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Stochastic gradient Markov chain Monte Carlo (SGMCMC) has become a popularmethod for scalable Bayesian inference. These methods are based on sampling adiscrete-time approximation to a continuous time process, such as the Langevindiffusion. When applied to distributions defined on a constrained space thetime-discretization error can dominate when we are near the boundary of thespace. We demonstrate that because of this, current SGMCMC methods for thesimplex struggle with sparse simplex spaces; when many of the components areclose to zero. Unfortunately, many popular large-scale Bayesian models, such asnetwork or topic models, require inference on sparse simplex spaces. To avoidthe biases caused by this discretization error, we propose the stochasticCox-Ingersoll-Ross process (SCIR), which removes all discretization error andwe prove that samples from the SCIR process are asymptotically unbiased. Wediscuss how this idea can be extended to target other constrained spaces. Useof the SCIR process within a SGMCMC algorithm is shown to give substantiallybetter performance for a topic model and a Dirichlet process mixture model thanexisting SGMCMC approaches.

Large-Scale Stochastic Sampling from the Probability Simplex