Multilevel Langevin Pathwise Average for Gibbs Approximation
提出一种基于过阻尼朗之万扩散的多层级方法，通过组合不同步长的欧拉格式的占据测度来近似吉布斯分布，并给出ε-逼近的复杂度上界，适用于强凸势函数及贝叶斯学习场景。
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 2 -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].