Title: Scalable Metropolis-Hastings for Exact Bayesian Inference with Large Datasets,
Abstract: Bayesian inference via standard Markov Chain Monte Carlo (MCMC) methods such
as Metropolis-Hastings is too computationally intensive to handle large
datasets, since the cost per step usually scales like $O(n)$ in the number of
data points $n$. We propose the Scalable Metropolis-Hastings (SMH) kernel that
exploits Gaussian concentration of the posterior to require processing on
average only $O(1)$ or even $O(1/\sqrt{n})$ data points per step. This scheme
is based on a combination of factorized acceptance probabilities, procedures
for fast simulation of Bernoulli processes, and control variate ideas. Contrary
to many MCMC subsampling schemes such as fixed step-size Stochastic Gradient
Langevin Dynamics, our approach is exact insofar as the invariant distribution
is the true posterior and not an approximation to it. We characterise the
performance of our algorithm theoretically, and give realistic and verifiable
conditions under which it is geometrically ergodic. This theory is borne out by
empirical results that demonstrate overall performance benefits over standard
Metropolis-Hastings and various subsampling algorithms.