  Langevin diffusion is a commonly used tool for sampling from a given
distribution. In this work, we establish that when the target density $p^*$ is
such that $\log p^*$ is $L$ smooth and $m$ strongly convex, discrete Langevin
diffusion produces a distribution $p$ with $KL(p||p^*)\leq \epsilon$ in
$\tilde{O}(\frac{d}{\epsilon})$ steps, where $d$ is the dimension of the sample
space. We also study the convergence rate when the strong-convexity assumption
is absent. By considering the Langevin diffusion as a gradient flow in the
space of probability distributions, we obtain an elegant analysis that applies
to the stronger property of convergence in KL-divergence and gives a
conceptually simpler proof of the best-known convergence results in weaker
metrics.

Machine Learning