Convex Modeling of Interactions with Strong Heredity
This work addresses the challenge of modeling interactions in high-dimensional regression for statisticians and data scientists, but it is incremental as it builds upon and generalizes prior methods.
The authors tackled the problem of fitting regression models with interactions while enforcing strong heredity constraints, proposing a convex optimization framework called FAMILY that generalizes existing methods and demonstrates performance in simulations and an HIV dataset.
We consider the task of fitting a regression model involving interactions among a potentially large set of covariates, in which we wish to enforce strong heredity. We propose FAMILY, a very general framework for this task. Our proposal is a generalization of several existing methods, such as VANISH [Radchenko and James, 2010], hierNet [Bien et al., 2013], the all-pairs lasso, and the lasso using only main effects. It can be formulated as the solution to a convex optimization problem, which we solve using an efficient alternating directions method of multipliers (ADMM) algorithm. This algorithm has guaranteed convergence to the global optimum, can be easily specialized to any convex penalty function of interest, and allows for a straightforward extension to the setting of generalized linear models. We derive an unbiased estimator of the degrees of freedom of FAMILY, and explore its performance in a simulation study and on an HIV sequence data set.