Tamed Langevin sampling under weaker conditions

TL;DR

This paper introduces a taming Langevin sampling method capable of efficiently sampling from complex, weakly dissipative distributions that do not meet standard smoothness conditions, with rigorous non-asymptotic guarantees.

Contribution

It proposes a novel taming scheme for Langevin algorithms that handles superlinear growth in log-gradients under weaker assumptions than traditional methods.

Findings

Provides explicit non-asymptotic guarantees in KL, TV, and Wasserstein distances.

Handles distributions with superlinear gradient growth and weak dissipativity.

Extends Langevin sampling applicability beyond standard smoothness and dissipativity conditions.

Abstract

Motivated by applications to deep learning which often fail standard Lipschitz smoothness requirements, we examine the problem of sampling from distributions that are not log-concave and are only weakly dissipative, with log-gradients allowed to grow superlinearly at infinity. In terms of structure, we only assume that the target distribution satisfies either a log-Sobolev or a Poincar\'e inequality and a local Lipschitz smoothness assumption with modulus growing possibly polynomially at infinity. This set of assumptions greatly exceeds the operational limits of the "vanilla" unadjusted Langevin algorithm (ULA), making sampling from such distributions a highly involved affair. To account for this, we introduce a taming scheme which is tailored to the growth and decay properties of the target distribution, and we provide explicit non-asymptotic guarantees for the proposed sampler in terms of the Kullback-Leibler (KL) divergence, total variation, and Wasserstein distance to the target distribution.