A fast asynchronous MCMC sampler for sparse Bayesian inference
This addresses efficiency issues for researchers and practitioners in high-dimensional statistics, though it is incremental as an extension of existing asynchronous Gibbs sampling methods.
The paper tackles the computational challenge of sparse Bayesian inference by proposing a fast approximate MCMC sampler that reduces per-iteration cost to O(ns) and shows linear mixing time in high-dimensional regression, recovering the main signal with high probability under certain assumptions.
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.