Multilevel Langevin Pathwise Average for Gibbs Approximation

• 成果类型: Article
• 署名作者: Egea, Maxime; Panloup, Fabien
• 署名单位: Universite d'Angers; Centre National de la Recherche Scientifique (CNRS)
• 刊物名称: MATHEMATICS OF OPERATIONS RESEARCH
• ISSN/ISSBN: 0364-765X
• DOI: 10.1287/moor.2021.0243
• 发表日期: 2025
• 关键词: recursive computation; invariant measure; markov-chain; CONVERGENCE; sdes

摘要

We propose and study a new multilevel method for the numerical approximation of a Gibbs distribution pi on Rd, based on (overdamped) Langevin diffusions. This method relies on a multilevel occupation measure, that is, on an appropriate combination of R occupation measures of (constant -step) Euler schemes with respective steps gamma r � gamma 02-r, r � 0,.. .,R. We first state a quantitative result under general assumptions that guarantees an epsilon-approximation (in an L2 -sense) with a cost of the order epsilon-2 or epsilon-2|log epsilon |3 under less contractive assumptions. We then apply it to overdamped Langevin diffusions with strongly convex potential U : Rd -> R and obtain an epsilon-complexity of the order O(d epsilon-2log3(d epsilon-2)) or O(d epsilon-2) under additional assumptions on U. More precisely, up to universal constants, an appropriate choice of the parameters leads to a cost controlled by (lambda U proves 1)2 lambda-3U d epsilon-2 (where lambda U and lambda U respectively denote the supremum and the infimum of the largest and lowest eigenvalue of D2U). We finally complete these theoretical results with some numerical illustrations, including comparisons to other algorithms in Bayesian learning and opening to the non-strongly convex setting.