On the Absence of Spurious Local Trajectories in Time-Varying Nonconvex Optimization

In this article, we study the landscape of an online nonconvex optimization problem, for which input data vary over time and solution is a trajectory rather than single point. To understand complexity finding global introduce notion spurious (i.e., nonglobal) local trajectory as generalization to spurious in (time-invariant) optimization. We develop ordinary differential equation (ODE) associated with time-varying nonlinear dynamical system which, at limit, characterizes solutions problem. prove that absence closely related transient behavior developed system. particular, show if problem time-varying, variation may force all ODE trajectories initialized arbitrary minima initial gradually converge trajectory. Jacobian along minimum how its eigenvalues are manipulated by natural consequently trigger escaping poor time.