Accelerating Hopfield Network Dynamics: Beyond Synchronous Updates and Forward Euler

Cédric Goemaere, Johannes Deleu, Thomas Demeester
Proceedings of the 1st ECAI Workshop on "Machine Learning Meets Differential Equations: From Theory to Applications", PMLR 255:1-21, 2024.

Abstract

The Hopfield network serves as a fundamental energy-based model in machine learning, capturing memory retrieval dynamics through an ordinary differential equation (ODE). The model’s output, the equilibrium point of the ODE, is traditionally computed via synchronous updates using the forward Euler method. This paper aims to overcome some of the disadvantages of this approach. We propose a conceptual shift, viewing Hopfield networks as instances of Deep Equilibrium Models (DEQs). The DEQ framework not only allows for the use of specialized solvers, but also leads to new insights on an empirical inference technique that we will refer to as ’even-odd splitting’. Our theoretical analysis of the method uncovers a parallelizable asynchronous update scheme, which should converge roughly twice as fast as the conventional synchronous updates. Empirical evaluations validate these findings, showcasing the advantages of both the DEQ framework and even-odd splitting in digitally simulating energy minimization in Hopfield networks. The code is available at https://github.com/cgoemaere/hopdeq.

Cite this Paper

BibTeX
@InProceedings{pmlr-v255-goemaere24a, title = {Accelerating Hopfield Network Dynamics: Beyond Synchronous Updates and Forward Euler}, author = {Goemaere, C\'{e}dric and Deleu, Johannes and Demeester, Thomas}, booktitle = {Proceedings of the 1st ECAI Workshop on "Machine Learning Meets Differential Equations: From Theory to Applications"}, pages = {1--21}, year = {2024}, editor = {Coelho, Cecı́lia and Zimmering, Bernd and Costa, M. Fernanda P. and Ferrás, Luı́s L. and Niggemann, Oliver}, volume = {255}, series = {Proceedings of Machine Learning Research}, month = {20 Oct}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v255/main/assets/goemaere24a/goemaere24a.pdf}, url = {https://proceedings.mlr.press/v255/goemaere24a.html}, abstract = {The Hopfield network serves as a fundamental energy-based model in machine learning, capturing memory retrieval dynamics through an ordinary differential equation (ODE). The model’s output, the equilibrium point of the ODE, is traditionally computed via synchronous updates using the forward Euler method. This paper aims to overcome some of the disadvantages of this approach. We propose a conceptual shift, viewing Hopfield networks as instances of Deep Equilibrium Models (DEQs). The DEQ framework not only allows for the use of specialized solvers, but also leads to new insights on an empirical inference technique that we will refer to as ’even-odd splitting’. Our theoretical analysis of the method uncovers a parallelizable asynchronous update scheme, which should converge roughly twice as fast as the conventional synchronous updates. Empirical evaluations validate these findings, showcasing the advantages of both the DEQ framework and even-odd splitting in digitally simulating energy minimization in Hopfield networks. The code is available at https://github.com/cgoemaere/hopdeq.} }
Endnote
%0 Conference Paper %T Accelerating Hopfield Network Dynamics: Beyond Synchronous Updates and Forward Euler %A Cédric Goemaere %A Johannes Deleu %A Thomas Demeester %B Proceedings of the 1st ECAI Workshop on "Machine Learning Meets Differential Equations: From Theory to Applications" %C Proceedings of Machine Learning Research %D 2024 %E Cecı́lia Coelho %E Bernd Zimmering %E M. Fernanda P. Costa %E Luı́s L. Ferrás %E Oliver Niggemann %F pmlr-v255-goemaere24a %I PMLR %P 1--21 %U https://proceedings.mlr.press/v255/goemaere24a.html %V 255 %X The Hopfield network serves as a fundamental energy-based model in machine learning, capturing memory retrieval dynamics through an ordinary differential equation (ODE). The model’s output, the equilibrium point of the ODE, is traditionally computed via synchronous updates using the forward Euler method. This paper aims to overcome some of the disadvantages of this approach. We propose a conceptual shift, viewing Hopfield networks as instances of Deep Equilibrium Models (DEQs). The DEQ framework not only allows for the use of specialized solvers, but also leads to new insights on an empirical inference technique that we will refer to as ’even-odd splitting’. Our theoretical analysis of the method uncovers a parallelizable asynchronous update scheme, which should converge roughly twice as fast as the conventional synchronous updates. Empirical evaluations validate these findings, showcasing the advantages of both the DEQ framework and even-odd splitting in digitally simulating energy minimization in Hopfield networks. The code is available at https://github.com/cgoemaere/hopdeq.
APA
Goemaere, C., Deleu, J. & Demeester, T.. (2024). Accelerating Hopfield Network Dynamics: Beyond Synchronous Updates and Forward Euler. Proceedings of the 1st ECAI Workshop on "Machine Learning Meets Differential Equations: From Theory to Applications", in Proceedings of Machine Learning Research 255:1-21 Available from https://proceedings.mlr.press/v255/goemaere24a.html.

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