Adaptive penalization in high-dimensional regression and classification with external covariates using variational Bayes.

Penalization schemes like Lasso or ridge regression are routinely used to regress a response of interest on a high-dimensional set of potential predictors. Despite being decisive, the question of the relative strength of penalization is often glossed over and only implicitly determined by the scale of individual predictors. At the same time, additional information on the predictors is available in many applications but left unused. Here, we propose to make use of such external covariates to adapt the penalization in a data-driven manner. We present a method that differentially penalizes feature groups defined by the covariates and adapts the relative strength of penalization to the information content of each group. Using techniques from the Bayesian tool-set our procedure combines shrinkage with feature selection and provides a scalable optimization scheme. We demonstrate in simulations that the method accurately recovers the true effect sizes and sparsity patterns per feature group. Furthermore, it leads to an improved prediction performance in situations where the groups have strong differences in dynamic range. In applications to data from high-throughput biology, the method enables re-weighting the importance of feature groups from different assays. Overall, using available covariates extends the range of applications of penalized regression, improves model interpretability and can improve prediction performance.