The Zig-Zag Process and Super-Efficient Sampling for Bayesian Analysis of Big Data

TL;DR

This paper introduces a non-reversible, super-efficient Monte Carlo method based on the Zig-Zag process for scalable Bayesian inference in big data, achieving exact sampling with sub-sampling and variance reduction techniques.

Contribution

It develops a new Zig-Zag process-based Monte Carlo method that is exact, scalable, and super-efficient for Bayesian analysis of large datasets.

Findings

The method achieves super-efficiency with data-independent cost.

It provides an exact sampling scheme that targets the true posterior.

The approach demonstrates favorable convergence properties.

Abstract

Standard MCMC methods can scale poorly to big data settings due to the need to evaluate the likelihood at each iteration. There have been a number of approximate MCMC algorithms that use sub-sampling ideas to reduce this computational burden, but with the drawback that these algorithms no longer target the true posterior distribution. We introduce a new family of Monte Carlo methods based upon a multi-dimensional version of the Zig-Zag process of (Bierkens, Roberts, 2017), a continuous time piecewise deterministic Markov process. While traditional MCMC methods are reversible by construction (a property which is known to inhibit rapid convergence) the Zig-Zag process offers a flexible non-reversible alternative which we observe to often have favourable convergence properties. We show how the Zig-Zag process can be simulated without discretisation error, and give conditions for the process to be ergodic. Most importantly, we introduce a sub-sampling version of the Zig-Zag process that is an example of an {\em exact approximate scheme}, i.e. the resulting approximate process still has the posterior as its stationary distribution. Furthermore, if we use a control-variate idea to reduce the variance of our unbiased estimator, then the Zig-Zag process can be super-efficient: after an initial pre-processing step, essentially independent samples from the posterior distribution are obtained at a computational cost which does not depend on the size of the data.