Convex Modeling of Interactions with Strong Heredity

TL;DR

This paper introduces FAMILY, a convex optimization framework for modeling interactions with strong heredity constraints, generalizing existing methods and enabling efficient, globally optimal solutions with applications to real and simulated data.

Contribution

FAMILY is a flexible, convex optimization-based framework that unifies and extends existing interaction modeling methods with guaranteed convergence and easy extensions.

Findings

FAMILY outperforms existing methods in simulation studies.

The algorithm guarantees convergence to the global optimum.

Application to HIV data demonstrates practical utility.

Abstract

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.