  We consider the problem of sampling from a density of the form $p(x) \propto
\exp(-f(x)- g(x))$, where $f: \mathbb{R}^d \rightarrow \mathbb{R}$ is a smooth
and strongly convex function and $g: \mathbb{R}^d \rightarrow \mathbb{R}$ is a
convex and Lipschitz function. We propose a new algorithm based on the
Metropolis-Hastings framework, and prove that it mixes to within TV distance
$\varepsilon$ of the target density in at most $O(d \log (d/\varepsilon))$
iterations. This guarantee extends previous results on sampling from
distributions with smooth log densities ($g = 0$) to the more general composite
non-smooth case, with the same mixing time up to a multiple of the condition
number. Our method is based on a novel proximal-based proposal distribution
that can be efficiently computed for a large class of non-smooth functions $g$.
