Universal Hopfield Networks: A General Framework for Single-Shot Associative Memory Models (2202.04557v2)

Abstract: A large number of neural network models of associative memory have been proposed in the literature. These include the classical Hopfield networks (HNs), sparse distributed memories (SDMs), and more recently the modern continuous Hopfield networks (MCHNs), which possesses close links with self-attention in machine learning. In this paper, we propose a general framework for understanding the operation of such memory networks as a sequence of three operations: similarity, separation, and projection. We derive all these memory models as instances of our general framework with differing similarity and separation functions. We extend the mathematical framework of Krotov et al (2020) to express general associative memory models using neural network dynamics with only second-order interactions between neurons, and derive a general energy function that is a Lyapunov function of the dynamics. Finally, using our framework, we empirically investigate the capacity of using different similarity functions for these associative memory models, beyond the dot product similarity measure, and demonstrate empirically that Euclidean or Manhattan distance similarity metrics perform substantially better in practice on many tasks, enabling a more robust retrieval and higher memory capacity than existing models.

• The paper presents a unified framework that generalizes traditional associative memory models using core operations of similarity, separation, and projection.
• It demonstrates that using alternative metrics like Euclidean and Manhattan distances significantly enhances memory retrieval robustness and capacity.
• The work bridges classical associative memory with transformer attention mechanisms, paving the way for high-capacity, adaptable neural architectures.

Universal Hopfield Networks

Introduction to Universal Hopfield Networks

Associative memory networks, capable of retrieving data points based not on explicit addresses but through metric similarity to a query, are central to both biological and artificial intelligence systems. Historically, models like Hopfield Networks (HN) and Sparse Distributed Memory have been pivotal. Recently, these have been generalized to the modern design - Modern Continuous Hopfield Networks (MCHN), sporting exponential memory capacity and transformers attention relationships. In this development, a unified framework called Universal Hopfield Networks (UHNs) has been proposed, offering a mathematically elegant description of various associative memory models.

Theoretical Foundation

UHNs are based on three core operations: similarity, separation, and projection. This framework abstracts associative memory function as a sequence integrating these operations. Associative memory models can be encapsulated within this framework, showcasing their operations as instances of it. Applying recent advancements by Krotov and Hopfield (2020), an extension of energy-based models with local computation has been achieved for UHNs, alongside a derived general energy function acting as a Lyapunov function for the dynamics of the network. This is a significant step, demonstrating that the concepts behind HNs can be generalized to a broader context and potentially apply to various types of associative memory models.

Empirical Insights

Empirically, the exploration into the use of different similarity metrics has resulted in a substantial discovery. It was observed that metrics like Euclidean or Manhattan distance outperformed the traditional dot-product in many associative tasks. The performance boost was especially apparent with robust retrieval and higher memory capacity. This suggests that a shift in the perspective from dot-product-based memory retrievals can lead to more powerful associative memory frameworks, which is crucial when considering the requirements for robustness in practical applications.

Implications and Connected Work

Finally, the broader implications of UHNs are not overlooked. The distinction between feedforward and iterative associative memory models has been explored, broadening the understanding of how associative memories can operate in different regimes. The universality of UHNs offers insights into not only memory retrieval in artificial networks but also underlines a possible organizing principle in neural memory retrieval. Moreover, the presented work provides a compelling narrative connecting associative memory models to attention mechanisms in transformers, leading to an exciting intersection of foundational neural network models with modern machine learning architectures.

This research reinforces the assertion that associative memory models hold significant foundational value for AI. Universal Hopfield Networks offer a modular and adaptable approach bridging traditional models and contemporary machine learning constructs, elucidating a path towards more robust and high-capacity memory systems. The implications of adopting different similarity functions, as empirically demonstrated, pose an enticing avenue for enhancement of neural architectures, not just within the bounds of associative memory but in the broader landscape of neural networks and deep learning.