  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.
