Zhu Y. Convergence Rates for the
Constrained Sampling via Langevin Monte Carlo.
Entropy (Basel) 2023;
25:1234. [PMID:
37628264 PMCID:
PMC10453724 DOI:
10.3390/e25081234]
[Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/26/2023] [Revised: 08/08/2023] [Accepted: 08/15/2023] [Indexed: 08/27/2023]
Abstract
Sampling from constrained distributions has posed significant challenges in terms of algorithmic design and non-asymptotic analysis, which are frequently encountered in statistical and machine-learning models. In this study, we propose three sampling algorithms based on Langevin Monte Carlo with the Metropolis-Hastings steps to handle the distribution constrained within some convex body. We present a rigorous analysis of the corresponding Markov chains and derive non-asymptotic upper bounds on the convergence rates of these algorithms in total variation distance. Our results demonstrate that the sampling algorithm, enhanced with the Metropolis-Hastings steps, offers an effective solution for tackling some constrained sampling problems. The numerical experiments are conducted to compare our methods with several competing algorithms without the Metropolis-Hastings steps, and the results further support our theoretical findings.
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