Convergence of Langevin MCMC in KLdivergence
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Proceedings of Algorithmic Learning Theory, PMLR 83:186211, 2018.
Abstract
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 $\KL{\p}{\p^*}≤ε$ in $\tilde{O}(\frac{d}{ε})$ steps, where $d$ is the dimension of the sample space. We also study the convergence rate when the strongconvexity 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 KLdivergence and gives a conceptually simpler proof of the bestknown convergence results in weaker metrics.
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