Multilevel Langevin Pathwise Average for Gibbs Approximation

We propose and study a new multilevel method for the numerical approximation of a Gibbs distribution on Rᵈ, based on (overdamped) Langevin diffusions. This method inspired by mainPPlangevin and gilesₛzpruchᵢnvariant relies on a multilevel occupation measure, i. e. on an appropriate combination of R occupation measures of (constant-step) Euler schemes with respective steps ᵣ = ₀ 2^-r, r=0, , R. We first state a quantitative result under general assumptions which guarantees an -approximation (in a L²-sense) with a cost of the order ^-2 or ^-2| |³ under less contractive assumptions. We then apply it to overdamped Langevin diffusions with strongly convex potential U: Rᵈ and obtain an -complexity of the order O (d^-2³ (d^-2) ) or O (d^-2) under additional assumptions on U. More precisely, up to universal constants, an appropriate choice of the parameters leads to a cost controlled by (U 1) ²U^{-3} d^-2 (where U and 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 non strongly convex setting.