Generalized linear models with structured sparsity estimators

In this paper, we introduce structured sparsity estimators for use in Generalized Linear Models. Structured sparsity estimators in the least squares loss are introduced by Stucky and van de Geer (2018). Their proofs exclusively depend on their use of fixed design and normal errors. We extend their results to debiased structured sparsity estimators with Generalized Linear Model based loss through incorporating random design and non-sub Gaussian data. Structured sparsity estimation means that penalized loss functions with a possible sparsity structure in a norm. These norms include norms generated from convex cones. Our contributions are threefold: (1) We generalize the existing oracle inequality results in penalized Generalized Linear Models; (2) We provide a feasible weighted nodewise regression proof which generalizes the results in the literature; (3) We realize that norms used in feasible nodewise regression proofs should be weaker or equal to the norms in penalized Generalized Linear Model loss. & COPY; 2023 Elsevier B.V. All rights reserved.