Finite-Time Convergence in Continuous-Time Optimization

Orlando Romero, Mouhacine Benosman
Proceedings of the 37th International Conference on Machine Learning, PMLR 119:8200-8209, 2020.

Abstract

In this paper, we investigate a Lyapunov-like differential inequality that allows us to establish finite-time stability of a continuous-time state-space dynamical system represented via a multivariate ordinary differential equation or differential inclusion. Equipped with this condition, we successfully synthesize first and second-order dynamical systems that achieve finite-time convergence to the minima of a given sufficiently regular cost function. As a byproduct, we show that the p-rescaled gradient flow (p-RGF) proposed by Wibisono et al. (2016) is indeed finite-time convergent, provided the cost function is gradient dominated of order q in (1,p). Thus, we effectively bridge a gap between the p-RGF and the normalized gradient flow (NGF) (p=\infty) proposed by Cortes (2006) in his seminal paper in the context of multi-agent systems. We discuss strategies to discretize our proposed flows and conclude by conducting some numerical experiments to illustrate our results.

Cite this Paper

BibTeX
@InProceedings{pmlr-v119-romero20b, title = {Finite-Time Convergence in Continuous-Time Optimization}, author = {Romero, Orlando and Benosman, Mouhacine}, booktitle = {Proceedings of the 37th International Conference on Machine Learning}, pages = {8200--8209}, year = {2020}, editor = {III, Hal Daumé and Singh, Aarti}, volume = {119}, series = {Proceedings of Machine Learning Research}, month = {13--18 Jul}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v119/romero20b/romero20b.pdf}, url = {https://proceedings.mlr.press/v119/romero20b.html}, abstract = {In this paper, we investigate a Lyapunov-like differential inequality that allows us to establish finite-time stability of a continuous-time state-space dynamical system represented via a multivariate ordinary differential equation or differential inclusion. Equipped with this condition, we successfully synthesize first and second-order dynamical systems that achieve finite-time convergence to the minima of a given sufficiently regular cost function. As a byproduct, we show that the p-rescaled gradient flow (p-RGF) proposed by Wibisono et al. (2016) is indeed finite-time convergent, provided the cost function is gradient dominated of order q in (1,p). Thus, we effectively bridge a gap between the p-RGF and the normalized gradient flow (NGF) (p=\infty) proposed by Cortes (2006) in his seminal paper in the context of multi-agent systems. We discuss strategies to discretize our proposed flows and conclude by conducting some numerical experiments to illustrate our results.} }
Endnote
%0 Conference Paper %T Finite-Time Convergence in Continuous-Time Optimization %A Orlando Romero %A Mouhacine Benosman %B Proceedings of the 37th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2020 %E Hal Daumé III %E Aarti Singh %F pmlr-v119-romero20b %I PMLR %P 8200--8209 %U https://proceedings.mlr.press/v119/romero20b.html %V 119 %X In this paper, we investigate a Lyapunov-like differential inequality that allows us to establish finite-time stability of a continuous-time state-space dynamical system represented via a multivariate ordinary differential equation or differential inclusion. Equipped with this condition, we successfully synthesize first and second-order dynamical systems that achieve finite-time convergence to the minima of a given sufficiently regular cost function. As a byproduct, we show that the p-rescaled gradient flow (p-RGF) proposed by Wibisono et al. (2016) is indeed finite-time convergent, provided the cost function is gradient dominated of order q in (1,p). Thus, we effectively bridge a gap between the p-RGF and the normalized gradient flow (NGF) (p=\infty) proposed by Cortes (2006) in his seminal paper in the context of multi-agent systems. We discuss strategies to discretize our proposed flows and conclude by conducting some numerical experiments to illustrate our results.
APA
Romero, O. & Benosman, M.. (2020). Finite-Time Convergence in Continuous-Time Optimization. Proceedings of the 37th International Conference on Machine Learning, in Proceedings of Machine Learning Research 119:8200-8209 Available from https://proceedings.mlr.press/v119/romero20b.html.

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