Large-Scale Stochastic Sampling from the Probability Simplex
This addresses a critical bottleneck for scalable Bayesian inference in models like network or topic models, where existing methods struggle with sparse constraints, offering an incremental improvement over current SGMCMC approaches.
The paper tackles the problem of biased sampling in sparse simplex spaces for large-scale Bayesian models using stochastic gradient Markov chain Monte Carlo (SGMCMC), and proposes the stochastic Cox-Ingersoll-Ross (SCIR) process to eliminate discretization error, resulting in substantially better performance for topic and Dirichlet process mixture models.
Stochastic gradient Markov chain Monte Carlo (SGMCMC) has become a popular method for scalable Bayesian inference. These methods are based on sampling a discrete-time approximation to a continuous time process, such as the Langevin diffusion. When applied to distributions defined on a constrained space the time-discretization error can dominate when we are near the boundary of the space. We demonstrate that because of this, current SGMCMC methods for the simplex struggle with sparse simplex spaces; when many of the components are close to zero. Unfortunately, many popular large-scale Bayesian models, such as network or topic models, require inference on sparse simplex spaces. To avoid the biases caused by this discretization error, we propose the stochastic Cox-Ingersoll-Ross process (SCIR), which removes all discretization error and we prove that samples from the SCIR process are asymptotically unbiased. We discuss how this idea can be extended to target other constrained spaces. Use of the SCIR process within a SGMCMC algorithm is shown to give substantially better performance for a topic model and a Dirichlet process mixture model than existing SGMCMC approaches.