Title: Continuous-Time Accelerated Methods via a Hybrid Control Lens,
Abstract: Treating optimization methods as dynamical systems can be traced back
centuries ago in order to comprehend the notions and behaviors of optimization
methods. Lately, this mind set has become the driving force to design new
optimization methods. Inspired by the recent dynamical system viewpoint of
Nesterov's fast method, we propose two classes of fast methods, formulated as
hybrid control systems, to obtain pre-specified exponential convergence rate.
Alternative to the existing fast methods which are parametric-in-time second
order differential equations, we dynamically synthesize feedback controls in a
state-dependent manner. Namely, in the first class the damping term is viewed
as the control input, while in the second class the amplitude with which the
gradient of the objective function impacts the dynamics serves as the
controller. The objective function requires to satisfy a certain sharpness
criterion, the so-called Polyak--{\L}ojasiewicz inequality. Moreover, we
establish that both hybrid structures possess Zeno-free solution trajectories.
We finally provide a mechanism to determine the discretization step size to
attain an exponential convergence rate.