Cédric Goemaere

Cédric Goemaere, Johannes Deleu, Thomas Demeester

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

Felix Koulischer, Cédric Goemaere, Tom van der Meersch et al.

The recent discovery of a connection between Transformers and Modern Hopfield Networks (MHNs) has reignited the study of neural networks from a physical energy-based perspective. This paper focuses on the pivotal effect of the inverse temperature hyperparameter $β$ on the distribution of energy minima of the MHN. To achieve this, the distribution of energy minima is tracked in a simplified MHN in which equidistant normalised patterns are stored. This network demonstrates a phase transition at a critical temperature $β_{\text{c}}$, from a single global attractor towards highly pattern specific minima as $β$ is increased. Importantly, the dynamics are not solely governed by the hyperparameter $β$ but are instead determined by an effective inverse temperature $β_{\text{eff}}$ which also depends on the distribution and size of the stored patterns. Recognizing the role of hyperparameters in the MHN could, in the future, aid researchers in the domain of Transformers to optimise their initial choices, potentially reducing the necessity for time and energy expensive hyperparameter fine-tuning.