L\'evy Langevin Monte Carlo for sampling from heavy-tailed target distributions

TL;DR

This paper introduces a Le9vy Langevin Monte Carlo method driven by a compound Poisson process, enabling efficient sampling from heavy-tailed, non-smooth, and multimodal distributions with exponential convergence.

Contribution

It extends the Le9vy Langevin Monte Carlo framework to heavy-tailed targets, proving convergence and demonstrating advantages over classical Langevin diffusion.

Findings

Proves convergence of the method for subexponential distributions.

Allows sampling from non-smooth and multimodal heavy-tailed distributions.

Offers a simple implementation due to compound Poisson noise.

Abstract

We extend the L\'evy Langevin Monte Carlo method studied by Oechsler in 2024 to the setting of a target distribution with heavy tails: Choosing a target distribution from the class of subexponential distributions we prove convergence of a solution of a stochastic differential equation to this target. Hereby, the stochastic differential equation is driven by a compound Poisson process - unlike in the case of a classical Langevin diffusion. The method allows one to sample from non-smooth targets and distributions with separated modes with exponential convergence to the invariant distribution, which in general cannot be guaranteed by the classical Langevin diffusion in presence of heavy tails. The method is promising due to the possibility of a simple implementation because of the compound Poisson noise term.