Abstract:
We study the problem of sampling from a probability distribution
on $\mathbb R^p$ defined via a convex and smooth potential function.
We first 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 approximation. As a consequence, in the case
of low-temperature limit we infer a new result on the convergence of the
penalized gradient flow for the optimization problem.
