Accelerating Hopfield Network Dynamics: Beyond Synchronous Updates and Forward Euler (2311.15673v2)

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

• The paper introduces a novel approach by integrating deep equilibrium models with even-odd splitting to significantly reduce convergence times in HAMs.
• The experiments on a 3-layer HAM using the MNIST dataset demonstrate rapid convergence without compromising test accuracy.
• The methodology converts synchronous updates into efficient parallel asynchronous steps, enabling swift local energy minimization and computational savings.

Introduction to Hierarchical Associative Memory

The concept of Hierarchical Associative Memory (HAM) models, which expand upon continuous Hopfield networks, provide a promising avenue for memory retrieval applications. Despite their potential, optimizing the efficiency of these models, particularly within their energy minimization process, has been a relatively uncharted area of research.

Efficient Energy Minimization Approaches

One of the methods explored in this paper involves interpreting HAMs as Deep Equilibrium Models (DEQs). This perspective enables the application of advanced numerical solvers, which are both faster and more stable compared to traditional methods used in Hopfield networks. Moreover, through even-odd layer splitting—a technique previously proposed for Hopfield networks—HAMs demonstrated improved convergence speeds. Notably, when both techniques are merged, energy minimization in HAMs is executed significantly faster, as confirmed by experimental evidence.

Even-Odd Splitting Methodology

The process of even-odd splitting effectively transforms synchronous state updates in HAMs to parallel asynchronous ones, enhancing computational efficiency. Pivotal to HAMs, unlike traditional Hopfield networks, is that this method does not lead to a self-dependent optimal neuron value, facilitating a swift calculation for the local energy minimum. Furthermore, this splitting allows for the omission of part of the state representation, either the even or odd components, depending on the output layer's parity, which contributes to computational savings.

Experimental Verification

Experiments conducted on the MNIST dataset with a 3-layer HAM demonstrated that both strategies significantly accelerate the energy minimization process without impacting test accuracy. Specifically, the integration of advanced DEQ solvers and the practice of even-odd splitting, when paired together, yielded the most rapid convergence. These results hold promise for scaling up the practical application of Hopfield networks and propel forward the research in efficient memory retrieval.