  We present an improved analysis of the Euler-Maruyama discretization of the
Langevin diffusion. Our analysis does not require global contractivity, and
yields polynomial dependence on the time horizon. Compared to existing
approaches, we make an additional smoothness assumption, and improve the
existing rate from $O(\eta)$ to $O(\eta^2)$ in terms of the KL divergence. This
result matches the correct order for numerical SDEs, without suffering from
exponential time dependence. When applied to algorithms for sampling and
learning, this result simultaneously improves all those methods based on
Dalayan's approach.
