Scalable Bayesian Inference via Particle Mirror Descent

Bayesian methods are appealing in their flexibility in modeling complex data and their ability to capture uncertainty in parameters. However, when Bayes’ rule does not result in closed-form, most approximate Bayesian inference algorithms lacks either scalability or rigorous guarantees. To tackle this challenge, we propose a scalable yet simple algorithm, Particle Mirror Descent (PMD), to iteratively approximate the posterior density. PMD is inspired by stochastic functional mirror descent where one descends in the density space using a small batch of data points at each iteration, and by particle filtering where one uses samples to approximate a function. We prove result of the first kind that, after T iterations, PMD provides a posterior density estimator that converges in terms of KL-divergence to the true posterior in rate O(1/ √ T ). We show that PMD is competitive to several scalable Bayesian algorithms in mixture models, Bayesian logistic regression, sparse Gaussian processes and latent Dirichlet allocation. 1 ar X iv :1 50 6. 03 10 1v 1 [ cs .L G ] 9 J un 2 01 5