Center‐augmented ℓ2‐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 ℓ1 -type penalty. In this paper, by introducing the group centers and ℓ2 -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(n2)$O(n^2)$ of the conventional pairwise-penalty method to only O(nK)$O(nK)$ , 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 R2 is produced and three additional significant variables are identified compared to those of the existing methods.