Langevin diffusion is a commonly used tool for sampling from a given
distribution. In this work, we establish that when the target density pβ is
such that logpβ is L smooth and m strongly convex, discrete Langevin
diffusion produces a distribution p with KL(pβ£β£pβ)β€Ο΅ in
O~(Ο΅dβ) 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
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