Regularized higher-order Taylor approximation methods for nonlinear least-squares

Abstract: In this paper, we develop a regularized higher-order Taylor based method for solving composite (e.g., nonlinear least-squares) problems. At each iteration, we replace each smooth component of the objective function by a higher-order Taylor approximation with an appropriate regularization, leading to a regularized higher-order Taylor approximation (RHOTA) algorithm. We derive global convergence guarantees for RHOTA algorithm. In particular, we prove stationary point convergence guarantees for the iterates generated by RHOTA, and leveraging a Kurdyka-{\L}ojasiewicz (KL) type property of the objective function, we derive improved rates depending on the KL parameter. When the Taylor approximation is of order $2$, we present an efficient implementation of RHOTA algorithm, demonstrating that the resulting nonconvex subproblem can be effectively solved utilizing standard convex programming tools. Furthermore, we extend the scope of our investigation to include the behavior and efficacy of RHOTA algorithm in handling systems of nonlinear equations and optimization problems with nonlinear equality constraints deriving new rates under improved constraint qualifications conditions. Finally, we consider solving the phase retrieval problem with a higher-order proximal point algorithm, showcasing its rapid convergence rate for this particular application. Numerical simulations on phase retrieval and output feedback control problems also demonstrate the efficacy and performance of the proposed methods when compared to some state-of-the-art optimization methods and software.