  Classically, the continuous-time Langevin diffusion converges exponentially
fast to its stationary distribution $\pi$ under the sole assumption that $\pi$
satisfies a Poincar\'e inequality. Using this fact to provide guarantees for
the discrete-time Langevin Monte Carlo (LMC) algorithm, however, is
considerably more challenging due to the need for working with chi-squared or
R\'enyi divergences, and prior works have largely focused on strongly
log-concave targets. In this work, we provide the first convergence guarantees
for LMC assuming that $\pi$ satisfies either a Lata\l{}a--Oleszkiewicz or
modified log-Sobolev inequality, which interpolates between the Poincar\'e and
log-Sobolev settings. Unlike prior works, our results allow for weak smoothness
and do not require convexity or dissipativity conditions.
