Escaping Spurious Local Minimum Trajectories in Online Time-varying Nonconvex Optimization

This paper is concerned with solving online nonconvex optimization problems using simple gradient-based algorithms with an arbitrary initialization. The main objective is to understand how the natural data variation of an online optimization problem affects finding its time-varying global minima. To this end, we investigate the properties of a time-varying gradient flow system with inertia, which can be regarded as the continuous-time limit of the online tracking scheme obtained by working through the optimality conditions for a discretized sequential optimization problem with a proximal regularization. We introduce the notion of the dominant trajectory and show that the inherent temporal variation of the problem could re-shape the landscape and help a proximal algorithm escape the spurious local minimum trajectories if the global minimum trajectory is dominant. By studying the three notions of jumping, tracking and escaping for nonlinear dynamical systems, sufficient conditions are derived to guarantee that no matter how the local search method is initialized, it will find and track a time-varying global solution after some time.