Authors: Ali Dabouei, Sobhan Soleymani, Fariborz Taherkhani, Jeremy Dawson, Nasser M. Nasrabadi Description: Recently, ensemble models have demonstrated empirical capabilities to alleviate the adversarial vulnerability. In this paper, we exploit first-order interactions within ensembles to formalize a reliable and practical defense. We introduce a scenario of interactions that certifiably improves the robustness according to the size of the ensemble, the diversity of the gradient directions, and the balance of the member's contribution to the robustness. We present a joint gradient phase and magnitude regularization (GPMR) as a vigorous approach to impose the desired scenario of interactions among members of the ensemble. Through extensive experiments, including gradient-based and gradient-free evaluations on several datasets and network architectures, we validate the practical effectiveness of the proposed approach compared to the previous methods. Furthermore, we demonstrate that GPMR is orthogonal to other defense strategies developed for single classifiers and their combination can further improve the robustness of ensembles.