An ADMM approach for multi-response regression with overlapping groups and interaction effects

The constraints ensure that the interaction term can be nonzero only if the corresponding main term is nonzero. Even though the idea is still young, it has been applied in different areas, for example to multinomial logistic regression (Asenso et al., 2022b), Cox's proportional hazards model (Du and Tibshirani, 2018) and support vector machines (Asenso et al., 2022a). However, in all the above studies, the block-wise coordinate descent procedure was used in solving the problem which includes overlapping groups. The algorithm involves multiple "if" statements and a generalized gradient at the final stage. This implies that extending the model to a multi-response case would require rigorous computations like the case of Li et al. 2015, which might be difficult to handle. In this paper, we introduce the alternating direction method of multipliers (ADMM) to handle this problem and extend the results from the single response model to a multi-response problem. We provide a publicly available software package MAD-MMplasso (Asenso and Zucknick, 2022) implemented in R. We present a brief review on the ADMM algorithm in what follows.