  We introduce Group Spike-and-slab Variational Bayes (GSVB), a scalable method
for group sparse regression. A fast co-ordinate ascent variational inference
(CAVI) algorithm is developed for several common model families including
Gaussian, Binomial and Poisson. Theoretical guarantees for our proposed
approach are provided by deriving contraction rates for the variational
posterior in grouped linear regression. Through extensive numerical studies, we
demonstrate that GSVB provides state-of-the-art performance, offering a
computationally inexpensive substitute to MCMC, whilst performing comparably or
better than existing MAP methods. Additionally, we analyze three real world
datasets wherein we highlight the practical utility of our method,
demonstrating that GSVB provides parsimonious models with excellent predictive
performance, variable selection and uncertainty quantification.
