{"title":"Multilevel Langevin Pathwise Average for Gibbs Approximation","authors":"Maxime Egéa, Fabien Panloup","doi":"10.1287/moor.2021.0243","DOIUrl":null,"url":null,"abstract":"We propose and study a new multilevel method for the numerical approximation of a Gibbs distribution π on [Formula: see text], 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 [Formula: see text]. We first state a quantitative result under general assumptions that guarantees an ε-approximation (in an L<jats:sup>2</jats:sup>-sense) with a cost of the order [Formula: see text] or [Formula: see text] under less contractive assumptions. We then apply it to overdamped Langevin diffusions with strongly convex potential [Formula: see text] and obtain an ε-complexity of the order [Formula: see text] or [Formula: see text] under additional assumptions on U. More precisely, up to universal constants, an appropriate choice of the parameters leads to a cost controlled by [Formula: see text] (where [Formula: see text] and [Formula: see text] respectively denote the supremum and the infimum of the largest and lowest eigenvalue of [Formula: see text]). 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.Funding: The authors are grateful to the SIRIC ILIAD Nantes-Angers program, supported by the French National Cancer Institute [INCA-DGOS-Inserm Grant 12558].","PeriodicalId":1,"journal":{"name":"Accounts of Chemical Research","volume":null,"pages":null},"PeriodicalIF":16.4000,"publicationDate":"2024-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Accounts of Chemical Research","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1287/moor.2021.0243","RegionNum":1,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"CHEMISTRY, MULTIDISCIPLINARY","Score":null,"Total":0}
引用次数: 0
Abstract
We propose and study a new multilevel method for the numerical approximation of a Gibbs distribution π on [Formula: see text], 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 [Formula: see text]. We first state a quantitative result under general assumptions that guarantees an ε-approximation (in an L2-sense) with a cost of the order [Formula: see text] or [Formula: see text] under less contractive assumptions. We then apply it to overdamped Langevin diffusions with strongly convex potential [Formula: see text] and obtain an ε-complexity of the order [Formula: see text] or [Formula: see text] under additional assumptions on U. More precisely, up to universal constants, an appropriate choice of the parameters leads to a cost controlled by [Formula: see text] (where [Formula: see text] and [Formula: see text] respectively denote the supremum and the infimum of the largest and lowest eigenvalue of [Formula: see text]). 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.Funding: The authors are grateful to the SIRIC ILIAD Nantes-Angers program, supported by the French National Cancer Institute [INCA-DGOS-Inserm Grant 12558].
我们提出并研究了一种新的多级方法,基于(过阻尼)朗格文扩散,对[公式:见正文]上的吉布斯分布π进行数值逼近。该方法依赖于多级占优度量,即具有各自步长的(恒定步长)欧拉方案的 R 级占优度量的适当组合[公式:见正文]。我们首先给出一个一般假设下的定量结果,它保证了ε 近似(在 L2 意义上),其代价为[公式:见正文]或[公式:见正文]。然后,我们将其应用于具有强凸势的、过阻尼的朗格文扩散[公式:见正文],并在 U 的额外假设下得到[公式:见正文]或[公式:见正文]阶的ε复杂性。更确切地说,在不超出普遍常数的情况下,参数的适当选择会导致[公式:见正文]所控制的代价(其中[公式:见正文]和[公式:见正文]分别表示[公式:见正文]的最大和最小特征值的上峰和下峰)。最后,我们通过一些数值说明完成了这些理论结果,包括与贝叶斯学习中其他算法的比较,以及向非强凸设置的开放:作者感谢法国国家癌症研究所[INCA-DGOS-Inserm Grant 12558]支持的 SIRIC ILIAD Nantes-Angers 计划。
期刊介绍:
Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance.
Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.