Nesterov smoothing for sampling without smoothness

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 more challenging and 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. With a properly chosen smoothing intensity, the accuracy of the algorithm is guaranteed. For strongly log-concave distributions, our algorithm can achieve $\mathcal{E}$ error in Wasserstein-2 distance with complexity $ \widetilde{\mathcal{O}} \left( \frac{ d^{1/3}}{ \mathcal{E}^{5/3}} \right) .$ For log-concave distributions, we achieve $\mathcal{E}$ error in total variation with complexity $\mathcal{O} \left(\frac{ M_\pi d }{ \mathcal{E}^{3}} \right) $ in expectation with $M_\pi$ being the second moment of the target distribution. For target distributions satisfying the logarithmic-Sobolev inequality, our algorithm has complexity $\widetilde{\mathcal{O}} \left( \frac{ d }{\mathcal{E}}\right)$.