  We study sampling from a target distribution $\nu_* = e^{-f}$ using the
unadjusted Langevin Monte Carlo (LMC) algorithm when the potential $f$
satisfies a strong dissipativity condition and it is first-order smooth with a
Lipschitz gradient. We prove that, initialized with a Gaussian random vector
that has sufficiently small variance, iterating the LMC algorithm for
$\widetilde{\mathcal{O}}(\lambda^2 d\epsilon^{-1})$ steps is sufficient to
reach $\epsilon$-neighborhood of the target in both Chi-squared and Renyi
divergence, where $\lambda$ is the logarithmic Sobolev constant of $\nu_*$. Our
results do not require warm-start to deal with the exponential dimension
dependency in Chi-squared divergence at initialization. In particular, for
strongly convex and first-order smooth potentials, we show that the LMC
algorithm achieves the rate estimate $\widetilde{\mathcal{O}}(d\epsilon^{-1})$
which improves the previously known rates in both of these metrics, under the
same assumptions. Translating this rate to other metrics, our results also
recover the state-of-the-art rate estimates in KL divergence, total variation
and $2$-Wasserstein distance in the same setup. Finally, as we rely on the
logarithmic Sobolev inequality, our framework covers a range of non-convex
potentials that are first-order smooth and exhibit strong convexity outside of
a compact region.
