B0590
Title: Adaptive Langevin Monte Carlo methods for heavy-tailed sampling via weighted functional inequalities
Authors: Tyler Farghly - University of Oxford (United Kingdom) [presenting]
Jun Yang - University of Oxford (United Kingdom)
Abstract: The non-asymptotic analysis of Langevin Monte Carlo (LMC) is a subject that has received increased attention within computational statistics, with a focus on theoretical guarantees in high-dimensional settings. However, most existing works require that the target distribution is sufficiently light-tailed. When applied to heavy-tailed targets, guarantees for LMC are limited. In fact, existing analyses of the Langevin diffusion, from which LMC is derived, have only established rates with exponential dependence on the dimension. In this paper, we propose a simple generalisation of LMC that employs a weighting function that is chosen according to the tail-growth of the target. The algorithm is based on an adaptive Euler-Maruyama (EM) discretisation of the natural diffusion associated with the weighting. Using weighted logarithmic Sobolev inequalities, we establish non-asymptotic rates in both Wasserstein distance and KL divergence that depend polynomially on dimension. Our analysis applies to a class of heavy-tailed targets interpolating between Gaussian and generalised Cauchy distributions. As part of our analysis, we develop a framework for analysing LMC-type algorithms that use adaptive EM schemes based on the theory of random time changes.
