The Zig-Zag process and super-efficient sampling for Bayesian analysis of big data

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 and 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 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. * The authors acknowledge the EPSRC for support under grants EP/D002060/1 (CRiSM) and EP/K014463/1 (iLike).MSC 2010 subject classifications: Primary 65C60; secondary 65C05, 62F15, 60J25

Estimation and uncertainty quantification for the output from quantum simulators

“…A number of recently developed methods require only unbiased estimators of (9). These include the stochastic gradient Langevin dynamics (SGLD) [49], the zig-zag sampler [4], or the bouncy particle sampler [6,39]. We propose to construct an unbiased estimator of (9) using the debiasing technique from the recent works [32,42], and then use this unbiased estimator within such MCMC methods.…”

“…Debiasing. Some MCMC methods, such as SGLD [49], as well as some recently developed piecewise-deterministic Markov processes (PMDP) [12,4,6,39], require only an unbiased estimator of the gradient of the log-likelihood for implementation. Others, such as pseudo-marginal MCMC, require an unbiased and non-negative estimator of the likelihood itself.…”

Comment: A brief survey of the current state of play for Bayesian computation in data science at big-data scale

“…-On a single CPU, in our view the most promising approach with big n and tiny p, when a gradient is available, involves methods based on creating continuous-time stochastic processes without discretization that sample correctly from the posterior of interest-these include ScaLE (Pollock et al, 2016), ZigZag sampling (Bierkens, Fearnhead and Roberts, 2016), and the Bouncy Particle Sampler (BPS, Bouchard-Côté, Vollmer and Doucet, 2015). The key advantage of these methods is the ability to run while evaluating one data point at a time.…”

Ergodicity of the zigzag process

“…The zigzag process can be defined in multiple dimensions [5] but here we consider only one spatial dimension. In this setting the zigzag process is a Markov process (X(t), Θ(t)) in the state space E := R×{−1, +1}.…”