  In this paper, we introduce structured sparsity estimators in Generalized
Linear Models. Structured sparsity estimators in the least squares loss are
introduced by Stucky and van de Geer (2018) recently for fixed design and
normal errors. We extend their results to debiased structured sparsity
estimators with Generalized Linear Model based loss. Structured sparsity
estimation means penalized loss functions with a possible sparsity structure
used in the chosen norm. These include weighted group lasso, lasso and norms
generated from convex cones. The significant difficulty is that it is not clear
how to prove two oracle inequalities. The first one is for the initial
penalized Generalized Linear Model estimator. Since it is not clear how a
particular feasible-weighted nodewise regression may fit in an oracle
inequality for penalized Generalized Linear Model, we need a second oracle
inequality to get oracle bounds for the approximate inverse for the sample
estimate of second-order partial derivative of Generalized Linear Model.
  Our contributions are fivefold: 1. We generalize the existing oracle
inequality results in penalized Generalized Linear Models by proving the
underlying conditions rather than assuming them. One of the key issues is the
proof of a sample one-point margin condition and its use in an oracle
inequality. 2. Our results cover even non sub-Gaussian errors and regressors.
3. We provide a feasible weighted nodewise regression proof which generalizes
the results in the literature from a simple l_1 norm usage to norms generated
from convex cones. 4. We realize that norms used in feasible nodewise
regression proofs should be weaker or equal to the norms in penalized
Generalized Linear Model loss. 5. We can debias the first step estimator via
getting an approximate inverse of the singular-sample second order partial
derivative of Generalized Linear Model loss.
