or compromise sampling accuracy, making it challenging to achieve both efficiency and fidelity. This work introduces a localization principle that enables surrogate-based Metropolis–Hastings proposals to exploit gradient-level geometric information without computing gradients of either the target or the surrogate. By integrating regularization and temperature tuning, the authors construct an efficient proposal mechanism. The resulting DART framework—Delayed Acceptance with Regularization and Temperature tuning—provides the first theoretical mixing-time guarantees for surrogate-based MCMC methods: for strongly log-concave distributions, it achieves a mixing time of $O(\kappa \cdot \max\{\kappa, d\})$ from a warm start, matching MALA’s $O(\kappa d)$ rate when $d \geq \kappa$, and otherwise converging at a dimension-independent rate of $O(\kappa^2)$.

Abstract

Most approaches for accelerating Markov chain mixing either rely on incorporating expensive geometric information in the proposals, or reduce the per-step cost of sampling via surrogate densities. We propose a localisation principle that allows a surrogate-based Metropolis-Hastings proposal to exploit gradient-level geometric information of the target density, without evaluating either the target gradient or the surrogate gradient. The construction relies on regularisation and tempering of the proposal measure. We show that the expected proposal displacement coincides with the Langevin drift up to controlled error. The resulting framework, Delayed Acceptance with Regularisation and Tempering (DART), achieves an $O(κ\max\{κ, d\})$ mixing time from warm start for strongly log-concave targets with condition number $κ$ in $d$ dimensions. This matches the known $O(κd)$ rate for MALA when $d \ge κ$, and scales as $O(κ^2)$, independent of dimension, otherwise. This is, to our knowledge, the first mixing time guarantee for a surrogate-transition-based MCMC method. We demonstrate DART on a hierarchical spatial generalised linear mixed model. In this setting, the Dirichlet-Neumann averaging parametrisation, originally introduced for the efficient simulation of Gaussian processes, is repurposed to supply the surrogate, and its linear memory and log-linear arithmetic scaling in the number of observation sites carry over to inference.

Problem

• Markov chain mixing
• gradient-free sampling
• surrogate-based MCMC
• mixing time
• Metropolis-Hastings

Innovation

• surrogate-based MCMC
• gradient-free Langevin drift
• regularisation and tempering
• mixing time guarantee
• Delayed Acceptance