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Bayesian methods are appealing in their flexibility in modeling complex dataand ability in capturing uncertainty in parameters. However, when Bayes' ruledoes not result in tractable closed-form, most approximate inference algorithmslack either scalability or rigorous guarantees. To tackle this challenge, wepropose a simple yet provable algorithm, \emph{Particle Mirror Descent} (PMD),to iteratively approximate the posterior density. PMD is inspired by stochasticfunctional mirror descent where one descends in the density space using a smallbatch of data points at each iteration, and by particle filtering where oneuses samples to approximate a function. We prove result of the first kind that,with $m$ particles, PMD provides a posterior density estimator that convergesin terms of $KL$-divergence to the true posterior in rate $O(1/\sqrt{m})$. Wedemonstrate competitive empirical performances of PMD compared to severalapproximate inference algorithms in mixture models, logistic regression, sparseGaussian processes and latent Dirichlet allocation on large scale datasets.

Provable Bayesian Inference via Particle Mirror Descent