Heavy-tailed Sampling via Transformed Unadjusted Langevin Algorithm

TL;DR

This paper introduces a transformed Unadjusted Langevin Algorithm to efficiently sample from heavy-tailed distributions, analyzing its complexity and establishing conditions for polynomial-order convergence.

Contribution

It develops a class of diffeomorphic transformations for heavy-tailed densities, enabling efficient Langevin-based sampling with proven complexity bounds.

Findings

Polynomial-order oracle complexity for certain heavy-tailed densities

Construction of diffeomorphic transformations suited for diffusion-based samplers

Connection between functional inequalities and heavy-tailed equilibrium densities

Abstract

We analyze the oracle complexity of sampling from polynomially decaying heavy-tailed target densities based on running the Unadjusted Langevin Algorithm on certain transformed versions of the target density. The specific class of closed-form transformation maps that we construct are shown to be diffeomorphisms, and are particularly suited for developing efficient diffusion-based samplers. We characterize the precise class of heavy-tailed densities for which polynomial-order oracle complexities (in dimension and inverse target accuracy) could be obtained, and provide illustrative examples. We highlight the relationship between our assumptions and functional inequalities (super and weak Poincar\'e inequalities) based on non-local Dirichlet forms defined via fractional Laplacian operators, used to characterize the heavy-tailed equilibrium densities of certain stable-driven stochastic differential equations.