Title: Adaptive Optimizers with Sparse Group Lasso. Abstract: We develop a novel framework that adds the regularizers to a family of adaptive optimizers in deep learning, such as MOMENTUM, ADAGRAD, ADAM, AMSGRAD, ADAHESSIAN, and create a new class of optimizers, which are named GROUP MOMENTUM, GROUP ADAGRAD, GROUP ADAM, GROUP AMSGRAD and GROUP ADAHESSIAN, etc., accordingly. We establish theoretically proven convergence guarantees in the stochastic convex settings, based on primal-dual methods. We evaluate the regularized effect of our new optimizers on three large-scale real-world ad click datasets with state-of-the-art deep learning models. The experimental results reveal that compared with the original optimizers with the post-processing procedure which use the magnitude pruning method, the performance of the models can be significantly improved on the same sparsity level. Furthermore, in comparison to the cases without magnitude pruning, our methods can achieve extremely high sparsity with significantly better or highly competitive performance.