  Modern statistical learning algorithms are capable of amazing flexibility,
but struggle with interpretability. One possible solution is sparsity: making
inference such that many of the parameters are estimated as being identically
0, which may be imposed through the use of nonsmooth penalties such as the
$\ell_1$ penalty. However, the $\ell_1$ penalty introduces significant bias
when high sparsity is desired. In this article, we retain the $\ell_1$ penalty,
but define learnable penalty weights $\lambda_p$ endowed with hyperpriors. We
start the article by investigating the optimization problem this poses,
developing a proximal operator associated with the $\ell_1$ norm. We then study
the theoretical properties of this variable-coefficient $\ell_1$ penalty in the
context of penalized likelihood. Next, we investigate application of this
penalty to Variational Bayes, developing a model we call the Sparse Bayesian
Lasso which allows for behavior qualitatively like Lasso regression to be
applied to arbitrary variational models. In simulation studies, this gives us
the Uncertainty Quantification and low bias properties of simulation-based
approaches with an order of magnitude less computation. Finally, we apply our
methodology to a Bayesian lagged spatiotemporal regression model of internal
displacement that occurred during the Iraqi Civil War of 2013-2017.
