  There has been recent interest in developing scalable Bayesian sampling
methods such as stochastic gradient MCMC (SG-MCMC) and Stein variational
gradient descent (SVGD) for big-data analysis. A standard SG-MCMC algorithm
simulates samples from a discrete-time Markov chain to approximate a target
distribution, thus samples could be highly correlated, an undesired property
for SG-MCMC. In contrary, SVGD directly optimizes a set of particles to
approximate a target distribution, and thus is able to obtain good
approximations with relatively much fewer samples. In this paper, we propose a
principle particle-optimization framework based on Wasserstein gradient flows
to unify SG-MCMC and SVGD, and to allow new algorithms to be developed. Our
framework interprets SG-MCMC as particle optimization on the space of
probability measures, revealing a strong connection between SG-MCMC and SVGD.
The key component of our framework is several particle-approximate techniques
to efficiently solve the original partial differential equations on the space
of probability measures. Extensive experiments on both synthetic data and deep
neural networks demonstrate the effectiveness and efficiency of our framework
for scalable Bayesian sampling.
