Convergence of Langevin MCMC in KL-divergence
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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 p^* is L smooth and m strongly convex, discrete Langevin diffusion produces a distribution p with KL(p||p^*)≤ϵ in Õ(d/ϵ) 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.
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