Approximation of heavy-tailed distributions via stable-driven SDEs

Abstract

Constructions of numerous approximate sampling algorithms are based on the well-known fact that certain Gibbs measures are stationary distributions of ergodic stochastic differential equations (SDEs) driven by the Brownian motion. However, for some heavy-tailed distributions it can be shown that the associated SDE is not exponentially ergodic and that related sampling algorithms may perform poorly. A natural idea that has recently been explored in the machine learning literature in this context is to make use of stochastic processes with heavy tails instead of the Brownian motion. In this paper we provide a rigorous theoretical framework for studying the problem of approximating heavy-tailed distributions via ergodic SDEs driven by symmetric (rotationally invariant) $\alpha$-stable processes.

arXiv comment21 pages

References in corpus (6)

• Non-asymptotic convergence analysis for the Unadjusted Langevin Algorithm
• Strong convergence of the Euler--Maruyama approximation for a class of Lévy-driven SDEs
• Fractional Underdamped Langevin Dynamics: Retargeting SGD with Momentum under Heavy-Tailed Gradient Noise
• On the Convergence of Langevin Monte Carlo: The Interplay between Tail Growth and Smoothness
• Non-Asymptotic Analysis of Fractional Langevin Monte Carlo for Non-Convex Optimization
• Exponential ergodicity for SDEs and McKean-Vlasov processes with Lévy noise

Cited by in corpus (1)

• Exponential ergodicity for SDEs and McKean-Vlasov processes with Lévy noise