Variational Bayes for high-dimensional linear regression with sparse priors
This work addresses efficient Bayesian variable selection for high-dimensional data, but it is incremental as it builds on existing variational methods with a novel updating scheme.
The authors tackled the problem of high-dimensional linear regression with sparse priors using a mean-field spike and slab variational Bayes approximation, showing that it converges to the sparse truth at the optimal rate and performs comparably to state-of-the-art Bayesian methods in simulations.
We study a mean-field spike and slab variational Bayes (VB) approximation to Bayesian model selection priors in sparse high-dimensional linear regression. Under compatibility conditions on the design matrix, oracle inequalities are derived for the mean-field VB approximation, implying that it converges to the sparse truth at the optimal rate and gives optimal prediction of the response vector. The empirical performance of our algorithm is studied, showing that it works comparably well as other state-of-the-art Bayesian variable selection methods. We also numerically demonstrate that the widely used coordinate-ascent variational inference (CAVI) algorithm can be highly sensitive to the parameter updating order, leading to potentially poor performance. To mitigate this, we propose a novel prioritized updating scheme that uses a data-driven updating order and performs better in simulations. The variational algorithm is implemented in the R package 'sparsevb'.