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Abstract This paper studies a statistical learning model where the model coefficients have a pre-determined non-overlapping group sparsity structure. We consider a combination of a loss function and a regularizer to recover the desired group sparsity patterns, which can embrace many existing works. We analyze directional stationary solutions of the proposed formulation, obtaining a sufficient condition for a directional stationary solution to achieve optimality and establishing a bound of the distance from the solution to a reference point. We develop an efficient algorithm that adopts an alternating direction method of multiplier (ADMM), showing that the iterates converge to a directional stationary solution under certain conditions. In the numerical experiment, we implement the algorithm for generalized linear models with convex and nonconvex group regularizers to evaluate the model performance on various data types, noise levels, and sparsity settings.