Center-augmented ℓ 2 -type regularization for subgroup learning.

The existing methods for subgroup analysis can be roughly divided into two categories: finite mixture models (FMM) and regularization methods with an ℓ 1 -type penalty. In this paper, by introducing the group centers and ℓ 2 -type penalty in the loss function, we propose a novel center-augmented regularization (CAR) method; this method can be regarded as a unification of the regularization method and FMM and hence exhibits higher efficiency and robustness and simpler computations than the existing methods. In particular, its computational complexity is reduced from the O ( n 2 ) $O(n^2)$ of the conventional pairwise-penalty method to only O ( n K ) $O(nK)$ , where n is the sample size and K is the number of subgroups. The asymptotic normality of CAR is established, and the convergence of the algorithm is proven. CAR is applied to a dataset from a multicenter clinical trial, Buprenorphine in the Treatment of Opiate Dependence; a larger R 2 is produced and three additional significant variables are identified compared to those of the existing methods.