Fast Gradient-Based Methods with Exponential Rate: A Hybrid Control Framework
Arman Sharifi Kolarijani, Peyman Mohajerin Esfahani, Tamas Keviczky
Abstract
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–{Ł}ojasiewicz inequality. Furthermore, a discretization method is introduced such that the resulting discrete dynamical system possesses an exponential rate of convergence.
BibTeX
@InProceedings{pmlr-v80-kolarijani18a,
title = {Fast Gradient-Based Methods with Exponential Rate: A Hybrid Control Framework},
author = {Kolarijani, Arman Sharifi and Esfahani, Peyman Mohajerin and Keviczky, Tamas},
booktitle = {Proceedings of the 35th International Conference on Machine Learning},
pages = {2728--2736},
year = {2018},
editor = {Dy, Jennifer and Krause, Andreas},
volume = {80},
series = {Proceedings of Machine Learning Research},
month = {10--15 Jul},
publisher = {PMLR},
pdf = {http://proceedings.mlr.press/v80/kolarijani18a/kolarijani18a.pdf},
url = {https://proceedings.mlr.press/v80/kolarijani18a.html},
abstract = {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–{Ł}ojasiewicz inequality. Furthermore, a discretization method is introduced such that the resulting discrete dynamical system possesses an exponential rate of convergence.}
}