Summary: In this paper, we present a thorough study of the regularized submodular maximization problem, in which the objective \(f := g-\ell\) can be expressed as the difference between a submodular function and a modular function. This problem has drawn much attention in recent years. While existing works focuses on the case of \(g\) being monotone, we investigate the problem with a nonmonotone \(g\). The main technique we use is to introduce a distorted objective function, which varies weights of the submodular component \(g\) and the modular component \(\ell\) during the iterations of the algorithm. By combining the weighting technique and measured continuous greedy algorithm, we present an algorithm for the matroid-constrained problem, which has a provable approximation guarantee. In the cardinality-constrained case, we utilize random greedy algorithm and sampling technique together with the weighting technique to design two efficient algorithms. Moreover, we consider the unconstrained problem and propose a much simpler and faster algorithm compared with the algorithms for solving the problem with a cardinality constraint.