🤖 AI Summary
This work addresses the slow convergence and strong dimension dependence of traditional MCMC methods in high- and infinite-dimensional Bayesian posterior sampling by proposing and analyzing two novel multi-proposal preconditioned Crank–Nicolson algorithms, termed mpCN and MTpCN. Leveraging parallelized proposal mechanisms, these algorithms achieve enhanced sampling efficiency and are shown to converge under non-convex, high-dimensional settings. The study establishes, for the first time, rigorous dimension-independent and proposal-number-uniform exponential convergence rates for both methods. By innovatively constructing two coupling schemes, the authors derive Wasserstein contraction, an $L^2$ spectral gap, and non-asymptotic statistical guarantees. The theory demonstrates that dimension-independent mixing is attainable without convexity assumptions, provided the log-likelihood is bounded and Lipschitz. Numerical experiments confirm faster warm-up and more robust parameter tuning, significantly outperforming standard pCN and independent parallel-chain approaches.
📝 Abstract
We study two recently discovered "dimension-free" Monte Carlo sampling algorithms, the multiproposal and multiple-try preconditioned Crank-Nicolson methods (mpCN and MTpCN). These methods were designed to address certain non-parametric (i.e. infinite-dimensional) sampling problems, defined relative to a Gaussian reference measure, by combining proposal and acceptance mechanisms that take non-trivial advantage of parallel computing architectures. We provide the first rigorous analysis of both algorithms, establishing exponential convergence to the target measure through the weak Harris framework, both for a finite number of proposals and in the infinite-proposal limit. The resulting mixing rates are independent of the dimension and uniform in the number of proposals, and apply to targets with bounded, Lipschitz log-likelihoods, without requiring convexity. At the center of the analysis are two new coupling constructions, together with analytical tools of independent interest, yielding Wasserstein contraction estimates, $L^2$ spectral gaps, and associated statistical guarantees (laws of large numbers, central limit theorems, and non-asymptotic concentration bounds) for the corresponding Monte Carlo estimators. These theoretical results are complemented by a numerical study on benchmark problems with complex posterior geometries and high-dimensional structure, comparing mpCN and MTpCN against standard pCN and independent parallel-chain implementations. The experiments indicate that the multiproposal methods can offer a shorter warm-up phase and greater robustness to the choice of tuning parameters as the number of proposals grows.