Penalized Langevin dynamics with vanishing penalty for smooth and log-concave targets

TL;DR

This paper introduces Penalized Langevin dynamics (PLD), a diffusion process with a vanishing penalty term, providing new convergence guarantees for sampling from smooth, log-concave distributions and for optimization in low-temperature regimes.

Contribution

It proposes a novel PLD process with a vanishing penalty, deriving bounds on its convergence to the target distribution and offering new nonasymptotic guarantees for penalized gradient flow.

Findings

Upper bound on Wasserstein-2 distance between PLD and target

Influence of penalty decay rate on approximation accuracy

Nonasymptotic convergence guarantees for penalized gradient flow

Abstract

We study the problem of sampling from a probability distribution on $\mathbb R^p$ defined via a convex and smooth potential function. We consider a continuous-time diffusion-type process, termed Penalized Langevin dynamics (PLD), the drift of which is the negative gradient of the potential plus a linear penalty that vanishes when time goes to infinity. An upper bound on the Wasserstein-2 distance between the distribution of the PLD at time $t$ and the target is established. This upper bound highlights the influence of the speed of decay of the penalty on the accuracy of the approximation. As a consequence, considering the low-temperature limit we infer a new nonasymptotic guarantee of convergence of the penalized gradient flow for the optimization problem.