Taming under isoperimetry

Abstract

In this article we propose a novel taming Langevin-based scheme called $\mathbf{sTULA}$ to sample from distributions with superlinearly growing log-gradient which also satisfy a Log-Sobolev inequality. We derive non-asymptotic convergence bounds in $KL$ and consequently total variation and Wasserstein-$2$ distance from the target measure. Non-asymptotic convergence guarantees are provided for the performance of the new algorithm as an optimizer. Finally, some theoretical results on isoperimertic inequalities for distributions with superlinearly growing gradients are provided. Key findings are a Log-Sobolev inequality with constant independent of the dimension, in the presence of a higher order regularization and a Poincare inequality with constant independent of temperature and dimension under a novel non-convex theoretical framework.