  In high-dimensions, the prior tails can have a significant effect on both
posterior computation and asymptotic concentration rates. To achieve optimal
rates while keeping the posterior computations relatively simple, an empirical
Bayes approach has recently been proposed, featuring thin-tailed conjugate
priors with data-driven centers. While conjugate priors ease some of the
computational burden, Markov chain Monte Carlo methods are still needed, which
can be expensive when dimension is high. In this paper, we develop a
variational approximation to the empirical Bayes posterior that is fast to
compute and retains the optimal concentration rate properties of the original.
In simulations, our method is shown to have superior performance compared to
existing variational approximations in the literature across a wide range of
high-dimensional settings.
