Strengthened stability analysis of discrete-time Lurie systems involving ReLU neural networks

Carl Richardson, Matthew Turner, Steve Gunn, Ross Drummond
Proceedings of the 6th Annual Learning for Dynamics & Control Conference, PMLR 242:209-221, 2024.

Abstract

This paper addresses the stability analysis of a discrete-time (DT) Lurie system featuring a static repeated ReLU nonlinearity. Such systems often arise in the analysis of recurrent neural networks and other neural feedback loops. Custom quadratic constraints, satisfied by the repeated ReLU, are employed to strengthen the standard Circle and Popov Criteria for this specific Lurie system. The criteria can be expressed as a set of linear matrix inequalities (LMIs) with less restrictive conditions on the matrix variables. It is further shown that if the Lurie system under consideration has a unique equilibrium point at the origin, then this equilibrium point is in fact globally stable or unstable, meaning that local stability analysis will provide no additional benefit. Numerical examples demonstrate that the strengthened criteria achieve a desirable balance between reduced conservatism and complexity when compared to existing criteria.

Cite this Paper

BibTeX
@InProceedings{pmlr-v242-richardson24a, title = {Strengthened stability analysis of discrete-time {L}urie systems involving {ReLU} neural networks}, author = {Richardson, Carl and Turner, Matthew and Gunn, Steve and Drummond, Ross}, booktitle = {Proceedings of the 6th Annual Learning for Dynamics & Control Conference}, pages = {209--221}, year = {2024}, editor = {Abate, Alessandro and Cannon, Mark and Margellos, Kostas and Papachristodoulou, Antonis}, volume = {242}, series = {Proceedings of Machine Learning Research}, month = {15--17 Jul}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v242/richardson24a/richardson24a.pdf}, url = {https://proceedings.mlr.press/v242/richardson24a.html}, abstract = {This paper addresses the stability analysis of a discrete-time (DT) Lurie system featuring a static repeated ReLU nonlinearity. Such systems often arise in the analysis of recurrent neural networks and other neural feedback loops. Custom quadratic constraints, satisfied by the repeated ReLU, are employed to strengthen the standard Circle and Popov Criteria for this specific Lurie system. The criteria can be expressed as a set of linear matrix inequalities (LMIs) with less restrictive conditions on the matrix variables. It is further shown that if the Lurie system under consideration has a unique equilibrium point at the origin, then this equilibrium point is in fact globally stable or unstable, meaning that local stability analysis will provide no additional benefit. Numerical examples demonstrate that the strengthened criteria achieve a desirable balance between reduced conservatism and complexity when compared to existing criteria.} }
Endnote
%0 Conference Paper %T Strengthened stability analysis of discrete-time Lurie systems involving ReLU neural networks %A Carl Richardson %A Matthew Turner %A Steve Gunn %A Ross Drummond %B Proceedings of the 6th Annual Learning for Dynamics & Control Conference %C Proceedings of Machine Learning Research %D 2024 %E Alessandro Abate %E Mark Cannon %E Kostas Margellos %E Antonis Papachristodoulou %F pmlr-v242-richardson24a %I PMLR %P 209--221 %U https://proceedings.mlr.press/v242/richardson24a.html %V 242 %X This paper addresses the stability analysis of a discrete-time (DT) Lurie system featuring a static repeated ReLU nonlinearity. Such systems often arise in the analysis of recurrent neural networks and other neural feedback loops. Custom quadratic constraints, satisfied by the repeated ReLU, are employed to strengthen the standard Circle and Popov Criteria for this specific Lurie system. The criteria can be expressed as a set of linear matrix inequalities (LMIs) with less restrictive conditions on the matrix variables. It is further shown that if the Lurie system under consideration has a unique equilibrium point at the origin, then this equilibrium point is in fact globally stable or unstable, meaning that local stability analysis will provide no additional benefit. Numerical examples demonstrate that the strengthened criteria achieve a desirable balance between reduced conservatism and complexity when compared to existing criteria.
APA
Richardson, C., Turner, M., Gunn, S. & Drummond, R.. (2024). Strengthened stability analysis of discrete-time Lurie systems involving ReLU neural networks. Proceedings of the 6th Annual Learning for Dynamics & Control Conference, in Proceedings of Machine Learning Research 242:209-221 Available from https://proceedings.mlr.press/v242/richardson24a.html.

Related Material