A fast asynchronous MCMC sampler for sparse Bayesian inference

TL;DR

This paper introduces a fast, approximate asynchronous MCMC sampler tailored for sparse Bayesian inference, significantly reducing computational costs and effectively recovering signals in high-dimensional regression tasks.

Contribution

It extends the asynchronous Gibbs sampler to a Bayesian context, providing theoretical guarantees on signal recovery and mixing time in high-dimensional models.

Findings

Linear mixing time in number of regressors

High probability of correct signal recovery

Reduced computational complexity per iteration

Abstract

We propose a very fast approximate Markov Chain Monte Carlo (MCMC) sampling framework that is applicable to a large class of sparse Bayesian inference problems, where the computational cost per iteration in several models is of order $O(ns)$, where $n$ is the sample size, and $s$ the underlying sparsity of the model. This cost can be further reduced by data sub-sampling when stochastic gradient Langevin dynamics are employed. The algorithm is an extension of the asynchronous Gibbs sampler of Johnson et al. (2013), but can be viewed from a statistical perspective as a form of Bayesian iterated sure independent screening (Fan et al. (2009)). We show that in high-dimensional linear regression problems, the Markov chain generated by the proposed algorithm admits an invariant distribution that recovers correctly the main signal with high probability under some statistical assumptions. Furthermore we show that its mixing time is at most linear in the number of regressors. We illustrate the algorithm with several models.