# Variational Bayes for high-dimensional linear regression with sparse priors

## TL;DR

This paper develops a mean-field variational Bayes approach for high-dimensional sparse linear regression, providing theoretical guarantees and practical improvements over existing methods.

## Contribution

It introduces a novel prioritized updating scheme for variational inference that enhances performance and offers theoretical oracle inequalities for the approximation.

## Key findings

- VB approximation converges at the optimal rate under certain conditions
- The proposed updating scheme outperforms standard coordinate-ascent in simulations
- The method performs comparably to state-of-the-art Bayesian variable selection techniques

## Abstract

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'.