  Discretization of continuous-time diffusion processes is a widely recognized
method for sampling. However, it seems to be a considerable restriction when
the potentials are often required to be smooth (gradient Lipschitz). This paper
studies the problem of sampling through Euler discretization, where the
potential function is assumed to be a mixture of weakly smooth distributions
and satisfies weakly dissipative. We establish the convergence in
Kullback-Leibler (KL) divergence with the number of iterations to reach
$\epsilon$-neighborhood of a target distribution in only polynomial dependence
on the dimension. We relax the degenerated convex at infinity conditions of
\citet{erdogdu2020convergence} and prove convergence guarantees under
Poincar\'{e} inequality or non-strongly convex outside the ball. In addition,
we also provide convergence in $L_{\beta}$-Wasserstein metric for the smoothing
potential.
