  We study the problem of sampling from a target distribution in $\mathbb{R}^d$
whose potential is not smooth. Compared with the sampling problem with smooth
potentials, this problem is much less well-understood due to the lack of
smoothness. In this paper, we propose a novel sampling algorithm for a class of
non-smooth potentials by first approximating them by smooth potentials using a
technique that is akin to Nesterov smoothing. We then utilize sampling
algorithms on the smooth potentials to generate approximate samples from the
original non-smooth potentials. We select an appropriate smoothing intensity to
ensure that the distance between the smoothed and un-smoothed distributions is
minimal, thereby guaranteeing the algorithm's accuracy. Hence we obtain
non-asymptotic convergence results based on existing analysis of smooth
sampling. We verify our convergence result on a synthetic example and apply our
method to improve the worst-case performance of Bayesian inference on a
real-world example.
