Structured Sparse Aggregation

We introduce a method for aggregating many least squares estimator so that the resulting estimate has two properties: sparsity and structure. That is, only a few candidate covariates are used in the resulting model, and the selected covariates follow some structure over the candidate covariates that is assumed to be known a priori. While sparsity is well studied in many settings, including aggregation, structured sparse methods are still emerging. We demonstrate a general framework for structured sparse aggregation that allows for a wide variety of structures, including overlapping grouped structures and general structural penalties defined as set functions on the set of covariates. We show that such estimators satisfy structured sparse oracle inequalities --- their finite sample risk adapts to the structured sparsity of the target. These inequalities reveal that under suitable settings, the structured sparse estimator performs at least as well as, and potentially much better than, a sparse aggregation estimator. We empirically establish the effectiveness of the method using simulation and an application to HIV drug resistance.