Generalized Linear Model Regression under Distance-to-set Penalties
This addresses the issue of unwanted shrinkage in high-dimensional GLM estimation for researchers and practitioners, though it is incremental as it builds on existing penalized likelihood methods.
The paper tackles the problem of constrained estimation in generalized linear models by penalizing squared distance to constraint sets instead of using shrinkage-inducing penalties like lasso, resulting in strong empirical performance in applications such as shape constraints and sparse regression.
Estimation in generalized linear models (GLM) is complicated by the presence of constraints. One can handle constraints by maximizing a penalized log-likelihood. Penalties such as the lasso are effective in high dimensions, but often lead to unwanted shrinkage. This paper explores instead penalizing the squared distance to constraint sets. Distance penalties are more flexible than algebraic and regularization penalties, and avoid the drawback of shrinkage. To optimize distance penalized objectives, we make use of the majorization-minimization principle. Resulting algorithms constructed within this framework are amenable to acceleration and come with global convergence guarantees. Applications to shape constraints, sparse regression, and rank-restricted matrix regression on synthetic and real data showcase strong empirical performance, even under non-convex constraints.