A Gradient-Based Optimization Method Using the Koopman Operator

Abstract: In this paper, we propose a novel approach to solving optimization problems by reformulating the optimization problem into a dynamical system, followed by the adaptive spectral Koopman (ASK) method. The Koopman operator, employed in our approach, approximates the evolution of an ordinary differential equation (ODE) using a finite number of eigenfunctions and eigenvalues. We begin by providing a brief overview of the Koopman operator and the ASK method. Subsequently, we adapt the ASK method for solving a general optimization problem. Moreover, we provide an error bound to aid in understanding the performance of the proposed approach, marking the initial step in a more comprehensive numerical analysis. Experimentally, we demonstrate the applicability and accuracy of our method across a diverse range of optimization problems, including min-max problems. Our approach consistently yields smaller gradient norms and higher success rates in finding critical points compared to state-of-the-art gradient-based methods. We also observe the proposed method works particularly well when the dynamical properties of the system can be effectively modeled by the system's behaviors in a neighborhood of critical points.