🤖 AI Summary
Traditional MCMC methods suffer from low sampling efficiency in complex distributions, while multi-proposal MCMC (MP-MCMC) offers parallelization potential but lacks a clear theoretical understanding of its behavior and optimization mechanisms under a large number of proposals. This work establishes a general theoretical framework for multi-proposal involutive MCMC in abstract state spaces, systematically analyzing the properties of transition kernels under various proposal and acceptance schemes. It introduces three novel algorithms (Algs. 1.1, 3.3, 3.4), unifies existing MP-MCMC approaches by revealing their intrinsic connections, and eliminates ineffective strategies. Through asymptotic analysis in the large-proposal limit and a unified modeling perspective, the study clarifies convergence and efficiency properties under high parallelism, providing both theoretical foundations and practical algorithms for large-scale parallel MCMC.
📝 Abstract
Multiproposal MCMC (MP-MCMC) algorithms use clouds of proposals to efficiently traverse state spaces and overcome complex target geometries. While MCMC methods are embarrassingly parallel by nature, the non-trivial forms of parallelism provided by the MP-MCMC formalism sometimes leads to significant improvements over a naive approach. Here, one important tuning parameter is the number of proposals p used by a single MP-MCMC iteration. While a number of computational strategies have been proposed to efficiently leverage large numbers of proposals within the MP-MCMC paradigm, much remains unknown about these algorithms, particularly in the large p-regime. In this contribution, we discover surprising results by identifying and studying several promising new methods (Algorithm 1.1, Algorithm 3.3, Algorithm 3.4), ruling out other extant approaches and discovering new relationships between different MP-MCMC methodologies. Our analysis is centered on a general state space multiproposal involutive theory recently constructed by the authors
combined with the consideration of the large p-limit kernels for MP-MCMC algorithms within a variety of different classes of proposal and acceptance structures.