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Accelerating Hopfield Network Dynamics: Beyond Synchronous Updates and Forward Euler

Cédric Goemaere, Johannes Deleu, Thomas Demeester

TL;DR

This work reframes Hopfield network dynamics by casting CHN and HAM as Deep Equilibrium Models (DEQs), enabling fixed-point solvers to replace traditional forward Euler integration for finding the equilibrium $\vec{s}^*$. It introduces even-odd splitting as a parallel asynchronous update within the DEQ framework, proving that CHN can be transformed into a HAM under mild conditions and showing that EO splitting accelerates convergence. Theoretical analysis derives a correspondence between CHN and HAM and provides a DEQ formulation for EO splitting, with practical benefits demonstrated on MNIST/EMNIST-MNIST datasets when combined with DEQ solvers. Empirical results show that DEQ solvers plus EO splitting can significantly reduce iteration counts while maintaining or improving accuracy, particularly in HAMs, suggesting a scalable and memory-efficient path to accelerating energy-minimization in Hopfield-like networks.

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

Accelerating Hopfield Network Dynamics: Beyond Synchronous Updates and Forward Euler

TL;DR

This work reframes Hopfield network dynamics by casting CHN and HAM as Deep Equilibrium Models (DEQs), enabling fixed-point solvers to replace traditional forward Euler integration for finding the equilibrium . It introduces even-odd splitting as a parallel asynchronous update within the DEQ framework, proving that CHN can be transformed into a HAM under mild conditions and showing that EO splitting accelerates convergence. Theoretical analysis derives a correspondence between CHN and HAM and provides a DEQ formulation for EO splitting, with practical benefits demonstrated on MNIST/EMNIST-MNIST datasets when combined with DEQ solvers. Empirical results show that DEQ solvers plus EO splitting can significantly reduce iteration counts while maintaining or improving accuracy, particularly in HAMs, suggesting a scalable and memory-efficient path to accelerating energy-minimization in Hopfield-like networks.

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
Paper Structure (34 sections, 6 theorems, 30 equations, 4 figures, 2 tables)

This paper contains 34 sections, 6 theorems, 30 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Our contributions:
  3. Preliminaries
  4. Continuous Hopfield network (CHN)
  5. Hierarchical Associative Memory (HAM)
  6. Deep Equilibrium Model (DEQ)
  7. Even-odd splitting
  8. Even-odd splitting from a Deep Equilibrium perspective
  9. From Hopfield network to DEQ
  10. Splitting the states into even & odd
  11. Finding the local energy optimum
  12. DEQ formulation of even-odd splitting in HAMs
  13. Advantages of even-odd splitting in Hopfield networks
  14. Faster convergence
  15. Guaranteed convergence
  16. ...and 19 more sections

Key Result

theorem 1

Even-odd splitting rearranges the layered structure of $\vec{s}$ using a permutation matrix $\vec{P}$, such that $\vec{s} = [\vec{s}_1; \vec{s}_2; \vec{s}_3; \ldots]$ is converted into $\vec{P}\vec{s} = [\vec{s}_\textup{even}; \vec{s}_\textup{odd}]$, with $\vec{s}_\textup{even} = [\vec{s}_2,\vec{s}_

Figures (4)

  • Figure 1: List of symbols
  • Figure 2: A view of synchronous updates across time reveals two separate even-odd DEQs (solid & dashed)
  • Figure 3: Density heatmap of the state trajectories for a 3-layer CHN (left) and HAM (right), and the impact of using DEQ solvers ('DEQ') and even-odd splitting ('EO'). The horizontal axis represents the number of iterations of the DEQ. The vertical axis represents the relative residual, which is used to determine the state convergence (the lower, the more converged). The limit of $10^{-4}$ as chosen criterion for convergence is indicated with a white dashed line. For every setting, we show the cumulative results of 5 different seeds, run on the entire MNIST test set. In cyan, we show the mean number of iterations corresponding to a given convergence criterion. The white circular marker at the limit of $10^{-4}$ corresponds to the value reported in \ref{['table_results']}.
  • Figure 4: Density heatmap of the state trajectories for a 5-layer (left) and 7-layer HAM (right), and the impact of using DEQ solvers ('DEQ') and even-odd splitting ('EO'). The horizontal axis represents the number of iterations of the DEQ. The vertical axis represents the relative residual, which is used to determine the state convergence (the lower, the more converged). The limit of $10^{-4}$ as chosen criterion for convergence is indicated with a white dashed line. For every setting, we show the cumulative results of 5 different seeds, run on the entire MNIST test set. In cyan, we show the mean number of iterations corresponding to a given convergence criterion. The white circular marker at the limit of $10^{-4}$ corresponds to the value reported in \ref{['table_results']}.

Theorems & Definitions (12)

  • remark 1
  • theorem 1
  • theorem 2
  • remark 2
  • proposition 1
  • definition 1
  • lemma 1
  • proof
  • lemma 2
  • proof
  • ...and 2 more