  We study the Unadjusted Langevin Algorithm (ULA) for sampling from a
probability distribution $\nu = e^{-f}$ on $\mathbb{R}^n$. We prove a
convergence guarantee in Kullback-Leibler (KL) divergence assuming $\nu$
satisfies a log-Sobolev inequality and the Hessian of $f$ is bounded. Notably,
we do not assume convexity or bounds on higher derivatives. We also prove
convergence guarantees in R\'enyi divergence of order $q > 1$ assuming the
limit of ULA satisfies either the log-Sobolev or Poincar\'e inequality. We also
prove a bound on the bias of the limiting distribution of ULA assuming
third-order smoothness of $f$, without requiring isoperimetry.
