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Poster
Fast Gradient-Based Methods with Exponential Rate: A Hybrid Control Framework
Arman Sharifi Kolarijani · Peyman Mohajerin Esfahani · Tamas Keviczky

Fri Jul 13 09:15 AM -- 12:00 PM (PDT) @ Hall B #17
Ordinary differential equations, and in general a dynamical system viewpoint, have seen a resurgence of interest in developing fast optimization methods, mainly thanks to the availability of well-established analysis tools. In this study, we pursue a similar objective and propose a class of hybrid control systems that adopts a 2nd-order differential equation as its continuous flow. A distinctive feature of the proposed differential equation in comparison with the existing literature is a state-dependent, time-invariant damping term that acts as a feedback control input. Given a user-defined scalar $\alpha$, it is shown that the proposed control input steers the state trajectories to the global optimizer of a desired objective function with a guaranteed rate of convergence $\mathcal{O}(e^{-\alpha t})$. Our framework requires that the objective function satisfies the so called Polyak--{\L}ojasiewicz inequality. Furthermore, a discretization method is introduced such that the resulting discrete dynamical system possesses an exponential rate of convergence.