  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.
